plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 1, issue 2, 2018, pp. 111-120 issn:2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame1802119p * corresponding author. e-mail addresses: ljubisa.preradovic@aggf.unibl.org(lj. preradović), sinapsavla@yahoo.com (v. đajić), gordana.jakovljevic@aggf.unibl.org (g. jakovljević) gender and age structure as risk factors of carotid artery stenosis and specific themes areas of cartography ljubiša preradović1*, vlado đajić2, gordana jakovljević1* 1 university of banja luka, faculty of architecture, civil engineering and geodesy, bosnia and herzegovina 2 university of banja luka, faculty of medicine, bosnia and herzegovina received: 6 march 2018 accepted: 2 september 2018 available online: 2 september 2018. original scientific paper abstract: the stroke prevention project was implemented in the period between 2012 –2017 in the republic of srpska when 38,863 patients of both genders were examined. each of the patients underwent an ultrasound examination of the blood vessels of the neck on both sides. all the examinations were standardized and carried out by specially trained researchers. the presentation of the research results included descriptive statistics and a certain statistical test, which showed a statistically significant difference in carotid artery stenosis in male and female patients. the geographic information system was used for mapping carotid artery stenosis with the aim of determining the susceptibility of the population of a particular area, city and/or municipality to this disease and predicting it. the created epidemiological patterns show correlation between age structure and a particular area. key words: carotid artery; gis; mapping; prevention; risk factors 1 introduction annually, about 4,5 million people die of a stroke, as one of the toughest and most common diseases of modern man. the stroke, regarding its consequences, is the first cause of disability of modern man and, therefore, its prevention is very important (primatesta et al., 2007). it requires detection of the people with stroke risk factors (high blood pressure, diabetes, heart disease, high blood lipids, overweight people, smokers, people with a family history of stroke and people exposed to stress), as well as detection of pathological changes in the blood vessels of the neck and the head, whose treatment can lead to stroke prevention (autret et al., 1987; hennerici et al., mailto:ljubisa.preradovic@aggf.unibl.org mailto:sinapsavla@yahoo.com mailto:gordana.jakovljevic@aggf.unibl.org preradovic et al./decis. mak. appl. manag. eng. 1 (2) (2018) 111-120 112 1987; o'holleran et al., 1987; norris et al., 1991; inzitari et al., 2000; thom et al., 2008; đajić et al., 2015). in the republic of srpska there is a great number of citizens with so-called stroke risk factors, who cannot afford an ultrasound examination. this project provides citizens with a free and fast ultrasound screening of the blood vessels in the neck and the head thus contributing to stroke prevention. the geographic information system (gis) enables identification of epidemiological connection patterns between the risk factors and a particular area. the aim of this research is to detect pathological changes in the blood vessels of the head and the neck in the people having stroke risk factors as well as to prevent a stroke in order to determine the asymptomatic carotid disease prevalence in general population on the basis of a random sample of patients who underwent an ultrasound examination of the blood vessels in the neck. therefore, the mapping is carried out of carotid artery stenosis by using the gis with the aim of determining the susceptibility of the population of a particular area to a given disease and of predicting it. 2 material and methods in the period between 2012-2017, 38,863 patients were examined, i.e. 24,411 (62,8%) females and 14,452 (37,2%) males. all the examinees who had asymptomatic stroke (mu) and transient ischemic attack (tia) were not included in the project. before the examination, each patient filled in the standardized questionnaire asking for the following information: gender, age, height, weight, education, personal and family anamnesis of previous mu or tia, heart disease, diabetes, hypertension, hyperlipidemia, smoking, alcoholism. after filling in the questionnaire each of them underwent an ultrasound examination of the blood vessels in the neck on both sides. all these examinations were standardized and carried out by specially trained researchers. the stroke prevention project on the territory of the republic of srpska is carried out with the aim of determining the prevalence of the asymptomatic carotid disease in a representative sample of citizens in the republic of srpska. according to the last census (published in 2017), 1,228,423 citizens live in the republic of srpska (popis bih, 2013), that is, 1,170,342 citizens (rezultati popisa u bih, 2013) (the difference in the number of citizens is due to different methodologies that were applied to conducting the census). the previous census was published in 1991, but, due to the war, there was a big migration of the population. this census could not be used for calculating the number of patients who needed to be examined in certain municipalities; however, the sample was formed on the basis of the list of voters. local media and family doctors were previously informed about the project, as well as the local population, through a campaign which consisted of flyers, billboards, posters, media appearances, and so on. each project participant was invited to come for an examination by a nurse or a family doctor, or he checked in at the local medical institution on his own. tabular presentation was carried out using descriptive statistics and the mannwhitney u test, by applying analytic-statistic tools of the spps (originally called: statistical package for the social sciences), version 20, while for conducting graphical presentation, the spss, version 20 and microsoft excel 2007 were used. creating thematic maps was done in the software arcmap 10.2. the statistical data, on the basis gender and age structure as risk factors of carotid artery stenosis and specific themes areas… 113 of which mapping was carried out, were prepared in microsoft excel 2007 (.csv format). 3 research results on the territory of the republic of srpska, starting from 2012, the stroke prevention project has been carried out, with 38,863 examined patients (table 1). table 1 examined patients in the period from 2012 – 2017 year of examination gender of examinee total male female 2012 2284 4095 6379 2013 2 743 4281 7024 2014 2416 4421 6837 2015 3466 5931 9397 2016 2283 3957 6240 2017 1260 1726 2986 total 14452 24411 38863 the degree of carotid artery stenosis (blockage) ranged from 0 to 100% (in patients of both genders). median (md) of stenosis for all patients is 17,00% (in female patients median is less by 5,00% as compared to male patients), table 2.the average carotid artery stenosis for all patients is 19,03% (but in female patients average stenosis is less by 3,95% as compared to male patients). table 2 degree of carotid artery stenosis fig. 1 shows a degree of the carotid artery stenosis according to the gender of the patient. by applying the mann-whitney u test, a statistically significant difference is calculated (z= -27,485, p = 0,000) between carotid artery stenosis in female patients (n = 24,411, md = 15,00) and male patients (n = 14,452, md = 20,00). the carotid artery stenosis which is less than 20%, and, therefore, does not require any treatment was found in 21,408 (55,1%) patients (14,631 or 59,9% of all female patients and 6,777 or 46,9% of all male patients). by observing the percentage of the carotid artery stenosis representation according to gender, one can notice a higher frequency of carotid artery stenosis in male patients (table 3), as follows:  stenosis ranging from 20 49%: 47,5% in male patients and 37,3% in female patients,  stenosis ranging from 50 69%: 4,1% in male patients and 2,1% in female patients,  stenosis ranging from 70 99%: 1,1% in male patients and 0,5% in female patients, and, gender of examinee n minimum maximum median mean std. dev. male 14452 0 100 20.00 21.51 15.008 female 24411 0 100 15.00 17.56 12.556 total 38863 0 100 17.00 19.03 13.654 preradovic et al./decis. mak. appl. manag. eng. 1 (2) (2018) 111-120 114  stenosis of 100%: 0,4% in male patients and 0,2% in female patients. gender and age structure as risk factors of carotid artery stenosis and specific themes areas… 115 fig.1 degree of carotid artery stenosis according to the patient's gender table 3 degree of carotid artery stenosis /groups/ according to the patient’s gender carotid artery stenosis (%) gender of patient total male female 0-19 n 6777 14631 21408 % 46.9% 59.9% 55.1% 20-49 n 6870 9109 15979 % 47.5% 37.3% 41.1% 50-69 n 588 519 1107 % 4.1% 2.1% 2.8% 70-99 n 157 113 270 % 1.1% 0.5% 0.7% 100 n 60 39 99 % 0.4% 0.2% 0.3% total n 14452 24411 38863 % 37.2% 62.8% 100.0% percentage of the presence of carotid artery stenosis /group/ according to the patients’ gender is shown in fig. 2. preradovic et al./decis. mak. appl. manag. eng. 1 (2) (2018) 111-120 116 fig. 2 degree of stenosis of carotid artery /groups/ according to the patient’s gender the majority of patients, who underwent an examination, were between 55 and 64 years of age (13,642 or 35,1%); of these 6,679 had carotid artery stenosis ranging from 20 to 49%. every fourth patient (10,207 or 26,3%) was older than 64, and 779 of them had carotid artery stenosis ranging from 50-69% (70,4% of all patients had carotid artery stenosis ranging from 50-69%); in 189 patients carotid artery stenosis was between 70-99% (70,0% of all patients had carotid artery stenosis between 70 and 99%), and 59 patients had complete blockage of the carotid artery (59,6% of all patients with complete blockage of the carotid artery). the patients who belonged to young age categories had smaller carotid artery stenosis (table 4). table 4 degree of carotid artery stenosis /groups/ according to the patients’ age age group carotid artery stenosis (%) total 0-19 20-49 50-69 70-99 100 <= 24 n 264 2 0 0 0 266 % 1.2% 0.0% 0.0% 0.0% 0.0% 0.7% 25 34 n 1638 6 0 0 0 1644 % 7.7% 0.0% 0.0% 0.0% 0.0% 4.2% 35 44 n 3915 212 1 1 0 4129 % 18.3% 1.3% 0.1% 0.4% 0.0% 10.6% 45 54 n 6895 2036 33 5 6 8975 % 32.2% 12.7% 3.0% 1.9% 6.1% 23.1% 55 64 n 6560 6679 294 75 34 13642 % 30.6% 41.8% 26.6% 27.8% 34.3% 35.1% >= 65 n 2136 7044 779 189 59 10207 % 10.0% 44.1% 70.4% 70.0% 59.6% 26.3% total n 21408 15979 1107 270 99 38863 % 55.1% 41.1% 2.8% 0.7% 0.3% 100.0% gender and age structure as risk factors of carotid artery stenosis and specific themes areas… 117 degree of carotid artery stenosis /groups/ according to the patients’ age is shown in fig. 3. fig. 3 degree of carotid artery stenosis /groups/ according to the patients’ age groups 4 creation of thematic maps of the carotid artery thematic cartography is a cartographic discipline that enables presentation of spatial arrangement of objects, phenomena and processes that are under study. the geographic information system development ensured simpler collecting, processing and visualizing of spatial and associated data. the geographic information systems (giss) and spatial analysis techniques are powerful tools for describing epidemiological patterns, as well as for detecting, explaining and predicting clusters of diseases in space and time (grobusch et al., 2016). the gis application to mapping anatomic features and clinical events has been infrequent in the gis and medical literature (garb et al., 2007). the greatest potential of the gis is its ability to clearly show the results of complex analyses through maps (mullner et al., 2004). unlike tables and spreadsheets with seemingly endless numbers, maps produced by the gis have the ability to transform data into information that can be quickly and easily communicated. likewise, these systems also extend the range of problems that can this technology can help solving by allowing the users to more efficiently deal with complex problems (melnick&flemming, 1999; preradović et al., 2017). the creation of thematic maps of the carotid artery stenosis (blockage) is done using software of the company ersi, arcgis 10.2. based on data basis. arcgis uses an preradovic et al./decis. mak. appl. manag. eng. 1 (2) (2018) 111-120 118 object-relational data base. simple tables and defined types of attributes allow the storage of spatial data, and sql (structural query language) enables creating, modifying and querying the tables. data are saved in shapefile format. geometry of the object in .shp file can be presented by a dot, line or polygon. apart from the data on geometry, .shp file also contains attributive table which stores descriptive information, such as: the name of municipality, postcode, etc. spatial objects (political borders of municipalities in the republic of srpska) were used as spatial references for the carotid artery blockage presentation. the borders of municipalities are presented by polygons in .shp format. the cartogram method is used to show prevalence of a certain degree of the carotid artery blockage by the patients’ age groups while the average age of population is presented by the coloring method with the category borders defined by the method of natural borders. data on patients’ age and carotid artery blockage are downloaded in .xlsx format. the carotid artery blockage is shown by percentage and sorted in 5 categories (0-19, 20-49, 5079, 80-99, 100). average age of population is downloaded from the official site of the 2013 census of population, households and dwellings in bosnia and herzegovina in .xlsx format [10]. as the data in their original form were not suitable for further processing, they were harmonized and sorted. sorted data were saved in .csv format. .csv format stores tabular data as plain text and ensures data exchange between different programs and, therefore, it is used in this paper. connecting spatial and statistical data is carried out on the basis of mutual field (name of the municipality), by using option join. fig. 4 shows carotid artery blockages (separately for each category of carotid artery blockage and age group) by municipalities in the republic of srpska. gender and age structure as risk factors of carotid artery stenosis and specific themes areas… 119 fig. 4 carotid artery blockage by municipalities in the republic of srpska fig. 5 shows percentages of patients with carotid artery stenosis higher than 50% by municipalities in the republic of srpska. preradovic et al./decis. mak. appl. manag. eng. 1 (2) (2018) 111-120 120 fig. 5 percentage of patients with carotid artery stenosis higher than 50% by municipalities in the republic of srpska 5 conclusion on the basis of these results, it is evident that the minimal (0 to 19%) carotid artery stenosis in percentage (in relation to the number of examined patients) is more prevalent in female patients, and while the carotid artery stenosis which needs to be treated (conservatively and/or surgically) is more prevalent in male patients. the created epidemiological patterns indicate that the examinees in certain regions (cities and municipalities) have a high risk of a stroke. in accordance with the obtained and presented research results, it is necessary to do an analysis of equipment of medical institutions in vulnerable regions, purchase additional medical equipment and educate health care workers and population, with the aim of reducing the risk of this, very common, disease, with a high mortality rate, whose consequences are very severe – for the patient, family and whole society. references primatesta, p., allender, s., ciccarelli, p., doring, a., graff-iversen, s., holub, j., panico, s., trichopoulou, a. &verschuren, w.m. 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(2017). creating epidemiological patterns of connection between risk factors and particular. proceedings of icmnee 2017, regional association for security and crisis management and european centre for operational research, 1, 230-241. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). http://www.popis.gov.ba/popis2013/doc/rezultatipopisa_sr.pdf http://www.rzs.rs.ba/front/article/2369/ plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 3, issue 2, 2020, pp. 149-161. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame2003149z * corresponding author. e-mail addresses: zizovic@gmail.com (m. žižović), bole@ravangrad.net (b. miljković), dragan.marinkovic@tu-berlin.de (d. marinković) objective methods for determining criteria weight coefficients: a modification of the critic method mališa žižović1*, boža miljković1 and dragan marinković1 1 university of kragujevac, faculty of technical sciences in cacak, cacak, serbia 2 university of novi sad, faculty of education sombor, novi sad, serbia 3 technische universität berlin, faculty of mechanical and transport systems, berlin, germany received: 25 july 2020; accepted: 30 september 2020; available online: 10 october 2020. original scientific paper abstract: determining criteria weight coefficients is a crucial step in multicriteria decision making models. therefore, this problem is given great attention in literature. this paper presents a new approach in modifying the criteria importance through intercreteria correlation (critic) method, which falls under objective methods for determining criteria weight coefficients. modifying the critic method (critic-m) entails changing the element normalization process of the initial decision matrix and changing data aggregation from the normalized decision matrix. by introducing a new normalization process, we achieve smaller deviations between normalized elements, which in turn causes lower values of standard deviation. thus, the relationships between data in the initial decision matrix are presented in a more objective way. by introducing a new process of aggregation of weight coefficient values in the critic-m method, a more comprehensive understanding of data in the initial decision matrix is made possible, leading to more objective values of weight coefficients. the presented critic-m method has been tested in two examples, followed by a discussion of results via comparison to the classic critic method. key words: critic, criteria weights, multi-criteria decision making. 1. introduction determining criteria weight is one of the key problems of multi-criteria analysis models. methodologies for determining criteria weight have been the topic of intensive research and scientific discussions for many years. generally, most mailto:zizovic@gmail.com mailto:bole@ravangrad.net mailto:dragan.marinkovic@tu-berlin.de žižović et al./decis. mak. appl. manag. eng. 3 (2) (2020) 149-161 150 approaches to determining weight criteria can be divided into subjective and objective. subjective approaches are based on determining criteria weight using information from decision makers or experts included in the decision process. subjective approaches reflect the subjective opinion and intuition of decision makers which means that decision makers influence the decision making process. contrary to subjective approaches, objective approaches are based on determining criteria weight using data that is present in the initial decision matrix. objective approaches disregard the opinion of decision makers. with the subjective approach, the decision maker or expert gives their opinion on the significance of criteria for a given process in accordance with their preferences. there are multiple ways of determining criteria weights using a subjective approach and they differ in the number of participants in the process of determining weights, applied methods and the way of forming final criteria weights. subjective models used for aggregating partial values include: trade-off method (keeney and raiffa 1976); swing method (von winterfeldt and edwards 1986); smart method (the simple multi-attribute rating technique) (edwards and barron, 1994); the new version of smart method: smarter (smart exploiting ranks) developed by edwards and barron (1994). smarter uses the centroid method for determining criteria weight coefficients. apart from the listed subjective approaches for determining criteria weights, there are also approaches based exclusively on pairwise comparisons. these approaches are called pairwise comparison methods. the pairwise comparison method was developed by thurstone (1927) and it requires that comparisons be made by one or a team of experts. the pairwise comparison method is used for presenting relative significance of m alternatives in situations where it is not possible or meaningful to grade alternatives based on criteria. in pairwise methods, one or a team of experts compare an alternative to other alternatives from a set, in relation to a considered criterion. one of the best known methods for determining criteria weights using pairwise comparison is the analytical hierarchy processes (ahp) method (saaty, 1980). the ahp method is based on mutual comparison of criteria significance using saaty’s nine level scale. apart from ahp, other pairwise comparison methods include: decision-making trial and evaluation laboratory (dematel) method (gabus and fontela, 1972), step-wise weight assessment ratio analysis (swara) method (keršuliene et al., 2010); best worst method (bwm) (rezaei, 2015); full consistency method (fucom) (pamucar et al., 2018); level based weight assessment (lbwa) (žižović and pamučar, 2019); non-decreasing series at criteria significance levels (ndsl) (žižović et al., 2020) resistance to change method (roberts and goodwin, 2002) which contains elements of the swing and pairwise comparison methods. contrary to subjective methods, objective approaches eliminate, in a way, the decision maker, i.e. criteria weights are determined based on criteria values of alternatives. the emphasis is on the analysis of the decision matrix, i.e. values of alternatives are considered in relation to a set of criteria, followed by reaching data about values of criteria weights. the decision matrix allows cross referencing alternatives and criteria based on qualitative and quantitative values of each alternative in relation to each criteria. the best known models include: entropy method (shannon and weaver, 1947), critic method (criteria importance through intercriteria correlation), (diakoulaki, et al, 1995), fanma method, named for its authors (fan, 1996; ma et al,1999) and data envelopment analysis (dea) (charnes et al, 1978). objective methods for determining criteria weight coefficients: a modification of the critic ... 151 the entropy method entails determining objective criteria weights based on shannon’s concept of entropic grading of data in the decision matrix (shannon and weaver, 1947). the method focuses on measuring lack of definition of data in the decision matrix. the entropy method generates the set of weight coefficients based on mutual contrast of individual criteria value alternatives for each criterion and then for all criteria. determining criteria weights using the fanma method is based on using the principle of distance from the ideal point and the so-called early weight normalization (srdjevic et al., 2003). objective determination of criteria weights using the dea method (charnes et al, 1978) is based on solving linear optimisation models for alternatives and measuring efficiency of each alternative in relation to defined criteria. criteria are categorized as input and output criteria. then, a number of linear models equal to the number of options is solved. dea objectively ranks options which is the end goal of a multi-criteria analysis, and features groups of criteria weight values for all options as a step to reaching the end goal. the critic method is part of the best known and most widely used objective methods. the critic method is a correlation method, which uses standard deviation of ranked criteria values of options per column, as well as correlation coefficients of all paired columns to determine criteria contrasts. this paper identifies certain limitations when applying the classic critic method and suggests a modification of the critic method (critic-m) that entails: 1) changing the normalization process of the initial matrix elements and 2) changing the function for aggregating data that represents values of weight coefficients. the presented modifications to the critic method are aimed at reaching more objective values of weight coefficients. the remainder of the paper is organized as follows. in the next section (section 2) is presented the mathematical basis of the classic critic method while sections 3 shows the motivation for developing the critic-m method and the steps of the developed methodology. in the fourth section of the paper, we present the application of the critic-m method on two examples and compare the results with the classic critic method. final observations and the direction of future research are presented in section 5. 2. the critic method the critic method (criteria importance through intercriteria correlation), (diakoulaki, et al, 1995) is a correlation method. standard deviations of ranked criteria values of options in columns, as well as correlation coefficients of all paired columns are used to determine criteria contrasts. step 1: starting from an initial decision matrix, ij m n x       , we normalize the element of the initial decision matrix and form the normalized matrix ij m n x       . 1 2 11 12 11 2 21 22 2 1 2 n n n m m m mn m n c c c a a x a                          (1) the normalization of matrix elements ij m n x       is done by applying (2) and (3): žižović et al./decis. mak. appl. manag. eng. 3 (2) (2020) 149-161 152 a) for maximizing criteria: min max min , 1, 2,..., ; 1, 2,..., ; ij j ij j j i n j m           (2) b) for minimizing criteria: max max min , 1, 2,..., ; 1, 2,..., ; j ij ij j j i n j m           (3) where    max min1 2 1 2max , ,..., ; min , ,...,j j j mj j j j mj jj          . upon normalizing criteria of the initial decision matrix, all elements ij  are reduced to interval values [0, 1], so it can be said that all criteria have the same metrics. step 2: for criterion j c  1, 2,...,j n we define the standard deviation j , that represents the measure of deviation of values of alternatives for the given criterion of average value. standard deviation of a given criterion is the measure considered in the further process of defining criteria weight coefficients. step 3: from the normalized matrix ij m n x       we separate the vector  1 2, ,. .., j j j mj    that contains the values of alternatives  1, 2,..,ia i m for the given criterion j c  1, 2,...,j n . after forming the vector  1 2, ,. .., j j j mj    , we construct the matrix jk n n l l      , that contains coefficients of linear correlation of vectors j  and k  . the bigger the discrepancy between criteria values of options for criteria j and k , the lower the value of coefficient jkl . in that sense, the expression (4) represents the measure of conflict of criterion j in relation to other criteria in the given decision matrix. 1 (1 ) n j jk k l    (4) the quantity of data j w contained within criterion j is determined by combining previously listed measures j  and jk l as follows: 1 (1 ) n j j j j kj k w l        (5) based on the previous analysis we can conclude da a higher value j w means a larger quantity of data received from a given criterion, which in turn increases the relative significance of the given criterion for the given decision process. step 4: objective weights of criteria are reached by normalizing measures j w : 1 j j m k k w w w    (6) diakoulaki et al. (1995) and deng et al. (2000) recommend determining criteria weights based on values of standard vector deviation, expression (7): objective methods for determining criteria weight coefficients: a modification of the critic ... 153 1 j j m k k w      (7) where j  stands for standard deviation defined in step 2.. 3. modification of critic method: critic-m method the modification of the critic method presented in this section of the paper is based on two assumptions: 1) modification of normalizing data in the initial decision matrix and 2) modification of expressions for determining final values of criteria weights. 1) motivation for modifying the normalization of data in the initial decision matrix. in the original critic method we apply linear normalization that entails that each column of a normalized matrix contains at least one element with values 0 and 1. an exception would only be a column in which all values are the same (which rarely happens), in which case this criterion has no influence on the final decision. distribution of normalized values in the interval [0, 1] increases root-mean-square deviations, which in turn significantly influences values of criteria weight coefficients. if the standard deviation is close to zero for a certain criterion, then all elements regarding that criterion are centred around the average value of the element as per this criterion. in this situation, all values regarding this criterion are approximately equal so this criterion does not influence choice. in the modified critic method, normalization of the elements in the initial decision matrix entails dividing all the elements of the initial decision matrix with the maximum value in that column, expression (8). max , 1, 2,..., ; 1, 2,..., ; ij ij j i n j m       (8) where  max 1 2max , ,...,j j j mj j     . by applying expression (8) we normalize maximized criteria in the initial decision matrix. normalization of the minimized criteria is done in two steps. in the first step, values are normalized as with maximized criteria, i.e. by applying expression (8). in this way, we arrive at values * ij  . in the second phase, we normalize values by applying expression (9). * * max * min ; 1, 2,..., ; 1, 2,..., ; ij ij j j i n j m         (9) where     * * *max * * *min * * 1 12 2 max , ,..., ; min , ,..., j jj j mj j j mj jj          . this normalization process decreases the root-mean-square deviation and resulting values of criteria weight coefficients better reflect the relationship between data in the initial decision matrix. 2) motivation for modifying the expression for determining final criteria weight values. if the standard deviation is close to zero for a certain criterion, then all the elements regarding that criterion are centred around the average value of elements for this criterion. therefore, all the values for this criterion are approximately equal and this criterion does not influence choice. keeping this in mind, we adjust the expression for determining objective criteria weight values, expression (10) žižović et al./decis. mak. appl. manag. eng. 3 (2) (2020) 149-161 154 1 1 1 j j j j n j j j j w                  (10) where j  stands for the arithmetic average of elements of the normalized decision matrix as per criterion j, i.e. 1 1 n j ij im      . expression (10) represents an extension of expression (7) by introducing average values which favores criteria with average values closer to the ideal value, i.e. closer to one. this means that regarding this criterion, many alternatives have maximum values. in this way, we introduce a certain amount of subjectivity in the objective methodology of the critic method. the following section presents the steps of the modified critic (critic-m) method: step 1: starting with the initial decision matrix, ij m n x       , we normalize the elements of the initial decision matrix and form a normalized matrix ij m n x       . normalization of the elements of the matrix ij m n x       is done by applying expressions (8) and (9). maximized criteria (higher values is better) are normalized by applying expression (8), while minimized criteria (lower values are better) are normalized by applying expression (9). step 2: calculation of standard deviation of elements of the normalized matrix ij m n x       . as with the classic critic method, for each criterion j c  1, 2,...,j n we define standard deviation j  . step 3: constructing the matrix of linear correlations jk n n l l      . for each criterion j c from the normalized matrix ij m n x       we define the vector  1 2, ,. .., j j j mj    and calculate linear vector correlations j  and k  . summing linear correlations per criteria results in measure of criteria conflict: 1 (1 ) n j jk k l    (11) quantity of data j w in the criterion j is determined by applying expression (12): 1 (1 ) n j j kj k w l    (12) step 4: determining weight coefficients of criteria. objective weights of criteria are reached by applying expression (13) 1 1 1 j j j j n j j j j w w w                (13) weights of criteria can be determined based on values of standard vector deviation, expression (14): objective methods for determining criteria weight coefficients: a modification of the critic ... 155 1 1 1 j j j j n j j j j w                  (14) where j  stands for standard deviation. 4. determining criteria weights using the critic-m method example 1: the following section demonstrates the application of the critic-m method on an example that considers the evaluation of five alternatives ( 1, 2,...,5) i a i  in relation to four criteria ( 1, 2,..., 4) j c j  . all criteria in the initial decision matrix are maximized (max). the initial decision matrix ( ij m n x       , 1,2,...,i m , 1,2,...,j n ) is presented using expression (15). 1 2 3 4 8 4 10 2 7 6 4 6 5 5 6 7 6 6 7 8 1 2 3 4 5 65 7 6 c c c c a a x a a a                  (15) in the following section we present the application of the critic-m method in steps defined in the previous section of the paper: step 1: normalization of the initial decision matrix (15). since all criteria are maximized, we used expression (8) for normalizing elements. the normalized matrix is presented using expression (16). 1 2 3 4 1.000 0.571 1.000 0.250 0.875 0.857 0.400 0.750 0.625 0.714 0.600 0.875 0. 750 0.857 0.700 1.000 0.625 1.5 000 0.60 0 0.7 50 max max ma 1 2 3 4 x c a a x a a a c c c                  max (16) normalization of elements a1-c2 in matrix (16) was done in the following way: 12 12 max 2 4 0.571 7       ;   2 max 2 max 4, 6,5, 6, 7 7 c    . normalization of the remaining elements of matrix (16) was done in a similar way. step 2: calculation of standard deviation of elements of normalized matrix (16). we arrive at standard deviation for criteria  0.1630, 0.1629, 0.2191, 0.2850j  . step 3: matrix of linear correlation 4 4 jk l l      is presented using expression (17). 1 2 3 4 1.000 0.605 0.473 0.740 00.605 1.000 .681 0.635 0.473 0.681 1.000 0.671 0.740 0.635 0.67 0 1 3 1 1.00 2 4 c c c c c c l c c                   (17) žižović et al./decis. mak. appl. manag. eng. 3 (2) (2020) 149-161 156 by applying expression (11) and matrix (17), we arrive at the measure of criteria conflict:  3.873, 3.651, 3.878, 3.776j  element 1  for criterion 1 c is reached in the following way: 1 (1 1) (1 0.605) (1 0.473) (1 0.740) 3.873          . remaining values j  we reach in a similar way. by applying expression (12) we define quantity of data j w :  0.6312, 0.5947, 0.8497, 1.0763jw  quantity of data j w for criterion 1 c is reached in the following way: 1 1 1 0.1630 3.873 0.6312w       . remaining values j w are calculated in a similar way. step 4: determining objective values of criteria weights. by applying expression (13) we arrive at criteria weight coefficients  0.2405, 0.2632, 0.1825, 0.3139jw  . the value of criteria weight coefficient 1 w is reached in the following way: 1 0.775 0.6312 1 0.775 0.2405 0.775 0.800 0.660 0.725 0.631 0.594 0.849 1.076 1 0.775 1 0.800 1 0.660 1 0.725 w                                     in a similar way we arrive at the remaining criteria weight values. criteria weights can also be calculated by applying expression (14), i,e, based on the standard deviation j  . by applying expression (14) we arrive at weight coefficients:  0.2349, 0.2726, 0.1780, 0.3145jw  by applying expression (14), we arrive at the value of the weight coefficient of criterion 1 w in the following way: 1 0.775 0.1630 1 0.775 0.2349 0.775 0.800 0.660 0.725 0.1630 0.1629 0.2191 0.2850 1 0.775 1 0.800 1 0.660 1 0.725 w                                     the values of weight coefficients of the remaining criteria we reach in a similar way. example 2: in the following section, we present the application of critic-m method on an example that considers the evaluation of six alternatives ( 1, 2,...,6) i a i  in relation to three criteria ( 1, 2,3) j c j  . criteria c1 and c3 are maximized (max), while criterion c2 is minimized (min). the initial decision matrix ( 6 3 ij x       , 1,2,...,6i  , 1, 2,3j  ) is presented using the expression (18). 1 2 3 15 525 7 30 400 5 0 0 1 2 3 4 50 210 8 3 350 5 3 400 15 6 20 350 3 a a a x a c a c a c                     (18) objective methods for determining criteria weight coefficients: a modification of the critic ... 157 application of critic-m method on example 2 is presented in the following section: step 1: normalization of elements of matrix (18) is done by applying expressions (8) and (9). the normalized matrix is presented using the expression (19). x 1 2 3 0.300 0.400 0.875 0.600 0.638 0.625 1.000 1.000 1.000 0.600 0.733 0.625 0.600 0.638 0.125 0.400 0.7336 1 2 3 4 5 min m 0.375 x ama a a a x a a a c c c                     (19) normalization of elements a1-c1 in matrix (19) is done by applying the expression (8): 11 11 max 1 15 0.300 50       ;   1 max 1 max 15,30,50330,30, 20 50 c    . normalization of elements a1-c2 in matrix (19) is done by applying expression (9): * * max * min 12 12 2 2 1.00 1.00 0.400 0.400            , where   2 * max 2 1.000, 0.762, 0.4 000, 0.667, 10.762, 0 0.max 667 . c    ;   2 * min 2 1.000, 0.762, 0.40 00, 0.667, 0 ..762, 00.6min 67 0 4 c    . normalization of the remaining elements of matrix (19) was done in a similar way. step 2: from the normalized matrix (19) we get standard deviations for criteria ( 1, 2,3) j c j  :  0.240, 0.195, 0.320j  . step 3: matrix of linear correlations 4 4 jk l l      is presented using expression (20). 1 2 3 1.000 0.8 2 6 66 0.3 0 0.86 1.000 0 0.189 .3 9 1 20 0.18 1 2 .0003 c c c c l c c           (20) by applying expression (11) and matrix (20), we get the measure of criteria conflict  0.814, 0.945, 1.491j  , while by applying expression (12) we define the quantity of data  0.196, 0.184, 0.478jw  . step 4: determining objective values of criteria weight coefficients. by applying expressions (13) and (14) respectively, we reach criteria weight coefficients:  0.2670, 0.3447, 0.3883jw  and  0.1937, 0.2903, 0.5160jw  . table 1 presents criteria weight coefficients reached using the classic critic method and the critic-m method in examples 1 and 2. žižović et al./decis. mak. appl. manag. eng. 3 (2) (2020) 149-161 158 table 1. criteria weight coefficients by applying critic and critic-m criteria critic critic-m, expression (13) critic-m, expression (14) example 1 c1 0.2221 0.2405 0.2349 c2 0.3994 0.2632 0.2726 c3 0.1979 0.1825 0.1780 c4 0.1805 0.3139 0.3145 example 2 c1 0.2468 0.1937 0.2670 c2 0.2708 0.2903 0.3447 c3 0.4824 0.5160 0.3883 table 1 presents two groups of data reached using the critic-m method. the first group of data was reached using expression (13), while the second group of data was reached using expression (14). based on data from table 1 we note a very small difference between application of critic-m, expressions (13) and (14). we note that determining conflict between criteria through coefficients of linear correlation, expression (13), does not identify significant differences that influence final values of criteria weights. however, the calculation of linear correlation matrix elements and the introduction of that data to the calculation of criteria weights significantly complicates the calculation of criteria weight coefficients. therefore, we recommend the application of standard deviation (expression (14)) for calculating criteria weights, because it presents quite well the relationships between criteria in the initial decision matrix. by comparing weight coefficients reached by using critic and critic-m methods we note that there are significant differences between resulting values. differences in the weights are due to 1) different way of data normalization (critic linear normalization and critic-m percentual normalization) and 2) application of different aggregation functions used for final values of criteria weight coefficients. applying linear normalization in the critic method results in higher values of standard deviation because normalization distributes all values in the interval [0, 1]. on the other hand, by applying percentual normalization in the critic-m method, all normalized values are distributed in the interval min max ,1 j j          . this shifts the distribution of all values towards the ideal value, i.e. towards one. as a consequence, standard deviation values are lower. both examples in this paper show that criteria weight coefficients centre around average values. also, we can point out that the critic-m method contributes to a better objectivity of results. this can be noted in the second example and criteria c1 and c2. by applying the classic critic method, there are very small differences between weight coefficients of criteria c1 and c2. on the other hand, by applying the critic-m method, the differences between these criteria are clearly marked. further, in the critic-m method, the function for aggregating values of weight coefficients has been changed by introducing average values. the reason for introducing average values and presenting their influence on criteria weights is favoring criteria whose average values are closer to the ideal value. by introducing this type of subjectivity to the critic-m method, we eliminate one of the bad characteristics of the classic critic method: assigning low values of criteria weight coefficients to criteria that, for most alternatives, have values close to the ideal value. objective methods for determining criteria weight coefficients: a modification of the critic ... 159 5. conclusion weight coefficients are a calibration tool for decision models and the quality of their definition directly influences the quality of the decision. the reason for studying this problem lies in the fact that each of the subjective and objective methods for determining criteria weights has its advantages and flaws. this paper considers certain limitations of the critic method and puts forward a modification with its new critic-m algorithm. the modification of the critic method presented in this paper is based on a new approach to normalization of values in the initial decision matrix and on a new approach to aggregation of data from the initial decision matrix. the new normalization process in the critic method makes it possible to reach lower standard deviation values for normalized values, which contributes to more objective representation of relationships between data in the initial decision matrix. apart from modifying the critic method using a new normalization process, we also present a new approach for aggregating values of weight coefficients. aggregation of weight coefficient values in the critic-m entails average values of normalized elements. introduction of average values aims to favor criteria per which alternatives have values close to ideal values. although this approach introduces a certain degree of subjectivity to this critic methodology, authors maintain that this approach enables a more comprehensive understanding of data in the initial decision matrix and a more objective set of weight coefficient values. it is clear that values reached by using objective and subjective methods can lead to completely different results, i.e. to completely different weight coefficient values. keeping this in mind, objective methods for determining criteria weights can be used to correct criteria weights determined using subjective methods or based on subjective preferences of decision makers. therefore, the presented critic-m methodology can be a useful tool for correcting criteria weights. further, future research can be directed towards defining absolute, ideal and anti-ideal values in the initial decision matrix. this would eliminate rank reversal problems in the case of adding new alternatives to the initial decision matrix and reduce its indirect influence on significant changes to criteria weights. also, future research should also be directed towards application of uncertainty theories in the critic-m method, such as fuzzy theory. this is supported by the significant position of fuzzy theory in the field of multi-criteria decision making, and as far as the authors are aware, there has, so far, been no presentation of expanded critic methods in fuzzy environments. author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research received no external funding. 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(2019). new model for determining criteria weights: level based weight assessment (lbwa) model. decision making: applications in management and engineering, 2(2), 126-137. objective methods for determining criteria weight coefficients: a modification of the critic ... 161 žižović, m., pamučar, d., ćirović, g., žižović, m.m., & miljković, b. (2020). a model for determining weight coefficients by forming a non-decreasing series at criteria significance levels (ndsl). mathematics, 8(5), 745. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 1, issue 2, 2018, pp. 153-163 issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame1802160p * corresponding author. e-mail addresses: milenap@fon.bg.ac.rs (m. popovic), marija.kuzmanovic@fon.bg.ac.rs (m. kuzmanovic), gordana.savic@fon.bg.ac.rs (g. savic). a comparative empirical study of analytic hierarchy process and conjoint analysis: literature review milena popović1*, marija kuzmanović1, gordana savić1 1 university of belgrade, faculty of organizational sciences, belgrade, serbia received: 5 april 2018; accepted: 26 september 2018; available online: 28 september 2018. original scientific paper abstract: this paper is based on the main difference between conceptual and theoretical frameworks as well as literature review of comparative studies of two multi-criteria decision making methods (mcdm): analytic hierarchy process (ahp) and conjoint analysis. the ahp method represents a formal framework for solving complex multiatributive decision making problems, as well as a systemic procedure for ranking multiple alternatives and/or for selecting the best from a set of available ones. conjoint analysis is an experimental approach used for measuring individual’s preferences regarding the attributes of a product or a service. it is based on a simple premise that individuals evaluate alternatives, with these alternatives being composed of a combination of attributes whose part-worth utilities are estimated by researchers. bearing in mind the quality of desired results, it must be dependent on the problems and aspects of research: knowledge of the mcdm methods, level of complexity (number of criteria), order effects, level of consistency, chooses the appropriate method. key words: analytic hierarchy process, conjoint analysis, multi-criteria decision making (mcdm) methods, literature review. 1. introduction decision making refers to the process of selecting an alternative, from a set of available ones, which resolves a given problem. the following elements can be distinguished in the decision-making issue: goals to be achieved by making a decision, criteria that measure the achievement of the goals, weights of the criteria that reflect their importance and alternatives within which the most desirable is to be selected (anderson et al., 2012). a goal is to understand as the state of the system that is to be reached by making a decision. criteria are the attributes describing alternatives and usually in the given decision-making issue not all the criteria are mailto:milenap@fon.bg.ac.rs mailto:marija.kuzmanovic@fon.bg.ac.rs mailto:gordana.savic@fon.bg.ac.rs popovic et al./decis. mak. appl. manag. eng. 1 (2) (2018) 153-163 154 equally important. their relative importance stems from the preferences of a decision maker, respectively, a respondent. decision making has increasingly been present in scientific research projects around the world recently, as it has become clear that the success of companies largely depends on the decisions made. when we say that a manager makes quality decisions, this means that these decisions are well thought out, made at the right time, and the realization of such decisions is precisely planned, all in order to maximize the effects that the decisions need to achieve. generally, a decision maker is exposed to an environment that is extremely complex and dynamic, being burdened with his paradigms and a series of influences which he, sometimes knowingly and sometimes unconsciously, includes into the decision-making process. the situation changes when a decision maker disposes with enough information about the problem and when the events related to the problem are certain, which implies full knowledge of the event or knowledge of the probability of the occurrence of an event. the methods used in decision making can be classified into the two basic groups: 1. single-criterion optimization methods 2. multi-criteria optimization methods multi-criteria decision making can be divided into (figueira et al., 2005): 1. madm (multiple attribute decision making), and 2. modm (multiple objective decision making). basic difference between the multiple attribute and the multiple objective decision making is reflected in the fact that in the multiple attribute decision making the best action is selected from the final set of previously defined actions described by explicit attributes, while in the multiple objective decision making the final set of objectives is defined on the basis of which the action which will fulfill defined objectives is selected. primarily because of their similarity, but also because of the wide applicability in the last years, in this paper, two techniques of multi attribute valuation are selected: the ahp method and the conjoint analysis. the ahp method is designed for a subjective assessment of multiple alternatives compared to multiple criteria, organized into a hierarchical structure. at the upper level the criteria are assessed, and alternatives based on the criteria are evaluated at the lower level. a decision maker gives its subjective assessment separately for each level and sub-level. according to these estimates pair comparison matrices are formed, which are based exclusively on subjective assessments. the ahp is a technique used to rank more alternatives and/or to select the best one from a set of available ones. ranking/selection is performed in relation to the overall goal which is described through multiple criteria. conjoint analysis is based on the assumption that complex decisions are made not based on a single attribute, but on several attributes and their levels considered jointly, hence the term conjoint. the technique can establish the relative values of particular attributes and identify the trade-offs the customers are likely to make in choosing a product and service and the price they are willing to pay for it. the paper is organized as follows: the sections 2 and 3 describe conjoint analysis and the ahp method, basic concepts, goals and the methodology of performance. conceptual comparison and overview of the applications of the selected methods will be described in chapters 4 and 5. finally, the main conclusions are summarized in section 6. a comparative empirical study of analytic hierarchy process and conjoint analysis… 155 2. conjoint analysis conjoint analysis is a multivariate technique used specifically to understand how a respondent’s preferences are developed (hair et al., 1995). more precisely the technique is used to gain insights into how individuals evaluate the total worth of a profile by combining the separate amounts of utility for each attribute level. there are three basic major phases for conducting a conjoint study. the first phase involves determining relevant attributes and the levels of each attribute. lists of attributes describing single alternatives are called profiles (real or hypothetical) being presented to respondents who are invited to express their preference by rating or ranking these profiles. the second phase involves design data collection of measuring individual preference and estimating respondent’s utility functions. to determine the relative importance of different attributes to respondents, a relationship between the attributes’ utility and the rated responses must be specified. the most commonly used model is the linear additive model. this model assumes that the overall utility derived from any combination of attributes of a given good or service is obtained from the sum of the separate part-worths of the attributes. thus, respondent i's (i= 1,…, i) predicted conjoint utility for profile j (j = 1 ,…, j)can be specified as follows (kuzmanović et al., 2013a): 1 1 klk ij ikl jkl ij k l u x      (1) where: xjkl is a (0,1) variable that it equals 1 if profile j has attribute k at level l, otherwise it equals 0 βikl– respondent i’s utility with respect to level l (lk – the number of levels of attribute k)of attribute k (k – the number of attributes) ij – stochastic error term. the parameters βikl (also known as part-worth utilities) are estimated by a regression analysis. the value of beta coefficients can be used: to indicates the amount of any effect that an attribute has on overall utility of the profiles; for preference-based segmentation; to calculate the relative importance of each attribute (importance value). importance values are calculated by taking the utility range for each attribute separately, and then dividing it by the sum of the utility ranges for all of the factors (2). the results are then averaged to include all of the respondents (kuzmanović et al., 2013). error! objects cannot be created from editing field codes. (2) where fiik is the relative importance that ith respondent assigned to the factor k. the results are then averaged to include all the respondents: 1 , 1,..., i k ik i fi fi i k k    (3) if the market is characterized by heterogeneous customer preferences, it is possible to determine the importance of each attribute for each isolated market segment. popovic et al./decis. mak. appl. manag. eng. 1 (2) (2018) 153-163 156 the last (third) phase involves market simulation to predict how buyers will choose among competing products and how their choices are expected to change as product features and/or price are varied. 3. the analytic hierarchy process (ahp) the analytic hierarchy process – ahp is a multi-criteria decision making method that was developed by saaty (1980). this method considers a given set of qualitative and/or quantitative criteria combines them through the decomposition of complex problems into a model that has the form of a hierarchy (goal, criteria, sub-criteria and alternatives). the main objective of ahp is ranking/selection of several alternatives made in relation to the set goal, as well as the choice of the best one from a set of available ones, in situations where decision-making involves a larger number of experts and criteria (popovic et al., 2018). the generalized method can be simply described as follows (bhushan & rai, 2007): data are collected from decision makers in the pairwise comparison of alternatives on a qualitative scale. decision makers can rate the comparison as equal, marginally strong, strong, very strong, and extremely strong. the pairwise comparisons of various criteria are organized into a square matrix. the diagonal elements of the matrix are 1. the criterion in the i-th row is better than criterion in the j-th column if the value of element (i, j) is more than 1; otherwise the criterion in the j-th column is better than that in the i-th row. the (j, i) element of the matrix is the reciprocal of the (i, j) element. the principal eigenvalue and the corresponding normalized right eigenvector of the comparison matrix give the relative importance of the various criteria being compared. the elements of the normalised eigenvector are termed weights with respect to the criteria or sub-criteria and ratings with respect to the alternatives. therefore a comparisons made by ahp are subjective this method tolerates inconsistency through the amount of redundancy in the approach. if this consistency index (ci) fails to reach a required level then answers to comparisons may be reexamined (4) (sener et al., 2010). max( ) / ( 1)ci n n   (4) where maxλ max is the maximum eigenvalue of the judgment matrix. ahp calculates a consistency ratio (cr) comparing the consistency index (ci) with a random matrix (ri). saaty (1980) suggests the value of cr should be less than 0.1. finaly, the rating of each alternative is multiplied by the weights of the subcriteria and aggregated to get local ratings with respect to each criterion. the local ratings are then multiplied by the weights of the criteria and aggregated to get global ratings. it should be noted that ahp is a method that orders the priorities in a given situation, incorporating the element of subjectivity and intuition so that a final decision can be reached by experts for part-issues in a consistent way and gradually move up levels to deal with the given situation have clear idea of what it entails (alharbi, 2001). 4. conceptual comparison of ahp and conjoint analysis both the conjoint analysis and the ahp method can be used to measure preferences of respondents and determine relative importance of attributes (criteria), a comparative empirical study of analytic hierarchy process and conjoint analysis… 157 but having in mind the quality of the desired results, a more appropriate method should be selected based on the specific problem and the research conditions. basic theoretical differences between the traditional conjoint analysis and the ahp method are provided in the table 1. table 1. conceptual comparison of ahp and conjoint analysis (mulye, 1998; helm et al., 2004; scholl et al., 2005; kallas et al., 2011) conjoint analysis ahp pre-condition preferential independence of the attributes preferential independence of the attributes survey form decompositional compositional scale used ordinal or interval scale ratio scale utility model additive part-worth model weighted additive utility model applicability up to six attributes with two to four levels many attributes possible with up to seven to eight attribute-levels respondents market segment on basis of individual customers individual decision makers interview expense ranking, rating or paired comparisons paired comparisons the basic aim of application measuring preferences decision making application range design problems selection problems and/or design problems results part-worths of all attributelevels relative preferences of attribute-levels and attributes although both techniques were developed with a different aim, they can be used in the same study. fundamental assumption on which both methods are based is the preferential independence of the attributes, i.e., one level of attributes (for example, a brand) has no influence on the characteristics of another level of attributes (for example, on color). conjoint analysis can function also in some cases of mutual interaction of attributes, but at least basic preferential independence is required. considering the ahp evaluation task is based on direct paired comparisons of single attributes and attribute levels, it is possible to survey tasks consisting of many attributes and their levels. but, conjoint analysis asks the respondents to evaluate complete profiles. therefore, the number of profiles and the number of attributes and their levels are limited as cognitive resources of the respondents are restricted. the differences in the scales used to evaluate the criteria cause differences in the evaluation steps. both the ahp method and the conjoint analysis are based on comparative analysis, but in the conjoint analysis other evaluation steps are also possible. both methods are applicable for studies which use ‘pen and paper’ method, however, in the case of application of the ahp method, it is recommended the use of commercial softwares (www.expertchoice.com) which, during the evaluation process itself, determine consistency level of the responses and require that the responses to popovic et al./decis. mak. appl. manag. eng. 1 (2) (2018) 153-163 158 the same questions are repeated in case of too large inconsistencies. the number of respondents is not limited, and the only difference is that the target group in the ahp method are the respondents representing individual decision makers (most often they are experts in a given field of research), and in the conjoint analysis, these are arbitrarily chosen market segment. there are several factors such as the motivation of respondents, the scope of information that a questionnaire contains, the clarity of a questionnaire, the knowledge of the method which can influence the results of empirical research using the ahp method and the conjoint analysis. these factors determine practical applicability of the method; so for example, the questionnaires that are difficult to answer can reduce the validity of the results (hartmann & sattler, 2004). likewise, the time needed to complete the questionnaire affects the results obtained. longer questionnaires can exhaust the respondents, cause response distortion or provoke deviations in the study. time is also a factor that affects total costs, as the total costs of conducting research increase by increasing the time required. the question arises as to what was the influence of the factors, such as the knowledge of the methods by the respondents, the complexity of the study (number of criteria) and the problem of research, to the result of the comparison of these methods. 5. overview of the research projects based on the comparison of the ahp method and the conjoint analysis in the research projects based on the comparison of the conjoint analysis and the ahp method are obtained contradictory conclusions regarding the conditions of application of these methods. therefore, in order to compare them (during the application procedure), it is necessary to control all the factors that can favor one against the other method. further in the paper, comparative overview of basic concepts of eight studies aimed at comparing the results of the conjoint analysis and the ahp method (table 2) will be presented. table 2. overview of basic concepts of the research of comparison of the conjoint analysis and the ahp decision problem number of attributes and attribute levels respondents complexity of the decision problem tscheulin (1991) ship travels 5 attributes (4 with 3 and 1 with 4 levels) no knowledge of the methods relatively complex mulye (1998) i study running shoes 4 attributes (2 with 3 and 2 with 4 levels) knowledge of the methods (students) relatively simple mulye (1998) ii study rental accomodation 8 attributes (each consisting 3 levels) knowledge of the methods (students) relatively complex helm et al. (2004) universities 6 attributes (5 with 3 and 1 with 2 levels) knowledge of the methods (students) relatively complex helm et al. (2008) mountain bikes 4 attributes (po 3 levels) two groups– with/without relatively simple a comparative empirical study of analytic hierarchy process and conjoint analysis… 159 decision problem number of attributes and attribute levels respondents complexity of the decision problem knowledge of the methods ijzerman et al. (2008) treatment preferences in people with neurological disorders 7 attributes (2-4 levels) no knowledge of the methods relatively complex kallas et al. (2011) rabbit meat in menus in spain 4 attributes (each consisting 3 levels) no knowledge of the methods relatively simple ijzerman et al. (2012) stroke rehabilitation 8 attributes (2-4 levels) no knowledge of the methods relatively complex danner et al. (2017) age-related macular degeneration 5 attributes (1 with 4, 2 with 3 and 2 with 2 levels) no knowledge of the methods relatively complex danner et al., (2017) claim that common application of the ahp method and the conjoint analysis is the broadest in the field of health care system. however, on the basis of comparative overview of fundamental concepts of the research carried out so far, as shown in the table 3.6, it can be noted that the spectrum of the decision making issues is broad. according to the research issue, the studies conducted differ in complexity of the decision-making issue. authors use four to eight attributes with two, three, four, or even five levels to describe their research issue. taking into consideration the limitations of the application of the conjoint analysis based on the number of attributes, certain decision-making issues can be characterized as relatively complex. although the study conducted by kallas et al., (2011) did not have as the primary goal determining which method was better, the results obtained allowed them to see the advantages and disadvantages of each of the method. the ahp method proved to be easier in this study, while the conjoint analysis allowed combining the obtained preferences with socio-demographic variables. an important prerequisite for the quality of the obtained empirical results, stated by the authors in their papers, is the knowledge of the method (procedure) of the research by the respondents. in the table 3 is provided an overview of the effects of comparison of the conjoint analysis and the ahp based on the knowledge of the research methods and the complexity of the questionnaires found in the previous studies (table 2). as can be seen from the table 3, the studies showed that different results were obtained if respondents knew the methods and understood the procedure: the conjoint analysis appeared to be better when the respondents were not familiar with the research methodology, while the ahp should be opted for when respondents understand the steps of the method. tscheulin (1991) suggests explaining some of the relevant methodological aspects of the ahp and the conjoint analysis before the popovic et al./decis. mak. appl. manag. eng. 1 (2) (2018) 153-163 160 interview itself. this can be performed as a "pre-research" through several minor and simpler common decision-making issues. table 3. influence of knowledge of methods and complexity of questionnaires on the results of research (helm et al., 2008; ijzerman et al., 2012) complexity of the evaluation task high/medium low knowledge in preference measurement yes ahp better (ii study -mulye, 1998; helm et.al., 2004) similar results (i study mulye, 1998) conjoint analysis slightly better (helm et.al., 2008) no conjoint analysis better (tscheulin, 1991; ijzerman et al. 2012) conjoint analysis remarkably better (helm et.al., 2008) given the consistency level achieved with the conjoint analysis and the ahp method in all studies, the lower levels are less preferred. if sensitivity and consistency level are observed, the obtained results disagree. although helm et al., (2004) found in the first study that the ahp was less sensitive compared to the conjoint analysis, in the second study (helm et al., 2008) they came to the opposite conclusion. the conjoint analysis proved to be less sensitive to changes and required a lower minimum level of consistency than the ahp, hence a large number of insufficiently consistent respondents in the study. the explanation of this difference is not obvious, but it may again result from a change in the complexity of the decisionmaking issues, because the inconsistency in the conjoint analysis has much more direct impact on the final result than the local inconsistency in the ahp, which only applies to one attribute. considering other factors that influence the result of the comparison, it can be said that the conjoint analysis leads to better results when applied after the ahp (mulye, 1998). helm et al., (2004), in contrast to mulye, obtains opposite results, which is probably the consequence of the complexity of the problem, in the first study, however, in the second study based on somewhat simpler issues, slightly better effects can be observed when the conjoint analysis is applied after the ahp (helm et al., 2008). the conclusion of a former research summarize the four aspects may influence the quality of the results of conjoint analysis and ahp as technique for measuring preferences:  knowledge of the mcdm methods,  level of complexity (number of criteria),  order effects,  level of consistency. it can be said that conjoint analysis is a better choice in relatively simple decisionmaking issues. in case of complex decision-making problems and/or respondents with prior knowledge of the method of research, the ahp seems to be more convenient method. having in mind practical applicability, the ahp method has a a comparative empirical study of analytic hierarchy process and conjoint analysis… 161 potential advantage because it requires less time to complete the survey and achieve a higher level of satisfaction of the respondents (helm et al., 2008; ijzerman et al., 2012). both methods require certain level of consistency in respondents' responses, with the conjoint analysis being more resistant in simple, and the ahp in more complex issues. in any case, any "pre-research" performed before starting evaluation could have positive effects. these findings could have an influence on future practice of measuring preferences, since more than 65% of all conjoint analysis surveys include more than six attributes. therefore, researchers need a new method that supports operating with multiple attributes. many of the newly developed variants of the conjoint analysis have failed in practice because there have been no commercial softwares to support them. today, currently available adaptive conjoint analysis softwares are so far the most dominant commercial softwares that can compensate these deficiencies of the traditional conjoint analysis. additionally, with the professional ahp-based softwares, more advanced options for measuring preferences appear in practice. another advantage of the conjoint analysis in relation to the ahp is that it offers the possibility of segmentation based on the results obtained, as well as the prediction of market share, which has not been taken into account by the authors of the previous studies. 6. conclusions the findings of this paper are significant on both a theoretical and an applied level. on a theoretical level, both methods can be applied in the measurement of the preferences of respondents and determining relative importance of attributes (criteria), but considering the quality of the required results, it is necessary based on the specific issue and the aspect of research (knowledge of the mcdm methods, level of complexity (number of criteria), order effects, level of consistency) to choose the adequate method. on the applied level, the results provide information to policy makers to help them make decisions more effectively. in fact, although these two methods were originally developed with different objectives, they can still be used independently in similar or the same research projects. acknowledgment: this research was partially supported by the ministry of science and technological development, republic of serbia, project numbers: tr33044 and iii44007. references al-harbi, k. m. a. s. (2001). application of the ahp in project management. international journal of project management, 19(1), 19-27. anderson, d.r., sweeney, d.j., williams, t.a., camm, j.d., & martin, k., (2012). an introduction to management science: quantitative approaches to decision making, south-western cengage learning. bhushan, n., & rai, k. (2007). strategic decision making: applying the analytic hierarchy process. springer science & business media. popovic et al./decis. mak. appl. manag. eng. 1 (2) (2018) 153-163 162 danner, m., vennedey, v., hiligsmann, m., fauser, s., gross, c., & stock, s., (2017). comparing analytic hierarchy process and discrete-choice experiment to elicit patient preferences for treatment characteristics in age-related macular degeneration. value in health, 20(8), 1166-1173. figueira, j., greco, s., ehrgott, m. 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(1980), the analytic hierarchy process, mcgraw-hill, new york. http://inderscience.metapress.com/content/110876/?p=f0d2592e07734bf2b09ca05d00e296f6&pi=0 http://inderscience.metapress.com/content/110876/?p=f0d2592e07734bf2b09ca05d00e296f6&pi=0 a comparative empirical study of analytic hierarchy process and conjoint analysis… 163 scholl, a., manthey, l., helm, r., & steiner, m. (2005). solving multiattribute design problems with analytic hierarchy process and conjoint analysis: an empirical comparison. european journal of operational research, 164(3), 760-777. şener, ş., şener, e., nas, b., & karagüzel, r. (2010). combining ahp with gis for landfill site selection: a case study in the lake beyşehir catchment area (konya, turkey). waste management, 30(11), 2037-2046. tscheulin, d. k. (1991). ein empirischer vergleich der eignung von conjoint-analyse und analytic hierarchy process (ahp) zur neuproduktplanung. zeitschrift für betriebswirtschaft, 61(11), 1267-1280. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 1, number 2, 2018, 81-92 issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame1802079s * corresponding author. e-mail addresses: sremacs@uns.ac.rs (s. sremac), ilijat@uns.ac.rs (i. tanackov), miloskopic@uns.ac.rs (m. kopić), radovic93@yahoo.com (d. radović) anfis model for determining the economic order quantity siniša sremac1*, ilija tanackov1, miloš kopić1, dunja radović2 1 faculty of technical sciences, university of novi sad, novi sad, serbia 2 faculty of transport and traffic engineering, university of east sarajevo, doboj, bosnia and hercegovina received: 4 april 2018; accepted: 27 august 2018; available online: 30 august 2018. original scientific paper abstract: the determination of the economic order quantity is important for the rational realization of the logistics process of transport, manipulation and storage in the supply chain. in this paper an expert model for the determination of the economic order quantity has been developed. the model has been developed using the hybrid method of artificial intelligence adaptive neuro-fuzzy inference systems anfis. it has been used for modeling a complex logistics process in which it is difficult to determine the interdependence of the presented variables applying classical methods. the hybrid method has been applied to take advantages of the individual methods of artificial intelligence: fuzzy logic and neural networks. experience of an experts and information on the operations of the company for a certain group of items have been used to form the model. analysis of the validity of the model results was performed on the basis of the average relative error and it has showed that the model imitates the work of the expert in the observed company with great accuracy. sensitivity analysis has been applied which indicates that the model gives valid results. the proposed model is flexible and can be applied to various types of goods in supply chain management. key words: adaptive neuro-fuzzy inference systems, economic order quantity, supply chain management, logistics processes. 1. introduction the economy is largely in the phase of intense globalization. this does not mean only increasing the interdependence of regional economies and levels of technological integration, but also significant structural changes in the field of science, highly developed technique and its way of functioning. scientific and technological progress, in coordination with economic development, covers all areas mailto:sremacs@uns.ac.rs mailto:ilijat@uns.ac.rs mailto:miloskopic@uns.ac.rs mailto:radovic93@yahoo.com sremac et al./decis. mak. appl. manag. eng. 1 (1) (2018) 81-92 82 of the economy and its possibilities are used in the search for solutions for better organization and efficiency of flow of goods (sremac, 2013). the determination of the economic order quantity (eoq) is a logistics process that has a significant influence on the successful operation of a company (melis teksan & geunes, 2016). from the logical aspect, the determination of the eoq requires an adequate attention, since inadequate purchase can additionally burden the company’s business (abraham, 2001). on the other hand, in order to achieve a high level of service for the client, all purchase should be realized independently of their value (maddah & noueihed, 2017). many phenomena in nature, society and the economy cannot be described and it is not possible to predict their behavior by traditional mathematical methods (griffis et al., 2012). due to the lack of flexibility of this approach, the human factor compensates for the uncertainty of mathematical model using knowledge based on experience (negnevitsky, 2005) and make decisions based on data that are difficult to enter into a mathematical model (efendigil, 2014). a modern approach to determining eoq is the application of adaptive neuro-fuzzy inference systems (anfis), as one of the hybrid methods of artificial intelligence. the basic hypothesis of this paper is that it is possible to design a model on hybrid neuro-fuzzy approach of artificial intelligence to determine eoq. the next goal is to effectively use such a system in the observed company in a highly dynamic and changing business environment. one of the objectives is that the proposed system shall be flexible and applicable in other companies for other types of goods in supply chain management (scm). the basic motive for the design of such a decision support system is the development of the tool for eoq that will be able to perform complex and real processes of scm using a hybrid artificial intelligence technique. the rest of this paper is organized as follows. the relevant literature review is classified and reviewed in section 2. section 3 describes anfis used in the proposed methodology. section 4 presents proposed models and a sensitive analysis for different membership functions. conclusion remarks are drawn in section 5. 2. literature review the problem often arising and being examined is determining the amount of goods needed to meet customers' demands (lagodimos et al., 2018). a century ago, harris (1913) introduced eoq inventory model. most of the companies apply eoq model to determine the maximum level of inventory or ordering lot size (abdelaleem et al., 2017). the application of classical methods for eoq is based on limited assumptions that cannot cover the nature of modern complex logistics processes such as demand is constant in unit time, lead time is deterministic and stationary, constant price etc. (maddah & noueihed, 2017). but, making decisions in scm takes place in an environment where objectives and constraints are not and cannot often be precisely defined (latif et al., 2014; taleizadeh et al., 2016). therefore a certain approximation is required in order to obtain a high quality model of a real system where the application of artificial intelligence has an important role. consequently, individual methods of artificial intelligence (keshavarz ghorabaee et al., 2016) or their combination in the form of hybrid method are increasingly used in solving real and complex problems (teksan & geunes, 2016, zavadskas et al., 2016). some researchers (davis-sramek & fugate, 2007) interviewed a few visionaries in the field of scm and recognized the irresistible call of these individuals for modeling and simulation to be involved in the research (wallin et al., 2006). modeling of the anfis model for determining the economic order quantity 83 scm seeks for the best possible system configurations to minimize costs and increase operational efficiency in order to meet customer expectations (bowersox et al., 2010). important issue in scm is the need to make the right decision, despite the occurrence of significant ambiguity (giannoccaro et al., 2003). in addition to fluctuations in demand and delivery times, vagueness is associated with the lack of information from the production and distribution processes in scm (chatfield et al., 2013). some authors expressed uncertainty of market demand and inventory costs in the model theory of fuzzy sets (azizi et al., 2015). hereinafter, there is a review of some works from the field of scm based on neurofuzzy approach. jang (1993) first introduced the anfis method by embedding the fuzzy inference system into the framework of adaptive networks. demand uncertainty is considered in the optimization model of gupta and maranas (2003) in which by a two-stage stochastic programming model they consider all production decisions in the first stage and all the supply chain decisions in the second. yazdanichamzini et al. (2012) used anfis and artificial neural network (ann) model for modeling the gold price. guneri et al. (2011) developed a new method using anfis for the supplier selection problem. vahdani et al. (2012) presented numerous quantitative methods for supplier selection and evaluation in the literature, where the most current technique is hybrid approaches. later ozkan and inal (2014) employed anfis in supplier selection and evaluation process. several methods for eoq in scm have appeared in literature, including approaches based on a neuro-fuzzy (yazdani-chamzini et al., 2017). paul et al. (2015) presents the application of anfis and ann in inventory management problem to determine optimum inventory level. abdel-aleem et al. (2017) study and analyze the optimal lot size in a real production system to obtain the optimal production quantity. anfis has a wide application in the fields of finance, marketing, distribution, business planning, information systems, production, logistics etc. (ambukege et al., 2017; mardani et al., 2017; rajab & sharma, 2017). the route guidance system developed by pamučar & ćirović (2018) is an adaptive neuro fuzzy inference guidance system that provides instructions to drivers based upon "optimum" route solutions. 3. description adaptive neuro-fuzzy inference systems anfis are the modern class of hybrid systems of artificial intelligence. they are described as artificial neural networks characterized by fuzzy parameters. by combining two different concepts of artificial intelligence it is tried to exploit the individual strengths of fuzzy logistics and artificial neural networks in hybrid systems of homogeneous structure (figure 1). such engineered systems are increasingly used to solve everyday complex problems and with assistance of logistics experts and historical data, this approach can be designed on the basis of computer aided systems. sremac et al./decis. mak. appl. manag. eng. 1 (1) (2018) 81-92 84 figure 1. basic characteristics of fuzzy logistics and neural networks the possibility of displaying the fuzzy model in the form of a neural network is most often used in the methods of automatic determination of the parameters of the fuzzy model based on the available input-output data. the structure of adaptive neuro-fuzzy inference systems is similar to the structure of neural networks. the membership functions of the input data are mapped to the input data of the neural networks and the input-output laws are defined through the output data of the neural networks (figure 2). figure 2. the basic structure of adaptive neuro-fuzzy inference systems parameters characteristic of the corresponding membership functions change through the network learning process. calculation of these parameters is usually done on the basis of the gradient of the vector, which is a measure of the accuracy of the transfer of the fuzzy inference system of the input set into the output set for the given set of verified parameters (cetisli, 2010). basic idea of the adaptive neuro-fuzzy inference system is based on fuzzy modelling and learning methods according to the given dataset. based on the inputoutput data set, an appropriate fuzzy inference system is formed and the parameters of the membership function are calculated. the parameters of the membership functions of the fuzzy system are set using the backpropagation algorithm or a combination of the algorithm and the method of least squares. this setting allows fuzzy systems to learn on the basis of input-output data set. this learning method is similar to the method of learning neural networks. anfis model for determining the economic order quantity 85 4. the development of anfis model for determining eoq 4.1. designing the model this paper develops an adaptive neuro-fuzzy inference system model for determining the economic order quantity (anfis model eoq) based on the inputoutput data in the observed company. the formation of the proposed model consists of the following steps:  determination of input-output data set in the form customized for training of the neuro-fuzzy inference system.  the model structure with parameters is assumed, which by the rules reflects the input membership functions into output functions. the model is trained on the training data. in doing so, the parameters of the membership functions are modified according to the selected error criterion in order to get the valid model results. this way of modeling is appropriate if the training data are fully representative for all the properties that anfis model should have. in some cases, the data used to train the network contain measurement errors so they are not fully representative for all features that should be included in the model. therefore, the model should be checked using the testing data. there are two ways of testing the model. the first way is to check the model when input data are those that are not used for training. this procedure shows how accurately the model predicts the output value set and it is implemented in the paper. another way to test the model is a mathematical procedure when the data that were used for training are now used as a data set for testing and it is necessary to obtain the output with a minimal error. the model presented here was developed in the matlab version r2007b using anfis editor, included in the fuzzy logic toolbox. anfis editors only support sugeno-type fuzzy systems (tahmasebi & hezarkhani, 2010). benefits of sugeno type are that it is computationally more efficient, suitable for mathematical analysis, works well with linear, optimization and adaptive techniques. the course of the anfis model formation is presented in figure 3. figure 3. the model formation flowchart the anfis model eoq has the following structure. the input variables are: the size of demand, the level of inventory and price, while the output variable is eoq. the number of membership functions of the input variables is three, except for the input variable the size of demand which has five values. input membership functions are gaussian. the structure of the neural network is shown in figure 4. sremac et al./decis. mak. appl. manag. eng. 1 (1) (2018) 81-92 86 figure 4. fuzzy model mapped into a neural network the developed model has the form of a multilayer neural network with the propagation of the signal forwards. the first layer represents the input variable, the hidden (middle) layer represents the fuzzy rule, and the third layer is the output variable. fuzzy sets are defined in the form of link weights between nodes. settings are performed in adaptive nodes to reduce the error that occurs at the exit of the model. the error is the difference between the known output values and the values obtained at the exit from the neuro-fuzzy network. the signals on the network are spreading forwards and the bugs are spreading backwards. thus, the output numerical value approaches the optimal, i.e. the required value. the basic characteristics of the model are shown in table 1. table 1. basic characteristics of anfis model eoq the key model characteristics are: number of nodes 118 number of linear parameters 45 number of nonlinear parameters 22 total number of parameters 67 number of training data pairs 50 number of testing data pairs 10 number of fuzzy rules 45 the data set for the training of the neural network was obtained on the basis of concrete data on business operations and the survey of the logistics expert in the observed company. for training (figure 5), a hybrid optimization method was used consisting of: • backpropagation algorithms, by which the errors of variables are determined recursively from the output layer to the input layers • the methods of least squares for determining the optimal set of consequential parameters. anfis model for determining the economic order quantity 87 figure 5. training of the neural network in order to train the network, 50 input-output procurement data sets were used in the observed company, while model testing was conducted on the basis of 10 inputoutput data sets. a grid partition technique was applied to generate one model output and a hybrid optimization method as well. it was assumed that the output membership functions are of a constant type. the number of training cycles (epochs) is 500. at the output of the neural network, there is an error of 2.15 (figure 6). figure 6. results of training of anfis model eoq after the training phase, the anfis model eoq was tested on the basis of 10 inputoutput datasets, which were not used in the training of the model. the average error in testing the model is 4.03 (figure 7). sremac et al./decis. mak. appl. manag. eng. 1 (1) (2018) 81-92 88 figure 7. results of testing of anfis model eoq testing makes it possible to check the functioning of the model. output data, generated by the network, are compared with known company data. the model is not expected to function without an error, but deviations must be within the limits of the predicted tolerance. if there are large deviations, a new training network needs to be done, or it is sometimes necessary to exclude problematic data. the validity analysis of the model's results was carried out on the basis of the average relative error of the tested data (figure 8). on the basis of the testing of 10 examples of eoq determination, an average relative error of 3.28% was obtained. on the basis of this analysis it can be said that anfis model eoq gives valid results. figure 8. relative error of anfis model eoq in % 4.2. sensitive analysis one of the basic requirements when modeling is to achieve a satisfactory sensitivity of the model. this means that with certain small changes in input variables, the output from the model must also have small changes in value. the sensitive analysis of the anfis model eoq was carried out by changing the shape of the membership functions of the input variables and the number of values of the input variables as well. instead of the gaussian curves applied in the basic model, triangular, trapezoidal and bell-shaped curves were tested (table 2). in the analysis the "prod "(product of array elements) method was used for the operator "and" and "prob" (probably) method for the operator "or". two cases were tested: first, where all input variables have three values, and the other one where the first input variable, size of demand, has five values, while the other two input variables, the level of inventory and price, have three values (table 3). anfis model for determining the economic order quantity 89 table 2. sensitive analysis by changing the form of membership functions membership function triangular trapezoidal bell eoq 120 124 125 42 32 24 220 225 228 60 57 59 132 133 135 table 3. sensitive analysis by changing the number of input values* membership function triangular trapezoidal bell number of the variable values 3-3-3 5-3-3 3-3-3 5-3-3 3-3-3 5-3-3 training error 4,16 2,21 8,40 2,82 3,55 1,77 testing error 7,02 6,58 8,56 6,99 6,36 2,83 * number of epochs is set to 500. for defined cases of model sensitivity testing, the obtained results are the same or with negligible differences. this shows that the proposed anfis model eoq gives valid results. 5. conclusion the applied concept of artificial intelligence is utilized for presenting, manipulating and implementing human knowledge on the efficient management for determining the economic order quantity. adaptive neuro-fuzzy inference systems has proven to be a valuable artificial intelligence concept in determining eoq that is designed using intuition and assessment of a logistics expert. hybrid concept of artificial intelligence enabled the explanation of the system dynamics via a linguistic presentation of knowledge on a logistics process. it was used for modeling a complex linguistic system in which it is difficult to determine the interdependence of the presented variables applying other classical methods. in the paper, anfis model eoq for solving a concrete problem in a business practice was developed, following the tendency in contemporary scientific research. the model was tested and verified, and hence it can be practically applied. a sensational analysis was conducted and it gave the results of a model with negligible differences. the advantage of the proposed model is that with some minor modification, it can be applied in any company dealing with the flow of goods realization. during the research it was observed that in addition to the advantages, the applied hybrid concept of artificial intelligence also had certain flaws, and that none of the tools was universally applicable. the observed flaws are that the selection and adjustment of the membership functions of the variables are very sensitive area that has a significant impact on the results of the model. therefore, it is necessary to precisely and carefully form the logical base of the fuzzy rules. during development of the model, the neuro-fuzzy training time usually requires a large amount of data and can be very long, and therefore the need for frequent repetitions of training can make sremac et al./decis. mak. appl. manag. eng. 1 (1) (2018) 81-92 90 the application unusable. a small number of input parameters gives rough and inaccurate results, so the survey sample must be representative. in further research, current methods of multiple-criteria decision-making can be applied (pamučar et al., 2018; stević et al., 2017, yazdani-chamzini et al., 2017) and the flexibility of the proposed model can be used for determining the amount of procurement of other types of goods. acknowledgement: the paper is a part of the research done within 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(2016). hybrid multiple criteria decision-making methods: a review of applications for sustainability issues. economic research-ekonomska istraživanja, 29, 857–887. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 2, issue 1, 2019, pp. 1-12. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame1901001a * corresponding author. e-mail addresses: krish.math23@gmail.com (k. adhikary), jaga.math23@gmail.com (j. roy), kar_s_k@yahoo.com (s. kar) newsboy problem with birandom demand krishnendu adhikary 1, jagannath roy 1* and samarjit kar 1 1 department of mathematics, national institute of technology durgapur, durgapur, india received: 15 october 2018; accepted: 12 december 2018; available online: 18 december 2018. original scientific paper abstract: estimation of accurate product demand in a single period inventory model (spim) is an essential prerequisite for successfully managing the supply chain in large and medium merchandise. managers/ decision makers (dms) often find it difficult to forecast the exact inventory level of a product due to complex market situations and its volatility caused by several factors like customers uncertain behavior, natural disasters, and uncertain demand information. in order to make fruitful decisions under such complicated environment, managers seek applicable models that can be implemented in profit maximization problems. many authors studied spim (also known as newsboy problem) considering the demand as a normal random variable with fixed mean and variance. but for more practical situations the mean demand also varies time to time yielding two-folded randomness in demand distribution. thus, it becomes more difficult for dms to apprehend the actual demand having two-folded random/birandom distribution. a blend of birandom theory and newsboy model has been employed to propose birandom newsboy model (bnm) in this research to find out the optimal order quantity as well as maximize the expected profit. the practicality of the projected bnm is illustrated by a numerical example followed by a real case study of spim. the results will help dms to know how much they should order in order to maximize the expected profit and avoid potential loss from excess ordering. finally, the bnm will enhance the ability of the managers to keep parity of product demand and supply satisfying customers’ needs effectively under uncertain environment. key words: newsboy problem, uncertain variable, birandom variable, expectation. 1. introduction the classical newsboy problem (cnp) aims at determining the optimal order quantity of products, which minimize the expected total cost and / or maximize the mailto:krish.math23@gmail.com mailto:jaga.math23@gmail.com adhikary et al./decis. mak. appl. manag. eng. 2 (1) (2019) 1-12 2 expected profit in a spim. thus, the characteristics of a spim become more complicated due to complex market situations and its volatility caused by several factors like customers uncertain behavior. in response to handle such problems, the cnp is extended in many directions and several researchers have proposed modified versions of cnp. some of them made proper and thoughtful use of probability theory to solve newsboy problems where product demands follow poisson and normal distribution (hadley and whitin, 1963), weibull distribution (tadikamalla, 1978), erlang distribution (mahoney and sivazlian, 1980), compound poisson distribution (dominey and hill, 2004). gallego and moon (1993, 1994) analyzed distribution-free newsboy problems where dms have no idea about the demand distribution. the only information they have are mean and variance of demand. their newsboy models can be used as strategic tools in deciding the stock of products that have a limited selling period. this paragraph is dedicated to articulate the recent developments of spim and find the literature gap. agarwal and seshadri (2000) worked in cnp where they assumed the demand distribution as a function of selling price and the objective of the riskaverse retailers. to maximize dms expected utility they presented two models for comparing the risk-neutral retailers (who charge a higher price for less order) with a risk-averse retailer (who charge a low price). the distribution-free newsboy problem under the worst-case and best-case scenario was revealed by kamburowski (2014). further, kamburowski (2015) studied a newsboy problem where the distribution of the random variable is only known when to be non-skewed with given support, mean and variance. for the distribution-free newsboy problem, gler (2014) extended the model developed by lee and hsu (2011). here, the authors showed the expected profit increases with a proper advertisement policy while an unorganized advertising policy can have its backfire effect or make a very small improvement of the optimal profit value. ding (2013) proposed a chance constraint multi-product newsboy problem with uncertain demand and uncertain storage capacity. abdel-aal et al. (2017) studied a multi-product newsboy problem assuming the service level as a constraint to offer the dms to select the market to serve. watt and vzquez (2017) considered newsboy problem under two new assumptions. first, they assumed that the wholesaler is an expert who sets the wholesale price optimally and a newsboy can return the unsold item with some salvage value. in the second one, the salvage value acts as a standard insurance demand. sun and guo (2017) built a newsboy model with fuzzy random demand based on fuzzy random expected value model. vipin and amit (2017) proposed a loss aversion spim under alternative option and proved the rationality of the decision maker to predict the order quantity by imposing loss aversion in the newsboy model with the change of selling price and purchase cost factors. additionally, they showed the models based on utility functions perform better in forecasting the rational behavior due to loss aversion. natarajan et al. (2017) allowed asymmetry and ambiguity in newsvendor models. the effects of decision makers emergency order in spim are discussed and analyzed by pando et al. (2013) and zhang et al. (2017). zhang et al. (2017) compared two ways to treat the excess demand and came up with the better one. in the aforementioned works, the product demand is assumed to be either normally distributed with 𝑁(𝜇, 𝜎 2) or exponentially distributed with constant mean (𝜆) or somewhere distribution free. but the dms face difficulties to forecast the exact demands of products in many practical problems. the demand distribution changes dynamically from time to time, which yields randomness in the mean demand. for example, the demand (𝐷) is normally distributed with 𝐷 ~ 𝑁(𝜇 ; 400) where 𝜇 ~ 𝑈(3000; 4000) or 𝐷 ~ 𝑁(𝜇 ; 400) with 𝜇 ~ 𝑒𝑥𝑝(0.0003). however, in newsboy newsboy problem with birandom demand 3 problems, it is largely appreciated to consider demand variable having the standard normal distribution. from the probabilistic viewpoint and the above arguments, it would be more realistic to assume the values of 𝜇 and 𝜎 2 should also be treated as random variables. for such cases, it is more convincing and practical to represent the product demand as birandom variable effectively captures two-folded randomness. traditional probabilistic approaches cannot handle such complicated real world problems. in response, peng and liu (2007) developed a birandom theory to tackle such problems. zhaojun and liu (2013) showed the common formula on birandom variable and further, several researchers used this theory to solve inventory problems in the birandom environment. in the recent years, birandom theory is widely accepted as the mathematical language of uncertainty. some more notable extensions and applications of newsboy problem can be found in the recent literature (abdel-malek and otegbeye 2013; chen and ho 2013). from its inception until today, birandom theory has progressed and been applied to different areas. xu and zhou (2009) introduced a class of multiple objective decision making problems using birandom variables and by transforming the birandom uncertain problem into its crisp equivalent form through expected value operator and used it in the flow shop scheduling problem. a portfolio selection problem is analysed by yan (2009) assumed the security returns as birandom variables. xu and ding (2011) developed the general chance constrained multi objective linear programming model with birandom parameters for solving a vendor selection problem. they presented a crisp equivalent model for a special case and gave a traditional method to solve the crisp model. wang et al. (2012) established a class of job search problem with birandom variables, where the job searcher examined job offers from a finite set jobs having equivalent probability. a multi-mode resource constrained project scheduling problem (zhang and xu, 2013) of drilling grounding construction projects considering the uncertain parameters as birandom variables. in the earlier year, xu et al. (2012) used birandom theory to develop the nonlinear multi objective bi-level models for finding the minimum cost in a network flow problem dealing a large scale construction project. tavana et al. (2013) measured the efficiencies of decision making units after developing a data envelopment analysis (dea) model with birandom input and output data. nevertheless, many more real life applications can be found in the following literature: a multi-objective birandom inventory problem (tao and xu, 2013), a birandom multi-objective scheduling problem (xu et al., 2013) in ship transportation, optimal portfolio selection with birandom returns(cao and shan, 2013), a modified genetic algorithm (maity et al., 2015), chance-constrained programming model for municipal waste management with birandom variables (zhou et al., 2015), and the ccus (carbon capture, utilization, and storage) management system in birandom environment (wang et al., 2017). to the best of our knowledge, no researcher has investigated the spim with birandom demand till date. with these considerations, we discuss an inventory model with single period considering the demand for the birandom variable. we solve a real problem using the presented model in searching the optimum order quantity for maximum profit by using the expected value model. the principal aim of this paper is to deliver basic knowledge and suggest precise results of a complex inventory practical problem for the management. the remaining part of our paper is presented in the following way. section 2 presents some basic knowledge of birandom variable and related theorems. section 3 introduces the birandom simulation for finding the expected value of the birandom http://www.dmame.org/index.php/dmame/article/view/24 adhikary et al./decis. mak. appl. manag. eng. 2 (1) (2019) 1-12 4 variable. in section 4, we provide a mathematical model for newsboy problem with birandom demand. a numerical example is discussed in section 5. to validate the applicability of the proposed model we discuss a real case study in section 6. finally, the conclusion and future research directions are presented in section 7. 2. preliminaries in this section we discuss the basic notations of birandom variables. 2.1. birandom variable roughly speaking a birandom variable is a random variable of a random variable, i.e., a function defined from a probability space to a collection of random variables is said to be a birandom variable. the formal definition of birandom variable and related theorems are defined in the following way. definition 1 (peng and liu, 2007). a birandom variable 𝜉 is a mapping from a probability space (𝛺, 𝐴, 𝑃𝑟) to a collection 𝑆 of a random variable such that for any borel subset 𝐵 of the real line ℜ the induced function 𝑃𝑟 {𝜉(𝜔) ∈ 𝐵} is a measurable function with respect to 𝜔. for each given borel subset 𝐵 of the real line ℜ, the function 𝑃𝑟 {𝜉(𝜔) ∈ 𝐵} is a random variable defined on the probability space (𝜔, 𝐴, 𝑃𝑟). lemma 1 (peng and liu 2007). let an n dimensional birandom vector 𝜉 = (𝜉1, 𝜉2, … , 𝜉𝑛 ) and 𝑓: ℜ 𝑛 → ℜ be a measurable function. then 𝑓(𝜉) is a birandom variable. let two probability spaces (ω1, 𝐴1, pr1) and (ω2, 𝐴2, pr2), 𝜉1 and 𝜉2 be two birandom variables respectively taken from that probability spaces. then 𝜉 = 𝜉1 + 𝜉2 is a birandom variable on (ω1 × ω2, 𝐴1 × 𝐴2, pr1 × pr2) defined by        1 2 1 1 2 2 1 2 1 2, , , ω ω             widely, for the n-tuple operation on birandom variables defined as follows. let a borel measurable function defined as 𝑓: ℜ𝑛 → ℜ and 𝜉𝑖 be birandom variable defined on (ω𝑖 , 𝐴𝑖 , pri), 𝑖 = 1,2, … , 𝑛 respectively. then 𝜉 = 𝑓(𝜉1, 𝜉2, … , 𝜉𝑛 ) is birandom variable on (ω1 × ω2 × … × ω𝑛 , 𝐴1 × 𝐴2 × … × 𝐴𝑛, pr1 × pr2 × … × prn), defined by            1 2 1 1 2 2 1 2 1 2 , , , , , ., , , ., ω ω ω n n n n n f                     2.2. expected value of birandom variables we can transform the complex uncertain problems into their equivalent crisp models, which will be easier to solve. generally, expected value operator is applied to transform the birandom problem into its deterministic value model for calculating the objective functional value. first, we present the definition of the expected value operator of a birandom variable and then the expected value model of spim. the effective tool of an uncertain variable is expectation, which is applied in a different field of applications. therefore, the idea of expected value of birandom variable is useful. the expected value operator of birandom variable is defined as follows. newsboy problem with birandom demand 5 definition 2 (peng and liu, 2007). let 𝜉 be a birandom variable defined on the probability (ω, 𝐴, 𝑃𝑟). then the expected value of birandom variable 𝜉 is defined as         0 0 pr ω| pr ω|e e t dt e t dt                       (1) provided that at least one of the above two integrals is finite. lemma 2 (peng and liu, 2007). let 𝜉 be a birandom variable defined on the probability (𝜔, 𝐴, 𝑃𝑟). if the expected value 𝐸[𝜉(𝜔)] of the random variable 𝜉(𝜔) is finite for each 𝜔, then 𝐸[𝜉(𝜔)] is a random variable on (𝜔, 𝐴, 𝑃𝑟). lemma 3 (peng and liu, 2007). let us consider two birandom variable 𝜉 and 𝜂 with finite expected value, then for any two real numbers a and b, we have      e a b ae be      we know that a function from a probability space (ω, 𝐴, 𝑃𝑟) to a collection of random variables is called a birandom variable, from the definition of birandom variable. birandom variables are two type. they are either discrete birandom variable or continuous birandom variable. expectation theory of birandom variables will be discussed in the current subsection. definition 3 (xu et al., 2009). for a birandom variable 𝜉, in the probability space (𝛺, 𝐴, 𝑃𝑟), if 𝜉(𝜔) is a random variable with a continuous distribution function when 𝜔 ∈ 𝛺 and its expected value 𝐸[𝜉(𝜔)] is a birandom variable. then we call 𝜉 continuous birandom variable. definition 4 (xu et al., 2009). suppose 𝜉 is a birandom variable, then 𝜉(𝜔) is a random variable. if 𝑓(𝑥, 𝜉) be the density function of 𝜉(𝜔), and     ωx e xf x dx        (2) then the density function of birandom variable 𝜉 is 𝑓(𝑥, 𝜉). definition 5 (xu et al., 2009). for the continuous birandom variable 𝜉, if its density function is 𝑓(𝑥), we can define the expected value of 𝜉 as follows       0 0 ω ωx x e pr xf x r dr pr xf x r dr                              (3) definition 6 (xu et al., 2009). if the density function of a birandom variable 𝜉 is 𝑓(𝑥, 𝜉) and 𝑔(𝑥) is a continuous function. then expectation for the birandom variable 𝑔(𝜉) is defined as           0 0 ω ωx x e g pr g x f x r dr pr g x f x r dr                                 (4) theorem 1 (xu et al., 2009). let 𝑓(𝑥, 𝜉) is the density function of the birandom variable 𝜉. then the expected value of 𝜉 exists if only if the expected value of random variable 𝜉(𝜔) exists. theorem 2 (xu et al., 2009). the expectation of a birandom variable. 𝜉~𝑁(𝜇, 𝜎 2), where, 𝜇~u(a, b) is 𝑎+𝑏 2 . http://www.dmame.org/index.php/dmame/article/view/24 adhikary et al./decis. mak. appl. manag. eng. 2 (1) (2019) 1-12 6 theorem 3. the expectation of a birandom variable 𝜉~𝑁(𝜇, 𝜎 2) where 𝜇~exp (𝜆) is 1 𝜆 . proof: by definition (2), we know         0 0 pr | pr | e e t dt e t dt                       since 𝜇~ exp(𝜆), and obviously 𝐸[𝜉(𝜔)] = 𝜇, by definition 4 and 5, the above function can be transformed as follows,       0 0 pr pre t dt t dt          since, 𝜇~ exp(𝜆), and we know that the density function and the distribution function of exponential distribution are as follows,    , 0, xf x e x and       1 , 0, .xf x e x    according to the definition of the distribution function we can obtain the following two functions from the distribution function pr(𝜇 ≤ 𝑥) = 1 − 𝑒 −𝜆𝑥 , 0 ≤ 𝑥 < ∞ 𝑎𝑛𝑑 pr(𝜇 ≥ 𝑥) = 𝑒 −𝜆𝑥 , 0 ≤ 𝑥 < ∞ obviously, ∫ pr{𝜇 ≤ 𝑡} 𝑑𝑡 = 0 0 −∞ therefore 𝐸[𝜉] = ∫ pr{𝜇 ≥ 𝑡} 𝑑𝑡 = ∫ 𝑒 −𝜆𝑡 𝑑𝑡 = [ 𝑒 −𝜆𝑡 𝜆 ] 0 ∞ = 1 𝜆 ∞ 0 ∞ 0 . however, it is very hard to accomplish the mathematical expression of expected value for all types of birandom variables. but using birandom simulation, we could calculate the expected value of birandom variables, with the help of strong number law. 3. birandom simulation let (𝛺, 𝐴, 𝑃𝑟), be a probability space and a 𝑓: ℜ𝑛 → ℜ be a measurable function. consider that 𝜉 is an 𝑛 − dimension birandom vector on the given probability space. now we have to find the expectation 𝐸[𝑓(𝜉)] of birandom variable. using stochastic simulation, we can find the expected value for every ω ∈ ω. here, we have used an algorithm for birandom simulation to find the expectation of 𝐸[𝑓(𝜉(𝜔))], which is defined as follows. algorithm: step 1. start step 2. set l=0 and n= number of iteration step 3. sample 𝜔 from 𝛺 according to the probability measure 𝑃𝑟 newsboy problem with birandom demand 7 step 4. 𝐸[𝑓(𝜉(𝜔))] is find by the stochastic simulation. step 5. then 𝑙 ← 𝑙 + 𝐸[𝑓(𝜉(𝜔))] step 6. repeat the steps from second to fifth steps 𝑁 ttimes. step 7. 𝐸[𝑓(𝜉(𝜔))] = 𝑙 𝑛 . step 8. stop 4. mathematical formulation we are assuming a single period inventory problem with single product. here all the costs (buying cost and selling cost) are deterministic. salvage value is taken which is also deterministic. but the demand is birandom variable. the mathematical notation of a birandom newsboy problem is defined as follows: 𝜉 ̅ : the demand of market, a birandom variable 𝑥 : the quantity which to be order, a decision variable 𝑝 = 𝑐(1 + 𝑚) : selling price per unit 𝑠 = 𝑐(1 − 𝑑) : salvage value per unit 𝑐 : purchasing cost per unit 𝑔 (𝑥, 𝜉 ̅) : the profit for the order quantity 𝑥 and demand 𝜉 ̅ 𝜇 : expected value of birandom demand 𝜉 ̅ 𝜎 2 : variance of the birandom demand 𝜉̅ 𝑚 : mark-up, i.e., return per dollar on unit sold 𝑑 : discount rate, i.e., loss per dollar on unit unsold 𝑥 + = max{𝑥, 0} : the positive part of x then the profit can be expressed as      , min ,g x p x s x cx            (5) now min (𝑥, 𝜉 ̅ ) = 𝜉̅ − (𝜉̅ − x) + where      x x x          (6)       , g x p s x cx p s x            (7) since the demand for the product is birandom, the profit function 𝑔 (𝑥, 𝜉̅ ) is also consists of birandom variable. hence, the expectation criteria is used for handling the birandom variable. therefore, to find the optimal quantity, the decision maker will maximize the total expected value. we can write the expected profit as          π x p s s c x p s e x          (8) http://www.dmame.org/index.php/dmame/article/view/24 adhikary et al./decis. mak. appl. manag. eng. 2 (1) (2019) 1-12 8 or using the definition of m and d, as        π [x c m d xd m d e x         (9) the information of 𝜉 ̅is known. to maximize the profit function, we need the following lemma. lemma 4. for given a birandom variable 𝜉,̅ we have the following inequality,       1 22 2 2 x x e x               (10) proof: notice that (𝜉̅ − 𝑥) + = |�̅̃�−𝑥|+(�̅̃�−𝑥) 2 the result follows by taking expectations and by using the cauchy-schwarz inequality 𝐸 [𝜉̅ − 𝑥] ≤ [𝐸 |𝜉̅ − 𝑥| 2 ] 1 2 = [𝜎 2 + (𝑥 − 𝜇)2] 1 2 by using the lemma (4) the equation (9) will be rewritten as           1 22 2 π 2 x x x c m d xd m d                          (11) it is easy to validate that equation no (11) is strictly convex in 𝑥. upon setting the derivative to zero and solving for 𝑥 we obtain the ordering rule 1 1 2 2 * 2 m d x d m                     (12) 5. numerical example assume that the unit purchase price of a perishable product is 𝑐 = $40, the unit selling price is 𝑝 = $60, and there is no salvage value (𝑠 = 0). thus, 𝑚 = 𝑝 𝑐 − 1 = 60 40 − 1 = 1 2 . discount rate 𝑑 = 1 − 𝑠 𝑐 = 1. further assume that the product demand is a birandom variable with normal distribution 𝑁(µ1, 400), and µ1 ∼ 𝑒𝑥𝑝(0.0003). from theorem (3), we have 𝜉 ∼ 𝑁(µ1, 400), where, 𝜆 = 0.0003. hence, by theorem (3) we can say that the mean (µ) of the birandom variable (𝜉) is 1 𝜆 = 1 0.0003 = 3333.33, and 𝜎 2 = 400. now it remains to calculate the optimal order quantity and expected profit. for this purpose, we apply equation (12) and obtained the optimal order quantity, 𝑥 ∗ = 3326. hence, the expected profit is, 𝛱 = $66100. newsboy problem with birandom demand 9 6. a case study to endorse the model developed in this study, we sent our projected framework to five leading fish merchants in west bengal, india. they sell only the freshest and best quality fishes, and maintains quality control at every stage of packaging and delivery in many different parts or areas of west bengal. among them two firms positively responded to explore this research proposal and we conducted necessary preliminary tasks on these companies. we selected a reputed fish merchant (“lakshmi fish enterprise”, name changed), situated in the ”southern” west bengal, which has several operational units nationwide. our objective is to incorporate the perceptions of all participants (customer/retailer/company mangers) in the fish industry and to achieve its comprehensive outcomes since this research is purely grounded on birandom product demand information obtained from experts in the business. in this paper, we consider the perspectives of a wholesale fish merchant who buys fishes from the company, having a large market share in kolkata zone. in west bengal, fish merchants generally sell a special fish in monsoon. the name of this fish is hilsa. the business of this fish is a good example of spim. the business of this fish totally depends on its demand and supply in the monsoon season. merchants have to decide how many fishes should be purchased from his or her supplier depending on the customer’s demand. buying more amount of hilsa may not bring him more profit. rather it can cause him a great loss since it cannot be preserved for long periods and the expired fish has no market value. if he/she buys too few amount of hilsa he/she will lose the opportunity of making a higher profit. thus, the actual inventory level cannot be determined precisely in such complex situation. it may be assumed for simplicity that the fish demand follows normal distribution. but under such circumferences, the manager looks after of some previous data, and finds the mean demand of ”hilsa” is also a random variable. this leads us to consider the ”hilsa” demand as a birandom variable. each such fish sells for $60 and costs for the shop owner $40. investigating the previous year data, the decision maker decides the demand follows the two types twofolded random variable (birandom). scenario 1: normal distribution 𝑁~(𝜇1, 400), with 𝜇1~𝑈(3000, 4000). scenario 2: normal distribution 𝑁~(𝜇1, 400), with 𝜇1~𝑁(3500, 500). therefore, according to our proposed model we have 𝑐 = 40, 𝑝 = 60, and there is no salvage value i.e., 𝑠 = 0. in scenario 1, by theorem (2) the mean of the birandom variable is µ = 3500, and 𝜎 2 = 400. therefore the optimal order quantity 𝑥 ∗ = 3492. expected profit 𝛱 = $69434. and in scenario 2, using birandom simulation we get the mean of the birandom simulation µ = 3508, and 𝜎 2 = 400. therefore, the optimal order quantity 𝑥 = 3500, expected profit = $69594. finally, we shared the outcomes of this research work with the managers of our case enterprise, they are satisfied with the outcomes and willing to accept this result for their monsoon business of ”hilsa”. 7. conclusion in this paper we have proposed a newsboy problem where the demand is considered as birandom variable. the market volatility and uncertainty in customers’ behavior make the demand of the product a birandom variable. we use the expected value model for handling this birandom variable and to convert the bnm into its equivalent deterministic model. we discuss a case study of fish merchant to validate the usefulness and applicability of the proposed model. in this proposed model 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(2000). impact of uncertainty and risk aversion on price and order quantity in the newsvendor problem. manufacturing & service operations management, 2(4), 410-423. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 4, issue 2, 2021, pp. 241-256. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame210402241n * corresponding author. e-mail addresses: gnegiji@gmail.com (g. negi), anuj4march@gmail.com (a. kumar), pant.sangeet@gmail.com (s. pant), mangeyram@gmail.com (m. ram) optimization of complex system reliability using hybrid grey wolf optimizer ganga negi1, anuj kumar2, sangeeta pant2*, mangey ram3,4 1 department of mathematics, graphic era deemed to be university, dehradun, india 2 department of mathematics, university of petroleum and energy studies, dehradun, india 3 department of mathematics, computer science and engineering, graphic era, dehradun, india 4 institute of advanced manufacturing technologies, peter the great st., petersburg polytechnic university, saint petersburg, russia received: 30 march 2021; accepted: 20 july 2021; available online: 18 august 2021. original research paper abstract: reliability allocation to increase the total reliability has become a successful way to increase the efficiency of the complex industrial system designs. a lot of research in the past have tackled this problem to a great extent. this is evident from the different techniques developed so far to achieve the target. metaheuristics like simulated annealing, tabu search (ts), particle swarm optimization (pso), cuckoo search optimization (cs), genetic algorithm (ga), grey wolf optimization technique (gwo) etc. have been used in the recent years. this paper proposes a framework for implementing hybrid pso-gwo algorithm (hpsogwo) for solving reliability allocation and optimization problems of complex bridge system and life support system in space capsule. the supremacy/competitiveness of the proposed framework are demonstrated from the numerical experiments. comparison of the results obtained by hpsogwo with previously used algorithms named pso and gwo shows that in one problem named the complex bridge system, the hpsogwo uses lesser number of function evaluations as compared to pso and gwo. hence, the overall solutions obtained by hpsogwo are not only comparable to the previously obtained results by some of the other well-known optimization methods, but also better than that. keywords: cost function, metaheuristics, reliability allocation problems, particle swarm optimization (pso), grey wolf optimizer (gwo), hybrid psogwo algorithm (hpsogwo). mailto:gnegiji@gmail.com mailto:anuj4march@gmail.com mailto:pant.sangeet@gmail.com mailto:mangeyram@gmail.com negi et al./decis. mak. appl. manag. eng. 4 (2) (2021) 241-256 242 1. introduction the present-day real-world problems of engineering have reached to an advanced level that have motivated the researchers to find ways to increase the efficiency of complex systems. reliability being the main criteria for this task and the attention of the researchers is diverted towards allocating reliability to the complex systems and the components. nowadays there has been lot of research in the field of reliability optimization considering its wide applications in real life and industry (atiqullah & rao, 1993; pham et al., 1995; eiben & schippers, 1998; kishor et al., 2009; jayabarathi et al., 2016;). such improvisations increase the efficiency and give better results for stochastic nonlinear optimization problems (ramírez-rosado & bernal-agustín, 2001). the constraints of weight, budget, volume, can be appropriately set, in reliability allocation problem (rap) to optimize reliability of the system (kishor et al., 2007; pant et al., 2015). due to the immense applications, rap problems have attracted the attention of many researchers to explore this technology (mohan & shanker, 1987; majety et al., 1999; pant & singh, 2011; kumar et al., 2016). basically, reliability optimization problems can be classified into three categories depending upon the decision variables involved. these are (i) reliability allocation (li et al., 2008; mirjalili et al., 2016; pant et al., 2017; kumar et al., 2019a, 2019b;) (ii) redundancy allocation (atiqullah & rao, 1993; misra & sharma, 1991a, 1991b; yang & deb, 2009) and (iii) reliabilityredundancy allocation (sakawa, 1978; coelho, 2009; deep & deepti, 2009). going by the concept of mathematical programming reliability allocation is a continuous nonlinear programming problem (nlp). redundancy allocation is a pure integer nonlinear programming problem (inlp) for nonlinear polynomial hard problems. reliability-redundancy allocation is covered under mixed integer nonlinear programming problem (minlp) for solving problems of nonconvex nature and combinatorial search space. last few decades have witnessed much research in the field of reliability allocation problem (rap) and reliability optimization by researchers to solve single objective and multiple objective optimization problem. basically, the solutions techniques used so far to solve rap and optimization problems are approximation, exact, heuristic and metaheuristic methods. among these are exact solution techniques for rap like the cutting plane algorithm was proposed by majety et al. (1999) with discrete-cost reliability data for components and other such techniques by hikita et al. (1986, 1992). random search algorithm for rap presented by mohan & shankar (1987) for complex system reliability optimization. three levels decomposition approach the khun tucker multiplier method for rap was given by salazar et al. (2006). among the metaheuristic techniques for rap ant colony technique applied by shelokar et al. (2002); nsga 2 by kishore et al. (2007, 2009); pso by pant et al. (2011); csa by kumar et al. (2016). these optimization techniques yield solutions for problems of convex nature and monotonicity. 2. literature review in order to solve complex reliability allocations problems and reliability redundancy allocation problems which are nonlinear optimization problems of nonconvex nature and combinatorial search spaces more advanced algorithms called the metaheuristics have been formulated. these require lot of computational effort to find optimal solutions. as proposed by wolpert & macready (1997) that one type of optimization algorithm is not enough for all optimization problems. so, some researchers are constantly working on developing different types of nature inspired optimization of complex system reliability using hybrid grey wolf optimizer 243 meta-heuristics technique. some of them recently developed are evolutionary algorithm (ea) (ramírez-rosado & bernal-agustín, 2001), ant colony optimization (aco), (zha et al., 2007; dorigo & gambardella, 1997) particle swarm optimization algorithm (pso) (eberhart & kennedy, 1995; kennedy & eberhart, 1997; hu & eberhart, 2002, pant & singh, 2011) grey wolf optimization technique (gwo) (mirjalili et al. 2014; fouad et al., 2015; jayabarathi et al., 2016; mosavi et al., 2016; kumar et al., 2017; kumar et al., 2019a, 2019b; pant et al., 2019;), flower pollination algorithm (pant et al., 2017) and cuckoo search algorithm (csa) (yang & deb, 2009). the detailed reviews of reliability optimization especially gwo, pso optimization techniques are given by kuo and prasad (2000); negi et al. (2020); padhye et al. (2009); uniyal et. al. (2020). these previous researches have led to the development of some of the recent researches in the field of metaheuristic algorithms and hybrid metaheuristic algorithms and their applications. hassan & rashid (2021) proposed a new evolutionary clustering algorithm (eca) based on social class ranking and meta-heuristic algorithms for stochastically analysing heterogeneous and multifeatured datasets. rahman & rashid (2020) presented the idea of learner performance-based behavior algorithm lpb inspired from the process of accepting graduated learners from high school in different departments at university and has a greater ability to deal with the large optimization problems. a collaborative working approach to path finding was introduced by shamsaldin et al. (2019) in the form of donkey and smuggler optimization algorithm to solve different problems such as tsp, packet routing, and ambulance routing. abdullah & ahmed (2019) proposed fitness dependent optimizer inspired by the bee swarming reproductive process which uses the problem fitness function value to produce weights for guiding during the exploration and exploitation phases. some of the modified and hybrid algorithms have also been in the recent years to solve many real-world engineering problems. a new k-means grey wolf algorithm was developed by mohammed et al. (2021) to enhance the limitations of the wolves’ searching process of attacking gray wolves. a novel hybrid woa-gwo presented by mohammed & rashid (2020) by embedding the hunting mechanism of gwo into the woa exploitation phase with the enhanced exploration for global numerical optimization and to solve the pressure vessel design problem. mohammed et al. (2019) introduced a systematic and meta-analysis survey of whale optimization algorithm modifying and hybridizing woa algorithm with bat algorithm in order to avoid local stagnation as well as increase the rate of convergence to achieve the global optimum solution. ibrahim et al. (2020) presented a hybrid meta-heuristic algorithm of shuffled frog leaping algorithm and genetic algorithm (sfga), an energy efficient service composition mechanism consuming minimum cost, response time and energy in a mobile cloud environment as compared to other algorithms. muhammed et al. (2020) proposed an improved fitness-dependent optimizer algorithm ifdoa by first doing the randomization and then minimization of the weight fitness values using it in aperiodic antenna array designs. to forecast students’ outcomes by improving the faculty and students’ learning experiences rashid et al. (2019) presented a hybrid system a multi hidden recurrent neural network with a modified grey wolf optimizer. mukherjee et al. (2021) presented the idea of a multi-objective antlion optimizer for the ring tree problem with secondary sub-depots (mortpssd), to overcome the problems of telecommunication and logistics networks by minimizing the circuits’ total routing cost. in addition to the above optimizer for secondary depots mukherjee et al. (2021) introduced a modified discrete antlion optimizer for the ring star problem https://www.sciencedirect.com/science/article/pii/s1110866520301419#! javascript:; javascript:; https://ieeexplore.ieee.org/author/37086811234 https://ieeexplore.ieee.org/author/37086812554 https://ieeexplore.ieee.org/author/37086812554 https://ieeexplore.ieee.org/author/37086812554 https://ieeexplore.ieee.org/author/37086812554 https://ieeexplore.ieee.org/author/37086812554 https://ieeexplore.ieee.org/author/37086812554 https://www.emerald.com/insight/search?q=hardi%20m.%20mohammed https://ieeexplore.ieee.org/author/37086812554 https://link.springer.com/article/10.1007%2fs00521-020-04823-9#auth-hardi-mohammed https://link.springer.com/article/10.1007%2fs00521-020-04823-9#auth-tarik-rashid https://ieeexplore.ieee.org/author/37086812554 https://ieeexplore.ieee.org/author/37086812554 https://ieeexplore.ieee.org/author/37086812554 https://ieeexplore.ieee.org/author/37087408603 https://link.springer.com/article/10.1007/s12351-021-00623-8#auth-anupam-mukherjee https://link.springer.com/article/10.1007/s12351-021-00623-8#auth-anupam-mukherjee negi et al./decis. mak. appl. manag. eng. 4 (2) (2021) 241-256 244 (rspssd) which overcomes the challenge of minimizing the cost by selecting the suitable primary and secondary subdepots. in this paper, we present use of hybrid optimization technique called hybrid psogwo for reliability allocation problems. the idea is to apply the hpsogwo algorithm to minimize the cost of the complex bridge system and life support system in a space capsule resulting in better performance in terms of number of cost function evaluations and number of search agents used in pso and gwo algorithms individually. both pso and gwo are population-based swarm intelligence (si) techniques. both involve less and only suitable parameters, which have easy application and execution together with optimum convergence to the global solution. that’s why it yields better results than other metaheuristics. section 2 consists of detailed explanation of the particle swarm optimization technique. section 3 involves description of the grey wolf optimization and section 4 gives an overview of hybrid algorithms and describes hybrid pso gwo algorithm (hpsogwo). the formulation of the mathematical models for the proposed problems have been presented in section 5. section 6 analyses the results of the optimization techniques used. section 7 presents the conclusion and scope for further results. 3. particle swarm optimization technique (pso) particle swarm optimization (pso) simulates the social behaviour of birds of a flock. (kennedy & eberhart, 1997; hikita et al., 1992; pant & singh, 2011; abd-elazim & ali, 2015). it is a population-based optimization technique. the randomly generated population of the initial swarm or the particles and their random velocities start the initial process of algorithm. pbest represents the personal best position of each particle whereas gbest denotes the particle with the best value of fitness and hence called the global best particle. in the 𝐷-dimensional search space 𝑋𝑖 = (𝑥𝑖 1, 𝑥𝑖 2 , … … … … . 𝑥𝑖 𝐷)r and 𝑉𝑖 = (𝑣𝑖 1, 𝑣𝑖 2 , … … … … . 𝑣𝑖 𝐷 )r denote the position and velocity of the 𝑖𝑡ℎ particle whereas the previous best position of the 𝑖𝑡ℎ particle is denoted by 𝑃𝑖 = (𝑝𝑖 1, 𝑝 , … … … … . 𝑝𝑖 𝐷 )r . according to the fitness the best particle is denoted by 𝑃𝑔 = (𝑝𝑔 1 , 𝑝𝑔 2 , … … … … . 𝑝𝑔 𝐷 )r which is the global best particle. the change in the position and velocities are expressed by the equations: (kennedy & eberhart,1997; hikita et al., 1992; abd-elazim & ali, 2015) 𝑉𝑖𝑑 𝑘+1 = { 𝑉𝑚𝑎𝑥 , 𝑖𝑓 𝑉𝑖𝑑 𝑘+1 > 𝑉𝑚𝑎𝑥 −𝑉𝑚𝑎𝑥 , 𝑖𝑓 𝑉𝑖𝑑 𝑘+1 < − 𝑉𝑚𝑎𝑥 𝑉𝑖𝑑 𝑘+1, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (1) 𝑥𝑖𝑑 𝑘+1(𝑡 + 1) = 𝑥𝑖𝑑 𝑘 (𝑡) + 𝑣𝑖𝑑 𝑘+1 (𝑡 + 1) (2) here 𝑖 = 1, 2, 3 … . . 𝑁; 𝑁 = swarm size, 𝑘 =iteration number, 𝑑 =1, 2, 3…. 𝐷, 𝑤 = inertia weight, (for controlling the momentum of the particle by weighing the contribution of the previous velocity), 𝑐1 and 𝑐2 are the positive acceleration coefficients; 𝑟1and 𝑟2 are the random numbers between 0and 1. the variations in 𝑐1and 𝑐2 with the time are represented by following equations respectively 𝑐1(𝑡) = 𝑐1𝑖 + (𝑐1𝑓 − 𝑐1𝑖 ) ∗ 𝐼𝑇𝐸𝑅/𝐼𝑇𝐸𝑅𝑀𝐴𝑋 (3) 𝑐2(𝑡) = 𝑐2𝑖 + (𝑐2𝑓 − 𝑐2𝑖 ) ∗ 𝐼𝑇𝐸𝑅/𝐼𝑇𝐸𝑅𝑀𝐴𝑋 (4) here, initially value of 𝑐1is kept large and value of 𝑐2 is kept small to ensure enough exploration of the search space to avoid local stagnation. this will lead to the global best solution in the long run. then, small value of 𝑐1and large value 𝑐2 leads to the optimization of complex system reliability using hybrid grey wolf optimizer 245 population best that is the global optimum solution. the maximum velocity and position that the particle can attain in each dimension are given by the equation as follows: 𝑉𝑖𝑑 𝑘+1 = { 𝑉𝑚𝑎𝑥 , 𝑖𝑓 𝑉𝑖𝑑 𝑘+1 > 𝑉𝑚𝑎𝑥 −𝑉𝑚𝑎𝑥 , 𝑖𝑓 𝑉𝑖𝑑 𝑘+1 < − 𝑉𝑚𝑎𝑥 𝑉𝑖𝑑 𝑘+1, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (5) 𝑥𝑖𝑑 𝑘+1 = { 𝑋𝑚𝑎𝑥, 𝑖𝑓 𝑥𝑖𝑑 𝑘+1 > 𝑋𝑚𝑎𝑥 𝑋𝑚𝑖𝑛 , 𝑖𝑓 𝑥𝑖𝑑 𝑘+1 < 𝑋𝑚𝑎𝑥 𝑥𝑖𝑑 𝑘+1 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑤𝑒 (6) 4. grey wolf optimization 4.1. the guiding factor for the algorithm gwo presented by mirjalili et al. (2014) is an optimization technique which is based on the hierarchical behaviour and social intelligence of the wolves. the entire mechanism of hunting is carried out by the four categories of wolves together. each category of wolf has a particular role. alpha, the leader category takes the decisions regarding the whole process and the others follow them. thus, gwo algorithm based on this very principle is used to find the global optimum solution. the next in the hierarchy are the beta followed by delta and omega. these four initially become the candidates of solution and which are improved in the gradually in further iterations. 4.2. mathematical model formulation of the gwo algorithm the model comprises of:  surveying  surrounding  attacking the whole process of change of position of the attacking wolves is shown by the following equations constructed to carry out the simulation are as follows. (mirjalili et al. 2014) 𝐷 = |𝐶. 𝑋𝑝(𝑡) − 𝑋(𝑡)| (7) 𝑋(𝑡 + 1) = 𝑋(𝑡) − 𝐴. 𝐷 (8) note that in the equations, vectors are used so they are applicable to any number of dimensions. 𝑋(𝑡) , 𝑋(𝑡 + 1) show the present and the new locations of the wolf. the location of the prey is represented by the vector d. following equations are useful to calculate the value of a and c: 𝐴 = 2𝑎. 𝑟1 − 𝑎 (9) 𝐶 = 2. r2 (10) where, r1 .and r2 are random vectors in the interval [0,1]. the components of vector a are linearly decreased from 2 to 0 over the course of iterations. the value of a ranges from -2 to 2 as there are random variables in the expression. it is supposed that, alpha, beta and delta are the three best solutions in gwo as they have good idea of the negi et al./decis. mak. appl. manag. eng. 4 (2) (2021) 241-256 246 location because they are the strongest in the entire population. so, the other wolf should try to update their position as follows. 𝑋(𝑡 + 1) = 1 3 𝑋1 + 1 3 𝑋2 + 1 3 𝑋3 (11) where, 𝑋1, 𝑋2, 𝑋3 are calculated with the equations: 𝑋1 = 𝑋𝛼 (𝑡) − 𝐴1. 𝐷𝛼 𝑋2 = 𝑋𝛽 (𝑡) − 𝐴2. 𝐷𝛽 𝑋3 = 𝑋𝛿 (𝑡) − 𝐴3. 𝐷𝛿 (12) here, 𝐷𝛼 , 𝐷𝛽 , 𝐷𝛿 are calculated as follows: 𝐷𝛼 = |𝐶1. 𝑋𝛼 − 𝑋| 𝐷𝛽 = |𝐶2. 𝑋𝛽 − 𝑋| 𝐷𝛿 = |𝐶3. 𝑋𝛿 − 𝑋| (13) pseudo code of gwo is given in figure 1 (mirjalili et al., 2014). figure1. pseudo code of the gwo algorithm 4.3. balancing of the effective hunting mechanisms: it is very essential to do enough surveying before attacking the prey so as to make the hunting mechanism a success. the leading wolves decide and the wolves following the leaders can then take the appropriate positions to encircle the prey. for this parameter a has to be chosen so as to get the suitable value of a correspondingly which should be between -1 and 1. exploration is followed by exploitation. to stimulate proper exploitation of the available conditions, the parameter setting requires iai < 1. the success of the exploitation is dependent to a great extent on rigorous and balanced exploration so that the result is not stagnated and unrefined. gwo efficiently helps in achieving this. optimization of complex system reliability using hybrid grey wolf optimizer 247 5. hybrid algorithms to use the best qualities of some metaheuristics together the attention of the researchers has now been attracted by hybrid of two metaheuristic together solve s the purpose of reaching the global best solution with results much better than the individual metaheuristics as such in terms of quality, time, better convergence rate. generally, the phenomena of exploration and exploitation (eiben & schippers, 1998) are regarded as if they cannot go hand in hand and one disturbs the progress of the other. but a balance of these two phenomena actually, leads to the global optimum which is the best solution in terms of avoiding local stagnation, appropriate convergence rate and better result. in using a hybrid of two metaheuristic techniques, they can be used at two levels. one could be low level and other could be high level. along with this the hybridization could be done in two ways. one is as relay that is one after the other and the other method is coevolutionary which means the techniques hybridized are run parallelly and not one after the other. since the two different techniques are used in generating the final solution of the problem so it is said that a hybrid is a mixed kind of a technique. now here the challenge lies in choosing the appropriate level to which the techniques are used. as well as, the suitable method used either relay of parallel. a slight difference in the choices made could lead to the global best solution or no better solution at all. some of the hybrid techniques used successfully so far by the researchers are gwo-aco (ab rashid, 2017), gwo-ga (singh & singh, 2017), gwo-ann (tawhid & ali, 2017) and pso-aco (holden & freitas, 2008). to enable the process of exploitation, hybrids of pso have been developed by many researchers. mirjalili & hashmi, (2010) proposed hybrid pso with gravitational search algorithm (gsa) that is pso-gsa to combine the advantages of the pso with those of the gsa for better performance to escape from local convergence. hpso by ahmed et al., (2013) aims at using particle swarm optimization with (pso) with genetic algorithm (ga) mutation technique which give much better result than the pso. abd-elazim & ali, (2015) introduced a hybrid of bacterial foraging optimization algorithm (bfoa) and pso called bacterial swarm optimization (bso) which has proved to be testifier in tuning with svc. in order to avoid local stagnation and obtain better quality in terms of global best and stability factor gwo has been hybridized with many other optimization techniques. 5.1 hybrid pso-gwo algorithm it is clear that to improve the convergence behaviour of the metaheuristic technique researchers have started developing hybrids of some of the meta heuristics as mentioned earlier. one of those is hybrid pso gwo technique (hpsogwo) (singh & singh, 2017). the advantage of hybridization of the pso and gwo technique is that with gwo, the exploration technique is improved as the wolves do enough exploration of the search space. whereas, pso helps in improving the exploitation so that the convergence to the solution can be achieved timely as well as to the global optimum. proper exploitation and exploration with a balance is maintained. this ultimately complements and strengthens the performance of both the techniques taken together avoiding the influence of the shortcomings in terms of local stagnation or suitable convergence rate. the modifications done in the related equations are shown by the use of an inertia weight constant. for this, the positions of the search agents are to be improved first so that the searching and the exploring process can be bettered. this will automatically control the exploitation and exploration phenomena as a whole. negi et al./decis. mak. appl. manag. eng. 4 (2) (2021) 241-256 248 introduction of the inertia constant to control the surveying and attacking processes of the wolves can be expressed as follows. (singh & singh, 2017). 𝐷𝛼 = |𝐶1. 𝑋𝛼 − 𝑤 ∗ 𝑋| 𝐷𝛽 = |𝐶2. 𝑋𝛽 − 𝑤 ∗ 𝑋| 𝐷𝛿 = |𝐶3. 𝑋𝛿 − 𝑤 ∗ 𝑋| (14) to enhance the exploitation capacities of the pso the velocity and upgraded locations of the search agents are expressed by the equations as follows: 𝑉𝑖 𝑘+1 = 𝑤 ∗ {𝑉𝑖 𝑘 + 𝑐1𝑟1(𝑥1 − 𝑥𝑖 𝑘 ) + 𝑐2𝑟2(𝑥2 − 𝑥 𝑖 𝑘 ) + 𝑐3𝑟3(𝑥3 − 𝑥𝑖 𝑘 )} (15) 𝑥𝑖 𝑘+1 = 𝑥𝑖 𝑘 + 𝑣𝑖 𝑘+1 (16) pseudo code of the hpsogwo algorithm is given in figure 2. (singh & singh, 2017). figure 2. pseudo code of the hpsogwo algorithm 6. formulation of the mathematical model for the problems last few decades have witnessed lot of research in formulation of mixed configuration as pure series or parallel configuration are not enough to design complex system of the real-world engineering problems. the following problems of mixed configuration with both series and parallel structures based on the reliability allocation have been solved using hpsogwo technique. in this paper, the two problems considered are complex bridge system and life support system in space capsule. these are nonlinear optimization problems subject to respective constraints of component reliability and system costs. 6.1 problem of complex bridge system: complex bridge system (padhye et al., 2009; pant & singh, 2011, kumar; pant & ram, 2017) has a mixed configuration of series and parallel. the system has a total of optimization of complex system reliability using hybrid grey wolf optimizer 249 . 2 1 4 1 4 2 3 five components (fig.3). the system reliability (rs) and system cost (cs) of a complex bridge network are given below. 𝑅𝑠 = 𝑟1𝑟4 + 𝑟2𝑟5 + 𝑟2𝑟3𝑟4 + 𝑟1𝑟3𝑟5 + 2𝑟1𝑟2𝑟3𝑟4𝑟5 − 𝑟1𝑟2𝑟4𝑟5 − 𝑟1𝑟2𝑟3𝑟4 − 𝑟2𝑟3𝑟4𝑟5 − 𝑟1𝑟2𝑟3𝑟5 𝑟1𝑟3𝑟4𝑟5 (17) 𝐶𝑠 =∑ 𝑎𝑖 5 𝑖=1 exp[ 𝑏 (1−𝑟𝑖) ] (18) the optimization problem in mathematical form is as under: minimize 𝐶𝑠 subject: 0 ≤ 𝑟𝑖 ≤ 1 𝑖 = 1, 2, 3, 4, 5 0.99 ≤ 𝑅𝑠 ≤ 1 𝑎𝑖 = 1, and 𝑏𝑖 = 0.0003, for 𝑖 = 1,2, 3, 4, 5 where, 𝑅𝑖 is 𝑖 𝑡ℎ component’s reliability. 6.2. problem of space capsule: life support system in space capsule (anthony, 2006) presented below is composed of 4 components (fig. 4). this mixed seriesparallel system is used for space exploration and the related equations are as follows: (kumar et al., 2017) 𝑅𝑠 = 1 − 𝑟3 [(1 − 𝑟1)(1 − 𝑟4)] 2 − (1 − 𝑟3)[1 − 𝑟2{1 − (1 − 𝑟1)(1 − 𝑟4)}] 2 (19) 𝐶𝑠 =2𝐾1 𝑟1 𝛼1 + 2 𝐾2 𝑟2 𝛼2 + 𝐾3 𝑟3 𝛼3 + 2 𝐾4 𝑟4 𝛼4 (20) where, 𝐾1 = 100, 𝐾2 = 100, 𝐾3 = 200, 𝐾4 = 150 and 𝛼𝑖 = 0.6, 𝑖 = 1, 2, 3, 4, 5 minimize 𝐶𝑠 subject: 0.5 ≤ 𝑟𝑖 ≤ 1 𝑖 = 1, 2, 3, 4, 5 0.9 ≤ 𝑅𝑠 ≤ 1. figure 3. complex bridge system figure 4. life support system in a space capsule 7. result analysis for the above-mentioned problems of reliability allocation, we employed the simplest penalty functions method for constraints handling and the hpsogwo algorithm has been implemented in matlab with number of grey wolves fixed same as gwo and the best results obtained are reported in table 1 & table 2. outin 1 4 52 3 outin 1 4 52 3 negi et al./decis. mak. appl. manag. eng. 4 (2) (2021) 241-256 250 analysis of the results of problem 6.1 complex bridge systems shows that with hpsogwo for 400 iterations and population size of 100 (total number of grey wolf) the number of function evaluations is 40,000 with same reliability as with pso and gwo individually (table 1). so, this result is better than that of gwo with regards to number function evaluations (figure 5). table 1: result comparison for complex bridge system complex bridge system pso gwo hpsogwo 𝒓𝟏 0.9348210000 0.9341000000 0.9308565080 𝒓𝟐 0.9350280000 0.9363500000 0.9399944690 𝒓𝟑 0.7919480000 0.7913700000 0.8094644730 𝒓𝟒 0.9350050000 0.9338800000 0.9354764350 r5 0.9347350000 0.9356500000 0.9313288850 no. of iterations 300 300 200 𝑹𝒔 0.99000500000 0.99002800000 0.99000033494 𝑪𝒔 5.01991800000 5.01990000000 5.066228730000 fe 1,20,000 9000 6000 figure 5. search history for problem 6.1 analysis of the result of problem 6.2 life support system in a space capsule shows that with hpsogwo for 200 iterations and population size of just 30 (total number of grey wolf) the result is quite competitive in terms of higher reliability cost minimisation and the number of function evaluations is just 6000 which is far better than the results obtained with pso and gwo individually (figure 6). the results obtained are shown in table 2. optimization of complex system reliability using hybrid grey wolf optimizer 251 table 2: result comparison for life support system in a space capsule life support system in a space capsule pso gwo hpsogwo 𝒓𝟏 0.500000000 0.500000000 0.500000000 𝒓𝟐 0.838924024 0.838920000 0.838924024 𝒓𝟑 0.500000000 0.500000000 0.500000000 𝒓𝟒 0.500000000 0.500000000 0.500000000 no. of iterations 300 500 400 𝑹𝒔 0.900000000 0.900000000 0.900000000 𝑪𝒔 641.823562000 641.823600000 641.823562000 fe 2040 50,000 40000 figure 6. search history for problem 6.2 thus, hybrid of pso and gwo gives an overall better performance than the individual optimization technique. a comparison of the results clarifies that though results are better than other optimization techniques, but hpsogwo results better in one or the other form all the previously obtained results. from the results, it is clear, that hpsogwo gives comparatively much better results than other metaheuristics like pso, gwo individually used earlier for such complex reliability allocation problems in terms of the lesser number of function evaluations (table 1 & table 2). 8. conclusion and further scope nature inspired optimization algorithms have extended their roots in almost all complex optimization problems of modern-day industries. reliability allocation problems which are usually np-hard in nature are one of them. negi et al./decis. mak. appl. manag. eng. 4 (2) (2021) 241-256 252 in this article, a hybrid algorithm named hpsogwo has been used to solve two complex reliability allocation problems named complex bridge system and life support system in space capsule. hpsogwo algorithm has proved superior or comparable overall in terms of lesser number of function evaluation as compared to gwo and pso. also, the technique can serve the solutions to the various reliability allocation problems (raps) and reliability-redundancy allocation problem (rraps) with the use of a proper penalty function. as further scope, the decision makers can decide the allocation of the desired reliability of the components as well as the whole complex system which can be optimized using hpsogwo. together with this the repair and maintenance of the components also can be a part of decision as the reliability of the whole system can be managed better to get competitive results with the hpsogwo technique. currently, the authors are working on numerous improvements related to the benchmark problems in reliability allocation problems (raps) and reliability-redundancy allocation problem (rraps). author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research received no external funding. declaration of conflicting interests: the authors have no conflict of interests. references ab rashid, m.f.f. 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(2007). reliability optimization using multi-objective ant colony system approaches, reliability engineering & system safety. 92, 109-120. © 2021 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). https://www.researchgate.net/profile/nitin_uniyal/publication/342123987_an_overview_of_few_nature_inspired_optimization_techniques_and_its_reliability_applications/links/5ee36805299bf1faac4e8998/an-overview-of-few-nature-inspired-optimization-techniques-and-its-reliability-applications.pdf https://www.researchgate.net/profile/nitin_uniyal/publication/342123987_an_overview_of_few_nature_inspired_optimization_techniques_and_its_reliability_applications/links/5ee36805299bf1faac4e8998/an-overview-of-few-nature-inspired-optimization-techniques-and-its-reliability-applications.pdf plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 4, issue 2, 2021, pp. 126-139. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame210402126g *corresponding author. e-mail addresses: ghosal.suman987@gmail.com (s.ghosal), swatidey@yahoo.com (s.dey), ppc@metal.becs.ac.in (p.p.chattopadhyay), shu.datt@gmail.com (s.datta), pb_etc_besu@yahoo.com (p. bhattacharyya) designing optimized ternary catalytic alloy electrode for efficiency improvement of semiconductor gas sensors using a machine learning approach suman ghosal1*, swati dey2, partha pratim chattopadhyay3, shubhabrata datta4 and partha bhattacharyya1 1 department of electronics and telecommunication engineering, indian institute of engineering science and technology, shibpur, west bengal, india. 2 department of aerospace engineering and applied mechanics, indian institute of engineering science and technology, shibpur, west bengal, india. 3 national institute of foundry and forge technology, hatia, ranchi 834003, india. 4 department of mechanical engineering, srm institute of science and technology, kattankulathur, india. received: 19 december 2020; accepted: 17 may 2021; available online: 13 june 2021. original scientific paper abstract: catalytic noble metal (s) or its alloy (s) has long been used as the electrode material to enhance the sensing performance of the semiconducting oxide-based gas sensors. in the present paper, optimized ternary metal alloy electrode has been designed, while the database is in pure or binary alloy compositions, using a machine learning methodology is reported for detection of ch4 gas as a test case. pure noble metals or their binary alloys as the electrode on the semiconducting zno sensing layer were investigated by the earlier researchers to enhance the sensitivity towards ch4. based on those research findings, an artificial neural network (ann) model was developed considering the three main features of the gas sensor devices, viz. response magnitude, response time and recovery time as a function of zno particle size and the composition of the catalytic alloy. a novel methodology was introduced by using ann models considered for optimized ternary alloy with enriched presentation through the multi-objective genetic algorithm (ga) wherever the generated pareto front was used. the prescriptive data analytics methodology seems to offer more or less convinced evidence for future experimental studies. designing optimized ternary catalytic alloy electrode for efficiency improvement of semiconductor gas sensors using a machine learning approach 127 keywords: oxide based gas sensor, ternary alloy catalyst design, sensing parameters, artificial neural network, genetic algorithm, multi-objective optimization. 1. introduction noble metals like palladium, platinum, silver, gold etc. were investigated for a long time due to their contribution towards improving the performance of semiconductor gas sensor devices (acharyya et al., 2016). it was found that the catalytic activities of these metals can further be reinforced by judiciously alloying it with a secondary metal (acharyya & bhattacharyya, 2016; roy et al., 2012). such binary alloys, often utilized in the form of electrode, act as the potential adsorption size for the target gas species, either through chemical sensitization or through electronic sensitization (roy et al., 2012). they help in lowering down the activation energy requirement for gas dissociation. due to subsequent spill-over effect, the surface activity of the main sensing layer (i.e., semiconducting oxide) is also significantly enhanced, which often leads to lower operating temperature, better sensitivity and faster response/recovery kinetics of the device (acharyya & bhattacharyya, 2016; roy et al., 2012; quaranta et al., 1999; bhattacharyya et al., 2007; bhattacharyya et al., 2015). moreover, in some cases, the alloying element often retards the degradation of the catalytic metal electrodes, thereby improving the long-term stability of the device dramatically (roy et al., 2012; quaranta et al., 1999; bhattacharyya et al., 2007; bhattacharyya et al., 2015; wollenstein et al., 2003; lee et al., 2003). for example, pd electrode on zno sensing layer was found to offer improved sensitivity, but at the cost of poor stability towards ch4 (bhattacharyya et al., 2008; bhattacharyya et al., 2007; basu et al., 2008). as long-term exposure to ch4 often leads to formation of palladium hydrate which, due to lattice mismatch with zno, degrades the stability of the sensor (particularly at high temperature) (bhattacharyya et al., 2007). further studies revealed that if pd is alloyed with 25-30% of ag, the probability of such hydrate formation is reduced significantly, which leads to better stability of the metal semiconductor junction and the device as a whole (maity et al., 2018). however, not a large variety of such binary alloys on different oxide surface has so far been investigated (bhattacharyya et al., 2008; basu et al., 2008; mishra et al., 2010; bhattacharyya et al., 2008; ghosal et al., 2019). moreover, most of the approaches are based on trial-and-error method which is time consuming, expensive, and even without any guarantee of success. to improve the performance of such catalytic electrode material, computational design of the alloy, before experimentation, is of immense importance to avoid the above limitations. the non-availability of constitutive models for complex materials systems has prompted researchers to rely on data-driven design approaches (datta & chattopadhyay, 2013). however, the artificial intelligence (ai) and machine learning (ml) have been found to be effective tools for the purpose of designing the alloys, using the experimental findings published by the earlier researchers (datta & chattopadhyay, 2013). kumar et al. (2011) used ml techniques on the raw data attained from four different odours/gases, responses of an oxygen plasma treated thick film tin oxide sensor array. pławiak and rzecki (2015) employed similar methods to study the effect of gas concentration on the performance of a sensor. in an earlier work, by the present authors (ghosal et al., 2019) aimed at oxide-based gas sensor to sense methane gas competently, ai based methodology was incorporated ghosal et al./decis. mak. appl. manag. eng. 4 (2) (2021) 126-139 128 magnificently to design ternary catalytic alloy systematically as per the data set of pure or binary alloys. this was the first attempt to design ternary electrode materials using ai. however, in that work, the ternary alloys were designed without any constraint in the combinations (weight percentage) of elements in the alloys or the amount of each element in it. on the contrary, in the present work, a noble approach has been employed using ai techniques to design ternary alloy catalysts with improved performance, where the experience of the earlier researchers in selection of elements and the maximum and the minimum allowable limit on the amount of a particular alloy element was incorporated in the database through restructuring the data, and thus incorporating the system knowledge in the models (the method of data restructuring is explained in database section). the objectives of the present work are to improve the three pivotal performance parameters of the gas sensor device, viz. response magnitude, response time and recovery time, simultaneously. as the objectives (performance parameters of the sensors) are repeatedly contradictory in nature, multi-objective optimization using genetic algorithm (ga) has been employed for designing alloys with conflicting requirement (deb, 2001; goldberg, 2002; dey et al., 2016; datta, 2016). as we have already mentioned above, a model through the data made by the past researchers on binary alloys was established for three different aspects through artificial neural network (ann) (kumar, 2004; anderson, 1995). ann has proven its capability to plot the input-output relationship of compound materials systems (longo et al., 2017; ray et al., 2009) aimed at optimization process. the ann models are used as the objective functions. the ann models as objective functions while using for ga based optimizations techniques in materials structures had been fruitfully designed by the earlier researchers (datta & chattopadhyay, 2013; sinha et al., 2013). while developing the pareto fronts (goldberg, 2002) as a consequence in the optimization procedure comprising non-dominated solutions was established for identifying the nature of combinations and structures to design the optimized alloy with potential to improve the gas sensor performance in a predetermined and tailor-made fashion. 2. problem formulation the traditional trial and error methodology of designing new materials particularly, binary or ternary complex, with improved performance is a timewasting process, which might often lead to a whole sewerage of materials. the concept of designing computational materials using intelligent data analytics techniques can search solutions computationally, which can later be experimentally validated. in the present case, three attributes, viz. response magnitude, response time and recovery time, were used via the measure of performance of the gas sensor. these attributes were described as the compositional utilities of the catalytic metal electrode, particle size of zno, ch4 concentrations and the optimum temperature of sensing (quaranta et al., 1999). the designed catalyst alloy was targeted to overcome the issues related to the slow response and recovery kinetics, and not higher response magnitude. due to this, for enhancing the device performance can be defined as dropping the response time and recovery time and increasing the response magnitude simultaneously. as revealed from the earlier reports, these objects might often be contradictory and that is why multi-objective optimization through the ga was engaged in the present approach. for describing the overhead three aspects, three distinct ann models were developed, were recycled, and used as objective functions for studying the optimization techniques. as multiobjective complications do not lead to single optimum solution, non-dominant solutions were achieved, proposing designing optimized ternary catalytic alloy electrode for efficiency improvement of semiconductor gas sensors using a machine learning approach 129 the best suitable cooperation between the objectives, known as pareto front (dey et al., 2015). these solutions can be analyzed and utilized to design optimized ternary alloy electrode, suitable for enhancing the performance of semiconducting oxidebased gas sensor devices. 3. construction of database as mentioned earlier, the database was generated from experimental results published by the earlier researchers (bhattacharyya et al., 2008; basu et al., 2008; mishra et al., 2010; bhattacharyya et al., 2008; ghosal et al., 2019). primary analysis of the database revealed that the noble metals used as catalytic electrode can be put into three clusters, pure metals, binary alloys with  50-50 ratio of the elements and binary alloys with  75-25 ratio of elements. table 1. the list of inputs parameters for three different output variables with their minimum, maximum, average and standard deviation (bhattacharyya et al., 2008; basu et al., 2008; mishra et al., 2010; bhattacharyya et al., 2008; ghosal et al., 2019). aimed at designing ternary alloy, the database was derived by dividing the compositional parameters (or components) into three different components, based on the weight percent (%) of pt, pd, rh, au and ag while they are used as pure or binary form as an electrode in the device via taking the clue from the experience of alloy development by the earlier researchers (bhattacharyya et al., 2008; basu et al., 2008; mishra et al., 2010; bhattacharyya et al., 2008). among the three components of the elements, component 1 consists of the elements having more than 50 weight% in the alloy including the pure metals, component 2 consists the presence of metals within the range of 30 to 50 weight%, and component 3 has the elements with less than 30 weight% in the alloy. this restructuring of the database was carried out to divide the elements into three groups, as per the stoichiometry maintained by earlier researchers. such division of the alloying elements into three components, made the input variables min max mean standard deviation ch4 conc. (%) 0.01 1.5 0.524754 0.426081395 temperature (oc) 100 350 225.4098 64.54517205 zno particle size (nm) 20 60 57.72131 15.47054314 pt (wt%) 0 100 8.196721 27.65912729 pd (wt%) 0 74 34.42623 31.57660474 rh (wt%) 0 100 22.95082 42.40063924 ag (wt%) 0 70 27.86885 27.30779828 au (wt%) 0 100 6.557377 24.95898275 output variable response magnitude (%) 20 83.6 45.93888 18.74628351 response time (s) 2.69 86 36.11855 18.91098114 recovery time (s) 16 102 55.48555 22.39324979 ghosal et al./decis. mak. appl. manag. eng. 4 (2) (2021) 126-139 130 database suitable for designing ternary alloy(s). after division of the elements, it was observed that for designing ternary alloy (as per their atomic number) three probable arrangements may be made, i.e. pt-pd-ag, au-pd-ag and rh-pd-ag. in this way, the basic components of the three alloy systems could be finalized using the prior understanding of the alloys used for the purpose. in addition to above, three components, zno particles size, optimum temperature for sensing and the concentration of methane (ch4) were also included as input parameters. basically, the three output parameters are response magnitude, response time and recovery time as the performance indicators of a gas sensor device. 4. computational techniques the processing unit of artificial neural network (ann) bears a resemblance to that of a human brain. inputs generated from other processing units are accepted by the multiple layered processing units that constitute the network architecture. the concluding output of the scheme is the productivity that is generated on the completion of processing from different units. the processing and modelling of materials are provided a profoundly sound and innovative approach through the ann’s. ann can recognize the outline of inputs and outputs commencing the earlier encounters and even the present forms without any previous assumption of their characteristics and interconnections, and this is the most important characteristic of ann. when compared with the traditional approaches, the network has the capacity to calculate and determine further difficult relations in the data of material properties. a quantity of inputs (experimental variables), an only output, and an inbetween hidden layer are contained by the network. at individual hidden unit  ih , the weighted amalgamation of the standardized inputs  njx is functioned on a hyperbolic tangent transfer function which is presented in eq. (1), creates definite that each input subsidizes to each and every hidden unit. (1) (1) tanh n i ij j j j h w x            (1) after that, output neuron at that time computes a linear weighted sum of the outputs of the hidden units, as indicated in eq. (2): (2) 2 i i i y w h   (2) in the directly above for both equations, y is the output, n jx defined as normalized inputs, ih defined as the outputs through hidden units, ijw , iw defined as weights, and i and  defined as bias. therefore, which is likely to achieve dissimilar outputs via changing the weights, ijw ((equations (1) and (2)). the optimal values of these weights are achieving through “training” the network on a set of normalized input– output data. for that, the input–output data can be first standardized in the sort of -1 to +1 from the eq. (3):  min max min 2 1 jn j x x x x x      (3) designing optimized ternary catalytic alloy electrode for efficiency improvement of semiconductor gas sensors using a machine learning approach 131 here n jx is denoted by normalized value, jx signifies the input or output variable, minx and maxx are the minimum and the maximum values of the variable, respectively. the network is trained by adjusting the weights ( ijw ) to minimize an error function, which is basically a regularized sum of square errors. this ultimately leads to an optimal description of the input–output relationship. table 2. parameters used in the multi-objective genetic algorithm for resolving the back propagation (bp) algorithm, is intended of determination of weights and biases for a multilayer ann using feed forward connections through the input layer to the hidden layers and then to the output layer. for minimizing the mean square error among the projected output and the preferred output, the algorithm is an iterative gradient lineage algorithm, which is intended. scaled conjugate gradient back propagation algorithm was used for the contemporary work. via determination, the effect of the input parameters on the final things is impossible, as artificial neural network (ann) is a very high compound and non-linear model. (a) (b) (c) figure 1. representative scatter plot showing the prediction by the trained ann models for (a) response magnitude for the three alloy combinations under consideration (b) response time for the three alloy combinations under consideration and (c) recovery time for the three alloy combinations under consideration. crossover probability 0.95 random seed value 0.19 mutation probability 0.05 parental selection strategy tournament selection ghosal et al./decis. mak. appl. manag. eng. 4 (2) (2021) 126-139 132 (a) (b) (c) figure 2. sensitivity studies of the independent variables for all three output parameters, for (a) au-pd-ag, (b) pt-pd-ag, (c) rh-pd-ag alloys. this makes it imperative to conduct a sensitivity study to reveal the complex hidden connection in the ann model. sensitivity analysis would reveal the gross virtual prominence of the parameters on the belongings. researchers have used several approaches of sensitivity study. for this work, the connection weight method was selected (dolden, 2004). to compute the sensitivity, weights of the input-hidden and hidden-output associates in the competent ann models have been used. the philosophy of advancement of classes promoted by charles darwin has inspired the improvement of the unconventional optimization technique recognized as genetic algorithm (ga). selection, crossover and mutation are the main biological principles which are followed via the simple genetic algorithm (sga) (pławiak & rzecki, 2015; singh et al., 2020). in case of selection process, the candidates aimed at next generation which were recognized. through the crossover process via exchanging and transmission of genetic statistics among two parents for the birth of offspring is transported. finally, a small, probabilistic variation in the genetic makeup is made by the mutation process. in another circumstance, one of the objective conflicting in character, then method is termed as multi-objective optimization (moo) (malbašić & đurić, 2020). unlike single objective optimization where a single prime solution evolves, in circumstance of moo, a non-dominated set of solutions, termed as the ‘pareto front’, evolve (dey et al., 2015; milosevic et al., 2021). this enables a decision producer to select the utmost suitable solution out of the numerous alike finest solutions trading off amongst the differing objectives. in the present work, the developed ann models which are describing the three features of the sensor system are recycled as the objective functions and non-dominated sorting genetic algorithm (nsga-ii) code (deb et al., 2002; albu et al., 2019; gharib, 2020; messinis & vosniakos, 2020), while using for the multi-objective optimization. the constraints used for optimization are specified in table 2. 5. results and discussion figure1 displays the performances of some of the developed artificial neural network (ann) models for predicting response magnitude, response time and recovery time for all the three alloy combinations under consideration. the target versus achieved output plots for the ann models show that the performance of most of the ann models are satisfactory. but in some cases, particularly for the response magnitude models, the prediction by the ann models are not as expected. the sensitivity analyses of the variables based on the trained ann models are shown in fig. 2. it is seen that in the pd-ag-au alloy, au has the most significant role in reducing the recovery time. the role of the other alloying elements is not that designing optimized ternary catalytic alloy electrode for efficiency improvement of semiconductor gas sensors using a machine learning approach 133 significant in improving the output parameters of the sensor device. while, the other two ternary alloy systems, pt in pt-pd-ag alloy and pd and rh in pd-rh-au alloy seem to have positive effect in increasing the response magnitude. the multi-objective optimization using genetic algorithm generates non-dominated solutions in the form of ‘pareto front’. the pareto solutions are generated with different combinations of objectives, viz. (i) response magnitude and response time, (ii) response magnitude and recovery time, and (iii) response magnitude, response time and recovery time for designing all three alloy systems for three different zno particle sizes separately. different combinations of objective functions are used for the multi-objective optimization processes to study the role of the variables in the optimum solutions for different conflicting situations. the results from the optimizations taking three objectives together provide the solutions which can be processed for further investigations, even experimental trials, as those consider all the factors required for the improvement of the catalytic alloy performance. some representative results are selected for all three alloy systems and shown as fig. 3 and fig. 4 and tables 3-5. (a) (b) (c) (d) (e) figure 3. the multi-objective optimization results for the au-pd-ag alloy (a) pareto front for response magnitude and response time, b) pareto front for response magnitude and recovery time, (c) pareto front for response magnitude, response time and recovery time, (d) variation of pd in the pareto solution with increasing response magnitude, and (e) variation of ag in the pareto solution with increasing response magnitude for different search processes. fig. 3(a-c) shows the pareto fronts of au-pd-ag alloy system generated from multi-objective optimization of different combinations of objectives for zno particle size of 45 nm. the conflicting nature of the two objectives are clearly visible when response magnitude is maximized with simultaneous minimization of recovery time 60 70 80 90 100 110 120 38 40 42 44 46 48 50 r e s p o n s e m a g n it u d e , % recovery time, s ghosal et al./decis. mak. appl. manag. eng. 4 (2) (2021) 126-139 134 (fig. 3b). the conflict between the objectives of maximizing response magnitude with minimization of response time is not that significant, as evident from the spread of the pareto front (fig. 3a). the same phenomenon can be observed in the pareto surface developed using all three objectives together (fig. 3c). the non-dominated solutions for all three cases of optimization are arranged along with increasing response magnitude and numbered. the maximum and minimum values of the amounts of elements used in the ternary alloys with optimized performance, as shown in the pareto solutions for all three optimization conditions, are described in table 3. it is seen that the au content does not vary significantly for any optimization condition and remains close to 50 wt%. the other two elements, i.e. pd and ag, have varied to a certain extent during tradeoff between the objectives. the variation in pd and ag for the bi-objective optimization of response magnitude and response time are lesser than the other two cases. this is in conformance of the finding of the pareto front (fig. 3a). the variations of the pd and ag in the solutions for three different optimization conditions are plotted in figs. 3(d-e), where increasing number of pareto solution depicts increase in response magnitude. table 3. the ranges of the weight percentages of the elements in the optimum ternary (au-pd-ag) alloys in the pareto solutions of three different optimization process. objectives au (wt%) pd (wt%) ag (wt%) minimum maximum minimum maximum minimum maximum rm –res.t 50.93941 53.87713 33.33568 37.9842 10.20336 15.14193 r.m –rec.t 50 50.53885 31.07032 45.603958 4.375566 18.855211 rm –res.t-rec. t 50.00015 52.65257 30.72845 44.444019 5.47625 18.394596 table 4. the ranges of the weight percentages of the elements in the optimum ternary (pt-pd-ag) alloys in the pareto solutions of three different optimization process. objectives pt (wt%) pd (wt%) ag (wt%) minimum maximum minimum maximum minimum maximum rm –res.t 50 50.00064 30 30.000008 20.09934 20.099768 r.m –rec.t 50.00336 50.00866 30 30.000065 20.09123 20.091375 rm –res.t-rec. t 50.0002 50.10892 30 30.000731 19.98956 20.09416 it is clear from the figure that increase in pd with simultaneous decease in ag increase the response magnitude, with the expense of increase in recovery and response times. the trends of disparity of ag and pd in the pareto solution are same in the bi-objective optimization response magnitude and recovery time and the triobjective optimization for all objectives. this gives clear indication that to achieve a certain performance level, i.e. to achieve a specific tradeoff between the three properties, the user can get the required alloy composition easily from those to pareto solutions. the other pareto solution, developed from optimization of response magnitude and response time, may provide a different solution to some extent. the minimum and the maximum values of the alloying elements in the pareto solutions of designing optimized ternary catalytic alloy electrode for efficiency improvement of semiconductor gas sensors using a machine learning approach 135 pt-pd-ag alloy system for zno particle size of 60 nm are given in table 4. the values clearly indicate that the optimum weight percentages for the three elements are almost same throughout the pareto front, and even for all three types of multiobjective optimizations (a) (b) (c) (d) (e) figure 4. the multi-objective optimization results for the rh-pd-ag alloy (a) pareto front for response magnitude and response time, b) pareto front for response magnitude and recovery time, (c) pareto front for response magnitude, response time and recovery time, (d) variation of pd in the pareto solution with increasing response magnitude, and (e) variation of ag in the pareto solution with increasing response magnitude for different search processes. for all cases a pt0.5pd0.3ag0.2 alloy provides the best performance. this leads to the fact that the three objectives do not have any significant conflict between them, at least within the used search space. some of the important results coming out from the multi-objective optimizations for the rh-pd-ag alloy system for zno particle size 30 nm are shown in fig.4. the pareto fronts of the alloy system generated from three different optimization processes, as before, are given in fig. 4(a-c). table 5. the ranges of the weight percentages of the elements in the optimum ternary (rh-pd-ag) alloys in the pareto solutions of three different optimization process objectives rh (wt%) pd (wt%) ag (wt%) minimum maximum minimum maximum minimum maximum rm –res.t 50 50 36.32003 46.794388 3.304113 13.77707 r.m –rec.t 50 51.16267 32.45868 44.862404 5.232672 17.638802 rm –res.t-rec. t 50 52.85969 33.32817 44.906982 5.137318 15.753006 ghosal et al./decis. mak. appl. manag. eng. 4 (2) (2021) 126-139 136 it is seen in this case that the variations of the properties are quite high in the nondominated solution, which means the objectives are quite conflicting in this alloy. table 5 gives the ranges of the alloying elements required to achieve optimum solutions for improved performance, as shown in the pareto solutions of the three optimizations. here it is seen again that the major element of this alloy (rh) has not varied for all the solutions and remained at around 50 wt%, same as previous two alloy systems. when the variations of other two elements in the solutions are plotted, it is seen that pd content decreases and ag increases with increasing in response magnitude (fig. 4(d-e)). this is contrary to the trend shown in au-pd-ag alloy. the amount of pd for different trade off options the pareto front shows almost similar values in case of optimization of response magnitude and recovery time as well as in case where all three objectives are considered together. but the solutions from the optimization of response magnitude and response time show higher values of pd. in case of addition of ag as the third element, the trend is similar for the two cases as above, whereas the third optimization expectedly shows lower values of ag. the results of the data-driven modelling using ann and multi-objective optimization using ga for designing three alloys, viz. au-pd-ag, pt-pd-ag and rh-ptag, have shown some clear trends in most of the cases. in the earlier work (ghosal et al., 2019), the ga based searching for ternary alloys with improved performance was completely random in nature, which was necessary to gather primary idea about the probable set of ternary alloys. but in this work, a systematic search has been carried out, where the prior knowledge regarding the role of the various elements has been incorporated meticulously in the searching process. in the process the three alloy systems have one base element each (au, pt and rh), one major alloying element (pd) and one addition of comparatively lower amount (ag). this makes the variations in compositions more perceptible, compared to the randomness in variation of composition previously, and thus the decision-making process for experimental trial becomes more logical. during the experimental validation, one should remember that all the results reported here based on data-driven models generated from data collected from secondary source, and hence it is better to consider the general trends of the findings, not a particular solution. 6. conclusion catalytic noble metals in the form of electrode element were found to improve the semiconductor gas sensor device performance. earlier experimental findings revealed that alloy of such noble metals improved the sensing parameters, viz. response magnitude, response time, recovery time, operating temperature and selectivity. however, optimizing all these parameters simultaneously is a crucial challenge, as the requirement for such individual optimization is often mutually conflicting. in the present paper, a systematic approach based on ann and ga was reported to design an optimized ternary alloy electrode from the constructed database of the earlier reported binary ones. it was found with such ga based multiobjective optimization, the gas sensor device performance can be judiciously optimized employing the properly designed ternary alloy, as the electrode material. it was found that, pt-pd-ag (wt % ratio of pt: 50%, pd: 30 %, ag: 20%) offered most promising results for output parameters among the lot. the experimental verification for the present theoretical simulation has been considered as future work. designing optimized ternary catalytic alloy electrode for efficiency improvement of semiconductor gas sensors using a machine learning approach 137 acknowledgement: this publication is a product of the research and development work commenced in the project under the visvesvaraya ph.d. scheme of ministry of electronics and information technology, government of india, being instigated by digital india corporation (formerly media lab asia). 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(2003). cobalt oxide based gas sensors on silicon substrate for operation at low temperatures. sensors and actuators b, 93 (1-3), 442–448. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 4, issue 2, 2021, pp. 225-240. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame210402215s e-mail addresses: pinki.fet@mriu.edu.in (p. sagar), prinima@mru.edu.in (p. gupta), rohit.tanwar.cse@gmail.com (r. tanwar). a novel prediction algorithm for multivariate data sets pinki sagar1, prinima gupta 1 and rohit tanwar 2* 1 computer science and technology, manav rachna university, haryana, india 2 school of computer science, university of petroleum & energy studies, dehradun uttarakhand, india received: 26 april 2021; accepted: 14 july 2021; available online: 15 july 2021. original scientific paper abstract: regression analysis is a statistical technique that is most commonly used for forecasting. data sets are becoming very large due to continuous transactions in today's high-paced world. the data is difficult to manage and interpret. all the independent variables can’t be considered for the prediction because it costs high for maintenance of the data set. a novel algorithm for prediction has been implemented in this paper. its emphasis is on the extraction of efficient independent variables from various variables of the data set. the selection of variables is based on mean square errors (mse) as well as on the coefficient of determination r2p, after that, the final prediction equation for the algorithm is framed based on of deviation of the actual mean. this is a statistical-based prediction algorithm that is used to evaluate the prediction based on four parameters: root mean square error (rmse), mean absolute error (mae), mean absolute percentage error (mape), and residuals. this algorithm has been implemented for a multivariate data set with low maintenance costs, preprocessing costs, lower root mean square error and residuals. for one dimensional, two-dimensional, frequent stream data, time series data and continuous data, the proposed prediction algorithm can also be used. the impact of this algorithm is to enhance the accuracy rate of forecasting and minimized the average error rate. keywords: coefficient of determination, mean square error, actual means, multiple linear regression (mlr), root mean square error (rmse), mean square error (mse). 1. introduction regression techniques come under the category of supervised learning methods in which the existing training data sets can be used as guidance and to supervise the mailto:rohit.tanwar.cse@gmail.com pinki et al./decis. mak. appl. manag. eng. 4 (2) (2021) 225-240 226 complete learning and prediction process. the results of supervised learning approaches are dependent on algorithms and their complexity. in the regression techniques, new values are predicted for future analysis, which will be calculated based on historical or previous data sets. linear regression fits a straight line and it has two components b0 (intercept) and coefficient b1 and one predictor termed as the independent variable. in today’s scenarios, data sets are maintained with multiple attributes and it requires so much processing time and costs for prediction. the cost of preprocessing and maintenance is depending on the type of data sets but at the time of analysis, it is not necessary to consider all attributes. in this paper prediction algorithm is introduced based on actual means which improve the prediction rate and reduce the cost of maintenance of the data. the regression line includes the following properties: the line restricts the aggregate of squared differentiation between observed values (y dependent variable) and foreseen characteristics (the ŷ values enrolled from the regression line). the regression line experiences the mean of the ‘x’ and mean of the ‘y’. in the linear regression, the b0 is considered as the intercept of the regression equation and it is the incline of the regression line. the regression coefficient b1 is the average change in the dependent variable ‘y’ for a per-unit change in the independent variable ‘y’. 1.1. regression coefficients regression coefficients are estimates of the unknown population parameters and describe the relationship between a dependent and the independent variable. these are identified using methods, such as least square and matrix form. the least square method is a modest linear forecasting approach, in which there is only a binary dependent variable and the other one is a neutral or independent variable. the equation for prediction using regression is: ŷ = b0 + x ∗ b1 (1) least-squares regression coefficients daniya et al. (2020): 𝑏1 = ∑[(𝑥(𝑖−𝑛)− 𝑥 ̅)((𝑦(𝑖−𝑛)− �̅�))] ∑[((𝑥(𝑖−𝑛)− �̅� 2) (2) 𝑏0 = 𝑦 − 𝑥 ∗ 𝑏1 (3) in equation (1) ŷ is the projected value of the reliant variable in linear regression two variables are used b0 and b1. the x(i---n) is the value of the independent variable for observation or new predicted values. for observed values y(i---n) is used, in which y is a dependent variable. x implies x score, and y is the mean y score. equation (2) and (3) daniya et al. (2020) represents the method of coefficients calculation. with multiple linear regression, things are getting progressively jumbled and confused. in multiple linear regression, ‘n’ free factors and ‘n + 1’ relapse coefficients and ordinary conditions are used. finding the least-squares arrangement includes solving ‘n + 1’ condition with ‘n + 1’ questions. equation (1) is eligible only for prediction in the onedimensional data set. if the prediction is done in a multivariate data set then there will be many independent variables but all are not required for prediction. so, the proposed algorithm will work on the selection of attributes that have the highest weightage and more suitable for prediction, after selection of variable prediction equation will be formed based on the actual mean. a novel prediction algorithm for multivariate data sets 227 1.2. research contribution in this paper, an algorithm (mipa) is explained that will be applicable for multivariate data sets. in the preprocessing part, irrelevant variables are reduced. based on the selected variables actual mean has been calculated for identifying the coefficients. the selection of variables is based on the coefficient of determination (r2p) and mean square errors (mse). this algorithm can be applied on various types of data set and reduced the errors like rmse, mae, and mape so that the accuracy rate of prediction can be improved. 2. related work regression techniques are important tools for prediction and analysis. it indicates the significant associations between the dependent variable and independent variable and the strength of the impact of multiple independent variables on a dependent variable. chai et al. (2007) introduced two prediction algorithms that are applicable for one-dimensional and two-dimensional stream data: frequent item prediction method (fipm) and frequent temporal pattern data stream (ftpds). stream data converted into discrete data to get dependent and independent variables so that regression models can be applied. in these algorithms, there were some limitations that were recovered by sequence forecast algorithm plane regression (sfapr) introduced. this plane regression algorithm is based on linear regression for twodimensional data sets and reduces the error rate. kavitha et al. (2016) discussed that, after the advancement of technologies in big data, data analytic has been developed wonderfully in today’s environment. the measurable strategies are utilized for the assessment of prescient models; the choice of accurate systems depends on the prerequisites of the information. the expectation and determining are done generally with time-series data sets. the majority of the applications of prediction are: climate determining, account and securities exchange join recorded information with the present gushing information for better exactness. in this paper, the author divided the time arrangement information using a regression model. linear and multiple linear regression models are connected using the training data set for applying and also for preparation of informational assortment so that it can operate the right model for improvement. ostertagova et al. (2016) presented the application of linear regression algorithm for processing of stress state data which were collected through drilling into a harmonic star method (hsm) it was used for the collection of final data. the noncommercial software based on the harmonic star method enables us to automate the process of measurement for the direct collection of experiment data. such programming empowered us to gauge worries in a specific purpose of the analyzed surface and, simultaneously, separate these anxieties. for example, a camera was utilized to move the image of its chromatic edges legitimately to a computer. mustapha & fadzil (2015) presented a regression algorithm for vendors to forecast their yearly profit and it is based on their historical data. using a forecasting approach vendor can prepare their evaluation exercise. in this article, the author used various regression techniques that analyze the vendor’s performance. the performance report demonstrates the capability of data mining tasks in helping the entrepreneur development unit (edu) to predict vendors’ performance and to identify groups of on performance and under-performance. the entrepreneur development unit was responsible for managing a big group of vendors that hold contracts with the company. pinki et al./decis. mak. appl. manag. eng. 4 (2) (2021) 225-240 228 khan et al. (2016) discussed a non-linear regression by assuming that the data depend on a variety of folds. they divided the data space into multiple areas to construct a partitioned linear regression analysis as an estimation of the non-linearity among the experiential and the expected data, in place of setting up the range and limitations of the particular category, the algorithm was exposed to immediately adapt to the differences in the data and it was very successful for high dimensional data as well as for small data sets. saptawati et al. (2015) stated that the major activity in the mineral industry was most significant and costly in drilling. although finding targets for drilling, geologists were using qualitative study, which resulted in a lot of failures in drilling. the authors worked on the analysis of methods, used for mining, and used to retrieve the facts in the outcome, the categorization of informed data, and mining of common item sets that can maintain the forecasting of drilling targets. the objective of the work is to reduce the threat of failure of drilling and hold the industry’s decision and decide on a new target in drilling. ilayaraja & meyyappan (2015) discussed the data mining techniques and applied them in many areas of medicine for different objectives. they partitioned a process to estimate the risk factors of the patients who were having symptoms of heart disease through the collected frequent data sets. data sets for the heart patients were collected from the medical institutes or hospitals. frequent data item sets are produced and depend on the selected symptoms and minimum support value. the frequent or common data sets which were extracted could help the doctors to make decisions in diagnostics and to predict the level of risk at an early stage so that immediate treatment could be provided. the projected approach could be applied to a data set of medical fields which helps in predicting the factors that affect risk with the level of risk and the patients based on selective factors item sets. yang et al. (2019) used both a linear model and a nonlinear model to predict the future cash flow. a hybrid model integrating linear and nonlinear was constructed to enhance the prediction effect and calculate the fund reserve ratio and improved the accuracy rate of prediction. in the literature survey, it has been discussed that the prediction algorithm discussed by zhao & li (2005) is used for two-dimensional stream data that was based on plane regression. chai et al. (2007) discussed the prediction algorithm for one-dimensional and two-dimensional stream data. later on, a non-linear regression algorithm has been proposed for two-dimensional and onedimensional stream data. after those algorithms for multi-dimensional data sets were discussed but it takes a lot of cost and time for the maintenance and preprocessing of data. in this paper, we minimize the cost of storing and preprocessing data sets and increase the accuracy rate of prediction and decrease the error rate via the proposed method. antoniadis et al. (2021) reviewed the sector of sensitivity analysis and targeted the link between random forest and global sensitivity analysis (gsa). the concept is to use the random forest technique as an effective non-parametric method for building a meta-model that permits effective sensitivity analysis. in addition to its straightforward relevance to regression problems, the random forest methods additionally have the flexibility to implicitly handle correlation and high-dimensional data. authors have used the rank-based random forest (rf) variable index to define sensitivity indexes. the author further reviewed the acceptable tool set for quantifying the importance of variables and used these tools to cut back the spatial property of the model, thereby conducting sensitivity analysis studies that might not be performed. yıldırım et al. (2021) used a preferred deep learning tool known as long short-run memory (lstm), which has been shown to be terribly effective in several time-series prognostication problems. they projected the hybrid model using two data sets that a novel prediction algorithm for multivariate data sets 229 mix two separate lstms to improve the prediction, and it was found that the model gave good results for real data. mukherjee et al. (2019) proposed a model to predict the images based on spatio temporal sequence forecasting problems. they trained the convolutional long short term memory (conv-lstm) to learn the temporal relationships while preserving the spatial data and present in the latent space. in this method first, the encoder and decoder network are trained to learn the spatial features of the data. after that, the conv-lstm is inserted between the encoder and the decoder. the weights of the encoder-decoder are freezed and then the conv-lstm is trained. in the experiment loss function is used to predict the next set of frames for a given set of frames in a video. gauba et al. (2017) proposed a novel approach to predict the rating of video advertisements based on a multimodal framework combining physiological analysis of the user and global sentiment-rating available on the internet. in the framework, they record the eeg signals while the users were asked to watch the video advertisement simultaneously. to predict the rating of an advertisement using eeg data, they used the regression technique based on random forest and then eeg-based rating is combined with nlp-based sentiment score to improve the overall prediction. 3. proposed work in the literature survey, it has been identified that various algorithms have been introduced by authors for prediction using regression. some existing algorithms and methods discussed in section 2 are used for the prediction process for forecasting. the proposed algorithm is based on the selection of efficient variables and prediction of new dependent variables with low residuals, root mean square errors, mean absolute error, and mean absolute percentage error. 3.1. problem formulation forecasting is the estimation of a dependent variable ‘y’, based on the independent variable ‘x’. some algorithms are implemented for one-dimensional and two-dimensional data sets. these data sets can be stream, continuous or discrete data. most of the data sets have multiple independent variables and in the prediction equation, all variables are used for prediction, which may cause the extra cost of maintenance of the data set, more execution time, and low accuracy rate of prediction. the proposed algorithm focused on the selection of relevant variable (independent) from multivariate data sets and improving accuracy rates because multivariate data set consist of various independent variable but all are not required for prediction model and it is very difficult to maintain huge data set with multiple variables due to high cost of maintenance and pre-processing of data set. in the literature survey, it has also been found that existing algorithms are restricted to the number of independent variables for oneand two-dimensional stream data. the objective of the proposed algorithm is to reduce the errors with more accuracy in prediction. the proposed algorithm is used for data sets that have numerous independent variables and reduce the cost of maintenance by selecting the appropriate variables for prediction then the prediction equation is framed based on assumed means. the selection of variables is based on the coefficient of determination and the prediction equation is based on actual means. https://ieeexplore.ieee.org/author/37086234303 https://www.sciencedirect.com/topics/engineering/regression-technique https://www.sciencedirect.com/topics/computer-science/random-decision-forest pinki et al./decis. mak. appl. manag. eng. 4 (2) (2021) 225-240 230 this algorithm is applied on “energy data prediction”, from the uci repository, (https://archive.ics.uci.edu/ml/datasets/appliances+energy+prediction). in the proposed prediction algorithm, there is one dependent variable and 'n’ independent variables. the dependent variable is humidity (average of all areas of the building) and the independent variables are average temperature (average of all areas of the building), pressure, rh_out, wind speed, visibility, etc. 3.2. multivariate item prediction algorithm (mipa) in the algorithm coefficients are calculated for independent variables using the actual mean of dependent and independent variables, in step 4 formula for coefficient calculations has been explained. algorithm name: multivariate item prediction algorithm (mipa) input: multivariate data set output: low residuals, rmse, mae, mape during the prediction step 1: take 2n regression equations ‘n’ is the number of independent variables. step 2: categorize all the equations into models, model 1= no independent variable, model 2 = 1 independent variable, model 3 = 2 independent variables and so on. step 3: using anova compare r2p and mse for each regression model in each model. if (r2p ↑ mse ↓) /*select the regression model if this condition is true highest value of r2p and lowest value of mse */ for i=0 to 2n /*select the appropriate regression model which has the highest r2p and lowest mse out of 2n regression model*/ step 4: select the regression model from each model which have highest r2p and lowest mse. step 5: compare all selected regression equation from each model consider the regression model with lowest mse and highest r2p /*selection of independent variables.*/ step 6: find the deviation of actual mean for each selected independent variable. 𝑑𝑥(𝑖 = 1. . . . 𝑛) = 𝑥 – x̅ 𝑑𝑦(𝑖 = 1. . . . 𝑛) = 𝑦 − �̅� 𝑏𝑦𝑥(𝑖=1 𝑡𝑜 𝑛) = 𝑛∑[(𝑑𝑥(𝑖=1 𝑡𝑜 𝑛) ∗ 𝑑𝑦 ) − ∑(𝑑𝑦(𝑖=1 𝑡𝑜 𝑛) ∗ 𝑑𝑦)] 𝑛∑(𝑑𝑥(𝑖=1 𝑡𝑜 𝑛)) 2 − (∑(𝑑𝑥(𝑖=1 𝑡𝑜 𝑛))) 2 step 7: put value of byx(i=1 to n)in the prediction model ŷ = 𝑏𝑦𝑥1 (𝑥1 – x̅1) + 𝑏𝑦𝑥2 (𝑥2 – x̅2) + 𝑏𝑦𝑥3 (𝑥3 – x̅3) + ⋯ + 𝑏𝑦𝑥𝑛 (𝑥 – x̅) + y̅ step 8: analysis of prediction algorithm (rmse and residuals) 𝑅𝑀𝑆𝐸 = [∑((y − ŷ) 𝑛 𝑘=0 ∗ (y − ŷ))/n] residuals=actual valuespredicted values( yŷ) algorithm mipa has been discussed in detail as follows: a novel prediction algorithm for multivariate data sets 231 step 1: find the 2n regression models in which ’n’ is the number of independent variables. e.g. in the case of 4 independent variables total possible equations will be 16. in table 1, model 1 has not considered any independent variable, for model 2 one independent variable is considered, for model 3 two independent variables are considered, and so on. in table 1 all possible regression models are shown. table 1. the 2n possible regression equations model. mode1 model 2 model 3 model 4 model 5 y=b0+e y=b0+b1x1 y= b0+b1x1+b2x2 y=b0+b1x1+b2x 2+b3x3 y= b0+b1x1+b2x2 +b3x3+ b4x4 y=b0+b2x2 y= b0+b1x1+b3x3 y=b0+b1x1+b2x 2+b4x4 y=b0+b3x3 y= b0+b1x1+b4x4 y=b0+b1x1+b3x 3+b4x4 y=b0+b4x4 y= b0+b2x2+b3x3 y=b0+b2x2+b3x 3+b4x4 y=b0+b2x2+b4x4 y= b0+b3x3+b4x4 step 2: find the analysis of variance (anova table) of each regression model (2n). select the regression model from each model which has the highest r2p (coefficient of determination) and lowest mse. table 2 includes the values of r2p and mse from each model of table 1. table 2. mse and r2p (coefficient of determination) model 2 model 3 model 4 model 5 r2p (%) mse square r2p (%) mse square r2p (%) mse square r2p (%) mse square 85.35 0.24 95.74 0.07 95.87 0.07 95.91 0.08 12.08 1.47 90.91 0.16 95.83 0.07 88.11 0.19 93.96 0.11 93.96 0.11 3.7 1.61 88.14 0.21 91.33 0.16 76.92 0.4 88.35 0.2 step 3: model 2 consists of 1 independent variable, model 3 consists of 2 independent variables, and model 3 consists of 4 independent variables, and so on. the condition must be r2p ↑ mse ↓ means that if the value of r2p is increasing then the value of mse will decrease. so that model 4 is selected in table 2, which has values 95.87 and 0.07 belong to the first regression model of model 4 in table-1. it includes x1, x2, and x3 independent variables; it means that these three variables are most relevant for the prediction algorithm. pinki et al./decis. mak. appl. manag. eng. 4 (2) (2021) 225-240 232 figure 1(a). highest coefficient of determination r2p; figure 1(b). lowest mean square error (mse) in figure 1(a) values of coefficient of determination are plotted which are selected as the highest value from each model and in figure 1(b) plotted the values of lowest mse of table 2. in figure 1(a) value 95.91 is highest but mse is high corresponding to it so that 95.87 will be selected which is corresponding to the lowest mse i.e. 0.07. step 4: derivation of proposed regression algorithm on the basis of deviation from actual mean is as follows: find the value of dx and dy on the basis of actual mean: 𝑑𝑥(𝑖 = 1. . . . 𝑛) = 𝑥 – x̅ (4) 𝑑𝑦(𝑖 = 1. . . . 𝑛) = 𝑦 − �̅� (5) 𝑏𝑦𝑥(𝑖=1 𝑡𝑜 𝑛) = 𝑛∑[(𝑑𝑥(𝑖=1 𝑡𝑜 𝑛)∗𝑑𝑦 )−∑(𝑑𝑦(𝑖=1 𝑡𝑜 𝑛)∗𝑑𝑦)] 𝑛∑(𝑑𝑥(𝑖=1 𝑡𝑜 𝑛)) 2−(∑(𝑑𝑥(𝑖=1 𝑡𝑜 𝑛))) 2 (6) ŷ = 𝑏𝑦𝑥1 (𝑥1 – x̅1) + 𝑏𝑦𝑥2 (𝑥2 – x̅2) + 𝑏𝑦𝑥3 (𝑥3 – x̅3) + ⋯ + 𝑏𝑦𝑥𝑛 (𝑥 – x̅) + y̅ (7) step 5: find the rmse. using equation (8) rmse is calculated. if rmse is low during the prediction means that accuracy of prediction is high. 𝑅𝑀𝑆𝐸 = [∑ ((y − ŷ) 𝑛 𝑘=0 ∗ (y − ŷ))n] (8) step 6: analysis of residual error (actual yŷ) in the proposed algorithm step 1, 2 and 3 are preprocessing steps, used for the selection of efficient variables which are required for the prediction equation (7). in step 4 equations (6) represent the coefficients calculations and prediction equation (7) is drafted for forecasting of new values. as we have mentioned that this algorithm is also valid for different types of data sets, the above-explained approach was extended to a multivariate data set. here, this method is also applied to onedimensional data set selected from the uci repository https://archive.ics.uci.edu/ml/datasets/parking +birmingham), data set contain four attributes parking id (system code number), the capacity of parking (capacity), parking rates, and updated details. in this data set parking occupancy is an independent variable and parking rates are the dependent variable. the coefficient for independent variables is calculated using equations (4), (5), and (6) and then places a novel prediction algorithm for multivariate data sets 233 the values of byx for each independent variable in equation (7). this proposed prediction equation can also be applicable to the stream data set. for the prediction of stream data first, the stream data need to convert into a form of discrete data sets. 4. implementation and result the algorithm is implemented in “r 3.3.2” version. one-dimensional and multivariate data sets are collected from an online repository. 4.1. implementation one dependent variable (humidity) and 4 independent variables (wind speed, average, temperature, t out, press m hg) are included in multivariate data collection. in one dimensional data set four attributes parking id (system code number), the capacity of parking (capacity), parking rates are available. the capacity of parking is an independent variable and parking rate is a dependent variable. for multivariate datasets, all possible equations for independent variables are considered in this process. for the prediction system, appropriate independent variables can be identified using preprocessing. in this process, 2n equations are considered as n is the number of independent variables, and these equations are shown in table-1. in the multivariate data sets, it is not needed to consider all variables in the prediction algorithm. coefficient of determination and mse are used for finding relevant variables. all independent variables have no equal significance or priority so few variables can be eliminated. for one-dimensional data set, there is no need to process the data set, only the regression equation will be applied on attributes x and y. 4.2. results in this paper, a mipa algorithm is compared with mlr because it deals with multiple independent variables. rmse values for mlr and mipa algorithms are plotted in figure 4. as compared with mlr, it has been evaluated that rmse's are low for the mipa algorithm. in table 3 residuals are analyzed or compared which are generated through mlr and mipa algorithms. these values are corresponding to humidity. by analyzing table 3 it can be easily observed that the improved algorithm gives low error rates during the prediction of humidity based on temperature, wind speed, etc. independent variables. pinki et al./decis. mak. appl. manag. eng. 4 (2) (2021) 225-240 234 table 3. residuals using mlr and mipa independent variables dependen t variable humidity average(t emp)=x1 press_mm _hg=x2 rh_out=x3 windspee d=x4 residuals mipa residuals mlr 50.91 17.1674074 733.5 92 7 0.42107573 0.99597124 50.83 17.1496296 733.6 92 6.66666666 0.00879531 1.17010151 50.63 17.1037037 733.7 92 6.33333333 0.55354625 1.48437316 50.57 17.0670370 733.8 92 6 0.95598559 1.64734098 50.73 17.0707407 733.9 92 5.66666666 1.10794818 1.56379616 50.79 17.0485185 734 92 5.33333333 1.37981300 1.59155517 50.79 17.0407407 734.1 92 5 1.70193451 1.6768479 50.8 17.0185185 734.166666 91.8333333 5.16666666 1.63454412 1.67028356 50.9 17.0185185 734.233333 91.6666666 5.33333333 1.46779281 1.56250345 51.05 17.0396296 734.3 91.5 5.5 1.21976676 1.38058893 51.23 17.0667592 734.366666 91.3333333 5.66666666 0.93158178 1.15983686 51.47 17.1103703 734.433333 91.1666666 5.83333333 0.58419480 0.87670657 51.85 17.1851851 734.5 91 6 0.04521525 0.41176678 52.68 17.2149074 734.616666 90.5 6 0.65000026 0.44031096 53.52 17.2522222 734.733333 90 6 1.37553362 1.32006378 53.5 17.2866666 734.85 89.5 6 1.24106697 1.33981661 53.38 17.3107407 734.966666 89 6 0.98628249 1.24189434 53.38 17.3133333 735.083333 88.5 6 0.83118018 1.24629699 52.97 17.3196296 735.2 88 6 0.27623678 0.84953717 54.37 17.3748148 735.233333 87.8333333 6 1.67429658 2.2952218 55.07 17.465 735.266666 87.6666666 6 2.42625301 3.0850061 54.9 17.4588888 735.3 87.5 6 2.19335931 2.90766546 4.3. analysis with existing algorithms mipa algorithm can be used for various types of data sets such as stream data sets (one dimensional, two-dimensional), time-series data set, and multivariate data sets. it identifies the relevant variables from the data set which are the best suitable for the prediction. in fipm, ftpds preprocessing of data is done with the use of sliding window protocol and it is applicable only for one-dimensional and two-dimensional stream data sets. mlr is used for data sets where multiple independent variables are used for prediction, but it consumes lots of time and cost. here in table 4, mipa is analyzed with mlr, fipm, and ftpds, based on residuals parameters. residuals are the errors or difference values of the dependent variable and the observed values. in table 3 independent variables are mentioned, humidity is predicted based on the selected independent variables. a novel prediction algorithm for multivariate data sets 235 table 4. analysis of residuals of existing algorithm with proposed algorithm. humidity mipa mlr fipm ftpds 50.91 0.421075736 0.9959712 1.3323720 0.58600286 50.83 0.008795312 1.1701015 1.4105585 0.786391724 50.63 0.553546256 1.4843731 1.6086900 1.114464984 50.57 0.955985598 1.6473409 1.6668764 1.297415314 50.73 1.107948187 1.5637961 1.5050629 1.252681248 50.79 1.379813003 1.5915551 1.4431944 1.313070113 50.79 1.70193451 1.6768479 1.4413808 1.430897512 50.8 1.634544126 1.6702835 1.4323151 1.420897512 figure 2 plotting of residuals during prediction in figure 2 residuals of existing algorithms and mipa algorithm are plotted compared to mipa and have low residuals in comparison of mlr, fipm, and ftpds. figure 3. analysis of average of residuals pinki et al./decis. mak. appl. manag. eng. 4 (2) (2021) 225-240 236 in figure 3 we have compared the average residuals of algorithms mipa has low average residual values 0.71525 and average residual values of mlr, fipm, and ftpds are 0.7664, 0.9615, and 0.8799 respectively. 5. analysis of results the analysis of mipa has been done on the basis of parameters such as root mean square error (rmse), mean absolute error (mae), mean absolute percentage error (mape) and residuals. 5.1. analysis of rmse rmse is the standard deviation of residuals that are the prediction errors. it is a measure of how these residuals are spaced out. residuals are a measure of how far data points are out from the regression line. by analysis of mlr and mipa, it has been identified that mipa has low rmse values in comparison to mlr. in figure 4, mipa’s rmse values are 0.647776757, 0.582948804, 0.516691943, and 0.496299287 are respectively plotted which are low in comparison to rmse values calculated by mlr. figure 4. analysis of root mean square errors (rmse) 5.2. analysis of residuals in figure 5, the red line represents the plotting of residuals 0.99597124, 1.17010151, 1.48437316, respectively which are generated by the existing algorithm mlr, the blue line represents the plotting of residuals 0.421075736, 0.008795312, 0.553546256, respectively which are generated by mipa. these values are corresponding to humidity. by analyzing figure 5 it can be easily observed that the improved algorithm gives low error rates during the prediction of humidity based on temperature, wind speed, etc. independent variables. a novel prediction algorithm for multivariate data sets 237 figure 5. analysis of residuals during prediction 5.3. analysis of mean absolute error (mae) mae takes the absolute difference between the values that are actual and predicted and finds the average. mae is crucial to identify the absolute value because it doesn't allow for any form of error value cancellation. for example, the average value of 0 if the average of 1 and -1 is considered because 1 and -1 will cancel out each other. in figure 6 mae for algorithms mlr and mipa are compared, mipa gives a better result. mipa prediction algorithm gives a better selection of important predictors or independent variables out of many independent variables in data sets. it reduces the cost of maintenance and collection of the data sets. figure 6. analysis of mean absolute error (mae) 5.4. analysis of mean absolute percentage error (mape) mape is often referred to as the mean absolute percentage deviation (mapd), a measure of the prediction accuracy of a statistical forecasting system. in figure 7 mape for algorithms mlr and mipa are compared, it gives a better result. 0 0.5 1 1.5 2 2.5 3 3.5 50 51 52 53 54 55 56 r e si d u a ls humidity residual analysis of mlr and mipa residuals(mlr) residuals(mipa ) 0.7 0.72 0.74 0.76 0.78 0.8 mipa mlr m e a n a b so lu te e rr o rs (m a e ) mean absolute error of algorithms analysis of mae pinki et al./decis. mak. appl. manag. eng. 4 (2) (2021) 225-240 238 figure 7. analysis of mean absolute percentage error (mape) 5.5. low cost in execution and maintenance of data sets in the mipa algorithm, only relevant variables are considered for data analysis and prediction process. the irrelevant variables get eliminate after the process of selection of variables or preprocessing, by eliminating irrelevant variables maintenance costs of data get reduce and it takes less time in the execution. initially, in the data set four independent variables x1, x2, x3 and x4 are used, after the preprocessing of data set x1, x2, x3 selected as relevant independent variables. the irrelevant variable x4 gets eliminated. for further prediction process, we do not need to maintain the x4 as the independent variable, so it takes less time in the prediction process. 6. conclusion and future scope in this paper, a mipa algorithm is based on actual mean values. the analysis of the algorithms is done based on the parameters such as rmse and residuals, mae, mape. the accuracy of the prediction algorithm is measured with low rmse and residuals. in this algorithm deviation for actual means is estimated for each relevant independent variable, using this estimated value, the prediction algorithm is framed. prediction algorithm for ‘n’ independent variables is framed and it predicts the “humidity” based on the rest of the independent variables. in figure 3 it can be easily analyzed that average residuals of mipa are 5.11% less than mlr, 24.62% are less than fipm and 16.46 % are less than ftpds. in section 5 it has been observed that mipa is better than mlr. mipa is the regression based algorithm that can also be used in medical areas for the prediction of diseases based on the symptoms of patients. through analysis, it can be found that values of error rate are more reduced in the implemented regression algorithm rather than mlr. it reduces the cost of data maintenance and reduces the execution time. it can help in the forecasting of diseases, revenues of the company, production, and weather, and in other areas. author contributions: each author has participated and contributed adequately to take open accountability for suitable portions of the content. 1.34 1.36 1.38 1.4 1.42 1.44 1.46 1.48 1.5 mipa mlrm e a n a b so lu te p e rc e n ta g e e rr o r (m a p e ) mean absolute percentage error of algorithms analysis of mape a novel prediction algorithm for multivariate data sets 239 funding: this research received no external funding. conflicts of interest: the authors declare no conflicts of interest. references antoniadis, a., lambert-lacroix, s., & poggi, j.-m. 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(2005). a plane regression-based sequence forecast algorithm for stream data. international conference on machine learning and cybernetics(icmlc), (1559-1562). ieee. © 2021 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 4, issue 2, 2021, pp. 140-162. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame210402140g * corresponding author. e-mail addresses: omer.gorcun@khas.edu.tr (o.f. gorcun), itssenthil@yahoo.com (s. senthil), hkucukonder@bartin.edu.tr (h. küçükönder) evaluation of tanker vehicle selection using a novel hybrid fuzzy mcdm technique omer faruk görçün1, s. senthil2, and hande küçükönder 3* 1 faculty of business, department of business management, kadir has university, istanbul, turkey 2 department of mechanical engineering, kamaraj college of engineering and technology, virudhunagar, india 3 department of business administration, faculty of economics and administrative sciences, bartin university, bartın, turkey received: 26 april 2021; accepted: 28 may 2021; available online: 13 june 2021. original scientific paper abstract: petroleum products transportation considered as one of the crucial parts of dangerous material transportation is a risky logistics activity. the selection of the appropriate tanker vehicles may be a suitable solution to reduce the risks and increase the efficiency and performance of the fuel transportation companies. however, the selection of a suitable road tanker vehicle is not an easy task for decision-makers as there are many conflicting criteria and many decision alternatives. in addition, decision-makers may have to decide with insufficient information since collecting crisp values may not be possible at all times. hence, many ambiguities affecting the evaluation results exist in an assessment process performed to select the best tanker vehicle option. this paper suggests a novel integrated fuzzy approach to solve these decision-making problems. sensitivity analysis is conducted to test the validation of the proposed integrated fuzzy approach and its results was performed by forming 130 scenarios. the results of sensitivity analysis prove that the proposed model can be applied to solve these kinds of decisionmaking problems. key words: road tanker vehicle, fuzzy swara, fuzzy codas, dangerous goods transportation, mcdm. 1. introduction in recent years, the flammable liquid transportation industry is an important part of the dangerous goods logistics industry has grown quickly depending on the increase of energy needs of both industries and individuals. the number of car evaluation of tanker vehicle selection using a novel hybrid fuzzy mcdm technique 141 ownership per 1000 inhabitants has averagely increased at the rate of 17.61% around the world between 2001 and 2020 (oecd 2021). increase in car ownership has caused to increase energy needs of individuals. hence, the volume of fuel transported has shown a sharp increase in both urban and rural areas. the reports published by market insight companies and international institutions suggest that this trend will continue increasingly in near future and the analysts forecast that the global oil transportation market will grow at the rate of 7.00% annually. the risks associated with the transport activities will also increase. selection of the most appropriate tanker vehicle is a very crucial task for decisionmakers with respect to reducing the risks in addition to the efficiency and performance of the oil and petroleum products transportation. however, there is no study on the selection of proper road tanker vehicles in the literature, even though dangerous goods transportation is a very important issue for all parties of logistics such as international and national policymakers, operators, companies, local authorities, and ordinary citizens. previous work on dangerous goods transportation focused on route selection (bęczkowska, 2019; wang & liang, 2020; li, 2018). a multi-objective mathematical model is used for selection of dangerous goods transportation (samanlıoğlu, 2013; pamučar et al. 2016; jassbi & makvandi, 2010). also, many studies are available on dangerous materials transportation such as designing the terminal layout for safety (santarremigia et al., 2018; hervás-peralta, 2020), evaluating safety systems in airfreight operations (huang et al., 2020), risk assessment (huang et al., 2021; kanj et al., 2019; galieriková et al., 2018; gul et al., 2019; raemdonck et al., 2013), evaluation of the work of advisors in the transport of hazardous goods (pamucar et al., 2019; milosevic et al., 2021). a limited study is carried on the selection of freight trailers (görçün, 2019a; görçün, 2019b) but they are not related to tanker selection. equipment and vehicles used in ordinary freight transportation are quite different from equipment and vehicles used in dangerous goods transportation. hence it is not possible to connect between previous works dealing with freight trailer selection and the current paper focuses on the road tanker selection. to the best of our knowledge, there is no study dealing with the selection of transport vehicle. the selection of the appropriate road tanker vehicle has vital importance and is crucial for companies, governments, local authorities, and individuals. it can also be a determinative factor for effectivity and performance of a hazmat (hazardous materials) transport company. to analyze the significant criteria affecting the selection of proper tanker vehicles, a set of research questions were determined. a team of experts is formed. all of them are also experts and advisors of hazardous materials transportation certificated by public authority. the researchers organized many face-to-face interviews with each expert and directed these research questions to the experts. at the end of the face-toface interviews, preparing a list for decision alternatives and criteria was requested from each professional. these lists were collected and researchers eliminated the repetitive criteria and the final list for criteria and options has been determined by providing full consensus among these experts. hence, the determined selection criteria are realistic and suitable to real-life and they can also be used by future works and it can be taken into consideration by practitioners in a reel assessment process. as another finding of the research process, experts indicate that there is no mathematical model implemented to determine the best tanker vehicle in the field of hazardous materials transportation, furthermore, decision-makers, who responsible görçün et al./decis. mak. appl. manag. eng. 4 (2) (2021) 140-162 142 to decide on this issue, make decision based on their own experiences, and they mostly consider each process as a case. it proves that using a mathematical model is a crucial requirement for solving these kinds of decision-making problems encountered in the field of dangerous goods transportation. in order to respond to these requirements, the current paper proposes an integrated fuzzy approach consisting of the fuzzy step-wise weight assessment ratio analysis (f-swara) and the combinative distance-based assessment (f-codas) techniques. although the fuzzy codas technique that is a part of the proposed integrated fuzzy model is a novel multi-criteria decision-making (mcdm) approach, it has been observed that it has been applied in some studies for solving decision-making problems in various fields. for example; evaluation of it technology alternatives for a university (dahooie et al., 2020), evaluation of environmental quality (ouhibi & frikha, 2020), the selection of vehicle shredding facility location (simic et al., 2021), evaluation of renewable energy alternatives (deveci et. al., 2020), evaluation of personnel selection problem (yalçın & pehlivan, 2019), market segment evaluation (keshavarz ghorabaee et al., 2016a). the number of studies using the f-codas technique is very limited and none of these papers dealt with decision-making problems encountered in the field of logistics. the classical decision-making techniques use crisp values to assess the options however these evaluations are not working and they often fail in real life. there are many uncertainties in an assessment process and decision-makers may have to make decisions with insufficient information (pamucar & savin, 2020). the proposed integrated fuzzy approach can enable to deal with ambiguities because it has the ability to include the ambiguities to the scope of the evaluation process. in addition, this technique takes into consideration the combinative form of euclidean distance and hamming distance in the aspect of the intangibility of decision-maker (dm). hence it can present accurate and reasonable results by considering ambiguities (wang et al., 2020; ali et al., 2021). the proposed method can be used as a methodological frame for both future works and practitioners who responsible to decide in the field of dangerous goods logistics. after the proposed integrated fuzzy approach was applied, a comprehensive sensitivity analysis consisting of two stages was performed to test the validation of the model and its results by forming different 130 scenarios. according to the results of the analysis, a5 has remained for all scenarios and its ranking performance has never changed. in addition, it has been observed minor changes, which did not change the overall results, in the ranking performances of some alternatives. the results of the analysis prove that, the proposed fuzzy model is a very strong approach and its ranking results are reasonable, accurate and realistic. hence, it can be applied to solve these kinds of decision-making problems encountered in various fields of logistics in addition to the dangerous goods logistics industry. the major contribution of this work is:  it presents a set of criteria, which are novel and suitable to real life, for evaluating the road tanker vehicle selection. there is no criterion defined by previous studies in this field.  the proposed integrated fuzzy approach is a novel mcdm technique and it can contribute to future work that will be carried out on this issue as well as it can help to practitioners in the field of hazardous material transportation.  it presents a methodological frame which can enable to deal with many ambiguities existing in an assessment process on the tanker vehicle selection. evaluation of tanker vehicle selection using a novel hybrid fuzzy mcdm technique 143 the rest of the paper is organized as follows. in section 2, the proposed integrated fuzzy approach consisting of f-swara and f-codas and its basic algorithm consisting of four stages are presented. section 3 describes the numerical analysis for calculating the selection criteria and determines the ranking performances of the road tanker vehicle alternatives. in section 4, a comprehensive sensitivity analysis is performed to test the validation of the proposed integrated fuzzy model and its results. section 5 describes the overall results and conclusions. the limitations encountered during the research process, and a set of suggestions for future works are presented. 2. the proposed integrated fuzzy approach the proposed integrated fuzzy approach and its basic algorithm are presented in this section. the proposed model consists of three stages. the first stage is organized as the preparation process. the main problems were determined, the board of experts was formed, after the criteria and decision alternatives were defined fuzzy data were collected in this phase. the weights of the criteria were computed by applying the f-swara technique. finally, the ranking performances of the decision alternatives were determined by using the f-codas technique. the basic algorithm of the proposed model is presented in figure 1. figure 1. the basic algorithm of the proposed integrated fuzzy model 2.1. preliminaries the fuzzy set theory introduced by zadeh (1965) is a useful technique enabling to deal with ambiguities for decision-makers. the fuzzy sets have degrees of membership and the fuzzy set theory uses triangular fuzzy numbers (tfns) to convert the linguistic evaluations. görçün et al./decis. mak. appl. manag. eng. 4 (2) (2021) 140-162 144 a fuzzy number a on r to be an ftn if its membership function ( ) a x : r→ [0,1] is equal to the following equation (1) (zadeh, 1965): ( ) 0 a x l l x m m l u x m x ux u m otherwise                (1) as is seen from equation 1, the values of l, m, and u symbolize tfns and while l is the minimum value, u denotes the maximum value of the fuzzy number, and m represents the moderate value of the fuzzy number. the fuzzy set theory has been used by many studies (petrovic et al., 2019; deveci et al., 2020; pamucar & ecer, 2020; ecer & pamucar, 2020; alosta et al., 2021). 2.2. the fuzzy swara technique the fuzzy swara (f-swara) technique is an extended version of the traditional swara technique developed by kersuliene et al. (2010). this technique is preferred over other traditional mcdm techniques such as ahp and anp. this can estimate the decision makers preferences considering the significances of the criteria (mardani et al., 2017). it does not require an additional consistency analysis as this fuzzy technique is maximally consistent. the basic algorithm of the fuzzy swara technique consisting of five implementation steps is given below: (mavi et al., 2017; perçin 2019; sumrit et. al., 2012; zolfani & saparauskas 2013). step 1. rank the selection criteria: in a group decision process, each expert ranks the criteria. next, the ranking position of each criterion is determined by computing the arithmetic mean of given ranking scores by experts. step 2. determine the relative importance ratio: after the criteria are ranked, the criterion is compared with the next criterion by each expert. the criterion j is compared with the criterion of j-1. to make these comparisons, decision-makers use the linguistic terms given in the linguistic evaluation scale which is given in table 4. these evaluations are converted to the corresponding triangular fuzzy numbers (tfns) in the evaluation scale. by calculating the arithmetic mean of these values, the final fuzzy relative importance ratio of each criterion is determined. step 3. calculate the coefficient jk as eq. (2): 1, =1 1, >1 j j j k s j     (2) step 4. compute the intermediated weight jq as eq. (3): 1 1, =1 , >1 j j j j q q j k        (3) where;  , , l m u j j j j q q q q evaluation of tanker vehicle selection using a novel hybrid fuzzy mcdm technique 145 step 5. compute the relative weights of the evaluation criteria as eq. (4): 1 j j n j k q w q    (4) where jw donates the fuzzy weight of criterion j. step 6. compute non-fuzzy value of jw as eq. (5): ( ) 3 l m u j j j j w w w w    (5) 2.3. the fuzzy codas technique the combinative distance-based assessment (codas) technique developed by keshavarz ghorabaee et al. (2016) is a quite novel mcdm technique and is a very useful approach. this technique considers a combinative form of the euclidean distance and taxicab distance to determine the ranking performances of the decision alternatives. however, the codas technique gives results when crisp values are available. in order to present an mcdm technique, which enables to deal with uncertainties, this approach was extended in later studies (keshavarz ghorabaee et al., 2016a; yalçın & pehlivan, 2019; vinodh & wankhede, 2020; roy et. al., 2019) with the help of the fuzzy set theory introduced by zadeh (1965). the basic algorithm of the fuzzy codas (f-codas) technique consisting of nine implementation steps are given as follows: (keshavarz ghorabaee et al., 2016a; yalçın & pehlivan, 2019; roy et. al., 2019; katrancı & kundakcı, 2020) step 1. generate the fuzzy decision matrix: in a group decision process, k number of experts evaluate m number of decision alternatives by considering n number of criteria as seen in equation 6. the decision-makers perform linguistic evaluations for options considering the linguistic terms given in table 5. 1 1 1 2 2 2 11 12 1 11 12 1 11 12 1 1 1 2 2 2 1 221 22 2 21 22 2 21 22 1 1 1 2 2 2 31 32 3 31 32 3 31 32 1 1 1 2 2 2 1 2 1 2 1 2 ... ... ... ... ... .. , ,..., ... ... ... ... k k n n k k kn n k k n n k k m m mn m m mn m m x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x                               1 2 3 . ... ... k n k n k n k mn x x x x               (6) next, these evaluations are converted to the corresponding tfns in the linguistic evaluation scale. as a result, the k numbers of initial fuzzy decision matrices are obtained. finally, these matrices are combined and the initial aggregated fuzzy matrix is formed as follows (equation 7). görçün et al./decis. mak. appl. manag. eng. 4 (2) (2021) 140-162 146 11 12 1 21 22 2 1 2 ... ... ...... ... ... ... n n m m mn x x x x x x x x x x                (7) where the element of the matrix ijx denotes the average fuzzy rating score of ith option with respect to jth criterion and k symbolizes the number of experts. step 2. determine the weights of criteria: the fuzzy weight of each criterion is determined in the previous stage of the suggested integrated fuzzy approach with the help of the f-swara technique. step 3. generate the fuzzy normalized decision matrix: the fuzzy aggregated matrix is normalized by applying equation 8.     , max 1 , max ij ij i ij ij ij i x j b k x n x j c k x            (8) b denotes the benefit criteria; c symbolizes the cost criteria. step 4. form the fuzzy weighted normalized decision matrix: the fuzzy normalized matrix ij mxn n n    is weighted by using equation 9 and the weighted normalized fuzzy matrix ij mxn r r    is constructed. ij ij jr n w  (9) step 5. determine fuzzy negative-ideal solution as eqs. (10-11) 1 j xm ns ns    (10) min ijj i ns r (11) where       min min , k 1, 2,...nij kj kj kj i i r r r r     step 6. calculate the euclidean distance (edi) and hamming distance (hdi): the distance values for each alternative are computed by using equations 12 and 13.   1 , m ij ji e j ed d r ns    (12)   1 , m ij ji h j hd d r ns   (13) step 7. determine relative assessment matrix  ik mxnra p eq. (14): evaluation of tanker vehicle selection using a novel hybrid fuzzy mcdm technique 147       ik i k i k i kp ed ed t ed ed x hd hd     (14)   1 0 if x t x if x        (15) the threshold parameter (  ) of this function can be set by decision-maker. it can take values between 0 and 1. step 8. compute the assessment score (asi) of each alternative by using equation 16. 1 m i ik k as p   (16) step 9. rank the decision alternatives: decision alternatives are ranked with respect to their assessment scores in descended order. the best alternative is determined as the option having the highest assessment score. 3. evaluation of the road tanker vehicle alternatives in this section, the proposed integrated fuzzy approach is implemented to solve the decision-making problems related to the selection of road tanker vehicles. by following the basic algorithm of the integrated fuzzy model, the best solution was tried to obtain. details of the selected professionals are given in table 1. table 1. details of the members of the board of experts no graduate duty exp. country dm-1 logistics management advisor of dg 14 turkey dm-2 business management advisor of dg 18 turkey dm-3 logistics management advisor of dg 17 netherlands dm-4 business management advisor of dg 18 turkey dm-5 mechanical engineering advisor of dg 21 bulgaria dm-6 industrial engineering advisor of dg 27 india the research questions determined in the first step of the research process were directed to these experts and the obtained answers were recorded. as one of the most significant findings of these research process, it has been ascertained that there is no mathematical model or computational tool used for solving the road tanker selection problem. according to the opinions of the members of the board of experts, decisionmakers mostly decide considering their own experiences and, knowledge, and competence furthermore they consider these kinds of problems as singly case. at the end of the face-to-face interviews and well-attended meetings with the members of the boards, researchers requested to prepare a list from each expert to determine the selection criteria. next, these lists were combined and the final criteria list was formed as in the table 2. görçün et al./decis. mak. appl. manag. eng. 4 (2) (2021) 140-162 148 table 2. the selection criteria for road tanker vehicles code criteria code criteria c1 purchase price c8 empty weight c2 maintenance cost c9 tanker length c3 number of authorized services c10 number of divisions c4 capacity c11 design pressure value c5 material thickness c12 design temperature value max c6 safety c13 design temperature value min c7 loaded weight according to the opinions of the experts, technical and economic criteria are crucial to select the appropriate road tanker vehicle. these vehicles should be in good condition and they have to respond some requirements related to dangerous goods transportation. as a result of negotiations taken long time with the experts, the selection criteria were determined by providing full consensus among these experts. the set of the selection criteria are defined as follows: c1 purchase price: it is the price an individual or companies, which operates dangerous goods transportation, pays for purchasing a road tanker vehicle. it can also be defined as the acquisition cost. c2 maintenance cost: it refers to the expense incurred to ensure that a road tanker vehicle continues to operate healthily. the maintenance cost is also defined as all kinds of costs which are bear by individuals or businesses for keeping their vehicles in good working condition. c3 number of authorized services: it refers to the number of authorized service suppliers, which can provide for regular or irregular maintaining and repairing support at the global level. c4 capacity: it refers to the total weight capacity that can be carried by a road tanker vehicle at a minimum or zero risk level in terms of the liter, m3, or tons. c5 material thickness: it defines the thickness of materials used for manufacturing a road tanker. unece has determined minimum standards on metal plate thickness used as a semi-finished product for manufacturers and transport operators since it is very crucial for safety and security. c6 safety: it refers to the availability of the safety requirements, which determined by international institutions on dangerous transportation, for a road tanker vehicle. all of them may not be installed on each road tanker or all of the safety equipment such as electronic stability control systems, safety valves, manholes, internal bulkheads, and so on may not have the same quality and abilities. due to these differences, the safety level of a road tanker is one of the crucial factors for selecting the appropriate vehicle. c7 loaded weight: it means to total weight of the loaded tanker vehicle. it equals the sum of the weight of liquid cargo and tare weight of a vehicle. overweighting can cause to rollover of road tanker vehicles as there is a negative correlation between the total weight and vehicle stability. hence, it is a crucial factor that affects road, vehicle, and cargo security directly. c8 empty weight: empty weight means the weight of a road tanker vehicle including the operating body and accessories. there is a negative correlation between the empty weight of a tanker vehicle and its carrying capacity. hence, it can be accepted as both a technical and economic criterion since it is a determinative factor with respect to the carriage capacity of a vehicle. evaluation of tanker vehicle selection using a novel hybrid fuzzy mcdm technique 149 c9 tanker length: it refers to the maximum length of a road tanker vehicle that is fixed or extendable. it is one of the crucial selection criteria since can affect the maneuverability of a tanker vehicle and utility of these kinds of vehicles in all dangerous goods transport operations. c10 number of divisions: it means the number of internal baffles in a tanker truck. while it causes to reduce carriage capacity of a vehicle, allows carrying different types of liquid cargo (i.e. oil, diesel, gasoline, and etc.) in a single transport operation. c11 design pressure value: it refers to technical requirements for loading, unloading liquid cargo in terms of bar or psi. standards for these requirements at a minimum level have determined by international institutions such as unece. c12 design temperature value max: providing a certain temperature value may be required when special liquid cargoes are carried. hence, it means the temperature value that can be provided by a road tanker vehicle at a maximum degree. c13 design temperature value min: it means the temperature value that can be provided by a road tanker vehicle at a minimum degree. next, the decision alternatives were determined together with experts and given in table 3. table 3. decision alternatives for road tanker vehicles code options code options a1 brand ok. a5 brand ko a2 brand ot. a6 brand tr a3 brand ta a7 brand rh a4 brand tş a8 brand iz while the decision alternatives were determined, road tanker vehicle market of europe and turkey was taken into consideration and the products manufactured by key players of the market were included into the scope of the research process. then, the criteria and options were determined, researchers progressed to the next stage of the proposed fuzzy model. 3.1. calculation of the weights of the criteria next, experts performed linguistic evaluations for criteria and alternatives and these evaluations were converted to the corresponding tfns in the linguistic evaluation scale as given in the table 4. table 4. linguistic weighting scale for criteria (perçin, 2019) triangular fuzzy number (tfns) for tangible linguistic terms abbr. l m u very low vl 0.00 0.00 0.30 low l 0.00 0.25 0.50 medium m 0.30 0.50 0.70 high h 0.50 0.75 1.00 very high vh 0.70 1.00 1.00 step 1. expert rank the criteria by considering own experiences and judgments. then the final ranking scores of the criteria were determined by computing the görçün et al./decis. mak. appl. manag. eng. 4 (2) (2021) 140-162 150 geometric mean of ranking scores given by experts for each criterion as shown in table 5. table 5. ranking of the criteria and the final ranking scores code dm1 dm2 dm3 dm4 dm5 dm6 geo mean c1 1 2 1 2 2 3 1.698 c4 4 5 5 6 5 5 4.966 c6 7 6 6 5 7 7 6.287 c5 2 3 3 3 3 2 2.621 c2 5 4 4 4 1 1 2.615 c3 3 1 2 1 4 4 2.140 c10 9 10 8 8 9 9 8.807 c9 10 9 10 9 10 10 9.655 c7 8 8 9 7 8 8 7.979 c8 6 7 7 10 6 6 6.878 c11 11 11 11 12 11 11 11.161 c12 12 13 12 11 12 12 11.986 c13 13 12 13 13 13 13 12.828 step 2. experts perform linguistic evaluation for criteria to determine the relative importance ratio of each criterion by making comparison between criterion j and j-1 criterion. the determined linguistic evaluations are presented in table 6. table 6. linguistic evaluations for the relative importance ratios code dm1 dm2 dm3 dm4 dm5 dm6 c1 c4 vl vl vl l vl l c6 m h m m m vl c5 vl vl vl vl vl vl c2 vl l vl vl vl l c3 vl m vl vl m m c10 h h h vh l m c9 l vl l l vl vl c7 l vh m l vh vh c8 vl vl vl vl l l c11 vh h vh m vh vh c12 vl l vl l l l c13 vl l vl l l l the weights of the factors were computed and presented in table 7. table 7. the results obtained by applying the fuzzy swara technique code js jk jq jw 𝑑𝑓𝑢𝑧𝑧𝑖𝑒 𝑑 j w c1 1.000 1.000 1.000 1.000 1.000 1.000 0.137 0.191 0.318 0.215 0.172 c4 0.000 0.083 0.367 1.000 1.083 1.367 0.732 0.923 1.000 0.100 0.176 0.318 0.198 0.158 c6 0.283 0.458 0.683 1.283 1.458 1.683 0.435 0.633 0.779 0.060 0.121 0.248 0.143 0.114 c5 0.000 0.000 0.300 1.000 1.000 1.300 0.334 0.633 0.779 0.046 0.121 0.248 0.138 0.110 c2 0.000 0.083 0.367 1.000 1.083 1.367 0.245 0.584 0.779 0.034 0.111 0.248 0.131 0.105 c3 0.150 0.250 0.500 1.150 1.250 1.500 0.163 0.467 0.678 0.022 0.089 0.215 0.109 0.087 c10 0.417 0.667 0.867 1.417 1.667 1.867 0.087 0.280 0.478 0.012 0.053 0.152 0.072 0.058 c9 0.000 0.125 0.400 1.000 1.125 1.400 0.062 0.249 0.478 0.009 0.047 0.152 0.069 0.055 c7 0.400 0.667 0.783 1.400 1.667 1.783 0.035 0.150 0.342 0.005 0.028 0.109 0.047 0.038 evaluation of tanker vehicle selection using a novel hybrid fuzzy mcdm technique 151 c8 0.000 0.083 0.367 1.000 1.083 1.367 0.026 0.138 0.342 0.004 0.026 0.109 0.046 0.037 c11 0.600 0.875 0.950 1.600 1.875 1.950 0.013 0.074 0.214 0.002 0.014 0.068 0.028 0.022 c12 0.000 0.167 0.433 1.000 1.167 1.433 0.009 0.063 0.214 0.001 0.012 0.068 0.027 0.022 c13 0.000 0.167 0.433 1.000 1.167 1.433 0.006 0.054 0.214 0.001 0.010 0.068 0.026 0.021 3.2. determining the preference ratings of the alternatives the example is related to road tanker vehicles used in the field of dangerous goods transportation. hazardous transportation firms need to assess all potential tanker vehicles and have to choose the appropriate alternative among them to reach safe, effective, and productive transport operations. in order to conduct successful and productive research and reach accurate, reasonable, and realistic results, a board of experts was constructed by researchers, and they took on a task as advisors and experts during the research process as mentioned in the previous section. step 1. experts performed linguistic evaluations for decision alternatives considering the linguistic terms given in table 8. these evaluations were converted to the corresponding tfns in the linguistic evaluation scale. table 8. linguistic scale for alternatives (chen, 2000) triangular fuzzy number (tfns) for tangible linguistic terms abbr. l m u very poor vp 0 0 1 poor p 0 1 3 medium poor mp 1 3 5 medium m 3 5 7 medium good mg 5 7 9 good g 7 9 10 very good vg 9 10 10 after, these evaluations were converted to the tfns, k number of initial fuzzy decision matrices were generated and these matrices were combined and the aggregated fuzzy matrix (table 9) was constructed as follows. step 2. the fuzzy weight of each criterion was computed by using the f-swara technique. the obtained fuzzy weights of the criteria are given in table 7. table 9. the initial fuzzy matrix code a1 a2 a3 a4 a5 l m u l m u l m u l m u l m u c1 2.57 3.71 5.14 5.14 6.86 8.14 3.86 5.29 6.57 3.00 4.57 6.29 1.43 2.71 4.43 c2 5.57 7.14 8.14 4.57 6.14 7.57 7.00 8.43 9.14 6.29 7.57 8.43 1.71 2.86 4.43 c3 1.57 2.43 3.86 4.14 5.43 6.71 2.29 2.86 3.86 6.00 7.57 8.57 2.57 3.71 5.14 c4 6.57 7.86 8.57 6.14 7.71 8.71 7.00 8.43 9.29 7.57 9.14 9.86 6.43 8.14 9.14 c5 4.86 6.29 7.57 3.71 4.71 5.86 4.00 5.71 7.29 2.57 4.29 6.00 6.86 8.29 9.00 c6 6.14 8.00 9.14 7.29 8.86 9.57 7.57 9.00 9.57 7.00 8.71 9.57 6.71 8.43 9.43 c7 4.43 5.71 6.86 5.57 6.86 7.86 5.14 7.00 8.43 6.43 8.00 8.86 5.86 7.57 8.86 c8 7.29 8.86 9.57 3.57 4.86 6.00 4.86 6.43 7.71 5.57 6.86 7.71 4.29 5.86 7.43 c9 5.14 6.57 7.71 4.43 5.71 6.86 4.86 6.43 7.71 6.00 7.57 8.57 7.00 8.71 9.71 c10 7.86 9.29 9.86 6.29 7.71 8.71 8.14 9.57 10.00 6.43 8.29 9.43 6.86 8.29 9.00 c11 0.71 1.14 2.43 1.14 1.86 3.29 3.00 3.57 4.43 7.86 9.14 9.57 6.71 8.43 9.43 c12 3.29 4.71 6.14 5.14 7.00 8.43 7.57 9.14 9.86 5.14 7.00 8.43 5.57 7.43 8.86 c13 3.57 5.14 6.86 4.86 6.71 8.14 4.86 6.43 7.86 5.86 7.43 8.57 7.29 8.86 9.71 code a6 a7 a8 görçün et al./decis. mak. appl. manag. eng. 4 (2) (2021) 140-162 152 l m u l m u l m u c1 3.43 4.57 5.86 2.86 3.71 4.71 6.14 7.00 7.43 c2 1.43 2.86 4.71 5.86 6.86 7.43 5.29 6.43 7.29 c3 3.57 5.29 7.00 3.57 5.14 6.57 1.71 2.14 3.14 c4 5.29 6.57 7.57 7.57 9.00 9.57 6.29 7.71 8.57 c5 6.71 8.29 9.14 7.00 8.57 9.43 4.86 6.14 7.14 c6 6.43 8.14 9.14 6.71 8.43 9.43 5.71 7.14 8.14 c7 7.29 8.71 9.57 5.00 6.29 7.43 4.86 6.71 8.29 c8 2.71 4.14 5.86 3.86 5.14 6.43 3.71 5.29 6.71 c9 4.71 6.00 7.14 7.00 8.57 9.43 5.14 6.57 7.71 c10 8.14 9.57 10.00 7.86 9.43 10.00 6.86 8.14 8.71 c11 5.14 5.71 6.14 3.43 4.43 5.57 4.29 5.14 6.00 c12 5.14 7.00 8.43 7.29 8.86 9.57 7.43 8.43 8.71 c13 8.71 9.86 10.00 7.29 8.86 9.71 7.29 8.71 9.29 step 3-5. by implementing the equation 8, the initial fuzzy matrix was normalized. afterward, with the help of equation 9, the matrix was weighted (table 10) as follows in the step 4. next, by applying equation 10-11, the fuzzy negative-ideal solutions( jns ) were computed. table 10. the weighted normalized fuzzy matrix code a1 a2 a3 a4 a5 l m u l m u l m u l m u l m u c1 0.07 0.12 0.24 0.03 0.06 0.15 0.05 0.09 0.20 0.05 0.10 0.22 0.08 0.14 0.27 c2 0.01 0.03 0.11 0.01 0.04 0.13 0.00 0.02 0.07 0.01 0.03 0.09 0.02 0.08 0.21 c3 0.00 0.02 0.08 0.01 0.05 0.14 0.01 0.03 0.08 0.01 0.07 0.18 0.01 0.03 0.11 c4 0.07 0.14 0.27 0.06 0.14 0.28 0.07 0.15 0.30 0.08 0.16 0.31 0.06 0.14 0.29 c5 0.02 0.08 0.19 0.02 0.06 0.14 0.02 0.07 0.18 0.01 0.05 0.15 0.03 0.10 0.22 c6 0.04 0.10 0.23 0.04 0.11 0.24 0.05 0.11 0.24 0.04 0.11 0.24 0.04 0.10 0.23 c7 0.00 0.02 0.07 0.00 0.02 0.09 0.00 0.02 0.09 0.00 0.02 0.10 0.00 0.02 0.10 c8 0.00 0.00 0.03 0.00 0.01 0.07 0.00 0.01 0.06 0.00 0.01 0.05 0.00 0.01 0.06 c9 0.00 0.02 0.07 0.00 0.02 0.08 0.00 0.02 0.08 0.00 0.01 0.06 0.00 0.01 0.05 c10 0.01 0.05 0.15 0.01 0.04 0.13 0.01 0.05 0.15 0.01 0.04 0.14 0.01 0.04 0.14 c11 0.00 0.00 0.02 0.00 0.00 0.02 0.00 0.01 0.03 0.00 0.01 0.06 0.00 0.01 0.06 c12 0.00 0.01 0.04 0.00 0.01 0.06 0.00 0.01 0.07 0.00 0.01 0.06 0.00 0.01 0.06 c13 0.00 0.01 0.04 0.00 0.00 0.03 0.00 0.00 0.03 0.00 0.00 0.03 0.00 0.00 0.02 table 10. the weighted normalized fuzzy matrix (continue) code a6 a7 a8 jns l m u l m u l m u l m u c1 0.06 0.10 0.21 0.07 0.12 0.23 0.04 0.06 0.12 0.025 0.057 0.123 c2 0.02 0.08 0.21 0.01 0.03 0.10 0.01 0.04 0.12 0.003 0.017 0.074 c3 0.01 0.05 0.15 0.01 0.05 0.14 0.00 0.02 0.07 0.004 0.019 0.068 c4 0.05 0.12 0.24 0.08 0.16 0.30 0.06 0.14 0.27 0.053 0.116 0.241 c5 0.03 0.10 0.23 0.03 0.10 0.23 0.02 0.07 0.18 0.012 0.052 0.145 c6 0.04 0.10 0.23 0.04 0.10 0.23 0.03 0.09 0.20 0.034 0.086 0.202 c7 0.00 0.02 0.10 0.00 0.02 0.08 0.00 0.02 0.09 0.002 0.016 0.074 c8 0.00 0.02 0.08 0.00 0.01 0.07 0.00 0.01 0.07 0.000 0.003 0.029 c9 0.00 0.02 0.08 0.00 0.01 0.05 0.00 0.02 0.07 0.000 0.006 0.046 c10 0.01 0.05 0.15 0.01 0.05 0.15 0.01 0.04 0.13 0.008 0.041 0.132 c11 0.00 0.01 0.04 0.00 0.01 0.04 0.00 0.01 0.04 0.000 0.002 0.016 evaluation of tanker vehicle selection using a novel hybrid fuzzy mcdm technique 153 c12 0.00 0.01 0.06 0.00 0.01 0.06 0.00 0.01 0.06 0.000 0.006 0.042 c13 0.00 0.00 0.01 0.00 0.00 0.02 0.00 0.00 0.02 0.000 0.000 0.009 step 6. the euclidean distance (edi) and hamming distance (hdi) were calculated by using equations 12 and 13. the computed values are presented in table 11. table 11. the euclidean distance (edi) and hamming distance (hdi) for each option code a1 a2 a3 a4 a5 a6 a7 a8 edi 0.213 0.408 0.223 0.294 0.376 0.349 0.320 0.150 hdi 0.184 0.173 0.187 0.245 0.322 0.292 0.279 0.121 step 7. the relative assessment matrix was constructed by applying equation 1415. the threshold parameter (  ) was taken as 0.02 in the current paper. step 8. the assessment score (asi) of each alternative was calculated by using equation 16. the relative assessment matrix and the assessment scores of options are presented in table 12. table 12. the relative assessment matrix and the assessment scores of options code a1 a2 a3 a4 a5 a6 a7 a8 i as rank a1 0.00 -0.18 -0.01 -0.14 -0.30 -0.24 -0.20 0.13 -0.95 7 a2 0.18 0.00 0.17 0.04 -0.12 -0.06 -0.02 0.31 0.52 4 a3 0.01 -0.17 0.00 -0.13 -0.29 -0.23 -0.19 0.14 -0.86 6 a4 0.14 -0.04 0.13 0.00 -0.16 -0.10 -0.06 0.27 0.17 5 a5 0.30 0.12 0.29 0.16 0.00 0.06 0.10 0.43 1.45 1 a6 0.24 0.06 0.23 0.10 -0.06 0.00 0.04 0.37 0.99 2 a7 0.20 0.02 0.19 0.06 -0.10 -0.04 0.00 0.33 0.65 3 a8 -0.13 -0.31 -0.14 -0.27 -0.43 -0.37 -0.33 0.00 -1.97 8 step 9. at the end of the f-codas technique, the decision alternatives were ranked with respect to their assessment scores. 4. the validation test a comprehensive sensitivity analysis consisting of two stages was performed to test the validation of the proposed integrated novel fuzzy model. first, impact of changing the weights of the input and output factors on the ranking results were examined. secondly, the results of the proposed model were compared to the results of different fuzzy techniques. a) examination of impacts of changing the weights of the criteria on the ranking performances of the alternatives: in the first stage of the sensitivity analysis, the weight of each criterion is modified to examine its impacts on the preference ratings of the option by forming 130 scenarios. previous works suggested changing the weights of criteria that are in the first three ranks (stankovic et al., 2020). this kind of approach can give a limited result since it did not consider the potential impacts of changes in weights of the remained criteria. this work takes into consideration the potential effects of all criteria having impacts on the results more or less. therefore, while the weight of each factor is modified at the rate of 10% in each scenario, the weights of the remaining factors are corrected to meet the condition of the sum of görçün et al./decis. mak. appl. manag. eng. 4 (2) (2021) 140-162 154 weights should be equal to 1. new weight values of the criteria are determined for each scenario with the help of equations 17, 18, and 19 respectively.  1 1 1 .nv pv pv vw w w   (17)  1 2 2 1 1 nv rfv pv w w w n     (18) 1 2 1nv rfvw w  (19) here, 1nvw denotes new value of modified weight of j th factor, 1 pvw is the previous values of the criterion, v is the modification degree in terms of percentage (i.e. 10%, 20%,...,100%). also, 2 1rfvw symbolizes new values of remaining factors, n is the number of factors, 2 pvw is the previous values of the remaining criteria. to examine the effects of modified weights on the preference ratings of the options, the new ranking performances of the alternatives are calculated by using the changed new weights of the criteria and the obtained results are presented in figure 2. figure 2. impacts of changes of criteria weight on the ranking performance of the alternatives evaluation of tanker vehicle selection using a novel hybrid fuzzy mcdm technique 155 when impacts of modification of the criteria weight on preference ratings of the options are evaluated, the same ranking result has been obtained for 51 scenarios and average correlation coefficient value among the results of the scenarios has been determined as 95.67% for all scenarios. a5 has remained as the best option for 105 scenarios (80.77%). when the weight value of criterion 1th is changed at the rate of 50%, the ranking position of a5 has also changed. in addition, it has been observed the same situation, when the weights of the criteria c11 (at over 40%) and c12 (at over 60%) has been changed. a6 which has been determined as the second-best alternative by applying the proposed fuzzy model has also remained at the same ranking position for 106 scenarios (81.54%). the ranking performance of a3 has changed for only 4 scenarios, and it has remained at the same rank for 126 scenarios (90%). changes in preference ratings of other alternatives are presented in the following table13. table 13. ranking of the options with respect to 130 scenarios code 1th 2th 3th 4th 5th 6th 7th 8th similarity (%) a1 0 0 0 0 0 4 117 9 90.00 a2 15 4 30 54 27 0 0 0 41.54 a3 0 0 0 0 0 126 4 0 96.92 a4 0 0 1 39 90 0 0 0 69.23 a5 105 14 11 0 0 0 0 0 80.77 a6 9 106 9 3 3 0 0 0 81.54 a7 1 6 79 34 10 0 0 0 60.77 a8 0 0 0 0 0 0 9 121 93.08 when the results of the first stage of the sensitivity analysis are evaluated in general, slight changes which cannot change the overall results in the preference ratings of the alternatives depending on modification of the weight values of the criteria. these changes occur when the weights of the criteria were changed. although modifications were made in the weights of criteria at excessive level, the obtained results show that the proposed integrated fuzzy approach is a very strong technique giving accurate, realistic and reasonable results even in adverse conditions which have low possibility of emergence. b) making comparisons with other fuzzy approaches: in this stage, the results of the proposed fuzzy model is compared with fuzzy approaches such as f-mabac (jokić et al., 2021), f-edas (keshavarz-ghorabaee et al., 2016b), f-marcos (stanković et al., 2020), f-topsis (chen et al., 2000), and f-mairca (boral et al., 2020). the obtained results of the comparisons is shown in figure 4. görçün et al./decis. mak. appl. manag. eng. 4 (2) (2021) 140-162 156 figure 4. re-ranking of the decision alternatives with respect implemented other fuzzy decision models. it is observed that a5 determined as the best option by using the proposed fuzzy approach is also the best alternative for all applied fuzzy techniques. a6 which is the second-best alternative and a7 that is the third-best option have remained in the same rank for all implemented fuzzy techniques except the fuzzy mabac. also, the ranking position of a8 has not changed for all applied fuzzy approaches. it has been observed that there are slight changes in ranking performances of remained alternatives. the spearman correlation coefficients between the proposed model and other fuzzy mcdm techniques were calculated and the obtained results are given in table 14. table 14. spearman's correlation (ssc) coefficients among the fuzzy mcdm methods code proposed hybrid model fuzzymabac fuzzyedas fuzzymarcos fuzzytopsis fuzzymairca proposed hybrid model 1 ,833* ,952** ,929** ,929** ,929** fuzzy-mabac ,833* 1 ,881** ,952** ,952** ,952** fuzzy-edas ,952** ,881** 1 ,929** ,929** ,929** fuzzy-marcos ,929** ,952** ,929** 1 1,000** 1,000** fuzzy-topsis ,929** ,952** ,929** 1,000** 1 1,000** fuzzy-mairca ,929** ,952** ,929** 1,000** 1,000** 1 * correlation is significant at the 0.05 level (2-tailed). ** correlation is significant at the 0.01 level (2-tailed). since the average correlation value is high at the rate of 0.914, the results of the sensitivity analysis validate the proposed model. evaluation of tanker vehicle selection using a novel hybrid fuzzy mcdm technique 157 5. overall results and conclusion dangerous goods transportation is one of the special types of logistics activities requiring a set of special implementations for reducing the risks. since there is no tolerance to any mistake, each process related to transport operations should be planned. road tanker vehicle selection is also one of the very crucial decision-making problem since it can affect almost all process related to dangerous goods transportation. the number of studies is limited and there are serious gaps in the literature. there is no mathematical and systematic model applied to solve these kinds of decision-making problems encountered in the field of dangerous goods logistics. decision-makers make decisions considering their own experiences and individual judgments and they consider almost all selection process as a special case and try to produce solution for each problem individually. this work proposes a hybrid fuzzy approach that can be applied as a methodological and systematic frame for both scientific works carried out in the future and practitioners who responsible to decide in the field of dangerous goods logistics. the proposed integrated fuzzy technique consists of the swara and the codas technique and while the f-swara is applied to calculate the weights of the criteria, the f-codas method is implemented for determining the preference ratings of the decision alternatives. the proposed integrated fuzzy approach is a novel hybrid technique and it has many advantages compared to traditional mcdm methods. the extended codas technique with the help of fuzzy set theory can provide more reasonable, applicable, and accurate results, which appropriate to real life as it uses the euclidean distance and the hamming distance, and utilizes only the negative-ideal solution in the evaluation process (yalçın & pehlivan, 2019). the sensitivity analysis performed to validate the proposed model and its results have verified the applicability of the model. according to the obtained results, as well as the hybrid fuzzy technique can be applied to solve these kinds of decision-making problems, the results obtained by implementing the integrated fuzzy technique proposed in the current paper are accurate, realistic and reasonable. consequently, the potential contributions of the paper can be summarized as follows:  the current paper suggests a novel integrated fuzzy approach having an applicable basic algorithm that can also be implemented by decisionmakers who are in the field of logistics industry.  it presents a methodological frame which can enable to deal with many ambiguities existing in an assessment process on the tanker vehicle selection.  it determines a set of novel criteria, which can be considered by practitioners in an assessment processes on the selection of appropriate road tanker vehicles.  it can deal with ambiguities existing in an evaluation process.  the proposed hybrid approach provides flexibility to the decision-makers  it can also be applied to solve decision-making problems encountered in various fields. when the main findings of the paper focusing on the selection of road oil tankers are examined, c1 "purchasing price" has been determined as the most significant criterion. c4 "capacity" and c6 "safety" criteria follow the first ranked criterion respectively. the importance and sequence of the criteria are aligned from most critical to least critical is c1>c4>c6>c5>c2>c3>c10> c9> c7> c8> c11> c12> c13. görçün et al./decis. mak. appl. manag. eng. 4 (2) (2021) 140-162 158 according to the results of the analysis, a5 brand ko is the best alternative that has the highest performance score, and a6 brand tr and a7 brand rh follow it. other options are ranked as follows: a2>a4>a3>a1>a8. the current work has also some limitations. first of all, experts should be selected carefully to obtain reasonable and realistic results. therefore, selecting experts who are highly experienced, having deep knowledge, and certificated by authority may beneficial for researchers who will carry out research on this issue in the future. also, selecting the right and appropriate criteria is crucial, and performing only a literature review for determining the criteria may not be sufficient hence fieldwork performed together with experts may a beneficial way to describe the proper selection criteria. in addition, the f-codas technique can be extended with the help of different operators such as the normalized weighted and normalized weighted geometric bonferroni aggregate functions (ecer & pamucar, 2020), heronian mean (hm) operators (yu et al., 2012), hybrid weight power heronian operator (wphap,q) and hybrid weight geometric power heronian operator (wgpha p,q) (pamucar & jankovic., 2020). also, it can be examined comparatively with different approaches based on 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(2013). new application of swara method in prioritizing sustainability assessment indicators of energy system, engineering economics, 24(5), 408-414. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). decision making: applications in management and engineering vol. 1, number 1, 2018, pp. 97-120 issn: 2560-6018 doi: https://doi.org/10.31181/dmame180197k * corresponding author. e-mail addresses: zkaravidic@gmail.com (z. karavidić), damirpro@yahoo.com (d. projović) a multi-criteria decision-making (mcdm) model in the security forces operations based on rough sets zoran karavidić1*, damir projović1 1 university of defence in belgrade, military academy, department of management, belgrade, serbia received: 3 january 2018; accepted: 18 february 2018; published: 15 march 2018. original scientific paper abstract: the paper points to a multi-criteria decision-making model based on the rough set theory application. the model demonstrates exceptional importance of the software application of the rough sets to decision-making in the security forces operations. applying the rough sets represents a useful tool when the data, needed for the decision-making process, include vagueness and uncertainty. by applying the model based on the applicative use of the rough sets, specific decision-making rules are formulated. these rules guide the decision-makers through the complete process of planning the security operations. key words: multi-criteria decision-making, rough sets, course of action, rosetta, rose2. 1. introduction modern international relations are very unpredictable in the political, economic and social life. in such an environment, there is a frequent need for engaging security forces due to the demand for protection of national interests or democratic order. the security forces are engaged in various operations. in recent years, the security forces have often been involved in counterterrorism and counter-insurgency assignments in the world. however, the objective of these operations could also be to support civilians in the case of natural disasters, fight against crime or have various other combat and non-combat engagements involving military, police and other security forces (slavkovic et al., 2012, 2013). the complexity of managing security forces operations, especially of deciding how to use the security forces, represents a major challenge. choosing one from a set of available courses of action (coa) is a part of multi-criteria decision-making (mcdm) process which cannot be avoided. in this respect, the problem is how to choose a coa based on incomplete, inaccurate mailto:zkaravidic@gmail.com mailto:damirpro@yahoo.com karavidić & projović/decis. mak. appl. manag. eng. 1 (1) (2018) 97-120 98 and inseparable data in the security forces operations with the help of various decision-making support models. the significance of this problem is reflected in possible major losses of resources, both human and material. in that sense, every model contributing to a well-timed and better decision made by the managing authorities, will contribute to a more efficient implementation of the security forces operation. so far, the following have been considered for the needs of the security forces: a fuzzy logical system in support to the decision-making process in military organization (pamučar et al., 2011), a hybrid model fahp-mabac in selecting locations for the preparation of laying-up positions (božanić et al., 2016), as well as combined gis and multi-criteria techniques in the selection of sites which are suitable for ammunition depots (gigović et al., 2016). due to secrecy and licenses, various world experiences are rather difficult to access. they are also limited to learning about general settings of functioning. in other areas, applying the rough sets theory (rst) in decision-making focuses its use on modern business environment (shen & chen, 2013, shen, et al., 2017), estimation of bridges construction (kuburić et al., 2012), performance improvement of transportation systems (deshpande & bajaj, 2017) and mining for underground deep-hole mining (jiang et al., 2009). applying the rst is significant in the medical field in preventing diseases (chowdhary & acharjya, 2016) diagnostics (stokić et al., 2010; ji et al., 2012; burney & abbas, 2015), and processing medical data (durairaj & sathyavathi, 2013). the rst has also been used in data mining (greco et al., 2002; jia et al., 2007; chen et al., 2015), with different computer models (dobrilovic et al., 2012). the methods dealing with support in the decision-making operations of security forces choose a coa based on different methodologies of attribute comparison and suggest a given solution to the decision-makers. the application of the model based on the rough sets uses the previously performed security forces operations. by using the software systems with reduction principle, the most important attributes for decision-making are discovered. through decision algorithms, guidelines are given to decision-makers in the decision-making process. the advantage of this model is not only in providing support to the decision-making process in choosing a coa but also in guiding the whole decision-making process. at the same time, a great amount of time is saved. the paper is divided into several sections, namely: section 2 explains the problems of decision-making in a modern security environment, while section 3 presents the basics of the rst. section 4 refers to the existing software systems based on the rst, while section 5 shows the use of the proposed model based on the rst. section 6 gives a discussion of the model results. finally, section 7 presents the conclusions highlighting directions for further research. 2. problems of decision-making in security operations in a modern environment a modern security environment does not represent a precisely defined set of variables. it is an extremely complex part of the society that expresses all its interactions. the use of the security forces in operations is certainly susceptible to the impact of such an environment. each of the possible impacts consists of a subsystem spectrum and contains different interconnections. a great number of factors, which could more or less affect the operation results, emerge from a complex a multi-criteria decision-making (mcdm) model in the security forces operations ... 99 and unpredictable environment. those factors can be observed as criteria or attributes in the decision-making process. persons who decide on the use of force are trying in various ways to make the most appropriate choice among the coas offered. the appropriate decision is often reflected in human lives, and the proper approach is extremely important. such problems represent a major challenge for decision-makers. they are semistructured and unstructured which makes it difficult to solve them. therefore, there is space for implementing different decision-making support models that need to improve the decision-making process. they represent symbiosis of information systems, the application of a set of functional knowledge and the ongoing decisionmaking process (suknović & delibašić, 2010). for their work, they search for a database that forms the source of information, certain model solutions, and the corresponding user interface. the models should improve the knowledge of the decision-maker in order to help him make the right decision. supporting the choice of the coa in security forces operations is a very complex process. in addition to a large number of inseparable factors, there is a constant time constraint as well as the need for a quick response of the entire system. time constraint is one of the biggest problems since it affects, directly or indirectly, different parts of the planning process and the organization of operations. in the process of preparing such operations, time limits the implementation of various expert methods and disables the complete analysis of the environment. the time for decision-making, usually measured in hours, is very brief and it can be even shorter. the short time can make the entire decision-making process even harder. these problems are often expanded by a large number of contradictory, unclear and inseparable pieces of information, which arise in the later stages of the decisionmaking process. the time frame in those situations does not allow a detailed analysis and precise classification. various software systems have been developed for the needs of the entire decision-making process in security forces operations. these systems provide different types of support to the decision-making process. one of such systems is topfas (tamai, 2009) developed especially for the needs of the comprehensive approach to planning the use of nato forces. it contains support for all levels of planning. it enables a detailed and rapid system analysis, support to decision-making and assistance in monitoring the implementation of the decision. 3. the basis of the rough sets imperfect knowledge has always been the subject of study in various fields of science. many approaches to the problem, such as how to understand imperfect knowledge and how to handle it, have been developed. one of the approaches to the problem is the rst. the rough set theory is a mathematical theory presented by the polish scientist zdzisław pawlak at the beginning of the 80’s in the 20th century (pawlak, 1982). this theory has found a number of interesting applications and it is essential for artificial intelligence and cognitive sciences, especially in the areas of machine learning, knowledge acquisition, decision analysis, knowledge discovery from databases, expert systems, inductive reasoning, and pattern recognition. the rough set theory starts from the assumption that each object in the universe (u) is described by some characteristic information. different objects that are described by the same piece of information are considered to be inseparable, i.e. karavidić & projović/decis. mak. appl. manag. eng. 1 (1) (2018) 97-120 100 similar to each other. the indiscernibility relation (i) created in this way represents the mathematical foundation of the rst and in certain sense describes our lack of knowledge about the universe. every rough set contains an appropriate boundary area with objects. these objects cannot be regarded, with any certainty, as belonging to any observed set or its complement. accordingly, it is assumed that a rough set can be represented by a pair of classical sets, which we call its upper and lower approximation. the lower approximation contains objects which certainly belong to the set, while the upper approximation contains objects which possibly belong to the observed set. these two basic operations can be displayed in the following way: upper approximation 𝐈∗(𝐗) = {𝐱 ∈ 𝐔: 𝐈(𝐱) ∩ 𝐗 ≠ 𝟎} and (1) lower approximation 𝐈∗(𝐗) = {𝐱 ∈ 𝐔: 𝐈(𝐱) ⊆ 𝐗}, (2) where x is a subset of u. the difference between the upper and lower approximation is the boundary region of the rough set (figure 1). the specified operation can be displayed as follows: boundary region 𝐁𝐑𝐈(𝐗) = 𝐈 ∗(𝐗) − 𝐈∗(𝐗) (3) figure 1. graph view of the rough set with upper and lower approximation rough sets are defined by approximations. approximations have the following properties: 𝐈∗(𝐗) ⊆ 𝐗 ⊆ 𝐈 ∗(𝐗) (4) 𝐈∗(ø) = 𝐈 ∗(ø) = ø, 𝐈∗(𝐔) = 𝐈 ∗(𝐔) = 𝐔 (5) 𝐈∗(𝐗⋂𝐘) = 𝐈∗(𝐗) ⋂𝐈∗(𝐘) (6) 𝐈∗(𝐗⋃𝐘) ⊇ 𝐈∗(𝐗) ⋃𝐈∗(𝐘) (7) 𝐈∗(𝐗⋂𝐘) ⊆ 𝐈∗(𝐗) ⋂𝐈∗(𝐘) (8) 𝐈∗(𝐗⋃𝐘) = 𝐈∗(𝐗) ⋃𝐈∗(𝐘) (9) if x ⊆ 𝐘, then 𝐈∗(𝐗) ⊆ 𝐈∗(𝐘) and 𝐈 ∗(𝐗) ⊆ 𝐈∗(𝐘) (10) a multi-criteria decision-making (mcdm) model in the security forces operations ... 101 𝐈∗(−𝐗) = −𝐈 ∗(𝐗) (11) 𝐈∗(−𝐗) = −𝐈∗(𝐗) (12) 𝐈∗(𝐈∗(𝐗)) = 𝐈 ∗(𝐈∗(𝐗)) = 𝐈∗(𝐗) (13) 𝐈∗(𝐈∗(𝐗)) = 𝐈∗(𝐈 ∗(𝐗)) = 𝐈∗(𝐗) (14) it is concluded that the upper and the lower approximation were, in a sense, created under the influence of the indiscernibility relation. the pieces of information we have about the objects in the boundary region are often inconsistent or even unclear. when the boundary region is empty (bri = 0), i.e. when the lower and upper approximations match, the case is about crisp (precise) set. the larger the boundary region, the rougher the set becomes. this can be shown by using the accuracy approximation coefficient: 𝛂𝐈(𝐗) = |𝐈∗(𝐗)| / |𝐈 ∗(𝐗)| (15) where |x| is the cardinality of х. for𝛂𝐈(𝐗)=1 the set is precise. for all the values 𝟎 ≤ 𝛂𝐈(𝐗) ≤ 𝟏 the set is rough. therefore, the cardinality of the border region can be used to determine the measure of vagueness, that is, the uncertainty in relation to the observed set (čupić & suknović, 2010). the uncertainty is connected to the elements that belong to the set. because of the above, rough sets can be also defined by the rough membership function. it defines the uncertainty through indiscernibility relation 𝐈: 𝝁𝑿 𝐈 (x) = |𝐗 ∩ 𝐈(𝐱)| / |𝐈(𝐱)| (16) where 𝟎 < 𝝁𝑿 𝐈 (𝐱) < 𝟏. if 𝝁𝑿 𝐈 (𝐱) < 1, the set x is rough due to i for every x ∈ x, in the case 𝝁𝑿 𝐈 (𝐱) = 𝟏, the set is precise. rough membership function has the following properties: 𝝁𝑿 𝐈 (𝐱)= 1, iff x ∈ 𝐈∗(𝐗) (17) 𝝁𝑿 𝐈 (𝐱)= 0, if x ∈ 𝐔 − 𝐈∗(𝐱) (18) 𝟎 < 𝝁𝑿 𝐈 (𝐱) < 𝟏, iff x ∈ 𝐁𝐑𝐈 (𝐗) (19) 𝝁𝑼−𝑿 𝐈 (𝐱)= 1-, if x ∈ 𝝁𝑿 𝐈 (𝐱), for any x ∈ 𝐔 (20) 𝝁𝑼∩𝑿 𝐈 (𝐱) ≤ min (𝝁𝑿 𝐈 (𝐱), 𝝁𝒀 𝐈 (𝐱)), for any x ∈ 𝐔 (21) 𝝁𝑼⋃𝑿 𝐈 (𝐱)≥ max (𝝁𝑿 𝐈 (𝐱), 𝝁𝒀 𝐈 (𝐱)), for any x ∈ 𝐔 (22) generally, the rough membership function represents a coefficient which expresses the uncertainty of element x, where x ∈ 𝐗. the rough membership function can be used to define approximations and the boundary region of a set, as follows: 𝐈∗(𝐗) = {𝐱 ∈ 𝐔: 𝐈𝝁𝑿 𝐈 (𝐱) > 𝟎} (23) 𝐈∗(𝐗) = {𝐱 ∈ 𝐔: 𝐈𝝁𝑿 𝐈 (𝐱) = 𝟏} (24) 𝐁𝐑𝐈(𝐗) = {𝐱 ∈ 𝐔: 𝟎 < 𝝁𝑿 𝐈 (𝐱) < 𝟏} (25) when solving the problem by using the rst, the rules having different decisions for more elements of the same kind can be noticed. these rules are called inconsistent and, when used, they lead to an inability to make the right decision. the problem of inconsistent rules is solved by using consistency factor c. based on the decision rule δ(x), this factor is defined as follows: karavidić & projović/decis. mak. appl. manag. eng. 1 (1) (2018) 97-120 102 c(δ(x)) = { 1, for μx i (x) = 0 or 1 μx i (x), for 0 < μx i (x) < 1 (26) the closer the value of the consistency factor gets to one, the more authentic the rule becomes. should the factor be equal to one, the rule is consistent. in the rough set theory, there is a strict link between vagueness and uncertainty (boričić, 2004). vagueness relates to sets while uncertainty to objects. due to that, approximations are necessary when speaking about vagueness of the set while the rough membership function is necessary when speaking about uncertainty of the given objects’ belonging to the observed set. input data can be quantitative and qualitative. output data represent decisive rules in the form of the statement "if ... then ...", which can be exact or approximate. based on these rules, decisions relating to the observed objects are made. 4. software systems for applying the rough set theory in order to apply the rst to data sets, a large number of software systems, which support rst, has been developed (abbas, 2016). this development can be attributed to the successful application of rough sets to data mining and knowledge discovery. for the purposes of this paper, two applications, namely rosetta and rose2, will be presented. these applications enable the work with the data needed to support the decision-making of the security forces. 4.1. rosetta rosetta was developed by the joint efforts of two groups of researchers from the norwegian university of science and technology and the mathematical institute of the university of warsaw. the project leaders were jan komorovski and andrej skovron (komorowski, 2002). the application design and the graphical user interface were developed by a norwegian group led by alexander ohrn. the rough set algorithms were applied in the software and further developed in the polish group. the rosetta system is a software package based on the concept of rough sets. the system includes a large number of algorithms for discretization and attribute reduction and data classification. it also generates if-then rules and allows data sharing for training, testing and validating of the induced rules and patterns. all these features in used version 1.4.41, are supported by the graphical user interface available for windows systems. the system is widely used in different areas. 4.2. rose2 rose2 is a software system that implements a large number of tools for working with rough sets. the system includes pre-processing (addition of missing values and discretization), approximation of values (determination of upper and lower approximation and boundary regions), calculating the core, attribute reduction, generating decision rules, classification and validation (predki et al., 1998). the basic version of the rose software system has been upgraded several times, adapted to various operating systems, and is now up-to-date as rose2. graphically and visually in the windows environment, this system in used version 2.2, does not seem to be intuitive when presenting solutions like rosetta, but it contains different algorithms that can be applied to the reduction and generation of decision rules. a multi-criteria decision-making (mcdm) model in the security forces operations ... 103 it was developed at the laboratory for intelligent support to decision-making of the institute of computer sciences in poznan, poland. 5. model application based on rough sets in security forces operations the support to the decision-making process in the security forces operations will be included in the proposed model. the phases of the model are as follows: 1) selecting the coa and defining the attribute values, 2) determining the attribute values of the selected coa and forming the decision table, 3) attribute reduction, and 4) generating decision-making rules. the model (figure 2) will be elaborated through the application of two software systems, and the results will be compared and analyzed. figure 2. decision-making support model in security forces operations 5.1. selection of coa and defining the attribute values in order to apply the rst and the proposed model, a source of data on security forces operations is required. the data of coa can be obtained in two ways: (1) from a previously conducted security force operation, and (2) from different simulated operations. the experiences from the conducted security forces operations are a good base for guiding the decision-making process. by analyzing the aforementioned operations, the data that will be used is found. project number 98-98 of the university of defence in belgrade rationalization of the military decision-making process, 2011 is especially significant for the data source. simulations of security forces operations contribute to the checking of selected coas and represent an experienced basis that leads to the improvement of the decision-making process. the university of defence simulation center simulates the operations of the jcats and karavidić & projović/decis. mak. appl. manag. eng. 1 (1) (2018) 97-120 104 janus programs and presents the data source that will be used in this paper. the coa data is entered into the model through the criteria attributes. in the evaluation process, it is necessary to assign certain values to each of these attributes. therefore, it will be necessary to specifically describe or define the values for every attribute. the application of rough sets does not exclusively require quantitative values, and the attributes in this section will be presented in a descriptive or linguistic way (table 1). however, for the needs of a more compact display and later for easier software data processing, the values of the attributes can be replaced by the corresponding numerical or letter substitutions. one of the ways to evaluate attributes is presented in the following text. table 1. overview of the attributes with values in security forces operations attribute description values the strength of our forces (a1) it represents the number of people and units through doctrinal principles for performing various security forces operations 3 – more than needed; 2 adequate (sufficient forces according to the doctrinal principles); 1 insufficient the strength of enemy forces (a2) in terms of the strength and sufficiency of the enemy, the location of the operation is examined. the number and sufficiency of the enemy are viewed through the environment in which the operation is carried out (e.g. the number of enemies in the urban environment or in the classical frontal operation is seen). 3 – very strong forces; 2 – adequate for the planned operation (sufficient forces and strength); 1 weaker enemy forces operations preparing time(a3) a time determination showing the total time available for planning the operation at all levels. in case of decrease the time for planning, harmful consequences can arise because the enemy's action will not be prevented. 3 – sufficient time; 2 limited time, which requires greater and faster approximations; 1 – insufficient time combat environment (a4) it is considered through the prism of organization complexity and the limitation of the use of our various forces in different environments. 3 favorable unpopulated (unlimited use of our forces); 2 usual (poorly populated, the terrain is different); 1 complex (most often urban) our forces casualties (a5) the losses are perceived in accordance with the principles of conducting the operation. 3 – big losses; 2 –average losses; 1 – small losses civilian assessed based on the scope of the 3 – big losses; a multi-criteria decision-making (mcdm) model in the security forces operations ... 105 attribute description values casualties (a6) operation and the complexity of the environment in which the operation is performed. 2 –average losses; 1 – small losses maneuver (a7) skillful use of movement and fire in order to bring our own forces into a more favorable position in relation to the enemy. the success of the maneuver realization greatly contributes to the realization of the operation’s goal. 3 – completely successful; 2 partially successful; 1 unsuccessful combat support (a8) reflected in the sufficiency of combat support resources in different environment. it represents the fire and operational support of our forces that conduct the operation. 2 adequate or sufficient; 1 -inadequate or insufficient protection of our forces (a9) includes various activities that are planned and undertaken in order to reduce the ability of detecting our own forces and preventing or reduce the effects of the enemy's actions. 2 – sufficient; 1 -insufficient sustainability of our forces (a10) for efficiency and autonomy of forces during their use, it combines various activities, measures and procedures of logistical support, personnel and financial security in operations. 2 – favorable; 1 unfavorable simplicity of action (a11) it implies the complexity of the conducted operation. greater complexity in accordance with doctrinal principles leads to a more difficult achievement of the planned goal. it is related to the success of the maneuver. 3 – simple; 2 partly complex; 1 – fully complex morale (a12) it implies the moral-psychological state and the determination to carry out the task. it refers to our forces that participate in the operation, but also to the condition and readiness of civilian structures to accept the consequences of the operation. the extraordinary significance of the moral aspect is manifested in unforeseen situations when it can bring a dominance over the enemy. 3 – favorable; 2 partly favorable; 1 unfavorable intelligence system (a13) collecting, processing and using intelligence data is inseparably linked to the success of the operation. quality work of the services will contribute to more precise data and reduce the uncertainty in the decision-making process 2 – adequate; 1 -inadequate command and control – c2 (a14) it implies the expertise and experience of persons who manage the operation, their organization, operability, efficiency and elasticity in conducting the operation. it is 3 – high level; 2 – adequate; 1 insufficient karavidić & projović/decis. mak. appl. manag. eng. 1 (1) (2018) 97-120 106 attribute description values related to the speed of information transmission and timely response to emerging situations. coordination with civil structures (a15) cooperation with civil administration authorities in the operations zone 2 – adequate; 1 inadequate decision attribute success of the operation (d) the result of the operation 2 – successful with minor or greater losses 1 unsuccessful 5.2. assigning values to the attributes of the selected coa and forming a decision table the decision table is a data table that distinguishes two attribute classes condition attributes (a1, a2 ... a15) and decision (action) attributes (d).table 2 shows the overview of the coa and attributes. in each row, one coa is described, and in each column, one attribute is described. the records in the table are the values of the attribute. attribute values can be expressed linguistically, but due to a more compact display, they will be replaced by numerical substitutions. in this way, each row can provide a piece of information on a particular coa in the operation. table 2. decision-making table coa a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 a 10 a 11 a 12 a 13 a 14 a 15 d 1. 3 3 3 3 2 2 3 2 2 2 3 2 2 3 2 2 2. 3 1 2 1 2 3 2 2 2 2 2 2 1 2 1 2 3. 1 2 1 1 3 3 1 2 1 1 2 2 2 2 2 1 4. 2 2 2 2 2 2 1 2 2 2 1 1 1 1 1 1 5. 3 2 2 2 2 2 3 2 1 2 2 2 2 3 1 2 6. 3 3 2 2 1 1 3 2 2 2 2 2 2 3 2 2 7. 1 2 1 2 2 1 1 2 1 2 2 2 2 2 2 1 8. 3 1 3 1 2 3 3 2 2 2 3 2 2 3 1 2 9. 1 2 1 3 2 2 1 1 1 1 3 1 1 1 1 1 10. 3 1 3 1 2 2 2 2 2 2 2 2 2 3 2 2 11. 2 2 3 2 3 2 3 2 2 2 2 2 2 2 1 2 12. 2 1 2 1 2 2 3 1 2 1 2 2 2 3 1 2 13. 3 1 3 2 1 2 3 2 2 2 2 2 2 2 2 2 14. 2 3 1 2 1 2 2 2 2 2 2 2 2 2 2 1 15. 3 1 2 2 1 2 3 2 1 2 2 2 2 3 1 2 16. 3 2 3 1 2 3 2 2 2 2 2 2 1 1 1 2 17. 3 3 3 2 3 2 3 2 2 2 2 2 2 2 2 2 18. 2 2 1 3 1 2 1 2 2 2 3 2 2 2 2 1 19. 1 2 1 2 3 2 1 2 2 2 3 2 2 2 2 1 20. 3 1 3 3 2 2 3 2 1 2 3 2 2 1 2 2 21. 2 2 1 2 2 2 2 2 2 1 1 2 2 3 2 1 22. 2 2 1 3 2 1 1 1 2 2 2 2 2 2 1 1 a multi-criteria decision-making (mcdm) model in the security forces operations ... 107 coa a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 a 10 a 11 a 12 a 13 a 14 a 15 d 23. 2 2 3 2 2 2 3 2 2 1 1 2 2 3 2 2 24. 2 2 3 2 1 1 3 1 2 2 2 2 2 2 1 2 25. 1 1 3 2 1 1 3 1 2 2 2 2 2 2 1 2 26. 2 3 1 2 1 2 2 2 2 2 2 2 2 2 2 1 27. 3 1 2 2 1 2 3 2 1 2 2 2 2 3 1 2 28. 3 2 3 1 2 3 2 2 2 2 2 2 1 1 1 2 29. 3 3 3 2 3 2 3 2 2 2 2 2 2 2 2 2 30. 2 2 1 3 1 2 1 2 2 2 3 2 2 2 2 1 31. 1 2 1 2 3 2 1 2 2 2 3 2 2 2 2 1 32. 3 1 3 3 2 2 3 2 1 2 3 2 2 1 2 2 33. 2 2 1 2 2 2 2 2 2 1 1 2 2 3 2 1 34. 2 2 1 3 2 1 1 1 2 2 2 2 2 2 1 1 35. 2 2 3 2 2 2 3 2 2 1 1 2 2 3 2 2 36. 2 2 3 2 1 1 3 1 2 2 2 2 2 2 1 2 37. 1 1 3 2 1 1 3 1 2 2 2 2 2 2 1 1 in figures 3 and 4 screen review decision table in software systems rosetta and rose2 can be seen. in software system rosetta, the names of the attributes are given linguistically, while in rose2 they are written in symbols. figure 3. decision table in software system rosetta figure 4. decision table in software system rose2 karavidić & projović/decis. mak. appl. manag. eng. 1 (1) (2018) 97-120 108 the software system rose2 can be used for determining the upper and lower approximation of the sets ''coa that was successful (i.e. the security forces operation is successful according to the selected coa)'' and sets ''coa that was unsuccessful (i.e. security forces operation was unsuccessful according to the selected coa'' (figure 5). the software system displays the number of objects by the decision attribute, upper and lower approximation and accuracy approximation coefficient. the software system rosetta does not have such possibilities. figure 5. determining the upper and lower approximation and the accuracy approximation coefficient in the software system rose2 accuracy approximation coefficient αi(x) in the software system rose2 is shown as accuracy. it can be seen that α (successful) = 0,875 and α (unsuccessful)= 0,9130. based on equations (15), in case of αi(x) → 1through equations (3) bri(x) → 0is obtained. it means that the upper and lower approximations are approaching each other. for αi(x) = 1 follows bri(x) = 0. the above implies that the combinations of attributes in coa are unique, i.e. there are no identical condition attributes for different decision attributes. in that case, the set is crisp. by increasing the number of coa, the given sets would increase their degree of vagueness. the set would become more "rough". then, the coefficient of approximation accuracy would be smaller, and the available knowledge would be more difficult to classify, but this would not affect the capabilities of these software systems. the work with reduced consistency of the rules is a fundamental advantage of the rst when working with incomplete and unspecified data. 5.3. attribute reduction the next step is to assemble a minimal subset of independent attributes, i.e. reductions. these reductions guarantee the quality of classifications as a whole set. output data form the attribute core. reduction of attributes implies a decrease in volume of the core or the number of all attributes that influence the decision-making process. the aim is to identify those attributes, which according to the requirements of the decision-maker, significantly influence the decision-making process. attribute reduction is used only in the case when it does not disturb the quality of the approximation. finding the reductor will be perceived through the rosetta and rose2 software systems by using the most important reduction algorithms offered. a multi-criteria decision-making (mcdm) model in the security forces operations ... 109 5.3.1. attribute reduction with software system rosetta rosetta offers more various reductors or reduction algorithms which can be applied to data. one part of the reductors is implemented as a variant of the original form of the algorithm, and the other as customized and perfected reductors regarding existing algorithms for application in the software system. perfected reductors for applying the rst are developed for the needs of the rosetta software system and they have the prefix rses. johnson reducer is a variant of the simple ''greedy'' algorithm (johnson’s algorithm) used for calculating only one shorter reduction. the algorithm tends to find the main implication of a minimum length (johnson, 1974). it always selects the most frequent attribute in the decision-making function or a row of decision-making matrices and it continues until the reducts are obtained. this algorithm considers the attribute that most often appears as the most significant one. even though this is not true in all cases, an optimal solution is usually found (abbas, 2016). the result of the application on the decision-making table is shown in figure 6. figure 6. rosetta reduction johnson reducer rses exhaustive reducer calculates the reductions by the principle of rough computer power without approximations, comparing all the given combinations of attributes with one another. the output gives more reductions that significantly affect the decision attribute (dobrilovic et al., 2012; romański, 1988). the result of the application on the decision-making table is shown in figure 7. karavidić & projović/decis. mak. appl. manag. eng. 1 (1) (2018) 97-120 110 figure 7. rosetta reduction rses exhaustive reducer rses johnson reducer is an advanced version of a simple johnson algorithm adapted to the rosetta software system (li, 2014). the result of the application on the decision-making table is shown in figure 8. figure 8. rosetta reduction rses johnson reducer rses genetic reducer implements a variant of the genetic algorithm (jaddi & abdullah, 2013; wroblewski, 1995) to search for reductions until the search area is exhausted, i.e. until the maximum number of reductions is noticed. as the aforementioned, the reducer is adapted to the rosetta software system and it provides various options for selecting the parameters depending on the search speed requirements and the coverage of the reduction. the result of the application on the decision-making table is shown in figure 9. a multi-criteria decision-making (mcdm) model in the security forces operations ... 111 figure 9. rosetta reduction rses genetic reducer 5.3.2. attribute reduction with software system rose2 the rose2 software system also offers multiple reducers based on different algorithms. the lattice search reducer attempts to reduce search space by extracting a part that has no potential to include reduction of including a reduct (grabowski, 2016; prędki & wilk, 1999). the result of the application on the decision-making table is shown in figure 10. figure 10. rose2reduction lattice search discernibility matrix reductor is a more computer-efficient algorithm for generating reductions based on an open matrix (skowron & rauszer, 1992). the result of the application on the decision-making table is shown in figure 11. karavidić & projović/decis. mak. appl. manag. eng. 1 (1) (2018) 97-120 112 figure 11. rose2reduction discernibility matrix heuristic search reducer implements a strategy based on adding attributes to the core. it determines approximately the reduction value when it is not possible to accurately determine other algorithms. because of this characteristic, heuristic search reducer is significant when other methods fail (liang et al., 2014).the result of the application on the decision-making table is shown in figure 12. figure 12. rose2reduction heuristic search 5.3.3. review of reduced attributes it is important to highlight that, due to the essence of the decision-making support process, finding a shorter coordinated core is of crucial importance. such a need arises from the demand that the time for considering attribute conditions be a multi-criteria decision-making (mcdm) model in the security forces operations ... 113 shortened because analyzing every additional attribute takes additional time and that is always a limiting factor. some reducers offer only shorter cores while others offer cores of different lengths sorted by the quality of reduction. because of this, only the cores of the shortest length (in our case, two attributes) and the highest quality reduction will be considered. comparison of various components of rosetta and rose2 software systems and attributes obtained by reduction are given in table 3. table 3. results of attributes’ reduction software system reductor attributes obtained by reduction 1. reduction 2. reduction rosetta johnson reducer а1, а3 rses exhaustive reducer а1, а3 а1, а7 rses johnson reducer а3, а1 rses genetic reducer а1, а3 а1, а7 rose2 lattice search а1,а7 а1, а3 discernibility matrix а1, а7 а1, а3 heruistic search а1, а3 а1, а7 it can be seen from the previous table that different reducers give very similar results. the mild differences are the result of the applied algorithms, their way of attribute reduction and limitations in the reduction process, but also of the number of coas being considered. with the increase in the number of coas, it is expected that there would be equalization of different algorithm reduction results. looking at the results of all the obtained reductions, it can be concluded that there is no unique combination of two attributes around which the offered algorithms are completely "compatible". the most compatible attribute combination is a1 and a3 (the strength of our forces and operations preparing time). however, it is noticeable that three attributes are repeated in the results of all reducers both on the first and the second reduction. therefore, the final reduction cannot be performed by using the shortest combination of two attributes. instead, three attributes will be used: а1, а3, а7 that is, the strength of our forces, operations preparing time and maneuver. these attributes essentially represent the core of the attributes required for decision-making. other attributes are rejected because their values will not have a significant effect on classifying coa and generating the decision-making rules. 5.4. generating decision-making rules the obtained attributes are sufficient to form a reduced decision-making table. the rosetta software system allows the consideration of the harmonized reduced decision-making table through the manual reducer and generating decision-making rules for the specified attributes (figure 13). also, each of the aforementioned reducers generates its decision-making tables. however, the above will be used due to a more comprehensive view of the selected condition attributes. karavidić & projović/decis. mak. appl. manag. eng. 1 (1) (2018) 97-120 114 figure 13. rosetta –reduced decision-making table it can be noticed that besides generating complete decision-making algorithms, the rosetta software system also generates various other data related to certain probability properties. the most important characteristics for observation and further consideration of the attributes are support, strength, certainty and coverage (pawlak, 2002). these factors have various names in different software systems, and therefore, direct translations can be diverse, but for the purposes of this work, previously given property names will be kept. the support factor represents the number of coa with all identical attributes. in figure 13, it is presented as the rhs support. the software system also offers the lhs support feature, which represents the number of coas with equal attribute conditions. this is less important for further consideration. by reducing the consistency of the decision-making rules, differences between the two properties indicated would be made. the strength factor represents the participation of the coa determined by the observed attributes in the total number of the monitored coas and the sum of all must be 100%. basically, it represents the support factor in percentages compared to the total number of coas considered. it gives an important indicator of the coa towards which should be strived. it is a significant statistic prediction indicator if the values are higher. the strength factor is most often calculated from data, but it can be also obtained by estimation (pawlak, 2002). it is obtained by estimation when an expert in a particular field estimates that the appropriate combination of attributes in coa is more significant than a simple percentage participation in the sum of all coas. in figure 13, it is presented in the lhs coverage column and it is derived from the existing table data. the certainty factor is at a high level, due to different combinations of condition attributes in a reduced decision-making table. this feature represents practically the participation of the support factor of a particular condition attribute combination in the total support of that condition attribute combination. it gives knowledge on certainty of the observed coа. the value will decrease if there are identical condition attributes with different decision attributes. because of its importance, this property of probability leads us to consider the coas that have a higher value of certainty, i.e. closer to the number 1.00. in this sense, the certainty factor can be identified with the previously defined consistency factor c (δ (h)) and it should be the first property and the most important factor to be considered in the analysis of the further offered algorithms. the coa with a smaller consistency factor will further focus a multi-criteria decision-making (mcdm) model in the security forces operations ... 115 consideration of the generated rules. this property is shown in figure 13 in the rhs accuracy column. the coverage factor provides significant information about the participation of a particular value of the decision attribute. it implies percentage of one attribute combination in the given decision attribute. the sum of all factor values must be 100% by one value of the decision attribute. it is particularly emphasized in considering a single decision attribute in a large number of coas. this is shown in figure 13 in rhs coverage column. further reduced decision-making table can be presented by the following decision-making algorithms and prominent probability properties (table 4). the generated decision rules for c(δ(х))=1 were taken into account. table 4. rosetta -decision-making algorithms for c(δ(х))=1 if then strength factor (%) coverage factor (%) condition attributes decision attribute strength of our forces operations preparing time maneuver success of the operation more than needed sufficient completely successful successful 18,9 31,8 more than needed limited partially successful successful 2,7 4,5 insufficient insufficient unsuccessful unsuccessful 13,5 33,3 adequate limited unsuccessful unsuccessful 2,7 6,6 more than needed limited completely successful successful 13,5 22,7 more than needed sufficient partially successful successful 8,1 13,6 adequate sufficient completely successful successful 13,5 22,7 adequate insufficient partially successful unsuccessful 10,8 26,6 adequate insufficient unsuccessful unsuccessful 10,8 26,6 the mentioned prominent properties of probability in the decision-making algorithm are directed to the specific if-then rules, which, due to the above properties, further emphasize their significance. the rose2 software system offers a different approach to generating decision rules. it uses a modified lem2 (modlem) algorithm that recognizes extreme differences in rules and separates the most positive and most negative attributes from the impact on decision attributes. all the offered variants of this algorithm have a "greedy" approach and give short decision-making rules. for the purposes of this paper, the rule generator will be considered with the extended minimum coverage as can be seen in figure 14. karavidić & projović/decis. mak. appl. manag. eng. 1 (1) (2018) 97-120 116 figure 14. rose2–decision rules the obtained data can be used. however, due to the lack of a certain number of probabilities and the combination of condition attributes, they are less important in the further decision-making process than the results of the rosetta program. they represent a shortened lead with the described coverage factor, which should be sought in the further decision-making process. the software system also directs to the decision rules with consistency factor c (δ (h)) = 1. in accordance with the possibilities of the rst, it shows the rules for which c (δ (h)) <1, but do not give a precise value. those rules are called approximate rules. the decision-making algorithms, derived from the software system rose2 for c(δ(х)) = 1, are presented in table 5. table 5. rose2 -decision-making algorithms for c(δ(х))=1 if then coverage factor (%) condition attributes decision attribute strength of our forces operations preparing time maneuver success of the operation insufficient unsuccessful 86,6 unsuccessful unsuccessful 66,6 more than needed or adequate sufficient or limited completely or partially successful successful 95 rose2 directs with the coverage factor. in this way, the shorter coverage of the rules in percentages as the only property of the probability of the given rule, as given by the rose2 software system, is not sufficiently strong to lead to the desired decision attribute. however, even this coverage of the rule can be significant in the a multi-criteria decision-making (mcdm) model in the security forces operations ... 117 decision-making process where it gives certain knowledge about processes that are shaped and at least partially directs the decision-makers. 6. discussion of results the obtained attribute core is essential for the success of the operation, but also for other condition attributes. the influence of the attribute core on the success of the operation can be considered through other condition attributes (us army, 2015). for example, operations preparing time (core attribute а3) affects the quality of planning all elements of the operation. it also affects combat support (а8), protection of our forces (а9) and sustainability of our forces (а10). time also has an effect on all activities that completely or partially precede performing of the operation. some of those activities are intelligence system (а13) and coordination with civil structures (а15). within the sufficient time frame, shortcomings in command and control c2 (a14) can be compensated. additionally, our forces casualties (a5) can be reduced through greater preparation of the protection of our forces (a9). similarly, other core attributes dominantly affect other condition attributes. the strength of our forces (core attribute a1) can compensate for different negativities in other attributes. on the other hand, there is a certain feedback between all attributes. moreover, there is a mutual influence which is impossible to fully comprehend due to the stated complexity of the environment. such feedback is also present between the core attributes, but less significant than with other attributes. an example for that is the influence of operations preparing time (core attribute а3) on maneuver (core attribute а7). in practice it can have a positive influence, but not necessarily. by using this decision-making support model, the complexity of the mutual influence of all condition attributes can be partially overcome. this is one of its biggest advantages. the obtained decision algorithm, especially the one from the rosetta software system, directs and manages the authorities that plan the coa of the security forces operations to the rules that bring success in operations in a complex environment (gordic et al., 2013). they also provide information on combinations of attributes that will lead to unsuccessful operation. guided by these rules in different situations, time spent on certain options in entire planning and decision-making process is reduced. it is a necessary time-saving. the application of the decision-making support system based on the rst enables an additional source of information to the decision-maker and the persons who take part in the entire decision-making process. thus, the purpose of such a system is fulfilled. 7. conclusion the rst in the decision-making support model uses entirely internal knowledge, unlike other methods whose application requires additional assumption models or some form of preprocessing. the internal knowledge represents the existing operational data, and there is no need to rely on modeling assumptions. the advantage of the decision-making model based on the rst in the decisionmaking process is the ability to use qualitative-quantitative data, as well as the ifthen decision-making algorithms. these algorithms can be applied to the whole decision-making process by directing the decision-maker in every moment of the process, and not just at the moment of selecting a coa. karavidić & projović/decis. mak. appl. manag. eng. 1 (1) (2018) 97-120 118 using the proposed decision-making support model makes it possible to reach extremely valuable indicators in a rather simple way, which can help in the decisionmaking process. the paper presents one method of use; however, due to the complexity of the environment in which security forces operations are planned and implemented, it is possible to apply the rough set concept to lower levels the sublevels of these attributes. simultaneous application of the rough set concept to lower and higher levels of attributes in security forces operations, complemented by classifying and/or clustering at lower levels, can be a challenge for future work. in this way, the support for decision-making in security forces operations in the modern security environment would be raised to a higher level. acknowledgements the work reported on in this paper is a part of the investigation in the research projects va-dh/2/18-20 supported by the university of defence in belgrade and muo-in supported by the university of defence in belgrade, ministry of defence, republic of serbia and ministry of education, science and technological development, republic of serbia. this support is gratefully acknowledged. references abbas, z., & burney, a. 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(1995). finding minimal reducts using genetic algorithms. in proccedings of the second annual join conference on infromation science, 2, 186-189. plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 3, issue 2, 2020, pp. 131-148. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame2003131r * corresponding author. e-mail addresses: markoradovanovicgdb@yahoo.com (m. radovanović), aca.r.0860.ar@gmail.com (a. ranđelović), antras1209@gmail.com (ž. jokić) application of hybrid model fuzzy ahp vikor in selection of the most efficient procedure for rectification of the optical sight of the longrange rifle marko radovanović1*, aca ranđelović1 and željko jokić1 1 university of defence, military academy, belgrade, serbia received: 12 july 2020; accepted: 25 september 2020; available online: 10 october 2020. original scientific paper abstract: the paper presents a decision support model when choosing the most efficient rectification procedure of the optical sight of the long range rifle. the model is based on the fuzzy ahp method and the vikor method. using the fuzzy ahp method, coefficient values of the criteria were defined. fuzzification of the ahp method was performed by combining data obtained from experts comparison of criteria in pairs and the degree of confidence in the comparison. using the vikor method, the best alternative was selected. through the paper, the criteria that condition this choice are elaborated and the application of the method in a specific situation is presented. also, the paper presents the sensitivity analysis of the developed model. key words: fuzzy ahp, vikor, multi-criteria decision-making, rectification, long-range rifle. 1. introduction the serbian army is a complex organizational system, where the decision-making process is a very important element. therefore, the application of multi-criteria decision-making methods is an indispensable segment in this process. this paper presents a model for selecting the most efficient rectification method of a 12.7 mm m93 long range rifle optical sight. a long-range rifle is a weapon to support infantry platoons in attack and defense. it is a type of small arms that is specially designed for fire action on people, noncombat and lightly armored combat vehicles, at distances up to 1800 m (randjelovic et al. 2019a). it is a weapon of high accuracy and precision and achieves its firepower on targets by direct shooting. successful rectification of sights achieves the accuracy and precision of a longrange rifle. based on accuracy and precision, the probability of hitting the target is mailto:markoradovanovicgdb@yahoo.com mailto:aca.r.0860.ar@gmail.com mailto:antras1209@gmail.com radovanović et al./decis. mak. appl. manag. eng. 3 (2) (2020) 131-148 132 determined, which affects the efficiency of long-range rifle 12.7 mm m93 solving fire tasks in operations. having in mind the importance of rectification of the optical sight of a long range rifle for performing combat actions, the most efficient rectification procedure was selected by applying the method of multi criteria decision making. 2. problem description through this paper, a model is presented which determines the most efficient and most economical procedure of rectification of the optical sight of a long range rifle. procedures for rectification of the optical sight of the 12.7 mm m93 long-range rifle are defined on the basis of the provisions of the technical and temporary instructions for the optical sight of the long-range rifle and the instructions for use for the optical sight of the long-range rifle (long-range rifle 12.7 mm m93 (description, handling and maintenance), 2010; purpose, description and handling of the 12.7 mm longrange rifle, 1999; the long-rifle optical sight on m93 for the long-range rifle "zastava" 12.7 mm m93, 1998). in addition to the above, as one alternative, a modeled rectification procedure was taken, which was reached on the basis of the results of previous research in this area, presented in detail in radovanović (2016), radovanović et al. (2016) and randjelovic et al. (2019a). the aim of this paper is to select the most efficient rectification procedure using the method of multi-criteria decision-making in order to indirectly increase the efficiency of realization of fire tasks with a long-range rifle. the results used for the analysis were obtained on the basis of realized shootings at the training field "pasuljanske livade". most units of the serbian army for the process of rectification of the optical sight of the long-rifle 12.7 mm m93, use the model shown in the temporary instructions for long-range rifle (purpose, description and handling of long-range rifle 12.7 mm, 1999). to a lesser extent, other methods of rectification are used in the units. according to the above, it can be concluded that there is no universality regarding the rectification of the optical sight of a long-range rifle. comparisons regarding quality, but also other parameters of rectification have not been performed so far. in other words, there are several satisfactory ways of rectification, but so far no detailed analysis has been made as to which way (model) would be the most acceptable from several aspects (quality, price, required resources, etc.). accordingly, it is clear that the presented problem is an ideal field for the application of multi-criteria decisionmaking methods. in the literature available to the authors, it was found that there is not a large number of papers dealing with this issue. radovanović (2016) models a new rectification procedure and the software program correction of sights. in the paper radovanović et al. (2016) performed a numerical analysis of different ways of rectification in relation to certain criteria such as ammunition consumption, time and price of rectification. randjelovic et al. (2019a) show the dependence of the rectification procedure on the execution of fire tasks in a counter-terrorist operation. the available literature describes only a part of the criteria on the basis of which the most efficient rectification procedure is selected. 3. description of applied methods the hybrid model, applied when solving the problem of choosing the most efficient rectification method of the long range rifle optical sight, was defined by a application of hybrid model fuzzy ahp vikor in selection of the most efficient procedure ... 133 combination of the fuzzy ahp and vikor methods. this part of the paper describes the methods used in the paper. the fuzzy ahp method was used to define the coefficient values, while the vikor method was used to select the best alternative. figure 1 shows the phases through which this model was realized. figure 1. appearance of the model for rectification of the optical sight of a long-range rifle 3.1. fuzzy ahp method the ahp method was developed by thomas saaty (1980). to date, this method has undergone a large number of modifications (božanić et al., 2013; stević et al., 2017; petrović et al., 2018; chatterjee et al., 2019; afriliansyah et al., 2019; osintsev et al., 2020; zhu et al., 2020;), but in some cases it is still used in its original form (radovanović et al., 2019; radovanović and stevanović, 2020; ranđelović et al., 2019b) both in the individual (badi and abdulshahed, 2019) and in group decision making (srđević and zoranović, 2003). analytical hierarchical process is a method based on the decomposition of a complex problem into a hierarchy, with the goal at the top, criteria, sub-criteria and alternatives at the levels and sublevels of the hierarchy (saaty, 1980). for comparisons in pairs, which is the basis of the ahp method, the saaty’s scale is usually used, table 1. table 1. saaty’s pair-wise comparison scale standard values definition inverse values 1 same meaning 1 3 weak dominance 1/3 5 strong dominance 1/5 7 very strong dominance 1/7 9 absolute dominance 1/9 2, 4, 6, 8 intermediate values 1/2, 1/4, 1/6, 1/8 the comparison in pairs leads to the initial decision matrices. the saaty’s scale is most commonly used to determine the coefficient values of the criteria, but can also be used to rank alternatives. radovanović et al./decis. mak. appl. manag. eng. 3 (2) (2020) 131-148 134 very often when taking values from the saaty’s scale in the pair-wise comparison process, decision makers hesitate between the values they will assign to a particular comparison. in other words, it happens that they are not sure of the comparison they are making. due to the above, various modifications of the saaty’s scale are often made. one of them is the application of fuzzy numbers. there are different approaches in the fuzzification of the saaty scale, and in principle they can be divided into two groups: sharp (hard) and soft fuzzification (božanić et al., 2015b). fasification can be done with different types of fuzzy numbers, and is most often done using a triangular fuzzy number figure 2. t1 t2 t3 1  t x    2 1, t x x t     1 1 2 2 1 , t x x t t x t t t         3 2 3 3 2 , t x t x t x t t t         1 0, t x x t     3 0, t x x t    0 αt1 αt2 figure 2. triangular phase number t (pamučar et al., 2016b) by "sharp" fuzzification is meant that a fuzzy number  1 2 3, ,t t t t is a predetermined confidence interval, that is, it is predetermined that the value of the fuzzy number will not be greater than 3 t or less than 1 t (božanić et al., 2015b). based on the predefined fuzzy saaty’s scale, a comparison is made in pairs. in soft fuzzification, the confidence interval of the values in the saaty’s scale is not predetermined, but is defined during the decision-making process, based on additional parameters. the definition of the coefficient values of the criteria in this paper was performed by applying the phased saaty’s scale presented in the works of božanić et al. (2016), pamučar et al. (2016a), božanić (2017), božanić et al. (2018), bojanic et al. (2018) and bobar et al. (2020). the starting elements of this fuzzification are (bobar et al., 2020): 1) introducing the fuzzy numbers instead of classic numbers of the saaty scale, 2) introducing the degree of confidence of decision makers/analysts/experts (dm/a/e) in the statements they make when comparing in pairs  . the degree of confidence () is defined at the level of each comparison in pairs. the value of the degree of confidence belongs to the interval 0,1, where =1 describes the absolute confidence of dm/a/e in the defined comparison. the decrease in the confidence of dm/a/e in the performed comparison is accompanied by a decrease in the degree of confidence ji. forms for calculating fuzzy numbers are given in table 2. application of hybrid model fuzzy ahp vikor in selection of the most efficient procedure ... 135 table 2. fuzzification of the saaty's scale using the degree of confidence (bobar et al., 2020) definition standard values fuzzy number inverse values of fuzzy number same meaning 1 (1, 1, 1) (1, 1, 1) weak dominance 3   3 , 3, 2 3 ji ji   1 2 3,1/ 3,1 3  ji ji strong dominance 5   5 , 5, 2 5 ji ji   1 2 5,1/ 5,1 5  ji ji very strong dominance 7   7 , 7, 2 7 ji ji   1 2 7 ,1/ 7,1 7  ji ji absolute dominance 9   9 , 9, 2 9 ji ji  1 (2 )9 ,1 / 9,1 9  ji ji intermediate values 2, 4, 6, 8   , , 2 , ji jix x x 2, 4, 6, 8x    1 2 ,1/ ,1  ji jix x x 2, 4, 6, 8x  an example of the appearance of a fuzzy number with different degrees of confidence is given in figure 3. for example, the value of low dominance from the saaty’s scale and degrees of confidence =1, =0.7 and =0.3 are taken. 0 1 0.7  53.5 6.5 b) 0 1 0.3  51.5 8.5 c ) 0 1 1  5 a) figure 3. dependence of fuzzy number on degree of confidence by introducing different values of the degree of confidence, the left and right distributions of fuzzy comparisons change according to the expression (bobar et al., 2020):           1 2 1 2 1 2 1 2 3 2 2 2 3 2 3 2 2 3 , , , 1 / 9, 9 , , , 1 / 9,9 2 , , , 1 / 9, 9                    t t t t t t t t t t t t t t t t t t t (1) where the value of t2 represents the value of the linguistic expression from the classical saaty’s scale, which in the fuzzy number has the maximum affiliation t2=1. fuzzy number     1 2 3, , , , 2   t t t t x x x ,  1, 9x  is defined by expressions (božanić, 2017): 1 , 1 1, 1             x x x t x x (2)  2 , 1, 9t x x   (3) radovanović et al./decis. mak. appl. manag. eng. 3 (2) (2020) 131-148 136    3 2 , 1, 9   jit x x (4) inverse fuzzy number     1 1 2 31/ ,1/ ,1/ 1 2 ,1/ ,1     ji jit t t t x x x ,  1, 9x  is defined as (božanić, 2017):          3 1 2 , 1 2 1 1 / 1 2 , 1, 9 1, 1 2 1                 ji ji ji ji x x t x x x (5) 21/ 1/ , 1/ 1, 9  t x x (6)  31 / 1 , 1 / 1, 9  jit x x (7) accordingly, the initial decision matrix has the following form (božanić et al., 2015a): 1 2 1 11 11 12 12 1 1 2 21 21 22 22 2 2 1 1 2 2 ; ; ; ; ; ; ; ; ;                       n n n n n n n n n n nn nn c c c c a a a a c a a a c a a a (8) where ji=ij. reaching the final results implies further application of the standard steps of the ahp method. at the end of the application, the fuzzy number is converted to a real number. numerous methods are used for this procedure (herrera and martinez, 2000). some of the known terms for defuzzification are (liou and wang, 1992; seiford, 1996): 3 1 2 1 1 (( ) ( )) / 3    a t t t t t (9)  3 2 11 / 2      a t t t (10) where  represents the optimism index, which can be described as the belief/ratio dm/a/e in decision-making risk. most often, the optimism index is 0, 0.5 or 1, which corresponds to the pessimistic, average or optimistic view of the decision maker (milićević, 2014). 3.2. vikor method vikor (višekriterijumsko kompromisno rangiranje) is a method of multicriteria decision-making whose use is very common. it was developed by serafim opricović (1986). it is suitable for solving various decision-making problems. it is especially emphasized for situations where criteria of a quantitative nature prevail. the vikor method starts from the "boundary" forms of lp metrics, where the choice of the solution that is closest to the ideal is made. the presented metric represents the distance between the ideal point f* and the point f (x) in the space of criterion functions (opricović, 1986). minimizing this metric determines a compromise solution. as a measure of the distance from the ideal point, the following is used: application of hybrid model fuzzy ahp vikor in selection of the most efficient procedure ... 137  *pl (f , f)       1/p pn * j jj=1 f -f (x) ,1 p (11) the vikor method has been applied in a large number of papers in its original form (nisel, 2014; kuo and liang, 2011; opricović and tzeng, 2004; jokić et al., 2019, radovanović et al. 2020), but also in fuzzy (chatterjeea and chakrabortyb, 2016; ince, 2007; shemshadi et al., 2011;) and a rough (li and song, 2016; wang et al. 2018) environment. when applying the vikor method, the following terms are used:  n – number of criteria  m – number of alternatives for multicriteria ranking  fij – the values of the i criterion function for the j alternative,  wj – the value of the j criterion function,  v – the weight of the strategy, meeting most of the criteria,  i – ordinal number of the alternative, i = 1, ..., m.,  j – ordinal number of the criteria, j = 1, ..., n,  qi – measure for multi-criteria ranking of the j alternative. for each alternative, there are qi values, after which the alternative with the lowest qi value is selected. the measure for multi-criteria ranking of the i action (qi) is calculated according to the expression (opricović, 1998): 1 i i q v* qs ( v )* qr   (12) where: * i i * s s qs s s     (13) * i i * r r qr r r     (14) by calculating the qsi, qri, and qi values for each alternative, it is possible to form three independent rankings. the qsi value, is a measure of deviation that displays the requirement for maximum group benefit (first ranking list). qri value is a measure of deviation that shows the requirement to minimize the maximum distance of an alternative from the "ideal" alternative (second ranking list). qi value represents the establishment of a compromise ranking list that combines qsi and qri values (third ranking list). by choosing a smaller or larger value for v (the weight of strategies to meet most criteria), the decision maker can favor the influence of qsi value or qri value in the compromise ranking list. for example, higher values for v (v > 0.5) indicate that the decision maker gives greater relative importance to the strategy of satisfying most of the criteria (nikolić et al., 2010). modeling the preferential dependence of criteria usually includes the weights of individual criteria. if the given values are weights w1,w2,…..,wn, the multi-criteria ranking by the vikor method is realized by using the measure si and ri. in the previous terms, the labels used have the following meanings:    * * 1 1 / n n i i i ij i i j ij j j s w f f f f w d         (15)    * *max / maxi i i ij i i j ij j j r w f f f f w d        (16) radovanović et al./decis. mak. appl. manag. eng. 3 (2) (2020) 131-148 138 i = 1,2, ..., m, j=1,2,...,n, and where: * * * min max min max max min i i i i i i i i ij i ij i s s s s r r r r f f f f          alternative ai is better than alternative ak according to j criterion, if:  ij kj f f (for max fj, that is when the criterion has a maximum requirement),  ij kj f f (for min fj, that is when the criterion has a minimum requirement). in multi-criteria ranking by the vikor method, alternative ai is better than alternative ak (in total, according to all criteria), if: qi <qk. a compromise ranking list for the value v = 0.5 is taken as an acceptable ranking list according to the vikor method. if an alternative is in the first position on such a compromising ranking list, it still does not mean that this alternative is considered the best. in order for an alternative to be adopted as the best, it must be first on the compromise ranking list and meet two conditions: condition c1 and condition c2. condition c1: the first alternative on the compromise ranking list for the value v = 0.5, must have a "sufficient advantage" over the action from the next position. “advantage” is calculated as the difference of measures qi for the value v = 0.5. alternative a' has a "sufficient advantage" over the following a" from the ranking list, if fulfilled: ( `) ( ``) q a q a dq , (17) 1 min(0.25, ) 1   dq m (18) where:  dq – “sufficient advantage” threshold value  m – number of alternatives,  0,25 – a “sufficient advantage” threshold value that limits the threshold value for cases with a small number of alternatives. condition c2: the first alternative on the compromise ranking list for the value v = 0.5, must have a "sufficiently stable" first position with a change in value v. the first alternative on the compromise ranking list has a "sufficiently stable" position, if it meets at least one of the following conditions:  has the first position on the ranking list according to qs,  has the first position on the ranking list according to qr,  has the first position on the ranking list according to q for v = 0.25 and v = 0.75. if the first action from the compromise ranking list does not meet one or both conditions (c1 and c2), it is considered that it is not "sufficiently" better than the action from the second position and possibly some more actions. in such cases, a set of compromise solutions is formed, which consists of the first, second and possibly application of hybrid model fuzzy ahp vikor in selection of the most efficient procedure ... 139 some other actions (third, fourth ...). if the first action does not meet only the condition c2, then only the first and second actions are included in the set of compromise solutions. however, if the first action does not meet condition c1 (or both conditions, both c1 and c2), then the set of compromise solutions contains actions from the compromise ranking list to the action that meets condition c1, that is to the one over which the first action has a "sufficient advantage" via dq. the results of the vikor method are:  ranking lists according to qsi, qri and qi measures,  a set of compromise solutions (in case the conditions c1 and/or c2 are not met). these results represent the basis for deciding and adopting the final solution. 4. description of the criteria and calculation of the coefficient values of the criteria through the first phase of the model application, the criteria that influence the selection of the optimal alternative, that is the rectification procedure, were defined. when defining the criteria for the selection of rectification methods, it is necessary to include all relevant facts of the optimized system, which is further important for determining the weight coefficients of the criteria. the criteria are defined on the basis of a study of the available literature and the views of experts. six criteria are defined and presented in this part of the paper. the rectification time (c1) represents the total time from the moment of placing the long-range rifle in the firing position, settings for shooting, shooting, and setting the optical sight to the end of rectification, and is expressed in units of time or minutes (radovanović et al., 2016). the stated criterion is of numerical character and "cost" type (smaller values are more desirable). ammunition consumption (c2) represents the number of bullets needed to perform shooting in order to realize the rectification of the optical sight of a long-range rifle (radovanović, 2016). the specified criterion is of the "cost" type. ammunition consumption directly affects the economic characteristics of long-range rifle rectification, such as the cost of the rectification procedure. the criterion is numerical and is expressed by the number of bullets required for the realization of the rectification of the optical sight. shooting accuracy (c3) represents the measured size of the image of scattering hits limited by four probable deflections (vs) in each side of the middle hit (kokelj and randjelovic, 2018). the smaller the scatter, the smaller the image of the beam trajectory, which makes the weapon more accurate. shooting accuracy is prescribed in accordance with the size of the image of the hits, and is expressed in millimeters (mm). the criterion is of the "cost" type. the number of engaged persons (c4) is a criterion that affects the economic characteristics of rectification, and is expressed in the minimum number of people needed to realize the rectification of a long-range rifle. the criterion is of the "cost" type. the type of target (c5) is related to the characteristics of the target and represents a criterion that is directly related to the rectification model and directly affects the efficiency of the rectification performed based on the target model (radovanović, 2016). depending on the appearance and size of the target point on the target, the shooting results may also differ. the criterion is of a linguistic character, where higher values are more desirable. qualitative scores are quantified using the scale shown in table 3. radovanović et al./decis. mak. appl. manag. eng. 3 (2) (2020) 131-148 140 table 3. scale of converting quantitative criteria into qualitative ones quantitative type of target note 0.9 target 1 modeled target (figure 4a) 0.7 target 2 target defined by technical instruction (figure 4b) 0.5 target 3 target defined by the instructions for use of the m93 optical sight (figure 4c) 0.3 target 4 target defined by the temporary instruction for the 12.7 mm m93 long-range rifle (figure 4d) the appearance of these targets is given in figure 4. a) target 1 b) target 2 c) target 3 d) target 4 figure 4. rectification target models shooting accuracy (c6) is the measured value between the scattering of the beam trajectory and the target being shot. the criterion is numerical and is expressed in millimeters. it is defined by the distance of the image of scattering hits and the image of the target at a certain distance. the conclusion about accuracy is made on the basis of the magnitude of the deviation of the middle hit (mh) from the center of the target (ct). the shooting is more accurate because the deviation of the middle goal from the center of the target is smaller, and vice versa. the criterion is of the "cost" type. the accuracy of shooting depends on the work of the shooter, the meteorological conditions in which the shooting takes place, the completeness and correctness of accessories and instruments and ammunition (randjelovic et al., 2019a). using the fuzzy ahp method shown in the previous section, the coefficient values of the criteria were defined. the coefficient values were calculated for each expert separately, and the obtained values were aggregated into one. the obtained coefficient values of the criteria are given in table 4. application of hybrid model fuzzy ahp vikor in selection of the most efficient procedure ... 141 table 4. coefficient values of the criteria criteria coefficient values c1 0.143 c2 0.110 c3 0.376 c4 0.026 c5 0.049 c6 0.295 5. choosing the best alternative the analysis of the literature defines seven alternatives, that is seven models of rectification of the optical sight of the 12.7 mm m93 long-range rifle, which is in use in the units of the serbian army:  a1 model defined in optical sight on pd 12.7 m93 for long-range rifle "zastava" 12.7 mm m93, temporary instruction, (1998).  a2 model defined in optical sight on pd 12.7 m93 for long-range rifle "zastava" 12.7 mm m93, temporary instruction, (1998) with the use of rectifiers (r) defined in radovanović, (2016).  a3 model defined in long-range rifle 12.7 mm m93, technical manual, (2010).  a4 model defined in long-range rifle 12.7 mm m93, technical manual, (2010), with the use of rectifiers (r) defined in radovanović, (2016).  a5 model defined in purpose, description and handling of the 12.7 mm m93 long-range rifle, temporary instruction, (1999).  a6 model defined in purpose, description and handling of a 12.7 mm m93 long-range rifle, temporary instruction, (1999), with the use of rectifiers (r) defined in radovanović, (2016).  a7 model defined in radovanović, (2016). assessments of alternatives according to the criteria are given in the initial decision matrix (table 5). table 5. initial decision matrix c1 (min) c2 (min) c3 (min) c4 (min) c5 (max) c6 (min) w=0.143 w=0.110 w=0.376 w=0.026 w=0.049 w=0.295 a1 90 20 47 1 target 3 22 a2 70 16 38 2 target 3 18 a3 52 18 40 1 target 2 29 a4 31 12 33 2 target 2 19 a5 90 20 52 1 target 4 45 a6 70 16 41 2 target 4 40 a7 23 8 36 2 target 1 15 table 6 shows a quantified initial decision matrix, where for the criterion of target type (c5) the conversion from qualitative to quantitative assessments was performed based on the scale shown in table 3. radovanović et al./decis. mak. appl. manag. eng. 3 (2) (2020) 131-148 142 table 6. quantified initial decision matrix c1 (min) c2 (min) c3 (min) c4 (min) c5 (max) c6 (min) w=0.143 w=0.110 w=0.376 w=0.026 w=0.049 w=0.295 a1 90 20 47 1 0.5 22 a2 70 16 38 2 0.5 18 a3 52 18 40 1 0.7 29 a4 31 12 33 2 0.7 19 a5 90 20 52 1 0.3 45 a6 70 16 41 2 0.3 40 a7 23 8 36 2 0.9 15 using the expression 11-16, the final values of the alternatives were obtained, table 7. table 7. calculated values for qsi, qri, qi (v=0,5), qi (v=0,75), (v=0,25) qsi qri qi(v=0,5) qi(v=0,75) qi(v=0,25) a1 0.615 0.688 0.651 0.633 0.669 a2 0.310 0.129 0.220 0.265 0.175 a3 0.406 0.250 0.328 0.367 0.289 a4 0.090 0.030 0.060 0.075 0.045 a5 1.000 1.000 1.000 1.000 1.000 a6 0.639 0.589 0.614 0.627 0.601 a7 0.000 0.000 0.000 0.000 0.000 based on the obtained values from table 7, the ranking of alternatives was performed, table 8. as can be seen from table 8, the best alternative is a7. in order to choose a certain alternative as the best, it is necessary that it meets the conditions c1 and c2. testing of condition c1 was performed, which was not fulfilled because: q(a4) – q(a7)= 0,060 -0,00= 0,060 < dq= 1/(7-1)=0,167 alternative a4 enters a set of compromise solutions, because the first alternative from the ranking list a7 does not have a "sufficient advantage" over the secondranked alternative a4. other alternatives are not included in the set of compromise solutions, because alternative a7 has a "sufficient advantage" over the third-ranked alternative a2, other alternatives do not need to be tested according to the stated condition. table 8. ranking lists of alternatives based on qsi, qri, qi values qsi qri qi(v=0,5) qi(v=0,75) qi(v=0,25) a1 5 6 6 6 6 a2 3 3 3 3 3 a3 4 4 4 4 4 a4 2 2 2 2 2 a5 7 7 7 7 7 a6 6 5 5 5 5 a7 1 1 1 1 1 condition c2 is met if alternative a7 has a "sufficiently stable" first place according to two criteria:  alternative a7 has the first position on the ranking list according to qr and application of hybrid model fuzzy ahp vikor in selection of the most efficient procedure ... 143  alternative a7 takes the first position on the ranking list for q (v = 0.25) and q (v = 0.75) based on the obtained results, the final solution is defined by a set of compromise solutions in which there are alternatives a7 and a4. in this case, the decision maker can choose the alternative a7 rectification model described in radovanović (2016), and as the first back-up rectification procedure, the alternative a4 model defined in 12.7 mm m93 long-range rifle, technical instruction, (2010), is proposed, with the use of rectifiers (r) defined in radovanović, (2016). 6. sensitivity analysis during the last phase, the sensitivity of the applied mathematical model was examined, in order for the decision maker to receive confirmation of the rationality and quality of the obtained solution, that is to determine how changes in the significance of criteria lead to changes in the ranks of alternatives (tešić and božanić, 2018). checking the stability of the mcdm methods used is an indispensable step in the process of developing a model to support decision-making (pamučar et al., 2017). table 9 shows six scenarios (from s1 to s6) of changing the coefficient values of the criteria, on the basis of which the alternatives were ranked using the vikor method. table 9. criteria for changing the significance of the criteria c1 c2 c3 c4 c5 c6 s1 0,30 0,14 0,14 0,14 0,14 0,14 s2 0,14 0,30 0,14 0,14 0,14 0,14 s3 0,14 0,14 0,30 0,14 0,14 0,14 s4 0,14 0,14 0,14 0,30 0,14 0,14 s5 0,14 0,14 0,14 0,14 0,30 0,14 s6 0,14 0,14 0,14 0,14 0,14 0,30 table 10 shows the ranks of alternatives obtained by applying different scenarios. table 10. ranks of alternatives obtained by applying different scenarios s1 s2 s3 s4 s5 s6 a1 6 6 6 2 5 5 a2 4 3 4 6 4 4 a3 3 4 3 1 2 3 a4 2 2 2 5 3 2 a5 7 7 7 3 7 7 a6 5 5 5 7 6 6 a7 1 1 1 4 1 1 in order to establish the correlation of the ranks obtained by different types of scenarios, the spiraman’s coefficient was used: 2 1 2 6 1 ( 1) n i i d s n n     (19) where di represents the difference of rank according to the given scenario and rank in the corresponding scenario, and n the number of ranked elements. the spiraman's radovanović et al./decis. mak. appl. manag. eng. 3 (2) (2020) 131-148 144 coefficient belongs to the value interval [-1,1]. when the ranks of the alternatives completely match the spirman's coefficient is 1 (“ideal positive correlation”), when the ranks are completely opposite the spiraman's coefficient is -1 (“ideal negative correlation”), is when s = 0 the ranks are uncorrelated. the values of the spirman's coefficient in this case are shown in table 11. table 11. spirman's coefficient values s0 s1 s2 s3 s4 s5 s6 s0 1 0.964 1 0.964 -0.286 0.857 0.929 s1 1 0.964 1 -0.107 0.929 0.964 s2 1 0.964 -0.286 0.857 0.929 s3 1 -0.107 0.929 0.964 s4 1 0.214 0.071 s5 1 0.964 s6 1 based on the results in table 12, it is concluded that the values of the spiraman's coefficient are extremely high for most cases, that is there is an ideal positive correlation of ranks in most cases. deviation from the ideal positive correlation is observed in scenario s4, compared to other scenarios. negative correlation in scenario s4 occurs as a consequence of two causes: 1) criterion c4 has the lowest coefficient value in scenario s0, 2) criterion c4 can have values of 1 or 2, which directly affects the negative correlation of scenario s4. during the change of the coefficient values of the criteria of the first-ranked alternative a7, not counting the scenario s4 did not change its rank by changing the significance of the criteria. based on all the above, it is possible to conclude that the model has sufficient sensitivity. 7. conclusion the paper successfully applied the fuzzy ahp-vikor hybrid model to the selection of the most efficient rectification method of the 12.7 mm m93 long-range rifle optical sight. in this way, a more detailed review of the presented problem was performed. the paper presents the phases of development and application of multicriteria decision-making models. the definition of criteria of importance for the selection of the rectification model and the calculation of coefficient values using the fuzzy ahp method was performed. the selection of the most efficient model of rectification of the optical sight of a long range rifle was performed using the vikor method. the final results indicate a set of compromise solutions (alternatives a7 and a4). the paper analyzes the sensitivity of the presented model, changes in the significance of coefficients of the criteria (through several scenarios by favoring one criterion). the results of the analysis indicate sufficient sensitivity of the model. the contribution of this work is reflected in the selection of the most efficient model of rectification of the optical sight of a long-range rifle, whose application would increase the efficiency of long-range rifle squad, reduce rectification time, reduce rectification cost and achieve universality of long-range rifle rectification. the presented model can be further improved by a more detailed analysis of the criteria and the application of other methods of multicriteria analysis in the problem of choosing the most efficient rectification procedure and defining the values of the criteria. application of hybrid model fuzzy ahp vikor in selection of the most efficient procedure ... 145 acknowledgement: this paper was written under the va-dh/1/18-20 project, financed by the ministry of defence of the republic of serbia. author contributions: each author has participated and contributed 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(2018). application of the mairca method in the selection of places for crossing tanks under water. tehnika, 68 (6), 860 -867. wang, z., gao, j., wang, r., chen, k., gao, z., & zheng, w. (2018). failure mode and effects analysis by using the hous of reliability – based rough vikor approach. ieee transactions on reliability, 67 (1), 230-248. zhu, g. n,, hu j., & ren h., (2020). a fuzzy rough number-based ahp-topsis for design concept evaluation under uncertain environments. applied soft computing, 91, 106228. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 3, issue 2, 2020, pp. 49-69 issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame2003049c * corresponding author. e-mail addresses: s_chakraborty00@yahoo.co.in (s. chakraborty), sohinirits@gmail.com (r. chattopadhyay), santonabchakraborty@gmail.com (s. chakraborty), an integrated d-marcos method for supplier selection in an iron and steel industry ritwika chattopadhyay 1, santonab chakraborty 2, shankar chakraborty 1* 1 department of production engineering, jadavpur university, kolkata, west bengal, india 2 industrial engineering and management department, maulana abul kalam azad university of technology, west bengal, india received: 2 june 2020; accepted: 30 august 2020; available online: 30 august 2020. original scientific paper abstract: the modern era of manufacturing has recognized the importance of a sustainable supply chain management (scm) system in order to attain the desired level of stability and productivity for fulfillment of the customers’ requirements. selection of the most suitable set of suppliers is an integral part of scm which can be effectively solved with the deployment of different multicriteria decision making (mcdm) techniques. this paper endeavors to resolve the uncertainty involved in the decision making process for supplier selection with the application of d numbers. a relatively new mcdm technique in the form of measurement alternatives and ranking according to compromise solution (marcos) is later employed for ranking of a set of competing suppliers. this integrated approach is finally applied to choose the best performing supplier in a leading indian iron and steel making industry based on seven selective evaluation criteria and opinions of three decision makers. it would provide more generic and unbiased results while addressing uncertainty and ambiguity involved in the supplier selection process. key words: supplier selection; d numbers; marcos; iron and steel industry. 1. introduction in the modern day highly competitive manufacturing environment, a sustainable supply chain management (scm) system has been recognized as one of the predominant issues for survival and long-term prosperity of any organization. a sustainable scm system ensures supply of the best quality products at reduced costs to the customers, hence helping a manufacturing organization capturing its superior chattopadhyay et al./decis. mak. appl. manag. eng. 3 (2) (2020) 49-69 50 position over its competitors in the market. in case of expensive products, it focuses on quick delivery in order to minimize inventory and associated holding cost. thus, an efficient sc should take care of a wide range of objectives keeping in mind the welfares of both the organization and its customers. the present day manufacturing organizations should focus on devising a reliable as well as flexible sc based on a proper opinionated research. in sc, an essential responsibility bestowed on the purchasing department is to identify a set of compatible suppliers based on their capabilities to fulfill the primary requirements of cost, quality, delivery, technological capability, production capacity, financial strength etc. thus, with the adoption and advancement of scm, supplier selection has started playing a pivotal role. the supplier selection process mainly focuses on the following tasks, i.e. a) identification of the products to be procured, b) assimilation of a list of potential suppliers, c) shortlisting of the key factors (criteria) based on which the suppliers need to be evaluated, d) formation of a team of experts/decision makers to extensively analyze and strategize this selection process, e) choosing of the most apposite supplier while disposing off the inefficient ones, and f) continuous performance evaluation of the finally sleected supplier (de boer et al., 2001). over the course of development, supplier selection process has undergone a gradual transition from an intuitionistic approach to a more tangible strategic one, hence characterizing its further complication (parkhi, 2015). it has already been well acknowledged that scs form the backbone of most of the manufacturing industries for selection of the reliable suppliers who can provide continuous stock of quality raw materials in order to fulfill the basic objectives of productivity and profitability with economic justification to the manufacturing processes. in supplier selection process, the main challenge and mathematical complexity lies in the identification of disparate evaluation criteria with varying degrees of importance, requiring a sensible trade-off amongst them. the manufacturing sector, heavily relying on scs to achieve its goals, finds strong dependence on the application of different multi-criteria decision making (mcdm) techniques to choose the best fit supplier from a pool of competing alternatives based on the shortlisted evaluation criteria. the mcdm has become interesting among the researcher community over a long time, whereby, it has come across innovative methodologies to help the decision makers to weigh multiple alternatives to choose the best option, while taking into account a set of conflicting qualitative and quantitative criteria. application of any of the mcdm techniques in supplier selection has two basic objectives, i.e. a) deriving the preferential weights (relative importance) of the considered criteria by evaluating one against the others, and b) ranking of the candidate suppliers based on the accumulative score with respect to each criterion. in this direction, an unlimited number of mcdm techniques, like analytic hierarchy process (ahp), technique of order preference similarity to the ideal solution (topsis), vikor (vlsekriterijumska optimizacija i kompromisno resenje), grey relational analysis (gra), preference ranking organization method for enrichment evaluation (promethee), combinative distance-based assessment (codas), weighted aggregated sum product assessment (waspas) etc. has been deployed for solving the supplier selection problems in diverse manufacturing industries. recently, stević et al. (2020) proposed a new mcdm tool, called measurement of alternatives and ranking according to compromise solution (marcos) involving ranking of the alternatives based on a compromised solution. in this approach, the ranking procedure is based on the distance of the alternatives from the ideal and anti-ideal solutions with respect to the considered criteria and their aggregated score reflected by a utility function. an integrated d-marcos method for supplier selection in an iron and steel industry 51 however, the biggest challenge in decision making lies in the underlying uncertainty of the decision makers while evaluating the alternatives with respect to a set of qualitative criteria based on some predefined benchmarks and linguistic judgements. a linguistic judgement cannot always be ascertained, especially when there is not a single decision maker, rather an entire team, introducing chances of biasness in the decision making process. in real life situations, it becomes difficult for the decision makers to ascertain a particular degree or rating to a specific criterion owing to their varied backgrounds and experiences. various mathematical tools, like fuzzy set theory, intutionistic fuzzy set etc. have already been employed to deal with the uncertainty and ambiguity involved in the supplier selection process. deng (2012) introduced another tool in the form of d numbers to successfully account for uncertainty involved in the decision making processes. this paper aims at addressing the issue of uncertainty involved in the supplier selection process when the concerned decision makers assign relative scores to the competing suppliers with respect to different evaluation criteria, which if ignored, may result in highly ambiguous results. though marcos method itself is a robust yet mathematically simple model, it still does not address the issue of uncertainty often involved in group decision making where the team of experts comes from different backgrounds and experiences. while there are alternatives, like fuzzy theory, dempster-shafer (d-s) theory etc. to deal with such uncertainty, they often have constraints, like elements in the frame of discernment should be mutually exclusive whereby the sum of the basic probability of mass function should be one. however, d numbers, free from such constraints, provide more generalized solutions. thus, combining d numbers with marcos gives a more holistic and impactful model covering the major loopholes involved in group decision making by accounting for uncertainties using a mathematically simpler formulation. this paper thus deals with implementation of the proposed methodology for supplier selection in a fully operational large scale iron and steel industry in india which has to compete with other stalwarts to carve its own position in the global market. the organization of this paper is as follows. section 2 provides a brief literature review on the applications of different mcdm techniques in supplier selection. section 3 deals with the mathematical details of d numbers and marcos method. section 4 illustrates the application of the propsoed methodology for supplier selection in an indian iron and steel industry. finally, conclusions are drawn in section 5 along with the future dierctions of research. 2. literature review the importance of supplier selection can be proved by the humungous extent of researches conducted based on the applications of various mcdm techniques under both certain and uncertain manufacturing environments. table 1 provides a concise list of different evaluation criteria and mathematical approaches considered for resolving supplier selection problems, with a special emphasis on steel making industries. it can be clearly noticed from table 1 that different mathematical techniques have mainly been employed for two purposes, i.e. a) determination of weights (relative importance) to be assigned to various evaluation criteria and b) ranking of the competing suppliers. the ahp, best-worst method etc. have been deployed for criteria weight estimation, while anp, topsis, gra, promethee, vikor, codas etc. have been augmented for supplier ranking. it is also noticed that some of those mcdm chattopadhyay et al./decis. mak. appl. manag. eng. 3 (2) (2020) 49-69 52 techniques have been combined with fuzzy set, intuitionistic fuzzy set, d-s theory etc. for providing more accurate solutions to supplier election problems dealing with qualitative information under group decision making environment. for effective supplier selection, the primary task is to shortlist the appropriate set of evaluation criteria. back in the 1960s, dickson (1966) stressed on the dependence of supplier selection on various evaluation criteria, while enlisting an exhaustive set of criteria. however, with rapid technological advancements and involvement of global economic parameters, a shift in the criteria for supplier selection has been observed. in the 1990s, the decision makers emphasized on the introduction of more qualitative criteria in the supplier selection process making it more complicated and prone to variation due to human involvement. stević (2017) performed a comprehensive review on various criteria and sub-criteria considered for dealing with the supplier selection problems. however, these specific sets of evaluation criteria vary from one manufacturing organization to another. with every organization thriving hard to develop the best sustainable scm system, importance of a perfect set of evaluation criteria cannot be thus ignored. based on the literature review, it is observed that maximum importance has been provided on price, delivery, quality and production capacity. the extravagant research shows the importance of supplier selection in manufacturing industries. however, there has been relatively less light reflected on the uncertainty involved in the decision making process due to expensive computational steps. for industries seeking a robust decision, it has now become mandatory that the adopted technique should be both exhaustive and efficient eradicating any chance of mistake. most of the past research works have weighed the participating decision makers equally, not accounting for their varied level of expertise and experience. those studies have also been based on the assumption that human preference can be linearly determined. in order to overcome the drawbacks of the previously adopted techniques, in this paper, a new approach for supplier selection integrating d numbers and marcos method is proposed. it is numerically easier to implement, yet provides more reliable ranking results, making it attractive for the manufacturing industries. finally, it is applied to an indian iron and steel making industry while considering the opinions of three experts/decision makers based on five alternative suppliers and seven evaluation criteria. 3. methods 3.1. d numbers the d numbers are an extension of the d-s theory, accounting for uncertainty of information. it can be defined as follows (deng, 2012; deng et al., 2014b): let ω be a finite non-empty set, d number is a mapping formulated by: ]1,0[: d (1) with   b φdbd 0)(and1)( (2) where ϕ is an empty set and b is a subset of ω. an integrated d-marcos method for supplier selection in an iron and steel industry 53 table 1. list of criteria and methods considered for supplier selection author(s) criteria method(s) tahriri et al. (2008) quality, delivery, direct cost, trust, financial position, management and organization ahp gnanasekaran et al. (2010) quality, delivery, cost, financial position, service fuzzy ahp liu (2010) price, delivery, quality, relationship, financial position normalization ying-tuo and yang (2011) quality of products, environmental friendship, price, development capability vague sets vimal et al. (2012) minimum quantity, maximum quantity, defective item, late delivery, product price, order quantity topsis parthiban et al. (2013) quality, delivery, productivity, service, cost, technological capability, application of conceptual manufacturing, environment management, human resource management, manufacturing challenges fuzzy logic, strengthweaknessopportunitythreat (swot) analysis, data envelopment analysis dargi et al. (2014) quality, price, production capacity, technical capability and facility, service and delivery, reputation, geographical location fuzzy analytic network process kar (2015a) product quality, delivery compliance, price, technological capability, production capability, financial strength, electronic transaction capability ahp, fuzzy set theory, neural network kar (2015b) product quality, delivery compliance, price, technological capability, production capability, financial strength, electronic transaction capability delphi method, fuzzy ahp kamath et al. (2016) quality, cost, delivery, vendor relationship management ahp abdulshahed et al. (2017) quality, direct cost, lead time, logistics service grey system theory azimifard et al. (2018) economic sustainability, environmental sustainability, social sustainability ahp, topsis badi et al. (2018) quality, direct cost, lead time, logistics service codas banaeian et al. (2018) service level, quality, price, environmental management system fuzzy topsis, fuzzy vikor, fuzzy gra jain and singh (2018) quality, delivery, performance history, cost ahp, waspas kumar et al. (2018) cost, delivery capability, quality, performance, reputation fuzzy topsis chattopadhyay et al./decis. mak. appl. manag. eng. 3 (2) (2020) 49-69 54 abdullah et al. (2019) cost of product, quality of product, service provided, ontime delivery, technology level, environmental management system, green packaging promethee jain and singh (2019) economic, environmental, social fuzzy modified kano model javad et al. (2020) collaborations, environmental investment and economic benefit, resource availability, green competency, environmental management initiative, research and design initiatives, green purchasing capability, regulatory obligations, pressures and market demand best worst method, fuzzy topsis jain and singh (2020) economic sustainability, environmental sustainability, social sustainability fuzzy interference system with fuzzy kano model the d numbers have the leverage over the d-s theory according to which all elements of set ω need to be mutually exclusive and ,1)(  b bd i.e. the information should be complete. however, d numbers are also capable of dealing with incomplete information, i.e. when 1)(  b bd . on considering a set ω = {b1, b2,…,bi,…,bn}, where rbi  and bi ≠ bj, d numbers can be represented as: d({b1}] = v1, d({b2}] = v2,…,d({bi}] = vi,…,d({bn}] = vn it can also be expressed as d = {(b1,v1), (b2,v2),…,(bi,vi),…,(bn,vn)} where vi > 0 and .1 1   n i iv there are certain properties which are important for performing different operations on d numbers. property 1: (permutation invariability) (deng et al., 2014a; 2014b) assuming two different d numbers, i.e. d1 = {(b1,v1),…,(bi,vi),…,(bn,vn)}and d2 = {(bn,vn),…,(bi,vi),…,(b1,v1)}, then 21 dd  . property 2: (deng, 2012; deng et al., 2014b). if d = {(b1,v1), (b2,v2),…,(bi,vi),…,(bn,vn)}, the integrated value of d can be defined as : i n i ivbdi    1 )( (3) property 3: (deng, 2012; deng et al., 2014a) assuming two different d numbers, d1 and d2 such that )},(),...,,(),...,,{( 11111 1 1 11 nnii vbvbvbd  and )},,(),...,,(),...,,{( 222221 2 12 mmjj vbvbvbd  the combination of d1 and d2 can be expressed as 21 ddd  which can be further defined as follows: d(b) = v (4) where an integrated d-marcos method for supplier selection in an iron and steel industry 55 2 21 ji bb b   (5) c vv v ji           2 21 (6) 1 2 1 1 1 2 1 2 1 1 1 1 2 1 2 1 1 1 1 2 1 2 1 2 1 2 1 1 1 1 , 2 , 2 2 , 2 2 , 2 2 2 2 i jm n j i i j c jm n m j i j i jm n n i c j i i i j c jm n m n i c c c j i j i v v v v v v c v v v v v v v v v v v v                                            (7) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 and 1; 1 and 1; 1 and 1; 1 and 1; m n i ji j m n i ji j m n i ji j m n i ji j v v v v v v v v                         where 1 1 1 n c ii v v    and 2 2 1 1 m c jj v v    it is worthwhile to mention here that the combination operation is not associative in nature. hence, a further operation can be formulated to combine multiple d numbers. property 4: (deng et al., 2014a) if d1, d2,…,dn are n d numbers, µj is an order variable for each dj, indicated by the tuple  jj dμ , , then the function fd represents the combination operation of multiple d numbers, 1 21 2 ( , ,..., ) [...[ ] ... ] nd n f d d d d d d        (8) where 1λ d is equal to jd in the tuple  jj dμ , in which the value of jμ is the least. 3.2. marcos method it is a recently developed mcdm technique used for ranking of the candidate alternatives (stević et al., 2020). consideration of the reference ideal and anti-ideal solutions at the initial stages of analysis makes it advantageous over the other ranking techniques. in this method, each alternative receives a particular value of utility function depending on its relation with the ideal and anti-ideal solutions. preference is provided to those alternatives which are closest to the ideal solution and farthest from the anti-ideal solution. its computation starts with the formation of a decision chattopadhyay et al./decis. mak. appl. manag. eng. 3 (2) (2020) 49-69 56 matrix showing the performance of the alternatives with respect to different criteria. in this matrix, the ideal solution (having maximum values for benefit criteria and minimum values for cost criteria) and anti-ideal solution (with maximum values for cost criteria and minimum values for benefit criteria) are defined. the initial matrix is normalized with respect to the reference value and the corresponding weighted normalized matrix is derived by multiplying all the elements of the normalized matrix with the weight coefficients of the considered criteria. this matrix is finally employed to evaluate the utility degree for each of the alternatives based on which they are subsequently ranked. 3.3. d-marcos method it has already been mentioned that this paper deals with integration of d numbers with marcos method for selection of the most apposite supplier in an indian iron and steel making industry while taking into account the uncertainty prevalent in human judgement to make the decision more robust. for its successful implementation, a set of n criteria is recognized along with determination of their weights (relative importance) using a suitable criteria weight measurement technique. a versatile team of r experts is then formulated where each expert is assigned a weight λk > 0 (i = 1,2,…,r) such that 1 1   r i kλ based on his/her level of experience and expertise. the procedural steps of d-marcos method are presented as below: step 1: in this step, the evaluation matrices for all the participating experts are formulated. due to different backgrounds and variation in human judgements, there exists certain extent of uncertainty while evaluating the alternatives with respect to each of the criteria, which can be taken care of by the implementation of d numbers. for kth expert, the performance score assigned to ith alternative against jth criterion is represented by d number kijd . hence, the decision matrix with m alternatives and n criteria for kth expert is represented as below:                k mn k m k m k n kk k n kk k m k k k ddd ddd ddd a a a t      21 22221 11211 2 1 ' (9) step 2: the aggregated decision matrix for all the experts in the team is now computed based on the properties of d numbers, keeping in mind the weight assigned to each expert. if there are two matrices evaluated by experts e1 and e2: , 11 2 1 1 1 2 1 22 1 21 1 1 1 12 1 11 1 1 2 1 1 ' 1                mnmm n n m ddd ddd ddd a a a t                     22 2 2 1 2 2 2 22 2 21 2 1 2 12 2 11 2 2 2 2 1 ' 2 mnmm n n m ddd ddd ddd a a a t      then the aggregated decision matrix is presented as follows: 1 11 12 1 2 21 22 2 1 2 ' n n m m m mn a d d d a d d d t a d d d             (10) an integrated d-marcos method for supplier selection in an iron and steel industry 57 such that , 21 ijijij ddd  where mi 1 and nj 1 . for more than two experts in the decision making team, the aggregated decision matrix is developed using eq. (8). step 3: in order to rank the candidate alternatives applying marcos method, a consolidated m×n matrix is formulated, integrating each of the d numbers assigned to a particular alternative against each criterion.              mnmm n n m xxx xxx xxx a a a x      21 22221 11211 2 1 (11) where xij = i(dij). step 4: all the considered evaluation criteria are now grouped into two categories, i.e. benefit (larger-the-better) (represented by b) and cost (smaller-the-better) (denoted by c). step 5: the consolidated matrix is extended by defining two additional rows, indicating the ideal (ai) and anti-ideal (aai) solutions. the anti-ideal solution reflects the worst alternative, whereas, the ideal solution reflects the best possible alternative. n21 ccc                       anaa mnmm n n aanaaaa m xxx xxx xxx xxx xxx x        21 21 22221 11211 21 2 1 ai a a a aai (12) where cjxbjxaai ijiiji  if max and if min (13) cjxbjxai ijiiji  if min and if max (14) step 6: the x' matrix is then normalized to form another matrix n of (m + 2)×n dimension, i.e.   nmij nn   )2( , based on the following equations: if bj x x n ai ij ij  (for benefit criterion) (15) c if  j x x n ij ai ij (for cost criterion) (16) step 7: the final weighted matrix   nmij yy   )2( is obtained while multiplying the elements of the normalized matrix by the corresponding criteria weights. jijij wny  (17) where nij is an element of matrix n and wj is the weight assigned to jth criterion. step 8: the positive and negative degrees of utility for each alternative with respect to the ideal and anti-ideal solutions are respectively determined using the following equations: ai i i t t k  (18) chattopadhyay et al./decis. mak. appl. manag. eng. 3 (2) (2020) 49-69 58 aai i i t t k  (19) where )...,2,1( 1 miyt n j iji   (20) step 9: the utility function hence used to evaluate the compromise of each alternative with respect to the ideal and anti-ideal solutions can be defined as follows: )( )(1 )( )(1 1 )(            i i i i ii i kf kf kf kf kk kf (21) where the utility function with respect to the ideal and anti-ideal solutions can be respectively defined using the following equations:      ii i i kk k kf )( (22)      ii i i kk k kf )( (23) step 10: the final ranking order of the alternatives can be obtained while assigning the best rank to the alternative having the highest utility function value. 4. application of d-marcos method for supplier selection as mentioned earlier, this paper deals with the application of d-marcos method for selecting the most apposite supplier for an iron and steel making industry. the steel industry being considered here is located in an industrial town of west bengal, india and procures the requisite materials from various organizations across the globe. it is a leading producer of steel with annual production of around 2.4 million tonnes of crude steel. it came into existence in the year of 1959 and has been growing ever since. although some of its primary raw materials are arranged from its own captive mines or from the parent organization, there are a lot of other materials need to be acquired from other suppliers. it is a gigantic unit which houses a large number of equipments and machineries, requiring huge indenting volume. apart from the semi-finished products, its product basket consists of structural, merchant and railway items. in this plant, there is large number of furnaces and reheating units continuously in action, involving huge refractory consumption. these refractory materials are mostly procured from the external suppliers. this unit needs to be managed to stand the test of time while satisfying its clients across the globe. the importance of sc in such a big unit thus cannot be ignored. there is a dedicated team continuously working to evaluate its wide range of suppliers and choosing the most eligible ones. based on the humongous set of criteria available in the literature for iron and steel industry (kar, 2015b), seven most important criteria are shortlisted for evaluation of the competing suppliers engaged in supply of refractory materials to the considered plant. table 2 provides the list of those criteria which are again weighed by the participating experts using the best-worst method (rezaei, 2015). it is worthwhile to mention here that amongst the criteria, delivery compliance (c2) and price (c3) are the cost criteria always preferred with their minimum values. it is also noticed from table an integrated d-marcos method for supplier selection in an iron and steel industry 59 2 that product quality (c1) and delivery compliance (c2) are the two most important criteria for this supplier selection, whereas, electronic transaction capability (c7) is the least important criterion. table 3 represents details of the five major suppliers among whom the most competent one needs to be identified using d-marcos method. these five suppliers are now appraised by a team consisting of three decision makers from the steel melting unit, materials management and finance department having more than 15 years of industrial experience. based on their varying expertise and knowledge, they are assigned weights with 0.4, 0.35 and 0.25 respectively. they are asked to assess the relative performance of the considered suppliers with respect to each criterion using a 1-9 scale, where 1-2 represent the least scores, 8-9 mark the highest scores, 4-6 denote medium scores, and 3 and 7 are intermediate scores. table 2. list of the criteria for supplier selection criterion description weight product quality (c1) it takes into account worth of a product in compliance to a particular threshold value for minimum assured life and guaranteed performance. 0.312 delivery compliance (c2) it accounts for the time within which delivery is met. scheduled delivery of materials is much needed to ensure proper inventory level such that production never gets disrupted due to unavailability of resources. 0.223 price (c3) it is the monetary value of an item to be paid by the organization to the concerned supplier. 0.208 technological capability (c4) with the advancements of cutting edge technology, product and service must be proficient enough to meet various requirements of the organization even beyond maintaining the delivery schedule. it deals with the compatibility of a supplier to upkeep with the advanced technology. 0.125 production capability (c5) it primarily deals with the ability of a supplier to deliver the required quantity of material at the specified time keeping in mind the fluctuating requirements. it is often graded with respect to standard certifications. 0.114 financial strength (c6) it stresses on the overall financial stability of a supplier with respect to changing market scenario. it is ranked based on a particular supplier’s annual turnover. 0.009 electronic transaction capability (c7) with technological advancements, electronic transaction capability is a much needed sophistication for a supplier to ensure online payment with reduction of other additional costs. 0.006 chattopadhyay et al./decis. mak. appl. manag. eng. 3 (2) (2020) 49-69 60 table 3. list of the shortlisted suppliers supplier description s1 an almost new organization with presence in different countries is well preferred by the steel industries due to its capability to deliver functional refractory at reasonably low price. s2 it was established in early 70s as an msme and proceeded towards adapting better technology of late, but has already succeeded in carving its name amongst the top suppliers of refractory materials. s3 it started its journey in early 70s and has become a well-known supplier of regular refractory materials. with the introduction of state-of-the-art technology, it has also collaborated with other international manufacturers to sustain through the competitive race. s4 it was established in 80s with modern technology and management. it has always been adaptive to the latest technologies grabbing the steel industry’s attention. s5 established in late 90s, it grossly depends on outsourcing of materials with high variation in product quality and hence, is supposed to be a risky supplier. tables 4-6 respectively show the corresponding evaluation matrices developed by the participating decision makers (dm1, dm2 and dm3) while assessing the performance of each of the five suppliers with respect to each criterion in terms of d numbers. for example, in table 4, using the 1-9 scale, dm1 assigns scores 7 and 8 with 50% assurance in each case while appraising supplier s1 with respect to criterion c1. similarly, in table 5, dm2 is 80% confident to assign a score of 6 to supplier s1 with respect to criterion c1. the dm2 is in a dilemma (20% chance) while appraising supplier s1 with respect to criterion c1, i.e. in 20% cases, dm2 is not assured to provide any score to supplier s1 against c1. in table 6, dm3 is 100% assured to assign a score of 6 to supplier s1 against criterion c1. now, based on the individual evaluation matrices by the three decision makers and using properties (2)-(4) of d numbers, the aggregated d number scores are computed in table 7. it is observed that the scores assigned to supplier s1 with respect to criterion c1 by dm1, dm2 and dm3 are respectively d1 = {(7, 0.5), (8, 0.5)}, d2 = {(6,0.8)} and d3 = {(6,1)}. therefore, the aggregated score for supplier s1 against criterion c1 is derived as: 1 2 3 ( ( )) {(6.5, 0.35), (7, 0.35)}d d d d    . an integrated d-marcos method for supplier selection in an iron and steel industry 61 t a b le 4 . e v a lu a ti o n m a tr ix b y d m 1 s u p p li e r c ri te ri a c 1 c 2 c 3 c 4 c 5 c 6 c 7 s 1 {( 7 ,0 .5 ), (8 ,0 .5 )} {( 2 ,0 .5 ), (3 ,0 .5 )} {( 1 ,1 )} {( 7 ,1 )} {( 8 ,0 .2 ), (7 ,0 .8 )} {( 8 ,1 )} {( 7 ,1 )} s 2 {( 8 ,1 )} {( 3 ,1 )} {( 5 ,1 )} {( 7 ,0 .2 ), (8 ,0 .8 )} {( 7 ,1 )} {( 8 ,1 )} {( 7 ,0 .8 ), (8 ,0 .2 )} s 3 {( 7 ,0 .8 ), (8 ,0 .2 )} {( 1 ,0 .8 )( 2 ,0 .2 )} {( 3 ,0 .5 ), (4 ,0 .5 )} {( 9 ,1 )} {( 9 ,1 )} {( 7 ,0 .6 ), (8 ,0 .4 )} {( 7 ,1 )} s 4 {( 6 ,0 .5 ), (7 ,0 .5 )} {( 3 ,0 .8 )} {( 4 ,1 )} {( 8 ,1 )} {( 8 ,1 )} {( 9 ,1 )} {( 8 ,1 )} s 5 {( 7 ,0 .5 )} {( 2 ,1 )} {( 3 ,0 .2 ), (4 ,0 .8 )} {( 8 ,0 .4 ), (7 ,0 .6 )} {( 7 ,0 .8 ), (8 ,0 .2 )} {( 7 ,0 .8 ), (8 ,0 .2 )} {( 9 ,1 )} chattopadhyay et al./decis. mak. appl. manag. eng. 3 (2) (2020) 49-69 62 t a b le 5 . e v a lu a ti o n m a tr ix b y d m 2 s u p p li e r c ri te ri a c 1 c 2 c 3 c 4 c 5 c 6 c 7 s 1 {( 6 ,0 .8 } {( 2 ,1 )} {( 1 ,1 )} {( 7 ,0 .6 ), (6 ,0 .4 )} {( 7 ,1 )} {( 7 ,1 )} {6 ,1 )} s 2 {( 7 ,1 )} {( 2 ,0 .7 ), (3 ,0 .3 )} {( 4 ,1 )} {( 9 ,1 )} {( 8 ,1 )} {( 7 ,0 .6 ), (8 ,0 .4 )} {( 8 ,1 )} s 3 {( 7 ,1 )} {( 1 ,1 )} {( 3 ,0 .7 ), (2 ,0 .3 )} {( 8 ,1 )} {( 9 ,0 .8 ), (8 ,0 .2 )} {( 8 ,0 .7 ), (7 ,0 .3 )} {( 7 ,1 )} s 4 {( 6 ,0 .3 ), (7 ,0 .7 )} {( 2 ,1 )} {( 4 ,0 .7 ), (5 ,0 .3 )} {( 8 ,0 .5 ), (7 ,0 .5 )} {( 8 ,0 .8 ), (9 ,0 .2 )} {( 9 ,1 )} {( 8 ,0 .5 ), (7 ,0 .5 )} s 5 {( 6 ,0 .8 )} {( 2 ,0 .9 )} {( 3 ,0 .6 ), (4 ,0 .4 )} {( 7 ,0 .2 ), (6 ,0 .8 )} {( 7 ,1 )} {( 7 ,1 } {( 9 ,1 )} an integrated d-marcos method for supplier selection in an iron and steel industry 63 t a b le 6 . e v a lu a ti o n m a tr ix b y d m 3 s u p p li e r c ri te ri a c 1 c 2 c 3 c 4 c 5 c 6 c 7 s 1 {( 6 ,1 )} {( 2 ,1 )} {( 1 ,1 )} {( 7 ,0 .6 ), (8 ,0 .4 )} {( 8 ,1 )} {( 7 ,1 )} {( 7 ,1 )} s 2 {( 8 ,1 )} {( 2 ,0 .5 ), (3 ,0 .5 )} {( 5 ,1 )} {( 9 ,0 .6 ), (8 ,0 .4 )} {( 8 ,1 )} {( 7 ,0 .5 ), (8 ,0 .5 )} {( 9 ,1 )} s 3 {( 7 ,0 .7 )} {( 1 ,1 )} {( 3 ,0 .4 ), (4 ,0 .6 )} {( 9 ,0 .3 ), (8 ,0 .7 )} {( 9 ,1 )} {( 8 ,1 )} {( 9 ,1 )} s 4 {( 7 ,0 .6 ), (8 ,0 .4 )} {( 3 ,1 )} {( 4 ,0 .6 )} {( 8 ,1 )} {( 8 ,0 .8 ), (9 ,0 .2 )} {( 8 ,1 )} {( 8 ,1 )} s 5 {( 6 ,0 .7 ), (7 ,0 .3 )} {( 1 ,0 .3 ), (2 ,0 .7 )} {( 3 ,0 .8 ), (4 ,0 .2 )} {( 7 ,1 )} {( 7 ,1 )} {( 7 ,0 .8 ), (8 ,0 .2 )} {( 9 ,1 ) chattopadhyay et al./decis. mak. appl. manag. eng. 3 (2) (2020) 49-69 64 t a b le 7 . a g g re g a te d d e ci si o n m a tr ix f o r th e s u p p li e r se le ct io n p ro b le m s u p p li e r c ri te ri a c 1 c 2 c 3 c 4 c 5 c 6 c 7 s 1 {( 6 .5 ,0 .3 5 ), (7 ,0 .3 5 )} {( 2 ,0 .5 ), (2 .5 ,0 .5 )} {( 1 ,1 )} {( 7 ,0 .3 7 5 ), (6 .7 5 ,0 .3 1 2 5 ), (7 .2 5 ,0 .3 1 2 5 )} {( 7 .7 5 ,0 .4 ), (7 .2 5 ,0 .6 )} {( 7 .5 ,1 )} {( 6 .7 5 ,1 )} s 2 {( 7 .7 5 ,1 )} {( 2 .5 ,0 .3 2 5 ), (2 .7 5 ,0 .3 7 5 ), (3 ,0 .3 )} {( 4 .7 5 ,1 )} {( 8 ,0 .1 8 3 2 5 ), (8 .5 ,0 .3 3 3 2 5 ), (8 .2 5 ,0 .3 1 6 5 ), (7 .7 5 ,0 .1 6 6 5 )} {( 7 .5 ,1 )} {( 8 .2 5 ,0 .3 7 5 ), (7 .5 ,0 .3 1 8 7 5 ), (8 ,0 .3 0 6 2 5 )} {( 7 .7 5 ,0 .6 ), (8 .2 5 ,0 .4 )} s 3 {( 7 ,0 .3 4 2 5 ), (7 .5 ,0 .1 9 2 5 )} {( 1 ,0 .6 ), (1 .5 ,0 .4 )} {( 3 ,0 .2 ), ( 3 .5 ,0 .2 ), (3 .2 5 ,0 .3 ), (3 .7 5 ,0 .1 6 5 ), (2 .7 5 ,0 .1 3 5 )} {( 8 .5 ,0 .5 2 2 ), (8 .7 5 ,0 .4 7 7 )} {( 9 ,0 .5 3 3 3 ), (8 .7 5 ,0 .4 6 6 6 )} {( 7 .5 ,0 .2 9 ), (8 ,0 .2 4 ), (7 .2 5 ,0 .2 6 ), (7 .7 5 ,0 .2 1 )} {( 7 .5 ,1 )} s 4 {( 6 .2 5 ,0 .1 4 5 ), (6 .7 5 ,0 .3 ), (7 .2 5 ,0 .1 5 5 ), (7 ,0 .2 ), (6 .5 ,0 .2 )} {( 2 .7 5 ,0 .6 )} {( 4 ,0 .3 3 1 2 5 ), (4 .2 5 ,0 .3 0 6 2 5 )} {( 8 ,0 .5 ), (7 .7 5 ,0 .5 )} {( 8 .2 5 ,0 .3 7 5 ), (8 .5 ,0 .2 7 5 ), (8 ,0 .3 5 )} {( 8 .7 5 ,1 )} {( 8 ,0 .5 ), (7 .7 5 ,0 .5 )} s 5 {( 6 .5 ,0 .1 7 5 ), (6 .7 5 ,0 .1 5 5 )} {( 1 .7 5 ,0 .3 2 5 ), (2 ,0 .3 5 )} {( 3 ,0 .2 7 5 ), (4 ,0 .1 9 ), ( 3 .5 ,0 .3 ), (3 .7 5 ,0 .2 6 ), (3 .2 5 ,0 .1 4 )} {( 7 .5 ,0 .2 ), (7 ,0 .2 5 ), (6 .7 5 ,0 .3 ), (7 .2 5 ,0 .2 5 )} {( 7 ,0 .6 ), (7 .5 ,0 .4 )} {( 7 .5 ,0 .2 ), (7 ,0 .3 5 ), (7 .7 5 ,0 .1 5 ), (7 .2 5 ,0 .3 )} {( 9 ,1 )} an integrated d-marcos method for supplier selection in an iron and steel industry 65 it is worthwhile to mention here that in this supplier selection problem, the participating decision makers have been assigned weights with 0.4, 0.35 and 0.25 respectively depending on their varying experience and expertise. thus, the combination operation for d numbers is first performed between dm2 and dm3 with minimum weights, and then the corresponding d number for dm1 is taken into consideration for the combination operation. now, based on the developed aggregated decision matrix in terms of d numbers, the corresponding consolidated matrix x is formulated using eq. (3). for instance: x11 = ((6.5×0.35) + (7×0.35)) = 4.72. in the similar direction, dm1, dm2 and dm3 respectively evaluate the performance of supplier s2 against criterion c4 as d1 = {(7,0.2), (8,0.8)}, d2 = {(9,1)}and d3 = {(9,0.6),(8,0.4)} in terms of d numbers. the aggregated score for supplier s2 with respect to criterion c4 is calculated as: ))(( 321 dddd  = {(8.0, 0.18325), (8.5, 0.33325), (8.25, 0.3165), (7.75, 0.1665)} thus, the value of element x24 in the consolidated matrix becomes: 2.8))1665.075.7()3165.025.8()33325.05.8()18325.08((24 x based on the procedural steps of d-marcos method, another matrix x' (extended matrix) is formulated from the consolidated matrix by defining two additional rows, indicating the ideal (ai) and anti-ideal (aai) solutions at the bottom and top of the consolidated matrix respectively. now, based on the type of the considered criterion and employing eqs. (15)-(16), the related normalized decision matrix is obtained. 7654321 ccccccc                        1111111 183.081.082.041.094.028.0 87.0193.091.038.073.087.0 83.087.01131.0149.0 88.090.084.095.021.044.01 75.086.084.081.0153.061.0 75.083.081.081.021.044.028.0 5 4 3 2 1 ai s s s s s aai n                  929.72.709.706.427.118.2 87.776.823.887.763.265.175.6 5.760.788.861.826.320.184.3 95.793.75.72.875.474.275.7 75.65.745.77125.272.4 x 7654321 ccccccc                        976.888.861.8120.175.7 929.72.709.706.427.118.2 87.776.823.887.763.265.175.6 5.760.788.861.826.320.184.3 95.793.75.72.875.474.275.7 75.65.745.77125.272.4 75.629.72.7775.474.218.2 5 4 3 2 1 ai s s s s s aai x chattopadhyay et al./decis. mak. appl. manag. eng. 3 (2) (2020) 49-69 66 the weighted normalized decision matrix is then computed by multiplying each element of the normalized matrix with the corresponding criteria weights. 7654321 ccccccc                        0060.00090.01140.01250.02080.02230.03120.0 0060.00074.00923.01025.00853.02096.00874.0 0052.00090.01060.01138.00790.01628.02714.0 0050.00078.01140.01250.00645.02230.01529.0 0053.00081.00958.01188.00436.00981.03120.0 0045.00078.00958.01013.02080.01182.01903.0 0045.00074.00923.01013.00436.00981.00874.0 5 4 3 2 1 ai s s s s s aai y using eqs. (18)-(23), the positive and negative degrees of utility, and value of the utility function for all the competing suppliers are estimated, as shown in table 8. the detailed computational steps for determining the utility function value for supplier s1 are explained as below: for ideal solution: tai = 0.3120 + 0.2230 + 0.2080 + 0.1250 + 0.1140 + 0.0090 + 0.0060 = 0.9970 for anti-ideal solution: tai = 0.0874 + 0.0981 + 0.0436 + 0.1013 + 0.0923 + 0.0074 + 0.0045 = 0.4346 for supplier s1: t1 = 0.1903 + 0.1182 + 0.2080 + 0.1013 + 0.0958 + 0.0078 + 0.0045 = 0.7259 7281.0 9970.0 7259.0 1   k ; 6702.1 4346.0 7259.0 1   k ; ;696410.0 7281.06702.1 6702.1 )( 1    kf 303590.0 7281.06702.1 7281.0 )( 1    kf 0.643001 303590.0 303590.01 696410.0 696410.01 1 6702.17281.0 )( 1      kf in order to identify the most apposite supplier for providing refractory materials to the considered iron and steel making industry, they are now ranked based on the computed values of utility function. it is observed that supplier s4 with the maximum utility value of 0.661829 is ranked first, closely followed by supplier s1. the performance of suppliers s2 and s3 is almost similar. on the other hand, supplier s5 would be considered with least preference. table 8. estimation of utility functions for the candidate suppliers supplier ti  ik  ik )(  ikf )(  ikf )( ikf rank s1 0.7259 1.6702 0.7281 0.303590 0.696410 0.643001 2 s2 0.6817 1.5686 0.6838 0.303587 0.696412 0.603879 4 s3 0.6922 1.5927 0.6943 0.303585 0.696414 0.613154 3 s4 0.7472 1.7193 0.7494 0.303560 0.696439 0.661829 1 s5 0.5905 1.3587 0.5923 0.303588 0.696412 0.523066 5 an integrated d-marcos method for supplier selection in an iron and steel industry 67 5. conclusions this paper proposes integration of d numbers with marcos method for effective selection of suppliers for refractory materials in an iron and steel industry in india. for this purpose, the relative performance of five competing suppliers is evaluated with respect to seven conflicting criteria using d numbers based on the opinions of three decision makers with varying knowledge and expertise. the marcos method is later employed for ranking of the considered suppliers. it has already been acknowledged that accounting for uncertainty involved in supplier selection process for effective scm system development is an important task in today’s manufacturing environment. although there are several approaches, like fuzzy set theory, d-s theory etc. to deal with uncertainty in decision making processes, the concept of d numbers supersedes the others with respect to its ability to provide more robust and flexible results while taking into consideration varied opinions of individual decision makers who can evaluate the relative performance of the participating suppliers with varying degrees of uncertainty. thus, this integrated mcdm tool can be efficiently adopted in other domains of decision making, like selection of optimal maintenance strategy, plant layout, inventory control policy, machine tool etc. in uncertain manufacturing environment. author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research received no external funding. conflicts of interest: the authors declare no conflicts of interest. references abdullah, l., chan, w., & afshari, a. 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(2011). iron and steel companies green suppliers’ selection model based on vague sets group decision-making method. in: proceedings of the international conference on electronics, communications and control, 2702-2705. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 1, issue 2, 2018, pp. 121-130 issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame1802128l * corresponding author. e-mail addresses: feng.liu@gmail.com (f. liu), gaiwuzh@126.com (g. aiwu) lukovacvesko@yahoo.com (v. lukovac), milena.vukic12@gmail.com (m. vukic) a multicriteria model for the selection of the transport service provider: a single valued neutrosophic dematel multicriteria model feng liu1*, guan aiwu2, vesko lukovac 3, milena vukić4 1 business school, zhejiang wanli university, ningbo, china 2 school of management, jiangsu university, zhenjiang, china 3 university of defence in belgrade, military academy, department of logistics, belgrade, serbia 4 the college of hotel management, belgrade, serbia received: 5 april 2018; accepted: 2 september 2018; available online: 2 september 2018. original scientific paper abstract: the decision-making process requires, a priori, defining and considering certain factors, especially when it comes to complex areas such as transport management in companies. one of the most important items in the initial phase of the transport process that significantly influences its further flow is decision-making about the choice of the most favorable transport provider. in this paper a model for evaluating and selecting a transport service provider based on a single valued neutrosophic number (svnn) is presented. the neutrosophic set concept represents a general platform that extends the concepts of classical sets, fuzzy sets, intuitionistic fuzzy sets, and an interval valued intuitionistic fuzzy sets. the application of the svnn concept made a modification of the dematel method (decision-making trial and evaluation laboratory method) and proposed a model for ranking alternative solutions. the svnn-dematel model defines the mutual effects of the provider's evaluation criteria, while, in the second phase of the model, alternative providers are evaluated and ranked. the svnn-dematel model was tested on a hypothetical example of evaluation of five providers of transport services. key words: multicriteria decision-making, dematel, single valued neutrosophic numbers, provider selection mailto:feng.liu@gmail.com mailto:gaiwuzh@126.com mailto:lukovacvesko@yahoo.com mailto:milena.vukic12@gmail.com liu et al./decis. mak. appl. manag. eng. 1 (2) (2018) 121-130 122 1. introduction outsourcing approach is widely present in all logistic aspects of business, especially in the transport domain, which is distinguished by its significant and direct participation in overall logistics costs. after making a decision on accepting outsourcing for certain logistical activities of the organization, the management is facing the issue of selecting the provider that will implement these activities for the organization needs. the problem of selecting a transport service provider is conceptually similar to the choice of providers in most other logistics activities. in that sense, when it comes to the models of selection of the transport service provider, of relevance are those research studies that are focused on the selection of carrier, suppliers, vendor, or independent logistics providers (third party logistics provider selection). regardless of differences in the views on the structuring of the providers selection problem (ordoobadi & wang, 2011; shen yu, 2012), as well as on the structure of the selection process itself (snir & hitt, 2004; monczka et al., 2005; cao & wang, 2007), when it comes to the nature of this process, its multidimensional character is often mentioned (vinodh et al., 2011; senthil et al., 2014). in that sense, numerous multicriteria decision-making methods have been used to select providers. various examples of combining different approaches that treat uncertainty (fuzzy access, etc.) with traditional multicriteria techniques, such as topsis (zouggari & benyoucef, 2011; senthil et al., 2014), vikor (sanayei et al., 2010), ahp (singh & sharma, 2011; senthil et al., 2014), anp (nobar et al., 2011) etc. can be found in the literature. an example of the dematel method application to the recognition of the relevant criteria as well as to the identification of their significance and causal relationships in the process of structuring a model for the supplier selection with carbon management competencies can be seen in (hsu et al., 2011). as can be seen in a review of the referential literature given here, most approaches prefer the use of traditional multicriteria decision-making (mcdm) models in combination with fuzzy techniques (senthil et al., 2014). however, in the real world, the decision-maker may prefer attribute assessment by using linguistic variables instead of crisp values either due to his partial knowledge about attributes or the lack of information from the problem domain. the fuzzy set presented by zadeh (1965) is one of the tools used to present such imprecision in mathematical form. however, the fuzzy set can focus only on the degree of affiliation of unclear parameters or events. the fuzzy set cannot represent the degree of non-affiliation and the degree of imprecision of uncertainty parameters. in order to partially overcome the difficulties in defining parameters that are imprecise, atanassov (1986) introduced intuitionistic fuzzy sets (ifs) that are characterized by the degree of affiliation and non-affiliation simultaneously. however, in the ifs, the sum of the affiliation degree and non-affiliation degree of the unclear parameter is less than one (unity). in order to eliminate these shortcomings, smarandache (1999) introduced a neutrosophic concept in order to deal with unspecified or inconsistent information that usually exists in reality. the concept of a neutrosophic set represents a general platform that extends the concepts of classical sets, fuzzy sets (zadeh, 1965), intuitionistic fuzzy sets (atanassov, 1986), and interval valued intuitionistic fuzzy sets (atanassov & gargov, 1989). unlike intuitionistic fuzzy sets and interval valued intuitionistic fuzzy sets, in the neutrosophic set indeterminacy is explicitly characterized. using the advantages of the neutrosophic sets mentioned above, the original svnn-dematel model for the transport service provider evaluation was proposed in this paper. in the next section of work (section 2), the basic items of the svnn are multicriteria model for the selection of the transport service provider: single valued… 123 presented. thereafter, in the third section of the paper, an original vko model based on svnn was presented. testing of the presented model was performed in the fourth section of the work. 2. neutrosophic sets according to the definition of a neutrosophic set, neutrosophic set a is a universal set x characterized by function of affiliation describing truth-membership function ta(x), indeterminacy-membership function ia(x) and the function of falsitymembership fa(x). where ta(x), ia(x) and fa(x) are real standard or non-standard subsets of [-0,1+], each of the three neutrosophic components satisfy the condition that ta(x)→ [-0,1+], ia(x)→ [-0,1+] and fa(x)→ [-0,1+]. set ia(x) can be used to present not only indeterminacy, but also unclearness, uncertainties, inaccuracies, errors, contradictions, the undefined, the unknown, incompleteness, redundancy, etc. (biswas et al, 2016). in order to cover all unclear information, the degree of affiliation to the indeterminacy-membership degree can be subdivided into sub-components, such as "contradiction," "uncertainty," and "unknown" (smarandache, 1999). the sum of these three neutrosophic set affiliation functions ta(x), ia(x) and fa(x) should satisfy the following condition 0 ( ) ( ) ( ) 3a a at x i x f x       (biswas et al, 2016). the component of neutrosophic set a for all values x x is determined by ac so that ( ) 1 ( )ca at x t x    , ( ) 1 ( )ca ai x i x    and ( ) 1 ( )ca af x f x    . neutrosophic set a is contained in another neutrosophic set b ( a b ) if and only if for each value x x the following conditions are satisfied inf ( ) inf ( )a bt x t x , sup ( ) sup ( )a bt x t x , inf ( ) inf ( )a bi x i x , sup ( ) sup ( )a bi x i x , inf ( ) inf ( )a bf x f x , and sup ( ) sup ( )a bf x f x . single valued neutrosophic sets (svns) are a special case of the neutrosophic set that can be used more successfully in modern scientific and engineering applications, compared to the classical neutrophic set. basic arithmetic operations on svnn that are significant for the mathematical background of the mcdm model can be looked in detail in (wang et al., 2010; deli & şubaş, 2017). 3. single valued neutrosophic dematel method the dematel method is a very suitable tool for designing and analyzing the structural model. and it can be achieved through the definition of cause-effect relationships between factors that are complex (pamučar & ćirović, 2015; gigović et al., 2016). in order to comprehensively take into account the imprecision that exists in group decision-making, this paper performs a modification of the dematel method by using the svns. in the next section the steps of the svn-dematel method are elaborated, namely: step 1: factors expert analysis. assuming that there are m experts and n factors (criteria) that are observed, each expert should determine the degree of influence of factor i on factor j. a comparative analysis of the pair of the i -th and j -th factor by the k-th expert is marked by dije, where , , e e e e ij ij ij ijd t i f ,  1,..., ; 1,...,i n j n  represents a neutrophic number that is being compared in the pairs of factors. the value of each liu et al./decis. mak. appl. manag. eng. 1 (2) (2018) 121-130 124 pair dije takes the values from a previously defined single valued neutrosophic linguistic scale. the response of the e-th expert is displayed by a single valued neutrosophic matrix of , , e e e e e ij ij ij ij n n n n d d t i f     ,  1 e m  rank, where m represents the total number of experts. 12 12 12 1 1 1 21 21 21 2 2 2 1 1 1 2 2 2 0 , , , , , , 0 , , , , , , 0 e e e e e e n n n e e e e e e n n ne e e e e e e n n n n n n nxn t i f t i f t i f t i f d t i f t i f                 (1) where , , e e e ij ij ijt i f represents single valued neutrosophic linguistic expressions from a predefined linguistic scale which the expert e uses to represent his comparison in the pairs of criteria. thus we get matrices d1, d2, …, dm which represent the matrices of responses from each of the m experts. step 2: determination of weight coefficients of experts. it starts from the assumption that m experts  1 2, ,..., me e e with assigned weight coefficients 1 2{ , ,..., }m   , 0 1, ( 1, 2,..., )e e m   participate in the decision-making process. suppose that: (1) each expert from the group of m has his own weighting coefficient, (2) the weight coefficients of the experts differ in value, and (3) condition 1 1 m e e    is satisfied. then we can present the significance of each expert using linguistic variables from a predefined single valued neutrosophic linguistic scale. if we denote a single valued neutrosophic number with ( ), ( ), ( )e e e ee t x i x f x which evaluates the significance of the e-expert, then the weight coefficient of the e-th expert can be determined using the expression (2), [17]               2 2 2 2 2 2 1 1 1 ( ) ( ) ( ) 3 1 1 ( ) ( ) ( ) 3 e e e e m e e e e t x i x f x t x i x f x                  (2) where 1 1 m e e    ,  1 e m  . step 3: determination of the average responses matrix of the experts. on the basis of individual matrices of the answer of the m experts, we obtain a matrix of aggregated sequences of experts * , , e e e e ij ij ij ij n n n n d d t i f     ,  1 e m  , where  1 1 1 2 2 2, , , , , ,..., , ,e m m mij ij ij ij ij ij ij ij ij ijd t i f t i f t i f represent sequences which describe the relative importance of criterion i in relation to criterion j . using the expression (3), an aggregation of values is made at each position of matrix *d multicriteria model for the selection of the transport service provider: single valued… 125         1 1 1 1 1 1 , , e e e m m mm e e e e ij e ij ij ij ij e e e e d d t i f                         (3) where , ,ij ij ijijd t i f represents aggregated svnn. that is how we obtain an aggregated single valued neutrosophic matrix of the average response of the experts (4) 12 12 12 1 1 1 21 21 21 2 2 2 1 1 1 2 2 2 0 , , , , , , 0 , , , , , , 0 n n n n n n n n n n n n t i f t i f t i f t i f d t i f t i f                 (4) matrix d shows the initial effects that factor j causes, as well as the initial effects that factor j receives from the other factors. the sum of each i-th row of matrix d represents the total direct effects that factor i handed over to the other factors, and the sum of each i j--th column of matrix d represents the total direct effects that factor j receives from the other factors. step 4: determine the svn total relation matrix. using expression (5) we calculate a single valued neutrosophic total relation matrix ( ), ( ), ( )ij ij ij ij n n n n t t t t i t f t     . element ( ), ( ), ( )ij ij ij ijt t t i t f t represents the direct effect of factor i on factor j, while matrix t reflects the overall relationship between each pair of factors. since each single valued neutrosophic number consists of three sequences ( ), ( )ij ijt t i t and ( )ijf t then the svn matrix can be divided into three submatrices, i.e. , , n n d t i f   , where, ij n n t t      , ij n n i i      and ij n n f f      . furthermore,  lim m m t o   ,  lim m m i o   and  lim m m f o   , where 0 represents zero matrix. based on the defined settings, we obtain the svn matrix of total t effects by calculating the following elements             2 1 1 1 2 2 ( ) ( ) ( ) ( ) ( ) ( ) lim lim lim m ij n n m ij n n m m n nm ij m t t t t t t t t i t i i i i i t and f t f f f f f t i i i i i i                                                (5) sub-matrices ( )t t , ( )i t and ( )f t together represent a svn matrix of total impact      , , n n t t t i t f t   . based on expression (5) the svn matrix of total impacts is obtained liu et al./decis. mak. appl. manag. eng. 1 (2) (2018) 121-130 126 11 11 11 12 12 12 1 1 1 21 21 21 22 22 22 2 2 2 1 1 1 2 2 2 ( ), ( ), ( ) ( ), ( ), ( ) ( ), ( ), ( ) ( ), ( ), ( ) ( ), ( ), ( ) ( ), ( ), ( ) ( ), ( ), ( ) ( ), ( ), ( ) ( ), ( ), ( ) n n n n n n n n n n n n nn nn nn t t i t f t t t i t f t t t i t f t t t i t f t t t i t f t t t i t f t t t t i t f t t t i t f t t t i t f t              (6) where ( ), ( ), ( )ij ij ij ijt t t i t f t is a single valued neutrosophic number which expresses indirect effects of factors i on factor j . then matrix t reflects the interdependence of each pair of factors. step 5: calculating the sum of the rows and columns of the total impact t matrix. in the total impact t matrix the sum of rows and that of columns is represented by vectors r and c of n×1: 1 1 1 1 ( ), ( ), ( ) n n i ij ij ij ij j j n n r t t t i t f t                        (7) 1 11 1 ( ), ( ), ( ) n n i ij ij ij ij i in n c t t t i t f t                      (8) step 6: determination of the weighting coefficients of the criteria. the weighting coefficients of the criteria are determined using the expression                             2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 ( ) ( ) ( ) 3 1 ( ) ( ) ( ) 3 1 ( ) ( ) ( ) 3 1 ( ) ( ) ( ) 3 i i i j i i i i i i i i i t r i r f r w t c i c f c t r i r f r t c i c f c                                     (9) step 7: forming the initial decision matrix (n). as in dematel method, the evaluation of alternatives by the criteria is being done by m experts  1 2, ,..., me e e with assigned weighting coefficients 1 2{ , ,..., }m   , 1 1 m e e    . in order to make a final ranking of alternatives ia a ( 1,2,..,i b ), each expert ee ( 1, 2,...,e m ) evaluates alternatives by a defined set of criteria  1 2, ,... nc c c c . in that way, correspondent initial decision matrix ( )( ) ee ij b n n       is being constructed for each expert where elements of matrix ( )en ( ( )eij ) represent svn numbers from a predefined neutrosophic linguistic scale. final aggregated decision matrix n is obtained by centering matrix elements ( ) ( ) ( ) ( ), ,e e e eij ij ij ijt i f    of matrix ( )e n . that is how we obtain matrix ij b n n       , where elements , ,ij ij ij ijt i f    are obtained by applying the swnswaa operator, the expression (10) multicriteria model for the selection of the transport service provider: single valued… 127       (1) (2) ( ) (1) 1 ( ) ( ) ( ) 1 1 1 ( , ,.., ) 1 1 , , e e e m m ij ij ij ij e ij b m m m e e e ij ij ij b b b svnswaa t i f                           (10) where e is weighting coefficient, 0 1, ( 1, 2,..., )e e m   , 1 1 m e e    . step 8: calculation of the elements of the difficult matrix (d). the elements of difficult matrix , ,ij dij dij dij b n b n d d t i f           are obtained by applying the expression (11)  , , 1 1 , , j j j w w w ij dij dij dij j ij ij ij ij d t i f w t i f          (11) step 9: ranking alternatives. on the basis of the value of criterion functions iq ( 1, 2,...,i b ) ranking of alternatives is carried out. the criteria functions are obtained by applying expression (12), 1 , 1, 2,..., ; 1, 2,..., . n i j j q d i b j n     (12) 4. numerical example the svnn-dematel vko model for selecting providers was tested on a hypothetical example of the selection of five providers of transport services. as a result of the use of the model, the weighting coefficients of the evaluation criteria were determined and the ranking of the transport providers was performed. four experts in the field of transport participated in the testing of the model; they got weighting coefficients assigned by using the expression (2) e1=0.2864, e2=0.2741, e3=0.2170 and e4=0.1673. experts evaluated the criteria using a linguistic scale: very important – vi (0.90,0.10,0.10); important – i (0.75,0.25,0.20); medium – m (0.50,0.50,0.50); unimportant – ui (0.35,0.75,0.80); very unimportant – vu (0.10,0.90,0.90). five criteria were used to evaluate the provider: c1 – reliability, c2 – business excellence, c3 – total cost, c4 – customer service, c5 – green image. expert evaluations of the criteria are shown in table 1. table 1 expert analysis of the criteria criteria c1 c2 c3 c4 c5 c1 0 vi;vi;vi;i i;m;m;i vi;vi;ivi i;i;m;ui c2 i;m;m;i 0 m;m;vi;vi m;m;m;m vi;i;i;vi c3 m;m;m;m m;m;i;i 0 m;i;m;m vi;vi;vi;vi c4 i;i;ivi m;m;m;m m;m;m;m 0 m;m;m;m c5 m;vu;vu;ui i;i;i;i m;m;m;m i;m;m;i 0 by summing up the elements of the total relation matrix (6) by rows, equation (7), and by columns, equation (8), the values of the total direct and indirect effects of criterion j on the other criteria and the other criteria on criterion j are obtained. liu et al./decis. mak. appl. manag. eng. 1 (2) (2018) 121-130 128 these values together with the threshold value (α) of the total relation matrix are used for defining the cause-and-effect relationship diagram. the cause and effect relationship (cer) diagram (fig. 1) is formed to visualize the complicated causal relationship of criteria in a visible structural model. ri+ci ri-ci 0.00 -0.9 0.9 c3 0.20 0.8 c1 c5 c2 c4 figure 1 cerd diagram the elements in matrix t with a value higher than the threshold value α will be identified and mapped on the diagram (fig. 1) where the x-axis denotes (ri+ci), and yaxis denotes (ri-ci). these values will be used for demonstrating the relationship between two factors. in the course of the demonstration, the arrow denoting the cause-effect membership is directed from the element with a value lower than α towards the element characterized by a higher value than α. using the expression (9), we obtain the weight coefficients of the criteria: c1 (0.828,0.156,0.145), c2 (0.606,0.381,0.364), c3 (0.873,0.129,0.147), c4 (0.641,0.372,0.329) and c5 (0.709,0.307,0.318). expert evaluation of providers by the criteria (table 2) was carried out using a linguistic scale: extremely good/high – eg/eh (1,0,0); very very good/high – vvg/vvh (0.9,0.1,0.1); very good/high – vg/vh (0.8,0.15,0.2); good/high – g/h (0.7,0.25,0.3); medium good/high – mg/mh (0.6,0.35,0.4); medium /fair – m/f (0.5,0.5,0.5); medium bad/low – mb/ml (0.4,0.65,0.6); bad/low – b/l (0.3,0.75,0.7); very bad/low – vb/vl (0.2,0.85,0.8). table 2 expert evaluation of providers according to the evaluation criteria alternative/ criteria c1 c2 c3 c4 c5 a1 vg;mg;vg;g g;g;mg;g mg;mg;m;m g;m;mg;m m;mh;vh;m a2 g;vg;mg;mg vg;mg;m;mg vg;g;vg;vg vg;vg;m;g vh;m;h;h a3 m;gmg;m m;vg;g;g m;g;mg;mg mg;mg;mg;mg h;h;m;mh a4 g;mg;g;mg mg;m;vg;m g;mg;g;mg m;mb;mg;vg m;m;mh;h a5 g;g;mg;vg g;g;mg;vg mg;g;vg;g mg;g;vg;g h;vh;vh;vh applying expressions (10) (12) we get the final rank of the provider: a1 (0.622,0.330,0.374)> a2 (0.571,0.384,0.425)> a3> (0.504,0.457,0.497)>a4 (0.499,0.457,0.497)> a5(0.344,0.643,0.637). the 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(2011). simulation based fuzzy topsis approach for group multi-criteria supplier selection problem, engineering applications of artificial intelligence, doi:10.1016/j.engappai.2011.10.012. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). refinement of acyclic-and-asymmetric payoff aggregates of pure strategy efficient nash equilibria in finite noncooperative games by maximultimin and superoptimality decision making: applications in management and engineering vol. 4, issue 2, 2021, pp. 178-199. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame210402178r * corresponding author. e-mail address: v.romanuke@amw.gdynia.pl (v. romanuke) refinement of acyclic-and-asymmetric payoff aggregates of pure strategy efficient nash equilibria in finite noncooperative games by maximultimin and superoptimality vadim v. romanuke 1* 1 polish naval academy, 69 śmidowicza street, gdynia, poland received: 25 may 2021; accepted: 5 july 2021; available online: 12 july 2021. original scientific paper abstract: a theory of refining pure strategy efficient nash equilibria in finite noncooperative games under uncertainty is outlined. the theory is based on guaranteeing the corresponding payoffs for the players by using maximultimin, which is an expanded version of maximin. if a product of the players’ maximultimin subsets contains more than one efficient nash equilibrium, a superoptimality rule is attached wherein minimization is substituted with summation. the superoptimality rule stands like a backup plan, and it is enabled if just a single refined efficient equilibrium (a metaequilibrium) cannot be produced by maximultimin. the number of the refinement possible outcomes is 10. there are 3 single-metaequilibrium cases, 3 partial reduction cases, and 4 fail cases. despite successfulness of refinement drops as the game gets bigger, efficient equilibria in games with no more than four players are successfully refined at no less than a 54 % rate. key words: finite noncooperative games, efficient equilibria, refinement, maximultimin, superoptimality, metaequilibrium, uncertainty partial reduction. 1. introduction game theory allows making decisions that would be appropriate simultaneously for multiple sides (or players, persons, subjects, etc.). this is achieved by persuading players that holding at the equilibrium is the best rule. there are many types of equilibria but mostly they are modifications of nash equilibrium. either in pure or mixed strategies, nash equilibrium is a stable state, at which the player cannot increase one’s payoff by acting alone. in a finite noncooperative game (fncg), however, a few nash equilibria in pure strategies can exist (harsanyi & selten, 1988; osborne, 2003; vorob’yov, 1984). such equilibria may be nonequivalent: despite the mailto:v.romanuke@amw.gdynia.pl refinement of acyclic-and-asymmetric payoff aggregates of pure efficient nash equilibria 179 equilibrium stability itself, the players’ payoffs at some equilibrium can be greater that those at other equilibria (leinfellner & köhler, 1998; osborne, 2003; vorob’yov, 1984). the equilibria which are strictly and non-strictly dominated are out of interest for players. then, only efficient equilibria remain. a question is how to persuade players to hold at a certain efficient equilibrium, when there are multiple equilibria (two or more)? for fncgs with two or three players (bimatrix and trimatrix games) this question is yet not trivial (vorob’yov, 1985; vorob’yov, 1958). fncgs with more than three players are rather complicated, so searching for the most efficient equilibrium, if any, in pure strategies requires a non-trivial theoretic basis. in fncgs having multiple equilibria, the subsequent decision-making problem lies in refining those equilibria. the refinement is required, in the first turn, for realworld practice applications, wherein messing around with multiple equilibria makes no sense. those applications emerge from economics, ecology, politics, jurisprudence, computer networking, and other related fields, where multiple subjects come to interaction (harsanyi & selten, 1988; leyton-brown & shoham, 2008; myerson, 1997; romanuke, 2010b; vorob’yov, 1984). 2. background and motivation in game theory, the refinement is understood as the selection of a subset of such equilibria, which are believed to be more plausible than other equilibria (belhaiza et al., 2012; myerson, 1978; vorob’yov, 1985). sometimes, it is called the identification of actualized equilibria (dopfer & potts, 2007; liu & forrest, 2010). thus, the refinement does not necessarily imply selecting the best efficient nash equilibria. nor does it imply finding a single efficient equilibrium. in a wider sense, the refinement implies narrowing down the game model decisions. nash equilibria, for instance, are a particular example of a set of such decisions) (harsanyi & selten, 1988; vorob’yov, 1984; vorob’yov, 1985). the essential obstacle is that the plausibility does not straightforwardly imply profitability. for repeatable fncgs, it means that the refined equilibrium can be unstable: a player cannot be “captured” on a non-profitable payoff, although such payoff issues from a plausible equilibrium (belhaiza et al., 2012; fudenberg & tirole, 1991; myerson, 1978). so, in the course of game repetitions, the player may spring off its non-profitable strategy (leinfellner & köhler, 1998; liu & forrest, 2010; osborne, 2003; romanuke, 2010b). thus, the plausibility determining the refinement without primordial profitability is not stable itself. in particular,  -equilibria are that kind of instability, wherein players may constantly keep searching for profits rather than stop at a moment (belhaiza et al., 2012; fudenberg & tirole, 1991; vorob’yov, 1984; vorob’yov, 1985). in non-repeatable fncgs, profitability mainly defines the plausibility. it simplifies persuasion of selecting definite equilibrium strategies. on the other hand, a great deal of the existing refinements, e. g. mertens-stable equilibrium (kohlberg & mertens, 1986), trembling hand perfect equilibrium (selten, 1975), proper equilibrium (myerson, 1978; van damme, 1984), sequential equilibrium (fudenberg & tirole, 1991; gerardi & myerson, 2007), quasi-perfect equilibrium (bajoori et al., 2013; mertens, 1995; van damme, 1984), become inapplicable due to mixed strategies lose their reason. strong nash equilibrium (suh, 2001; tian, 2000) remaining as the most remarkable refinement is efficient itself (romanuke, 2016b; romanuke, 2016a). romanuke/decis. mak. appl. manag. eng. 4 (2) (2021) 178-199 180 in practice, the existing approaches to refining rarely provide just a single refined equilibrium. commonly, there are still multiple (or even a continuum of) equilibria after the refinement (gerardi & myerson, 2007; harsanyi & selten, 1988; myerson, 1978; romanuke, 2018a). moreover, sometimes equilibria can be just nonrefinable. such nonrefinability is easily shown and seen by examples of the bimatrix (even 2 2 ) game whose efficient nash equilibria produce either identical or symmetric payoffs (myerson, 1997; osborne, 2003; romanuke, 2010b; vorob’yov, 1985; vorob’yov, 1984). bigger fncgs with the nonrefinability are easily built. at that, symmetric equilibria are even worse than identical. in other cases, efficient equilibria are nonrefinable as there is no additional information that could have helped to understand which equilibrium is better and which is worse (romanuke, 2018a; suh, 2001; tian, 2000). this is equivalent to uncertainty of equilibria, where players may not comprehend a distribution of the plausibility over definite equilibria, except for their profitability at one-step action. an effort made in (romanuke, 2018b) for refining pure strategy efficient nash equilibria in trimatrix games had been developed on a base of expanding the refinement approach for bimatrix games. for a player, that novel approach exploits the maximin, expanded to the maximinimin principle, and a superoptimality rule using summing over pure strategy subsets of the other players. the maximin rule is intended for guaranteeing payoffs. the superoptimality rule (romanuke, 2018a) is enabled if just a single refined efficient nash equilibrium cannot be produced by maximin. so, the present goal is to finalize the theory of refining pure strategy efficient nash equilibria in fncgs under uncertainty. 3. denotations of fncg and its efficient nash equilibria we consider a non-repeatable fncg (1) of players, in which n x is a set of pure strategies of the n -th player, nk is its payoff -matrix by and for the set of indices   1 n i i j j   ,  1,i ij m 1,i n  . (2) the non-repeatability means that fncg (1) models a process, in which the player upon one’s decision is made is able to implement it only once (or a few times at most). suppose that, in fncg (1),   1 q q q e e   is a set of efficient nash equilibria in pure strategies, where (a refinement is needless if 1q  , i. e. there is a single equilibrium): by . (3) pure strategy efficient nash equilibrium (3) produces a payoff aggregate (4) refinement of acyclic-and-asymmetric payoff aggregates of pure efficient nash equilibria 181 whose n elements are the respective elements of n payoff matrices . (5) nash equilibria, which are strictly and non-strictly dominated by other nash equilibria, bear no utility (in fact, they are inefficient). this is why they are not further considered (harsanyi & selten, 1988; liu & forrest, 2010; osborne, 2003; romanuke, 2018b; romanuke, 2018a; vorob’yov, 1985; vorob’yov, 1984). all pure strategy efficient nash equilibria in fncg (1) with payoff matrices (5) constitute a subset of all pure strategy situations which contain equilibrium strategies of every player: (6) by 1,n n  , (7) where the indices’ subsets (8) are such that for every element of set  1,q q  such that , 1,n n . it should be noted that every element of set is not necessarily an equilibrium point. this means that some aggregates may not be the equilibria – see an example sketch in figure 1 and also refer to figure 1 in (romanuke, 2018b). theoretically, inefficient equilibria make no sense. a situation producing payoffs 0  by some  1,m n (9) cannot be an efficient nash equilibrium, whichever situation producing payoffs (4) is. however, if the weak pareto efficiency is acceptable to supplement the set of the efficient nash equilibria, then both situations producing payoffs (4) and (9) are efficient, if the situation (4) is an efficient nash equilibrium. in general, the weak efficiency, if accepted, implies that a situation producing payoffs 0 n    for all    1, \n n m by some  1,m n (10) is efficient as well (kumano, 2017; marden, 2017). nevertheless, if nash equilibria are weakly efficient in bimatrix games, then they are fairly senseless due to the player whose payoff is decreased by  will avoid one’s weakly efficient strategy. in romanuke/decis. mak. appl. manag. eng. 4 (2) (2021) 178-199 182 figure 1. the efficient nash equilibria set (highlighted via dashed rectangles) and its relation (shown by arrows) to subsets (7) in a trimatrix 7 4 5  game over a player’s payoff three-dimensional matrix of size 7 4 5  (the view is the same for every player); see the similar visualizations in (romanuke, 2018b; romanuke, 2018a) trimatrix games, where the situation producing payoffs (11) is an efficient equilibrium, a weakly efficient situation producing payoffs , or , or (12) by pretty great 1  and 2  , or 1  and 3  , or 2  and 3  , respectively, is fairly senseless as well. however, if   3 1r r  are sufficiently (or even negligibly) small, weakly efficient nash equilibria producing payoffs (11) and (12) may become (more) significant. for fncgs with four players and more, weak efficiency of nash equilibria gets more important, especially if payoffs (9) differ from efficient payoffs (4) by a negligible  . 4. properties of payoff matrices and distinguishability of efficient nash equilibria first, none of n payoff matrices (5) of fncg (1) can have elements, whose values are the same. in particular, matrices (5) of fncg (1) cannot be null matrices. in addition, these matrices cannot contain strictly dominated  1n  -dimensional slices. otherwise, if there are strictly dominated  1n  -dimensional slices, the slices are deleted. for example, a 3 2 2  game with matrices (a similar example can be found in (romanuke, 2018b) but matrices’ values here have been slightly changed) refinement of acyclic-and-asymmetric payoff aggregates of pure efficient nash equilibria 183 1 7 4 6 2 5 5 3 6 6 3 5 1                           k , 2 7 6 4 3 7 6 5 7 6 7 9 6                           k , 3 7 6 6 6 4 8 4 8 8 9 7 8                           k (13) is reduced to the 2 2 2  game with payoff 2 2 2  matrices 1 7 4 6 2 5 5 3 6                 k , 2 7 6 4 3 7 6 5 7                 k , 3 7 6 6 6 4 8 4 8                 k . (14) indeed, the third slice of matrix 1 k in (13) is strictly dominated by its first slice (the first row values are greater than the third row values), and so the first player will never use one’s pure strategy which is strictly dominated. there are two efficient nash equilibria in the trimatrix game with payoff matrices (14):  1 1 1 1, ,e x y z and  2 2 2 2, ,e x y z . these equilibria produce payoffs  7, 7, 7 and  6, 7, 8 , respectively. note that the collective payoffs are the same. meanwhile, payoffs  6, 7, 8 are produced by the situation involving the non-strictly dominated second slice of matrix 3 k (where the left flat submatrix non-strictly dominates the right flat submatrix). another peculiarity is that the second player does not care about difference of one’s payoffs at these equilibria, whereas equilibrium 2 e imparts a little advantage to the third player (as opposed to the first player). thus, equilibrium 1 e is more attractive for the first player, and equilibrium 2 e is more attractive for the third player. the strength of the attractiveness is the same. distinguishability of efficient nash equilibria is a principally important property before considering possible refinement. identical payoff aggregates produced in symmetric situations are non-distinguishable. an example is a trimatrix 2 2 2  game with payoffs matrices (romanuke, 2018b) 0 0 1 0 0                       k and 0 0 2 0 0                       k and 0 0 0 3 0 0 0                       k by 0    , 0   , 0    . (15) unlike the slices (left and right flat submatrices) of matrices 1 k and 2 k , the slices (left and right ones) of matrix 3 k are different. this game has two efficient nash equilibria, each of which produces the same payoff triplet  , ,   . thus, these equilibria are absolutely non-distinguishable. therefore, cyclicity and symmetry of payoff aggregates (with respect to situations in which they are produced) is opposed to distinguishability of efficient nash equilibria. so, only refinement based on acyclicand-asymmetric payoff aggregates is possible. 5. maximultimin and superoptimality for refining efficient nash equilibria in fncgs considering the inclusion of the set of the efficient nash equilibria in (6), fncg can be reduced to an fncg defined on product . in the reduced fncg, payoff matrices are romanuke/decis. mak. appl. manag. eng. 4 (2) (2021) 178-199 184 at (16) by 1,r n  and re-indexing with a set of corresponding indices. if matrices (16), being the corresponding submatrices of matrices (5), still have n dimensions, then they constitute the reduced game by (7) and (8). the uncertainty amongst the equilibria to be refined (or selected) is still the same as it is for the initial fncg (1). to reduce the uncertainty, the maximin principle can be applied to guarantee the corresponding “no-less-than” payoffs for the players. for games, this principle is the maximultimin being a generalization of maximin and maximinimin by romanuke (2018a, 2018b): by , 1,r n (17) and re-indexing with a set of corresponding indices. set (17) guarantees that the r -th player gets a payoff not less than , 1,r n . (18) however, not all situations in the set (19) are efficient nash equilibria. moreover, set (19) may not contain any equilibria. if set (20) is nonempty then it contains the refined efficient nash equilibria. in particular, if set (20) is a singleton (in other words, it contains just a single element), then the refinement is finished, and the single element is the single efficient nash equilibrium. if r   or 1r  then the superoptimality rule originally introduced to distinguish optimal player strategies (game situations) in matrix games (e. g., see romanuke, 2010a) can be applied just as well as it is applied for bimatrix and trimatrix games in (romanuke, 2018a) and (romanuke, 2018b), respectively. if set r   then using strategies from subsets (17) involves players into an unstable (wandering) process: the players will search for new pure strategies beyond these subsets for every game round (as there is no a single equilibrium). to guarantee the best payoffs for players under uncertainty of the efficient equilibria in this case, one of the best actions is to use strategies from subsets refinement of acyclic-and-asymmetric payoff aggregates of pure efficient nash equilibria 185 by . (21) subsets (21) are those guarantors. this uncertainty reduction concerns as fncg (1), as well as the reduced fncg with payoff matrices (16), whether matrices (16) have n dimensions or fewer. in the case of 1r  there are at least two equilibria in product (19) of maximultimin subsets (17). thus, we still have an uncertainty of which equilibrium to select. let (22) by and 1,r n  and the respective indices’ subsets by which . (23) if then, according to superoptimality, by (24) and re-indexing with a set of corresponding indices, 1,r n . otherwise, if , then and . note that finding sets (24) does not guarantee that * 1 n r r x r            , (25) where a case * 1 1 n r r x r           (26) is as ideal as the case 1r  . statement (25) is only assuredly true for a case, when for all    01, \r n r , and or . (27) romanuke/decis. mak. appl. manag. eng. 4 (2) (2021) 178-199 186 hence, the refinement is finished by processing the reduced game with payoff matrices (16), wherein primarily maximultimin subsets (17) are found along with maximultimin payoffs (18). the process of refining is continued to payoff aggregate maximizations if 1r  . the refinement is counted perfectly finished when 1r  or (26) is true. if set (19) is a singleton and it contains a single efficient equilibrium, then using strategies from other situations is unreasonable. in this case, it can be said that product (19) of the maximultimin subsets has perfectly responded, and the players will stick to the single efficient equilibrium. however, if there are at least two equilibria with different payoffs, the single efficient equilibrium is simultaneously unprofitable (or disadvantageous) for 0 n players, where  0 1, 1n n  . the nonprofitability (disadvantageousness) follows from the pareto efficiency definition. suppose that a player, who loses some payoff at the single refined efficient equilibrium, tries to increase one’s payoff. the payoff increment is possible by when other players do not change their strategies (which are still profitable for them) upon the equilibrium is changed. consequently, the payoff increment is unlikely even for a few rounds of the game. this attitude can be re-formulated in the terms of the noncooperative game equilibrium: while at the nash equilibrium a player cannot improve one’s payoff by acting oneself, at the single refined efficient equilibrium it is unlikely for a player to increase one’s payoff at least for a few rounds of the game (barelli & duggan, 2015; kumano, 2017; romanuke, 2010b). thus the refined efficient equilibrium called the metaequilibrium becomes an attractive point for all the players (romanuke, 2018b), although with probably different strengths of the attraction. obviously, there can be a few refined efficient equilibria. despite such metaequilibria are considered nonrefinable, the refinement may have a positive impact by some conditions that will be explained in the section right below. 6. a generalized algorithm of refinement in fncgs the stated approach of maximultimin and superoptimality for refining efficient nash equilibria in fncgs with subsets (17), (21), or (24) does not guarantee the best outcome, i. e. that there will be a single metaequilibrium. in general, this approach gives us one of the seven possible final outcomes: 1. the refinement is quite impossible (it totally fails) when r   and . (28) both the maximultimin principle and superoptimality rule miss all the equilibria here. this is the case when both the maximultimin and superoptimality do not work at all. this is the worst case. 2. a few metaequilibria are returned when r   but . (29) this outcome has two mutually exclusive interpretations. if (30) refinement of acyclic-and-asymmetric payoff aggregates of pure efficient nash equilibria 187 then the uncertainty amongst the equilibria is nonetheless partially reduced. otherwise, if (31) then the superoptimality rule hits the whole set of the equilibria. clearly, this does not reduce the uncertainty amongst the equilibria as those metaequilibria remain the same as all those efficient equilibria. so, this is a fail of the refinement, where the superoptimality rule works in vain. 3. a few metaequilibria are returned when 1r  and equality (26) is false by * 1 n r r x r            . (32) such an outcome has two mutually exclusive interpretations. if r e , that is 1 r e  (33) here, then the uncertainty of equilibria is nonetheless partially reduced. otherwise, if r e by (32), then the superoptimality rule does not hit an equilibrium, whereas the maximultimin hits the whole set of the equilibria (factually, this is a fail of the refinement, where the maximultimin principle works in vain). there are r metaequilibria anyway, found by just subsets (17). 4. a few metaequilibria are returned when 1r  and equality (26) is false by * 1 1 n r r x r           . (34) such an outcome has two mutually exclusive interpretations also. if * 1 1 n r r x r e            (35) then the uncertainty of equilibria is nonetheless partially reduced (either owing to maximultimin or superoptimality). otherwise, if * 1 n r r x r e           (36) then both the maximultimin and superoptimality hit the whole set of the equilibria. as those metaequilibria remain the same as all those efficient equilibria, the uncertainty amongst the equilibria is not reduced at all. so, this is a fail of the refinement, where both the maximultimin and superoptimality work in vain. 5. a single metaequilibrium is returned when 1r  by finding just subsets (17). this is the perfect and fastest outcome of the refinement. 6. a single metaequilibrium is returned when 1r  and equality (26) is true. here, subsets (17) and (24) are involved. this is the perfect refinement outcome as well. 7. a single metaequilibrium is returned when r   but romanuke/decis. mak. appl. manag. eng. 4 (2) (2021) 178-199 188 . (37) although the maximultimin principle misses all the efficient equilibria, the superoptimality rule perfectly hits the single metaequilibrium. therefore, a generalized algorithm of refinement in fncgs under uncertainty consists of four key branches (figure 2): 1r  start find maximultimin subsets (17) false true intersection r by (20) contains a single equilibrium false true find subsets (23) find subsets (21) the refinement totally fails return return a single metaequilibrium is in set (20) false true intersection (38) is nonempty find subsets (24) false true equality (37) holds return a single metaequilibrium is in set (38) return uncertainty of equilibria has been partially reduced: the players’ metaequilibrium strategies are in intersection (38) false true inequality (30) holds a fail of the refinement return false true equality (26) holds return a single metaequilibrium is in set (39) return uncertainty of equilibria has been partially reduced: the players’ metaequilibrium strategies are in intersection (39) false true inequality (35) holds a fail of the refinement return false true intersection (39) is nonempty return uncertainty of equilibria has been partially reduced: the players’ metaequilibrium strategies are in intersection r by (20) false true inequality (33) holds a fail of the refinement return figure 2. an algorithmic generalized scheme for the nash equilibria refinement in fncgs (romanuke, 2018b; romanuke, 2018a) refinement of acyclic-and-asymmetric payoff aggregates of pure efficient nash equilibria 189 1. find maximultimin subsets (17) over payoff matrices (16). 2. return a single metaequilibrium if 1r  , i. e. product (19) of maximultimin subsets (17) contains the single equilibrium. 3. if r   then find subsets (21) which maximize the respective players’ payoffs over subsets (7). finally, return the resulting set , (38) whether containing metaequilibria or not. 4. otherwise, if 1r  then maximize the respective players’ payoffs over subsets (23), and find those of subsets (24) (by 1,r n ) whose corresponding subsets (23) contain more than one strategy. finally, determine a set * 1 n r r x r           . (39) if set (39) is not empty, return the resulting metaequilibrium or metaequilibria in set (39). more precisely, if to distinguish the partial equilibrium uncertainty reduction and the refinement fail in outcomes ## 2 – 4, the number of the refinement possible outcomes is 10. they all are highlighted with three different colors in figure 2. statistics of those outcomes bear some similarity at least for bimatrix and trimatrix games by increasing the number of players’ pure strategies (romanuke, 2018b). statistics for other fncgs will be revealed in the section right below. 7. statistics of refinement for getting more real examples, fncgs are simulated by a pseudorandom matrix generator suggested in (romanuke, 2018b). according to the generator, we take 1 n r r m   payoff matrix equal to    1 n r r b m     (40) by a function   1 n r r m   returning a pseudorandom 1 n r r m   matrix whose elements are drawn from the standard uniform distribution on the open interval  0; 1 and a function    returning the integer part of number  , where 0b  , 0  . constants b and  are taken such that the payoffs would be moderately scattered (leinfellner & köhler, 1998; leyton-brown & shoham, 2008; myerson, 1997; romanuke, 2018b; vorob’yov, 1985; vorob’yov, 1984). the purpose is to count statistics of refinement along with how many fncgs need refinement and need not, having either a single equilibrium or no equilibria at all. for obtaining confident results, each type of fncg will be generated for 100,000 times. generator (40) will be used to produce the same number of pure strategies for players. such simplification does not influence much on common inferences from romanuke/decis. mak. appl. manag. eng. 4 (2) (2021) 178-199 190 results of the simulation. bimatrix games are generated by   1 110 , 40m m   for 1 2, 10m  . (41) so, statistics for bimatrix games will be re-examined for 2 2 up to 10 10 games. trimatrix games are generated in the same way:   1 1 110 , , 40m m m   for 1 2, 10m  . (42) processing fncgs with a greater number of players consumes much more resources, so their topmost number of pure strategies is decreased:   1 1 1 110 , , , 40m m m m   , 1 2, 8m  , (43)   1 1 1 1 110 , , , , 40m m m m m   , 1 2, 7m  , (44)   1 1 1 1 1 110 , , , , , 40m m m m m m   , 1 2, 6m  , (45) for fncgs with  4, 5, 6n  players, respectively. the statistics of refinement should reflect the number of fncgs which include the following features: 1. have no nash equilibria. 2. have a single equilibrium. 3. need refinement. 4. a single metaequilibrium is returned by maximultimin in set (20). 5. a single metaequilibrium is returned by superoptimality in set (38). 6. the refinement totally fails. 7. the uncertainty of equilibria is partially reduced owing to superoptimality: a few metaequilibria are returned in set (38). 8. a fail of the refinement: maximultimin does not hit an equilibrium, although superoptimality hits all the equilibria at once. 9. a single metaequilibrium is returned by superoptimality in set (39). 10. the uncertainty of equilibria is partially reduced owing to maximultimin: a few metaequilibria are returned in set (20), whereas superoptimality does not hit an equilibrium. 11. a fail of the refinement: maximultimin hits all the equilibria at once, whereupon superoptimality does not hit an equilibrium. 12. the uncertainty of equilibria is partially reduced: a few metaequilibria are returned in set (39). 13. a fail of the refinement: maximultimin hits all the equilibria at once, as well does superoptimality. additional commentaries to the list required for unambiguous interpretation are as follows. those fncgs counted by feature #3 have two or more efficient equilibria. feature #5 implies that r   , i. e. maximultimin does not hit an equilibrium. feature #6 implies that r   and equality (28) holds (maximultimin does not hit an equilibrium, nor does superoptimality as well), so both sets (20) and (38) are empty. maximultimin does not work in features #7 and #8 (again, r   ). statistics of refinement for bimatrix games generated by (41) are shown in figure 3. whichever the number of pure strategies is, features ## 7 – 12 are statistically negligible. meanwhile, as this number increases, the number of fncgs having a single equilibrium decreases (feature #2) along with the increasing number of fncgs wherein the refinement is needed (feature #3). fail cases, when both maximultimin and superoptimality do not hit an equilibrium (feature #6), refinement of acyclic-and-asymmetric payoff aggregates of pure efficient nash equilibria 191 considerably increase. eventually, they constitute a significant part for 10 10 games. the fail of the refinement caused by the opposite event (feature #13) grows but much slower. the superoptimality rule hits the single metaequilibrium quite rarely (feature #5). nonetheless, maximultimin does that effectively growing up as 1 m increases. the ratio of the single-metaequilibrium cases (features ## 4, 5, 9) to the refinement fail cases (features ## 6, 8, 11, 13) varies from 1.2 to 2.04, where the worst refinable bimatrix games are 9 9 ones (54.6 % of successful refinement into single-metaequilibrium), and the best refinable bimatrix games are 3 3 ones (67.1 %). considering positively also the partial reduction of the equilibria uncertainty (adding features ## 7, 10, 12), the ratio varies from 1.22 to 2.05: the worst and best refinable 9 9 and 3 3 bimatrix games respectively constitute 55 % and 67.2 % of the successful refinement. figure 3. statistics of refinement for bimatrix games generated by (41) statistics of refinement for trimatrix games generated by (42) and shown in figure 4 resemble that in figure 3. however, a few differences are distinctly visible. the increasing number of fncgs, wherein the refinement is needed (feature #3), is from 30.2 % to 192.6 % greater. fail cases (feature #6) are stronger, whereas the fail of the refinement caused by the opposite event (feature #13) slowly descends. maximultimin and superoptimality separately hit the single metaequilibrium (feature #4 and #5) by 61.7 % better. the ratio of the single-metaequilibrium cases (features ## 4, 5, 9) to the refinement fail cases (features ## 6, 8, 11, 13) varies from 1.06 to 2.09, where the worst refinable trimatrix games are 10 10 10  ones (51.5 % of successful refinement into single-metaequilibrium), and the best refinable trimatrix games are 2 2 2  ones (67.7 %). adding the partial reduction of the equilibria uncertainty (features ## 7, 10, 12), the ratio varies from 1.08 to 2.13: the worst and best refinable 10 10 10  and 2 2 2  trimatrix games respectively constitute 52 % and 68.1 % of the successful refinement. on average, trimatrix games have better refinability (considering features ## 4, 5, 7, 9, 10, 12) by 63.3 %. 1 2 3 4 5 6 7 8 9 10 11 12 13 23 45 67 89 10 0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000 55000 60000 7 8 9 10 11 12 2 3 4 5 6 7 8 9 10 0 50 100 150 200 250 300 1 m number of fncgs features 1 m features romanuke/decis. mak. appl. manag. eng. 4 (2) (2021) 178-199 192 statistics for fncgs with four players (figure 5) much differ from those with five (figure 6) and six players (figure 7). fncgs with 4n  are only 2.87 % refinable better than trimatrix games. then, nevertheless, refinability of fncgs worsens from 4n  to 5n  by 29.1 %, and from 5n  to 6n  by 63.5 %. figure 4. statistics of refinement for trimatrix games generated by (42) figure 5. statistics of refinement for fncgs with 1 1 1 1 m m m m   payoff matrices generated by (43) 1 2 3 4 5 6 7 8 9 10 11 12 13 23 45 67 8 0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000 55000 60000 1 m number of fncgs features 1 2 3 4 5 6 7 8 9 10 11 12 13 23 45 67 89 10 0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000 1 m number of fncgs features 1 m features 7 8 9 10 11 12 2 3 4 5 6 7 8 9 10 0 50 100 150 200 250 300 350 400 450 500 550 refinement of acyclic-and-asymmetric payoff aggregates of pure efficient nash equilibria 193 figure 6. statistics of refinement for fncgs with 1 1 1 1 1 m m m m m    payoff matrices generated by (44) figure 7. statistics of refinement for fncgs with 1 1 1 1 1 1 m m m m m m     payoff matrices generated by (45) 1 2 3 4 5 6 7 8 9 10 11 12 13 2 3 4 5 6 0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000 55000 60000 65000 70000 75000 80000 85000 90000 1 m number of fncgs features 1 2 3 4 5 6 7 8 9 10 11 12 13 2 3 4 5 6 7 0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000 55000 60000 65000 70000 75000 1 m number of fncgs features romanuke/decis. mak. appl. manag. eng. 4 (2) (2021) 178-199 194 a conspicuous fact of fncgs by  4, 5, 6n  is that feature #11 starts badly growing. finally, 85.5 % of 6 6 6 6 6 6     games needing refinement come with the fail: maximultimin hits all the equilibria at once, whereupon superoptimality does not hit an equilibrium. they have only 10 % successful refinement including the uncertainty partial reduction. dyadic games with that many players, i. e. 2 2 2 2 2 2     games, have 60.2 % of that measure, though. in general, except for 2 2 bimatrix games, dyadic games are the best refinable. their ratio of the successful refinement decreases from 68.1 % down to 60.2 %. in total ratio, the part of successful refinement decreases: it is 58.6 %, 57 %, 54.4 %, 39.9 %, 26.1 % for 2, 6n  . 8. discussion obviously, the presented theory does not assure a perfect refinement. a total number of fails increases as fncg (1) gets of a bigger size. the perfect outcome (features ## 4, 5, 9) strongly depends on the key parameters of an fncg: the number of players and the players’ numbers of pure strategies. that way how payoff matrices are given influences also, but a regular structure of payoff scattering, like the considered (40), is far less influential. if payoffs are non-moderately (illogically) scattered then a single metaequilibrium, if even it exists, may be significantly disadvantageous for one or more players (romanuke, 2018b). this disadvantage, meaning “someone always has less”, is common for the known refinement concepts. bimatrix, trimatrix, and 1 2 3 4 m m m m   games are the most widespread ones in practice. and for them the stated approach of maximultimin and superoptimality for refining efficient nash equilibria works very good. based on the statistics by generators (41) – (45), the best refinability (considering only features ## 4, 5, 7, 9, 10, 12; do not confuse it with the part of successful refinement) is expected to be a distinctive property of trimatrix games and fncgs with four players (whose numbers of pure strategies do not differ much from each other). another merit is a possibility to partially reduce the uncertainty of equilibria by achieving one of conditions (30), (33), (35). however, if payoffs are not scattered illogically, even a partial reduction may lead to an eventually successful single-metaequilibrium refinement. consider an example of a trimatrix 8 6 3  game (figure 8) generated by (42). this game has five equilibria, four of which are efficient:   4 1 q q e e            15 26 32 18 21 32 18 26 32 18 23 33, , , , , , , , , , ,x x x x x x x x x x x x and their respective payoff aggregates are  49, 48, 48 ,  49, 47, 49 ,  49, 47, 49 ,  47, 49, 48 . equilibrium  16 21 34, ,x x x producing payoff aggregate  49, 45, 44 is nonstrictly dominated by equilibria  1 2 3, ,e e e . subsets (7) are , , . (46) maximultimin subsets (17) are as follows: refinement of acyclic-and-asymmetric payoff aggregates of pure efficient nash equilibria 195 45 46 40 48 42 42 45 44 46 41 44 42 49 43 42 41 46 48 46 43 48 46 43 49 48 43 44 43 48 47 46 42 41 47 47 41 43 42 40 48 44 48 42 44 40 41 47 47 41 41 45 41 41 41 47 47 46 49 41 43 44 41 42 43 46 46 46 44 49 40 45 46 40 46 46 41 48 49 41 40 44 49 46 42 45 44 44 48 48 46 48 44 41 49 47 45 44 43 42 43 45 42 45 44 40 40 47 41 41 48 47 47 45 49 42 48 41 47 47 43 47 45 47 47 46 40 44 46 40 44 47 49 43 41 46 45 44 44 45 43 44 46 43 47 45 40 45 41 42 49 48 40 41 43 44 40 41 46 42 40 42 42 44 47 43 41 49 41 49 47 42 42 42 41 41 43 48 42 48 42 46 49 49 43 41 40 40 40 46 45 43 44 41 48 47 46 47 41 43 41 47 48 40 44 49 42 49 41 49 45 44 46 48 44 43 49 49 44 49 45 49 49 42 42 47 41 44 40 45 43 48 48 43 43 43 42 41 41 48 48 42 42 44 40 49 42 40 43 46 49 46 40 46 40 43 49 42 48 44 40 45 45 47 41 40 40 42 42 44 45 48 46 43 40 41 45 46 43 41 48 42 43 49 47 48 47 45 45 40 44 44 41 47 40 46 49 43 44 46 48 47 48 49 49 43 48 41 46 47 45 45 47 46 45 48 48 40 45 42 42 43 46 44 44 47 40 47 47 41 42 47 43 46 45 41 45 48 44 40 43 40 44 40 44 41 47 49 42 48 49 49 46 46 43 49 40 43 48 47 46 41 47 44 47 44 45 45 42 40 45 43 48 44 42 40 41 41 49 48 44 47 49 42 41 46 45 46 44 46 40 45 47 44 47 41 41 40 41 43 47 44 44 47 48 49 44 48 43 45 41 40 43 45 41 48 46 41 47 40 43 47 40 45 41 46 45 41 47 41 42 45 47 41 47 43 45 42 48 47 41 48 41 43 45 45 42 48 43 43 46 48 49 46 47 44 42 47 41 40 43 46 47 49 45 49 40 41 49 46 46 49 42 40 47 41 43 47 43 42 46 40 45 47 45 42 40 42 42 43 42 45 49 49 44 47 45 46 43 48 44 49 40 44 45 46 44 48 42 43 45 41 48 46 48 42 48 43 45 45 46 45 48 40 44 48 48 46 40 47 43 41 49 49 42 44 49 44 42 41 46 47 45 42 44 43 40 40 43 43 40 47 48 44 48 48 41 41 43 45 49 41 42 43 46 44 46 41 40 46 40 41 40 46 49 43 42 48 49 44 40 48 46 45 48 40 41 44 41 45 40 42 43 44 40 48 48 48 46 45 45 47 40 43 42 42 47 49 48 45 40 41 48 44 49 43 43 44 48 45 41 42 42 48 41 40 42 45 43 40 47 40 46 48 42 42 46 40 47 47 41 43 45 42 48 42 42 45 43 44 41 40 44 48 49 44 43 47 42 46 42 47 40 41 43 40 43 45 44 44 40 40 48 44 43 41 46 41 44 40 46 49 46 41 44 46 44 42 41 41 45 43 45 45 42 48 43 49 42 43 48 40 49 42 45 48 43 45 41 44 44 43 42 46 48 40 40 42 46 46 40 figure 8. a stack of three payoff 8 6 3  matrices of a trimatrix 8 6 3  game, where efficient payoffs are highlighted bold (the second and third columns referring to the respective pure strategies of the third player) , , . so, here set (20) is     15 26 32 18 26 32, , , , ,r x x x x x x e  (47) and case (27) is true. now, by the superoptimality rule, there is a summing over two singletons in (24): , that still gives us two metaequilibria  15 26 32, ,x x x and  18 26 32, ,x x x in set (47) with their respective payoffs  49, 48, 48 and  49, 47, 49 (figure 9). consequently, this is outcome #4 and feature #12, wherein inequality (35) is true 3 * 1 1 2 4 r r x r e              . (48) romanuke/decis. mak. appl. manag. eng. 4 (2) (2021) 178-199 196 despite the superoptimality rule works here in vain, maximultimin allows to have reduced the uncertainty of equilibria, according to (48), twice as less. 44 40 49 43 46 44 49 49 49 42 47 40 45 40 48 44 40 49 47 40 47 49 49 49 49 45 48 44 43 48 49 43 49 42 48 41 figure 9. a stack of three 2 3 2  submatrices of matrices in figure 8, which are defined on a product of subsets (46), where payoffs at metaequilibria are highlighted squared at first view, it seems that the obtained result with two equilibria is not yet perfect refinement. however, the first player will not care of either of those two equilibria. and, nonetheless, the second player is guaranteed a payoff of 47, and the third one is guaranteed a payoff of 48. furthermore, the second and third players do not have a choice. that is a perfect result of decision-making. while refining, a researcher should be aware of that both numbers of games without pure nash equilibria and having a single pure nash equilibrium decrease as fncg (1) gets of a bigger size. at the same time, the number of games having multiple efficient equilibria (feature #3) increases up to fncgs with six players. multiple equilibria are more probable in bigger size games. these factors define necessity of refinement, which strengthens for bigger fncgs. but despite the refinement statistics by figures 3 – 7 are still weakly promising, notice that generators (41) – (45) are yet “pessimistic” giving us only integer-valued repetitions of a set of payoffs (for instance, see them in figure 8). therefore, those percentages in the range of successful refinement (58.6 %, ..., 26.1 %) are expected to be higher for fncgs which model real-world practice interactions (such fncgs have a way less repeating payoffs). 9. conclusion finding maximultimin subsets (17) and either superoptimality subsets (21) or (24) constitute the core of the theory of refining pure strategy efficient nash equilibria in fncgs under uncertainty. this theory exploits the maximin, expanded via the maximinimin to maximultimin principle, and a superoptimality rule wherein minimization is substituted with summation. the maximultimin rule guarantees definite payoffs for players. the superoptimality rule stands like a backup plan, and it is enabled if maximultimin cannot produce just a single refined efficient nash equilibrium (now-called a metaequilibrium). an average effectiveness of refinement by maximultimin and superoptimality, which is exemplarily visualized in figures 3 – 7 (for a pessimistic approach analysis), appears satisfactory. the theory reckons on dealing with efficient equilibria producing acyclic-andasymmetric payoff aggregates. non-distinguishable equilibria produce identical payoff aggregates, and also aggregates with mirror-like symmetry and cyclic symmetry. such equilibria cannot be principally refined. however, this initial nonrefinability is quite rare in practice due to payoff matrices are hardly ever estimated “symmetrically”. refinement of acyclic-and-asymmetric payoff aggregates of pure efficient nash equilibria 197 nominally, the stated approach to the refinement is a contribution to the field of the equilibria refinement game theory. the contribution still can be advanced for cases when the uncertainty amongst the equilibria is reduced only partially, that is when one of conditions (30), (33), (35) holds. in a way, the reduction can be evaluated as a ratio of the number of initial efficient equilibria to the number of metaequilibria. for instance, this ratio is 2 in the example with inequality (48). if the refinement fails, a method of finding approximate nash equilibrium situations with possible concessions can be attached and used (romanuke, 2016b; romanuke, 2016a) for the case of situations in pure strategies. for this, the smallest possible concessions are to be found to refine equilibria at least partially. at that, both cases of refinement fails and non-distinguishable equilibria may be rectified. funding: this research received no external funding. acknowledgments: this work was technically supported by the faculty of navigation and naval weapons, polish naval academy, poland. conflicts of interest: the author declares no conflicts of interest. references bajoori, e., flesch, j., & vermeulen, d. 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(1985). game theory for economist-cyberneticians. moscow: nauka. © 2021 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 4, issue 1, 2021, pp. 19-32. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame2104019k * corresponding author. e-mail addresses: ckaramasa@anadolu.edu.tr (ç. karamaşa), edemir@pirireis.edu.tr (e. demir), salih.memis@giresun.edu.tr (s. memiş), selcuk.korucuk@giresun.edu.tr (s. korucuk). weighting the factors affecting logistics outsourcing çağlar karamaşa1*, ezgi demir2, salih memiş3 and selçuk korucuk3 1 anadolu university faculty of business department of business administration, turkey 2 piri reis university faculty of economics and administrative sciences department of management information systems, turkey 3 giresun university bulancak kadir karabaş vocational school department of international trade and logistics, turkey received: 20 september 2020; accepted: 30 october 2020; available online: 9 november 2020. original scientific paper abstract: today, growing and changing competitive conditions, products, and services, free movement of labor, and businesses with the information they develop strategies that create value to obtain a competitive advantage. now, final buyers have the convenience of purchasing the products they demand with the features and conditions they want and at the price they accept. in such an environment, businesses use their supply chain and logistics activities more effectively and efficiently than their competitors. today, achieving a strategic superiority in a global market where the content and quality of the products are the same is only possible by delivering the desired products to the customer at the desired price, at the desired time, in the desired amount, through the right channel, as quickly as possible and without any damage. in such a situation, the desire to focus on the main activities of the enterprises, the need for effective logistics operations, etc. logistics outsourcing has increased rapidly for reasons. businesses can carry out logistics activities requiring expertise thanks to third party logistics (3pl) service providers in the field such as transportation, storage, customs clearance, without investing in logistics. for logistics outsourcing to be beneficial, a correct logistics service provider must be selected under the needs of the business. selecting the right logistics service provider is important in increasing the benefit of outsourcing. in this study neutrosophic ahp was used to prioritize the factors. key words: logistics, outsourcing, neutrosophic ahp. mailto:ckaramasa@anadolu.edu.tr mailto:edemir@pirireis.edu.tr mailto:salih.memis@giresun.edu.tr mailto:selcuk.korucuk@giresun.edu.tr karamaşa et al./decis. mak. appl. manag. eng. 4 (1) (2021) 19-32 20 1. introduction nowadays, with the effects of the competitive environment and the effect of globalization, businesses transfer their work areas other than their main products to the enterprises that carry out their main activity products under the name of outsourcing to reduce costs and focus on their core competencies. in this way, businesses develop that product by focusing on their main products and at the same time, they can carry outside activities more systematically with the help of specialized enterprises. one of the issues where businesses transfer business to a structure outside the business with the help of outsourcing other than their main products is logistics services. businesses that feel the intensity of competition are extremely strong and think that it is difficult to allocate resources and time for logistics elements, get help from logistics companies specialized in their field to carry out one or more of their activities (such as warehousing, transportation, and inventory management) and thus, this situation provides them to become professional. logistics services have been transferred to businesses that provide 3pl services to provide better quality and less cost. at this point, it is important for businesses that will receive 3pl services to be able to select and eliminate the companies that provide this service and to make a decision to agree with the right one. the selection process of the 3pl business has played an important role in determining the performance of logistics activities. if a 3pl business that is not suitable for the business is selected, the quality of the logistics service is low, the efficiency of the logistics activities decreases, the customer and the business are damaged so there is a disconnection between the 3pl business and the customer and the business, etc. serious problems may be encountered. due to these problems, the customer can reduce the reputation of the business in the sector and lead to the loss of trust in the business. in the face of increasing competition with globalization and rising customer expectations, businesses that produce products and services focus on their main abilities and skills by leaving their logistics activities to 3pl. during this process, the relations between the logistics service provider and the enterprises receiving the service have come to the fore regarding service standards. the relationships that businesses establish with logistics service providers contribute to the increasing efficiency of the buyer businesses in operational and financial matters. the purpose of this study is to determine and weigh the factors of outsourcing in logistics companies operating in giresun. for the solution of these extremely complicated and complex elements, neutrosophic ahp as a multi-criteria decision making method was used. in the following part of the study, the literature has been reviewed, and the methods applied in the third part has been explained. in the fourth part, the method of the study has been applied to the problem, and in the last part, the study has been ended by making suggestions regarding the results and future studies. 2. literature review if an enterprise chooses the external source from which it will receive services for the realization of its logistics activities and transfer its activities to it, it will be important that it starts by choosing 3pl enterprises that are specialized in their fields. in recent years, there are many logistics service providers and they provide advantages to their customers by effectively performing logistics activities in today's competitive environment. while choosing the 3pl businesses, the business manager, who will receive logistics services, handles all other units of the business and chooses weighting the factors affecting logistics outsourcing and selecting the most ideal company 21 the 3pl business that will provide the best integration to this process, suitable for the technological infrastructure, offer the most advantages, and will add positive values to the reputation and financial power of the enterprise. when the literature has been examined; dapiran et al. (1996) and bhatnagar et al. (1999) have revealed that service delivery and price are the most important factors for outsourcing criteria. boyson et al. (1999) stated three headings as service costs, customer service, and financial stability as the most important criteria. petroni and braglia (2000) introduced different criteria such as reputation, geographical location, financial stability, experience, technological competence, flexibility, production capacity, and management competence. menon et al. (1998) suggested nine criteria for 3pl evaluation and selection, such as price, timely delivery, error rate, financial stability, creative management, fulfillment or exceeding promises, the presence of senior management, responding to unforeseen problems, meeting performance and quality requirements. prater and ghosh (2005) determined that smes operate abroad with the need to follow their customers in their research. besides, the international communication problem between overseas facilities poses a major obstacle for smes. another finding obtained from the research is that smes engaged in operational commercial activities in european countries, especially in germany, started to be interested in environmental issues such as reverse logistics. bottani and rizzi (2006) developed the fuzzy topsis approach in their studies and determined nine criteria such as compatibility, financial stability, service flexibility, performance, price, physical equipment, and information systems, quality, strategic attitude, trust, and justice to select the most suitable 3pl business. it has been worked on research by arbore and ordanini (2006), in which the outsourcing strategies of smes were examined in terms of the size of the enterprise and the geographical region to which these enterprises are affiliated. in this research, it has been determined that the size of the enterprise is a relative dimension in the choice of outsourcing strategy for smes in terms of internal resources. işıklar et al. (2007) used an integrated approach combining cbr, rbr, and consensus programming to address the 3pl selection problem. the evaluation criteria include cost, quality, technical ability, financial stability, successful track record, service category, personnel qualification, information technology, comparable culture, region, etc. jharkharia and shankar (2007) used the anp approach in their study to select the most suitable 3pl according to four main determinative criteria such as compatibility, cost, quality, and reputation. fu and liu (2007) determined the weights of the criteria with the ahp technique by considering five factors for outsourcing, including cost, quality, project, certification, and delivery performance in their study. qureshi et al., (2008) developed an interpretative structural modeling-based approach to define and classify key criteria and to examine their roles in 3pl evaluation. in the study, quality of service, size, and quality of fixed assets, management quality, computing capability, delivery performance, information sharing, and trust, operational performance, compliance, financial stability, geographic spread and range, long-term relationship, reputation, the optimal cost in operation and delivery, volatility and flexibility as 15 criteria were determined. liu and wang (2009) presented a three-step approach for the evaluation and selection of 3pl enterprises. in the first stage, the fuzzy delphi method was used to define important evaluation criteria. next, a fuzzy inference method has been applied to estimate non-eligible 3pl businesses. at the last stage, a fuzzy linear assignment approach has been developed for final selection. mickaitis et al. (2009) obtained from their study, on the outsourcing of smes in lithuania; it has been karamaşa et al./decis. mak. appl. manag. eng. 4 (1) (2021) 19-32 22 observed that outsourcing is widely used in smes in lithuania, and the main factors for these enterprises to prefer outsourcing are to reduce costs, increase efficiency and productivity, and increase quality. the short-term benefits of outsourcing have been identified as reducing the need for personnel, allowing better concentration on basic activities, and reducing costs. ji-fan ren et al. (2010) obtained from their study examining the determinants of the partnership quality of smes in china on outsourcing; it has been determined that perceived benefits of outsourcing, outsourcing competency management, the correct definition of outsourcing, and senior management's attitude towards outsourcing are significantly related to the quality of outsourcing. in light of the findings obtained from a study by o'regan and kling (2011) examining whether outsourcing is a competitive factor for smes operating in the production sector; it has been observed that small enterprises with low r&d investments tend to outsource. özbek and eren (2013) adopted the analytical network process approach for the selection of 3pl companies in their studies and considered quality, long-term relationships, company image, and operational performance as the basic criteria. govindan et al. (2016) used the dematel method in their examinations and determined that the most important criteria in the selection of 3pl companies are delivery performance, technology level, financial stability, human resources management, service quality, and customer satisfaction, respectively. sremac et al. (2018) applied with a total of 10 logistics providers for the transport of dangerous goods for chemical industry companies in their study. the importance of the eight criteria underlying the study in which the assessment was made, it was determined using the rough-swara method. korucuk (2018) evaluated the criteria to be used in the selection of third-party logistics (3pl) in cold chain transportation companies in istanbul and made the selection of the best 3pl. as a result of the study, it has been determined that “business performance” is the most important main criterion in the selection of 3pl companies, and the “c” option is the best 3pl provider company. erdoğan and tokgöz (2020) investigated the role of formal and relational governance in the success of outsourcing in information technology (it) in the aviation industry. according to the results of the research contracts and relationship norms in the success of it outsourcing business aviation operating in turkey, it is effective individually. also, they stated that formal and relational governance mechanisms are complementary rather than substitutes for each other. when businesses want to work with a company that provides logistics services, it is not an easy process to decide which company to be in the market. there is more than one third party logistics company with different competencies and skills in the market. outsourcing for logistics operations should be determined by experts. the increasing demand for outsourcing also increases the service alternatives offered. logistics service provider companies, which increase the level of competition, enable access with high quality and the most affordable cost level. the factors affecting the choice of outsourcing of enterprises have been listed as follows: weighting the factors affecting logistics outsourcing and selecting the most ideal company 23 table 1. 3pl main selection criteria criteria authors customer service quality menon et al. (1998), boyson et al. (1999), bottani and rizzi (2006), işıklar et al. (2007), jharkharia and shankar (2007), fu and liu (2007), qureshi et al. (2008), bhatti et al. (2010), chen and wu (2011), erkayman et al. (2012), li et al. (2012), govindan et al. (2012), bansal and kumar (2013), cirpin and kabadayi (2015), hwang et al. (2016), sremac et al. (2018) computing capability bottani and rizzi (2006), işıklar et al. (2007), bhatti et al. (2010), rajesh et al. (2011), erkayman et al. (2012), bansal and kumar (2013), hwang et al. (2016), sremac et al. (2018) operational performance chen and wu (2011), wong (2012), korucuk (2018) cost menon et al. (1998), boyson et al. (1999), bottani and rizzi (2006), işıklar et al. (2007), fu and liu (2007), jharkharia and shankar (2007), qureshi et al. (2008), vijayvargiya and dey (2010), rajesh et al. (2011), chen and wu (2011), govindan et al. (2012), erkayman et al. (2012), bansal and kumar (2013), cirpin and kabadayi (2015), hwang et al. (2016), sremac et al. (2018), korucuk(2018) supply chain capability bhatti et al. (2010), sremac et al. (2018) sustainability bansal and kumar (2013), cirpin and kabadayi (2015) risk management capability perçin (2009), rajesh et al. (2011), sremac et al. (2018), korucuk (2018) confidence petroni and braglia (2000), bottani and rizzi (2006), jharkharia and shankar (2007), qureshi etal. (2008), bansal and kumar (2013), sremac et al. (2018) geographical location suitability petroni and braglia (2000), qureshi et al. (2008), govindan et al. (2012), bansal and kumar (2013) history of performance petroni and braglia (2000), fu and liu (2007), qureshi et al. (2008), govindan et al. (2012) value added service vijayvargiya and dey (2010), bansal and kumar (2013) on time delivery rajesh et al. (2011), erkayman et al. (2012), govindan et al. (2012) flexibility petroni and braglia (2000), bottani and rizzi (2006), erkayman et al. (2012), govindan et al. (2012), korucuk (2018) karamaşa et al./decis. mak. appl. manag. eng. 4 (1) (2021) 19-32 24 in the literature review, it has been determined that there are limited studies on logistics outsourcing and choosing the most ideal company. the study differs from other studies in terms of the method used and the province applied. it is thought that it will contribute to the literature with this aspect. table 2. 3pl selection criteria main criteria sub-criteria cost transportation / storage and documentation cost, payment terms, discount offers customer service quality scope of services, flexibility and reliability, timeliness and ease of transaction / communication, customer satisfaction and cooperation, value added service computing capability edi presence and compatibility, computing network availability, data integrity and reliability, system stability operational performance delivery performance and reliability, relationship management, geolocation compliance, performance history, document accuracy supply chain capability trained logistics staff, infrastructure / equipment and logistics technology, efficiency capacity, risk management capability, reverse logistics function sustainability economic responsibility, social responsibility, environmental responsibility 3. methodology in this section firstly neutrosophic set (ns) is introduced then a single-valued neutrosophic set (svns) as a special case of ns is explained too. besides the steps of the neutrosophic ahp as newly developed multi-criteria, decision-making method are stated. 3.1. neutrosophic set smarandache (1998) introduced the concept of neutrosophic sets (ns) having with a degree of truth, indeterminacy, and falsity membership functions in which all of them are independent. let u be a universe of discourse and 𝑥 ∈ 𝑈. the neutrosophic set (ns) n can be expressed by a truth membership function 𝑇𝑁(𝑥), an indeterminacy membership function 𝐼𝑁(𝑥) and a falsity membership function 𝐹𝑁(𝑥), and is represented as 𝑁 = {< 𝑥:𝑇𝑁(𝑥),𝐼𝑁(𝑥),𝐹𝑁(𝑥) >,𝑥 ∈ 𝑈}. also the functions of 𝑇𝑁(𝑥), 𝐼𝑁(𝑥) and 𝐹𝑁(𝑥) are real standard or real nonstandard subsets of ]0 −,1+[ and can be presented as 𝑇,𝐼,𝐹:𝑈 → ]0−,1+[ . there is not any restriction on the sum of the functions of 𝑇𝑁(𝑥), 𝐼𝑁(𝑥) and 𝐹𝑁(𝑥) so 0 − ≤ 𝑠𝑢𝑝𝑇𝑁(𝑥) + 𝑠𝑢𝑝𝐼𝑁(𝑥) + 𝑠𝑢𝑝𝐹𝑁(𝑥) ≤ 3 +. the complement of an ns n is represented by 𝑁𝐶 and described as below: 𝑇𝑁 𝐶(𝑥) = 1+ ⊖ 𝑇𝑁(𝑥) (1) 𝐼𝑁 𝐶(𝑥) = 1+ ⊖ 𝐼𝑁(𝑥) (2) 𝐹𝑁 𝐶(𝑥) = 1+ ⊖ 𝐹𝑁(𝑥) for all𝑥 ∈ 𝑈 (3) weighting the factors affecting logistics outsourcing and selecting the most ideal company 25 a ns, n is contained in other ns p in other words , 𝑁 ⊆ 𝑃 if and only if 𝑖𝑛𝑓𝑇𝑁(𝑥) ≤ 𝑖𝑛𝑓𝑇𝑃(𝑥), 𝑠𝑢𝑝𝑇𝑁(𝑥) ≤ 𝑠𝑢𝑝𝑇𝑃(𝑥), 𝑖𝑛𝑓𝐼𝑁(𝑥) ≥ 𝑖𝑛𝑓𝐼𝑃(𝑥), 𝑠𝑢𝑝𝐼𝑁(𝑥) ≥ 𝑠𝑢𝑝𝐼𝑃(𝑥), ), 𝑖𝑛𝑓𝐹𝑁(𝑥) ≥ 𝑖𝑛𝑓𝐹𝑃(𝑥), 𝑠𝑢𝑝𝐹𝑁(𝑥) ≥ 𝑠𝑢𝑝𝐹𝑃(𝑥), for all 𝑥 ∈ 𝑈 (biswas et al., 2016). 3.2. single valued neutrosophic sets (svns) single valued neutrosophic set (svns) as a special case of ns for considering indeterminate, inconsistent, and incomplete information was developed by wang, smarandache, zhang, and sunderraman (2010). they acquire the interval [0,1]instead of/for solving the real-world problems. let u be a universe of discourse and 𝑥 ∈ 𝑈. a single-valued neutrosophic set b in u is described by a truth membership function𝑇𝐵(𝑥), an indeterminacy membership function 𝐼𝐵(𝑥)and a falsity membership function 𝐹𝐵(𝑥). when u is continuous an svns, b is depicted as𝐵 = ∫ <𝑇𝐵(𝑥),𝐼𝐵(𝑥),𝐹𝐵(𝑥)> 𝑥 :𝑥 ∈ 𝑈 𝑥 . when u is discrete a svns b can be represented𝐵 = ∑ <𝑇𝐵(𝑥𝑖),𝐼𝐵(𝑥𝑖),𝐹𝐵(𝑥𝑖) 𝑥𝑖 :𝑥𝑖 ∈ 𝑈 𝑛 𝑖=1 as (mondal, pramanik, & smarandache, 2016) the functions of 𝑇𝐵(𝑥),𝐼𝐵(𝑥) and 𝐹𝐵(𝑥) are real standard subsets of [0,1] that is 𝑇𝐵(𝑥):𝑈 → [0,1], 𝐼𝐵(𝑥):𝑈 → [0,1]and 𝐹𝐵(𝑥):𝑈 → [0,1]. also the sum of 𝑇𝐵(𝑥),𝐼𝐵(𝑥) and 𝐹𝐵(𝑥) are in [0,3] that0 ≤ 𝑇𝐵(𝑥) + 𝐼𝐵(𝑥) + 𝐹𝐵(𝑥) ≤ 3 (biswas et al., 2016). let a single-valued neutrosophic triangular number �̃� = 〈(𝑏1,𝑏2,𝑏3);𝛼�̃�,𝜃�̃�,𝛽�̃�〉 is a special neutrosophic set on r. additionally 𝛼�̃�,𝜃�̃�,𝛽�̃� ∈ [0,1] and 𝑏1,𝑏2,𝑏3 ∈ 𝑅 where 𝑏1 ≤ 𝑏2 ≤ 𝑏3. truth, indeterminacy, and falsity membership functions of this number can be computed as below (abdel-basset et al., 2017). t�̃�(𝒙) = { α�̃� ( x−𝑏1 𝑏2−𝑏1 ) α�̃� 𝛼�̃� ( 𝑏3−𝑥 𝑏3−𝑏2 ) 0 (𝑏1 ≤ 𝑥 ≤ 𝑏2) (x = 𝑏2) (𝑏2 < 𝑥 ≤ 𝑏3) otherwise (4) i�̃�(𝒙) = { ( 𝑏2−x+𝜃�̃�(x−𝑏1) 𝑏2−𝑏1 ) θ�̃� ( 𝑥−𝑏2+𝜃�̃�(𝑏3−𝑥) 𝑏3−𝑏2 ) 1 (𝑏1 ≤ 𝑥 ≤ 𝑏2) (x = 𝑏2) (𝑏2 < 𝑥 ≤ 𝑏3) otherwise (5) f�̃�(𝒙) = { ( 𝑏2−x+𝛽�̃�(x−𝑏1) 𝑏2−𝑏1 ) β�̃� ( 𝑥−𝑏2+𝛽�̃�(𝑏3−𝑥) 𝑏3−𝑏2 ) 1 (𝑏1 ≤ 𝑥 ≤ 𝑏2) (x = 𝑏2) (𝑏2 < 𝑥 ≤ 𝑏3) otherwise (6) according to the eqs. (4)-(6) 𝛼�̃�,𝜃�̃�, 𝑎𝑛𝑑 𝛽�̃� denote maximum truth membership, minimum indeterminacy membership, and minimum falsity membership degrees respectively. suppose �̃� = 〈(𝑏1,𝑏2,𝑏3);𝛼�̃�,𝜃�̃�,𝛽�̃�〉 and �̃� = 〈(𝑐1, 𝑐2, 𝑐3);𝛼𝑐̃,𝜃𝑐̃,𝛽𝑐̃〉 as two singlevalued triangular neutrosophic numbers and 𝜆 ≠ 0 as a real number. considering the addition of the abovementioned condition of two single-valued triangular neutrosophic numbers are denoted as follows (abdel-basset et al., 2017). �̃� + �̃� = 〈(𝑏1 + 𝑐1,𝑏2 + 𝑐2,𝑏3 + 𝑐3);𝛼�̃� ∧ 𝛼𝑐̃,𝜃�̃� ∨ 𝜃𝑐̃,𝛽�̃� ∨ 𝛽𝑐̃〉 (7) subtraction of two single-valued triangular neutrosophic numbers are defined as eq.(8): karamaşa et al./decis. mak. appl. manag. eng. 4 (1) (2021) 19-32 26 �̃� − �̃� = 〈(𝑏1 − 𝑐3,𝑏2 − 𝑐2,𝑏3 − 𝑐1);𝛼�̃� ∧ 𝛼𝑐̃,𝜃�̃� ∨ 𝜃𝑐̃,𝛽�̃� ∨ 𝛽𝑐̃〉 (8) the inverse of a single-valued triangular neutrosophic number (�̃� ≠ 0)can be denoted as below: �̃�−1 = 〈( 1 𝑏3 , 1 𝑏2 , 1 𝑏1 );𝛼�̃�,𝜃�̃�,𝛽�̃�〉 (9) multiplication of a single-valued triangular neutrosophic number by a constant value is represented as follows: 𝜆�̃� = { 〈(𝜆𝑏1,𝜆𝑏2,𝜆𝑏3);𝛼�̃�,𝜃�̃�,𝛽�̃�〉 𝑖𝑓 (𝜆 > 0) 〈(𝜆𝑏3,𝜆𝑏2,𝜆𝑏1);𝛼�̃�,𝜃�̃�,𝛽�̃�〉 𝑖𝑓 (𝜆 < 0) (10) division of a single-valued triangular neutrosophic number by a constant value is denoted as eq.(11): �̃� 𝜆 = { 〈( 𝑏1 𝜆 , 𝑏2 𝜆 , 𝑏3 𝜆 );𝛼�̃�,𝜃�̃�,𝛽�̃�〉 𝑖𝑓 (𝜆 > 0) 〈( 𝑏3 𝜆 , 𝑏2 𝜆 , 𝑏1 𝜆 );𝛼�̃�,𝜃�̃�,𝛽�̃�〉 𝑖𝑓 (𝜆 < 0) (11) multiplication of two single-valued triangular neutrosophic numbers can be seen as follows: �̃��̃� = { 〈(𝑏1𝑐1,𝑏2𝑐2,𝑏3𝑐3);𝛼�̃� ∧ 𝛼𝑐̃,𝜃�̃� ∨ 𝜃𝑐̃,𝛽�̃� ∨ 𝛽𝑐̃〉 𝑖𝑓 (𝑏3 > 0,𝑐3 > 0) 〈(𝑏1𝑐3,𝑏2𝑐2,𝑏3𝑐1);𝛼�̃� ∧ 𝛼𝑐̃,𝜃�̃� ∨ 𝜃𝑐̃,𝛽�̃� ∨ 𝛽𝑐̃〉 𝑖𝑓 (𝑏3 < 0,𝑐3 > 0) 〈(𝑏3𝑐3,𝑏2𝑐2,𝑏1𝑐1);𝛼�̃� ∧ 𝛼𝑐̃,𝜃�̃� ∨ 𝜃𝑐̃,𝛽�̃� ∨ 𝛽𝑐̃〉 𝑖𝑓 (𝑏3 < 0,𝑐3 < 0) (12) division of two single-valued triangular neutrosophic numbers can be denoted as eq.(13): �̃� 𝑐̃ = { 〈( 𝑏1 𝑐3 , 𝑏2 𝑐2 , 𝑏3 𝑐1 );𝛼�̃� ∧ 𝛼𝑐̃,𝜃�̃� ∨ 𝜃𝑐̃,𝛽�̃� ∨ 𝛽𝑐̃〉 𝑖𝑓 (𝑏3 > 0,𝑐3 > 0) 〈( 𝑏3 𝑐3 , 𝑏2 𝑐2 , 𝑏1 𝑐1 );𝛼�̃� ∧ 𝛼𝑐̃,𝜃�̃� ∨ 𝜃𝑐̃,𝛽�̃� ∨ 𝛽𝑐̃〉 𝑖𝑓 (𝑏3 < 0,𝑐3 > 0) 〈( 𝑏3 𝑐1 , 𝑏2 𝑐2 , 𝑏1 𝑐3 );𝛼�̃� ∧ 𝛼𝑐̃,𝜃�̃� ∨ 𝜃𝑐̃,𝛽�̃� ∨ 𝛽𝑐̃〉 𝑖𝑓 (𝑏3 < 0,𝑐3 < 0) (13) score function (𝑠𝑏) for a single-valued triangular neutrosophic number 𝑏 = (𝑏1,𝑏2,𝑏3) can be found as below (stanujkic et al., 2017). 𝑠𝑏 = (1 + 𝑏1 − 2 ∗ 𝑏2 − 𝑏3)/2 (14) where 𝑠𝑏 ∈ [−1,1]. 3.3. neutrosophic ahp steps of neutrosophic ahp are depicted as below (abdel-basset et al., 2017): step 1: decision problem is arranged in terms of hierarchical viewpoint composed of goal, criteria, sub-criteria, and alternatives respectively. step 2: pairwise comparisons are constructed to form a neutrosophic evaluation matrix consisting of triangular neutrosophic numbers showing the experts’ views. neutrosophic pairwise evaluation matrix (�̃�)can be written as follows: �̃� = [ 1̃ �̃�12 ⋯ �̃�1𝑛 ⋮ ⋮ ⋮ ⋮ �̃�𝑛1 �̃�𝑛2 ⋯ 1̃ ] (15) according to eq.(15) �̃�𝑗𝑖 = �̃�𝑖𝑗 −1 is valid. step 3: neutrosophic pairwise evaluation matrix is formed by applying a transformed scale for neutrosophic environment seen as table 3: weighting the factors affecting logistics outsourcing and selecting the most ideal company 27 table 3. ahp transformed scale related to neutrosophic triangular numbers (abdel-basset et al., 2018) value explanation neutrosophic triangular scale 1 equally influential 1̃ = 〈(1,1,1);0.5,0.5,0.5〉 3 slightly influential 3̃ = 〈(2,3,4);0.3,0.75,0.7〉 5 strongly influential 5̃ = 〈(4,5,6);0.8,0.15,0.2〉 7 very strongly influential 7̃ = 〈(6,7,8);0.9,0.1,0.1〉 9 absolutely influential 9̃ = 〈(9,9,9);1,0,0〉 2 4 6 8 intermediate values between two close scales 2̃ = 〈(1,2,3);0.4,0.65,0.6〉 4̃ = 〈(3,4,5);0.6,0.35,0.4〉 6̃ = 〈(5,6,7);0.7,0.25,0.3〉 8̃ = 〈(7,8,9);0.85,0.1,0.15〉 step 4: neutrosophic pairwise evaluation matrix is transformed into a deterministic pairwise evaluation matrix for calculating the weights of criterion as below: let �̃�𝑖𝑗 = 〈(𝑑1,𝑒1,𝑓1),𝛼�̃�,𝜃�̃�,𝛽�̃�〉 be a single-valued neutrosophic number, then the score and accuracy degrees for �̃�𝑖𝑗 can be calculated computed as below: 𝑆(�̃�𝑖𝑗) = 1 16 [𝑑1 + 𝑒1 + 𝑓1]𝑥(2 + 𝛼�̃� − 𝜃�̃� − 𝛽�̃�) (16) 𝐴(�̃�𝑖𝑗) = 1 16 [𝑑1 + 𝑒1 + 𝑓1]𝑥(2 + 𝛼�̃� − 𝜃�̃� + 𝛽�̃�) (17) score and accuracy degrees for �̃�𝑖𝑗 are obtained according to the following equations. 𝑆(�̃�𝑗𝑖) = 1/𝑆(�̃�𝑖𝑗) (18) 𝐴(�̃�𝑗𝑖) = 1/𝐴(�̃�𝑖𝑗) (19) the deterministic pairwise evaluation matrix is formed with compensation by the score value of each triangular neutrosophic number related to the neutrosophic pairwise evaluation matrix. the deterministic matrix can be written as below: 𝐷 = [ 1 𝑑12 ⋯ 𝑑1𝑛 ⋮ ⋮ ⋮ ⋮ 𝑑𝑛1 𝑑𝑛2 ⋯ 1 ] (20) ranking of priorities as eigenvector x is obtained according to the following steps: a)firstly column entries are normalized by dividing each entry by the sum of column b)then row averages are summed. step 5: consistency index (ci) and consistency ratio (cr) values are calculated for measuring the inconsistency for decision-makers’ judgments in the entire pairwise evaluation matrix. if cr is greater than 0.1, the process should be repeated because of unreliable judgments. ci is calculated as below: a)each value in the first column of the pairwise evaluation matrix is multiplied by the priority of the first criterion and this process is repeated for all columns. values are summed across the rows to obtain the weighted sum vector. b) the elements of the weighted sum vector are divided by corresponding the priority for each criterion. then the average of values are obtained and showed by 𝜆𝑚𝑎𝑥. c) the value of ci is computed as eq. (21): 𝐶𝐼 = 𝜆𝑚𝑎𝑥−𝑛 𝑛−1 (21) karamaşa et al./decis. mak. appl. manag. eng. 4 (1) (2021) 19-32 28 according to eq.(21), the number of elements being compared is denoted by n. after the value of ci is calculated, cr can be acquired as below: 𝐶𝑅 = 𝐶𝐼 𝑅𝐼 (22) where ri represents the consistency index for randomly generated pairwise evaluation matrix and shown as table 4. table 4. ri table considered for obtaining cr value (abdel-basset et al., 2017) order of random matrix (n) 0 2 3 4 5 6 7 8 9 10 related ri value 0 0 0.58 0.9 1.12 1.24 1.32 1.4 1.45 1.49 step 6: overall priority values for each alternative are calculated and ranking is executed. 4. case study and analysis in this study, six criteria (cost, customer service quality, data processing ability, operational performance, supply chain ability, and sustainability) considered for factors affecting outsourcing related to 3pl are weighted via neutrosophic ahp firstly. for this purpose evaluations of five decision-makers in 3pl are considered. neutrosophic evaluation matrix in terms of factors affecting outsourcing related to 3pl is constructed through decision-makers’ linguistic judgments which are seen as table 1. neutrosophic evaluation matrix is transformed into a crisp one by using equation (16) and taking the geometric means of 5 decision-makers’ views. the crisp evaluation matrix for criteria is shown in table 5. table 5. the crisp evaluation matrix for criteria related to outsourcing criteria cost custome r service quality data processing ability operational performance supply chain ability sustainability cost 1 1.995 0.988 1.337 1.337 0.976 customer service quality 0.501 1 1.226 0.895 0.654 0.895 data processing ability 1.012 0.815 1 1.226 0.895 0.895 operational performance 0.747 1.116 0.815 1 1.226 1.337 supply chain ability 0.747 1.528 1.116 0.815 1 1.995 sustainability 1.023 1.116 1.116 0.747 0.501 1 the normalized evaluation matrix for criteria is constructed as table 6. table 6. the normalized evaluation matrix for criteria related to outsourcing weighting the factors affecting logistics outsourcing and selecting the most ideal company 29 criteria cost customer service quality data processing ability operational performance supply chain ability sustainability cost 0.596 0.203 0.116 0.176 0.169 0.113 customer service quality 0.079 0.101 0.144 0.118 0.082 0.104 data processing ability 0.161 0.083 0.117 0.161 0.113 0.104 operational performance 0.119 0.113 0.096 0.131 0.155 0.155 supply chain ability 0.119 0.155 0.131 0.107 0.126 0.232 sustainability 0.163 0.113 0.131 0.098 0.063 0.116 finally, the priorities for criteria as the eigenvector x are obtained by taking the overall row averages and presented as follows: table 7. priorities for criteria related to outsourcing criteria priorities cost 0.1528 customer service quality 0.1030 data processing ability 0.1139 operational performance 0.1348 supply chain ability 0.1354 sustainability 0.1244 according to table 7, while cost was found as the most important criterion having a value of 0.1529, customer service quality was obtained as the least important one having a value of 0.103. then the consistency of decision-makers’ judgments is checked by computing ci and cr values. ci value is found as 0.018 and by using equation (22) cr value is acquired as 0.012. decision-makers’ evaluations are consistent because of having cr value smaller than 0.1. 5. conclusion in this study factors affecting outsourcing related 3pl determined by extensive literature review process are ranked by using neutrosophic ahp. single valued neutrosophic sets are preferred compared to crisp, fuzzy, interval-valued, and intuitionistic sets due to efficiency, flexibility, and easiness for explaining decisionmakers’ indeterminate judgments. furthermore ranking of factors affecting outsourcing related to 3pl as a complex real-world decision making problem can be efficiently solved under neutrosophic sets based environment. for further researches factors affecting outsourcing related to 3pl can be expanded and results can be compared with different multi-criteria decision-making methods. also, various hybrid techniques can be proposed and applied for real-world complex decision-making problems. karamaşa et al./decis. mak. appl. manag. eng. 4 (1) (2021) 19-32 30 author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research received no external funding. conflicts of interest: the authors declare no conflicts of interest. references abdel-basset, m., mohamed, m., & smarandache, f. 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(2012). dss for 3pl provider selection in global supply chain: combining the multi-objective optimisation model with experts’ opinions, journal of intelligent manufacturing, 23(3), 599-614. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 4, issue 1, 2021, pp. 104-126. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame2104104s * corresponding author. e-mail addresses: sahu_r@rediffmail.com; (r. sahu), sdashfca@kiit.ac.in; (sr. dash), sujit.das@nitw.ac.in (s. das). career selection of students using hybridized distance measure based on picture fuzzy set and rough set theory rekha sahu 1, satya r. dash 2 and sujit das 3* 1 school of computer engineering, kiit university, bhubaneswar, india 2 school of computer applications, kiit university, bhubaneswar, india 3 department of computer science and engineering, national institute of technology warangal, warangal, india received: 20 october 2020; accepted: 2 february 2021; available online: 13 february 2021. original scientific paper abstract: since the future of the society depends upon the role of students, so suitable career selection methods for the students are considered to be an important problem to explore. it is assumed that if a student has the required capability and positive attitudes towards a subject, then the student will achieve more in that subject. to consider the uncertain issues involved with students’ career selection, picture fuzzy set (pfs) and rough set based approaches are proposed in this study as they are found to be appropriate due to their inherent characteristics to deal with incomplete and imprecise information. for the purpose of selecting a suitable career, the article analyzes student's features in terms of career, memory, interest, knowledge, environment and attitude. we propose two hybridized distance measures using hausdorff, hamming and euclidian distances under picture fuzzy environment where the evaluating information regarding students, subjects and student's features are given in picture fuzzy numbers. then we present an algorithmic approach using the proposed distance measures and rough set theory. we apply rough set theory to determine whether a particular subject is suitable for a student even if there is controversy to select a stream. lower and higher approximation with boundary region of rough set theory is used to manage the inconsistent situations. finally, two case studies are demonstrated to validate the applicability of the proposed idea. key words: picture fuzzy set, distance measure, rough set, students’ career. mailto:sahu_r@rediffmail.com mailto:sdashfca@kiit.ac.in career selection of students using hybridized distance measure based on picture fuzzy set and rough set theory 105 1. introduction selection of the subject for a better career of a student is a vital task since it is concerned with future employment which influences the whole life of the student and ultimately leads to social development. it is a delegate decision for the students as they are ambitious about it (batool et al., 2020, van dinh et al., 2019, pratiwi et al., 2020, mckenzie et al., 2020, wang et al., 2020, babajide et al., 2020, orewere et al., 2020). some students are in confusion or careless to choose a stream. the major confusion is that whether the stream is suitable for their future establishment and their eligibilities are sufficient or not for analyzing and studying efficiently and sincerely. sometimes many students have no faith in them as they do not know clearly what they will study or what is the content of a particular subjects or stream. as found in various literature related to different colleges and streams (wen et al., 2018, nehmeh et al., 2018), we have observed that students often face difficulty in choosing proper stream and make it as a career. to solve the students’ career selection problem, many authors have contributed in the last few decades. we have summarized some of the significant contributions which are narrated below. in (wen et al., 2018), authors have discussed the career choice issue related to choosing accountant as the career, which influenced the researchers to search the various methods for choosing a career for the student by selecting an appropriate subject. the survey of 216 students has shown that both of the internal and external factors influence the career selection process (babajide et al., 2020). a survey of three hundred students has shown that parents' economic status or social class, financial support, decision making and learning abilities have a huge impact on career decision of students (batool et al., 2020). research has also revealed that in order to fulfil parents' expectation and for family or cultural values, few students choose their career in medical school in spite of having lower academic performance (griffin et al., 2019) which in future hampers their career. students' engagement, family encouragement, family capital and various scientific matters also encourage the students to study science (silseth et al., 2018) and select their career accordingly. although the scientific beliefs, teachers’ and parents’ expectations, sense of encouragements, and academic prediction, motivate the students to choose science as their career but there is an explicit gap between the motivation factors and the career selection which has been illustrated in (ramentol et al., 2019). in the rural areas, the environment such as family poverty and rurality often influence the choices of students’ career (carrico et al., 2019). in (holloway-friesen et al., 2018), the study found that academic persistence, pursuit of career goals, and high career expectations are significantly influenced by the college environment. again, it is observed that the environmental impact in terms of more guidance and counselling centers influence the student to choose the right career (orewere et al., 2020). hence it can be concluded that the impact of environmental factors like the parent, teacher, family, place, and the institute is a major issue to choose a career. along with the environmental factors, few other factors are also there. the study of 502 students showed that the different factors like self-efficacy, outcome expectation and career intention have more impact on career choice (pratiwi et al., 2020, mckenzie et al., 2020). a survey on student mentioned in (madden et al., 2018) specified that hardness and softness of a subject do not matter for a student if he/she has an interest in that subject. interest in choosing profession influence the performance in the service and activities (alkaya et al., 2018). goel et al. (2018) investigated that the decisions to join the medical profession are mainly dependent on the factors like scientific (interest in medicine), socials (respect/prestige) and humanitarian (desire to help others). hannula et al. (2002) observed four ways to evaluate the attitude of a student which are emotions aroused sahu et al./decis. mak. appl. manag. eng. 4 (1) (2021) 104-126 106 in the situation, emotions associated with the stimuli, expected consequences and related situation to personal values. positive attitude towards a subject plays an important role, where the positive attitude influences the expected achievement (guido et al., 2018, burns et al., 2018). based on the above discussion, it can be concluded that extrinsic motivation, intrinsic interest and perceived support and encouragement to a particular subject strongly aspire the students to study that subject. hence by analyzing the related concerns, we find that the specific characteristics of a student may be the cause for the success in a stream. various studies mentioned above unpack that attitude, knowledge, interest, career, memory and environment are might be considered to be the key factors for students’ success. two computational intelligence techniques, i.e., picture fuzzy set (pfs) and rough set (rs) theory have a big role to predict the career for students and other decisionmaking process (kumbhar et al., 2020, si et al., 2020, das et al., 2018, de et al., 2019, si et al., 2019). in (dutta, 2018), pfs is proposed for medical investigation and diagnosis. pfs is an extension of intuitionist fuzzy set (ifs) to deal with uncertainty in the situations involving more answers of the types: yes, abstain, no (cuong et al., 2013), whereas a rough set is a pair of crisp sets, i.e., the lower approximation of set and upper approximation of the original set and these sets may contain fuzzy values (pawlak, 1995). a detailed study on picture fuzzy set is found in (cuong, 2014). picture fuzzy clustering algorithm can be developed for exploiting and investigating hidden knowledge from data. hierarchical picture clustering is proposed in (son, 2016) which is an integration of generalized picture distance measure and hierarchical picture fuzzy clustering. the pfs are also useful for computational intelligence problem (cuong et al., 2013). in the other side, a probability-based rough set theory is proposed in (ramentol et al., 2019) to predict how likely a student is to succeed in the academic year. an algorithm for decision making based on rough fuzzy set with 𝞭-clustering and upper-lower (𝞬, 𝞭) – approximation is modelled in (ramentol et al. 2019). the intuitionistic fuzzy rough set (tan et al., 2018) is a combination is of intuitionistic fuzzy (if) and rs theory, where if relations are defined and characterized by the lower and upper approximations of the rough set. then the measures are developed to evaluate approximation quality and ability of classification. the notion of picture fuzzy rough soft set is introduced in (cuong et al., 2018) which the combination of pfs and rs, and this formulation is used for classification, decision making, and knowledge discovery. in (cuong et al., 2018), picture fuzzy rough soft sets and picture fuzzy dynamic systems are introduced and these are the extensions of pfs with its applications. van et al. (2019) defined the distance measure between pfs with similarity and dissimilarity which are useful for image segmentation, decision making and pattern recognition. as found in literature, none of the researchers has contributed to the student career consideration based on the hybridization of distance measure using pfs and rs. choosing a stream as the career is a research area of applied fuzzy set theory. after having the basic knowledge of different subjects in school, the student is in a situation to make decisions on the career for whole life by choosing a stream. but due to the imprecise and incomplete nature of information regarding different streams, the concept of a fuzzy set is inevitable for decision-making purpose. in (dutta, 2018), authors discussed the application of pfs in medical diagnosis, but as per our knowledge, a very few researchers have applied fuzzy set theory in career selection although the deciding on choosing stream is a vital and critical task (wen et al., 2018). our proposed model, pfs with hybrid distance measures, has the intension to find the subject having the minimum distance from the student and help to decide to choose the right stream. some students have positive potential towards a particular subject like a student has high caliber in computer science and so the degree of positive career selection of students using hybridized distance measure based on picture fuzzy set and rough set theory 107 potential is to be taken into consideration when choosing a stream as a career. again, some students have less or negative potential towards a particular subject like having high caliber with literature or art may have less or negative knowledge in mathematics. hence, the degree of negative potential towards the stream is to be considered. it is observed in a career choice that some calibers may have a neutral effect on acquiring subject, i.e., the degree of neutrality can be considered for the subjects like physics, mathematics, or computer science, say neither negative nor positive. therefore, it is convenient to consider the degree of neutrality in career considering. in this article, we present a pfs-based approach which uses the hybridized distance measure for choosing a suitable and appropriate career for a student, where the rough set is used to avoid any kinds of confusion in choosing a stream. when students complete school education, they need to choose a suitable and particular stream as their career or to fulfil their ambitions. to find out the student's ability and then assign a stream to him/her according to suitability and appropriateness is a major challenge. to resolve the issue, we have quantified two qualitative concepts i.e., the requirement of the student to understand the subjects and the student's abilities towards the subject. hence, we have considered the subjects with its features concerning the student's interest, knowledge, memory, career, attitude and environment impact as the general requirement. we have also considered a student's features as interest, knowledge, memory, career, attitude and environment impact towards the specific stream. with these two quantities, we investigate the distance between the student and the subjects using the proposed hybridized distance measures and then select the subject as the career which has a minimum distance from the student. in the process, we may face some situations as defined below. a) eligible for more than one stream b) neutral for more than one stream. c) perfect for one subject. d) not eligible for the subjects. there will be no problem for case (c), but for cases (a), (b) and (d), the decision is critical. these cases can be concluded when choosing a stream using rough set (rs) theory. for example, if computer science, mathematics, physics have the same distance and also minimum distance, there will be fuzziness or inconsistency. again, for choosing computer science with a specialty in data science as a career, the student must have caliber on computer science, mathematics, statistics and in this case fuzziness or inconsistency may arise, which is resolved using rough set theory. we have proposed two hybridized distance measures namely hybridization of hausdorff and hamming distance measures, and hybridization of hausdorff and euclidean distance measures based on hausdorff, hamming and euclidian distance for measuring the distance between student's features and stream's features. the student's features are summarized based on the degree of positive potential, the degree of neutral potential and the degree of negative potential towards the subject whereas subject’s features are identified with interest, subject knowledge, memory, career, attitude and environmental impact as the potential’s requirement. also, the refusal degree in sense of neither positive, negative and neutral is taken into consideration. again, for interest, subject knowledge, memory, career, attitude and environment impact, the degree of positive potential, the degree of neutral potential, the degree of negative potential and the refusal degree are considered. thus, to choose a career we have taken the degree of positive potential, the negative potential, the degree of neutrality and refusal degree of student, interest, subject knowledge, sahu et al./decis. mak. appl. manag. eng. 4 (1) (2021) 104-126 108 memory, career, attitude and environment impact. for each student, we have a fourdimensional vector of information towards a particular subject and for each subject, we have a set of four-dimensional vectors where the set contents six elements. we have illustrated the proposed methods using two case studies. the workflow diagram of our proposed model is depicted in figure 1. subjects' requirement concerning a student characteristic student’s characteristic towards a subject distance measurement between subjects’ requirement and student’s characteristic whether multiple subjects combinedly consider selecting a subject? consider the subject for a student having a minimum distance measure rough set theory with picture fuzzy theory model is used to consider the subject yesno figure 1. workflow diagram of the proposed model. rest of this paper is structured as follows. we have noted the basic concepts used in the proposed models in section 2. section 3 presents the proposed model followed by its application in deciding a stream as a career of a student in section 4, where two case studies are illustrated for choosing the stream. the first case study is implemented using pfs and the second case study is implemented using both pfs and rs. comparative study is given in section 5. finally, in section 6, we have given our conclusion with possible future works. 2. preliminaries in this section, we have discussed, some basic concepts related to this paper. career selection of students using hybridized distance measure based on picture fuzzy set and rough set theory 109 2.1. picture fuzzy set (pfs) (cuong, 2014) pfs a on x is an object of the form 𝐴 = {(𝑥, 𝜇𝐴(𝑥), 𝜏𝐴 (𝑥), 𝛾𝐴(𝑥)): 𝑥𝜖𝑋}, where μa(x), τa(x) and γa(x) belongs to the close interval [0, 1]and ( )a x , 𝜏𝐴(𝑥) 𝑎𝑛𝑑 𝛾𝐴(𝑥) are satisfied with the conditions 0 ≤ 𝜇𝐴(𝑥) + 𝜏𝐴(𝑥) + 𝛾𝐴(𝑥) ≤ 1, 𝜌𝐴(𝑥) = 1 − (𝜇𝐴(𝑥) + 𝜏𝐴(𝑥) + 𝛾𝐴(𝑥)). here 𝜇𝐴(𝑥), be the degree of positive membership of x in a, 𝜏𝐴(𝑥), be the degree of neutral membership of x in a, and 𝛾𝐴(𝑥), be the degree of negative membership of x in a, and 𝜌𝐴(𝑥) be the refusal degree of x in a. 2.2. picture distance measure (dutta, 2018) for p, q ∈ pfs(x), d (p, q) is called the picture fuzzy distance measure if it satisfies the following criteria: (i) 𝑑(𝑃, 𝑄) ≥ 0 & 𝑑(𝑃, 𝑄) ≤ 1, (ii) 𝑑(𝑃, 𝑄) = 𝑑(𝑄, 𝑃), (iii) 𝑑(𝑃, 𝑄) = 0 <=> 𝑃 = 𝑄, (iv) 𝜇𝑃𝑄 × 𝑑(𝑃, 𝑄) + 𝜇𝑃𝑅 × 𝑑(𝑃, 𝑅) ≥ 𝜇𝑄𝑅 × 𝑑( 𝑄, 𝑅), 𝑓𝑜𝑟 𝑅 ∈ 𝑃𝐹𝑆(𝑋) . the symbol “x” indicates the arithmetic product, 𝜇𝑃𝑄 , 𝜇𝑃𝑅 𝑎𝑛𝑑 𝜇𝑄𝑅 are composition operations of p, q, and r, and the min-max composition formulae to calculate 𝜇𝑃𝑄, 𝜇𝑃𝑅 𝑎𝑛𝑑 𝜇𝑄𝑅 are as follows: μpq = mini {max{μp(xi), μq (xi)}} μqr = mini {max{μq (xi), μr(xi)}} μpr = mini{max{μp(xi), μr(xi)}} 2.3. hamming distance (tugrul et al., 2017) for p ∈ pfs(x), q ∈ pfs(y), d (p, q) is called the picture fuzzy hamming distance measure defined as follows. p = {(x, μp(x), τp(x), γp(x)): xϵx} & q = {(y, μq (y), τq(y), γq (y)): y ϵy} d(p, q) = |x − y| + | μp(x) − μq(y)| + |τp(x) − τq(y)| + | γp(x) − γq(y)| 2.4. euclidean distance (tugrul et al., 2017) for p ∈ pfs(x), q ∈ pfs(y), d (p, q) is called the picture fuzzy euclidian distance measure defined as follows. p = {(x, μp(x), τp(x), γp(x)): xϵx} & q = {(y, μq (y), τq(y), γq (y)): y ϵy} 𝑑(𝑃, 𝑄) = √(𝑥 − 𝑦)2 + ( μp(x) − μq (y)) 2 + (τp(x) − τq(y)) 2 + ( γp(x) − γq(y)) 2 sahu et al./decis. mak. appl. manag. eng. 4 (1) (2021) 104-126 110 2.5. hausdorff distance (aspert et al., 2002) for p ∈ pfs(x), q ∈ pfs(y), hausdorff distance from set p to set q is a maximum function, defined as ℎ(𝑃, 𝑄) = max 𝑎∈𝑃 {min 𝑏∈𝑄 { 𝑑(𝑎, 𝑏)}} where a and b are points of sets p and q respectively and d (a, b) is any distance metric between the points of p and q. 2.6. rough set (rs)theory (pawlak, 1995) let (u, a) be the information system (is), where u be a set of objects and a be a finite set of attributes such that, ꓯ α є a, α: u -> vα, where vα is the value set of α and u ≠ ø, a ≠ ø. t = (u, au{γ}) is the decision system, where the attributes contained in a are condition attributes and γ is the decision attribute. the rs theory deals with imperfect knowledge which is expressed by boundary region of a set, and defined with topological operations, interior and closure approximation. 2.7. the indiscernible relation r⊆ x × x is a binary relation satisfying reflexive, symmetric and transitive property. for x ∈x, the equivalence class is [x]r = {y| x r y for y ∈ x}. is = (u, a) is an information system and for b ⊆a, the associated equivalence relation is defined as: indis(b) = {(x, x’) ∈ u2 | ꓯ a ∈ b, a(x) = a(x’)}, where indis (b) is called the bindiscernible relation. the equivalence classes of the bindiscernible relation are denoted by [x]b. 2.8. set approximation t= (u, a), b ⊆ a & x ⊆ u is the decision system (ds) and we have. a. x is approximated by constructing the b-lower approximation and b-upper approximations of x using the information of b as b𝑋 = {𝑥|[𝑥]𝐵 ⊆ 𝑋}, and �̅�𝑋 = {𝑥|[𝑥]𝐵 ∩ 𝑋 ≠ ∅} respectively. b. b-boundary region of x, bnb(x) = �̅�x – bx consists of the objects which are not classified into x in b. c. b-outside region, u-�̅�x consists of the objects which are not belonging to x. d. if boundary reason is non-empty, then we have the rough set. 2.9. rough membership function the rough membership function quantifies the degree of relative overlap between the set x and the equivalence class r(x) to which x belongs and is defined as follows. 𝜇𝑋 𝑅 : 𝑈 → [0,1], where 𝜇𝑋 𝑅 (𝑥) = |𝑋∩𝑅(𝑥)| |𝑅(𝑥)| , and |x| denotes the cardinality of x. the membership function of the rough set is expressed as (i) the conditional probability that x belongs to x given r. (ii) a degree that x belongs to x given information about x expressed by r. career selection of students using hybridized distance measure based on picture fuzzy set and rough set theory 111 using rough membership function, approximations and boundary region of a set are as follows. r(x) = {x ∈u|𝜇𝑋 𝑅 (𝑥) = 1}. �̅�(x) = {x ∈u |𝜇𝑋 𝑅 (𝑥) = 0}. rnr(x)= {x ∈u |0 < 𝜇𝑋 𝑅 (𝑥) < 1}. 2.10. dealing with inconsistency situations if the inconsistency situations are shown then it may be solved using one of the following actions. a) consult the expert for taking actions. b) make different tables for the conflicting situation. c) the examples having with less support should be removed. d) basing on upper approximation set and lower approximation set, the quality method can be used to solve inconsistency. e) the method of generating new decision attributes. 3. proposed method this section presents the proposed method. initially, we present two hybridized distance measure namely hybridization of hausdorff and hamming distance and hybridization of hausdorff and euclidean distance measures. then we present an algorithmic approach using these two distance measures. the hybridization of hausdorff and hamming distance measure d1(a, b), and hybridization of hausdorff and euclidean distance measure d2(a, b) are defined as below. for a, b є pfs(x), d1(a, b) = ( 1 n ∑ ∆μi+∆τi+∆γi+∆ρi 4 +max(∆μi,∆τi,∆γi,∆ρi)) n i=1 ( 1 n ∑ ∆μi+∆τi+∆γi+∆ρi 4 +max(∆μi,∆τi,∆γi,∆ρi)) n i=1 + ∑ (max{φ i an i=1 ,φ i b}+|φ i a−φ i b| n +1 (1) for a, b є pfs(x), d2(a, b) = ( 1 n ∑ ∆μi 2+∆τi 2+∆γi 2+∆ρi 2 4 +max(∆μi 2,∆τi 2,∆γi 2,∆ρi 2))n i=1 ( 1 n ∑ ∆μ i 2+∆τ i 2+∆γ i 2+∆ρ i 2 4 +max(∆μi 2,∆τi 2,∆γi 2,∆ρi 2))n i=1 + 1 n ∑ [max{φi a,φi b}+|φi a−φi b| 2 ] 1 2n i=1 +1 (2) ∆𝜇𝑖 = |𝜇𝐴(𝑥𝑖 ) − 𝜇𝐵 (𝑥𝑖 )|, ∆𝜏𝑖 = |𝜏𝐴(𝑥𝑖 ) − 𝜏𝐵 (𝑥𝑖 )|, ∆𝛾𝑖 = |𝛾𝐴(𝑥𝑖 ) − 𝛾𝐵 (𝑥𝑖 )|, ∆𝜌𝑖 = |𝜌𝐴(𝑥𝑖 ) − 𝜌𝐵 (𝑥𝑖 )|, ∅𝑖 𝐴 = |𝜇𝐴(𝑥𝑖 ) + 𝜏𝐴(𝑥𝑖 ) + 𝛾𝐴(𝑥𝑖 )|, 𝑖 = 1,2, … , 𝑛 a distance measure is an objective score that summarizes the relative difference between two objects in a problem domain. hausdorff distance is the greatest of all distances from a point in one set to the closest point in the other set. hamming distance calculates the sum or the average differences between the two values. to calculate distance from a vector data to set of vectors data, we have used a set of distance values derived using hamming distance and a final distance is concluded using hausdorff distance. thus, we have the hybridization for x ∈ x, sahu et al./decis. mak. appl. manag. eng. 4 (1) (2021) 104-126 112 𝑑(𝑥, 𝑌) = 𝑓 (|𝑥 − 𝑦|, 𝑚𝑎𝑥|𝑥 − 𝑦|), 𝑦 ∈ 𝑌. euclidean distance calculates the distance between two real-valued vectors and it is the square root of the sum of the squared differences between the two vectors. using hamming distance, from a vector data to set of vectors data, we have a set of distance values and the final distance is concluded using hausdorff distance. thus, we have the hybridization for x ∈ x as 𝑑(𝑥, 𝑌) = 𝑓 ((𝑥 − 𝑦)2, max ((𝑥 − 𝑦)2)), 𝑦 ∈ 𝑌. where x and y are a set of vectors. algorithmic approach step 1. degree of measurements with pfs is noted for each subject. for each subject, interest, subject knowledge, memory, career, attitude, environment impact requirement are the characteristics. each characteristic value is quantified with four grounds i.e., the degree of positive potential, the degree of neutral potential, the degree of negative potential and the refusal degree. thus, we have 𝐴 = {(𝑥, 𝜇𝐴(𝑥), 𝜏𝐴(𝑥), 𝛾𝐴(𝑥)): 𝑥𝜖 𝑋}, where x be the set of subjects, 𝜇𝐴(𝑥), 𝜏𝐴(𝑥), 𝛾𝐴(𝑥) represent the degree of positive potential, the degree of neutral potential, and the degree of negative potential towards the stream respectively. the values of 𝜇𝐴(𝑥), 𝜏𝐴(𝑥), 𝛾𝐴(𝑥) lie in the close interval [0, 1] and satisfy the following condition. 0 ≤ 𝜇𝐴(𝑥) + 𝜏𝐴(𝑥) + 𝛾𝐴(𝑥) ≤ 1 the refusal degree for the stream x denoting 𝜌𝐴(𝑥) as follows. 𝜌𝐴(𝑥) = 1 − (𝜇𝐴(𝑥) + 𝜏𝐴(𝑥) + 𝛾𝐴(𝑥)) step 2. degree of measurements with pfs is noted for the students. thus, we have 𝐵 = {(𝑠, 𝜇𝐵 (𝑠), 𝜏𝐵 (𝑠), 𝛾𝐵 (𝑠)): 𝑠 𝜖 𝑆} where, s is the set of students, 𝜇𝐵 (𝑠), 𝜏𝐵 (𝑠), 𝛾𝐵 (𝑠) represent the degree of positive potential, the degree of neutral potential, and the degree of negative potential towards the subject respectively. the values of 𝜇𝐵 (𝑠), 𝜏𝐵 (𝑠), 𝛾𝐵 (𝑠) lie in the close interval [0, 1] and satisfy the following condition. 0 ≤ 𝜇𝐵 (𝑠) + 𝜏𝐵 (𝑠) + 𝛾𝐵 (𝑠) ≤ 1 the refusal degree for the subject s denoting 𝜌𝐵 (𝑠) is evaluated as follows. 𝜌𝐵 (𝑠) = 1 − (𝜇𝐵 (𝑠) + 𝜏𝐵 (𝑠) + 𝛾𝐵 (𝑠)) step 3. hybridized distance measures i.e., hybridization of hausdorff and hamming distance measures and hybridization of hausdorff and euclidean distance measures are calculated between student and subject. the subject features in terms of pfs are defined in step 1 and student caliber towards the subject in terms of pfs is defined in step 2. step 4. step 3 is repeated for all the subjects. then the subject having minimum distance is noted. step 5. if the subject distance measure from the student is sufficient to decide the career, then the subject having minimum distance is chosen for the career. otherwise, we have to follow the next step. https://en.wikipedia.org/wiki/euclidean_distance career selection of students using hybridized distance measure based on picture fuzzy set and rough set theory 113 step 6. if multiple subjects have the minimum distance or more than one subject is required to consider for choosing a career, we use rs theory for solving the fuzziness or inconsistency. step 6.1. all subjects (say set s) are approximated by constructing the b-lower approximation and b-upper approximations of s according to distance measures set b (the information as the distance measures from student to subjects) which are respectively stated as b𝑆 = {𝑥|[𝑥]𝐵 ⊆ 𝑆} and �̅�𝑆 = {𝑥|[𝑥]𝐵 ∩ 𝑆 ≠ ∅}. step 6.2. b-boundary region of s, bnb(s) = �̅�s – bs consists of the subjects which are not classified into s in b. step 6.3. b-outside region, u-�̅�s consists of the subjects which are not suitable for a student. step 6.4. rs theory is used when the boundary region is non-empty, where we the membership function values of the subjects are considered using probability and subjects with high membership function values are selected. 4. case study we have explained two case studies. in case study 1, we have illustrated the distance between students and subjects, then find out the minimum distance for choosing a stream. in case study 2, we have illustrated the inconsistency situation for choosing and to solve its rough set theory is implemented. case study 1: this case study is based on students’ distance measures from a stream for selecting the career using two different approaches i.e., hybridization of hausdorff and hamming distance measures and hybridization of hausdorff and euclidean distance measures. steps to be followed for the purpose is as follows. step 1. find the required calibers for the subjects. step 2. find the calibers of the student towards different subjects. step 3. measure the distance between different subjects from the student using the approaches hybridization of hausdorff & hamming distance measures and hybridization of hausdorff & euclidean distance measures. step 4. a suitable stream is selected for the student having minimum distance measure. in this case study, we have taken five subjects such as computer science(𝑥1), physics (𝑥2), chemistry (𝑥3), mathematics (𝑥4) and statistic (𝑥5) with their features as interest (𝑦1), subject knowledge (𝑦2), memory (𝑦3), career (𝑦4), attitude (𝑦5 ) and environment impact (𝑦6 ) as a potential requirement towards respective subjects as summarized in table-1. the evaluating values of the features concerning the subjects are represented in terms of pfs as 𝐴 = {(𝑥, 𝜇𝐴(𝑥), 𝜏𝐴(𝑥), 𝛾𝐴(𝑥)): 𝑥𝜖 𝑋}, where x be the set of streams, 𝜇𝐴(𝑥), 𝜏𝐴(𝑥), 𝛾𝐴(𝑥) represent the degree of positive potential, the degree of neutral potential, and the degree of negative potential towards the stream respectively. the values of 𝜇𝐴(𝑥), 𝜏𝐴(𝑥), 𝛾𝐴(𝑥) lie in the close interval [0, 1] and satisfy the following condition. 0 ≤ 𝜇𝐴(𝑥) + 𝜏𝐴(𝑥) + 𝛾𝐴(𝑥) ≤ 1 again, we have evaluated the refusal degree for the stream x denoting 𝜌𝐴(𝑥) as follows. 𝜌𝐴(𝑥) = 1 − (𝜇𝐴(𝑥) + 𝜏𝐴(𝑥) + 𝛾𝐴(𝑥)) sahu et al./decis. mak. appl. manag. eng. 4 (1) (2021) 104-126 114 table 1 presents the potentials of the stream towards the students on basis of features 𝑦1 , 𝑦2, 𝑦3, 𝑦4, 𝑦5𝑎𝑛𝑑 𝑦6 where each cell value of the table is the element of picture fuzzy set a and table 1 is called as feature-subject picture fuzzy relation r: s→ x. then, we have considered four students 𝑠1, 𝑠2, 𝑠3𝑎𝑛𝑑 𝑠4 and their potentials towards the subjects 𝑥1, 𝑥2, 𝑥3, 𝑥4𝑎𝑛𝑑 𝑥5 in terms of pfs b as follows. 𝐵 = {(𝑠, 𝜇𝐵 (𝑠), 𝜏𝐵 (𝑠), 𝛾𝐵 (𝑠)): 𝑠 𝜖 𝑆} where, s is the set of students, 𝜇𝐵 (𝑠), 𝜏𝐵 (𝑠), 𝛾𝐵 (𝑠) represent the degree of positive potential, the degree of neutral potential, and the degree of negative potential towards the stream respectively. t a b le 1 . f e a tu re -s u b je ct p ic tu re f u z z y r e la ti o n r e la ti o n r : (x y ) → [ 0 , 1 ] y 1 (i n te re st ) y 2 (s u b je ct k n o w le d g e ) y 3 (m e m o ry ) y 4 (c a re e r) y 5 (a tt it u d e ) y 6 (e n v ir o n m e n t im p a ct ) x 1 (c o m p u te r s ci e n c e ) (0 .4 ,0 ,0 ) (0 .3 ,0 .2 ,0 .4 ) (0 .1 ,0 .3 5 ,0 .5 ) (0 .4 ,0 .3 ,0 .2 ) (0 .1 ,0 .2 5 ,0 .5 ) (0 .7 ,0 .1 ,0 ) x 2 ( p h y si cs ) (0 .7 ,0 ,0 ) (0 .2 ,0 .4 ,0 .3 5 ) (0 ,0 .4 ,0 .5 ) (0 .7 ,0 .1 ,0 ) (0 .1 ,0 .3 ,0 .5 ) (0 .4 ,0 .2 ,0 .3 ) x 3 ( c h e m is tr y ) (0 .3 ,0 .4 ,0 .3 ) (0 .6 ,0 .2 ,0 .1 ) (0 .2 ,0 .3 ,0 .4 ) (0 .2 ,0 .3 5 ,0 .3 ) (0 .1 ,0 .2 ,0 .6 ) (0 .2 ,0 .1 ,0 .3 ) x 4 (m a th e m a ti cs ) (0 .1 ,0 .3 ,0 .5 ) (0 .2 ,0 .4 ,0 .3 ) (0 .8 ,0 ,0 ) (0 .2 ,0 .4 ,0 .3 ) (0 .2 ,0 .3 5 ,0 .3 ) (0 .6 ,0 ,0 .2 ) x 5 ( s ta ti st ic ) (0 .1 ,0 .3 ,0 .5 ) (0 ,0 .5 ,0 .3 5 ) (0 .2 ,0 .3 ,0 .5 ) (0 .2 ,0 .3 5 ,0 .4 ) (0 .8 ,0 ,0 .1 ) (0 ,0 .5 ,0 .3 5 ) career selection of students using hybridized distance measure based on picture fuzzy set and rough set theory 115 the values of 𝜇𝐵 (𝑠), 𝜏𝐵 (𝑠), 𝛾𝐵 (𝑠) lie in the close interval [0, 1] and satisfy the following condition. 0 ≤ 𝜇𝐵 (𝑠) + 𝜏𝐵 (𝑠) + 𝛾𝐵 (𝑠) ≤ 1 the refusal degree for the stream x denoting 𝜌𝐵 (𝑠) is evaluated as follows. 𝜌𝐵 (𝑠) = 1 − (𝜇𝐵 (𝑠) + 𝜏𝐵 (𝑠) + 𝛾𝐵 (𝑠)) ∆𝛾1 = |𝛾𝐴(𝑥1) − 𝛾𝐵 (𝑥1)| = |0.2 − 0.1| = 0.1 ∆𝜌1 = |𝜌𝐴(𝑥1) − 𝜌𝐵 (𝑥1)| = |0.1 − 0.5| = 0.4 for 𝑦5, ∆𝜇1 = |𝜇𝐴(𝑥1) − 𝜇𝐵 (𝑥1)| = |0.1 − 0.4| = 3, ∆𝜏1 = |𝜏𝐴(𝑥1) − 𝜏𝐵 (𝑥1)| = |0.25 − 0.4| = 0.15 ∆𝛾1 = |𝛾𝐴(𝑥1) − 𝛾𝐵 (𝑥1)| = |0.5 − 0.1| = 0.4 ∆𝜌1 = |𝜌𝐴(𝑥1) − 𝜌𝐵 (𝑥1)| = |0.15 − 0.5| = 0.35 for 𝑦6, ∆𝜇1 = |𝜇𝐴(𝑥1) − 𝜇𝐵 (𝑥1)| = |0.4 − 0.4| = 0, ∆𝜏1 = |𝜏𝐴(𝑥1) − 𝜏𝐵 (𝑥1)| = |0.3 − 0.4| = 0.1 ∆𝛾1 = |𝛾𝐴(𝑥1) − 𝛾𝐵 (𝑥1)| = |0.2 − 0.1| = 0.1 ∆𝜌1 = |𝜌𝐴(𝑥1) − 𝜌𝐵 (𝑥1)| = |0.1 − 0.5| = 0.4 1 6 (∑ ∆μi + ∆τi + ∆γi + ∆ρi 4 + max (∆μi, ∆τi, ∆γi, ∆ρi)) 6 i=1 = 1 6 { 0 + .4 + .1 + .5 4 + max(0, .4, .1, .5) + . 1 + .2 + .3 + 0 4 + 𝑚𝑎𝑥(. 1, .2, .3, 0) + . 3 + .05 + .4 + 0 4 + max(0.3, .05, 0.4, 0) + 0 + .1 + .1 + .4 4 + max(0, .1, .1, .4) + . 3 + .15 + .4 + .35 4 + max(. 3, .15, .4, .35) + 0 + .1 + .1 + .4 4 + max(0, .1, .1, .4) = 1 6 {. 25 + .5 + .15 + .3 + .1875 + .4 + .15 + .4 + .3 + .4 + .15 + .4} = 0.5979166667 ∑ (max{φi a6 i=1 , φi b} + |φi a − φi b| ) 6 = 1 6 {max(. 4, .9) +|. 4 − . 9| + max(. 9, .9) + |. 9 − .9| + max(. 95, .9) + |. 95 − .9| + max(. 9, .9) + |. 9 − .9| + max(. 85, .9) + |. 85 − .9| + max(. 8, .9) + |. 8 − .9|} = 1 6 {. 9 + .5 + .9 + .0 + .95 + .05 + .9 + 0 + .9 + .05 + .9 + .1} = 1.025 sahu et al./decis. mak. appl. manag. eng. 4 (1) (2021) 104-126 116 d1(a, b) = 1 n (∑ ∆μi + ∆τi + ∆γi + ∆ρi 4 + max(∆μi, ∆τi, ∆γi, ∆ρi)) n i=1 1 n (∑ ∆μi + ∆τi + ∆γi + ∆ρi 4 + max(∆μi, ∆τi, ∆γi, ∆ρi)) n i=1 + ∑ (max{φi an i=1 , φi b} + |φi a − φi b| ) n + 1 = 0.5979166667 0.5979166667 + 1.025 + 1 = 0.2278703839 thus, for student 𝑠1 and stream 𝑥1, the hausdorff and hamming distance is 0.2278703839. similarly, for other students and subjects, we have summarized in table 3. table 2. student-subject hybridization of hausdorff and hamming distance measures d1(a, b) 𝑥1 𝑥2 𝑥3 𝑥4 𝑥5 𝑠1 0.22787 0.218241 0.142857 0.193277 0.148241 𝑠2 0.224556 0.186441 0.172414 0.168111 0.183673 𝑠3 0.187817 0.201331 0.142857 0.145907 0.165217 𝑠4 0.170984 0.170887 0.211823 0.210526 0.215686 now we measure the distance of students’ potentials from streams’ required potentials using the metric of hausdorff and euclidean distance measure. for a, b belongs to pfs x, 𝑑2(𝐴, 𝐵) = ( 1 n ∑ ∆μi 2 + ∆τi 2 + ∆γi 2 + ∆ρi 2 4 + max(∆μi 2, ∆τi 2, ∆γi 2, ∆ρi 2))n i=1 ( 1 n ∑ ∆μi 2 + ∆τi 2 + ∆γi 2 + ∆ρi 2 4 + max(∆μi 2, ∆τi 2, ∆γi 2, ∆ρi 2))n i=1 + 1 n ∑ [max{φi a, φi b} + |φi a − φi b| 2 ] 1 2n i=1 + 1 where, for i = 1 to 4 ∆𝜇𝑖 = |𝜇𝐴(𝑥𝑖 ) − 𝜇𝐵 (𝑥𝑖 )|, ∆𝜏𝑖 = |𝜏𝐴(𝑥𝑖 ) − 𝜏𝐵 (𝑥𝑖 )|, ∆𝛾𝑖 = |𝛾𝐴(𝑥𝑖 ) − 𝛾𝐵 (𝑥𝑖 )|, ∆𝜌𝑖 = |𝜌𝐴(𝑥𝑖 ) − 𝜌𝐵 (𝑥𝑖 )|, ∅𝑖 𝐴 = |𝜇𝐴(𝑥𝑖 ) + 𝜏𝐴(𝑥𝑖 ) + 𝛾𝐴(𝑥𝑖 )|, ∅𝑖 𝐵 = |𝜇𝐵 (𝑥𝑖 ) + 𝜏𝐵 (𝑥𝑖 ) + 𝛾𝐵 (𝑥𝑖 )|. for student 𝑠1 ∈ 𝑆 let us consider for stream 𝑥1 ∈ 𝑋 where 𝑥1 stands for computer science. 𝑄(𝑠1, 𝑥1) = (0.4, 0.4, 0.1) 𝑤ℎ𝑒𝑟𝑒, 𝜇𝐵 (𝑥1) = 0.4, 𝜏𝐵 (𝑥1) = 0.4, 𝑎𝑛𝑑 𝛾𝐵 (𝑥1) = 0.1, 𝜌𝐵 (𝑥1) = 1 − (0.4 + 0.4 + 0.1) = 0.1 ∅1 𝐵 = |𝜇𝐵 (𝑥1) + 𝜏𝐵 (𝑥1) + 𝛾𝐵 (𝑥1)| = 0.9. 𝑅(𝑥1, 𝑦1) = (0.4, 0, 0) 𝑤ℎ𝑒𝑟𝑒, 𝜇𝐴(𝑥1) = 0.4, 𝜏𝐴(𝑥1) = 0, 𝑎𝑛𝑑 𝛾𝐴(𝑥1) = 0, 𝜌𝐴(𝑥1) = 1 − (0.4 + 0 + 0) = 0.6 ∅11 𝐴 = |𝜇𝐴(𝑥1) + 𝜏𝐴 (𝑥1) + 𝛾𝐴(𝑥1)| = 0.4. 𝑅(𝑥1, 𝑦2) = (0.3, 0.2, 0.4) 𝑤ℎ𝑒𝑟𝑒, career selection of students using hybridized distance measure based on picture fuzzy set and rough set theory 117 𝜇𝐴(𝑥1) = 0.3, 𝜏𝐴(𝑥1) = 0.2, 𝑎𝑛𝑑 𝛾𝐴(𝑥1) = 0.4, 𝜌𝐴(𝑥1) = 1 − (0.3 + 0.2 + 0.4) = 0.1 ∅12 𝐴 = |𝜇𝐴(𝑥1) + 𝜏𝐴 (𝑥1) + 𝛾𝐴(𝑥1)| = 0.9. 𝑅(𝑥1, 𝑦3) = (0.1, 0.34, 0.5) 𝑤ℎ𝑒𝑟𝑒, 𝜇𝐴(𝑥1) = 0.1, 𝜏𝐴(𝑥1) = 0.35, 𝑎𝑛𝑑 𝛾𝐴(𝑥1) = 0.5, 𝜌𝐴(𝑥1) = 1 − (0.1 + 0.35 + 0.5) = 0.05 ∅13 𝐴 = |𝜇𝐴(𝑥1) + 𝜏𝐴(𝑥1) + 𝛾𝐴(𝑥1)| = 0.95. 𝑅(𝑥1, 𝑦4) = (0.4, 0.3, 0.2) 𝑤ℎ𝑒𝑟𝑒, 𝜇𝐴(𝑥1) = 0.4, 𝜏𝐴(𝑥1) = 0.3, 𝑎𝑛𝑑 𝛾𝐴(𝑥1) = 0.2, 𝜌𝐴(𝑥1) = 1 − (0.4 + 0.3 + 0.2) = 0.1 ∅14 𝐴 = |𝜇𝐴(𝑥1) + 𝜏𝐴 (𝑥1) + 𝛾𝐴(𝑥1)| = 0.9. 𝑅(𝑥1, 𝑦5) = (0.1, 0.25, 0.5) 𝑤ℎ𝑒𝑟𝑒, 𝜇𝐴(𝑥1) = 0.1, 𝜏𝐴(𝑥1) = 0.25, 𝑎𝑛𝑑 𝛾𝐴(𝑥1) = 0.5, 𝜌𝐴(𝑥1) = 1 − (0.1 + 0.25 + 0.5) = 0.15 ∅15 𝐴 = |𝜇𝐴(𝑥1) + 𝜏𝐴 (𝑥1) + 𝛾𝐴(𝑥1)| = 0.85. 𝑅(𝑥1, 𝑦6) = (0.7, 0.1, 0) 𝑤ℎ𝑒𝑟𝑒, 𝜇𝐴(𝑥1) = 0.7, 𝜏𝐴(𝑥1) = 0.1, 𝑎𝑛𝑑 𝛾𝐴(𝑥1) = 0, 𝜌𝐴(𝑥1) = 1 − (0.7 + 0.1 + 0) = 0.2 ∅16 𝐴 = |𝜇𝐴(𝑥1) + 𝜏𝐴 (𝑥1) + 𝛾𝐴(𝑥1)| = 0.8. for 𝑦1, ∆𝜇1 = |𝜇𝐴(𝑥1) − 𝜇𝐵 (𝑥1)| = |0.4 − 0.4| = 0, ∆𝜏1 = |𝜏𝐴(𝑥1) − 𝜏𝐵 (𝑥1)| = |0 − 0.4| = 0.4 ∆𝛾1 = |𝛾𝐴(𝑥1) − 𝛾𝐵 (𝑥1)| = |0 − 0.1| = 0.1 ∆𝜌1 = |𝜌𝐴(𝑥1) − 𝜌𝐵 (𝑥1)| = |0.6 − 0.1| = 0.5 for 𝑦2, ∆𝜇1 = |𝜇𝐴(𝑥1) − 𝜇𝐵 (𝑥1)| = |0.3 − 0.4| = 0.1, ∆𝜏1 = |𝜏𝐴(𝑥1) − 𝜏𝐵 (𝑥1)| = |0.2 − 0.4| = 0.2 ∆𝛾1 = |𝛾𝐴(𝑥1) − 𝛾𝐵 (𝑥1)| = |0.4 − 0.1| = 0.3 ∆𝜌1 = |𝜌𝐴(𝑥1) − 𝜌𝐵 (𝑥1)| = |0.1 − 0.1| = 0 for 𝑦3, ∆𝜇1 = |𝜇𝐴(𝑥1) − 𝜇𝐵 (𝑥1)| = |0.1 − 0.4| = 0.3, ∆𝜏1 = |𝜏𝐴(𝑥1) − 𝜏𝐵 (𝑥1)| = |0.35 − 0.4| = 0.05 ∆𝛾1 = |𝛾𝐴(𝑥1) − 𝛾𝐵 (𝑥1)| = |0.5 − 0.1| = 0.4 ∆𝜌1 = |𝜌𝐴(𝑥1) − 𝜌𝐵 (𝑥1)| = |0.05 − 0.1| = 0.05 for 𝑦4, ∆𝜇1 = |𝜇𝐴(𝑥1) − 𝜇𝐵 (𝑥1)| = |0.4 − 0.4| = 0, ∆𝜏1 = |𝜏𝐴(𝑥1) − 𝜏𝐵 (𝑥1)| = |0.3 − 0.4| = 0.1 sahu et al./decis. mak. appl. manag. eng. 4 (1) (2021) 104-126 118 ∆𝛾1 = |𝛾𝐴(𝑥1) − 𝛾𝐵 (𝑥1)| = |0.2 − 0.1| = 0.1 ∆𝜌1 = |𝜌𝐴(𝑥1) − 𝜌𝐵 (𝑥1)| = |0.1 − 0.5| = 0.4 for 𝑦5, ∆𝜇1 = |𝜇𝐴(𝑥1) − 𝜇𝐵 (𝑥1)| = |0.1 − 0.4| = 3, ∆𝜏1 = |𝜏𝐴(𝑥1) − 𝜏𝐵 (𝑥1)| = |0.25 − 0.4| = 0.15 ∆𝛾1 = |𝛾𝐴(𝑥1) − 𝛾𝐵 (𝑥1)| = |0.5 − 0.1| = 0.4 ∆𝜌1 = |𝜌𝐴(𝑥1) − 𝜌𝐵 (𝑥1)| = |0.15 − 0.5| = 0.35 for 𝑦6, ∆𝜇1 = |𝜇𝐴(𝑥1) − 𝜇𝐵 (𝑥1)| = |0.4 − 0.4| = 0, ∆𝜏1 = |𝜏𝐴(𝑥1) − 𝜏𝐵 (𝑥1)| = |0.3 − 0.4| = 0.1 ∆𝛾1 = |𝛾𝐴(𝑥1) − 𝛾𝐵 (𝑥1)| = |0.2 − 0.1| = 0.1 ∆𝜌1 = |𝜌𝐴(𝑥1) − 𝜌𝐵 (𝑥1)| = |0.1 − 0.5| = 0.4 ( 1 6 ∑ ∆μi 2 + ∆τi 2 + ∆γi 2 + ∆ρi 2 4 + max(∆μi 2, ∆τi 2, ∆γi 2, ∆ρi 2)) n i=1 = 1 6 { 0 + .16 + .01 + .25 4 + max(0, .16, .01, .25) + . 01 + .04 + .09 + 0 4 + 𝑚𝑎𝑥(. 01, .04, .09, 0) + . 09 + .0025 + .16 + 0 4 + max(0.09, .0025, 0.16, 0) + 0 + .01 + .01 + .16 4 + max(0, .01, .01, .16) + . 09 + .0225 + .16 + .1225 4 + max(. 09, .0025, .16, .1225) + 0 + .01 + .01 + .16 4 + max(0, .01, .01, .16) = 1 6 {. 105 + .25 + .035 + .09 + .063125 + .16 + .045 + .16 + .09875 + .16 + .045 + .16} = .2286458333 1 6 ∑[max{φi a, φi b} + |φi a − φi b| 2 ] 1 2 n i=1 = 1 6 {{max(. 4, .9) +|. 4 − . 9|2} 1 2⁄ + {max(. 9, .9) + |. 9 − .9|2} 1 2⁄ + {max(. 95, .9) + |. 95 − .9|2} 1 2⁄ + {max(. 9, .9) + |. 9 − .9|2} 1 2⁄ + {max(. 85, .9) + |. 85 − .9|2} 1 2⁄ + {max(. 8, .9) + |. 8 − .9|2} 1 2⁄ } = 1 6 {1.15 1 2⁄ +. 9 1 2⁄ +. 9525 1 2⁄ +. 9 1 2⁄ +. 9025 1 2⁄ +. 91 1 2⁄ } = 0.974941232 d2(a, b) = career selection of students using hybridized distance measure based on picture fuzzy set and rough set theory 119 ( 1 n ∑ ∆μi 2 + ∆τi 2 + ∆γi 2 + ∆ρi 2 4 + max(∆μi 2, ∆τi 2, ∆γi 2, ∆ρi 2))n i=1 ( 1 n ∑ ∆μi 2 + ∆τi 2 + ∆γi 2 + ∆ρi 2 4 + max(∆μi 2, ∆τi 2, ∆γi 2, ∆ρi 2))n i=1 + 1 n ∑ [max{φi a, φi b} + |φi a − φi b| 2 ] 1 2n i=1 + 1 = . 2286458333 . 2286458333 + 0.974941232 + 1 = 0.103760744 thus, for student s1 and stream 𝑥1, the hausdorff and euclidean distance is 0.103760744. similarly, for other students and subjects, we have summarized in table 4. table 3. student-subject hybridization of hausdorff and euclidean distance measures distance ‘s’ to x 𝑥1 𝑥2 𝑥3 𝑥4 𝑥5 𝑠1 0. 10376 0.087871 0.038433 0.06533 0.051477 𝑠2 0.114139 0.061914 0.055452 0.068413 0.073627 𝑠3 0.071581 0.075949 0.041148 0.047902 0.0591 𝑠4 0.059973 0.058362 0.087786 0.086236 0.090134 finally, from table 3 and table 4, it is found students 𝑠1, 𝑠2, 𝑠3 have efficiency for chemistry (𝑥3) where student 𝑠4 have efficiency for physics (𝑥2 ) since they have minimum distance in respective subjects as highlighted in boldfaced. case study 2: this case study is the illustration of the extension approach of case study 1. we have used pfs with hybrid distance measure as well as rs theory for choosing a career to consider whether a stream is suitable for a student or not. pfs and hybrid distance measure have used for finding the distance between student's potential from the required potential for a stream. then the rs theory is implemented to choose the best stream from different options and criteria. first, we have used picture fuzzy set and hybrid distance measure and then rs theory with the distance measures to finalize the suitable stream. the steps have followed is summarized below. step 1. noted the required efficiency for the subjects. step 2. find out the data of students according to their efficiency towards the subjects. step 3. measure the distance between different subjects from the student using the approaches hybridization of hausdorff & hamming distance measures and hybridization of hausdorff & euclidean distance measures. step 4. use rs theory to categorize the students according to their distance measure and also found out the students are in fuzziness. steps 5. solve the fuzziness using the rule (c) of section 2.5. step 6. finalize the stream for the students. this case study is the study of 10 students ‘efficiency towards computer science say x1. it is found that choosing computer science as stream not only depends upon computer science (𝑥1), it also depends on efficiency in mathematics say 𝑥4and statistic say 𝑥5. it is noted in the previous case study that the requirements of efficiency for 𝑥1, 𝑥4𝑎𝑛𝑑 𝑥5 are defined in table 1. the sahu et al./decis. mak. appl. manag. eng. 4 (1) (2021) 104-126 120 efficiency of students towards 𝑥1, 𝑥4𝑎𝑛𝑑 𝑥5 of 10 students say 𝑠1, 𝑠2, 𝑠3, 𝑠4, 𝑠5, 𝑠6, 𝑠7, 𝑠8, 𝑠9, 𝑎𝑛𝑑 𝑠10 are summarized in table 5. table 4. students’ information towards computer science, mathematics and statistics. relation q (s, x) → [0,1] 𝑥1 (computer science) 𝑥4(mathematics) 𝑥5(statistic) 𝑠1 [0.33, 0.2414, 0.32] [0.2,0.4,0.3] [0.264,0.386,0.381] 𝑠2 [0.1,0.1,0.5] [0.4,0.2,0.2] [0.4,0.4,0.1] 𝑠3 [0.23,0.3,0.27] [0.1,0.4,0.5] [0.001,0.009,0.001] 𝑠4 [0.35,0.15,0.2] [0.4,0.3,0.3] [0.364,0.356,0.281] 𝑠5 [0.4,0.4,0.3] [0.38,0.4,0.2] [0.001,0.005,0.01] 𝑠6 [0.2,0.4,0.1] [0.4,0.4,0.2] [0.001,0.003,0.0] 𝑠7 [0.2,0.1,0.2] [0.4,0.4,0.1] [0.005,0.004,0.001] 𝑠8 [0.1,0.5,0.4] [0.1,0.1,0.4] [0.001,0.001,0.004] 𝑠9 [0.8,0.4,0.2] [0.05,0.1,0.04] [0.001,0.001,0.0] 𝑠10 [0.9,0.6,0.4] [0.0,0.95,0.05] [0.1,0.1,0.1] using hybridization of hausdorff & hamming distance measures and hybridization of hausdorff & euclidean distance measures on the tables 1 and 5 (as illustrated in case study 1), the distance of the students towards 𝑥1, 𝑥4 and 𝑥5 are calculated and they are summarized in table 6 and table 7 respectively. table 5. hybridization of hausdorff & hamming distance measures hamming distance 𝑥1 (computer science) 𝑥4 (mathematics) 𝑥5 (statistic) 𝑠1 0.16637 0.141324 0.16122 𝑠2 0.210526 0.168111 0.223301 𝑠3 0.177519 0.20398 0.402435 𝑠4 0.182283 0.175258 0.189107 𝑠5 0.215686 0.180608 0.401033 𝑠6 0.223301 0.185059 0.404215 𝑠7 0.230769 0.187817 0.402747 𝑠8 0.230769 0.23445 0.403653 𝑠9 0.340659 0.343365 0.404718 𝑠10 0.454545 0.347826 0.32299 career selection of students using hybridized distance measure based on picture fuzzy set and rough set theory 121 table 6. hybridization of hausdorff & euclidean distance measures euclidian distance 𝑥1(computer science) 𝑥4(mathematics) 𝑥5 (statistic) 𝑠1 0.05796 0.063871 0.054504 𝑠2 0.090645 0.04828 0.093741 𝑠3 0.059574 0.09732 0.356978 𝑠4 0.054177 0.053721 0.06565 𝑠5 0.091938 0.060448 0.354451 𝑠6 0.089632 0.061675 0.360454 𝑠7 0.110617 0.067327 0.357518 𝑠8 0.094258 0.111358 0.359446 𝑠9 0.227657 0.24884 0.361444 𝑠10 0.391903 0.268739 0.216263 here u = {𝑠1, 𝑠2, 𝑠3, 𝑠4, 𝑠5, 𝑠6, 𝑠7, 𝑠8, 𝑠9 , 𝑠10} be the set of students, a ={𝑥1, 𝑥2, 𝑥3} be the subjects namely computer science, mathematics and statistic respectively, and y stand for computer science as the decision variable. for α ∈ a, define α (𝑠𝑖 ), i = 1 to 10 as distance measure. our decision attribute y = computer science has criteria that for α = 𝑥1, vα =𝑡1, for α =𝑥4, vα =𝑡2 and for α = 𝑥5, vα = 𝑡3 where 𝑡1, 𝑡2 and 𝑡3 are the threshold values of corresponding subjects. the relation is defined as r1: x → y if vy ≤ 𝑡𝑐 . vy is the distance measure from x, 𝑡𝑐 is threshold value and if vy is less than equal to 𝑡𝑐 then it is interpreted that eligible for computer science. first considering for hybridization of hausdorff & hamming distance measures, when we have taken as r1: x → y with vy ≤ 0.3, y as computer science, vy ≤ 0.2, y as mathematics and vy ≤ 0.4, y as statistics then the conclusion is as follows. bx = {𝑠1, 𝑠2, 𝑠4} �̅�x = {𝑠1, 𝑠2, 𝑠3, 𝑠4, 𝑠5, 𝑠6, 𝑠7, 𝑠8, 𝑠10} bnb(x) = �̅�x – bx = {𝑠3, 𝑠5, 𝑠6, 𝑠7, 𝑠8, 𝑠10} b-outside region of, u �̅�x = {𝑠9} 𝜇𝑋 𝑅 : 𝑈 → [0,1] , where 𝜇𝑋 𝑅 (𝑥) = |𝑋∩𝑅(𝑥)| |𝑅(𝑥)| , and x ∈ x. |𝑅(𝑥)| = 3 r*(x) = {x ∈ u|𝜇𝑋 𝑅 (𝑥) = 1} = {𝑠1, 𝑠2, 𝑠4} |𝑋 ∩ 𝑅(𝑥)| = 3 r*(x) = {x ∈ u |𝜇𝑋 𝑅 (𝑥) = 0} = {𝑠1} |𝑋 ∩ 𝑅(𝑥)| = 0 rnr(x)= {x ∈ u |0 < 𝜇𝑋 𝑅 (𝑥) < 1} = {𝑠3, 𝑠5, 𝑠6, 𝑠7, 𝑠8, 𝑠10} |𝑋 ∩ 𝑅(𝑥)| = 1 𝑜𝑟 2 for 𝑠3, 𝑠8 𝑎𝑛𝑑 𝑠10, we have |𝑋 ∩ 𝑅(𝑥)| = 1 and for 𝑠5, 𝑠6 𝑎𝑛𝑑 𝑠7 , |𝑋 ∩ 𝑅(𝑥)| = 2. hence the membership function values for 𝑠3, 𝑠5, 𝑠6, 𝑠7, 𝑠8, 𝑎𝑛𝑑 𝑠10 are as follows. 𝜇𝑋 𝑅 (𝑠3) = 1 3 , 𝜇𝑋 𝑅 (𝑠5 ) = 2 3 , 𝜇𝑋 𝑅 (𝑠6) = 2 3 , 𝜇𝑋 𝑅 (𝑠7) = 2 3 , 𝜇𝑋 𝑅 (𝑠8) = 1 3 and 𝜇𝑋 𝑅 (𝑠10) = 1 3 again, considering for hybridization of hausdorff & euclidean distance measures, when we have taken as r1: x →y with vy ≤ 0.2 y as computer science, vy ≤ 0.09, y as mathematics and vy ≤ 0.3, y as statistics then the conclusion is as follows. bx = {𝑠1, 𝑠2, 𝑠4} sahu et al./decis. mak. appl. manag. eng. 4 (1) (2021) 104-126 122 �̅�x = {𝑠1, 𝑠2, 𝑠3, 𝑠4, 𝑠5, 𝑠6, 𝑠7, 𝑠8, 𝑠10} bnb(x) = �̅�x – bx = {𝑠3, 𝑠5, 𝑠6, 𝑠7, 𝑠8, 𝑠10} b-outside region of, u-�̅�x = {𝑠9} 𝜇𝑋 𝑅 : 𝑈 → [0,1] , where 𝜇𝑋 𝑅 (𝑥) = |𝑋∩𝑅(𝑥)| |𝑅(𝑥)| , and x ∈ x. |𝑅(𝑥)| = 3 r*(x) = {xєu|𝜇𝑋 𝑅 (𝑥) = 1} = {𝑠1, 𝑠2, 𝑠4} |𝑋 ∩ 𝑅(𝑥)| = 3 r*(x) = {xєu |𝜇𝑋 𝑅 (𝑥) = 0} = {𝑠9} |𝑋 ∩ 𝑅(𝑥)| = 0 rnr(x)= {xєu |0 < 𝜇𝑋 𝑅 (𝑥) < 1} = {𝑠3, 𝑠5, 𝑠6, 𝑠7, 𝑠8, 𝑠10} |𝑋 ∩ 𝑅(𝑥)| = 1 𝑜𝑟 2 for 𝑠3, 𝑠8 𝑎𝑛𝑑 𝑠10, we have |𝑋 ∩ 𝑅(𝑥)| = 1 and for 𝑠5, 𝑠6 𝑎𝑛𝑑 𝑠7, |𝑋 ∩ 𝑅(𝑥)| = 2. hence the membership function values for 𝑠3, 𝑠5, 𝑠6, 𝑠7, 𝑠8, 𝑎𝑛𝑑 𝑠10 are as follows. 𝜇𝑋 𝑅 (𝑠3) = 1 3 , 𝜇𝑋 𝑅 (𝑠5) = 2 3 , 𝜇𝑋 𝑅 (𝑠6) = 2 3 , 𝜇𝑋 𝑅 (𝑠7) = 2 3 , 𝜇𝑋 𝑅 (𝑠8) = 1 3 and 𝜇𝑋 𝑅 (𝑠10) = 1 3 thus, in both distance measure illustration, we have found, for students 𝑠3, 𝑠5, 𝑠6, 𝑠7, 𝑠8, 𝑎𝑛𝑑 𝑠10 are in inconsistent situations. we may follow the different rules summarized in section 2.4 for the situations. by removing fewer support cases (defined in section 2.2.4 rule c) and following the membership function values for 𝑠3, 𝑠5, 𝑠6, 𝑠7, 𝑠8, 𝑎𝑛𝑑 𝑠10, we are in conclusion that 𝑠5, 𝑠6 𝑎𝑛𝑑 𝑠7 are eligible for computer science whereas 𝑠3, 𝑠8 𝑎𝑛𝑑 𝑠10 are not. finally, we are in conclusion 𝑠1, 𝑠2, 𝑠4, 𝑠5, 𝑠6, 𝑎𝑛𝑑 𝑠7 may choose the stream and eligible for computer science whereas 𝑠3, 𝑠8, 𝑠9, 𝑎𝑛𝑑 𝑠10 are not. 5. comparative study fuzzy set-based approaches have also a good contribution on the students’ career selection process (natividad et al., 2019). in this paper, we have attempted to improve the students’ career selection process by incorporating more attributes regarding students’ career selection which are represented using pfns. these greater number of attributes are required to predict a more suitable decision in comparison to proposed fuzzy-based approach given in (natividad et al., 2019). we have interpreted the attributes using four concepts i.e., the degree of positive potential, the degree of neutral potential, the degree of negative potential and the refusal degree for each attribute. in (nguyen et al., 2018), authors studied fuzzy linguistic approach for multicriteria decision making by considering the interest of the student but practically along with interest other factors are also there. the proposed approach has considered the other factors also like subject knowledge, memory, career, attitude and environmental impact. in (peker et al., 2017), the authors have claimed that the students’ prior educational successes and teachers’ views are combinedly important to identify the students’ professional interests and capacities. in the process, the authors proposed a web-based system, namely web-cgs, which is modelled using mamdani fuzzy model where students’ interest are interpreted using the traditional methods of question-answering and evaluation by teachers, which may not be always accurate for selecting a career. in (nie et al., 2018), the authors have worked on the students' information like skill, regularity, economic status and subject interest, and trained that information using machine learning techniques for future forecasting, but the study does not interpret the skill, regularity, economic and interest exclusively which major an issue for accurate prediction. our proposed method analyses students’ information in terms of attitude, knowledge, interest, career, memory and career selection of students using hybridized distance measure based on picture fuzzy set and rough set theory 123 environment and expressed that information using pfs with the degree of positive potential, the negative potential, the degree of neutrality and refusal degree for both of the students and subjects. again, rs is implemented in the proposed approach when confusion arises in choosing stream, whereas none of the mentioned methods used it in the same context. 6. conclusion students are in confusion and feel difficulty in choosing the stream as their career after basic schooling. pfs and rs based approaches are useful in selecting the stream that will be appropriate for a student. the hybridization of hausdorff & hamming distance measures and hybridization of hausdorff & euclidean distance measures are proposed to find the distance between the student and subjects with the attributes interest, subject knowledge, memory, career, attitude and environment impact. the subject having the minimum distance from the student is chosen as the suitable and appropriate for the student. the rough set theory with lower approximation, higher approximation and boundary region is proposed to find out the inconsistency situations when a particular stream is taken into consideration. thus, we have proposed two models for choosing a career one for selecting a subject and another for selecting a subject in inconsistency situation and for both two case studies are illustrated. our proposed models have taken the subjects’ attributes values with the degree of positive membership, the degree of neutral membership, and the degree of negative membership when considering pfs. also, the models have considered the students’ attributes values with the degree of positive membership, the degree of neutral membership, and the degree of negative membership when considering pfs. finding the degree of positive membership, the degree of neutral membership and the degree of negative membership values are also challenging jobs. hence our future work will focus on to make a model to generate the degree of positive, the degree of neutral, and the degree of negative values when considering pfs. again, it is possible to extend with adding more attributes that influence the students on choosing streams. author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research received no external funding. conflicts of interest: the authors declare no conflicts of 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(2020). a novel approach to predictive analysis using attribute-oriented rough fuzzy sets. expert systems with applications, 161, 113644. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 4, issue 1, 2021, pp. 153-173. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame2104153m * corresponding author. e-mail addresses: zizovic@gmail.com (m. zizovic), dragan.pamucar@va.mod.gov.rs (d. pamucar), bole@ravangrad.net (b, miljkovic), karan972a@gmail.com (a, karan). multiple-criteria evaluation model for medical professionals assigned to temporary sars-cov-2 hospitals mališa zizovic 1, dragan pamucar 2*, boža d. miljković 3 and aleksandra r. karan 4 1 faculty of technical sciences in cacak, university of kragujevac, cacak, serbia 2 department of logistics, military academy, university of defence in belgrade, belgrade, serbia 3 faculty of education sombor, university of novi sad, sombor, serbia 4 general hospital "dr radivoj simonović", sombor, serbia received: 18 december 2020; accepted: 27 february 2021; available online: 13 march 2021. original scientific paper abstract: hospitals around the world, as health institutions with a key role in the health system, face problems while providing health services to patients with various types of diseases. currently, those problems are intensified due to the pandemic caused by sars-cov-2 virus. this pandemic has caused an extreme spread of the disease with constantly changing needs of patients which impacts the capacities and overall functioning of hospitals. in order to meet the challenge of the covid-19 (coronavirus disease2019) pandemic, health systems must adjust to new circumstances and establish separate hospitals exclusive for patients infected with sars-cov-2 virus. in the process of creating covid-19 hospitals, health systems face a shortage of medical professionals trained for work in covid-19 hospitals. using this as a starting point, this study puts forward a two-phase model for the evaluation and selection of nurses for covid-19 hospitals. each phase of the model features a separate multiple-criteria model. in the first phase, a multiple-criteria model with a dominant criterion is formed and candidates who meet the defined requirements are evaluated. in the second phase, a modified multiple-criteria model is formed and used to evaluate medical professionals who do not meet the requirements of the dominant criterion. by applying this model, two groups of medical professionals are defined: 1) medical professionals who completely meet the requirements for working in covid-19 hospitals and 2) medical professionals who require additional training. the criteria for evaluation of medical professionals in this multiple-criteria model are defined based on research conducted on medical professionals assigned to the covidmailto:zizovic@gmail.com mailto:zizovic@gmail.com mailto:dragan.pamucar@va.mod.gov.rs mailto:bole@ravangrad.net mailto:karan972a@gmail.com zizovic et al./decis. mak. appl. manag. eng. 4 (1) (2021) 153-173 154 19 crisis response team during the covid-19 pandemic in the republic of serbia. the model was tested on a real example of evaluating medical professionals assigned to the covid-19 hospital in sombor. the model for evaluating medical professionals presented in this paper can help decision makers in hospitals and national policy makers to determine the readiness level of hospitals for working in the conditions of the covid-19 pandemic, as well as underline the areas in which hospitals are not ready to meet the challenges of the pandemic. key words: covid-19 pandemic; health care service; multicriteria decision making. 1. introduction covid-19 (coronavirus disease2019) is a disease caused by a novel virus from the group of coronaviruses, first isolated in 1962. since then, it is known that some viruses from the coronavirus group, infect only certain animals, some humans, while some can breach the barrier between species, causing states ranging from a mild cold to severe acute respiratory syndrome (sars). a new, so far unknown coronavirus, sars-cov-2, the cause if covid-19 disease, belongs to the same subgroup as merscov and sars-cov, and was first detected in the chinese city of wuhan, in the province of huubei, the ground zero of the epidemic. who (world health organisation) declared a global pandemic in march 2020 and started a research mission into it. the covid-19 pandemic came as a surprise and brought with it many challenges to health systems all over the world, including our own. the immediate demand for staff training, repurposing structures, equipment acquisition, communication management, continuous supervision, healthcare availability to special groups of population under new conditions etc. has created numerous operational, logistical, organizational and ethical tasks for managers, medical professionals and associates. success and weaknesses of the health systems during the pandemic, is graded by the global health safety index, used for evaluating readiness in prevention, detection, fast response to high-risk environments, and following international protocols in new conditions. due to the specifics of the covid-19 pandemic, continuous analysis of all advantages and weaknesses of health systems and planning based on adaptive models of operation and providing services are necessary because the immediate demands and the unpredictability of the covid-19 infection requires agility and flexibility of all services in regards to timely responses, especially in the case of medical professionals. previous research on evaluating the readiness of hospitals and medical professionals for performance in crises and disasters is very limited. this applies especially to the application of multiple-criteria decision models in this field, which limited the analysis of already available literature. most authors consider the application of two concepts for solving this problem: 1) application of multiple-criteria decision making for evaluating readiness of hospitals in crises and 2) application of research based on surveys and statistical analysis. nekoie-moghadam et al. (2016) present a comprehensive review of literature with different methodologies used for evaluating hospitals for work in crises. in their review, they considered the most important topics that relate to evaluating health systems, such as logistics, planning, human resources, communication, management and control, training, evacuation, disaster recovery, coordination, transport, safety (fallah-aliabadi et al. 2020; verheul and duckers, 2020; nekoie-moghadam et al., 2016). of the 15 papers considered in multiple-criteria evaluation model for medical professionals assigned to temporary… 155 total by nekoie-moghadam et al. (2016), all consider the application of research based on statistical analysis of data collected in surveys. in some situations, authors used statistical methodologies, such as the delphi technique and similar tools (rezaei and mohebbi-dehnavi, 2019). a comprehensive review of literature that considers the application of statistical tools for hospital readiness analysis in crises is presented in fallah-aliabadi et al. (2020), verheul and duckers (2020), alruvaili et al. (2019) and nekoie-moghadam et al. (2016). in their study tabatabaei and abbasi (2016) carried out risk assessment during crises based on the hospital safety index. the safety index was defined based on comprehensive research with statistical data processing. samsuddin et al. (2018) identified key factors to determine the readiness level of hospitals for work in crises. the results have demonstrated that human resources, their training and ability to adapt in a timely fashion, are the crucial factor. marzaleh et al. (2019.) put forward an approach where the delphi technique is used to evaluate the readiness of emergency services in hospitals for working in crises. their study identified 31 criteria grouped in 3 clusters. the results demonstrate that training of medical professionals has the highest priority. a similar approach for analyzing hospital capacities using the delphi technique was demonstrated by shabanikiya et al. (2019.). however, aside from statistical processing of factor weight used for evaluating hospitals in crises, previous research also contains a number of papers based on multiple-criteria techniques. for example, mulyasari et al. (2013) have applied multiple-criteria techniques for ranking eight hospitals in iran. the ranking was carried out using factors grouped in four clusters, based on which hospital structural and functional readiness in crises was evaluated. following this study, hosseini et al. (2019.) developed a model for ranking hospitals based on the topsis (technique for order of preference by similarity to ideal solution) multiple-criteria technique. the study identifies 21 factors grouped in 4 clusters. however, this study used direct evaluation of survey participants instead of subjective/objective methods for determining weight coefficients of criteria. also, ortiz-barrios et al. (2017) demonstrate the possibility of applying analytical multiple-criteria approach for analyzing the readiness of particular wards/clinics in hospitals during crises. the approach is a hybrid model that consists of: 1) applying the ahp (analytic hierarchy process) and the dematel (decision making trial and evaluation laboratory) methods for determining weight coefficients of criteria and their mutual relationships and 2) applying the topsis method for evaluating hospital capacities. unlike previously listed studies, ortiz-barrios et al. (2017), in addition to defining weight coefficients of criteria, also put forward a methodology for defining mutual relationships and impact of factors used for evaluation. following ortiz-barrios et al. (2017), roy et al. (2018) identified key factors for evaluating hospital capacities using the dematel method. however, unlike ortiz-barrios et al. (2017), roy et al. (2018) used rough numbers to exploit the lack of certainty and precision found in expert preferences. considering that the crisis caused by the covid-19 pandemic is still ongoing, there is no research that considers the problem of evaluating the training of medical professionals for working in covid-19 hospitals. there is a limited number of papers that consider the application of multiple-criteria tools for solving problems caused by the covid-19 pandemic. sarkar (2020) maps the areas susceptible to covid-19 infection in bangladesh. the ahp method, in conjunction with gis (geographic information system) spatial analysis were used for area mapping. sangiorgio and parisi (2020) used artificial neural network and gis for mapping covid-19 infection risks in urban zones in italy. their study showed spatial analyses of a total of 257 city zizovic et al./decis. mak. appl. manag. eng. 4 (1) (2021) 153-173 156 districts. nardo et al. (2020) demonstrated the application of multi-criteria decision analysis for determining weights for eleven criteria in order to prioritize covid-19 non-critical patients for admission to hospitals in healthcare settings with limited resources. yildirim et al. (2020) evaluated the available covid-19 treatment options in hospitals. for their evaluation, they used modified promethee (preference ranking organization method for enrichment of evaluations) and vikor (visekriterijumska opitimizacija i kompromisno resenje in serbian) technique by applying fuzzy numbers. as we can see from the above-mentioned studies, there are numerous approaches relating to evaluating readiness of hospitals in crises. most papers contribute by putting forward methodological frames that require application of research featuring statistical processing of data collected via surveys. on the other hand, based on the above-mentioned literature, we can see the significance of multiple-criteria techniques for researching topics pertaining to evaluating readiness of hospital capacities in crises. we can note a wide spectrum of multiple-criteria techniques used in literature and applied in various fields. however, the number of papers that apply these methods for evaluating hospital capacities is limited. furthermore, the number of papers that consider the application of multiple-criteria methods for evaluating hospitals and medical professionals in the conditions of the covid-19 pandemic is especially limited. therefore, it is the aim of this paper to develop a multiple-criteria model for selection and evaluation of medical professionals for working in covid-19 hospitals in the conditions of a pandemic. the suggested model deals with assessing the training of medical professionals for working in the conditions of the covid-19 pandemic. furthermore, the suggested model has the following advantages that improve the literature relating to the application of multiple-criteria techniques in the field of healthcare: 1. an original, multiple-criteria methodology for evaluating propriety of medical professionals for working in covid-19 hospitals has been developed. the demonstrated methodology is conducted in two phases. a separate multiplecriteria model has been developed for each phase. the criteria and criteria value scales have been defined following months of research with the participation of medical professionals from the covid-19 crisis response team of the republic of serbia. 2. the demonstrated methodology is not limited to application in the healthcare field and is applicable in other fields due to its adaptability 3. the suggested methodology provides a new, clear and concise frame for resource management. in order to illustrate the effectiveness of the suggested methodology, an empirical study of the application of this multiple-criteria methodology is presented in the paper. 4. the approach presented in this paper can solve the problem of selection of medical professionals in the conditions of the covid-19 pandemic in a systematic and analytical way. the developed model was implemented and tested in a case study of the covid-19 hospital in sombor (serbia). 5. the model for evaluating medical professionals presented in this paper can help decision makers in hospitals and national policy makers to determine the readiness level of hospitals for working in the conditions of the covid-19 pandemic, as well as underline the areas in which hospitals are not ready to meet the challenges of the pandemic. the paper is structured into four sections. after the introduction that presents the problem and analyzes the existing literature, the second section mathematically formulates the multiple-criteria model for evaluating medical professionals in the multiple-criteria evaluation model for medical professionals assigned to temporary… 157 conditions of the covid-19 pandemic. the third section presents the implementation of the multiple-criteria model on a real example of evaluating medical professionals in the covid-19 hospital in sombor, the republic of serbia. the fourth section of the paper presents the conclusion and directions for future research. 2. multiple-criteria evaluation model for medical professionals assigned to covid hospitals let us assume a multiple-criteria model with defined criteria  1 2, ,...,j nc c c c where n stands for the number of criteria used in the multiple-criteria model. also let us assume a set of alternatives  1 2, ,...,i na a a a where m stands for the number of alternatives to be ranked in the model. we can define the decision matrix mxn whose elements ij a stand for value of j criterion for i alternative. table 1. decision matrix 1 c 2 c n c 1 a 11 a 12 a 1n a 2 a 21 a 22 a 2n a m a 1m a 2m a mn a all criteria of the set  1 2, ,...,j nc c c c were assigned weight coefficients  1 2, ,...,j nw w w w that meet the requirement of 1 1 n j j w   . let sc stand for the dominant criterion of the set  1 2, ,...,j nc c c c . in case of the dominant criterion not being met, then the alternative that does not meet it cannot be considered a solution to the problem. ranking of alternatives in the conditions of meeting or partially meeting the dominant criterion s c was considered by žižović et al. (2019). in multiplecriteria models with a dominant criterion, there is a problem where it is necessary to choose more than one alternative, for example p alternative (p<m) while criterion cs is met only with q alternatives, where q<p. this case is characteristic of crises where it is necessary to choose a certain number of candidates (alternatives) who are not sufficiently prepared for working in such conditions. this example considers the crisis caused by sars-cov-2 virus. this paper develops a model that makes it possible to evaluate medical professionals for assignment in covid-19 hospitals. the multiplecriteria evaluation model is conducted in two phases, figure 1. zizovic et al./decis. mak. appl. manag. eng. 4 (1) (2021) 153-173 158 experts opinion identification of criteria defining a modified set of criteria for evaluation p h a se 1 : m u lt ip le -c ri te ri a m o d e l w it h a d o m in a n t c ri te ri o n p h a se 2 : m u lt ip le -c ri te ri a m o d e l fo r a d d it io n a l e v a lu a ti o n o f m e d ic a l p ro fe ss io n a ls identification of alternatives defining criteria set for evaluation identifying the dominant criterion for evaluation ndsl model determining weight coefficients normalization of initial decision matrix evaluation of medical professionals additional training of candidates forming a initial decision matrix ndsl model determining weight coefficients normalization of preliminary decision matrix evaluation of medical professionals candidate meets the dominant criterion yes no ranking medical professionals selection of candidates and start of work in covid-19 hospitals forming a list of replacement candidates ready for assignment figure 1. model for evaluating medical professionals for assignment in covid-19 hospitals the model is conducted in two phases. a separate multiple-criteria model has been developed for each phase. for evaluating the training of medical professionals in the first phase, a multiple-criteria method with a dominant criterion has been developed. medical professionals who meet the requirements of the evaluation defined by the dominant criterion are directly assigned to covid-19 hospitals. in the second phase, medical professionals who do not meet the requirements of the first phase are evaluated. for evaluating medical professionals in the second phase, a simple additive weighting operator was used. medical professionals selected during the second phase undergo additional training for work in a covid-19 hospital. below, we present the multiple-criteria models in the first and second phases. first phase: the first phase requires defining the dominant criterion s c of the set  1 2, ,...,j nc c c c and formation of multiple-criteria model with a dominant criterion. on the basis of the defined multiple-criteria model, we carry out a selection of medical professionals who meet the criterion cs who are directly assigned to a covid-19 hospital upon completed evaluation. the multiple-criteria model with a dominant criterion is conducted in five steps below. multiple-criteria evaluation model for medical professionals assigned to temporary… 159 step 1: defining the set of criteria  1 2, ,...,j nc c c c for evaluating medical professionals and identifying the dominant criterion s c from the criteria set jc ( 1, 2,...,j n ). step 2: determining weight coefficients of the criteria j w ( 1, 2,...,j n ), 1 1 n j j w   from the defined criteria set j c ( 1, 2,...,j n ). in this paper, the authors used nondecreasing series at criteria significance levels (ndsl) (zizovic et al., 2020; pamucar et al., 2020a) model for determining weight coefficients of criteria, figure 2. the determination of the most significant criterion from within the set of the criteria  1 2, , , ns c c c the ranking of the criteria in accordance with the experts’ preferences: 1 2 ... n c c c   the grouping of the criteria according to the significance levels: level l1 : level l2 : level lk : ...  [1, 2), 1, 2, ,ic i n   [2, 3), 1, 2, ,ic i n   [ , 1), 1, 2, ,ic k k i n   defining the boundary values of the criteria significance(s): level l1 : level l2 : level lk : ... 3 2 i n n  0 3 i n  ( 1) ( 1) ( 2) x n k k n k k       defining the functions of the criteria significance(s) the calculation of the optimal values of the weight coefficients of the criteria figure 2. the ndsl model (zizovic et al., 2020) zizovic et al./decis. mak. appl. manag. eng. 4 (1) (2021) 153-173 160 a detailed presentation of phases of the ndsl model can be found in (zizovic et al., 2020). step 3: forming a preliminary decision matrix ij m n x x      where we chose p ( 1 p m  ) candidates for assignment to a covid-19 hospital. step 4: forming a normalized decision matrix ij m n x x      ( 1, 2,...,i m , 1, 2,...,j n ). by applying expressions (1) and (2) matrix elements x are normalized, i.e. translated into intervals [0,1], as per the following: a) for criteria j c ( 1, 2,...,j n ) max type (higher value is better):  max ij ij ij i x x x  (1) b) for criteria jc ( 1, 2,...,j n ) min type (lower value is better):  min ij i ij ij x x x  (2) where ijx stands for elements of the preliminary decision matrix ij m n x x      . step 5: evaluation and selection of candidates that meet additional criterion s c . the evaluation is conducted by applying expression (3) (pamucar et al., 2020b, 2020c). if we assume that we chose as the dominant criterion, the final grade of the candidate is defined by applying the following expression:   1 1 2 31 2 3( ... )i i i i ini nv a x w x x w x w x w          (3) where ijx ( 1, 2,...,i m , 1, 2,...,j n ) stands for elements of the normalized decision matrix ij m n x x      . also, in the expression (3) values ijx are values which alternative i a has according to criterion jc ( 1, 2,...,j n ); jw is the weight coefficient of criterion jc ( 1, 2,...,j n );  is the coefficient that determines the importance relation between the first dominant criterion and other criteria. if the number of candidates who meet the dominant criterion is higher than the number of needed candidates, then the first p of candidates ranked by  iv a , expression (3), is selected, and there is no need for the second phase of the model. however, if the number of the needed candidates who meet the dominant criterion is lower than p, the second phase of the model is carried out. the remaining (q) candidates, who do not meet the dominant criterion, are ranked in the second phase, presented below. second phase: forming a modified multiple-criteria model. in the modified multiple-criteria model, we adapt the starting set of criteria and evaluate q candidates as per newly defined set of criteria. let us assume that the first criterion is dominant, i.e. that 1 s c c . then we form an adapted set of criteria where the dominant criterion 1 c is replaced with * 1 c . the remaining criteria from the set jc ( 1, 2,...,j n ) can also be replaced with new criteria or remain unchanged. thus, we get a newly defined set of criteria  * * * *1 2, ,...,j nc c c c . the modified multiple-criteria model with a newly defined set of criteria is conducted in four steps presented below. multiple-criteria evaluation model for medical professionals assigned to temporary… 161 step 1: defining the criteria set  * * * *1 2, ,...,j nc c c c and defining the weight coefficients of the newly formed criteria set * j w ( 1, 2,...,j n ), * 1 1 n j j w   . as in the first phase (step 2), here too, we use the ndsl model for determining weight coefficients of criteria (behzad et al., 2020), slika 1. step 2: forming the preliminary decision matrix . ij m n y y      . where we select q ( 1 q m  ) candidates for assignment to a covid-19 hospital. step 3: forming the normalized decision matrix ij m n y y      ( 1,2,...,i m , 1,2,...,j n ). elements of the normalized decision matrix y are determined by applying expressions (4) and (5): a) for criteria *jc ( 1,2,...,j n ) max type (higher value is better):  max ij ij ij i y y y  (4) b) for criteria *jc ( 1,2,...,j n ) min type (lower value is better):  min ij i ij ij y y y  (5) where ij y stands for elements of the preliminary decision matrix ij m n y y      . step 4: evaluation and selection of candidates. each candidate is assigned a value of the criteria function based on which the evaluation is carried out. the criteria function is defined by applying the following expression (6):  * * * 1 m i jij i v a y w    (6) where ij y ( 1,2,...,i m , 1,2,...,j n ) stands for elements of the normalized decision matrix ij m n y y      . in case that    x yv a v a we determine the order of the criteria (1) (2) (8) ... r r r c c c   according to their importance and we reevaluate without the worst criterion (8)r c , while considering the remaining criteria. if, after eliminating the worst criterion, we arrive at  xv a >  yv a , we select the candidate  xv a . 3. results this chapter considers the case study of organizing hospitals specially prepared for the admission of patients suffering from a sars-cov-2 infection. such hospitals in the republic of serbia were organized as separate parts of existing hospital capacities or were repurposed as entire hospitals to admit only patients diagnosed with covid19 during the crisis. simultaneously, there was a need for medical professionals trained for working in the newly formed medical institutions. there was a need for priority treatment of patients suffering from a largely unknown, high-risk disease, with too few qualified medical professionals in the field. additionally, there was a need for selection of qualified medical professionals to carry out the listed duties, as well as simultaneously train other medical professionals that, at the moment, were not trained for assignment to covid-19 hospitals. zizovic et al./decis. mak. appl. manag. eng. 4 (1) (2021) 153-173 162 this study presents the application of the model for evaluating medical professionals for working in covid-19 hospitals on the example of the covid-19 hospital in sombor, republic of serbia. the study features part of the general hospital in sombor repurposed for treatment of covid-19 patients. eight criteria were identified that were used for evaluating nurses in two phases of the model. criteria and criteria value scales for evaluation of medical professionals were defined based on a survey with members of the covid-19 crisis response team and with medical doctors who participated in the operation of the covid-19 hospital in sombor. all candidates for assignment in covid-19 hospitals were psychologically tested and interviewed by teams from the covid-19 crisis response team (teams of medical doctors). data obtained in this way served to form evaluations for criteria. below, we present eight criteria with their value scales. the presented criteria are applied for evaluation in the first phase of the model. c1 experience in working with infectious and pulmonary diseases in hospital treatment (c1). value scale:  nurse with work experience at an infectious or pulmonary diseases ward in hospital treatment 5 points;  nurse with work experience at internal medicine ward in hospital treatment 4 points;  nurse with work experience at a different ward in hospital treatment 2 points;  nurse with no work experience in hospital treatment 1 point. c2 professional training of the candidate for working with covid-19 diagnosed (covid+) patients in “covid-19” zone:  professional training for immediate work with covid-19+ patients, use of personal protective equipment and materials in covid-19 conditions, knowledge of work organization in covid-19 conditions, training for movement through safety zones 5 points  professional training for immediate work with covid-19+ patients, use of personal protective equipment and materials in covid-19 conditions, knowledge of work organization in covid-19 conditions 4 points;  professional training for immediate work with covid-19+ patients, use of personal protective equipment and materials in covid-19 conditions 3 points;  professional training for immediate work with covid-19+ patients 2 points;  no professional training 0 points. c3 health risk of the candidate:  no health risk 10 points;  diseases and injuries that do not significantly affect ability 8 points;  physical injuries that partially affect mobility 7 points;  lower risk chronic diseases in covid-19 conditions 6 points;  single parent of a child up to 12 years old 3 points;  parent of a child up to 3 years old 2 points;  age (over 60) 1 point;  chronic diseases like: diabetes, psychosomatic diseases, autoimmune diseases, malignancy, diseases or treatment with a negative influence on the immune system, pregnancy 0 points. c4 physical evaluation of candidates for work in difficult working conditions and in shifts: multiple-criteria evaluation model for medical professionals assigned to temporary… 163  very capable 5 points;  capable 4 points;  partially capable 3 points;  barely capable 1 point. c5 motivation of candidates for working in covid-19 conditions:  very motivated 5 points;  motivated 4 points;  somewhat motivated 2 points;  barely motivated 1 point;  not motivated 0 points. c6 availability of candidate to the workplace:  easily available 5 points;  available 4 points;  poorly available 3 points;  very poorly available 1 point. c7 candidate reliability:  very reliable 5 points;  reliable 4 points;  somewhat reliable 2 points;  unreliable 1 point. c8 candidate’s vaccination history:  mandatory vaccination (with bcg+polio) 5 points;  mandatory vaccination (without bcg/polio) 4 points;  basic vaccination – 2 points;  none 1 point. after defining the criteria, below we present the application of the model on the selection of medical professionals. first phase: defining an additional criterion from the set of criteria and forming a multiple-criteria model with a dominant criterion. step 1: from the defined set of criteria  1 2 8, ,...,jc c c c for evaluating medical professionals, criterion 1 c was defined as the dominant criterion. all candidates who do not meet the conditions defined by criterion 1 c move on to the second phase of evaluation, while ranking of candidates who meet criterion 1 c is done in the first phase. step 2: determining criteria weight coefficients jw ( 1, 2,...,8j  ). as noted above, for determining criteria weight coefficients we used the ndsl model (žižović et al., 2020). determining criteria weight coefficients using the ndsl model is presented below in steps 2.1 2.5. step 2.1: determining the most important criterion from the criteria set  1 2 8, ,...,jc c c c and ranking criteria. as c1 was defined as the dominant criterion, c1 is also the most important criterion in the set jc . based on evaluations of experts, the criteria from the set jc were ranked as follows: c1>c7>c3>c4>c5>c6>c2>c8. step 2.2: grouping criteria into significance levels. the criteria were grouped into four levels, as follows: level 1 l : 1c , level 2 l : 3, 4, 5, 7c c c c , zizovic et al./decis. mak. appl. manag. eng. 4 (1) (2021) 153-173 164 level 3 l : 2, 6, 8c c c . step 2.3: based on relations for defining border values of the criteria ( ) the values i  were defined for criteria relating to significance levels: 5 1 2 1 7 3 4 6 2 83 0 17; 19; 20; 24; 26; 28; 29. : : : level l level l level l                 the values of border values of criteria significance ( i  ) were defined for the value 50n  . step 2.4: criteria significance functions ( ) i f c , 1, 2,...,8i  , were defined based on relation    ( ) /i i if c n n   : 1 3 7 4 5 6 2 8 ( ) 1.000; ( ) 0.493; ( ) 0.449; ( ) 0.429; ( ) 0.351; ( ) 0.316; ( ) 0.282; ( ) 0.266. f c f c f c f c f c f c f c f c         step 2.5: based on defined values ( ) i f c , 1, 2,...,8i  we arrive at values of criteria weight coefficients, table 2. table 2. criteria weight coefficients criterion wj c1 0.279 c2 0.079 c3 0.137 c4 0.119 c5 0.098 c6 0.088 c7 0.125 c8 0.074 step 3: forming the preliminary decision matrix. the crisis response team carried out the evaluation of medical professionals from the pulmonology and infectious wards of the general hospital in sombor. these two wards counted in total 43 nurses who were candidates for working in the covid-19 hospital, i.e. for working in the “red” and “orange” zones. a selection of 29 nurses was considered of the available 43 candidates. based on the defined evaluation criteria and available number of medical professionals, the preliminary decision matrix was formed 43 8 ij x x      , table 3. table 3. preliminary decision matrix – first phase alt c1 c2 c3 c4 c5 c6 c7 c8 a1 5 5 10 5 4 4 5 5 a2 2 5 10 5 5 4 5 5 a3 5 5 10 5 5 4 4 5 a4 1 2 0 1 1 5 1 5 a5 5 5 10 4 5 5 5 4 a6 5 5 10 3 5 5 5 5 a7 5 5 7 3 5 3 5 5 multiple-criteria evaluation model for medical professionals assigned to temporary… 165 alt c1 c2 c3 c4 c5 c6 c7 c8 a8 2 5 7 5 2 4 4 5 a9 5 5 6 4 4 5 5 2 a10 5 5 8 4 5 5 5 5 a11 5 5 10 5 2 4 2 5 a12 5 5 2 5 4 4 5 5 a13 1 5 10 5 5 5 5 5 a14 5 5 6 5 0 3 2 5 a15 5 5 10 4 5 3 5 5 a16 1 0 0 1 0 5 5 5 a17 5 5 2 5 2 5 2 5 a18 1 5 3 5 2 1 4 5 a19 5 5 10 5 5 5 5 2 a20 5 5 6 3 2 3 4 5 a21 5 5 10 5 5 5 5 5 a22 1 4 8 4 2 5 5 5 a23 2 4 7 3 0 5 5 4 a24 2 3 7 3 1 5 5 5 a25 2 3 7 3 1 5 5 5 a26 5 5 6 4 5 3 4 5 a27 5 3 1 1 1 5 5 5 a28 5 2 1 3 1 5 5 2 a29 2 3 1 1 1 5 5 2 a30 5 5 2 3 2 1 5 5 a31 2 5 10 5 5 5 5 5 a32 5 5 7 3 5 5 1 4 a33 5 5 10 5 5 5 5 4 a34 2 5 8 5 5 1 4 4 a35 5 5 10 5 4 5 2 5 a36 5 5 10 5 5 5 5 5 a37 5 5 10 5 5 4 4 5 a38 2 5 6 4 4 3 4 5 a39 1 5 10 5 5 5 5 5 a40 2 5 10 5 5 4 5 5 a41 2 5 6 4 2 5 4 5 a42 5 5 10 5 5 1 5 5 a43 5 5 10 5 5 5 5 5 step 4: normalization of elements of the preliminary decision matrix. since all the considered criteria fall under the max type (higher value is better), for the normalization of values we used the expression (1). the elements of the normalized matrix are shown in table 4. table 4. normalized decision matrix – first phase alt. c1 c2 c3 c4 c5 c6 c7 c8 a1 1.0 1.0 1.0 1.0 0.8 0.8 1.0 1.0 a2 0.4 1.0 1.0 1.0 1.0 0.8 1.0 1.0 a3 1.0 1.0 1.0 1.0 1.0 0.8 0.8 1.0 a4 0.2 0.4 0.0 0.2 0.2 1.0 0.2 1.0 a5 1.0 1.0 1.0 0.8 1.0 1.0 1.0 0.8 a6 1.0 1.0 1.0 0.6 1.0 1.0 1.0 1.0 zizovic et al./decis. mak. appl. manag. eng. 4 (1) (2021) 153-173 166 alt. c1 c2 c3 c4 c5 c6 c7 c8 a7 1.0 1.0 0.7 0.6 1.0 0.6 1.0 1.0 a8 0.4 1.0 0.7 1.0 0.4 0.8 0.8 1.0 a9 1.0 1.0 0.6 0.8 0.8 1.0 1.0 0.4 a10 1.0 1.0 0.8 0.8 1.0 1.0 1.0 1.0 a11 1.0 1.0 1.0 1.0 0.4 0.8 0.4 1.0 a12 1.0 1.0 0.2 1.0 0.8 0.8 1.0 1.0 a13 0.2 1.0 1.0 1.0 1.0 1.0 1.0 1.0 a14 1.0 1.0 0.6 1.0 0.0 0.6 0.4 1.0 a15 1.0 1.0 1.0 0.8 1.0 0.6 1.0 1.0 a16 0.2 0.0 0.0 0.2 0.0 1.0 1.0 1.0 a17 1.0 1.0 0.2 1.0 0.4 1.0 0.4 1.0 a18 0.2 1.0 0.3 1.0 0.4 0.2 0.8 1.0 a19 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.4 a20 1.0 1.0 0.6 0.6 0.4 0.6 0.8 1.0 a21 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 a22 0.2 0.8 0.8 0.8 0.4 1.0 1.0 1.0 a23 0.4 0.8 0.7 0.6 0.0 1.0 1.0 0.8 a24 0.4 0.6 0.7 0.6 0.2 1.0 1.0 1.0 a25 0.4 0.6 0.7 0.6 0.2 1.0 1.0 1.0 a26 1.0 1.0 0.6 0.8 1.0 0.6 0.8 1.0 a27 1.0 0.6 0.1 0.2 0.2 1.0 1.0 1.0 a28 1.0 0.4 0.1 0.6 0.2 1.0 1.0 0.4 a29 0.4 0.6 0.1 0.2 0.2 1.0 1.0 0.4 a30 1.0 1.0 0.2 0.6 0.4 0.2 1.0 1.0 a31 0.4 1.0 1.0 1.0 1.0 1.0 1.0 1.0 a32 1.0 1.0 0.7 0.6 1.0 1.0 0.2 0.8 a33 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.8 a34 0.4 1.0 0.8 1.0 1.0 0.2 0.8 0.8 a35 1.0 1.0 1.0 1.0 0.8 1.0 0.4 1.0 a36 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 a37 1.0 1.0 1.0 1.0 1.0 0.8 0.8 1.0 a38 0.4 1.0 0.6 0.8 0.8 0.6 0.8 1.0 a39 0.2 1.0 1.0 1.0 1.0 1.0 1.0 1.0 a40 0.4 1.0 1.0 1.0 1.0 0.8 1.0 1.0 a41 0.4 1.0 0.6 0.8 0.4 1.0 0.8 1.0 a42 1.0 1.0 1.0 1.0 1.0 0.2 1.0 1.0 a43 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 step 5: evaluation and selection of medical professionals who meet the dominant criterion 1 c . medical professionals who have the value of the dominant criterion 1 4c  enter the evaluation process in the first phase of selection. border value for the dominant criterion is defined based on the evaluation of experts. the remaining candidates who do not meet the requirement move on to the second phase of selection. based on the analysis of the preliminary decision matrix (table 3) we see that 26 candidates meet the requirement of 1 4c  . by applying the expression (3) we evaluate and rank the candidates, table 5. multiple-criteria evaluation model for medical professionals assigned to temporary… 167 table 5. grades of candidates who meet the requirement of the dominant criterion evaluation function v(ai) rank v(a21)=0.279+λ 0.72 0.3690 1 v(a36)=0.279+λ 0.72 0.3690 1 v(a43)=0.279+λ 0.72 0.3690 1 v(a33)=0.279+λ 0.705 0.3672 4 v(a1)=0.279+λ 0.683 0.3644 5 v(a5)=0.279+λ 0.681 0.3642 6 v(a3)=0.279+λ 0.677 0.3637 7 v(a37)=0.279+λ 0.677 0.3637 7 v(a19)=0.279+λ 0.676 0.3635 9 v(a6)=0.279+λ 0.672 0.3631 10 v(a10)=0.279+λ 0.669 0.3626 11 v(a15)=0.279+λ 0.661 0.3616 12 v(a42)=0.279+λ 0.65 0.3602 13 v(a35)=0.279+λ 0.625 0.3572 14 v(a7)=0.279+λ 0.596 0.3535 15 v(a26)=0.279+λ 0.581 0.3517 16 v(a9)=0.279+λ 0.577 0.3512 17 v(a12)=0.279+λ 0.573 0.3507 18 v(a11)=0.279+λ 0.569 0.3501 19 v(a32)=0.279+λ 0.517 0.3436 20 v(a20)=0.279+λ 0.499 0.3413 21 v(a17)=0.279+λ 0.477 0.3386 22 v(a14)=0.279+λ 0.457 0.3361 23 v(a30)=0.279+λ 0.434 0.3332 24 v(a27)=0.279+λ 0.392 0.3279 25 v(a28)=0.279+λ 0.379 0.3264 26 v(a31)=0.112+λ 0.288 0.1476 27 v(a2)=0.112+λ 0.281 0.1467 28 v(a40)=0.112+λ 0.281 0.1467 28 v(a34)=0.112+λ 0.233 0.1407 30 v(a8)=0.112+λ 0.231 0.1405 31 v(a38)=0.112+λ 0.225 0.1397 32 v(a41)=0.112+λ 0.223 0.1395 33 v(a24)=0.112+λ 0.209 0.1377 34 v(a25)=0.112+λ 0.209 0.1377 34 v(a23)=0.112+λ 0.201 0.1367 36 v(a29)=0.112+λ 0.139 0.1290 37 v(a13)=0.056+λ 0.144 0.0738 38 v(a39)=0.056+λ 0.144 0.0738 38 v(a22)=0.056+λ 0.119 0.0707 40 v(a18)=0.056+λ 0.094 0.0675 41 v(a16)=0.056+λ 0.062 0.0636 42 v(a4)=0.056+λ 0.052 0.0624 43 zizovic et al./decis. mak. appl. manag. eng. 4 (1) (2021) 153-173 168 an example of defining a candidate grade  1v a , on the condition that 0.125  (expression (3)), is given below:   1.0 0.279 1.0 0.125(1.0 0.079 1.0 0.137 ... 1.0 0.074) 0.364iv a            in a similar way, the remaining candidates’ grades were attained, as shown in table 5. the mandatory criteria were met by 26 candidates, who were ranked in table 5. since 29 candidates were needed for the covid-19 hospital, the remaining 17 candidates were ranked in the second phase of the model with the aim of selecting 3 additional, most appropriate candidates. second phase: forming a modified multiple-criteria model. in the modified multiple-criteria model, we adjust the starting set of criteria. criteria c1 and c2 are eliminated and in their place, we insert new ones: * 1 c evaluation of speed of acquiring knowledge and skills for working in covid-19 institutions:  very quick and safe start of work activities – 5 points;  sufficiently quick and safe start of work activities – 4 points;  satisfactorily quick and safe start of work activities – 3 points;  slow but safe start of work activities – 2 points;  slow and barely safe start of work activities – 1 point. * 2 c school grades in subjects close to the needs of the position. for this criterion, an average grade was taken, from the interval [2, 5]. the remaining criteria were unchanged, same as in the first phase, i.e. * 1 c , * 2 c , * 3 3 c c , * 4 4 c c , * 5 5 c c , * 6 6 c c , * 7 7 c c , * 8 8 c c . this forms the final set of criteria used for the evaluation of medical professionals in the second phase  * * * *1 2 8, ,...,jc c c c . the modified multiple-criteria model with a newly defined criteria set * j c ( 1, 2,...,8j  ) is conducted in four steps presented below. step 1: defining the criteria set  * * * *1 2 8, ,...,jc c c c and calculation of newly formed set of criteria weight coefficients * j w ( 1, 2,...,8j  ). similar to the first phase (step 2), weight coefficients are defined using the ndsl model (žižović et al., 2020). step 1.1: based on the evaluation of experts, the criteria from the set  * * * *1 2 8, ,...,jc c c c were ranked as follows: c1*>c7*>c4*>c6*>c3*>c8*>c5*>c2*. step 1.2: the criteria were grouped into sets of four levels, as follows: level 1 l : * * *1 7 4, ,c c c , level 2 l : * *6 3,c c , level 3 l : * *8 5,c c , level 4 l : *2c . step 1.3: based on relations for defining border values of criteria significance ( i  ) we define the values i  for criteria in significance levels: 2 1 7 4 6 3 8 5 2 1 3 4 0; 5; 10 18; 20; 26; 28; 32. : : : : level l level l level l level l                 multiple-criteria evaluation model for medical professionals assigned to temporary… 169 the values of border values of criteria significance ( i  ) were defined for the value 50n  . steps 1.4 and 1.5: based on the defined values * ( ) i f c , 1, 2,...,8i  we arrive at the values of criteria weight coefficients, table 6. table 6. criteria weight coefficients criterion * j w c1* 0.238 c2* 0.052 c3* 0.102 c4* 0.158 c5* 0.067 c6* 0.112 c7* 0.194 c8* 0.077 step 2: forming the preliminary decision matrix. since the minimum criteria were not met by 17 of 43 candidates, we form the preliminary decision matrix of rank , table 7. table 7. preliminary decision matrix second phase alt. c1* c2* c3* c4* c5* c6* c7* c8* a2 5 5 10 5 5 4 5 5 a4 5 2 0 1 1 5 1 5 a8 2 5 7 5 2 4 4 5 a13 2 5 10 5 5 5 5 5 a16 4 0 0 1 0 5 5 5 a18 5 5 3 5 2 1 4 5 a22 4 4 8 4 2 5 5 5 a23 4 4 7 3 0 5 5 4 a24 2 3 7 3 1 5 5 5 a25 5 3 7 3 1 5 5 5 a29 4 3 1 1 1 5 5 2 a31 4 5 10 5 5 5 5 5 a34 2 5 8 5 5 1 4 4 a38 4 5 6 4 4 3 4 5 a39 4 5 10 5 5 5 5 5 a40 4 5 10 5 5 4 5 5 a41 5 5 6 4 2 5 4 5 steps 3 and 4: normalization of elements of the preliminary decision matrix is done by applying the expression (4). after the normalization of the decision matrix elements, by applying expression (6) we define the grade for each candidate. the final grades and candidate ranking are shown in table 8. zizovic et al./decis. mak. appl. manag. eng. 4 (1) (2021) 153-173 170 table 8. candidate grades after second evaluation phase alt. v(ai) rank v(a2) 0.9776 1 v(a31) 0.9524 2 v(a39) 0.9524 2 v(a40) 0.9300 4 v(a13) 0.8572 5 v(a22) 0.8498 6 v(a41) 0.8486 7 v(a25) 0.8318 8 v(a38) 0.7830 9 v(a23) 0.7658 10 v(a18) 0.7600 11 v(a8) 0.7252 12 v(a34) 0.6930 13 v(a24) 0.6890 14 v(a29) 0.6136 15 v(a16) 0.6050 16 v(a4) 0.5316 17 after the evaluation of candidates (table 8), three best ranked candidates were selected, and after completing a training cycle, were assigned to a covid-19 hospital. the training program is defined by the crisis response team. the remaining 14 candidates also completed the training program but are currently not assigned to the covid-19 hospital. they are available for assignment in case of assigned staff being removed from the team for self-isolation. self-isolation may be a consequence of accidental exposure (human error, breakdown of equipment, etc.) or disease. 5. conclusions management of human resources is a key segment that affects the efficacy of the health system of every country. this is especially obvious in crises, like the covid-19 pandemic. this is why it is necessary to efficiently manage human resources in hospitals, to reduce as much as possible the dangers caused by the covid-19 pandemic. as far as the authors are aware, there are no current models for considering the evaluation of medical professionals’ training for working in crises, so the motivation for a study such as this is logical. in this paper, we put forward a multiple-criteria model that allows decision makers in medical institutions and national crisis response teams to evaluate the training of medical professionals for working in the conditions of the covid-19 pandemic. for the needs of this multiple-criteria model, we defined criteria based on which we evaluate medical professionals. the criteria and criteria evaluation scales were defined after months of research with participation from health institution managers and members of the crisis response team of serbia. the developed multiple-criteria model is conducted in two phases. the first phase evaluates medical professionals according to one or more dominant criteria. medical professionals who meet the conditions defined in the first phase, meet the conditions necessary for working in a covid-19 hospital. medical professionals who do not meet the conditions defined in multiple-criteria evaluation model for medical professionals assigned to temporary… 171 the first phase, move on to the second phase of evaluation. after completing the second phase of evaluation, the staff who partially meet the conditions are identified and they undergo training for working in covid-19 hospitals. this methodology was applied to the example of the covid-19 hospital in sombor. the suggested methodology can be used for other decision problems, by adapting the criteria according to the nature of the decision problem. the basic advantage of this study is application, i.e. testing of the suggested methodology on objective data in the conditions of the covid-19 pandemic. this demonstrates, on a real example, all advantages of this methodology. future research should be directed towards implementing the suggested methodology in the conditions of uncertain input model parameters (ecer and pamucar, 2020). uncertainty in future research can be exploited by applying various uncertainty theories such as fuzzy theory or rough theory. author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research received no external funding. conflicts of interest: the authors declare that there is no conflict of interest. references 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(2020). a clinical decision support system for the treatment of covid-19 with multi-criteria decisionmaking techniques. jmir medical informatics, 22, 88-106. zizovic, m.m, albijanic, m., jovanović, v., & zizovic, m (2019). a new method of multicriteria analysis for evaluation and decision making by dominant criterion. informatica, 30(4), 819-832. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 2, issue 1, 2019, pp. 49-65 issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame1901065f * corresponding author. e-mail addresses: hamed.hero@gmail.com (h. fazlollahtabar), aldina12345aldina@gmail.com (a. smailbašić) zeljkostevic88@yahoo.com and zeljko.stevic@sf.ues.rs.ba (ž. stević) fucom method in group decision-making: selection of forklift in a warehouse hamed fazlollahtabar1, aldina smailbašić2 and željko stević2* 1 department of industrial engineering, school of engineering, damghan university, damghan, iran 2 university of east sarajevo, faculty of transport and traffic engineering, doboj, bosnia and herzegovina received: 29 september 2018; accepted: 13 february 2019; available online: 19 february 2019. original scientific paper abstract. a warehouse system as a time transformation of the flows of goods plays an essential role in a complete logistics chain. the efficiency of a complete warehouse system largely depends on the efficiency of carrying out transport and handling operations. therefore, it is essential to have adequate means of internal transport that will influence the efficiency of the warehouse system by its performance. in this paper, the evaluation and selection of side loading forklift using the fucom-waspas model, which has been used for the first time in the literature in this paper, is performed. the fucom method was used to obtain the weight values of the criteria, while waspas was applied for the evaluation and ranking of forklifts. a possibility to apply the fucom method in group decision-making was presented. a comparative analysis, in which other methods of multi-criteria decision-making were applied, was carried out. the analysis showed the stability of the results obtained. key words: fucom method, forklift, waspas method, warehouse, group decision-making 1. introduction in the day-to-day performance of various activities and processes, logistics as an integral and indispensable part of each business system plays a very important role (stević et al., 2017a). there is a need to rationalize activities and processes that may significantly affect a company's competitive position (stević et al., 2017b). a warehouse as a special logistics subsystem and transport represent the major cause of logistics costs and there is a constant search for potential places of savings in these subsystems. in the very beginning, a warehouse was just a place used to separate fazlollahtabar et al./decis. mak. appl. manag. eng. 2 (1) (2018) 49-65 50 surplus products, while today its function is completely different (stojčić et al., 2018). compared to the former static function, today's warehouses represent dynamic systems in which the movement of goods is dominant. taking into account the above considerations, it is necessary to perform transport and handling operations as rationally as possible. from this aspect, forklifts within internal transport and warehousing operations play an important role. internal transport is the basis of every production process, both in functional and organizational terms. accordingly, rationalizing the movement of the means of transport and selecting the most convenient means of transport would lead to more efficient exploitation and reduction of costs. forklifts are the most widely used, most useful and most practical means of internal transport. forklifts are transport work machines for unloading, transport, warehousing and loading of various freight. there are a number of forklifts of different characteristics on the market. the side-loading forklift is intended for handling all types of freight. in this paper, seven criteria that could be taken into account when selecting a sideloading forklift were chosen. the aim of the paper is to obtain the best solution, i.e. an appropriate side-loading forklift that will meet the requirements of the euro-roal company where the research was carried out using multi-criteria decision-making. the choice of a specific side-loading forklift is conditioned by the optimality of the criteria that refer to the purchase price, age, working hours, maximum load capacity, maximum lift height, ecological factor and the supply of spare parts. in the paper, the fucom (full consistency method) and waspas (weighted aggregated sum product assessment) method were used to enable the evaluation and selection of a used sideloading forklift at the euro-roal company. using the fucom method, the determination of relative weights was performed, while using the waspas method, the ranking was completed. the remainder of the paper is organized as follows. in the second section of the paper, the methods used in the work, the fucom and waspas methods, are presented. fucom provides a possibility to determine accurately the weight coefficients of all the elements that are mutually compared. waspas represents a relatively new method of multi-criteria decision-making (mcdm) that is derived from two methods: weighted sum model (wsm) and weighted product model (wpm). the third section of the paper demonstrates the applicability of fucom method in group decision-making. based on the expert assessment of three decision-makers, the weight values of criteria are obtained. the fourth section is the evaluation and selection of forklifts using the waspas method, while in the fifth section, a comparative analysis is carried out using other methods. the paper ends with conclusions and directions for future research. 2. methods by applying multi-criteria decision-making methods, it is possible to select adequate strategies, rationalize certain logistics and other processes, and make appropriate decisions that affect the company's business or their subsystems, as evidenced by the following research (tzeng and huang, 2012; prakash and barua, 2016; żak and węgliński, 2014; hanaoka and kunadhamraks, 2009; zavadskas et al., 2018; stojić et al., 2018; radović et al., 2018; sremac et al., 2018) 2.1. fucom (full consistency method) fucom (pamučar et al., 2018) is a new mcdm method for determination of criteria weights. the problems of multi-criteria decision-making are characterized by the fucom method in group decision-making: selection of forklift in a warehouse 51 choice of the most acceptable alternative out of a set of the alternatives presented on the basis of the defined criteria. a model of multi-criteria decision-making can be presented by a mathematical equation      1 2max , ,..., , n 2nf x f x f x    , with the condition that  1 2, ,..., mx a a a a  ; where n represents the number of the criteria, m is the number of the alternatives, fj represents the criteria ( 1, 2,...,ј n ) and a represents the set of the alternatives ai ( 1, 2,...,i m ). the values ijf of each considered criterion j f for each considered alternative i a are known, namely    , , ; 1, 2,..., ; 1, 2,...,ij j if f a i j i m j n    . the relation shows that each value of the attribute depends on the jth criterion and the ith alternative. real problems do not usually have the criteria of the same degree of significance. it is therefore necessary that the significance factors of particular criteria should be defined by using appropriate weight coefficients for the criteria, so that their sum is one. determining the relative weights of criteria in multi-criteria decision-making models is always a specific problem inevitably accompanied by subjectivity. this process is very important and has a significant impact on the final decision-making result, since weight coefficients in some methods crucially influence the solution. therefore, particular attention in this paper is paid to the problem of determining the weights of criteria, and the new fucom model for determining the weight coefficients of criteria is proposed. this method enables the precise determination of the values of the weight coefficients of all of the elements mutually compared at a certain level of hierarchy, simultaneously satisfying the conditions of comparison consistency. in real life, pairwise comparison values / ij i j a w w (where aij shows the relative preference of criterion i to criterion j) are not based on accurate measurements, but rather on subjective estimates. there is also a deviation of the values ija from the ideal ratios /i jw w (where iw and jw represents criteria weights of criterion i and criterion j). if, for example, it is determined that a is of much greater significance than b, b of greater importance than c, and c of greater importance than a, there is inconsistency in problem solving and the reliability of the results decreases. this is especially true when there are a large number of the pairwise comparisons of criteria. fucom reduces the possibility of errors in a comparison to the least possible extent due to: (1) a small number of comparisons (n-1) and (2) the constraints defined when calculating the optimal values of criteria. fucom provides the ability to validate the model by calculating the error value for the obtained weight vectors by determining deviation from full consistency (dfc). on the other hand, in other models for determining the weights of criteria (the bwm, the ahp models), the redundancy of the pairwise comparison appears, which makes them less vulnerable to errors in judgment, while the fucom methodological procedure eliminates this problem. in the following section, the procedure for obtaining the weight coefficients of criteria by using fucom is presented. step 1. in the first step, the criteria from the predefined set of the evaluation criteria  1 2, ,..., nc c c c are ranked. the ranking is performed according to the significance of the criteria, i.e. starting from the criterion which is expected to have the highest weight coefficient to the criterion of the least significance. thus, the criteria ranked according to the expected values of the weight coefficients are obtained: fazlollahtabar et al./decis. mak. appl. manag. eng. 2 (1) (2018) 49-65 52 (1) (2) ( ) ... j j j k c c c   (1) where k represents the rank of the observed criterion. if there is a judgment of the existence of two or more criteria with the same significance, the sign of equality is placed instead of “>” between these criteria in the expression (1) step 2. in the second step, a comparison of the ranked criteria is carried out and the comparative priority ( / ( 1)k k   , 1, 2,...,k n , where k represents the rank of the criteria) of the evaluation criteria is determined. the comparative priority of the evaluation criteria ( / ( 1)k k   ) is an advantage of the criterion of the ( )j k c rank compared to the criterion of the ( 1)j k c  rank. thus, the vectors of the comparative priorities of the evaluation criteria are obtained, as in the expression (2):  1/ 2 2 / 3 / ( 1), ,..., k k     (2) where / ( 1)k k   represents the significance (priority) that the criterion of the ( )j k c rank has compared to the criterion of the ( )j k c rank. the comparative priority of the criteria is defined in one of the two ways defined in the following part: a) pursuant to their preferences, decision-makers define the comparative priority / ( 1)k k   among the observed criteria. thus, for example, if two stones a and b, which, respectively, have the weights of 300aw  grams and 255bw  grams are observed, the comparative priority ( /a b ) of stone a in relation to stone b is / 300 / 255 1.18 a b    . additionally, if the weights a and b cannot be determined precisely, but a predefined scale is used, e.g. from 1 to 9, then it can be said that stones a and b have weights 8 a w  and 7 b w  . respectively. then the comparative priority ( /a b  ) of stone a in relation to stone b can be determined as / 8 / 7 1.14 a b    . this means that stone a in relation to stone b has a greater priority (weight) by 1.18 (in the case of precise measurements), i.e. by 1.14 (in the case of application of measuring scale). in the same manner, decision-makers define the comparative priority among the observed criteria / ( 1)k k   . when solving real problems, decision-makers compare the ranked criteria based on internal knowledge, so they determine the comparative priority / ( 1)k k   based on subjective preferences. if the decision-maker thinks that the criterion of the ( )j k c rank has the same significance as the criterion of the ( 1)j k c  rank, then the comparative priority is / ( 1) 1 k k    . b) based on a predefined scale for the comparison of criteria, decision-makers compare the criteria and thus determine the significance of each individual criterion in the expression (1). the comparison is made with respect to the first-ranked (the most significant) criterion. thus, the significance of the criteria ( ( )j kc  ) for all of the criteria ranked in step 1 is obtained. since the first-ranked criterion is compared with itself (its significance is (1) 1 jc   ), a conclusion can be drawn that the n-1 comparison of the criteria should be performed. for example: a problem with three criteria ranked as c2>c1>c3 is being subjected to consideration. suppose that the scale   ( ) 1, 9 j kc   is used to determine the priorities of the criteria and that, based on the decision-maker’s preferences, the fucom method in group decision-making: selection of forklift in a warehouse 53 following priorities of the criteria 2 1 c   , 1 3.5 c   and 3 6 c   are obtained. on the basis of the obtained priorities of the criteria and condition / ( 1) 1 k k k k w w     we obtain following calculations 2 1 3.5 1 w w  i.e. 2 1 3.5w w  , 1 3 6 3.5 w w  i.e. 1 3 1.714w w  . in that way, the following comparative priorities are calculated: 2 1/ 3.5 / 1 3.5 c c    and 1 3/ 6 / 3.5 1.714 c c    (expression (2)). as we can see from the example shown in step 2b, the fucom model allows the pairwise comparison of the criteria by means of using integer, decimal values or the values from the predefined scale for the pairwise comparison of the criteria. step 3. in the third step, the final values of the weight coefficients of the evaluation criteria  1 2, ,..., t n w w w are calculated. the final values of the weight coefficients should satisfy the two conditions: (1) that the ratio of the weight coefficients is equal to the comparative priority among the observed criteria ( / ( 1)k k   ) defined in step 2, i.e. that the following condition is met: / ( 1) 1 k k k k w w     (3) (2) in addition to the condition (3), the final values of the weight coefficients should satisfy the condition of mathematical transitivity, i.e. that / ( 1) ( 1) / ( 2) / ( 2) k k k k k k          . since / ( 1) 1 k k k k w w     and 1 ( 1) / ( 2) 2 k k k k w w       , that 1 1 2 2 k k k k k k w w w w w w       is obtained. thus, yet another condition that the final values of the weight coefficients of the evaluation criteria need to meet is obtained, namely: / ( 1) ( 1) / ( 2) 2 k k k k k k w w         (4) full consistency, i.e. minimum dfc (  ) is satisfied only if transitivity is fully respected, i.e. when the conditions of / ( 1) 1 k k k k w w     and / ( 1) ( 1) / ( 2) 2 k k k k k k w w         are met. in that way, the requirement for maximum consistency is fulfilled, i.e. dfc is 0  for the obtained values of the weight coefficients. in order for the conditions to be met, it is necessary that the values of the weight coefficients  1 2, ,..., t n w w w meet the condition of / ( 1) 1 k k k k w w       and / ( 1) ( 1) / ( 2) 2 k k k k k k w w           , with the minimization of the value  . in that manner, the requirement for maximum consistency is satisfied. based on the defined settings, the final model for determining the final values of the weight coefficients of the evaluation criteria can be defined. fazlollahtabar et al./decis. mak. appl. manag. eng. 2 (1) (2018) 49-65 54 ( ) / ( 1) ( 1) ( ) / ( 1) ( 1) / ( 2 ) ( 2 ) 1 min . . , , 1, 0, j k k k j k j k k k k k j k n j j j s t w j w w j w w j w j                          (5) by solving model (5), the final values of the evaluation criteria  1 2, ,..., t n w w w and the degree of dfc (  ) are generated. 2.2. waspas method the weighted aggregated sum product assessment (waspas) method (zavadskas et al., 2012) represents a relatively new mcdm method that is derived from two methods: weighted sum model (wsm) and weighted product model (wpm). the waspas method consists of the following steps: step 1. forming the initial decision-making matrix ( x ). the first step is to evaluate m alternatives according to n criteria. the alternatives are shown by vectors  1 2, ,...,i i i ina x x x where ijx is the value of ith alternative according to jth criterion ( 1, 2,..., ; 1, 2,...,i m j n  ). 1 2 1 11 12 1 2 21 22 2 1 2 ... ... ... ... ... ... ... ... n n n m m m mn c c c a x x x a x x x x a x x x             (6) where m denotes the number of the alternative, and n denotes the total number of criteria. step 2. in this step, normalization of the initial matrix is required by applying the following equations: 1 2 , ,..., max ij ij n ij i x n for c c c b x   (7) 1 2 min , ,..., ij i ij n ij x n for c c c c x   (8) step 3. weighting the normalized matrix, so that the previously obtained matrix needs to be multiplied by the weight values of criteria: fucom method in group decision-making: selection of forklift in a warehouse 55 n ij m n v v      (9) , 1, 2,..., , ij j ij v w n i m j   (10) step 4. summing all the values of the alternatives obtained (summing by rows): 1 i ij m q q      (11) 1 n ij ij j q v    (12) step 5: determining a weighted product model by applying the following equation: 1 i ij m p p      (13)   1 j n w ij ij j p v   (14) step 6. determining the relative values of alternatives ai: 1 i ij m a a      (15)  1i i ia q p      (16) the coefficient λ ranges from 0, 0.1, 0.2,….1.0 step 7. ranking the alternatives. the highest value of alternatives implies the bestranked one, while the smallest value refers to the worst alternative. 3. fucom method in group decision-making processes the optimal choice of overhaul mechanization, in this case a forklift, depends solely on the precise determination and selection of appropriate criteria and their evaluation. the weights of the selected criteria were determined on the basis of their importance and needs of "euro-roal", doboj jug,, which were presented by experts and employees responsible for overhaul mechanization. table 1 gives the name, label and description of the criteria used for the selection of a forklift. table 1. criteria for forklift selection name and label of criteria criterion description purchase price (c1) forklift prices on the market are different and depend on manufacturers. when making an investment decision, the purchase price should not be decisive to the buyer, but it has a significant impact on the final decision. in an unsystematic approach, once the basic conditions are met, the purchase price is often a decisive factor. fazlollahtabar et al./decis. mak. appl. manag. eng. 2 (1) (2018) 49-65 56 age (c2) the age or year of production characterizes the production period of a forklift. forklifts manufactured recently have better specifications and options for adjustment to the requirements. working hours (c3) forklift utilization time is one of the most important criteria when selecting a forklift. the less the hours of the forklift utilization are, the lesser possibility of its breakdown is. maximum load capacity (c4) maximum load capacity is a criterion that represents the load capacity that a forklift can lift and it is expressed in kilograms. maximum lift height (c5) maximum lift height is a criterion that represents the height that a forklift can reach when lifting. ecological factors (c6) impact of forklift operation on the environment. supply of spare parts (c7) in experience, some representatives working in the market of the republic of serbia do not have in stock all necessary spare parts that are subject to frequent replacements, and their delivery is being waited for weeks, so the repairs of the means are long lasting. this criterion is in a group of qualitative criteria and is expressed by a fuzzified likert scale. table 2 shows seven criteria that were evaluated by three decision-makers. the decision-makers evaluated the criteria according to their importance to the company. table 2. comparison of criteria dm1 dm2 dm3 c1 5 5 5 c2 4 2 2 c3 1 1 1 c4 2 3 3 c5 3 4 4 c6 7 7 7 c7 6 6 6 determining the significance of criteria according to petrović et al. (2017) is one of the most important stages in a decision-making process. 3.1. determining the weight values of criteria for dm1 step 1. in the first step, the decision-makers rank the criteria: c3>c4>c5>c2>c1>c7>c6. step 2. in the second step (step 2b), the decision-maker performs a parwise comparison of ranked criteria from step 1. the comparison is made with respect to the first-ranked criterion c1. the comparison is based on the scale  1, 9 . thus, we obtain the significance of the criteria ( ( )j kc  ) for all the criteria ranked in step 1 (table 3). fucom method in group decision-making: selection of forklift in a warehouse 57 table 3. the significance of criteria criteria c3 c4 c5 c2 c1 c7 c6 ( )j kc  1 2.2 3,8 4.5 5 6,5 7 based on the obtained significance of the criteria, the comparative significance of the criteria is calculated: 3 4/ 2.20 / 1.0 2.20 c c    ; 4 5/ 3.8 / 2.20 1.73 c c    ; 5 2/ 4.50 / 3.8 1.18 c c    ; 2 1/ 5.00 / 4.50 1.11 c c    ; 1 7/ 6.50 / 5.00 1.30 c c    ; 7 3/ 7.00 / 6.50 1.08 c c    step 3. the final values of weight coefficients should meet two conditions: (1) the final values of weight coefficient should meet the condition (3), i.e. that: 3 4 4 5 5 2 2 1 1 7 7 6 / 2.20; / 1.73; / 1.18; / 1.11; / 1.30; / 1.08 w w w w w w w w w w w w       (2) in addition to the condition (3), the final values of weight coefficients should meet the condition of mathematical transitivity, i.e. that: 3 54 5 2 1 2 1 7 6 2.20 1.73 3.81; 1.73 1.18 2.04; 1.18 1.11 1.31; 1.11 1.30 1.44; 1.30 1.08 1.40 w ww w w w w w w w                using the expression (5), we can define the final model for determining weight coefficients: 8 5 74 2 1 4 5 2 1 7 6 8 54 2 1 5 2 1 7 6 7 1 min . . 2.20 , 1.73 , 1.18 , 1.11 , 1.30 , 1.08 , 3.81 , 2.04 , 1.31 , 1.44 , 1.40 , 1, 0, j j j s t w w ww w w w w w w w w w ww w w w w w w w w w j                                        by solving this model, we obtain the final values of weight coefficients for: purchase price, age, working hours, maximum load capacity, maximum lift height, ecological factor, supply of spare parts (0.082, 0.091, 0.410, 0.186, 0.108, 0.059, 0.068)τ and the deviation from a complete consistency, a result 𝑥 = 0.001. after calculating, it can be concluded that the most important criterion is working hours. for this element, the final value of the weight coefficient is 0.410. 3.2. determining the weight values of criteria for dm2 step 1. in the first step, the decision-makers ranked the criteria: c3>c2>c4=c5>c1>c7>c6. step 2. in the second step (step 2b), the decision-maker performs a pairwise comparison of ranked criteria from step 1. the comparison is made with respect to the fazlollahtabar et al./decis. mak. appl. manag. eng. 2 (1) (2018) 49-65 58 first-ranked criterion c1. the comparison is based on the scale  1, 9 . thus, we obtain the significance of the criteria ( ( )j kc  ) for all the criteria ranked in step 1 (table 4). table 4. the significance of criteria criteria c3 c2 c4 c5 c1 c7 c6 ( )j kc  1 2.8 3.5 3.5 4.2 5.5 6.5 based on the obtained significance of the criteria, the comparative significance of the criteria is calculated: 3 2/ 2.80 / 1.0 2.80 c c    ; 2 4/ 3.5 / 2.80 1.25 c c    ; 4 5/ 3.50 / 3.50 1.00 c c    ; 5 1/ 4.20 / 3.50 1.20 c c    ; 1 7/ 5.50 / 4.20 1.30 c c    ; 7 6/ 6.50 / 5.50 1.18 c c    step 3. the final values of weight coefficients should meet two conditions: (1) the final values of weight coefficient should meet the condition (3), i.e. that: 3 2 2 4 4 5 5 1 1 7 7 6 / 2.80; / 1.25; / 1.00; / 1.20; / 1.30; / 1.18 w w w w w w w w w w w w       (2) in addition to the condition (3), the final values of weight coefficients should meet the condition of mathematical transitivity, i.e. that: 3 2 4 4 5 1 5 1 7 6 2.80 1.25 3.50; 1.25 1.00 1.25; 1.00 1.20 1.20; 1.20 1.30 1.56; 1.30 1.18 1.53 w w w w w w w w w w                using the expression (5), we can define the final model for determining weight coefficients. 3 5 72 4 1 2 4 5 1 7 6 3 52 4 1 4 5 1 7 6 7 1 min . . 2.80 , 1.25 , 1.00 , 1.20 , 1.30 , 1.18 , 3.50 , 1.25 , 1.20 , 1.56 , 1.53 , 1, 0, j j j s t w w ww w w w w w w w w w ww w w w w w w w w w j                                        by solving this model, we obtain the final values of weight coefficients: purchase price, age, working hours, maximum load capacity, maximum lift height, ecological factor, supply of spare parts (0.094, 0.140, 0.398, 0.115, 0.116, 0.064, 0.077)τ and the deviation from a complete consistency, a result 𝑥 = 0.004. after calculating, it can be concluded that the most important criterion is working hours. for this element, the final value of the weight coefficient is 0.398. 3.3. determining the weight values of criteria for dm3 step 1. in the first step, the decision-makers ranked the criteria: c3>c2>c4=c5>c1>c7>c6. fucom method in group decision-making: selection of forklift in a warehouse 59 step 2. in the second step (step 2b), the decision-maker performs a pairwise comparison of ranked criteria from step 1. the comparison is made with respect to the first-ranked criterion c1. the comparison is based on the scale  1, 9 . thus, we obtain the significance of criteria ( ( )j kc  ) for all the criteria ranked in step 1 (table 5). table 5. the significance of criteria criteria c3 c2 c4 c5 c1 c7 c6 ( )j kc  1 2.8 3.5 3.5 4.5 6 7 based on the obtained significance of the criteria, the comparative significance of the criteria is calculated: 3 2/ 2.80 / 1.0 2.80 c c    ; 2 4/ 3.5 / 2.80 1.25 c c    ; 4 5/ 3.50 / 3.50 1.00 c c    ; 5 1/ 4.50 / 3.50 1.29 c c    ; 1 7/ 6.00 / 4.50 1.34 c c    ; 7 6/ 7.00 / 6.00 1.17 c c    step 3. the final values of weight coefficients should meet two conditions: 1) the final values of weight coefficients should meet the condition (3), i.e. that: 3 2 2 4 4 5 5 1 1 7 7 6 / 2.80; / 1.25; / 1.00; / 1.29; / 1.34; / 1.17 w w w w w w w w w w w w       (2) in addition to the condition (3), the final values of weight coefficients should meet the condition of mathematical transitivity, i.e. that: 3 2 4 4 5 1 5 1 7 6 2.80 1.25 3.50; 1.25 1.00 1.25; 1.00 1.29 1.29; 1.29 1.34 1.73; 1.73 1.17 2.02 w w w w w w w w w w                using the expression (5), we can define the final model for determining weight coefficients: 3 5 72 4 1 2 4 5 1 7 6 3 52 4 1 4 5 1 7 6 7 1 min . . 2.80 , 1.25 , 1.00 , 1.29 , 1.34 , 1.17 , 3.50 , 1.25 , 1.29 , 1.73 , 2.02 , 1, 0, j j j s t w w ww w w w w w w w w w ww w w w w w w w w w j                                        by solving this model, we obtain the final values of weight coefficients: purchase price, age, working hours, maximum load capacity, maximum lift height, ecological factor, supply of spare parts (0.095, 0.170, 0.418, 0.110, 0.112, 0.050, 0.065)τ and the deviation from a complete consistency, a result 𝑥 = 0.001. after calculating, it can be concluded that the most important criterion (table 6) is working hours. for this element, the final value of the weight coefficient is 0.418. fazlollahtabar et al./decis. mak. appl. manag. eng. 2 (1) (2018) 49-65 60 table 6. the criterion values for each decision-maker and values obtained by applying a geometric mean dm1 dm2 dm3 the values obtained by applying a geometric mean 0.082 0.094 0.095 0.090 0.091 0.140 0.170 0.129 0.410 0.398 0.418 0.409 0.186 0.115 0.110 0.133 0.108 0.116 0.112 0.112 0.059 0.064 0.050 0.057 0.068 0.077 0.065 0.070 the final values of weight coefficients were obtained by lingo software. from the table of results, it is clear that in this case working hours (c3) and maximum load capacity (c4) are the most important criteria. 4. the selection of forklift in a warehouse using the waspas method the euro-roal company owns several forklifts over 20 years of age and, in order to improve and refine their fleet, 10 alternatives (figure 1) (side-loading forklifts) will be evaluated. one of them, which would be suitable for the euro-roal, will be selected. figure 1. the alternatives in a multi-criteria model table 7 shows a formed multi-criteria model consisting of ten alternatives and seven criteria. fucom method in group decision-making: selection of forklift in a warehouse 61 table 7. initial decision-making matrix alternatives criteria c1 c2 c3 c4 c5 c6 c7 forklift 1 7.950 10 5012 4000 5400 5 7.67 forklift 2 12.900 10 7140 3000 3500 7 7.67 forklift 3 17.800 9 6500 5000 4500 7 5 forklift 4 19.300 19 4312 3000 6000 3 3.67 forklift 5 10.870 18 12000 3000 4000 5 3 forklift 6 30.400 7 4800 4000 4000 7.67 9 forklift 7 8.093 25 12000 4000 5900 3 5 forklift 8 29.800 11 3720 3000 5100 9 9 forklift 9 13.750 17 15350 4500 4800 3 5 forklift 10 18.297 13 6122 3000 4000 5 7 min min min max max max max 7.950 7 3720 5000 6000 5 7 the criteria that prefer minimal values are normalized by applying the following procedure: 11 21 31 41 51 10 1 7950 7950 7950 7950 1; 0.616; 0.446; 0.411; 7950 12900 17800 19300 7950 7950 0.731 . . . 0.434; 10870 18297 x x x x x x              the criteria that prefer maximum values are normalized by applying the following procedure: 14 24 34 44 54 10 4 4000 3000 5000 3000 0.80; 0.60; 1.00; 0.60; 5000 5000 5000 5000 3000 3000 0.60; . . . 0.60; 5000 5000 x x x x x x              table 8. normalized matrix alternatives criteria c1 c2 c3 c4 c5 c6 c7 forklift 1 1.000 0.700 0.742 0.800 0.900 0.556 0.852 forklift 2 0.616 0.700 0.521 0.600 0.583 0.778 0.852 forklift 3 0.447 0.778 0.572 1.000 0.750 0.778 0.556 forklift 4 0.412 0.368 0.863 0.600 1.000 0.333 0.408 forklift 5 0.731 0.389 0.310 0.600 0.667 0.556 0.333 forklift 6 0.262 1.000 0.775 0.800 0.667 0.852 1.000 forklift 7 0.982 0.280 0.310 0.800 0.983 0.333 0.556 forklift 8 0.267 0.636 1.000 0.600 0.850 1.000 1.000 forklift 9 0.578 0.412 0.242 0.900 0.800 0.333 0.556 forklift 10 0.434 0.538 0.608 0.600 0.667 0.556 0.778 w 0.090 0.129 0.409 0.133 0.112 0.057 0.070 weighting the normalized matrix, so that the previously obtained matrix needs to be multiplied by the weight values of criteria: fazlollahtabar et al./decis. mak. appl. manag. eng. 2 (1) (2018) 49-65 62 11 21 10 1 0.090 1.000 0.090; 0.090 0.616 0.055 . . . 0.090 0.434 0.039x x x           in table 9, after obtaining the values vij, the matrix is weighted, so that obtained values are multiplied by the values of weight coefficients. table 9. weighted normalized matrix 1 0.090 0.090 0.304 0.106 0.101 0.032 0.060 0.783q         determining a weighted product model using the following equation:               0.090 0.129 0.409 0.133 0.112 0.557 1 0.070 1.000 0.700 0.742 0.800 0.900 0.556 0.852 0.776 p         determining the relative values of alternatives ai :  1 0.5 0.783 1 0.5 0.782 0.779a       ranking the alternatives. the highest value of alternatives shows the best-ranked one, while the smallest value refers to the worst alternative. table 10 presents the results of ranking of forklifts based on the previous calculation. table 10. results and ranking the forklifts p a rank forklift 1 0.776 0.779 2 forklift 2 0.600 0.604 6 forklift 3 0.656 0.666 4 forklift 4 0.630 0.653 5 forklift 5 0.426 0.439 10 forklift 6 0.734 0.752 3 forklift 7 0.458 0.492 8 forklift 8 0.768 0.793 1 forklift 9 0.412 0.442 9 forklift 10 0.593 0.595 7 determining the relative weights of criteria was performed by the fucom method, while the ranking was performed using the waspas method. based on the results of alternatives criteria c1 c2 c3 c4 c5 c6 c7 forklift 1 0.090 0.090 0.304 0.106 0.101 0.032 0.060 forklift 2 0.055 0.090 0.213 0.080 0.065 0.044 0.060 forklift 3 0.040 0.100 0.234 0.133 0.084 0.044 0.039 forklift 4 0.037 0.048 0.353 0.080 0.112 0.019 0.029 forklift 5 0.066 0.050 0.127 0.080 0.075 0.032 0.023 forklift 6 0.024 0.129 0.317 0.106 0.075 0.049 0.070 forklift 7 0.088 0.036 0.127 0.106 0.110 0.019 0.039 forklift 8 0.024 0.082 0.409 0.080 0.095 0.057 0.070 forklift 9 0.052 0.053 0.099 0.120 0.090 0.019 0.039 forklift 10 0.039 0.069 0.249 0.080 0.075 0.032 0.054 fucom method in group decision-making: selection of forklift in a warehouse 63 the applied model, a solution that meets the current needs of the euro-roal company has been found, which is alternative 8, i.e. the baumann ehx 30/14/51 forklift 5. sensitivity analysis and discussion a logical sequence in most processes of multi-criteria decision-making is sensitivity analysis. for the sensitivity analysis of this model, the results of the saw method (maccrimon, 1968), the waspas method and the aras method (zavadskas and turskis, 2010) were compared. table 11 and figure 2 show the results and ranking the forklifts according to saw, waspas and aras methods. table 11. the results of sensitivity analysis according to saw, waspas and aras methods saw waspas aras a1 0.782 2 0.779 2 0.779 2 a2 0.608 6 0.604 6 0.607 6 a3 0.675 5 0.666 4 0.666 5 a4 0.677 4 0.653 5 0.671 4 a5 0.452 10 0.439 10 0.445 10 a6 0.769 3 0.752 3 0.768 3 a7 0.526 8 0.492 8 0.508 8 a8 0.817 1 0.793 1 0.817 1 a9 0.471 9 0.442 9 0.453 9 a10 0.598 7 0.595 7 0.594 7 alternative 1 according to saw, waspas and aras has the same rank (2). alternative 2 according to saw, waspas and aras has the same rank (6). alternative 3 according to the saw and aras methods is ranked fifth, whereas according to the waspas method, it is positioned fourth. alternative 4 according to the saw and aras methods is ranked fourth, whereas according to the waspas method, the fifth position is taken. alternative 5 according to saw, waspas and aras has the same rank (10). alternative 6 according to saw, waspas and aras has the same rank (3). alternative 7 according to saw, waspas and aras has the same rank (8). alternative 8 is the best solution according to all methods. alternative 9 according to saw, waspas and aras has the same rank (9). alternative 10 according to saw, waspas and aras has the same rank (7). fazlollahtabar et al./decis. mak. appl. manag. eng. 2 (1) (2018) 49-65 64 figure 2. sensitivity analysis 6. conclusion in this paper, a selection of transport and handling means was carried out in a warehouse system applying a combined fucom-waspas model. fucom was implemented throughout a group decision-making process where an expert team was formed to evaluate the significance of the criteria. obtaining the final weight values of the criteria was achieved using a geometric mean. the research has been conducted in a company whose primary task is to trade and distribute aluminum profiles. the applied model allows for an objective consideration of input parameters that have an impact on making a final decision. comparative analysis, which implies the application of two additional mcdm methods, presents the stability of originally obtained results if the model is generally observed throughout all possible variants. if individual positions are taken into account then the model shows the sensitivity to certain changes. future research regarding this paper relates to the formation of a model for determining the efficiency of using the selected side-loading forklift. references hanaoka, s., & kunadhamraks, p. 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(2010). a new additive ratio assessment (aras) method inmulticriteria decision making. technological and economic development of economy, 16(2),‐159-172. zavadskas, e. k., stević, ž., tanackov, i., & prentkovskis, o. (2018). a novel multicriteria approach–rough step-wise weight assessment ratio analysis method (r-swara) and its application in logistics. studies in informatics and control, 27(1), 97-106. zavadskas, e. k., turskis, z., antucheviciene, j., & zakarevicius, a. (2012). optimization of weighted aggregated sum product assessment. elektronika ir elektrotechnika, 122(6), 3-6. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 1, issue 2, 2018, pp. 34-50 issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame1802034v * corresponding author. e-mail addresses: veskos@sf.bg.ac.rs (s. vesković), zeljkostevic88@yahoo.com (ž. stević), gordan@uns.ac.rs (g. stojić), drmarkovasiljevic@gmail.com (m. vasiljević), s.milinkovic@sf.bg.ac.rs (s. milinković) evaluation of the railway management model by using a new integrated model delphi-swaramabac slavko vesković1, željko stević2*, gordan stojić3, marko vasiljević2, sanjin milinković1 1 university of belgrade, faculty of transport and traffic engineering, serbia 2 university of east sarajevo, faculty of transport and traffic engineering doboj, bosnia and herzegovina 3 university of novi sad, faculty of technical science, serbia received: 13 april 2018; accepted: 26 august 2018; available online: 26 august 2018. original scientific paper abstract the functioning of each traffic system depends to a great extent on the way the rail transport system operates. taking into account the aspect of market turbulence and the dependence on adequate delivery when it comes to freight transport and traffic in accordance with a yearly timetable in passenger traffic, transport policies are changing with time. therefore, this document is considering the railway management models on the territory of bosnia and herzegovina. for the purpose of evaluating these models, a new hybrid model has been applied, i.e. the model which includes a combination of the delphi, swara (step-wise weight assessment ratio analysis) and mabac (multi-attributive border approximation area comparison) methods. in the first phase of the study, the criteria ranking was determined based on 16 expert grades used in the delphi method. after that, a total of 14 decisionmakers determined the mutual criteria impact, which is a prerequisite for the application of the swara method used to determine the relative weight values of the criteria. the third phase involves the application of the mabac method for evaluating and determining the most suitable variant. in addition, a sensitivity analysis involving the application of the aras, waspas, saw and edas methods has been performed, thus verifying the previously obtained variant ranking. key words: railways, transport policy, delphi, swara, mabac. evaluation of the railway management model by using a new integrated model delphiswara-mabac 35 1 introduction although the railway has significant advantages which are reflected in a high level of safety, considerably less energy consumption per unit of transport and minimal impact on the environment, as well as the least impact on external transport costs comparing to other modes of transport, its participation in transport market has decreased significantly in the second half of the 20th century. to a large extent, it has been caused by historical, traditional and national influences on railway companies, and above all: a high level of government intervention in the business operations of national railway companies railway companies, through state control and intervention were used to meet political and social goals rather than to function in accordance with market principles, and, costs subsidizing and lack of incentives for change – a high proportion of passenger transport, which was unprofitable and politically supported, placed railway companies in the public service area, and they often transported passengers without an adequate compensation. in europe, all national railway administrations used to be state owned organizations which, for the sake of economic and social policy, were obliged to execute public passenger transport services. due to lower prices, the revenues did not cover actual costs, resulting in their inability to finance exploitation and infrastructure development. the lack of financial resources further led to economic weakening of the railway companies and their position on the market. national railway companies are integrated, i.e. they perform both functions of the infrastructure manager and operator. the regulatory framework is national with no competition in the form of foreign railways while there is no domestic market. due to non-profitability of the railway companies, there was a debt accumulation process in most european countries, especially in the late 1980s. the loss of railway competitiveness in the transport market in intermodal competition, a growing deficit and an increasing debt burden of the state-owned companies have triggered off reforms. in the eu member states and beyond, views and directives concerning the restructuring of the rail system have been adopted. prior reforms did not allow complete railway's liberalization and meeting the requirements of transport market, the expected positive operation of the railway system, the necessary level of rail services quality, satisfaction of the interests of the social community at the national, regional and local level. positive business results were partly achieved on the main railways (pan-european corridors), primarily in transit traffic. although the quality of services on railway system has improved, it is still far from the level required by transport market. defining the method of national railway companies restructuring, and thus the way of infrastructure management in europe, was mainly based on experts opinions, and it depended on the defined traffic policy, the country's level of development, and the readiness to accept changes (political, social and others). determination of the reforming method, or the most acceptable model of restructuring, is based on experiences, intuitions and subjective attitudes of individual institutions and experts. however, the countries have undertaken reforms aimed at easing the debt burden on national rail companies, reducing demands for high subsidies, mitigating and halting the fall of railways in market share comparing to other modes of transport. there was a need to create an efficient integrated railway system in the eu and to vesković et al./decis. mak. appl. manag. eng. 1 (2) (2018) 34-50 36 facilitate border crossing of goods within a single european market with the ultimate aim to: establish a railway transport market, develop competition in the railway sector, and, reduce state subsidies in the railway sector. the first task of railway restructuring is to transform the state organization into a business organization capable of carrying out transport operations both on the national and international transport market. in this process, the state has a role to create appropriate conditions for the development of a transport system that functions with the maximum application of market mechanisms and meets the transport needs of the society. in order to establish a harmonized market environment in which transporters functioning in different types of transport are affirmed on the basis of equal conditions of competition, it is necessary to calculate the total transport costs generated. the total costs of transport company include not only direct transport costs, infrastructure costs, traffic management and accident compensation, but also compensation for damage to the environment (cer, 2005). the actual situation is that in such conditions the railway has significant advantages over other modes of transport. in order to fully evaluate these facts, it is necessary to reform traditional railway companies and establish optimal models for their organization and functioning. this paper examines four different models of organization and structure of the railways of the republic of srpska (žrs), which are defined on the base of existing solutions for the reform of national rail companies in europe (predominantly in the european union member states). 2 literature review many studies in the domain of railway transport rely on the application of multicriteria decision-making methods. in (krmac & djordjević, 2017) the group analytical hierarchical process (ahp) was used to determine the key performance indicators for assessing intelligent transport systems. an integrated model consisting of the delphi, group analytical hierarchical process and promethee methods in (nassereddine & eskandari 2017) was applied in the field of public passenger transport, where, as a result, the metro is the most important passenger transport system. also, the integrated mcdm model (dematel, anp and vikor) was used to choose the transport mode in hualien (kuo & chen, 2015). aydin, (2017) commenced a three-year research in istanbul for measuring performances of the railway transit lines. for this purpose he used the topsis method. the performance evaluation of the railway zones in india (ranjan et al. 2016)) was conducted by combining the dematel and vikor methods, while in their research sang et al. (2015) used the fuzzy ahp method for selection and evaluation of railway freight third-partylogistics. leonardi (2016) applied a combination of fuzzy logics with multiplecriteria decision-making (ahp method) to plan a railway infrastructure, while in (santarremigia et al. 2018) the ahp was also applied in the safety area during the railway transport of dangerous materials. a combination of the bwm and saw methods was used in (stević et al. 2017a) to determine the importance of criteria in purchasing wagons in a logistics company. according to hashemkhani zolfani & bahrami (2014), the swara method is suitable for decision-making at a high level of decision-making and also instead of policy-making. its convenience in a decision-making process is reflected in the advantages it has in comparison to other methods for obtaining the weight values of evaluation of the railway management model by using a new integrated model delphiswara-mabac 37 criteria. these advantages are primarily seen in a significantly smaller number of comparisons in relation to other criteria, and the possibility to evaluate the opinions of experts on the significance of criteria in a process of determining their weights. over the few past years since this method came into existence, it has been used in a number of publications to determine weight values of the criteria. the swara was used to assess the relation between the floods and influencing parameters in (hong et al. 2017), while the anfis model is applied to flood spatial modeling and zonation, and it is used for the r&d project evaluation in (hashemkhani zolfani et al. 2015). using the swara method in (heidary dahooie et al. 2018), it is concluded that subject competency is the main criteria in it personnel selection. in (keshavarz ghorabaee et al. 2018), it is used to determine the significance of criteria in a process of evaluating construction equipment in sustainable conditions, while ruzgys et al. (2014) apply it to the evaluation of external wall insulation in residential buildings. it is successfully applied to risk assessment (valipour et al. 2017), for selection of a basic shape of the single-family residential house's plan (juodagalvienė et al. 2017), while karabašević et al. (2017) used the adapted swara with the delphi method for selection of personnel. the combination of the swara and waspas is used for solar power plant site selection in (vafaeipour et al. 2014), as well as in (ghorshi nezhad et al. 2015) where the combination of these two methods is applied in the nanotechnology industry. this combination is also integrated in (urošević et al. 2017) where it is used for the selection of personnel in tourism. the integration of the swara, fuzzy kano model and rov methods is proposed in (jain & singh, 2017) to solve supplier selection. the fuzzy swara is used to determine the significance of criteria, and the fuzzy copras for ranking and selecting sustainable 3prlps in the presence risk factors. the suggested model was applied to a case study from automotive industry (zarbakhshnia et al. 2018). a combination of the fuzzy swara and the fuzzy moora is used for sustainable third-party reverse logistic provider selection in plastic industry (mavi et al. 2017). the authors in (panahi et al. 2017) use the swara method for prospecting copper in the anarak region, central iran, while the authors in (ighravwe & oke, 2017) use it for sustenance of zero-loss on production lines from a cement plant. 3 methods 3.1 delphi method the delphi method does the study of and gives projections of uncertain or possible future situations for which we are unable to perform objective statistical legalities, in order to form a model, or apply a formal method. these phenomena are very difficult to quantify because they are mainly qualitative in their nature, i.e. not enough statistical data about them exist that could be used as the basis for our studies. the delphi method is one of the basic forecasting methods, the most famous and most widely used expert judgment method. methods of expert's assessments are representing significant improvement of the classical ways of obtaining the forecast by joint consultation of an expert's group for a given studied phenomenon. in other words, this is a methodologically organized use of the expert's knowledge to predict future states and phenomena. a typical group in one delphi session ranges from a few to thirty experts. each interviewed expert, participant in the method, relies on vesković et al./decis. mak. appl. manag. eng. 1 (2) (2018) 34-50 38 knowledge, experience and his / her own opinion. the goal of the delphi method is to exploit the collective, group thinking of experts about certain field. the goal is to reach a consensus on an event by group thinking. this is a method of indirect collective testing but with a return link. it consists of eight steps: 1: selection of the prognostic task, defining basic questions and fields for it; 2: selection of experts; 3: preparation of questionnaires; 4: delivery of questionnaires to experts; 5: collecting responses and their evaluating; 6: analysis and interpretation of responses; 7: re-exams; 8: interpretation of responses and setting up final forecast. the advantages of the delphi method • it covers the large number of respondents; • expert's statements are objective because they do not know the statements of others until the end of the circle; • it is possible to examine the opinion and attitude of an individual according to a task; • the method strengthens the sense of community and encourages thinking about the future of the organization. delphi method disadvantages: the success of the method depends exclusively on the participants in the expert panel; complicated implementation process; absence of the possibility to exactly identify the number of participants in the expert panel; long duration of research. according to the rules of the delphi method, the submitted forecasts of the first circle are statistically processed and sent to the experts again to make possible corrections if they consider other opinions. it is characteristic that most experts remain in their first-round prognosis. 3.2 swara method the swara (step-wise weight assessment ratio analysis) method is one of the methods for determining weight values that play an important role in a decisionmaking process. the method was developed by kersuliene et al. (2010) and, in their opinion, its basic characteristic is the possibility of assessing the opinion of experts on the significance of criteria in the process of determining their weights. after defining and forming a list of criteria involved in a decision-making process, the swara method consists of the following steps: step 1: criteria need to be sorted according to their significance. in this step, the experts perform the ranking of the defined criteria according to the significance they have; for example, the most significant is in the first place, the least significant is in the last place, while the criteria in-between have ranked significance. step 2: determine sj comparative importance of average value. starting from the second ranked criterion, it is necessary to determine their significance, that is, how much criterion cj is more important than criterion cj+1. evaluation of the railway management model by using a new integrated model delphiswara-mabac 39 step 3: calculate coefficient kj as follows: 1 1 1 1 j j j k s j      (1) step 4: determine recalculated weight qj as follows: 1 1 1 1 jj j j qq j k         (2) step 5: calculate the weight values of the criteria with the sum that is equal to one: 1 j j m j k q w q    (3) where wj represents the relative weight value of the criteria. 3.3 mabac method the mabac method (multi-attributive border approximation area comparison) is one of the recent methods. the mabac method was developed by dragan pamučar in the defense research center for defense logistics in belgrade and was first presented to the scientific public in 2015 (pamučar & ćirović, 2015). to date, it has found very wide application and modifications solving numerous problems in the field of multi-criteria decision-making. the basic setting of the mabac method is reflected in defining the distance of the criterion function of each observed alternative from the boundary approximation domain. in the following section, the procedure for implementing the mabac method consisting of 6 steps is shown: step 1: forming initial decision matrix ( )x . as a first step, m alternatives are evaluated by n criteria. alternatives are shown with vectors  1 2, ,...,i i i ina x x x , where ij x is the value of i-… alternative by j-… criteria ( 1, 2,..., ; 1, 2,..., )i m j n  . 1 2 1 11 12 1 2 21 22 2 1 2 ... ... ... ... ... ... ... ... n n n m m m mn c c c a x x x a x x x x a x x x             (4) step 2: normalization of elements of starting matrix (x). 1 2 1 11 12 1 2 21 22 2 1 2 ... ... ... ... ... ... ... ... n n n m m m mn c c c a t t t a t t t n a t t t             (5) the elements of normalized matrix (n) are determined using the expression: for criteria belonging to a "benefit" type (greater value of criteria is more desirable) vesković et al./decis. mak. appl. manag. eng. 1 (2) (2018) 34-50 40 ij i ij i i x x t x x       (6) for criteria belonging to a "cost" type (lower value of criteria is more desirable) ij i ij i i x x t x x       (7) where ij x , i x  and i x  are representing elements of the starting matrix of making decision (x), where i x  and i x  are defined as:  1 2max , ,...,i mx x x x   and representing maximal values of the observed criteria by alternatives.  1 2min , ,...,i mx x x x   and representing minimal values of the observed criteria by alternatives. step 3: calculation of the element of more difficult matrix (v). elements of more difficult matrix (v) are being calculated on the basis of expression (8) ij i ij i v w t w   (8) where ij t are representing the elements of normalized matrix (n), i w represents weighting coefficients of the criteria. by applying expression (8) we will get more difficult matrix v 11 12 1 1 11 1 2 12 2 1 21 22 2 1 21 1 2 22 2 2 1 2 1 1 1 2 2 2 ... ... ... ... ... ... ... ... ... ... ... ... ... n n n n n n n n m m mn m m n mn n v v v w t w w t w w t w v v v w t w w t w w t w v v v v w t w w t w w t w                                          where n represents the total number of the criteria, m represents the total number of the alternatives. step 4: determining the matrix of bordering approximative fields (g). bordering approximative field (gao) is being determined by expression (9) 1/ 1 m m i ij j g v          (9) where ij v are representing the elements of weighted matrix (v), m represents the total number of the alternatives. after calculating value i g the matrix of bordering approximative fields is being formed according to criteria g (10) in format 1n x (n represents the total number of the criteria by which the offered alternatives are being chosen).   1 2 1 2 ... ... n n c c c g g g g (10) step 5: the calculation of the distance matrix element is an alternative to boundary approximative area (q) evaluation of the railway management model by using a new integrated model delphiswara-mabac 41 11 12 1 21 22 2 1 2 ... ... ... ... ... ... n n m m mn q q q q q q q q q q             (11) distance of alternatives from boundary approximative area ( ) ij q is being determined as a difference of elements of heavier matrix (v) and values of bordering approximative areas (g).   11 12 1 21 22 2 1 2 1 2 ... ... ... ... ... ... ... n n n m m mn v v v v v v q v g g g g v v v                (12) 11 1 12 2 1 11 12 1 21 1 22 2 2 21 22 2 1 1 2 2 1 2 ... ... ... ... ... ... ... ... ... ... ... ... ... n n n n n n m m mn n m m mn v g v g v g q q q v g v g v g q q q q v g v g v g q q q                                 (13) where i g represents the bordering approximative areas for criterion i c , ij v represents elements of heavier matrix (v), n represents the number of the criteria, m represents the number of the alternatives. alternative i a may belong to a bordering approximative area (g), upper bordering approximative area ( )g  or lower bordering approximative area ( )g  , i.e.  ia g g g      . upper approximative area ( )g  represents the area in which ideal alternative (a+) is located, while lower approximative area ( )g  represents the area in which the anti-ideal alternative is located ( )a  (fig. 1). g  g  a  a  1 a 3 a 4 a 2 a 5 a 6 a 7 a g gornja aproksimativna oblast donja aproksimativna oblast granična aproksimativna oblast 0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 fig. 1 display of the upper, lower and bordering approximative areas (pamučar & ćirović, 2015) vesković et al./decis. mak. appl. manag. eng. 1 (2) (2018) 34-50 42 affiliation of alternative i a to approximative area (g, g+ or g-) is determined on the basis of expression (14) ij i i ij i ij i g if q g a g if q g g if q g          (14) in order for an alternative to be selected as the best from a given set, it is necessary for it to belong to the upper approximating field by as many criteria as possible ( )g  . if, for example, an alternative i a belongs to the upper approximative area by 5 criteria (out of 6 in total), and to the lower approximative area by one criterion, ( )g  that means that, by 5 criteria, this alternative is close to or equal with the ideal one, while by one criterion it is close to or equal to the anti-ideal one. if value 0 ij q  , i.e. ij q g   , then alternative i a is close or equal to the ideal alternative. value 0 ij q  , i.e. ij q g   , shows that alternative i a is close or equal to the anti/ideal alternative. step 6: alternatives ranking. calculation of values of the criteria functions by alternatives (15) is obtained as the sum of distance of the alternatives from bordering approximative fields ( ) i q . by summarizing the elements of the q matrix by rows, we obtain the final values of the criterion functions of alternatives (15) 1 , 1, 2,..., , 1, 2,..., n i ij j s q j n i m     (15) where n represents the number of the criteria, and m represents the number of the alternatives. 3 case study four variants of the management model for railway companies were considered: 1) variant 1 model of a single (independent) legal entity with a simple organizational structure and a high degree of centralization. fig. 2 variant 1 – model of unique (independent) legal subject evaluation of the railway management model by using a new integrated model delphiswara-mabac 43 2) variant 2 clear holding is a company exclusively dealing with management activities: establishment, financing and management of companies. this type of holding does not have any other special activities. clear holding does not deal with production or sale; neither does it perform any other business functions, even those that are common to companies daughters or members of the holding. fig. 3 variant 2 clear holding 3) varianta 3 mixed holding in addition to management tasks, mixed holding also performs other types of activities in the field of production, trade, research, finance or service activities. within the mixed-activity holding company there is a parent company (infrastructure) and companies engaged in the transport and traction of trains. fig. 4 variant 3 mixed holding 4) variant 4 – mixed holding – model of three independent companies: infrastructure, transport of passengers and transport of goods. criteria for selecting the most favorable model of restructuring and organization of railway companies are: k1 – model’s efficiency; k2 – the attractiveness of the model to attract an operator; k3 – satisfying the needs of transport market; k4 – compliance with eu directives; k5 – financial independence of the model; vesković et al./decis. mak. appl. manag. eng. 1 (2) (2018) 34-50 44 k6 – possibility of model realization. k1 – efficiency is the ability to achieve results and business goals. this means that the offered model should enable its efficient exploitation and maintenance. this criterion refers to management and functionality as well as the ability to use all the resources of the model in order to achieve the necessary effectiveness. the criterion should be maximized. k2 –“the attractiveness of the model to attract an operator” implies the ability of the model to provide an open access to infrastructure operators, the use of railway infrastructure by operators under equal conditions without discrimination. in this way, preconditions for multiple operators will be created. the criterion should be maximized. k3 – it refers to the possibility of the offered model to satisfy the needs of operators in the transport market in relation to the state and capacity of railway infrastructure capacities (permitted speed, throughput, electrification, permissible axial load, etc.). regardless of the operator's capability (transport time, prices, frequency, reliability, etc.), the state of the infrastructure significantly influences the definition of customers' demands on the market (population and economy). the criterion should be maximized. k4 – certain models can be fully or to some extent harmonized with eu directives aimed at the creation of a single transport market, its liberalization and ensuring the independence of the management of railway undertakings. the criterion should be maximized. k5 – the infrastructure manager should be a functionally sound and financially stable company. the state allocates financial resources to infrastructure managers only for the development of railway infrastructure, and not for workers' salaries. the k5 criterion should assess the extent to which the model can satisfy these requirements. the criterion should be maximized. k6 – it refers to the possibility of realization of the observed model from the aspect of legislation, environment, support of political, social and other participants, etc. the criterion should be maximized. in the first phase of the study, the ranking of criteria was determined based on 16 expert grades in the delphi method. after that, a total of 14 decision-makers determined the mutual impact of the criteria, which is a prerequisite for the application of the swara method used to determine relative weight values of the criteria. after applying eqs. (1) (3), we have obtained weight values of the criteria shown in table 1. table 1 calculation procedure and results of weight values of criteria obtained using swara method sj kj=sj+1 qj wj k3 1.000 1.000 1.000 0.224 k1 0.100 1.100 0.909 0.203 k5 0.148 1.148 0.792 0.177 k2 0.179 1.179 0.672 0.150 k4 0.168 1.168 0.575 0.129 k6 0.102 1.102 0.522 0.117 4.471 1.000 table 1 shows, in the first column, the alternative's ranking that was previously determined using the delphi method, while the second column represents the effect of the previous one in relation to the next criterion, which is the average value of the evaluation of the railway management model by using a new integrated model delphiswara-mabac 45 response of the decision-makers. based on the obtained results using the swara method, the most important is the first criterion of the model's efficiency, while the second criterion is the attractiveness of the model to attract operators elsewhere with a slightly lower value. the general conclusion when it comes to the value of the criteria considered in this study is that all the criteria have sufficient influence on the decision-making with respect to their values. in future research related to determining the significance of the criteria, it is recommended to use the rough swara method developed in (zavadskas et al. 2018). after obtaining the relative criteria values, it is necessary to determine the most favorable variant of railways management in bosnia and herzegovina. for this purpose, the mabac method is used. all 14 decision-makers who had previously determined the mutual impact of the criteria have also carried out the evaluation of the alternatives. by applying the geometric middle of all the answers, the initial decision matrix is shown in table 2. table 2 starting matrix of decision-making based on the responses from 14 decision-makers c1 c2 c3 c4 c5 c6 a1 4.238 3.918 4.530 3.710 4.502 4.810 a2 5.142 4.786 4.698 5.433 5.174 6.706 a3 6.470 4.909 5.463 6.069 6.020 6.392 a4 4.341 7.471 4.900 7.796 5.051 3.580 after the initial decision matrix, eqs. (6) and (7) must be applied in order to start normalization. since in this study all the criteria belong to a group of benefits for normalization, equation (6) is used, and the normalized matrix shown in table 3 is obtained. table 3 normalized matrix c1 c2 c3 c4 c5 c6 a1 0.000 0.000 0.000 0.000 0.000 0.393 a2 0.405 0.244 0.180 0.422 0.442 1.000 a3 1.000 0.279 1.000 0.577 1.000 0.899 a4 0.046 1.000 0.396 1.000 0.361 0.000 table 4 shows a more difficult normalized matrix obtained by multiplying the normalized matrix from table 3 with the weight values of the criteria obtained using the swara method. equation (8) is used to aggravate the normalized matrix. in addition, in the integral part of table 4, the values of the bordering approximative area are obtained by applying equation (9). table 4 weighted normalized matrix v c1 c2 c3 c4 c5 c6 a1 0.224 0.203 0.177 0.150 0.129 0.163 a2 0.314 0.253 0.209 0.214 0.186 0.234 a3 0.447 0.260 0354 0.237 0.257 0.222 a4 0.234 0.407 0.247 0.301 0.175 0.117 g 0.293 0.272 0.239 0.219 0.181 0.177 vesković et al./decis. mak. appl. manag. eng. 1 (2) (2018) 34-50 46 table 5 shows the distance matrix of the alternative from the bordering approximative area (q) obtained by applying eqs. (12) and (13) and the ranking of the model variant using equation (15). table 5 the distance matrix is an alternative to bordering approximative area (q) and alternative's range q=v-g c1 c2 c3 c4 c5 c6 is rank a1 -0.069 -0.068 -0.062 -0.068 -0.052 -0.014 -0.334 4 a2 0.021 -0.019 -0.030 -0.005 0.004 0.056 0.029 3 a3 0.154 -0.012 0.116 0.018 0.076 0.045 0.398 1 a4 -0.059 0.135 0.009 0.082 -0.006 -0.060 0100 2 after executing the budget and applying the hybrid model, the best-ranked variant of the railway management is a variant number 1 which implies that the model of a unified (independent) legal entity has a simple organizational structure with a high degree of centralization, while the worst ranking option is number 3. 4 sensitivity analysis in order to determine the stability of the previously obtained results using the hybrid delphi-swara-mabac model, the budget calculation for the multi-criteria model was carried out with four more aras methods (zavadskas and turksis, 2010), waspas (zavadskas et al. 2012), saw (maccrimmon, 1968, stević et al. 2017a), and edas (keshavarz ghorabaee et al., 2015; stević et al. 2016; stević et al. 2017b). the results of the sensitivity analysis are shown in table 6. table 6 the results of the sensitivity analysis mabac aras waspas saw edas v1 -0.334 4 0.644 4 0.381 4 0.652 4 0.652 4 v2 0.029 3 0.787 3 0.463 3 0.793 3 0.793 3 v3 0.398 1 0.884 1 0.521 1 0.891 1 0.891 1 v4 0.100 2 0.836 2 0.486 2 0.833 2 0.833 2 based on the obtained results of the sensitivity analysis, the model's stability and obtained levels of variant solutions are confirmed because in applying all the four methods in the analysis of sensitivity, the levels do not change, that is, each variant retains its initial level. 5 conclusion evaluation of the level of railway market restructuring and reforms is an important process that shows the phase in which a country is. level alignment is of great importance to the countries in the environment because in this way a more stable transport market can be established. this is especially important for the railways located in strong transit directions and pan-european corridors. the european rail system should not be "scraped" on the non-synchronized rail national reform levels since this does not contribute to the creation of a single european transport market, and thus to the desired open rail market. in evaluation of the railway management model by using a new integrated model delphiswara-mabac 47 addition, such a situation inevitably leads to a reduction in the quality of rail services and an uncompetitive position of the railways in the transport market. eu directives provide no unique solution in terms of selecting rail management models. the issue this document deals with is the development of a general model that provides a solution to the institutional management of rail national companies. quantified relevant criteria have been identified for the choice of management model. the synchronization of railway reforms has been promoted through various institutions, and the implementation of reforms and liberalization has often been carried out on the basis of experts' opinions or the application of inadequate methods. this document presents a new way of determining adequate restructuring model for railway national companies, which implies the integration of the delphi, swara and mabac methods. the three-phase hybrid model takes into account all the relevant facts and aspects that need to be considered in such research, and the integration of the above-mentioned methods is also one of the contributions of the work. in order to determine the stability of the model, a sensitivity analysis was performed in which four other methods of multicriteria analysis were applied, the results of which have confirmed the obtained results using the hybrid model proposed in this document. acknowledgements this paper is supported by ministry of science and technological development of the republic of serbia (project no. 36012). references aydin, n. 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(2012). optimization of weighted aggregated sum product assessment. elektronika ir elektrotechnika, 122(6), 3-6. zavadskas, e. k., stević, ž., tanackov, i., pretkovskis, o., (2018). a novel multi-criteria approach – rough step-wise weight assessment ratio analysis method (r-swara) and its application in logistics, studies in informatics and control, 27, 1. 97-106. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 4, issue 1, 2021, pp. 85-103. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame2104085g * corresponding author. e-mail addresses: gozde.koca@bilecik.edu.tr (g. koca), sdayldrmm@gmail.com (s. yıldırım) bibliometric analysis of dematel method gözde koca 1* and seda yıldırım 1 1 bilecik seyh edebali university, department of business administration, bilecik, turkey received: 11 january 2021; accepted: 3 february 2021; available online: 11 february 2021. original scientific paper abstract: in this study, a bibliometric analysis of the studies evaluated with dematel (decision making experiment and evaluation laboratory method), one of the mcdm methods in web of science, was performed according to various performance indicators. the total number of dematel publications examined is 1963 documents. when dematel studies are evaluated in terms of countries, it is seen that china is the leader (553 documents; 28.17%). the most cooperative country is china. the country with the highest h-index is taiwan (62). journal of cleaner production is the most efficient journal (96; 4.88%). national chiao tung university (102, 5.19%) is ranked as the most efficient institution in dematel research. among the most used words are "model", "dematel", "selection", "management", "fuzzy dematel". key words: multi-criteria decision making, bibliometric, web of science, dematel. 1. introduction decision making can be defined as individuals and organizations choosing the best alternative under current conditions to achieve their goals. decision making is an interdisciplinary field of research that attracts researchers and academics in almost every field. while intelligence, intuition and experience are important in decision making, it is equally important to use scientific methods. mcdm methods (multi-criteria decision-making methods) have been developed for the correct evaluation of multiple different criteria in solving complex problems. mcdm methods refer to the process of evaluating many criteria in a problem at the same time and assigning numerical evaluation to alternatives. mcdm allows decisionmakers to make evaluations and make decisions in multiple dimensions by bringing together multiple disciplines such as mathematics, management, social sciences, and economics (yıldırım & önder, 2018: 15). each method has solution logic in itself (çelikbilek, 2018: 3). the mcdm process consists of two stages. in the first of these stages, all the objectives and provisions given according to the alternatives are mailto:gozde.koca@bilecik.edu.tr mailto:sdayldrmm@gmail.com koca and yıldırım/decis. mak. appl. manag. eng. 4 (1) (2021) 85-103 86 brought together, in the second stage; the most appropriate decision is made by evaluating the alternatives among the combined provisions. (aytaç & gürsakal, 2015: 250). dematel (the decision making trial and evaluation laboratory), one of the mcdm methods, was developed in 1972 by the battelle memorial institute of geneva research center. the method is used in solving complex problem groups (shgeh et al., 2010: 277-282). the advantage of the dematel method is that it separates the distributor and receiver groups in the problem and determines the relationships between the criteria based on graph theory (impact-directional diagram) (lin & tzeng, 2009: 9686). the dematel method, which assumes that all criteria determined for the decision-making problem are in interaction with each other, evaluates the effect levels among the criteria. in the method, factors that are higher than the other criteria are called distributive, and criteria whose exposure level is higher than the effect on the system are called buyers (karaoğlan, 2016: 13). the increasing interest in mcdm methods has caused the publication of dematel method to increase continuously. in this study, bibliometric analysis was performed on the studies related to the method to interpret and summarize the information confusion caused by the continuous increase of the publications made with the dematel method. the reason why the dematel method is examined in this study is that it covers a very different literature that contributes from different disciplines. apart from this, it is to show how the method is examined in different disciplines by revealing causality and by revealing the importance of its differentiation from other mcdm methods. bibliometric analysis is an analysis method that examines scientific studies with the help of numerical analysis and statistics and shows the activities and current status of scientific studies in the field (çetinkaya bozkurt & çetin, 2016: 32). accordingly, bibliometric analysis reveals the productivity of countries, institutions and authors, citation analysis of countries, institutions and authors, which type of documents are used more, and how much the documents are distributed, and cooperation maps. for the research, the 1963 document searched from the web of science database with the subject "dematel" on 12.12.2020 was found in the bibliometrix library of the r studio program and analyzed with biblioshiny. all studies on the dematel method between 1999 and 2020 in the web of science database were included in the analysis. along with the analysis, annual studies and total citation rates, the productivity of countries, the number of citations and the cooperation map between countries, the most used journals and the number of citations in the studies conducted on the subject, the most efficient universities, the fields of science in which the dematel method is used and in which journals the studies were published the most, the most productive authors and citation rates, the most cited articles and the most used words in the articles written on the subject and the conceptual structure of the field were shown. 2. literature overview the study conducted by cole and eales in 1917 in the literature is known as the first bibliometric study. in this study; analyzes of studies published in the field of anatomy between 1550-1860 were made. after this study, an analysis was made in the field of historical science by e.wyndham hulme, a librarian at the british patent office in 1923. later, in 1927, p.l.k. gross and e.m. a citation analysis study was conducted by gross to evaluate the bibliography of the articles published in the journal of the biblometric analysis of dematel method 87 american chemical society. the first two studies were based on bibliographic features, not citations, and in gross & gross's study, citation analysis was performed (lawani, 1981: 295, hotamışlı & erem, 2014: 3). on the subject of mcdm, there are many studies conducted in the related literature. popular tools such as vosviewer, rbibliometric package were used in some of these studies. bibliometric studies made using popular tools in the field of mcdm are summarized in table 1 below. table 1. bibliometric studies using popular tools in the mcdm field authors year keyword used time span number of publications reviewed bragge et al. 2010 multi objective, multi criteria 19702007 15198 guerrero-baena et al. 2014 mcdm 19802012 347 zavadkas et al. 2014 mcdm review papers 19902013 71 tramarico et al. 2015 analytic hierarchy process and supply chain 19902014 116 blanco-mesa et al. 2017 fuzzy decision-making 19702014 8135 liu & liao 2017 fuzzy decision 19702015 13901 zyoud and funchs-hanusch 2017 ahp ve topsis 19762016 10188 ahp 2412 topsis peng & dai 2018 neutrosophic set 19982017 137 yu et al. 2018 multiple criteria decisionmaking 19772016 4464 liao et al. 2019 hesitant fuzzy sets 20092018 484 morkūnaitė et al. 2019 cultural heritage buildings with mcdm 1994– 2018 1039 there are literature reviews in the field of mcdm without using popular bibliometric tools. abu-taha (2011) reviewed more than 90 publications on mcdm in the field of renewable energy. he summarized both the application areas and the methodologies used in these publications. as a result of the literature review, it is revealed that ahp is the most used method among all mcdm methodologies. kahraman et al. (2015) examined the mcdd literature by dividing it into two parts as multi-specific and multi-purpose. in particular, they focused on multi-purpose decision making. they provided tables and graphs for each method (fuzzy ahp, fuzzy vikor, fuzzy topsis, fuzzy electre, etc.). mardani, et al. (2015) examined a total of 393 articles published in more than 120 peer-reviewed journals between 2000 and 2014. especially in the fields of energy, environment, and sustainability, they found that mcdm methods are frequently used. gül et al. (2016) conducted a literature review on vikor and fuzzy vikor applications and reviewed 343 publications in total. this comprehensive literature review they have done provides insight into vikor applications for researchers and practitioners. in their study, renganath & suresh (2016) analyzed the literature of mcdm methods used in supplier selection. after all, they said that the most popular method was fuzzy topsis. stojčić et al. (2019) koca and yıldırım/decis. mak. appl. manag. eng. 4 (1) (2021) 85-103 88 reviewed the literature on the application of mcdm methods in the field of sustainable engineering. they analyzed 108 articles scanned in the web of science (wos) database between 2008-2018. as a result, they found that sustainable engineering is a very suitable field for the use of mcdm. liu et al. (2019) conducted a comprehensive review of fmea (error type and effects analysis) studies using mcdm approaches to evaluate and prioritize error types. they reviewed 169 articles published between 1998-2018. this research sup0ports and provides insight into academics and practitioners in effectively adopting mcdm methods to overcome the shortcomings of traditional fmea. chowdhury and paul (2020) conducted a literature analysis of mcdm methods used in corporate sustainability between 2007 and 2019. as a result of this analysis, in which they examined 52 publications, they determined that the most used methods were ahp and topsis. 3. method bibliometric analysis is to make the scope of research in a particular area of interest both quantitatively and qualitatively (ellegaard and wallin, 2015: 1809). bibliometry developed for library and information sciences is used to classify research according to publications, times, and journals (merigo & yang, 2017: 37). in other words, bibliometry strengthens the scientific literature by understanding the research literature better (osareh, 1996: 149). stevens (1953) divided bibliometric studies into two main areas as seen below. descriptive studies contribute to authors, journals, years, and discipline by categorizing publications by country, while evaluators show where and how many publications are cited. 1. descriptive studies  country or geographic location  timespan discipline or subject area 2. evaluative studies  source  citation the analysis made allows identifying early trends in studies conducted in any field (ellegaard and wallin, 2015: 1809). in general terms, it describes scientific collaboration through collaborations between researchers, institutions, and countries. some new tools have been introduced to generate more broadcast data and provide a wide variety of indicators as listed in table 2. in this study, r-biblioshiny was used. table 2. popular tools for bibliometric analysis tools practitioners bibexcel olle persson authors authors ' frequency tables pajek vladimir batagelj and andrej mrvar citespace chaomeichen vosviewer nees jan van eck and ludowaltman r-bibliometric package massimo aria and carrado cuccurullo biblometric analysis of dematel method 89 4. results 1963 dematel publications in 800 sources (journals, books, etc) between 1999 and 2020 in the wos database were examined. dematel publications mostly consist of articles, book chapters, early access, proceedings papers and, review publications. average citations per document are 15,39 and average citations per year per doc is 3.274. figure 1 shows the annual number of citations of the studies conducted with the dematel method. the most citations to dematel's work took place in 2015 and 2018. it is seen that dematel studies get quite high citations. this shows that the method has a very dynamic structure. the distribution of the examined publications by years is given in figure 2. as can be understood from figure 2, the studies made with the dematel method have increased over the years. especially after 2015, the number of studies conducted with the method has increased. it is seen that most work on the method is in 2020. figure 1. number of citations by years figure 2. number of articles by years 0 1000 2000 3000 4000 1995 2000 2005 2010 2015 2020 total number of citations 0 50 100 150 200 250 300 350 400 1995 2000 2005 2010 2015 2020 total number of articles koca and yıldırım/decis. mak. appl. manag. eng. 4 (1) (2021) 85-103 90 table 3 shows the 20 most productive countries in the dematel method. according to the table, it is seen that the most productive country is china (553; 28.171%). after china, respectively, taiwan (519; 26.439%), iran (251; 12.787%), india (241; 12.277%) and turkey (184; 9.373%) are ranked. with the highest h-index of 62, and it was recorded by taiwan china (41), india (29), iran (27), turkey (24), and the united states (24) respectively. considering the citation rates of the countries, it is seen that the most cited country is taiwan (12884). after taiwan, respectively, china (6228), india (2892), iran (2878), and turkey (2499) are ranked. according to the number of studies of the countries, it is seen that the country with the highest citation average is denmark with 50.87%. table 3. ranking of top twenty most productive countries country no. of documents % h-index no. of citations average citations china 553 28,171 41 6228 11,26 taiwan 519 26,439 62 12884 24,82 iran 251 12,787 27 2878 11,42 india 241 12,277 29 2892 12,00 turkey 184 9,373 24 2499 13,58 usa 74 3,770 24 1710 23,11 england 63 3,209 17 839 13,32 malaysia 57 2,904 16 679 11,91 australia 41 2,089 11 492 12,00 spain 34 1,732 11 460 13,53 serbia 32 1,630 16 1039 32,47 denmark 31 1,579 23 1577 50,87 poland 31 1,579 7 217 7,00 lithuania 30 1,528 13 796 26,53 canada 29 1,477 8 354 12,21 italy 27 1,375 11 467 17,30 philippines 24 1,223 9 476 19,83 south korea 24 1,223 7 284 11,83 japan 23 1,172 8 524 22,78 indonesia 21 1,070 3 128 6,10 the world density map is shown in figure 3 below. the countries where dematel studies are carried out the most are listed from dark to light. countries with gray color do not have studies on the method. biblometric analysis of dematel method 91 figure 3. the world density map the most cooperating twenty countries according to the number of documents are shown in table 4. according to the table, among the countries with the highest cooperation, taiwan-china is the first with 74 documents, the usa-china is the second with 31 documents, and the uk-china is the third with 22 documents. table 4. the twenty most cooperative countries according to the number of documents from to frequency taiwan china 74 usa china 31 united kingdom china 22 india united kingdom 20 turkey china 20 china australia 17 iran lithuania 16 iran malaysia 14 india denmark 11 iran usa 11 malaysia saudi arabia 11 china denmark 10 china canada 9 india china 9 india usa 9 iran australia 9 taiwan usa 9 india lithuania 8 india spain 8 taiwan philippines 8 world cooperation map is given in figure 4. the countries where the lines are concentrated are determined as the countries that cooperate most with other koca and yıldırım/decis. mak. appl. manag. eng. 4 (1) (2021) 85-103 92 countries. accordingly, china the country with the highest cooperation with other countries, india, iran, taiwan, turkey, the uk and the us appear to be. figure 4. world cooperation map table 5 shows the sources of dematel publications. as shown in table 5 in this study, journal of cleaner production (96; 4,888%) has been the most comprehensive source of dematel research. then, sustainability (90; 4,582%) and expert system applications (77; 3,921%) journals follow. the most cited journal was determined to be the expert system applications journal with 7074 citations. besides, expert system applications journal has the highest h-index (48) and the highest average citation rate (91.87). then, it was seen that journal of cleaner production ranked second with 2895 citations. the journals with the highest h-indexes after the expert system application journal are journal of cleaner production (28), sustainability (16), computers & industrial engineering (16), applied soft computing (16), respectively. table 5. sources of dematel publications sources articles % hindex total citations average citations journal of cleaner production 96 4,888 28 2895 30,16 sustainability 90 4,582 16 741 8,23 expert systems with applications 77 3,921 48 7074 91,87 computers & industrial engineering 32 1,629 16 844 26,38 applied soft computing 26 1,324 16 917 35,27 benchmarking-an international journal 21 1,120 8 167 7,59 international journal of fuzzy systems 20 1,018 10 387 19,35 mathematical problems in engineering 20 1,018 7 216 10,80 biblometric analysis of dematel method 93 international journal of environmental research and public health 19 0,967 5 91 4,79 symmetry-basel 19 0,967 5 154 8,11 resources conservation and recycling 18 0,916 12 573 31,83 ieee access 17 0,866 4 40 2,35 international journal of production research 17 0,866 11 483 28,41 journal of intelligent & fuzzy systems 17 0,866 4 71 4,18 international journal of information technology & decision making 16 0,815 8 192 12,00 soft computing 16 0,815 6 288 18,00 international journal of production economics 15 0,764 13 1004 66,93 safety science 15 0,764 9 500 33,33 energies 14 0,713 5 68 4,86 technological and economic development of economy 14 0,713 8 331 23,64 table 6 shows the 20 most active universities in dematel research. accordingly, it is seen that the most productive university in dematel studies is national chiao tung university in taiwan with 102 documents (5,196). islamic azad university in iran ranks second with 90 documents (4,585) and nan kai university technology in china is third with 86 documents (4,381). the most cited university is national chiao tung university with 4344 citations and an average citation rate of 42.59%. also, national chiao tung university has the highest h-index (37). table 7 shows the ranking of the twenty most common areas in dematel studies. most of the published studies are in the field of computer science artificial intelligence (332; 16,904) and it was seen that the most used journal in this field was expert system with applications (77; 23,193%). following this area, the most common areas are environmental sciences (288; 14.664%), operations research management science (285; 14.511%), management (272; 13.849%), green sustainable science technology (235; 11.965%). table 8 shows the most productive twenty authors on dematel research. according to the table, with 121 documents (6.161%), tzeng g.h. seems to be. also, tzeng g.h is the author with the highest h-index (34) and the highest number of citations (4117). tzeng g.h. it is seen that the most prolific authors are tseng m.l. (38), dincer h. (36), and liou j.j.h (36). also, tseng m.l. is the second most cited author (1605). koca and yıldırım/decis. mak. appl. manag. eng. 4 (1) (2021) 85-103 94 t a b le 6 . t h e 2 0 m o st a ct iv e u n iv e rs it ie s in d e m a t e l r e se a rc h n a m e o f th e i n s ti tu ti o n n o . o f d o c u m e n ts % h -i n d e x t o ta l c it a ti o n s a v e r a g e c it a ti o n s c o u n tr y n a ti o n a l c h ia o t u n g u n iv e rs it y 1 0 2 5 ,1 9 6 3 7 4 3 4 4 4 2 ,5 9 t a iw a n is la m ic a z a d u n iv e rs it y 9 0 4 ,5 8 5 1 7 9 7 1 1 0 ,6 7 ir a n n a n k a i u n iv e rs it y t e ch n o lo g y 8 6 4 ,3 8 1 3 3 3 6 3 1 4 2 ,2 2 c h in a n a ti o n a l t a ip e i u n iv e rs it y 6 6 3 ,3 6 2 1 9 1 2 3 8 1 8 ,7 6 t a iw a n n a ti o n a l t a ip e i u n iv e rs it y o f t e ch n o lo g y 5 8 2 ,9 5 5 1 9 1 2 8 5 2 2 ,1 6 t a iw a n u n iv e rs it y o f t e h ra n 5 4 2 ,7 5 1 1 6 6 4 8 1 2 ,0 0 ir a n in d ia n i n st it u te o f t e ch n o lo g y s y st e m i it s y st e m 5 3 2 ,7 0 0 1 6 6 6 3 1 2 ,5 1 in d ia d a li a n u n iv e rs it y o f t e ch n o lo g y 5 1 2 ,5 9 8 1 8 1 1 1 4 2 1 ,8 4 c h in a n a ti o n a l t a iw a n n o rm a l u n iv e rs it y 4 0 2 ,0 3 8 1 2 5 0 9 1 2 ,7 3 t a iw a n is ta n b u l m e d ip o l u n iv e rs it y 3 6 1 ,8 3 4 8 1 6 6 4 ,6 1 t u rk e y a si a u n iv e rs it y t a iw a n 3 5 1 ,7 8 3 1 0 5 9 1 1 6 ,8 9 t a iw a n u n iv e rs it y o f e le ct ro n ic s ci e n ce t e ch n o lo g y o f c h in a 3 4 1 ,7 3 2 2 0 9 4 0 2 7 ,6 5 c h in a c h u n g h u a u n iv e rs it y 2 9 1 ,4 7 7 1 0 3 3 5 1 1 ,5 5 t a iw a n u n iv e rs it y o f s o u th e rn d e n m a rk 2 8 1 ,4 2 6 2 2 1 4 6 4 5 2 ,2 9 d e n m a rk v il n iu s g e d im in a s t e ch n ic a l u n iv e rs it y 2 8 1 ,4 2 6 1 3 7 8 8 2 8 ,1 4 l it h u a n ia c h in e se c u lt u re u n iv e rs it y 2 7 1 ,3 7 5 9 2 5 2 9 ,3 3 c h in a t a m k a n g u n iv e rs it y 2 7 1 ,3 7 5 1 6 7 5 9 2 8 ,1 1 t a iw a n n a ti o n a l c e n tr a l u n iv e rs it y 2 5 1 ,2 7 4 1 5 1 1 3 4 4 5 ,3 6 t a iw a n n a ti o n a l t a iw a n u n iv e rs it y o f s ci e n c e t e ch n o lo g y 2 4 1 ,2 2 3 9 8 1 0 3 3 ,7 5 t a iw a n s h a n g h a i ji a o t o n g u n iv e rs it y 2 4 1 ,2 2 3 1 1 3 6 0 1 5 ,0 0 c h in a biblometric analysis of dematel method 95 t a b le 7 : t h e t w e n ty m o st c o m m o n a re a s in d e m a t e l s tu d ie s s u b je c t a r e a n o . o f d o c u m e n ts % m o s t u s e d j o u r n a l n o . o f d o c u m e n ts % c o m p u te r s ci e n ce a rt if ic ia l in te ll ig e n c e 3 3 2 1 6 ,9 0 4 e x p e rt s y st e m s w it h a p p li ca ti o n s 7 7 2 3 ,1 9 3 e n v ir o n m e n ta l s ci e n ce s 2 8 8 1 4 ,6 6 4 jo u rn a l o f c le a n e r p ro d u ct io n 9 6 3 3 ,3 3 3 o p e ra ti o n s r e se a rc h m a n a g e m e n t s ci e n c e 2 8 5 1 4 ,5 1 1 e x p e rt s y st e m s w it h a p p li ca ti o n s 7 7 2 7 ,0 1 8 m a n a g e m e n t 2 7 2 1 3 ,8 4 9 b e n ch m a rk in g a n i n te rn a ti o n a l jo u rn a l 2 2 8 ,0 8 8 g re e n s u st a in a b le s ci e n ce t e ch n o lo g y 2 3 5 1 1 ,9 6 5 jo u rn a l o f c le a n e r p ro d u ct io n 9 6 4 0 ,8 5 1 e n g in e e ri n g e le ct ri ca l e le ct ro n ic 1 8 7 9 ,5 2 1 e x p e rt s y st e m s w it h a p p li ca ti o n s 7 7 4 1 ,1 7 6 e n g in e e ri n g i n d u st ri a l 1 8 7 9 ,5 2 1 c o m p u te rs i n d u st ri a l e n g in e e ri n g 3 2 1 7 ,1 1 2 c o m p u te r s ci e n ce i n te rd is ci p li n a ry a p p li ca ti o n s 1 6 1 8 ,1 9 8 c o m p u te rs i n d u st ri a l e n g in e e ri n g 3 2 1 9 ,8 7 6 e n v ir o n m e n ta l s tu d ie s 1 4 4 7 ,3 3 2 s u st a in a b il it y 9 0 6 2 ,5 0 0 e n g in e e ri n g e n v ir o n m e n ta l 1 3 1 6 ,6 7 0 jo u rn a l o f c le a n e r p ro d u ct io n 9 6 7 3 ,2 8 2 b u si n e ss 1 2 4 6 ,3 1 4 a fr ic a n j o u rn a l o f b u si n e ss m a n a g e m e n t 9 7 ,2 5 8 c o m p u te r s ci e n ce i n fo rm a ti o n s y st e m s 1 1 4 5 ,8 0 4 ie e e a cc e ss 1 7 1 4 ,9 1 2 e n g in e e ri n g m u lt id is ci p li n a ry 1 1 1 5 ,6 5 2 m a th e m a ti ca l p ro b le m s in e n g in e e ri n g 2 0 1 8 ,0 1 8 e co n o m ic s 1 0 5 5 ,3 4 6 e ch n o lo g ic a l a n d e co n o m ic d e v e lo p m e n t o f e co n o m y 1 4 1 3 ,3 3 3 e n g in e e ri n g m a n u fa ct u ri n g 8 8 4 ,4 8 1 in te rn a ti o n a l jo u rn a l o f p ro d u ct io n r e se a rc h 1 7 1 9 ,3 1 8 c o m p u te r s ci e n ce t h e o ry m e th o d s 7 9 4 ,0 2 2 jo u rn a l o f m u lt ip le v a lu e d l o g ic a n d s o ft c o m p u ti n g 5 6 ,3 2 9 e n e rg y f u e ls 7 0 3 ,5 6 4 e n e rg ie s 1 4 2 0 ,0 0 0 a u to m a ti o n c o n tr o l s y st e m s 6 1 3 ,1 0 6 in te rn a ti o n a l jo u rn a l o f f u z z y s y st e m s 2 0 3 2 ,7 8 7 e n g in e e ri n g c iv il 5 8 2 ,9 5 3 e n g in e e ri n g c o n st ru ct io n a n d a rc h it e ct u ra l m a n a g e m e n t 7 1 2 ,0 6 9 m u lt id is ci p li n a ry s ci e n ce s 5 4 2 ,7 4 9 s y m m e tr y b a se l 1 9 3 5 ,1 8 5 koca and yıldırım/decis. mak. appl. manag. eng. 4 (1) (2021) 85-103 96 table 8. the most productive twenty authors on dematel research authors articles % h-index total citations average citations tzeng gh 121 6,161 34 4117 34,02 tseng ml 38 1,935 19 1605 42,24 dincer h 36 1,833 8 165 4,58 liou jjh 36 1,833 17 1115 30,97 huang cy 35 1,782 7 394 11,26 yuksel s 32 1,629 8 161 5,03 kumar a 26 1,324 9 230 8,85 pamucar d 23 1,171 13 826 35,91 govindan k 22 1,120 16 1202 54,64 liu hc 21 1,069 16 1054 50,19 mangla sk 21 1,069 11 440 20,95 tsai sb 21 1,069 14 464 22,10 chuang yc 20 1,018 8 315 15,75 luthra s 20 1,018 12 478 23,90 lee yc 17 0,866 8 279 16,41 zavadskas ek 17 0,866 13 741 43,59 sarkis j 16 0,815 12 686 42,88 wu kj 16 0,815 9 429 26,81 wu hh 15 0,764 8 484 32,27 hsu cc 14 0,713 11 390 27,86 in table 9, the most cited ten articles about the dematel method are given. the most cited article in dematel with 570 citations is tzeng g.h., et al "evaluating intertwined effects in e-learning programs: a novel hybrid mcdm model based on factor analysis and dematel" (2007). in this article, the factors of the e-learning program are analyzed. the second most cited article with 500 citations, wu, w.w. & lee, y.t. "developing global managers' competencies using the fuzzy dematel method" (2007). the article by buyukozkan & cifci (2012) titled "a novel hybrid mcdm approach based on fuzzy dematel, fuzzy anp, and fuzzy topsis to evaluate green suppliers" is ranked third with 444 citations biblometric analysis of dematel method 97 t a b le 9 : t h e m o st c it e d t w e n ty a rt ic le s a b o u t th e d e m a t e l m e th o d a u th o r s t it le p u b li c a ti o n y e a r s o u r c e t it le t o ta l c it a ti o n s a v e r a g e p e r y e a r t z e n g g .h ., e t a l. e v a lu a ti n g i n te rt w in e d e ff e c ts i n e -l e a rn in g p ro g ra m s: a n o v e l h y b ri d m c d m m o d e l b a se d o n f a ct o r a n a ly si s a n d d e m a t e l 2 0 0 7 e x p e rt s y st e m s w it h a p p li ca ti o n s 5 7 0 4 0 ,7 1 w u , w .w . & l e e , y .t . d e v e lo p in g g lo b a l m a n a g e rs ' co m p e te n ci e s u si n g t h e f u z z y d e m a t e l m e th o d 2 0 0 7 e x p e rt s y st e m s w it h a p p li ca ti o n s 5 0 0 3 5 ,7 1 b u y u k o z k a n , g . & c if ci , g . a n o v e l h y b ri d m c d m a p p ro a ch b a se d o n f u z z y d e m a t e l , fu z z y a n p a n d f u z z y t o p s is t o e v a lu a te g re e n s u p p li e rs 2 0 1 2 e x p e rt s y st e m s w it h a p p li ca ti o n s 4 4 4 4 9 ,3 3 w u , w .w . c h o o si n g k n o w le d g e m a n a g e m e n t st ra te g ie s b y u si n g a co m b in e d a n p a n d d e m a t e l a p p ro a ch 2 0 0 8 e x p e rt s y st e m s w it h a p p li ca ti o n s 2 8 6 2 2 l in , r .j . u si n g f u z z y d e m a t e l t o e v a lu a te t h e g re e n s u p p ly c h a in m a n a g e m e n t p ra ct ic e s 2 0 1 3 jo u rn a l o f c le a n e r p ro d u ct io n 2 6 6 3 3 ,2 5 l in , c .j . & w u , w .w . a c a u sa l a n a ly ti ca l m e th o d f o r g ro u p d e ci si o n -m a k in g u n d e r fu z z y e n v ir o n m e n t 2 0 0 8 e x p e rt s y st e m s w it h a p p li ca ti o n s 2 6 0 2 0 h su , c .w .,e t a l. u si n g d e m a t e l t o d e v e lo p a c a rb o n m a n a g e m e n t m o d e l o f su p p li e r se le ct io n i n g re e n s u p p ly c h a in m a n a g e m e n t 2 0 1 3 jo u rn a l o f c le a n e r p ro d u ct io n 2 5 2 3 1 ,5 c h a n g , b ., e t a l. f u z z y d e m a t e l m e th o d f o r d e v e lo p in g s u p p li e r se le ct io n cr it e ri a 2 0 1 1 e x p e rt s y st e m s w it h a p p li ca ti o n s 2 4 9 2 4 ,9 s h ie h , j ., e t a l. a d e m a t e l m e th o d i n i d e n ti fy in g k e y s u cc e ss f a ct o rs o f h o sp it a l se rv ic e q u a li ty 2 0 1 0 k n o w le d g e -b a se d s y st e m s 2 4 7 2 2 ,4 5 t se n g , m .l . a c a u sa l a n d e ff e ct d e ci si o n m a k in g m o d e l o f se rv ic e q u a li ty e x p e ct a ti o n u si n g g re y -f u z z y d e m a t e l a p p ro a ch 2 0 0 9 e x p e rt s y st e m s w it h a p p li ca ti o n s 2 2 2 1 8 ,5 koca and yıldırım/decis. mak. appl. manag. eng. 4 (1) (2021) 85-103 98 the most commonly used keywords in dematel method are shown in figure 5. keyword analysis shows common keywords used by authors. accordingly, the most used keyword in dematel is seen as "model". in addition, the words "dematel", "selection", "management", "performance", "anp", "decision making" are the most common keywords. figure 5. the most commonly used keywords in dematel method 5. conclusion the focus of this study was to conduct a bibliometric analysis of global studies on the dematel method, one of the mcdm methods. 1963 documents obtained from the wos database between 1999-2020 were analyzed with the r studio program. in the study, the annual research outputs of the researches published on the dematel method, document types, countries, important journals and authors contributing to the field, the most efficient universities, and which fields of science the method is used in are shown. in the dematel method, china (553), taiwan (519), iran (251), india (241), turkey (184) are among the top five countries. the most cited country in his studies was observed as taiwan (12884). with the cooperation of taiwan and china 74, it is in the main position of international cooperation. in the analysis, it was seen that he was actively participating in researches related to the dematel method in other countries. the most prolific authors in the field are tzeng g.h. was seen as. next comes tseng m.l. (38), dincer h. (36), liou j.j.h. (36), huang c.y. (35). when we look at the web of science categories, it is seen that studies are concentrated in fields such as computer science and artificial intelligence, environmental science, operations research and management science, management, green sustainable technologies, electrical electronics engineering, and industrial engineering. biblometric analysis of dematel method 99 in the studies related to the field, the journal "journal of cleaner production" ranks at the top with 96 studies. then, the magazine "sustainability" takes second place with 90 studies, and the magazine "expert systems with applications" takes third place with 77 studies. the most cited journal is “expert systems with applications” with 7074 citations. the most productive university is national chiao tung university (taiwan) with 102 studies. next is islamic azad university (iran) with 90 studies, followed by nan kai university technology (china) in three with 86 studies. when we look at the conceptual structure of the studies, it is seen that they concentrate on words such as model, dematel, selection, management, performance, anp, decision making, fuzzy dematel. the findings of the study show the development of the studies in the dematel method, which is the mcdm method. as a result of the evaluations, it was seen that the studies on the dematel method were quite dynamic. it is possible to say that the studies on this method will increase in the following years. the methodology used can be applied to other methods and other topics. overall, the findings of this analysis provide a general picture of the evolution of the dematel method. this can assist practitioners and academics in identifying and evaluating efforts to advance research in these areas. this will help develop new lines of research for the future and advance the use of these methods in more applications. the methodology used can be applied to other mcdm methods or other topics. using the relative advantages of different bibliometric tools, the use of variables can be expanded. author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research received no external funding. conflicts of interest: the authors declare no conflicts of interest. references abu-taha, r. 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(2017). “a bibliometric-based survey on ahp and topsis techniques”, expert systems with applications, 78, 158-181. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 3, issue 2, 2020, pp. 97-118 issn: 2560-6018 eissn: 2620-0104 doi:_ https://doi.org/10.31181/dmame2003097v * corresponding author. e-mail addresses: m.j.vilela-ibarra@rgu.ac.uk (m. vilela), g.f.oluyemi@rgu.ac.uk (g. oluyemi), a.petrovski@rgu.ac.uk (a. petrovski) a holistic approach to assessment of value of information (voi) with fuzzy data and decision criteria martin vilela*1, gbenga oluyemi 1 and andrei petrovski 2 1 school of engineering, robert gordon university, aberdeen, united kingdom 2 school of computing, robert gordon university, aberdeen, united kingdom received: 15 august 2020; accepted: 28 september 2020; available online: 30 september 2020. original scientific paper abstract: classical decision and value of information theories have been applied in the oil and gas industry from the 1960s with partial success. in this research, we identify that the classical theory of value of information has weaknesses related with optimal data acquisition selection, data fuzziness and fuzzy decision criteria and we propose a modification in the theory to fill the gaps found. the research presented in this paper integrates theories and techniques from statistical analysis and artificial intelligence to develop a more coherent, robust and complete methodology for assessing the value of acquiring new information in the context of the oil and gas industry. the proposed methodology is applied to a case study describing a value of information assessment in an oil field where two alternatives for data acquisition are discussed. it is shown that: i) the technique of design of experiments provides a full identification of the input parameters affecting the value of the project and allows a proper selection of the data acquisition actions, ii) when the fuzziness of the data is included in the assessment, the value of the data decreases compared with the case where data are assumed to be crisp; this result means that the decision concerning the value of acquiring new data depends on whether the fuzzy nature of the data is included in the assessment and on the difference between the project value with and without data acquisition, iii) the fuzzy inference system developed for this case study successfully follows the logic of the decision-maker and results in a straightforward system to aggregate decision criteria. key words: value of information, fuzzy logic, design of experiments, uncertainty, decision making. mailto:m.j.vilela-ibarra@rgu.ac.uk mailto:g.f.oluyemi@rgu.ac.uk mailto:a.petrovski@rgu.ac.uk vilela et al./decis. mak. appl. manag. eng. 3 (2) (2020) 97-118 98 1. introduction the classical methodology for the value of information (voi) assessment has been used in the oil and gas industry since the 1960s, even though it is only recently that more applications have been published. it is commonly acknowledged that, due to a large number of data acquisition actions and the capital investment associated with it, the oil and gas industry is an ideal domain for developing and applying the voi assessments. the current methodology for the voi has several weaknesses for its applicability in oil and gas projects, and the objective of this research is to present a complete theory for voi that overcomes those weaknesses. the weaknesses found in the current voi theory are the following: 1) typically, the classical approach for voi assessment is carried out when it has been identified that the value of the project depends on an uncertain input variable that may be better defined if a specific piece of data is acquired. this approach lacks a complete analysis of the project uncertainties and the impact that the different inputs and their interactions have on the project’s value. this procedure to assess the value of acquiring data can limit the opportunities to improve the project’s value. 2) the classical approach to voi does not provide an integrated assessment of the impact that a specific data-gathering activity may have on the uncertainty of more than one variable. 3) voi does not consider that the data to be acquired may carry uncertainties that are due not only to randomness but also to fuzziness. 4) although the utility value is a well-known concept, most of the times, it is not used in voi assessments. 5) the criteria used by decision-makers for making decisions (e.g. to reject a project or to accept a data acquisition proposal) are fuzzy. however, the results from the classical voi assessment are crisp numbers; the handling of this dichotomy requires different tools from the ones used in the classical approach for voi. the aim of this research is to address the gaps identified in the classical methodology for the voi by integrating three existing techniques from other domains. firstly, the research identifies that the design of experiments (doe) approach can be used in the voi for providing a holistic assessment of the complete set of uncertain parameters, selecting the ones that have the most impact on the value of the project, and supporting the selection of the data acquisition actions for evaluation. secondly, the fuzziness of the data is captured through membership functions, and the expected utility value of each financial parameter is estimated using the probability of the states conditioned to the membership functions (in the classical methodology, this is conditioned to crisp values of the data). thirdly, a fuzzy inference system is developed for making the voi evaluation, with the human decision-making logic integrated into the assessment process, and several financial parameters aggregated into one. a case study, taken from the oil and gas industry, is discussed to show a successful application of the proposed methodology. 2. literature review value of information is a theory for deciding whether it is worthwhile to acquire information in the frame of a project’s value; this will happen when new data is used a holistic approach to assessment of value of information (voi) with fuzzy data and decision... 99 to change a decision that would be made differently without that information and when the value of the project increases after data is acquired. voi theory was developed by schlaifer (1959) and later developed further by grayson (1960), raiffa and schlaifer (1961), newendorp (1967) and raiffa (1968) in the context of business administration. one of the first references of voi in the oil and gas industry is grayson’s (1960) application of voi to uncertain drilling decisions. newendorp (1967) discusses a voi problem including the risk attitude of the decision-maker described through the use of the exponential utility function; this same author (newendorp, 1972) reviews in great detail the bayes’ theorem and its application for voi assessment. a series of works from several authors in the oil and gas industry shows an increasing interest in using voi as a tool for making decisions. dougherty (1971) shows several straightforward applications of voi for the oil and gas industry. warren (1983) discusses the case study of a field development decision regarding initiating, rejecting or postponing a project decision until more information is gathered; lohrenz (1988) reviews four examples of the value of data acquisition using decision trees; silbergh and brons (1972) debate several methods of project valuation, utility functions, and voi. moras, lesso, and macdonald (1987) show the value associated with different numbers of observation wells to monitor underground gas storage. gerhardt and haldorsen (1989) show several applications of voi for typical examples of decisions in subsurface problems; dunn (1992) discusses the voi of well logs while stibolt and lehman (1993) do the same for seismic data. demirmen (1996) broadens the use of voi by using it in the two types of appraisal activities: screening, and optimization; this is one of the first references that discuss the use of voi on a complete oil and gas project and, open the possibility to use this tool as a means for ranking subsurface appraisal activities. koninx (2000) reviews voi from a methodological perspective and discuss important criteria that should be taken into consideration when data is proposed to be acquired such as the value of assurance and value of creation; bratvold, bickel, and lohne (2007) show how to make a voi assessment and discuss a statistical review of the published work about voi which indicates that it is still far from being a standard application in the oil and gas industry and conclude with identification and discussions of the possible causes of the limited use of voi in the oil and gas industry. new insight on voi is shown by kullawan, bratvold, and bickel (2014) by applying voi to real-time geosteering operations and by vilela, oluyemi and petrovski (2018, 2019a) by introducing the fuzzy nature of the data in the voi assessment. in the previous works, voi was applied on “isolated” data-gathering activities related to one of the project uncertainties, by assessing the impact that acquiring such data had on the value of the project; however, from a project standpoint, the essential objective is the identification and quantification of the benefits that are likely to come from any possible data acquisition activities that maximize the project value and not just one of the possible data acquisition activities, without considering the uncertainties in the complete project. the identification and definition of the data acquisition activities that maximize the project’s value can be made using the technique of doe. uncertainty can be aleatoric (related with noise inherent in the observations; it is unavoidable) and epistemic (related with models used to mimic the reality; it is feasible to be reduced by additional data acquisition). in problems characterized by epistemic uncertainty in the input and/or output variables, it is important to know vilela et al./decis. mak. appl. manag. eng. 3 (2) (2020) 97-118 100 what are the ranges of variability and the relative importance that each of the input variables has on the range of variability and values of the output variables. design of experiments is a structured and organized methodology to conduct and analyze experiments by defining each one by a specific set of values for the input parameters; the experiments (or simulation runs) should be performed to assess the impact that input parameters and their interactions have on the output variables (montgomery, 2005). doe has been used for improving the performance of processes and reducing result variability and cost (telford, 2007). doe is used to understand a system or process by means of experimentation; figure 1 shows that the input parameters, combined by the system or process under consideration, are affected by factors (controllable and uncontrollable) which produce the output parameters. figure 1. diagram of the design of experiments approach doe was invented by statistician ronald fisher (1935) to understand the factors involved in increasing the crop yield in the uk, and its foundations were completed thanks to the work of box and wilson (1951), box, hunter and hunter (1978) and box and draper (1987). law and kelton (1991) and myers and montgomery (2002) develop doe methods for simulations proposes; doe has expanded its applications to several domains such as the chemical industry (yang, bi, and mao, 2002; sjoblom et al., 2005; ruotolo and gubulin, 2005), materials (suffield, dillman, and haworth, 2004; liao, 2003; hoipkemeier-wilson et al., 2004), industrial engineering (tong, kwong, and yu, 2004; galantucci, percoco, and spina, 2003; du et al., 2002), electronic (ogle and hornberger, 2001) or mechanical engineering (passmore, patel and lorentzen, 2001; nataraj, arunachalam, and dhandapani, 2005; farhang-mehr and azann, 2005; cervantes and engstrom, 2004), aerospace (zang and green, 1999) and the analysis and optimization of nonlinear systems (sacks et al., 1989). computational deterministic experimentation (e.g. used for dynamic reservoir simulation) differs from real-world experimentation in the fact that the former does not have a random error as the latter has; in practical terms, that means we always get the same output using a specific set of input parameters. similar to real-world experimentation, the objective of simulation experimentation is to determine the a holistic approach to assessment of value of information (voi) with fuzzy data and decision... 101 factors that have a large impact on the response, getting the results with the least number of simulation runs (law, 2015). the first applications of doe in the oil and gas industry were by damsleth, hage, and volden (1992), egeland et al. (1992) and larsen, kristoffersen, and egeland (1994); after those applications, doe has been used for identifying the main geological parameters responsible for oil recovery (white et al., 2001); for uncertainty integration to quantify their impact on original oil in place, recoverable reserves and production profiles (corre, de feraudy and vincent, 2000); for assessing uncertainties in production profiles (venkataraman, 2000); for investigating the impact of geologic heterogeneities and uncertainties in different development schemes (wang and white, 2002); and for defining the minimum number of reservoir simulation runs needed to identify and quantify the factors responsible for the uncertainties of the reservoir performance (peake, abadah and skander, 2005). additionally, studies on production forecasting and ultimate recovery estimates representing the numerical reservoir simulation by a surrogate response surface model are discussed by friedmann, chawathe and larue (2001) and murtha et al. (2009), while dejean and blanc (1999) discuss doe, dividing the uncertain factors into uncontrollable and controllable and adapting doe accordingly, and law (2017) discuss the workflow for applying doe to simulation modelling. capturing all the uncertainties that the project may have and their impact on the output variables is of great importance in order to determine which data is worthwhile to acquire. in 1965, lotfi zadeh published the paper “fuzzy sets” where he describes the mathematics of fuzzy numbers and how fuzzy logic can be used to describe events with a partial degree of belonging to sets. founded on this work, bellman and zadeh (1970), lakoff (1978), dunn (1992), bezděk (1993, 2014), negoita and ralescu (1977), goguen (1967), bandler and kohout (1978), sugeno and murofushi (1987), sugeno and kang (1988), mizumoto and tanaka (1976, 1981), tanaka, taniguchi, and wang (1999), zimmermann and sebastian (1994), zimmermann (1996), etc. continue the development of the new theory. zadeh (1968) showed how fuzzy events could be described using fuzzy set theory. in 1971, zadeh published “quantitative fuzzy semantics”, where he developed the formal elements of the fuzzy logic and its applications. fuzzy inference is the process of mapping a set of input variables onto a set of output variables using fuzzy logic; in general, there are two ways of doing that: mamdani and sugeno, depending on the way the outputs are determined. the first fuzzy inference system (fis) was a fuzzy controller for a steam engine developed by assilian and mamdani (1974) where fuzzy logic was used to convert heuristic control rules into an automatic control strategy; the first real implementation of a fuzzy controller was made by lauritz peter holmblad, and jens-jørgen østergaard (1980), who developed the commercial system of fuzzy control working for f.l. smidth & co. in a cement factory in denmark (larsen, 1980; umbers and king, 1980), which resulted in one of the first successful tests runs on a full-scale industrial process. subsequent applications of fuzzy logic in several domains have been reported: the assessment of water quality in rivers (ocampo, 2008); improvements in the quality of image expansion (sakalli, yan and fu, 1999); the differential diagnosis of non-toxic thyropathy (guo and ling, 2008); the development of a fuzzy logic controller for a traffic junction (pappis and mamdani, 1997); the design of a sensor-based fire monitoring system for coal mines using fuzzy logic (muduli, jana and mishra, 2018); estimation of the impact of tax legislation reforms on potential tax (musayev, madatova, and rustamov, 2016); pipeline risk assessment (jamshidi et al., 2013); the vilela et al./decis. mak. appl. manag. eng. 3 (2) (2020) 97-118 102 diagnosis of depression (chattopadhyay, 2014); the assessment of predicted river discharge (jayawardena et al., 2014); calculation of geological strength indices and slope stability assessments (sonmez, gokceoglu and ulusay, 2004); regulation of industrial reactors (ghasem, 2006); the use of a fuzzy logic approach for file management and organization (gupta, 2011). similarly, in the oil and gas industry, fuzzy logic has been used for a streamline-based fuzzy logic workflow to redistribute water injection by accounting for operational constraints and number of supported producers in a pattern (bukhamseen et al., 2017); the identification of horizontal well placement (popa, 2013); estimating the strength of rock using fis (sari, 2016); and predicting the rate of penetration in shale formations (ahmed et al., 2019). fuzzy logic has been used in combination with other artificial intelligence techniques such as adaptative neuro-fuzzy inference system (anfis) in practical applications, e.g. to predict the inflow performance of vertical wells producing two-phase flow (basfar et al., 2018) or to predict geomechanical failure parameters (alloush et al., 2017); fis has also been used in conjunction with analytical hierarchical processes to evaluate the water injection performance in heterogeneous reservoirs (oluwajuwon and olugbenga, 2018) and to make decisions in the application of fuzzy inference systems for voi in the oil and gas industry (vilela, oluyemi, and petrovski, 2019b). from a methodological perspective, a fis can be understood as a general procedure that transforms a set of input variables into a set of outputs, following the dataflow shown in figure 2. figure 2. fuzzy inference system dataflow a holistic approach to assessment of value of information (voi) with fuzzy data and decision... 103 3. case study 3.1. reservoir information this case study is based on a clastic reservoir; four explorations and appraisal wells have already been drilled, the first three wells showing good production test results while the fourth well, located in the south of the reservoir, shows inferior results; these test results correlate well with the reservoir quality observed in the four wells; the differences in reservoir quality are attributed to diagenesis processes that occurred in the reservoir. figure 3 shows the four wells in the dynamic simulation model. figure 3. structural map of the field with the exploration and appraisal wells 3.2. project subsurface uncertainties the technical team agreed that six parameters are carrying most of the subsurface uncertainty of this project: i)horizontal permeability distribution (pxy), ii)vertical permeability (pze), iii)relative permeability (rpe), iv)aquifer strength (aqu), v)oilto-water contact (owc) and, vi)well productivity index (pi) value multiplier (wpi); these parameters and their range of uncertainty are shown in table 1. vilela et al./decis. mak. appl. manag. eng. 3 (2) (2020) 97-118 104 table 1. uncertain parameters: low, medium, and high values uncertain parameters low medium high horizontal permeability extended diagenesis medium case diagenesis local diagenesis vertical permeability (md) 0.01 0.50 10.00 relative permeability co=3.1 / cw=3.3 / sorw=0.15 co=2.5 / cw=4.4 / sorw=0.17 co=1.8 / cw=5.5 / sorw=0.20 aquifer strength, aqu vol. / aqi pi stb/(stb/d/psi) 9 2.5e / 217 11 2.52e / 434 13 2.52e / 868 oil/water contact (m) 1,070 1,075 1,080 well pi multiplier 0.90 8.90 18.40 the uncertainty associated with the distribution of the reservoir quality is captured by three different scenarios built to represent the high, medium and low cases for the property distribution to the south of the reservoir; due to the lack of data in this field, the range of variability in vertical permeability, relative permeability curves, and aquifer strength are taken from analogous fields. the range of values for oil-to-water contact is defined by the values observed in the three wells drilled, and the well pi multipliers are the figures used to history match the test results. the dynamic model used to generate the production profiles was made using the eclipse software (schlumbergertm). the operator company responsible for this field must decide whether to proceed with or to terminate, the project; however, the acquisition of new data can change the value of the project and impact that decision. acquiring data carries a cost and possible delay in the project start; these negative impacts may be worthwhile if compensated by the positive effects of risk reduction and an increase in the project’s value. 3.3. assessment of project value of information the assessment of the value of data acquisition starts with the screening phase, which consists of the identification of the input variables that have the most impact on the objective variable, which in this case is the utility of the net present value (unpv). in this case, study, having six input variables (the uncertain variables described in table 1), sixty-six dynamic simulation cases should be set and run (each variable is evaluated at its low and high values). figure 4 shows the cumulative oil production of these sixty-six simulations runs. a holistic approach to assessment of value of information (voi) with fuzzy data and decision... 105 figure 4. uncertainty in cumulative oil production the financial model used to evaluate the project benefits is built using excel software (windows office); this model includes the oil production forecast resulting from the simulation runs and the capex (capital expenditure or investment), opex (operational expenditure), oil price forecast; for this analysis, no other financial factor was included. as discussed by walls (2005), the utility function used is exponential, which in this case study will have a tolerance factor (tf) of $ 4,000 mm; this tf is representative of the company’s historic attitude toward risk for oil and gas exploitation projects. for a reference on utility function in the oil and gas industry, see vilela, oluyemi, and petrovski (2017). in figure 5, a pareto plot of the effects shows that the variables with the larger impact on the objective variable are a (owc), e (aqu), c (rep), b (pxy), ab (owc/pcy), and ac (owc/rep), where the last two correspond to the interaction effect of the first four variables; in conclusion, the most relevant parameters for the study are a, e, c and b, which correspond to owc, aqu, rep and pxy. vilela et al./decis. mak. appl. manag. eng. 3 (2) (2020) 97-118 106 figure 5. pareto chart of the effects of the parameters, with a significance level of 0.05 this interpretation is confirmed by using the normal plot, as shown in figure 6. figure 6. normal plot of the effects of parameters, with a significance level of 0.05 based on these four relevant variables already identified, sixteen dynamic models need to be evaluated corresponding to running each input variable to the low and high values while keeping the rest of variables at the medium level; the outcomes of those models should be further assessed in terms of values (npv, irr) and utility values (unpv, uirr). a holistic approach to assessment of value of information (voi) with fuzzy data and decision... 107 the technical team estimates the prior probabilities of occurrence for each of these sixteen cases. two alternatives for data acquisition are considered: i) drilling a new well and performing an extended well test, and ii) performing an extended well test on an existing well. 1) drilling a new well and performing an extended well test 2) by drilling a new well and performing an extended well test, the four uncertain input variables will be impacted; the new well should be located between the three wells with good properties and the well with bad properties; this well will de-risk the pxy distribution and the owc; in addition, a new core sample can be taken to assess the relative permeability; the extended well test will be used to obtain the aquifer parameters. 3) performing an extended well test on an existing well 4) by using one of the existing wells for performing an extended well test, only the uncertainty related to the aquifer strength can be investigated, keeping the remaining uncertainties at the same level as in the case without data acquisition. 5) bayes’ theorem should be applied to incorporate the value of the new data in the project value; to do that, reliability probability for all the combinations state-data outcome should be estimated, and those values are converted by means of the bayes’ theorem in the posterior probabilities, which are used for calculating the project value for each alternative. 6) in this research study, two different cases are assessed: the case where the data are treated as crisp, and the case where the data are treated as fuzzy. in the latter case, the uncertainty in the data due to imprecision is captured by using membership functions for doing that, three membership functions are defined: m1 or high, m2 or medium, and m3 or low. here, high means that the compound effect of data acquisition over the four variables is high, although in one or more variables that may not be the case. the same applies to medium and low membership functions. the value assigned to each compound state for each membership function describes the degree of membership that the compound state has in the respective membership function and, the compound value of the four variables in the membership function is the average value. tables 2a. and 2b. show the membership functions m1, m2 and m3 for each potential data outcome in the case of drilling a new well and performing an extended well test alternative. table 2a. membership functions for the first eight compound parameters for drilling a new well and performing an extended well test (hhhh) (hhhl) (hhlh) (hhll) (hlhh) (hlhl) (hllh) (hlll) m1 0.638 0.550 0.525 0.438 0.500 0.413 0.388 0.300 m2 0.250 0.263 0.275 0.288 0.250 0.263 0.275 0.288 m3 0.113 0.188 0.200 0.275 0.250 0.325 0.338 0.413 vilela et al./decis. mak. appl. manag. eng. 3 (2) (2020) 97-118 108 table 2b. membership functions for the last eight compound parameters for drilling a new well and performing an extended well test (lhhh) (lhhl) (lhlh) (lhll) (llhh) (llhl) (lllh) (llll) m1 0.525 0.438 0.413 0.325 0.388 0.300 0.275 0.188 m2 0.263 0.275 0.288 0.300 0.263 0.275 0.288 0.300 m3 0.213 0.288 0.300 0.375 0.350 0.425 0.438 0.513 for the case of using an existing well, the membership functions m1, m2 and m3 corresponding to high, medium and low are calculated. in this case (using an existing well), the only parameter that is evaluated is the aquifer strength. the tables 3a. and 3b. show the value assigned to each compound state within the three membership functions, which reflects the degree of membership that the state has in the corresponding membership function for the “performing an extended well test on an existing well” alternative. table 3a. membership functions for the first eight compound parameters for performing an extended well test on an existing well (hhhh) (hhhl) (hhlh) (hhll) (hlhh) (hlhl) (hllh) (hlll) m1 0.700 0.700 0.700 0.700 0.150 0.150 0.150 0.150 m2 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 m3 0.100 0.100 0.100 0.100 0.650 0.650 0.650 0.650 table 3b. membership functions for the last eight compound parameters for performing an extended well test on an existing well (lhhh) (lhhl) (lhlh) (lhll) (llhh) (llhl) (lllh) (llll) m1 0.700 0.700 0.700 0.700 0.150 0.150 0.150 0.150 m2 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 m3 0.100 0.100 0.100 0.100 0.650 0.650 0.650 0.650 in the decision phase, on the top of the unpv already used in the screening phase, the internal rate of return (irr) and its utility value (uirr) are used. the fis was built using matlab® r2015a software with triangular and truncated triangular functions. the values involved in the decision are unpv and uirr, and their fuzziness is represented with three membership functions for each one: unpv_ high, unpv_medium, unpv_low, uirr_high, uirr_medium, and uirr_low. the decision options are “to endorse”, “not to endorse” or “to reframe” the project. if…then rules are designed to reflect the imprecision in the decision process. for this case study, nine rules were created, as shown in table 4. a holistic approach to assessment of value of information (voi) with fuzzy data and decision... 109 table 4. decision-making rules for the fis decision rules # if then rule 1 (unpv is unpv_low) and (uirr is uirr_high) (decision is reframing) rule 2 (unpv is unpv_low) and (uirr is uirr_medium) (decision is no_endorsement) rule 3 (unpv is unpv_low) and (uirr is uirr_low) (decision is no_endorsement) rule 4 (unpv is unpv_medium) and (uirr is uirr_high) (decision is endorsement) rule 5 (unpv is unpv_medium) and (uirr is uirr_medium) (decision is reframing) rule 6 (unpv is unpv_medium) and (uirr is uirr_low) (decision is no_endorsement) rule 7 (unpv is unpv_high) and (uirr is uirr_high) (decision is endorsement) rule 8 (unpv is unpv_high) and (uirr is uirr_medium) (decision is endorsement) rule 9 (unpv is unpv_high) and (uirr is uirr_low) (decision is reframing) 3.4. case study results expected value assessment using crisp and fuzzy data 1) the expected value for drilling a new well and performing an extended well test data acquisition table 5 shows the results of the evaluation for the case of drilling a new well and performing an extended well test table 5. expected value assessment for drilling a new well and performing an extended well test data acquisition proposal values no data crisp data fuzzy data npv (mm $) 3.02 12.19 −9.78 irr (%) −2.30 −2.49 −2.49 unpv −0.0069 0.0006 −0.0074 uirr −0.2536 0.2665 −0.2741 unpv analysis when unpv is used as a decision criterion, table 6 shows that when the classical methodology has used the value of “drilling a new well and performing an extended well test alternative is higher than the value of “do not acquire data” alternative; however, when the fuzzy nature of the data is included in the analysis, the value of “drilling a new well and performing an extended well test” alternative is reduced, and indeed the “no data acquisition” alternative is better than data acquisition. this change in the decision when the fuzzy characteristics of the data are included in the analysis is maintained in the case of using values instead of utility values. uirr analysis vilela et al./decis. mak. appl. manag. eng. 3 (2) (2020) 97-118 110 when uirr is used as a decision criterion, the “drilling a new well and performing an extended well test” alternative has a lower value than the value of “do not acquire data” alternative in both cases, crisp and fuzzy data; and this assessment holds in case values are used instead of utilities. 2) the expected value for performing an extended well test on an existing well data acquisition for this alternative, table 6 shows the results of the evaluation. table 6. expected value assessment for performing an extended well test on an existing well data acquisition proposal values no data crisp data fuzzy data npv (mm $) 3.02 98.04 97.31 irr (%) −2.30 −1.53 −1.53 unpv −0.0069 0.0197 0.0174 uirr −0.2536 −0.1752 −0.1798 unpv analysis using unpv as a decision criterion, and with the classical methodology for the data acquisition case of performing an extended well test on an existing well, the data acquisition alternative has a higher value than the value of ”do not acquire data” alternative whether the data is crisp or fuzzy. the same conclusion is reached using values instead of utilities. uirr analysis when the uirr is used as a decision criterion to assess the case of performing an extended well test on an existing well, the classical methodology shows that the best project is the data acquisition alternative, because both crisp and fuzzy acquisition have higher values than the value of “do not acquire data” alternative; a similar conclusion is reached when values are used instead of utilities. fuzzy inference system assessment using crisp and fuzzy data tables 7 and 8 show the outcomes of the fuzzy inference assessments of the case of drilling a new well and performing an extended well test and performing an extended well test on an existing well. table 7. fis assessment for drilling a new well and performing an extended well test data acquisition proposal values no data crisp data fuzzy data fis values −0.217 −0.170 −0.359 fis utility values −0.274 −0.268 −0.289 table 8. fis assessment for performing an extended well test on an existing well data acquisition proposal values no data crisp data fuzzy data fis values −0.217 0.444 0.444 fis utility values −0.274 −0.171 −0.178 a holistic approach to assessment of value of information (voi) with fuzzy data and decision... 111 considering the results shown in table 7 for drilling a new well and performing an extended well test, using crisp data, both values and utility values (bring about through the utility function) indicate the best alternative is “acquire data” or drill the well and perform an extended well test”; however, when the fuzzy characteristics of the data is included in the assessment, the best alternative switched to “do not acquire data”. table 8 shows that for “performing an extended well test on an existing well” alternative, both objective functions, fis values, and fis utility values, indicate that the best alternative is “acquire data” or “perform an extended well test on an existing well”. the inclusion of the fuzzy characteristics of the data in the analysis does not change the results. 4. conclusions and recommendations the inclusion of the fuzzy characteristics of the data that deal with aleatoric, but also affect epistemic uncertainties, in the voi assessment is very important because it can have a significant impact on the final decisions. in the case study discussed in this paper for “drilling a new well and performing an extended well test” alternative, the decision switched from “do not acquire data” to “acquire data” when the fuzzy data nature of the data is included in the analysis. it was observed that in “performing an extended well test on an existing well” alternative, that switch does not occur. that happens because of two reasons: i) the difference in values and utility values between the two alternatives: “performing an extended well test on an existing well” and “do not acquire data” is not large and, ii) the degree of fuzziness or the level of vagueness assigned to the data as described by the membership functions. in general, it is observed that when the fuzzy characteristic of the data is included in the analysis, the value of the data acquisition is reduced. using a fuzzy inference system allows for the aggregation of two or more decision criteria (npv, irr, etc.) within only one decision criterion that summarizes the result of the assessment; the several decision criteria can be weighted as desired into the fis. doe is a robust theory suitable for analysis of voi problems and steering the decision process for selecting the data acquisition actions that provide the optimum value to the project; proceeding in this way ensures that the decision process fits the needs of the oil and gas industry. however, the membership functions and utility functions still carry a large degree of subjectivity and further work is required to assess the level of subjectivity and how this might impact voi analysis. in the near future, additional research efforts should be dedicated to the use of machining learning to support the decision-making process by integrating the normative and descriptive elements of the decision process in a coherent and rational manner. author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research received no external funding. conflicts of interest: the authors declare no conflicts of interest. vilela et al./decis. mak. appl. manag. eng. 3 (2) (2020) 97-118 112 references ahmed, a., elkatatny, s., ali, a., mahmoud, m. & 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(1996). fuzzy logic on the frontiers of decision analysis and expert systems. in: proceedings of the 1996 biennial conference of the north american fuzzy information processing society – nafips, berkeley, ca, usa, june 19–22, 1996, pp. 65–69. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 3, issue 2, 2020, pp. 119-130 issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame2003119m * corresponding author. e-mail addresses: km.18ma1103@phd.nitdgp.ac.in (k. mohanta), arindam84nit@gmail.com (a. dey), anita.buie@gmail.com (a. pal) a study on picture dombi fuzzy graph kartick mohanta 1*, arindam dey 2 and anita pal 1 1 department of mathematics, national institute of technology durgapur, india 2 department of computer sciences and engineering, saroj mohan institute of technology, hooghly, india received: 3 june 2020; accepted: 4 september 2020; available online: 20 september 2020. original scientific paper abstract: the picture fuzzy graph is a newly introduced fuzzy graph model to handle with uncertain real scenarios, in which simple fuzzy graph and intuitionistic fuzzy graph may fail to model those problems properly. the picture fuzzy graph is used efficiently in real world scenarios which involve several answers to these types: yes, no, abstain and refusal. in this paper, the new idea of dombi picture fuzzy graph is introduced. we also describe some operations on dombi picture graphs, viz. union, join, composition and cartesian product. in addition, we investigated many interesting results regarding the operations. the concept of complement and isomorphism of picture dombi fuzzy graph are presented in this paper. some important results on weak and co-weak isomorphism of picture dombi fuzzy graph are derived. key words: t-norm, s-norm, picture dombi fuzzy graph, union, composition, cartesian product, join, complement, homomorphism, isomorphism. 1. introduction menger (1942) presented triangular norms (t-norms) and triangular co-norms (tconorms) in the framework of probabilistic metric spaces which were later defined and discussed by schweizer and berthold (2011). alsina et al. (1983) proved that t-norms and t-conorms are standard models for intersecting and unifying fuzzy sets, respectively. since then, many other researchers have presented various types of t-operators for the same purpose (hamacher, 1978). zadeh’s conventional t-operators, min and max, have been used in almost every application of fuzzy logic particularly in decision-making processes and fuzzy graph theory. it is a well-known fact that from theoretical and experimental aspects other t-operators may work better in some situations, especially in the context of decision-making processes. for example, the product operator may be preferred to the min operator (dubois et al., 2000). for the selection of appropriate tmailto:km.18ma1103@phd.nitdgp.ac.in mailto:arindam84nit@gmail.com mailto:anita.buie@gmail.com mohanta et al./decis. mak. appl. manag. eng. 3 (2) (2020) 119-130 120 operators for a given application, one has to consider the properties they possess, their suitability to the model, their simplicity, their software and hardware implementation, etc. as the study on these operators has widened, multiple options are available for selecting t-operators that may be better suited for given research. there are various reallife problems that we cannot explain with the concept of fuzzy set theory. for solving these kinds of problem, k. atanassov (1986) proposed the idea of an intuitionistic fuzzy set (ifs). in ifs we consider membership function and non-membership function such that their sum is lying in [0; 1]. in ifs theory, the idea of neutrality membership value is not considering. in many real-life situations, the neutral membership degree is needed, like a democratic election station. human beings generally give opinions having more answers of the type: yes, no, abstain and refusal. for example, in a democratic voting system, 1000 people participated in the election. the election commission issues 1000 ballot paper and one person can take only one ballot for giving his/her vote and a is only one candidate. the results of the election are generally divided into four groups came with the number of ballot papers namely “vote for the candidate (500)”, “abstain in the vote (200)”, “vote against a candidate (200)” and “refusal of voting (100)”. the “abstain in the vote” describes that ballot paper is white which contradicts both “vote for the candidate” and “vote against a candidate” but it considers the vote. however, “refusal of voting” means bypassing the vote. this type of real-life scenarios cannot be handled by intuitionistic fuzzy set. if we use intuitionistic fuzzy sets to describe the above voting system, the information of voting for non-candidates may be ignored. to solve this problem, cuong and kreinovich (2014) proposed the concept of picture fuzzy set which is a modified version of the fuzzy set and intuitionistic fuzzy set. picture fuzzy set (pfs) allows the idea degree of positive membership, degree of neutral membership and degree of negative membership of an element. graph theory is an important mathematical tool for handling many real-world problems. graph theory has various application in different areas like computer science, social sciences, economics, physics, system analysis, chemistry, neural networks, electrical engineering, control theory, transportation, architecture, and communication. kaufmann (1975) introduces the basic concept of fuzzy graph theory and after that rosenfeld (1975) describes more idea on the fuzzy graph-theoretic concept. krassimir t atanassov introduces the concept of intuitionistic graph theory. shovan dogra (2015) describes different types of product of fuzzy graphs. havare, özge çolakoğlu, n.d. discussed on the coronary product of two fuzzy graphs. in this paper we present the concept of picture dombi fuzzy graph (pdfg) and discussed the operations like union, join, composition, cartesian product, h-morphism, isomorphism, complement of dpfg’s. we also introduce some theorems and examples on pdfg’s. 2. preliminaries t-norm a t-norm is a binary mapping : [0,1] [0,1] [0,1]t   which is satisfies the following conditions: , , , [0,1]a b c d  1. (boundedness property) (0, 0) 0, ( ,1) (1, )t t a t a a   ; 2. (monotonicity property) ( , ) ( , )t a b t c d , if a c and b d ; 3. (commutativity property) ( , ) ( , )t a b t b a ; 4. (associativity property) ( , ( , )) ( ( , ), )t a t b c t t a b c . a study on picture dombi fuzzy graph 121 t-conorm or s-norm a t-conorm is a binary mapping : [0,1] [0,1] [0,1]s   which is satisfies the following conditions: , , , [0,1]a b c d  (boundedness property) (1,1) 1s  , ( , 0) (0, )s a s a a  ; (monotonicity property) ( , ) ( , )s a b t c d , if a c and b d ; (commutativity property) ( , ) ( , )s a b s b a ; (associativity property) ( , ( , )) ( ( , ), )s a s b c s s a b c . hamacher norm hamacher define t-norm and s-norm as follows: , [0,1]a b  (t-norm) ( , ) (1 )( ) ab t a b a b ab       , 0  . (s-norm) ( 1) ( , ) 1 ab a b s a b ab        , 1   . dombi norm the dombi norm is given by , [0,1]a b  (t-norm) 1 1 ( , ) 1 1 1 [( ) ( ) ] t a b a b a b         ; (s-norm) 1 1 ( , ) 1 1 1 [( ) ( ) ] s a b a b a b            . remark 1: if we put 1  in dombi t-norm, we have ( , ) ab t a b a b ab    , , [0,1]a b  . if we put 1  in dombi s-norm, we have 2 ( , ) 1 a b ab s a b ab     , , [0,1]a b  . fuzzy set let x be a universal set. a fuzzy set m of x is the collection of elements  in x s. t., ( ) [0,1]t   . here t is called a membership function of m i.e., : [0,1]t x  . fuzzy graph a f-graph of the graph ( , ) g g g v e  is a pair ( , )g    , where : [0,1]v  is a fuzzy set on g v and : [0,1] g g v v   is a fuzzy relation on g v s. t., ( , ) ( ) ( )x y x y     , ( , ) g g x y v v   (zadeh, 1965). picture fuzzy set (pfs) let be an universal set. a pfs is defined as follows { , ( ), ( ), ( ) : 0 ( ) ( ) ( ) 1, }                     . here : [0,1]  , : [0,1]  and : [0,1]  are called positive membership degree, neutral membership degree and negative membership degree respectively. for all   , 1 ( ( ) ( ) ( ))          is called refusal function of  in . mohanta et al./decis. mak. appl. manag. eng. 3 (2) (2020) 119-130 122 picture fuzzy relation (pfr) let and be two universal sets. a pfr is subset of  s. t., { ( , ), ( , ), ( , ), ( , ) : 0 ( , ) ( , ) ( , ) 1, ( , ) }                               , where : [0,1]   , : [0,1]   and : [0,1]   are called positive membership function, neutral membership function and negative membership function respectively. dombi graph let ( , ) g g g v e be a crisp undirected graph contain no self-loop and parallel edges. also, let : [0,1]v  membership degree on v and : [0,1]v v   be the membership degree on the symmetric fuzzy relation e v v  . then ( , , )v   , is said to be a dombi graph if ( ) ( ) ( , ) ( ) ( ) ( ) ( ) a b a b a b a b           , ( )ab e  . picture dombi fuzzy graph (pdfg) let ( , ) g g g v e be a crisp undirected graph contain no self-loop and parallel edges. also, let ( , , )        s. t., : [0,1]v   , : [0,1]v   and : [0,1]v   be the positive membership degree, neutral membership degree and negative membership degree respectively on the pfs v . we consider ( , , )        s. t., : [0,1]v v    , : [0,1]v v    and : [0,1]v v    as the positive membership degree, neutral membership degree and negative membership degree respectively, in the symmetric pfr e v v  . then ( , , )v   , is said to be a pdfg if 1. ( ) ( ) ( ) ( ) ( ) ( ) ( ) a b ab a b a b                  , ( ) g ab e  ; 2. ( ) ( ) ( ) ( ) ( ) ( ) ( ) a b ab a b a b                  , ( ) g ab e  ; 3. ( ) ( ) 2 ( ) ( ) ( ) 1 ( ) ( ) a b a b ab a b                   , ( ) g ab e  . 3. some operation on pdfg’s union the union of two pdfg's ( , , )v   and ( , , )v   of the graphs ( , ) g g g v e  and ( , ) h h h v e  respectively, is denoted by  and is defined as ( , , ) g h v v       , where ( , , )                   and ( , , )                  s. t., ( )( )      ( ), if g h v v       ( ), if h g v v       a study on picture dombi fuzzy graph 123 ( ) ( ) , if ( ) ( ) ( ) ( ) g h v v                         ( )( )      ( ), if g h v v       ( ), if h g v v       ( ) ( ) , if . ( ) ( ) ( ) ( ) g h v v                         ( )( )      ( ), if g h v v       ( ), if h g v v       ( ) ( ) 2 ( ) ( ) , if 1 ( ) ( ) g h v v                          . ( )( )ab     ( ), if ( ) g h ab ab e e     ( ), if ( ) h g ab ab e e     ( ) ( ) , if ( ) ( ) ( ) ( ) g h ab ab e e ab ab ab ab                   ( )( )ab     ( ), if ( ) g h ab ab e e     ( ), if ( ) h g ab ab e e     ( ) ( ) , if ( ) ( ) ( ) ( ) ( ) g h ab ab ab e e ab ab ab ab                  ( )( )ab     ( ), if ( ) g h ab ab e e     ( ), if ( ) h g ab ab e e     ( ) ( ) 2 ( ) ( ) , if ( ) 1 ( ) ( ) g h ab ab ab ab ab e e ab ab                   . example 1: we consider two pdfg's ( , ) a a a    (shown in fig. 1(a) ) and ( , ) b b b    (shown in fig. 1(b) )of the graphs ( , ) a a a v e  and ( , ) b b b v e  respectively, where { , , } a v x y z , { , , } a e xy yz zx , { , , } b v y z w and { , , } b e yz yw zw . then the union of a and b are shown in figure 1(c). mohanta et al./decis. mak. appl. manag. eng. 3 (2) (2020) 119-130 124 figure 1(a). pdfg a figure 1(b). pdfg b figure 1(c). pdfg a b join the join of two pdfg's ( , , )v   and ( , , )v   of the graphs ( , ) g g g v e  and ( , ) h h h v e  respectively, is denoted by  and is defined as ( , , , ) g h v v e       , where ( , , )                   , ( , , )                   , g h v v   , g h e e e e   (e  set of all edges joining the nodes of g v and ) h v s. t., ( )( ) ( )( ), if g h v v                ( )( ) ( )( ), if g h v v                ( )( ) ( )( ), if g h v v                ( )( )ab     ( )( ), if ( ) g h ab ab e e        ( ) ( ) , if ( ) ( ) ( ) ( ) ( ) a b ab e a b a b                 ( )( )ab     ( )( ), if ( ) g h ab ab e e        ( ) ( ) , if ( ) . ( ) ( ) ( ) ( ) a b ab e a b a b                 ( )( )ab     ( )( ), if ( ) g h ab ab e e        a study on picture dombi fuzzy graph 125 ( ) ( ) 2 ( ) ( ) , if ( ) 1 ( ) ( ) a b a b ab e a b                  theorem 1: the join of two pdfg's is a pdfg. composition the composition of two pdfg's ( , , )v   and ( , , )v   of the graphs ( , ) g g g v e  and ( , ) h h h v e  respectively, is denoted by and is defined as ( , , , ) g h v v e     , where ( , , )               , ( , , )               and 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 1 0 1 0 1 {(( , )( , )) : , ( ) } {(( , )( , )) : ( ) , } {(( , )( , )) : ( ) , } g h g h g e s t s t s v t t e s t s t s s e t v s t s t s s e t t          s. t., ( )( , )      ( ) ( ) ( ) ( ) ( ) ( )                      ( )( , )      ( ) ( ) ( ) ( ) ( ) ( )                      ( )( , )      ( ) ( ) 2 ( ) ( ) 1 ( ) ( )                       v  and ( , ) e   , ( ) ( ) ( )(( , )( , )) ( ) ( ) ( ) ( )                               ( ) ( ) ( )(( , )( , )) ( ) ( ) ( ) ( )                               ( ) ( ) 2 ( ) ( ) ( )(( , )( , )) 1 ( ) ( )                                v  and ( , ) e   , ( ) ( ) ( )(( , )( , )) ( ) ( ) ( ) ( )                               ( ) ( ) ( )(( , )( , )) ( ) ( ) ( ) ( )                               ( ) ( ) 2 ( ) ( ) ( )(( , )( , )) 1 ( ) ( )                                ( , ) g e   , and h v   , ( )(( , )( , ))          ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 ( ) ( ) ( )                                               ( )(( , )( , ))         ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 ( ) ( ) ( )                                                mohanta et al./decis. mak. appl. manag. eng. 3 (2) (2020) 119-130 126 ( )(( , )( , ))         ( ) ( ) ( ) 2 ( ) ( ) 2 ( ) ( ) 2 ( ) ( ) 4 ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) ( ) 2 ( )                                                                                        example 2: we consider two pdfg's ( , ) a a a    and ( , ) b b b    of the graphs ( , ) a a a v e  and ( , ) b b b v e  respectively, where { , , } a v x y z , { , } a e xy yz , { , } b v a b and { } b e ab . then the composition of a and b are shown in fig. 2(c). figure 2(a). pdfg a figure 2(b). pdfg b figure 2(c). pdfg a b cartesian product the cartesian product of two pdfg's ( , , )v   and ( , , )v   of the graphs ( , ) g g g v e  and ( , ) h h h v e  respectively, is denoted by  and is defined as ( , , ) g h v v       , where ( , , )                   and ( , , )                   s. t., ( , ) v v    , ( ) ( ) ( )( , ) ( ) ( ) ( ) ( )                             ( ) ( ) ( )( , ) ( ) ( ) ( ) ( )                             ( ) ( ) 2 ( ) ( ) ( )( , ) 1 ( ) ( )                              v  and ( , ) e   , a study on picture dombi fuzzy graph 127 ( ) ( ) ( )(( , )( , )) ( ) ( ) ( ) ( )                               ( ) ( ) ( )(( , )( , )) ( ) ( ) ( ) ( )                               ( ) ( ) 2 ( ) ( ) ( )(( , )( , )) 1 ( ) ( )                                v  and ( , ) e   , ( ) ( ) ( )(( , )( , )) ( ) ( ) ( ) ( )                               ( ) ( ) ( )(( , )( , )) ( ) ( ) ( ) ( )                               ( ) ( ) 2 ( ) ( ) ( )(( , )( , )) 1 ( ) ( )                                ( , )( , ) ( )v v e       , ( )(( , )( , )) 0          , ( )(( , )( , )) 0          , ( )(( , )( , )) 0          . remark 1: the cartesian product of two pdfg's is not necessarily a dfg. complement of a pdfg let ( , , ) g v   be a pdfg of the graph ( , ) g g g v e . then the complement of is represented as ( , , ) c c c g v   and is defined as follows: c      , c      and c      ( ) c ab  ( ) ( ) , if ( ) 0 ( ) ( ) ( ) ( ) a b ab a b a b                   ( ) ( ) ( ), if 0 ( ) 1 ( ) ( ) ( ) ( ) a b ab ab a b a b                       ( ) c ab  ( ) ( ) ( ) , if ( ) 0 ( ) ( ) ( ) b b a ab a b a                   ( ) ( ) ( ) ( ), if 0 ( ) 1 ( ) ( ) ( ) b a b ab ab a b a                       ( ) c ab   ( ) ( ) 2 ( ) ( ) , if ( ) 0 1 ( ) ( ) a b a b ab a b                   ( ) ( ) 2 ( ) ( ) ( ) , if 0 ( ) 1 1 ( ) ( ) a b a b ab ab a b                        mohanta et al./decis. mak. appl. manag. eng. 3 (2) (2020) 119-130 128 example 3: we consider the pdfg ( , ) a a    of the graph ( , ) g g g v e  where { , , } g v x y z , { } g e yz . then complement c of shown in fig. 3(a) and fig. 3(b) respectively. figure 3(a). pdfg figure 3(b). c theorem 2: let ( , , ) g v   be a pdfg of the graph ( , ) g g g v e . then ( ) c c  . homomorphism, isomorphism, weak isomorphism, co-weak isomorphism let us consider two pdfg's ( , , )v   and ( , , )v   of the graphs ( , ) g g g v e  and ( , ) h h h v e  , where ( , , )        , ( , , )        , ( , , )        and ( , , )        . (homomorphism) a mapping :  is said to be a homomorphism, if g v  ( ) ( ( ))        , ( ) ( ( ))        and ( ) ( ( ))        ; ( ) g ab e  ( ) ( ( ))ab ab      , ( ) ( ( ))ab ab      and ( ) ( ( ))ab ab      . (isomorphism) a mapping :  is said to be an isomorphism, if g v  ( ) ( ( ))        , ( ) ( ( ))        and ( ) ( ( ))        ; ( ) g ab e  ( ) ( ( ))ab ab      , ( ) ( ( ))ab ab      and ( ) ( ( ))ab ab      . if and are isomorphism, then we write  . (weak-isomorphism) a mapping :  is said to be a weak isomorphism, if  homomorphism; g v  ( ) ( ( ))        , ( ) ( ( ))        and ( ) ( ( ))        . (co-weak isomorphism) a mapping :  is said to be a co-weak isomorphism, if  is a homomorphism; ( ) g ab e  ( ) ( ( ))ab ab      , ( ) ( ( ))ab ab      and ( ) ( ( ))ab ab      . self-complementary let ( , , ) g v   be a pdfg of the graph ( , ) g g g v e . then is said to be selfcomplementary if c . a study on picture dombi fuzzy graph 129 theorem 3: let ( , , ) g v   be a self-complementary pdfg of the graph ( , ) g g g v e . then 0 0 0 0 0 0 0 0 0 0 0 0 ( ) ( )1 ( ) , 2 ( ) ( ) ( ) ( )s t s t s t s t s t s t                      0 0 0 0 0 0 0 0 0 0 0 0 ( ) ( )1 ( ) 2 ( ) ( ) ( ) ( )s t s t s t s t s t s t                      0 0 0 0 0 0 0 0 0 0 0 0 ( ) ( ) 2 ( ) ( )1 ( ) . 2 1 ( ) ( )s t s t s t s t s t s t                       proof: let be a self-complementary graph. so,  an isomorphism : c   s. t., g v  ( ) ( ( )) c         , ( ) ( ( )) c         ( ) g ab e  , ( ) ( ( )) c ab ab      , ( ) ( ( )) c ab ab      and ( ) ( ( )) c ab ab      . now, we know that, ( ( )) ( ( )) ( ( ) ( )) ( ( ) ( )) ( ( )) ( ( )) ( ( )) ( ( )) c c c c c c c a b a b a b a b a b                               or, ( ) ( ) ( )) ( ( ) ( )) ( ) ( ) ( ) ( ) a b ab a b a b a b                       or, ( ) ( ) ( )) ( ( ) ( )) ( ) ( ) ( ) ( )a b a b a b a b ab a b a b a b                             or, ( ) ( ) 2 ( )) ( ) ( ) ( ) ( )a b a b a b ab a b a b                      or, ( ) ( )1 ( )) 2 ( ) ( ) ( ) ( )a b a b a b ab a b a b                      . in similar way we can proof the remaining two results. this completes the proof. 4. conclusion in this paper, we have introduced the new concept of picture dombi fuzzy graph. we have proposed some operators of union, join, composition and cartesian product of any two dombi picture fuzzy graphs and investigate many interesting properties of dombi picture fuzzy graph. finally, we define the complement picture dombi fuzzy graph and the isomorphic properties on it. the concept of picture dombi fuzzy graphs can be used to model in several areas of expert systems, transportation, artificial neural networks, pattern recognition and computer networks. author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research received no external funding. conflicts of interest: the authors declare no conflicts of interest. mohanta et al./decis. mak. appl. manag. eng. 3 (2) (2020) 119-130 130 references alsina, c., trillas, e., & valverde, l. 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(n.d.). on corona product of two fuzzy graphs. kaufmann, a. (1975). introduction à la théorie des sous-ensembles flous à l’usage des ingénieurs (fuzzy sets theory). menger, k. (1942). statistical metrics. proceedings of the national academy of sciences of the united states of america, 28(12), 535. rosenfeld, a. (1975). fuzzy graphs††the support of the office of computing activities, national science foundation, under grant gj-32258x, is gratefully acknowledged, as is the help of shelly rowe in preparing this paper. in fuzzy sets and their applications to cognitive and decision processes. https://doi.org/10.1016/b9780-12-775260-0.50008-6 schweizer, berthold, and a. s. (2011). probabilistic metric spaces. courier corporation. zadeh, l. a. (1965). fuzzy sets. information and control. https://doi.org/10.1016/s0019-9958(65)90241-x © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 1, issue 2, 2018, pp. 1-15 issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame1802001m selection of the railroad container terminal in serbia based on multi criteria decision-making methods milan milosavljević1*, marko bursać1, goran tričković1 1 the school of railway applied studies, zdravka čelara 14, 11000 belgrade, serbia received: 6 november 2017; accepted: 25 may 2018; available online: 27 may 2018. original scientific paper abstract: intermodal transport is one of the key elements for sustainable freight transport at large and medium distances. however, its efficiency in many cases depends on the location of the railroad container terminals (ct). the favorable position of serbia provides an opportunity to establish a large number of container trains, which can lead to a more developed intermodal transport system in the entire balkans and beyond. in this paper the problem of the container terminal location in serbia has been considered and resolved. the aim of this paper is to determine the potential macro location of the ct in serbia, which will be most suitable for different stakeholders in the transport chain. choosing the most suitable alternative is a complex multi-criteria task. for this reason, a multi criteria decision-making model has been formulated which consists of a number of alternatives and criteria. alternatives represent potential areas for a site, while some of the criteria are: cargo flows, infrastructure, economic development, social and transport attractiveness and environmental acceptability. for defining weights of the criteria two approaches are used, namely, the delphi and the entropy method. in this paper three methods of the multi criteria decision-making, namely, topsis, electre and mabac are used. by comparing the results of these three methods, an answer to the question where to locate ct will be presented. this is the first step in determining the location of the container terminal. the next phase should respond to the issue of micro location of the terminal. also, after certain customization, the model can be used for solving other categories of location problems. key words: location model, container terminal, topsis, electre, mabac. * corresponding author. e-mail addresses: mimilan89@gmail.com (m. milosavljević), markobursac1987@gmail.com (m. bursać), tricko86@gmail.com (g. tričković) mailto:mimilan89@gmail.com mailto:markobursac1987@gmail.com mailto:tricko86@gmail.com milosavljević et al./decis. mak. appl. manag. eng. 1 (2) (2018) 1-15 2 1. introduction the efficiency of intermodal transport largely depends on the location of the container terminals. the sustainability of transport in europe requires an increasing reallocation between different modes of transport in order to reduce traffic congestion and environmental protection. therefore, the choice of the most favorable location of the railroad terminal is one of the most important strategies for optimization of the entire transport chain. due to its favorable geographical position and important transport corridors located on its territory, the republic of serbia has a great potential for developing intermodal transport. considering that there is almost no such type of terminal in serbia, along with the tendency to join the european transport network, the aim of this paper is to determine the potential location of ct. there is a number of developed methods used for finding the most suitable location of the terminal, such as standard methods for finding the optimal location defined as the p-median problem (limbourg & jorquin, 2009). klose & drexl (2005) deals with different location problems formulated as optimization ones. in addition, a large number of location problems are solved using multi criteria decision-making methods. unlike conventional methods and techniques of operational research, these methods do not provide for an „objectively the best” solution. these methods are based on mathematical algorithms that are developed to help decision-makers in choosing the most suitable variant. there is a large number of papers devoted to this issue, such as determining the location of the logistic center based on electre method (žak & weglinski, 2014), location of logistic center on the black sea in turkey (uysal &yavuz, 2014), electre i method (maroi et al., 2017), determining the location of the main postal center using topsis method (miletić, 2007), logistic center location in the area of western serbia (tomić et al., 2014), location problem based on ahp method (stević et al., 2015). some authors have compared several multi criteria methods, doing, for example, a comparative analysis of two weighting criteria methods entropy and critic for air conditioner selection using moora and saw (vujičić et al., 2017). more recently, combinations of multi criteria decision-making techniques and fuzzy logic are used for solving location problems (tadić et al., 2015), fuzzy-topsis method for selecting hospital locations (senvar et al., 2016), fuzzy-ahp method for determining solar fields location (asakereh et al., 2017). in addition to conventional methods, there are also others such as the mabac for solving location problems of wind farms in vojvodina (gigović et al., 2017), copras-g method for container terminal operations optimization (barysiene, 2012), hybrid fuzzy-aph-mabac model for selecting the location of masking bindings (božanić et al., 2016), selection of transport and handling resources in logistics centers (pamučar et al., 2015) and the like. 2. problem formulation the observed problem lies in the selection of the most suitable location/region on which the railroad terminal will be located. as a potential location for this terminal, railway sections from serbia are used, as well as the areas in which these sections are located. total numbers of variants are 11, although the serbian railway network is divided into 12 sections: požarevac, lapovo, niš, zaječar, kraljevo, užice, pančevo, zrenjanin, novi sad, subotica and ruma. belgrade railway section was not taken into selection of the railroad container terminal in serbia based on multi criteria decision-making… 3 consideration due to the existence of a container terminal in belgrade in belgrade marshalling yard „makiš“. 2.1. definition of variants for each variant, a railway section is associated with a particular area in which the section is located although the boundaries of the section are different in terms of administrative division. the data about loading and unloading railway freight cars are based on the real railway sections although they cross the administrative boundaries of the area, while the other data used in this paper are taken from the areas in which the section is located. variant 1 subotica is a railway section located in the northern part of serbia and it is the administrative center of severna bačka district. its total area is close to 1784 km2, and its population amounts to 186 906 people. the region is characterized as average in many regards. it is characterized by an average level of economic development, annual gdv per capita of 429 000 rsd and logistical and transport activities imply one important road and rail corridor. the main advantage of this variant is high investment attractiveness because of two free zones, subotica and apatin. the unemployment rate in this region ranks among the lowest in the country (10,7%). the volume of transported goods and number of freight cars are the lowest (4599 freight cars 126 277 t), while in the case of unloading goods in domestic and international traffic region it is in the pre-position. variant 2 novi sad is the capital of južna bačka district. population in this area amounts to 615 371 people, while the total area is close to 4026 km2. the economic potential is high, when considering gdv per capita of 608 000 rsd, which is of the highest value in the whole territory of serbia, without belgrade. novi sad offers a great opportunity for education of younger people with the highest number of high schools and faculties. the total volume of all transported goods in this section is average and close to 890 819 t, and 23675 used freight cars. through the novi sad pass international road corridor e75 and railway corridor e85. the weakness of variant 2 is a high unemployment rate of 15,9% and existence of one free zone novi sad. the region is attractive is terms of environment-friendliness with low noise emission and national park fruška gora. variant 3 zrenjanin is the capital of srednji banat district, located in the northeast part of serbia. its total area is 3257 km2 and its population amounts to 187 667 people. the region is characterized by a high unemployment rate of 14,1% which places this variant at the very top according to this criterion. gdv per capita is 416 000 rsd, while transport and logistic competitiveness is small because there is no large number of economic entities. although the volume of railway transport has been growing in recent years, this section is at very bottom for number of loaded freight cars. with 5644 unloaded cars and 152 492 t of transported goods this region occupies the lowest position. transport infrastructure in variant 3 is in a very poor condition. there is only one international railway line, while there are no state ia roads. this area is environment-friendly. variant 4 pančevo is the capital of južni banat district, with population of 293 730 people and an area close to 4246 km2. the economic potential of this variant is slightly lower than average because of gdv per capita which is 384 000 rsd, and a huge unemployment rate of 20,9%. another weakness of this variant is a very poor condition of transport infrastructure and connection with other nearby cities. availability of transport infrastructure is lower than average with two international railway lines and no state ia roads. investment attractiveness is low because there is milosavljević et al./decis. mak. appl. manag. eng. 1 (2) (2018) 1-15 4 a large number of business subjects. azotara, petrohemija and oil refinery in combination with the port are some of the subjects that can contribute positively to this variant. unfortunately, it does not possess free zones. the total number of loaded and unloaded freight cars in domestic and international transport is 43849 with 1 600 600 t of transported goods. variant 5 ruma is located in the north-eastern part of the country, and it is the capital of srem district. its area is around 3485 km2 and its population amounts to 312 278 people. the region is characterized by a higher than average level of gdv per capita is 411 000 rsd, and a higher unemployment rate of 18,3%. near to this region is šabac free zone which increases investment attractiveness. variant 5 is environment-friendly with a low level of noise. the industrial attractiveness of this variant is reflected in the number of transported goods, which amounts to 1 102 168 t in 2016 and 30 398 used freight cars. variant 6 požarevac is located in the region of braničevo. its total area is 3857 km2, and its population amounts to 183 625 people. this variant has a low unemployment rate of 11% and large industrial attractiveness. with 89 877 freight cars and 3 154 202 t transported cargo, this is the first of all the variants. the reason for this is a steel company in smederevo, which uses two railway stations radinac and smederevo. near to smederevo passes european corridor e75 as well as state ia road and railway lines e70 and e85. variant 7 zaječar is located in the eastern part of the country in the region of zaječar. the total area of the region is 3624 km2 and the population is close to 119 967 people. gdv per capita is 314 000 rsd which is the second lowest value. the unemployment rate is 18,3%, but this variant has a big potential which is evident in a small number of logistic and transport companies and business subjects. the weakness of this variant is that both road and rail transport infrastructures are undeveloped; there are no state ia roads while there is only one railway line. industrial attractiveness is good because of the mines in bor and majdanpek, and the total number of used freight cars is 58602 with 1 508 932 t. no free zones are in this region, either. variant 8 lapovo is the railway section which is located in the central part of serbia in the region of pomoravlje. the total area of this section is 2614 km2 and its population amounts to 71 231 people. this section is located near two state ia roads, and railway corridors e70 and e85. the unemployment rate is huge (19%) and gdv per capita is 322 000 rsd. investment attractiveness is average. svilajnac free zone is located in this region. number of used freight cars is 23562, and total volume of transported goods is 946 831 t. variant 9 niš is the railway section which covers the southern part of the country; it is the center of the region of niš. the total population of the region is 376 319 people while the total area is 2728 km2. this variant has the highest unemployment rate in serbia 24,7%. gdv per capita is 348 000 rsd, and there are two free zones, pirot and vranje. with 14 faculties and higher schools this region attracts a lot of young people and offers them a great opportunity for education. volume of loaded and unloaded cargo is very small amounting to 202 385 t loaded cargo and 499 144 t unloaded cargo. there are two road corridors and three important railway lines. variant 10 kraljevo section is located in the region of raška. population of this region is 309 258 people and the total area of the region is 3923 km2. it is characterized by a low level of gdv per capita of 240 000 rsd. the region is attractive from the logistic and transport point of view. its benefits are big industrial companies and centers located in kragujevac as well as the existence of two free selection of the railroad container terminal in serbia based on multi criteria decision-making… 5 zones, kruševac and kragujevac. total volume of transported goods in 2016 was 765 523 t. the weaknesses of this region are: a relatively poor condition of the transport infrastructure and serious social problems, including a very high unemployment rate of 21,6%. no highways in this region; the railway line in this variant is in a very bad condition. the region is considered to be environment-friendly because of national park kopaonik and a low level of noise. variant 11 užice is the railway section which is located in western part of serbia in the region of zlatibor. this region has the largest area close to 6140 km2. total population is 286 549 people according to 2011 population census. unemployment rate is 15% and gdv per capita is 369 000 rsd. the level of logistics and transport competitiveness is small which makes this region favorable only in terms of its location. volume of transport is 1 051 473 t in 2016. railway line belgrade bar is in a very bad condition while a highway from belgrade to bar is under construction. 2.2. formulation of criteria c1 availability of transport infrastructure (points). this maximized criterion is defined as number of state ia roads and international railway lines that pass through each region or section of the railway network. it measures region accessibility and transport efficiency for distributing goods. also, it shows the condition of the road and rail infrastructure, taking into account water traffic in the case there is a port of terminal in the same region. the criterion is measured on the scale 1-6, whereby point 1 is given for a region with the lowest numbers of corridors and the worst infrastructure condition; point 6 is given, consequently, for the best region. c2 economic development (in thousand rsd). this maximized criterion is defined as an annual value of gdv per capita for each region in serbia. based on this criterion, we can measure the economic potential of each region, i.e. it can be determined whether an investor would like to invest in the given region or not. c3 investment attractiveness (points). this maximized criterion uses the measurement scale of 1 to 10 points for assessment of the overall level of attractiveness of the region. it is defined as a total number of free zones in regions and close to regions. c4 level of transport and logistics competitiveness (points). this minimized criterion is defined on the scale of 1 to 10 and it shows share of logistic and transport companies and business subjects in the region compared to their total number in serbia. this criterion is minimized because any new investor shall first opt for the region with no competition whatsoever. the data necessary for this criterion were based on experience and interviews with experts. c5 transport and logistics attractiveness (t). this criterion measures the industry attractiveness of each region (max). it is expressed in total loaded and unloaded weight and transported by rail in domestic and international transport. unfortunately, this criterion does not include data about transported goods by road. also, given that statistics about transported containers and volume of transport goods in transit on the serbian railway network are only conducted for the whole network, this data are not relevant and have not been taken into account when settling the problem. c6 unemployment rate (%). this minimized criterion is defined as a percentage of unemployed residents in the region. the level of social satisfaction affects the region. this criterion can be defined by the components such as opportunities for education and career development (number of state faculties and high schools). milosavljević et al./decis. mak. appl. manag. eng. 1 (2) (2018) 1-15 6 c7 – environment-friendliness (points). this criterion (max) defines the environment-friendliness of each region. it includes an average daily and night level of noise in the centers of regions and the number of fully protected territories like national parks. 3. a multi criteria decision-making model existence of a multi criteria analysis means existence of more variants and criteria, of which some have to be minimized or maximized, where decisions are made in conflict conditions with the application of instruments that are more flexible than the mathematical method of pure optimization. criteria that are to be maximized are in the profit criteria category although they may not necessarily be profit criteria. similarly, the criteria that are to be minimized are in the cost criteria category. an ideal solution would maximize all the profit criteria and minimize all the cost criteria. normally, this solution is not obtainable. in literature a large number of methods of multi criteria analysis can be found. however, not all the methods are equally theoretically and practically represented and important. there are two types of multi criteria decision-making methods. one is compensatory and the other is a non-compensatory one. compensatory methods are those which calculate the final solution by tolerating some of bad features of a variant under the condition that all other features of this variant are favorable. they actually permit „tradeoffs“ between attributes. a slight decline in one attribute is acceptable if it is compensated by some enhancement in one or more of other attributes. some of these methods are (dimitrijević, 2016):  simple additive weighting (saw),  technique for order preference by similarity to ideal solution (topsis),  preference ranking organization method of enrichment evaluation (promethee),  analytic hierarchy process (ahp), and,  elimination et choix traduisant la realite (electre). in addition to these conventional methods, the following methods are increasingly used:  evaluation based on distance from average solution (edas),  complex proportional assessment (copras),  evaluation of mixed data (evamix),  combinative distance-based asessment (codas),  weighted aggregated sum product assessment (waspas), and,  multi-attribute border approximation area comparison (mabac). the presented model of macro location of the container terminal was done using three compensatory methods, i.e., topsis, electre and mabac, after which the results are compared by methods, and the most favorable variant was adopted for the macro location of the container terminal in serbia. these methods are used because of their common use in solving this type of problem in addition to their simple use and easy definition of input parameters. models are solved by microsoft excel, i.e. its addition for a multi criteria analysis which is called sanna. the aim of this paper is to compare 11 variants, which represent sections on the railway network, in order to find an optimal solution for the railroad container terminal location. these sections are district control offices, from which the management of a certain part of the railway network is performed. there are twelve sections on the serbian railway network, but in this model section belgrade is not selection of the railroad container terminal in serbia based on multi criteria decision-making… 7 used because there is already a railroad container terminal in belgrade marshaling yard. the criteria for comparison and selecting the best variant are described in the previous section and their values are shown in table 1. table 1 the values of the criteria for the observed variants variants c1 c2 c3 c4 c5 c6 c7 subotica 2 429 2 6 441268 10,7 7,00 novi sad 2 608 1 10 890819 15,9 4,25 zrenjanin 1 416 1 2 386899 14,1 8,00 pančevo 2 384 0 9 1592715 20,9 3,75 ruma 3 411 1 1 1102168 18,3 8,00 požarevac 2 405 1 8 3154202 11,0 6,00 zaječar 1 316 0 7 1508932 15,5 7,50 lapovo 5 322 1 5 946831 19,0 5,50 niš 6 348 2 10 701979 24,7 3,25 kraljevo 1 245 2 4 765523 21,6 6,00 užice 1 369 1 3 1051473 15,0 4,75 according to table 1, each of the above criteria needs to be maximized, except for criterion 4 (level of transport and logistic competitiveness) and criterion 6 (unemployment rate, which is a logical conclusion because a lower unemployment rate is more favorable for the development of each region). data about transported goods by railway and number of freight cars (c5) are obtained thanks to the statistics from sector for freight transport „serbian railways“ and nowadays „serbia cargo“. criterion 1, availability of transport infrastructure, is covered by the data from the statistical office of the republic of serbia and working timetable which we use for calculation the number of railway lines. data from the statistical office of the republic of serbia are used for the following criteria: economic development (c2), investment attractiveness (c3) and unemployment rate (c6). yearly statistic handbook from the statistical office of the republic of serbia and statistics of local government are used for defining criterion 7, environmentfriendliness. 3.1. criteria weighting one of the main problems in multi criteria problems belong to criteria (vuković, 2014). taking into account that the weight of criteria can significantly affect the decision-making process, special attention must be paid to the criteria weighting, which, unfortunately, is not always present in problem-solving. for that reason we use two methods, the delphi and the entropy. 3.1.1. delphi method weights of criteria are defined through interviews with experts in the field of railway transport. the final values of weight coefficients, based on experts’ answers and using the delphi method are given in table 2. weight criteria are calculated through three iterations. mean values, standard deviation and coefficient of variation for each criterion are made, and the obtained average value of the coefficient of variation is 12,81%. in the next section, models for milosavljević et al./decis. mak. appl. manag. eng. 1 (2) (2018) 1-15 8 location railroad container terminal using topsis, electre and mabac methods are shown. table 2 weight of criteria by the delphi method criteria c1 c2 (thou. rsd) c3 c4 c5 (t) c6 (%) c7 normalized weights of criteria 0,27 0,13 0,10 0,12 0,23 0,08 0,07 3.1.2. entropy method determination of the objective criteria weights according to the entropy method is based on the measurement of uncertain information contained in the decision matrix. it directly generates a set of weights for a given criteria based on mutual contrast of individual criteria values of variants for each criteria and then for all the criteria at the same time (vuković, 2014). determination of objective criteria weights wj according to the entropy method is carried out in three steps (dimitrijević, 2016). step one involves the normalization of criteria values of variants xij from decision matrix mxn ij xx  : ji x x p m i ij ij ij ,, 1    , (1) entropy ej of all variants is calculated as: jppe m i ijijj   1 ,ln , (2) a constant ε, ε=1/ln m, is used to guarantee that 0≤ej≤1. the degree of divergence dj is calculated as: njed jj ,...,1,1  , (3) since the value of dj is a specific measure of the intensity of a criteria contrast cj, the final relative weight of the criteria, in the third step of the method, can be obtained by simple additive normalization: j d d w n i j j j    , 1 , (4) final values of weight coefficients, based on entropy method are given in table 3. table 3 weight of criteria by the entropy method criteria c1 c2 c3 c4 c5 c6 c7 ej 0,915 0,990 0,977 0,938 0,928 0,987 0,984 dj 0,085 0,010 0,023 0,062 0,072 0,013 0,016 selection of the railroad container terminal in serbia based on multi criteria decision-making… 9 wj 0,301 0,036 0,083 0,220 0,256 0,046 0,058 3.2. application of the topsis method topsis method is the one which compares variants based on their distance from a positive and negative ideal solution (hwang & yoon, 1981). the method is characterized by calculation of the weighted normalized decision matrix and formulation of the positive and negative ideal solution. also, this method is based on the concept that the chosen variant should have the shortest distance from the positive ideal solution and the longest distance from the negative ideal solution (čičak, 2003). weighted criterion matrix is shown in table 4. table 4 weighted criterion matrix with the delphi method variants c1 c2 c3 c4 c5 c6 c7 di+ dici subotica 0,05692 0,04243 0,04714 0,02843 0,02258 0,03842 0,02448 0,18392 0,07634 0,29332 novi sad 0,05692 0,06013 0,02357 0,00000 0,04559 0,02415 0,01486 0,17721 0,06257 0,26095 zrenjanin 0,02846 0,04114 0,02357 0,05687 0,01980 0,02909 0,02797 0,20338 0,07209 0,26171 pančevo 0,05692 0,03798 0,00000 0,00711 0,08151 0,01043 0,01311 0,16217 0,07050 0,30300 ruma 0,08538 0,04065 0,02357 0,06397 0,05641 0,01756 0,02797 0,14032 0,10041 0,41711 požarevac 0,05692 0,04005 0,02357 0,01422 0,16143 0,03760 0,02098 0,12823 0,15291 0,54389 zaječar 0,02846 0,03125 0,00000 0,02132 0,07723 0,02525 0,02623 0,17998 0,06826 0,27499 lapovo 0,14230 0,03184 0,02357 0,03554 0,04846 0,01564 0,01923 0,12780 0,12635 0,49716 niš 0,17076 0,03442 0,04714 0,00000 0,03593 0,00000 0,01136 0,14919 0,15112 0,50321 kraljevo 0,02846 0,02423 0,04714 0,04265 0,03918 0,00851 0,02098 0,19463 0,06769 0,25803 užice 0,02846 0,03649 0,02357 0,04976 0,05381 0,02662 0,01661 0,18280 0,07124 0,28042 weights 0,27000 0,13000 0,10000 0,12000 0,23000 0,08000 0,07000 ideal 0,17076 0,06013 0,04714 0,06397 0,16143 0,03842 0,02797 basal 0,02846 0,02423 0,00000 0,00000 0,01980 0,00000 0,01136 3.3. application of the electre i method evaluation matrix for the electre method is the same as in the case with the topsis method. the only difference is in the steps leading to the final solution. in this method, the variants are compared with each other as a couple; dominant and weak (or dominant and recessive) variants are identified and then weak and defeated alternatives are removed. in the electre method, it is also necessary to define the concordance and discordance index which can be defined as the average values of all values ckl and dkl calculated according to the following equations (5) and (6) (dimitrijević, 2016).   lk mm c c m k m s kl       , 1 1 1 , (5)   lk mm d d m k m s kl       , 1 1 1 , (6) milosavljević et al./decis. mak. appl. manag. eng. 1 (2) (2018) 1-15 10 based on value of concordance index ckl which represents domination of variant vk relative to vl based on weight criteria, we calculate preference threshold value ( c ) and its value is 0,5596. index where variant vk is worse than variant vl shows another index discordance index dkl. in that case we calculate dispreference threshold value ( d ) and its value is 0,7364. 3.4. application of the mabac method the basic setting of the mabac method is reflected in the definition of the distance of the criterion function of each of the observed alternatives from the approximate border area (pamučar & ćirović, 2015). mathematical computation of this method is presented through six steps as follows (božanić & pamučar, 2016): step 1 creating initial decision matrix x. step 2 normalization of the elements of initial decision matrix x. step 3 calculation of weighted matrix elements v. step 4 border approximate area for each criterion is determined by expression: m m j iji vg /1 1            , (7) matrix of approximate border areas g in both variants is given in table 5. table 5 matrix of approximate border areas weight of criteria c1 c2 c3 c4 c5 c6 c7 g delphi method 0,3342 0,1782 0,1507 0,1698 0,2873 0,1217 0,1051 entropy method 0,3726 0,0494 0,1251 0,3113 0,3198 0,0700 0,0871 step 5 calculation of the matrix elements distance from the border approximate area q step 6 ranking variant calculation of the criteria function values by variants is obtained as the sum of the distances of the variants from the border approximate areas qi. summing up the elements of matrix q by rows gives the final values of the criteria function variants:    n j iji minjqs 1 ,...,2,1,,...,2,1, , (8) where n represents the number of criteria, and m represents the number of variants. 4. results based on the previously defined input parameters and weight criteria, the results of the considered methods show which of the given variants is the best for the container terminal location. selection of the railroad container terminal in serbia based on multi criteria decision-making… 11 4.1. results obtained by topsis method complete ranking of the variants using topsis method is shown in table 6. the best variant for micro location of the railroad container terminal in both the variants is variant v6 railway section požarevac. table 6 complete order of variants with the topsis method variant delphi method entropy method r.u.v. rank r.u.v. rank subotica 0,29332 6 0,26737 10 novi sad 0,26095 10 0,18773 11 zrenjanin 0,26171 9 0,32506 6 pančevo 0,30300 5 0,28655 8 ruma 0,41711 4 0,48188 3 požarevac 0,54389 1 0,81239 1 zaječar 0,27499 8 0,27463 9 lapovo 0,49716 3 0,50997 2 niš 0,50321 2 0,47136 4 kraljevo 0,25803 11 0,29766 7 užice 0,28042 7 0,33564 5 4.2. results obtained by the electre method using electre i method two variants are dominant and much better than the others. these variants are 5 and 6, railway sections ruma and požarevac. this method gave 40 preference relations of all the variants, and nine inefficient variants when using the delphi method for weight criteria, and 42 preference relations when using the entropy method. the final results are shown through aggregate dominance matrix in table 7, where the first number means variant one, delphi method and the second number means variant two, entropy method. table 7 aggregate dominance matrix v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v1 0/0 0/0 1/1 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 v2 1/1 0/0 1/1 0/0 0/0 0/0 0/0 0/0 1/1 1/1 0/0 v3 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 v4 0/0 1/1 0/0 0/0 0/0 0/0 1/1 0/0 1/1 1/1 1/1 v5 1/1 1/1 1/1 0/0 0/0 0/0 0/0 1/0 1/1 1/1 1/1 v6 0/0 1/1 1/1 1/1 0/0 0/0 1/1 1/0 1/1 1/1 1/1 v7 0/0 0/1 0/0 0/0 0/0 0/0 0/0 0/0 0/1 1/1 1/1 v8 1/1 1/1 1/1 0/0 0/0 0/0 0/0 0/0 0/1 1/1 0/0 v9 1/1 0/0 1/1 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 v10 0/0 0/0 1/1 0/0 0/0 0/0 0/0 0/0 1/1 0/0 0/0 v11 0/0 1/1 1/1 0/0 0 0 0 1/1 1/1 1/1 0/0 milosavljević et al./decis. mak. appl. manag. eng. 1 (2) (2018) 1-15 12 4.3. results obtained by mabac method ranking of all variants using mabac method is shown in table 8. table 8 rank of the variants using mabac method variant delphi method entropy method si rank si rank subotica 0,0659 5 0,0208 5 novi sad -0,0062 7 -0,1098 11 zrenjanin 0,0014 6 0,0116 6 pančevo -0,1007 11 -0,1066 10 ruma 0,1564 2 0,2083 1 požarevac 0,1897 1 0,1658 3 zaječar -0,0732 9 -0,0689 9 lapovo 0,1254 3 0,1749 2 niš 0,0860 4 0,0881 4 kraljevo -0,0774 10 -0,0268 8 užice -0,0266 8 0,0014 7 4.4. comparison between methods based on the obtained results using the electre method, the best variants and only efficient variants in both the variants are v5 and v6 požarevac and ruma. by comparison the topsis and mabac method, in both variants, in three of four cases the best variant is v6. also, in all situations the first four variants are always the same, požarevac (v6), ruma (v5), lapovo (v8) and niš (v9). rank of variants is given in table 9. table 9 comparison of topsis and mabac method variant mabac topsis mabac topsis delphi method delphi method entropy method entropy method subotica 5 6 5 10 novi sad 7 10 11 11 zrenjanin 6 9 6 6 pančevo 11 5 10 8 ruma 2 4 1 3 požarevac 1 1 3 1 zaječar 9 8 9 9 lapovo 3 3 2 2 niš 4 2 4 4 kraljevo 10 11 8 7 užice 8 7 7 5 selection of the railroad container terminal in serbia based on multi criteria decision-making… 13 general conclusion is that the railroad container terminal should be first located in the area of the railway section požarevac, in the region of braničevo. the best region for location is požarevac. this variant is high in terms of its volume of transported goods and high investment attractiveness. the transportation infrastructure of this region represents an average level, while the unemployment rate is very low. a clear advantage of this region is great connectivity with other regions and the existence of main road and rail corridors. by looking at the complete range of variants, with all the methods, and variants of weighting criteria it can be concluded that those with a high volume of transport and accessibility of infrastructure can be potential locations. regions (railway sections) like kraljevo or zrenjanin should not be taken into further consideration because they would not justify terminal existence by any parameter. 5. conclusion a railroad container terminal location problem, like any other location problem, is a very complex task, which requires a detailed analysis of different segments and parameters. using multi criteria decision-making methods, the model presented in this paper was developed. the macro location of the terminal is defined, which represents the first phase of determining its potential location. the proposed methodology has a universal character and can be applied to different types of location models, both for the selection of the location of railroad terminals, as well as for other railway logistics location problems. a further model development is based on a more detailed analysis of all input parameters. in particular, it is necessary to analyze the flows of goods more closely, including the volume of transported goods from road or water transport. also, the analysis of transport infrastructure can be expanded, using water transport and its impact on potential locations. in addition, an analysis of environmental parameters as well as transport safety in each region can be approached in more detail. market analysis, investment attractiveness and other economic criteria are another direction in the development of the model. the model can be improved using more relevant data for weight criteria, using some other methods for its calculation. for a more detailed analysis, and comparison of the results, other methods such as electre iii/iv, saw and some newer ones can be applied. the next step in our research and development is the formulation and solving of the second phase of the observed problem, that is, micro location of the railroad container terminal. this approach requires an analysis of the micro plan, within the region, in order to find the most suitable field for the location of the railroad container terminal. 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(2014). the selection of the logistics center location based on mcdm/a methodology. transport research procedia, 3, 555-564. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 4, issue 2, 2021, pp. 163-177. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame210402163s * corresponding author. e-mail addresses: haresh_fosc@sgtuniversity.org (haresh. sharma), kriti.kri89@gmail.com (kriti. kumari), kar_s_k@yahoo.com (samarjit. kar) forecasting sugarcane yield of india based on rough set combination approach haresh kumar sharma1, kriti kumari 2 and samarjit kar 3 1 department of mathematics, shree guru gobind singh tricentenary university, gurugram, india 2 department of mathematics, banasthali vidyapith, jaipur, rajasthan, india 3 department of mathematics, national institute of technology durgapur, west bengal, india received: 26 january 2021; accepted: 8 june 2021; available online: 17 june 2021. original scientific paper abstract: this study applied a combination approach using rough set approach for forecasting sugarcane production in india. the rough set is a new mathematical approach that can deal with qualitative time series data. the method of combining forecast values based on rough set, the original time series and single forecasts obtained from single models are taken as condition and decision attributes. finally, the decision table based on actual time series and single forecasting results are used to calculate the weights for the combination of forecasts. moreover, dependency, importance and weights are also calculated for time series through condition and decision attributes. the paper uses autoregressive integrated moving average (arima), double exponential smoothing (des) and grey model (gm) to generate the single forecasts. to validate our proposed analysis, sugarcane production data from 1950 to 2011 was used for the overall empirical analysis and out-sample forecasts were generated from 2012 to 2021 for the comparative analysis. also, arima (2, 1, 1) model was found more appropriate for forecasting sugarcane production. key words: sugarcane, forecast, time series models, rough set combination. 1. introduction india produces the largest amount of sugarcane and thus lands second on the list of top sugarcane-producing nations just after brazil according to foreign agriculture service (fsc) 2020. it reported uttar pradesh has the largest contribution amounting to 38.61% of the overall sugarcane production in the fiscal year 2020-21(icarsugarcane report and molasses production, 2019). then come maharashtra and mailto:kriti.kri89@gmail.com sharma et al./decis. mak. appl. manag. eng. 4 (2) (2021) 163-177 164 karnataka as the second and third largest sugarcane producing states. some other contributors on the list are bihar, tamil nadu, haryana, gujarat and andhra pradesh. thus sugarcane being such a precious commodity it becomes really significant for the indian economy to have highly reliable and accurate forecast of its productions. there exists an effective relationship between the productivity and the price of the crop. with an unanticipated fall in the production, the market stock of the crop declines, thereby reducing the income of the farmer which is followed by the price rise as its consequence. oppositely, if the market is flooded with the crop, it leads to a sudden fall in the prices and thus affecting the income of the farmer. therefore, it can be concluded that these repercussions due to the variations in the prices of the commodity play a significant role in the formulation of the significant economic policies like inflation rate, gdp, wages, salaries etc. apart from this it also affects the production level of other industries which further processes sugarcane and its byproducts thereby affecting their profit margins. from the last few years, various literature has been applied single time series models in the area of time series forecasting. arima models are very popular to forecast sugarcane production (bajpai and venugopalan, 1996; kumar and madhu, 2014). for example, venugopalan and srinath (1998), suresh at al. (2011) and tsitsika et al. (2007) applied arima models for modelling and forecasting of fish catches. also, hanson et al. (1999) compare the forecasting efficiency of neural network models with arima models. arima models were used to forecast the production and productivity of a variety of crops of tamilnadu (balanagammal et al. 2000). boken (2000) used arima models for the forecasting of wheat production in pakistan and canada. balasubramanian and dhanavanthan (2002) have applied arima models to forecast seasonal paddy in tamilnadu and food grains in india. ravichandran and prajneshu (2001) and prajneshu et al. (2002) were compared the accuracy of structural time series models with arima models. maccioitta et al. (2002) use arma models to forecast milk, fat and protein yields of italian simmental cows. state level agricultural production forecasting was also done by applying arima models (indira and datta, 2003). also, chandran and prajneshu (2005) compare the forecasting performance of arima models with nonparametric regression approach for the forecasting of oilseed production in india. forecasting of irrigated crops like potato, mustard and wheat were forecasted by employing arima models (sahu, 2006). milk production in india was also predicted using different time series methods (pal et al. 2007). also, there are different time series models, such as econometric, smoothing models and different combination approaches. in recent years, rough set (rs) approach has been widely used in combine forecasting approach. for example, bao et al. (2006) employed a combination approach based on rough set theory to determine the weighting coefficient in predicting the future of electric power load from 19942000 in zhejiang. xiao et al. (2009) examined the forecasting of international trade in the chongqing municipality of china using a combined approach based on the rough set, which he goes on to compare with individual models. ahmed et al. (2009) applied a combination of forecasts based on rough set. suo et al. (2013) evaluated the weight coefficient by using rough set theory to combine the forecasts of the quadratic curve, grey, and cubic exponent smooth models for forecasting agriculture machinery total power from the period of 1996 to 2008. they explain that rough set combination approach is higher than the individual forecasting methods. zhou and zhang (2013) employed rough set combination method by using support vector machine and neural network to predict the chinese co2 emissions from 1990 to 2011. sharma et al. (2019) proposed hybrid rough set based forecasting model and applied on tourism demand of air transportation passenger data set in australia tourism demand. tang et al. forecasting sugarcane yield of india based on rough set combination approach 165 (2021) applied hybrid fuzzy rough set models in missing traffic data. patra and barman (2021) employed rough set based dependency measure to reduce dimensionality of hyperspectral images. ala'raj et al. (2021), proposed seird dynamic model for forecasting of covid-19 data and applied arima correction model for validation of data set. for instance, jahangir (2020) employed rough set based artificial neural networks model to predict multimodal short-term wind speed. li and wang (2019) proposed hybridized nmgm-arima and nmgm-bp models to forecast india's dependence on foreign oil. sharma et al. (2018) applied rough set based forecasting methods in airline data. wang et al. (2018) used hybrid arima and metabolic nonlinear grey model to forecasting u.s. shale gas monthly production. also sharma et al. (2020) applied rough set theory for forecasting model’s ranking. rough set theory has been successfully applied to various real life decision making problems (karavidić, and projović, 2018; roy et al., 2018; roy et al., 2019; sharma et al., 2018a; stanković et al. 2019). other soft computing approach use to tackle imprecision and vagueness of a data, which has been successfully applied to various real life problems (karavidić and projović, 2018; žižović and pamucar, 2019, vasiljević et al. 2018; mukhametzyanov and pamucar, 2018). moreover, elgabbanni et al. (2014) applied rough set combination model (rsc) with an appropriate weight coefficient to forecast traffic accident time series data for washington dc in the us from 1982-2008. they reveal that the combination method outperforms other individual methods. additionally, the main concern in the combination of forecasts is that how to evaluate some appropriate weight coefficient to combine the forecasts of various single time series models. there have been various ways of determining the weight coefficient in the combination approach such as simple average, the inverse of mape, variancebased, the inverse of mean square error etc. however, previous researchers have not been yet studied the rough set theory in sugarcane production literature, to the best of our knowledge. hence, the main objective of our study is to forecast sugarcane production in india using a novel rough set combination (rsc) approach. the study aims to apply an appropriate way to combine the different single models to improve the forecasting accuracy of single time series models. arima, des and gm models have been combined by applying rough set theory to forecast sugarcane production in india for the period 1950 to 2011. we also study the comparative analysis of single time series and rough set combination methods by underlying mean absolute percentage error (mape) criterion. the remaining of the study is organized as follows. a methodology section discusses the rough set theory. the next section illustrates the procedures of rough set combination method to the study of sugarcane production. data section explains the data. empirical results section describes the results of the empirical study, which includes time series models and their combination. performance comparison of different models section presents the different performance criteria used in the forecasting comparison and the last section gives the conclusions. 2. research methodology rough set is a very useful classification technique for categorical variables like low, average and high. in this method, time series data has been arranged in an information table (decision table) with their objects (data points) by using a dependent and independent time series variables. then, time series variables are transformed into condition and decision variables (attributes). table 1 shows the hypothetical example sharma et al./decis. mak. appl. manag. eng. 4 (2) (2021) 163-177 166 of decision table based on actual time series and single forecasts. these attributes are categorized into different grades like low, average or high and true or false etc. hence, the applications of rough set are applied to generate the weights by establishing the relationships between single forecasts and actual time series. the method of combining forecast values based on rough set, the original time series and single forecasts obtained from single models are taken as condition and decision attributes. finally, the decision table based on actual time series and single forecasting results is used to calculate the weights for the combination of forecasts. table 1. hypothetical example of decision table is a table. condition attributes decision attribute time 𝑋𝑡 (1) ̂ 𝑋𝑡 (2) ̂ 𝑋𝑡 (3)̂ ⋯ 𝑋𝑡 (𝑁)̂ 𝑦 (𝑡) 𝑡1 𝑡2 𝑡3 ⋯ ⋯ 𝑡𝑚 𝑋1(1) ̂ 𝑋1(2) ̂ 𝑋1(3)̂ ⋯ 𝑋1(𝑁)̂ 𝑋2(1) ̂ 𝑋2(2) ̂ 𝑋2(3)̂ ⋯ 𝑋2(𝑁)̂ 𝑋3(1) ̂ 𝑋3(2) ̂ 𝑋3(3)̂ ⋯ 𝑋3(𝑁) ̂ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ 𝑋𝑚 (1) ̂ 𝑋𝑚 (2) ̂ 𝑋𝑚 (3) ̂ ⋯ 𝑋𝑚 (𝑁) ̂ 𝑦(1) 𝑦(2) 𝑦(3) ⋯ ⋯ 𝑦(𝑚) in table 1, 𝑋𝑡 (1) ̂ , 𝑋𝑡 (2) ̂ 𝑋𝑡 (3), ̂ … , 𝑋𝑡 (𝑁) ̂ and 𝑦(𝑡) are single forecasts called independent variable sand actual time series called dependent variables, respectively and𝑋𝑚 (1) ̂ , 𝑋𝑚 (2) ̂ 𝑋𝑚 (3), ̂ … , 𝑋𝑚 (𝑁) ̂ 𝑎𝑛𝑑 𝑦 (𝑚) are the objects (data points) of time series variables. further, these variables are transformed into condition and decision attributes using normalization. the normalization is used to convert quantitative data into qualitative data. the normalization (𝑁𝑡) technique is defined as: 𝑁𝑡 = 𝑍𝑘𝑡−𝑍𝑚𝑖𝑛 𝑍𝑚𝑎𝑥−𝑍𝑚𝑖𝑛 where, 𝑍𝑘𝑡 is the set of actual and single forecasts time series variables, 𝑍𝑚𝑖𝑛 𝑎𝑛𝑑 𝑍𝑚𝑎𝑥 are the minimum and maximum values of 𝑍𝑘𝑡 , k =1,2, . . . . further, actual and single forecasts are transformed into qualitative normalized values (nv) like low, average and high which are defined as; low(l) (if 0 < nv <0.4), average(a)(if 0.4 < nv≤0.8), high(h)(if nv > 0.8). 3. rough set theory (rst) the rough set theory (rst) is a new mathematical technique to handle imprecision, vagueness, and uncertainty (pawlak, 1982). for the evaluation of a vague description of the objects, it is the excellent mathematical tool. the adjective vague express the information quality that is uncertainty or ambiguity that chase from information granulation. the main aim of the rough set theory is the approximation of a set by a pair of two crisp sets called the lower and upper approximations of the sets. forecasting sugarcane yield of india based on rough set combination approach 167 let 𝑈 be the non-empty finite set of objects referred to as universe and a be a nonempty finite set of attributes, then 𝑆 = (𝑈, 𝐴, 𝐶, 𝐷) is called an information system, where 𝐶 and 𝐷 are condition and decision attribute, respectively. for 𝑆 = (𝑈, 𝐴, 𝐶, 𝐷) and p ⊆ a, r ⊆ u can be approximated based on the knowledge having in 𝑃by assembling the p-lower and p-upper approximation of r, represents by 𝑃(𝑅) and 𝑃(𝑅) respectively; where p(r) = {𝑥|[𝑥]𝑃 ⊆ r} (1) p(r) = {𝑥|[𝑥]𝑃 ∩ r ≠ ∅} (2) the objects in 𝑃(𝑅) is known as the set of all members of 𝑈 which can be certainly classified as an object of r in the knowledge p whereas objects in 𝑃(𝑅) is the set of all elements of 𝑈 that can be possibly classified as an object of 𝑅 involving knowledge 𝑃. the boundary region of 𝑅 is expressed as: 𝐵𝑁𝑝 (𝑅) = p(r) − 𝑃(𝑅) is the set of a member which cannot decisively classify into r consisting knowledge p. if lower approximation and upper approximation set are similar then boundary region of set is empty set. in the opposing case, if the boundary region having some member (object) than the set r is referred as rough set concerning p. rst gives an accuracy measure on the quality of classification (pawlak, 1982; pawlak and skowron, 2007). the quality of classification illustrates the ratio of all correctly classified objects of the data set, is calculated in the following manner: 𝛾𝐶 (𝐷) = |𝑃𝑂𝑆𝐶(𝐷)| |𝑈| where, |𝑈| is the cardinality of the universal set (objects) and |𝑃𝑂𝑆𝐶 (𝐷)| is the cardinality of union the of all lower approximation of 𝐷 𝑜𝑛 𝐶. 4. autoregressive integrated moving average (arima) box-jenkins (1976) introduced arima model for modeling a time series with the trend and seasonal component. it is the combination of autoregressive (ar) and moving average (ma) models. arima model for a time series, say arima model for a time series, say 𝑋𝑡(𝑡 = 1, 2. . . . 𝑇), is given by 𝜑𝑝(𝐵)∆ 𝑑 𝑋𝑡 = 𝜃𝑞 (𝐵)𝑎𝑡 , where 𝜑𝑝(𝐵) = 1 − 𝜑1(𝐵)−, … , −𝜑𝑝(𝐵 𝑝 ). 𝑎𝑛𝑑 𝜃𝑞 (𝐵) = 1 + 𝜃1(𝐵) + ⋯ + 𝜃𝑞 (𝐵 𝑝 ) are ar and ma models, respectively, b is the backshift operator, ∆𝑑 ∆𝑠 𝐷 𝑋𝑡 = (1 − 𝐵)(1 − 𝐵𝑠 )𝑋𝑡,𝜑𝑝 < 1, 𝜃𝑞 < 1. 5. grey model grey model developed by deng (1982). in this model, future trend is estimated using linear differential equation of order one. the parameters involved in the model can be estimated using the ordinary least squares (ols) method wang (2004) and xu et al. (2016)). the grey model of first order linear differential equation is written as 𝑑𝑋𝑡 𝑑𝑡 + 𝑎 ∗ 𝑋𝑡 = 𝑏. where 𝑋𝑡 is a time series and 𝑎 & 𝑏 are the parameters. sharma et al./decis. mak. appl. manag. eng. 4 (2) (2021) 163-177 168 6. combination forecast based on rough set because the combination method yields better results than a single method, the modelling, and forecasting approach with high accuracy is adopted in this study. there are three main steps involved in the combined approach, i.e. single forecasts, computation of weight coefficient and forecast combination. let: 𝑋𝑡 (𝑡 = 1, 2, …,n) is an actual time series with time 𝑡 and 𝑋1𝑡 , 𝑋2𝑡 , 𝑋3𝑡 , . . , 𝑋𝑗𝑡 (𝑗 = 1, 2, . . , 𝑚) respectively, are 𝑚𝑡ℎ , forecasting value of single forecasts at time 𝑡 and 𝑊𝑗 (𝑗 = 1, 2, … , 𝑚) is the 𝑚𝑡ℎ weight coefficient of single forecasts 𝑋𝑚𝑡 and then the combined forecasting approach can be written as follows: 𝑍𝑐,̂(𝑡) = ∑ 𝑊𝑖 𝑚 𝑖=1 ∗𝑋𝑚𝑡 ∑ 𝑊𝑖 𝑚 𝑖=1 where, 𝑍𝑐,̂(𝑡) rough set combination forecasted values. also, the overall procedure is described in figure 1. figure 1. the framework of the proposed work. 6.1. weight determination based on rough set let 𝑆 = (𝑈, 𝐴, 𝐶, 𝐷) denote the rough set decision table, where 𝑈 represents the universal set of time points in time series, and 𝐶 = 𝑋1𝑡 , 𝑋2𝑡 , 𝑋3𝑡 , . . , 𝑋𝑗𝑡 (𝑗 = 1, 2, . . , 𝑚) indicate the set of single forecasts, condition attributes and 𝐷 is the decision attribute, 𝑋𝑡 in order to determine the weight coefficient. the overall procedures of deriving a weight coefficient are as described below1 as the several steps: step 1: input the actual time series 𝑋𝑡 = (𝑡 = 1, 2, … , 𝑛) and forecasts it with respective single forecasts, 𝑋1𝑡 , 𝑋2𝑡 , 𝑋3𝑡 , . . , 𝑋𝑗𝑡 (𝑗 = 1, 2, . . , 𝑚). step 2: to construct the rough set decision table, 𝑆 by arranging the condition and decision attributes. step 3: compute the dependence (ahmed et al., 2009) of decision attribute (𝐷) on the condition attributes (𝐶). forecasting sugarcane yield of india based on rough set combination approach 169 step 4: evaluate the dependence of each attribute concerning 𝐷 using the following expression. 𝛾𝐶−{𝐶 ′}(𝐷) = |𝑃𝑂𝑆 𝐶−{𝐶′} (𝐷)| |𝑈| where 𝐶 ′ is the subset of 𝐶. step 5: calculate the importance and weight coefficients of each single forecasts and then combine each single forecasts. 6.2. data our empirical analysis uses yearly sugarcane production from 1950 to 2011 time series data in india. time series data are obtained from sugar and molasses production (2019). the r-3.0.3 software is used for the overall empirical analysis of arima, des and grey models. also, the weight coefficient has been calculated for rough set combination method via rough set data explorer (rose2) software (predki et al. 1998). the whole time series is divided into two periods (1) 1950-2011, in-sample, consists of 61 observations for the modelling process of the several methods; (2) data of 2012-2021 are used to generate the out-of-sample forecasts for different models. 6.3. comparison criterion the performance of all respective models for seasonal time series has been evaluated using mean absolute percentage error (mape) criterion for measuring level prediction accuracy, as follows: 𝑀𝐴𝑃𝐸 = ∑ (|𝐴𝑐𝑡−𝑃�̂�𝑡|)/𝐴𝑐𝑡 𝑛 𝑡=1 𝑛 (3) where 𝐴𝑐𝑡 (𝑡 = 1, 2, … , 𝑛) is the actual value, 𝑃�̂�𝑡 (t = 1, 2,…, n) represents the predicted values and n is the total number of observations. lewis (1982) demonstrates that the value of mape being less than 10% indicates the high accuracy of forecasting. moreover, when it lies between 10-20% forecasting is good, 20-50% is reasonable, and more than 50% denotes inaccuracy in forecasting. 7. results according to the forecasting results of every single model (see table 1), we apply the discretization method to discrete data into three grades (0, 1, 2). the transformed discrete value, expressed as an attribute value is exhibited in table 3. consequently, decision table 4 can be a build-up for the evaluation of weights 𝑊𝑗 (𝑗 = 1, 2, … , 𝑚) by using rough set in the combination forecast. further, dependence and importance can be calculated through equation 1 and 2. in next step weights are computed by normalizing the importance of each single model. finally, the combined forecasting model can be established as: 𝑋�̂� = 0.2683 𝐴𝑅𝐼𝑀𝐴 + 0.4756 𝐷𝐸𝑆 + 0.2561 𝐺𝑅𝐸𝑌. (4) the evaluated results of dependence, importance and weights are given in table 4. moreover, we applied the actual and forecasts values from 1950 to 2011 for the evaluating the weight coefficient by using rough set. table 2 gives the sugarcane production and forecasts obtained from arima, des and grey models. further, arima, des and grey models are considered the three condition attributes and actual values is the decision attributes in order to establish the rough set theory and then sharma et al./decis. mak. appl. manag. eng. 4 (2) (2021) 163-177 170 these attributes have been normalized (yuan and xu, 2013), such as shown in table 1. where 𝑋1𝑡 represents the forecasts of the arima model, 𝑋2𝑡 , indicates the forecasts of des model and 𝑋3𝑡 , denote the forecasts of grey model. also, 2012-2018 out-ofsample forecasts were generated for different models. further simple average and inverse of mape combination methods (bates and granger 1969, menezes et al. 2000, armstrong 2001, aiolfi and timmermann 2006, andrawis et al. 2011) are also employed for the prediction of sugarcane production in india. the forecasting results of arima, des and grey models are combined using the weight coefficient 𝑊𝑗 (𝑗 = 1, 2, … , 𝑚) obtained from simple average and the inverse of mape combination methods. according to the forecasting results of each single model weights are computed by the inverse of mape obtained from each single model. in the simple average method, each single forecast has equal weight. finally, the combined forecasting model using simple average and inverse of mape methods can be established as: 𝑋�̂� = 0.5 𝐴𝑅𝐼𝑀𝐴 + 0.5 𝐷𝐸𝑆 + 0.5 𝐺𝑅𝐸𝑌. (5) 𝑋�̂� = 0.2309 𝐴𝑅𝐼𝑀𝐴 + 0.2477 𝐷𝐸𝑆 + 0.1663𝐺𝑅𝐸𝑌. (6) table 2. actual and forecasts values of models year arima des gm actual 1952 59.69924 66.21 80.05650072 51 1953 44.34212 55.58 82.16917439 44.41 1954 49.33245 48.99 84.33760107 58.74 1955 68.02704 63.32 86.56325205 60.54 1956 51.24382 65.12 88.84763749 69.05 1957 71.69061 73.63 91.19230735 71.16 1958 66.22566 75.74 93.59885255 73.36 1959 73.1471 77.94 96.06890595 77.82 1960 78.06447 82.4 98.60414353 110 …… ………… ……… …………….. ……. 2002 302.09588 303.01 294.450717 281.58 2003 273.44244 286.16 302.2212075 233.87 2004 227.62409 238.45 310.1967595 237.09 2005 269.76906 241.67 318.3827847 281.18 2006 293.25304 285.76 326.7848373 355.52 2007 353.68756 360.1 335.4086182 348.19 2008 297.20754 352.77 344.2599788 285.03 2009 269.79121 289.61 353.3449249 292.31 forecasting sugarcane yield of india based on rough set combination approach 171 table 3. decision table for rough set u arima des gm actual values 1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 4 0 0 0 0 5 0 0 0 0 6 0 0 0 0 7 0 0 0 0 8 0 0 0 0 9 0 0 0 0 …….. …….. …….. …….. …….. 51 2 2 2 1 52 2 2 2 2 53 1 2 2 2 54 2 2 2 2 55 2 2 2 2 56 2 2 2 2 57 2 2 2 2 58 2 2 2 2 table 4. estimated parameters for models models dependency importance weight arima 0.6207 0.3793 0.268284057 des 0.3276 0.6724 0.47559768 gm 0.6379 0.3621 0.256118263 7.1. discussion 7.1.1. performance comparison of different models the forecasting performance of different models has been evaluating using mape criterion. table 5 reports the mape results for each of the individual models and rough set combination (rsc) method for the out-of-sample from the period of 2012 to 2018. regarding mape values, the forecasting accuracy of rsc, 𝑤1 = 0.2685, 𝑤2 = 0.4756 𝑎𝑛𝑑 𝑤3 = 0.2561 is better than the individual forecasting methods. the combining method inverse of mape outperforms the arima, des, grey, simple average and rough set combination based on mape (lewis 1982) for out-of-sample forecasts. also, the forecasting performance of arima model is highly accurate than des, grey and combination methods. for better understanding figure 2 compares the actual and forecasting values for each model based on out-of-sample forecasts. the sharma et al./decis. mak. appl. manag. eng. 4 (2) (2021) 163-177 172 forecasting curves of all models are good fitted with the actual curve but the arima suggest the best fit for the prediction of sugarcane production. all these confirmed that the forecasting results of imape and arima models are higher than the other single and combination time series models. since arima can forecast more accurately for the sugarcane production according to the out-of-sample forecasts. consequently, arima is used to predict the sugarcane production for the next three years from 2019 to 2021 with the forecasting results of des, grey, a simple average, imape and rough set combination models. results of forecasting using hybrid model (rsc model) are presented in table 6. the obtained out-of-sample forecasts results of mape for the next three years are shown in table 5. table 5. mape of forecasting models models in-sample out-of-sample arima 7.8 0.85 des 9.2 12.1 gm 13.8 7.85 sa 8.1 1.35 imape 7.6 2.59 rsc 8.2 3.71 figure 2. comparative analysis of different models. forecasting sugarcane yield of india based on rough set combination approach 173 table 6. forecasts for the next ten years year arima des gm sa imape rsc 2012 319.5522 306.05 382.0637366 335.888646 2485.14905 329.140 2013 305.7307 310.63 392.1463156 336.169005 2497.903017 330.193 2014 318.7837 315.21 402.4949718 345.496224 2564.058917 338.524 2015 331.7398 319.79 413.1167267 354.882176 2630.777165 346.898 2016 327.1773 324.37 424.0187875 358.522029 2663.934317 350.645 2017 317.3611 328.95 435.2085512 360.50655 2687.587675 353.055 2018 317.3513 333.53 446.6936104 365.858303 2731.273146 358.172 2019 323.7363 338.11 458.4817579 373.442686 2788.326094 365.083 2020 325.6805 342.69 470.5809919 379.650497 2837.53279 370.881 2021 322.1153 347.27 482.9995222 384.128274 2876.82159 375.284 8. conclusions since single forecasting performs always well. hence, this paper applied a novel combination of forecasts by underlying rough set (rs) approach for the prediction of sugarcane production to india for the period of 1950 to 2011 in order to improve the performance of single models. the forecasting results of autoregressive integrated moving average (arima), des and grey models are combined using the weight coefficient obtained from simple average, the inverse of mape (imape) and rough set combination methods. moreover, the performance of several forecasts has been evaluated under mean absolute percentage error (mape) criterion. our empirical study suggests the following outcomes. first, all of the single forecasting models appeared to provide the accurate and reliable forecasting results according to the less than 10% mape values. secondly, the arima and imape models have better accuracy than the other models according to mape values. further, arima performance is highly accurate among all different approaches. in addition, combination methods are found to be effective for the forecasting of sugarcane production in india. the contribution of the article is that the combination of the forecast with the rough set approach firstly opts in agriculture empirical study. the obtained results suggest that arima a combination method is an effective way for sugarcane forecasting. it is important to describe the importance of single model and dependency of sugarcane production for better forecasting performance. it is expected that future study would benefit from concentrating on other single methods for agriculture forecasting. author contributions: haresh kumar sharma contributed to the research designing, detailed data analysis through selected methodology, structuring, writing, and editing of the manuscript. kriti kumari participated in the data collection and preliminary analysis. samarjit kar has supervised the research and editing of the manuscript. funding: this research received no external funding. sharma et al./decis. mak. appl. manag. eng. 4 (2) (2021) 163-177 174 acknowledgements: we wish to express our most profound appreciation to the editors and the anonymous reviewers. conflicts of interest: the authors reported no potential conflict of interest. references ahmed, e. f., yang, w. j., & abdullah, m. y. 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(2019). new model for determining criteria weights: level based weight assessment (lbwa) model. decision making: applications in management and engineering, 2(2), 126-137. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 4, issue 1, 2021, pp. 194-214. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame2104194c * corresponding author. e-mail addresses: s_chakraborty00@yahoo.co.in (s. chakraborty), crewvidyapati@gmail.com (v. kumar). development of an intelligent decision model for non-traditional machining processes shankar chakraborty 1* and vidyapati kumar 1 1department of production engineering, jadavpur university, kolkata, india, 700032 received: 10 january 2021; accepted: 7 march 2021; available online: 13 march 2021. original scientific paper abstract. in order to fulfil the ever increasing requirements of various hard and difficult-to-machine materials in automobile, turbine, nuclear, aviation, tool and die making industries, the conventional material removal processes are now being continuously substituted by an array of non-traditional machining (ntm) processes. the efficient and improved capabilities of these ntm processes have made them indispensible for the present day manufacturing industries. while deploying a particular ntm process for a specific machining application, the concerned process engineer must be aware of its capability which is influenced by a large number of controllable parameters. in this paper, an intelligent decision model is designed and developed in vbasic to guide the concerned process engineer to have an idea about the values of various ntm process responses for a given parametric combination. it would also advise about the tentative settings of different ntm process parameters for achieving a set of target response values. it would thus aim in assisting the process engineers and designers to efficiently identify the technically feasible ntm processes in the early design and machining stages, focusing more on developing the required product functionalities and appearance with the feasible processes in mind, utilizing the process characteristics more effectively. the operational procedure of this developed decision model is demonstrated with the help of three real time examples. key words: non-traditional machining process; parameter; response; decision model; vbasic. 1. introduction the emerging need for generating intricate and precise shape features in various advanced engineering materials, like high-strength-temperature-resistant alloys, tungsten carbide, titanium and its alloys, ceramics, fibre-reinforced composites, mailto:s_chakraborty00@yahoo.co.in mailto:crewvidyapati@gmail.com development of an intelligent decision model for non-traditional machining processes 195 stainless steel, refractories etc. has resulted in development of a group of new machining processes, collectively known as non-traditional machining (ntm) processes. these advanced materials, having higher strength, toughness, hardness, low machinability and other varied properties, are in demand in various manufacturing industries, like automobile, nuclear, aviation, missile, tool and die making etc. in conventional machining processes, material removal usually takes place in the form of chips while applying forces on the workpiece using a wedge-shaped cutting tool which is harder than the work material. these processes usually incur higher cost with respect to tool wear and poor quality due to the generation of residual stresses in work material. they are also incapable to attain the dimensional accuracy and surface finish as desired by the modern day manufacturing industries. in these processes, as the relative motion between the tool and workpiece is typically rotary or reciprocating, the generated shape is thus restricted to only circular or flat features, and except in computer-numerical control (cnc) tools, machining of threedimensional surfaces is extremely difficult. thus, in order to cater the needs of higher dimensional accuracy (in microor nano-level), surface finish, capability to machine difficult-to-cut materials with high strength-to-weight ratio, low surface damage, minimum tolerance, automated data transmission and miniaturization, the conventional machining processes are now being gradually replaced by the ntm processes (jain, 1980; pandey & shan, 1980; el-hofy, 2005). in ntm processes, instead of sharp cutting tools, energy in its direct form is employed to remove material from the workpiece surface. these processes usually adopt mechanical, thermal, electrical and chemical energies or any combination of them for removing materials in the shape of micro-chips or atoms to achieve the desired accuracy and machined surface without any burr. in these processes, there is also no physical contact between the tool and workpiece, and the related material removal mechanism is not dependant on the mechanical properties of the work materials. some of the ntm processes can also machine workpieces in areas inaccessible for the conventional machining techniques. thus, these enhanced and efficient capabilities of ntm processes have made them almost indispensible and popular at the shop floor (rajurkar et al., 2017). over the years, more than 20 different ntm processes have been successfully developed and deployed to meet the diverse needs of the present day manufacturing industries. in order to make efficient use of the ntm processes, it thus becomes necessary to understand the exact nature of the machining problem. they can never replace the conventional machining processes, and a particular ntm process may be highly acceptable for a given set of requirements, but it may sometimes fail to prove its acceptability under different machining conditions. thus, an extensive knowledge regarding the capabilities of various ntm processes is crucial in order to select the most suitable ntm process to generate the desired shape feature on a given work material. existence of a large number of ntm processes with diverse uniqueness and capabilities has compelled the process engineers to develop structured approaches for ntm process selection for assorted machining applications. in this direction, development of a decision making framework in the form of an intelligent knowledgebased system is worth demanding. the developed intelligent decision model would help the process engineers in multi-directional ways, like (a) selecting the most suitable ntm process for a given problem, (b) managing a huge volume of machining data and responding quickly, (c) standardizing the conclusions drawn from a given data set, and (d) capturing the scarce expertise and making it available for subsequent use. the remainder of the paper is organized as follows. section 2 presents the applications of different methodologies adopted by the past researchers for ntm chakraborty and kumar/decis. mak. appl. manag. eng. 4 (1) (2021) 194-214 196 processes selection. section 3 describes the developmental framework of the decision model. section 4 presents three examples to demonstrate the applicability and usefulness of the developed model. finally, section 5 concludes the paper, highlighting its assumptions, limitations and future directions. 2. literature review yurdakul & cçogun (2003) proposed an ntm process selection method for a given application requirement while combining analytic hierarchy process (ahp) and technique for order of preference by similarity to the ideal solution (topsis). the alternative ntm processes were first narrowed down to a set of feasible solutions which were subsequently ranked based on their suitability for the desired application. chakraborty & dey (2006) developed an ahp method-based expert system with a graphical user interface to find out the most apposite ntm process with the highest acceptability index value. chakraborty & dey (2007) applied quality function deployment (qfd) methodology for identification of the most suitable ntm process for a given industrial application based on the development of a house of quality matrix for comparing the considered product and process characteristics. chakladar & chakraborty (2008) integrated ahp and topsis methods in order to select the most appropriate ntm process for a specific work material and shape feature combination. chandraseelan et al. (2008) developed a web-based knowledge base system for identification of the most suitable ntm process based on some input parameters and process capability requirements. chakladar et al. (2009) proposed a digraph-based approach to entirely automate the ntm process selection procedure. based on the capabilities of the considered ntm processes to generate a desired shape on a given material, they were subsequently ranked in decreasing order of preference. sugumaran et al. (2010) presented a neural network-based approach to help the process engineers in preparing a list of feasible ntm processes for a specific machining operation on a given work material. karande & chakraborty (2012) solved four real time ntm process selection problems while applying an integrated promethee (preference ranking organization method for enrichment evaluation) and gaia (geometrical analysis for interactive aid) approach. temuçin et al. (2013) employed both fuzzy and crisp-based approaches to solve ntm process selection problems, and developed a decision support model to assist the process engineers to arrive at the correct ntm process selection decision. roy et al. (2014) first applied fuzzy ahp method for estimating the relative importance of various ntm processes based on several product and process characteristics, and later adopted qfd methodology for evaluating the performance scores of various ntm processes to choose the best one. sarkar et al. (2015) proposed a multi-objective optimization on the basis of ratio analysis (moora) method-based decision support system for selection of ntm processes having a given set of quantitative and qualitative selection attributes. madić et al. (2015b) combined ahp, moora and topsis methods for determination of the relative significance of various quality criteria, and hence, selection of the most suitable ntm process for a given application. based on a hybrid multi-criteria decision making (mcdm) framework, azaryoon et al. (2015) developed a knowledge-based system for identification of ntm processes. the developed approach employed the combined applications of decision making trial and evaluation laboratory (dematel), analytic network process (anp) and vikor (višekriterijumsko kompromisno rangiranje) methods to evaluate various performance measures, such as applicability of workpiece material and shape development of an intelligent decision model for non-traditional machining processes 197 features, process capabilities, and cost-related factors. madić et al. (2015a) applied operational competitiveness rating analysis (ocra) method as an mcdm tool for selection of the ntm processes from a large number of candidate alternatives. saenz et al. (2015) proposed a novel method for selection and comparison of non-traditional sheet metal cutting processes. khandekar & chakraborty (2016) applied fuzzy axiomatic design principles for selection of the most appropriate ntm processes for generating cavities on ceramics, and micro-holes on hardened tool steel and titanium materials. chatterjee et al. (2017) presented a novel hybrid approach consisting of factor relationship (fare) and multi-attributive border approximation area comparison (mabac) methods for selection and evaluation of ntm processes. the fare method was first applied to determine the corresponding criteria weights, and the alternative ntm processes are later ranked using mabac method. roy et al. (2017) proposed a combined application of fuzzy ahp and qfd methods for investigating the relative significance of different technical requirements in an ntm process selection approach, and also identified the suitability of electrochemical machining process for a specified industrial application. prasad & chakraborty (2018) developed a decision guidance framework to assist the process engineers in choosing the most suitable ntm process for a given machining application and identifying the ideal process parameter settings for the said process. yurdakul et al. (2019) presented intuitionistic and triangular fuzzy-based models for ranking of the suitable ntm processes for machining of some specific shape features on the given work materials. the performance of those models was later compared with that of crisp-based models. amalnik (2019) proposed a feature-based expert system for optimization of design of an abrasive waterjet machining process. the corresponding database would consist of lists of 20 work materials, eight abrasive types and four machine types. the developed expert system would aid the process engineers while providing information with respect to machining cycle time, machining cost and cutting rate. rohith et al. (2019) first adopted a data envelopment analysis (dea)-based model for shorlisting the efficient ntm processes for a given shape feature and work material combination, and then employed ahp, topsis and ocra methods for ranking and selection of the efficient ntm processes. yurdakul & i̇ç (2019) presented the applications of fuzzybased models of ahp and topsis methods for ntm process selection for a particular work material and shape feature combinations. talib and asjad (2019) developed a model using ahp method for prioritizing as well as ranking of various ntm processes based on 27 evaluation criteria. chakraborty et al. (2020) integrated rough numbers with mabac method to identify the most feasible ntm processes for generation of standard through holes in glass and deep through cavities in titanium work materials. based on firefly algorithm, singh & shukla (2020) developed a graphical user interface for selecting the optimal input parameters for electrochemical machining, electrochemical micro-machining and electrochemical turning processes. from an extensive review of the existing literature, it has been observed that varieties of expert systems have already been developed so as to help the process engineers in identifying the most competent ntm processes for different work material and shape feature combinations. those developed expert systems have also been integrated with several other mathematical techniques, like ahp, anp, topsis, dea, qfd etc. for arriving at the best courses of action. those expert systems have been so designed and developed that they could only identify the most apposite ntm processes for varying machining applications. the expert system developed by prasad & chakraborty (2018) could only advise the concerned process engineers about the tentative parametric settings of the chosen ntm processes, apart from selecting the most competent ntm processes to fulfil different application requirements. chakraborty and kumar/decis. mak. appl. manag. eng. 4 (1) (2021) 194-214 198 no research work has been carried out till date so as to predict the most probable values of various responses based on a given combination of different ntm process parameters or envisage the tentative settings of various ntm process parameters so as to achieve the most desired values of the considered responses. in this paper, an attempt is thus put forward so as to design and develop an intelligent decision model in visual basic (vbasic) that would help the process engineers in advising about the achievable values of different responses for a specific set of ntm process parameters or selecting the optimal parametric mix in order to attain a set of target responses. the developed system is supposed to be flexible and versatile enough as it encompasses all the available ntm processes, work materials and shape features, and also userfriendly and interactive as it always guides the end users in arriving at the optimal selection decision. 3. development of the decision model the procedural steps in the form of a flowchart in order to run this developed intelligent decision model without any error are exhibited in figure 1. figure 1. flowchart exhibiting procedural steps of the developed decision model at first, it would ask the end user to select the type of the ntm operation to be performed along with the combination of work material to be machined and shape/sub-shape feature to be generated. once the ntm operation is specified, two lists containing the controllable parameters and responses associated with the selected ntm process would now appear in the next screen of the developed system. the end user would then be directed to preselect the machining parameters as available in that ntm process along with the set of desired responses in order to fulfil the end product requirements. the end user can also choose all the available ntm development of an intelligent decision model for non-traditional machining processes 199 process parameters and responses while pressing the ‘select all’ functional key. it would then direct the end user to input the desired values of the identified ntm process parameters based on which the tentative values of the selected responses would be provided on pressing of the ‘ok’ button. here, only the available values of the selected process parameters for a particular ntm process would appear in the screen in the form of drop-down menus. the reverse approach can also be augmented in this decision model while selecting the ‘response to pp’ functional key. the end user can also choose all the available ntm process parameters and responses while pressing the ‘select all’ functional key, and select the ‘response to pp’ for having the tentative values of the ntm process parameters. it would then direct the end user to input the desired ranges of the identified ntm responses as beneficial and nonbeneficial criteria based on which the conditional values of the selected process parameters would be provided on pressing the ‘ok’ button. here, only the available values of the selected responses for a particular ntm process would appear in the screen in the form of drop-down menu. if the end user opts for generating infeasible shape features or machining unsuitable materials using any of the considered ntm processes, an error message would appear indicating the incapability of that ntm process to generate the chosen shape feature on the given material. in this case, the end user has to repeat the procedural steps of ntm process selection from the beginning. in this paper, 17 ntm processes, i.e. (a) abrasive jet machining (ajm), (b) abrasive water jet machining (awjm), (c) electron beam machining (ebm), (d) electrochemical grinding (ecg), (e) electrochemical machining (ecm), (f) electrochemical discharge machining (ecdm), (g) electro-discharge machining (edm), (h) electro jet drilling (ejd), (i) focused ion beam machining (fib), (j) hot chlorine machining (hm), (k) laser beam machining (lbm), (l) magnetorheological finishing (mrf), (m) plasma arc machining (pam), (n) photochemical milling (pcm), (o) ultrasonic machining (usm), (p) wire electro-discharge machining (wedm) and (q) water jet machining (wjm) are considered for subsequent development of the intelligent decision model. similarly, the list of the considered work materials consists of (a) alumina, (b) aluminium, (c) boron carbide, (d) ceramics, (e) composites, (f) cemented tungsten carbide, (g) duralumin, (h) inorganic glass, (i) inconel 718, (j) inconel 800, (k) inconel 825, (l) incoloy, (m) monel 400, (n) monel k-500, (o) nickel, (p) nimonics, (q) plastics, (r) refractories, (s) silicon nitride, (t) silicon carbide, (u) steel, (v) stainless steel, (w) titanium (astm grade i), (x) tungsten carbide and (y) titanium-based super alloys. this system takes into consideration the following shape and sub-shape features for subsequent generation on the selected work material: (a) holes (i) precision holes (d ≤ 0.25 mm) (where d = diameter) (ii) precision holes (d > 0.25 mm) (iii) standard holes (l/d ≤ 20) (where l/d = length/diameter = slenderness ratio) (iv) standard holes (l/d > 20) (b) cavities (i) precision (aspect ratio ≤ 5) (ii) standard (c) surfacing (i) double contouring (ii) surface of revolution (d) through cutting (i) shallow (depth of cut < 40 µm) (ii) deep (depth of cut > 40 µm) chakraborty and kumar/decis. mak. appl. manag. eng. 4 (1) (2021) 194-214 200 (e) finishing. the entire database containing the capabilities of all the considered ntm processes with respect to workpiece material, and shape and sub-shape features to be generated is stored in ms-access linked with vbasic, and the decisions regarding values of various responses and settings of the ntm process parameters are arrived at based on sets of simple if-then rules. 4. illustrative examples in order to demonstrate the applicability and usefulness of the developed decision model in the domain of ntm processes, the following three examples are cited. 4.1 example 1: electro-discharge machining in this example, it is supposed that precision cavities with aspect ratio ≤ 5 need to be generated on inconel 718 alloy using edm process. for this machining application, the corresponding input window in the form of a graphical user interface is shown in figure 2. figure 2. input window for the first example pressing of the ‘ok’ functional key then leads the end user to the next window, as exhibited in figure 3, where the lists of all the important edm process parameters, i.e. peak current, open circuit voltage, pulse-on time, duty factor, flushing pressure, pulseoff time, dielectric level, tool electrode lift time, polarity, type of the tool and flushing speed, and responses, like surface crack density, tool wear ratio (twr), perpendicularity error (pe), material removal rate (mrr), surface roughness (sr), overcut (oc), electrode wear rate, edge deviation, white layer thickness and microharness are displayed. in this example, at first, the end user selects peak current, open circuit voltage, pulse-on time, duty factor, polarity and type of the tool as the controllable process parameters as available in the considered edm set-up. on the other hand, based on the end product requirements, surface crack density, twr, pe, mrr, sr and micro-hardness are treated with utmost importance. the end user can also choose all the edm process parameters and responses while pressing the ‘select all’ key. now, when the ‘next’ button is pressed, in the subsequent window, as depicted in figure 4, the end user is opted to enter the appropriate values for the preselected edm process parameters based on which the approximate values of the shortlisted responses would be predicted. in this example, the end user chooses the options as peak current = 9 a, open circuit voltage = 60v, pulse-on time = 100 µs, duty factor = 70%, polarity = positive and tool material = copper. now, when the ‘ok’ button is pressed, this decision model would guide the end user to have an idea about various responses envisaged as surface crack density = 0.0055-0.0057 µm/µm2, mrr = 89.12-89.18 mm3/min, sr = 6.2-6.7 µm, pe = 0.09-1.11%, micro-hardness = 392.40392.50 hv and twr = 0.0012-0.0016. development of an intelligent decision model for non-traditional machining processes 201 figure 3. window for selection of edm process parameters and responses in figure 3, when the end user presses the ‘response to pp’ functional key, the settings of the preselected edm process parameters can be predicted based on the chosen values of the shortlisted edm responses. as exhibited in figure 5, the end user desires to have high value of mrr (59.881-89.164 mm3/min), and low values of electrode wear rate (0.011-0.059 mm3/min), sr (2.133-4.866 µm), oc (0.03-0.19 mm), surface crack density (0.008-0.011 µm/µm2), white layer thickness (16.64617.866 µm) and micro-hardness (352.600-407.755 hv). based on these input response values, the developed decision model predicts the related edm process parameters as open circuit voltage = 47-50 v, peak current = 10.5-11.5 a, pulse-on time = 190-200 µs, duty factor = 78-82%, flushing pressure = 0.15-0.25 bar, type of the tool = copper, polarity = positive and pulse-off time = 25-35 µs. it is worthwhile to mention here that among the considered responses, material removal rate is the sole beneficial attribute requiring its higher value, whereas, lower values for the remaining non-beneficial responses are preferred. based on the past experimental data on edm processes (ray, 2016; datta et al., 2017), all the related response values are classified into three groups, i.e. low, medium and high so as to relieve the end user in providing an exact value for a specific response which may sometimes be a difficult task. according to the end product requirements, the end user can now be able to opt for only low, medium or high value for a particular response of interest. the derived parametric settings of the considered edm process are only tentative. in order to achieve most accurate target values of the responses, fine-tuning of those parameters may often be needed. chakraborty and kumar/decis. mak. appl. manag. eng. 4 (1) (2021) 194-214 202 figure 4. prediction of responses based on edm process parameters figure 5. prediction of edm process parameters based on the responses in figure 6, when the end user opts for performing deep through cutting operation (depth of cut > 40 µm) on ceramic materials using the edm process, an error message would appear indicating the incapability of edm process to generate the chosen shape feature on ceramics. it can be interestingly noticed that with increasing values of all the edm process parameters, mrr would also increase. higher values of gap voltage, peak current and pulse-on time are all responsible for the available discharge energy to increase, resulting in more melting and vaporization of material from the workpiece. the impulsive force in the spark gap also increases, which is responsible for higher mrr (gopalakannan et. al., 2012). increments in gap voltage and peak current generate stronger discharge energy, creating higher temperature and formation of larger craters on the machined surface, resulting in poor surface quality (kiyak & çakır, 2007) similarly, twr increases with higher values of gap voltage, peak current and cycle time. at these higher parametric settings, there are micro tool wears development of an intelligent decision model for non-traditional machining processes 203 due to availability of higher spark energy density at the machining zone. generally, lower settings of these edm process parameters tend to enhance the possibility of carbon deposition on the tool surface, which finally helps in lowering twr value (lin & lee., 2008). the pe in the machined components occurs due to non-uniform undercut and oc which can be effectively controlled by proper settings of different edm process parameters. with increasing values of gap voltage and peak current, pe shows an increasing trend. at higher gap voltage and peak current, there are occurrences of secondary spark discharges caused by poor flushing as well as sporadic machining which are responsible for inferior pe. during edm operation, oc occurs due to side erosion and removal of debris. at higher settings of voltage, peak current and pulse-on time, availability of higher gap voltage and gap width allows breakdown of the dielectric at a wider gap due to higher electric field. at higher gap voltage and peak current, spark energy density would be more with a faster machining rate, which is also responsible for higher oc. hence, the predicted parametric intermix for the edm process would minimize the oc of the machined components. the above parametric setting can also be validated based on the observations of the past researchers (ray, 2016; prasad & chakraborty, 2018). figure 6. an error message for edm process 4.2 example 2: ultrasonic machining here, the end user desires to generate standard holes with slenderness ratio of less than equal to 20 on titanium (astm grade i) work material while utilizing usm process. figure 7 exhibits the input window for this example. in figure 8, from a list of the available controllable parameters for the usm process, type of the abrasive material, abrasive grit size, amplitude of vibration, machining time, type of the tool material, power rating and slurry concentration are first shortlisted. on the other hand, conicity, mrr, sr, tool wear rate (tw) and micro-harness are opted as the important responses. depending on the requirements, the entire lists for the available usm process parameters and responses can also be selected. the entire information related to these usm process parameters and responses are accumulated from (kumar & khamba, 2010; kataria et al., 2017). chakraborty and kumar/decis. mak. appl. manag. eng. 4 (1) (2021) 194-214 204 figure 7. input window for example 2 now, in figure 8, when the ‘next’ functional key is pressed, the developed decision model would seek for the values of the shortlisted usm process parameters in another window, as portrayed in figure 9. figure 8. window for selection of usm process parameters and responses in this case, the end user chooses the values of different usm parameters as type of the abrasive material = boron carbide, abrasive grit size = 280, amplitude of vibration = 25 µm, machining time = 8.70 min, type of the tool material = tungsten carbide, power rating = 550 w and slurry concentration = 45%. the drop-down menu attached with each of the process parameters guides the user to opt for the most apposite value as available in a particular usm set-up. development of an intelligent decision model for non-traditional machining processes 205 figure 9. prediction of responses based on usm process parameters based on these requirements, the developed system predicts the responses as conicity = 0.023-0.038º, sr = 0.78-0.85 µm, mrr = 0.025-0.035 mm3/min, tw = 0.981.05 mm3/min and micro-hardness = 155-160 hv. now, when the ‘response to pp’ functional key is pressed in figure 8, this system would jump to a new window, as shown in figure 10, where the end user is asked to input the desired values of the preselected responses in order to guide him/her about the tentative settings of different usm process parameters. figure 10. prediction of usm process parameters based on the responses here, high value of mrr (0.066-0.870mm3/min), and low values for conicity (0.014-0.032º), out-of-roundness (0.200-0.285 mm), sr (0.48-0.87 µm) and hole oversize (0.075-0.265mm) are sought by the end user. depending on these requirements, it advises the user to set the corresponding usm parameters as type of chakraborty and kumar/decis. mak. appl. manag. eng. 4 (1) (2021) 194-214 206 the abrasive material = silicon carbide, abrasive grit size = 400, type of the tool material = hss, power rating = 400-450 w, slurry concentration = 35-37%, slurry flow rate = 6.5-7.5 l/min and feed rate = 1.12-1.28 mm/min. in order to achieve more accurate machining performance, fine-tuning of the settings of the considered usm process parameters may often be required. when the end user chooses the same usm process for generation of precision holes (d ≤ 0.25 mm) to be machined on aluminium work material, an error message, as shown in figure 11, would automatically be generated by the system indicating the fact that it cannot machine precision holes on aluminium material. figure 11. a typical error message for usm process in usm process, when the amplitude of vibration increases, energy at the tool tip also increases, resulting in higher sr due to increased impact of the abrasive particles on the workpiece. furthermore, tw also increases due to increase in the slurry flow rate containing harder abrasive particles, which are bombarded on the tool tip. the cavitation effects also lead to an increase in tw. with increase in amplitude of vibration, there is an increment in mrr as higher amplitude attributes to higher momentum imparted to the abrasive particles before striking the workpiece. it raises the energy with which the abrasive particles collide on the work surface and hence, the micro-crack or micro-crater created by each impact facilitates the material removal process. on the other hand, mrr decreases because the successive impacts between the abrasive grains and the work material may lead to large amount of plastic deformation resulting in the formation of a work-hardened layer, causing reduction in mrr (bhosale et al., 2014). an increment in slurry concentration is responsible for more impact on the work surface leading to higher sr. this also causes an increase in tw since more abrasive particles come into contact with the tool over a given period of time. however, the material removal tendency decreases because of the loss of energy possessed by the abrasives in the slurry. as the number of particles between the tool and the work surface increases due to higher slurry concentration, loss of energy due to interparticle collision may prevail during this phenomenon (kataria et al., 2017; chakraborty et al., 2020). 4.3 example 3: plasma arc machining in this example, deep through cutting operation with depth of cut > 40 µm needs to be performed on a workpiece made of stainless steel using pam process. in order to satisfy these requirements, the corresponding ntm process, work material, shape feature and sub-shape feature are accordingly selected in figure 12. development of an intelligent decision model for non-traditional machining processes 207 figure 12. input window for example 3 for the pam process, based on an extensive survey of the existing literature (xu et al., 2002; das et al., 2014; adalarasan et al., 2015; ramakrishnan et al., 2018), arc voltage, cutting current, cutting speed, feed rate, torch stand-off distance, plasma gas pressure and pierce height are identified as the predominant control parameters influencing its machining performance. on the other hand, the important responses are shortlisted as conicity, chamfer, dross, heat affected zone (haz), kerf width, mrr and sr. now, in figure 13, the end user preselects arc voltage, cutting current, feed rate and torch stand-off distance as the available pam process parameters, and chamfer, dross, kerf width and sr as the desired responses. figure 13. window for selection of pam process parameters and responses the values of these four shortlisted pam process parameters are set as arc voltage = 120 v, cutting current = 42.5 a, feed rate = 945 mm/min and torch stand-off distance = 2.5 mm, as exhibited in figure 14. based on this parametric combination, the developed decision model predicts the shortlisted responses as chamfer = 1.80-1.85 mm, dross = 3.60-3.64 mm2, kerf width = 2.70-2.75 mm and sr = 0.76-0.85 µm. thus, chakraborty and kumar/decis. mak. appl. manag. eng. 4 (1) (2021) 194-214 208 this system would help the process engineers to have an idea about the achievable values of different responses based on a preselected set of parametric combinations. figure 14. prediction of responses based on pam process parameters in figure 13, if the end user presses the ‘response to pp’ functional key, a new input window, as shown in figure 15, would now be available where the ranges of values for different responses can be set according to the end product requirements. in this example, the end user opts for high value of mrr (2.06-2.80 mm3/min), and low values for conicity (0.009-0.021º), haz (325-400 µm), chamfer (1.00-1.32 mm), dross (0.45-3.49 mm2), kerf width (1.93-2.53 mm) and sr (0.724-0.875 µm). now, based on these response requirements, the developed system would advise the process engineer to set different parameters of the pam process as feed rate = 930950 mm/min, cutting speed = 2260-2280 mm/min, plasma gas pressure = 4.57-4.92 kg/cm2, arc voltage = 115-125 v and torch stand-off distance = 2.5-4.5 mm. these are only the tentative settings of the considered pam setup. the process engineer may require to fine-tune these settings in order to achieve more accurate results. development of an intelligent decision model for non-traditional machining processes 209 figure 15. prediction of pam process parameters based on the responses as shown in figure 16, when the end user wants to machine precision holes on refractories using the available pam process, the system would automatically generate an error message highlighting its inability to machine the specified work material. figure 16. an error message for pam process in pam process, torch stand-off distance has the strongest effect on the quality characteristics. stand-off distance is one of the crucial parameters in pam process as it controls sr and conicity of the cut edge. it has also been observed that cutting current also influences the haz of the cut edge. it greatly influences sr of the cut due to the fact that the plasma gas beam is not of cylindrical shape but resembles the shape of a reversed candle flame. therefore, depending on the relative position of the plasma to the workpiece surface, the surface quality is drastically affected due to thermal properties of the material (salonitis & vatousianos, 2012). the mrr increases with an increase in gas pressure and high gas flow because it leads to an increase in mean arc voltage and its fluctuations as more heat is transferred into the workpiece, and chakraborty and kumar/decis. mak. appl. manag. eng. 4 (1) (2021) 194-214 210 consequently, sr reduces. however, mrr remains constant with an increase in standoff distance as there is a slight fluctuation in energy. for higher plasma gas flow rate, arc voltage also becomes higher. as the gas flow rate increases, more energy is needed to ionize the gas, therefore the arc voltage should be higher. the kerf is narrower at the top, it widens at the middle, and again becomes narrower at the bottom, making heat distribution along the cut to be irregular. during pam operation, dross formation at the bottom of the workpiece needs to be minimized while controlling the corresponding process parameters. at low speed, input energy to the workpiece is high, causing melting of more materials. dross is formed when adequate force of the plasma jet is not available. to obtain a dross-free cut surface, plasma force and energy input to the workpiece need to be balanced properly. plasma power increases with plasma gas flow rate and arc current. to achieve a square cut of narrow kerf with minimal dross, the decision model can efficiently predict the tentative ranges of the process parameters (mittal & mahajan, 2018). the same parametric combination for the pam process is also well derived by the past researchers (ramakrishnan et al., 2018; prasad & chakraborty, 2018; chakraborty et al., 2020). 5. conclusions in this paper, an attempt is made to design and develop an intelligent decision model in vbasic so as to help the concerned process engineers in the domain of ntm processes. based on the availability of a particular ntm process, and selected workpiece and shape feature combination, it can identify values of different responses for a given set of parametric combinations. on the other hand, it has also the capability of predicting the tentative settings of different ntm process parameters while meeting the specified values of a given set of responses. in this system, the decision making procedure is based on an exhaustive set of if-then rules, and it consists of all the possible combinations of different ntm processes, work materials and shape features. it is easy to operate as the graphical user interface continuously interacts with the end users restricting them to commit any error. it has also the flexibility to cater any combination of ntm process, work material and shape feature. it warns the end user when a particular machining operation cannot be performed by a specific ntm process. the developed decision model assists the process engineers and designers to efficiently identify the technically feasible ntm processes in the early design and machining stages, enabling in developing the required product functionalities and appearance with the feasible processes in mind, while utilizing the process characteristics more effectively. after the detailed design is complete, the feasible processes identified in the earlier steps can be reevaluated, reassessing their technical feasibilities for manufacturing the designed product. the design can also be modified accordingly, if needed, to ensure manufacturability of the product. the main advantage of this decision model is that it does not require any in-depth technical knowledge regarding the applicability of the ntm processes. it also acts as an expert system to ease out and automate the ntm process selection procedure. this decision model has also some limitations. firstly, it does not take into account the presently available hybrid machining and additive manufacturing processes. moreover, it is developed based on a static database. it would be worth investigating the possibility of integrating the decision model into the ‘cloud’ under the industry 4.0 context, allowing prompt feedback and rapid update. it is also assumed that the developed decision model has no maintenance and operation costs. it lacks the creative responses of the human experts, and is also not able to explain the logic and development of an intelligent decision model for non-traditional machining processes 211 reasoning behind a decision to the end user. it opens opportunities to include micromachining, hybrid machining and additive manufacturing technology selection modules as well as improving selection results while incorporating more selection criteria and work materials in the model. it is expected that the developed model would be well accepted by the manufacturing industries for arriving at the prompt ntm process selection decisions. it can also be implemented in a group decision making environment involving opinions of different process engineers having varying background knowledge and expertise for more pragmatic results. its capability, reach and usability may further be enhanced while making it entirely web-based to become accessible to its end users through an internet network. author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research 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(2019). development of an intuitionistic fuzzy ranking model for nontraditional machining processes. soft computing, 24(1), 1-16. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 4, issue 1, 2021, pp. 1-18. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame2104001d * corresponding author. e-mail addresses: i.naric@yahoo.com (i. djalic), terzic_svetlana@yahoo.com (s. terzic) violation of the assumption of homoscedasticity and detection of heteroscedasticity irena djalic1* and svetlana terzic1 1 university of east sarajevo, faculty of transport and traffic engineering, doboj, republic of srpska, bosnia and herzegovina received: 1 september 2020; accepted: 20 october 2020; available online: 24 october 2020. original scientific paper abstract: in this paper, it is assumed that there is a violation of homoskedasticity in a certain classical linear regression model, and we have checked this with certain methods. model refers to the dependence of savings on income. proof of the hypothesis was performed by data simulation. the aim of this paper is to develop a methodology for testing a certain model for the presence of heteroskedasticity. we used the graphical method in combination with 4 tests (goldfeld-quantum, glejser, white and breusch-pagan). the methodology that was used in this paper showed that the assumption of homoskedasticity was violated and it showed existence of heteroskedasticity. key words: economic phenomena; heteroskedasticity; homoskedasticity; random errors. 1. introduction econometrics is a discipline that determines the connection between economic phenomena and confirms or does not confirm economic theory, starting from mathematical equations and forming econometric models suitable for testing. regression analysis is one of the most commonly used tool in econometrics to describe the relationships between economic phenomena. one of the classic assumptions of linear regression is homoskedasticity. homoskedasticity implies that the variance of random error is constant and equal for all observations. when the random errors of the classical linear regression model are not homoskedastic, then they are heteroskedastic (mladenović & petrović, 2017). the main goal of the paper is to show how the linear regression model behaves in conditions of violating the assumption of homoskedasticity and how this violation is detected. the basic contribution of the paper is that in one place it gives a developed method of detecting violating of homoskedasticity, ie the existence of mailto:i.naric@yahoo.com mailto:terzic_svetlana@yahoo.com djalic and terzic/decis. mak. appl. manag. eng. 4 (1) (2021) 1-18 2 homoskedasticity in linear regression models. this paper presents a methodology for detecting heteroskedasticity in linear regression models by a combination of a graphical method and four tests. after the introduction, a review of the literature was performed, after which the basics of heteroskedasticity were presented. in this part of the paper, the goldfeldquantum, glejser, white and breusch-pagan tests are presented. at the end of the paper, concluding remarks were made and recommendations for further research were given. 2. literature review aue et al. (2017) state that heteroskedasticity is a common characteristic of financial time series and most often refers to the process of model development using autoregressive conditional heteroskedastic and generalized autoregressive conditional heteroskedastic processes. ferman & pinto (2019) formed a model of inference that works with adjusting differences in differences with several treated and many controlled groups in the presence of heteroskedasticity. charpentier et al. (2019) developed the gini-white test, which shows greater strength in solving the problem of heteroskedasticity than the ordinary white test in cases when external observations affect the data. moussa (2019) analyzes cases in which heteroskedasticity is the result of individual effects or idiosyncratic errors, or both. linton & xiao (2019) study the effective estimation of nonparametric regression in the presence of heteroskedasticity and conclude that in many popular nonparametric regression models their method has a lower asymptotic variance than the usual unweighted procedures. a large number of authors pay attention to heteroskedasticity and develop models for solving certain problems (baum & schaffer, 2019; brüggemann et al., 2016; lütkepohl & netšunajev, 2017; cattaneo et al., 2018; ou et al., 2016; sato & matsuda, 2017). taşpınar et al. (2019) investigate the properties of finite samples of the heteroskedasticity-robust generalized method of moments estimator (rgmme), ie develop a robust spatial econometric model with an unknown form of heteroskedasticity. crudu et al. (2017) propose a new inference procedures for models of instrumental variables in the presence of many, potentially weak instruments that are robust to the presence of heteroskedasticity. lütkepohl & velinov (2016) compare models of long-term restriction that are widely used to identify structural shocks in vector autoregressive (var) analysis based on heteroskedasticity. harris & kew (2017) test adaptive hypotheses for a fractional differential parameter in a parametric arfima model with unconditional heteroskedasticity of unknown shape. in the case of heteroskedasticity, there are occasionally precise theoretical reasons for assuming that the errors have different variances for different values of the independent variable. very often, arguments for the presence of heteroskedasticity are so well defined, and sometimes there is a vague suspicion that the assumption of homoskedasticity is too strong (barreto & howland, 2006). it is important to note that heteroskedasticity is a common occurrence in spatial samples due to the nature of collection of data. obvious sources of heteroskedasticity are associated with different dimensions for different regions in the study area, unequal concentrations of population and economic activity in rural and urban areas (arbia, 2006). baum & lewbel (2019) provide advice and guidance to researchers who wish to use tests to check heteroskedasticity. violation of the assumption of homoscedasticity and detection of heteroscedasticity 3 3. methodology the simplest form of linear regression, which shows a linear relationship between two phenomena, is a simple linear regression: y x     (1) 𝜀 is a random error that we make during linear regression, and α and β are unknown parameters. to estimate the unknown parameters, we use a sample. for fixed n values of the independent variable 𝑋 the values of the variable 𝑌 are determined. in this way, n pairs (𝑋1, 𝑌1), (𝑋2, 𝑌2), … , (𝑋n, 𝑌n) are obtained, which forming the model of the simple linear regression sample: ,i i iy x     i = 1,2, …, 𝑛 (2) the assumption of homoskedasticity for the random variable 𝜀i is: 2 2 ( ) ( ) .,i iv e const     for each i = 1,2, …, n (3) when this assumption is violated, that is, when the random errors of the classical linear regression model do not satisfy this characteristic, then they are heteroskedastic. if the assumption of homoskedasticity (jovičić, 2011):   2 2 2 ( ) ( ) ( )i i i iv e e e        , for each 𝑖 (4) is not met, but the variances are different and valid: 2 ( )i iv   i = 1, . . ., 𝑛, (5) respectively (mladenović & petrović, 2017), 2 2 2 1 1 2 2( ) , ( ) ,..., ( ) ,n nv v v        2 1  ≠ 2 2  ≠ … ≠ 2 n  (6) it can be said that the errors are heteroskedastic or there is heteroskedasticity in the model. figure 1 presents a model where heteroskedasticity of the error is assumed. the growth of savings with increasing income is shown, where the variance of savings is smaller with different income levels. the variance is not constant, but increases with the growth of income, which corresponds to real economic relations (mladenović & petrović, 2017). djalic and terzic/decis. mak. appl. manag. eng. 4 (1) (2021) 1-18 4 figure 1. heteroskedastic errors source: mladenović & petrović (2017) heteroskedasticity can also be caused by errors of specification . for example, by omitting an important regressor whose influence will be covered by the error, a different variation of the error for different observations can be obtained. similarly, the wrong functional form of the model can lead to heteroskedasticity of the error. as data collection techniques are advancing, which implies the provision of representative samples for statistical processing, so do errors and thus their dispersions are decreasing. and this may be another reason for the occurrence of heteroskedasticity. 3.1. consequences of heteroskedasticity the presence of heteroskedasticity in the model of dependence of savings on income can be represented on the basis of the following point scatter diagram (figure 2): violation of the assumption of homoscedasticity and detection of heteroscedasticity 5 figure 2. diagram of distribution of points (mladenović & petrović, 2017) estimates of unknown parameters using the ordinary least squares method are determined from the condition that the residual sum of squares, 2ie , is minimal. in that case, all squares of the residuals have the same weight, ie they give the same information when forming the necessary estimates. this condition is not precise enough for the sample presented in figure 2. data that are far from the sampling regression line provide less useful information about its position than those that are closer to it. higher residual values in absolute terms correspond to more distant data. these residues dominate in the total residual sum of squares. therefore, it is realistic to expect that the application of ordinary least squares method does not provide estimates with desirable statistical properties. suppose that in the model (mladenović & petrović, 2017): 0 , i i i y x     (7) there is heteroskedasticity: 2 ( ) ,i iv   i = 1 , 2, …, n (8) the estimate b of the parameter β, obtained using the ordinary least squares method, is unbiased, because the corresponding proof does not use the assumption of the stability of the variance of the random error. to determine the variance of the estimate b we start from the expression: b – β = 1 n i i i w    , (9) 2 1 i i n i i x w x    , (10) djalic and terzic/decis. mak. appl. manag. eng. 4 (1) (2021) 1-18 6 based on which the variance is:                 2 2 2 1 2 2 2 2 2 2 1 1 2 2 1 2 1 2 1 3 1 3 2 2 2 2 2 2 1 1 2 2 1 2 1 2 1 3 1 3 2 2 2 2 n i i i n n n n v b e b e b e b e w e w w w w w w w w e w e w e w w e w w e                                                (11) in the eq. (11), all elements of the form  i je   , i ≠ j are equal to zero. the expression for the variance of the estimate b is:        2 2 2 2 2 21 1 2 2 n nv b w e w e w e     2 2 2 2 2 2 1 1 2 2 n nw w w     . 2 2 1 n i i i w    2 2 2 2 1 2 2 21 1 1 n n i i i i in n i i ii i xx x x                           (12) the variance of the estimate b, in a simple linear regression model, is given by the following expression:   2 2 1 n i i v b x     (13) when the existence of heteroskedasticity is neglected, the estimate of the variance of the estimate b is obtained by the following formula: 2 2 2 1 2 2 1 1 1 2 n i i b n n i i i i es s n x x          (14) when the variance of the random error grows in parallel with the explanatory variable then the estimate 2bs underestimates the actual variance of the estimate b. this arises because the estimate of the random error variance, 2 s , underestimates the actual random error variance of the initial model. thus, the properties of the estimates of parameter obtained by applying the ordinary least squares method in the presence of heteroskedasticity are: 1. the ratings are unbiased, 2. estimates do not have minimal variance, that is, they are ineffective. violation of the assumption of homoscedasticity and detection of heteroscedasticity 7 3. the assessment of the variance of a random error underestimates, in most cases, the actual variance. therefore, the estimate of the variance of the estimate of slope, , also underestimates the variance . 4. confidence intervals and tests based on the assessment of the variance of a random error are unreliable. 3.2. testing of heteroskedasticity the true nature of heteroskedasticity is usually unknown, so the choice of the appropriate test depends on the nature of the data. but as the amount of error variation around the mean value typically depends on the height of the independent variables, all tests rely on examining whether the error variance is some function of the regressor. certain methods for testing the existence of heteroskedasticity are presented below. 3.2.1. graphic method one of the simplest methods for examining the existence of heteroskedasticity consists in visually viewing the residuals of the estimated model. it is common to form point scattering diagram of residual 𝑒𝑖 or their absolute value, ie , and independent variable ix . since the variance of a random error  2ie  , there is an opinion that on the point scattering diagram of residual values should be replaced by their square, 2ie . based on the point scatter diagrams, we can conclude about whether heteroskedasticity exists, and if so, in what form it occurs, ie how the variance of random error is generated. figure 3 presents graphs of some of the possible point scatter diagrams (mladenović, 2011). figure 3. point scatter diagrams (mladenović, 2011) djalic and terzic/decis. mak. appl. manag. eng. 4 (1) (2021) 1-18 8 the first graph corresponds to a model in which there is no systematic dependence between the variances of random errors and the independent variable .ix in such a model, random errors can be considered homoskedastic. other graphs show the regularity in the position of the points on the scatter diagram, suggesting possible heteroskedasticity. the second graph indicates a linear dependence, while the third and fourth graphs represent the dependence expressed in square form, in the sense that the variance of the random error is correlated with 2 ix . graphic methods are only a means of preliminary analysis. in order to get a more precise answer to the question of whether heteroskedasticity is present or not, it is necessary to use appropriate tests. 3.2.2. goldfeld-quandt test one of the earliest, which is very simple and often used is the goldfeld-quandt test (kalina & peštová, 2017). this test tests the null hypothesis of random error constancy versus alternative that the variance of a random error is a linear function of the independent variable. it is assumed that the random error is non-autocorrelated and with normal distribution. the test procedure is as follows (mladenović & petrović, 2017):  observations from the sample are arranged according to the increasing values of the independent variable.  from the set of n observations, c central observations are omitted, so that further analysis is based on two sets of observations: the first 2 n c and the last 2 n c observations where is necessary to ensure that 2 n c k   , and 𝑘 is the number of evaluated parameters.  we individually evaluate the two regressions based on the first 2 n c and the last 2 n c observations. the obtained sums of the residual squares are denoted by 21e and 2 2 2 2 1 1 e e e   ( 2 1e corresponds to the regression with the lower, аnd 2 2 22 1 1 e e e   to the regression with the higher values of the independent variable). the homoskedasticity of a random error implies the same degree of variation in two subsets of observations, which is manifested by approximately the same values of the variable sums 21e and 2 2 2 2 1 1 e e e   . in this case, the quotient of these two sums is close to the value of 1. on the contrary, the existence of heteroskedasticity results in a higher value of the residual sum 2 2 22 1 1 e e e   . the purpose of the test is to check whether 2 2 2 1 e e   is statistically significantly different from 1. assuming that the null hypothesis of constant variance is correct, the following holds: 2 21 22 2 : n c k e x     (15) violation of the assumption of homoscedasticity and detection of heteroscedasticity 9 2 22 22 2 : n c k e x     (16) where the k is the number of parameters for evaluation in the known model. it follows that the observed relationship: 2 2 2 1 e e   has an f – distribution with 2 2 n c k  and 2 2 n c k  degrees of freedom. therefore, the goldfeld-quandt test statistic is in a form: 2 2 2 1 e f e    (17) if the calculated value of f – statistics is higher than the corresponding critical value at a given level of significance, we conclude that there is heteroskedasticity in the model. 3.2.3. glejser test the application of this test does not require a priori knowledge of the nature of heteroskedasticity, but it is reached during the testing. the test procedure is as follows (im, 2000):  the initial regression 0 1 1 i i k ik iy x x       is estimated by the method of ordinary least squares and the residuals ie are calculated .  the next regression is estimated: 0 1 error h i ie x    (18) the values 1, −1, and 1/2 are usually assigned to the parameter h so that regressions are evaluated: 0 1 errori ie x    (19) 0 1 / errori ie x    (20) 0 1 error i i e x    (21)  the statistical significance of the evaluation of the parameter 1 is tested using the t-test.  the coefficients of determination obtained for different values of the parameter h are compared. the statistical significance of the estimate 1 leads to the conclusion that there is heteroskedasticity. the very character of heteroskedasticity is determined according to the regression with the highest value of the coefficient of determination. djalic and terzic/decis. mak. appl. manag. eng. 4 (1) (2021) 1-18 10 3.2.4. white test the test is based on the comparison of the variance of the estimators obtained by the method of ordinary least squares in the conditions of homoskedasticity and heteroskedasticity. if the null hypothesis is correct, the two estimated variances would differ only due to fluctuations in the sample. the null hypothesis about the homoskedasticity of a random error is tested against the widely placed alternative hypothesis that the variance of a random error depends on the explained variables, their squares and intermediates, ie. the variation of the residuals under the combined action of the regressors is examined. the white test consists of the following steps (white, 1980): step 1: the model 0 1 1 i i k ik iy x x       should be estimated to obtain a series of residuals ie ie their squared values. step 2: evaluate the auxiliary regression in which the squares of the residuals of the function of all regressors of the model, their squares and intermediate products, ie apply the method of ordinary least squares on 2 0 1 1 2 2 i i i p ipe z z z error        , i = 1, 2, …, n, where for simple regression 2 1 2 2, i i i ip z x and z x   so the test is based on analysis of model 2 2 0 1 2 i i ie x x error      , i = 1, 2, …, n. the significant influence of the independent variables ix and 2 ix at 2 ie results in a high value of the coefficient of determination 2 r . the significant influence of the independent variables 1ix and 2ix , the specification is 5p  , 21 1 2 2 3 1 2 4 1 , , , i i i i i i i i iz x z x z x x z x    and 2 5 2 i iz x . due to the possible large loss of degrees of freedom, it is possible to use instead of individual values of the regressors, their linear combination: 2, i iy y . step 3: based on the value of the coefficient of determination from the auxiliary regression, 2wr , form the white test 2 wnr , where n is the sample volume. asymptotically, under the null hypothesis of homoskedasticity, the test statistic 2wnr leads to 2 distribution with the number of degrees of freedom equal to the number of regressors in the auxiliary regression: 2 2 ~ w pnr x . step 4: if the calculated value of the test statistics is greater than the tabular value, ie if the coefficient of determination in the auxiliary function of the residual square is high enough, the homoskedasticity hypothesis is rejected. the white test is not sensitive to the deviation of errors from normal and it is simpler, so it is more often used to test the existence of heteroskedasticity. in the case that there are multiple regressors, the introduction of squares and all intermediates in the auxiliary regression can mean a large loss in the number of degrees of freedom. that is why the white test is often performed without intermediates. 3.2.5. breusch-pagan test this test is based on the idea that the estimates of the regression coefficients obtained by the least squares method should not differ significantly from the maximally plausible estimates, if the homoskedasticity hypothesis is true (halunga et al., 2017). the null hypothesis about the homoskedasticity of random error is tested against the broadly set alternative hypothesis about the influence of a number of violation of the assumption of homoscedasticity and detection of heteroscedasticity 11 factors on the variance of random error. for simplicity, assume that test examines the influence of the explanatory variable ix in simple regression. the testing procedure is as follows (mladenović & nojković, 2017): residuals ie are formed from the regression iy at a constant and ix . the average value of the sum of the squares of the residual is determined: 2 2 e sp n   , and then forms a new variable: 2 1 2 ,i e g sp  i = 1, 2, … , n. from regression ig at ix the explained sum of the squares of the dependent variable is obtained  2ˆig . the relationship 2 2 ˆ ig has 2 x distribution with one degree of freedom. the heteroskedasticity hypothesis will be accepted when the value of the calculated ratio 2 2 ˆ ig is greater than the critical value of 2 x distribution with one degree of freedom. 4. application of the model: data simulation table 1 shows the data so as to simulate the next deviation 2 2 0.01i ix  . the population straight line is 2 3y x  , where y is savings and x is income. in line ,iy 1 , , 30i   , there are values y to which errors i have been added. table 1. display of simulated data no. xi y i yi 1 10 32 -0.13677 31.86323 2 10 32 1.045263 33.04526 3 10 32 0.324248 32.32425 4 10 32 -1.80589 30.19411 5 10 32 0.568473 32.56847 6 10 32 -0.17024 31.82976 7 10 32 0.676169 32.67617 8 10 32 -0.57257 31.42743 9 10 32 -1.53944 30.46056 10 10 32 -0.38377 31.61623 11 20 62 -4.85783 57.14217 12 20 62 -1.66701 60.33299 13 20 62 9.513881 71.51388 14 20 62 0.817791 62.81779 15 20 62 -11.1762 50.82381 16 20 62 -6.47024 55.52976 djalic and terzic/decis. mak. appl. manag. eng. 4 (1) (2021) 1-18 12 17 20 62 9.51661 71.51661 18 20 62 2.045394 64.04539 19 20 62 5.286107 67.28611 20 20 62 7.451416 69.45142 21 30 92 19.59112 111.5911 22 30 92 33.2486 125.2486 23 30 92 -23.2211 68.7789 24 30 92 -28.8606 63.13944 25 30 92 38.95497 130.955 26 30 92 -1.97921 90.02079 27 30 92 -36.9439 55.05615 28 30 92 12.09004 104.09 29 30 92 41.06767 133.0677 30 30 92 -38.0374 53.96263 based on the values iy and ix from table 1 evaluate the linear regression model is evaluated: , 2 : (0, 0.01 ), 1, 2,..., 30 i i i i i y x n x i         after evaluation, the following results were obtained (table 2): table 2. coefficients мodel estimated value standard error 𝑝 value �̂� �̂� 1.022 3.090 8.880 0.411 0.909 0.000 after the obtained coefficients, the analysis of model variance was performed (table 3): table 3. analysis of variance sum of quares no. of degrees freedom average value of sum 𝑝 value regressional residual total 19090.319 9462.096 28552.414 1 28 29 19090.319 337.932 0.000 the coefficient of determination was determined, 2 r = 0.669. figure 4 graphically shows the simulation model. violation of the assumption of homoscedasticity and detection of heteroscedasticity 13 figure 4. graphic representation of the population model figure 4 shows the population line 2 3y x  by interrupted line, while the sample line 1.022 .0ˆ 3 90y x  is shown by full line. the graph clearly shows that the scatterings are higher for higher values of the independent variable x and that sample line ŷ slightly deviates from the line y . after the graphical representation of the model, it can be assumed that certain deviations exist, so we will test the heteroskedasticity with the previously described tests. 4.1. graphic method figure 5 in graph (a) clearly shows the relationship between the residuals and the independent variable x (the larger x , the larger residuals), while in diagram (b) the dependence of the squared residuals with respect to x can be seen (the dependence in the square form). figure 5. diagrams of residual scattering djalic and terzic/decis. mak. appl. manag. eng. 4 (1) (2021) 1-18 14 4.2. goldfeld-quandt test after the order of observations in ascending order of magnitude x two models (for the first 15 and last 15 observations) of linear regression i i iy x     are evaluated. the first 15 observations: 2 3.075 2.873 0.92ˆi iy x r   . (3.324) (0.235) the last 15 observations: 2 2ˆ 9.516 2.803 0.22i iy x r   . (39.369) (1.454) the residual sum for the first 15 observations is 18.411, and for the last 15 observations it is 704,495. based on these residuals, the value of the test statistic is: 704.495 38.26 18.411 f   . as the critical value of the 𝐹 distribution with 13 and 13 degrees of freedom and a significance level of 0.05 is 2.58, this test shows that heteroskedasticity is present (the value of the test statistics is higher than the critical value). 4.3. glejser test three linear regression models are being tested: model 1: i ie x error    model 2: /i ie x error    model 3: i ie x error    the results are shown in table 5. тable 5. the results of the glejser test estimated parameters estimated values standard error 𝑝 -value 𝑹𝟐 model 1 �̂� �̂� -15.419 1.335 4.169 0.193 0.001 0.000 0.631 model 2 �̂� �̂� 31.510 -331.137 4.500 66.813 0.000 0.000 0.467 model 3 �̂� �̂� -37.469 11.151 7.804 1.745 0.000 0.000 0.593 the estimated parameters that stand next to the regressors are statistically significant. all parameters are suitable for testing the hypothesis of heteroskedasticity, and based on the coefficient of determination, the first is preferred because it is the largest. this test also shows the presence of heteroskedasticity. violation of the assumption of homoscedasticity and detection of heteroscedasticity 15 4.4. white test auxiliary linear regression was estimated: 2 2 0 1 2 i i i ie x x       and the values are shown in the following table 4: тable 4. coefficients мodel estimated value standard error 𝑝 value �̂� 𝟎 �̂� 𝟏 �̂� 𝟐 766.923 -117.142 4.053 484.031 54.964 1.360 0.125 0.042 0.006 where the coefficient of determination is 2 0.607wr  . it can be observed that the coefficients along with ix and 2 ix are statistically significant while the constant is not. white's test statistic is 2 30 0.607 1 8.21wnr   which is greater than the tabular value of the 2 istribution with two degrees of freedom, 5.991. it is the same conclusion as before, that heteroskedasticity is present. 4.5. breusch-pagan test based on the linear regression equation 1.022 3.090i iy x  the estimated value of the error variance is obtained: 2 9462.10 315.403ˆ 30    the new regression equation: 2 5ˆ 1.8 2 0.143 ip x  (0.609) (0.028) where is: 2 2 2 315.403ˆ i i i e e p    test statistics is: 2 40.660 20.33 2 ˆ 2 ig   the critical value of the 𝜒2 distribution with one degree of freedom and a significance level of 0.05, is 3.841, so it is also concluded that heteroskedasticity is present. djalic and terzic/decis. mak. appl. manag. eng. 4 (1) (2021) 1-18 16 4.6. discussion after testing, it is clear that all four tests show the presence of heteroskedasticity in a given model. the goldfeld-quandt test shows that the f – distribution is equal to 2.58 and it is higher than the corresponding critical value at a given level of significance (0.05). based on this we can conclude that heteroskedasticity is present in the model. in the glejser test the parameter 1 is tested and the coefficients of determination obtained for different values of the parameter h are compared. in this model (table 5) all parameters are suitable for testing the hypothesis of heteroskedasticity, and based on the coefficient of determination, the first is preferred because it is the largest. this test also shows the presence of heteroskedasticity. white test shows that the calculated value (18.21) of the test statistics is greater than the tabular value and we can conclude heteroskedasticity is present. in the breusch-pagan test the value of the calculated ratio is 20.33 and it is greater than the critical value of 2 x distribution that is 3.841 with one degree of freedom, and we also can conclude that heteroskedasticity is present. 5. conclusion one of the classic assumptions of linear regression is homoskedasticity, and when it is disturbed, heteroskedasticity occurs. graphical methods and heteroskedasticity tests are used to detect heteroskedasticity, although it is not possible to say with certainty which test is the best. in this paper, we explained and applied the graphical method and four tests (goldfeld-quantum, glejser, white and breusch-pagan test). through a review of the literature, it can be seen that many authors have addressed this issue and used various tests to detect heteroskedasticity. the tests were applied by data simulation. it can be seen that the graphical method and all four applied tests confirm the presence of heteroskedasticity, so we can conclude that all four tests showed a good result and that it can be confirmed the assumption of the existence of heteroskedasticity in the model. future researchers are left with the question of solving heteroskedasticity, ie the question of removing heteroskedasticity from the model. when eliminating heteroskedasticity, care must be taken which method can be used depending on the form 2 i . author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research received no external funding. conflicts of interest: the authors declare no conflicts of interest. references arbia, g. 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(2019). heteroskedasticity-consistent covariance matrix estimators for spatial autoregressive models. spatial economic analysis, 14(2), 241-268. white, h. (1980). a heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. econometrica: journal of the econometric society, 817-838. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). decision making: applications in management and engineering vol. 4, issue 2, 2021, pp. 106-125. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame210402106a * corresponding author. e-mail address: taliparsu@aksaray.edu.tr (t. arsu) investigation into the efficiencies of european football clubs with bi-objective multi-criteria data envelopment analysis talip arsu 1* 1 vocational school of social sciences, university of aksaray, turkey received: 15 march 2021; accepted: 24 may 2021; available online: 12 june 2021. original scientific paper abstract: a financially successful football club can achieve sporting achievements as well as become financially stable. in other words, the success of football clubs depends on both financial and sportive success. contrary to the studies in the literature that focus on financial and sportive success separately, the present study aimed to examine the 5-season activities of 10 football clubs in the big-five league, which are the top leagues of europe, by using financial and sports criteria. bi-objective multi criteria data envelopment analysis (bio-mcdea) was used for the efficiency analysis. in the study, the number of social media followers, the average number of viewers and total market value were used as input, and the uefa club score and total revenues were used as output. as a result, arsenal, paris saint-germain, and juventus were determined as efficient in the 2015-2016 season, paris saintgermain and liverpool in the 2016-2017 season, manchester united, paris saint-germain and chelsea in the 2016-2017 season, manchester united, real madrid, bayern munich and arsenal in the 2018-2019 season, manchester united, paris saint-germain and chelsea. the reasons why psg was the most successful club in the efficiency analysis (efficient in four out of five seasons) were examined. in addition, in the sensitivity analysis conducted to determine the effect of inputs and outputs on the model, it was concluded that efficiency was highly related to financial data. keywords: european football clubs, efficiency, multi-criteria data envelopment analysis, bi-objective multi-criteria data envelopment analysis mailto:taliparsu@aksaray.edu.tr investigation into the efficiencies of european football clubs with bi-objective multi-criteria data envelopment analysis 107 1. introduction football is the most popular sport in the world. although there are many factors that underlie this popularity, the simplicity of the rules and the low cost can be considered as the most important factors (galariotis et al., 2018). however, in professional football, which has undergone a great transformation since the early 1990s, footballer salaries have started to increase exponentially (dobson & goddard, 2011). the bosman ruling introduced by the european court of justice in 1995 had a significant impact on the future of european football. the bosman ruling was named after the belgian midfielder jean-marc bosman's lawsuit that was filed for blocking his transfer from belgium to france at the end of his contract. the bosman ruling included the liberalization of the immigration of professional athletes within the eu and the abolition of transfer fees after the expiry of contracts. in addition, restrictions on the number of eu players that clubs can have playing on the field were also considered illegal according to bosman ruling (marcén, 2019). after the bosman ruling was recognized by uefa in march 1996, the transfers of football players between teams began to be carried out at astronomical figures. in addition, the fact that broadcasting contracts yielded an unimaginable scale of revenue just a few years ago, the complete reconstruction of many football fields, and the immeasurable increase in the importance of commercial sponsorship and merchandising increased the importance of football's financial infrastructure (dobson & goddard, 2011). football clubs are no longer organizations that only provide emotional and symbolic satisfaction to their supporters and focus on sporting success without profit. instead, football clubs have become a complex system in which investors invest capital and expect financial returns (miragaia et al., 2019). this development in professional football has turned football from being not only a sport branch in europe but also an industry branch. the revenues and brand values of football clubs have become competitive with many industries and brands. spain (la liga), england (premier league), italy (serie a), germany (bundesliga) and france (ligue 1), which are called the “big five league”, constitute a large part of the world football industry. the big five league, which ha s gained great value in the last 20 years, increased its total value from eur 2.95 billion in 1998 to eur 26.8 billion in 2021 (transfermarkt.com, 2021). however, these financial values are not governed by all the clubs in the big five league, but only the top 10 clubs in europe in terms of both sporting success and financial standing. manchester united, real madrid, fc barcelona, bayern munich, manchester city, arsenal, paris saint-germain (psg), chelsea, liverpool and juventus have a value of eur 7.96 billion, which is almost one third of all other clubs total value of the big five league (transfermarkt.com, 2021). the growth of the football industry at this scale in as little as 20 years has brought along both control and financial difficulties. although football clubs have many financial resources, these resources are largely related to sporting success. in other words, football clubs must be continuously successful in order to avoid experiencing financial difficulties, which is not possible. as the financial difficulties experienced by football clubs are beginning to become continuous, uefa has brought some restrictions on clubs with a regulation called financial fair play (ffp). ffp, which entered into force in 2009 and is updated every three years, basically aims to improve the economic and financial capabilities of the clubs and increase both their transparency and reliability. at the same time, thanks to the ffp, which aims to bring more discipline and rationality to club football financing, the ratio of net debts of clubs to their income has decreased from 65% to 35% in a short period of time (uefa, arsu/decis. mak. appl. manag. eng. 4 (2) (2021) 106-125 108 2021). in order to achieve this financial success, uefa has imposed many restrictions on clubs and imposed severe penalties such as the deletion of points, transfer restrictions and ban from tournaments, for those who do not comply with these restrictions. football clubs, which are suppressed by uefa, are also trying to meet the sportive success expectations of the stakeholders. it does not seem realistic to evaluate these two processes independently from each other in football clubs where financial success supports sportive success. some studies in the literature have carried out financial evaluations by only considering the financial data of clubs (pradhan et al., 2016; chelmis et al., 2019), some have focused only on sportive success (rossi et al., 2019; salabun et al.2020) and others have tried to associate financial success with sportive success (sakınç et al., 2017; galariotis et al., 2018). however, the success of football clubs is possible with the realization of the financial and sportive success together in this cycle. the aim of this study, which was designed with the motivation of the idea of realizing financial and sportive success together and the lack in the literature, was to investigate the efficiency values for three seasons of 10 football clubs that are at the top in terms of both sport and finance in europe. when conducting the effectiveness analysis, the bi-objective multiple criteria data envelopment analysis (bio-mcdea) method, recommended by ghasemi (2014) to eliminate the low discrimination problem of the classical data envelopment analysis (dea), was used. the paper begins with a detailed literature review in section 2. in the section 3, firstly, the classical dea and multiple criteria data envelopment analysis (mcdea) methods that form the basis of the bio-mcdea model are introduced and the biomcdea model is shown. in the data title at the end of the section 3, how the criteria used in this study were determined, the source of the data used as criteria and the criteria values are shown. in section 4, the findings of the study are presented and in section 5, the findings are discussed. in section 6, the sensitivity analysis is given to determine the contribution of each criterion to the model. in the last section, the conclusions, advantages and limitations of the study and managerial implications are given. 2. literature review the popularity of football around the world and the huge budgets managed by football clubs have made the football industry the subject of many academic studies conducted to examine the sportive or financial performances of national and international leagues, clubs and even players. in many of these studies, mcdm methods have been used for performance evaluation. pradhan et al. (2016) investigated the financial performance of italian clubs using gray relation analysis (gra), galariotis et al. (2018) determined the business, financial and sports performance of clubs in the french league using the promethee ii method, sakınç et al. (2017) studied the financial and sporting performance of 22 european clubs using the topsis method, chelmis et al. (2019) investigated the financial, commercial and sporting performance of clubs in the greek league using promethee ii and salabun et al. (2020) determined the performance of football players using the characteristic objects method (comet) and topsis method. in addition to these methods used, the most used mcdm method is dea which was developed by charnes et al. (1978). in recent years dea has been used in many decision problems such as the effectiveness of agricultural practices (angulo-meza et al., 2019), financial performance assessment (anthony et al., 2019), hospital efficiency assessments (kohl et al., 2019), sustainability assessment of the water sector (lombardi et al., 2019), bank activities investigation into the efficiencies of european football clubs with bi-objective multi-criteria data envelopment analysis 109 assessments (kamarudin et al., 2019), sustainable supplier selection (rashidi & cullinane, 2019), efficiency assessment of railway enterprises (blagojević et al., 2020), assessment of medium and large-sized industries in the diversity sector (hassanpour, 2020). dea studies in the literature generally consist of efficiency analyses conducted for all teams in the league of a specific country. dea was used to determine the efficiencies of the teams in england’s premier league (pestana barros & leach, 2006; guzman & morrow, 2007; haas, 2003a; kern et al., 2012), germany’s bundesliga (haas et al., 2004), france’s ligue 1 (jardin, 2009), usa’s major league soccer (mls) (haas, 2003b), italian serie a (rossi et al., 2019), and brazil’s serie a (pestana barros et al., 2010). in addition, the efficiencies of european clubs (halkos & tzeremes, 2013; miragaia et al., 2019) and national teams participating in euro 2012 (rubem & brandao, 2015) were determined using dea. however, no study examining the 5season efficiency values of 10 top european clubs which make up half of the total value of the big five league were found in the literature. furthermore, classical dea was used in almost all efficiency studies in the literature. although classical dea is a widely used nonparametric efficiency instrument, it has the disadvantage of low discrimination power. in order to avoid this disadvantage, the bio-mcdea model, which was developed by ghasemi et al. (2014) and has been used in decision problems such as the equipment efficiency assessment for automotive industry (da silva et al., 2017), port efficiency assessment (de andrade et al., 2019), electrical distribution units efficiency assessment (ghofran et al., 2021), was used in this study. 3. material and methods bio-mcdea is a goal programming based efficiency determination model developed by ghasemi et al. (2014) in which the dea model aims to improve the discrimination power. bal et al. (2010) proposed the goal programming data envelopment analysis (gpdea) model which is based on goal programming that would eliminate the problem of discrimination power and weight distribution of the dea model. the gpdea model is based on solving unwanted deviations using equal weight. bio-mcdea was used in the present study to exclude classical dea and thus avoid the disadvantage of low discrimination power and because its solution steps are easier. 3.1. multiple criteria data envelopment analysis (mcdea) classical dea is a widely used non-parametric analysis for efficiency analysis, used especially in social sciences. the conversion of the classical dea method into a linear programming form proposed by charnes et al. (1978) is shown below. 𝑀𝑎𝑥 ℎ0 = ∑ 𝑢𝑟 𝑦𝑟𝑗 𝑠 𝑟=1 𝑠. 𝑡. ∑ 𝑣𝑖 𝑥𝑖𝑗 = 1 (1) 𝑚 𝑖=1 arsu/decis. mak. appl. manag. eng. 4 (2) (2021) 106-125 110 ∑ 𝑢𝑟 𝑦𝑟𝑗 𝑠 𝑟=1 − ∑ 𝑣𝑖 𝑥𝑖𝑗 ≤ 0 𝑚 𝑖=1 , 𝑗 = 1, … , 𝑛 𝑢𝑟 ≥ 0 𝑣𝑖 ≥ 0 where; j is the number of decision-making units (dmu), r is the number of outputs, i is the number of inputs, 𝑦𝑟𝑗 is the value of the rth output for the jth dmu, 𝑥𝑖𝑗 is the value of the ith input for the jth dmu, 𝑢𝑟 is the weight of the rth output, 𝑣𝑖 is the weight of the ith input and ℎ0 refers to relative efficiency. in this model, any dmu must be ℎ0 = 1 in order to be effective (charnes et al., 1978; despic et al., 2019) . although classical dea is an efficiency measurement method, li & reeves’ (1999) mcdea model is based on ineffectiveness. 𝑑0, which is limited to the [0, 1] range can be considered a measurement of “ineffectiveness” and is defined as ℎ0 = 1 − 𝑑0. therefore the smaller the 𝑑0 value, the less ineffective (and therefore more effective) dmu is. in the method of li & reeves (1999), besides the minimization of d0, which is the measure of ineffectiveness, there are two independent objective functions, namely, minimizing maximum deviation and minimizing the sum of deviations. their model is as follows: 𝑀𝑖𝑛 𝑑0 (𝑜𝑟 max ℎ0 = ∑ 𝑢𝑟 𝑦𝑟𝑗0) 𝑠 𝑟=1 𝑀𝑖𝑛 𝑀 𝑀𝑖𝑛 ∑ 𝑑𝑗 (2) 𝑛 𝑗=1 𝑠. 𝑡. ∑ 𝑣𝑖 𝑥𝑖𝑗0 = 1 𝑚 𝑖=1 ∑ 𝑢𝑟 𝑦𝑟𝑗 − ∑ 𝑣𝑖 𝑥𝑖𝑗 + 𝑑𝑗 = 0 𝑚 𝑖=1 𝑠 𝑟=1 𝑀 − 𝑑𝑗 ≥ 0 𝑗 = 1, … . . , 𝑛 𝑢𝑟 , 𝑣𝑖 , 𝑑𝑗 ≥ 0 the mcdea model was proposed primarily as a tool for improving the discrimination power of the classical dea model. in the solution procedure, mcdea was proposed as an interactive approach to solve three objectives. the first objective accommodates the classical ria solution within a set of mcdea solutions. the other two objectives, minimax and minsum, provide more restrictive or lax efficiency solutions, respectively. this model proves that a wider solution is possible to achieve more reasonable input and output weights (ghasemi et al., 2014). 3.2. a bi-objective multiple criteria data envelopment analysis (bio-mcdea) the mcdea model consists of three independent objective functions: 𝑀𝑖𝑛 𝑑0 , 𝑀𝑖𝑛 𝑀 and 𝑀𝑖𝑛 ∑ 𝑑𝑗𝑗 as defined in model 2. in a weighted model, these three independent objective functions can be weighted as 𝑤1𝑑0 + 𝑤2𝑀 + 𝑤3 ∑ 𝑑𝑗𝑗 into a single-objective problem. different efficiency scores can be achieved by changing the investigation into the efficiencies of european football clubs with bi-objective multi-criteria data envelopment analysis 111 weights 𝑤𝑖 (𝑖 = 1,2,3). however, since the first objective function (𝑀𝑖𝑛 𝑑0 ) has the same meaning as the classical dea model, it can be removed from the mcdea model as the discrimination power of the second (𝑀𝑖𝑛 𝑀) and third (𝑀𝑖𝑛 ∑ 𝑑𝑗𝑗 ) objective functions has been proved to be higher than the 𝑀𝑖𝑛 𝑑0 objective (li & reeves, 1999; san cristobal, 2011; hatami-marbini & toloo, 2017). therefore, only the 𝑀𝑖𝑛 𝑀 and 𝑀𝑖𝑛 ∑ 𝑑𝑗𝑗 objectives are weighted in the bio-mcdea model, which is shown below: 𝑀𝑖𝑛 ℎ = (𝑤2𝑀 + 𝑤3 ∑ 𝑑𝑗 𝑗 ) 𝑠. 𝑡. ∑ 𝑣𝑖 𝑥𝑖𝑗0 = 1 𝑚 𝑖=1 (3) ∑ 𝑢𝑟 𝑦𝑟𝑗 − ∑ 𝑣𝑖 𝑥𝑖𝑗 + 𝑑𝑗 = 0 𝑚 𝑖=1 𝑠 𝑟=1 𝑀 − 𝑑𝑗 ≥ 0 𝑗 = 1, … . . , 𝑛 𝑢𝑟 ≥ 𝜀, 𝑟 = 1,2, … , 𝑠 𝑣𝑖 ≥ 𝜀, 𝑖 = 1,2, … , 𝑚 𝑑𝑗 ≥ 0 𝑗 = 1,2, … , 𝑛 the constraints of the bio-mcdea model consist of the same constraints as the mcdea model of li & reeves (1999). only the 𝑢𝑟 and 𝑣𝑖 variables are constrained by the constant 𝜀. although ghasemi et al. (2014) used 𝜀 = 0,0001 in the samples they solved, they did not suggest an approach to find a suitable value for the constant 𝜀. in addition, the bio-mcdea model is still robust if 𝜀 = 0 in the sample solved using a different data set. in this study, 𝜀 = 0 was used as in the original model. 3.3. data the data of this study was obtained from the 2020, 2019, 2018, 2017 and 2016 deloitte football money league reports and transfermrkt.com, which regularly collects data on the european football industry every year. in the study, the number of social media followers (v1), average number of viewers (v2) and total market value (v3) were used as input, while the uefa club score (u1) and total revenue (u2) variables were used as output. in their studies aichner (2018), alaminos et al. (2020) and weimar et al. (2021) used number of social media followers, haas (2003a), pestana barros et al. (2010), kern et al. (2012), alaminos et al. (2020) used average number of viewers, kulikova & goshunova (2014) and rubem & brandao (2015) used total market value, rubem & brandao (2015) used uefa club score, halkos & tzeremes (2013), kulikova & goshunova (2014), jardin (2009), guzman & morrow (2007), pestana barros et al. (2010), kern et al. (2012), chelmis et al. (2019) and miragaia et al. (2019) used the total revenue of the club as input or output variable. the definitions of the input and output variables are shown in table 1. arsu/decis. mak. appl. manag. eng. 4 (2) (2021) 106-125 112 table 1. bio-mcdea model input and output variable definitions variables definition the number of social media followers (𝑣1) the number of people following the clubs on facebook, instagram and twitter (*106) average number of viewers (𝑣2) the average number of people who came to the stadium as a spectator in matches hosted by clubs total market value(𝑣3) the sum of the market values of the club's footballers (*106 €) uefa club score (𝑢1) the total points the club has obtained from all matches during a season total revenue (𝑢2) the sum of club's matchday revenues, broadcasting revenues and commercial revenues (*106 €) pearson correlation coefficients are widely used when choosing input and output in dea (lewin et al., 1982; thanassoulis et al., 1987; golany & roll, 1989; friedman & sinuany-stern, 1998; dyson et al., 2001). lewin et al. (1982) argued that inputs should not be highly correlated with other inputs and outputs should not be highly correlated with other outputs. they also stated that if the inputs and outputs are negatively correlated with each other, these variables may be excluded from the model since the increase in inputs will affect the output negatively. the pearson correlation coefficients of the data used in the present study are shown in table 2. table 2. pearson correlation coefficients for bio-mcdea model inputs and outputs v1 v2 v3 u1 u2 v1 1 v2 0.578 1 v3 0.448 0.136 1 u1 0.179 0.209 0.219 1 u2 0.781 0.712 0.534 0.205 1 according to the results given in table 2, none of the pearson correlation coefficients had a very high, very low or negative value. therefore, no input or output variable was excluded from the model. in this study, an analysis of the efficiency of five seasons of 10 top european football clubs was performed. the values of input and output variables selected to determine the effectiveness of the 2015-2016, 2016-2017, 2017-2018, 2018-2019 and 20192020 seasons are shown in table 3. investigation into the efficiencies of european football clubs with bi-objective multi-criteria data envelopment analysis 113 table 3. input and output values of the bio-mcdea model 2019-2020 season football clubs v1 v2 v3 u1 u2 manchester united 127.2 74698 670.45 22000 711.5 real madrid 226.7 61040 913.75 17000 757.3 fc barcelona 216.5 76104 930.93 24000 840.8 bayern munich 74.4 75865 777.33 36000 660.1 manchester city 62.9 54130 1048.6 25000 610.6 arsenal 69.7 59897 607.65 10000 445.6 psg 73.7 46911 874.15 31000 635.9 chelsea 82.2 40445 705.85 17000 513.1 liverpool 71.9 53053 1002.7 18000 604.7 juventus 83.4 39101 661.88 22000 459.7 2018-2019 season manchester united 117.2 75102 797.6 19000 666 real madrid 207.8 66337 1033.1 19000 750.9 fc barcelona 195.5 70872 1201.4 30000 690.4 bayern munich 68.9 75354 784.88 20000 629.2 manchester city 53.1 54054 1203.4 25000 568.4 arsenal 64.7 59323 659.05 26000 439.2 psg 60.4 46929 1009.9 19000 541.7 chelsea 74.4 41281 1166.6 30000 505.7 liverpool 54.8 52958 1172.4 29000 513.7 juventus 63.1 36510 871.05 21000 394.9 2017-2018 season manchester united 110.2 75305 645.10 28985 676.3 real madrid 189.7 69426 716.2 37028 674.6 fc barcelona 184.3 78678 772.5 27028 648.3 bayern munich 59.5 75000 610.25 24914 587.8 manchester city 41 54019 616.35 20985 527.7 arsenal 61.2 59957 633.90 21985 487.6 psg 49.9 45160 581.10 22883 486.2 chelsea 69.9 41532 642.15 2985 428 liverpool 45.3 53094 495 2985 424.2 juventus 45.2 37195 540.53 35850 405.7 2016-2017 season manchester united 97.4 75327 533.25 15850 689 real madrid 158.4 71280 743.1 37785 620.1 fc barcelona 159.1 79724 787.2 30785 620.2 bayern munich 52.3 75017 595.4 32285 592 manchester city 30.7 54013 621.4 28850 524.9 arsenal 55 59980 522.75 17850 468.5 psg 37.5 46160 502.05 26216 520.9 chelsea 63.3 41500 603.3 20850 447.4 arsu/decis. mak. appl. manag. eng. 4 (2) (2021) 106-125 114 liverpool 39.8 44108 394.15 24850 403.8 juventus 34.6 39106 463.78 20300 341.1 2015-2016 season manchester united 83.1 75335 374.15 2714 519.5 real madrid 128.9 72969 700.75 33042 577 fc barcelona 132.8 77632 618.5 38042 560.8 bayern munich 41.5 72882 608.5 31171 474 manchester city 25.3 45345 452.75 17714 463.5 arsenal 46.4 59992 408.6 22714 435.5 psg 28.9 45789 433.3 23183 480.8 chelsea 56.1 41546 579.8 23714 420 liverpool 34.5 44675 325 12714 391.8 juventus 26.3 36292 394.33 32800 323.9 the reason why the 10 clubs were included in the efficiency evaluation is that these 10 clubs were ranked in the top 10 for five seasons in the deloitte football money league report, which was the main data source of this study. the deloitte football money league report publishes data for the 20 top financially successful clubs each season. however, 10 clubs other than the top 10 change at a certain rate each year. as data of some of the clubs other than the top 10 clubs from different sources could harm the homogeneity of the data, the clubs not included in the top 10 clubs were excluded from the scope of the study. 4. results the mcdea and bio-mcdea efficiency scores of the football clubs were calculated using lindo w32 software. the first three columns in table 4 are the efficiency results of the mcdea model solution. the fourth column consists of efficiency values obtained as a result of the bio-mcdea model solution. the last column refers to the ranking of the football clubs according to the results of the efficiency values obtained with the bio-mcdea model solution. the efficient football clubs (eff. 1) were ranked first, while the other clubs were ranked in order after that. table 4. bio-mcdea model efficiency scores. football clubs classical dea/min d0 min m min ∑d biomcdea rank 2 0 1 9 -2 0 2 0 s e a so n manchester united 1 1 1 1 1 real madrid 1 0.902 0.852 0.890 5 fc barcelona 1 0.911 0.886 0.917 2 bayern munich 0.988 0.890 0.834 0.834 7 manchester city 0.996 0.878 0.859 0.859 6 arsenal 0.826 0.843 0.800 0.800 8 psg 1 1 1 1 1 chelsea 1 1 1 1 1 liverpool 0.924 0.929 0.908 0.908 3 investigation into the efficiencies of european football clubs with bi-objective multi-criteria data envelopment analysis 115 juventus 0.928 0.852 0.891 0.891 4 2 0 1 8 -2 0 1 9 s e a so n manchester united 1 1 1 1 1 real madrid 1 1 1 1 1 fc barcelona 0.965 0.940 0.931 0.941 5 bayern munich 1 1 1 1 1 manchester city 1 0.943 0.922 0.922 7 arsenal 1 0.996 0.994 1 1 psg 1 0.977 0.995 0.981 3 chelsea 1 1 0.986 0.986 2 liverpool 0.996 0.931 0.934 0.934 6 juventus 0.953 0.943 0.953 0.946 4 2 0 1 7 -2 0 1 8 s e a so n manchester united 1 1 1 1 1 real madrid 1 0.965 0.972 0.965 4 fc barcelona 0.866 0.863 0.866 0.863 6 bayern munich 1 0.883 0.896 0.896 5 manchester city 1 0.979 0.990 0.990 2 arsenal 0.849 0.835 0.827 0.827 8 psg 1 1 1 1 1 chelsea 1 0.864 1 1 1 liverpool 1 0.979 0.969 0.969 3 juventus 1 0.946 0.828 0.828 7 2 0 1 6 -2 0 1 7 s e a so n manchester united 1 0.810 0.992 0.992 2 real madrid 0.944 0.844 0.844 0.844 4 fc barcelona 0.731 0.731 0.731 0.731 9 bayern munich 1 0.922 0.922 0.922 3 manchester city 1 0.837 0.837 0.837 5 arsenal 0.825 0.783 0.783 0.783 6 psg 1 1 1 1 1 chelsea 0.955 0.744 0.744 0.744 7 liverpool 1 1 1 1 1 juventus 0.913 0.738 0.738 0.738 8 2 0 1 5 -2 0 1 6 s e a so n manchester united 1 0.671 0.562 0.662 8 real madrid 0.798 0.791 0.798 0.798 5 fc barcelona 0.936 0.903 0.936 0.936 3 bayern munich 0.789 0.766 0.788 0.788 6 arsu/decis. mak. appl. manag. eng. 4 (2) (2021) 106-125 116 manchester city 1 0.871 0.852 0.852 4 arsenal 1 0.909 1 1 1 psg 1 1 1 1 1 chelsea 1 0.726 0.673 0.673 7 liverpool 1 0.934 0.983 0.983 2 juventus 1 1 1 1 1 it can be seen from table 4 that psg is the only club that was efficient for all three seasons. however, the ranking of other clubs according to the bio-mcdea model differed for each season. for instance, manchester united ranked first in the 20172018 season, second in the 2016-2017 season and eighth in the 2015-2016 season. this shows that the financial and sporting success of the clubs affects their rankings in different seasons. spearman rank correlation was commonly used in the literature to test the relationship between dmu rankings (haas et al., 2004; bal et al., 2010; örkcü & bal, 2011). in this study, the relationship between the ranks determined for three different seasons as a result of the bio-mcdea model was tested with spearman rank correlation. when the results of the spearman rank correlation were examined, no statistically significant relationship was found between the rankings for the three seasons. this result supports the idea that the financial and sporting achievements of the clubs affect their rankings in different seasons. in other words, the clubs achieved a ranking according to how successful they were in sports or financial terms. although the model was created for three consecutive seasons, the rankings differed greatly. the spearman correlation values are shown in table 5. table 5. biomcdea efficiency ranking spearman rank correlations values seasons 2015-2016 2016-2017 2017-2018 2018-2019 2019-2020 2015-2016 1.000 2016-2017 0.529 1.000 2017-2018 -0.503 -0.080 1.000 2018-2019 -0.354 0.132 -0.076 1.000 2019-2020 -0.242 0.160 0.714 -0.146 1.000 as can be seen from the figure, fc barcelona ranked third in 2015-2016, ninth in 2016-2017 and sixth in 2017-2018. fc barcelona was the second club with the highest total market value among the clubs included in the analysis of the 2015-2016 season. it was also the second club with the highest total revenue in the same season. this was reflected in their sporting success as they reached the highest uefa score among the clubs involved in the analysis. total revenue, total market value and uefa club points placed fc barcelona in third place in the bio-mcdea model. however, although fc barcelona seemed to be the most valuable club in terms of total market value in the 2016-2017 and 2017-2018 seasons, it was observed that the quality of the footballers was not sufficient to increase their uefa club points and total revenue. due to this result, fc barcelona ranked lower in the 2016-2017 and 2017-2018 seasons according to the bio-mcdea model. investigation into the efficiencies of european football clubs with bi-objective multi-criteria data envelopment analysis 117 5. sensitivity analysis these remarkable results raise the question of how much input and output variables contribute to the model when determining the bio-mcdea model ranking. therefore, a sensitivity analysis was performed to determine which input or output contributed to the model. to determine the contribution of each input and output variable, bio-mcdea efficiency values including all the variables and bio-mcdea efficiency values calculated by excluding each input and output variable were examined. in addition, pearson correlation coefficients were examined to determine the relationship between the efficiency values of the model including all input and output variables and the efficiency values when each variable was excluded from the model. the sensitivity analysis results and pearson correlation coefficients are shown in table 6. table 6. sensitivity analysis results for bio-mcdea model variables football clubs biomcdea without v1 (r1= 0.943*) without v2 (r2= 0.334) without v3 (r3= 0.302) without u1 (r4= 0.760*) without u2 (r5= 0.220) 2 0 1 9 -2 0 2 0 s e a so n manchester uni 1 1 1 0.795 1 0.660 real madrid 0.890 1 0.687 0.880 0.864 0.476 fc barcelona 0.917 1 0.757 0.821 0.996 0.600 bayern munich 0.834 0.893 1 0.621 0.877 1 manchester city 0.859 0.804 0.758 0.931 0.818 0.675 arsenal 0.800 0.725 0.822 0.702 0.750 0.350 psg 1 0.990 0.883 1 1 0.992 chelsea 1 0.954 0.811 1 0.965 0.663 liverpool 0.908 0.809 0.758 0.998 0.836 0.504 juventus 0.891 0.918 0.757 0.809 0.908 0.902 2 0 1 8 -2 0 1 9 s e a so n manchester uni 1 0.968 0.839 0.774 1 0.589 real madrid 1 1 0.454 0.839 1 0.361 fc barcelona 0.941 0.965 0.511 0.730 0.854 0.780 bayern munich 1 0.945 1 0.723 1 0.620 manchester city 0.922 0.888 0.654 0.887 0.913 0.750 arsenal 1 1 1 0.616 0.855 0.988 psg 0.981 0.925 0.684 1 1 0.665 chelsea 0.986 1 0.654 0.997 0.895 0.990 liverpool 0.934 0.893 0.653 0.794 0.842 0.870 juventus 0.946 0.937 0.663 0.887 0.867 0.882 2 0 1 7 -2 0 1 8 manchester uni 1 1 0.996 0.849 1 0.522 real madrid 0.965 0.972 0.781 0.861 0.998 0.259 fc barcelona 0.863 0.866 0.742 0.779 0.867 0.189 bayern munich 0.896 0.896 0.990 0.741 0.891 0.349 manchester city 0.990 0.990 0.925 0.913 0.961 0.504 arsenal 0.827 0.827 0.801 0.751 0.829 0.528 psg 1 1 0.866 1 1 0.595 chelsea 1 1 0.763 1 0.870 0.069 liverpool 0.969 0.969 1 0.775 0.860 0.090 juventus 0.828 0.828 0.687 0.959 0.952 1 2 0 1 6 -2 0 1 7 manchester uni 0.992 0.993 1 0.810 1 0.380 real madrid 0.844 0.844 0.719 0.771 0.789 0.680 fc barcelona 0.731 0.731 0.664 0.678 0.726 0.677 bayern munich 0.922 0.923 0.940 0.699 0.819 0.775 manchester city 0.837 0.838 0.782 0.861 0.835 0.750 arsenal 0.783 0.784 0.764 0.692 0.776 0.536 psg 1 1 0.940 1 1 0.996 arsu/decis. mak. appl. manag. eng. 4 (2) (2021) 106-125 118 chelsea 0.744 0.744 0.672 0.955 0.808 0.375 liverpool 1 1 0.992 0.811 0.898 0.994 juventus 0.738 0.739 0.704 0.773 0.737 0.711 2 0 1 5 -2 0 1 6 manchester uni 0.662 0.662 0.895 0.639 0.657 0.041 real madrid 0.798 0.798 0.803 0.763 0.745 0.628 fc barcelona 0.936 0.936 0.937 0.705 0.775 0.761 bayern munich 0.788 0.788 0.790 0.631 0.676 0.623 manchester city 0.852 0.852 0.859 0.961 0.937 0.478 arsenal 1 1 0.995 0.693 0.864 0.704 psg 1 1 1 0.992 1 0.649 chelsea 0.673 0.673 0.682 0.957 0.719 0.427 liverpool 0.983 0.983 0.969 0.822 1 0.470 juventus 1 1 0.996 0.889 0.769 0.984 when the results of the sensitivity analysis were examined, a significant correlation was observed between the bio-mcdea efficiency scores, which included all inputs and outputs, and the bio-mcdea efficiency scores, where two inputs and one output were excluded from the model. in particular, when the number of social media followers (v1) input variable was excluded from the model, an excellent correlation (r1=0.943) was observed between the obtained efficiency values and the activity values in which all variables were included in the model. that is to say, this variable did not contribute to the model. in the same way, a statistically significant and strong relationship (r4=0.760) was observed between the efficiency values obtained by excluding the uefa club score output (u1), and the bio-mcdea model in which all variables were included. it was found that these variables did not contribute to the model. the biggest contribution to the model was average number of viewers (v2) and the total market value (v3) inputs and the total revenue (u2) output. in other words, mainly the financial variables influenced the ranking of the bio-mcdea model of the clubs. 6. discussion in the analysis made using the data of the 2015-2016, 2016-2017, 2017-2018, 2018-2019 and 2019-2020 seasons, a comprehensive assessment was made by using the number of social media followers, average number of viewers, total market values, uefa club scores and total revenues. when the mcdea model min d0, which gives the same results with classical output-oriented dea efficiency values, and bio-mcdea model efficiency values were compared, it was concluded that the bio-mcdea model improved the discrimination power. this was because while seven clubs in the 20152016 season, five clubs in the 2016-2017 season, eight clubs in the 2017-2018 season, seven clubs in the 2018-2019 season and five clubs in the 2019-2020 season were efficient according to the classical dea model, only three clubs in the 2015-2016 season, two clubs in the 2016-2017 season, three clubs in the 2017-2018 season, four clubs in the 2018-2019 season and three clubs in the 2019-2020 season were efficient according to the bio-mcdea model. according to the results, arsenal, psg and juventus emerged as the efficient clubs in the 2015-2016 season, psg and liverpool emerged as the efficient clubs in the 2016-2017 season, manchester united, psg and chelsea emerged as the efficient clubs in the 2017-2018 season, manchester united, real madrid, bayern munich and arsenal emerged as the efficient clubs in the 20182019 season and manchester united, psg and chelsea emerged as the efficient clubs in the 2019-2020 season. psg was determined as an efficient club in four out of five seasons included in the analysis. in other words, psg was the most successful club among the analyzed clubs. investigation into the efficiencies of european football clubs with bi-objective multi-criteria data envelopment analysis 119 this may be attributed to the sale of the club to a qatari fund group in 2011. correspondingly, the market value of the club increased with the large expenditures made for transfers immediately after the sale. with this acceleration, the club which had only won two championships in the france ligue 1 since 1970, became the champion seven times in eight seasons after the 2012 season. the club, which became the champion almost every season after the 2012 season, increased its uefa club points by joining the champions league every year and achieved a financially stable structure. in other words, the club, which was financially supported after the 2011 season, increased its sporting achievements, which in turn stabilized its financial support. among the 10 clubs included in the analysis, manchester city, which was purchased by a funding organization like in the case of psg, was not as efficient as psg according to the bio-mcdea model. in the 2008 season when manchester city was acquired, it was financially supported, similar to psg. however, manchester city has not been as successful as psg. the reason for this is that manchester city cannot dominate the premier league as psg dominates ligue 1, as 5 of the top 10 clubs at the top of the big-five league compete in the premier league. this suggests that financial support alone does not have an impact on success. in this study, a sensitivity analysis was performed to measure the sensitivity of efficiency measurement results according to different input/output combinations. each input and output was removed from the model, which was then resolved and the behavior of the model against the extracted variable was monitored. while a perfect correlation was found between the model created by subtracting the “the number of social media followers” input and the original bio-mcdea model, statistically significant correlations were observed between the model created by subtracting the “uefa club scores” output from the model and the bio-mcdea model. this means that while the “the number of social media followers” input makes almost no contribution to the model, the “uefa club scores” output provide relatively less contribution to the model than other inputs and outputs. no relationship was found between the original model and the bio-mcdea model created by excluding the “average number of viewers” and the “total market values” inputs and the “total revenues” output from the model. these variables were determined as the main determinants of the model. this suggests that the variables that contribute to the model are mainly financial variables. however, especially considering the use of social media in the 21st century, it is noteworthy that the “the number of social media followers” variable is not determinative in terms of the model. 7. conclusion the purpose of this study was to determine the efficiencies of the top 10 clubs in the big-five league, which make up the largest share of the world football industry. the analysis of efficiency for only 10 clubs can be counted among the limitations of this study. the reason for the inclusion of these 10 clubs in the efficiency review was that although the rankings of the clubs have changed, they are still in the top 10. the deloitte football money league report, from which most of the data in this study was obtained, publishes data on the top 20 clubs in terms of finance every year. while the clubs in the top 10 almost never change, the clubs in the last 10 can enter and exit the list. only 10 clubs were included in the analysis to obtain consistent data over the entire five years of the analysis. another limitation of this study was that the analysis was carried out with only quantitative data. however, this analysis could be supported arsu/decis. mak. appl. manag. eng. 4 (2) (2021) 106-125 120 by qualitative data obtained from football professionals including club managers, sponsors, etc. in future studies, the number of football clubs included in the analysis can be increased by using more resources and time, and the obtained quantitative findings can be supported by qualitative findings. the bio-mcdea model, which is an efficiency determination method based on linear programming, was used in the efficiency analysis. it can be said that using this model was the most obvious advantage of the study. the reason for selecting the biomcdea model was to prevent the low discrimination problem of classical dea. the findings of the study also included the results obtained with classical dea. when the classical dea findings were examined, it was concluded that a very high number of clubs were efficient. in this case, it will be difficult to distinguish between clubs. moreover, useful information for decision-makers cannot be obtained. "super efficiency" models can be used to determine which of the efficient clubs are more efficient. in this case, it will produce more complex results for both decision makers and analysts. in addition, the bio-mcdea model has easier solution steps compared to other methods such as mcdea and gpdea that aim to eliminate the low discrimination power problem of classical dea. another advantage of this study is that sports and financial data were used together. this is because financial success is to be used as leverage for sportive success. in this respect, instead of evaluating and associating financial and sports data separately, this study included both in the same model. as financial and sportive success can only be achieved through successful management practices, some managerial implications were made in line with the findings of the study. in order to examine the contribution of the criteria to the model, a sensitivity analysis was conducted in which each criterion was removed from the model, which was then solved again. according to the results of this analysis, the criteria of average audience number, total market value and total income were determined as the criteria that made the greatest contribution to the model. although the criteria for total market value and total income are direct financial criteria, the average number of viewers seems to be a non-financial criterion. this is because the matchday revenues are at the lower ranks among the revenue items of football clubs. however, bringing fans to the stadium does not only contribute to the clubs as ticket revenue but also to the sales in commercial products and to sponsors spending more on stadium advertisements. in addition, the fact that football clubs achieve more sportive success in the home field can be explained with the support of the fans. from this point of view, club management can implement various practices to make stadiums more attractive to the fans. among the practices that increase the attractiveness of stadiums are the club management selling tickets at lower prices, facilitating access to the stadium, and creating areas where families can spend time in the stadium. although the 10 clubs analyzed are not in the same league, they are constantly in competition as they participate in international tournaments every year. in order to keep competition alive, the financial resource must be sustainable. in order for the financial resource to be sustainable, clubs want to continuously participate in international tournaments, which are one of the most revenue generating elements of the industry. the financial benefits of a successful season will only benefit the club in the next season. although real madrid was champion in the champions league in the 2017-2018 season, according to the analysis conducted in this study, it was found to be an efficient club in the 2018-2019 season, not the 2017-2018 season. similarly, chelsea, which was champion in the european league in 2018-2019, was only found to be an efficient club in the 2019-2020 season. as these examples show, clubs can investigation into the efficiencies of european football clubs with bi-objective multi-criteria data envelopment analysis 121 only provide sustainable financial resources with sustainable sportive success. moreover, they can transfer talented players who can participate in international tournaments every year to make financial resources sustainable, or they can invest in their academies to produce their own qualified football players. clubs that make their financial resources sustainable are referred to as "big clubs". it can be said that achieving sportive success is easier for these clubs compared to other clubs, as big clubs are more advantageous in terms of attracting talented and qualified players. however, financial sustainability depends on a number of factors that are not constantly under control. for example, some penalties imposed by uefa in accordance with ffp policies harm the financial sustainability of clubs. in addition to the clubs' efforts to cope with the ffp limitations, the covid-19 pandemic, which emerged in the province of wuhan in china in december 2019 and spread all over the world in a short time, led to huge decreases in the revenues of the clubs. due to covid19, some countries have suspended their leagues for a long period of time, broadcasting agreements were interrupted and stadium revenues were not obtained. to avoid the effects of factors such as these that could harm financial sustainability, clubs sometimes turn to finding new sources of funding. for example, clubs may try to provide additional financing with initiatives such as the "european super league", which was established on april 19, 2021 and was dissolved after only 48 hours. however, for a sport whose rules and organizations are deeply rooted, such initiatives may cause clubs to disconnected from other clubs. therefore, in order to make the financial resource 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(2019). https://www.transfermarkt.com/spielerstatistik/wertvollstemannschaften/marktwertetop accessed 26 december 2019 uefa (2021). financial fair play. https://www.uefa.com/insideuefa/protecting-thegame/financial-fair-play/ accessed 2 may 2021. appendix 1. example of lindo codes for bio-mcdea model (manchester united 2019-2020 season) min 0.5m+0.5d1+0.5d2+0.5d3+0.5d4+0.5d5+0.5d6+0.5d7+0.5d8+0.5d9+0.5d10 subject to 127.2x1+74698x2+670.45x3=1 22000y1+711.5y2-127.2x1-74698x2-670.45x3+d1=0 17000y1+757.3y2-226.7x1-61040x2-913.75x3+d2=0 24000y1+840.8y2-216.5x1-76104x2-930.93x3+d3=0 36000y1+660.1y2-74.4x1-75865x2-777.33x3+d4=0 25000y1+610.6y2-62.9x1-54130x2-1048.6x3+d5=0 10000y1+445.6y2-69.7x1-59897x2-607.65x3+d6=0 31000y1+635.9y2-73.7x1-46911x2-874.15x3+d7=0 17000y1+513.1y2-82.2x1-40445x2-705.85x3+d8=0 18000y1+604.7y2-71.9x1-53053x2-1002.7x3+d9=0 22000y1+459.7y2-83.4x1-39101x2-661.88x3+d10=0 m-d1>=0 m-d2>=0 m-d3>=0 m-d4>=0 m-d5>=0 m-d6>=0 m-d7>=0 m-d8>=0 m-d9>=0 m-d10>=0 end © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 1, issue 2, 2018, pp. 131-152 issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame1802138e * corresponding author. e-mail addresses: hami.ebrahami@yahoo.com (h. ebrahimi), milos.tadic@gmail.com (m. tadic). optimization of dangerous goods transport in urban zone hamid ebrahimi1*, tadic milos2 1 lahore university of management sciences, lahore, pakistan 2 university of defence in belgrade, department of logistics, belgrade, serbia received: 18 february 2018; accepted: 9 september 2018; available online: 9 september 2018. original scientific paper abstract: due to the specificity of the transport of dangerous goods, as well as the obligations arising from the legislation regulating this field, all the actors of this process are obliged to take special measures in order to avoid undesired consequences. special attention is paid to the planning of the transport of dangerous goods. one of the most important planning elements is choosing a route for the transport of dangerous goods in urban areas. in order to take preventive measures, risk assessment is carried out on the routes and the minimum risk route is defined. in this paper, a new model for selection of the routes for the transport of dangerous goods (hazmat) on the network of urban roads is proposed. the model is based on a multi-criteria risk analysis and the traditional dijkstra algorithm (d-r model). the d-r model is a new approach for minimizing the cost and a variety of risk criteria in hazmat routing, which adequately takes into account and minimizes a number of risks on potential routes. the model is based on route selection based on the absolute risk size. the proposed routing model was tested in a real case and in a real urban hazmat routing problem, in serbia. key words: multi-criteria decision-making, hazardous materials routing, risk, dijkstra’s algorithm. 1 introduction in transport management, mitigation of the negative consequences of transport, especially those related to safety and environmental impact, is often emphasized. due to the harmfulness and the extent of the possible consequences, managing the transport of dangerous goods, especially in urban areas, is an issue gaining more and more attention. one of the main problems in managing the transport of dangerous goods is the problem of route selection. the problem of dangerous goods routing is manifested in numerous variations. the formulation of the problem depends on mailto:hami.ebrahami@yahoo.com mailto:milos.tadic@gmail.com ebrahimi & tadic/decis. mak. appl. manag. eng. 1 (2) (2018) 131-152 132 whether the selected is one route (between two nodes in the network) or multiple ones (in general, among multiple destinations), whether the parameters of the network are of a static or dynamic character, whether they are stochastic or deterministic, whether choosing the route is from a local or global perspective, etc. a large number of factors are involved in the process of solving this problem, and, consequently, solutions require numerous compromises. the essence of the compromise is reflected in the set of criteria for route selection that are present in the decision-making model. also, a major problem for decision-makers is the availability and reliability of the data that are needed for decision-making, as well as models of risk assessment in transport hazmat. the main objective of this paper is to propose a model that can serve as a useful tool for decision-making in planning hazmat transport routes in urban areas. with the model proposal that deals with the problem of hazmat rutting in a comprehensive way, with respect to both cost aspects and various risk aspects, as well as numerous uncertainties in the decision-making process, it is shown that academic research models can be more practical and useful for real hazmat routes planning. the rest of the paper is organized as follows. in addition to the introduction and conclusion, the paper is structured through three more chapters. in second chapter, a review of the literature with an emphasis on the application of the rutting models used for the transport of dangerous goods is given, while the third unit is a description of the model used in this paper. in the third chapter the dijkstra-risk (d-r) routing model algorithm is presented in detail. the fourth chapter presents the implementation of the d-r routing model in the real case of transporting dangerous goods in the ministry of defense. 2 literature review a large number of international studies have shown that the risk originating from mobile sources (vehicles transporting dangerous goods) has the same significance as the risk originating from fixed sources (ormsby& le, 1988; brockoff, 1992; vilchez et al., 1995; bonvicini and spadoni, 2005), so that it is necessary to reduce the size of the risk originating from mobile sources and keep it within the limits of acceptable values. a number of different methodologies have been developed in the literature for the selection of routes for the movement of vehicles transporting dangerous goods: from case studies that include risk analysis (bubbico at al., 2000; rao madala, 2000; milazzo et al., 2002; scenna& santa cruz, 2005; govan, 2005; wang et al., 2015), through studies where the choice of route is based on the data obtained from statistical analysis and research of a number of incident situations (fabiano et al., 2002; anderson &barkan, 2004; hamouda, 2004; ohtani& kobayashi, 2005), to solving the choice of a route through algorithms for routing vehicles (fu, 2001; bonvicini et al., 2002; akshay&prozz, 2004; zografos and androutsopoulos, 2004; bahar&verter, 2004; godoy, 2007; zografos, 2008; batarlienė, 2008; wang et al., 2015; androutsopoulos & zografos, 2010;pamučar et al., 2016). the methods that are very easy to use, that are understandable and with a high level of reliability of risk level determination have been developed by (rao et al., 2004; bubbico et al., 2004; huang, 2005; ghazinoory&kheirkhah, 2008): also, there are methods that are adapted to support decision-making process and are intended for spatial planning (spadoni et al., 2000; lin, 2001; gheorghe et al., 2005;jovanović, 2009). in the last ten years, special attention has been devoted to developing methodologies for determining the level of risk of transporting dangerous goods in tunnels; these methodologies have been optimization of dangerous goods transport in urban zone 133 developed by (oecd, 2001; saccomanno and haastrup, 2002; knoflacher, 2002; van den horn et al., 2006; kohl et al., 2006). from the above, it can be concluded that there are numerous methodologies developed with the aim of selecting a route for the movement of vehicles transporting dangerous goods from the aspect of risk management. the hybrid methodologies, which represent the application of a multi-criteria analysis in combination with the conventional routing models, in spite of their simplicity, have not been considered in the literature so far. this paper presents a new model named a d-r model for hazmat vehicle routing problem (hvrp) in urban zones based on the application of the dijkstra algorithm and the multi-criteria minimization of risk. one of the advantages of this model comparing to the existing ones lies in its complex consideration of a number of parameters which affect the risk of dangerous goods transport in urban areas. in this sense, in addition to the carrier's operating costs, as criteria for the convenience of routes for the transport of dangerous goods on the network of urban roads, six parameters which define the level of risk are considered: emergency response, environmental risk, risk of an accident, consequences of an accident, risk associated with infrastructure and risks of terror attack / hijack. a risk (r) value is introduced as a convenience measure for the transport of dangerous goods. by optimizing the routes for the transport of dangerous goods in urban areas with the help of the proposed model the safety of residents in urban areas is improved and the risk of accidents is reduced. in general, since in most models for solving the hazmat routing problem as criterion functions there are cost and / or risk functions that are related to randomness and uncertainty, here a soft computing approach is desirable, as it is desirable to use a more comprehensive set when selecting a route criteria. a comprehensive approach to the risk analysis when planning the route for the transport of dangerous goods adds a new value to the decision-making process and evaluates the problems associated with the urban hazmat routing. the second advantage of this model is its processing of group knowledge in the process of selecting vehicle routes since this model was formed on the basis of an expert knowledge base which stems from the heuristic management experience. the third advantage is the adaptability of the model, which is reflected in the possibility of adjusting the model depending on the specificity of a concrete problem, thus achieving risk management in an uncertain environment. 3 d-r routing model the d-r model is realized through two phases. in the first phase of the d-r model, a transport network is formed in the urban area and the input parameters (criteria) are identified, based on which the r values of the branch network are determined. defining r values of the branch network is done using the term (1) min 1 n j j j f y w   (1) where jy represents the value of the criterion for the observed network branch, jw represents the weighting coefficient of the optimization criteria, while n represents the total number of optimization criteria. the input parameters in expression (1) are presented through seven criteria that influence the definition of the r value of the transport network branch: the carrier's operating costs, emergency response, environment risk, risk of an accident, the ebrahimi & tadic/decis. mak. appl. manag. eng. 1 (2) (2018) 131-152 134 consequences of an accident, risk associated with infrastructure and risks of terror attack / hijack. as the output from the i phase of the d-r model, r values are obtained for each specific link of the transport network. after defining the r values on the network, in the second phase, using the dijkstra algorithm, the routes for the transport of dangerous goods are defined. the criterion function minimized by means of the dijkstra algorithm is the sum of the r values of the branch network that are on the routes. the routing model in urban zones is realized through the following steps: step 1 a network of roads is defined. within the network of roads, network nodes containing the customers to which dangerous goods are delivered are defined. step 2 input parameters of the adaptive neural network that influence the determination of r values on the branches of the transport network are identified. in the d-r model, seven parameters are set, representing the aggregated value of costs and risks during the transport of dangerous goods in urban areas. step 3 input parameters are calculated (, j = 1, 2,…7), expression (1), for each branch of the transport network. this defines r values for all branches of the observed transport network. step 4 using the dijkstra algorithm, the routes for the transport of dangerous goods in urban areas are designed. 3.1 criteria for minimizing risk in the d-r model as stated in the previous chapter, seven criteria are identified on the basis of which r values are determined on the observed transport network (table 1). the selection of criteria and their indicators was carried out on the basis of the recommendations of pamučar et al., (2016) research and expert recommendations. table 1. criteria for defining r values on the transport network of urban roads no. criteria criterion description c1 costs of transport transport costs are proportional to the values of the variables: travel time, distance, fuel costs, etc. the value of the criterion is presented as the length of the branch expressed in kilometers (km). c2 emergency response in the event of an accident emergency response is the time for which city services (fire services and emergency services) react in the event of an accident. the average response time is taken as an input parameter, which is determined based on the distance of these services from the middle of the branch network. the value of this criterion is expressed in minutes (min). c3 environment risk it is determined based on the number of sensitive areas of the environment (water surfaces, green areas) located in the branch belt. the branch belt is defined as a critical area that can be contaminated in the event of an accident. the width of the branch belt depends on the type of dangerous goods and covers an area of 800 meters from the branch. the value of this criterion is determined on a scale of 1-10, where the value 1 represents a very small number of optimization of dangerous goods transport in urban zone 135 no. criteria criterion description sensitive areas, and the value of 10 is a very large number of sensitive areas. c4 risk of traffic accidents the risk of a traffic accident is defined based on the number of traffic accidents (f1) at the branch in the last 10 years, the number of traffic lanes (f2), the percentage of freight vehicles in the traffic flow (f3) and the signaling (f4). the total risk of an accident (x) is obtained using the expression x = 0.3 f1 + 0.2 f2 + 0.2 f3 + 0.3 f4. the values of f1, f2, and f3 are presented with quantitative indicators, while the quantification of indicators f4 is done using the scale: 1 traffic rules, 2 traffic signal regulation, 3 –light signals regulation. the value of this criterion is determined on a scale of 1-10, where value 1 represents a very small risk of a traffic accident, and the value of 10 is a very high risk of a traffic accident. c5 implications for the population in the event of an accident it is represented by the number of inhabitants (affected population) living in the belt of the branch. the belt of the branch is space of 800 meters from the branch. the value of this criterion is determined on a scale of 1-10, where value 1 represents a very small number of affected population, and the value of 10 is a very large number of affected population. c6 infrastructure and important facilities risk infrastructure and important facilities risk is the number of important infrastructure facilities in the branch belt (railways, electrical installations, industry, business and transport facilities, schools, hospitals, historic buildings, official buildings). the value of this criterion is determined on a scale of 110, where the value 1 represents a very small number of infrastructure objects, and the value of 10 a very large number of infrastructure objects. c7 the risk of a terrorist attack the risk of a terrorist attack is an assessment of the threat to the branch as a potential site of a terrorist attack, with the aim of endangering the population, significant infrastructure facilities and vulnerable areas of the environment. the risk is proportional to the significance of potential objectives in the branch belt. the value of this criterion is determined on a scale of 1-10, where value 1 represents a very small risk of a terrorist attack, and the value of 10 is a very high risk of a terrorist attack. weight coefficients ( j w ). the weight criteria of these criteria are defined by interviewing experts. in the next section of the paper, a model for estimating the reliability of the results and its application in this study is presented. the significance of the criteria was determined using the 1-10 scale, where 1 is a little important and ebrahimi & tadic/decis. mak. appl. manag. eng. 1 (2) (2018) 131-152 136 10 is a very important criterion. the results of the survey of experts are shown in table 2. table 2. weight coefficients of the criteria no. criterion middle value weight coefficient 1. costs of transport 6.2 0.109 2. emergency response in the event of an accident 8.7 0.153 3. environmental risk 9.1 0.160 4. risk of traffic accidents 9.2 0.162 5. implications for the population in the event of an accident 9.5 0.168 6. infrastructure and important facilities risk 8.1 0.143 7. the risk of a terrorist attack 5.9 0.105 the final values of weight coefficients have been normalized using additive normalization. an example of the calculation of the final value of the weight coefficient for the criteria "transport costs" is shown in the following expression 1 1 7 1 6.2 8.7 9 9 .1 9.2 9.5 8 . 6.2 0 0 .1 . 5 1 9 j j x w x            where 1x represents the mean value of the criteria transport costs, while the 7 1 j j x   represents the sum of the median value of all the criteria obtained by interviewing the experts. similarly the weight criteria for the remaining criteria were obtained, table 2. 3.2 dijkstra algorithm dijkstra (1959) has developed one of the most efficient and most used algorithms for determining the shortest paths from one node to all other nodes in the network. this algorithm presents a special case of the exposed generic algorithm. in the dijkstra's algorithm, a node i corresponding to the minimum value of the shortest known path is removed from the list of candidates v in each iteration. step 1 in the first step it is necessary to determine the initial node in the network. in the model presented in this paper, the initial node in the network is defined in advance and represents the location of the clc. we begin the process from node l . since gp from node l to node l is equal to zero we assign the initial node with 0p lg  . we give predecessor node l the symbol +, and so lq   (where iq is the node in front of node i, at the shortest distance from node l to node i). step 2 since the paths from node l to all of the remaining nodes are for now undiscovered, we designate them temporarily as ,p lig   for i l . since i precursor nodes to nodes i l are unknown on the shortest paths we designate them iq   for optimization of dangerous goods transport in urban zone 137 all i l . the only node currently in a closed state is node l . therefore, we can say that c l . step 3 in order to transform some of the temporary designations into actual ones, it is necessary to examine all of the branches (c,i) coming out of the last node that is in a closed state (node c). if node i is in a closed state, then examination of the next node begins. if node j is in an open state, we obtain its designation as an euf vehicle on the basis of the relation   , , ,max , ,p cj p j p ac pg g g g c j  (2) if node j is in an open state, we obtain its designation on the basis of the relation   , , ,min , ,p cj p j p ac pg g g g c j  (3) step 4 to determine which node is next to move from an open to a closed state, the size of all of the nodes in an open state is compared. we choose the node with the lowest size value gp. let it be node j. node j passes from an open to a closed state, since there is no value of gp from a to j that is less than ,p ajg (4). the link performance through any other node would be higher.  , ,maxp aj p ajg g (4) step 5 since the next node which passes from an open to a closed state is node j we determine the predecessor node for node j, on the shortest path which leads from node a to node j. the performances of the links of all of the branches (i,j) which lead from the nodes in a closed state to node j are tested until we determine that the relation is fulfilled (5)  , , ,p ai p aj pg g g i j  (5) let this relation be fulfilled for node t. this means that node t, the predecessor node to node j, is on the shortest path that leads from node a to node j. this means that we can say that iq t . step 6 if all the nodes in the network are in a closed state, then we have finished with the process of finding the optimal routes for vehicles. if there are still any nodes that are in an open state, then we go back to step 3. 4 testing of the d-r model for dangerous goods routing in urban zones the model has been tested in the case of the transport of dangerous goods for the needs of the ministry of defense of the republic of serbia. the transport of dangerous goods was considered on the route: the vasa čarapić barracks warehouse – knic warehouse of propulsion assets (leskovac) and return to the knic warehouse of propulsion assets (leskovac) – the vasa čarapić barracks. the transport of dangerous goods is carried out in both directions, which additionally complicates the set task. by looking at the road networks and determining possible road directions for the realization of the assigned task, it comes to the road network that is shown in figure1. ebrahimi & tadic/decis. mak. appl. manag. eng. 1 (2) (2018) 131-152 138 figure 1. display of the road network for the realization of the task display of the road network for realization based on figure1, important knots and branches related to the city zones of the cities of kragujevac and belgrade cannot be seen, so these zones need to be shown separately. figure 2 shows the road network for the city of kragujevac. figure 2. display of the road network of the city of kragujevac optimization of dangerous goods transport in urban zone 139 the same thing has to be done for the city zone of belgrade. the enlarged view is shown in figure 3. figure 3. display of road network of the city of belgrade for a simpler view of the transport network, a schematic representation of all nodes and branches of the road network shown in figures 1, 2 and 3 is shown in figure 4. the schema is not in ratio but only shows the transport network and the connection of the nodes on it. the transport network in figure 4 was used to solve the dijkstra algorithm. 1 2 3 4 6 5 7 98 101112 1718 16 15 14 13 1921 2022 23 26 2425 27 ebrahimi & tadic/decis. mak. appl. manag. eng. 1 (2) (2018) 131-152 140 figure 4. the network where it is necessary to determine the optimal route for the transport of dangerous goods 4.1 evaluation of the transport network branch determination of the value of the branch was made on the basis of the criteria described in the previous chapter. for each branch, the values of the criteria are individually determined in the following way: criterion k1 (transport costs) is determined on the basis of the length of the branch and is expressed in kilometers. the k2 criterion (emergency response in the event of an accident) was determined on the basis of the proximity of the branch from the emergency services and is expressed in minutes. -the criterion k3 (environmental risk) was determined on the 1-9 scale in the following way: the values 1 and 2 were assigned to city zones in which there are few green areas, 3, 4, and 5 were assigned to urban and populated areas in which there are green areas, 6 and 7 were assigned to zones in which the branch of large length stretches along the agricultural land or next to a protected property, 8 and 9 were assigned to zones in which the branch passes by or across rivers and lakes, and often in combination with green areas and agricultural land. the k4 criterion (risk of a traffic accident) is determined on the basis of road characteristics that directly affect the safety of traffic and the possibility of a traffic accident. it was determined on the 1-9 scale in the following way: the values 1, 2, and 3 were assigned to freeways and roads without curves, the values 4, 5, and 6 were assigned to roads with multiple crossing points, traffic roundabouts, curves and intensive traffic, value 7, 8 and 9 were assigned to road directions with many curves, poor road transparency, high-intensity traffic and travel loops. the k5 criterion (consequences for the population in the case of an accident) is determined based on the number of inhabitants living near the branch. it is determined on the 1-9 scale in the following way: the values 1, 2, and 3 were assigned to branches that pass through uninhabited and poorly populated places, the values 4, 5, and 6 were assigned to the branches that pass through villages and suburban zones, values 7, 8 and 9 were assigned to branches that pass through urban settlements. criterion k6 (infrastructure and important facilities risk) is determined based on the number of infrastructure and important facilities located near the branch. it was set on the 1-9 scale in the following way: the values 1, 2, and 3 were assigned to branches in the vicinity of not many important objects, the values 4, 5, and 6 were assigned to the branches in the vicinity of infrastructural objects of minor importance (smaller factories, ambulances), values 7, 8 and 9 were assigned to branches in the vicinity of large plants, factories, schools, hospitals, embassies, state facilities. the k7 criterion (the risk of a terrorist attack) is directly related to the number of infrastructure and important facilities. it was set on the 1-9 scale in the following way: the values 1, 2 and 3 were assigned to branches that go through smaller urban areas, the values 4, 5, and 6 were assigned to the branches in the vicinity of tourist sites, police stations, hospitals, schools, the values 7, 8 and 9 were assigned to branches in the vicinity of tourist sites, embassies, state buildings, factory plants, military facilities, institutions, etc. the values of the criteria by branches are shown in table 3. optimization of dangerous goods transport in urban zone 141 table 3. displaying the value of the criteria by branch network branch k1(km) k2(min) k3-(1-9) k4-(1-9) k5-(1-9) k6-(1-9) k7-(1-9) (1,2) 10.90 12 2 5 3 1 2 (2,3) 13.70 9 3 3 6 6 4 (3,4) 4.30 5 5 6 9 8 8 (3,6) 3.90 5 6 7 9 8 8 (4,6) 1.50 3 5 6 9 9 8 (6,7) 1.20 5 5 4 8 6 5 (4,5) 1.10 8 7 3 7 6 4 (5,7) 2.00 10 4 3 6 5 3 (5,9) 102.00 18 2 2 2 2 2 (24,26) 27.00 17 4 5 6 4 4 (8,9) 9.50 15 4 5 5 3 2 (9,10) 22.40 10 3 3 5 4 3 (8,11) 35.80 14 4 7 6 4 4 (10,11) 3.80 10 3 3 3 3 2 (10,17) 10.90 10 4 5 8 6 6 (17,18) 5.70 8 6 6 9 9 9 (16,17) 1.70 8 5 7 8 8 8 (16,18) 1.20 7 6 6 9 9 8 (18,19) 0.28 6 6 6 9 9 8 (15,16) 2.30 10 7 4 9 6 6 (14,15) 0.55 6 5 5 9 5 5 (14,20) 0.29 3 3 4 8 8 8 (19,20) 0.60 2 4 4 7 9 9 (19,21) 0.45 5 5 4 9 9 9 (21,22) 0.60 5 5 4 9 9 9 (20,22) 0.45 4 4 4 7 9 9 (11,15) 7.20 10 4 5 7 5 5 (11,12) 7.10 14 5 7 7 6 6 (12,13) 3.00 14 5 8 9 6 5 (13,21) 1.60 8 6 9 9 8 8 (13,14) 2.20 9 4 7 9 7 6 (2,23) 36 20 9 9 7 7 9 (23,25) 38 20 9 8 7 6 4 (25,27) 31 20 9 9 6 5 5 (12,27) 45.3 19 7 8 9 6 4 (23,24) 28.1 20 9 9 6 5 3 (24,25) 21.5 20 8 7 9 6 4 (24,27) 47.9 20 9 9 6 5 3 (7,26) 36.6 20 7 5 7 6 4 (8,26) 31.8 20 9 7 6 4 5 ebrahimi & tadic/decis. mak. appl. manag. eng. 1 (2) (2018) 131-152 142 by normalizing the values shown in table 3 the values of the comparable nondimensional size on the basis of which they are calculated are obtained, the expression (1), the final value of the branches, and the total value of the risk. the normalization of the value of the criterion was made using the percentage normalization, i.e. by dividing the values of the criteria with the highest value of the observed criterion. table 4 shows the normalized values of the criteria and the value of each branch is determined using the expression (1), min 1 1 2 2 7 7...f w y w y w y    ; where minf represents the final value of risk on the branch, 1 2 7, ...w w w represent the weight coefficients of the criteria, while jy represent the normalized values of the criteria for the observed network branch. table 4. normalized branch network values branch mark criterions k1 (0.109) k2 (0.153) k3 (0.160) k4 (0.162) k5 (0.168) k6 (0.143) k7 (0.105) σ (1,2) 0.107 0.600 0.222 0.556 0.333 0.111 0.222 0.32 (2,3) 0.134 0.450 0.333 0.333 0.667 0.667 0.444 0.48 (3,4) 0.042 0.250 0.556 0.667 1.000 0.889 0.889 0.57 (3,6) 0.038 0.250 0.667 0.778 1.000 0.889 0.889 0.65 (4,6) 0.015 0.150 0.556 0.667 1.000 1.000 0.889 0.64 (6,7) 0.012 0.250 0.556 0.444 0.889 0.667 0.556 0.54 (4,5) 0.011 0.400 0.778 0.333 0.778 0.667 0.444 0.53 (5,7) 0.020 0.500 0.444 0.333 0.667 0.556 0.333 0.43 (5,9) 1.000 0.900 0.222 0.222 0.222 0.222 0.222 0.43 (24,26) 0.265 0.850 0.444 0.556 0.667 0.444 0.444 0.49 (8,9) 0.093 0.750 0.444 0.556 0.556 0.333 0.222 0.45 (9,10) 0.220 0.500 0.333 0.333 0.556 0.444 0.333 0.44 (8,11) 0.351 0.700 0.444 0.778 0.667 0.444 0.444 0.49 (10,11) 0.037 0.500 0.333 0.333 0.333 0.333 0.222 0.39 (10,17) 0.107 0.500 0.444 0.556 0.889 0.667 0.667 0.53 (17,18) 0.056 0.400 0.667 0.667 1.000 1.000 1.000 0.68 (16,17) 0.017 0.400 0.556 0.778 0.889 0.889 0.889 0.63 (16,18) 0.012 0.350 0.667 0.667 1.000 1.000 0.889 0.69 (18,19) 0.003 0.300 0.667 0.667 1.000 1.000 0.889 0.67 (15,16) 0.023 0.500 0.778 0.444 1.000 0.667 0.667 0.64 (14,15) 0.005 0.300 0.556 0.556 1.000 0.556 0.556 0.51 (14,20) 0.003 0.150 0.333 0.444 0.889 0.889 0.889 0.54 (19,20) 0.006 0.100 0.444 0.444 0.778 1.000 1.000 0.54 (19,21) 0.004 0.250 0.556 0.444 1.000 1.000 1.000 0.62 (21,22) 0.006 0.250 0.556 0.444 1.000 1.000 1.000 0.62 (20,22) 0.004 0.200 0.444 0.444 0.778 1.000 1.000 0.55 (11,15) 0.071 0.500 0.444 0.556 0.778 0.556 0.556 0.50 (11,12) 0.070 0.700 0.556 0.778 0.778 0.667 0.667 0.59 (12,13) 0.029 0.700 0.556 0.889 1.000 0.667 0.556 0.65 (13,21) 0.016 0.400 0.667 1.000 1.000 0.889 0.889 0.70 (13,14) 0.022 0.450 0.444 0.778 1.000 0.778 0.667 0.65 optimization of dangerous goods transport in urban zone 143 branch mark criterions k1 (0.109) k2 (0.153) k3 (0.160) k4 (0.162) k5 (0.168) k6 (0.143) k7 (0.105) σ (2,23) 0.353 1.000 1.000 1.000 0.778 0.778 1.000 0.82 (23,25) 0.373 1.000 1.000 0.889 0.778 0.667 0.444 0.79 (25,27) 0.304 1.000 1.000 1.000 0.667 0.556 0.556 0.74 (12,27) 0.444 0.950 0.778 0.889 1.000 0.667 0.444 0.79 (23,24) 0.275 1.000 1.000 1.000 0.667 0.556 0.333 0.71 (24,25) 0.211 1.000 0.889 0.778 1.000 0.667 0.444 0.79 (24,27) 0.470 1.000 1.000 1.000 0.667 0.556 0.333 0.72 (7,26) 0.359 1.000 0.778 0.556 0.778 0.667 0.444 0.75 (8,26) 0.312 1.000 1.000 0.778 0.667 0.444 0.556 0.67 a schematic representation of the transport network with the previously calculated values is shown in figure 5. ebrahimi & tadic/decis. mak. appl. manag. eng. 1 (2) (2018) 131-152 144 1 2 3 4 6 5 7 98 101112 1718 16 15 14 13 1921 2022 0.32 0.48 0.57 0.65 0.64 0.53 0.54 0.43 0.43 0.45 0.49 0.44 0.39 0.53 0.59 0.65 0.510.65 0.54 0.70 0.50 0.64 0.69 0.63 0.68 0.67 0.62 0.54 0.55 0.62 2624 23 25 27 0.75 0.67 0.49 0.82 0.79 0.71 0.79 0.72 0.74 0.79 figure 5. display of transport network with branch values 4.2 application of the dijkstra's algorithm to calculating the optimal route using the dijkstra algorithm described in section 3.4 of this paper the shortest paths from node 1 to all other nodes in the network are calculated. since the values of the transport network branch are the risk calculated using the criteria determining the shortest paths from node 1 to all other nodes, an optimal route (the safest) for the transport of dangerous goods will be obtained. on the given transport network, node 1 is the warehouse of propulsion assets of clob "knic" (leskovac), and node 22 is the barrack vasa čarapić. by determining the shortest route between these two nodes, an optimum route for the transport of dangerous goods is obtained. optimization of dangerous goods transport in urban zone 145 the process of searching for the shortest paths starts from node 1. since the length of the shortest path from node 1 to node 1 is equal to 0, that is 1,1 0d  . the precursor to the starting node 1 is indicated by the + symbol, therefore 1q   . the lengths of all the shortest paths from node 1 to all other nodes 1i  for now are unexplored, and that is why it is for all other nodes 1i  putted that 1,id  . since the nodes are the precursors to the nodes 1i  on the shortest paths it is putted iq   for all 1i  . the only node that is currently closed is node 1. that's why it is 1c  . in addition to the labels of the node 1 the sign  0, the sign‘ is placed to indicate that node 1 is in a closed state. this completes the first step of the algorithm. in the second step of the algorithm, the lengths of all branches that come out of node 1 that is in a closed state are examined. it follows that:  1,2 min , 0 0,32d    , i.e. 1,2 0,32d  . in the third step, since the branch (1, 2) is the only branch leaving node 1, this means that the next node that goes into the closed state is node 2. since it is 1,2 1,1(1, 2) 0,32 0,32 0d d d     , it follows that in the fourth step, node 1 is precursor to node 2 on the shortest path, that is, 2 1q  . in the fifth step, it can be noticed that there are still nodes in the transport network that are in an open state, so the second step is repeated according to the algorithm. the last node that is in a closed state is node 2, which means that 2c  . by examining all branches that go from node 2 to nodes in the open state, it follows that:      1,3 1,2min , (2,3) min , 0,32 0, 48 min , 0,8 0,8d d d              1,23 12min , (2, 23) min , 0,32 0,82 min ,1,14 1,14d d d         since it is 1,3 1,23d d , this means that node 3 goes from an open to a closed state. also, since it is: 1,3 1,2(2,3) 0,8 0, 48 0,32d d d     , this means that node 2 is the node-precursor of node 3, i.e. that 3 2q  . in the fifth step after the second pass through the algorithm, it is determined that there are still open nodes on the transport network and, therefore, the algorithm is repeated. in the third pass through the algorithm follows:      1,4 1,3min , (3, 4) min , 0,8 0,57 min ,1,37 1,37d d d              1,6 1,3min , (3, 6) min , 0,8 0, 65 min ,1, 45 1, 45d d d         1,23 1,14d  , so it is  1,23 1,23 1,4 1,6min , , 1,14d d d d  , and 1,23 1,2(2, 23) 1,14 0,82 0,32d d d     ; then it follows that node 2 is the node precursor for node 23 in the shortest path, so it is 1,2 2q  , which means that the next node that goes to the closed state is node 23. in the last 26th pass, we got the following results:      1,20 1,14min , (14, 20) min , 4,17 0,54 min , 4, 71 4, 71d d d         ,      1,21 1,13min , (13, 21) min , 4, 01 0, 7 min , 4, 71 4, 71d d d         , ebrahimi & tadic/decis. mak. appl. manag. eng. 1 (2) (2018) 131-152 146       1,22 1,21 1,20min (21, 22), (20, 22) min 4, 71 0, 62, 4, 71 0,55 min 5,33,5, 26 5, 26 d d d d d        so it is 1,22 1,20(20, 22) 5, 26 0,55 4,71d d d     , and from this it follows that node 20 is the node-precursor of node 22 on the shortest path, so it is 1,20 22q  , which means that the next node that goes into the closed state is node 22. after 26 passes it can be determined that there are no open nodes on the network, which means that the algorithm is finished. the shortest paths are displayed in figure 6. 1 2 3 4 6 5 7 98 101112 1718 16 15 14 13 1921 2022 0.32 0.48 0.57 0.65 0.53 0.54 0.43 0.45 0.44 0.39 0.53 0.65 0.51 0.54 0.70 0.50 0.63 0.68 0.67 0.55 2624 23 25 27 0.49 0.82 0.79 0.71 0.72 0.79 figure 6. display of the shortest paths from node 1 to all other nodes the optimal route for the transport of dangerous goods is: 1-2-3-4-5-9-10-1115-14-20-22. the total value of the risk on the optimal route is obtained using the following expression:       1,22 1,21 1,20min (21, 22), (20, 22) min 4.71 0.62, 4.71 0.55 min 5.33,5.26 5.26 d d d d d        optimization of dangerous goods transport in urban zone 147 4.3 analysis of the obtained result the d-r model sets the minimum risk values for transporting dangerous goods from node 1 to all other nodes. the optimal route for the transport of dangerous goods is: the barrack vasa čarapić bulevar jna jajinaci – bubanj potok e-75 – batočcina kragujevac leskovac. in return, the same route was used. in the ministry of defense this task has been solved in a different way. the route for transporting dangerous goods in the rural areas is the same as the optimal route obtained in the operation. the difference between the routes is in the city zone of belgrade. in the urban zone, the criteria that are either not considered in practice or are not given enough importance come to the fore. for these reasons, in practice, most often there are mistakes when choosing a route for the transport of dangerous goods. the difference between the route obtained by the dr model and the route used in practice is best seen in the schematic representation, figure 7. in figure 7, the red color indicates the route in which the transport of dangerous goods is carried out in practice, while the blue color presents the optimal route for transport dangerous goods obtained by the dr model. the risk on the route used for the transport of dangerous goods in the ministry of defense is: min (1, 2) (2,3) (3, 4) (4,5) (5,9) (9,10) (10,17) (17,18) (18,19) (19, 21) (21, 22) 0.32 0.48 0.57 0.53 0.43 0.44 0.53 0.68 0.67 0.62 0.62 5.89 f d d d d d d d d d d d                         while the risk in the d-r model is represented by the following term min (1, 2) (2,3) (3, 4) (4,5) (5,9) (9,10) (10,11) (11,15) (15,14) (14, 20) (20, 22) 0.32 0.48 0.57 0.53 0.43 0.44 0.39 0.50 0.51 0.54 0.55 5.26 f d d d d d d d d d d d                        it is evident that the risk of the route used in practice is higher than that of the route obtained by applying a routing model for  100 1 10.7 %dijk vsx x x      this means that the solution obtained by the d-r model is significantly safer for the transport of dangerous goods than the one used in practice. ebrahimi & tadic/decis. mak. appl. manag. eng. 1 (2) (2018) 131-152 148 55 1 2 3 4 6 5 7 98 1112 17 16 15 14 13 21 2022 0.32 0.48 0.57 0.65 0.64 0.53 0.54 0.43 0.43 0.45 0.49 0.44 0.39 0.53 0.59 0.65 0.510.65 0.70 0.50 0.64 0.69 0.63 0.68 0.67 0.62 0.54 0. 0.62 2624 23 25 27 0.75 0.67 0.49 0.82 0.79 0.71 0.79 0.72 0.74 0.79 0.54 19 18 10 figure 7. comparing the used and the optimal transport routes 5 conclusions the paper presents a new approach to the application of the dijkstra algorithm and the multi-criteria model in solving urban hvrp. the multi-criteria model was used to determine r values when transporting dangerous goods on urban roads. the authors’ opinion is that this new approach to hazmat routing (d-r model) represents a optimization of dangerous goods transport in urban zone 149 qualitative move towards improving the methodology of routing dangerous goods in urban zones. the proposed d-r model extends the theoretical framework of knowledge in the field of dangerous goods routing. the problem of routing dangerous goods is considered by the new methodology and thus forms the basis for further theoretical and practical upgrading. also, the presented model highlights the multiple aspects of the risk assessment on the network of roads that have not been unified in the models so far, and they are important for this issue. by introducing and combining those with the criterion of operational transport costs, what is stressed is the need for a more versatile approach in further analysis of hazmat vehicle routing and similar problems. the proposed d-r model has three main advantages over other methods. firstly, it can reflect a variety of decision-making criteria in times of need. the system has the ability of adaptability, which is reflected in the ability to adjust the weight of the criteria depending on the problem under consideration. secondly, it can be implemented as a computer-based system and, therefore, it supports a dynamic decision-making process in hazmat routing. thirdly, the proposed model allows for relatively fast and objective estimations of cost and risk factors in hazmat transport under the conditions of a changing environment. the direction of future research should move towards the identification of additional parameters that influence the identification of risks on the network of urban roads and the implementation of additional decision criteria in the proposed model. in this sense, the methods of fuzzy linear and dynamic programming in combination with heuristic and metaheuristic methods find their place of application. one of the recommendations is the consideration of the strategy of scheduling vehicles that transport different quantities of dangerous goods to selected routes, using genetic algorithms, while defining the limits that are considered with fuzzy linear programming and visualizing the solutions obtained using the geographic information system. acknowledgments: 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(2015). bayesian network-based risk assessment for hazmat transportation on the middle route of the south-to-north water transfer project in china, stochastic environmental research and risk assessment, 30 (3), 841–857. http://link.springer.com/journal/477 http://link.springer.com/journal/477 ebrahimi & tadic/decis. mak. appl. manag. eng. 1 (2) (2018) 131-152 152 zografos, k. (2008). a decision support system for integrated hazardous materials routing and emergency response decisions, transportation research board part c, 16, 684-703. zografos, k., androutsopoulos, k. (2004). a heuristic algorithm for solving hazardous materials distribution problems, european journal of operational research, 152 (2), 507-519. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 6, issue 1, 2023, pp. 1-17. issn: 2560-6018 eissn:2620-0104 doi:_ https://doi.org/10.31181/dmame181221015k * corresponding author. e-mail addresses: fayzal.phd@nitap.ac.in (m. kabir), sahadevroy@nitap.ac.in (s. roy) hazard perception test among young inexperienced drivers and risk analysis while driving through a t-junction md faysal kabir1* and sahadev roy1 1department of electronics & communication engineering, national institute of technology arunachal pradesh, jote, india received: 2 july 2021; accepted: 24 october 2021; available online: 19 december 2021. original scientific paper abstract: in this article, the hazard-based accident duration model and the reaction time due to various types of distractions are considered. this analysis is mainly considered t-junction or t-junctions like all other forms of road geometries which are more prone to accident. the model proposed is constructed by using a electronics devices along with driving simulator, to study the behavior of the yang inexperienced drivers as well as an experienced driver about their reaction by calling their phone from unknown numbers intentionally. the results show that drivers distracted by mobile phones uses the hard breaking due to least available time to respond after identification of an event. some of the research and theory bearing on decision making and risk perception, driver situation awareness, and possible mediators of risk-taking is also analyzed here. key words: accident analysis, decision making, driver situation awareness, intersection design, modeling of driver reaction time, t-junction, traffic accidents. 1. introduction traffic accidents are the single most common cause of injury (dewan et al. 2016) and death among young people in all over the world (kabir & roy, 2020). traffic signs and signals are necessary to meet the specifications set forth all over the country, and may differ from one another like driving lane of india and usa. federally standardized signals and road signs provide pedestrians and drivers with the clear instructions sets they need to ensure minimum hazard on the highways. these instructions include about the sign and picture is made of, its color, and its shape (fang et al. 2003). benbassat, et al. (2019) performed about twenty-seven ergonomics and human factors considered from 10 countries for analyzed the conventional sign relative to their expectation with the conventional sign with the same meaning. for example, diamondkabir et al./decis. mak. appl. manag. eng. 6 (1) (2023) 1-17 2 shaped signs indicate a warning, whereas a circular sign with crossing indicates a railroad crossing (almutairy et al. 2019). this is not always case, in the europe countries warning is indicated by the signs with the triangle shape, and circular signs with white background and red bordure can indicate some limitation (e.g. speed limit) or can be used as information sign if it have circular shape, bluer background and white symbol. the color red is generally used dedicatedly for danger signs and lights, like the yield sign and the stop sign. yellow indicates a warning message and cautionary message of some sort. green with or without arrow provides directional guidance. pavement markings like zebra crossing, side line, middle line, etc. provide pedestrians and drivers with information in the similar fashion that road signs do. this standardization and frequent use of symbols may be useful for those who can’t understand the local language of the territory where they’re driving, even for drivers who are color-blind. there have been a number of reviews published in the last few years which have focused on accident risk (eboli et al. 2020) and the possible causal factors underlying it (schlögl, 2020). along with geometric characteristics (demasi et al., 2018), road surface conditions (fancello et al. 2019), traffic status and driver behavior (chang et al. 2019), weather conditions (drosu et al. 2020) are among the most significant variables along with other causes (mukherjee & mitra, 2020) potentially influencing the incidence of highway crashes. consequently, accidents due to the presence of different junctions and traffic on road have been in the focus of research efforts for several decades. traffic disruption is another possibility, and is a rising concern amongst legislators. much analysis and interest in this field is linked to driver disruption, due in large part to the growing usage of cell phones and other technology by drivers (backer-grøndahl & sagberg, 2011). the attention is about the mobile phone usage by drivers, highlighting the significance of the existing study, even though it is vital to recognize that many other disruption factors are also hazard triggers. having this in view, the use of cell phones could be seen as an example of the wider driver interruption issue. driver disturbance continues an ambiguous and poorly described concept and may define as a diversion from tasks which are essential for driving safety to a competitive activity (shahzad, 2020). most of the research work that has already been published is basically based upon the braking behavior of the distracted drivers due to the usage of both hand held as well as hand free mobile phones during driving (mcevoy et al. 2006). the main objective is to study the behavior of the drivers due to the combined effects of mobile phone usage like tjunction, yjunction, 4 arms junction, staggered junction and round shape junction. in order to design the proposed model effects of some external variables on the result has also been considered, such as speed of the vehicle, driving experience of the drivers, age and sex of the driver, duration and type of mobile phone usage during driving etc. to carry out this study a group of young distracted drivers within the age group of 18-25 years has been exposed to a t-junction where they interacts with incoming traffic from the other side of the road while driving. this study represents detailed analysis about the impacts of mobile phone conversation during driving through a t-junction as well as measures the changes in the reaction time required by the drivers to stop the vehicle after detection of an event. the aim of this study is also to examine the reaction time (rt) taken by the drivers to stop the vehicle after detection of an event. to realize the effect of hazard exposure is on the driver situation awareness and effect with age and roadway complexity, the experiments done with the help of young (18–25 years), middle (26-35 years) and older (above 35 years) hazard perception test among young inexperienced drivers and risk analysis while … 3 participants drove in a simulator under rural and city environments. the studies investigated driver behavior using assistive device in yield-controlled t-junctions based on the reaction time. to identify the unique elements of the risk factors associated with fatal t junction crashes. 2. research motivation accidents caused by distraction results in more number of injuries. distraction can be various types such asdistraction caused by mobile phone conversations, typing during driving, looking the road side peoples etc. while driving, typing in mobile phones has more effect on drivers' actions as typing takes drivers' eyes off the road for more time than when the drivers engage in a call. young drivers are more susceptible to distraction-related collisions, because the younger generation is more susceptible to use cell phones while driving than older ones. as can be seen from the literature review section, most of the research work based on the disruptions caused by cell phones has been conducted in developed countries like the us,, canada, china, korea etc. despite being the second populous country in the world, only very few research work is carried out in developing countries such as india on the basis of disruptions caused by cell phones or other factors. this paper explores the effects of different types of distracted driving on accident severity and reaction time in indian roads, particularly on rural roads. the current literatures as most of the studies are focused on disturbances caused by different forms of cell phone uses while driving (such as chat, text or multimedia). in addition, the present study is performed on several simulator-designed t-junctions, as this type of junction is the most dangerous to drive between all other types of junctions. in addition, a comparative study of distracted and non-distracted driving and its effects on reaction time and braking activity is also presented in this work. furthermore, in this work the impacts on the braking time due to the specific duration of cell phone usage and the amount of drug consumption while driving are also considered. although some significant research work were conducted on the distractions caused by mobile phone usage during driving, the study findings explaining the cumulative impact of drug consumption and cell phone use are scarce available. the probability of drugs intake at night is unexpectedly high in comparison with day time (das et al. 2012). several researchers were conducted over the mobile phone usage during driving that describes the variation in the reaction time due to distracted driving in comparison with the non-distracted driving. it is found that both the conversation and texting during has severe impact on the reaction time, as both conversation and texting distracts driver's focus from driving. accident at the intersections is another major cause of road accidents. the standard shapes of t un-signalized intersection and four-leg un-signalized intersections there are on the fig 1 (highway capacity manual , 2000) to flow the traffic smoothly. kabir et al./decis. mak. appl. manag. eng. 6 (1) (2023) 1-17 4 figure 1. traffic streams at four leg intersection and t junction of right hand side driving. rank indicates the flow of traffic among all the types of junctions (t-junction, y-junction, round about junction, four arm junction, staggered junction), accidents took place at the t-junction contributes the most number of accidents in comparison with the accidents took place at all other junction (robbins et al. 2019). in major cases of accidents at t-junction the drivers are not aware of the incoming traffic from the other side of the road due to the presence of obstacles like walls, slow or stand by cars, road vendors, etc. as shown in the figure 2. (a) (b) (c) (d) figure 2. real images of accidents capture from cctv footage (collected from web) occurred due to jumping red light also (a) due to the driver vision blocked by wall, and (b) slow moving truck, (c) driving on wrong side lane and (d) over speeding hazard perception test among young inexperienced drivers and risk analysis while … 5 if the driver is drunk or using mobile phone during driving then the combined effects of these can be very severe. a hazard based statistical approach has been utilized over here, in order to consider the impacts of mobile phone usage, video calling during driving with different ages and experience the non-distracted drivers. combine effects of alcohol and mobile phone is not considered here due to the legal complexity and also not encourage, which is also considered criminal offence in india under section 185 of the motor vehicles act, 1988 (india, 2021). 3. experimental setup the experimental technique was performed in a ps4 driving simulator by considering the health concerns of the participants. logitech g29 force racing wheel is used as real time controller. the driving simulator is able to take input from accelerator, gear, brake, and steering wheel (fig 3) for realistic driving experience with real jerking using force feedback. all participants completed an informed consent form and were voluntarily agreed for the said experiments. they were offered two different 10-min trial driving that would enable them to become familiar with the driving simulator. a helping team was instructed to create the experimental scenario and hazards. the choice route has multiple t-junctions. the participant experiences the near about same hazardous events while driving through the connecting arm of the t-junctions. behavior of the drivers when they face the hazardous event is collected with the help of the driving simulator that is used to measure the reaction time of the drivers. the difference between the time the hazards were noticed and the time the driver responded to the event, either by accelerating or by pressing the brake pedal is considered as the time of reaction. (a) (b) figure 3. driving simulator (a) real view with full camera setting, (b) logitech g29 force racing wheel for realistic driving experience the number of participants were 20 including 14 male and 6 are female. the driving experience variable it can be observed that 2 participants holds a driving experience of 0-1 year, whereas 9 participants holds minimum 2 years and maximum 4 years of experience and rest 7 participants hold more than 4 years of experience as details analysis is tabulated in table 1. kabir et al./decis. mak. appl. manag. eng. 6 (1) (2023) 1-17 6 table 1. sample breakdown of participants age group and number of drivers gender 18-25 years 26-35 years 36-50 years number % number % number % female 3 50 2 33.33 1 16.67 male 5 35.71 4 28.58 5 35.71 driving experience in years gender 0 1 2 – 4 above 4 number % number % number % female 2 33.33 2 33.33 2 33.33 male 2 14.29 7 50 5 35.71 period of mobile phone usage during driving short (< 5s) longer time (> 5s) gender number % number % female 4 66.67 2 33.33 male 8 57.14 6 42.86 impact time stamp measurements were performed by using filimora 9 video editing and analyzing software as shown in figure 4. (a) (b) (c) figure 4. experimental setup for impact time stamp measurement using filimora 9 video editing and analyzing software (a) hazard detected and (b) just before collision and (c) just after the collision hazard perception test among young inexperienced drivers and risk analysis while … 7 the arduino range of microcontrollers is able to provide analog inputs that can be used to measure the break instance by using the simple program. data are captured and sent to excel sheet with time stamp for breaking behavior analysis. details configuration, pin connection and sample output is shown in figure 6. (a) (b) (c) figure 5. circuit configuration to study breaking behavior (a) circuit configuration, (b) pin connection and (c) sample output in excel 4. modeling procedure and data analysis to observe the driving characteristic in the t junction crossing, three preset conditions were tested which are categorized as no distraction, distraction by mobile phone calling, and distraction by video calling. in all three situations, the tested motor vehicle begins to increase speed from the initial position to the desired speed of 25, 40, or 55 km/h. for each trial, the drivers are instructed to achieve the desired speed and maintained the desired speeds and also avoid hazard as much as possible. the tested vehicle’s driver is directed to keep the speed fixed as much as possible all over the trial. at the intersection of t junction, the hazard vehicle appears at a certain position and speed in each type of proposed scenario. in the entire three scenarios, the hazard vehicle appears at the moments when distance of tested vehicle’s driver from hazard vehicles is about of 10m, 15m, and 20m and travels at a speed as taken from selected speeds of 10-25, 26-40 and 41-55 km/h. here nine combination of scenario is considered as shown in figure 6. kabir et al./decis. mak. appl. manag. eng. 6 (1) (2023) 1-17 8 figure 6. description of the scenario considered for testing the reaction time data were collected for three scenarios: no distraction, distraction by mobile phone calling, and distraction by video calling. the reaction time data were analyzed using ibm spss statistics for a 3 x 2 x 3 x 2 x 9 x 3 repeated measures anova: (a) (i) with gender two levels i.e. male and female, (ii) with age three levels (iii) with driving experience three levels i.e. learner, new driver (1-2 years driving experiences) and rest (iii) mobile usage (two levels) as between-subjects factor and (b) with the different scenarios as shown in figure 6 i.e. nine levels different scenario type along with three levels of effects within-predetermined subject factors. there was a considerable variation in the means of the reaction time at each level of no distraction, distraction by mobile phone calling, and distraction by video calling scenarios. hence, an independent repeated measures anova (kim, 2015) was performed for every driving condition. the result analysis revealed that the independent variables that have considerable consequences on the reaction time for all scenarios. it also examined that hazard distance, speed, gender, and age were important factors that directly affect driver reaction time in all kind of driving situations. basically modeling of reaction time due to distractions is conducted through two approaches which are field study and simulation based study (choudhary & velaga, 2017). although several researchers adopted the field study approach for the collection of data for the measurement of reaction time caused by distractions but by considering the safety of the participants and the accuracy in the data collection, simulation based study is considered as the more suitable approach for the measurement of the reaction time due to distracted driving. in order to analyze the braking behavior of the distracted drivers various methods has been adopted by the researchers such as t-test, anova test, linear regression test etc. whereas only a few researchers used the statistical approach such as hazard based duration model and linear mixed model to find out the impacts of external variables on the reaction time of the distracted drivers. at first, a brief of the present work is shared through a seminar to all the participants. after that all the participants was asked to fill a questionnaire that contains all the valuable information that is required to predict their habitants during driving (details of questionnaire form is illustrated in participants section). after the completion of paper work all the participants goes through a test drive on the simulator, to make them familiar with the simulator. the next day after another test drive, the participants (as grouped in the participants' section) finally drove on the simulator in 3 conditions: (i) non-distracted, (ii) distracted by mobile phone usage by calling from an unknown number, and fake calls hazard perception test among young inexperienced drivers and risk analysis while … 9 and (iii) distraction by video calling. the variables and their modeling procedure are shown in tables 2. table 2. statistical descriptive details of explanatory variables used for modeling proposed model external variables descriptions of the variables demographic details of the participants age age of the participants gender if the participant is a male = 1, otherwise = 0 if the participant is a female = 1, otherwise = 0 driving experience (in years) 0 1 if the participant has driving experience of 0 to 1 year = 1, otherwise = 0 2 4 if the participant has driving experience of minimum 2 year and maximum 4 year = 1, otherwise = 0 above 4 if the participant has driving experience of more than 4 years = 1, otherwise = 0 driving history type of license if the participant holds an open license =1, otherwise = 0 if the participant holds an provisional license =1, otherwise = 0 traffic offense if the participant received any kind of violation notice (due to mobile phone usage) in real life driving =1, otherwise = 0 if the participant does not received any kind of violation notice in real life driving =1, otherwise = 0 crash involvement history if the participant involved in any type of traffic collision in real life driving =1, otherwise = 0 if the participant does not involved in any type of traffic collision in real life driving =1, otherwise = 0 distractions history of the participants (during driving) type of mobile phone if the participant use hand held mobile phone during driving = 1, otherwise = 0 if the participant use hand free mobile phone during driving = 1, otherwise = 0 period of mobile phone usage if the participant involved in a conversation for less than 5s = 1, otherwise = 0 if the participant involved in a conversation for more than 5s but less than 10s = 1, otherwise = 0 if the participant involved in a conversation for more than 10s = 1, otherwise = 0 the reaction of the participants when they drove through the connecting arm of the t-junction, were noted and analyzed, as soon as the event enters into the visual range of the driver from the main road. the reaction time of the drivers was calculated with no-distractions, distractions due to mobile phone usage and distraction by video calling during driving with the help of hazard based duration model of a survival rate kabir et al./decis. mak. appl. manag. eng. 6 (1) (2023) 1-17 10 of the drivers were developed by considering some external variables such as demographic detail of the drivers (age and gender), driving characteristics of the drivers (driving experience in years, mobile phone usage history of the participants (conversation/text, period of usage)of the road. accelerated failure time (aft) models are a class of mainly parametric models, which generates from the observation that time to event variable (t) cannot be negative; hence this can be work with its logarithm. the generalized aft model is as follows, 𝑙𝑜𝑔𝑇𝑖 = 𝑥𝑖 / 𝛽 + 𝜖𝑖 where, 𝑥𝑖 / , 𝛽, and𝜖𝑖 are covariates, coefficients and error term with given distribution (mean 0, sd 1). the distribution of the log survival model is just shift of the base line distribution of the error term. 5. hazard function in order to measure the reaction time (time taken by drivers to react after detection of an event) of the distracted drivers a hazard based duration model has been utilized in this present study. as reaction time is defined as the time difference between detection of an event and reacting against that event, so the hazard based probabilistic method is perfectly suited for the for the measurement of reaction time (duration model). the reaction time of the driver is taken as the duration variable in this study. because of the promising characteristics and appropriateness to the modeling of duration variable, aft based model was applied in this present study. the reaction time (i.e., duration model) is a continuous time variable 𝜏, with respect to a probability density function f (t) and cumulative function f(t) as shown in eq. (1): f(t) = p (τ < 𝑡) = ∫ f(t)dt t 0 (1) hazard function (h(t)) of the used aft model can be expressed as a function of conditional probability and mathematically written as shown in eq. (2): h(t) = f(t) [1 − f(t)] (2) as time passes the probability of detection of the event by the drivers will also increases, so it can be concluded that the h(t) is a monotonously increasing function with time. the survival model duration model is also known hazard based as where the rate of survival 𝜓(𝑡), can be expressed as the probability 𝑃(𝜏) of wrong detection and can be formulated as sown in eq. (3): ψ(t) = p(τ ≥ t) = 1 − f(t) (3) in this paper, measurement of reaction time is conducted with the help 𝜓based on statistical analysis of data, where the impacts of the explanatory variables on the result, are also considered in the analysis. basically the explanatory variables integrates with the model through two approaches, named as: (1) aft and (2) proportional hazard based (ph), but as the aft model provides better results than the ph model when more than one variable is introduced in the model analysis, so aft model is more suitable than the ph model for this present work. in this present study, weibull distributional function is considered as the hazard perception test among young inexperienced drivers and risk analysis while … 11 required distributional function for the complete analysis of reaction time (duration variable), as the h(t) is a monotonically increasing function and weibull distributional function is applicable to both monotonically increasing as well decreasing h(t). the hazard function of the weibull distributional function can be presented as a function of location parameter λ and scale parameter p, as shown in eq. (4): h(t) = pλ p tp−1 (4) and the corresponding survival rate of the weibull distributional function can be represented as shown in eq. (5): s(t) = exp(−λt)p (5) to incorporate the weibull distributional function with the aft model, natural logarithm of the duration variable (t) is represented as a linear function of two vectors: explanatory variables (x) and estimated parameter (β), as shown in eq. (6): ln(t) = βx + ∆ (6) where ∆ represents the error term. in this present study is repeated observations for each participant were collected so the reaction time of each of the participants might be correlated at individual level. the measurement of reaction time of each of the participants is possible through two approaches: one is the clustered model of heterogeneity and another one is based on the estimation of standard error. standard error estimation approach is considered as the best suited approach in this study as this study measures the reaction of each of the individual separately. 6. result analysis and discussion all the obtained values of scale parameter from the designed model were greater than 1, which indicates that the probability of detecting an event got increased with passage of time for all the events. the explanatory variables which affect the designed model are as follows: driving experience of the participants, type of license hold by the participants, traffic offense received by the participants, and period of cell phone usage while driving. the driving experience of the participants has significant impact on the reaction time. type of license holds by the participants is also an important factor in the calculation of reaction time as it is found that the provisional license holders are less prone to involved in a collision with a less reaction time in comparison with the open license holders. in case of driver’s distraction due to mobile phone usage, all the three periods of phone usage (short, moderate and longer) were analyzed. it is found that moderate periods of mobile phone usage during driving have maximum impacts on the reaction time. as in the shorter period drivers are more concentrated in driving other than the distractions whereas in case of the participants who involved in longer periods of mobile phone usage, adapt themselves with the cell phone usage during driving. increment in the reaction time due to moderate period of cell phone usage during driving is due to the fact that in this case the driver’s focus shifts from the roadway to the mobile, which results in wrong judgment and ended with a fatal collision. actually this situation is responsible for the delay in detection of the event which in turn increases the reaction time. from the estimated parameter it can be observed that kabir et al./decis. mak. appl. manag. eng. 6 (1) (2023) 1-17 12 with the increment of driving experience of 1 year, the reaction time is increased as shown in table 3. table 3. statistical analysis of reaction time s ce n a ri o distraction significance by a liner mixed model estimated marginal means normal phone calling video calling mean std. error 95% confidence interval r f (10%) lower bound upper bound 1 1.066 0.807 0.561 0.927 35.52 0.811 0.011 0.789 0.833 2 1.343 0.968 0.660 0.954 59.10 0.989 0.011 0.966 1.011 3 1.938 1.111 0.962 0.888 21.53 1.334 0.017 1.299 1.368 4 1.042 0.797 0.580 0.915 29.71 0.806 0.011 0.784 0.828 5 1.304 1.027 0.710 0.939 43.34 1.019 0.011 0.996 1.042 6 1.604 0.971 0.836 0.904 26.04 1.139 0.012 1.115 1.163 7 1.097 0.805 0.699 0.871 18.16 0.867 0.011 0.845 0.888 8 1.388 0.994 0.785 0.834 13.23 1.056 0.021 1.014 1.097 9 1.935 1.527 0.844 0.919 31.37 1.435 0.024 1.387 1.483 impacts on the estimated reaction time due to the uses of mobile phone during driving are shown in figure 7. figure 7. mean reaction time of different scenario the participants who receives traffic rules violation notice exhibits slowest reaction time after the detection of any event than the open license holders and the participants with driving experience of (0-1) year. when the traffic rules violators are distracted due to mobile phone usage they achieved best result to react against that event, whereas the open license holder got moderate value and the un-experienced participants got more time to stop the vehicle after detection of an event. from the simulation data it can be seen that in any unknown city, the pedestrian fatality risk is increased. the results emphasize the importance of intersection design to increase drivers’ compliance to the traffic rules. experienced drivers exhibited greater speed hazard perception test among young inexperienced drivers and risk analysis while … 13 control and lane control in response to all kind of hazards mostly in the t-junctions and blind turn with heavy traffic; whereas, inexperienced and young driver maintained higher speed than the recommended speed, as compared to usual driving, when faced blind junctions. the average increment in the reaction time due to the distractions caused by mobile phones is of in comparison with the non-distracted driving. in their study, it is found that intake of alcohol affects the actions of drivers in many wayssuch as control over steering, visual ability, braking time, taking a decision while driving etc. expert practice divides the driver's response into simple and complex. as a rule, during automated technical testing, forensic experts encounter a complex response, such as diagnosing a person whose response to a threat is not prepared in advance. a road accident will be occurred where participants reach the point of collision. the general options given in these recommendations for determining the various values (whittaker, 2020) of the driver's response time during an emergency do not fully cover the possible mechanisms of accident development. in particular, the driver's response to the danger of exceeding the speed limit, which was introduced at the scene of an accident, when a pedestrian or car was hit, was not determined. a complex response is usually associated with choosing the best solution out of several possible solutions. when a driver detects an object (danger or obstacle) it depends on how likely it is. the more the object appears, the more closely the driver has to monitor the traffic situation, the sooner he will be able to detect it. if the object is not likely to appear (danger or obstruction), the driver may be distracted by direct observation of the road to perform other functions, such as the control device, reading observation of other objects. in such traffic situations, the detection time of the object (obstacle) in the carriageway, which appears suddenly, can be much longer than in the first case. 6.1. advantages and limitation of the study studies have revealed that two major issues involved in most of the rear-end clashes are drivers’ failure to recognize and respond to the actions of preceding vehicle and going after a lead vehicle too closely. the perception and the reaction of the driver to the lead vehicle’s action is the primary influencing cause in rear-end clashes. in this modern era hand-held-phone usage while driving is a crucial factor to divert drivers’ concentration to react against the action of lead vehicle. in this article exclusively three different states of distances and driving speeds were experimented. these states are classified as no distraction, distraction by mobile phone calling, and distraction by video calling. in field experiments, engaging drivers in surprised or emergency situations on the road for measuring the perception and movement durations is very tough job to accomplish. it is also not an easy task to propose a scenario that can comprise all the factors simultaneously. this study prevails over these hindrances by modeling nine different experimental car-following situational combinations using acquired data. the approach in this study is advantageous in implicating car-following hazard perception test of the drivers to develop rear end clash prevention algorithms and mimicking traffic simulation models specifically in the driving through a t-junction while using hand held phones. the most significant factor used to calibrate carfollowing models for risk analysis is driver’s typical reaction time considering particular influencing constraints such as usage of phone. the brake-reaction time in reaction to deceleration of the preceding vehicle conclusively varies with the response for obstruction on road or with the response to a passing vehicle at the crossroads. kabir et al./decis. mak. appl. manag. eng. 6 (1) (2023) 1-17 14 the present study is advantageous for exploration of the result of driver demography and circumstantial issues on the driver reaction time while driving through a t-junction using hand held phones and presentation of analytical models which have immense contributions for car-following analysis algorithms. in particular, the reaction time model for the no distraction scenario (without phone usage), on phone call and on video call conditions will be necessary to model the rear-end collision warning systems. the difference between the times, the hazards was introduced and the driver response time to the event either accelerating or pressing the brake pedal is known as the time of reaction. several researchers adopted the field study approach, keeping in view that riding machine tests can vary with realistic riding environments and thus the findings can undoubtedly involve some bias but by considering the safety of the participants and the accuracy in the data collection, simulation based study is considered as the more suitable approach for the measurement of the reaction time due to distracted driving. as reaction time is defined as the time difference between detection of an event and reacting against that event, so the hazard based probabilistic method is perfectly suited for the measurement of reaction time. with the help of hazard based duration model a survival rate of the drivers were developed by considering some external variables such as demographic detail of the drivers (age and gender), driving characteristics of the drivers, mobile phone usage history of the participants as well as the geometry (straight/t-junction) and environment in the both urban and rural road. in this present study 9 repeated observations of reaction time for each participant were collected so the reaction time of each of the participants might be correlated at individual level. standard error estimation approach is considered as the best suited approach in this study as this study measures the reaction of each of the individual separately. all the obtained values of scale parameter from the designed model were greater than 1, which indicates that the probability of identifying an event enhanced with the elapse of time for all the three events. few limitations in this study must be accounted for. the hazard-based reaction test was obtained using only 20 participants. however, a development of this research could be with the more drivers but due to the extensive scenarios performed with every participant the sample size is quite healthy to run this study. in this research the knowledgebase of the route was not evaluated. this variable could be a major factor impacting drivers’ activities while driving through a t-junction. if a rider has prior knowledge about the route, may have a greater chance of risk management from the one who is not familiar with this route. in future study, an extension of the research should be planned to validate the impacts of variables such as road geometry, traffic volume on the risk analysis. at the start of the experiment, the variations in reaction levels of the drivers (too slow or very fast) produce some biased data. the test did not deliver the reaction time data concerned with the deceleration and the acceleration. in the next study investigation to be designed to exclude the outliers while verifying the variations for the reaction times at the acceleration and deceleration phases. finally, considering that the drivers’ consciousness for involving in the experiment could affect the reaction time, a future study has been aimed for collecting data from an instrumented vehicle and for analyzing the behavior of unaware drivers. this research may be taken as a directive study and an initial step towards future advancement on drivers’ behavior at t-junction maneuver. in addition, although with the limitations above acknowledged, the outcomes of this study could aid to the hazard perception test among young inexperienced drivers and risk analysis while … 15 improvement of advanced driver supporting systems for the risk analysis while driving through a t-junction based on existing driving situations. 7. conclusion this study measured the reaction time of the young inexperienced drivers distracted by mobile phone in various hazardous conditions. we examine the reaction time of the participants against an event, in both distracted and non-distracted condition. the impacts of distractions on the reaction time were most evident for the participants who violate the traffic rules in their real life driving, than the unexperienced drivers and the participants with open license holders. the proposed model was designed with the help of weibull distributional function with clustered heterogeneity. driving performance may consider in terms of reaction time, speed control and lane maintenance. results demonstrated hazards to cause declines in various conditions awareness and speed control or liability to various hazard type was dependent on drivers experience and age. these observations are applicable to modeling driver behavior under hazardous environments and may helpful for the design of smart vehicle assistive devices for collision predictions with over speeding, overtaking procedure, study of over loaded vehicle characteristics and its effects on the reaction time or on the braking behavior of the drivers before issuing the driver licenses and post accident analysis. author contributions: conceptualization, faysal kabir and sahadev roy; methodology, faysal kabir and sahadev roy; software, faysal kabir; validation, faysal kabir and sahadev roy; formal analysis, faysal kabir and sahadev roy; investigation, faysal kabir; resources, faysal kabir and sahadev roy; data curation, faysal kabir; writing, faysal kabir; – review & editing, sahadev roy; visualization, faysal kabir and sahadev roy. funding: national institute of technology, arunachal pradesh. conflicts of interest: the authors declare no conflicts of interest. references almutairy, f., alshaabi, t., nelson, j., & wshah, s. 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(2020). review of road accident analysis using gis technique. international journal of injury control and safety promotion, 27(4), 472-481 hazard perception test among young inexperienced drivers and risk analysis while … 17 whittaker, s., khalfan, m. m., &ulhaq, i. (2020). developing community disaster resilience through preparedness. international journal of critical infrastructures, 16(1), 53-76. © 2021 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 4, issue 1, 2021, pp. 33-50. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame2104033d * corresponding author. e-mail addresses: rishidwivedi12@yahoo.co.in (r. dwivedi), kprasad.prod@nitjsr.ac.in (k. prasad), nabankur2009@gmail.com (n. mandal), shwetasinghaka12@gmail.com (s. singh), mayank1293vardhan@gmail.com (m. vardhan), dragan.pamucar@va.mod.gov.rs (d. pamucar). performance evaluation of an insurance company using an integrated balanced scorecard (bsc) and best-worst method (bwm) rishi dwivedi 1, kanika prasad 2*, nabankur mandal 3, shweta singh 1, mayank vardhan 1 and dragan pamucar 4 1 department of finance, xavier institute of social service, ranchi, india 2 department of production and industrial engineering, national institute of technology, jamshedpur, india 3 department of mechanical engineering, mckv institute of engineering, west bengal, india 4 department of logistics, military academy, university of defence in belgrade, belgrade, serbia received: 19 october 2020; accepted: 5 december 2020; available online: 13 december 2020. original scientific paper abstract: recent economy and financial business environment is undergoing a quick and accelerating revolution and paradigm shift, resulting in growing uncertainty and complexity. therefore, the need for an all-inclusive and farreaching performance measurement model is universally felt as it can provide management-oriented information and act as a supporting tool in developing, inspecting, and interpreting policy-making strategies of an enterprise to achieve competitive advantages. hence, this paper proposes an application of the balanced scorecard (bsc) model in an insurance organization for coordinating and regulating its corporate vision, mission, and strategy with organizational performance through the interrelation of different layers of business perspectives. in the next stage, a framework to unify both bsc and best-worst method (bwm) models is implemented for the very first time in the insurance domain to assess its performance over two-time periods. the integrated bsc-bwm model can help managers and decision-makers to figure out and interpret competing strength of the said enterprise and consecutively expedite inefficient and compelling decision making. nevertheless, this integrated model is embraced and selected for a certain categorical business and there is enough future scope for its application to distinct industries. mailto:rishidwivedi12@yahoo.co.in mailto:kprasad.prod@nitjsr.ac.in mailto:nabankur2009@gmail.com mailto:shwetasinghaka12@gmail.com mailto:mayank1293vardhan@gmail.com mailto:dragan.pamucar@va.mod.gov.rs r. dwivedi et al./decis. mak. appl. manag. eng. 4 (1) (2021) 33-50 34 key words: service sector, insurance industry, bsc model, best-worst method, bwm, performance evaluation. 1. introduction insurance is a contract or an agreement between an individual and a company, where the company pledges to pay an amount of sum assured to its customers against the insurance based on its terms and policies. the most significant reason for having insurance is that it provides security in many terms, to name a few: it generates financial resources, promotes economic growth, keeps commerce moving, ensures business and family stability, and encourages savings thereby securing future goals. a well-designed insurance plan helps in managing risks. in today's competitive world every company in the insurance sector tries hard to satisfy its customers. every day the companies are introducing new plans and policies to attract new customers and retain their initial customer base. the insurance sector plays a vital role in the development of the economy of any country. it is the only sector that garners long terms savings and generates funds for the development of the capital market and infrastructure, and hence providing stability to the growth of the economy. besides, insurance is a vital component in carrying out smooth operational processes of national economies throughout the world. in developed countries, such as germany, england, switzerland, france, etc. insurance has become an important component of the economy as it contributes majorly to the global market. the indian insurance sector has faced many stumbling blocks to attain its current position. liberalization, privatization, and globalization (lpg) reforms in the year 1991 have played an important part in the development of this sector in india. based on a report published by india brand equity foundation in 2019, the overall insurance industry in india is expected to reach us$ 280 billion by 2020. the life insurance industry in the country is expected to grow by 12 % to 15 % annually for the next three to five years. hence, the insurance sector is one of the booming sectors of the indian economy. traditionally, the performance evaluation of an insurance organization’s progress is carried out based on financial and customer factors only, which are mostly past indicators of success. the future drivers of progress for the enterprise are generally encompassed in the learning and growth and internal business perspectives of the balanced scorecard (bsc) model and are not taken into consideration in the traditional models of performance evaluation. bsc model considers both lagging and leading factors. the leading factors are also termed as futuristic parameters. they include the internal business perspective and the learning and growth perspective. the bsc model works with a holistic and integrated view of the business. it is a present-day performance measurement technique designed to overcome the drawback of traditional performance measurement systems. it is a performance measurement tool that consists of a set of measures that facilitates the enterprise's view of its overall performance. however, with time the bsc model has also been used as a strategic management tool as it can identify the key performance indicators (kpis) of a company. in this paper, a bsc model is proposed for an insurance company operating in india while considering multiple factors such as profit after tax (pat), operating profit ratio, asset under management (aum), number of products, market share, etc. impacting the performance of the considered organization. next, the best-worst method (bwm) is employed in this paper to solve the multi-criteria decision-making (mcdm) problem. according to the bwm technique, the best or the most important and the worst or the least important business performance evaluation criteria of an performance evaluation of an insurance company using an integrated bsc and bwm 35 insurance company are identified first by the decision-makers. further, pair-wise comparisons are carried out for best to others identified criteria, and others chosen criteria to worst criterion. a maxi-min problem is then formulated and solved to determine the optimal weights of the different business performance parameters. a consistency ratio is estimated in the next step of bwm methodology to check and verify the reliability of the comparisons (vesković et al., 2020). furthermore, an integrated bsc-bwm structure is also designed in this paper to evaluate and appraise the progress of selected insurance enterprises over two time periods concerning key performance measures of the proposed bsc model to demonstrate the efficiency and effectiveness of management strategies applied. the results derived from the implementation of the above models would not only facilitate business performance measurement of the organization during a time period in quantitative terms but would also help the decision-makers to recognize what should be carried out and measured in an organization to reinforce and boost its productivity, performance, and progress. the results obtained from the application of the above model also contribute to the long term sustenance of the organization. 2. review of literature rezaei (2015) proposed a new technique bwm to solve multi-criteria decisionmaking problems. rezaei et al. (2016) developed a supplier selection model for food industry using the bwm tool. ahmadi et al. (2017) employed bwm in manufacturing companies to analyze the social sustainability of their supply chains. gupta (2018) applied hybrid bwm to identify the service quality parameters in the airline industry and then prioritized them based on customers’ needs. rezaei et al. (2018) developed a weighted logistics performance index based on the bwm method. salimi and rezaei (2018) proposed a multi-criteria decision-making method called bwm to calculate the weightage of research and development measures in small and medium-sized enterprises (smes). zhao et al. (2018) and kushwaha et al. (2020) implemented a hybrid framework on the basis of the bwm technique to assess the comprehensive benefit of eco-industrial parks in terms of circular economy and sustainability. pamucar (2020) employed bwm for determining criteria weights in supply chain management. brunelli and rezaei (2019) examined the bwm methodology for the mcdm problem from a more mathematical perspective. kheybari et al. (2019) developed a bwm model for bioethanol facility location selection in provinces of iran. zolfani and chatterjee (2019) not only investigated the weighting of important and related criteria for sustainable design but also evaluated the similarities and differences between the step-wise weight assessment ratio analysis (swara) and bwm techniques. chen et al. (2011) proposed an effective performance evaluation model based on the bsc technique which helped decision-makers to understand apt actions and achieve a competitive advantage. mendes et al. (2012) applied the bsc model in the urban hygiene and solid waste division of the loulé municipality in the south of portugal. pesic and dahlgaard (2013) suggested that there were strong positive correlations between the bsc perspectives and european foundation for quality management (efqm) excellence model criteria. hakkak and ghodsi (2014) concluded that the bsc model helped organizations in achieving sustainable competitive advantage. staš et al. (2015) designed a conceptual framework for creating the green transport (gt) and bsc models from the viewpoint of industrial companies and supply chains using an appropriate mcdm method. dwivedi and chakraborty (2016) developed a bsc model for an indian thermal power plant to help the organization to r. dwivedi et al./decis. mak. appl. manag. eng. 4 (1) (2021) 33-50 36 make strategic and tactical decisions. anjomshoae et al. (2017) proposed a dynamic bsc model in the humanitarian supply chain. dwivedi et al. (2018) designed a technique based on bsc for an indian seed manufacturing organization to know about its performance so that the managers can align a company’s operation with the existing business environment. abdelghany and abdel-monem (2019) applied the bsc model that provided the utilities' managers with a fast but comprehensive view of the utilities’ performance. mohammadi et al. (2019) developed a feasibility test model for creative experiences based on the bsc technique through qualitative content analysis. it can be comprehended from the ongoing literature review that earlier researchers have individually developed tools either by applying bsc or bwm for different real-world scenarios. the previous research works by past researchers have not consolidated bsc and bwm together. thus, by applying the two methodologies together, this research work focuses on doing the performance evaluation of an insurance organization in india while understanding the leading and lagging parameters that contribute to the success and failure of the selected organization. consequently, this integrated bsc and bwm model would fill the present gap for the real-time practical implementation of combined bsc as a bwm tool. this model would provide the latest perspective and motion for addressing real-world business performances and their assessment. 3. bsc model traditional performance measurement systems study and review the progress of an organization basically on the basis of a short-term financial objective. those are no longer relevant to conquer the challenges faced by the organizations in recent times. furthermore, with dynamic and growing business environment organizations have to make sure that their strategies are transformed into subsequent actions through a more meticulous and precise consideration of the objectives of related stakeholders. bsc model is often suggested and approved as an inclusive management tool linking critical strategic and short-term action planning (kaplan, 1994; badi et al., 2019). this technique is developed in such a way that it cancels the most typical and trivial mistake of the existing traditional systems of performance management, i.e. describing and reporting only on the basis of financial data. in today’s cut-throat aggressive and competitive market, it is even more critical to attaining a balance between financial and non-financial data in management reporting and recording. therefore, bsc is developed as a modern performance evaluation procedure to overcome the defect of the formerly adopted performance measurement systems through introducing four perspectives, i.e. financial perspective, customer perspective, internal perspective, and learning and growth perspective on which development and pace of an organization would be assessed. financial and customer measures take into account the past performance of the organization and they are termed as lagging indicators. on the other hand, internal business process, and learning and growth perspective are leading indicators. thus, a bsc model provides a holistic and integrated view of the business concerning four perspectives to the management. all the kpis that are identified for the bsc model under each perspective must fit the sequence of cause and effect relationships within them. 3.1. designing of the bsc model for an insurance sector organization developing a suitable bsc model for an enterprise needs a subtle evaluation of the organization’s foundations, core values, beliefs, ideas, opportunities, financial performance evaluation of an insurance company using an integrated bsc and bwm 37 position, short-term and long-term goals, and operating business. the confidentiality of the studied organization is maintained here and hence, the name of the insurance enterprise is not being revealed and hereafter, it is referred to as abc limited. it is an organization that is growing at a rapid pace for a strong presence in the domestic insurance market. abc limited has exclusive, remarkable, and innovative insurance products that help it to compete in the market. all the data required for the development of a performance measurement tool based on the bsc model pertains to the financial years 2016-2017 and 2017-2018. here, a focused group is selected constituting of subject experts to develop a bsc model for abc limited, while keeping a balanced representation of kpis from each perspective. figure 1 displays the developed bsc model for abc limited. it can be observed that the developed bsc model identified 20 business performance indicators that provide the management with a concise summary of the kpis of abc limited. these performance measures also assist in appropriately coordinating and regulating the business processes of abc limited with its overall policy. each identified performance parameter signifies a particular goal and competence of abc limited for all the four perspectives of the developed bsc model. for example, the key performance measure indicator of pat suggests how much the organization really earns after paying interest and tax and how much it can utilize for its day to day business activities, whereas the operating profit ratio explains the profit a company generates after paying for variable costs of production, such as wages, raw materials, etc. the earning per share (eps) is a critical and essential financial measure, which illustrates and reveals the profitability of a company. it is of utmost relevance and vital for people and investors trading in the stock market. eps of a company is directly related to its profitability. on the other hand, gross written premium (gwp) is the total revenue from a contract expected to be received by an insurance company before deductions for reinsurance and ceding commissions. moreover, average claim settlement time (in days), number of insurance products, number of branches, market share, and number of agents reveal the relationship of abc limited with its customers, which acts as a stimulant for its future growth. further, the number of policies issued, aum, number of employees, the amount spent on corporate social responsibility (csr) activities, and numbers of transactions managed (in lakhs) are the internal performance measures that show the operational competence of abc limited to deliver its products and provide services consistently. next, net promoter score (nps), training expenses, transactions managed and policies issued within the turnaround time (tat), management expenses, and employees' remuneration and welfare benefits represent the learning and growth aspect in the organization, which are of predominant importance for its long-term success while operating in a competitive insurance sector. thus, the developed bsc model can expedite an effective monitoring of the organization's overall progress. this model successively helps the decision-makers in future strategy formulation. r. dwivedi et al./decis. mak. appl. manag. eng. 4 (1) (2021) 33-50 38 figure 1: developed bsc model for abc limited 4. bwm technique bwm is a multi-criteria decision-making (mcdm) tool developed to overcome the drawbacks of previously existent methods, like the analytical hierarchy process (ahp) in 2015 (rezaei, 2015). bwm can be used in various decision-making fields, such as health, information technology, engineering, business, economics, and agriculture. in fundamental, wherever the objective is to rank and select a preference among a set of options, this method can be used. the pertinent features of bwm as compared to most current mcdm methods are that it requires fewer comparisons of data and leads to more dependable, steady, logical, and rational comparisons, which implies that the bwm model produces more decisive and stable results (bozanic et al., 2020). the bwm technique is employed to examine a set of alternatives with respect to some chosen criteria. it is based on a systematic pairwise comparison of the decision criteria, i.e. after identifying the decision criteria, two criteria are selected as the best decision criterion and the worst decision criterion by the decision-makers or experts. the best criterion is the one that has the most important and crucial role in making the decision, while the worst criterion has the opposite significance (pamucar & ecer, 2020). the decision-makers then give their preferences of the best decision criterion over all the others and also their preferences of all the other criteria over the worst decision criterion using a number from a predefined scale of 1 to 9. these two sets of pairwise comparisons and correlation are used as input for an optimization problem, the optimal results of which are the weights of the criteria. performance evaluation of an insurance company using an integrated bsc and bwm 39 5. development of a business performance measurement tool for an insurance sector organization employing bsc and bwm tools the performance evaluation of an insurance company is dependent upon various factors. here, the important drivers impacting the performance of selected insurance organization is already identified as the 20 kpis of the bsc model. next, in order to determine the relative importance of selected kpis, it is worthwhile to employ an mcdm method. several mcdm methods have been applied in earlier studies for different selection problems. in this study, bwm is implemented for the evaluation of identified kpis for the chosen insurance organization. this methodology has been successfully applied in several distinct real-world problems. moreover, compared to other similar existing mcdm methods, bwm needs less pairwise comparison of data and the results achieved by it are more consistent. 5.1. steps in implementation of bwm technique the steps followed in bwm are as follows: step 1. identify a set of evaluation criteria. in this step, a set of criteria {c1, c2, c3,…..,cn} is chosen for making a decision. step 2. the best criterion (most imperative or most important or most significant) and the worst criterion (least imperative or least important or least influential) are determined based on the decision-maker(s) or expert(s) opinion. step 3. the preference of the best decision criterion over all the other decision criteria is determined based on a score between 1 and 9, where a score of 1 means equal preference between the best criterion and another criterion and a score of 9 means the extreme preference of the best criterion over the other criterion. the result of this step is the vector of best-to-others (bo) and is represented by vector ab: ab = (ab1, ab2, ab3,….., abn), where abj indicates the preference of the best criterion b over criterion j, and it can be deduced that abb = 1. step 4. the preference of all the other decision criteria over the worst criterion is determined based on a score between 1 and 9. the result of this step is the vector of others-to-worst (ow) and is indicated by vector aw: aw = (a1w, a2w, a3w,……………., anw)t, where ajw shows the preference of the criterion j over the worst criterion w. it also can be deduced that aww = 1. step 5. the optimal weights (w1*, w2*, w3*,………, wn*) are calculated. the optimal weight of the criteria satisfies the following requirements: for each pair of wb/wj and wj/ww, the ideal situation is where wb/wj = abj and wj/ww = ajw. to satisfy these conditions for all j, a solution should be found where the maximum absolute differences |wb/wj – abj| | 𝑤𝐵 𝑤𝑗 − 𝑎𝐵𝑗| | 𝑤𝐵 𝑤𝑗 − 𝑎𝐵𝑗| and |wj/ww ajw| for all j is minimized. considering the non-negativity and sum condition for the weights, the following problem is resulted: r. dwivedi et al./decis. mak. appl. manag. eng. 4 (1) (2021) 33-50 40 , / – , , / – , 1 0, b j bj j w jw j j minx subject to w w a x for all j w w a x for all j w w for all j      (1) after solving problem equation (1), the optimal weights (w1*, w2*, w3*, …..wn *) and ξ* are obtained. ξ* can be seen as a direct indicator of the comparison system’s consistency. the closer the value of ξ* is to zero, the higher the consistency, and, therefore, the more reliable and steadier the comparisons become. 5.2. application of the bwm in insurance organization through the literature review and expert's view, 20 kpis impacting the performance of the selected insurance enterprise under four perspectives of the bsc model is already identified in the previous section. here in this research work, each perspective of the bsc model is treated as the main business performance criteria. next, under all main business performance parameter for the chosen organization, five different kpis are considered as sub-business performance criteria as shown in table 1. table 1. performance criteria of abc limited main business performance criteria sub business performance criteria customer perspective market share (%) average claims settlement time (in days) number of branches number of products number of agents financial perspective pat (in rs 000) operating profit ratio (%) earnings per share (eps) gwp (in rs 000) claims paid (in rs 000) internal business perspective number of employees number of policies issued aum (in rs crore) amount spent on csr activities (in rs) number of transactions managed (in rs lakhs) learning and growth perspective nps training expenses (in rs 000) transactions managed and policies issued within tat (%) management expenses (in rs 000) employees' remuneration and welfare benefits (in rs 000) performance evaluation of an insurance company using an integrated bsc and bwm 41 after the four main business parameters and five sub-business performance criteria under each of those main parameters are listed, the next step is to estimate the relative weights of all sub-business performance parameters. this is carried out by first finding the global weights of four main business performance parameters. after that, the local weights of sub-business performance criteria under each main business parameter are computed. final weights of each sub-business performance criteria are determined by utilizing equation (2). final weight of sub business performance criteria local weight of sub performance criteria global weight of main performance parameter   (2) after carefully evaluating four main business performance parameters with respect to the identified organization’s mission and operational constraints, the financial perspective is identified as the best criteria whereas the learning and growth perspective is selected as the worst one for abc limited. the experts specified the preference of best criterion with respect to selected other main business performance criteria as shown in table 2. table 3 displays the other main business performance criteria’s preference over the least important criterion. table 2. preference of best criterion with respect to selected other main business performance criteria best to others financial perspective customer perspective internal business perspective learning and growth perspective most important : financial perspective 1 2 4 5 table 3. other main business performance criteria’s preference over the least important criterion others to the worst least important: learning and growth financial perspective 5 customer perspective 4 internal business perspective 2 learning and growth perspective 1 next, the optimal global weights of the main business performance criteria for abc limited are calculated through solving equation (1) while implementing the steps of the bwm technique as discussed in the earlier section. table 4 shows the estimated global weights for all main business performance criteria with the consistency ratio (zolfani et al., 2020). r. dwivedi et al./decis. mak. appl. manag. eng. 4 (1) (2021) 33-50 42 table 4. global weights for main business performance criteria main business performance criteria global weight customer perspective 0.2796 financial perspective 0.4946 internal business perspective 0.1398 learning and growth perspective 0.0860 consistency ratio, ξ* 0.0645 it can be observed from table 4 that the financial perspective has the highest weight of 0.4946. the financial perspective’s influence is the most critical and essential criterion when abc limited attempts to accomplish its mission and objectives. this is followed by the customer perspective and internal business perspective with weights of 0.2796 and 0.1398 respectively. the learning and growth perspective has the lowest criterion weight of 0.0860 showcasing its minimal relevance and importance in the comprehensive business growth of abc limited. the consistency ratio (ξ*) is close to zero, i.e. 0.0645, which shows the high reliability and authenticity of the comparisons. furthermore, the sub-business performance criteria under the main business performance parameter of a financial perspective are examined to compute their local weights. here, pat is identified as the best sub-criterion whereas claims paid are selected as the worst one for abc limited. the best sub-criterion to other related subcriteria comparisons and others sub-criteria to worst sub-criterion comparisons of the financial perspective of abc limited is shown in table 5 and table 6. table 5. preference of best sub-criterion of financial perspective with respect to other sub-business performance criteria of financial perspective best to others pat operating profit ratio eps gwp claims paid most important : profit after tax 1 2 4 5 9 table 6. other sub-business performance criteria’s preference over the least important criterion of financial perspective others to the worst least important: claims paid pat 9 operating profit ratio 8 eps 6 gwp 4 claims paid 1 local weights for the five sub-business performance criteria under the main business performance parameter of financial perspective are estimated while following steps of bwm implementation as discussed in the earlier section and are displayed in table 7. performance evaluation of an insurance company using an integrated bsc and bwm 43 table 7. local weights for sub business performance criteria of financial perspective sub business performance criteria local weight pat 0.4457 operating profit ratio 0.2713 eps 0.1357 gwp 0.1085 claims paid 0.0388 consistency ratio, ξ* 0.0969 it can be interpreted from table 7 that pat is the most crucial sub-business performance criteria under the financial perspective with a local weight of 0.4457, followed by an operating profit ratio with the local weight of 0.2713. claims paid under financial perspective with a local weight of 0.0388, is having minimal significance. the consistency ratio, ξ* for this particular estimation is 0.0969 showing its high consistency and reliability. in a similar manner, the relative significance of sub-business performance criteria encompassed under the other three main business performance criteria, i.e. customer, learning and growth, and internal business perspectives are calculated as shown in table 8. table 8. local weights for sub business performance criteria of customer, learning, and growth, and internal business perspectives main business performance criteria sub business performance criteria local weight customer perspective market share 0.4416 average claims settlement time 0.2589 number of products 0.1726 number of agents 0.0863 number of branches 0.0406 consistency ratio, ξ* 0.0761 internal business perspective aum 0.4615 number of policies issued 0.2706 number of employees 0.1353 number of transactions managed 0.0902 amount spent on csr activities 0.0424 consistency ratio, ξ* 0.0796 learning and growth perspective nps 0.4560 transactions managed and policies issued within tat 0.2606 training expenses 0.1303 employees' remuneration and welfare benefits (rs 000) 0.1042 management expenses 0.0489 consistency ratio, ξ* 0.0651 r. dwivedi et al./decis. mak. appl. manag. eng. 4 (1) (2021) 33-50 44 after the global weights for all four main business performance criteria and local weights of 20 sub business performance parameters are estimated, final weights of those sub-business performance criteria need to be estimated using equation 2. for example, to estimate the final weight of pat, its local weight, i.e. 0.4457 is multiplied with the global weight of main business performance criteria of financial perspective under which it falls, i.e. 0.4946. thus, the final weight of the pat is calculated as 0.2205. the computed final weight of all sub-business performance criteria of abc limited is shown in table 9. table 9. the final weight of all sub-business performance criteria of abc limited main business performance criteria global weight (i) sub business performance criteria local weight (ii) final weight (i) * (ii) customer perspective 0.2796 market share 0.4416 0.1235 average claims settlement time 0.2589 0.0724 number of products 0.1726 0.0483 number of agents 0.0863 0.0241 number of branches 0.0406 0.0114 financial perspective 0.4946 pat 0.4457 0.2205 operating profit ratio 0.2713 0.1342 eps 0.1357 0.0671 gwp 0.1085 0.0537 claims paid 0.0388 0.0192 internal business perspective 0.1398 aum 0.4615 0.0645 number of policies issued 0.2706 0.0378 number of employees 0.1353 0.0189 number of transactions managed 0.0902 0.0126 amount spent on csr activities 0.0424 0.0059 learning and growth perspective 0.0860 nps 0.4560 0.0392 transactions managed and policies issued within tat 0.2606 0.0224 training expenses 0.1303 0.0112 employees' remuneration and welfare benefits 0.1042 0.0090 management expenses 0.0489 0.0042 5.3. designing a performance measure index for abc limited after the comparative importance of all sub-business performance, criteria are determined for abc limited, an index is developed to assess the enterprise's overall business performance. to compute the performance measure index and examine the progress of abc limited concerning the performance evaluation parameters of the bsc model, the related data for all the 20 recognized sub-business performance criteria for two different time periods are required. the first set of actual data related to the identified 20 performance measure indicators for the initial period is set as the baseline value, which in this case is derived from abc limited's annual report for the financial year 2016-17. next, the current period value illustrates the second data set for those selected 20 kpis and is drawn performance evaluation of an insurance company using an integrated bsc and bwm 45 from the annual report of abc limited for the financial year 2017-18. an initial value of 100 is assigned to all sub-business performance criteria and afterward, their weighted points are calculated using equation (3).   weighted point normalized performance measure weight npmw initial point   (3) afterward, the business performance measure index for the initial period is estimated by taking the summation of all the initial period weighted points computed for each sub-business performance criteria. table 10 shows a detailed computation of the performance measure index of abc limited for the initial period. table 10. the initial period performance index for abc limited sub business performance criteria baseline value npmw initial point an initial period weighted point pat 427973 0.2205 100 22.05 operating profit ratio 2 0.1342 100 13.42 market share 2 0.1235 100 12.35 average claims settlement time 46 0.0724 100 7.24 eps 0.97 0.0671 100 6.71 aum 2355 0.0645 100 6.45 gwp 18426964 0.0537 100 5.37 number of products 114 0.0483 100 4.83 nps 26 0.0392 100 3.92 number of policies issued 1373056 0.0378 100 3.78 number of agents 6000 0.0241 100 2.41 transactions managed and policies issued within tat 92 0.0224 100 2.24 claims paid 10131714 0.0192 100 1.92 number of employees 1702 0.0189 100 1.89 number of transactions managed 18 0.0126 100 1.26 number of branches 128 0.0114 100 1.14 training expenses 162191 0.0112 100 1.12 employees' remuneration and welfare benefits 1264666 0.0090 100 0.90 amount spent on csr activities 1109000 0.0059 100 0.59 management expenses 4646140 0.0042 100 0.42 performance index 100 to inspect and analyze the growth and advancement of the organization over the considered period, the performance measure index for the consequent period is also needed. so, the first step in calculating the performance measure index for the current year is to calculate the current period points by the following equations. equation (4) r. dwivedi et al./decis. mak. appl. manag. eng. 4 (1) (2021) 33-50 46 and equation (5) are employed for beneficial and non-beneficial performance parameters respectively.   / 100current period point current year value baseline value  (4)   200 – / 100 current period point current year value baseline value     (5) applying equation (6), the value of the current period weighted point for the individual sub-business performance parameter is now similarly estimated.   weighted point normalized performance measure weight npmw initial point   (6) next, the performance index for the current period is calculated by adding up all the current period weighted points for all the individual sub-business performance measures. the comprehensive estimation of the current period index for abc limited is shown in table 11. a judgment of the performance measure index values for the current period with those for the initial period, derived from table 10 and table 11 respectively, helps in assessing and understanding the comprehensive business performance of abc limited over the two different time periods and thus allowing the policymakers to understand the effectiveness of various applied policies. it can be noticed that abc limited has significantly improved its performance measure index to 129.98 for the current period in comparison to 100 for the initial period. this implies that the said enterprise is advancing in an appropriate direction to its organizational objectives, vision, and mission. furthermore, a comparison of the current period weighted point with that of initial period weighted point values for individual sub-business performance criteria suggests abc limited’s progression over two time periods with respect to that particular criteria. it can be observed that abc limited has significant enhancement and development in current period weighted point of pat, operating profit ratio, aum, number of products, nps, number of policies issued, number of agents, and number of transactions managed with respect to their corresponding values in initial period weighted point. the standard operating procedures of the organization to achieve this advancement in the said criteria should be maintained in the future also. besides, it can also be noticed that although the organization is progressing in the right direction, there is some sub-business performance parameter on which abc limited is not advancing in the correct preposition. the sub-business performance criteria on which abc limited is not performing well are market share, average claims settlement time, eps, transaction managed and policies issued within tat, training expenses, number of branches and employees’ remuneration and welfare benefits. these identified parameters are the area where the resources need to be deployed by the management of abc limited in order to convert them from nonperforming criteria to performing criteria in the future. this can help the organization in achieving long term sustenance. performance evaluation of an insurance company using an integrated bsc and bwm 47 table 11. performance index for current period for abc limited sub-business performance criteria current period value current period point current period weighted point pat 786281 183.72 40.51 operating profit ratio 3 150.00 20.13 market share 1.65 98.21 12.13 average claims settlement time 50 91.30 6.61 eps 0.57 58.76 3.94 aum 2992 127.05 8.20 gwp 19507884 105.87 5.68 number of products 130 114.04 5.50 nps 40.5 155.77 6.11 number of policies issued 2012574 146.58 5.54 number of agents 7500 125.00 3.02 transactions managed and policies issued within tat 90.4 97.94 2.20 claims paid 9221366 108.99 2.09 number of employees 1769 103.94 1.97 number of transactions managed 29 161.11 2.03 number of branches 127 99.22 1.13 training expenses 203982 74.23 0.83 employees' remuneration and welfare benefits 1351988 93.10 0.83 amount spent on csr activities 2104468 189.76 1.13 management expenses 4650523 99.91 0.42 performance index 129.98 6. conclusion the insurance sector is one leading sector for corporate houses for business operation. the proposed model is inclusive of both lagging factors and leading factors related to the business performance for an insurance sector organization. the integrated bsc and bwm technique is a comprehensive and broad model for realworld insurance sector application. through this study, it can be comprehended that a combined bsc-bwm model can be successfully applied to design a quantitative tool to measure the business performance of an organization and monitor the efficiency and effectiveness of the implemented strategies, which may drive the pathway for future policymaking. in this combined model both past (lagging) and futuristic (leading) parameters are taken into account with two different periods. this provided better analysis and scrutiny of business operating data to the management of an organization. the management can take corrective action to enhance the business potential if there is a significant deviation from the desired and expected business performance of the organization. there is enormous competition among insurance sector organizations to enhance their profit margin, market share, and customer base while minimizing operating r. dwivedi et al./decis. mak. appl. manag. eng. 4 (1) (2021) 33-50 48 expenses and developing and launching new insurance products that are widely accepted by customers. in this paper, a bsc model is first developed for an insurance sector enterprise in india to identify a relevant range of financial and non-financial parameters that supports effective and impressive business management. next, a business performance measurement tool combining bsc and bwm methods is developed and applied in the said organization to exhibit how it can be engaged to monitor the performance of the firm, which can be finally utilized as a driver for the organization's future growth. this unified bsc and bwm model can also be employed by other organizations of different sectors with minor adjustments. it requires time and attention while defining the kpis for the insurance sector which is a limitation of the proposed work. author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research received no external funding. conflicts of interest: the authors declare no conflicts of interest. references abdelghany, m., & abdel-monem, m. 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(2020). a vikor and topsis focused reanalysis of the madm methods based on logarithmic normalization. facta universitatis, series: mechanical engineering. 18(3), 341-355. plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 1, number 1, 2018, pp. 13-37 issn: 2560-6018 doi: : https://doi.org/10.31181/dmame180113p * corresponding author. e-mail addresses: dpamucar@gmail.com (d. pamučar), cirovic@sezampro.rs (g. ćirović). vehicle route selection with an adaptive neuro fuzzy inference system in uncertainty conditions dragan pamučar1*, goran ćirović2 1 university of defence in belgrade, military academy, department of logistics, belgrade, serbia 2 the belgrade university college of civil engineering and geodesy, belgrade, serbia received: 13 november 2018; accepted: 19 january 2018; published: 15 march 2018. original scientific paper abstract: a useful routing system should have the capability of supporting the driver effectively in deciding on an optimum route to his preference. this paper describes the problem of choice of road route under conditions of uncertainty which drivers are faced with as they carry out their task of transportation. the choice of road route depends on the needs stated in the transport requirements, the location of the users and the conditions under which the transport task is performed. the route guidance system developed in this paper is an adaptive neuro fuzzy inference guidance system (anfigs) that provides instructions to drivers based upon "optimum" route solutions. a dynamic route guidance (drg) system routes drivers using the current traffic conditions. anfigs can provide actual routing advice to the driver in light of the real-time traffic conditions. in the drg system for the choice of road route, the experiential knowledge of drivers and dispatchers is accumulated in a neuro-fuzzy network which has the capability of generalizing a solution. the adaptive neuro-fuzzy network is trained to select an optimal road route on the basis of standard and additional criteria. as a result of the research, it is shown that the suggested adaptable fuzzy system, which has the ability to learn, has the capability of imitating the decision making process of the drivers and dispatchers and of showing a level of competence which is comparable with the level of their competence. key words: neuro-fuzzy model, vehicle assignment problem, route selection. 1. introduction the vehicle routing problem (vrp) has played a very important role in the distribution and supply chain management, in addition to many other areas. during mailto:dpamucar@gmail.com mailto:cirovic@sezampro.rs pamučar & ćirović./decis. mak. appl. manag. eng. 1 (1) (2018) 13-37 14 the past five decades, many have engaged in research on various types of vehicle routing problems and have had a lot of success. most of them have aimed at static vrps, and all their information is assumed to be known and not to be changed during the whole process. however, most vehicle routing problems are dynamic in the real world. dispatchers often need to readjust the vehicles routes to improve vehicle efficiency and enhance service quality when accidents or unexpected incidents occur. with advances in modern communication technology to enable people to quickly access and process real-time data, the dynamic vehicle routing problem (dvrp) is being given more and more attention. in dynamic vehicle routing problems, the situation is essentially different. transport requests arrive in time according to a stochastic pattern, and the task is to route the vehicles in an orderly fashion to satisfy the demand. relative to the static problem, the dynamic problem has many notable features (bertsimas & simchi-levi, 1996). they include that the time dimension is essential, future information is imprecise or unknown, rerouting and reassignment decisions may be warranted, faster solution speed is necessary and so on. in particular, it must be dynamic, given that the decision-making is based on incomplete, uncertain and changing information. thus, it is not possible for the decision maker to solve the entire problem at once (gendreau et al., 2006). reviews on the problem can be found in bertsimas and simchi-levi (1996) and ghiani et al. (2003). in the last decades, there have been many attempts to solve the problem of assigning vehicles to transportation routes. in its simplest form, the vehicle assignment problem (vap) can be formulated as a linear programming problem (yongheng & grossmann, 2015) and solved with an application of the simplex method (pilla et al., 2012), the assignment algorithm often called the hungarian method (rais et al., 2014), network algorithms (salari & naji-azimi, 2012) or the transportation method (veluscek et al., 2015), as well as its extensions (masson et al., 2015). in real life situations, however, vap is more complicated and requires more advanced methods to be solved. some authors (pilla et al., 2012; lobel, 1997; rouillon et al., 2006), formulate vap in terms of the linear, integer or mixed integer programming problem. some others (milenković et al., 2015) transform the linear, discrete model into a non-linear, continuous form. in both cases, the problems are formulated either in a deterministic or non-deterministic form. many models are based on the queuing theory, too (werth et al., 2014) and they consider either a homogeneous (masson et al., 2015) or a non-homogeneous fleet (milenković et al., 2015). some of the models combine vap with other fleet management problems, such as fleet sizing and vehicle routing (maalouf et al., 2014) or vehicle scheduling (lobel, 1997; pillac et al., 2011) within time and capacity constraints. the models usually refer to specific transportation environments, such as urban transportation (pamučar & ćirović, 2015; pamučar et al., 2016) rail transportation (shi et al., 2015) or air transportation (teodorović & pavković, 1996; yuzhen et al., 2015). in the majority of cases, the proposed vehicle assignment models have a single objective character, however, different objective functions are considered. the most popular are: total transportation costs (rouillon et al., 2006), profit (rouillon et al., 2006; maalouf et al., 2014) or empty rides (flows) (lobel, 1997). depending on the specific characteristics of vap and the complexity of the decision models, various solution procedures and algorithms are applied to solve specific instances of vap. yuzhen et al. (2015) present interesting considerations on the assignment of airplanes to particular transportation routes. they formulated vap in terms of mixed integer mathematical programming with price-wise linear constraints. the decision vehicle route selection with an adaptive neuro fuzzy inference system in uncertainty ... 15 problem is solved by a cplex solver for the gams system and a heuristic procedure for the rounding of non integer solutions. the most up to date approaches to modeling and solving vap involve: stakeholders' analysis leading to multiple objective formulations of the problem (ćirović et al., 2014), analysis of uncertainty and imprecision of data (pamučar et al., 2016a; shi et al., 2015) and the application of artificial intelligence methods in the problem solving procedures (maalouf et al., 2014; pamučar & ćirović, 2015; ćirović et al., 2014). ćirović and pamučar (2013) claim that multiple criteria formulations of different categories of transportation decision problems are more realistic than their single criterion equivalents. preetvanti & saxena (2003) investigate another variant of a transportation problem focused on optimization of the total transportation time between certain origins and destinations. the authors consider three non-linear, time oriented criteria, such as: riding time, loading time and unloading time, and a set of numerous constraints. the problem is solved by a heuristic procedure that utilizes a specific and original structure of the problem. the optimal solution defines the minimum flow of materials in the transportation network and the minimum time required to distribute this flow in a network. the computational efficiency of the proposed algorithm is analyzed on a real life case study focused on the transportation of iron ore in the steel industry. teodorović and pavković (1996) formulate a vap for a road transportation company. the authors consider a heterogeneous fleet operating from a central depot and define types of vehicles allocated to specific transportation jobs. the decision problem is formulated in terms of fuzzy mathematical programming and solved by an original heuristic procedure. fuzzy numbers are applied to model the dispatcher's preferences and different categories of constraints associated with fleet assignment. further extension of this research is presented in the articles of vukadinovic et al. (1999) in which neural networks are applied to generate a set of fuzzy decision rules allocating vehicles to transportation routes. due to the fact that in many real life situations vap is characterized by high computational complexity, especially when it is combined with other fleet management problems, several authors apply heuristic procedures to solve the analyzed problems. in some cases heuristics are combined with other well-known techniques, such as branch-and-bound algorithms (piu et al., 2015). in the last several years metaheuristic algorithms have earned great popularity as solution procedures for an assignment problem (sicilia et al., 2015; ying et al., 2015). however, as can be seen from the presented literature, there is no available literature dealing with the selection of a road route under the conditions of uncertainty. this paper describes the problems of choice of road route under the conditions of uncertainty which are faced by transport units as they carry out their transportation task. the choice of road route depends on the needs expressed in the transport requirements of transport units in the petroleum industry of serbia (pis) and the location the users themselves. transport units receive a high number of transport requests from other users. one of the characteristics of a transport request is the choice of route by which the vehicle is required to carry out the request which is given to the transport unit. the route guidance system developed in this paper is an adaptive neuro fuzzy inference guidance system (anfigs) that provides instructions to drivers based upon "optimum" route solutions. a driver can make the destination known to the system. a dynamic route guidance (drg) system can route drivers using the current traffic conditions. the system can provide actual routing advice to the driver in light of the pamučar & ćirović./decis. mak. appl. manag. eng. 1 (1) (2018) 13-37 16 real-time traffic conditions. it is based on real-time information regarding conditions and incidents in the traffic network, and it is conceived so as to integrate the routing and the traffic control functions. one objective of such a dynamic route guidance system is to balance the level of service on all major network links so as to increase the efficiency, speed, safety and quality of travel (e.g. to minimize travel time). this system could prove to be extremely useful when transportation needs to be carried out under conditions, when a traffic accident has taken place or when work is being carried out on damaged roads. in the drg system for selecting a road route, which is presented in this paper, the experiential knowledge of drivers who run transport vehicles in transport units is accumulated in a neuro-fuzzy network which has the capacity to generalize solutions. the driver's preference is modeled as a fuzzy expert system, and his reaction to the advice and information provided by the drg system is stored. the previous choices of the driver, in particular deviations from the recommendation, are then used for training the system so that it is made adaptive to the driver. the adaptive neuro-fuzzy network is trained to select the optimal travel route on the basis of criteria (type of road surface, travel distance, travel time, route capacity, traffic capacity, road capacity, the existence of alternative roads along the length of the route). the paper is organized as follows. at the end of introduction, the problem of selecting a road route under conditions of uncertainty is described, and the available literature which considers the issues described above is presented. the second section shows the modeling of the anfis model, the training algorithm and the data set which is used for training the model. in the third section of the paper, the developed model is tested on the example of choice of transport route based on the stated transport requirements of a users. 2. development of an anfis model for selecting a transport route under conditions of uncertainty one of the most important functions of transport management in the pis is transport and supply. supply means the procurement, deployment, storage and care of material reserves, including determining the type and amount of reserves at each level. each day, transport units receive a large number of transport requests from other users who want to carry different types of load (liquefied petroleum gas, oil, gasoline etc.) to various destinations. each transportation request is characterized by a large number of attributes, among which the most significant are type of goods, quantity of goods (weight and volume), place of loading and unloading, the preferred time of loading and/or unloading, and the distance to which the goods are shipped. since the transport fleet has many different types of vehicles, dispatchers have to make decisions every day about which type of vehicle is most suitable to perform the task. one of the essential prerequisites for the choice of vehicle is the choice of route for carrying out the transport request. the criteria by which the transport manager selects and makes a decision regarding which route the vehicle should use for the task are: type of road surface, travel distance, travel time, route capacity, traffic capacity, road capacity and the existence of alternative roads along the length of the route. experienced dispatchers have constructed criteria which they use for selecting a route for carrying out a transport assignment. when selecting routes, vehicles are chosen with the structural and technical characteristics which satisfy the conditions for transporting particular types of load. fuzzy sets can quantify linguistic i.e. qualitative and imprecise information that occurs when making decisions. thus, fuzzy vehicle route selection with an adaptive neuro fuzzy inference system in uncertainty ... 17 reasoning can be used as a technique by which descriptive heuristic rules are translated into automatic management strategy i.e. decision-making. by developing a fuzzy system it is possible to transform the deployment strategy for vehicles on specific routes into an automatic control strategy. 2.1. description of the problem the problem being considered is the daily assignment of available vehicles for a specified number of transportation requests and transport routes. the vehicle for carrying out a transportation task comes from a base to which it is returned when the task is completed. the reasons for this tactical method are that transporting different types of load in the same vehicle is not allowed and the fact that different types of loads belong to different users. the problem under consideration belongs to the task of scheduling (assignment). the problem of scheduling belongs to the problem of linear programming, that is, the problem of transport. it consists of allocating n activities or resources to m the individual carrying out the action or the place, with the purpose of achieving maximum efficiency. in our case it means that it is necessary to define the goal function i.e. to allocate vehicles to transport routes with minimum transport costs within the limitations, and treating the problem as a mathematical programming problem. the main deficiency of an approach based on mathematical programming is that it is not easy to formulate the goal function and to determine the "hard" constraints. besides this, the information available to the dispatcher and drivers is often imprecise and given in descriptive form. as a result of the above, a conventional approach cannot take into account all of the relevant imprecise parameters. in the majority of cases, this phase in the decisionmaking process of traffic support organs is reduced to the experiential knowledge of the decision-maker. however, a problem arises when a decision regarding the selection of routes needs to be made by an individual without sufficient experiential knowledge. a solution to the given problem is presented in this paper using an anfis model. 2.2. designing the anfis model an integral part of an anfis model is a fuzzy reasoning system. the problems which an analyst encounters when developing a fuzzy system are determining the set of linguistic rules used by the dispatcher and the parameters of the membership functions of the input/output pairs. generating the membership functions of fuzzy sets and the rules according to which dispatchers act involves much communication with a large number of experienced dispatchers. membership functions of fuzzy sets, which describe the same concept, and which are proposed by different dispatchers can be very different. for this reason the characteristics of the developed fuzzy system depend on the number of available dispatchers and the ability to formulate their deployment strategy. it is intended for the fuzzy system to comprise of seven input variables, which are type of road surface, travel distance, travel time, route capacity, traffic capacity, road capacity and the existence of alternative roads along the length of the route. in addition to the eight input variables, the fuzzy system has a single output variable, preference of the dispatcher to select a particular route for carrying out a particular transport assignment. anfis implements a takagi sugeno kang (pap et al., 2000; pamučar et al, 2016b) fuzzy inference system in which the conclusion of a fuzzy rule is constituted by a weighted linear combination of the crisp inputs rather than by a fuzzy set. the relative pamučar & ćirović./decis. mak. appl. manag. eng. 1 (1) (2018) 13-37 18 importance of criteria and the degree of their influence on the dispatchers’ preference of choice are gained by normalization of weights (wki) in the following way (jovanović et al., 2014): 1 1 1 1 n n j ki jj ki nn j k j n j ki i j i k ki w w w w w                 (1) where  , 0,1j j   is the preference of the j-th decision maker, i.e., the degree of confidence, kiw is the weight ratio of the i criteria to the k decision maker; kiw normalization of weights; n is the total number of decision makers participating in the research. the degree of confidence is specified for each decision maker individually, based on their degree of confidence. in order to define the weight ratios, ten (n=10) decision makers were interviewed. the described criteria are listed in table 1. table 1. criteria for evaluating transport vehicle routes criterion num. ling. k k+ weights k1 type of road surface (trs) • • 0.18 k2 travel distance (td) • • 0.15 k3 travel time (tt) • • 0.12 k4 route capacity (rcc) • • 0.10 k5 road capacity (rc) • • 0.14 k6 traffic capacity (tc) • • 0.15 k7 the existence of alternative roads along the length of the route (ear) • • 0.16 the composite of ci (i=1,2,...,7) is made of two subsets: c+, subset of the criteria of beneficial type, higher values desirable and c-, subset of the criteria of cost type, lower values desirable. the values of some input variables are described by means of linguistic descriptors. defuzzification of the linguistic variables (criteria) k4, k5, k6 and k7 is carried out using the scale shown in table 2a, while defuzzification of the linguistic variable k1 is carried out using the scale shown in table 2b. the main problem faced by the analyst when developing a fuzzy system is determining the base of fuzzy rules and the membership function parameters of the fuzzy sets which describe the input and output variables. for all the input variables of the fuzzy model, as well as the type of membership function, it is also necessary to determine the number of membership functions for each input. a larger number of membership functions requires an increase in the number of rules, which can make setting up the system more difficult. it is therefore recommended, in accordance with the nature of the variables, to begin with the lowest number of membership functions. however, reducing the number of membership functions must not result in an incomplete description of the input variables. starting from the given postulates it is defined that input variables in the fuzzy model have at least three linguistic values. the membership function parameters and their nature are shown in table 3. gaussian curves are used in the fuzzy system since they describe the entry variables well and vehicle route selection with an adaptive neuro fuzzy inference system in uncertainty ... 19 ensure that the sensitivity of the system is satisfactory. adjusting the membership function to the form of a gaussian curve ensures the smallest error at the output of the anfis model. table 2. linguistic variables (criteria) k1, k4, k5, k6 and k7 a) linguistic variables (criteria) k4, k5, k6 and k7 linguistic variables triangular fuzzy number very low (0;0;0,1) low (0;0,1;0,3) medium low (0,1;0,3;0,5) medium (0,3;0,5;0,7) medium high (0,5;0,7;0,9) high (0,7;0,9;1) very high (0,9;1;1) b) linguistic variables (criteria) k1 road surface triangular fuzzy number dirt (0;0;0,25) natural (0;0,25;0,55) gravel (0,25;0,55;0,75) metalled (0,55;0,75;0,1) contemporary (0,75;1;1) when a comparison between the output of the fuzzy system and the desired set of solutions was made, the designed fuzzy system did not give satisfactory results. the difference between the expected results and the value of the criteria functions obtained at the output of the system was not satisfactory i.e. it was not within the limits of tolerance. an attempt to gain satisfactory values by changing the type and parameters of the membership functions at the output did not give the expected results. table 3. parameters of the membership functions before training the anfis model mf/ input mf 1 mf 2 mf 3 k1 gmf (0,10;0,04;0,12;0,09) gmf (0,1404;0,478) gmf (0,16;0,88;0,10;1,08) k2 gmf (0,169;0,09178) gmf (0,1924;0,492) smf (0,3608;0,9748) k3 zmf (0,072;0,6053) gmf (0,139;0,5092) smf (0,3659;0,8917) k4 zmf (0,4249;2,34;0,0821) gmf (0,16;0,46;0,15;0,53) smf (0,147;0,856) k5 gmf (0,17;0,1618) gmf (0,21;1,462;0,534) gmf (0,201;0,8424) k6 gmf (0,246;2,247;6,94) gmf (0,214;0,50) gmf (0,18;0,89;0,14;1,04) k7 zmf (0,518;3,825;0,18) gmf (0,21;1,402;0,50) smf (0,5439;0,914) *zmf (z-shaped membership function), smf (s-shaped membership function), gmf (gaussian membership function). table 4, shows the comparative values of the criteria functions at the output of the fuzzy system (ffis) and the required criteria functions (fdispatcher). in addition to the criteria functions in table 4, it also shows the values of the criteria on the basis of pamučar & ćirović./decis. mak. appl. manag. eng. 1 (1) (2018) 13-37 20 which the transport route is chosen, which are at the same time the input variables of the fuzzy system. in the example given in table 4 a total of 25 transport requests were processed, and a schedule was completed for 25 transportation routes. table 4. characteristics of twenty five transportation routes route no. k1 k2 k3 k4 k5 k6 k7 *fdispatcher *ffis 1. 0.992 134.75 0.281 5 0.204 0.044 0.237 0.51 0.339 2. 0.795 119.07 0.147 8 0.787 0.437 0.329 0.56 0.487 3. 0.913 121.19 0.426 12 0.482 0.625 0.410 0.64 0.799 4. 0.071 91.66 0.076 12 0.058 0.583 0.283 0.22 0.197 5. 0.509 26.35 0.902 11 0.041 0.352 0.219 0.48 0.290 6. 0.638 114.95 0.374 9 0.844 0.568 0.119 0.60 0.418 7. 0.924 85.72 0.726 12 0.929 0.811 0.748 0.97 0.508 8. 0.087 119.79 0.308 7 0.954 0.761 0.817 0.94 0.758 9. 0.915 148.74 0.917 10 0.200 0.004 0.215 0.71 0.588 10. 0.270 139.12 0.44 5 0.133 0.999 0.788 0.72 0.658 11. 0.231 57.15 0.497 13 0.883 0.884 0.613 0.91 0.671 12. 0.066 147.34 0.888 10 0.537 0.529 0.712 0.96 0.462 13. 0.037 116.3 0.510 14 0.425 0.762 0.496 0.69 0.691 14. 0.905 134.38 0.128 7 0.471 0.678 0.712 0.67 0.731 15. 0.599 72.20 0.807 6 0.286 0.210 0.035 0.56 0.520 16. 0.567 126.18 0.576 11 0.285 0.757 0.588 0.76 0.812 17. 0.551 121.09 0.152 9 0.837 0.405 0.324 0.69 0.566 18. 0.954 28.01 0.143 14 0.555 0.446 0.947 0.82 0.631 19. 0.488 75.93 0.891 6 0.330 0.954 0.293 0.78 0.911 20. 0.636 113.79 0.587 12 0.464 0.250 0.116 0.61 0.496 21. 0.735 42.83 0.817 9 0.731 0.516 0.529 0.97 0.768 22. 0.666 96.42 0.971 8 0.223 0.052 0.800 0.95 0.650 23. 0.017 48.14 0.499 11 0.349 0.138 0.695 0.71 0.954 24. 0.950 62.12 0.666 11 0.929 0.029 0.367 0.98 0.611 25. 0.467 97.30 0.147 12 0.238 0.326 0.886 0.91 0.512 * ffis output of the fuzzy system and fdispatcher the required criteria functions by analysing the data given, an average error of 0.481 was obtained. since the desired values were not obtained by the fuzzy system, it was mapped into a fivelayered adaptive neural network (anfis), figure 1. vehicle route selection with an adaptive neuro fuzzy inference system in uncertainty ... 21 x1 x2 x3 x4 a2 a1 a3 b1 b2 b3 c1 c2 c3 d1 d2 d3 trs td rcc tt x5 x6 x7 e2 e1 e3 f1 f2 f3 g1 g2 rc tc ear y µ v 1 (y ) µ v2 (y) µ v3 (y) µv 8 (y ) µ v 9 (y ) preferential dispatcher µ v4 (y) µv5(y) µv6 (y) µv7 (y ) layer 1 layer 2 layer 3 layer 4 layer 5 o 1 i o 2 i o 3 i o 4 i o 5 i g23 figure 1. structure of the anfis system the fuzzy logic system was mapped into an adaptive neural network as the error which occurred at the output of the fuzzy system was unacceptable. in other words, the difference between the desired set of solutions and the set of solutions obtained by the fuzzy system was unacceptable. according to experts, acceptable error is less then or equal to 0.08. 50 100 150 200 2500 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 fdispatcher ffis figure 2. comparison between the fuzzy system output and the desired data set figure 2 shows a graph of the values of the criteria functions at the output of the fuzzy system and the required values of the criteria functions. as already mentioned, a neuro-fuzzy network consists of five layers. layer 1. the nodes of the first layer are verbal categories of the input variables quantified by fuzzy sets. each node of the first layer is an adaptive node, described by a membership function,  , 1, 2,...,5 ix x i  . the membership functions are described by the gaussian distribution characterised by two parameters c (the centre of the function) and σ (the width of the function) (pamučar et al, 2016c):   2 2 1 ,,           cx ecxgaussian (2) as the fuzzy rules are the if-then rules, „if premise, then consequence“, the categories of input variables quantified by fuzzy sets are shown as adaptive nodes of the first layer. pamučar & ćirović./decis. mak. appl. manag. eng. 1 (1) (2018) 13-37 22 layer 2. each node of this layer calculates the minimum value ( i ) of three input values of the adaptive neural network. the output values of the layer 2 nodes are the rule signification:      2 1 2 7...i i ii i a b co x x x        (3) layer 3. each i-node in this layer calculates the total weight ( i ) of the i-rule from the rule base by the following equation 3 1 , 1, 2,..., i i i n i i o i n         (4) layer 4. this layer has 8 adaptive nodes representing the preference that a certain link (node) of the network is allocated the highest "preferential dispatcher" value. each neuron in this layer is connected to the respective normalization neuron in the third layer, and also receives initial input signals x1, x2 ,... ,x7. a defuzzification neuron computed the "weighted consequent value" of a given rule as: 4 1 1 1 2 1 3 1( ), 1, 2,...,i i i io f p x q x r x s i n       (5) where n is the total number of rules in the rule base, and pi, qi, ri and si are consequent parameters of the rule i. layer 5. the only node of layer 5 is the fixed node where the anfis output result is calculated. this is a fuzzy set with defined degrees of membership of possible "preferential dispatcher" values of the given link (node) of the network. defuzzification is carried out in the fifth-layer node. the output result is a real number in the interval [0,1]: 1 1 1 2 1 3 1 8 8 5 1 1 8 1 1 ( )i i i i i i i p x q x o overalloutp r s t x fu                (6) where:       1 2 31 1 2 7 ...a b gx x x       (7)       1 1 22 1 2 7 ...a b gx x x       (8) ...       3 2 18 1 2 7 ...a a ax x x       (9) (  is the t-norm). 2.3. forming a data set for training the anfis model if with     ,l ux x r x r and     , , 0 1, ,l uy y r y r r x y x    , we define the fuzzy numbers which are used to evaluate the observed alternatives in relation to the defined optimization criteria, then for the fuzzy numbers x and y the following relationships are valid (pamučar et al, 2013):          , 0 1l l u ux y x r y r x r y r r       (10)         ,l l u ux y x r y r x r y r    (11) vehicle route selection with an adaptive neuro fuzzy inference system in uncertainty ... 23           , , 0 , , 0 l u u l kx r x r k kx kx r x r k        (12) let  1 2, ,..., na a a denote the set of routes evaluated by experts  1, 2,...,ge g k in relation to the observed set of criteria  1, 2,...,jc j n . then the problem of fuzzy multi-criteria decision making can be represented in the matrix form as: 11 12 13 1 21 22 23 2 31 32 33 3 1 2 3 ... ... ... ... ... ... ... ... ... n n n m m m mn x x x x x x x x d x x x x x x x x                   (13) where xij is the value of the criteria function of the given route ai in relation to a criterion cj. summarizing the values in the rows of the matrix d is carried out using the following transformation:     1 1 1 , kj m m m l u k kj kj k k k rs x x r x r                 (14) normalization of the summarized values in rows is carried out using the following transformation         1 1 1 1 1 1 1 , n nl u kj kj j ji k m m n m nl u kj kj k j k jk k x r x r rs s x r x r rs                          (15) the weight coefficient of each criteria is obtained by forming a matrix w in which comparison is made in pairs of criteria based on of decisions made by experts who participated in the study. 11 12 1 21 22 2 1 2 1 2 1 1 2 2 n n k k kn n k k c c c w w we w w w we w w e ww w w                         (16) by multiplying the entries of matrices d and w and by using the previously mentioned arithmetical operations, we obtain the final values of the criteria functions which describe the significance of each of the observed routes pamučar & ćirović./decis. mak. appl. manag. eng. 1 (1) (2018) 13-37 24 11 12 13 1 21 22 23 2 31 32 33 3 1 2 3 11 12 13 1 21 22 23 2 31 1 2 3 1 2 3 1 2 3 ... ... ... ... ... ... ... ... ... ... ... n n n m m m mn n n n n n x x x x w x x x x w wf d w x x x x wx x x x x w x w x w x w x w x w x w x w x                                              32 33 3 1 2 3 1 2 1 2 3 3 1 2 3 ... ... ... ... ... ... ... n m m m mn n nn f f w x w x w x w f fx w x w x w x w                                       (17) the next step is to determine the ideal solution from the given set of values of criteria functions. the ideal solution a  and the negative ideal solution a  are obtained using the relation (pamučar et al, 2017)  1 2, ,..., na f f f    (18)  1 2, ,..., na f f f    (19) where  1, 2,..., belongs to criteria which are maximizedj j m j  ,  1, 2,..., belongs to criteria which are minimizedj j m j   as the best alternatives, those which have the highest value fij in relation to the criteria which are maximized and the lowest fij in relation to the criteria which are minimized are chosen. the positive ideal and negative ideal solutions are represented by fuzzy numbers. the following relations describe the ideal positive solution ( a  )     , , 0 1 l u a a r a r r       (20) where                   1 1 1 1 , ,..., , , ,..., , 0 1 l l l l u u u u a r f r f r f r a r f r f r f r r             (21) the ideal negative solution a  is calculated in exactly the same way. the distance between the fuzzy numbers x and y is calculated as   1/ 2 1 2 2 0 , ( ) ( ) ( ) ( ) l l u u d x y x r y r x r y r dr                (22) the next step is to calculate n dimensional euclidean distances of all observed alternatives for the ideal and the negative ideal solution         1/ 2 1 2 2 0 + l u l u i ij j ij j j j j j d f r f r f r f r dr                            (23) vehicle route selection with an adaptive neuro fuzzy inference system in uncertainty ... 25         1/ 2 1 2 2 0 + u l u l i ij j ij j j j j j d f r f r f r f r dr                            (24) where 1, 2,..., .i m for each alternative the relative distance of the coefficients id  and id  is calculated according to the relation , 1, 2,...,i i i i d q i m d d        (25) where is the alternative ai closer to an ideal solution if iq  ( 0 1iq    ) is closer to a value of 1. 3. training the anfis model: supervised learning problem parameter learning for many control problems, system identification, adaptive control and classification problems can be reduced to a function approximation problem where, given a function, we want to adjust the fls parameters as to best approximate it. fls could be regarded as a nonlinear parametric mapping between the input and output. we can express it as ( , )y f x w where y is the scalar fls output, x is the n-dimensional input vector and w is the p -dimensional vector containing all the fls’s adjustable parameters. if there is a difference between the obtained and expected data, modifications are made to the connections between the neurons in order to reduce error i.e., membership functions are tuned into adaptive nodes. by training the neural net with numerical examples of made decisions, the initial forms of input/output functions of adherence to the phase of composites are readjusted. the values of the membership functions after training the anfis system are shown in table 5. after obtaining the values of the criteria functions, the processed experimental data are accessed using the clustering technique. by cluster we mean a finite number of similar points which can be classified into the same group, by one or more distinctive features. the center of the cluster can be considered as the representative of a group of data. in this way, a large amount of experimental data is reduced to a smaller number of representative cluster centers and the study continues with a smaller number of data. this processing of data is essential in order to remove unnecessary similar data, as well as contradictory data. table 5. values of function parameters after training the anfis system mf/input mf 1 mf 2 mf 3 k1 gmf(0,005;0,04;0,4;0,18) gmf (0,4588;0,478) gmf (0,39;0,86;0,1;1,07) k2 gmf (0,298;0,274) gmf (0,2853;0,555) smf (0,2896;1,27) k3 zmf (0,0232;0,8503) gmf (0,3851;0,376) smf (0,1705;1,01) k4 zmf (0,4464;1,13;0,0239) gmf (0,28;0,49;0,24;0,51) smf (0,0187;1,06) pamučar & ćirović./decis. mak. appl. manag. eng. 1 (1) (2018) 13-37 26 mf/input mf 1 mf 2 mf 3 k5 gmf (0,313;0,1793) gmf (0,367;2,022;0,66) gmf (0,2911;1,03) k6 gmf (0,598;2,07;0,1341) gmf (0,3305;0,607) gmf (0,346;0,0704;0,996) k7 zmf (0,634;1,99;0,183) gmf (0,346;1,11;0,535) smf (0,1682;1,01) *zmf (z-shaped membership function), smf (s-shaped membership function), gmf (gaussian membership function) the fuzzy clustering technique is used in this research [35]. nasibov and ulutagay developed an iteration procedure which is based on the minimization of the function representing the geometric distance from any given point to the cluster center, but with an additional weighting factor based on the membership function (µ) of the analyzed point (k). the distance between two points tested from the data set for variable v is calculated as the minimum negative value of similarity  mink v vd  (26) the degree of membership in a cluster (mik) for each point is defined as 2 1 1 ik q ik jk m d d           (27) where ik k id u c  is the distance of point k from the cluster centre ci and q weighting exponent. the two points which have the lowest value kd are considered to be the nearest points. since neuro-fuzzy networks have the ability to generalize the obtained data, for the study the set ( f) of 3550f  of the criteria functions was identified. by using the clustering techniques described and a toolbox developed in the matlab software package to implement the clustering techniques, the set fʹ was reduced to a total of fʹʹ=248 values of the criteria functions. a comparison of the set of criteria functions fʹ and fʹʹ can be seen in figure 3. 0 200 800 1400 2000 2600 3200 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 3800 4400 5000 figure 3. the set of criteria functions (points) before and after the application of clustering techniques vehicle route selection with an adaptive neuro fuzzy inference system in uncertainty ... 27 training of the adaptive neuro-fuzzy network is carried out with the set fʹʹ and a base of fuzzy rules is formed. if the initial set of criteria functions (f or fʹ) had been used for forming the base of rules and training the neuro-fuzzy system, all data would be treated with the same significance and it would be impossible to create a base of rules which, as output from the neuro-fuzzy system, give a result with the minimum deviation from the required values. with a type of system such as the one studied, it is possible to generate a neuro-fuzzy system with a minimum number of fuzzy rules. in addition, the time required for training the neuro-fuzzy system is significantly reduced. the proposed neural net is trained on 248 dispatcher decisions. when planning a trip or during a trip, the driver (dispatcher) can modify the relative importance of the various route attributes using some settings on a panel. this is a convenient way for specifying the driver's preference, which could be useful for planning a special purpose trip. the driver (dispatcher) inputs his origin and destination to the system, and a set of route candidates is obtained. for each route candidate, the attribute scores are inputs into the fuzzy-neural network, and the output is the overall score of that route candidate. with the computation of this overall score, a ranking of the set of route candidates is performed. the driver (dispatcher) can accept the recommendation from the system. alternatively, he can choose an alternate route. any derivation from the recommendation will be stored, and this information is used for forming the training pairs of the fuzzy-neural network. hence, the system can be made adaptive to the decision-making of the driver or dispatcher. the composite of data for training that neural net is gained by surveying the heads that have a minimum of 15 years’ working experience in the jobs of organizing transport support. the back propagation algorithm is used for training. the data form a training composite xk, k=1,2,…n, where n is the overall number of input values of the anfis model, which are periodically transmitted through the net. figure 4 shows a graph of the training process of the anfis model and the reduction in error at the output of the system. figure 4. error variation process during training of anfis while training the anfis model, the data from the training set are periodically passed through the network. training the anfis model was carried out in four phases, which lasted a total of 250 epochs. the first training phase of the anfis model was completed after 70 epochs. after completion of the first phase, an error of 0.250 was obtained at the output (figure 5a). in the following phase, after 120 epochs, an error of 0.1547 was obtained at the output (figure 5b), which compared to the previous phase is a 38.12 % reduction in error. the third phase of training the anfis model was completed after 200 epochs and an error of 0.089 was obtained (figure 5c), which in relation to the second phase is a reduction in error of 42.46 %. in the fourth and final pamučar & ćirović./decis. mak. appl. manag. eng. 1 (1) (2018) 13-37 28 phase, which was completed after 250 epochs, at the output from the model the error was 0.0353 (figure 5d), which compared to the third phase is a reduction in error of 60.33 %. upon completion of the fourth phase, it was concluded that the error obtained at the output of the anfis model was negligible (acceptable error ≤ 0.08). in addition, the conclusion is that the neuro-fuzzy network is trained and capable of generalizing to new entry data. the five-layered adaptive net is tested on twenty five dispatcher decisions. for each route, the data from the transport requirement are transmitted through the anfis system, hence gaining certain values of input functions. the transport route is chosen as:  maxr rf f (28) where r represents number of routes. after training, a sensitivity analysis of the anfis model was performed. the sensitivity analysis was conducted in seven phases. in each phase, the sensitivity of the system was analyzed on one input criterion. at the same time, in each phase of the sensitivity analysis each of the observed criteria were given values in the interval  min max,i ik k , where minik is the minimum value, and maxik is the maximum value of the input criterion. when changing the input parameters of the observed criterion the parameters of the remaining input criteria did not change. error = 0.0353 0 50 100 150 200 250 0 0.2 0.4 0.6 0.8 1 1.2 error = 0.250 a) 50 100 150 200 2500 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 error = 0.089 fdispatche r fanfis c) 50 100 150 200 2500 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 fdispatche r fanfis d) fdispatche r fanfis 0 50 100 150 200 250 0 0.2 0.4 0.6 0.8 1 1.2 error = 0.1547 b) fdispatche r fanfis figure 5. training data anfis output thus, different values of the output criteria functions of the anfis model were obtained. in each phase, a set of 40 input values of the criteria ik were passed through the anfis model as shown in table 6 (20 input values) and in table a.1 (20 input values). in this way, criteria function values were obtained which show response and sensitivity of the system to changing only one of the observed criteria. figure 6 shows the sensitivity of the anfis model and the values of the criteria functions obtained in phases i, ii, v and vii. vehicle route selection with an adaptive neuro fuzzy inference system in uncertainty ... 29 table 6. set of data used for analyzing the sensitivity of the anfis model no. trs td ... tc ear input k1 output f1 input k2 output f2 ... input k6 output f6 input k7 output f7 1. 0.788 0.871 8623.95 0.843 ... 0.343 0.605 0.502 0.933 2. 0.795 0.863 29478.21 0.289 0.606 0.790 0.889 0.489 3. 0.428 0.459 30419.01 0.286 0.783 0.871 0.910 0.382 4. 0.428 0.459 25087.84 0.422 0.511 0.709 0.838 0.705 5. 0.228 0.232 30889.4 0.272 0.796 0.888 0.917 0.355 6. 0.795 0.863 29635.01 0.301 0.588 0.777 0.881 0.534 7. 0.249 0.443 12857.52 0.767 0.356 0.617 0.595 0.916 8. 0.615 0.665 7055.96 0.887 0.335 0.598 0.463 0.942 9. 0.822 0.878 20383.87 0.518 0.404 0.668 0.741 0.811 10. 0.819 0.873 13171.12 0.422 0.503 0.703 0.821 0.724 11. 0.422 0.451 26342.23 0.355 0.538 0.723 0.866 0.641 12. 0.430 0.461 10819.13 0.671 0.379 0.636 0.685 0.859 13. 0.611 0.661 26655.83 0.347 0.549 0.738 0.866 0.605 14. 0.831 0.915 8153.55 0.887 0.341 0.601 0.481 0.938 15. 0.781 0.877 31359.8 0.712 0.362 0.620 0.619 0.902 16. 0.43 0.461 30262.21 0.689 0.375 0.635 0.677 0.866 17. 0.842 0.924 21167.87 0.546 0.402 0.665 0.733 0.822 18. 0.260 0.253 13798.31 0.789 0.351 0.613 0.569 0.926 19. 0.586 0.623 21167.87 0.258 0.83 0.897 0.923 0.297 20. 0.835 0.918 6271.96 0.887 0.321 0.585 0.401 0.945 the sensitivity of the model and the values of the criteria functions obtained in phases iii, iv and vi are shown graphically in appendix b (figure b.1.) by looking at the graph of the sensitivity analysis (figure 6 and figure b.1.) we can conclude that the output values of the criteria functions of the anfis model depend on the weight values of the criteria ki (table 1) and on the nature of the criteria themselves (benefit or cost criteria). figure 6 shows the four criteria which have the greatest weight as defined in the database of rules. sensitivity analysis showed that benefit-type criteria with higher input values correspond with higher values of the output functions. in addition, it was found that small changes in the values of input criteria with greater weight lead to proportional increase in the value of output functions. however, with cost-type criteria the value of the output functions is inversely proportional to the values of the input criteria. figure b.1. shows the four criteria with the lowest weight. by analyzing the graph in figure b.1. we can conclude that in the case of criteria with low weight, the conditions defined in table 1 are met. pamučar & ćirović./decis. mak. appl. manag. eng. 1 (1) (2018) 13-37 30 0 5 10 15 20 25 30 35 40 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 30 35 40 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 30 35 40 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 30 35 40 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 output f1 output f2 output f7output f5 b) anfis output (phases i, ii, v and vii) 0 5 10 15 20 25 30 35 40 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 input k1 0 5 10 15 20 25 30 35 40 0 0.5 1 1.5 2 2.5 3 3.5 x 10 4 input k2 a) anfis input (phases i, ii, v and vii) 0 5 10 15 20 25 30 35 40 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 30 35 40 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 input k7input k5 figure 6. sensitivity analysis of the anfis model (phases i, ii, v and vii) vehicle route selection with an adaptive neuro fuzzy inference system in uncertainty ... 31 4. results twenty five transport requirements are considered. the transport task is described in terms of the time of loading and unloading, location where the user is set, as well as the possibility of using alternative directions. table 7 shows a comparison between the results obtained using the developed anfis model and the preferences of the dispatcher when choosing a route for the transport requests. when selecting a route for each individual transportation request, four alternative routes were considered. based on the characteristics of the routes considered, the dispatchers and the anfis model identified the most suitable route for carrying out the given transport request. table 7. comparative review of decisions and anfis model number of transport requests selection of routes for the transport request dispatcher anfis 1. r3 r2, r3 2. r3 r3 3. r4 r4 4. r3 r3 5. r2 r2 6. r1 r1 7. r3 r3 8. r3 r3, r4 9. r3 r3 10. r3 r3, r4 11. r1 r1 12. r3 r3 13. r3 r3, r4 14. r3 r3, r4 15. r1 r1, r2 16. r3 r3 17. r3 r3, r4 18. r4 r4 19. r3 r3, r4 20. r1 r1, r2 21. r4 r4 22. r4 r4 23. r1 r1, r2 24. r1 r1 25. r4 r4 the numerical results of table 7 imply the applicability of the proposed model used as a decision-making tool for vehicle route assignment. as seen in table 7 the decisions regarding the selection of routes for transportation vehicles obtained at the output of the anfis model are identical to the decisions made by the dispatchers. for transport requests numbered 1, 8, 10, 13, 14, 15, 17, 19, 20 and 23, the anfis model offered other routes as an alternative, which is acceptable, and in some situations even desirable, since the pis has a hetergenous structure of its fleet. pamučar & ćirović./decis. mak. appl. manag. eng. 1 (1) (2018) 13-37 32 5. conclusion the hybrid neuro-fuzzy system briefly presented in this paper was successfully applied in designing an intelligent decision support system for route selection in uncertainty conditions. the research conducted proves that fuzzy neural networks are a very effective and useful instrument for the implementation of intelligent decision support systems for route selection. developing the anfis model enabled the deployment strategy for vehicles on transport routes to be transformed into an automatic control strategy. the performance of the developed system depends on the number of experienced transport support managers, and the ability of analysts, after long communication with them, to formulate a decision-making strategy. as a result of the research, it has been shown that the proposed adaptive fuzzy system, which has the ability to learn, can imitate the decision-making of transport support managers and demonstrate a level of expertise that is comparable to the level of their expertise. by reviewing the performance of the trained neural network i.e. the adjusted fuzzy system and the results obtained, we can conclude that the anfis model can reproduce the decisions of dispatchers with great accuracy, and thus allocate vehicles to meet transport requirements. this is particularly important in situations when a decision needs to be made by a transport support organ which has a lack of sufficient experiential knowledge and in conditions when making a quality decision is influenced by a large number of uncertainties. in addition, the results in table 7 imply the potential applicability of the proposed model used as a decision-making tool for route selection. the proposed methodology could be used to solve other complex traffic and transportation problems characterized by uncertainty and the need for on-line control. extension and modification of the proposed model for other operational cases may warrant more research. further effort in training the proposed neuro-fuzzy based model with more valid data is also needed for practical applications. acknowledgements the work reported in this paper is a part of the investigation within the research project tr 36017 supported by the ministry for science and technology, republic 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(2015). optimization of multi-fleet aircraft routing considering passenger transiting under airline disruption. computers & industrial engineering, 80,132–44. pamučar & ćirović./decis. mak. appl. manag. eng. 1 (1) (2018) 13-37 36 appendix a table a.1. set of data used for analyzing the sensitivity of the anfis model no. trs td ... tc ear input k1 output f1 input k2 output f2 ... input k6 output f6 input k7 output f7 1. 0.583 0.641 19913.47 0.598 ... 0.396 0.647 0.719 0.829 2. 0.845 0.925 22422.26 0.433 0.441 0.699 0.808 0.764 3. 0.830 0.912 4076.77 0.936 0.267 0.521 0.260 0.975 4. 0.770 0.862 4703.97 0.933 0.272 0.526 0.338 0.971 5. 0.430 0.261 21795.06 0.456 0.431 0.691 0.767 0.795 6. 0.620 0.675 21481.46 0.449 0.438 0.695 0.779 0.778 7. 0.621 0.676 10662.33 0.711 0.366 0.624 0.648 0.891 8. 0.423 0.238 25558.24 0.406 0.519 0.711 0.853 0.692 9. 0.605 0.482 20383.87 0.497 0.421 0.688 0.748 0.802 10. 0.601 0.637 13955.11 0.661 0.385 0.640 0.701 0.849 11. 0.644 0.762 29164.61 0.323 0.562 0.769 0.875 0.576 12. 0.810 0.842 17561.49 0.636 0.389 0.642 0.713 0.836 13. 0.621 0.676 11916.72 0.699 0.372 0.633 0.662 0.873 14. 0.600 0.643 30419.01 0.289 0.681 0.821 0.898 0.438 15. 0.661 0.811 6428.76 0.928 0.292 0.547 0.366 0.966 16. 0.661 0.811 6271.96 0.921 0.316 0.573 0.382 0.958 17. 0.405 0.255 16307.1 0.664 0.383 0.638 0.692 0.855 18. 0.260 0.253 9407.94 0.818 0.346 0.609 0.533 0.933 19. 0.589 0.643 6271.96 0.288 0.716 0.848 0.910 0.406 20. 0.830 0.912 4860.77 0.928 0.276 0.531 0.359 0.966 vehicle route selection with an adaptive neuro fuzzy inference system in uncertainty ... 37 appendix b a) anfis input (phases iii, iv and vi) b) anfis output (phases iii, iv and vi) 0 5 10 15 20 25 30 35 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 input k3 40 0 5 10 15 20 25 30 35 40 4 6 8 10 12 14 16 18 input k4 0 5 10 15 20 25 30 35 40 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 input k6 0 5 10 15 20 25 30 35 40 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 output f3 0 5 10 15 20 25 30 35 40 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 output f4 0 5 10 15 20 25 30 35 40 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 output f6 figure b.1. sensitivity analysis of the anfis model (phases iii, iv and vi) plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 4, issue 2, 2021, pp. 26-46 issn: 2560-6018 eissn: 2620-0104 doi:_ https://doi.org/10.31181/dmame210402026d * corresponding author. e-mail addresses: serif.demirdag@giresun.edu.tr (s.a. demirdag), selcuk.korucuk@giresun.edu.tr (s. korucuk), ckaramasa@anadolu.edu.tr (ç. karamaşa) evaluation of innovative management success criteria in hotel establishments: case study in giresun-turkey şerif ahmet demirdağ 1, selçuk korucuk 2 and çağlar karamaşa 3* 1 giresun university, bulancak kadir karabaş vocational school, department of tourism management, turkey 2giresun university, bulancak kadir karabaş vocational school, department of international trade and logistics, turkey 3anadolu university, faculty of business, department of business administration,turkey received: 21 december 2020; accepted: 17 april 2021; available online: 25 april 2021. original scientific paper abstract: the purpose of this study was to determine the success criteria for innovative management in hotel establishments that have a corporate identity in the turkish city of giresun as well as to find out the best hotel by ranking them based on the determined criteria. for the study, “multi-objective optimization by simple ratio analysis (moosra)” and “evaluation based on distance from average solution (edas)” methods were used. the results revealed that the success criteria for innovative management are beneficial in increasing operational efficiency, performance, and customer satisfaction. the results also show that managers are well aware of the success criteria for innovative management that should be prioritized, as well as those that are likely to lead to the success of their establishments and keep them ahead of the competition. while “presenting an innovative vision” was found as the most important success criteria, “use of in-hotel information sources” came out as the least important one. the hotels considered were then ranked following the identified criteria. key words: innovative management, success criteria, moosra, edas, hotels. mailto:serif.demirdag@giresun.edu.tr mailto:selcuk.korucuk@giresun.edu.tr mailto:ckaramasa@anadolu.edu.tr evaluation of innovative management success criteria in hotel establishments: case study... 27 1. introduction the concepts of organization and management date back to ancient times and cover every phase of human history. humans, as psychological and emotional beings, find it difficult to live alone, and always feel the need for others throughout their lives. it is for this reason that people have always come together in history leading to the emergence of organizations the need by humans to live in organized systems and the advances in technological innovations in the recent past has led globalization to occupy an important role in modern society. globalization is enhanced by, among other factors, the increasing global mutual relationships; the increasing mobility of commodity, service, money, information and culture and people; transnational technological advancements; the consideration of knowledge as a basic production component; the commitment to knowledge-based industries; the strength of international financial markets; and the rise of multinational companies. economic globalization can be defined as the development of economic relationships between countries occasioned by improved commodity, capital, and labor mobility as well as the condensation, and spread of mutual interactions across the world (fischer, 2003). in order to position themselves well with regard to the competition and improve their standing in the current market or venture into a new market, companies often find themselves faced with the prospect of introducing, or incorporating into their operations innovative methods and practices (tiftik, 2020). innovative activities, in this context, entail the efforts to develop new technologies, products, services and production processes (drucker, 2003). increased innovation activities call for innovative management within the firm. the success of innovative management depends on resources (people, equipment, technology, information, etc.) and the management capacity of the organization. strategically, innovative management is essential in drawing an innovation-friendly business model and keeping the firm on a competitive path (berghman et al., 2013). the economic structure brought about by the increased globalization has led to the heightening of the intensity of competition in the sectors in which the enterprises operate. in this economic structure, the enterprises need to increase their competitiveness in order to survive and this can only be effectively achieved if they start to act innovatively in ways that will reduce costs, and/or make a difference (memiş & korucuk, 2019). the highlighted changes have led to a move by the management systems in enterprises from a production-orientation to individualorientation, and then to a model that incorporates both approaches. today, the innovative management approach that is the subject of this study has started to dominate industry 4.0 and has adopted a management style focused on both the individual and production. emerging management styles and approaches to innovation have been found to impact different sectors differently, and this has led players in different sectors, in our case, the tourism sector which holds a major share in the gdps of most countries, to look for ways that suit their structures. the human factor is considered the most important in the tourism sector as it is humans who produce, provide and purchase services in the sector. it is therefore imperative that businesses in the sector give special attention to the human factor to attain improved efficiency and sustainability. the first step to achieving this is to strive to satisfy their employees and adopt a management style in this direction. various approaches are needed to realize the aforementioned issues. one of these approaches is undoubtedly an innovative management style. innovation plays an important role in giving businesses a competitive edge and survival in the market (burmaoğlu & şeşen, 2011). according to tucker et al. (2002), innovation is one of the demirdag et al./decis. mak. appl. manag. eng. 4 (2) (2021) 26-46 28 key performance indicators as an element of cumulative business success and is the responsibility of all business units. innovation is the process by which new ideas are transformed into results that create sustainable value through economic activity (baporikar, 2015). however, innovation is directly related to learning and change, innovation is uncertain, it has failures as well as successful results, and is often risky and costly (bayhan, 2004). it is widely accepted that an organization's ability to innovate is closely linked to its intellectual capital or ability to use information resources (subramaniam & youndt, 2005). innovation is an important factor that creates value and ensures sustainable growth for organizations. a study by tether (2005) finds that the focus of innovation differs with the sector. while players in the service sector focus on organizational innovation, manufacturing enterprises focus on product and process innovations. another study finds that innovation in the service sector is geared towards strengthening relationships with customers, increasing customer loyalty, reducing costs, and increasing market efficiency (bolton et al., 2007). innovative management is yet another way through which innovation can be implemented in an organization. hamel & bren (2007) defined the concept of innovative management as a structure that serves organizational purposes by greatly changing the usual organizational processes and how management is done. in other words, innovative management means that an organization manages technology, processes, and human relations in a way that supports and promotes innovation. this requires having certain strategic and organizational skills. an organization is said to have strategic skills if it has a long-term vision, the ability to identify and predict market trends, the ability to collect, process and absorb technological and economic information. and, organizational skills depend on the firm's ability to identify and manage risks, the level of collaborations between operational units, the level and quality of investment in research institutions, universities, consulting firms, customers and suppliers, and human capital (elçi, 2007). therefore, innovative management requires a systematic focus on many aspects. another key factor of innovation is the organizational culture. organizational culture emerges and develops following changes in different situations because the key component is influenced by changes in other elements (smith et al., 2008). culture plays an important role in the management processes as it highly influences what managers do and how (hamel, 2007). innovative management can generally be considered as an organizational practice that creates added value in organizational roles and structures (soylu & göl, 2010). it aides the organization in gaining a competitive advantage, adopting a flexible structure, ensuring internal and external customer satisfaction, and taking on an agile structure while reducing their costs. there are several success criteria for effective, efficient, and economical management in innovative management. some that have been identified in the research include “openness in information sources”, “use of in-hotel information sources”, “r&d expenditures / support for innovation”, “presenting an innovative vision”, “spreading innovative management to all units”, “new innovative management approaches”, “fundraising and tasks allocation”, “cooperation with other establishments and market size”, “participation in decisions and number of solutions”, and “training and idea generation” (tidd et al., 2001; feams et al., 2005; lukas & ferrell, 2000; de jong & hartog 2010; burmaoğlu & şenen, 2011). the purpose of this study was to determine the success criteria for innovative management and to find out the best hotel among the hotels that have a corporate identity in the city of giresun, turkey. the study examines the effect of the success criteria on the key components of competitive strength, business performance, and evaluation of innovative management success criteria in hotel establishments: case study... 29 timesaving. due to the complex structure of the problem, multi-criteria decisionmaking (mcdm) methods such as moosra and edas are selected. in the second part of the study, a literature review related to innovative management studies is presented. moosra and edas methods used in the study are explained in the third part. a case study is analyzed in the fourth part, and the conclusion and suggestions are presented in the last part. 2. literature review numerous academic studies have been conducted on innovation and innovative management across different sectors and businesses. the sectors and types of businesses considered include government institutions and information infrastructure (hendrick, 1994; young et al., 2001; sadriev & pratchenko, 2014, etc.), accounting systems (chenhall & langfield-smith, 1999), military systems (drezner et al., 1999), universities and different educational institutions (khoury & analoui, 2004; zhao & ordóñez de pablos, 2009; barnard & van der merwe, 2016), supply chains and distribution systems (soosay & sloan, 2005; majercak et al., 2016) and travel companies and the tourism sector (buhalis & o'connor, 2005; bolgova et al., 2016; chkalova et al., 2019). hendrick (1994), who investigates the design, development, and implementation of information systems, states that information systems are very important for the success of innovative management approaches at all levels of governments. young et al. (2001), who investigated the effects of top management and network in the adoption of innovative management practices in total quality management (tqm) in public hospitals, determined that institution factors were important determinants. sadriev & pratchenko (2014) examined the prerequisites of idea management systems in innovative management and analyzed modern management practices of different companies. they stated that production technologies need software support, evaluation, and the support of innovative ideas. chenhall & langfield-smith (1999) review innovations in management accounting systems (mas) in manufacturing firms operating in australia. they suggest that a commitment to the initiatives; a successful pilot application; incremental development; appropriate training; and integration with other processes and systems. drezner et al. (1999) examine innovative management in unmanned aerial vehicle programs related to military and defense systems and explain innovative management in terms of military systems. khoury & analoui (2004) set up an integrated and innovative model (sofia) to manage the performance evaluation process of full-time faculty members at palestinian public universities. they addressed a variety of topics including setting of clear institutional strategy, participation in goal setting, coaching, two-way communication between faculty members and their superiors, feedback, developing and rewarding faculty members. zhao & ordóñez de pablos (2009) looked at innovative management within an organizational learning model by analyzing innovative management as a school subject and the impact of organizational learning. they find that education plays an important role in promoting innovative power and encourages creative education. barnard & van der merwe (2015) examined the role of innovative management in institutional sustainability in higher education. they found that innovation in sustainable development is supported by decisive leadership on strategic direction, regular, flexible, and inclusive planning, regular culture climate surveys, constant monitoring of progress, and strategic agility that is essential to promote innovation among the entire workforce. soosay & sloan (2005), looking at the innovative demirdag et al./decis. mak. appl. manag. eng. 4 (2) (2021) 26-46 30 management approach in distribution centers, found that resistance to change is inevitable, individuals express resistance both secretly and explicitly, and a cycle of resistance and acceptance against an emotional change should be expected and actively managed. they also stated that employee involvement is an essential component for successful change management. in supply chains, majercak et al. (2016) stress the need to focus on innovation based on the position of the product life cycle. the literature on innovative management in the tourism and travel sector seem to have almost similar results. buhalis & o'connor (2005) addressed the changes in the tourism industry regarding information communication technologies (ict). they state that e-tourism and the internet have greatly improved the process of developing, managing, and marketing tourism products and destinations, and have led to the emergence of new opportunities and new challenges. they contend that only organizations that appreciate the opportunities offered by the advancement in these areas and that can successfully bring and manage their ict resources will be able to increase their innovation and competitiveness in the future. bolgova et al. (2016), who examined the innovative management in travel companies, stated that the search for innovative attitudes and management is very important for the transformation, change, and competitive advantage in the tourism sector. according to the study, it is the task of leaders to get the most out of the investment and innovative technologies used in the operations of travel agencies. chkalova et al. (2019) looked at the effects and functioning of an innovative mechanism to manage local systems in the tourism and recreation sector with a theoretical justification in the volga federal district of russia. they stated that innovative mechanisms are very important for the management of local tourism systems. chkalova et al. (2019) concluded that the suitability of the development of a local innovative system is largely determined by the current situation in the russian economy and the ever-increasing role of integration in innovative systems at regional and inter-regional levels. delanoy & kasztelnik (2020) studied the supporting role of innovative leadership and management decisions in canada. they argue that integrating innovative management processes such as demographic analysis, platform understanding, and communication methods are very important for any public businesses. they also state that in the age of social media, understanding innovative management and how consumers use open big data analytics resources will also help leadership practices. based on expert opinions on the management of innovative processes in agriculture and food safety they obtained, tokhayeva et al. (2020) emphasize that it is very important to encourage technological and innovative management practices regarding basic agriculture and food security, to encourage scientific development, and to contribute to information flows. in a study investigating the innovative management of green tourism and recreational agriculture, tao et al. (2021) argue that the development of leisure agriculture in the future will be based on experience, tourism, landscapes, and local cultures to highlight agricultural diversity, and therefore innovative management is important. zaika et al. (2020), who investigated the development of innovative management methods in a modern business environment, state that distinguished techniques emphasized by specific tools ensure the consistent application of innovative management at all levels of the information business environment. they also emphasized that it is very important to study and implement more innovative mechanisms and impact tools on the current economic processes in tourism. this study derives its unique value from the fact that in the literature review, no study was found that examines the success criteria for innovative management and which ranks evaluation of innovative management success criteria in hotel establishments: case study... 31 the hotels based on the determined criteria. it is also differentiated by methodology utilized as well as used and the field of application hence provides a substantial contribution to the literature. 3. case study in the study, a two-stage multi-criteria decision-making model was created to determine the success criteria for innovative management in hotel establishments and to evaluate the alternatives in the selection of the best hotel. figure 1. application steps of the study the processes in the study followed the order in the application steps shown in figure 1. first, success criteria for innovative management in hotels were determined using the literature review and expert opinions and supported by the decision model. since the criteria determined do not have the same level of importance, the criteria needed to be prioritized. this was achieved using the moosra method. based on the prioritized criteria, the selection for the best hotel based on the success criteria for the innovative management was done using the edas method. to determine the criteria, opinions were sought from a total of 13 experts: managers from hotels with three stars and above operating in giresun (10), officials from the directorate of culture and tourism (1), and academicians working in the field of tourism management at giresun university (2). the data were collected in january 2020, just before the decision to stop some services of hotels due to the coronavirus (covid-19). table 1 was created based on the studies (lukas & ferrell, 2000; tidd et al., 2001; de jong & hartog, 2010; burmaoğlu & şeşen, 2011; çapraz et al., 2014) in the related literature. literature view and expert opinions literature view and expert opinions literature view and expert opinions expert group moosra method edas method demirdag et al./decis. mak. appl. manag. eng. 4 (2) (2021) 26-46 32 table 1. decision criteria criteria openness in information sources (c1) use of in-hotel information sources (c2) r&d expenditures / support for innovation (c3) presenting an innovative vision (c4) spreading innovative management to all units (c5) new innovative management approaches (c6) fundraising and task allocation (c7) cooperation with other establishments and market size(c8) participation in decisions and number of solutions (c9) training and idea generation (c10) 4. methodology the methodology section consists of two sections. the first section includes moosra as a criteria-weighting method while the second section comprises edas as an approach for ranking alternatives. 4.1. moosra (multi-objective optimization by simple ratio analysis) the moosra method, developed by das et al. (2012), is one of the multi-purposes and optimization methods characterized by criteria, alternatives, or attributes of importance or individual weights (jagadish and ray, 2014). while, calculating the performance values of each alternative under the moosra method, the normalized performance values of the useful and non-useful criteria are obtained by the simple ratio method (baležentienė et al., 2013). the moosra method has been used in several areas such as machine selection (sarkar et al., 2015), project critical path selection (dorfeshan et al., 2018), waste disposal assessment (narayanamoorthy, 2020), laptop selection (adalı & işık, 2017), assessment of the quality of life (ömürbek et al. 2017) and optimization of edm process parameters (anitha & das, 2020). the steps of the moosra method are as follows (jagadish & ray, 2014): step i. creating the decision matrix: in this method, the process starts with the creation of a decision matrix seen as table 2 listing the alternatives and criteria, and the performance of the relevant criterion or alternative is created as in equation (1) below: 𝑋𝑖𝑗 = [ 𝑋11 𝑋12 ⋯ 𝑋1𝑛 𝑋21 𝑋22 ⋯ 𝑋2𝑛 𝑋𝑚1 𝑋𝑚2 ⋯ 𝑋𝑚𝑛 ] (1) evaluation of innovative management success criteria in hotel establishments: case study... 33 table 2. decision matrix for moosra criteria c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c1 7 9 3 5 7 7 7 7 5 7 c2 5 7 3 5 5 7 3 7 3 3 c3 7 5 9 7 7 9 5 5 3 5 c4 3 3 7 9 9 7 7 5 3 5 c5 3 5 5 7 3 5 7 7 9 3 c6 5 3 3 5 5 7 7 3 5 3 c7 5 5 7 7 9 9 3 3 3 5 c8 3 3 5 5 7 7 5 7 7 7 c9 7 3 5 3 7 5 3 5 7 5 c10 9 9 5 7 7 5 9 3 5 7 step ii. normalizing the decision matrix: the process of converting the attribute value to the 0-1 range is called normalization. in multi-criteria decision-making, the values in the decision matrix must be converted from different units to a uniform unit, and the normalization process is used for this purpose. 𝑋𝑖𝑗 ∗ = 𝑥𝑖𝑗 √∑ 𝑥𝑖𝑗 2𝑛 𝑖=1 (2) the value 𝑋𝑖𝑗 ∗ represents the normalized value of the ith alternative over jth. normalized decision matrix is seen as table 3. table 3. normalized decision matrix criteria c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c1 0.385 0.502 0.172 0.255 0.324 0.319 0.372 0.406 0.294 0.423 c2 0.275 0.390 0.172 0.255 0.232 0.319 0.159 0.406 0.176 0.181 c3 0.385 0.279 0.515 0.356 0.324 0.410 0.266 0.290 0.176 0.302 c4 0.165 0.167 0.400 0.510 0.417 0.319 0.372 0.290 0.176 0.302 c5 0.165 0.279 0.286 0.356 0.139 0.228 0.372 0.406 0.528 0.181 c6 0.275 0.167 0.172 0.255 0.232 0.319 0.372 0.174 0.294 0.181 c7 0.275 0.279 0.400 0.356 0.417 0.410 0.159 0.174 0.176 0.302 c8 0.165 0.167 0.286 0.255 0.324 0.319 0.266 0.406 0.411 0.423 c9 0.385 0.167 0.286 0.153 0.324 0.228 0.159 0.290 0.411 0.302 c10 0.495 0.502 0.286 0.356 0.324 0.228 0.478 0.174 0.294 0.423 step iii. defining alternative values: the performance values (yi) of all alternatives are calculated by taking the simple ratio of the weighted sum of useful and non-useful criteria. in this calculation, the following equation (3) is used. 𝑌𝑖 = ∑ 𝑤𝑗𝑥𝑖𝑗 ∗𝑔 𝑗=1 ∑ 𝑤𝑗𝑥𝑖𝑗 ∗𝑛 𝑗=𝑔+1 (3) step iv. sorting alternatives: in the last step, the alternatives are sorted. when alternatives are listed in descending order, the best alternative is the one with the highest value. alternative values determination and sorting are formed as table 4. demirdag et al./decis. mak. appl. manag. eng. 4 (2) (2021) 26-46 34 𝑌𝑖 = ∑ 𝑥𝑖𝑗 ∗𝑔 𝑗=1 ∑ 𝑥𝑖𝑗 ∗𝑛 𝑗=𝑔+1 (4) table 4. defining alternative values and sorting criteria ∑ 𝑤𝑗𝑥𝑖𝑗 ∗ 𝑔 𝑗=1 ∑ 𝑤𝑗𝑥𝑖𝑗 ∗ 𝑛 𝑗=𝑔+1 yi c1 2.485 0.967 0.389 c2 2.053 0.512 0.249 c3 2.220 0.847 0.382 c4 2.044 1.074 0.525 c5 2.101 0.839 0.399 c6 1.716 0.725 0.423 c7 2.087 0.861 0.413 c8 2.470 0.977 0.395 c9 1.958 0.747 0.381 c10 3.100 1.187 0.384 4.2. edas (evaluation based on distance from average solution) edas is one of the multiple criteria decision-making methods introduced to the literature by ghorabaee et al. (2015). edas method uses evaluations based on average solution distance in determining the most optimal alternative in the decision-making process. the authors who developed the method compared the edas method with other multiple criteria decision-making (mcdc) methods such as vikor, topsis, saw, and copras (complex proportional assessment) and tested the validity of the method (özbek & engür, 2018). edas has been used in several application areas such as information technology (stanujkić et al., 2018), textile (karabasević et al., 2018), transportation (vesković et al., 2020), hospital selection (gündoğdu et al., 2018), and supplier selection (ghorabaee et al. 2016). the steps followed in the edas method are as follows (ghorabaee et al., 2015). step i. creating the decision-making matrix (x): decision-making matrix is shown in equation (5) below. in the corresponding matrix, 𝑥𝑖𝑗; i represents the performance of the option based on criteria j. decision matrix for edas is seen as table 5. 𝑋 = 𝑋𝑖𝑗 = [ 𝑎11 𝑎12 ⋯ 𝑎1𝑛 𝑎21 𝑎22 ⋯ 𝑎2𝑛 ⋮ 𝑎𝑚1 ⋮ 𝑎𝑚2 ⋱ ⋯ ⋮ 𝑎𝑚𝑛 ] (5) table 5. decision matrix for edas alternatives c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 a1 3.70 4.20 3.40 3.30 4.40 4.10 4.05 3.10 3.15 3.25 a2 2.80 2.95 3.00 4.10 4.25 3.75 3.70 3.55 3.40 2.45 a3 4.15 4.00 3.70 3.60 3.15 2.45 2.30 2.15 2.20 2.75 a4 1.90 2.15 1.75 2.35 3.30 1.95 1.70 2.85 1.55 3.10 step ii. creating the mean values matrix (avij): in the second stage of the edas method, the average solutions matrix related to the evaluation criteria is determined with the help of equation (6). average values matrix is obtained as table 6. evaluation of innovative management success criteria in hotel establishments: case study... 35 𝐴𝑉𝑗 = ∑ 𝑥𝑖𝑗 𝑛 𝑖=1 𝑛 (6) table 6. average values matrix alternatives c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 a1 3.70 4.20 3.40 3.30 4.40 4.10 4.05 3.10 3.15 3.25 a2 2.80 2.95 3.00 4.10 4.25 3.75 3.70 3.55 3.40 2.45 a3 4.15 4.00 3.70 3.60 3.15 2.45 2.30 2.15 2.20 2.75 a4 1.90 2.15 1.75 2.35 3.30 1.95 1.70 2.85 1.55 3.10 average value 3.14 3.30 2.96 3.34 3.38 3.06 2.94 2.91 2.58 2.89 step iii. creating positive and negative distance matrices from average: a positive distance from average (pda) matrix and negative distance from average (nda) matrix are created for each criterion. the calculation of these values varies according to the benefit or cost characteristics of the criteria. 𝑃𝐷𝐴 = [𝑃𝐷𝐴𝑖𝑗 ]𝑛𝑥𝑚 (7) 𝑁𝐷𝐴 = [𝑁𝐷𝐴𝑖𝑗 ]𝑛𝑥𝑚 (8) in the equations given above, pda refers to the positive distance of the ith alternative to the average solution of jth criteria, and nda refers to the negative distance of the ith alternative to the average solution of jth criteria. if the evaluation criterion is benefit-oriented, equations (9) and (10) are used. 𝑃𝐷𝐴𝑖𝑗 = 𝑚𝑎𝑥(0,(𝑋𝑖𝑗 −𝐴𝑉𝑗) 𝐴𝑉𝑗 , 𝑗 ∈ 𝑏𝑒𝑛𝑒𝑓𝑖𝑡 𝑣𝑎𝑙𝑢𝑒 (9) 𝑁𝐷𝐴𝑖𝑗 = 𝑚𝑎𝑥(0,(𝐴𝑉𝑗−𝑋𝑖𝑗 ) 𝐴𝑉𝑗 , 𝑗 ∈ 𝑏𝑒𝑛𝑒𝑓𝑖𝑡 𝑣𝑎𝑙𝑢𝑒 (10) if the evaluation criterion is cost-oriented, equations (11) and (12) are used. 𝑃𝐷𝐴𝑖𝑗 = 𝑚𝑎𝑥(0,(𝐴𝑉𝑗−𝑋𝑖𝑗 ) 𝐴𝑉𝑗 , 𝑗 ∈ 𝑐𝑜𝑠𝑡 𝑣𝑎𝑙𝑢𝑒 (11) 𝑁𝐷𝐴𝑖𝑗 = 𝑚𝑎𝑥(0,(𝑋𝑖𝑗−𝐴𝑉𝑗) 𝐴𝑉𝑗 , 𝑗 ∈ 𝑐𝑜𝑠𝑡 𝑣𝑎𝑙𝑢𝑒 (12) average positive distance matrix and average negative distance matrix are formed as table 7 and 8 respectively. table 7. average positive distance matrix alternatives c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 a1 0.178 0.273 0.149 0 0.302 0.340 0.378 0.065 0.221 0.125 a2 0 0 0.014 0.228 0.258 0.226 0.259 0.220 0.318 0 a3 0.322 0.212 0.250 0.078 0 0 0 0 0 0 a4 0 0 0 0 0 0 0 0 0 0.073 demirdag et al./decis. mak. appl. manag. eng. 4 (2) (2021) 26-46 36 table 8. average negative distance matrix alternatives c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 a1 0 0 0 0.012 0 0 0 0 0 0 a2 0.118 0.106 0 0 0 0 0 0 0 0.152 a3 0 0 0 0 0.068 0.199 0.218 0.261 0.173 0.048 a4 0.395 0.349 0.409 0.296 0.023 0.363 0.422 0.021 0.399 0 step iv. calculating weighted total values: weighted total positive distances (spi) and weighted total negative (sni) distances are calculated using equations (13) and (14). the wj value in the equations expresses the importance of each evaluation criterion. 𝑆𝑃𝑖 = ∑ 𝑤𝑗 𝑥 𝑃𝐷𝐴𝑖𝑗 𝑚 𝑗=1 (13) 𝑆𝑁𝑖 = ∑ 𝑤𝑗 𝑥 𝑁𝐷𝐴𝑖𝑗 𝑚 𝑗=1 (14) whether alternatives are optimal or not varies depending on whether spi and sni values increase or decrease. weighted total positive spi and weighted total negative sni values are calculated and seen as table 9 and 10 respectively. table 9. weighted total positive spi values alterna tives c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 a1 0.069 0.068 0.057 0 0.121 0.144 0.156 0.026 0.084 0.048 a2 0 0 0.004 0.120 0.103 0.096 0.107 0.087 0.121 0 a3 0.125 0.053 0.096 0.041 0 0 0 0 0 0 a4 0 0 0 0 0 0 0 0 0 0.028 table 10. weighted total negative sni values alternatives c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 a1 0 0 0 0.006 0 0 0 0 0 0 a2 0.046 0.026 0 0 0 0 0 0 0 0.058 a3 0 0 0 0 0.027 0.084 0.090 0.103 0.066 0.018 a4 0.154 0.087 0.156 0.155 0.009 0.154 0.174 0.008 0.152 0 step v. normalizing weighted total distances: in step 5, the weighted and normalized nspi and nsni values of all alternatives are calculated with the help of equations (15) and (16). 𝑁𝑆𝑃𝑖 = 𝑆𝑃𝑖 𝑚𝑎𝑥𝑖(𝑆𝑃𝑖) (15) 𝑁𝑆𝑁𝑖 = 1 − 𝑆𝑁𝑖 𝑚𝑎𝑥𝑖(𝑆𝑁𝑖) (16) spi and spin values are computed for alternatives and seen as table 11. evaluation of innovative management success criteria in hotel establishments: case study... 37 table 11. spi and spin values alternatives 𝑆𝑃𝑖 𝑆𝑃𝑖n a1 0.773 1 a2 0.638 0.825 a3 0.315 0.407 a4 0.028 0.036 sni and snin values are calculated for alternatives and seen as table 12. table 12. sni and snin values alternatives sni snin a1 0.006 0.994 a2 0.130 0.876 a3 0.388 0.630 a4 1.049 0 step vi. calculating success scores for each alternative: in the last stage of the edas method, asi, which represents the success score to be used in performance evaluation, is obtained for each alternative by taking the average of the nspi and nsni values calculated in the previous stage. here, the alternative with the highest asi value is considered the best alternative (akbulut, 2019). 𝐴𝑆𝑖 = 1 2 (𝑁𝑆𝑃𝑖 + 𝑁𝑆𝑁𝑖) (17) evaluation scores and asi values are calculated and alternative ranking is obtained as table 13. table 13. evaluation scores and asi values alternatives 𝐴𝑆𝑖 ranking a1 0.977 1 a2 0.864 2 a3 0.519 3 a4 0.018 4 4.3. weighting criteria at this stage, using the moosra method, a dual comparison questionnaire was created to evaluate the criteria. the questionnaire was presented to 13 experts, who are stakeholders of the subject area: hotel managers (10), culture and tourism directorate officials (1), and academicians working in the field (2). the results of the analysis are presented below: table 14. criteria weights table c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 weight 0.389 0.249 0.382 0.525 0.399 0.423 0.413 0.395 0.381 0.384 ranking 6 10 8 1 4 2 3 5 9 7 demirdag et al./decis. mak. appl. manag. eng. 4 (2) (2021) 26-46 38 according to table 14, the most important criterion for innovative management success criteria is “presenting an innovative vision”. this was followed by “new innovative management approaches”, “fundraising and task allocation” and “spreading innovative management to all units”, respectively. on the other hand, the “use of in-hotel information sources” criterion was found as the least important criterion. the bottom of the list was rounded by “participation in decisions and number of solutions”, “r&d expenditures / support for innovation”, “training and idea generation”, “openness in information sources”, and “cooperation with other establishments and market size”, respectively as the least important criteria. 4.4. ranking of alternatives in determining the alternatives for the study, the opinions of the expert group were taken into consideration and four hotels with corporate identity were determined. for the determined alternatives, a 1-5 likert scale questionnaire was administered to the entire expert group. edas method was used to list these four alternatives. using the weights of the criteria determined using moosra, the best hotel with regard to success criteria for the innovative management was selected using the edas method. the ranking values obtained by the edas method are presented in the following table: table 15. ranking values with edas method a1 a2 a3 a4 value 0.977 0.864 0.519 0.018 ranking 1 2 3 4 according to table 15, a1 was chosen as the best hotel in giresun regarding success criteria for innovative management in hotel establishments. the overall ranking in the selection of the best hotel regarding the success criteria for innovative management was a1> a2> a3> a4. 5. sensitivity analysis it is important to review the results of the model according to the demands of decision-makers and different conditions. an essential component of the review is the detection of alternative ranking sensitivity in terms of varying decision makers’ judgments. for this study, a sensitivity analysis was done to present the alternative ranking according to the changes in criteria weight as per the judgments of the decision-makers (korucuk, 2019). if this level of rationality is demanded from an individual decision-maker, then mcdm methods used as a support to rational decision making should also satisfy the condition (pamučar et al., 2017) several scenarios are formed for examining the alternative rankings for sensitivity analysis. while the first scenario assigns equal criteria weights, others allow for the interchange of weights between criteria. the obtained criteria weights for six scenarios are given in the appendix a the results for the alternative ranking of the six different scenarios are presented in table 16. evaluation of innovative management success criteria in hotel establishments: case study... 39 table 16. sensitivity analysis results alternatives a1 a2 a3 a4 ranking 1 2 3 4 scenario 1. assigning equal weights to all criteria scenario ranking 1 2 3 4 scenario 2. the interchange between criteria having the highest weight and the lowest weight scenario ranking 1 2 3 4 scenario 3. the interchange between criteria having the second-highest weight and the second-lowest weight scenario ranking 1 2 3 4 scenario 4. the interchange between criteria having the third-highest weight and the third lowest weight scenario ranking 1 2 3 4 scenario 5. the interchange between criteria having the fourth highest weight and the fourth-lowest weight scenario ranking 1 2 3 4 the results of the sensitivity analysis show a similar alternative ranking for the six different scenarios, an indication of the strength of the study in terms of significance and validity. 6. conclusion and suggestions the innovative management approach has been considered to be effective in the execution of other innovation activities. therefore, the development of successful innovative management structures in organizations should be able to raise the performance levels of the organization and increase their productivity, while also positively impacting competitiveness. effective innovative management also clears the path for other innovative elements such as product/service, customer satisfaction, marketing, and process. successful innovative management ensures the satisfaction of internal and external customers and effective resource use. this is because successful management practices affect the competitiveness of a firm, its efficiency, customer satisfaction, the performance of its internal processes, and many more value-added factors. due to the range of factors involved, the determination of the success criteria for innovative management and the selection of the best hotel is considered a complex decision-making problem, and thus, the determination of the most suitable one among the alternatives requires the use of mcdm methods. the main purpose of this study was to determine the success criteria for innovative management and to select the best hotel in line with the established criteria. this process was performed using moosra and edas methods. the results show that the most important criterion for the determination of success criteria for innovative management in hotels was “presenting an innovative vision (c4)”. this was followed, respectively, by “new innovative management approaches (c6)”, “fundraising and task allocation (c7)” and, “spreading innovative management to all units (c5)”, respectively. the other end of the spectrum had “use of in-hotel information sources (c2)” as the least important criterion. this was followed by “participation in decisions and number of solutions (c9)”, “r&d expenditures / support for innovation (c3)”, “training and idea generation (c10)”, “openness in demirdag et al./decis. mak. appl. manag. eng. 4 (2) (2021) 26-46 40 information sources (c1)”, and “cooperation with other establishments and market size (c8)”, in that order. a1 was found to be the best hotel in giresun based on the success criteria for innovative management in hotel establishments outlined in this study. the overall ranking of the alternatives was a1> a2> a3> a4. with the effect of globalization, production approaches and management in the organizations have shown radical changes as mentioned in the previous sections of this study. as has been stated in the various news articles regarding the current pandemic (covid19), things in organizations are not going to go back to where they were. for this reason, establishments, operating in the service sector, such as hotels, should be very careful in this regard. this is because, for the hotel industry, most of the tourism activities are produced and consumed by people, and often require faceto-face relationships. with the opening of the hotels, the study recommends that the hotels should approach their production and management innovatively in the new order. establishments should adopt more human-oriented innovative management styles, and ensure that their employees are satisfied. to the best of our knowledge, the extant literature doesn’t have any similar studies. the methodology used and the area of application (hotels in the turkish province of giresun) give the study its originality edge. the application, on hotel businesses, of the numerous criteria compiled from the literature regarding the success criteria of innovative management can be considered as the basic limitation of this study. one of the limitations of the study was the smaller number of the expert group interviewed. the number could not be increased due to time constraints. another limitation of the study is that no similar criterion sets on the innovative management theme was found in both the opinions of the expert group and the literature review. the covid-19 pandemic could be cited as another factor that limited the study. due to the epidemic, the number of expert groups could not be increased. it is important to note that the results obtained with moosra and edas methods may change with the differentiation of experts. in future studies, these deficiencies could be eliminated by including the opinions of all relevant stakeholders and evaluating them using different mcdm methods. despite the limitations, it was determined, in the interviews with the expert group, that the results of the study supported the expectations of the decision-makers. however, modeling the real situation is both very difficult and complex because human decisions, expectations, and judgments cannot be expressed precisely in numerical terms and are ambiguous. in addition, the findings of the study only relate to businesses operating in the hotel sector. future studies may expand this by considering different sectors as well as use different multi-criteria decision-making methods in the selection and ranking. on the other hand, future studies could also look at how the success criteria for innovative management relate to each other, determine the nature of this relationship using appropriate mcdm and compare the results obtained with the results of the present study. comparisons could also be done using fuzzy and/or extensions-based mcdm methods. author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research received no external funding. evaluation of innovative management success criteria in hotel establishments: case study... 41 conflicts of interest: the authors declare no conflicts of interest. references adalı, e.a. & işık, a.t. 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(2009). school innovative management model and strategies: the perspective of organizational learning. information systems management, 26(3), 241-251. evaluation of innovative management success criteria in hotel establishments: case study... 45 appendix a table a1. scenario 1 alternative ranking c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 weight 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 a1 a2 a3 a4 value 0.998 0.819 0.447 0.096 ranking 1 2 3 4 table a2. scenario 2 alternative ranking c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 weight 0.389 0.525 0.382 0.249 0.399 0.423 0.413 0.395 0.381 0.384 a1 a2 a3 a4 value 0.999 0.768 0.526 0.017 ranking 1 2 3 4 table a3. scenario 3 alternative ranking c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 weight 0.389 0.249 0.382 0.525 0.399 0.381 0.413 0.395 0.423 0.384 a1 a2 a3 a4 value 0.997 0.856 0.522 0.018 ranking 1 2 3 4 table a4. scenario 4 alternative ranking c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 weight 0.389 0.249 0.413 0.525 0.399 0.423 0.382 0.395 0.381 0.384 a1 a2 a3 a4 value 0.997 0.851 0.524 0.019 ranking 1 2 3 4 table a5. scenario 5 alternative ranking c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 weight 0.389 0.249 0.382 0.525 0.384 0.423 0.413 0.395 0.381 0.399 a1 a2 a3 a4 value 0.997 0.848 0.520 0.019 ranking 1 2 3 4 demirdag et al./decis. mak. appl. manag. eng. 4 (2) (2021) 26-46 46 table a6. scenario 6 alternative ranking c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 weight 0.395 0.249 0.382 0.525 0.399 0.423 0.413 0.389 0.381 0.384 a1 a2 a3 a4 value 0.997 0.756 0.412 0.018 ranking 1 2 3 4 © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 1, issue 2, 2018, pp. 93-110 issn: 2560-6018 eissn: 2620-0104 doi:_ https://doi.org/10.31181/dmame1802091p * corresponding author e-mail addresses: ivanpetrovic1977@gmail.com (i. petrović), kankaras.milan@outlook.com (m. kankaraš) dematel-ahp multi-criteria decision making model for the determination and evaluation of criteria for selecting an air traffic protection aircraft ivan b. petrović 1 *, milan kankaraš 1 1 university of defence, military academy, pavla jurišića šturma 33,11000 belgrade, serbia received: 29 march, 2018; accepted: 31 august, 2018; available online: 31 august 2018. original scientific paper abstract: this paper describes an approach in the determination and evaluation of the criteria and attributes of criteria for selecting the air traffic protection aircraft. after collected initial criteria and attributes, the interaction between criteria and attributes of criteria for the selection of the aircraft especially for the protection of air traffic was evaluated by 45 respondents. data processing and criteria and attributes determination were carried out by the dematel method (by eliminating less significant criteria and attributes). furthermore, the weight values of each criterion and attribute were determined by the ahp method. prioritization was carried out using an eigenvector method. for determination reliability the consistency ratio was checked for each result. as a result the model for the selection of the aircraft was proposed. key words: aircraft; air traffic; attribute; criterion; consistency; protection. 1. introduction from an economic point of view air traffic can be one of the more profitable business activities of each country. the organization and implementation of air traffic is complex process, which includes the need for continuous improvement (menon, sweriduk & bilimoria, 2004; chen, chen & sun, 2017; menon & park, 2016; steiner, mihetec & božičević, 2010; durso & manning, 2008; abbass, tang, amin, ellejmi & kirby, 2014). but, the issue of improving the protection of air traffic from aircraft threats has become particularly important since 9/11 (petrović, kankaraš & cvetković, 2015). mailto:ivanpetrovic1977@gmail.com mailto:kankaras.milan@outlook.com petrović & kankaraš /decis. mak. appl. manag. eng. 1 (2) (2018) 93-110 94 there are many approaches to address air traffic protection issues. the basic way is in the existence of duty–aircrafts (it is essentially a fighter aircraft that provide a rapid reaction in the case of airspace violation and other situations of violation of air traffic safety). some of small countries (in quantitative and qualitative terms) (gordić & petrović, 2014) give another countries the jurisdiction for the conducting of this mission. in the case study of the republic of serbia, which is a synonym for the small country, it can be noticed what are the criteria and how to prioritize them for the needs of equipping the country with the aircraft whose main purpose is to protect air traffic and intercept the aircraft that violated the airspace. the small area of the republic of serbia, and unusual, elongated form of territory allow for a short flight time over the territory and a simple and rapid airspace violation (petrović, 2013). therefore, it is necessary to determine which criteria and attributes of criteria are significant for the needs of equipping the country with the aircraft. the general objective of this paper is: determination and evaluation of criteria and attributes within the determinated criteria for selecting the aircraft for the purpose of air traffic protection from the airspace violation and other aviation threats using the dematel and ahp methods. this multi-criteria model consists of criteria and attributes that are significant for the selection of combat aircraft. the above stated research objective gives rise to following general hypothesis: using the dematel and ahp methods, it is possible to determinate and to evaluate the criteria for selecting the aircraft for the purpose of air traffic protection from the airspace violation and other aviation threats. the scientific and methodological contribution of paper is reflected in the new approach of determinating significant and eliminating less significant criteria attributes for the needs of selecting system with special role. also, the scientific contribution is reflected in increase of theoretical fund, which refers to the systematization of previous knowledge by the method of content analysis, and the gathering of relevant data about the criteria and attributes of criteria for the selection of the aircraft for the needs of conducting the missions during peacetime. the practical contribution is reflected in the fact that in the paper the model was created that could improve the process of equipping the system of defence with new equipment. also, modification of the model (by changing of criteria) enables its application in cases of procurement a wide range of equipment for the needs of realization of various forms of human activity. 2. materials and methodes the research was carried out in three phases: identification of initial criteria and attributes (for selection of combat aircraft), determination of significant criteria and attributes of criteria (for selection of aircraft), and prioritization of selected criteria and attributes (figure 1). in the first, all measures have been identified that enable selection of the combat aircraft by analyzing the contents of the relevant scientific fund (čokorilo, gvozdenović, mirosavljević & vasov, 2010; kirby, 2001; dagdeviren, yavuz & kilinc, 2009; petrović, cvetković, kankaraš & kapor, 2017). the selection and conceptual evaluation of military aircraft characteristics by applying the overall evaluation criterion (oec) was done by mavris & delaurentis (1995). the selection and evaluation of the criteria for equipping the army with combat aircraft using the ahp method was done by vlačić (2012). the identified measures are divided into general dematel-ahp multi-criteria decision making model for the determination and evaluation… 95 and specific measures using the classification method (based on the level of generality). the general measures represent criteria, and attributes are specific. taking into consideration the number and different significance of the identified criteria and attributes it was necessary to eliminate irrelevant and to evaluate significant criteria and attributes. it was carried out using the questionnaire, the dematel and the ahp method. based on these results, the model that provides a multi-criteria analysis of the selection of the aircraft for the air traffic protection from the airspace violation and other aviation threats was developed. figure 1. algorithm of a multi-criteria selection of the aircraft petrović & kankaraš /decis. mak. appl. manag. eng. 1 (2) (2018) 93-110 96 using the questionnaire and contents analysis of literature, the criteria and the attributes of each criterion (initial criteria and attributes) for the combat aircraft were selected. the following criteria are selected: aaerodynamics and mechanics of the flight, b construction and general systems, c propulsion, d – avionics and sensors, e integrated logistics support, f – armament, g – reconnaissance equipment, h – concept of pilot training and i – economy. the initial attributes of criterion aerodynamics and mechanics of the flight are: a1 – weight, a2 airspeed, a3 – acceleration performance, a4 – length of take off landing, a5 ceiling of flight, a6 – rate of climb, a7 – range of flight, a8 – maneuvering and stability performance, a9 – ability of supercruise and a10 – reaction time. the initial attributes of criterion construction and general systems are: b1 wing mechanization and flight control system, b2 – obstacle avoidance system, b3 – gps terrain-following, b4 – voice command system, b5 – oxygen system, b6 – radar crosssection and infrared signature, b7 – potential for modernization, b8– durability, b9 – ability of aerial refueling and b10 – possibility of ejection of pilot's seat. the initial attributes of criterion propulsion are: c1 – reliability and maintainability, c2 – maximum engine's thrust with afterburning, c3 – maximum engine's thrust without afterburning, c4 – thermal emission and c5 – maintenance system. the initial attributes of criterion avionics and sensors are: d1 – radars and other sensors, d2 – communication equipment, d3 – fire-control radar, d4 – electronic warfare equipment, d5 – multi-function display, d6 – navigation equipment, d7 – multimedia link. the initial attributes of criterion integrated logistics support are: e1 – reliability of aircraft, e2 convenience of maintenance, e3 – maintenance of aircraft, e4 – maintainability, e5 – ability of maintenance staff, e6 – maintenance equipment and e7 – infrastructure. the initial attributes of criterion armament are: f1 – capacity of locations for mounting armament, f2 – variety of armament, f3 – standardization of armament, f4 – number hardpoints of armament, f5 – under-fuselage hardpoints, f6 – possibility of using armament, f7 – safety work with armament on the ground, f8 – air – to – air missiles and rockets, f9 – bombs and other air to surface armament and f10 guns (cannons). the initial attributes of criterion reconnaissance equipment are: g1 possibility of reconnaissance in different weather conditions, g2 sensors range, g3 dataprocessing of reconnaissance information, g4 data-processing of reconnaissance photos and g5 data-processing of reconnaissance video. the initial attributes of criterion concept of pilot training are: h1 pilot training abroad, h2 individual training, h3 collective training and h4 simulators of flight. the initial attributes of criterion economy are: i1 – acquisition cost, i2 – life cycle costs and i3 – aircraft disposal costs. from initial criteria and attributes, the determination of criteria and attributes for the selection of the air traffic protection aircraft was preformed using the dematel method (moghaddam, sahafzadeh, alavijeh, yousefdehi, & hosseini, 2010; sumrit & anuntavoranich, 2013). by applying this method (decision – making trial and evaluation laboratory), based on the determination of direct and indirect influences between each criterion (attribute) on each citerion (attribute), criteria, which mutual impact on other criteria being less significant, were eliminated (moghaddam et al, 2010). dematel-ahp multi-criteria decision making model for the determination and evaluation… 97 each of the respondents (45 specialists – military pilots and officers of the aviation technical service) indicated the degree of direct and indirect influences between each criterion on each citerion and each attribute on each attribute of the criterion using the questionnarie. this step was done according to dematel method (sumrit & anuntavoranich, 2013). pairwise comparison was done as follows. the value of each pair is ranked by a number whose value is from 0 to 4 (0 – no influence; 1 – low influence; 2 – middle influence; 3 – high influence; 4 – very high influence) the assessment of each respondent is shown by a nonnegative matrix n n (for criterion 9n  ). each element of the k-matrix which is calculated by the equation 1 is a non-negative number k ij x , where is 1 k m  . k k ij n n x x      (1) matrices 1x , 2x ,..., mx represent individual preference (pairwise comparison) matrices of the respondents. the diagonal values are 0 because there is no influence between same criterions (sumrit & anuntavoranich, 2013). by calculating the means of the individual gathered values, a matrix of direct influences was created (table 1). table 1. matrix of direct influences of criteria k a b c d e f g h i a 0 3.85 3.92 3.45 3.73 3.68 0.45 0.54 3.9 b 2.17 0 2.04 3.12 1.45 1.72 0.53 0.34 1.14 c 2.94 1.11 0 1.73 1.14 0.94 0.52 0.32 0.85 d 3.65 3.2 3.91 0 3.17 3.2 0.61 0.29 3.23 e 3.42 3.17 2.12 1.92 0 2.73 0.45 0.34 2.45 f 3.18 2.57 3.14 3.22 2.72 0 0.32 0.35 2.74 g 0.51 0.42 0.37 0.32 0.41 0.38 0 0.42 0.39 h 0.33 0.42 0.44 0.41 0.37 0.39 0.51 0 0.19 i 3.92 3.17 2.93 3.45 3.15 3.08 0.28 0.32 0 in the second phase, the normalization of the matrix of direct influences is calculated using the following equation: 1 1 1 1 max max , max n n i ij i n iji j x d x x                (2) d – normalized matrix of direct influences, x – element of the mean value matrix of estimation of mutual influence. each element of the matrix of direct influences of criteria is divided with the maximum value of the sum of the columns and rows of the matrix of direct influence and new matrix is formed – normalized matrix of direct influence of criteria (table 2). petrović & kankaraš /decis. mak. appl. manag. eng. 1 (2) (2018) 93-110 98 table 2: normalized matrix of direct influence of criteria k a b c d e f g h i a 0.000 0.164 0.167 0.147 0.159 0.156 0.019 0.023 0.166 b 0.092 0.000 0.087 0.133 0.062 0.073 0.023 0.014 0.048 c 0.125 0.047 0.000 0.074 0.048 0.040 0.022 0.014 0.036 d 0.155 0.136 0.166 0.000 0.135 0.136 0.026 0.012 0.137 e 0.145 0.135 0.090 0.082 0.000 0.116 0.019 0.014 0.104 f 0.135 0.109 0.134 0.137 0.116 0.000 0.014 0.015 0.116 g 0.022 0.018 0.016 0.014 0.017 0.016 0.000 0.018 0.017 h 0.014 0.018 0.019 0.017 0.016 0.017 0.022 0.000 0.008 i 0.167 0.135 0.125 0.147 0.134 0.131 0.012 0.014 0.000 in the next phase, all the relations between each pair of the criteria are expressed by the matrix of direct influences. elements of matrix of full direct/indirect influence of criteria were derived by the equation 3 and the matrix is shown in table 3.   1 t d i d    in (3) , , 1, 2,... ij nxn t t i j n    t – matrix of full influence, i – unit matrix of influence, ij t element of the matrix of full influence. table 3: matrix of full influence of criteria k a b c d e f g h i a 0.383 0.486 0.509 0.470 0.451 0.448 0.083 0.072 0.436 b 0.299 0.194 0.288 0.310 0.236 0.244 0.059 0.042 0.214 c 0.279 0.200 0.164 0.220 0.188 0.180 0.050 0.037 0.170 d 0.484 0.435 0.479 0.313 0.406 0.405 0.083 0.058 0.389 e 0.409 0.376 0.353 0.331 0.233 0.336 0.066 0.051 0.313 f 0.429 0.378 0.416 0.398 0.359 0.254 0.066 0.056 0.343 g 0.070 0.062 0.063 0.058 0.057 0.056 0.009 0.025 0.055 h 0.058 0.057 0.060 0.056 0.051 0.052 0.030 0.006 0.042 i 0.490 0.433 0.443 0.439 0.404 0.401 0.070 0.059 0.268 by comparing the values in the matrix of full influence of criteria with the calculated threshold value it is determined whether the criteria are significant or not. namely, if all the values of one criterion are less than the threshold value, this criterion is not significant for the selection of the aircraft. the threshold value is calculated using the equation 4 and is 0.232. 1 1 n n ij i j t n          (4)  threshold value, n – full number of elements of matrix t. dematel-ahp multi-criteria decision making model for the determination and evaluation… 99 table 4. comparison of the elements of matrix of full influence of criteria with the threshold values of criteria k a b c d e f g h i a 0.151 0.254 0.277 0.238 0.219 0.216 -0.149 -0.160 0.204 b 0.067 -0.038 0.056 0.078 0.004 0.012 -0.173 -0.190 -0.018 c 0.047 -0.032 -0.068 -0.012 -0.044 -0.052 -0.182 -0.195 -0.062 d 0.252 0.203 0.247 0.081 0.174 0.173 -0.149 -0.174 0.157 e 0.177 0.144 0.121 0.099 0.001 0.104 -0.166 -0.181 0.081 f 0.197 0.146 0.184 0.166 0.127 0.022 -0.166 -0.176 0.111 g -0.162 -0.170 -0.169 -0.174 -0.175 -0.176 -0.223 -0.207 -0.177 h -0.174 -0.175 -0.172 -0.176 -0.181 -0.180 -0.202 -0.226 -0.190 i 0.258 0.201 0.211 0.207 0.172 0.169 -0.162 -0.173 0.036 by observing the obtained results it is concluded that two criteria (g and h) are not significant for the selection of the aircraft (table 4). in the same way attributes of selected criteria that are not relevant for the selection of the aircraft were eliminated (table 5-11). table 5. comparison of the elements of matrix of full influence of attributes of criterion aerodynamics and mechanics of the flight with the threshold values a a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a1 -0.236 -0.144 -0.145 -0.213 -0.201 -0.148 -0.135 -0.133 -0.209 -0.151 a2 -0.162 0.225 0.363 -0.150 -0.148 0.159 0.371 0.421 -0.156 0.360 a3 -0.153 0.436 0.249 -0.140 -0.141 0.316 0.418 0.436 -0.152 0.401 a4 -0.214 -0.134 -0.140 -0.234 -0.208 -0.155 -0.137 -0.128 -0.208 -0.128 a5 -0.224 -0.169 -0.169 -0.221 -0.239 -0.180 -0.166 -0.164 -0.217 -0.167 a6 -0.165 0.383 0.360 -0.149 -0.148 0.135 0.364 0.399 -0.159 0.331 a7 -0.179 0.207 0.169 -0.175 -0.174 0.096 0.082 0.237 -0.182 0.069 a8 -0.165 0.248 0.181 -0.163 -0.166 0.172 0.282 0.160 -0.171 0.244 a9 -0.211 -0.121 -0.120 -0.200 -0.210 -0.142 -0.116 -0.109 -0.233 -0.120 a10 -0.147 0.443 0.425 -0.138 -0.138 0.346 0.445 0.456 -0.147 0.246 the attributes a1, a4, a5 and a9 are eliminated. petrović & kankaraš /decis. mak. appl. manag. eng. 1 (2) (2018) 93-110 100 table 6. comparison of the elements of matrix of full influence of attributes of criterion construction and general systems with the threshold values b b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b1 0.303 -0.111 -0.091 -0.092 0.390 0.442 0.174 -0.095 -0.072 0.523 b2 -0.073 -0.192 -0.162 -0.161 -0.086 -0.083 -0.106 -0.160 -0.157 -0.068 b3 -0.102 -0.174 -0.194 -0.163 -0.117 -0.114 -0.126 -0.166 -0.166 -0.097 b4 -0.085 -0.163 -0.157 -0.192 -0.103 -0.089 -0.127 -0.162 -0.171 -0.087 b5 0.532 -0.105 -0.084 -0.093 0.220 0.394 0.269 -0.093 -0.088 0.517 b6 0.335 -0.112 -0.099 -0.104 0.285 0.189 0.223 -0.114 -0.107 0.427 b7 0.494 -0.100 -0.078 -0.082 0.389 0.379 0.143 -0.091 -0.084 0.498 b8 -0.081 -0.162 -0.156 -0.156 -0.100 -0.099 -0.119 -0.191 -0.153 -0.078 b9 -0.090 -0.176 -0.171 -0.170 -0.106 -0.095 -0.111 -0.157 -0.192 -0.084 b10 0.518 -0.106 -0.089 -0.092 0.281 0.435 0.314 -0.094 -0.090 0.330 the attributes b2, b3, b4, b8 and b9 are eliminated. table 7. comparison of the elements of matrix of full influence of attributes of criterion propulsion with the threshold values c c1 c2 c3 c4 c5 c1 0.001 0.147 0.051 0.368 0.263 c2 0.056 -0.218 -0.214 0.082 -0.140 c3 0.123 0.002 -0.172 0.184 0.015 c4 -0.070 -0.101 -0.053 -0.094 -0.009 c5 -0.051 -0.008 -0.047 0.090 -0.174 all attributes are accepted. table 8. comparison of the elements of matrix of full influence of attributes of criterion avionics and sensors with the threshold values d d1 d2 d3 d4 d5 d6 d7 d1 -0.083 0.030 -0.001 0.011 0.028 0.070 -0.100 d2 0.006 -0.071 -0.015 -0.013 0.028 0.070 -0.048 d3 0.140 0.147 -0.034 0.146 0.162 0.188 0.099 d4 -0.065 -0.062 -0.095 -0.135 -0.088 -0.011 -0.106 d5 -0.099 -0.053 -0.134 -0.118 -0.144 -0.065 -0.111 d6 0.099 0.115 0.060 0.080 0.097 0.008 0.037 d7 0.019 0.026 -0.008 -0.006 0.028 0.081 -0.109 all attributes are accepted. dematel-ahp multi-criteria decision making model for the determination and evaluation… 101 table 9. comparison of the elements of matrix of full influence of attributes of criterion integrated logistics support with the threshold values e e1 e2 e3 e4 e5 e6 e7 e1 0.114 -0.091 0.360 0.375 0.305 0.348 0.279 e2 -0.079 -0.212 -0.082 -0.085 -0.103 -0.088 -0.125 e3 0.287 -0.121 0.127 0.280 0.206 0.270 0.214 e4 0.139 -0.133 0.117 0.021 0.085 0.077 0.037 e5 0.154 -0.142 0.053 0.113 -0.020 0.064 0.022 e6 0.219 -0.134 0.201 0.187 0.131 0.058 0.102 e7 0.360 -0.102 0.328 0.333 0.281 0.307 0.102 the attribute e2 is eliminated. table 10. comparison of the elements of matrix of full influence of attributes of criterion armament with the threshold values f f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f1 -0.010 0.073 0.171 0.057 -0.179 0.163 0.132 0.029 -0.155 0.122 f2 0.021 -0.023 0.052 0.061 -0.176 0.037 0.094 0.087 -0.174 0.086 f3 -0.052 -0.056 -0.087 -0.044 -0.200 -0.041 -0.036 -0.065 -0.192 -0.034 f4 0.012 0.023 0.030 -0.031 -0.178 0.104 0.096 0.047 -0.181 0.134 f5 -0.174 -0.175 -0.158 -0.173 -0.243 -0.154 -0.148 -0.180 -0.222 -0.168 f6 0.089 0.125 0.193 0.158 -0.164 0.056 0.160 0.078 -0.156 0.217 f7 0.105 0.085 0.152 0.040 -0.182 0.088 0.009 0.014 -0.166 0.076 f8 0.251 0.256 0.298 0.237 -0.141 0.279 0.283 0.080 -0.129 0.253 f9 -0.181 -0.175 -0.170 -0.182 -0.227 -0.170 -0.155 -0.176 -0.242 -0.165 f10 0.189 0.214 0.218 0.151 -0.140 0.240 0.266 0.193 -0.145 0.113 the attributes f5 and f9 are eliminated. table 11. comparison of the elements of matrix of full influence of attributes of criterion economy with the threshold values i i1 i2 i3 i1 0.075 0.275 0.317 i2 0.097 -0.332 -0.041 i3 0.055 -0.108 -0.334 all three attributes are accepted. the evaluation of the selected criteria and attributes of criteria was performed by the ahp method (the analytich hierarchy process). the gathering data was carried out using the questionnaire which was adapted to scale of relative importance (saaty, 1980). using the standard scale, each element of comparasion ij a of matrix a can get one of 17 numerical values from a discrete interval [1/9, 9]. prioritization is conducted using the eigenvector method – ev (saaty, 1980). the criteria and attributes of criteria are pairwise compared by respondents. by calculating the mode of the individual gathered values, a pairwise comparison matrix was created (table12). petrović & kankaraš /decis. mak. appl. manag. eng. 1 (2) (2018) 93-110 102 table 12.pairwise comparison matrix (for criteria) k a b c d e f i a 1 4 4 3 4 3 2 b 0.25 1 2 0.5 0.5 0.333 0.5 c 0.25 0.5 1 0.333 0.5 0.5 0.5 d 0.333 2 3 1 2 3 2 e 0.25 2 2 0.5 1 0.5 0.5 f 0.333 3 2 0.333 2 1 0.333 i 0.5 2 2 0.5 2 3 1 based on values from the pairwise comparison matrix, a normalized pairwise comparison matrix was calculated by the equation 5 (saaty, 1980). 1 ij nij ij j a a a     (5) table 13. normalized pairwise comparison matrix (for criteria) k a b c d e f i a 0.343 0.276 0.250 0.487 0.333 0.265 0.293 b 0.086 0.069 0.125 0.081 0.042 0.029 0.073 c 0.086 0.034 0.063 0.054 0.042 0.044 0.073 d 0.114 0.138 0.188 0.162 0.167 0.265 0.293 e 0.086 0.138 0.125 0.081 0.083 0.044 0.073 f 0.114 0.207 0.125 0.054 0.167 0.088 0.049 i 0.171 0.138 0.125 0.081 0.167 0.265 0.146 from the table 13, the weight values w were calculated by the equation 6, which are shown in table 14. 1 n ij j i a w n     (6) i w weight value, ij a element of normalized pairwise comparison matrix table 14. weight values of criteria ( 0.055cr  ) k a b c d e f i w rank a 0.343 0.276 0.250 0.487 0.333 0.265 0.293 0.321 1 b 0.086 0.069 0.125 0.081 0.042 0.029 0.073 0.072 6 c 0.086 0.034 0.063 0.054 0.042 0.044 0.073 0.057 7 d 0.114 0.138 0.188 0.162 0.167 0.265 0.293 0.189 3 e 0.086 0.138 0.125 0.081 0.083 0.044 0.073 0.090 5 f 0.114 0.207 0.125 0.054 0.167 0.088 0.049 0.115 4 i 0.171 0.138 0.125 0.081 0.167 0.265 0.146 0.156 2 dematel-ahp multi-criteria decision making model for the determination and evaluation… 103 it can be noted (table 14) that the highest weight value in the selection of the aircraft for air traffic protection has the criterion of aerodynamics and flight mechanics (a), while the lowest weight value has criterion propulsion (c). checking the consistency of the results was tested by the consistency ratio applying the following equation (pamučar, 2017): cicr ri  (7) where is: ci consistency index. max 1 n ci n     (8) max  maximum eigenvector of the matrix of comparison. this value was calculated as follows: max 1 1 n i i n      (9) i i i b w   (10) value i b was calcualted as follows: 11 12 11 1 2 21 22 2 2 1 2 n n n nn n nn a a ab w b a a a w b wa a a                           (11) ij a represents the value of the element from the pairwise comparison matrix. ri random index, which depends on the number of rows columns of the matrix n (pamučar, 2017). for example, if 2n  , than is 0ri  , if 3n   0.52ri  , if 4n   0.89ri  , if 5n   1.11ri  , if 6n   1.25ri  , if 7n   1.35ri  , if 8n   1.4ri  . if 0.10cr  then the result is consistent. in this case, the consistency ratio is 0.055 and it is lower then 0.1, so the result is consistent (there is no need for corrections of the comparison). the weight values for attributes are determined in the same way. weight values for the attributes of each criterion are shown in the following tables. table 15. weight values for attributes of criterion aerodynamics and mechanics of the flight ( 0.03cr  ) a a2 a3 a6 a7 a8 a10 w1 rank a2 0.185 0.222 0.273 0.222 0.254 0.147 0.217 2 a3 0.046 0.056 0.045 0.037 0.028 0.088 0.050 6 a6 0.092 0.167 0.136 0.148 0.169 0.147 0.143 3 a7 0.061 0.111 0.068 0.074 0.042 0.088 0.074 5 a8 0.061 0.167 0.068 0.148 0.085 0.088 0.103 4 a10 0.554 0.278 0.409 0.370 0.423 0.441 0.413 1 petrović & kankaraš /decis. mak. appl. manag. eng. 1 (2) (2018) 93-110 104 table 16. weight values for attributes of criterion construction and general systems ( 0.01cr  ) b b1 b5 b6 b7 b10 w2 rank b1 0.404 0.412 0.316 0.343 0.490 0.393 1 b5 0.058 0.059 0.053 0.057 0.061 0.057 5 b6 0.134 0.118 0.105 0.086 0.082 0.105 4 b7 0.202 0.176 0.211 0.171 0.122 0.177 3 b10 0.202 0.235 0.316 0.343 0.245 0.268 2 table 17. weight values for attributes of criterion propulsion ( 0.02cr  ) c c1 c2 c3 c4 c5 w3 rank c1 0.162 0.222 0.222 0.176 0.147 0.186 2 c2 0.081 0.111 0.148 0.118 0.117 0.115 3 c3 0.054 0.056 0.074 0.118 0.084 0.077 4 c4 0.054 0.056 0.037 0.059 0.065 0.054 5 c5 0.649 0.556 0.519 0.529 0.587 0.568 1 table 18. weight values for attributes of criterion avionics and sensors ( 0.03cr  ) d d1 d2 d3 d4 d5 d6 d7 w4 rank d1 0.152 0.133 0.170 0.190 0.194 0.100 0.218 0.165 3 d2 0.076 0.067 0.068 0.095 0.129 0.067 0.036 0.077 5 d3 0.304 0.333 0.339 0.286 0.258 0.400 0.327 0.321 1 d4 0.038 0.033 0.057 0.048 0.032 0.067 0.036 0.044 7 d5 0.051 0.033 0.085 0.095 0.065 0.067 0.055 0.064 6 d6 0.304 0.200 0.170 0.143 0.194 0.200 0.218 0.204 2 d7 0.076 0.200 0.113 0.143 0.129 0.100 0.109 0.124 4 table 19. weight values for attributes of criterion integrated logistics support ( 0.03cr  ) e e1 e3 e4 e5 e6 e7 w5 rank e1 0.374 0.261 0.357 0.350 0.329 0.462 0.355 1 e3 0.124 0.087 0.036 0.100 0.082 0.077 0.084 5 e4 0.075 0.174 0.071 0.100 0.055 0.058 0.089 4 e5 0.053 0.043 0.036 0.050 0.041 0.058 0.047 6 e6 0.187 0.174 0.214 0.200 0.164 0.115 0.176 3 e7 0.187 0.261 0.286 0.200 0.329 0.231 0.249 2 dematel-ahp multi-criteria decision making model for the determination and evaluation… 105 table 20. weight values for attributes of criterion armament ( 0.04cr  ) f f1 f2 f3 f4 f6 f7 f8 f10 w6 rank f1 0.032 0.024 0.031 0.029 0.025 0.023 0.053 0.024 0.030 8 f2 0.065 0.049 0.125 0.114 0.041 0.023 0.053 0.043 0.064 5 f3 0.065 0.024 0.063 0.114 0.062 0.034 0.074 0.071 0.063 6 f4 0.065 0.024 0.031 0.057 0.062 0.034 0.074 0.071 0.052 7 f6 0.161 0.146 0.125 0.114 0.124 0.136 0.122 0.107 0.130 3 f7 0.097 0.146 0.125 0.114 0.062 0.068 0.074 0.043 0.091 4 f8 0.226 0.341 0.313 0.286 0.373 0.341 0.368 0.428 0.334 1 f10 0.290 0.244 0.188 0.171 0.249 0.341 0.184 0.214 0.235 2 table 21. weight values for attributes of criterion economy ( 0.02cr  ) i i1 i2 i3 w7 rank i1 0.621 0.600 0.692 0.638 1 i2 0.310 0.300 0.231 0.280 2 i3 0.069 0.100 0.077 0.082 3 3. results on the basis of the first two phases of the research, less significant criteria and attributes are eliminated. these criteria are: reconnaissance equipment and concept of pilot training. in the same way attributes of criterion aerodynamics and mechanics of the flight are eliminated: weight, length of take off landing, range and ceiling of flight and ability of supercruise. eliminated attributes of criterion construction and general systems are: obstacle avoidance system, gps terrain-following, voice command system, durability and ability of aerial refueling. also, attribute convenience of maintenance of criterion integrated logistics support is eliminated. the following attributes of criterion armament are eliminated: under-fuselage hardpoints and bombs and other air to surface armament. other attributes of selected criteria are significant for selection the air traffic protection aircraft. their determination was the objective of the first part of the research. determining differences in significance between criteria and attributes of criteria was the objective of the second part of the research (using the ahp method). prioritization of the criteria determined that the most significant criterion (table 14 and figure 2) is aerodynamics and mechanics of the flight (rank 1, weight 0.321), while the least significant is the criterion propulsion (rank 7; 0.057). attributes are also evaluated by prioritizing. the the most significant attribute of the criterion aerodynamics and mechanics of the flight (table 15) is reaction time, and the least significant attribute is acceleration performance. furthermore, for criterion construction and general systems the most significant attribute is wing mechanization and flight control system, and least significant is oxygen system. the most significant attribute of the criterion propulsion (table 17) is maintenance system and the least significant attribute is thermal emission. for the criterion avionics and sensors the highest weight value (table 18) has fire-control radar and the lowest weight value has electronic warfare equipment. for the integrated logistics support the most significant is reliability of aircraft and the least significant is ability of maintenance staff (table19). the air – to – air missiles and petrović & kankaraš /decis. mak. appl. manag. eng. 1 (2) (2018) 93-110 106 rockets are most significant for the criterion armament and the least significant attribute for the same criterion is capacity of locations for mounting armament (table 20). prioritization for the criterion economy is determined (table 21) that highest weight value has acquisition cost, in the middle is life cycle costs and the lowest weight value has aircraft disposal costs. for each weight value calculation, the consistency of the results was checked. since all consistency ratio were less than 0.1, it is concluded that there is consistency for all results of prioritization. considering all aforementioned, it is concluded that the objective of the research is achieved and the general hypothesis is proven and the model is proposed (figure 2). figure 2. proposed model for selection of the air traffic protection aircraft with weight values for criteria and attributes of criteria dematel-ahp multi-criteria decision making model for the determination and evaluation… 107 4. discussion on the basis of the results it can be concluded that there are criteria and attributes which are significant for equipping the army with the combat aircraft (vlačić, 2012), but which are irrelevant in peacetime for the purpose of air traffic protection in the case of airspace violation. for example, the most significant criterion for the combat aircraft is aerodynamics and mechanics of the flight, but also because of the multi roles, very significant is criterion reconnaissance equipment. the need for equipping two or three squadrons with the combat aircraft is the reason for the significance of the criterion concept of pilot training. despite the aforementioned, the criterion economy is less significant for equipping with the combat aircraft than in the case of equipping with the air traffic protection aircraft (vlačić, 2012). this difference as well as the difference in the significance of the selected criteria and attributes is a consequence of the overall picture of the organization and functioning of air traffic over the territory of the republic of serbia. small area, elongated form of territory, high frequency of traffic, geostrategic position, number of air routes, financial capabilities of the country, availability and classes of airports are only several factors that have an impact on the determination and evaluation of criteria for selecting the aircraft (for example, it is easy to notice that due to the form of the territory and the area of the country, the reaction time is very significant for aircraft the time required by duty aircraft to take prescribed measures on the ground after receipt of an airspace endagering warning, to take off to be navigated and to intercept an aviation threat). the differences in the significance of the factors are also a consequence of the fact the combat aircraft conducts a wide range of tasks such as: air-to-air combat, aerial reconnaissance, forward air control, electronic warfare, air interdiction, suppression of enemy air defence and close air support. these missions would be conducted by aircraft in extremely specific conditions. therefore, for selection of the aircraft are significant the following four overall evaluation criteria: affordability, mission capability, operational readiness and operational safety (mavris & delaurentis, 1995). it might be concluded that there are a lot of factors which impact on the determination and evaluation of criteria and attributes of criteria for selecting the air traffic protection aircraft. also, those criteria are specific due to mission that is conducted by air traffic protection aircraft, although it is essentially the aircraft designed for use both in peacetime and wartime. 5. conclusion air traffic is not immune to numerous security threats, including aviation threats. in the modern age, the possibility of occurrence of the airspace violation and other aviation threats is a reality. therefore, the protection of air traffic from aviation threats is a very important security mission all around the world. in small countries, this task is conducted by their own aviation or aviation of some other countries. there is no doubt that for each country it is better to conduct this mission with its own aviation. it is also important to know that the aircrafts whose mission is to protect the air traffic from aviation threats have to meet the relevant international standards and technological criteria. bearing in mind aforementioned and price of modern military aircrafts, the small countries usually make the decision to equip only petrović & kankaraš /decis. mak. appl. manag. eng. 1 (2) (2018) 93-110 108 a few aircrafts for the conducting of this mission. therefore, it is necessary to determine very precisely according to the criteria of equipping, which depend on the set of factors mentioned in this paper and because of it precise determination and evaluation of the criteria for the selection of the air traffic protection aircraft on the example of the republic of serbia was the subject of this research. for the purposes of this paper, traditional multi-criteria decision making methods are used and the model is proposed that can be applied in practice (and for the purpose of other countries that have simmilar teritorial characterics). by determining the mutual influence of the criteria (attributes) using the dematel method, the final definition of the criteria (atributtes) and their weights are calculated by ahp. the applied methods, the obtained results and the proposed model make this research scientifically and methodologically justified. furthermore, it is possible to propose similar models for the needs of equipping the system of defence with other types of equippment. above mentioned makes this research practical justified. in the future research, it is possible to select a specific aircraft using some other the multi-criteria decision making methods (topsis, mabac, vikor, mairca, etc.). also, the models for designing certain technological solutions according to user requirements can be created. furthermore, the application of similar models is possible for the purpose of implementing organizational changes in some organizational systems. future research can also focus on the development of similar models using traditional methods in combination with methods that take into account uncertainty – fuzzy numbers tipe one-two or rough or interval-valued rough fuzzy numbers, intuitionistic fuzzy numbers, etc (vahdani, tavakkoli-moghaddam, meysam mousavi & ghodratnama, 2013; sizong & tao, 2016; zywica, stachowiak, & wygralak, 2016, pamučar, petrović & cirović, 2018), which would significantly improve the field of multi-criteria decision making. acknowledgement: the paper is a part of the research done within the project vadh/3/17-19. the authors would like to thank to the the ministry of defence, and the project manager. references abbass, h., tang, j., amin, r., ellejmi, m., & kirby, s. 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(2013). soft computing based on new interval-valued fuzzy modified multi-criteria decision-making method, applied soft computing, 13, 165–172. doi.org/10.1016/j.asoc.2012.08.020 vlačić, s. (2012). definisanje kriterijuma za izbor višenamenskog borbenog aviona za potrebe vazduhoplovstva i protivvazduhoplovne odbrane vojske srbije. beograd: univerzitet odbrane. [unpublished doctoral dissertation] [in serbian] zywica, p., stachowiak, a. & wygralak, m. (2016). an algorithmic study of relative cardinalities for interval-valued fuzzy sets. fuzzy sets and systems, 294, 105–124. doi.org/10.1016/j.fss.2015.11.007 plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 4, issue 1, 2021, pp. 127-152. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame2104127b * corresponding author. e-mail addresses: mahmut.bakir@samsun.edu.tr (m. bakır), oatalik@eskisehir.edu.tr (ö. atalık). application of fuzzy ahp and fuzzy marcos approach for the evaluation of e-service quality in the airline industry mahmut bakır 1* and özlem atalık 2 1 department of aviation management, school of civil aviation, samsun university, samsun, turkey 2 department of aviation management, faculty of aeronautics and astronautics, eskişehir technical university, eskişehir, turkey received: 28 december 2020; accepted: 29 january 2021; available online: 16 february 2021. original scientific paper abstract: airlines today use e-services extensively for marketing activities and the distribution of services. monitoring and evaluating e-service quality are essential for customers’ satisfaction and thus the success of airlines. this study aims to evaluate e-service quality in the airline industry from the point of view of the consumers. to achieve this, an integrated fuzzy analytical hierarchy process (f-ahp) and fuzzy measurement alternatives and ranking according to compromise solution (f-marcos) approach was proposed to handle the uncertain and imprecise nature of e-service evaluation. in the first stage, eservice quality criteria were prioritized using the f-ahp method. then, a realworld case study was carried out on scheduled airlines to demonstrate the applicability of the proposed approach using the f-marcos method, utilizing a total sample of 395 airline passengers in turkey. as a result, the top three eservice criteria were found as reliability, understandability, and security. a three-stage sensitivity analysis was also conducted to examine the credibility and stability of the results. this study is the first study to integrate f-ahp and f-marcos methods for the first time in literature. key words: e-service quality; airlines; fuzzy ahp; fuzzy marcos; fuzzy sets theory. 1. introduction the spread of internet and information technologies (it) has profoundly affected many industries. with the growth of the internet, firms started to set up their websites and have offered their marketing and distribution activities through this channel (cheng, 2011). one of the industries most affected by this development is the airline file:///c:/users/green/appdata/local/appdata/local/temp/mahmut.bakir@samsun.edu.tr file:///c:/users/green/appdata/local/appdata/local/temp/oatalik@eskisehir.edu.tr bakır and atalık/decis. mak. appl. manag. eng. 4 (1) (2021) 127-152 128 industry. until the 1990s, airlines had distributed their services through travel agencies, call centers, and global distribution systems (gds) while later adopting the internet as an important outlet for the distribution of e-services. airlines, especially low-cost carriers, have gained a significant presence and power owing to online channels and geographic barriers losing their importance (díaz & martín-consuegra, 2016). thus, airlines have bypassed intermediaries and have the chance to reach consumers directly and at lower prices. the elimination of intermediaries has also provided opportunities such as reducing costs, reducing uncertainty in the use of eservices, sharing detailed product information, and continuing marketing activities (tsai et al., 2011). today, airlines offer many services to current and potential consumers over the internet. an airline website is a pivotal channel where these e-services are offered, and it includes many functions such as providing flight routes, price information, an interactive communication channel, online booking, ticket purchase, and online checkin (díaz & martín-consuegra, 2016). moreover, in addition to providing core services, complementary services, such as hotel booking and car rental, are available on these websites (harison & boonstra, 2008). therefore, it is necessary to create costeffective, content-rich, and attractive websites to increase e-service use. by doing so, the online presence and effectiveness of airlines will increase, thus increasing business performance, customer satisfaction, and loyalty (díaz & martín-consuegra, 2016; shankar & datta, 2020). in the existing literature, many studies have addressed the evaluation of e-service quality in the airline industry (elkhani et al., 2014; güreş et al., 2015; tarkang et al., 2020). due to the multi-dimensional nature of e-services, this situation can also be considered as a multiple criteria decision-making (mcdm) problem (büyüközkan et al., 2020; tsai et al., 2011). mcdm methods are applied in complex problems involving conflicting criteria to assist decision-making processes. in this regard, service quality, measured by quantitative and qualitative criteria, can be handled with mcdm methods (mardani et al., 2015). however, human judgments are often uncertain and imprecise in service quality evaluation (hu & liao, 2011). this study aims to evaluate e-service quality in the airline industry considering the uncertain and imprecise environment. in doing so, we propose an integrated fuzzy analytical hierarchy process (f-ahp) and fuzzy measurement alternatives and ranking according to compromise solution (fmarcos) approach. this proposed approach provides a systematic and well-defined solution for e-service quality evaluation in the airline industry. the contribution of this study to literature is twofold. first, this study employs an integrated f-ahp and fmarcos methods for the first time in literature. second, the e-service performance of scheduled airlines in turkey has been successfully evaluated utilizing a comprehensive framework. the rest of this paper proceeds as follows: section 2 reviews the concept of eservice quality and the existing literature within the airline industry context. section 3 explains the detailed algorithms for the f-ahp and f-marcos methods. section 4 presents the application of the real-world case study, followed by a three-stage sensitivity analysis. next, section 5 discusses the results. this study ends with a discussion of the managerial and theoretical implications, presenting research limitations and avenues for future research. application of fuzzy ahp and fuzzy marcos approach for the evaluation of e-service… 129 2. literature review fassnacht and koese (2006) defined e-services as “services delivered via information and communication technology where the customer interacts solely with an appropriate user interface (e.g. automated teller machine or web site) in order to retrieve desired benefits”. in literature, this concept is also discussed under different labels such as website quality and online service quality. however, it should be noted that the main focus is the needs of consumers in the virtual environment and what they expect from online services. researchers have discussed that e-service quality has implications for many key marketing concepts such as customer satisfaction, loyalty, and purchase intention, and it consequently affects financial outcomes positively. starting from this point of view, the exploration of e-service quality and the question of which critical factors e-services cover have attracted researchers. eservices provided in a virtual environment have distinctive characteristics, and since service quality scales are incapable of measuring e-services, e-service-specific scales have been proposed in literature (shankar & datta, 2020). considering the main scales, yoo and donthu (2001) developed the sitequal scale to evaluate e-service quality of online shopping websites. wolfinbarger and gilly (2003) used the etailq scale, consisting of fulfillment/reliability, privacy/security, website design, and customer service dimensions for retail e-services. parasuraman et al. (2005) applied the e-s-qual scale to evaluate e-commerce websites. based on the multi-dimensional nature of e-services, different second-order measurement models with various sub-dimensions have also been proposed (blut, 2016). reviewing the related studies, e-service quality is generally discussed in different contexts such as e-services, e-retailing, e-banking, and website-based services (shankar & datta, 2020). in addition, it is observed that proposed models are predominantly based on the technology acceptance model (tam) or the theory of reasoned action (tra). thus, the e-service quality scales are generally a combination of usefulness and ease of use concepts, as well as other related beliefs such as entertainment (cheng, 2011; loiacono et al., 2002). in the context of the airline industry, many studies have addressed e-service quality. generally speaking, the methodology used in these studies can be grouped as conventional statistical methods, usability testing, content analysis, and mcdm methods (chong & law, 2019). in the first group, elkhani et al. (2014) examined the effect of e-service quality on e-satisfaction and e-loyalty through e-servqual, emarketing, and expectancy disconfirmation theory (edt) frameworks. llach et al. (2013) analyzed the impact of e-service quality on perceived value and loyalty in spanish airline services. lee and wu (2011) added hedonics to the e-s-qual model and examined the relationships between website quality, perceived value, and satisfaction for 30 different airlines. vuthisopon and srinuan (2017) reported the positive impact of e-service quality on customer satisfaction in low-cost airlines in thailand. güreş et al. (2015) studied the relationship between e-service quality, passenger satisfaction, and passenger loyalty of domestic and international passengers in turkey. in a recent study, tarkang et al. (2020) examined the associations between airline website quality, electronic word of mouth, and purchase intention in turkey. in the second group, economides and apostolou (2009) introduced a holistic airline site evaluation framework (asef) model and analyzed the websites of 30 major airlines using multiple quality criteria from the perspectives of the consumers. díaz and martín-consuegra (2016) conducted a content analysis using 240 airline websites through six dimensions, including informativeness, usability, involvement, bakır and atalık/decis. mak. appl. manag. eng. 4 (1) (2021) 127-152 130 inspiration, credibility, and reciprocity. ceballos hernandez et al. (2020) evaluated the web-based e-service quality of 25 chinese airlines with content analysis. there are also some studies examining airline e-services using usability testing. ekşioğlu et al. (2013) analyzed the website quality of airlines in turkey through usability testing and heuristic evaluation. murillo et al. (2017) evaluated latam airlines’ critical website functions through usability testing and heuristic evaluation. although these studies provide powerful tools that incorporate qualitative and quantitative methods, they only aimed to identify usability problems on the websites (chong & law, 2019). finally, limited studies in the extant literature have evaluated e-service quality of airlines using mcdm methods. in fact, the mcdm approach is frequently preferred in the airline industry. badi and abdulshaded (2019) analyzed the overall performance of libyan airlines using ahp and the full consistency method (fucom). on the other hand, e-services are also very suitable to be evaluated using mcdm methods. it also enables the imprecise information that may arise in quality evaluations to be easily overcome using fuzzy logic (pamucar & ecer, 2020). for example, pamucar et al. (2018) applied ahp and mabac (multi-attributive border approximation area comparison) methods using interval rough numbers (irn) and evaluated faculty web pages. in the airline industry, tsai et al. (2011) examined e-marketing and e-service performance of airlines in taiwan using the decision making trial and evaluation laboratory (dematel), anp, and vikor methods. çelik and gök kısa (2017) presented an e-service quality evaluation in the turkish civil aviation industry by employing ahp and promethee methods in a fuzzy environment. abbasi et al. (2018) evaluated website quality of 5 airlines in iran using the methods of f-ahp and fuzzy technique for order preference by similarity to ideal solutions (f-topsis). similarly, bakır and atalık (2019) prioritized the factors affecting website quality of airline firms. more recently, büyüközkan et al. (2020) developed a digital service quality model for airlines using the interval-valued intuitionistic fuzzy ahp (ivif-ahp) method. apart from these, some other studies have analyzed the website performance of airlines using ordinary or fuzzy mcdm methods (dominic & khan, 2014; vatansever & akgül, 2018). however, it should be noted that these studies use many criteria such as website traffic and broken link based on web diagnostic tools. therefore, these studies focusing on technical issues do not reflect the consumer perspective. built on the consumer perspective, the present study contributes to the evaluation of e-service quality in airlines using a larger sample size. 3. research methodology this section covers the computation steps of the proposed f-ahp and f-marcos methods. the f-ahp method was used since it is the most frequently and successfully used methodology in service quality evaluation (mardani et al., 2015). the f-marcos is also a recent and reasonable method that combines the rate and the reference point approaches, thus providing more robust results under uncertainty (stanković et al., 2020). 3.1. preliminaries for triangular fuzzy sets in many real-world problems, human judgments and perceptions are not certain or precise. the fuzzy sets theory was proposed by zadeh (1965) to handle the uncertainty of judgments. fuzzy sets are sets with membership degrees defined as real application of fuzzy ahp and fuzzy marcos approach for the evaluation of e-service… 131 numbers in the interval [0;1] (lin, 2010). a triangular fuzzy number (tfn) can be defined as ( , , )a l m u and membership functions are found in eq. (1) as follows: 1 , ( ) , 0, x m l u x u ma l x m x m x u otherwise              (1) where l and u stand for the lower and upper bounds of the fuzzy number a , and m is the mid-value of a . if it is assumed that 1 1 1 1 ( , , )a l m u and 2 2 2 2 ( , , )a l m u are two tfns, the basic arithmetic operations for these two sets can be shown in eq. (2)(6) (lin, 2010; stanković et al., 2020): 1 2 1 1 1 2 2 2 1 2 1 2 1 2 ( , , ) ( , , ) ( , , )a a l m u l m u l l m m u u       (2) 1 2 1 1 1 2 2 2 1 2 1 2 1 2 ( , , ) ( , , ) ( , , )a a l m u l m u l l m m u u       (3) 1 2 1 1 1 2 2 2 1 2 1 2 1 2 ( , , ) ( , , ) ( , , )a a l m u l m u l u m m u l       (4) 1 1 1 1 1 1 1 2 2 2 2 2 22 ( , , ) , , ( , , ) a l m u l m u l m u l m ua         (5) 1 1 1 1 1 1 1 1 1 1 1 1 ( , , ) , ,a l m u u m l           (6) 3.2. the f-ahp method the ahp method is used to calculate criteria weights based on decision-maker (dm) judgments (saaty, 1980). the ahp method uses pairwise comparisons and allows both qualitative and quantitative criteria to be evaluated. to handle uncertainty, the ordinary ahp method was extended to the fuzzy sets theory, and fahp was introduced. in literature, the f-ahp method has been successfully applied in many studies, including in-flight service quality evaluation (li et al., 2017), the classification of container terminals (adenso-díaz et al., 2019), and the prioritization of traffic accessibility criteria (stanković et al., 2019). in this study, we adopted buckley’s (1985) approach, which received the least criticism in the existing literature (kahraman et al., 2018). the application steps can be summarized as follows (havle & kılıç, 2019; singh & prasher, 2019): step 1. construct a fuzzy pairwise comparison matrix. in this step, dms construct a pairwise comparison matrix of criteria using linguistic terms. in doing so, we adopted the nine-point conversion scale of anagnostopoulos et al. (2007) to convert responses into fuzzy numbers (see table 1). the resulting comparison matrix is given in eq. (7). 12 1 21 2 1 2 1 1 1 n n m m a a a a a a a             (7) bakır and atalık/decis. mak. appl. manag. eng. 4 (1) (2021) 127-152 132 table 1. the nine-point fuzzy conversion scale (anagnostopoulos et al., 2007) linguistic terms crisp scale tfs scale reciprocal tfn scale equally preferred 1 (1,1,1) (1/1,1/1,1/1) equally to moderately preferred 2 (1,2,3) (1/3,1/2/,1/1) moderately preferred 3 (2,3,4) (1/4,1/3,1/2) moderately to strongly preferred 4 (3,4,5) (1/5,1/4,1/3) strongly preferred 5 (4,5,6) (1/6,1/5,1/4) strongly to very strongly preferred 6 (5,6,7) (1/7,1/6,1/5) very strongly preferred 7 (6,7,8) (1/8,1/7,1/6) very strongly to extremely preferred 8 (7,8,9) (1/9,1/8,1/7) extremely preferred 9 (8,9,9) (1/9,1/9,1/8) step 2. aggregate the fuzzy pairwise comparison matrix. in case of group decisionmaking, the judgments of the dms are aggregated using eq. (8). 1/ 1 k k ij ijk k l l          , 1/ 1 k k ij ijk k m m          , 1/ 1 k k ij ijk k u u          (8) where ( , , ) ij ij ij a l m u and k denotes the number of dms. step 3. calculate the fuzzy weights matrix. in this step, the fuzzy comparison values are first calculated using eq. (9), as buckley (1985) suggested. 1/ 1 n n i ij j r a          , 1, 2,...,i n (9) then, fuzzy weights i w of criteria are calculated using eq. (10). 1 1 2 ( ... ) i i n w r r r r       (10) where i r represents the geometric mean of the fuzzy comparison values, while i w represents criteria weights. step 4. defuzzy fuzzy weights i w . since i w is a fuzzy number, it is defuzzified with the center of area (coa) method using eq. (11). ( ) / 3 i i i i w lw mw uw   (11) step 5. normalize the crisp weights. the calculated crisp values are normalized using eq. (12) and crisp criteria weights are obtained. 1 i r n i i w w w    (12) 3.3. the f-marcos method the marcos method has been proposed by stević et el. (2020) more recently. the basic principle of the marcos method is to find a solution based on the relationship between alternatives and reference values. accordingly, the utility functions of the alternatives are calculated based on the ideal and anti-ideal solutions indicating best application of fuzzy ahp and fuzzy marcos approach for the evaluation of e-service… 133 and worst values according to the criteria. similar to the topsis method, the best alternative is located closest to the ideal solution and farthest from the anti-ideal solution (puška et al., 2020). in literature, the marcos method has successfully been applied in sustainable supplier selection (stević et al., 2020), the evaluation of human resource (stević & brković, 2020), and the selection of software (puška et al., 2020). it also has been applied in fuzzy numbers (stanković et al., 2020), d numbers (chakraborty et al., 2020), and grey numbers (badi & pamucar, 2020). the steps of the f-marcos method can be summarized as follows (stanković et al., 2020): step 1. formulate a fuzzy aggregated initial matrix. first, a decision matrix, which includes m alternatives and n criteria, is established. in this step, the fuzzy anti-ideal ( )a ai and the fuzzy ideal ( )a id solutions are also determined using eq. (13). 1 2 11 12 1 1 2 1 2 ai ai ain n m m mn id id idn x x x x x x x x x x x x x                 (13) the fuzzy ideal ( )a id solution marks the desirable alternative, while the fuzzy anti-ideal ( )a ai solution shows the undesirable alternative. considering the criterion type, solutions ( )a id and ( )a ai are defined using eq. (14) and (15). ( ) min i îj a ai x if j b and maxi ijx if j c (14) ( ) max i îj a id x if j b and min i ijx if j c (15) where b denotes the benefit type criteria, and c denotes the cost type criteria. step 2. create the fuzzy normalized decision matrix. in this step, the fuzzy decision matrix including ( )a id and ( )a ai solutions is normalized using eq. (16) and (17). ( , , ) , , l l l l m u id id id ij ij ij ij u m l ij ij ij x x x n n n n x x x           if j c (16) ( , , ) , , l m u ij ij ijl m u ij ij ij ij u u u id id id x x x n n n n x x x           if j b (17) where the elements , , l m u ij ij ij x x x and ,l u id id x x are subtracted from the fuzzy decision matrix x . step 3. construct the fuzzy weighted-normalized decision matrix. the elements of the fuzzy normalized matrix are multiplied by weight coefficients using eq. (18). ( , , ) l l m m u u ij ij j ij j ij j v n w n w n w    (18) step 4. calculate the fuzzy summation matrix ( ) i s . in this step, the fuzzy weightednormalized matrix elements are summed using eq. (19). bakır and atalık/decis. mak. appl. manag. eng. 4 (1) (2021) 127-152 134 1 n i ij i s v    (19) for , , l m u ij ij ij n n n elements, the calculation is repeated and the column values are summed. step 5. determine the utility degree for each alternative. two different matrices ( i k  and i k  ) are constructed, taking into account the ideal and anti-ideal solutions. this procedure is performed using eq. (20) and (21). , , l m u i i i i i u m l ai ai aiai s s s s k s s ss          (20) , , l m u i i i i i u m l id id idid s s s s k s s ss          (21) step 6. construct the fuzzy combined matrix  it . the utility degree scores of the alternatives are summed using eq. (22). ( , , ) l l m m u u i i i i i i i i i t k k k k k k k k               (22) in this step, i t elements also need to be converted to a new fuzzy number  d using eq. (23). note that in doing so, the maximum values of the columns are used. ( , , ) max l m u i ij d d d d t  (23) in this step, the fuzzy number d is finally defuzzified by applying the formula 4 6 l m u crisp df   . by doing so, a crisp number is obtained. step 7. determine the utility functions of alternatives. based on the formula crisp df , the utility functions are calculated according to the ideal ( ) i f k  and anti-ideal ( ) i f k  solutions. in doing so, eq. (24) and (25) are applied. ( ) , , u m l i i i i i crisp crisp crisp crisp k k k k f k df df df df                (24) ( ) , , l m u i i i i i crisp crisp crisp crisp k k k k f k df df df df                (25) step 7 is finalized with the defuzzification of i k  , i k  , ( ) i f k  and ( ) i f k  values through the same defuzzification formula. step 8. calculate the defuzzified utility function ( ) i f k . using eq. (26), the final utility function score for each alternative is calculated. ( ) 1 ( ) 1 ( ) 1 ( ) ( ) i i i i i i i k k f k f k f k f k f k             (26) application of fuzzy ahp and fuzzy marcos approach for the evaluation of e-service… 135 step 9. alternatives are ranked according to the decreasing values of utility function ( ) i f k . 4. empirical case study and findings this section presents an empirical real-world case study in the turkish airline industry employing the proposed approach. in this approach, we aimed to demonstrate the applicability of the integrated f-ahp and f-marcos methods in eservice quality evaluation. to achieve this, we focused on the web-based e-services provided by scheduled airlines in turkey. a three-stage approach was adopted to address this problem, and the research framework of this study is given in figure 1. 4.1. the proposed hierarchical model the existing literature proposes many hierarchical models for measuring e-service quality from the perspective of the consumers. one of these models is delone and mclean’s (2003) updated information systems (is) success model. according to this model, information systems consist of three quality dimensions, namely: information quality, system quality, and service quality. in this study, this three-dimensional hierarchical model based on the is success model was used, which has since been successfully applied in many studies (chou & cheng, 2012; ecer, 2014; nilashi et al., 2012; tsai et al., 2011). the definitions of the hierarchical model elements are presented in table 2. define research objective literature review extract evaluation criteria and alternatives determine the decision-makers (dms) create a decision hierarchy decisionmakers (dms) construct pairwise comparison matrix get dms judgments through the fuzzy ahp method obtain criteria weights conduct a survey on the criteria by passenger aggregate and establish a fuzzy decision matrix evaluate alternatives through the fuzzy marcos method rank alternatives stage 1 stage 2 stage 3 conduct a sensitivity analysis figure 1. framework for research methodology bakır and atalık/decis. mak. appl. manag. eng. 4 (1) (2021) 127-152 136 table 2. e-service quality criteria and definitions criteria definition references informatio n quality (c1) it refers to the appropriateness of the information provided by the website. relevancy (c11) it means that the information on the website complies with customer needs and expectations. (chou & cheng, 2012; tsai et al., 2011; ustasüleyman, 2013) understan dability (c12) it expresses that the information on the website is clear and easy to understand. (blut, 2016; ecer, 2014; kim & stoel, 2004; y. lee & kozar, 2006) currency (c13) it means that the information on the website is current and timely. (ecer, 2014; lin, 2010; nilashi et al., 2012; tsai et al., 2011) richness (c14) it means how detailed the information is about the service provided by the website. (chou & cheng, 2012; ecer, 2014; ustasüleyman, 2013) system quality (c2) it refers to the technological equipment and infrastructural competence of the website. security (c21) it means the level of confidentiality and protection of customer information on the website. (blut, 2016; hu & liao, 2011; tsai et al., 2011) response time (c22) it expresses how quickly the website loads. (alptekin et al., 2015; lin, 2010; nilashi et al., 2012) personaliza tion (c23) it refers to the level of the website's ability to be personalized for the users. (blut, 2016; ecer, 2014; hu & liao, 2011) navigabilit y (c24) it means how easy it is to navigate around the website. (ecer, 2014; lin, 2010; tsai et al., 2011) accessibilit y (c25) it refers to how easily the website can be accessed. (alptekin et al., 2015; chou & cheng, 2012; lin, 2010) service quality (c3) it refers to the overall support to the users offered by the website. empathy (c31) it refers to how caring the information is and the attention paid to the users. (ecer, 2014; nilashi et al., 2012; tsai et al., 2011) responsive ness (c32) it expresses the level of willingness to provide service promptly and helpfully to online customers. (chou & cheng, 2012; hu & liao, 2011; tsai et al., 2011) reliability (c33) it expresses how accurate the services offered by the website are and the fulfillment of the promised service. (chou & cheng, 2012; fassnacht & koese, 2006; hu & liao, 2011) trust (c34) it means how well the reputation of the e-services is perceived and that using the website is reassuring and relaxing. (chou & cheng, 2012; ecer, 2014; lin, 2010; nilashi et al., 2012) 4.2. data collection process the questionnaire technique was employed as a primary data source in both the weighting and evaluation stages. in the first questionnaire, dms made pairwise application of fuzzy ahp and fuzzy marcos approach for the evaluation of e-service… 137 comparisons for dimensions and criteria, while in the second questionnaire, passengers evaluated airlines based on the same criteria. questionnaires for the fahp application were collected from 11 dms in the june-july period in 2017. the dms encompassed airline managers, website designers, and aviation academics. in the second step, the questionnaire was distributed to passengers to evaluate their perceptions about airline e-services. the responses were collected in the antalya airport, turkey during the period 6-10 november 2017. since five domestic airlines operate at the airport, the weighted stratified sampling approach was adopted based on the number of passengers they carried. a total of 395 valid questionnaires were obtained. demographic characteristics and the number of airline-based observations for the sample are given in table 3. table 3. characteristics of sample and strata variables alternatives percentage (%) gender male 65 female 35 age 18-25 27 26-35 41 36-45 24 46-55 6 56+ 2 educational status high school 22 university 60 postgraduate 18 travel purpose business 35 leisure 28 education 13 vfr (visiting friends and relatives) 18 other 6 airline passengers carried size of strata (n) x1 2,682,682 153 x2 1,915,506 110 x3 1,117,509 67 x4 564,745 32 x5 589,551 33 total 6,869,993 395 4.3. application of the f-ahp method in this section, pairwise comparison matrices for dimensions and criteria are constructed using eq. (7). then, using eq. (8), the responses from dms are aggregated through anagnostopoulos et al.’s (2007) scale into fuzzy numbers. the comparison results and local weights of e-service quality dimensions obtained using eq. (9)-(12) are given in table 4. bakır and atalık/decis. mak. appl. manag. eng. 4 (1) (2021) 127-152 138 table 4. fuzzy comparison matrix of dimensions c1 c2 c3 weight c1 (1.00,1.00,1.00) (0.87,1.28,1.85) (0.68,0.91,1.25) 0.348 c2 (0.54,0.78,1.15) (1.00,1.00,1.00) (0.51,0.72,1.08) 0.276 c3 (0.80,1.09,1.47) (0.93,1.39,1.98) (1.00,1.00,1.00) 0.386 table 4 shows that the most important dimension is service quality 3 ( 0.386) c w  , followed by information quality 1 ( 0.348) c w  and system quality 2 ( 0.276) c w  . after the analysis of the dimensions, the evaluation criteria were subjected to pairwise comparison. the aggregated comparison matrices expressing local criteria weights are given in tables 5-7. table 5. fuzzy comparison matrix for information quality criteria c11 c12 c13 c14 weight c11 (1.00,1.00,1.00) (0.57,0.72,0.96) (0.56,0.76,1.05) (1.22,1.75,2.36) 0.234 c12 (1.04,1.38,1.76) (1.00,1.00,1.00) (1.13,1.38,1.75) (1.86,2.66,3.59) 0.350 c13 (0.95,1.32,1.77) (0.57,0.72,0.88) (1.00,1.00,1.00) (1.72,2.33,2.86) 0.281 c14 (0.42,0.57,0.70) (0.28,0.38,0.52) (0.36,0.44,0.78) (1.00,1.00,1.00) 0.135 table 6. fuzzy comparison matrix for system quality criteria c21 c22 … c25 weight c21 (1.00,1.00,1.00) (3.50,4.39,5.22) … (1.18,1.63,2.09) 0.431 c22 (0.19,0.23,0.29) (1.00,1.00,1.00) … (0.54,0.72,0.99) 0.147 c23 (0.16,0.20,0.25) (0.31,0.37,0.45) … (0.26,0.36,0.55) 0.072 c24 (0.24,0.28,0.34) (0.65,0.83,1.12) … (0.46,0.58,0.79) 0.130 c25 (0.48,0.61,0.85) (1.01,1.40,1.86) … (1.00,1.00,1.00) 0.221 table 7. fuzzy comparison matrix for service quality criteria c31 c32 c33 c34 weight c31 (1.00,1.00,1.00) (0.75,0.97,1.21) (0.29,0.39,0.54) (0.40,0.53,0.71) 0.155 c32 (0.83,1.03,1.34) (1.00,1.00,1.00) (0.31,0.38,0.53) (0.46,0.70,1.11) 0.172 c33 (1.84,2.59,3.42) (1.90,2.61,3.27) (1.00,1.00,1.00) (0.94,1.21,1.58) 0.388 c34 (1.41,1.88,2.49) (0.90,1.42,2.18) (0.63,0.83,1.07) (1.00,1.00,1.00) 0.285 the judgments of the dms are consistent since the consistency ratio of the comparison matrices presented above is below 0.10 (ecer, 2014). lastly, the synthesizing procedure is carried out to find the global weight of each criterion. the local and global weights of the criteria are given in table 8. as can be seen in table 8, the most important criterion is found to be reliability 33 ( 0.146) c w  . following this criterion, understandability 12 ( 0.122) c w  and security 21 ( 0.119) c w  rank second and third, respectively. application of fuzzy ahp and fuzzy marcos approach for the evaluation of e-service… 139 table 8. global weights of the criteria dimension weight criteria local weight global weight c1 0.348 c11 0.234 0.081 c12 0.350 0.122 c13 0.281 0.098 c14 0.135 0.047 c2 0.276 c21 0.431 0.119 c22 0.147 0.040 c23 0.072 0.020 c24 0.130 0.036 c25 0.221 0.061 c3 0.386 c31 0.155 0.058 c32 0.172 0.065 c33 0.388 0.146 c34 0.285 0.107 4.4. application of the f-marcos method in this subsection, a real-world case study on the turkish airline industry is presented to evaluate e-service quality of airlines. air traffic has steadily increased in turkey and a total of 11 airlines are already based in the country (shgm, 2020). these airlines are composed of different carriers, such as scheduled airlines, cargo carriers and charter airlines. as we focused on the consumer market, cargo carriers and charter airlines that usually do not sell directly and trade their seats with tour operators were excluded from this study (williams, 2011). therefore, the remaining five scheduled airlines (x1-x5) were the subject of the study. using a five-point scale (1=“strongly disagree” to 5=“strongly agree”), a total of 395 passengers were surveyed to evaluate the e-services offered by scheduled airlines. the triangular fuzzy numbers (tfns) corresponding to these values are presented in table 9 (pandey & shukla, 2019). after collecting the questionnaire forms, they were aggregated via the arithmetic mean and fuzzy aggregated decision matrix including the fuzzy anti-ideal ( )a ai and the fuzzy ideal ( )a id solutions are given in table 10. table 9. five-point fuzzy rating scale range linguistic terms tfns 1 sd strongly disagree (0.0,1.0,2.0) 2 d disagree (1.0,2.0,3.0) 3 n neutral (2.0,3.0,4.0) 4 a agree (3.0,4.0,5.0) 5 sa strongly agree (4.5,5.0,5.0) bakır and atalık/decis. mak. appl. manag. eng. 4 (1) (2021) 127-152 140 table 10. average performance ratings and reference values x1 … x5 ai id c11 (3.67,4.39,4.82) … (3.42,4.25,4.91) (3.17,4.00,4.67 (3.67,4.39,4.91) c12 (3.62,4.35,4.82) … (3.53,4.28,4.78) (3.37,4.17,4.77) (3.62,4.35,4.88) c13 (3.83,4.51,4.87) … (3.50,4.28,4.84) (3.50,4.28,4.76) (3.83,4.51,4.88) c14 (3.34,4.13,4.71) … (3.53,4.31,4.88) (3.00,3.88,4.65) (3.53,4.31,4.88 c21 (2.87,3.69,4.33) … (2.61,3.50,4.28) (2.29,3.18,3.97) (2.87,3.69,4.33) c22 (3.25,4.05,4.65) … (3.11,3.97,4.69) (2.85,3.73,4.48) (3.25,4.05,4.70) c23 (2.49,3.39,4.21) … (2.53,3.44,4.25) (1.98,2.94,3.84) (2.82,3.68,4.41) c24 (3.40,4.16,4.67) … (3.81,4.53,4.97) (3.17,4.00,4.67) (3.81,4.53,4.97) c25 (3.55,4.27,4.71) … (3.73,4.44,4.84) (3.42,4.19,4.71) (3.73,4.44,4.85) c31 (3.00,3.84,4.53) … (2.66,3.63,4.56) (2.66,3.63,4.48) (3.18,4.01,4.66) c32 (2.47,3.39,4.23) … (2.89,3.75,4.47) (2.15,3.09,3.97) (2.89,3.75,4.47) c33 (3.32,4.12,4.72) … (3.61,4.38,4.91) (2.84,3.72,4.47) (3.61,4.38,4.91) c34 (3.15,3.96,4.59) … (3.34,4.16,4.78) (2.88,3.76,4.52) (3.34,4.16,4.78) the next step was the normalization process to eliminate the anomalies in the decision matrix. since all criteria are of the benefit type, the normalization procedure of alternatives and reference values was completed using eq. (17). the fuzzy normalized decision matrix is given in table 11. table 11. fuzzy normalized decision matrix x1 … x5 ai id c11 (0.75,0.89,0.98) … (0.70,0.87,1.00 (0.65,0.82,0.95) (0.75,0.89,1.00) c12 (0.74,0.89,0.99) … (0.72,0.88,0.98) (0.69,0.86,0.98) (0.74,0.89,1.00) c13 (0.78,0.92,1.00) … (0.72,0.88,0.99) (0.72,0.88,0.97) (0.78,0.92,1.00) c14 (0.69,0.85,0.97) … (0.72,0.88,1.00) (0.62,0.80,0.95) (0.72,0.88,1.00) c21 (0.66,0.85,1.00) … (0.60,0.81,0.99) (0.53,0.73,0.92) (0.66,0.85,1.00) c22 (0.69,0.86,0.99) … (0.66,0.84,1.00) (0.61,0.79,0.95) (0.69,0.86,1.00) c23 (0.56,0.77,0.95) … (0.57,0.78,0.96) (0.45,0.67,0.87) (0.64,0.83,1.00) c24 (0.68,0.84,0.94) … (0.77,0.91,1.00) (0.64,0.81,0.94) (0.77,0.91,1.00) c25 (0.73,0.88,0.97) … (0.77,0.92,1.00) (0.71,0.86,0.97) (0.77,0.92,1.00) c31 (0.64,0.82,0.97) … (0.57,0.78,0.98) (0.57,0.78,0.96) (0.68,0.86,1.00) c32 (0.55,0.76,0.95) … (0.65,0.84,1.00) (0.48,0.69,0.89) (0.65,0.84,1.00) c33 (0.68,0.84,0.96) … (0.74,0.89,1.00) (0.58,0.76,0.91) (0.74,0.89,1.00) c34 (0.66,0.83,0.96) … (0.70,0.87,1.00) (0.60,0.79,0.94) (0.70,0.87,1.00) after normalization, the fuzzy weighted-normalized matrix was constructed. in doing so, the fuzzy normalized matrix elements were multiplied by criteria weights using eq. (18). the fuzzy weighted-normalized matrix is given in table 12. application of fuzzy ahp and fuzzy marcos approach for the evaluation of e-service… 141 table 12. fuzzy weighted-normalized decision matrix x1 … x5 ai id c11 (0.06,0.07,0.08) … (0.06,0.07,0.08) (0.05,0.07,0.08) (0.06,0.07,0.08) c12 (0.09,0.11,0.12) … (0.09,0.11,0.12) (0.08,0.10,0.12) (0.09,0.11,0.12) c13 (0.08,0.09,0.10) … (0.07,0.09,0.10) (0.07,0.09,0.10) (0.08,0.09,0.10) c14 (0.03,0.04,0.04) … (0.03,0.04,0.04) (0.03,0.04,0.04) (0.03,0.04,0.04) c21 (0.08,0.10,0.12) … (0.07,0.10,0.12) (0.06,0.09,0.11) (0.08,0.10,0.12) c22 (0.03,0.03,0.04) … (0.03,0.03,0.04) (0.02,0.03,0.04) (0.03,0.03,0.04) c23 (0.01,0.01,0.02) … (0.01,0.01,0.02) (0.01,0.01,0.02) (0.01,0.02,0.02) c24 (0.02,0.03,0.03) … (0.03,0.03,0.03) (0.02,0.03,0.03) (0.03,0.03,0.03) c25 (0.04,0.05,0.06) … (0.05,0.05,0.06) (0.04,0.05,0.06) (0.05,0.05,0.06) c31 (0.04,0.05,0.06) … (0.03,0.05,0.06) (0.03,0.05,0.06) (0.04,0.05,0.06) c32 (0.04,0.05,0.06) … (0.04,0.05,0.06) (0.03,0.04,0.06) (0.04,0.05,0.06) c33 (0.07,0.09,0.10) … (0.08,0.10,0.11) (0.06,0.08,0.10) (0.08,0.10,0.11) c34 (0.10,0.12,0.14) … (0.10,0.13,0.15) (0.09,0.12,0.14) (0.10,0.13,0.15) in the next step, the fuzzy summation matrix ( ) i s was calculated using eq. (19). after this step, the utility degree of each alternative was determined based on the ideal and anti-ideal solutions. in doing so, fuzzy numbers i k  and i k  were calculated applying eq. (20) and (21). following this, the ideal and anti-ideal utility degrees of the alternatives were summed to construct the fuzzy combined matrix ( ) i t using eq. (22). however, the i t elements needed to be converted to fuzzy number ( )d using eq. (23) and subsequently defuzzified. the results obtained by applying eq. (19)-(23) are given in table 13. table 13. calculation of steps 4-6 through the fuzzy marcos application i s ik  i k  i t d ai (0.61,0.79,0.94) x1 (0.69,0.85,0.97) (0.73,1.08,1.59) (0.69,0.97,1.35) (1.42,2.05,2.95) (1.42,2.06,3.01) 2.11 x2 (0.66,0.83,0.97) (0.70,1.05,1.59) (0.66,0.94,1.35) (1.35,1.99,2.94) x3 (0.66,0.84,0.97) (0.71,1.06,1.59) (0.66,0.95,1.35) (1.37,2.01,2.95) x4 (0.62,0.80,0.94) (0.66,1.01,1.55) (0.62,0.91,1.31) (1.28,1.92,2.86) x5 (0.69,0.86,0.99) (0.73,1.09,1.62) (0.69,0.98,1.38) (1.42,2.06,3.01) id (0.72,0.88,1.00) as seen in table 13, the defuzzification was carried out as the final step. the elements of the matrix i t were summed as follows using eq. (22): 1 (0.73 0.69, 1.08 0.97, 1.59 1.35) (1.42, 2.05, 2.95) x t      then, a new fuzzy number d shows the maximum values of the column it . the maximum elements in this column were calculated as  , , (1.42, 2.06, 3.01)l m ud d d d  . in the defuzzification of the number d , the formula 4 6 l m u crisp df   was used and 1.42 4 2.06 3.01 6 2.11 crisp df     . the remaining bakır and atalık/decis. mak. appl. manag. eng. 4 (1) (2021) 127-152 142 application steps were carried out based on the crisp df values. firstly, the utility functions of alternatives related to the ideal  if k  and anti-ideal  if k  solutions were calculated using eq. (24) and (25). the calculated i k  , i k  ,  if k  and  if k  values were also converted to crisp numbers with the defuzzification procedure above. in the last step, the utility functions  if k of the criteria were calculated using these crisp values applying eq. (26). the results and ranking applied by eq. (24)-(26) are depicted in table 14. table 14. ranking of alternatives via utility functions  if k   if k  i k  i k   if k   if k   if k rank x1 (0.33,0.46,0.64) (0.35,0.51,0.75) 1.107 0.987 0.467 0.524 0.686 2 x2 (0.31,0.45,0.64) (0.33,0.50,0.75) 1.081 0.964 0.456 0.511 0.649 4 x3 (0.31,0.45,0.64) (0.33,0.50,0.75) 1.088 0.969 0.459 0.514 0.658 3 x4 (0.29,0.43,0.62) (0.31,0.48,0.73) 1.040 0.927 0.438 0.492 0.593 5 x5 (0.33,0.46,0.65) (0.35,0.51,0.77) 1.118 0.996 0.471 0.529 0.702 1 table 14 provides the ranking of airline alternatives. based on the table, the best alternative is x5. on the other hand, x4 is the worst alternative. 4.5. sensitivity analysis in this section, a sensitivity analysis was conducted to ensure the robustness of the application and validate the calculation. in doing so, our sensitivity analysis consisted of three parts to perform a rigorous analysis. in the first part, we tested the effect of the change in criteria weights on the calculation. considering keshavarz ghorabaee et al.’s (2018) guidelines, 13 simulated scenarios (set1-set13) were employed to generate different criteria weights (see figure 2). the ranking of the alternatives resulting from the scenarios is given in figure 3. as figure 3 shows, the ranking is largely stable except for a slight change in x2 and x3. figure 2. simulated weights for scenarios 0 0.03 0.06 0.09 0.12 0.15 set1 set2 set3 set4 set5 set6 set7set8 set9 set10 set11 set12 set13 c11 c12 c13 c14 c21 c22 c23 c24 c25 c31 c32 c33 c34 application of fuzzy ahp and fuzzy marcos approach for the evaluation of e-service… 143 figure 3. ranking of airlines under different scenarios the rank reversal effect is considered to be one of the major shortcomings of mcdm methods. it means that adding or removing an alternative in the decision matrix affects the ranking (stanković et al., 2020; yazdani et al., 2019). a scenariobased solution on dynamic decision matrices is presented in the second part. as mentioned above, the ranking of airlines was obtained as x5>x1>x3>x2>x4. accordingly, x4 is clearly the worst alternative. therefore, if this alternative is eliminated, in a robust calculation, the ranking of the remaining alternatives is expected to remain the same. to ensure this, we created dynamic decision matrices based on eliminating the worst alternative in each round and progressing until the last alternative remained. the scenario-based rankings in this study are presented in table 15. as shown in table 15, the worst alternative (x4) was deleted first, and then the remaining four alternatives are ranked in scenario 1. the alternative x2, which was the worst in the new ranking, was then deleted, and the application was finalized in scenario 3. it is clear that the ranking remained unchanged. table 15. rank reversal effect in the application alternative initial rank scenario 1 scenario 2 scenario 3 x1 2 2 2 2 x2 4 4 ● ● x3 3 3 3 ● x4 5 ● ● ● x5 1 1 1 1 in the last part of the sensitivity analysis, the stability of the results was compared with the results of other alternative fuzzy mcdm techniques. in this context, some effective methods such as topsis, mabac, moora, waspas, and mairca were employed under a fuzzy environment. spearman’s rank-order correlation coefficients  sr were also employed for rankings in the analysis. as shown in the correlation heatmap in figure 4, the proposed airline ranking is highly credible. 0 1 2 3 4 5 6 set1 set2 set3 set4 set5 set6 set7 set8 set9 set10 set11 set12 set13 x1 x2 x3 x4 x5 bakır and atalık/decis. mak. appl. manag. eng. 4 (1) (2021) 127-152 144 figure 4. correlation heatmap for results of alternative methods 5. results and discussion in the application, the dimensions and criteria in the hierarchical model were first subjected to pairwise comparison. fuzzy pairwise comparison matrices are presented in tables 4-7. the results are consistent since the consistency ratios of the comparison matrices are below 0.10 (cr <0.10) (ecer, 2014). from the fuzzy pairwise comparison matrix the most important dimension was service quality (c3), which expresses the overall support offered by the airline website. service quality was followed by the information quality (c1) and system quality (c2) dimensions, respectively (see table 8). considering the existing literature, service quality comes to the fore in many studies (alptekin et al., 2015; ustasüleyman, 2013). with regards to information quality, lionella et al. (2020) reported that this is the most determining factor on overall e-service quality from the meta-analytical perspective. moreover, they argued that information quality is the factor most associated with perceived trust. in addition, similar to the findings of chou and cheng (2012), system quality was found to be the least important criteria. in light of these findings, considering the characteristics of the airline industry, it can be concluded that passengers prioritize; a) the delivery of services without any problems, b) smooth interactions, c) the acquisition of high-quality information about the service to be purchased. in terms of the global weight of the criteria, the most important criterion was reliability (c33), followed by the understandability (c12) and security (c21) criteria. the importance of these criteria has also been identified in previous studies (chou & cheng, 2012; lee & kozar, 2006). additionally, it is seen that the reliability criterion, which considers the degree to which the delivered services were as promised (i.e., flight delay due to increased air traffic, flight cancellation, and overbooking), was the most important criterion for airline passengers. in fact, reliability was also found to be the most important criterion in previous studies (lee & lin, 2005; shankar & datta, application of fuzzy ahp and fuzzy marcos approach for the evaluation of e-service… 145 2020; tsai et al., 2011). following the deregulation of the us airline industry in 1978, both flight frequencies and air traffic have increased considerably, which has caused some confusion for passengers. thus, in this context, the understandability criterion may represent the expectations of passengers that the information provided will be understandable and satisfactory. in addition, as seen in previous studies (çelik & gök kısa, 2017; hsu et al., 2012), in the airline industry where services are mainly distributed online, consumers are concerned about the confidentiality of their personal and financial data and, therefore, they pay particular attention to the security criterion. this issue has also been found in past studies as one of the main concerns of passengers during online booking (bigné et al., 2010; lee et al., 2019). in light of the above-mentioned findings, consumer expectations can also be associated with hofstede’s cultural dimensions theory (hofstede & bond, 1984). as has previously been established (blut et al., 2015), cultural dimensions exert a moderating effect on people’s perceptions of the quality of e-services. thus, it can be deduced that passengers care about receiving adequate general support and satisfactory information so as to minimize their risks in turkey, a country in which the population is generally considered to be characterized by high levels of uncertainty avoidance. after weighting, the web-based e-service quality of airlines was evaluated using fmarcos under uncertainty. as a result, the best and worst alternatives were x5 and x4, respectively. finally, a three-stage sensitivity analysis was conducted to check the robustness of the calculation. with regards to the rank reversal effect, it was seen that there is no rank reversal effect. in addition, figures 3 and 4 indicate that the weight changes do not affect ranking, thus indicating that the application presented is credible. 5. conclusions being aware of consumer expectations helps to ensure firms’ survival. yet, it is evident that firms cannot fully understand consumer expectations (kurtulmuşoğlu et al., 2016). from a different view, it is also not always possible to meet all consumer expectations in practice. therefore, the most sensible approach is to recognize those criteria which have been prioritized. as the importance of providing satisfactory eservices has been established in literature, the importance levels of the constituent criteria should be ascertained in order to satisfy consumers. in this study, the criteria affecting e-service quality in airlines have been prioritized, and a real-world case study based on passenger evaluations of scheduled airlines is presented. it should be noted, however, that service quality evaluations often suffer from imprecise judgments. therefore, we employed the proposed approach in a fuzzy environment in order to handle the subjective and imprecise judgments of people. theoretically, the key contributions of this study are twofold. this study is the first to integrate f-ahp and f-marcos methods in a case study. this approach, which was applied successfully in the area of airline e-service quality, can be used in different domains. additionally, e-service quality of scheduled airlines in turkey has been analyzed for the first time in literature. in this regard, the proposed approach is expected to contribute to the existing literature. the present study also has numerous implications for turkish airline managers. from a managerial perspective, the findings will assist airlines in providing more satisfactory online services by becoming more aware of the priorities of customers in terms of the e-service quality elements. moreover, since a hierarchical model with 13 criteria has been developed in this study, bakır and atalık/decis. mak. appl. manag. eng. 4 (1) (2021) 127-152 146 airlines can use these criteria to monitor their e-service processes. the success of airlines depends on their compatibility with the voice of the customer. therefore, this study, in which passengers evaluated the e-service quality of scheduled airlines, provides vital understanding for airline managers. in terms of the findings, reliability, understandability, and security criteria were the most important criteria. therefore, airlines should consider these in their web-based marketing strategies and strengthen the perception of reliability in the minds of the consumers. several recommendations for further research can be made based on the findings of this study. first, since a comprehensive e-service quality evaluation model is still required for the airline industry (chong & law, 2019), we suggest that the hierarchical model could be enriched by using focus groups, in-depth interviews, etc. prior studies concerning e-service quality have considered matters such as website quality and eservice quality, and they have predominantly focused on services delivered through a personal computer (pc). however, it has been shown that the use of mobile devices in relation to the airline industry is increasing dramatically. according to sita (2019), mobile applications for passenger services are one of the investment priorities of airlines. mobile devices are frequently used in many services such as booking, bag tracking, self-boarding, etc. in addition, due to the ubiquity and localization characteristics of mobile devices, consumer expectations can be shaped accordingly (lionello et al., 2020). therefore, future studies should address airline e-service quality from the perspective of mobile devices. second, when the literature regarding airline e-service quality is considered, it quickly becomes apparent that the issue of business model segmentation has previously been ignored. thus, future studies can provide deeper insights by focusing on different airlines such as full-service carriers (fscs) and low-cost carriers (lccs). it is important to recognize that this study has a few limitations. first, the survey data were only collected from a limited number of passengers using scheduled airlines in turkey. therefore, it is assumed that the sample represents the population. is it also worth mentioning that demographics (age/culture etc) of passengers could also affect which criteria are more important. in addition, the ratings reflect the period in which the data were collected. as airline websites are very dynamic, there is a possibility that different results would be obtained in future studies. second, the research data were collected at only a single airport due to procedural difficulties and time limitations. third, the proposed hierarchical model may have excluded some factors that affect eservice quality. therefore, the evaluation performed in this study reflects the investigated airlines’ performances according to only certain criteria. finally, the results should not be generalized because they were derived from data concerning just five scheduled airlines. acknowledgement: this paper was derived from mahmut bakır’s master’s thesis entitled “an integrated approach to the evaluation of e-service quality in airline companies” conducted at anadolu university, eskişehir. author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research received no external funding. conflicts of interest: the authors declare that there is no conflict of interest. application of fuzzy ahp and fuzzy marcos approach for the evaluation of 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(1965). fuzzy sets. information and control, 8(3), 338–353. https://doi.org/10.1016/s0019-9958(65)90241-x © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 4, issue 1, 2021, pp. 257-279. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame210402257b * corresponding author. e-mail addresses: mbaydas@erbakan.edu.tr (m. baydaş), oeelma@erbakan.edu.tr (o.e. elma). an objective criteria proposal for the comparison of mcdm and weighting methods in financial performance measurement: an application in borsa istanbul mahmut baydaş1* and orhan emre elma2 1 necmettin erbakan university, faculty of applied sciences, department of accounting and finance management, turkey 2 necmettin erbakan university, faculty of applied sciences, department of accounting and finance management, turkey received: 21 may 2021; accepted: 30 july 2021; available online: 3 september 2021. original scientific paper abstract: financial performance research with multi-criteria decision making (mcdm) methods, is a common subject of study not only for researchers in the finance literature, but also in the applied sciences. financial performance manifests itself in an internal universe that a firm can directly control, while the share return of the same firm is shaped synchronically in an external universe which cannot be controlled directly. on the other hand, preferring the most suitable mcdm and weighting method to use in measuring financial performance is often regarded as a source of uncertainty. in this study, share price is used as an external proxy and a tool for comparing mcdm methods, completely different from the previously proposed approaches based on the superiority of internal features. this study was conducted on 131 manufacturing companies in borsa istanbul, covering entire 20-quarter period between 2014 and 2018. the experimental findings of the study provides valid solutions for the mcdm and weighting selection problem, that can be proposed as a practical and indirect solution. the results show that preference ranking organization method for enrichment of evaluations (promethee) method used with hybrid weighting technique produced by far the best performance rankings in 19 out of 20 quarterly periods when compared to technique for order preference by similarity to ideal solution (topsis) and weighted sum approach (wsa). key words: financial performance, mcdm, share return, spearman’s correlation coefficient. mailto:mbaydas@erbakan.edu.tr mailto:oeelma@erbakan.edu.tr baydaş & elma/decis. mak. appl. manag. eng. 4 (2) (2021) 257-279 258 1. introduction performance measurement is vital for companies in order for them to make rational financial decisions at the right time, in the modern competitive markets. the multidimensional, complex and contradictory nature of financial performance, which can be described as the determination of firm success, requires the use of mcdm methods for measurement. mcdm provides simplicity and practicality in performance evaluation by purely converting the performance to a single score with the data consisting of multiple criteria. mcdm approaches summarize many aspects of firm-related behavior, fluctuation and output in a single score. in addition, performance measurement provides comparability with competitor companies, even in the most complex scenarios. thus, it can be said that mcdm has become a decision support system that can help financial information users in making more straightforward and accurate decisions in financial markets, when they are on the verge of investing. literature demonstrates that there are more than 100 mcdm methods to assist decision making processes (danesh et al., 2017). despite these benefits, there is still no consensus on choosing the right method, for a specific real-life scenario. when mcdm methods use the common decision matrix consisting of the same data, their goals are essentially similar in terms of choosing the best alternative. in fact, many mcdm types create similar rankings with significant correlation levels (karaoğlan & şahin, 2018). however, different weighting methods based on a fixed mcdm are more likely to create different rankings, whether they are objective or subjective. since different technical approaches and hypothetical limitations affect the ranking results of the methods, this may affect the whole order, not just the best alternative. this is a chronical and inherent problem in the mcdm paradigm as the optimal order is essentially unknown. in the uncertainty of the optimal, it can be said that the mcdm models which can picture the goal-oriented actual life scenarios with a sound mathematical background are more useful, practical, acceptable and reliable for decision makers. as a quantitative comparison measure, correlation coefficient between the ranking produced by mcdm methods and an independent proxy representing real life has not discussed comprehensively in the literature. for this purpose, it would be useful to analyze and conceptualize the proposal of this approach, specifically in financial performance studies. in this study, a comprehensive analysis will be made in order to test the novel approach mentioned. at this research, financial performance will be measured at a constantly developing market on the basis of 7 decision criteria for 131 manufacturing companies registered in borsa istanbul (bist), in 20 quarters between 2014 and 2018. in order to ensure comparability, 3 popular types of mcdm will be used with 3 types of weighting method thoroughly used. the relationship between the obtained mcdm ranking results and the simultaneous share-return rankings will be analyzed with the spearman correlation test. in this regard, the aim of the study is to reveal the practicality of rho correlation coefficient in mcdm methods, which expresses the relationship level of the financial performance of manufacturing firms measured by mcdm methods, with their reallife share returns as an indirect superiority criterion. based on this criteria, the hypotheses of the study are as follows: h1: there is a relationship between financial performance measured by mcdm and the share return of the respective firm. a novel and practical criteria proposal for selecting the suitable mcdm and weighting... 259 h2: in the given period, one mcdm method produces higher rho coefficient than another mcdm method. h3: in the given period, one weighting technique produces higher rho than another technique. these hypotheses could be regarded as an alternative and indirect solution to the selection problem of mcdm methods and weighting techniques, which have been studied in the literature for many years. 2. literature in line with the purpose of the study, the literature is divided into three segments at this section. studies on mcdm and weighting method selection, and research on the measurement of financial performance with mcdm methods and comparing ranking results with share returns were demonstrated, respectively. finally, although there is little research on the usability of rho production capacity to classify mcdm methods, the related literature will be examined. 2.1. mcdm and the selection of weighting methods there are many suggestions on which reference criteria should be selected for the mcdm methods that can perfectly fit for the given real life problem. when the approaches for the method selection in the literature are examined, it can be clearly seen that the process of determining the strengths and weaknesses of the mcdm methods to solve a specific problem is vital. it may be efficient to examine which advantages and disadvantages are more common for each method and then explore new methods that can effectively combine strengths while eliminating weaknesses (velasquez & hester, 2013). on the other hand, comparative analyzes of the methods show that none of the mcdm methods are regarded as perfect. ideally and whenever possible, more than one method should be applied to the same problem in order to provide a more comprehensive result for the decision maker (mulliner et al., 2016). in addition, choosing a criterion weighting technique among the alternatives is a problem that frequently arises in the mcdm methods. obviously, criteria weighting has a crucial role in obtaining accurate results (olson, 2004). weighting technique selection, like mcdm selection, is also an important problem. considering that the weights of criteria can significantly affect the result, it is important to pay particular attention to the objectivity factors of criteria weights in the decision-making process. subjective weighting methods are computationally convenient and more understandable than objective weighting methods. because, in objective methods, information is derived from each criterion by adopting a mathematical function to determine the weights without the input of the decision maker (odu, 2019). according to a research, the capacity of mcdm to represent real life scenarios become more vital than ever, as these models can now be evaluated more with other measures (munier, 2006). therefore, a rational mcdm should not only be based on the internal hypothetical scenario, but also model the actual life explicitly. in other words, a mcdm method which has a significant correlation with proxy actual life rankings can be adopted in method selection, surely if the results are not coincidental. this approach has been examined at previous studies in weighting for different versions of a mcdm type (yaakob et al., 2016). baydaş & elma/decis. mak. appl. manag. eng. 4 (2) (2021) 257-279 260 2.2. financial performance and an external real time proxy on the field of mcdms measuring financial performance, which should exceed the average human memory and intelligence, with mcdm methods in a complex multi-criteria environment, in order to help financial information users to make critical and more appropriate decisions has gained intensity since the early 2000s. ertuğrul and karakaşoğlu (2009) used a wide range of financial ratios to establish the performance ranking of companies in the cement sector registered on borsa istanbul. that study proposed a fuzzy topsis-based model with fuzzy ahp in financial performance evaluation. yalçın et al. (2012) proposed an innovative financial performance rating approach with using ahp-based topsis and vikor methods, which relies on accounting based financial performance (afp) and valuebased financial performance (vfp), in order to generate rankings for the firms in the manufacturing industry sub-sector in turkey. comparing the financial performance ranking results of mcdm with a relevant external ranking, such as share return, is useful for financial information users and can be more helpful in their decisions. özden et al. (2012) calculated the financial performances of cement industry companies using vikor method. whether there is a relationship between the financial performance rankings and the stock returns of these firms was calculated with the sperman rank correlation coefficient. öztürk (2017) preferred promethee method at his study on bist-50 index companies in order to generate performance scores. at that research, the relationship between the performance scores for each year and the prices of the stocks were examined. çalış and sakarya (2020) investigated the relationship between stock returns and financial performance of banks operating in the bist banking index, with the help of promethee method. analysis was performed utilizing the quarterly results of the selected banks in turkey. spearman rank correlation test was applied to determine whether there is a statistically significant relationship between stock returns and financial performance. 2.3. the usage of correlation coefficient between financial performance and share return as a benchmark to compare mcdms after finding significant and strong relationships between financial performance and stock return rankings obtained with mcdm, there are either no known studies or arguably very limited number of research using this method in classifying superiority of the models. however, the spearman rho coefficient, which expresses the relationship between financial performance results and share return, can be used to compare different mcdm models and weighting techniques. a recent comparative study on specific topsis types can be shown as an ambiguous example (yaakob et al., 2016). the financial performances of kuala lumpur stock exchange companies were measured using the topsis method, which is one of the popular mcdm techniques. along with classical topsis, non-rule based fuzzy topsis approaches were also used. the relationship between the obtained financial performance rankings and current share return rankings was investigated. according to the study, there is a correlation between financial performance rankings of topsis and share return rankings. the study demonstrated the rationale for comparing share return with financial performance in order to compare benchmarking results, validate rankings, prove practicality and effectiveness, and finally showcase the robustness of methods. based on the power of the correlation a novel and practical criteria proposal for selecting the suitable mcdm and weighting... 261 coefficient, that study showed a significantly weaker performance compared to other topsis models proposed by classical topsis. in comparison to the current literature, the distinctive feature of this study is to compare different mcdm methods comperehensively, in terms of data, number of periods and methodological diversity. the usability of rho will also be evaluated in comparing the performance of mcdms at this research. 3. research methodology in this paper, financial performance measurement is used as a tool to compare mcdm-based approaches. in the empirical study, three popular mcdms and weighting methods are used to measure the financial performance of manufacturing companies, as seen in table 1. in this section, the details of the proposed approach, performance metrics, weighting criteria and mcdm methods are explored. table 1. performance criteria, weighting and mcdm methods used in this study weighting methods mcdm methods criteria hybrid topsis altman-z entropy wsa roe & roa & ros equal weigthing promethee mva margin & spread mv/bv 3.1. comparison of mcdm and weighting methods for the evaluation of companies as mentioned in the literature, more than one criterion is required to evaluate the performance of companies. in other words, no single criterion performs best overall. therefore, a comprehensive procedure is required in order to model firms' performance as a mcdm problem. this study proposes a comparison process for weighting methods using topsis as a control element to compare the results of three weighting methods as shown in table 1. the reason for choosing topsis here is that it is a widely recognized and popular mcdm method. in weighting, again widely used entropy weighting and equally weigthing methods were used, as well as hybrid weighting technique as a suggestion. the same weighting coefficients were used as a control variable to compare the results of the three types of mcdm methods. this study proposes a comparative evaluation process for mcdm methods based on identical weighting. the procedure which is explained in section 3.1 applies here as well. but at this step, different types of mcdms will be compared. according to the rankings obtained using mcdm methods based on rho coefficient and share-returns, a method of choosing the most suitable mcdm model will be proposed. 3.2. performance metrics and preferred statistical measure at this study seven performance metrics are chosen in order to evaluate financial performance of the manufacturing companies with the help of mcdm, which are altman-z score, roe, roa, ros, mva margin, mva spread and mv/bv. these 7 criteria are all based on growth and will be explained below. in addition to these, baydaş & elma/decis. mak. appl. manag. eng. 4 (2) (2021) 257-279 262 spearman's rho correlation coefficient is also explained as a non-parametric rank correlation measure. 3.2.1. altman-z score in financial performance practices, this classic criterion is frequently used for evaluating the success, failure or risk of the companies. according to some studies, altman-z score growth, which expresses the change in this ratio, is the ratio that best represents shareholder value (carton, 2004: 281). altman-z score is a multidirectional and powerful indicator that can demonstrate success, risk and share return at the same time.altman-z score is a utility-based criterion, thus higher values in this ratio are always desired. 3.2.2. return on equity return on equity (roe) is defined as the ratio of net profit to equity, and it is a classic criterion recommended for evaluating financial performance (bodie et al., 2003: 456). roe growth is a utility-based criterion. 3.2.3. return on assets return on assets (roa) is an indicator which shows the degree of efficiency in the utilization of assets. it is a commonly used measure to evaluate financial performance. roa growth is also a benefit criterion (bodie et al., 2003: 457). 3.2.4. return on sales return on sales (ros) is a classic indicator commonly preferred to evaluate financial performance, which is focusing on how efficient sales are made. ros growth is a utility-based criterion (carton, 2004: 110). 3.2.5. market value added margin market value added (mva) margin is the ratio of mva to sales. mva is a classic value-based measure for evaluating financial performance. mva margin is a relatively new measure derived from the popular market value added ratio. mva margin growth is also a utility-based criterion (stewart, 2013: 306). 3.2.6. market value added spread market value added (mva) spread is the ratio of mva to invested capital. mva is a classic value-based measure for evaluating financial performance. mva spread is an old benchmark derived from market value added. mva spread growth is a utility criterion (stewart, 2013: 306). 3.2.7. market to book value ratio it is the ratio of market value to equity. in other words, it is the market value which is created by the book value. according to the literature, mva spread is an improved version of market to book ratio, and these two ratios essentially serve the same purpose (stewart, 2013: 118). this ratio is similar to mva derivatives and in that sense can be regarded as a value-based ratio. a novel and practical criteria proposal for selecting the suitable mcdm and weighting... 263 growth rates or rate of change is the equation best described as the change of ratio according to the base period. the financial ratios used in the study and their calculations are listed in table 2 below. table 2. calculation of the financial variables used in this study ratios formulas references mva spread (mv t) – (inv. cap. t-1) / inv. cap. t-1 stewart (2013) mva margin (mvt inv. cap. t-1) / sales t-1 stewart (2013) market to book market value / equity stewart (2013) roe net income t / equity t-1 damodaran (2007) roa net income / total assets bodie et al. (2003) ros net income / net sales carton (2004) altman-z 1.2 (working capital) + 1.4 (retained earnings / total assets) + 3.3 (ebit / total assets) + 0.6 (market value of equity / total liabilities) + 1.0 (sales / total assets) carton (2004) share return (ending stock price – initial stock price) / initial stock price carton (2004) 3.2.8. spearman’s (rho) rank correlation coefficient spearman's rank correlation is one of the most popular rank correlation coefficient measures (sałabun & urbaniak, 2020). it is denoted by rs and calculated by the following formula: 𝑟𝑠 = 1 − 6 ∑𝑑𝑖2 𝑛 (𝑛2 −1) (1) in the formula above, di denotes the difference in paired rankings, and n denotes the number of cases. the spearman coefficient is interpreted as the percentage of one variable's rank variance explained by the other variable. in this study, conceptualizing the external proxy subject and analyzing it with different data is highly regarded, in order to propose the rho coefficient as a clear criterion. some studies suggest that a positive incremental change in financial performance metrics should provide statistically significant increases for shareholders as a means of adjusted return in the market. previous work indicated that change scores could be used instead of or in addition to static criteria. (carton & hofer, 2006: 235). 3.3. mcdm methods in order to avoid using a single method among many mcdm methods and to obtain comparative evaluation results, three popular mcdm methods have been chosen in this study topsis, wsa, and promethee. topsis was chosen because it is the most widely recognized mcdm method in the utility theory school. promethee is a widely used and popular mcdm method from the european transitional and pairwise comparison school. wsa, on the other hand, was included in the comparison because it is the simplest method to model daily life. topsis vector normalization differs in ranking production with assumptions of euclidean distance to ideal values. topsis makes a preference that produces the best global benefit as baydaş & elma/decis. mak. appl. manag. eng. 4 (2) (2021) 257-279 264 the best alternative. promethee, on the other hand, differs in ranking production with the preference functions it uses while relying on the advantages of alternatives in pairwise comparison. finally, in wsa there are no assumptions or limitations other than the normalization. 3.3.1. technique for order preference by similarity to ideal solution technique for order preference by similarity to ideal solution (topsis) was based on the logic of choosing the alternative with the most relative closeness to the ideal alternative. the alternative chosen in this method is at the same time the closest to the ideal solution and the furthest to the nonideal solution. the steps and formulas of the method are as follows (wang & rangaiah, 2017: 561-562): step 1. creating a normalized decision matrix: 𝐹𝑖𝑗 = 𝑓𝑖𝑗 √∑ 𝑓𝑖𝑗 2𝑚 𝑖=1 (2) step 2. obtaining the weighted normalized matrix: 𝑣𝑖𝑗 = 𝐹𝑖𝑗 × 𝑤𝑗 (3) step 3. finding positive (a+) and negative (a–) ideal solutions: at first, find the biggest value or in other words the best value of each objective to maximize. 𝐴+ = {(𝑀𝑎𝑥𝑖 (𝑣𝑖𝑗 )│𝑗 ∈ 𝐽), (𝑀𝑖𝑛𝑖 (𝑣𝑖𝑗 )│𝑗 ∈ 𝐽 ′)│𝑖 ∈ 1, 2, … , 𝑚} = {𝑣1 +, 𝑣2 +, 𝑣3 +, … , 𝑣𝑗 +, … , 𝑣𝑛 +} (4) after, find the worst value of each objective, which can be find as the largest and smallest value at the objective matrix for minimization and maximization objective, respectively. 𝐴− = {(𝑀𝑖𝑛𝑖 (𝑣𝑖𝑗 )│𝑗 ∈ 𝐽), (𝑀𝑎𝑥𝑖 (𝑣𝑖𝑗 )│𝑗 ∈ 𝐽 ′)│𝑖 ∈ 1, 2, … , 𝑚} = {𝑣1 −, 𝑣2 −, 𝑣3 −, … , 𝑣𝑗 −, … , 𝑣𝑛 −} (5) step 4. calculating distance values for positive and negative ideals distance to positive ideal: 𝑆𝑖+ = √∑ (𝑣𝑖𝑗 − 𝑣𝑗 +) 2𝑛 𝑗=1 𝑖 = 1, 2, 3, … , 𝑚 (6) distance to negative ideal: 𝑆𝑖− = √∑ (𝑣𝑖𝑗 − 𝑣𝑗 −) 2𝑛 𝑗=1 𝑖 = 1, 2, 3, … , 𝑚 (7) step 5. computing relative proximity to ideal solution: 𝐶𝑖 = 𝑆𝑖− 𝑆𝑖−+𝑆𝑖+ (8) the optimal solution having the largest ci is the recommended solution. in the decision problem, the most preferred alternatives will be obtained when the values calculated by their proximity to the ideal solutions, which is mentioned in the step above, are placed in a descending order for each alternative. 3.3.2. weighted sum approach weighted sum approach (wsa) is a method that aims to determine the option that provides the maximum benefit from the set of alternatives. this method is based a novel and practical criteria proposal for selecting the suitable mcdm and weighting... 265 on the calculation of the alternatives’ global utilitization value, via taking normalized criterion weights into account. it basically consists of two stages. these are normalization and determination of the total utility (taşabat et al., 2015). it is the closest and simplest method to the average daily life usage with few subjective limitations. if the units of measure are different, the criteria values are normalized and the total score of each alternative is obtained, after summing up according to the criterion weight. this method consists of the following two stages (şen, 2014: 58): step 1. normalization of values: normalization step is applied with the practice of the formula below. 𝑟𝑖𝑗 = 𝑦𝑖𝑗−𝐷 𝐻𝑗−𝐷𝑗 (9) here; i shows the order of the alternative, j designates the order of criterion, yij refers to the original value of j criterion for the alternative i, hj denotes the maximum value of the j criterion which represents the ideal option, dj signalizes the minimum value of the j criterion, which represents the ideal option. accordingly, the maximum utility (rij) is achieved when it is equal to 1, concurrently the minimum benefit is achieved when it is 0. step 2. calculation of the total benefit: at this stage, the utility value of each alternative is calculated. this is found by multiplying the normalized values with the specified criteria weights. with the notion of k indicating the order of the criteria, the formula is as follows: 𝑢(𝑎𝑖 ) = ∑ 𝑟𝑖𝑗 . 𝑣𝑗 𝑘 𝑗=1 (10) after this stage, alternatives can be listed according to their utility values. the most appropriate solution to the decision makers’ problem is the alternative with the highest utility value. in cases where cost of the evaluations are made, reverse operations should be performed in normalization and utility value calculations. 3.3.3. preference ranking organization method for enrichment of evaluations preference ranking organization method for enrichment of evaluations ii (promethee ii) aims to provide a complete ranking of a finite set of viable alternatives, from best to worst. this method is essential for applying other promethee methods, and most researchers have generally resorted to this version of the promethee. the basic principle of promethee ii relies on binary comparison of the alternatives across their recognized criterion. alternatives are evaluated according to different criteria that need to be maximized or minimized. the application of promethee ii requires two additional types of information. these are criteria weighting and preference function selection, which are left to the user's discretion. for each criterion, the preference function distinguishes the difference between the evaluations obtained by the two alternatives into a degree of preference ranking from zero to one. six basic types are recommended in order to facilitate the selection of a particular preference function: (1) general criterion, (2) u-shape criterion, (3) v-shape criterion, (4) level criterion, (5) v-shape and (6) gaussian criterion. these six types are particularly easy to identify. for each criterion; an indifference threshold (q), value of a strict preference threshold (p), and value of an intermediate value (s) between indifference and strict preference treshold must be fixed. in any case, these parameters have vital significance for the decision maker. promethee ii with the baydaş & elma/decis. mak. appl. manag. eng. 4 (2) (2021) 257-279 266 first type as the preference function is used in this study. the stepwise procedure for promethee ii is as follows (behzadian et al., 2010: 199): step 1. determine the deviations according to pair-wise comparisons: 𝑑𝑗 (𝑎, 𝑏) = 𝑔𝑗 (𝑎) − 𝑔𝑗 (𝑏) (11) here dj (a, b) shows the difference between a and b evaluations for each criterion. step 2. implementation of the preference function: 𝑃𝑗 (𝑎, 𝑏) = 𝐹𝑗 [𝑑𝑗 (𝑎, 𝑏)] 𝑗 = 1, … , 𝑘 (12) here pj (a, b) demonstrates the election of a in respect to b for each criterion, as a function of dj (a, b). step 3. calculation of a preference index: ∀ 𝑎, 𝑏 ∈ 𝐴, 𝜋(𝑎, 𝑏) = ∑ 𝑃𝑗 (𝑎, 𝑏)𝑤𝑗 𝑘 𝑗=1 (13) here, preference indices are determined for each alternative pair. the weighted sum for each criterion is defined as π (a, b) and the weight associated with the jth criterion is denoted as wj. step 4. calculation of transition flows: 𝜙+(𝑎) = 1 𝑛−1 ∑ 𝜋(𝑎, 𝑥)𝑥∈𝐴 (14) 𝜙−(𝑎) = 1 𝑛−1 ∑ 𝜋(𝑥, 𝑎)𝑥∈𝐴 (15) here ϕ+(a) and ϕ-(a) indicate the positive outranking flow and negative outranking flow for each alternative, respectively. step 5. calculation of net outranking flow: 𝜙(𝑎) = 𝜙+(𝑎) − 𝜙−(𝑎) (16) here ϕ(a) expresses the net outranking flow for each alternative. the procedure is initiated in order to determine deviations based on binary comparisons. procedure follows this by using a corresponding preference function for each criterion at step 2, calculating the overall preference index at step 3, and calculating the positive and negative transition flows for each alternative at step 4. finally, the procedure ends with the calculation of the net outranking flow for each alternative and the complete ranking. 3.4. weighting methods weighting methods are important in choosing the best alternative and also influencing the ranking. three weighting methods were chosen in this study. entropy, which is a popular method among mcdm studies, was chosen as the objective weighting method. yet again, a common equal weighting and hybrid weighting methods are proposed. 3.4.1. equally weighting technique (mean weight method) it is a technique based on equally weighting of criteria, with the assumption that all criteria used in comparing the performance of alternatives are of equal importance. in order to better observe the importance of criterion weights among mcdm methods, the differences between the results obtained by using other weighting methods can be revealed by regarding the equally weighted criteria as the control group (şen, 2014: 77). this technique can be used in cases where sufficient knowledge and expert opinion are not available in determining the importance of the criteria. it is one of the widely used and recognized techniques. to better illustrate a novel and practical criteria proposal for selecting the suitable mcdm and weighting... 267 this technique, in the assumption of 8 criteria, the significance level of each criterion weight is determined as 0.125, considering the fact that the criteria weights are equal to each other. 3.4.2. entropy weighting technique many objective weighting techniques have been proposed by researchers in recent years. one of the most popular among them is the entropy technique. the method is built on the concept of entropy, which is defined by shannon (1948) as the measure of uncertainty. in information theory, entropy is a criterion for the measure of uncertainty given by the discrete probability distribution. the higher the uncertainty in the data group the higher the entropy value will get. if we have a decision matrix that contains a certain amount of information for alternatives, the entropy method is a tool that can be used to determine the importance ranking of the criteria, in other words the weighting values. it can be said that information value will be higher if the information required in the distribution of decision criteria is less likely, on the contrary information value will be lower if it is highly probable. according to literature, the technique is mathematically formulated as follows (alp et al., 2015: 69). step 1. normalization of the evaluation index: rij= 𝑥𝑖𝑗 ∑𝑗𝑥𝑖𝑗 (17) step 2. calculation of the measure of entropy for every index: ej= -k ∑ 𝑟𝑖𝑗 ln(𝑟𝑖𝑗 ) 𝑛 𝑗=1 (18) step 3. defining the measure of weigthing for each criteria: wij = 1−𝑒𝑗 ∑ (1−𝑛𝑡=1 𝑒𝑗 ) (19) 3.4.3. hybrid weighting the scoring technique, which is regarded as one of the simplest and basic among subjective criteria weighting techniques, is based on the point allocation approach. in this technique, decision makers are expected to estimate weights based on a previously defined numerical range. the scoring technique begins by assigning an arbitrary score to the most important criterion. to exemplify, a score out of 100 is assigned to the most important criterion, and thereafter lower scores are given to the less important criteria in the ranking, proportionately. this process continues until the least important criterion is scored (şen, 2014: 78). the decision makers can prefer equal weighting or subjectively assigning different values to the criteria that make up the problem, based on their own initiatives (kirkwood, 1997: 59). the decision makers can also give a direct weight value in the subjective weighting. ultimately, the selection of weights is based on the assumption that the importance of one criterion is more important than another criterion, for the users of the system in question (gade & osuri, 2014: 51). in hybrid weighting, weights were given by direct scoring for the top two criteria, using the expert opinion. the lower 7 criteria were equally weighted by the weight of the group, as there were not enough expert knowledge on change ratios and mva margin. for the top two criteria in the hybrid method, weighting is determined for vfp group as 71% while for afp group as 29%, based on the average of previous studies in the literature (yalçın et al., 2012; alvandi et al., 2013; esbouei et al., 2014; baydaş & elma/decis. mak. appl. manag. eng. 4 (2) (2021) 257-279 268 ghadikolaei et al., 2014; ünlü et al., 2017). since the sub-criteria were equally weighted, weighting for each member of the three vfp groups was determined as 23.7%, while for each member of the four afp groups was designated as 7.3%. hybrid method was approached as an alternative to equally weighting, because while there is sufficient information about the upper criteria, there is not enough expert knowledge about some sub criteria. in addition, hybrid method is proposed as a partial subjective technique. 4. application in this section, the application and the results of mcdm-based evaluation method in this study will be presented. first, the data set will be explained briefly, thereafter the experimental process will be expressed, and finally the results and discussion of this research will be discussed. 4.1. experimental design in accordance with the aim of this study, 131 manufacturing firms in turkey was selected as decision alternatives, and seven different ratio values of these companies is selected as decision criterion, in order to measure the financial performance of bist companies with mcdm methods. the period of the study consists of 20 quarter periods between 2014-2018. the performance metrics for each period was calculated separately. first of all, three different topsis-based weighting methods was used for each period in order to determine the effect of criterion weights. afterwards, the rho coefficients for each period was obtained by comparing the mcdm results with the stock return rankings. weighting models was ranked according to the average rho coefficient they produce, and the most suitable weighting model was selected. on the other hand, this chosen best weighting method was used as the fixed weight method in the selection of mcdm methods in the next step. according to the results of the mcdm method, the one which produces the highest average rho coefficient was selected as the most suitable mcdm method. experimental process is as follows: step 1. preparing a data matrix: finnet software was used in order to obtain ratios and stock returns, which are regarded as financial performance indicators of firms. the obtained data was integrated into the decision matrix to calculate mcdms. step 2. weighting calculation: three different weighting results are obtained by applying equal weighting, entropy and hybrid methods. while quarterly entropy weights produced different results for each quarter, fixed weight values were used in all periods for entropy values and other weighting methods. equal weighting method was used primarily because sufficient subjective expert opinion on some criteria and change values could not be obtained. in the hybrid method, which is based on point allocation and equal weighting, in order to develop an alternative in comparison, the weight value was directly given to the criteria, as there was sufficient information about the top two main criteria. in the sub-criteria, because sufficient subjective expert opinion on some criteria and change values could not be obtained, equal weighting is preferred. thus, three different weighting methods were obtained: partial subjective, objective and non-judgmental. step 3. calculation of mcdms: a total of 60 different mcdm results were obtained for 3 different mcdms of 131 companies in 20 periods. adding the topsis results calculated in the weighting, a total of 100 different mcdm calculations were made. in other words, a company's mcdm was calculated 100 times. just for this reason alone, a novel and practical criteria proposal for selecting the suitable mcdm and weighting... 269 the research has a comprehensive scope that has not been seen generally in previous mcdm studies. sanna excel extension was preferred to perform mcdm calculations. this program can analyze very large problems for decision makers in a very short time with various methods (jablonsky, 2014). for promethee, the general preference function was used. step 4. evaluation of mcdm ranking results: periodic stock return rankings that occur simultaneously with mcdm rankings have been subjected to spearman correlation analysis, in order to better understand and pinpoint mcdms in terms of producing superior results. this analysis was made in the minitab software. step 5. use of rho correlation results as a superiority function in mcdm methods: both mcdms and weighting methods are listed according to the 20-period average rho produced by mcdms. the method with the best average has been recommended to its decision maker. 4.2. findings and results undoubtedly, the weighting process has a vital place in mcdm theory. it is a classic and popular problem whether subjective or objective methods should be preferred in the weighting phase. these preferences are mostly left to the initiative of the decision maker who can surely involve in the process and consequently settle on a method. however, when decision maker chooses a subjective method, he or she has to seek expert opinion. thus, the choice of weighting method in the mcdm procedure continues to be a concern for decision makers. in this study, using share price as a proxy of financial performance indicator, weighting method selection becomes more practical and accessible. many studies consider end year financial statement data to represent that year. however, these results may also be effected by the cumulative actions of previous years. to bypass this type of unseen outcomes and reach genuine financial values for each year, performance change value can be taken. while the previous literature used static values to be used in the mcdm matrix, this study used the difference between two time periods as a performance indicator. with this method, the performance indicator used can better represent the whole time frame. this study focused on a comparison over the relative value and result created by mcdm methods, while the previous financial performance literature compared mcdm methods with an evaluation based on capacity and ability. correlation values created by each mcdm method as regards to the share returns are taken into account as results. as seen on table 3, weighting types with a fixed mcdm (topsis) are compared according to the rho coefficients they produce. according to the results based on the weighting method that yields the best, the hybrid weighting method was suggested by this study. the method suggested in this study, predominantly provided the best significant correlation results. two outcomes can be derived from these findings: firstly, the probability of obtaining correct results with the right experts is high; secondly the subjective evaluation of the top two criteria is easier and more verifiable than the bottom seven criteria. sub-criteria may be risky since there are more sub-criteria then main criteria. if decision makers are not confident enough in expert opinion, they can use the entropy weighting method. according to the findings of this study, long term (20 months) fixed weight value in entropy method is more efficient than short term (3 months) variable weight values. in terms of efficiency, there is an equal weighting method which is positioned almost in the middle of these two weighting methods. therefore, the following can be said for baydaş & elma/decis. mak. appl. manag. eng. 4 (2) (2021) 257-279 270 weighting methods: the efficiency potential and risk of subjective weighting is high. users can choose the weighting combination that can give the highest rho with sensitivity analysis. but from there on, they must obtain an order provided they model the results. table 3. rho coefficients of topsis rankings with share return rankings according to different weighting types equal weighted entropy 3month periods entropy 20month period hybrid q1 2014 0.409 0.309 0.392 0.407 q2 2014 0.501 0.545 0.591 0.757 q3 2014 0.606 0.695 0.668 0.777 q4 2014 0.565 0.734 0.595 0.667 q1 2015 0.50 0.458 0.48 0.609 q2 2015 0.449 0.584 0.475 0.581 q3 2015 0.19 0.047 0.217 0.493 q4 2015 0.485 0.18 0.52 0.594 q1 2016 0.351 0.069 0.356 0.557 q2 2016 0.50 0.20 0.59 0.747 q3 2016 0.632 0.714 0.699 0.777 q4 2016 0.548 0.766 0.640 0.757 q1 2017 0.368 0.507 0.381 0.665 q2 2017 0.554 0.531 0.591 0.779 q3 2017 0.565 0.324 0.616 0.719 q4 2017 0.479 0.593 0.506 0.462 q1 2018 0.436 0.478 0.469 0.635 q2 2018 0.384 0.30 0.37 0.40 q3 2018 0.654 0.589 0.637 0.414 q4 2018 0.562 0.269 0.572 0.611 p= 0,000 table 4 shows the superiority of hybrid weighting method, whose standard deviation is lower and the rho production is higher than alternatives. hybrid method can be recommended to researchers because it has a more efficient and stable structure. in this respect, hybrid method is chosen as the weighing method at this study. table 4. rho coefficients of weighting methods with share return rankings weighting method ranking mean rho. pairwise comparison st. dev. hybrid 1 0.6204 14 times best ranking 0.1075 entropy 20-month period 2 0.5182 0.2145 a novel and practical criteria proposal for selecting the suitable mcdm and weighting... 271 equally weighted 3 0.4869 2 times second ranking 0.1220 entropy 3-month periods 4 0.4460 4 times best ranking 0.1277 p= 0,000 figure 1 indicates the precedence of hybrid weighting method, when it comes to producing significantly higher rho compared to other methods. the exception is the 3rd quarter of 2018. it is striking that the sector's average return on shares is 0.0136, when economic volatility is in effect after a relatively long period of stability. hybrid method showed the poorest performance only in this period. figure 1. 3-d line chart graph showing rho coefficient rankings of topsis with share return rankings according to weighting methods. according to table 5 below, among the three mcdm methods that are analyzed above, promethee method can be suggested to decision makers who wants to measure financial performance, as it clearly generates the highest rankings in both spearman’s rho and kendall’s tau. 0 0.2 0.4 0.6 0.8 2 0 1 4 /0 3 2 0 1 4 /0 6 2 0 1 4 /0 9 2 0 1 4 /1 2 2 0 1 5 /0 3 2 0 1 5 /0 6 2 0 1 5 /0 9 2 0 1 5 /1 2 2 0 1 6 /0 3 2 0 1 6 /0 6 2 0 1 6 /0 9 2 0 1 6 /1 2 2 0 1 7 /0 3 2 0 1 7 /0 6 2 0 1 7 /0 9 2 0 1 7 /1 2 2 0 1 8 /0 3 2 0 1 8 /0 6 2 0 1 8 /0 9 2 0 1 8 /1 2 equal entropy (variable values) entropy (fixed value) hybrid baydaş & elma/decis. mak. appl. manag. eng. 4 (2) (2021) 257-279 272 table 5. spearman’s rho and kendall’s tau coefficients of different mcdm rankings with share return rankings mcdm methods ranking sprmn mean sprmn st. dev. pairwise comparison promethee 1 0.7128 0.1022 19 best ranking topsis 2 0.6320 0.1274 11 second ranking, 9 third ranking wsa 3 0.5973 0.1366 1 best ranking, 8 second ranking, 10 third ranking mcdm methods ranking k. tau mean k. tau st. dev. pairwise comparison promethee 1 0.5209 0.0889 19 best ranking topsis 2 0.4583 0.1032 1 best ranking, 11 second ranking wsa 3 0.4245 0.1236 12 second ranking, 8 third ranking as seen in the table 6 below, different mcdm types were compared according to the rho coefficient they produced, provided that the most efficient hybrid method was used in weighting. according to the findings, the mcdm method that dominantly gives the best results is the promethee method. table 6. rho coefficients of different mcdm rankings with share return rankings topsis wsa promethee march 2014 0.407 0.302 0.449 june 2014 0.757 0.672 0.808 september 2014 0.778 0.779 0.787 december 2014 0.667 0.703 0.792 march 2015 0.609 0.473 0.681 june 2015 0.581 0.508 0.710 september 2015 0.493 0.256* 0.654 december 2015 0.594 0.447 0.65 march 2016 0.557 0.390 0.632 june 2016 0.747 0.749 0.777 september 2016 0.777 0.787 0.851 december 2016 0.758 0.735 0.811 march 2017 0.665 0.579 0.668 june 2017 0.779 0.723 0.79 september 2017 0.719 0.721 0.785 december 2017 0.462 0.512 0.665 march 2018 0.635 0.571 0.684 june 2018 0.405 0.493 0.490 september 2018 0.414 0.609 0.629 december 2018 0.611 0.612 0.681 *p= 0,003. ** for the remaining coefficients of the table, p= 0,000. a novel and practical criteria proposal for selecting the suitable mcdm and weighting... 273 promethee produced the best results in 19 pairwise comparisons and came second only in one comparison. this method also produces best outcomes in terms of rho means. it can be said that topsis performs slightly better than wsa in rho means. in addition, at pairwise comparison, it can be said that wsa is similar to topsis. compared to previous studies, these results which cover 20 quarters are not random. the results explicitly show that promethee produces the best rho as a mcdm method. the findings of this research can give an idea about the success of promethee, which is based on binary comparison, does not include traditional normalization, and uses simple preference functions as 0 and 1. it has been determined that promethee is a more efficient mcdm model compared to other alternatives in modeling real life scenarios, as shown in figure 2. the results of this financial performance measurement study are in line with some previous research which states that mcdm methods like promethee that perform paired comparison in scoring are better and more suitable than others (kou et al., 2020). this study does not indicate which mcdm model is the best, but simply offers a benchmark that paves the way for it. promethee is found to be the most suitable mcdm method among 3 alternative mcdms, in the limitations and conditions of this study. figure 2. line graph showing the rho coefficient rankings of topsis with share return rankings according to mcdm methods. according to table 7, companies that rank first as regards to the order generated by promethee are different from other methods, in most cases. topsis and wsa produced generally harmonious first alternatives. the main purpose of mcdm methods is to choose the best alternative. in this case, the ranking produced by promethee, which shows the best performance in 19 periods, can be accepted as the reference. the only exceptional period is the second quarter of 2018, where wsa is more successful among others. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 2 0 1 4 /0 3 2 0 1 4 /0 6 2 0 1 4 /0 9 2 0 1 4 /1 2 2 0 1 5 /0 3 2 0 1 5 /0 6 2 0 1 5 /0 9 2 0 1 5 /1 2 2 0 1 6 /0 3 2 0 1 6 /0 6 2 0 1 6 /0 9 2 0 1 6 /1 2 2 0 1 7 /0 3 2 0 1 7 /0 6 2 0 1 7 /0 9 2 0 1 7 /1 2 2 0 1 8 /0 3 2 0 1 8 /0 6 2 0 1 8 /0 9 2 0 1 8 /1 2 r h o period mcdm performance topsis wsa promethee baydaş & elma/decis. mak. appl. manag. eng. 4 (2) (2021) 257-279 274 table 7. top performer companies according to different mcdms topsis wsa promethee march 2014 gedza gedza krdmb june 2014 ulker ulker knfrt september 2014 vesbe vesbe egeen december 2014 afyon afyon jants march 2015 kent kent kent june 2015 afyon afyon gerel september 2015 dogub dogub dogub december 2015 ekiz ekiz pengd march 2016 otkar otkar oylum june 2016 cemas ekiz sasa september 2016 kutpo katmr kutpo december 2016 dokta dokta snpam march 2017 afyon kent dirit june 2017 alka kaplm alka september 2017 yatas yatas yatas december 2017 dokta dokta gerel march 2018 dogub dogub cemas june 2018 bsoke bfren gents september 2018 izmdc dogub dmsas december 2018 nibas nibas nibas mcdm methods may not be compared directly, but with the help of market response these methods can be compared indirectly, which is a proxy solution in financial performance studies. the test results of the hypotheses proposed in this study shows that there is a significant relationship between financial performance and share return, also mcdm methods and weighting methods produced different correlation coefficients. spearman correlation analysis was conducted to test the first hypothesis which states that there is a significant relationship between financial performance rankings measured by mcdm and share return rankings. according to the analysis, the first hypothesis of the study was accepted (h0: rho = 0, h1: rho ≠ 0). in order to test the hypothesis that denotes one mcdm method produces a higher degree of relationship than the other mcdm method in the base period, the correlation coefficients produced by the methods were compared. according to the results, the second hypothesis of the study was accepted (h0: rho = 0, h2: rho ≠ 0). in order to test the hypothesis that expresses a weighting method produces a higher degree of relationship than another weighting method in the base period, the correlation coefficients produced by the methods were compared. according to the findings, the third hypothesis of the study was also accepted (h0: rho = 0, h3: rho ≠ 0). the choice of mcdm and weighting methods for the decision maker was a technical problem in itself and considered as a paradox. it can be said that this study suggests a verifiable practical criterion to overcome this problem, in the given setting. in other words, this research shows that it is possible to reduce the cost of a novel and practical criteria proposal for selecting the suitable mcdm and weighting... 275 not choosing the best alternative in financial decision making with integrating proxy solutions into mcdm methods. 4.3. conclusion determining the best alternative in financial performance studies measured by mcdm is a scientific problem. in order to solve this phenomenon, almost a hundred mcdm methods have been derived and proposed to the decision makers in general. although they have the same purpose, mcdm methods do not always offer the same best alternative and order. the alternative cost of not choosing the best alternative can be very high. in this respect, choosing the most suitable mcdm method becomes a serious concern for the decision maker. moreover, even if this situation is solved, which weighting method to use is another technical problem in itself. the relative capacity of the mathematical foundation of the methods is surely important in the reliability of the measurements. in addition, it is also vital to be able to model real life better (munier, 2006). in this study, the mcdm model which provides a better relationship between financial performance and stock return is assumed the most efficient for decision makers. the results showed that some mcdm models were able to produce explicitly higher rho coefficients, which leave no room for coincidence. no mcdm model can produce rho coefficients that are not available. some mcdm models predominantly provide an existing relationship better than others, and this can imply these methods advantage over others for decision makers, in the given real-life scenarios. the algorithms, assumptions, threshold values, preference functions and normalization methods used by mcdms may be different. all of these determinants contribute in the difference and significance of the correlation coefficient. thus, the market has shown a tendency to approve promethee clearly and strongly than topsis and wsa. promethee rankings have a lower standard deviation than other methods, meaning this mcdm method has more consistent and stable characteristic than topsis and wsa, in terms of financial performance measurement. on the other hand, according to the analyzes performed under the same conditions, assuming topsis as a fixed mcdm method, hybrid weigthing technique produced a stronger relationship than entropy and equal weighting methods. as for the weighting methods, hybrid method is more effective than entropy, which is one of the equal weighting and objective methods. this shows how important the critical touch of expert opinion is. for the entropy method, unlike previous literature, it can be derived that the accuracy of the results increases when the criteria data for the entropy method are observed for a longer time period. based on these results, promethee method and hybrid weighting technique is proposed as a financial performance measurement tool which is focused on shareholder value. the superiority of mcdm methods to each other is not absolute and the conditions are important. the number of alternatives, criteria and data types can affect the results. different conditions can make different types of mcdm methods more successful. mcdm methods dont perform randomly, and some methods are more successful under certain conditions. therefore, the superiority of mcdm methods over each other are demonstrated distinctively, from a financial decision making perspective. this research is aiming to contribute to the literature over showcasing certain advantages of different mcdm and weighting methods for financial information users. in order to accomplish that, an objective and practical comparison procedure for the comparison of mcdm methods is proposed. this benchmark is different from baydaş & elma/decis. mak. appl. manag. eng. 4 (2) (2021) 257-279 276 most of the previous work in terms of implementing share return as an external proxy. users of financial information, business owners, company managers, suppliers, investors, shareholders and creditors desire to learn more about the actual performance of the company in order to make an accurate financial decision. ultimately, it is an important step to find and develop an appropriate benchmark to choose the optimal company for financial information users. expanding the equivalents of this proxy into the other scientific fields may require detailed research. however, this criterion can be easily recommended for the comparison of mcdm methods for future financial performance measurement studies. 4.4. limitations this study was made on borsa istanbul which is an emerging capital market located in turkey. the time span of the study is 20 quarterly periods between 20142018 years. performance measurement system with 7 criteria was applied for 131 companies at borsa istanbul manufacturing index. the place, time, criteria and number of companies are the limitations of this study. research results should be evaluated under these constraints. 4.5. suggestions for future research researchers can use the rho coefficient proposed in this study to evaluate efficiency in comparing mcdm and weighting methods in future financial performance studies. also, researchers can explain the reason why some mcdm and weighting methods produce higher rho coefficients, by concentrating on the mathematical foundation behind them. in addition, it can be investigated why mcdms that can model complex real life have mathematically more simple background, like binary comparison of promethee. finally, if the existence of a real-life proxy sequence associated with mcdm results in quantitative fields such as operations research, engineering and informatics can be confirmed, the approach in this study can be tested on other scientific fields, and thus can be generalized. author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research received no external funding. data availability statement: the quarterly data of the analyzed companies are taken from the public disclosure platform of turkey (kap). the share prices of the firms are provided from borsa istanbul historic and reference data platform. conflicts of interest: the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. references alp, i̇., öztel, a., & köse, m. s. 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(2012). application of fuzzy multicriteria decision making methods for financial performance evaluation of turkish manufacturing industries. expert systems with applications, 39, 350–364. © 2021 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 5, issue 1, 2022, pp. 169-193. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame211221075g * corresponding author. e-mail addresses: gergin@gumushane.edu.tr (r. e. gergin), iskenderpeker@gumushane.edu.tr (i. peker), cansugok@hitit.edu.tr (a. c. gök kısa) supplier selection by integrated ifdemateliftopsis method: a case study of automotive supply industry ramazan eyüp gergin1, iskender peker2, a. cansu gök kısa3* 1 gumushane university, irfan can köse vocational school, transport services department, turkey 2 gumushane university, faculty of economics and administrative sciences, business administration department, turkey 3 hitit university, faculty of economics and administrative sciences, international trade and logistics department, turkey received: 29 april 2021; accepted: 17 december 2021; available online: 21 december 2021. original scientific paper abstract: selecting the best supplier emerges as a crucial subject for all sectors to achieve long term collaborations in supply chains. this study object to select the most suitable supplier for a company engage in activities in the automotive supply industry. for this purpose, a five-stage intuitionistic fuzzy multi-criteria decision making (ifmcdm) model is conducted. firstly, decision criteria are defined by literature research and expert group opinions. secondly, the importance weights of these criteria are obtained by if decision making trial and evaluation laboratory (ifdematel). followingly, the most suitable supplier is assessed by if technique for order preference by similarity to ideal solution (iftopsis). in the fourth stage, sensitivity analysis is utilized to analyze the effect of differentiation in criterion. lastly, a comparative analysis is carried out. the results of the study has pointed that “price” is the most important criterion in supplier selection and “supplier 4” is the best alternative for this case. main contribution of this study is to integrate ifdematel-iftopsis method for the first time in automotive supplier selection literature and propose a specific decision framework. in addition, proposed model is found robust and valid. key words: automotive supply industry, ifdematel, supplier selection, iftopsis. mailto:gergin@gumushane.edu.tr mailto:iskenderpeker@gumushane.edu.tr mailto:cansugok@hitit.edu.tr gergin et al./decis. mak. appl. manag. eng. 5 (1) (2022) 169-193 170 1. introduction enterprises working in automotive supply industry, one of the sectors in which the competitiveness is at the highest level in today’s developing world, need to pay special attention to supplier selection to continue their existence. supplier selection is agreed to be a strategically crucial subject in terms of maintaining competitive position of the companies (banaeian et al., 2018). enterprises need to rely on their suppliers to increase the product quality, to lower their costs, and to improve their economic activities. therefore, the right supplier selection has a great deal of importance for all businesses (gao et al., 2020). especially during the covid-19 pandemic that has been going on for the last 1 year, it has become even more prominent that businesses choose their suppliers correctly. it has been inevitable in all sectors to ensure economic sustainability, to maintain its place in the supply chain, and to ensure coordination with stakeholders. the way to achieve this is through long term collaborations and working with the right suppliers. automotive industry is among the leading sectors in the economies of the industrialized countries and requires and effective supply chain management. the reason why the automotive industry is in close relation with the other sectors of the economy is that this sector has a driving force in the business circle. the sector is composed of two subsectors, namely the main and supply industry. the sector in which the vehicles are produced is called the main industry. the supply industry is the sector that provides the production and supply of the spare parts, system, equipment etc. both for the enterprises in the field of vehicle production and for the part replacement requests of the existing vehicles according to the technical characteristics the vehicles have. automotive sector is accepted to be in the purchaser position for the main industry branches such as iron and steel, petroleum chemicals, and rubber. also, it is in the supplier position for the vital sectors of the economic system such as tourism, infrastructure, transportation, agriculture etc., in the sense of the vehicle types they require. besides, this sector provides basis for the development of the defense industry and the increase in the technological level. when the ranking of the automotive industry enterprises among the top ten enterprises in turkey’s 500 biggest industrial enterprises list for 2019 (iso 500, 2020) is analyzed; it is seen that the number of automotive firms ranked in the list is (i) 4 based on the productionbased sales, (ii) 2 based on the gross value added, (iii) 5 based on the export value, and (iv) 3 based on the number of employees. taking into account all of these rankings, automotive industry is regarded as a strategic industry branch within national economies and there is an increasing trend towards this sector day by day. in addition, the automotive export of turkey in 2019 is 31.2 billion dollars and automotive supply industry export volume is 10,618 million dollars. supplied products are categorized as safety glass, storage battery, engine, tube and outer tires, other components and parts. other components and parts category including vehicle body and lighting parts has the biggest share in automotive supply industry (kpmg, 2020). with respect to related literature, supplier selection can be considered as a multicriteria decision making (mcdm) problem for which numerous quantitative and qualitative factors (cost, price, reliability, geographical location, relations with the sellers, etc.) need to be taken into account together. supplier selection is an intuitive decision-making problem based on decision makers’ (dms’) opinions including ambiguity and vagueness. to handle this problem, atanassov (1986) presented intuitionistic fuzzy (if) sets which is the generalized form of fuzzy sets (boran et al., 2012) to address these weaknesses associated with sufficiently expressing dms’ supplier selection by integrated ifdematel-iftopsis method: a case study of automotive… 171 judgments (wei, 2018). in other words, the main benefit of an if set over the crisp or a traditional fuzzy set is to separate the positive and negative factors for the membership and non-membership of an element in the set (büyüközkan et al., 2017). in this context, this study purposes to evaluate suppliers, producing sub-industry products for a turkish enterprise that exports and imports vehicle body and lighting parts. with this aim, a five-step intuitionistic fuzzy multi-criteria decision making (ifmcdm) model is preferred to reach a solution in this study. in the first step, a group of experts working as the sector managers is created and literature review is conducted to determine the mostly used criteria for supplier selection. subsequently, if decision making trial and evaluation laboratory (ifdematel) is utilized to obtain the relations between these criteria and determine their weights. in the third step, the alternatives are identified based on the suggestions of the enterprise for choosing the best supplier in accordance with the aim of the study. afterwards, if technique for order preference by similarity to ideal solution (iftopsis) method is implied for the evaluation of these suppliers. then, one dimensional sensitivity analysis is used to reveal the impact of the changings in the criteria weights on the ranking of the alternatives. finally a comparative analysis is conducted to validate the results. ifdematel is a powerful mcdm technique (pilko et al., 2017), which can be effectively employed in subjective dm problems. therefore, this method is useful in determining the importance of criteria. when there are several conflicting criteria, iftopsis is utilized to rank the alternatives based on their closeness to the ideal solution and selecting a prominent one. we argue that the integrated model based on if theory is more robust in defining dms’ judgments than the crisp or the fuzzy arithmetic based approaches. the contributions of this paper can be summarized as follows. first, it proposes a framework for choosing and evaluating suppliers that operate in automotive supply chain. second, it analyzes a real case of an enterprise in automotive industry and this is the first study to integrate ifdematel-iftopsis-one dimensional sensitivity analysis into the related field. third, this study provides and effective decision model that contributes to the cooperation of manufacturers and suppliers in their management processes. this paper is organized in five parts. part 2 expresses the literature review. part 3 gives brief information about ifdematel, iftopsis and onedimensional sensitivity analysis methods whereas part 4 discusses supplier selection of a turkish enterprise in automotive spare parts sector within the supply industry. at the end, part 5 displays results and conclusions respectively. 2. literature review literature review of this research consists of two subsections as “studies on supplier selection criteria” and “studies on automotive supplier selection with mcdm methods”. 2.1. studies on supplier selection criteria supplier selection can be expressed as a mcdm problem that can be realized with more than one criterion. there are various criteria that have both qualitative and quantitative characteristics within supplier selection problems. the most commonly utilized criteria which are used in this study according to relevant literature are summarized in table 1. gergin et al./decis. mak. appl. manag. eng. 5 (1) (2022) 169-193 172 table 1. review of supplier selection criteria criteria notation author(s) geographical position c1 aksoy and öztürk, 2011; golmohammadi, 2011; rajesh and malliga, 2013; dargi et al., 2014; tosun and akyüz, 2015; vahdani et al., 2015; khan et al., 2016; prakash and barua, 2016; adalı and işık, 2017; jiang et al., 2018 providing demo products c2 eş and kocadağ, 2020 price c3 xia and wu, 2007; kasirian and yusuff, 2010; kuo et al., 2010; mafakheri et al., 2011; amindoust et al., 2012; huang and hu, 2013; junior et al., 2014; rezaei et al., 2014; zhong and yao, 2017; arabsheybani et al., 2018; jain et al., 2018; ; jiang et al., 2018; feng and gong, 2020; karabıçak et al., 2020; öztürk and paksoy, 2020 guaranty c4 kasirian and yusuff, 2010; keramati et al., 2014; khan et al., 2016; pitchipoo et al., 2015; jain et al., 2018 reliability c5 huang and keskar, 2007; kasirian and yusuff, 2010; chang et al., 2011; lin et al., 2011; adalı and işık, 2017; kumar et al., 2018; bai et al., 2019 velocity c6 chang et al., 2011; öztürk and paksoy, 2020 service c7 xia and wu, 2007; kasirian and yusuff, 2010; chang et al., 2011; huang and hu, 2013; fei et al., 2019; gupta et al., 2019 mold c8 karabıçak et al., 2020 quality c9 sarkis and talluri, 2002; xia and wu, 2007; dağdeviren and eraslan, 2008; lee et al., 2009; kasirian and yusuff, 2010; wu and weng, 2010; shemshadi et al., 2011; amindoust et al., 2012; magdalena, 2012; huang and hu, 2013; ghadimi and heavey, 2014; hruska et al., 2014; rezaei et al., 2014; adalı and işık, 2017; wan et al., 2017; jain et al., 2018; fei et al., 2019; gupta et al., 2019; hadian et al., 2020; karabıçak et al., 2020 risk factors c10 chan and kumar, 2007; hadian et al., 2020 design c11 chan, 2003; jiang et al., 2018 delivery c12 narasimhan et al., 2001; jadidi et al., 2009; fazlollahtabar et al., 2011; shahroudi and rouydel, 2012; junior et al., 2014; arabsheybani et al., 2018; jain et al., 2018; vasiljević et al. 2018 product return flexibility c13 sarkis and talluri, 2002; awasti et al., 2018; eş and kocadağ, 2020; hadian et al., 2020 product performance c14 kahraman et al., 2003; jadidi et al., 2009; liao et al., 2010; fazlollahtabar et al., 2011; vahdani et al., 2015; kumar et al., 2018; vasiljević et al. 2018; hadian et al., 2020 supplier selection by integrated ifdematel-iftopsis method: a case study of automotive… 173 innovation c15 chan and chan, 2004; fazlollahtabar et al., 2011; hashemi et al., 2018; vasiljević et al. 2018; hadian et al., 2020 in line with the literature review conducted in this paper, table 2 shows that price, reliability, service, quality, and delivery are the main evaluation criteria for selecting the suppliers. 2.2. studies on automotive supplier selection with mcdm methods mcdm methods that are often used to solve problems with multiple conflicting criteria, are utilized to handle supplier selection problems. in current literature, there are various studies that employ mcdm methods in supplier selection. some studies conducted in automotive industry are indicated in table 2. table 2. review of automotive supplier selection studies study method sensitivity analysis illustrative or case study kokangul and susuz (2009) ahp-mathematical programming case study kasirian et al. (2010) ahp and anp case study zeydan et al. (2011) fuzzy ahp-fuzzy topsis case study huang and hu (2013) fuzzy anp-goal programming case study dargi et al. (2014) fuzzy anp case study keramati et al. (2014) qfd (quality function deployment)-anp + case study ayağ and samanlıoğlu (2016) fuzzy anp case study dweiri et al. (2016) ahp + case study galankashi et al. (2016) balanced scorecardfuzzy ahp illustrative khan et al. (2016) ahp-qfd case study zimmer et al. (2017) fuzzy ahp + case study jain et al. (2018) fuzzy ahp and topsis + case study jiang et al. (2018) grey dematel based anp + case study vasiljević et al. (2018) rough ahp, fuzzy ahp case study gupta et al. (2019) fuzzy ahp with mabac, waspas, topsis + case study suraraksa and shin (2019) ahp illustrative hadian et al. (2020) vikor-ahp-bocr case study manupati et al. (2021) fuzzy ahp-fuzzy topsis-fuzzy vikor case study gergin et al./decis. mak. appl. manag. eng. 5 (1) (2022) 169-193 174 according to the current literature, it is noticed that mcdm methods are oftenly implemented in automotive industry. some of these studies prefer using only one of mcdm methods, whereas some studies prefer applying integrated and fuzzy mcdm methods. however supplier selection subject have been researched in many papers recently with mcdm methods (qin et al., 2017; banaeian et al., 2018; stević et al., 2019; biswas and das, 2020; stević et al., 2020; fazlollahtabar and kazemitash, 2021), few of them have been aimed to select supplier in automotive industries and use ifmcdm methods. to the best of authors’ knowledge, there is no paper that employs integrated ifdematel-iftopsis-one dimensional sensitivity analysis approach to solve a case of automotive supply industry. therefore, this study makes contribution by considering a real business case under if environment and proposing a new framework for automotive companies to decide their suppliers upon particular criteria. 3. methodology in some complex decision processes such as identifying cause and effect groups that involves fuzziness in dms’ opinions or insufficient knowledge about a problem, the fuzzy sets theory (zadeh, 1965) can be utilized in decision-making processes. on the other hand, the literature suggest that fuzzy sets can be insufficient in certain cases when they are used for processing human beings’ subjective judgments and the associated ambiguity such as the difficulty to formulate the degree of one alternative superior to the others (behret, 2014). to cope with such issues, intuitionistic fuzzy sets (ifs) can be employed in a practical way (büyüközkan et al., 2017). ifs is frequently used to represent dms’ opinions and handle the inherent ambiguity in human judgments more effectively. this study applies an ifmcdm framework including ifdematel-iftopsis, and one-dimensional sensitivity analysis for the aim of supplier selection in the automotive supply industry. this section presents these methods respectively. 3.1. ifdematel method dematel method, developed by geneva research center of battelle memorial institute (chang and chen, 2011: 115), is an effective method that provides analysis in terms of magnitude and types of the direct and indirect relations between factors (han and deng, 2018). dematel can provide an ideal way to better understand the structural relations through analysis of total relations among components and solve congruent system problems (li et al., 2014). supplier selection is a complicated system for multiple factors affecting one another. therefore, ifdematel method can be used to sort the factors influencing supplier selection and enhance the problem. the use of ifdematel method in automotive supply industry will provide the evaluation of supplier selection and define causality between the criteria taken into account during the selection process. the steps of ifdematel method are described as below (keshavarzfard and makui, 2015; büyüközkan et al., 2017): step 1: creating initial direct relation if matrix (�̌�𝑧 ): the evaluation scale (table 3) is used to generate a direct relation matrix for the pairwise comparisons to be realized by the experts. supplier selection by integrated ifdematel-iftopsis method: a case study of automotive… 175 table 3. pairwise comparison scale of ifdematel numerical values definitions of linguistic terms µ (membership) v (non membership) π (hesitancy) 4 very high effect (vh) 0.90 0.10 0.00 3 high effect (h) 0.75 0.20 0.05 2 medium effect (m) 0.50 0.45 0.05 1 low effect (l) 0.35 0.60 0.05 0 no effect (n) 0.00 1.00 0.00 as a result of obtained data with pairwise comparisons, direct relation if matrix ((�̌�𝑧) ) is created. step 2: normalized direct relation if matrix (�̌�𝑧 ): upon the creation of direct relations matrix (�̌�𝑧 ), equations (1) and (2) are utilized to obtain normalized direct relation matrix (�̌�𝑧 ). �̌�𝑧 = k x �̌�𝑧 (1) k = min ( 𝟏 𝒎𝒂𝒙 ∑ |�̌�𝑧𝒊𝒋| 𝒏 𝒋=𝟏 , 𝟏 𝒎𝒂𝒙 ∑ |𝒛�̌�𝑧𝒋 |𝒏𝒊=𝟏 ) i and j = 1,2,3,…,n (2) step 3: calculating total relation if matrix (�̌�𝑧 ): it is obtained by using unit matrix (i) via equation (3): �̌�𝑧 =�̌�𝑧 +�̌�𝑧 2+�̌�𝑧 3+ �̌�𝑧 4+…+ �̌�𝑧 m = �̌�𝑧 . (i – �̌�𝑧 ) -1 (m→ ∞) (3) step 4: calculating causal relations between factors: (�̌�𝑧) matrix is used to calculate the values of d and r. d values obtained from sum of rows, and r values obtained from sum of columns of (�̌�𝑧 ) matrix are calculated with equations (4) and (5), respectively. n d = t i i, j j=1  (i=1,2,…,n) (4) n r = t j i, j i=1  (i=1,2,…,n) (5) relations between criteria are defined according to the values of d-r, whereas the significance and total effects of the criteria are determined regarding to the values of 𝐷+𝑅. the fact that the factor has a higher d+r value means that it has more interaction with other factors. also, the criteria with positive values of d-r are classified in the “sender (cause) group” whereas criteria with negative values of d-r are in “receiver (effect) group”. positive valued criteria of d-r affect other criteria, in contrast negative valued criteria of d-r are affected by other criteria. defuzzied membership, nonmembership and hesitancy values are obtained by using the transformation formula given in equation (6). �̅�𝑖𝑗 = 𝜇𝑖𝑗 − 𝑣𝑖𝑗 + (2𝛼 − 1)𝜋𝑖𝑗 (6) step 5: determining criteria weights (w): criteria weights are calculated with equations (7) and (8). gergin et al./decis. mak. appl. manag. eng. 5 (1) (2022) 169-193 176 2 2 w = (d + r ) + (d r ) i i j i j (7) w iw = ni w i i  (8) 3.2. iftopsis method iftopsis method is applied to rank the available alternatives of this study. the steps of the method explained as follows (boran et al., 2009): step 1: constructing an iftopsis decision matrix: it is calculated by integrating the assessments of the dms for alternatives. in the assessment step, all opinions of the dms are aggregated as group data in order not to lose information. the linguistic terms presented in table 4 is utilized to reflect the dms’ preferences for each alternative. table 4. linguistic terms for assessing alternatives linguistic term if values µ υ π very poor 1 0.05 0.90 0.05 poor 2 0.25 0.70 0.05 fair 3 0.50 0.45 0.05 good 4 0.75 0.20 0.05 very good 5 0.90 0.05 0.05 rij = [1∏ (1 −𝑙𝑘=1 µij (k)) λk , ∏ .𝑙𝑘=1 ϑij (k)) λk , ∏ (1 −𝑙𝑘=1 µij (k)) λk ∏ .𝑙𝑘=1 ϑij (k)) λk ] (9) rij= (µij, ϑij, πij) , (i=1,2,…m; j=1,2,…n), where r is the member of the integrated decision matrix. r=[ µ11, ϑ11, π11 ⋯ µ1n,ϑ1n, π1n ⋮ ⋱ ⋮ µm1, ϑm1, πm ⋯ µ11,ϑ11, πmn ] = [ 𝑟11 ⋯ 𝑟1n ⋮ ⋱ ⋮ 𝑟m1 ⋯ rmn ] (10) step 2: calculating normalized and weighted iftopsis decision matrix: these matrices are calculated by using eq. (11-12), respectively. ŕ=( µ′ij , ϑ'ij ) = {(x, µij. µj , ϑij + ϑj ϑij . ϑj ), x ∈ x } πij= 1ϑij ϑj µij.. µj + ϑij . ϑj (11) ŕ= [ µ′11, ϑ′ 11, π′11 ⋯ µ′ 1n, ϑ′ 1n, π′ 1n ⋮ ⋱ ⋮ µ′ m1, ϑ′ m1, π′m1 ⋯ µ′ 11, ϑ′11, π′ mn ] = [ 𝑟′11 ⋯ 𝑟′1n ⋮ ⋱ ⋮ 𝑟′m1 ⋯ 𝑟′mn ] (12) step 3: specifying the positive and negative ideal solutions: a* refers positive ideal solution while arefers negative ideal solution as calculated by eq. (13-20), respectively. a* = (𝑟1 ′∗, 𝑟2 ′∗, … 𝑟𝑛 ′∗), 𝑟𝑗 ′∗ = (µ𝑗 ′∗ , 𝜗𝑗 ′∗, 𝜋𝑗 ′∗ ), 𝑗 = 1,2, … 𝑛 (13) a= (𝑟1 ′−, 𝑟2 ′−, … 𝑟𝑛 ′−), 𝑟𝑗 ′− = (µ𝑗 ′− , 𝜗𝑗 ′−, 𝜋𝑗 ′− ), 𝑗 = 1,2, … 𝑛 (14) µ𝑗 ′∗ = {(max { µ𝑖𝑗 ′ }, j ∈ j1 ), (min { µ𝑖𝑗 ′ }, j ∈ j2 )} (15) supplier selection by integrated ifdematel-iftopsis method: a case study of automotive… 177 𝜗𝑗 ′∗ = {(min { 𝜗𝑖𝑗 ′ }, j ∈ j1 ), (max { 𝜗𝑖𝑗 ′ }, j ∈ j2 )} (16) 𝜋𝑗 ′∗ = {(1max { µ𝑖𝑗 ′ } – min { 𝜗𝑖𝑗 ′ }, j ∈ j1 (17) = {(1min { µ𝑖𝑗 ′ } – max { 𝜗𝑖𝑗 ′ }, j ∈ j2 ) µ𝑗 ′− = {(min { µ𝑖𝑗 ′ }, j ∈ j1 ), (max { µ𝑖𝑗 ′ }, j ∈ j2 )} (18) 𝜗𝑗 ′− = {(max { 𝜗𝑖𝑗 ′ }, j ∈ j1 ), (min { 𝜗𝑖𝑗 ′ }, j ∈ j2 )} (19) 𝜋𝑗 ′− = {(1min { µ𝑖𝑗 ′ } – max { 𝜗𝑖𝑗 ′ }, j ∈ j1 )} (20) = {(1max { µ𝑖𝑗 ′ } – min { 𝜗𝑖𝑗 ′ }, j ∈ j2 )} step 4: calculation of positive (si*) and negative (si-) difference measurements: two methods such as hamming and euclidean can be used to obtain this measurement. in this application, hamming method is favored as calculated follows. si* = 1 2 ∑ [ |µ𝑖𝑗 ′ − µ𝑗 ′∗ |𝑛𝑗=1 + |ϑ𝑖𝑗 ′ − ϑ𝑗 ′∗ | + |π𝑖𝑗 ′− − π𝑗 ′∗| ], 𝑖 = 1,2, … 𝑚 (21) si= 1 2 ∑ [ |µ𝑖𝑗 ′ − µ𝑗 ′− |𝑛𝑗=1 + |ϑ𝑖𝑗 ′ − ϑ𝑗 ′− | + |π𝑖𝑗 ′ − π𝑗 ′−| ], 𝑖 = 1,2, … 𝑚 (22) step 5: determination of proximity coefficient for each alternative: it is obtained via using the eq. (23) below. ci* = ((si-) / (si*+ si-)), 0≤ ci*≤1, i=1,2,…,m (23) step 6: ranking the alternatives: alternatives are ordered according to the seniority of the proximity coefficients. 3.3. one dimensional sensitivity analysis sensitivity analysis is employed to examine the reliability and stability in the event of vagueness emerged by mcdm problems (karande et al., 2016). the criteria weights in mcdm problems are usually acquired with subjective assessments of the decision makers through different techniques. hence, conducting sensitivity analysis is a necessary stage of the decision making procedure for the certain interpretations of the obtained data. it provides (i) validation of the results acquired from the mcdm methods, (ii) detection of the most important factors creating differences in the ordering of the alternatives, and (iii) ranking regarding to the variations in the criteria weights (butler et al., 1997). this study applies one dimensional sensitivity analysis to acquire the impacts of the most important criteria on the ranking of the alternatives in case of weight differentiation. in this study, the weight of the most significant criterion is explained within an optimal interval and all the other criteria weights are identified equally so that the weight contribution limit could meet ∑ wj=1 n l=1 . wj is the most important criterion and it can be decreased to 0 and enhanced to wj ′. the wj ′ value that reflecting the highest criterion weights (wjmax) and lowest criterion weights (wjmin) is calculated with equation (16) (karande et al., 2016). wj ' =[(wjmax+(n-1)×(wjmin)] (16) gergin et al./decis. mak. appl. manag. eng. 5 (1) (2022) 169-193 178 4. application with the impact of increasing competition in supply chains, choosing the proper supplier has become more crucial for the enterprises. automotive sector is one of the key competition instrument for the countries in terms of in this globalized world. this study evaluates spare parts suppliers of automobiles, which are constantly active in business life. in the application part, a solution to supplier selection problem of an enterprise that sells automotive body and lighting parts is searched by ifdematel-iftopsis approach. this enterprise has a customer portfolio in 81 provinces of turkey in the automotive supply industry with a 20-year experience. it is one of the biggest five enterprises in the sector with regard to market share. accordingly, the flow chart of the application is displayed in figure 1. 4.1. establishing the expert group expert group used in this study consists of sector managers. data about the expert group of 5 sector managers participated in supplier selection procedure is presented in table 5. table 5. information of the expert group expert group title sector experience (year) working company expert 1 general manager 12 2 expert 2 r&d manager 8 3 expert 3 finance manager 10 4 expert 4 marketing manager 7 2 expert 5 purchasing manager 7 1 4.2. identifying the criteria firstly, a criteria pool is created following the literature research on supplier selection to determine the criteria utilized in this study. then, upon the interviews conducted with the expert group, the suggested criteria are taken into consideration (see table 1). 4.3. weighting the criteria the weights of the criteria for supplier selection defined in the previous stage are identified by conducting ifdematel method, which is also used for the analysis of the interactions between the criteria. the significance weights of the criteria are attained by evaluating the data obtained in accordance with the face-to-face interviews of expert group. according to the application steps of ifdematel, decision matrix is created using table 3. direct relation matrix is displayed in table 6 whereas interaction values between criteria and criteria weights are shown in table 7. supplier selection by integrated ifdematel-iftopsis method: a case study of automotive… 179 figure 1. application flow chart gergin et al./decis. mak. appl. manag. eng. 5 (1) (2022) 169-193 180 table 6. ifdematel direct relation matrix criteria c1 c2 c3 c4 c5 c1 0.90 0.10 0.00 0.50 0.45 0.05 0.90 0.10 0.00 0.90 0.10 0.00 0.90 0.10 0.00 c2 0.84 0.12 0.04 0.90 0.10 0.00 0.90 0.10 0.00 0.90 0.10 0.00 0.90 0.10 0.00 c3 0.14 0.81 0.05 0.07 0.88 0.05 0.90 0.10 0.00 0.35 0.60 0.05 0.50 0.45 0.05 c4 0.55 0.40 0.05 0.41 0.54 0.05 0.90 0.10 0.00 0.90 0.10 0.00 0.84 0.16 0.00 c5 0.50 0.45 0.05 0.38 0.57 0.05 0.78 0.22 0.00 0.81 0.19 0.00 0.90 0.10 0.00 c6 0.47 0.48 0.05 0.38 0.57 0.05 0.90 0.10 0.00 0.78 0.22 0.00 0.87 0.13 0.00 c7 0.70 0.25 0.05 0.65 0.30 0.05 0.90 0.10 0.00 0.87 0.13 0.00 0.81 0.19 0.00 c8 0.50 0.45 0.05 0.47 0.48 0.05 0.90 0.10 0.00 0.78 0.22 0.00 0.90 0.10 0.00 c9 0.28 0.67 0.05 0.21 0.74 0.05 0.90 0.10 0.00 0.44 0.51 0.05 0.55 0.40 0.05 c10 0.65 0.30 0.05 0.44 0.51 0.05 0.65 0.30 0.05 0.60 0.35 0.05 0.90 0.10 0.00 c11 0.84 0.16 0.00 0.65 0.30 0.05 0.90 0.10 0.00 0.90 0.10 0.00 0.90 0.10 0.00 c12 0.41 0.54 0.05 0.41 0.54 0.05 0.90 0.10 0.00 0.60 0.35 0.00 0.90 0.10 0.00 c13 0.78 0.22 0.00 0.70 0.25 0.05 0.90 0.10 0.00 0.78 0.22 0.00 0.90 0.10 0.00 c14 0.41 0.54 0.05 0.50 0.45 0.05 0.90 0.10 0.00 0.81 0.19 0.00 0.84 0.16 0.00 c15 0.60 0.35 0.05 0.60 0.35 0.05 0.90 0.10 0.00 0.75 0.20 0.05 0.84 0.16 0.00 criteria c6 c7 c8 c9 c10 c1 0.90 0.10 0.00 0.84 0.12 0.04 0.81 0.16 0.03 0.90 0.10 0.00 0.90 0.10 0.00 c2 0.90 0.10 0.00 0.90 0.10 0.00 0.84 0.12 0.04 0.90 0.10 0.00 0.90 0.10 0.00 c3 0.35 0.60 0.05 0.35 0.60 0.05 0.44 0.51 0.05 0.70 0.25 0.05 0.65 0.30 0.05 c4 0.78 0.22 0.00 0.70 0.25 0.05 0.78 0.22 0.00 0.90 0.10 0.00 0.90 0.10 0.00 c5 0.50 0.45 0.05 0.44 0.51 0.05 0.60 0.35 0.05 0.84 0.16 0.00 0.75 0.20 0.05 c6 0.90 0.10 0.00 0.65 0.30 0.05 0.84 0.16 0.00 0.90 0.10 0.00 0.90 0.10 0.00 c7 0.90 0.10 0.00 0.90 0.10 0.00 0.84 0.16 0.00 0.90 0.10 0.00 0.90 0.10 0.00 c8 0.65 0.30 0.05 0.65 0.30 0.05 0.90 0.10 0.00 0.90 0.10 0.00 0.90 0.10 0.00 c9 0.44 0.51 0.05 0.14 0.81 0.05 0.47 0.48 0.05 0.90 0.10 0.00 0.35 0.60 0.05 c10 0.65 0.30 0.05 0.60 0.35 0.05 0.75 0.20 0.05 0.78 0.22 0.00 0.90 0.10 0.00 c11 0.90 0.10 0.00 0.90 0.10 0.00 0.90 0.10 0.00 0.90 0.10 0.00 0.90 0.10 0.00 c12 0.70 0.25 0.05 0.50 0.45 0.05 0.60 0.35 0.05 0.78 0.22 0.00 0.78 0.22 0.00 c13 0.81 0.19 0.00 0.78 0.22 0.00 0.78 0.22 0.00 0.78 0.22 0.00 0.84 0.16 0.00 c14 0.65 0.30 0.05 0.55 0.40 0.05 0.70 0.25 0.05 0.78 0.22 0.00 0.84 0.16 0.00 c15 0.75 0.20 0.05 0.78 0.22 0.00 0.84 0.16 0.00 0.90 0.10 0.00 0.90 0.10 0.00 criteria c11 c12 c13 c14 c15 c1 0.78 0.18 0.04 0.81 0.16 0.03 0.81 0.16 0.03 0.90 0.10 0.00 0.81 0.16 0.03 c2 0.90 0.10 0.00 0.78 0.18 0.04 0.78 0.18 0.04 0.78 0.18 0.04 0.90 0.10 0.00 c3 0.14 0.81 0.05 0.65 0.30 0.05 0.44 0.51 0.05 0.55 0.40 0.05 0.65 0.30 0.05 c4 0.44 0.51 0.05 0.90 0.10 0.00 0.65 0.30 0.05 0.70 0.25 0.05 0.70 0.25 0.05 c5 0.47 0.48 0.05 0.75 0.20 0.05 0.50 0.45 0.05 0.75 0.20 0.05 0.50 0.45 0.05 c6 0.65 0.30 0.05 0.90 0.10 0.00 0.78 0.22 0.00 0.84 0.16 0.00 0.75 0.20 0.05 c7 0.65 0.30 0.05 0.90 0.10 0.00 0.78 0.22 0.00 0.90 0.10 0.00 0.81 0.19 0.00 c8 0.41 0.54 0.05 0.90 0.10 0.00 0.55 0.40 0.05 0.78 0.22 0.00 0.65 0.30 0.05 c9 0.28 0.67 0.05 0.65 0.30 0.05 0.35 0.60 0.05 0.50 0.45 0.05 0.47 0.48 0.05 c10 0.44 0.51 0.05 0.65 0.30 0.05 0.50 0.45 0.05 0.60 0.35 0.05 0.65 0.30 0.05 c11 0.90 0.10 0.00 0.90 0.10 0.00 0.84 0.16 0.00 0.84 0.16 0.00 0.84 0.16 0.00 c12 0.28 0.67 0.05 0.90 0.10 0.00 0.41 0.54 0.05 0.55 0.40 0.05 0.55 0.40 0.05 c13 0.70 0.25 0.05 0.90 0.10 0.00 0.90 0.10 0.00 0.81 0.19 0.00 0.75 0.20 0.05 c14 0.65 0.30 0.05 0.90 0.10 0.00 0.50 0.45 0.05 0.90 0.10 0.00 0.78 0.22 0.00 c15 0.84 0.16 0.00 0.90 0.10 0.00 0.50 0.45 0.05 0.87 0.13 0.00 0.90 0.10 0.00 supplier selection by integrated ifdematel-iftopsis method: a case study of automotive… 181 table 7. interaction values between criteria and criteria weights criteria d+r d-r group weights c1 1.3005 -0.8638 receiver 0.0732 c2 1.5456 -1.2483 receiver 0.0931 c3 1.5331 1.372 sender 0.0965 c4 1.0581 0.0033 sender 0.0496 c5 1.143 0.6956 sender 0.0627 c6 1.0237 -0.1667 receiver 0.0486 c7 1.0709 -0.5629 receiver 0.0567 c8 1.1143 -0.0228 receiver 0.0522 c9 1.6249 1.2369 sender 0.0957 c10 1.1539 0.6403 sender 0.0618 c11 1.1966 -0.9701 receiver 0.0722 c12 1.1959 0.6468 sender 0.0637 c13 1.3408 -0.5723 receiver 0.0683 c14 1.154 0.1043 sender 0.0543 c15 1.0379 -0.2923 receiver 0.0505 as seen in table 7, the most significant criterion for supplier selection in automotive supply industry is price (c3). following; quality (c9), and providing demo products (c2) are the other most important criteria regarding to their significance weights. speed (c6) is found as the least important criterion for supplier selection. according to d+r values, quality (c9) criterion has the highest interaction in terms of the degree of impact between criteria. other criteria having high interaction are respectively providing demo products (c2) and price (c3). considering the sending group, price (c3) criterion has the highest effect on other criteria. in addition, the most affected criterion is providing demo products (c2), whereas the least affected criterion is mold (c8). these criteria relations as a result of ifdematel analysis are illustrated in figure 2. figure 2. criteria relations of ifdematel analysis 4.4. determining the alternatives in line with the purpose of deciding optimal supplier, 5 suppliers that the mentioned company has worked at different times are chosen as the alternatives of this study, rest upon the opinions of the decision makers. thus, the alternatives are called s1, s2, s3, s4, and s5 to keep their names confidential. c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12 c13 c14 c15 -15 -10 -5 0 5 10 15 0 2 4 6 8 10 12 14 16 sender receiver gergin et al./decis. mak. appl. manag. eng. 5 (1) (2022) 169-193 182 4.5. supplier selection for automotive supply industry iftopsis method is utilized for supplier selection and the criteria weights displayed in table 7 are used while applying the method. the decision makers are required to evaluate each supplier for each criterion for building the decision matrix. in the evaluation process, table 4 is used. iftopsis decision matrix is composed with the mean values of the evaluations made by each decision maker. data of decision matrix of the alternatives are presented in table 8, while the order of the suppliers is shown in table 9. table 8. iftopsis decision matrix criteria a1 a2 a3 a4 a5 c1 0.65 0.30 0.05 0.45 0.50 0.05 0.60 0.35 0.05 0.50 0.45 0.05 0.65 0.30 0.05 c2 0.60 0.35 0.05 0.35 0.60 0.05 0.45 0.50 0.05 0.65 0.30 0.05 0.30 0.65 0.05 c3 0.25 0.70 0.05 0.75 0.20 0.05 0.60 0.35 0.05 0.65 0.30 0.05 0.65 0.30 0.05 c4 0.75 0.20 0.05 0.55 0.40 0.05 0.55 0.40 0.05 0.60 0.35 0.05 0.60 0.35 0.05 c5 0.81 0.14 0.05 0.60 0.35 0.05 0.65 0.30 0.05 0.75 0.20 0.05 0.84 0.11 0.05 c6 0.70 0.25 0.05 0.60 0.35 0.05 0.60 0.35 0.05 0.78 0.17 0.05 0.60 0.35 0.05 c7 0.75 0.20 0.05 0.45 0.50 0.05 0.35 0.60 0.05 0.65 0.30 0.05 0.60 0.35 0.05 c8 0.50 0.45 0.05 0.60 0.35 0.05 0.70 0.25 0.05 0.30 0.65 0.05 0.55 0.40 0.05 c9 0.78 0.17 0.05 0.45 0.50 0.05 0.50 0.45 0.05 0.50 0.45 0.05 0.75 0.20 0.05 c10 0.60 0.35 0.05 0.81 0.14 0.05 0.75 0.20 0.05 0.75 0.20 0.05 0.35 0.60 0.05 c11 0.50 0.45 0.05 0.40 0.55 0.05 0.30 0.65 0.05 0.65 0.30 0.05 0.30 0.65 0.05 c12 0.70 0.25 0.05 0.70 0.25 0.05 0.70 0.25 0.05 0.84 0.11 0.05 0.50 0.45 0.05 c13 0.13 0.82 0.05 0.81 0.14 0.05 0.60 0.35 0.05 0.60 0.35 0.05 0.40 0.55 0.05 c14 0.78 0.17 0.05 0.75 0.20 0.05 0.50 0.45 0.05 0.55 0.40 0.05 0.17 0.78 0.05 c15 0.30 0.65 0.05 0.55 0.40 0.05 0.25 0.70 0.05 0.50 0.45 0.05 0.17 0.78 0.05 table 9. order of the suppliers order alternatives proximity value 1 s4 0.6452 2 s2 0.5868 3 s3 0.4873 4 s1 0.4531 5 s5 0.4407 according to the ranking obtained by iftopsis method, s4 has the best supplier performance in pursuant of the criteria, whereas s5 has the worse supplier performance among all the suppliers of this decision problem. 4.6. sensitivity analysis one dimensional sensitivity analysis is employed to analyze the sensitivity of differentiation of criteria weights. with reference to the findings of this research, c3 (price) is the most important criterion with its highest precedence weight value of 0.096501. in the sensitivity analysis, criteria weight is set freely at an optimal interval, and the weights of all the other criteria are equally increased and decreased. accordingly, the most important criterion’s weight is decreased to 0.01 and increased to the upper limit. regarding to appendix-table a1, the weight of criterion c3 cannot be upraised over 0.77. if it is increased over 0.77, the least important criterion gets a supplier selection by integrated ifdematel-iftopsis method: a case study of automotive… 183 negative value. in this context, the weight of the criterion c3 is kept within the interval of 0.01 ≤ c3 ≤0.77 and new weight values shown in appendix-table a1 are obtained. changes in criteria weights are illustrated in graph 1. as a conclusion of the sensitivity analysis, changes in the alternatives ranking have been observed according to the weights obtained. appendix table a2 shows the changes in the alternatives. as seen in appendix-table a2, if the weight of the criterion c3 is decreased to 0.01 and increased to 0.23, there is no change in the order of the best supplier. on condition that the criterion c3 is increased over 0.23, then the result of the best supplier selection differs. figure 3. sensitivity analysis furthermore as seen in figure 3, if the weight of the criterion c3 is increased over 0.23, then the best supplier changes from supplier 4 (s4) to supplier 2 (s2). the main reason of this difference is that s2 has the best value of the criterion c3 in the evaluations made by the experts. 4.7. comparative analysis for testing the validity of proposed methodology comparative analysis with other mcdm methods is carried out in this section. edas (evaluation based on distance from average solution) and aras (additive ratio assessment) methods recently used in related studies, applied to rank the automotive suppliers mentioned in this study. obtained results are summarized in table 10. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 gergin et al./decis. mak. appl. manag. eng. 5 (1) (2022) 169-193 184 table 10. results of comparative analysis order edas result edas value aras result aras value 1 s4 0.776 s4 0.877 2 s2 0.539 s2 0.850 3 s1 0.478 s1 0.849 4 s3 0.252 s5 0.801 5 s5 0.173 s3 0.791 according to the rankings, results of aras and edas methods are very similar to each other, only the last row indicates a change. also, in comparison with the results of the ifdematel-iftopsis method in the study, it is seen that the order of the third rank supplier (s3) has changed, but the best performing suppliers remain in the same ranking. s4 is the best and s2 is the second in all methods, so the results appear to be valid. 5. conclusion and discussion fierce competition environment in the automotive supply industry obligates the enterprises in spare parts sector to put more emphasis on supplier selection decisions to survive. therefore, providing spare parts at the right time from the right supplier directly affects both the interests and competitive power of the enterprises in the market. from this point of view, supplier selection process in which various criteria play a crucial role, is seen as one of the necessary decision processes for the enterprises. additively, in recent covid-19 pandemic period, correct cooperation and selection in supply chain management has gained more importance both economically and socially. this paper contributes to the related field by suggesting an ifmcdm model to find out the best supplier in the automotive supply industry through considering the fuzziness and ambiguity of dms’ opinions. also, a case study of an enterprise supplying automotive spare parts is conducted effectively. within this framework, firstly, a group of experts working as the sector managers is created. following, a criteria pool is composed in line with a literature review about criteria utilized for supplier selection. then, upon the interviews realized with the expert group, the criteria to be used for the selection of the suppliers are clarified. then, ifdematel is applied to obtain both the relations between the criteria and the weights of these criteria for supplier selection. in the last stage, iftopsis method is performed to identify the best supplier. as a result of this research, the most important criterion for supplier selection is revealed as price (0.0965), followed by quality (0.0957), providing demo products (0.0931), and geographical location (0.0732) respectively. when the findings are compared with the previous studies in related literature, it is seen that weight values of the criteria are parallel with them. the order of priority for the criterion price which is selected as the most important criterion, is in line with the studies xia and wu (2007), kuo et al. (2010), mafakheri et al. (2011), amindoust et al. (2012), zhong and yao (2017). besides, in this case of the study, supplier 4 is identified as the best supplier among alternatives through evaluating with iftopsis method. in addition, with respect to the sensitivity and comparative analyzes this result came out to be valid and robust in other mcdm methods. an important limitation of mcdm techniques is the fact that as the criteria weights change, the results of the research might differ. according to the findings of one supplier selection by integrated ifdematel-iftopsis method: a case study of automotive… 185 dimensional sensitivity analysis presented to minimize the effect of mentioned limitation, it is determined that the weight of the most significant criterion can not be increased over 0.77 and if it is increased over 0.77, the least important criterion will have a negative value. furthermore, if the weight of the criterion c3 is reduced to 0.01 and increased to 0.23, there is no change in the order of the best supplier. in case this value is over 0.23, then the result of the best supplier changes from s4 to s2. in this paper, the information acquired from a group of 5 experts who have had business relations with suppliers, yet there is no information exchange with other enterprises in the same sector. because this study is concentrated on the case of an enterprise in automotive supply industry. therefore, a limitation of this paper is that the inferences of this study represent only one enterprise in the spare parts sector in which this study is carried out. besides, due to the subjectivity in the base of integrated ifdematel-iftopsis method, the results could be different in case different supplier selection criteria are included in or excluded from this study is another limitation. the conclusion of this research are shared with the decision makers involved in this study, and it is seen that the findings show parallelism with the enterprise behaviors. thus, an effective and usable decision-making approach is provided for a real case. also, suppliers analyzed in this research, try to maintain the spare parts in accordance with the requests of the enterprise in question. in future, a contribution may provide to the literature by examining the criteria used in this paper with delphi method or similar methods that ensure consensus in line with the opinions of the experts working in spare parts sector. another future research recommendation may be a study that proposes a new model for supplier selection by including various enterprises in automotive supply industry in turkey. last but not least, another future study could contribute to the literature in a way that determines the relations between criteria and helps the enterprises in automotive supply industry for supplier selection by integrating different mcdm methods and fuzzy logic approaches. author contributions: conceptualization, r. e. gergin; methodology, i. peker; validation, a. c. gök kısa; formal analysis, r. e. gergin; investigation, r. e. gergin and i. peker; data curation, i. peker and a. c. gök kısa; writing-original draft preparation, r. e. gergin; writing-review and editing, i. peker and a. c. gök kısa; supervision, i. peker and a. c. gök kısa. funding: this research received no external funding. conflicts of interest: the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. gergin et al./decis. mak. appl. manag. eng. 5 (1) (2022) 169-193 186 appendix table a1. weights of criteria in the range of 0.01-0.78 criteria c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12 c13 c14 c15 available weights .0732 .0932 .0965 .0496 .0628 .0486 .0567 .0523 .0958 .0619 .0723 .0638 .0684 .0544 .0506 .01 .0794 .0994 .0100 .0558 .0689 .0548 .0629 .0585 .1020 .0681 .0784 .0700 .0746 .0605 .0568 .05 .0766 .0965 .0500 .0530 .0661 .052 .0601 .0556 .0991 .0652 .0756 .0671 .0717 .0577 .0539 .10 .0730 .0929 .1000 .0494 .0625 .0484 .0565 .0520 .0955 .0616 .0720 .0635 .0681 .0541 .0503 .15 .0694 .0894 .1500 .0458 .0589 .0448 .0529 .0485 .0920 .0581 .0684 .0600 .0646 .0505 .0468 .20 .0658 .0858 .2000 .0422 .0554 .0413 .0494 .0449 .0884 .0545 .0649 .0564 .0610 .0470 .0432 025 .0623 .0822 .2500 .0387 .0518 .0377 .0458 .0413 .0848 .0509 .0613 .0528 .0574 .0434 .0396 .30 .0587 .0787 .3000 .0351 .0482 .0341 .0422 .0377 .0813 .0474 .0577 .0492 .0538 .0398 .0360 .35 .0551 .0751 .3500 .0315 .0447 .0305 .0386 .0342 .0777 .0438 .0541 .0457 .0503 .0362 .0325 .40 .0516 .0715 .4000 .0280 .0411 .0270 .0351 .0306 .0741 .0402 .0506 .0421 .0467 .0327 .0289 .45 .0480 .0679 .4500 .0244 .0375 .0234 .0315 .0270 .0705 .0366 .0470 .0385 .0431 .0291 .0253 .50 .0444 .0644 .5000 .0208 .0339 .0198 .0279 .0235 .0670 .0331 .0434 .0350 .0396 .0255 .0218 .55 .0408 .0608 .5500 .0172 .0304 .0163 .0244 .0199 .0634 .0295 .0399 .0314 .0360 .0220 .0182 .60 .0373 .0572 .6000 .0137 .0268 .0127 .0208 .0163 .0598 .0259 .0363 .0278 .0324 .0184 .0146 .65 .0337 .0537 .6500 .0101 .0232 .0091 .0172 .0127 .0563 .0224 .0327 .0242 .0288 .0148 .0110 .70 .0301 .0501 .7000 .0065 .0197 .0055 .0136 .0092 .0527 .0188 .0291 .0207 .0253 .0112 .0075 .75 .0266 .0465 .7500 .0030 .0161 .0020 .0101 .0056 .0491 .0152 .0256 .0171 .0217 .0077 .0039 .76 .0258 .0458 .7600 .0022 .0154 .0013 .0094 .0049 .0484 .0145 .0249 .0164 .0210 .0070 .0032 .77 .0251 .0451 .7700 .0015 .0147 .0005 .0086 .0042 .0477 .0138 .0241 .0157 .0203 .0062 .0025 .78 .0244 .0444 .7800 .0008 .0139 -.0002 .0079 .0035 .0470 .0131 .0234 .0150 .0196 .0055 .0018 table a2. ranking value of alternatives in the range of 0.01-0.78 weight supplier 1 supplier 2 supplier 3 supplier 4 supplier 5 .01 .5082 .5465 .4536 .6239 .3803 .05 .4930 .5568 .4626 .6296 .3977 .10 .4495 .5897 .4897 .6468 .4446 .15 .3943 .6348 .5265 .6711 .5029 .20 .3401 .6816 .5638 .6971 .5596 .21 .3299 .6906 .5708 .7021 .5702 .22 .3199 .6994 .5776 .7070 .5805 .23 .3103 .708 .5841 .7117 .5904 supplier selection by integrated ifdematel-iftopsis method: a case study of automotive… 187 weight supplier 1 supplier 2 supplier 3 supplier 4 supplier 5 .24 .3009 .7164 .5905 .7163 .6001 .25 .2918 .7246 .5965 .7208 .6094 .30 .2503 .7624 .6229 .7404 .6511 .35 .2150 .7948 .6433 .7558 .6852 .40 .1851 .8225 .6587 .7676 .7125 .45 .1595 .8463 .6700 .7764 .7342 .50 .1376 .8667 .6784 .7829 .7510 .55 .1186 .8844 .6845 .7876 .7641 .60 .1022 .8998 .6889 .7911 .7740 .65 .0879 .9133 .6921 .7936 .7814 .70 .0754 .9250 .6944 .7955 .7870 .75 .0646 .9354 .6961 .7968 .7910 .76 .0626 .9373 .6964 .7970 .7916 .77 .0607 .9391 .6966 .7972 .7922 .78 .0588 .9409 .6969 .7974 .7928 references adalı, e.a., & işık, a.t. 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(2017). assessing social risks of global supply chains: a quantitative analytical approach and its application to supplier selection in the german automotive ındustry. journal of cleaner production,149, 96109. © 2021 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 4, issue 2, 2021, pp. 47-75. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame210402047s * corresponding author. e-mail address: biswarupsamanta6@gmail.com (b. samanta), bcgiri.jumath@gmail.com (b.c. giri) a two-echelon supply chain model with price and warranty dependent demand and pro-rata warranty policy under cost sharing contract biswarup samanta 1* and bibhas c. giri 1 1 department of mathematics, jadavpur university, kolkata, india received: 31 december 2020; accepted: 25 march 2021; available online: 29 april 2021. original scientific paper abstract: in this article, a two-echelon supply chain model with a singlevendor a single-buyer is considered. the vendor's production process is imperfect and the market demand is assumed to be dependent on the buyer's selling price and warranty period. the vendor consents to return a definite portion of the buyer's purchase value, if any product is found defective within the length of warranty. the refund value or the warranty cost is considered as a function of the warranty period and the buyer's selling price of the item. this warranty cost is assumed to be fully borne by the vendor in the first model (model i) while in the second model (model ii), it is assumed that the buyer agrees to bear a portion of the warranty cost. the proposed models are solved under decentralized scenario. we also derive and optimize the average total profit of the supply chain in order to obtain the optimal decisions of the centralized model. we consider a stackelberg game between the vendor and the buyer in the decentralized scenario, where the vendor is assumed to be the leader and the buyer as the pursuer. through numerical study, it is observed that, with respect to all the key decisions of the models, model ii provides better outcomes than model i. sensitivity analysis is also carried out to examine the impacts of changes of parameter-values on the optimum decisions. key words: supply chain, optimal pricing, lot sizing, warranty, cost sharing contract. 1. introduction supply chain management (scm) can be defined as the management of flow of goods and services, beginning with the source of the product and ending with the use of the product of the user. the main purpose of scm is to monitor production, distribution and shipment of goods and services. supply chain managers use different mailto:biswarupsamanta6@gmail.com mailto:bcgiri.jumath@gmail.com samanta and giri/decis. mak. appl. manag. eng. 4 (2) (2021) 47-75 48 techniques and approaches to see that the entire chain works efficiently. in order to meet consumer expectations, merchants take inventory as a shared resource using distributed order management technology to fulfill orders from different nodes in a given chain. post-sales service is increasingly an essential factor in scm. failure of the product may occur due to faulty design, poor technical, age, use or increased operational and environmental pressure above the planned level. by guaranteeing after-sales service through warranty and service agreement, vendors can prevent or mitigate the impact of failure. generally, pro-rata warranty (prw) policy is applicable in the commercial enterprise because a single location for manufacturing activities and vast-scale devices commonly demand a long-term protection service program with the long warranty period is accepted by the customers, where a definite fraction of the warranty cost is borne by the customers. murthy and djamaludin (2002) observed that free repair warranty (frw) is erstwhile reflection of an unsavory technique, while prw is a savory technique that can bear the risk of behavior of warranty cost to each other. additionally, prw can be approvable in commercial uses (polatoglu and sahin, 1998). murthy and blischke (2000) mentioned that frw is generally applied for non-repairable products whereas prw is generally applied for repairable products. in this paper, we develop a two-echelon supply chain model consisting of a vendor and a buyer where the defective items in each lot are rejected at the end of the buyer's 100% screening process. this model considers a simple and practical situation where the delivery quantity to the buyer at each shipment is identical. the market demand depends on warranty period and selling price of the product. the product is sold with warranty under pro-rata warranty policy and the warranty cost is taken as a function of warranty period and the buyer's selling price. this paper explores the process of the cost sharing agreement between the vendor and the buyer. the main objectives of this study are to find the answers of the following questions: i) what would be the selling price of each good item from the buyer's side? ii) how many shipments are to be made by the vendor to meet the buyer's order? iii) what would be the size of each shipment from the vendor to the buyer? iv) what warranty period would be offered by the vendor to customers to buy the product? this article incorporates the view of the integrated vendor-buyer approach into the supply chain model with warranty and price dependent demand, and an agreement between the vendor and the buyer to share warranty cost herein. this model also considers that the delivery quantity to the buyer at each shipment is identical as in huang (2004). the rest of the paper is arranged as follows: in the following section, the related literature is reviewed. section 3 presents assumption and notations for developing the proposed model. sections 4 and 5 discuss the mathematical model and its solution procedure, respectively. numerical examples are provided in section 6. the optimal results are analyzed in section 7. section 8 summarizes the paper and indicates some future research directions. 2. literature review in this section, we review the related literature across three research domains – imperfect production, pricing and warranty policy in scm, and cost-sharing contract. the common unrealistic assumption of the joint inventory models is that all units produced are of perfect quality. however, the process may deteriorate and produce a two-echelon supply chain model with price and warranty dependent demand and ….. 49 poor quality items. hill (1997) considered a general type of policy for a single-vendor, single-buyer integrated production-inventory model, based on successive shipments to the buyer, within a single production batch, increased by a fixed factor. later, hill (1999) considered the problem of a vendor supplying a product to a buyer. the vendor manufactures the product in batches at a finite rate and ships the output to the buyer. the buyer then consumes the product at a fixed rate. goyal and nebebe (2000) considered the problem of determining economic production and shipment policy of a product supplied by a vendor to a buyer. jaber (2008) extended the work of salameh and jaber (2000) by assuming that the percentage defective per lot reduces according to a learning curve, which was empirically validated by data from the automotive industry. khan et al. (2011) summarized the body of research that extended salameh and jaber’s (2000) eoq model for imperfect items. there are more works in this directions (goyal & szendrovits, 1986; lee & rosenblatt, 1987; cheg, 1991; hoque & goyal, 2000; goyal & cardenas-barron, 2002; ertogral et al., 2007; taleizadeh et al., 2012; cheng et al., 2018; prez & torres, 2019; despic et al., 2019pamucar & savin, 2020). in this paper, the production process is assumed to be imperfect; it produces a certain number of defective items as considered by huang (2004). blischke and murthy (1992) formulated a taxonomy for warranty to assist the manager responsible for product warranty in choosing appropriate alternatives for evaluation before a final choice was made. murthy and blischke (1992) focused their attention mainly on system characterization, the first step of the systems approach. thomasand rao (1999) reviewed the literature on warranty models and analysis methods which were provided, along with some suggestions for further research. yeh et al. (2000) studied the optimal production run length for a deteriorating production system in which the products were sold with free minimal repair warranty. murthy and djamaludin (2002) carried out a review of the literature that has appeared in the last ten years. they highlighted issues of interest to manufacturers in the context of managing new products from an overall business perspective. jung and park (2003) developed the optimal periodic preventive maintenance policies following the expiration of warranty. yeh et al. (2007) investigated the effects of a free-repair warranty on the periodic replacement policy for a repairable product. naeij and shavandi (2010) developed a two-echelon supply chain model with one supplier and multi-retailer for a single product. chen and zhou (2012) presented a review of the issues associated with a manufacturer's pricing strategies in a two-echelon supply chain that comprises one manufacturer and two competing retailers, with warranty period-dependent demand. park et al. (2013) considered a renewable minimal repairreplacement warranty policy and proposed an optimal maintenance model after the warranty is expired. wu (2014) proposed three warranty return policies which decide whether new items should be sent to warranty claimants or not. wei et al. (2015) explored the optimal strategies on price and warranty period of two complementary products in a supply chain with two manufacturers and one common retailer from a two-stage game theoretic perspective. xie et al. (2016) studied a supply chain consisting of one supplier and n retailers. the market demand for each retailer was assumed to be dependent on the difference between the retail price and the average retail price. roy et al. (2016) considered a dual channel where the manufacturer uses e-tail channel and traditional retail channel to promote selling the items. mukhopadhyay and goswami (2016) developed an eoq type model showing the effect of newly launched hi-tech products with time and selling price dependent demand. maiti and giri (2017) presented a two-period supply chain model which was comprised of one manufacturer and one retailer who were involved in trading a single product. they assumed that the demand rate in each period is dependent on the selling samanta and giri/decis. mak. appl. manag. eng. 4 (2) (2021) 47-75 50 prices of the current period and the previous period. roy et al. (2018) studied a twoechelon supply chain model with single manufacturer and two competing retailers. the manufacturer announced wholesale price for the retailers and the retailers contest with each other declaring separate sales prices. a two-echelon closed-loop supply chain with one manufacturer and one retailer was considered by giri et al. (2018) and two game theoretic models were presented in which the first model (model i) considers demand dependent on selling price and warranty period while the second model (model ii) considers demand dependent on greening level in addition to the selling price and warranty period. sana (2020) investigated a price contest between green and non-green producers where the market demand depends on sales price, carbon emission and corporate social responsibility index. khorshidvand et al. (2021) developed a multi-level multi-channel supply chain considering the prices of sale channels, the advertisement level, and the green policy of the product. most of the above mentioned articles are contributed on pricing, warranty period and imperfect production. emphasize has not been given on warranty cost. here, we consider a function which balances between the warranty period and the warranty cost. the buyer sells all items with a pro-rata warranty (prw) policy. according to this argument, the vendor consents to pay a portion of the shopper's purchase value, if any product is found defective within the warranty period. this warranty cost decreases as the failure time of the product increases from the initial purchase. rogerson (2003) considered two-item menus where one item was a cost reimbursement contract and the other item was a fixed price contract. chu and sappington (2007) extended rogerson (2003) intriguing analysis of simple procurement contracts to settings where the supplier’s production cost was not necessarily distributed uniformly. huangand fang (2008) considered a decision problem under the policy of a pro-rata warranty (prw) and proposed a bayesian decision model in determining the optimal warranty proportion. chaoet al. (2009) discussed two contractual agreements by which product recall costs can be shared between a manufacturer and a supplier to induce quality improvement effort. leng and parlar (2010) considered a multisupplier, single manufacturer assembly supply chain by introducing appropriate buyback and lost-sales cost-sharing contracts, where the suppliers produce components of a short life-cycle product which is assembled by the manufacturer. tsao and sheen (2012) considered a two-echelon multi-retailer distribution channel under retailers promotional efforts and the sales learning curve incorporating the idea of the sales learning curve into the promotion cost. de giovanni and zaccour (2013) showed that a cost-revenue sharing is successful only under particular conditions, while the retailer is always willing to implement such a contract, the manufacturer is better off only when the product return and the remanufacturing efficiency are sufficiently large, and the sharing parameter is not too high. zhao et al. (2014) derived the optimal solutions of the nash equilibrium without cost sharing contract, and the stackelberg equilibrium with the integrator as the leader who partially shares the cost of the efforts of the supplier. a two-echelon supply chain model with price and warranty dependent demand and ….. 51 table 1. a comparison of the study at hand with the existing literature with respect to important model characteristics. although the concept of cost sharing in our model is similar to their models, our model construction and pricing decision are absolutely different from their models. some of the previous models cannot include warranty period on market demand. the models that considered the demand as a function of warranty period didn’t pay attention to the corresponding warranty cost as well as warranty cost sharing contract. we introduce the impact of warranty period and selling price simultaneously on demand in a two-echelon supply chain model. we consider pro rata warranty (prw) policy in our study. under this warranty, if an item is found defective before the deadline of the warranty period, it is replaced with some discount, which depends on the longevity of the item at the time of failure. the replacement item is then covered by an identical new warranty. this type of warranty is generally applied on nonrepairable products such as batteries, tires, etc. if an item is covered by a warranty, the vendor needs to set the warranty period and predict the corresponding warranty cost. sometimes the warranty period is influenced by the opponents in the market. for example, if a car company offers only a 1 year limited warranty, no one will intend to buy a new car, since there are so many car companies who offer 5, 7 or even 10 years warranty assurance. after settling the warranty policy, the vendor needs to predict the allotment to cover the future warranty cost. that is why, we construct a warranty cost function which calculates how much amount is discounted to the customer if an item needs to replace during warranty period. in this article, we investigate two scenarios. firstly, the vendor covers the whole warranty cost and the buyer acts as a mediator between the purchaser and the vendor if any item fails during the warranty period and secondly, the buyer agrees to share a portion of warranty cost keeping the warranty policy unchanged. table 1 compares the model developed in this study with the earlier works done in the relevant literature. vendorbuyer coordi nation imper -fect produ -ction scree ning optimal pricing warranty contract huang (2004) √ √ √ goyal & nebe (2000) √ √ jaber (2008) √ √ jung & park (2003) √ √ chen & zhou (2012) √ √ √ wu (2014) √ √ wei et. al. (2015) √ √ √ giri et. al. (2018) √ √ √ huang & frang (2008) √ √ √ chao et. al. (2009) √ √ leng & parlar (2010) √ √ √ cheng et. al. (2018) √ √ jaber et. al. (2008) √ √ samanta et. al. (2018) √ √ √ √ √ shah & chaudhuri (2016) √ √ √ this study √ √ √ √ √ √ samanta and giri/decis. mak. appl. manag. eng. 4 (2) (2021) 47-75 52 3. model assumptions and notations the notations used throughout the paper are given in table 2. table 2. notations 𝑄 ∶ size of each shipment from the vendor to the buyer (decision variable); 𝜔 ∶ length of the warranty period offered by the vendor (decision variable); 𝑛 ∶ total number of shipments per lot from the vendor to the buyer (decision variable); 𝑝 ∶ selling price per item for the buyer (decision variable); 𝑄𝑝 ∶ lot size; 𝐷(𝑝,𝜔): demand rate; 𝑃 ∶ production rate (𝑃 > 𝐷(𝑝,𝜔)); 𝑆𝑣 ∶ set up cost per production run for the vendor; 𝑆𝑏 ∶ ordering cost for the buyer; ℎ𝑣 ∶ stock-holding cost per item per year for the vendor; ℎ𝑏 ∶ stock-holding cost per item per year for the buyer; 𝑇 ∶ time interval between successive deliveries; 𝑇𝑐 ∶ cycle time; 𝐹 ∶ transportation cost per shipment; 𝐶 ∶ unit production cost; 𝑧 ∶ screening rate; 𝑥 ∶ unit screening cost; 𝑊 ∶ wholesale price per item for the vendor; 𝜆 ∶ failure intensity of a product; 𝜓 ∶ percentage of defective items; 𝜃 ∶ fraction of the total warranty cost borne by the buyer; 𝑡 ∶ variable time ∈ (0,𝑛𝑇); 𝑔(𝑡) ∶ failure density function; 𝐺(𝑡) ∶ cumulative failure distribution of a product associated with 𝑔(𝑡); 𝑅(𝑡) ∶ failure rate at any time 𝑡; 𝑟(𝑥) ∶ refund cost function of a failure item failed at any time 𝑥 ∈ (𝑡,𝑡 + 𝜔) from initial purchase; 𝑤(𝑡) ∶ warranty cost at any time 𝑡 ∈ (0,𝑛𝑇); 𝑤𝑚 ∶ total warranty cost in [0,𝑛𝑇]. the following assumptions are made to develop the proposed vendor-buyer model: (i) the model considers a single-vendor and a single-buyer for a single product. (ii) the time period is infinite and stock out is not allowed. (iii) the vendor's production rate p is constant whereas market demand 𝐷(𝑝,𝜔) depends on the buyer's selling price, satisfying the relation 𝐷(𝑝,𝜔) = 𝑎 − 𝑏𝑝 + 𝑐𝜔, where 𝑎,𝑏 > 0. (iv) successive deliveries are scheduled so that the next one arrives at the buyer when his/her stock from previous shipment has just been finished. (v) the product screening policy is performed to detect the defective items. the vendor delivers defective items in a single batch at the end of the buyer's 100% screening process with screening rate 𝑧 and unit screening cost 𝑥. it is assumed that there is no error in inspection and number of perfect units is at least equal to the demand during the screening time. a two-echelon supply chain model with price and warranty dependent demand and ….. 53 (vi) the buyer sells all items with a pro-rata warranty (prw) policy. according to this argument, the vendor consents to pay a portion of shopper's purchase value, if any product is found defective within the warranty stage proposed by the buyer. (vii) the warranty cost depends on the warranty period 𝜔. the refund cost function 𝑟(𝑥) of an item failed at any time 𝑥 ∈ (𝑡,𝑡 + 𝜔) from initial purchase is assumed as 𝑟(𝑥) = 𝑝(1 − 𝑥−𝑡 𝜔 );𝑡 ≤ 𝑥 ≤ (𝑡 + 𝜔). (samanta et al., 2018). 4. model formulation before formulating the objective value, we remember that only 𝜓% of total products are defective and these products must be rejected. therefore, during a production cycle, we have 𝐷 ≤ 𝑃(1 −𝜓 ) which means that 𝐷 1−𝜓 products must be produced to meet the whole demand. since, in a production cycle, the vendor produced 𝑛𝑄 number of products with production rate p, the total time of production is 𝑛𝑄 𝑃 . in a production cycle, the total time = 𝑛𝑇, the total demand = 𝑛𝐷𝑇 and the total acceptable products = 𝑛𝑄(1− 𝜓 ). since the total amount of acceptable products fulfills the buyer's total demand in a production cycle, so 𝑛𝐷𝑇 = 𝑛𝑄(1 − 𝜓), which implies that 𝑇 = 𝑄(1−𝜓) 𝐷 . for the vendor, sales revenue per unit time = 𝑊𝐷, production cost per unit time = 𝑛𝑄𝐶 𝑛𝑇 = 𝐶 (1−𝜓) (𝑎 − 𝑏𝑝 +𝑐𝜔), set-up cost per unit time = 𝑆𝑣 𝑛𝑇 = (𝑎−𝑏𝑝+𝑐𝜔)𝑆𝑣 𝑛𝑄(1−𝜓) , holding cost per unit time = [ 𝑄 2 + 𝑛−2 2 𝑄(1− 𝑎−𝑏𝑝+𝑐𝜔 𝑃(1−𝜓) )]ℎ𝑣 (huang, 2004), warranty cost per unit time = 𝜆 2 𝑝𝜔(𝑎 − 𝑏𝑝 + 𝑐𝜔) and discount cost per defective item per unit time = 𝑤𝑛𝑄(1−𝑅) 𝑛𝑇 = 𝑤𝐷(1−𝑅) 𝑅 . therefore, the vendor's total profit per unit time is given by 𝜋𝑣(𝑛,𝜔) = (𝑊 − 𝐶 1− 𝜓 − 𝑆𝑣 𝑛𝑄(1 − 𝜓) − 𝜆 2 𝑝𝜔)(𝑎 − 𝑏𝑝 +𝑐𝜔) −[ 𝑄 2 + 𝑛−2 2 𝑄(1 − 𝑎−𝑏𝑝+𝑐𝜔 𝑃(1−𝜓) )]ℎ𝑣 (1) for the buyer, sales revenue per unit time = 𝑝𝐷 = 𝑝(𝑎 − 𝑏𝑝 + 𝑐𝜔), purchase cost per unit time = 𝑊𝐷 = 𝑊(𝑎 − 𝑏𝑝 + 𝑐𝜔), holding cost per unit time = [ 𝑄 2 (1 − 𝜓) + 𝑝(𝑎−𝑏𝑝+𝑐𝜔)𝑄𝜓 𝑧(1−𝜓) ]ℎ𝑏 (huang,2004), transportation cost per unit time = 𝐹𝐷 𝑄(1−𝜓) = 𝑝(𝑎−𝑏𝑝+𝑐𝜔)𝐹 𝑄(1−𝜓) , screening cost per unit time = 𝑥𝐷 1−𝜓 = 𝑝(𝑎−𝑏𝑝+𝑐𝜔)𝑥 1−𝜓 , and ordering cost per unit time = 𝑆𝑏𝐷 𝑛𝑄(1−𝜓) = 𝑝(𝑎−𝑏𝑝+𝑐𝜔)𝑆𝑏 𝑛𝑄(1−𝜓) . therefore, the buyer's total profit per unit time is given by 𝜋𝑏(𝑝,𝑄) = (𝑝 − 𝑊)(𝑎 − 𝑏𝑝 + 𝑐𝜔) − 𝑝(𝑎 − 𝑏𝑝 + 𝑐𝜔)(𝐹 + 𝑆𝑏 𝑛 ) 𝑄(1 − 𝜓) − 𝑝(𝑎 − 𝑏𝑝 +𝑐𝜔)𝑥 1 − 𝜓 −[ 𝑄 2 (1 − 𝜓) + 𝑝(𝑎−𝑏𝑝+𝑐𝜔)𝑄𝜓 𝑧(1−𝜓) )]ℎ𝑏 (2) hence, the average total profit of the supply chain is given by 𝜋(𝑛,𝑝,𝑄,𝜔) = 𝜋𝑣(𝑛,𝜔) +𝜋𝑏(𝑝,𝑄) samanta and giri/decis. mak. appl. manag. eng. 4 (2) (2021) 47-75 54 = (𝑝 − 𝐶 + 𝑝𝑥 1 − 𝜓 − 𝜆 2 𝑝𝜔)(𝑎 − 𝑏𝑝 +𝑐𝜔) − 𝑝(𝑎 − 𝑏𝑝 + 𝑐𝜔)(𝐹 + 𝑆𝑏 + 𝑆𝑣 𝑛 ) 𝑄(1 − 𝜓) −[ 𝑄 2 (1 − 𝜓) + 𝑝(𝑎−𝑏𝑝+𝑐𝜔)𝑄𝜓 𝑧(1−𝜓) ]ℎ𝑏 −[ 𝑄 2 + 𝑛−2 2 𝑄(1− 𝑎−𝑏𝑝+𝑐𝜔 𝑃(1−𝜓) )]ℎ𝑣 (3) 4.1. centralized policy in the centralized scenario, the buyer and the vendor are regarded as a joint trade unit. they take decisions jointly on lot size in each delivery, retail price as well as warranty period systematically to encourage sales, and maximize the total profit of the supply chain. due to existence of a single decision maker, the interior parameter w (wholesale price) does not play any role. we have, 𝜕𝜋 𝜕𝜔 = − 1 2𝑛𝑃𝑄𝑧(1−𝜓) [ 𝑛𝑝(𝑎 − 𝑏𝑝)𝑃𝑄𝑧𝜆(1 − 𝜓) +2𝑐𝑛𝑃𝑄2ℎ𝑏𝜓 + 𝑐𝑧{2𝑃(𝑛𝐹 + 𝑆𝑏 + 𝑆𝑣)+ 2𝑛𝑃𝑄(𝐶 + 𝑥 + 𝑝(𝜔𝜆 − 1)(1 − 𝜓))}] 𝜕2𝜋 𝜕𝜔2 = −𝑐𝑝𝜆 (4) 𝜕𝜋 𝜕𝑝 = 𝑏𝑧{2(𝐶 +𝑥)𝑛𝑃𝑄 −ℎ𝑣𝑛(𝑛 − 2)𝑄 2 + 2𝑃(𝑛𝐹 + 𝑆𝑏 +𝑆𝑣)} + 2𝑏𝑛𝑃ℎ𝑏𝑄 2 2𝑛𝑃𝑄𝑧(1 − 𝜓) + {(𝑎 − 𝑏𝑝 + 𝑐𝜔)𝑧 − 2𝑝}(2− 𝜔𝜆) 2𝑧 𝜕2𝜋 𝜕𝑝2 = −𝑏(2 − 𝜔𝜆) (5) 𝜕𝜋 𝜕𝑄 = 1 2 [ {ℎ𝑣(𝑛 − 2)𝑛𝑄 2 + 2𝑃(𝑛𝐹 + 𝑆𝑏 +𝑆𝑣)}(𝑎 − 𝑏𝑝 + 𝑐𝜔) 𝑛𝑃𝑄2(1 − 𝜓) − ℎ𝑣(𝑛 −1) − ℎ𝑏{𝑧(1 −𝜓) 2 +2(𝑎 − 𝑏𝑝 + 𝑐𝜔)𝜓} 𝑧(1 −𝜓) ] 𝜕2𝜋 𝜕𝑄2 = − 2(𝑛𝐹+𝑆𝑏+𝑆𝑣)(𝑎−𝑏𝑝+𝑐𝜔) 𝑛𝑄3(1−𝜓) (6) 𝜕2𝜋 𝜕𝜔𝜕𝑝 = 0 𝜕2𝜋 𝜕𝑄𝜕𝑝 = − 𝑏 1 − 𝜓 𝐴0 𝜕2𝜋 𝜕𝜔𝜕𝑄 = 𝑐 1 −𝜓 𝐴0 where, a0 = [ (𝑛−2)ℎ𝑣 2𝑃 + 𝑛𝐹+𝑆𝑏+𝑆𝑣 𝑛𝑄2 − ℎ𝑏𝜓 𝑧 ]. from (4), (5) and (6), we have the following proposition: a two-echelon supply chain model with price and warranty dependent demand and ….. 55 proposition 1. (i) for fixed selling price (p), the integrated profit function 𝜋(𝑛,𝑝,𝑄,𝜔) is concave with respect to the warranty period (𝜔) whatever may be the lot size (𝑄) in each shipment. (ii) the integrated profit function 𝜋(𝑛,𝑝,𝑄,𝜔) is concave with respect to the selling price (p) for the warranty period (𝜔) satisfying the condition 0 ≤ 𝜔 ≤ 2 𝜆 . (iii) for fixed selling price (𝑝), if the vendor agrees to sell all the items with predefined warranty period (𝜔), then the integrated profit function 𝜋(𝑛,𝑝,𝑄,𝜔) is concave with respect to the shipment size (𝑄). we now examine the existence of unique optimal solution of the profit function 𝜋(𝑛,𝑝,𝑄,𝜔) in the following proposition: proposition 2. the profit function 𝜋(𝑛,𝑝,𝑄,𝜔) is jointly concave in 𝑝,𝑄 and 𝜔 if each of the following conditions is satisfied: (i) 𝑄 > 𝐴13 2 + 1 2 √ 12𝑛𝐴4−{3𝐴2 2+16𝐴1}𝐴13 3𝐴2𝐴13 (ii) 𝑝 > 𝑎 2𝑏 + √( 𝑎 2𝑏 )2 − 𝑏2{ℎ𝑣𝑛𝑧(𝑛−2)𝑄 2+2𝑃(𝑛𝐹+𝑆𝑏+𝑆𝑣)𝑧−2ℎ𝑏𝑛𝑃𝑄 2𝜓}2 8𝑏𝑐𝑛𝑝𝑃2𝑄(𝑛𝐹+𝑆𝑏+𝑆𝑣)𝑧 2(1−𝜓)𝜆 (iii) 1 c [ b2{hvnz(n-2)q 2+2p(nf+sb+sv)z-2hbnpq 2ψ} 2 8bcnpp2q(nf+sb+sv)z 2(1-ψ)λ -(a-bp)] < ω < 2 λ , where, 𝐴1 = 𝑏 3𝑛𝑃(𝑛𝐹 + 𝑆𝑏 + 𝑆𝑣)𝑧{ℎ𝑣(𝑛 − 2)𝑧 − 2ℎ𝑏𝑃𝜓} 𝐴2 = 𝑏 3𝑛2{ℎ𝑣(𝑛 −2)𝑧 − 2ℎ𝑏𝑃𝜓} 2 𝐴3 = 4𝑏3𝐴2𝐴4 𝑎2𝑐𝑛𝜆(1 − 𝜓) 𝐴4 = 𝑎 2𝑐𝑛𝑃2(𝑛𝐹 + 𝑆𝑏 + 𝑆𝑣)𝑧 2𝜆(1 − 𝜓) 𝐴5 = 27𝐴2𝐴4 2 − 64𝐴1𝐴3 𝐴11 = 2 2 3{𝐴5 + √𝐴5 2 − 1024𝐴3 3} 1 3 𝐴12 = 32𝐴3 + 2 1 3𝐴11 2 3× 2 2 3𝐴11𝐴2 𝐴13 = √𝐴12 − 8𝐴1 3𝐴2 proof: the hessian matrix associate with 𝜋(𝑛,𝑝,𝑄,𝜔) is given by 𝐻1 = ( 𝜕2𝜋 𝜕𝜔2 𝜕2𝜋 𝜕𝜔𝜕𝑝 𝜕2𝜋 𝜕𝜔𝜕𝑄 𝜕2𝜋 𝜕𝑝𝜕𝜔 𝜕2𝜋 𝜕𝑝2 𝜕2𝜋 𝜕𝑝𝜕𝑄 𝜕2𝜋 𝜕𝑄𝜕𝜔 𝜕2𝜋 𝜕𝑄𝜕𝑝 𝜕2𝜋 𝜕𝑄2 ) samanta and giri/decis. mak. appl. manag. eng. 4 (2) (2021) 47-75 56 = ( −𝑐𝑝𝜆 0 𝑐 1−𝜓 𝐴0 0 −𝑏(2 −𝜔𝜆) − 𝑏 1−𝜓 𝐴0 𝑐 1−𝜓 𝐴0 − 𝑏 1−𝜓 𝐴0 − 2(𝑛𝐹+𝑆𝑏+𝑆𝑣)(𝑎−𝑏𝑝+𝑐𝜔) 𝑛𝑄3(1−𝜓) ) here 𝜕2𝜋 𝜕𝜔2 = −𝑐𝑝𝜆 < 0 and the second order minor will be positive if 2 − 𝜔𝜆 > 0 i.e., if 0 < 𝜔 < 2 𝜆 . (7) and|𝐻1| = 𝑏(2−𝜔𝜆) 4𝑛2𝑃2𝑄4𝑧2(1−𝜓)2 [𝑏2{ℎ𝑣𝑛𝑧(𝑛 − 2)𝑄 2 + 2𝑃(𝑛𝐹 + 𝑆𝑏 + 𝑆𝑣)𝑧 − 2ℎ𝑏𝑛𝑃𝑄 2𝜓}2 − 8𝑐𝑛𝑝𝑃2𝑄(𝑛𝐹 + 𝑆𝑏 +𝑆𝑣)(𝑎 − 𝑏𝑝 +𝑐𝜔)𝑧 2𝜆(1 − 𝜓)]. it is clear that the profit function 𝜋(𝑛,𝑝,𝑄,𝜔) has unique solution if 𝐻1 is negative definite i.e., if |𝐻1| < 0 i.e.,if𝑏2{ℎ𝑣𝑛𝑧(𝑛 −2)𝑄 2 +2𝑃(𝑛𝐹 + 𝑆𝑏 + 𝑆𝑣)𝑧 − 2ℎ𝑏𝑛𝑃𝑄 2𝜓}2 < 8𝑐𝑛𝑝𝑃2𝑄(𝑛𝐹 + 𝑆𝑏 + 𝑆𝑣)(𝑎 −𝑏𝑝 + 𝑐𝜔)𝑧 2𝜆(1 −𝜓) i.e., if 𝑎 − 𝑏𝑝 + 𝑐𝜔 > 𝑏2{ℎ𝑣𝑛𝑧(𝑛−2)𝑄 2+2𝑃(𝑛𝐹+𝑆𝑏+𝑆𝑣)𝑧−2ℎ𝑏𝑛𝑃𝑄 2𝜓} 2 8𝑐𝑛𝑝𝑃2𝑄(𝑛𝐹+𝑆𝑏+𝑆𝑣)𝑧 2𝜆(1−𝜓) i.e, if 𝜔 > 1 𝑐 [ 𝑏2{ℎ𝑣𝑛𝑧(𝑛−2)𝑄 2+2𝑃(𝑛𝐹+𝑆𝑏+𝑆𝑣)𝑧−2ℎ𝑏𝑛𝑃𝑄 2𝜓} 2 8𝑏𝑐𝑛𝑝𝑃2𝑄(𝑛𝐹+𝑆𝑏+𝑆𝑣)𝑧 2(1−𝜓)𝜆 − (𝑎 − 𝑏𝑝)] (8) combining (7) and (8), we find the result (iii). to satisfy the condition (8), the right hand side must be positive and hence we can write 𝑏2{ℎ𝑣𝑛𝑧(𝑛 − 2)𝑄 2 + 2𝑃(𝑛𝐹 + 𝑆𝑏 +𝑆𝑣)𝑧 −2ℎ𝑏𝑛𝑃𝑄 2𝜓}2 > 8𝑏𝑐𝑛𝑝𝑃2𝑄(𝑛𝐹 + 𝑆𝑏 + 𝑆𝑣)𝑧 2(1 − 𝜓)𝜆(𝑎 − 𝑏𝑝) or, 𝑏𝑝2 − 𝑎𝑝 + 𝑏2{ℎ𝑣𝑛𝑧(𝑛−2)𝑄 2+2𝑃(𝑛𝐹+𝑆𝑏+𝑆𝑣)𝑧−2ℎ𝑏𝑛𝑃𝑄 2𝜓} 2 8𝑐𝑛𝑃2𝑄(𝑛𝐹+𝑆𝑏+𝑆𝑣)𝑧 2(1−𝜓)𝜆 > 0 the equation 𝑏𝑝2 −𝑎𝑝 + 𝑏2{ℎ𝑣𝑛𝑧(𝑛−2)𝑄 2+2𝑃(𝑛𝐹+𝑆𝑏+𝑆𝑣)𝑧−2ℎ𝑏𝑛𝑃𝑄 2𝜓} 2 8𝑐𝑛𝑃2𝑄(𝑛𝐹+𝑆𝑏+𝑆𝑣)𝑧 2(1−𝜓)𝜆 = 0 has two real roots 𝑝 = 𝑎 2𝑏 − √( 𝑎 2𝑏 ) 2 − 𝑏2{ℎ𝑣𝑛𝑧(𝑛 − 2)𝑄 2 + 2𝑃(𝑛𝐹 + 𝑆𝑏 + 𝑆𝑣)𝑧 − 2ℎ𝑏𝑛𝑃𝑄 2𝜓}2 8𝑏𝑐𝑛𝑝𝑃2𝑄(𝑛𝐹 +𝑆𝑏 + 𝑆𝑣)𝑧 2(1 − 𝜓)𝜆 and 𝑝 = 𝑎 2𝑏 + √( 𝑎 2𝑏 ) 2 − 𝑏2{ℎ𝑣𝑛𝑧(𝑛−2)𝑄 2+2𝑃(𝑛𝐹+𝑆𝑏+𝑆𝑣)𝑧−2ℎ𝑏𝑛𝑃𝑄 2𝜓}2 8𝑏𝑐𝑛𝑝𝑃2𝑄(𝑛𝐹+𝑆𝑏+𝑆𝑣)𝑧 2(1−𝜓)𝜆 but we are interested to find the selling price (𝑝) as real and positive. hence the result (ii) is obtained. again, the result (ii) can be accepted in real market only when the term in the square root is positive and hence it follows another relation which is ( 𝑎 2𝑏 ) 2 − 𝑏2{ℎ𝑣𝑛𝑧(𝑛 − 2)𝑄 2 + 2𝑃(𝑛𝐹 + 𝑆𝑏 + 𝑆𝑣)𝑧 − 2ℎ𝑏𝑛𝑃𝑄 2𝜓}2 8𝑏𝑐𝑛𝑝𝑃2𝑄(𝑛𝐹 + 𝑆𝑏 + 𝑆𝑣)𝑧 2(1 − 𝜓)𝜆 > 0 the left hand side is a function of 𝑄 only. the shipment size (𝑄) must be real and positive. remembering this, we can simplify the above relation and after some algebraic manipulations, we can prove the result (𝑖) by considering the relations as given in the proposition. hence the proposition is proved. using the first order conditions for optimality of the profit function 𝜋(𝑛,𝑝,𝑄,𝜔), the equilibrium solution can be obtained. we first consider the first order conditions: 𝜕𝜋 𝜕𝑝 = 0, 𝜕𝜋 𝜕𝑄 = 0 and 𝜕𝜋 𝜕𝜔 = 0. a two-echelon supply chain model with price and warranty dependent demand and ….. 57 from the above equations, we obtain 𝑝(𝜔,𝑄) = 𝑎+𝑐𝜔 2𝑏 + 2𝑛ℎ𝑏𝑃𝑄 2𝜓+𝑧{2𝑃(𝑛𝐹+𝑆𝑏+𝑆𝑣)+2𝑛𝑃𝑄(𝑥+𝐶)−ℎ𝑣𝑛(𝑛−2)𝑄 2} 2𝑛𝑃𝑄𝑧(2−𝜆𝜔)(1−𝜓) (9) 𝑄(𝜔,𝑝) = √ 2𝑃𝑧(𝑎−𝑏𝑝+𝑐𝜔)(𝑛𝐹+𝑆𝑏+𝑆𝑣) 𝑛[𝑃𝑧(1−𝜓){(1−𝜓)ℎ𝑏+(𝑛−1)ℎ𝑣}+(𝑎−𝑏𝑝+𝑐𝜔){2ℎ𝑏𝑃𝜓−(𝑛−2)𝑧ℎ𝑣}] (10) 𝜔(𝑝,𝑄) = 𝑧[2𝑛𝑃𝑄{2𝑝(1−𝜓)−(𝐶+𝑥)}+ℎ𝑣𝑛(𝑛−2)𝑄 2−2𝑃(𝑛𝐹+𝑆𝑏+𝑆𝑣)]−2𝑐𝑛ℎ𝑏𝑃𝑄 2𝜓 2𝑛𝑝𝑃𝑄𝑧𝜆(1−𝜓) − 𝑎−𝑏𝑝 2𝑐 (11) substituting the value of 𝑝 from (9) into equation (10), we have 𝑄(𝜔) = 𝑏2𝑛2𝐴7 4𝐴2 + 𝐴15 2 + 𝑏 2 √𝑛𝑏 2{𝑛3𝐴7 2(𝐴2𝐴15+𝑛 2𝑏4𝐴7)+16𝐴2 2𝐴8} 4𝐴2 3𝐴15 + 2 5 3𝐴2(𝐴10 2−2 2 3𝐴9) 3𝐴3𝐴10 (12) where, 𝐴6 = 2𝑏(𝐶 + 𝑥)− (𝑎 + 𝑐𝜔)(2 − 𝜔)(1− 𝜓) 𝐴7 = 𝑃 2𝑧2[(𝑛𝐹 +𝑆𝑏 + 𝑆𝑣) 𝐴2𝐴6 𝐴1 + 2(2 − 𝜔)(1 − 𝜓)2{(1 − 𝜓)ℎ𝑏 + (𝑛 − 1)ℎ𝑣}] 𝐴8 = 𝐴4𝐴6 𝑎2𝑐𝜆(1−𝜓) 𝐴9 = 6{ 𝐴3 𝑏4 + 𝐴4𝐴6𝑛 2𝑃2𝑧2 𝑎2𝑐𝜆(1−𝜓) } 𝐴10 = 27{ 𝑛4𝐴3𝐴7 2 𝑏2𝐴2 − 4𝐴2𝐴4 2𝐴6 2 𝑎4𝑏2𝑐2𝜆2(1−𝜓)2 } 𝐴14 = (𝐴10 +√𝐴10 2 +4𝐴9 3) 1 3 𝐴15 = √ 𝐴7 2𝑛4𝑏4 4𝐴2 2 + 2 1 3𝑏2𝐴9 𝐴2𝐴10 − 𝑏2𝐴10 3 ×2 1 3𝐴2 now, substituting the value of 𝑄(𝜔) given in (12) in the equation (9), we have 𝑝(𝜔) = 𝑎+𝑐𝜔 2𝑏 + 2𝑛ℎ𝑏𝑃(𝑄(𝜔)) 2𝜓+𝑧{2𝑃(𝑛𝐹+𝑆𝑏+𝑆𝑣)+2𝑛𝑃𝑄(𝜔)(𝑥+𝐶)−ℎ𝑣𝑛(𝑛−2)(𝑄(𝜔)) 2} 2𝑛𝑃𝑄(𝜔)𝑧(2−𝜆𝜔)(1−𝜓) (13) again, using (12) and (13), the equation (11) changes into the equation 𝜔 = 𝑧[ 2𝑛𝑃𝑄(𝜔){2𝑝(𝜔)(1 −𝜓) − (𝐶 +𝑥)} +ℎ𝑣𝑛(𝑛 − 2)𝑄(𝜔) 2 − 2𝑃(𝑛𝐹 + 𝑆𝑏 + 𝑆𝑣) ]− 2𝑐𝑛ℎ𝑏𝑃𝑄(𝜔) 2𝜓 2𝑛𝑝(𝜔)𝑃𝑄(𝜔)𝑧𝜆(1 −𝜓) − 𝑎 − 𝑏𝑝(𝜔) 2𝑐 solving the above equation, we can find the optimum value of 𝜔. let this optimum value be denoted by 𝜔∗. then the optimum values 𝑄∗,𝑝∗ of 𝑄 and 𝑝 can be found by putting 𝜔 = 𝜔∗in (12) and (13), respectively. these optimum values 𝑄∗,𝑝∗and 𝜔∗also give the optimum profit of the integrated supply chain model when 𝑝 = 𝑝∗,𝑄 = 𝑄∗ and 𝜔 = 𝜔∗are substituted in equation (3). 4.2. decentralized policy in the decentralized scenario, the vendor and the buyer are separate self-concerned members who intend to optimize their own profits. we assume that the vendor acts as the stackelberg leader and the buyer as the pursuer. at first the vendor sets the samanta and giri/decis. mak. appl. manag. eng. 4 (2) (2021) 47-75 58 warranty period and the number of shipments per lot, and then the buyer sets his/her selling price and the order quantity. the game is calculated with the help of backward policy. 4.2.1. model i if an item is found defective during warranty period from the moment of initial purchase, then the vendor agrees to repair or replace that item, whatever required. in this model, we consider that the cost due to this warranty is fully borne by the vendor. from (1) and (2), we have the following results: 𝜕𝜋𝑏 𝜕𝑝 = 𝑎 − 2𝑏𝑝 +𝑏𝑊 + 𝑐𝜔 + 𝑏(𝑛𝐹 + 𝑆𝑏 + 𝑥𝑛𝑄) 𝑛𝑄(1− 𝜓) + 𝑏ℎ𝑏𝑄𝜓 𝑧(1 − 𝜓) 𝜕2𝜋𝑏 𝜕𝑝2 = −2𝑏 (14) 𝜕𝜋𝑏 𝜕𝑄 = (𝑛𝐹 + 𝑆𝑏)(𝑎 − 𝑏𝑝 +𝑐𝜔) 𝑛𝑄2(1 − 𝜓) − [ 1 − 𝜓 2 + (𝑎 −𝑏𝑝 + 𝑐𝜔)𝜓 𝑧(1− 𝜓) ]ℎ𝑏 𝜕2𝜋𝑏 𝜕𝑄2 = − 2(𝑛𝐹+𝑆𝑏)(𝑎−𝑏𝑝+𝑐𝜔) 𝑛𝑄3(1−𝜓) (15) 𝜕2𝜋𝑏 𝜕𝑝𝜕𝑄 = − 𝑏 1 − 𝜓 { 𝑛𝐹 + 𝑆𝑏 𝑛𝑄2 + ℎ𝑏𝜓 𝑧 } 𝜕𝜋𝑣 𝜕𝜔 = − 𝑐[2𝐶𝑛𝑃𝑄−𝑛(𝑛−2)ℎ𝑣𝑄 2+2𝑃{𝑆𝑣−𝑛𝑄(𝑊−𝑝𝜔𝜆)(1−𝜓)}]+𝑛𝑝𝑃𝑄𝜆(𝑎−𝑏𝑝)(1−𝜓) 2𝑛𝑃𝑄(1−𝜓) 𝜕2𝜋𝑣 𝜕𝜔2 = −𝑐𝑝𝜆 (16) proposition 3. (i) for fixed lot size (𝑄) in each shipment, the buyer’s profit function 𝜋𝑏(𝑝,𝑄) is concave with respect to selling price (𝑝). (ii) for fixed selling price(𝑝) of a product, the buyer’s profit function 𝜋𝑏(𝑝,𝑄) is concave with respect to the lot size (𝑄). (iii) the vendor’s profit function 𝜋𝑣(𝑛,𝜔)is concave with respect to the warranty period (𝜔). proof: (i) this is obvious from the result (14), as 𝜕2𝜋𝑏 𝜕𝑝2 < 0. (ii) since 0 < 𝜓 < 1 and the demand function 𝐷(𝑝,𝜔) = (𝑎 − 𝑏𝑝 + 𝑐𝜔) > 0; we can conclude from the relation (15) that 𝜕2𝜋𝑏 𝜕𝑄2 < 0. hence the result. (iii) this is also obvious from the result (16), as 𝜕2𝜋𝑣 𝜕𝜔2 < 0. since, 𝜕2𝜋𝑣 𝜕𝜔2 = −𝑐𝑝𝜆 < 0, therefore, the existence of unique solution of the vendor's profit function is ensured. we now examine the existence of unique solution of the buyer's profit function 𝜋𝑏(𝑝,𝑄) in the following proposition: proposition 4.the profit function 𝜋𝑏(𝑝,𝑄)is jointly concave in 𝑝 and 𝑄 if the following conditions are satisfied: (i) 0 < min{± 𝐵2 2 ∓ 1 2 √ 4𝑧(𝑛𝐹+𝑆𝑏) ℎ𝑏𝑛𝜓 [1 + 2(𝑎+𝑐𝜔)𝑧(1−𝜓) 𝑏ℎ𝑏𝜓𝐵2 ]− 𝐵2 2} < 𝑄 < a two-echelon supply chain model with price and warranty dependent demand and ….. 59 𝐵2 2 + 1 2 √ 4𝑧(𝑛𝐹 +𝑆𝑏) ℎ𝑏𝑛𝜓 [1 + 2(𝑎 + 𝑐𝜔)𝑧(1 − 𝜓) 𝑏ℎ𝑏𝜓𝐵2 ]− 𝐵2 2 (ii) 0 < 𝑝 < 1 𝑏 [𝑎 + 𝑐𝜔 − 𝑏{(𝑛𝐹+𝑆𝑏)𝑧−𝑛𝑄 2ℎ𝑏𝜓} 2 4𝑛𝑄(𝑛𝐹+𝑆𝑏)𝑧 2(1−𝜓) ] where, 𝐵0 = 27𝑛(𝑎 + 𝑐𝜔) 2𝑧(1 −𝜓)2 +16𝑏2ℎ𝑏(𝑛𝐹 +𝑆𝑏)𝜓 𝐵1 = 16𝑏𝑛 3(𝑛𝐹 + 𝑆𝑏) 2𝑧3𝜓2ℎ𝑏 2 {𝐵0 − 8𝑏 2ℎ𝑏(𝑛𝐹 + 𝑆𝑏)𝜓 + (𝑎 + 𝑐𝜔)(1 − 𝜓)√𝐵0} 1 3 𝐵2 = √ 4𝑧(𝑛𝐹 + 𝑆𝑏) 3ℎ𝑏𝑛𝜓 + 4 ×2 1 3𝑏𝑧2(𝑛𝐹 + 𝑆𝑏) 2(1 +3ℎ𝑏 2 ) 3𝐵1 + 𝐵1 3× 2 1 3𝑏ℎ𝑏 2 𝑛2𝜓2 proof: the hessian matrix associated with 𝜋𝑏(𝑝,𝑄) is given by 𝐻2 = ( 𝜕2𝜋𝑏 𝜕𝑝2 𝜕2𝜋𝑏 𝜕𝑝𝜕𝑄 𝜕2𝜋𝑏 𝜕𝑄𝜕𝑝 𝜕2𝜋𝑏 𝜕𝑄2 ) = ( −2𝑏 − 𝑏 1 − 𝜓 { 𝑛𝐹 + 𝑆𝑏 𝑛𝑄2 + ℎ𝑏𝜓 𝑧 } − 𝑏 1 − 𝜓 { 𝑛𝐹 + 𝑆𝑏 𝑛𝑄2 + ℎ𝑏𝜓 𝑧 } − 2(𝑛𝐹 + 𝑆𝑏)(𝑎 − 𝑏𝑝 + 𝑐𝜔) 𝑛𝑄3(1− 𝜓) ) here, 𝜕2𝜋𝑏 𝜕𝑝2 = −2𝑏 < 0 and |𝐻2| = 𝑏[4𝑛𝑄(𝑛𝐹+𝑆𝑏)(𝑎−𝑏𝑝+𝑐𝜔)𝑧 2(1−𝜓)−𝑏{(𝑛𝐹+𝑆𝑏)𝑧−𝑛𝑄 2ℎ𝑏𝜓} 2] 𝑛2𝑄4𝑧2(1−𝜓)2 . for the existence of unique solution of 𝜋𝑏(𝑝,𝑄), |𝐻2| must be positive definite i.e., |𝐻2| > 0 i.e., 4𝑛𝑄(𝑛𝐹 + 𝑆𝑏)(𝑎 −𝑏𝑝 + 𝑐𝜔)𝑧 2(1 − 𝜓) − 𝑏{(𝑛𝐹 +𝑆𝑏)𝑧 −𝑛𝑄 2ℎ𝑏𝜓} 2 > 0 this will be true if 0 < 𝑝 < 1 𝑏 [𝑎 +𝑐𝜔 − 𝑏{(𝑛𝐹+𝑆𝑏)𝑧−𝑛𝑄 2ℎ𝑏𝜓} 2 4𝑛𝑄(𝑛𝐹+𝑆𝑏)𝑧 2(1−𝜓) ] provided that 𝑎 + 𝑐𝜔 − 𝑏{(𝑛𝐹 + 𝑆𝑏)𝑧 − 𝑛𝑄 2ℎ𝑏𝜓} 2 4𝑛𝑄(𝑛𝐹 + 𝑆𝑏)𝑧 2(1− 𝜓) > 0 i.e., 𝑏ℎ𝑏𝑄𝜓+2𝑧(𝑎+𝑐𝜔)(1−𝜓) 𝑏𝑧 − 𝑛𝐹+𝑆𝑏 2𝑛𝑄 − 𝑛𝑄3𝜓2ℎ𝑏 2 2𝑧2(𝑛𝐹+𝑆𝑏) < 0 i.e., 𝑏𝑛2ℎ𝑏 2 𝜓2𝑄4 − 2𝑏𝑛𝜓𝑧ℎ𝑏(𝑛𝐹 +𝑆𝑏)𝑄 2 − 4𝑛𝑄(𝑎 + 𝑐𝜔)(𝑛𝐹 + 𝑆𝑏)𝑧 2(1 −𝜓) + 𝑧2(𝑛𝐹 +𝑆𝑏) 2 < 0 (17) now, the equation 𝑛2ℎ𝑏 2 𝜓2𝑄4 − 2𝑏𝑛𝜓𝑧ℎ𝑏(𝑛𝐹 +𝑆𝑏)𝑄 2 − 4𝑛𝑄(𝑎 + 𝑐𝜔)(𝑛𝐹 + 𝑆𝑏)𝑧 2(1 −𝜓) + 𝑧2(𝑛𝐹 + 𝑆𝑏) 2 = 0 has four real roots, namely, 𝑄11 = − 𝐵2 2 − 1 2 √ 4𝑧(𝑛𝐹 + 𝑆𝑏) ℎ𝑏𝑛𝜓 [1 + 2(𝑎 + 𝑐𝜔)𝑧(1 − 𝜓) 𝑏ℎ𝑏𝜓𝐵2 ] − 𝐵2 2 𝑄21 = − 𝐵2 2 + 1 2 √ 4𝑧(𝑛𝐹 + 𝑆𝑏) ℎ𝑏𝑛𝜓 [1 + 2(𝑎 + 𝑐𝜔)𝑧(1 − 𝜓) 𝑏ℎ𝑏𝜓𝐵2 ] − 𝐵2 2 𝑄31 = 𝐵2 2 − 1 2 √ 4𝑧(𝑛𝐹 + 𝑆𝑏) ℎ𝑏𝑛𝜓 [1 + 2(𝑎 + 𝑐𝜔)𝑧(1 − 𝜓) 𝑏ℎ𝑏𝜓𝐵2 ] −𝐵2 2 samanta and giri/decis. mak. appl. manag. eng. 4 (2) (2021) 47-75 60 𝑄41 = 𝐵2 2 + 1 2 √ 4𝑧(𝑛𝐹 + 𝑆𝑏) ℎ𝑏𝑛𝜓 [1 + 2(𝑎 + 𝑐𝜔)𝑧(1 − 𝜓) 𝑏ℎ𝑏𝜓𝐵2 ] −𝐵2 2 it is clear that 𝑄 11 < 0 and 𝑄41 > 0. one of 𝑄21 and 𝑄31 must be positive while the other one is negative. the shipment size (𝑄) must be real and positive. remembering this, we can simplify (17) and say that min{𝑄21,𝑄31} < 𝑄 < 𝑄41. hence the proposition is proved. using the first order conditions for optimality of the profit function 𝜋𝑏(𝑝,𝑄), the equilibrium solution can be obtained. the first order conditions are 𝜕𝜋𝑏 𝜕𝑝 = 0 and 𝜕𝜋𝑏 𝜕𝑄 = 0. from these equations, we have 𝑝(𝑄) = 𝑎+𝑏𝑊+𝑐𝜔 2𝑏 + 𝑥+𝑊𝜓 2(1−𝜓) + 1 2𝑧(1−𝜓) [ℎ𝑏𝜓𝑄 + (𝑛𝐹+𝑆𝑏)𝑧 𝑛𝑄 ] (18) 𝑄(𝑝) = √ 2𝑧(𝑛𝐹+𝑆𝑏)(𝑎−𝑏𝑝+𝑐𝜔) 𝑛ℎ𝑏[𝑧(1−𝜓) 2+2𝜓(𝑎−𝑏𝑝+𝑐𝜔)] (19) substituting the value of 𝑝 from (18) into (19), we get an equation in 𝑄 only as follows: 𝑄 = √ 𝑏𝑧2(𝑛𝐹+𝑆𝑏) 2+𝑛𝑧𝑄(𝑛𝐹+𝑆𝑏)[𝑏(𝑥𝑧+𝑊𝜓+ℎ𝑏𝑄𝜓)−(𝑎−𝑏𝑊+𝑐𝜔)𝑧(1−𝜓)] 𝑛ℎ𝑏[𝑏𝑧𝜓(𝑛𝐹+𝑆𝑏)+𝑛𝑏𝑄𝜓(𝑥𝑧+𝑊𝜓+ℎ𝑏𝑄𝜓)−𝑛𝑄𝑧(1−𝜓){(𝑎−𝑏𝑊+𝑐𝜔)𝜓+𝑧 2(1−𝜓)2}] solving the above equation and remembering that q must be real and positive, we obtain: 𝑄𝑑1 = 𝐵5 4𝑏ℎ𝑏𝜓 2 + 𝐵11 2 + 1 2 √ 𝑛𝐵5 2(𝐵5+3𝑏ℎ𝑏𝜓 2𝐵11)−8𝑏ℎ𝑏𝜓 4𝐵3𝐵4 4𝑛𝑏3ℎ𝑏 3𝜓6𝐵11 − 𝐵11 2 (20) where, 𝐵3 = (𝑎 − 𝑏𝑊 + 𝑐𝜔)(1 − 𝜓) − 𝑏𝑥 𝐵4 = 𝑏(𝑛𝐹 +𝑆𝑏) 2𝑧2 𝐵5 = 𝑧(1− 𝜓) 3 + 𝜓𝐵3 𝐵6 = 𝑛𝐵3𝐵4 𝑏 𝐵7 = 𝑛 2𝑧ℎ𝑏𝐵5 𝐵8 = 3𝑛 2ℎ𝑏𝐵4 ( 𝑛𝑧𝐵3𝐵5 𝑏 − 4𝑏ℎ𝑏𝜓 2) 𝐵9 = 27𝑛 4ℎ𝑏 2 𝐵4 (𝑧 2𝐵5 2 − 𝜓2𝐵3 2𝐵4 𝑏 ) 𝐵10 = (𝐵9 +√𝐵9 2 − 4𝐵8 3) 1 3 𝐵11 = √ 𝐵5 2 4𝑏2ℎ𝑏 2 𝜓4 − 2 1 3𝐵8 3𝑏𝑛2𝜓2ℎ𝑏 2 𝐵10 − 𝐵10 3× 2 1 3𝑏𝑛2𝜓2ℎ𝑏 2 therefore, the optimum selling price is found by substituting 𝑄 = 𝑄𝑑1 of (20) in (18) and thus optimum selling price is 𝑝𝑑1 = 𝑎+𝑏𝑊+𝑐𝜔 2𝑏 + 𝑥+𝑊𝜓 2(1−𝜓) + 1 2𝑧(1−𝜓) [ℎ𝑏𝜓𝑄 𝑑1 + (𝑛𝐹+𝑆𝑏)𝑧 𝑛𝑄𝑑1 ] (21) a two-echelon supply chain model with price and warranty dependent demand and ….. 61 now, from equation 𝜕𝜋𝑣 𝜕𝜔 = 0, we have for model i, 𝜔 = 1 𝑝𝜆(1−𝜓) [ (𝑛−2)𝑄ℎ𝑣 2𝑃 − 𝑆𝑣 𝑛𝑄 + 𝑊(1 − 𝜓) − 𝐶]− 𝑎−𝑏𝑝 2𝑐 (22) the vendor's warranty period can be obtained by substituting the values of 𝑄𝑑1 and 𝑝𝑑1 from (20) and (21) into (22) as 𝜔𝑑1 = 1 𝑝𝑑1𝜆(1 −𝜓) [ (𝑛 − 2)𝑄𝑑1ℎ𝑣 2𝑃 − 𝑆𝑣 𝑛𝑄𝑑1 + 𝑊(1 −𝜓) − 𝐶]− 𝑎 − 𝑏𝑝𝑑1 2𝑐 4.2.2. model ii in this model, we assume that, at the beginning of the production period, the buyer offers a cost-sharing contract to encourage the vendor to actively carry out the quality production and promises to increase the length of warranty period. the buyer bears the warranty cost in proportion to 𝜃 at the beginning of the production period, and then according to 𝜃 selected, the vendor decides how much to increase the warranty period without affecting its own profit. we have, 𝜕𝜋𝑏 𝜕𝑝 = (1− 𝜔 2 )(𝑎 − 2𝑏𝑝 + 𝑐𝜔)+ 𝑏𝑊 + 𝑏(𝑛𝐹 + 𝑆𝑏 + 𝑥𝑛𝑄) 𝑛𝑄(1− 𝜓) + 𝑏ℎ𝑏𝑄𝜓 𝑧(1− 𝜓) 𝜕2𝜋𝑏 𝜕𝑝2 = −𝑏{2− 𝜆𝜔(1 −𝜃)} (23) 𝜕𝜋𝑏 𝜕𝑄 = (𝑛𝐹 + 𝑆𝑏)(𝑎 − 𝑏𝑝 +𝑐𝜔) 𝑛𝑄2(1 − 𝜓) − [ 1 − 𝜓 2 + (𝑎 −𝑏𝑝 + 𝑐𝜔)𝜓 𝑧(1− 𝜓) ]ℎ𝑏 𝜕2𝜋𝑏 𝜕𝑄2 = − 2(𝑛𝐹+𝑆𝑏)(𝑎−𝑏𝑝+𝑐𝜔) 𝑛𝑄3(1−𝜓) (24) 𝜕2𝜋𝑏 𝜕𝑝𝜕𝑄 = − 𝑏 1 − 𝜓 { 𝑛𝐹 + 𝑆𝑏 𝑛𝑄2 + ℎ𝑏𝜓 𝑧 } 𝜕𝜋𝑣 𝜕𝜔 = − 𝑐[2𝐶𝑛𝑃𝑄−𝑛(𝑛−2)ℎ𝑣𝑄 2+2𝑃{𝑆𝑣−𝑛𝑄(𝑊−𝑝𝜔𝜃𝜆)(1−𝜓)}]+𝑛𝑝𝑃𝑄𝜃𝜆(𝑎−𝑏𝑝)(1−𝜓) 2𝑛𝑃𝑄(1−𝜓) 𝜕2𝜋𝑣 𝜕𝜔2 = −𝑐𝑝𝜃𝜆 (25) from equations (23), (24) and (25), we can establish the following proposition for model ii: proposition 5. (i) the profit function 𝜋𝑏(𝑝,𝑄) is concave with respect to selling price 𝑝 if 𝜔 satisfies the relation 0 < 𝜔 < 2 𝜆(1−𝜃) . (ii) for known 𝑝, the profit function 𝜋𝑏(𝑝,𝑄) is concave with respect to 𝑄. (iii) the profit function 𝜋𝑣(𝑛,𝜔) is concave with respect to ω. since, 𝜕2𝜋𝑣 𝜕𝜔2 = −𝑐𝑝𝜃𝜆 < 0, therefore, the existence of unique solution of the vendor's profit function is ensured. we now examine the existence of unique solution of the buyer's profit function 𝜋𝑏(𝑝,𝑄) in the following proposition: proposition 6. the profit function 𝜋𝑏(𝑝,𝑄) is jointly concave in 𝑝 and 𝑄 if the following conditions are satisfied: (i) 0 < min{± 𝐵14 2 ∓ 1 2 √ 4𝑧(𝑛𝐹+𝑆𝑏) ℎ𝑏𝑛𝜓 [1 + {2−𝜆𝜔(1−𝜃)}(𝑎+𝑐𝜔)𝑧(1−𝜓) 𝑏ℎ𝑏𝜓𝐵14 ]− 𝐵14 2} < 𝑄 < samanta and giri/decis. mak. appl. manag. eng. 4 (2) (2021) 47-75 62 𝐵14 2 + 1 2 √ 4𝑧(𝑛𝐹+𝑆𝑏) ℎ𝑏𝑛𝜓 [1 + {2−𝜆𝜔(1−𝜃)}(𝑎+𝑐𝜔)𝑧(1−𝜓) 𝑏ℎ𝑏𝜓𝐵14 ]− 𝐵14 2 (ii) 0 < 𝑝 < 1 𝑏 [𝑎 + 𝑐𝜔 − 𝑏{(𝑛𝐹+𝑆𝑏)𝑧−𝑛𝑄 2ℎ𝑏𝜓} 2 2𝑛𝑄(𝑛𝐹+𝑆𝑏)𝑧 2(1−𝜓){2−𝜆𝜔(1−𝜃)} ] where, 𝐵12 = 4𝑏ℎ𝑏 2 𝑛2(𝑛𝐹 + 𝑆𝑏) 2𝑧3𝜓2[27𝑧(𝑎 + 𝑐𝜔)2{2 − 𝜆𝜔(1 −𝜃)}2(1 − 𝜓)2 +4𝑏(9 −𝑏)𝑛ℎ𝑏(𝑛𝐹 + 𝑆𝑏)𝜓] 𝐵13 = {𝐵12 + √𝐵12 2 − 4[4𝑏(𝑏 + 3)ℎ𝑏 2 𝑛2(𝑛𝐹 +𝑆𝑏) 2𝑧2𝜓2] 3 } 1 3 𝐵14 = √ 4𝑧(𝑛𝐹 + 𝑆𝑏) 3ℎ𝑏𝑛𝜓 + 4 ×2 1 3𝑏𝑧2(𝑛𝐹 + 𝑆𝑏) 2(𝑏 +3) 3𝐵13 + 𝐵13 3 ×2 1 3𝑏ℎ𝑏 2 𝑛2𝜓2 proof: the hessian matrix associate with 𝜋𝑏(𝑝,𝑄) is given by 𝐻3 = ( 𝜕2𝜋𝑏 𝜕𝑝2 𝜕2𝜋𝑏 𝜕𝑝𝜕𝑄 𝜕2𝜋𝑏 𝜕𝑄𝜕𝑝 𝜕2𝜋𝑏 𝜕𝑄2 ) = ( −2𝑏 + 𝑏𝜔𝜆(1− 𝜃) − 𝑏 1−𝜓 { 𝑛𝐹+𝑆𝑏 𝑛𝑄2 + ℎ𝑏𝜓 𝑧 } − 𝑏 1−𝜓 { 𝑛𝐹+𝑆𝑏 𝑛𝑄2 + ℎ𝑏𝜓 𝑧 } − 2(𝑛𝐹+𝑆𝑏)(𝑎−𝑏𝑝+𝑐𝜔) 𝑛𝑄3(1−𝜓) ) here, 𝜕2𝜋𝑏 𝜕𝑝2 = −𝑏[2 − 𝜆𝜔(1 − 𝜃)] < 0 and |𝐻3| = 𝑏[2𝑛𝑄(𝑛𝐹+𝑆𝑏)(𝑎−𝑏𝑝+𝑐𝜔)𝑧 2(1−𝜓){2−𝜆𝜔(1−𝜃)}−𝑏{(𝑛𝐹+𝑆𝑏)𝑧−𝑛𝑄 2ℎ𝑏𝜓} 2] 𝑛2𝑄4𝑧2(1−𝜓)2 . for the existence of unique solution of 𝜋𝑏(𝑝,𝑄), |𝐻3| must be positive definite i.e., |𝐻3| > 0 i.e., 2𝑛𝑄(𝑛𝐹 +𝑆𝑏)(𝑎 − 𝑏𝑝 +𝑐𝜔)𝑧 2(1 − 𝜓){2 −𝜆𝜔(1 − 𝜃)}− 𝑏{(𝑛𝐹 + 𝑆𝑏)𝑧 − 𝑛𝑄2ℎ𝑏𝜓} 2 > 0. this will be true if 0 < 𝑝 < 1 𝑏 [𝑎 + 𝑐𝜔 − 𝑏{(𝑛𝐹+𝑆𝑏)𝑧−𝑛𝑄 2ℎ𝑏𝜓} 2 2𝑛𝑄(𝑛𝐹+𝑆𝑏)𝑧 2(1−𝜓){2−𝜆𝜔(1−𝜃)} ] provided that 𝑎 + 𝑐𝜔 − 𝑏{(𝑛𝐹+𝑆𝑏)𝑧−𝑛𝑄 2ℎ𝑏𝜓} 2 2𝑛𝑄(𝑛𝐹+𝑆𝑏)𝑧 2(1−𝜓){2−𝜆𝜔(1−𝜃)} > 0 i.e., 𝑏𝑛2ℎ𝑏 2 𝜓2𝑄4 − 2𝑏𝑛𝜓𝑧ℎ𝑏(𝑛𝐹 + 𝑆𝑏)𝑄 2 −2𝑛𝑄(𝑎 +𝑐𝜔)(𝑛𝐹 +𝑆𝑏)𝑧 2(1 − 𝜓) {2 − 𝜆𝜔(1− 𝜃)} + 𝑧2(𝑛𝐹 + 𝑆𝑏) 2 < 0 (26) now, the corresponding equation of the above inequality has four real roots, namely, 𝑄12 = − 𝐵14 2 − 1 2 √ 4𝑧(𝑛𝐹 + 𝑆𝑏) ℎ𝑏𝑛𝜓 [1 + {2 − 𝜆𝜔(1− 𝜃)}(𝑎 + 𝑐𝜔)𝑧(1 −𝜓) 𝑏ℎ𝑏𝜓𝐵2 ]− 𝐵14 2 𝑄22 = − 𝐵14 2 + 1 2 √ 4𝑧(𝑛𝐹 + 𝑆𝑏) ℎ𝑏𝑛𝜓 [1 + {2 − 𝜆𝜔(1− 𝜃)}(𝑎 +𝑐𝜔)𝑧(1 −𝜓) 𝑏ℎ𝑏𝜓𝐵2 ]− 𝐵14 2 a two-echelon supply chain model with price and warranty dependent demand and ….. 63 𝑄32 = 𝐵14 2 − 1 2 √ 4𝑧(𝑛𝐹 +𝑆𝑏) ℎ𝑏𝑛𝜓 [1 + {2 −𝜆𝜔(1 − 𝜃)}(𝑎 +𝑐𝜔)𝑧(1 −𝜓) 𝑏ℎ𝑏𝜓𝐵2 ]− 𝐵14 2 𝑄42 = 𝐵14 2 + 1 2 √ 4𝑧(𝑛𝐹 +𝑆𝑏) ℎ𝑏𝑛𝜓 [1 + {2 −𝜆𝜔(1 − 𝜃)}(𝑎 +𝑐𝜔)𝑧(1 −𝜓) 𝑏ℎ𝑏𝜓𝐵2 ]− 𝐵14 2 it is clear that 𝑄 12 < 0 and 𝑄42 > 0. one of 𝑄22 and 𝑄32 must be positive while the other one is negative. the shipment size (𝑄) must be real and positive. simplifying (26), we can say that min{𝑄22,𝑄32} < 𝑄 < 𝑄42. hence the proposition is proved. we now consider the first order conditions 𝜕𝜋𝑏 𝜕𝑝 = 0 and 𝜕𝜋𝑏 𝜕𝑄 = 0, which give 𝑝(𝑄) = 𝑎+𝑐𝜔 2𝑏 + 𝑥+𝑊(1−𝜓) (1−𝜓){2−𝜆𝜔(1−𝜃)} + [ℎ𝑏𝜓𝑄+ (𝑛𝐹+𝑆𝑏)𝑧 𝑛𝑄 ] 𝑧(1−𝜓){2−𝜆𝜔(1−𝜃)} (27) 𝑄(𝑝) = √ 2𝑧(𝑛𝐹+𝑆𝑏)(𝑎−𝑏𝑝+𝑐𝜔) 𝑛ℎ𝑏[𝑧(1−𝜓) 2+2𝜓(𝑎−𝑏𝑝+𝑐𝜔)] (28) substituting the value of 𝑝 from (27) into (28), we get an equation in 𝑄 only as follows: 𝑄 = √ 𝑧(𝑛𝐹 + 𝑆𝑏)[2𝑏𝑧{𝑛𝐹 +𝑆𝑏 − 𝑛𝑄(𝑊(1 − 𝜓) + 𝑥)} + 2𝑏𝑛ℎ𝑏𝜓𝑄 2 − 𝑛𝑄𝑧(𝑎 + 𝑐𝜔)(1 −𝜓){2 −𝜆𝜔(1 − 𝜃)}] 𝑛ℎ𝑏[2𝑏𝑧𝜓{(𝑛𝐹 + 𝑆𝑏)+ 𝑛𝑄(𝑥 + 𝑊(1 − 𝜓))} + 2𝑏𝑛ℎ𝑏𝑄 2𝜓2 −𝑛𝑄𝑧(1 − 𝜓){2 − 𝜆𝜔(1− 𝜃)}{(𝑎 + 𝑐𝜔)𝜓 +𝑧(1− 𝜓)2}] solving the above equation and remembering that q must be real and positive, we obtain: 𝑄𝑑2 = 𝐶1 8𝑏ℎ𝑏𝜓 2 + 𝐶5 2 + 1 2 √ 2𝑏ℎ𝑏𝜓 2(3𝑛𝐶1 2𝐶5−16𝑏𝜓 2𝐶2)+𝑛𝐶1 3 32𝑛𝑏3ℎ𝑏 3𝜓6𝐶5 − 𝐶5 2 (29) where, 𝐶0 = (𝑎 +𝑐𝜔)(1 − 𝜓){2 − 𝜆𝜔(1 −𝜃)} − 2𝑏{𝑥 + 𝑊(1 − 𝜓)} 𝐶1 = 𝑧[𝑧{2 − 𝜆𝜔(1 − 𝜃)}(1 − 𝜓) 3 + 𝜓𝐶0] 𝐶2 = 3𝑛 2ℎ𝑏(𝑛𝐹 + 𝑆𝑏)𝑧 2[16𝑏2ℎ𝑏(𝑛𝐹 +𝑆𝑏)𝜓 2 + 𝑛𝐶0𝐶1] 𝐶3 = 27𝑏ℎ𝑏 2 𝑛4(𝑛𝐹 + 𝑆𝑏)(𝐶1 2 − 𝐶0 2𝜓2𝑧2) 𝐶4 = [2(𝑛𝐹 + 𝑆𝑏)𝑧 2𝐶3 + 2𝑧 3√𝑛3(𝑛𝐹 +𝑆𝑏) 3{27𝑛𝑏ℎ𝑏 2 𝐶3 − 𝐶0 3}] 1 3 𝐶5 = √ 𝐶1 2 16𝑏2ℎ𝑏 2 𝜓4 − 𝐶2 3×2 2 3𝑏𝑛2𝜓2ℎ𝑏 2 𝐶4 − 𝐶4 6×2 1 3𝑏𝑛2𝜓2ℎ𝑏 2 therefore, the optimum selling price is found by substituting 𝑄 = 𝑄𝑑2 from (29) in (27). thus the optimum selling price is 𝑝𝑑2 = 𝑎+𝑐𝜔 2𝑏 + 𝑥+𝑊(1−𝜓) (1−𝜓){2−𝜆𝜔(1−𝜃)} + ℎ𝑏𝜓𝑄 𝑑2+ (𝑛𝐹+𝑆𝑏)𝑧 𝑛𝑄𝑑2 𝑧(1−𝜓){2−𝜆𝜔(1−𝜃)} (30) now, from the equation 𝜕𝜋𝑣 𝜕𝜔 = 0, we have for model ii, samanta and giri/decis. mak. appl. manag. eng. 4 (2) (2021) 47-75 64 𝜔 = 1 𝑝𝜆𝜃(1−𝜓) [ (𝑛−2)𝑄ℎ𝑣 2𝑃 − 𝑆𝑣 𝑛𝑄 + 𝑊(1 − 𝜓) − 𝐶]− 𝑎−𝑏𝑝 2𝑐 (31) thus the vendor sets warranty period by substituting 𝑄𝑑2 and 𝑝𝑑2 from (29) and (30) into (31). then the vendor’s warranty period is obtained as 𝜔𝑑2 = 1 𝑝𝑑2𝜆𝜃(1− 𝜓) [ (𝑛 − 2)𝑄𝑑2ℎ𝑣 2𝑃 − 𝑆𝑣 𝑛𝑄𝑑2 + 𝑊(1 −𝜓) − 𝐶]− 𝑎 − 𝑏𝑝𝑑2 2𝑐 5. numerical example in this section, we illustrate the developed models through a numerical example. we assume the parameter-values for both the models as: 𝑃 = 160000; 𝑧 = 175200; 𝑆𝑣 = $ 300; 𝑆𝑏 = $ 100; ℎ𝑣 = $ 2; ℎ𝑏 = $ 5; 𝐹 = $25; 𝑥 = $ 0.5; 𝑎 = 3000;𝑏 = 40;𝑐 = 220; 𝜓 = 0.02;𝐶 = $ 5;𝜆 = 0.4;𝑊 = 40; 𝜃 = 0.25. table 3. optimal results for the centralized and decentralized models. optimal decisions centralized model decentralized model model i model ii 𝑛 𝑸 𝑷 𝝎 𝝅𝒃 𝝅𝒗 𝝅 5 148.011 32.0960 0.60484 39506.3 4 131.662 62.8366 0.25962 11604.5 16488.5 28093.0 4 131.828 62.2775 0.68030 12362.2 18020.7 30382.9 table 3 shows that the market demand, the total number of shipments per lot, and the shipment size are higher in the centralized model than those of the decentralized models. further, the time interval between successive deliveries and the buyer's selling price are lower in centralized model than those of the decentralized models. although the total number of shipments per lot remains the same, the warranty period and the market demand are higher in the decentralized model ii than those in decentralized model i. again, model i offers higher selling price than model ii. from the optimal results, it can be seen that if the vendor offers higher warranty period, then the market demand becomes higher and the buyer's selling price reduces. 6. sensitivity analysis in this section, we now discuss the sensibility of several leading parameters of the proposed models. we vary the value of one parameter at once and hold the other parameter-values unchanged to analyze its effect on the optimum solutions. the sensibility of the parameters 𝑎,𝑏,𝑐,𝜆,𝜓,𝑥 and 𝐶𝑝 are shown in table 4 and table 5. we also examine the remaining parameters but the models are insensitive with respect to these parameters. a two-echelon supply chain model with price and warranty dependent demand and ….. 65 6.1. sensitivity with respect to a as 𝑎 increases, the market demand increases. therefore, the buyer wants to receive bigger shipment size (𝑄∗). in this situation, the buyer’s selling price (𝑝∗) increases but the warranty period (𝜔∗) decreases for all the centralized and decentralized models (see table 4). as 𝑎 increases, the expected total profits of the buyer and the vendor and the whole system increase for the centralized and two decentralized models (see fig. 1, table 5). also, the warranty cost of the centralized model decreases. moreover, the value of 𝜔 decreases but the buyer’s selling price increases for both the decentralized models (see table 4). (a) 𝑎 vs total profit of the buyer (b) 𝑎 vs total profit of the vendor figure 1. change (%) in optimal results w.r.t. 𝑎. 6.2. sensitivity with respect to b when 𝑏 increases, the market demand decreases for all the models. as a result, the shipment size (𝑄∗) decreases. the selling price also decreases for all the models but the warranty period increases for both the decentralized models whereas it decreases for the centralized model. the selling price is highly sensitive for the centralized model whereas it is moderately sensitive for both the decentralized models (see table 4). as 𝑏 increases, the buyer's expected total profit slowly decreases for the centralized model, moderately decreases for decentralized model i and rapidly decreases for the decentralized model ii. the changes in selling price and warranty period together are responsible for this behavior of profits for all the models. the profit of the vendor rapidly decreases for both the decentralized models and moderately decreases for the centralized model when 𝑏 increases. total profits of the decentralized models i and ii rapidly decrease whereas the total profit of the centralized model decreases slowly (see fig. 2, table 5). the warranty cost of the centralized model increases and the warranty period decreases as the value of 𝑏 increases. when 𝑏 increases, the value of 𝜔 increases but the buyer's selling price decreases for both the decentralized models (see table 4). samanta and giri/decis. mak. appl. manag. eng. 4 (2) (2021) 47-75 66 (a) 𝑏 vs profit of the buyer (b) 𝑏 vs profit of the vendor figure 2. change (%) in optimal results w.r.t. b. 6.3. sensitivity with respect to c when 𝑐 increases, the market demand increases for all the centralized and decentralized models. then the buyer wants to receive a higher shipment size (𝑄∗). in this situation, the buyer's selling price (𝑝∗) and the warranty period (𝜔∗) also increase for all the centralized and decentralized models (see table 4). (a) 𝑐 vs profit of the vendor (b) 𝑐 vs total profit of the vendor and buyer figure 3.change (%) in optimal results w.r.t. 𝑐. as 𝑐 increases, total profits of the centralized and two decentralized models increase. the vendor's profit is insensitive in each of the decentralized models when the value of 𝑐 increases. the total profit of the centralized model decreases but it increases very slowly in each of the decentralized models (see fig. 3, table 5). 6.4. sensitivity with respect to 𝝍 the market demand in all the models decreases rapidly if the value of 𝜓 exceeds 0.6. so, we can say that, the vendor should not produce items more than 60% defective. the decentralized model ii has no impact for the changes in the value of 𝜓. the values of the optimum decisions are insensitive for the changes of the value of 𝜓 for all the centralized and decentralized models (see table 4). as 𝜓 increases, the expected total profits of the buyer, the vendor and the whole system decrease for all the models. the production of more defective items means less demand and more warranty cost. to meet up the market demand, the vendor has to produce more items since the defective items are rejected by the buyer after the completion of screening. here we observe a two-echelon supply chain model with price and warranty dependent demand and ….. 67 that, if the vendor produces more than 75% defective items, the profit of the vendor becomes negative (see figure 4, table 5). the production of more defective items of the vendor implies less inventory of the buyer since the buyer rejects these defective items after the completion of screening. therefore, the holding cost of the buyer decreases for all the centralized and decentralized models as the value of 𝜓 increases (see fig. 4, table 5). (a) 𝜓 vs. profit of the buyer (b) 𝜓 vs. profit of the vendor (c) 𝜓 vs. holding cost of the buyer figure 4. change (%) in optimal results w.r.t. 𝜓. 6.5. sensitivity with respect to 𝝀 figure 5. change (%) in optimal results w.r.t. 𝜆. samanta and giri/decis. mak. appl. manag. eng. 4 (2) (2021) 47-75 68 when the measure of 𝜆 increases, the market demand decreases for all the models. as a result, the shipment size (𝑄∗) decreases. the selling price and the warranty period also decrease for all the models (see table 4). as 𝜆 increases, the buyer’s expected total profit decreases for all the models. the profit of the vendor remains unchanged for both the decentralized models whereas it increases for the centralized model when 𝜆 increases (see fig. 5, table 5). 6.6. sensitivity with respect to x if the unit screening cost (𝑥) of the buyer increases, then it is obvious that the total screening cost of the buyer also increases and hence profit of the buyer decreases for all the centralized and decentralized models (see fig. 6, table 5). figure 6. change (%) in optimal results w.r.t. 𝑥. the decision variables i.e., lot size in each replenishment (𝑄∗), the buyer's selling price (𝑝∗) and warranty period (𝜔∗) have no change for the changes in 𝑥 (see table 4). 6.7. sensitivity with respect to 𝜽 figure 7. change (%) in optimal results w.r.t. 𝜃. the buyer will agree to bear a fraction of the total warranty cost (𝜃) to get more profit than ever before. on the other hand, the vendor will take off this proposal as the warranty cost gets reduced for him/her. in this contract, both the buyer and the a two-echelon supply chain model with price and warranty dependent demand and ….. 69 vendor are in a win-win situation. in our proposed model, the buyer will pay maximum 40% of the total warranty cost (see fig. 7, table 5). 6.8. sensitivity with respect to 𝑪𝒑 when the buyer agrees to share a portion of warranty cost, the vendor tries to produce more perfect items and wants to sell the products with more warranty. then naturally the production cost (𝐶𝑝) increases. then the vendor’s wholesale price obviously increases. as a result, the profit of the buyer in both the models decreases. but, the buyer’s profit in model-ii is always greater than that of model-i. we also observe that the buyer’s profit decreases at a higher rate in model-i than model-ii as 𝐶𝑝 increases (see fig. 8(a)) on the other hand, the warranty cost of the vendor also decreases as 𝐶𝑝 increases in both the models. the decreasing rate of warranty cost of the vendor is more beneficial for model-ii than model i (see fig.8 (b)). (a) 𝐶𝑝vs profit of the buyer (b) 𝐶𝑝 vs warranty cost of the vendor figure 8. change (%) in optimal results w.r.t. 𝐶𝑝. table 4. behaviour of optimal decisions with respect to change in some key parameters: decentralized model. para meter %change in parameter model i model ii 𝑄∗ 𝑝∗ 𝜔∗ 𝑄∗ 𝑝∗ 𝜔∗ 𝑎 +10 139.120 66.1064 ---141.551 64.2253 0.1219 +5 135.465 64.4635 0.0328 136.796 63.2416 0.4002 -5 127.693 61.2268 0.4893 126.621 61.3340 0.9624 -10 123.540 59.6355 0.7221 121.142 60.4126 1.2465 𝑏 +50 126.322 59.5018 0.5729 124.451 59.6765 1.0645 +25 129.007 61.0911 0.4171 128.183 60.9021 0.8727 -25 134.288 64.7629 0.1002 135.401 63.8244 0.4871 -50 136.889 66.9006 -----138.912 65.5702 0.2928 𝑐 +10 135.769 63.2506 0.3861 135.566 63.2269 0.8372 +5 133.726 63.0447 0.3261 133.686 62.7645 0.7637 -5 129.574 62.6260 0.1855 129.999 61.7622 0.5850 -10 127.463 62.4130 0.1025 128.207 61.2145 0.4754 𝜆 +10 127.848 62.4519 0.1075 128.530 61.3167 0.4514 +5 129.674 62.6361 0.1802 130.085 61.7875 0.5618 -5 133.834 63.0556 0.3468 133.785 62.7894 0.8082 samanta and giri/decis. mak. appl. manag. eng. 4 (2) (2021) 47-75 70 para meter %change in parameter model i model ii 𝑄∗ 𝑝∗ 𝜔∗ 𝑄∗ 𝑝∗ 𝜔∗ -10 136.220 63.2960 0.4429 135.987 63.3266 0.9470 𝜓 +50 132.978 62.8330 0.2573 133.158 62.2687 0.6769 +25 132.317 62.8348 0.2585 132.490 62.2731 0.6786 -25 131.013 62.8384 0.2608 131.173 62.2818 0.6820 -50 130.371 62.8401 0.2619 130.524 62.2861 0.6836 𝑥 +50 131.217 62.9738 0.2690 131.323 62.4350 0.6899 +25 131.439 62.9052 0.2643 131.576 62.3562 0.6851 -25 131.884 62.7680 0.2549 132.080 62.1987 0.6755 -50 132.106 62.6993 0.2549 132.332 62.1201 0.6708 table 5. behaviour of profits of the buyer’s and the vendor’s with respect to change in some key parameters: decentralized model. para mete r %change in parameter model i model ii 𝜋𝑏 ∗ 𝜋𝑣 ∗ 𝜋𝑏 ∗ 𝜋𝑣 ∗ 𝑎 +10 15141.4 22213.6 17054.9 24722.9 +5 13311.2 19217.5 14606.5 21226.2 -5 10020.0 14013.9 10317.6 15097.6 -10 8556.5 11781.5 8468.3 12448.8 𝑏 +50 9132.1 13577.7 9174.3 14688.6 +25 10300.3 14975.6 10679.0 16311.5 -25 13064.8 18129.8 14247.8 19824.6 -50 14705.6 19915.6 16364.9 21733.9 𝑐 +10 12267.9 16186.2 12931.9 17371.2 +5 11935.1 16324.3 12640.3 17667.9 -5 11276.1 16682.9 12099.8 18440.6 -10 10950.0 16912.5 11855.3 18941.5 𝜆 +10 11009.1 16867.9 11898.3 18843.7 +5 11291.7 16672.9 12111.9 18418.9 -5 11952.6 16316.5 12655.3 17650.9 -10 12342.2 16158.7 12998.5 17311.9 𝜓 +50 11591.5 16460.5 12352.6 17992.5 +25 11598.1 16474.5 12357.4 18006.6 -25 11610.9 16502.3 12367.0 18034.6 -50 11617.2 16516.0 12371.7 18048.4 𝑥 +50 11466.3 16314.8 12194.7 17834.2 +25 11535.3 16401.6 12278.3 17927.4 -25 11674.0 16575.6 12446.3 18114.1 -50 11743.6 16662.9 12530.7 18207.6 7. conclusions in this paper, we consider a supply chain model with a single vendor and a single buyer where the vendor delivers the buyer’s order in a number of shipments. the market demand depends on the selling price and the warranty period of the product. the buyer screens all the products after collecting from the vendor. the buyer deals a two-echelon supply chain model with price and warranty dependent demand and ….. 71 each item under pro rata warranty (prw) policy in which the vendor assents to pay back a portion of customer's purchase money, if a product goes wrong during warranty period interval provided by the buyer. we consider two decentralized models depending on warranty cost. in the first model, the warranty cost is completely borne by the vendor whereas in the second model, the buyer agrees to share a portion of warranty cost with the vendor. we optimize the profit of the supply chain with respect to the number of shipments from the vendor to the buyer, shipment size, buyer's selling price and the warranty period of a product. from the numerical study, we observe that profits of the vendor, the buyer and the whole supply chain increase if the vendor produces the items with more reliability. also, it is necessary for the vendor to produce items not more than 60% defective. the scaling constants 𝑎 and 𝑐 play an important role to increase the profit of the buyer, the vendor and the whole supply chain. since the market demand is higher in the centralized model than both the decentralized models, the sales revenue, expenditure and profits of both the vendor and the buyer as well as of the whole system for the centralized model are also higher than those of the decentralized models. we notice that this cost share not only increases the cost of the buyer but also increases his/her profit in model ii. again, the profits of the vendor and the whole system also increase in model ii than those of model i. thus we can conclude that the model ii provides the better result than model i. we have set up our model same as any other model, depending upon a set of assumptions. we have studied the market demand as deterministic, which has little uses in the global world. so, one can consider stochastic demand as an alternative of deterministic demand to extend the proposed model for future research. we have assumed a two-layer supply chain model with a single-buyer and a single-vendor. further research can develop the model by considering multi-layer supply chain model with multi-buyer and/or multi-vendor. we have considered prw policy when the buyer sells a product with warranty. one can improve this model by considering the items sold with frw policy or mixture of prw and frw policies. acknowledgments: the authors would like to thank the editor and the reviewers for their comments which led to considerable improvement in this article. author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research received no external funding. conflicts of interest: the authors declare no conflicts of interest. references blischke, w.r., & murthy, d.n.p. 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(2014). differential game analyses of logistics service supply chain coordination by cost sharing contract. journal of applied mathematics, https://doi.org/10.1155/2014/842409. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1155/2014/842409 plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 2, issue 1, 2019, pp. 35-48. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame1901035s * corresponding author. e-mail addresses: mirko.stojcic@sf.ues.rs.ba (m. stojčić), aleksandar.stjepanovic@sf.ues.rs.ba (a. stjepanović), djoles2014@gmail.com (dj. stjepanović) anfis model for the prediction of generated electricity of photovoltaic modules mirko stojčić1*, aleksandar stjepanović1 and đorđe stjepanović2 1 university of east sarajevo, faculty of transport and traffic engineering, doboj, bosnia and herzegovina 2 university of novi sad, faculty of technical sciences, novi sad, serbia received: 28 october 2018; accepted: 12 february 2019; available online: 19 february 2019. original scientific paper abstract: the fact that conventional energy sources are exhaustive and limited are increasingly encouraging research in the field of alternative and renewable energy sources. the electricity generated by solar photovoltaic modules and panels occupies an ever greater percentage in total electricity production, so it is clear that photovoltaic systems are increasingly integrating with the existing electricity network into one system or functioning as autonomous systems. the aim of the research is to create a model based on the principles of the fuzzy logic and artificial neural networks that will perform the task of predicting the maximum energy of photovoltaic modules as accurately as possible. the prediction should facilitate work in planning production and consumption, system management, economic analysis. the most important methods used in the research are modeling and simulation. input and output variables are selected and in the anfis (adaptive neuro fuzzy inference system) model a set of their values is presented. based on them it comes to the function of dependency. the prediction rating of the created model was performed on a separate data set for testing and a model with the lowest average test error value was selected. the performance of the model was compared with the mathematical model through sensitivity analysis, which led to the conclusion that the anfis model gives more accurate results. key words (bold): prediction, anfis (adaptive neuro fuzzy inference system), photovoltaic modules, artificial neural networks, fuzzy logic, rmse (root mean square error). 1. introduction today, huge attention is paid to the fact that conventional energy sources are exhaustive and limited, and their use is a major source of pollution. renewable mailto:mirko.stojcic@sf.ues.rs.ba mailto:aleksandar.stjepanovic@sf.ues.rs.ba mailto:djoles2014@gmail.com stojčić et al./decis. mak. appl. manag. eng. 2 (1) (2018) 35-48 36 energy sources, such as sun, wind, waves, etc., are increasingly being promoted. the electricity generated by solar photovoltaic modules and panels occupies an ever greater percentage in total electricity production. therefore, it is clear that solar systems increasingly integrate with the existing electricity network into one system or function as autonomous systems (ding et al., 2011). in order to supervise these systems, planning of production and consumption, management, economic analysis, it is necessary to make an accurate prediction of electricity generation. therefore, the aim of the research is to create a model based on the fuzzy logic and artificial neural networks that will perform the specified task as accurately as possible. generated power and energy of photovoltaic systems depend on several factors, and the intensity of the solar radiation and the temperature of the module (mahmodian et al., 2012), or ambient temperature (zhu et al., 2017), are indicated as the main sources. due to the dynamic nature of the energy generated over time and the non-linear dependence of the input and output variables, the prediction is a complex task. today there are two current prediction methods, physical and statistical. the physical method first predicts meteorological parameters that affect the generated power or energy, and then these same parameters are used in equations. the statistical method is based on a set of data from the past (zhu et al., 2017; zhu et al., 2015; wang et al., 2017). common statistical methods include svm (support vector machine) (zhu et al., 2017), markov chains and artificial neural networks (zhu et al., 2017). because of the properties that allow them to have a good non-linear approximation and generalization, artificial neural networks are most often used in the prediction of the performance of photovoltaic modules (antonanzas et al., 2016). in addition, using artificial neural networks avoids complicated mathematical principles, since the network learns from training data. various types of artificial neural networks are used, some of which are grnn, ffbr (feedforward back propagation) (saberian et al., 2014), k-nn (k-nearest neighbors), svm (wolff et al., 2016), rbfnn (radial basis function neural network), bpnn (back propagation neural network) (mandal et al., 2012), recurrent neural networks, etc. (mellit et al., 2009). hybrid approaches that involve a combination of physical and statistical methods are often used (wang et al., 2017). the paper presents the materials and methods used, as well as the individual steps during the research process. theoretical review of the neuro-fuzzy systems is particularly emphasized. the results of the research and discussion also take a special part in the work, where tables and graphs are given. the sensitivity analysis was performed in order to compare the values obtained with the anfis model with the mathematical model. based on the research presented by the work, appropriate conclusions were made. 2. materials and methods the most important methods used in the research are modeling and simulation. the task of the model, represented by this paper, is the prediction of the generated electricity of the photovoltaic modules at the daily level. it is necessary to first select the input and output variables and present the set of their values (obtained by simulation) on the basis of which the model itself will come to the function of dependance. the model represents the integration of the principle of fuzzy logic and artificial neural networks into an anfis (adaptive neuro fuzzy inference system). assessment of the ability of the prediction of the created model is done on a separate data set for testing and a model with the lowest average test error is chosen. anfis model for the prediction of generated electricity of photovoltaic modules 37 2.1. selection of variables and data collection the solar cell represents a pn coupling of a semiconductor which, on the basis of the photovoltaic effect, under the influence of the sunlight, releases the charge carriers, resulting in current in a closed circuit. more interconnected cells comprise a module, and more modules, a photovoltaic panel. according to its structure, the solar cell is most often constructed from semiconductor materials (si, ge, gaas), and represents a pn compound that absorbs photons from solar radiation and uses their energy to create electron-cavity pairs. the internal electric field that exists on an impoverished area of the pn junction separating couples holders that are created within or near the pn junction. from the front and back of the solar cell, contacts are collected that collect separate energies, and an electromotive force appears at the ends of the cell. the electrons and cavities in the semiconductor tend to move from a higher density region to a less-density region. when multiple solar cells transmit to a serial or parallel connection, a photovoltaic panel with the desired output voltage or current is obtained. due to the effect of solar radiation on the surface of the solar cell or panel, there are changes in several important parameters of the solar cell (photovoltaic panels), the concentration of free carriers and the width of the energy gap between the semiconductor. the change in these parameters is conditioned by the intensity of the radiation and the influence of the outside temperature. the increase in temperature on the panel affects the output parameters of the photovoltaic panel, the output current and voltage, and thus the power. a solar cell model based on two semiconductor diodes was selected to create a photovoltaic panel model. figure 1 shows the solar cell model (castaner et al., 2002): figure 1. solar cell model with two diodes (castaner et al., 2002) the dependence of the short-circuit current on the intensity of the incident solar radiation and the temperature of the environment and the module can be represented by the following pattern (castaner et al., 2002; chandani et al., 2014; guifang, 2014):  scmr scmscm c r i di i e t t 1000 dt           (1) and the open-circuit voltage is given by the approximate relation (castaner et al., 2002; chandani et al., 2014; guifang, 2014): stojčić et al./decis. mak. appl. manag. eng. 2 (1) (2018) 35-48 38  ocm scmocm ocmr c r t scmre v i v v t t v ln t i            (2) where: iscmr short-circuit current of the photovoltaic module at the reference temperature, e intensity of the solar radiation vocm the open-circuit voltage of the photovoltaic module, the vocmr the open-circuit voltage of the photovoltaic module at the reference temperature, t ambient temperature, tc temperature of the module, tr reference temperature. the output performance of photovoltaic modules depends to a large extent on the intensity of radiation and temperature. therefore, for the input variables of the model, the daily amount of solar radiation (measured in kwh/m2/day) and the average temperature of the module (in °c), are chosen. as the output variable is observed the virtual maximum energy that the photovoltaic modules can supply at the maximum power point for one day, measured in kwh/day. data collection for model training is carried out by the simulation method in pvsyst software, which is intended for engineers, architects and researchers involved in the analysis and construction of photovoltaic systems, i.e. systems that convert solar radiation into electricity. pvsyst is an industrial standard, but it is also very useful as an educational software. a simulation model of an autonomous photovoltaic system has been created which supplies consumers with electricity exclusively generated by solar modules, independent of the public electricity network. the selected location is in banja luka (rs, bih), and the model envisages a daily electricity consumption of 915 wh/day. the real components are available, so that two photovoltaic modules are specified (two modules are the panel) of lg electronics with a maximum power of 300 wp and a total area of 3 m2, batteries have a total capacity of 400 ah. once the system is defined, its block diagram can be seen in figure 2. it is important to point out that the regulator, belonging to this system, contains a maximum power point tracker (mppt maximum power point tracker) because of the low degree of conversion of solar radiation into electricity. the mppt is implemented as a microcontroller which, together with the dc-dc converter, transmits the maximum power from the module to the system. figure 2. block diagram of the photovoltaic autonomous system in the pvsyst program anfis model for the prediction of generated electricity of photovoltaic modules 39 figure 3 shows the general block diagram of the photovoltaic system. the a/d converter provides an input signal for the mppt that is connected to a pulse width modulator (pulse width modulator pwm). in pw modulation, the mean value of the signal changes depending on the length of the period and the duration of the rectangular pulse. a modulated signal represents an input signal for a dc-dc converter that transforms the dc voltage of one value to the dc voltage of the second value. figure 3. block diagram of photovoltaic system (gules roger et al., 2008) training data for the anfis model is obtained by performing simulations on the model of the photovoltaic system for a period of one year. the values of the input and output variables are selected for five days each month, which makes a total of 60 training vectors. in addition, a special set of data for testing and testing of the anfis model has been created, which consists of realized values of the variables for one day during each month 12 vectors. 2.2. artificial neural networks and fuzzy logic artificial neural networks represent an attempt to model the human brain. similarity with the work of the human brain is reflected through the structure, function and method of processing data and information. no matter what network it is, their common feature is the ability to learn. therefore, their main application is to look for dependencies between data that are not in a strict linear relationship. the training process is based on adjusting the weight of network connections. however, the network can be trained structurally, i.e. the correct choice of the number of neurons, layers, etc. (bašić et al., 2008). fuzzy logic is an extension of classical logic, allowing the work with uncertainties to make the computer adapt to the human way of thinking. the word fuzzy implies something unclear, indeterminate, but that does not mean that there is something unclear with the fuzzy logic itself, but that it has enabled the presentation of uncertainty. by applying the fuzzy logic in various fields, it was difficult to create a fuzzyinference system with good performance. tasks such as finding adequate membership functions and the fuzzy inference rules pose problems to experts in a stojčić et al./decis. mak. appl. manag. eng. 2 (1) (2018) 35-48 40 particular field. hence the idea that the principles of the fuzzy logic and neural networks are combined in a unique system called the neuro-fuzzy system, combining both the ability to learn and logical conclusion. anfis is one of the most commonly used architectures of the neuro-fuzzy system (figure 4). figure 4. architecture of the anfis model nodes of the first hidden layer define the fuzzy sets, i.e. fuzzy membership functions corresponding to the input variables. the nodes of this and the fourth layer are adaptive, which means that their parameters change during the training process. therefore, in figure 4 they have a rectangular shape in contrast to the fixed, circular nods. the nodes of the second hidden layer are fixed and perform an operation of multiplying the input signals (operation and), which determines the degree of consistency of the premise (if part) of each rule wi, which has a general shape: if x is a and y is b, then z=f(x,y), where a and b are fuzzy sets corresponding to the input variable, and z = f(x, y) function that is a consequence of the rule (salleh et al., 2016; rasit, 2009). the function z = f(x, y), can be a polynomial of zero or first order, i.e. constant or linear function. the third hidden layer normalizes the values obtained at the output of the nodes of the second hidden layer. in the case shown in figure 4, with two nodes in the second layer, the normalized value at the output of the node of the third hidden layer has the following mathematical form: 1 2 i i w w w w   (3) each node of the fourth hidden layer is an adaptive node with a function it realizes, which can be written as follows:  i i i i i iw f w p x q y r   (4) anfis model for the prediction of generated electricity of photovoltaic modules 41 where pi , qi and ri conclusion parameters. the fifth layer calculates the output as the sum of all input signals: i ii i ii ii w f f w f w      (5) the anfis model for the prediction of generated electricity of photovoltaic modules was created in the matlab software package. the anfis editor, thanks to the graphical user interface, allows easy definition and work with the model. the algorithm describing the process from creation to model evaluation can be written in the following steps: 1. loading data for training, checking and testing (60 training vectors, 12 for checking and testing), 2. defining the number and shape of the fuzzy membership functions of the input variables and the form of the membeship function of the output variable, 3. model training (hybrid training algorithm, tolerance error=0, epoch number=60), 4. testing the anfis model, rmse (root mean square error). 5. testing data allow you to evaluate the ability of the anfis model to execute a prediction of the value of the output variable. the outputs of the anfis model are compared with already known values and, based on this, rmse is calculated as follows:     2 1 1 n k rmse n k n k n       (6) where n is the number of individual observations (the number of data vectors for testing, n = 12), n(k) is the expected value (obtained by the simulation in the pvsyst program), ( )n k the value obtained by the model. the verification data is primarily aimed at preventing the occurrence of overfitting. 3. results and discussion table 1 shows the different values of the average testing error, depending on the shape of the membership function of the input variables, their number, and the output form of the anfis model. the model with the lowest value of the average testing error or rmse=0.209 is chosen. such an error value is achieved with the triangular form of the fuzzy membership functions (two of them for each input variable) and the linear function of the output of the model. stojčić et al./decis. mak. appl. manag. eng. 2 (1) (2018) 35-48 42 table 1. different values of the average testing error (rmse) for model shape of the membership functions of input variables form of output function of anfis model linear constant number of membership functions of the input variables 2 2 3 3 4 4 2 2 3 3 4 4 trimf 0.209 0.321 6.321 0.243 0.211 0.361 trapmf 0.245 0.938 0.442 0.483 0.292 0.981 gbellmf 0.223 1.047 39.464 0.348 0.220 0.396 gaussmf 0.212 0.642 36.433 0.288 0.215 0.335 gauss2mf 0.252 1.141 46.554 0.383 0.225 0.573 pimf 0.256 1.206 0.508 0.609 0.351 3.229 dsigmf 0.254 0.620 11.073 0.578 0.302 0.498 psigmf 0.254 0.620 11.073 0.578 0.302 0.498 5 5 2 3 3 2 5 5 2 3 3 2 trimf 10.450 0.221 0.240 0.335 0.217 0.256 trapmf 1.141 0.244 0.418 1.060 2.289 0.439 gbellmf 8.447 0.286 0.535 0.530 0.216 0.331 gaussmf 19.091 0.252 0.665 0.272 0.213 0.291 gauss2mf 280.229 0.399 0.296 16.590 0.245 0.344 pimf 3.839 0.240 0.727 6.537 0.330 0.558 dsigmf 34.961 0.226 0.661 6.608 0.307 0.518 psigmf 34.961 0.226 0.661 6.608 0.307 0.518 the graphical results of the prediction of the selected model are given in figure 5. red stars represent the values given by the model, while the blue points are known values (data for testing). from figure 5 it is evident that the generated daily electricity has the greatest value in the summer months, because it is presented on the apsis one day for each month of the year. figure 5. graphical representation of the prediction for the selected anfis model anfis model for the prediction of generated electricity of photovoltaic modules 43 figure 6 shows the shape of the membership functions of the input variables. as can be seen from table 1, membership functions have a triangular form. figure 6. forms of the membership functions of the input variables: a) average temperature of the module; b) daily amount of solar radiation figure 7 shows the structure of the selected anfis model where the number of nodes in each layer of the network is visible. figure 7. the structure of the selected anfis model the surface that represents the dependence of the output variable of the two inputs in the selected anfis model is shown in figure 8. it can be concluded that stojčić et al./decis. mak. appl. manag. eng. 2 (1) (2018) 35-48 44 when the energy of the solar radiation increases, the maximum virtual energy of the photovoltaic modules increases. in contrast, the average temperature adversely affects the output variable, as its increase results in a slight decrease in the maximum virtual energy of the photovoltaic modules. figure 8. the surface of the dependence of the maximum virtual energy from the solar radiation and the average temperature of the module 4. sensitivity analysis in order to perform the sensitivity analysis, in this section will be compared the values of the prediction created by the anfis model and the regression mathematical model. the regression was performed on a data set for training the anfis model. table 2 shows 15 different mathematical models together with the corresponding prediction correlation indexes. for simpler presentation, the selected variables carry the following tags: • daily amount of solar radiation e • average temperature of the module t • virtual maximum generated energy – electricity. table 2. regression mathematical models number model r2 pred (%) 1 electricity = 0.386 – 0.00128 t + 0.5045 e 87.90 2 electricity = 0.049 + 0.0446 t + 0.5027 e – 0.001189 t2 88.60 3 electricity = -0.146 + 0.0144 t + 0.882 e – 0.000764 t2 – 0.0407 e2 90.78 4 electricity = -0.164 + 0.0186 t + 0.882 e – 0.00101 t2 – 0.0407 e2 + 0.000004 t3 90.53 anfis model for the prediction of generated electricity of photovoltaic modules 45 number model r2 pred (%) 5 electricity = -0.184 + 0.0163 t + 0.923 e – 0.00091 t2 – 0.0522 e2 + 0.000003 t3 + 0.00089 e3 90.26 6 electricity = -0.065 + 0.842 e – 0.0502 e2 + 0.00130 e3 89.75 7 electricity = -0.046 – 0.01682 t + 1.032 e – 0.0769 e2 + 0.00250 e3 90.29 8 electricity = 0.039 – 0.01639 t + 0.920 e – 0.0447 e2 90.50 9 electricity = -0.020 + 0.7851 e – 0.0336 e2 90.01 10 electricity = 0.6886 + 0.1767 e2 – 0.01552 e3 87.75 11 electricity = 0.170 – 0.01507 t + 0.7518 e – 0.003306 e3 90.30 12 electricity = -0.122 + 0.986 e – 0.000454 t2 – 0.0649 e2 + 0.00167 e3 90.66 13 electricity = 0.069 + 0.7538 e – 0.000437 t2 – 0.003279 e3 90.75 14 electricity = 0.7324 – 0.000275 t2 + 0.1913 e2 – 0.01704 e3 87.87 15 electricity = -0.174 + 0.0134 t + 0.925 e – 0.000744 t2 – 0.0531 e2 + 0.00095 e3 90.52 as can be concluded from table 2, the largest correlation index has a model numbered at number 3, 90.78%. it is a second-degree model that has the following form: electricity = -0.146 + 0.0144 t + 0.882 e – 0.000764 t2 – 0.0407 e2 table 3 gives an overview of the expected values of the generated photovoltaic module energy (test data), as well as the values obtained by anfis and the regression model, for the same input values. values are expressed in kwh/day. table 3. the expected values of the generated energy and the values obtained by the prediction of anfis and the mathematical model expected value of generated energy value obtained by an anfis model value obtained by the regression mathematical model 0.48 0.56 0.57 2.64 2.19 2.09 2.36 2.60 2.49 3.23 3.39 3.41 4.19 4.17 4.22 3.50 3.41 3.42 3.68 3.81 3.83 4.12 3.93 3.92 2.24 2.28 2.28 0.58 0.76 0.71 1.19 0.88 0.76 0.75 0.56 0.56 in addition to a tabular display, these values can also be represented graphically, as shown in figure 9 stojčić et al./decis. mak. appl. manag. eng. 2 (1) (2018) 35-48 46 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 1 2 3 4 5 6 7 8 9 10 11 12 expected value anfis model mathematical model figure 9. diagram of expected values, values obtained by anfis and mathematical model by comparing the output variable from the test data with the values obtained by the specified mathematical model, the rmse = 0.235 value is obtained, which is more than the selected anfis model (0.209). therefore, it is clear that the anfis model shows better performance. 5. conclusion the research presented by the paper focuses on the development of a model for the prediction of maximum energy generated photovoltaic modules based on neurofuzzy principles. the model represents a simple solution that requires the value of the output variable for the given values of the energy of the sun's radiation and the average temperature of the module. model training was performed according to the data obtained by the simulation, so that it is possible to deviate if the values obtained by the prediction were compared with the actual measured values. nevertheless, pvsyst is a widely used software, so that data obtained by simulating the performance of photovoltaic modules can be considered relevant for the training of the anfis model. it is obvious that the selected model yields better results than mathematical model, although it has a high percentage of adequacy. the maximum energy generated depends to a large extent on the energy of the sun's radiation, while the influence of the temperature is considerably smaller and negative. future research may take into account other factors that influence the generation of energy in order to increase accuracy. references antonanzas, j., osorio, n., escobar, r., urraca, r., martinez-de-pison, f. j. & antonanzas-torres, f. 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(2017). an improved forecasting method for photovoltaic power based on adaptive bp neural network with a scrolling time window. energies, 10(10), 1-18. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). decision making: applications in management and engineering vol. 4, issue 2, 2021, pp. 200-224. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame210402200k * corresponding author. e-mail addresses: marija.kuzmanovic@fon.bg.ac.rs (m. kuzmanovic), milena.vukic12@gmail.com (m. vukic) incorporating heterogeneity of travelers’ preferences into the overall hostel performance rating marija kuzmanović1* and milena vukić2 1 university of belgrade, faculty of organizational sciences, belgrade, serbia; 2 the college of hotel management in belgrade, belgrade, serbia; received: 4 february 2021; accepted: 27 june 2021; available online: 12 july 2021. original scientific paper abstract: hostels have become a very popular form of accommodation and their varieties have grown steadily in recent years. to ensure the sustainability of this business model, it is necessary to understand the main drivers influencing travelers to choose a hostel accommodation. for this purpose, we conducted an online survey using convenience sampling and purposive sampling techniques. respondents' preferences to six hostel attributes (cleanliness, location, staff, atmosphere, facilities, and cancellation policy) were determined using discrete choice analysis. sample results showed that the most important attributes are cleanliness and location, while the atmosphere is the least important one. however, widespread heterogeneity in preferences was observed, and cluster analyzes identified three distinct groups of travelers: “cleanliness sticklers”, “location demanders” and “party seekers”. facilities and atmosphere were found to be very important attributes for particular clusters. these findings can help design a marketing strategy for each of the identified segments to ensure sustainable business. finally, we have proposed a new approach to calculating the hostel overall rating based on attribute importance, which shows much better discriminatory power compared to the traditional average-based approach. key words: hostel; discrete choice analysis; attribute importance scores; preference-based clustering; simulation; weighted performance rating. 1. introduction over the last six decades, tourism industry become one of the largest economic sectors in the world (mihalic, 2014). the importance of the tourism industry is evident in both developed and developing countries, which is best reflected through a number of direct and indirect impacts on national economies (world travel and tourism mailto:marija.kuzmanovic@fon.bg.ac.rs mailto:milena.vukic12@gmail.com incorporating heterogeneity of travelers’ preferences into the overall hostel performance… 201 council, 2018). further strong growth of the tourism industry around the world was expected, but the appearance of the covid-19 pandemic temporarily hindered it. farr (2020) states that after the end of the pandemic, it will take about 18 to 24 months for daily tourist activities to return to the level before the pandemic (nguyen, 2020). the structure of the tourism market has changed significantly and continues to change over the years, with increasing attention paid to the concept of sustainability and related topics such as circular economy, collaborative consumption, sharing economy, and low-income consumers targeting (lemus-aguilar et al., 2019). the development of the internet has significantly contributed to these trends. nowadays, tourists have access to more information, they are more mobile, and are more willing to experiment with unconventional forms of travel. the expansion of low-cost airlines has increased the number of both available destinations and flights between the two destinations, leading to further price reductions due to growing competition. a study carried out by eugenio-martin and inchausti-sintes (2016) shows that savings achieved by low-cost transportation are at least partially transferred to spending on the destination itself. positive changes are also noticeable in low-budget accommodations such as hostels, which show more online penetration than hotels and apartment rentals (muñoz-fernández et al., 2016). despite the outbreak of the covid19 pandemic in 2020 that negatively affected various industries, including tourism, travelers are expected to travel again, and their demands will affect the future of affordable accommodation. accordingly, hostels must be prepared to respond to these demands in the right way. although initially the cost of accommodation was the main reason for travelers to choose hostels, over the years, the type of hostel guests changed and their motives and preferences became more diverse. to increase guest satisfaction, some hostels have launched a number of specific services such as self-serving facilities, group social and sports activities, the ability to rent certain equipment. some hostels have recognized the importance of environmental sustainability and are taking action to promote such activities. the development of technology and digitization made it possible for different hostel visitors to exchange impressions and accommodation reviews. on the one hand, this provides the guests with a certain level of security when choosing an accommodation and helping them find accommodations that match their desires. hostel owners, on the other hand, receive feedback from their clients and can eliminate potential weaknesses on time, as in the internet era only a few negative reviews can have serious consequences on business success (martins et al., 2018). the fact is that hostels have become very attractive to investors in recent years, that their number and varieties are constantly growing, and that even more prosperity is expected in the future through improvement of product quality and product offer offerings. in such a competitive environment, the concept of sustainability becomes crucial. unlike traditional entrepreneurship, which focuses mainly on economic development, sustainable entrepreneurship and sustainable business models aim to balance economic, social and environmental goals (belz & binder, 2017). in this study, we focus on the key factors influencing hostel guests' satisfaction, which is closely related to the economic and social dimension of sustainable business. we sought to identify individuals' preferences for key hostel characteristics, identify the most important factors that influence their decision when choosing a hostel, and investigate whether these factors depend on the demographics of the respondents or their habits and attitudes. for that purpose, in this study discrete choice analysis (dca) was employed. dca is an approach for identifying the relative importance of attributes when the individuals choose between comparable products or services based on specific features. it has been successfully used for the analysis of individual m. kuzmanović et al./decis. mak. appl. manag. eng. 4 (2) (2021) 200-224 202 choice behavior in many fields such as economics, marketing, education, transportation, environmental management, and healthcare (rakotonarivo et al., 2016; popović et al., 2018; kuzmanovic et al., 2020). it has been applied also in the tourism industry, but primarily to determine guests' preferences and willingness to pay for a hotel room attributes and preferences towards tourist destination (capitello et al., 2017; chang et al., 2018; gonzález et al., 2018; kim & park, 2017; oppewal et al., 2015; vukic et al., 2015). however, so far dca has not been used to identify the tradeoffs that respondents are willing to make when it comes to different factors that affect their choice of hostel. furthermore, using dca it is possible to identify whether the guests’ preferences are heterogeneous, but also to calculate the real overall hostel rating, taking into account that not all factors are equally relevant to certain groups of guests. if there are differences in factors that affects the satisfaction of guests, hostel owners need to customize their offer to satisfy all guests. the rest of the paper is organized as follows. section 2 provides a literature review on hostel specifics and key hostel features that influence travelers’ choices. section 3 covers the study methodology i.e. overview of theoretical foundation of discrete choice modelling is outlined and design of present study is presented. the results of empirical study are given in section 4. both aggregated and segment level preferences are presented as well as simulation results. finally, section 5 provides concluding remarks and implications for further research. 2. literature review the hostel is a short-term accommodation, which focuses particularly on visitors who are most often referred to as backpackers (o׳regan, 2010). nevertheless, there is no precise definition of this type of accommodation. some discussion on this issue can be found in scientific literature (hory et al., 2017; oliveira-brochado & gameiro, 2013). difficulties in clear definition appear as a consequence of the global expansion of the spectrum of tourists and their heterogeneous demands. there are many ways in which the hostel business can contribute to the economic and social sustainability of areas, regions or countries. hostels are usually associated with people who are traveling on a tight budget and who do not mind to give up their privacy for the sake of lower cost accommodation. world trends and the financial crisis have led to hostels now attracting families and people on a business trip that is constrained by a relatively modest budget (cave et al., 2008). thus, one of the basic characteristics of the hostel is that it has to be budget-oriented. although budget tourists tend to spend less on a daily basis, their travel is generally extended, and total expenditures during their stay are often significantly higher than those of average tourists. moreover, hostel accommodation businesses in small cities where there are no hotels, contributing significantly to local economic development. a hostel typically has a more casual atmosphere than hotels and it is more adventure oriented, attracting the younger segment of travelers (de oliveira santos, 2016). hostels offer two types of accommodation, either shared rooms where individual travelers can book a single bed or private rooms with bathrooms when it comes to modern variants of hostels. consequently, they generally provide more and better opportunities for travelers who are staying in the short term to meet new people from different cultures, by virtue of shared facilities and a common rooms, such as living room, lounge, shower, and kitchen (lima & vicente, 2017). when it comes to the atmosphere and character of the hostel, the one of the famous hostel booking websites, hostelbookers, provides description of several different types of hostels incorporating heterogeneity of travelers’ preferences into the overall hostel performance… 203 (oliveira-brochado & gameiro, 2013): (a) ‘family hostels’, offering clean and comfortable low budget rooms in order to attract families; (b) ‘activity hostels’ focusing on providing a wide range of activities to their guests, such as surfing, skiing, hiking, biking etc.; (c) ‘party hostels’ designed for travelers who want to have fun and to experience the city's nightlife. these hostels do not have a police time, usually have a bar and organize parties. recently, a new type of hostel guests have been appeared, so called ‘flashpackers’, traveling with a higher budget, often using social media, and have a greater demand for comfort, quality and privacy than the backpackers (hory et al., 2017). flashpackers want the backpacking experience with the luxury of a comfortable hotel. while they want the friendly atmosphere, they also want some quiet downtime. an increasing number of such type of travelers steadily increases quality requirements towards the physical features of hostels, leading to a new type of hostel, the so-called ‘flashpacking hostels’. numerous studies have been conducted, both qualitative and quantitative, for determining the motives, factors and preferences of travelers towards hotels and hostels (lin et al., 2018; roy et al., 2019; puška et al., 2021). as expected, the main features of the hostel, a favorable accommodation price and value for money, have proven to be an important source of customer satisfaction (nash et al., 2006; lima & vicente, 2017; de oliveira santos, 2016; cró & martins, 2017; veríssimo & costa, 2018). however, hecht and martin (2006) find out that service preferences vary depending on the three key demographic categories: gender, age, and country of origin. in addition to the accommodation price, most studies emphasize two more hostel features, cleanliness and location, as very important factors that influence the guests' choice as well as their satisfaction (hecht & martin, 2006; nash et al., 2006; brochado et al., 2015; lima & vicente, 2017; martins et al., 2018; oliveira-brochado & gameiro, 2013; cró & martins, 2017; amblee, 2015). studies have also identified some other important tangible factors that increase the satisfaction of guests such as selfserving facilities and the possibility of renting certain equipment (nash et al., 2006; brochado et al., 2015; musa & thirumoorthi, 2011; oliveira-brochado & gameiro, 2013; veríssimo & costa, 2018). furthermore, the results show that hostel guests want to socialize, to participate in activities and events, to have local experiences and to feel at home (hecht & martin, 2006; brochado et al., 2015; martins et al., 2018; ban et al., 2019). this especially holds for young people (muñoz-fernández et al., 2016) and millennials (veríssimo & costa, 2018). lima and vicente (2017) revealed that the main drivers of the overall satisfaction of younger guests are staff’s competence and friendliness. musa and thirumoorthi (2011) found that politeness and friendliness, commitment, readiness to help and relevant knowledge are very important characteristics of a good staff. chitty et al. (2007) show that the brand image is a strong predictor of satisfaction of the low-budget guests with the hostel. cró and martins (2017) used a hedonic prices and guests' reviews to analyze the impact of the country's crime index on hostel accommodation prices and concluded that guests are willing to pay a higher price if a hostel has higher security and cleanliness level but also good location. similar conclusions were made by amblee (2015) who used a word-of-mouth approach. the most popular booking websites provide the opportunity for hostel guests to evaluate their satisfaction with accommodation on a variety of factors, primarily those that have been also identified as important through scientific literature. table 1 shows a list of factors that are usually evaluated on certain websites. m. kuzmanović et al./decis. mak. appl. manag. eng. 4 (2) (2021) 200-224 204 table 1. overview of guest satisfaction factors on hostel booking websites hоstelwоrld. cоm hоstelbооkers. cоm hоstels.c оm hostelz.c om bооking.c оm location      cleanliness      staff      safety/secu rity     atmosphere     value/price    facilities   comfort  services  character  3. materials and methods 3.1. discrete choice analysis discrete choice analysis (dca), often referred to as a choice-based conjoint analysis, is one of the most commonly used research techniques that helps reveal how individuals make choices and what they really value in products and services. instead of asking respondents what features they find most important, in discrete choice experiment, respondents are expected to choose between the concepts carefully assembled into choice sets. these concepts are potential real or hypothetical alternatives described by the most relevant tangible and intangible characteristics or attributes (ben-akiva et al., 2019). dca is theoretically grounded in random utility theory (rut) and relies on the assumptions of economic rationality and utility maximization (ben-akiva & lerman, 1985; mangham et al., 2009), but also assumes heterogeneity in decision-maker and choice alternatives (oppewal et al., 2015). the output from discrete choice analysis is a measure of utility scores that are numerical values that weight how each attribute and level attribute affected customer's decision to make that choice. there are five main steps in the process of discrete choice analysis: (1) the identification of key attributes and specification of their levels, (2) creation of the design of experiment, (3) data collection, (4) the choice model estimation, and (5) post-hoc segmentation. 3.2. identification of hostels attributes and levels at this stage of the research, the identification of characteristics that differentiate a certain set of hostels from others was done. five attributes have been identified based on the literature review: location, staff, atmosphere, cleanliness level and facilities. the appropriate levels have been assigned to these attributes (table 2). in addition, the attribute cancellation policy (refund) was also included in the survey. the need to include this attribute in the study was based on an analysis of the impressions guests leave on tripadvisor, the world's largest travel portal, as well as on the hostels' websites. namely, it was noted that one of the most frequent reasons why the guests complain is precisely the problem of refunding money in case of cancellation. this problem comes from three reasons. the first is that hostel reservations are made in about 80% of cases by means of internet platforms that have incorporating heterogeneity of travelers’ preferences into the overall hostel performance… 205 their own cancellation policy that has priority over the policy of hostel itself. the other is that these platforms see the cancellation as an opportunity to make additional profits, so they charge it as a special service, and the third reason is the insufficiently transparent policy of cancellation. two levels of attribute have been identified: deposit only and no refunds. the former means that if the guest cancel the reservation on time (usually 1-7 days before the start of the stay, depending on the hostel), the deposit will be refunded, while no refund option mean that the hostel retains the entire amount of the reservation. table 2. attributes and levels used in the study attribute levels location city center good connection to city center poor connection to city center staff friendly formal marvelous atmosphere homely party active traditional cleanliness level low moderate high facilities poor good superb cancellation policy deposit only non-refundable although crucial factor, the price of accommodation has not been included in the list of attributes in this research since prices generally varies from city to city and therefore are not comparable. 3.3. experimental design the next stage in discrete choice analysis is to decide which scenarios to present to individuals, i.e. to generate experimental design. based on selected attributes and attribute levels, a fractional factorial choice design was created. the design is efficient in terms of d-efficiency criteria and supports measurement of two-way interactions. in the cases when such designs are very complex and consist of a large number of choice tasks, so called blocked designs are often used. blocks are partitions of the choice tasks in the design of experiment that contain a limited number of choice questions for each respondent. in our study, an experimental design with 22 choice tasks was partitioned into two blocks so that each respondent evaluated only 11 choice tasks. each choice task consisted of three full profile alternatives (hostel 1, hostel 2 and hostel 3) and one "none of the above" option. each respondent evaluated a total of 33 alternatives. in this way, the survey covered a total of 66 (11×3×2) profiles from the possible 648 (=34×4×2). the example of choice task used in survey is given on figure 1. m. kuzmanović et al./decis. mak. appl. manag. eng. 4 (2) (2021) 200-224 206 figure 1. the example of the choice task 3.4. study implementation because dca calculates preferences for each single respondent, no large sample is required for the results to be valid. however, the generated experimental design has an influence on the sample size required for the trustworthy results (de bekker-grob et al., 2015). according to orme (2010), the rule of thumb for an acceptable sample size is: 𝐼 ≥ 500× largest product of levels of any two attributes number of tasks × number of alternatives per task , so the minimum sample size for this study was 182 (i.e. 500 × (3 × 4) / (11 × 3)). to collect responses from respondents, this study used a web-based survey created on the online platform conjoint.ly. it has been shown that web-based surveys are more suitable for discrete choice experiments than other forms of surveys for reasons such as ease of use, immediacy, time saving and high response rate (oppewal et al., 2015). according to the purpose and needs of the research, participants were recruited through convenience sampling and purposive sampling methods. the survey was shared in travel groups on social networks, travel-related forums, but also distributed directly by email. the intent was to effectively select participants who would be willing to provide the most relevant data to answer research questions defined. such were considered to be individuals who occasionally or frequently travel and stay in hotels , hostels or other paid accommodation. our questionnaire consisted of three parts: (1) questions concerning respondent demographics; (2) questions about general respondents’ habits and experience regarding accommodation while traveling, and (3) eleven hostel choice tasks. to test the questionnaire, the survey was piloted using a sample of 20 respondents. 3.5. estimation of the choice model discrete choice model can be derived from utility theory and specifies the probability that an individual chooses a particular hostel, with the probability expressed as a function of observed variables that relate both to the hostel and the individual. as mention before, the assumption is that the individuals tend to maximize utility by choosing those hostel that contain most desirable characteristics. in other words, given the set j of mutually exclusive hostel alternatives, it is assumed that an individual i (i = 1,...,i) will choose an alternative j (j = 1,...,j) if and only if the overall utility that alternative j provides to him at least equal to those associated with other incorporating heterogeneity of travelers’ preferences into the overall hostel performance… 207 alternatives in the same choice set. the choice of the individual i is designated by variables yij for each alternative j: 𝑦𝑖𝑗 = { 1,       𝑈𝑖𝑗 > 𝑈𝑖𝑚 ,    ∀𝑗 ≠ 𝑚 0,     otherwise (1) where uij represents overall utility and can be expressed as: uij = vij + ij . (2) random or stochastic component ij represents the unobserved sources of utility related to the characteristics of the individuals and/or hostel attributes and can represent both variations in preferences among population members and measurement errors. the deterministic component of utility, vij, is usually a linear additive model that maps the multidimensional attribute vector into the overall utility (kuzmanovic et al., 2020): 𝑉𝑖𝑗 = ∑ ∑ 𝛽𝑖𝑘𝑙 𝑥𝑗𝑘𝑙 𝐿𝑘 𝑙=1 𝐾 𝑘=1 (3) where 𝛽𝑖𝑘𝑙 is partial utility that respondent i attaches to lth level of attribute k, socalled part-worth, and xjkl is a binary variable that equals 1 if hostel j contains level l of attribute k, otherwise it equals 0. accordingly, the probability that individual i would choose hostel j from a set of three mutually exclusive hostels is given by: 𝑃𝑖𝑗 = 𝑒𝑥𝑝(𝑈𝑖𝑗) ∑ (𝑒𝑥𝑝(𝑈𝑖𝑗)) 3 𝑗=1 (4) hierarchical bayes (hb) approach can be used to estimate the parameters in model (4). hb models are hierarchical models analyzed by bayesian methods, which assume that probability is operationalized as a degree of belief, rather than frequencies as it is in classical statistics (rossi et al., 2012). the value of the hb model lies in its ability to describe heterogeneity in preferences while retaining its ability to study particular individuals. besides that, this approach allows more parameters to be estimated with less data collected from each respondent. estimated part-worths reflect how strongly that level influences the decision to choose the hostel. attributes with a large range of influence are consider more important. accordingly, relative importance of each attribute for each respondent are calculated by dividing the utility range for each attribute separately with the sum of the utility ranges for all attributes (vukic et al., 2015): 𝑊𝑖𝑘 = 𝑚𝑎𝑥 𝑙 𝛽𝑖𝑘𝑙−𝑚𝑖𝑛 𝑙 𝛽𝑖𝑘𝑙 ∑ (𝑚𝑎𝑥 𝑙 𝛽𝑖𝑘𝑙−𝑚𝑖𝑛 𝑙 𝛽𝑖𝑘𝑙) 𝐾 𝑘=1 . (5) individual importance scores can be aggregated for the sample as a whole or for clusters in the case of heterogeneous preferences. in addition to a priori segmentation based on socio-demographic variables, this study employs a post-hoc segmentation approach as well. this approach is expected to be more effective as segments will be isolated based on differences in respondents’ preferences (kuzmanovic & savic, 2020). clustering on individual preferences and behavioral differences has been found to be more robust and stable over time. clusters that differ in the behavioral drivers can be found using k-means cluster analysis (norris et al., 2014). m. kuzmanović et al./decis. mak. appl. manag. eng. 4 (2) (2021) 200-224 208 4. results 4.1. sample characteristics survey was conducted in november 2017. eligibility criteria were that respondents were 16 or older, and that travel at least once a year, staying in paid accommodation. of 1522 individuals who approached the survey, 273 fully completed the questionnaire, giving a response rate of 14.22%, out of which 218 questionnaires were valid (79.85%). slightly more than half of respondents (55.96%) were women. the average age of participants was 28.14 years (sd=8.33). over 60% of respondents had completed at least an undergraduate degree and almost 32% had a postgraduate degree or higher qualification. the majority of participants were european (76.6%). detailed demographics are given in table 3. table 3. demographic data category variable number (n=218) (%) gender male 96 43.78 female 122 55.96 education level high school or equivalent 71 32.57 bachelor's degree 75 34.40 master's degree 64 29.36 phd 5 2.29 other 3 1.38 employment status unemployed 11 5.04 full time employed 116 53.21 part time employed 18 8.26 student 49 22.48 employed students 14 6.42 other 10 4.59 average monthly income up to 300 € 15 6.88 300-600 € 40 18.35 600-1000 € 53 24.31 more than 1000 € 57 26.15 no answer 53 24.31 continent of residents europe 167 76.61 asia 13 5.96 north america 25 11.47 south america 5 2.29 africa 2 0.92 australia 6 2.75 the most common accommodation choice is hotel (38.25%), followed by hostel (27.65%) and airbnb (21.56%). bed & breakfast is the first choice for only 5.5% of the respondents, while the rest of sample prefer other accommodations. the primary guests of the hostel are considered to be backpackers, but only 8.72% of our sample perceived themselves as backpackers; the same percentage of sample considered themselves to be a tourist or traveler (41.28%), while 8.72% respondents did not match any of the listed categories. however, the majority of backpackers as the first choice of accommodation listed hostel (57.89% of all backpackers), while those incorporating heterogeneity of travelers’ preferences into the overall hostel performance… 209 who call themselves tourists are usually opt for hotel accommodation and very rare for hostel (15.56%). when choosing their accommodation, respondents mostly inform via the internet (41.74%) and by word-of-mouth from friends and acquaintances (34.86%). nearly 45% of respondents usually travel with friends. there is 24.37% respondents who travel with a partner, and 12.97% who travel and with a family member, while 16.46% respondents are usually solo travelers. respondents rarely stay in hostels for more than 7 nights (only 4.13%); 18.35% usually stay between 5 and 7 nights, while more than half of the respondents, 112 (51.38%) in hostels stays 2-4 nights. among the respondents there were 10.09% those who usually stay for only one night, as well as those who said they never stayed in hostels. among 35 respondents (16.06%) who never use hostel services there were 9 men and 26 women. respondents were also asked to indicate the lowest average hostel rating on booking websites, which is necessary to consider the booking accommodation. the distribution of the answers is shown in the figure 2. figure 2. distribution of the lowest acceptable hostel rating 4.2. aggregated respondents’ preferences the primary outputs of dca are the estimated part-worths (preferences) for various attribute levels. using hb method, the preferences were estimated for each single respondent, and then averaged for the whole sample (aggregated preferences). attribute levels have been coded using an effects coding procedure, which constrains the sum of part-worths of each attribute to be zero. part-worth utilities associated with particular attribute levels provide a deeper insight into which characteristics determine the consumer's choice. the aggregated preferences towards the key hostel characteristics are shown in table 4. the results indicate a high level of statistical significance for the attribute levels, with all levels having signs in line with a priori m. kuzmanović et al./decis. mak. appl. manag. eng. 4 (2) (2021) 200-224 210 expectations. namely, negative signs for some levels indicate that respondents considerably less prefer them than those with positive value. for example, respondents do not prefer hostels that have poor connections to the city center (partworth = -21.44), low level of cleanliness (-19.60), or poor offer of facilities (-7.48). table 4. part-worths estimates for the sample attribute attribute level relative importa nce partwort hs 90% confidence interval lower bound upper bound location 33.99% 32.35 % 35.61 % city center 12.55 11.76 13.42 good connection to city center 8.89 8.26 9.55 poor connection to city center 21.44 -22.40 -20.45 staff 6.43% 5.52 % 7.23 % friendly 3.03 2.60 3.38 formal -3.41 -3.92 -2.85 marvelous 0.38 0.01 0.78 atmosphere 3.38% 3.00 % 4.46 % homely 0.93 0.65 1.19 party 0.71 -0.15 1.65 active 0.81 0.26 1.34 traditional -2.45 -2.95 -2.00 cleanliness level 35.27% 33.91 % 36.35 % low 19.61 -20.15 -18.90 moderate 3.95 3.34 4.52 high 15.66 14.86 16.36 facilities 12.79% 11.83 % 13.58 % poor -7.48 -8.01 -6.87 good 2.18 1.78 2.57 superb 5.30 4.81 5.67 cancellation policy 8.15% 7.07 % 9.26 % deposit only 4.07 3.53 4.61 non-refundable -4.07 -4.61 -3.53 a 90% confidence interval is also reported under each parameter estimate. it identifies the range in which there is a 90% probability that the true parameter value falls. for all of the estimated parameters, a zero value fell outside of this 90% confidence interval, indicating that all independent variables have an influence on the dependent variable at the 90% level. an additional goodness of fit measure, called incorporating heterogeneity of travelers’ preferences into the overall hostel performance… 211 mcfadden's pseudo-r2 provides a measure of how well estimated results describe respondents' answers to the survey. a pseudo-r2 value of 67% in this survey indicates that the calculated part-worths well describe the respondents' choices (norris et al., 2014). the relative importance of the attributes are derived from corresponding partworth utilities and show the extent to which each of the attributes influence the decision of respondents, and which later affects their satisfaction. as it can been seen from table 4, the most important attributes for the sample as a whole are cleanliness and location, with the average importance values of 35.27% and 33.99% respectively. significantly less important is the attribute facilities (12.79%), while the atmosphere is the least important attribute with the average importance of just 3.38%, indicating that this attribute has almost no influence on the decision about the choice of hostel. the best-rated hostel concept according to the respondents' preferences is those with a high level of cleanliness, located in the city center, with superb facilities, friendly staff and a homely atmosphere, but also that returns whole amount of deposit in case of cancellation of the reservation. 4.3. preferences of predefined groups of respondents in order to determine whether the preferences of some predefined groups of respondents were homogeneous, a priori segmentation was performed based on the demographics and habits of the respondents. in particular, the analysis was done on the basis of three variables: (1) the continent of residence, (2) the first choice of accommodation, and (3) gender. both the partial utilities and the relative importance of the attributes for each subgroup are estimated separately. as it can be seen from figure 3, there is no significant differences between the segments in terms of the relative importance attached to the attributes. furthermore, the order of the levels according associated utilities within all attributes is almost the same, and only slight variations occur within the attribute atmosphere. namely, it has been found that males strongly prefer party hostels, while females prefer a hostel with an active and homely atmosphere, considering party hostels as undesirable for staying. similarly, respondents who usually stay in hostels give priority to the party hostels compared to other respondents. however, the atmosphere is negligible important attribute for all segments, and therefore differences in preferences to certain levels of this attribute have minor impact on the overall preferences. m. kuzmanović et al./decis. mak. appl. manag. eng. 4 (2) (2021) 200-224 212 figure 3. relative importance of attributes across a priori defined segments (in %) 4.4. preference-based clustering aggregated results often mask the real situation, so a cluster analysis based on respondents’ preference was made. namely, cluster analysis allows the detection of groups of respondents with homogenous preferences that remain latent in aggregated results. as the preferences are calculated for each respondent individually, preference-based clustering is enabled and for that purpose k-means cluster analysis is employed. relying on calinski-harabasz criteria and the dunn index, as well as on the bases of knowledge of the market and socio-demographic characteristics of the sample, three clusters were identified. the relative importance of the attributes for each of the clusters are shown in the table 5, while figure 4 shows the percentage of inhabitants of certain continents in each of the clusters. table 5. relative importance of the attributes for each of the clusters cluster 1 (cleanliness sticklers) cluster 2 (location demanders) cluster 3 (party seekers) location 21% 45% 36% staff 10% 2% 9% atmosphere 6% 4% 19% cleanliness level 38% 32% 21% facilities 19% 7% 8% cancellation policy (refund) 6% 10% 7% incorporating heterogeneity of travelers’ preferences into the overall hostel performance… 213 figure 4. comparison of clusters concerning the region of respondents’ residence 4.4.1. cluster 1 cleanliness sticklers the first cluster includes slightly more than a third, i.e. 35% of the total sample. for the respondents from this cluster, the cleanliness is the key decision factor when choosing a hostel (relative importance is 38%), whereby the respondents prefer only a high level of hostel cleanliness. a moderate level of cleanliness also has a positive sign, but a small value of utility. as it can be seen from table 5, location and facilities also appear as very important attributes, with an importance values of 21% and 19% respectively. when it comes to the location attribute, it can be noted that it is significantly less important than in the average. for the attribute facilities the opposite applies; this is the only cluster that attach to this factor so much importance. it can be noted that members of this cluster almost equally prefer hostels located downtown and outside the center with good connection to the center. furthermore, members of this cluster prefer hostels with homely or active atmosphere and friendly staff, although these characteristics are not crucial (figure 5). m. kuzmanović et al./decis. mak. appl. manag. eng. 4 (2) (2021) 200-224 214 figure 5. cleanliness sticklers' preferences a more detailed analysis has shown that this cluster consists mainly of female respondents (60%), who are on average slightly older than respondents in the remaining two segments. although most of them come from europe, this percentage is lower than in the other two clusters, as can be seen in figure 4. for respondents of this segment, the first choice when deciding on an accommodation is a hotel, while when choosing a hostel, the acceptable hostel rating is higher than for the other two groups. 4.4.2. cluster 2 – location demanders this cluster is the largest one (36% of the total sample) and is made up by respondents to whom location is by far the most important factor when choosing a hostel (relative importance of 45%). compared to the other two clusters, they much more prefer that hostel to be located in the city center, although they also largely accept a location with a good connection to the center (figure 6). incorporating heterogeneity of travelers’ preferences into the overall hostel performance… 215 figure 6. location demanders' preferences another very important attribute for this group of respondents is cleanliness (32%), while on the third place is the cancellation policy with relative importance of 10% which is higher value than for the other two clusters. this cluster also significantly more prefers the moderate level of cleanliness than the previous one. members of this cluster prefer the active and homely atmosphere, and superb or good facilities. the staff is the least important attribute for this cluster with an importance value of only 2%. it is interesting that the utility of the level of marvelous staff has a negative sign, and that the respondents showed indifference towards formal staff. the ratio of males and females is similar to that of the first cluster (58% females). average age of cluster members is 27.46 years and most of them are european (86%). compared to the "cleanliness sticklers", they accept a slightly lower average rating of the hostel, and to a large extent as the first choice of accommodation, next to the hotel, also quote airbnb. it is interesting that 33% of respondents from this segment are unemployed or part-time employed. 4.4.3. cluster 3 – party seekers the third cluster is the smallest one and covers 26% of the sample. as with the previous cluster, this group of respondents give the highest importance to the attribute location (36%), whereby they most prefer to be located out of the downtown but with a good connection to it. another interesting thing about this cluster is that the attribute atmosphere is much more important for members compared to other clusters (importance value is 19%). respondents belonging to this group prefer hostels with the party atmosphere, while the traditional and active atmosphere reduce their overall preferences (the partial utilities of these levels have a negative sign, which can be seen in figure 7). m. kuzmanović et al./decis. mak. appl. manag. eng. 4 (2) (2021) 200-224 216 figure 7. party seekers’ preferences cleanliness is the second most important factor with an importance value of 21%, but it is significantly less important than in the other two clusters. the remaining three attributes – staff, facilities, and cancellation policy are of almost equal importance. this segment is the only one with more males than females (33 versus 30) and with the average age of 26.44 represent the youngest cluster. even 84.13% of this group of individuals are single. it is interesting to point out that respondents from this cluster mostly stay in hostels when travelling (41.2% stated hostel as the first choice), and the hostel rating they find acceptable is slightly lower than for the other two segments. 4.5. simulation results 4.5.1. share of preference simulation using part-worths of the attribute levels (table 4) as input values, share of preference simulation was carried out. let us assume that three hostels are currently available on the market (scenario 1). table 6 shows the description of these three hostels as well as their share of preferences. in scenario 1, the highest share of preferences goes to hostel 2 (45.41%), while hostel 3 exhibits the lowest share of only 12.56%. a certain number of respondents would not opt for any of the offered hostels, in fact 3.07% of them (table 6). suppose that hostel 1 considers to change policy of cancellation in order to achieve a higher share of preferences and improve its market position. the simulation results show that this action will result in significantly higher share of preferences, reaching 56.26%, which makes this move justified. also, the percentage of respondents who did not opt for any of the offered hostels in the first scenario was lower in the scenario 2. incorporating heterogeneity of travelers’ preferences into the overall hostel performance… 217 table 6. share of preferences simulation location staff atmosphere cleanliness level facilities cancellation policy (refund) share of preference scenario 1 hostel 1 city center friendly party high good nonrefundable 38.97% hostel 2 good connection to city center marvelous homely high good deposit only 45.41% hostel 3 poor connection to city center friendly active high superb deposit only 12.56% none of the above 3.07% scenario 2 hostel 1 city center friendly party high good deposit only 56.26% hostel 2 good connection to city center marvelous homely high good deposit only 31.10% hostel 3 poor connection to city center friendly active high superb deposit only 11.13% none of the above 1.51% similar simulations can be performed for some other changes in hostel offerings, but it is also possible to simulate the share of preferences of a potential hostel that currently does not exist but is planning to run. furthermore, the impact of the newly introduced hostel on the preference share of existing hostels on the market can also be simulated. 4.5.2. overall weighted hostel rating as previously mentioned, a number of web portals offer guests the opportunity to evaluate hostels performance based on their key features. however, most estimates are based on a rating of each of the features individually from the set of features on the likert scale. evaluations of these features are then averaged to get the overall hostel performance score. this can greatly provide the wrong picture because not all features are equally important for all respondents. thus, in this paper we suggest that the relative importance of the attributes are used as weights. this would make the valuations more realistic and would better reflect the real satisfaction of the respondents. let bjk be the respondents’ rating for the jth hostel according to the kth criterion. the weights wk represent the feature importance obtained through dca. thus, the overall weighted score of the jth hostel is calculated as follows: 𝑊𝑅𝑗 = ∑ 𝑊𝑘 𝑏𝑗𝑘 𝐾 𝑘=1 ,    𝑗 = 1, . . . , 𝐽 (6) the advantages of the proposed approach will be presented on the example of four hypothetical hostels (see table 7). for each of the hostels, the ratings are given according to six features. averaged performance scores for all of these hostels is equal to 4, and it seems that they are equally good perceived (well-seen) by guests, but the question is whether it is really so. namely, some hostels with qualitatively different features achieve the same average score, indicating a weak discriminatory power of average-based approach. table 7 also presents the overall ratings obtained by the approach we proposed, which implies multiplying the ratings assigned to an individual hostel feature by the value of the feature importance to produce a weighted score. the results show that m. kuzmanović et al./decis. mak. appl. manag. eng. 4 (2) (2021) 200-224 218 there are clear differences in overall ratings of the hostel. it is especially interesting to compare hostel 3 and hostel 4 scores, where both hostels are rated by the same set of ratings, but these are assigned to different criteria. although the average scores are the same, the difference in the overall ratings is high and reaches a value of almost one. table 7. averaged versus weighted averaged overall hostel ratings features feature weight hostel 1 hostel 2 hostel 3 hostel 4 location 0.340 4 4 3 5 staff 0.064 4 4 5 3 atmosphere 0.034 4 3 5 3 cleanliness level 0.353 3 4 3 5 facilities 0.128 5 4 5 3 cancellation policy (refund) 0.081 4 5 3 5 average 4 4 4 4 weighted average 3.78 4.05 3.45 4.55 5. discussion and conclusions 5.1. key findings of the study bearing in mind that the hostel market is rapidly developing by increasing its diversity, and that it attracts a growing number of guests, usually millennials, but also investors, it is not surprising that the number of studies related to this market is also on the rise. in order for a hostel business to be sustainable, all three goals, namely economic, social and environmental, need to be addressed. as both the social and economic dimensions of a sustainable hostel business are conditioned by guest satisfaction, it is clear that understanding what guests expect to experience in a hostel is the best way to achieve it. to reveal travelers' habits and preferences for the factors they consider when choosing hostel accommodation, we conducted a survey. in a sample of 218 respondents, hostels occupy a significant second place when it comes to the type of accommodation they most stay in, which is in line with other studies. it was also found that most respondents book accommodation online, which is consistent with phocusvright report (2018). however, airbnb and other p2p platforms have been shown to be a threat to the hostel market, primarily because of the price, location and convenience of accommodation they offer, but also because they strongly promote social, economic and environmental sustainability (gössling & michael, 2019) . there are a number of research studies that have explored the motives of the traveler and the impact of different hostel characteristics on the choice of hostels, but they are mostly based on direct assessment of these characteristics individually. fewer research was done on the topic of how much users are willing to trade a certain hostel feature for another (kim & park, 2017). in this study, we used discrete choice analysis, a method that measures guests' individual preferences allowing them to make trade-offs between attributes such as cleanliness, location, staff, atmosphere, facilities, and cancellation policy. when it comes to the most important attributes, cleanliness and location, our findings are in line with the findings of many other studies as stated in the literature review. however, although all of the listed studies have also identified these two incorporating heterogeneity of travelers’ preferences into the overall hostel performance… 219 factors as very important, our study provides a clearer distinction of attributes in terms of the intensity of their influence on the choice of hostel. when it comes to facilities, numerous studies have highlighted the importance of this attribute for guest satisfaction. in our study, this attribute is third ranked, but its relative importance is almost three times lower than the importance of the two best ranked attributes. furthermore, other studies did not consider the cancelation policy as the attribute that affect individuals’ choice of hostel. our results have shown that this attribute is, on average, more important than the attributes of staff and atmosphere. it may be surprising that the staff is considerably less important attribute than cleanliness and location, even by about five times, as it contradicts the findings presented in (lima and vicente, 2017; martins et al., 2018). similarly, the atmosphere has less impact on hostel choice than cleanliness and location, for more than six times. the results also indicate that hostel guests are not a homogeneous group, pointing to the importance of revealing preferences of subgroups of travelers. the results of post hoc clustering clearly indicated differences in the three groups of respondents, which were not noticeable in an a priori segmentation. although cleanliness and location are high-ranking attributes in all three clusters, their impact on overall preferences is not the same. actually, some attributes that were significantly less important on the aggregate level, proved to be more important in some clusters. so, facilities and atmosphere attributes, with a relative importance of 19% each, are third-ranked ones among cleanliness sticklers and party seekers, respectively. also, the cancellation policy attribute ranks third in the location demanders cluster, with a relative importance of 10%. by performing simulations, we have shown that the results can be used for the decision making under uncertainty, in particular, to determine how the potential market share will change by changing some of the attribute levels, whether its own or those of the competitors. 5.2. implications, limitations and future work the theoretical implications of our study are reflected in the enrichment of the literature related to hostel stay motives and factors affecting guest preferences and satisfaction. as the impact of a hostel as a business model on the sustainability of the economy and society is reflected in higher consumption, greater employability, economic development of cities, regions, and even countries, the study findings have important implications for various stakeholders in tourism industry as well. study results may be useful for hostels to improve their service and increase their competitive advantage, but also for those who are considering investing in a sustainable hostel in the future. for example, in order to improve environmental performance, hostels should provide eco-friendly facilities to the first cluster (cleanliness sticklers) in addition to high cleanliness. given that the same cluster prefers an active atmosphere and agree to a location outside the city center, the possibility to rent a bike for free is another option that could have a positive effect on the environment. the satisfaction of these respondents would be even greater with friendly staff who could promote various social and ecological activities such as local sightseeing/attractions within walking distance. similar applies to the cluster of location demanders. for the third cluster (party seekers), hostels could organize green parties that could also become a trademark of hostel. the managers of hostel booking websites could use the results of this study to create a proper list of factors that guests could evaluate, but also to better understand the influences of those factors on the overall guests' perception of a particular hostel. because hostel online reviews can be used for sustainable strategic marketing m. kuzmanović et al./decis. mak. appl. manag. eng. 4 (2) (2021) 200-224 220 decisions against competitors it is important that guests have the opportunity to evaluate the hostel's environmental performance as well. however, our study has its limitations. first, not all aspects of a sustainable business model, such as the environmental one, have been considered. second, because of the sampling method used, the data collected may be biased and reflect the preferences of the study participants rather than the entire population. as the popularity of hostels is growing among family people and is increasingly being used for accommodation on business trips, to make the results more credible, the sample should also include those respondents who less use modern technology, social networking sites or forums, but also older respondents. third, the size of the sample itself could be more extensive by including significantly more respondents outside the european continent. regardless of the limitations, this study can be a good starting point for upgrading and drawing more general conclusions. future research could be directed towards conducting a survey that would cover a broader population and address all three pillars of a sustainability: economic, social and environmental. in this way, it would be possible to identify the respondents' preferences for each of the pillars, as well as to determine trade-offs that both businesses and tourists are willing to make in terms of giving up one sustainability component for another. author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research received no external funding. conflicts of interest: the authors declare no conflicts of interest references amblee, n. 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(2015). understanding the heterogeneity of generation y's preferences for travelling: a conjoint analysis approach. international journal of tourism research, 17(5), 482-491., doi:10.1002/jtr.2015. world travel and tourism council (2018). travel and tourism economic impact 2018: world.; world travel and tourism council: london. m. kuzmanović et al./decis. mak. appl. manag. eng. 4 (2) (2021) 200-224 224 © 2021 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 4, issue 1, 2021, pp. 174-193. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame2104174s logistics performances of gulf cooperation council’s countries in global supply chains ilija stojanović1 and adis puška2 1 college of business studies, al ghurair university, dubai, united arab emirates 2 institutes for scientific research and development, brčko district, bosnia and herzegovina received: 10 january 2021; accepted: 7 march 2021; available online: 13 march 2021. original scientific paper abstract: regional integration into the gulf cooperation council has enabled respective countries to effectively participate in global supply chains. to ensure effective integration of this region into global supply chains, logistics operations are a very important determinant. the aim of this study was to assess logistical performances of gcc countries, and to identify which country has the best conditions for establishing a regional logistic center. for this study, we used relevant data from logistics performance index (lpi) developed by the world bank. the research was conducted using a hybrid multi-criteria approach based on the critic and mabac methods. the findings of this study indicate that the united arab emirates has the best conditions for establishing a regional logistics center. this study also releveled the areas of logistics in which other gcc countries should make an improvement to improve their logistical performance. keywords: logistics center; logistics performance; global supply chains; gcc countries; multi-criteria analysis. 1. introduction global competitive pressure is forcing countries to strengthen their position in the world market through regional integration. with trade and customs agreements individual countries have been enabled to improve their competitive position within a single regional market towards other regions and countries globally. this was also the incentive for the gulf countries to establish a cooperation council for the arab states of the gulf in 1981, also known as gulf cooperation council (gcc), composed by bahrain, kuwait, oman, qatar, saudi arabia, and the united arab emirates. gulf integration has enabled facilitation of the movement of production, removing trade barriers, and coordinating economic policies, extending the size of the market for the estimated 35.65 million inhabitants who live in this region (fernandes and rodrigues, logistics performances of gulf cooperation council’s countries in global supply chains 175 2009). moreover, it has created the preconditions for establishment of supply chains with the aim of joint gcc exposure on the global market. according to the statistical centre for the cooperation council for the arab countries of the gulf (gcc-stat), total export of gcc countries was around 652 billion of usd in 2018 and rising. well known fact is that oil export is one of the key trade operations, but many other products take an important role in export activities of the gcc region. having this in mind, durugbo et al. (2020) pointed out the strategic global importance for supply chains for these countries. these scholars found that supply chains in the gcc region confront 3 main complexity management challenges including “strategically selecting and integrating network resources’, ‘reliably contracting and delivering high-quality solutions’, and ‘cost effectively controlling and financing operational expansions” (durugbo et al., 2020, p.13). they also proposed to gcc-based companies to work closely in enabling optimization of their export activities to maximize competitiveness and minimize operational risks and uncertainty. to create an effective supply chain, appropriate logistics operations are crucial. according to christopher (2017, p.4), “effective logistics and supply chain management can provide a major source of competitive advantage”. having in mind global market game and the necessity of gcc countries to be included effectively into global supply chains, we focused our academic curiosity to logistical performances of gcc countries. our main goal of this study is to see which gcc country provides the best conditions in terms of logistics to enable the gcc region to be effectively included into global supply chains. this study provides insight into areas of logistics for each gcc country where improvement is needed to enable more effective logistics operations. the selection of the logistics center was done using the logistics performance index (lpi) data developed by the world bank for the time periods 2012, 2014, 2016 and 2018. with the purpose of ranking gcc countries from their logistics performance, a combination of critic (criteria importance through intercriteria correlation) and mabac (multi-attributive border approximation area comparison) were applied. the critic method was used to determine the weight of the criteria in an objective way, while the mabac method was used to rank these countries. this approach allowed determination which of the gcc countries has the best characteristics in lpi over different time periods. this approach addressed the following questions: a) can combinations of mcda methods be used when choosing a logistics center? b) does the ranking of the gcc countries differ throughout different time periods? c) which of the gcc countries has the best lpi characteristics to be proposed for a joint logistics center? the contribution of this approach one can be found in the new way of ranking countries for other regions to determine those with the best lpi characteristics. thus, this study has paved the way for future investigation with the similar approach in other regions with the aim of selecting the logistics center's best location according to the country's logistics performances. in addition to the introduction section, this paper is organized as follows. section two is intended for literature review. in the third section the research methodology is explained, and the mcda methods to be used in this study. the fourth section is intended for research results and for the analysis of the obtained results. the fifth stojanović and puška/decis. mak. appl. manag. eng. 4 (1) (2021) 174-193 176 section is focused to discuss the obtained results, while the sixth selection is intended to conclude the obtained research results. 2. literature review global competition is a major characteristic of today’s marketplace where the race for a better position is constant. this is not only the race between companies, but also between countries that need constantly to evaluate their competitive position (önden at al., 2018). modern time is also characterized with dramatic increase in trade across borders (akkermans at al., 1999). according to kishore and padmanabhan (2016), globalization and global competition indicate the great importance of the logistics industry. however, klassen and whybark (1994), based on their study, found that the complexity of global logistics is one of key barriers to the effective management of international operations. with globalization, global supply chains become highly significant. „global supply chains are a mechanism by which firms can achieve a competitive advantage” (sundarakani at al., 2012, p.2). according to reyes et al. (2002), perceptive firms are increasingly pursuing global supply chain operations to reduce costs. some scholars identify some differences between logistics and supply chains (larson and halldorsson, 2004). according to memedovic et al. (2008, p. 355), “logistics commonly refers to organizing and coordinating the movements of material inputs, final goods and their distribution.” pham et al. (2017) argue that logistics is an important element of supply chains, putting focus especially on logistics centers. we especially emphasize the claim by stević et al. (2015), that logistical centers are key elements of logistical network. according to their opinion, the entire logistics system relies on logistics centers that have integrative function within logistics systems. zaralı and yazgan (2016) highlighted that logistics centers have key roles in streaming transport operation at national and international level; the selection of their position is of particular importance for their effectiveness and efficiency. one of the most rapid developing world regions by increasing worldwide circulation of commodities is the region of gulf cooperation council (gcc) countries which become a central node in global trade (ziadah, 2018). this region is composed by six araab countries: the kingdoms of bahrain and saudi arabia, the sultanate of oman, the states of kuwait and qatar, and the united arab emirates. durugbo et al. (2020) estimated that this region accounts for around 30% of the globally known oil reserves. the gcc region also has strategic geographic position along the asia–europe trade route. according to ziadah (2018), authorities in this region have recognized the possibility of economic diversification by making huge investments into logistics infrastructure: maritime ports, roads, rail, airports and logistics cities, and yet is to come from gcc development plans. fernandes and rodrigues (2009) particularly emphasized the importance of special economic zones that have been established as an instrument to boost employment, export, and foreign exchange. according to them, countries within this region are positioning themselves to be logistic hubs by strengthening transport, and connectivity, and this can lead to attracting foreign investments. durugbo et al. (2020) provided great insight into the existing literature of the supply chain management of the gcc region and found high levels of complexity and uncertainty within this regional business environment. one of the complexities found by these authors is related to strategically selecting and integrating network resources logistics performances of gulf cooperation council’s countries in global supply chains 177 within the gcc region, focusing attention on the views of multinational companies towards regional supply chains. according to these authors, those multinational companies located in the gcc region are very focused on regional supply chains. according to memedovic et al. (2008), oil-producing countries, with exception of the united arab emirates and bahrain, perform below their potential and their logistics systems usually focus on their main export commodities rather than focusing on diversification on trade logistics. these authors pointed to an example of dubai ports world that has become one of the most important global port operators, operating 42 port terminals in 27 countries. memedovic at al. (2008) also pointed out that countries with better logistics capabilities can attract more foreign direct investments, decrease transaction costs, diversify export structure, and have higher growth. very important issues in managing logistics operations arise among scholars. one of these issues, as stated by akkermans et al. (1999) is related to managing good flows between facilities in a chain of operations, thus putting focus on the importance of coordinated planning approach that can reduce costs. several scholars warned of the need to have an appropriate coordination in decision making on the design of international facility networks (scully and fawcett, 1993; meijboom and vos, 1997). coe at al. (2004) argued that with establishment of the global commodity chain approach, the importance of regions in economic activities arises. önden et al. (2018) argued that the location of the logistics centers is a key element of the transport system and location decisions should be done strategically. otherwise, opposite decisions could increase costs and create transport bottlenecks. however, due to undoubted advantages for the economy, regional authorities want their region to be considered for logistical centers and this could lead to rising logistics costs, increasing travel distances by trucks, and lacking multi-modal transportation possibilities. after analyzing the situation in the gcc region, ziadah (2018) found a large degree of duplication in port infrastructure in this part of the world. thus, analyzing which country in the gcc region provides the highest benefit for the economy of the region is fully justified and we are going to do this with this study. this is especially important due to the necessity to build long-term relationships between regions, which are according to li et al. (2011) critical factor to establish a successful logistics system. complex system of global value chains is dependent on efficient logistics (memedovic et al., 2008). thus, location of logistics centers has become an imperative of logistics and supply chain management because it contributes to the efficiency of supply chain (rao et al., 2015). memedovic et al. (2008) argued that characteristics of each supply chain logistics will affect decisions about the advantages and disadvantages of different locations, and especially costs, transport access, business environment for round-the-clock operations leads to a variety of location strategies. having in mind a trend of moving production in different global regions, this has affected changes in global distribution systems. according to coe et al. (2004), preferred locations for building large distribution centers became gateways and corridors with access to traditional trade gateways and to large consumer markets. based on this notion of the importance of location, these scholars highlighted the importance of enabling competitive logistics services at low rates. fernandes and rodrigues (2009) also argued that staying competitive for companies implies a strategy by which parts of the value chain are in countries where they can take advantage of lower costs due to location factors. at the same time, according to these scholars, companies search for multimodal hubs to optimize the cost efficiencies of sea freight with that of quicker but expensive air freight. stojanović and puška/decis. mak. appl. manag. eng. 4 (1) (2021) 174-193 178 martí et al. (2017) highlighted that international trade has been affected by increased competitiveness of lagged regions that in the past did not play such an important role in the world. thus, they believe that only those countries prepared to implement the advances that commercial globalization requires can benefit from improved logistics performance. according to chow et al. (1994), measurement of performance must recognize the role of an organization in a supply chain. önden et al. (2018) pointed out that logistics performance is an accelerator of the competitiveness of a country and thus, they need to evaluate their position using various indicators including logistics performance index (lpi). memedovic et al. (2008) indicated the usefulness of lpi as a composite index which shows that building the logistics capacity to connect firms, suppliers and consumers is even more important today than costs. thus, within logistics performance analysis for gcc countries we will use lpi data. biswas and anand (2020) performed a very interesting comparative analysis of the g7 and brics countries on the basis of logistical competitiveness, and they expanded the criteria by using the adoption of information and communication technologies and co2 intensity in addition to the lpi criteria. a very good insight in the literature dealing with the issue of selecting the best location of the logistics center was given by uyanik et al. (2018) who analyzed 35 different studies with the location selection problem. they found that different methods were applied by different authors, but what they found as common ground across these studies is that the selection decision was based on a different number of criteria by using multi-criteria decision-making models. kuo (2011) used ten selection criteria including port rate, import/export volume, location resistance, extension transportation convenience, transshipment time, one stop service, information abilities, port & warehouse facilities, port operation system, and density of shipping line, while ou and chou (2009) used six factors named valued added service, transportation and distribution systems, market potential, environment, infrastructure and culture to identify international distribution center from a foreign market perspective. elevli et al. (2014) believed that decision makers for selecting locations of logistic centers prefer to pursue more than one goal or consider more than one factor. this is where the justification for use of multi-criteria decision analysis with fuzzy logic lies. very significant studies can be found in the literature that deal with the problem of selection of logistics centers using multi-criteria decision analysis with fuzzy logic. kishore and padmanabhan (2016) argued that the fuzzy approach is capable of capturing vagueness associated with subjective perception of decision makers. li et al. (2011) analyzed among 15 regional logistics center cities and thirteen criteria to identify logistics center location, and they used linguistic variables instead of numerical values in this study applying fuzzy-set theory. these scholars believed that linguistic variables are more appropriate when performance values cannot be expressed with numerical values. elevli (2014) used fuzzy preference ranking organization method for enrichment evaluation. this method combined the concept of fuzzy sets to represent uncertain information with the promethee. kazançoğlu et al. (2019) applied sustainability benchmarking principles by using hybrid multicriteria decision-making method, fuzzy ahp and promethee methods in the selection process. sun et al. (2019) explored location problems in a three-stage logistics network that consists of suppliers, logistics centers, and customers and they put focus on the environmental sustainability. for their study, they applied two fuzzy mixed integer linear programming models. phamb et al. (2017) developed a benchmarking framework for selection of logistics centers by applying a hybrid of the logistics performances of gulf cooperation council’s countries in global supply chains 179 fuzzy method and the technique for order of preference by similarity to ideal solution (topsis). they found that freight demand, closeness to market, production area, customers, and transportation costs are most important factors for selection. biswas and anand (2020) applied the piv (proximity indexed value) method and the topsis method to perform a comparative analysis of the g7 and brics countries. with their study, wang et al. (2010) put their focus in selection of locations that maximize profits and minimize costs. they established a fuzzy multiple criteria decision-making model based on fuzzy ahp for the ldc assessment. few years later, wang et al. (2014) focused on the consistency and the historical assessments accuracy by introducing priority of consistency and historical assessments accuracy mechanism into a fuzzy multi-criteria decision making approach. focusing on several criteria, such as proximities to highway, railway, airports, and seaports; volume of international trade; total population; and handling capabilities of the ports, önden et al. (2018) combined the fuzzy analytic hierarchy process, spatial statistics and analysis approaches to evaluate suitable level for logistics center. one of most interesting studies we found in the literature is delivered by stević et al. (2015) who searched for the best location of logistics centers throughout the state of different facts important for selection of the best location. they used the ahp method of multi-criteria analysis. our research problem is focused on analyzing logistics performance of gcc countries to identify which country can provide the best logistical conditions to make this region even stronger within global supply chains. 3. methodology the identification of the most suitable location for the logistic center in this study was conducted at the first step with the analysis of logistic performances of selected countries. in this study, the identification of logistic center location was performed using a hybrid multi-criteria approach based on the critic and mabac methods. the selected countries that were examined under this study included 6 countries from the gulf cooperation council: bahrain, kuwait, oman, qatar, saudi arabia, and the united arab emirates (uae). to assess logistical performances of selected gcc countries, we used relevant data from the logistics performance index (lpi) developed by the world bank. based on the lpi, the following indicators were taken into consideration: customs, infrastructure, services, timeliness, tracking and tracing and international shipments (table 1). during the research, the following steps were conducted: 1. data collection 2. forming of decision matrix 3. determining weights for criteria 4. ranking of gcc countries 5. analysis of the results stojanović and puška/decis. mak. appl. manag. eng. 4 (1) (2021) 174-193 180 table 1. core components of lpi component definition c1 customs the efficiency of customs and border management clearing. c2 infrastructure the quality of trade and transport infrastructure. c3 services the competence and quality of logistics services. c4 timeliness the frequency with which shipments reach consignees within expected delivery times. c5 tracking and tracing the ability to track and trace consignments. c6 international shipments the ease of arranging competitively priced shipments source: world bank the first step of this research was data collection. for this study we collected data from the logistic performance indicators (lpi) from the world bank available from the website: lpi.worldbank.org. to obtain the most complete data on lpi trends for gcc countries, data for the years 2012, 2014, 2016 and 2018 were used. after the data were collected, a decision matrix is formed for the specified periods and selected countries. having in mind that we used data from 4 different periods, we formed five decision matrices to enable further analysis. these decision matrices are the basis for implementing methods for multi-criteria data analysis. the next step was to determine the importance of the used criteria. before we made rankings of selected countries, it was necessary to determine the importance of the criteria. in order to eliminate subjectivity in ranking of selected countries, the critic method was used to determine the weights of the criteria. the critic method has an approach to objectively calculate weight values based on standard deviation values and correlation coefficients. since we used data for four time periods, the weights for each of these decision matrices were calculated and reconciled. adjustment was done by applying the average value of the weights. these average values were used to rank gcc countries in terms of logistic center selection. after the initial decision matrix was formed and the weights of the criteria were calculated, the gcc countries were ranked in relation to the lpi. ranking was done using the mabac method. the results in previous studies obtained using the mabac method have shown that this method can be used as a support in decision making (božanić, et al., 2016) and in the ranking of alternatives (pamučar and ćirović, 2015). first, the data were normalized, then the normalized decision matrix was weighted, and the determination of the approximate border area matrix was calculated. following these steps, the alternatives were placed in relation to the value of the approximate border area, following with ranking of the alternatives. more details about the critic and mabac method are shown below. the ranking of alternatives was done for all selected time periods. after the selected countries were ranked, it was necessary to analyze the research results. the analysis of research results was applied in two ways. first, the results obtained by the mabac method were analyzed and compared with the results obtained by applying other methods of multi-criteria analysis. after confirming the results obtained by the mabac method, a sensitivity analysis was conducted. sensitivity analysis examines the extent to which a criterion has an impact on the ranking of alternatives. sensitivity analysis and comparison of results was performed for all time periods to get a complete insight of the lpi performance of gcc countries. logistics performances of gulf cooperation council’s countries in global supply chains 181 3.1. critic method the critic method was developed by diakoulaki, et al. (1995). this method serves to determine the objective values of the criteria weight, which includes the intensity of contrast and conflict that is contained in the structure of the decision problem (puška, et al., 2018). to determine the contrast of criteria, the standard deviations of the standardized values of the variants per column are used, as well as the correlation coefficients of all pairs of columns. the steps in implementing the critic method are as follows: step 1. defuzzification of the initial decision matrix. before the other steps of the critic method are performed, fuzzy numbers need to be converted to numerical values (kiani mavi, et al., 2016). defuzzied is performed using the following expression:     321 4 6 1~ mxmmmp  (1) where m1 is the first value of fuzzy number, m2 is the second value of fuzzy number and m3 is the third value of fuzzy number. step 2. normalization of the defuzzied initial decision matrix using the following expressions: for criteria to be maximized: (2) for criteria to be minimized: (3) where: x*j – the maximum value of the feature for a given criterion, x**j – the minimum value of the feature for a given criterion. step 3. calculation of the values of the standard deviation and the symmetric linear correlation matrix of all pairs per column. step 4. determination of the amount of information using the following expression. ,m j rc m k jkjj 1 )1( 1     (4) where j  standard deviation of criteria and jk r correlation coefficient for criteria. step 5. calculation of the final values using the following expression: 1 j j m j j c w c    (5) 3.2. mabac method the mabac method was developed by pamučar and ćirović (2015). the basic assumption of the mabac method is reflected in the definition of the distance of the alternative from the boundary approximate domain. the boundary approximate area represents the average value for all alternatives. if the alternative is above that value, its value will be positive and vice versa. the mabac method consists of the several steps. step 1. construct the initial decision matrix. as a first step, m alternatives are evaluated according to n criteria. the alternatives are represented with vectors ai = xi1, xi2,..., xin , where xij is the value of i alternative by j criterion (i  1,2,...,m; j = 1,2,...,n). *** ** jj rij ij xx xx r    *** ** 1 jj rij ij xx xx r    stojanović and puška/decis. mak. appl. manag. eng. 4 (1) (2021) 174-193 182 step 2. normalization of the elements of the initial matrix. the elements of the initial decision matrix are normalized by the following expressions: for benefit-type criteria: 𝑡𝑖𝑗 = 𝑥𝑖𝑗−𝑥𝑖 − 𝑥𝑖 +−𝑥𝑖 − (6) for cost-type criteria: 𝑡𝑖𝑗 = 𝑥𝑖𝑗−𝑥𝑖 + 𝑥𝑖 −−𝑥𝑖 + (7) where is 𝑥𝑖 − represents minimum values of the left distribution offuzzy numbers of the observed criterion by alternatives, and 𝑥𝑖 + represent the maximum values of the right distribution of fuzzy numbers of the observed criterion by alternatives. step 3. calculation of the weighted matrix (v) elements (božanić, et al, 2019). �̃�𝑖𝑗 = 𝑤𝑖 ∙ �̃�𝑖𝑗 + 𝑤𝑖 (8) where 𝑤𝑖 represents the weighted coefficients of the criterion. step 4. determination of the approximate border area matrix (g). 𝑔 = (∏ �̃�𝑖𝑗 𝑚 𝑗=1 ) 1/𝑚 (9) where m represents total number of alternatives step 5. calculation of the matrix elements of alternatives distance from the border approximate area. the distance of the alternatives from the border approximate area (�̃�𝑖𝑗 ) is defined as the difference between the weighted matrix elements (v) and the values of the border approximate areas (g). �̃� = �̃� − �̃� (10) now, border approximation area value for each criteria function serves as reference point/benchmark value for criteria-wise performance of an alternative 𝐴𝑖 . each individual candidate will belong to three different areas namely, the border approximation area (𝐺), upper approximation area (𝐺 +), and lower approximation area (𝐺 −). the ideal alternative (𝐴𝑖 +) can be found in the upper approximation area (𝐺 +) whereas the lower approximation area (𝐺 −) contains the anti-ideal alternative (𝐴𝑖 −) (božanić, et al., 2016). �̃�𝑖 ∈ { �̃� + 𝑖𝑓 �̃�𝑖𝑗 > 0 𝐺 ̃𝑖𝑓 �̃�𝑖𝑗 = 0 �̃� − 𝑖𝑓 �̃�𝑖𝑗 < 0 (11) for an alternative 𝐴𝑖 to be chosen as the best from the set, it is necessary for it to belong, by as many as possible criteria, to the upper approximate area (�̃� +). the higher the value �̃�𝑖 ∈ �̃� + indicates that the alternative is closer to the ideal alternative, while the lower the value q �̃�𝑖 ∈ �̃� − indicates that the alternative is closer to the anti-ideal alternative. step 6 ranking of alternatives. the calculation of the values of the criteria functions by alternatives is obtained as the sum of the distance of alternatives from the border approximate areas (�̃�𝑖 ). by summing up the matrix �̃� elements per rows, the final values of the criteria function of alternatives are obtained �̃�𝑖 = ∑ �̃�𝑖𝑗 , 𝑗 = 1,2, … , 𝑛, 𝑖 = 1,2, … , 𝑚 𝑛 𝑗=1 (12) 4. results before we ranked gcc countries according to lpi indicators, it was necessary to form an initial decision matrix. the initial decision matrix is presented in table 2. the logistics performances of gulf cooperation council’s countries in global supply chains 183 lpi data for gcc countries are comparable from the most recent data to the data from previous years. after the initial decision matrices for the observed time periods have been formed (table 2), the steps of the critic and mabac methods were performed. the example of data from 2018 explains the way in which gcc countries are ranked. table 2. lpi indicators for gcc countries in the period 2010-2018 2018 2016 country c1 c2 c3 c4 c5 c6 c1 c2 c3 c4 c5 c6 bahrain 2.67 2.72 3.02 2.86 3.01 3.29 3.14 3.10 3.33 3.38 3.32 3.58 kuwait 2.73 3.02 2.63 2.80 2.66 3.37 2.83 2.92 3.62 2.79 3.16 3.51 oman 2.87 3.16 3.30 3.05 2.97 3.80 2.76 3.44 3.35 3.26 3.09 3.50 qatar 3.00 3.38 3.75 3.42 3.56 3.70 3.55 3.57 3.58 3.54 3.50 3.83 saudi arabia 2.66 3.11 2.99 2.86 3.17 3.30 2.69 3.24 3.23 3.00 3.25 3.53 uae 3.63 4.02 3.85 3.92 3.96 4.38 3.84 4.07 3.89 3.82 3.91 4.13 2014 2012 country c1 c2 c3 c4 c5 c6 c1 c2 c3 c4 c5 c6 bahrain 3.29 3.04 3.04 3.04 3.29 2.80 2.67 3.08 2.83 2.94 3.42 3.42 kuwait 2.69 3.16 2.76 2.96 3.16 3.39 2.73 2.82 2.68 2.68 2.98 3.11 oman 2.63 2.88 3.41 2.84 2.84 3.29 3.10 2.96 2.78 2.73 2.59 3.17 qatar 3.21 3.44 3.55 3.55 3.47 3.87 3.12 3.23 2.88 3.25 3.50 4.00 saudi arabia 2.86 3.34 2.93 3.11 3.15 3.55 2.79 3.22 3.10 2.99 3.21 3.76 uae 3.42 3.70 3.20 3.50 3.57 3.92 3.61 3.84 3.59 3.74 3.81 4.10 since the critic and mabac methods use the same data normalization, the first step is the same for both methods and represents the normalization of the initial decision matrix (table 3). all criteria are of benefit type and expression 2 or 6 is used. after this step, the specific steps of the critic and the mabac methods are applied. since it is necessary to calculate the weights of the criteria at the first place, the steps in the critic method are explained first (table 4). after the normalization, the values of standard deviation and correlation coefficient are calculated. after that step, the amount of information and the weights of the criteria are determined. table 3. normalized decision matrix for lpi 2018 c1 c2 c3 c4 c5 c6 bahrain 0.01 0.00 0.32 0.05 0.27 0.00 kuwait 0.07 0.23 0.00 0.00 0.00 0.07 oman 0.22 0.34 0.55 0.22 0.24 0.47 qatar 0.35 0.51 0.92 0.55 0.69 0.38 saudi arabia 0.00 0.30 0.30 0.05 0.39 0.01 uae 1.00 1.00 1.00 1.00 1.00 1.00 stojanović and puška/decis. mak. appl. manag. eng. 4 (1) (2021) 174-193 184 table 4. steps in critic method c1 c2 c3 c4 c5 c6 st.dev 0.380 0.339 0.387 0.393 0.358 0.387 c1 c2 c3 c4 c5 c6 correlation 1.000 0.955 0.811 0.972 0.858 0.965 0.955 1.000 0.792 0.942 0.861 0.925 0.811 0.792 1.000 0.915 0.917 0.822 0.972 0.942 0.915 1.000 0.935 0.933 0.858 0.861 0.917 0.935 1.000 0.784 0.965 0.925 0.822 0.933 0.784 1.000 c1 c2 c3 c4 c5 c6 0.167 0.178 0.288 0.119 0.231 0.221 c1 c2 c3 c4 c5 c6 w 0.138 0.148 0.240 0.099 0.192 0.183 the same procedure is performed for other time periods, and individual weights are determined. to obtain one weight for the observed periods, the average weights are calculated. based on the obtained weights, one can conclude that criterion c3 has the highest importance (w = 0.238), while criterion c4 has the least importance (w = 0.120). table 5. weights of criteria by observed time periods c1 c2 c3 c4 c5 c6 2012 0.265 0.091 0.147 0.090 0.215 0.191 2014 0.189 0.124 0.285 0.103 0.124 0.174 2016 0.136 0.179 0.279 0.187 0.119 0.101 2018 0.138 0.148 0.240 0.099 0.192 0.183 w 0.182 0.136 0.238 0.120 0.163 0.162 after the weights have been calculated, the steps of the mabac method are applied and the gcc countries are ranked according to the lpi indicators. after the initial decision matrices are normalized (table 3), this matrix is aggravated (expression 8). after this, the average value of the criteria is calculated, which represents the expression of determination of the approximate border area matrix. the geometric mean is used here. the next step is focused to determine the distance of the alternatives from the arithmetic mean (table 6) and to calculate the sum of these values. based on the value, the ranking of alternatives is determined. the best alternative is the one that has the greatest value of �̃�𝑖 and vice versa. the obtained results have shown that the uae has the best lpi indicators, followed by qatar, while kuwait has the worst lpi indicators. in the same way, the calculation of the value of the mabac method and the ranking of orders for the observed time periods is performed.    m k jkjj rc 1 )1( logistics performances of gulf cooperation council’s countries in global supply chains 185 table 6. alternatives distance and the result of mabac method c1 c2 c3 c4 c5 c6 �̃�𝑖 rank bahrain -0.020 -0.036 -0.015 -0.013 -0.005 -0.028 -0.117 5 kuwait -0.009 -0.005 -0.091 -0.019 -0.049 -0.016 -0.188 6 oman 0.017 0.010 0.039 0.007 -0.010 0.048 0.113 3 qatar 0.042 0.033 0.127 0.047 0.064 0.034 0.346 2 saudi arabia -0.022 0.005 -0.021 -0.013 0.015 -0.026 -0.062 4 uae 0.160 0.100 0.146 0.100 0.114 0.135 0.755 1 the results have shown that the uae has the best lpi indicators for the entire observed time period, while kuwait has the worst lpi indicators for 3 years (2018, 2014, 2012), and saudi arabia has the worst lpi indicators in 2016. based on these findings, one can conclude that the uae has the best logistic indicators, thus suggesting that this country provides the best solution for establishing a logistical center in the gcc region. table 7. ranking of gcc countries using lpi indicators for the period 2012-2018 countries 2018 2016 2014 2012 �̃�𝑖 rank �̃�𝑖 rank �̃�𝑖 rank �̃�𝑖 rank bahrain -0.117 5 0.021 3 -0.003 4 0.003 4 kuwait -0.188 6 -0.063 4 -0.163 6 -0.198 6 oman 0.113 3 -0.072 5 -0.133 5 -0.118 5 qatar 0.346 2 0.350 2 0.479 2 0.264 2 saudi arabia -0.062 4 -0.140 6 0.004 3 0.149 3 uae 0.755 1 0.758 1 0.487 1 0.738 1 5. analysis of the results in order to confirm the results obtained using the combination of mabac and critic methods, the ranking of alternatives for all observed periods was performed using the methods: saw simple additive weighting technique, aras (additive ratio assessment), waspas (weighted aggregated sum product assessment), topsis (technique for order performance by similarity to ideal solution) and marcos (measurement alternatives and ranking according to the compromise solution). this represents the first step in analyzing the results. the second step is to examine the sensitivity analysis against the change in weight criteria. examination of the reliability of the results obtained by applying other methods showed that there is no deviation in the ranking of gcc countries according to lpi indicators. only for the indicators for 2014 there is a small deviation in the use of the topsis method. according to the results of this method, qatar has better results than the uae for this year. this result was to be expected because the results using the mabac method also showed that for this year there is a small difference between these two countries. based on the obtained results, it can be concluded that the results obtained by the mabac method are reliable and verified. stojanović and puška/decis. mak. appl. manag. eng. 4 (1) (2021) 174-193 186 figure 1. results of gcc countries ranking according to lpi indicators for the period 2012-2018 the second step of our results analysis was to conduct a sensitivity analysis. when conducting a sensitivity analysis, it is examined how the change in the weights of subcriteria affects the ranking order of the alternatives (puška et al., 2020). in accordance with this, scenarios were formed: the first scenario does not differentiate between criteria and gives the same importance to all criteria, the other scenarios give one of the criteria five times more importance compared to other criteria. since there are 6 criteria used in this study, 7 scenarios have been formed in order to perform the sensitivity analysis. table 8. scenarios in sensitivity analysis c1 c2 c3 c4 c5 c6 scenario 1 0.1667 0.1667 0.1667 0.1667 0.1667 0.1667 scenario 2 0.5000 0.1000 0.1000 0.1000 0.1000 0.1000 scenario 3 0.1000 0.5000 0.1000 0.1000 0.1000 0.1000 scenario 4 0.1000 0.1000 0.5000 0.1000 0.1000 0.1000 scenario 5 0.1000 0.1000 0.1000 0.5000 0.1000 0.1000 scenario 6 0.1000 0.1000 0.1000 0.1000 0.5000 0.1000 scenario 7 0.1000 0.1000 0.1000 0.1000 0.1000 0.5000 the sensitivity analysis has shown that the results for 2018 are the least sensitive to changes in the weight of the criteria. we found a change in rankings only in scenarios 3 and 7 in which kuwait showed better results compared to bahrain. this 0 2 4 6 bahrain kuwait oman qatar saudi arabia uae 2018 mabac saw aras waspas marcos topsis 0 2 4 6 bahrain kuwait oman qatar saudi arabia uae 2016 mabac saw aras waspas marcos topsis 0 1 2 3 4 5 6 bahrain kuwait oman qatar saudi arabia uae 2014 mabac saw aras waspas marcos topsis 0 1 2 3 4 5 6 bahrain kuwait oman qatar saudi arabia uae 2012 mabac saw aras waspas marcos topsis logistics performances of gulf cooperation council’s countries in global supply chains 187 can be justified with the fact that kuwait has better performances compared to bahrain in the infrastructure and international shipments indicators. the sensitivity analysis for 2016 has shown that the ranking does not change for the uae and qatar, while the ranking is changed for other countries. the highest oscillation we found for kuwait was due to the fifth place in three scenarios, the sixth place in three scenarios, and the third place in one scenario. saudi arabia also took the sixth place in 3 scenarios, and the fifth place in 3 scenarios. oman has ranked as sixth in one scenario. bahrain took the third place in five scenarios and the fourth place in two scenarios. figure 2. results of sensitivity analysis the sensitivity analysis for 2014 has shown the largest oscillations in the rankings. in almost all scenarios, the uae took first place, only in 2014 qatar took first place in two scenarios. the reason for this should be sought in the fact that qatar had better performances in services and timeliness indicators compared to the uae. oman was ranked at the last place in 5 scenarios while bahrain and kuwait were ranked at the last place in one scenario. the sensitivity analysis for 2012 has shown less oscillation in the rankings. saudi arabia had a better performance in services indicator compared to qatar and is ranked better in 4 scenarios. oman had a better performance of customs indicator compared to bahrain, and is ranked better ranked in scenario 2, while kuwait had better performance of tracking and tracing indicator compared to oman and it is ranked better in scenario 6. the sensitivity analysis has shown that the lpi indicators were the most conflicting in 2014 and 2016, while in 2018 they were the least conflicting. this caused the least change in the ranking in sensitivity analysis. this analysis has also shown that the uae 0 1 2 3 4 5 6 1 2 3 4 5 6 7 2018 bahrain kuwait oman qatar saudi arabia uae 0 1 2 3 4 5 6 1 2 3 4 5 6 7 2016 bahrain kuwait oman qatar saudi arabia uae 0 1 2 3 4 5 6 1 2 3 4 5 6 7 2014 bahrain kuwait oman qatar saudi arabia uae 0 1 2 3 4 5 6 1 2 3 4 5 6 7 2012 bahrain kuwait oman qatar saudi arabia uae stojanović and puška/decis. mak. appl. manag. eng. 4 (1) (2021) 174-193 188 has the best performances in lpi compared to other countries and it should be the first choice for a logistical center in this part of the world for trading and exporting goods globally. 6. discussion nowadays, it is fully accepted in theory and practice that logistics centers are extremely important for global supply chains. they represent a significant strategic tool in international trade to reduce costs, reduce duration of supply process, increase sustainability of supply, but also increase overall competitiveness. to achieve these mentioned benefits, decision-makers must find the best locations for logistics centers that will improve competitiveness in global supply chains. this leads to the conclusion that the evaluation of different locations is very important for the effectiveness of the logistics centers. we fully agree with the scholars who argue that selection of logistics center location should be based on the analysis of multiple criteria. uyanik et al. (2018) presented different approaches that can be found in the literature by which different criteria are used in the analysis of the best location for a logistics center. as these scholars found, there is no single approach in the literature about these criteria. in avoiding these challenges about which criteria should be used, we followed what was proposed by martí et al. (2017) who highlighted the benefits of the logistics performance index (lpi). we support their thesis that lpi can help countries to get to know their business partners and to understand what should be adjusted to stay competitive in the logistics sector. our findings have shown that the united arab emirates leads the gcc region when it comes to logistics performance. in this sense, this country can be perceived as the best choice for the location of a logistics center that will allow the gcc region to be connected into a single supply chain. several scholars indicated a highly developed and modern logistics infrastructure in this country (jacobs and hall,2007; memedovic et al., 2008; fernandes and rodrigues, 2009; ziadah, 2018). we are therefore fully convinced that this is one of the most important reasons why the uae is the best ranked with our study. as jacobs and hall (2007), we also believe that dubai is the most important transportation hub in the region. jebel ali port and port rashid, dubai airports, established free zones and regulatory reforms enabled dubai and the uae to become one of the most competitive transportation and logistics hubs in the gcc region. and development plans have yet to showcase new projects in dubai that will further strengthen their competitive position as a logistics hub. while we write this paper, dubai authorities announced full foreign share ownership of business in dubai that will directly influence many foreign companies to locate their business in dubai. thus, this will lead to higher demand for logistics services in dubai and the uae. furthermore, experience in logistics services makes an advantage when it comes to the position of the uae through logistics performance. when we point to experience, we mean the fact that dubai is home to world-renowned logistics and transportation companies such as dp world, emirates airlines jebel ali free zone (jafza), and dubai world central, that is also stressed out by fernandes and rodrigues (2009). in addition to the rapid progress of the united arab emirates in the field of logistics, other countries in the gcc region have made significant achievements. the gcc countries have learned about the importance of logistics and have started to build exceptional logistics infrastructure that gives this region the opportunity to become a logistics performances of gulf cooperation council’s countries in global supply chains 189 logistics hub for trade between east and west. however, certain overlaps accompanied by costly infrastructure investments may lead to some other regions in asia becoming more competitive in providing logistics services for global trade. therefore, the gcc countries region need to improve coordination in the governance process of regional supply chains and to maintain and improve its position in global trade of commodities in the long run. this paper points out the importance of regional cooperation between gcc countries. as coe at al. (2004) highlighted, we also support an integrated conceptual framework for ‘globalizing’ regional development. thus, to integrate gcc as a single supply chain into global trade, the gcc countries should work on global production networks and share regional assets, such as logistics centers. in this regard, we support durugbo et al. (2020) in their appeal to build long and profitable relationships with customers to replace traditionally fragmented approaches in the gcc region. the gulf cooperation council should have a special role in coordinating developmental activities of the region. the gcc administrative bodies should improve their governance capacity that is also indicated by dadush and falcao (2009) and create a joint development program for the region. the importance of joint development programs in the field of transport and logistics such as “new silk road” which should be the longest world road, or “traceca” which is an east-west transport corridor stretching from central asia to europe, was indicated by khassenovakaliyeva et al. (2017) as highly important developmental programs to increase competitiveness of some countries or regions. following similar patterns, the gcc can establish joint cooperation on the establishment of logistics centers that will serve in connecting the region into regional supply chains. dadush and falcao (2009) gave a very good proposal that gulf cooperation council (gcc) must work to improve logistics and reduce non-tariff barriers to trade. fernandes and rodrigues (2009) went even more deeply by proposing policy makers to aggressively pursue the monetary union in the gcc region that certainly can facilitate establishing logistics networks more easily the countries of the gcc region need to accept the fact of growing competitors in the field of logistics, such as singapore or some other asian countries pointed by fernandes and rodrigues (2009). thus, they should start to improve regional collaboration in the logistics chain. what some authors such as fernandes and rodrigues (2009) point out, refers to the need to carefully analyze what they called the logistics skill gap amongst the workforce, including high rents and costs of operation in dubai and the uae. despite the remarkable development of modern logistics infrastructure in this country, these issues need to be considered in order to maintain competitiveness. only in this way can this location maintain its long-term logistics performance compared to competitors. furthermore, some scholars such as sundarakani et al. (2012) favored adopting it solutions that can improve the effectiveness of logistics centers, but this should be followed with appropriate education of managers and employees to manage these systems in an efficient way. what we also noticed is that locations that allow multimodal access to transport and logistics are much better positioned in the context of logistics performance. this has already been discussed by some authors (fernandes and rodrigues, 2009; uyanik et al. 2018; kazançoğlu et al. 2019). we have to agree with fernandes and rodrigues (2009) who indicated that dubai represents an excellent world class integrated hub. this is where we find the main justification for the excellent results that dubai and the united arab emirates have achieved through this study. we must not forget certain strategic issues. the countries of the gcc region, as stated by memedovic et al. (2008) are still more focused on export of oil and similar energy resources. therefore, we stojanović and puška/decis. mak. appl. manag. eng. 4 (1) (2021) 174-193 190 support the strategic directions of individual countries, including the united arab emirates, to put focus on other sectors and to develop their logistics capacities. we must not forget the historical fact that the cities located on the main transport routes developed the most. it is noticeable that the countries of the gcc region, especially the united arab emirates, accept this fact and take big steps in the development of logistics infrastructure. but, as pham et al. (2017) suggested, this should be done in a systematic way by having master plans for the development of a logistics center system that will improve the practice of adequate selection and prioritizing of locations adequate for logistics centers. a crucial role at the regional level in the development of these plans and their coordination should be taken by the administrative bodies of the gulf cooperation council. 7. conclusions the selection of an adequate location for a logistics center is one of the most important issues in the field of logistics operations management. in the literature, location selection is largely based on multi-criteria decision models. in our study we used data from the logistics performance index developed by the world bank and applied a hybrid multi-criteria approach based on the critic and mabac methods. among the six gcc countries, we found that the united arab emirates are the best ranked in the observed period from 2012 to 2018. the exceptional logistics infrastructure built in this country certainly contributes to this result. in fact, the entire gcc region is taking big steps in the development of logistics infrastructure. our study showed that kuwait achieves poorer logistics performance compared to other countries in the region. some countries during the observed period had certain oscillations in the movement of logistics performance, such as saudi arabia. this study recommends that, based on logistics performance, the united arab emirates represent a country that can provide the best conditions for location of a regional logistics center that can connect the gcc region more efficiently into global supply chains. the significance of this study can be found in the study findings that indicates to gcc countries which areas should be improved to elevate overall logistics performance. this study did not deal with a detailed analysis of the structure of exports by countries and product types. therefore, transport and other relevant costs related to the inclusion of gcc countries in global supply chains through a single logistics center were not considered, which is one of the limitations of this study. furthermore, this study focused on the logistics performances of individual gcc countries to find which country provides the best logistical conditions but did not search for suitable locations for logistics centers in these countries. based on these limitations of the study, new areas are opened for future very interesting research endeavors. in future research, it can be examined the role of lpi in the competitiveness of individual countries and determined how important a particular lpi criterion is for the competitiveness. focus of future research can be also on other criteria and making decisions not with lpi only. however, the aim of this paper was to examine the trend of lpi for gcc countries to perceive which location provide best logistics performance for establishing logistics center. in addition, it is possible to consider other fuzzy methods when determining logistics centers and to establish hybrid methods. this research provides basic postulates for determining the location for logistics centers. logistics performances of gulf cooperation council’s countries in global supply chains 191 author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research received no external funding. conflicts of interest: the authors declare that there is no conflict of interest. references akkermans, h., bogerd, p., & vos, b. 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(2018). constructing a logistics space: perspectives from the gulf cooperation council. environment and planning d: society and space, 36(4), 666-682. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 5, issue 1, 2022, pp. 290-308. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame191221060l * corresponding author. e-mail addresses: rs_bulendralimboo@dibru.ac.in (b. limboo), palashdutta@dibru.ac.in (p. dutta) a q-rung orthopair basic probability assignment and its application in medical diagnosis bulendra limboo1* and palash dutta2 1 department of mathematics, dibrugarh university, assam, india received: 3 may 2021; accepted: 1 november 2021; available online: 20 december 2021. original scientific paper abstract: dempster-shafer theory is widely used in decision-making and considered as one of the potential mathematical tools in order to fuse the evidence. however, existing studies in this theory show disadvantage due to conflicting nature of standard evidence set and the combination rule of evidence. in this paper, we have constructed the framework of q-rung evidence set to address the issue of conflicts based on the q-rung fuzzy number due to its more comprehensive range of advantage compared to the other fuzzy or discrete numbers. the proposed q-rung evidence set has the flexibility in assessing a parameter through the q-rung orthopair basic probability assignment consisting of membership and non-membership belief degree. moreover, as the proposed q-rung orthopair basic probability assignment consists of pair of belief degrees, the possibility of conflicts cannot be ignored entirely. in this regard, a new association coefficient measure is introduced where each component of the belief degrees is modified through the weighted average mass technique. this paper uses various concept such as fuzzy soft sets, deng entropy, association coefficient measure and score function for decision-making problem. firstly, to obtain the initial q-rung belief function, we have implemented the intuitionistic fuzzy soft set to assess the parameter of the alternatives and deng entropy to find the uncertainty of the parameters. secondly, the association coefficient measure is used to avoid the conflict through the modified form of evidence. finally, we combined the evidence and found the score value of the intuitionistic fuzzy numbers for the ranking of the alternatives based on the score values of alternatives. this study is validated with the case study in the medical diagnosis problem from the existing paper and compared the ranking of alternatives based on the score function of belief measures of the alternatives. key words: fuzzy soft set, q-rung belief function, association coefficient measure, medical diagnosis. a q-rung orthopair basic probability assignment and its application in medical diagnosis 291 1. introduction decision-making in real life problems depends on the knowledge and information of the parameters, which affects the decision alternatives. in order to model the information with less uncertainty, various theories are discussed in literature to deal with the uncertainty in which probability theory, fuzzy set theory, and evidence theory are a few of them that are very popular in decision-making. fuzzy set theory was first introduced by l. zadeh in 1965, and it was further generalized into intervalvalued fuzzy set (gorzałczany, 1987)., intuitionistic fuzzy set (atanassov, 1986), pythagorean fuzzy set (yager, 2013), q-rung fuzzy set (yager, 2016), and picture fuzzy set (cuong, 2015), etc. the extended theories of fuzzy sets are applied in various directions by integrating with the fuzzy soft set theory, dempster-shafer theory, and multi-criteria decision-making. molodtsov, (1999) proposed soft set theory for the adequate parametrization of the parameters, and fuzzy soft set was later developed by maji et al. (2001). the proposed theories are widely used in decision-making problem. das et al. (2013) studied intuitionistic multi-fuzzy sets in group decision-making. çelik & yamak, (2013) applied fuzzy soft set theory in medical diagnosis using fuzzy arithmetic operations. peng et al. (2015) used intuitionistic fuzzy soft set in the decision-making after the introduction of intuitionistic fuzzy soft (maji, 2001). we have also developed the concept of bellshaped fuzzy soft sets (dutta & limboo, 2017) and applied it in medical diagnosis. das et al. (2017) proposed robust decision-making using intuitionistic fuzzy numbers. krishankumar et al. (2019) proposed the q-rung orthopair fuzzy set with partially known weight in the evaluation of renewable energy sources. hussain et al. (2020) proposed the q‐rung orthopair fuzzy soft average aggregation operator and their application in multicriteria decision‐making. mishra et al. (2021) extended the fuzzy decision-making framework using hesitant fuzzy sets for the drug selection of coronavirus disease (covid-19). dempster-shafer theory has been applied with its discrete basic probability assignment, a generalization of classical probability theory, and was first introduced in the year 1967 (dempster, 1967; shafer, 1976). in literature, this theory offers various applications in decision-making problems of many fields of sensor data fusion (jiang et al. 2016; xiao & qin, 2018; xiao, 2019), medical diagnosis (li et al. 2015; xiao, 2018; chen et al. 2019), target recognition (pan & deng, 2019), multicriteria decision-making (li & deng, 2019) etc., though it has produced some conflicts in the evidence combination rule (zadeh, 1978). the conflicts and counterinitiatives nature of evidence have been addressed for several decades. researchers developed another form of combination rule (dubois & prade, 1986; yager, 1987; inagaki, 1991; zhang, 1994) after realising that the conflicts can be resolved by either the method of pre-processing or modifying the basic probability assignment (bpa) before applying dempster’s combination rule (murphy, 2000; deng et al. 2004; jiang, 2016; xiao & qin, 2018). in the application of medical diagnosis, li et al. (2015) used fuzzy soft set (fss) and dempster-shafer theory (dst) in medical diagnosis problem and compared the result of the case study that put forward in basu et al. (2012) with the earlier technique. wang et al. (2016) proposed a method of using ambiguity measure, fuzzy soft set and dempster-shafer theory in the medical diagnosis. xiao, (2018) proposed the fuzzy preference relation and fuzzy soft set technique in medical diagnosis. chen et al. (2019) proposed the weighted average mass technique in medical diagnosis and compared with the earlier method. zhou et al. (2020) proposed a novel divergence measure of pythagorean fuzzy set (pfs) connecting the belief function and pfs with the application in the medical diagnosis. limboo and palash/decis. mak. appl. manag. eng. 5 (1) (2022) 290-308 292 in evidence theory, interval-valued and fuzzy form of bpa is more efficient than the precise bpa to reflect the uncertainty of a parameters. in this regard, yager, (2001) put forward interval-valued belief structure (ibs) and yager, (2014) proposed the intuitionistic view of the dempster-shafer belief structure. song et al. (2015) proposed a new distance measure between intuitionistic beliefs functions based on the new similarity measure of ifs with its properties. apart from this, li & deng, (2019) proposed the concept of intuitionistic evidence set in the form of support and non-support belief degree of bpas and applied the concept in multi-criteria decisionmaking (mcdm). gao & deng, (2020) introduced the quantum model of the mass function and established the relationship between quantum mass function and classical mass function with examples. xiao, (2020) generalizes the dst by introducing the complex basic probability assignment and applied it in medical diagnosis problem to show the efficiency of the proposed algorithm. kar et al. (2021) used picture fuzzy set based fusion operator and dempster-shafer theory in the medicine selection of covid 19 disease. in this paper, the main motivation and contributions of the paper are as follows:  to address the issue of conflict in evidence theory is still an open issue and the assessment of the alternatives from human cognitive subjective knowledge in more flexible way is always helpful in precise decision-making process.  we have introduced a new form of evidence theory called q-rung evidence set of classical evidence theory in the form of pair of support and nonsupport degree of basic probability assignment. the q-robpa is effective due to its flexibility offered to decision maker for the assessment of the alternatives based on the support as well as non-support degree of belief in the assessment of a problem.  since, the proposed q-rung orthopair bpa is based on the pair of two belief degree, therefore their conflict nature cannot be fully ignored. for this, we have further proposed an association coefficient measure to handle the uncertainty involved in the q-rung basic probability assignment (q-robpa).  the present method is based on the use of novel q-robpa and association coefficient measure in the medical diagnosis case study that put forward in basu et al. (2012) and compared its belief measure with the existing method to show its efficiency and superiority. the present paper is organised as follows: in the section 2, the basic overview of the preliminary concepts of dempster-shafer theory, and some theories related to uncertainty measure viz. deng entropy, distance measure and association coefficient measure with related property are put forward in section 3. section 4 carried out the fundamental ideas of fuzzy set theory and fuzzy soft set with related properties. in section 5, we have proposed the new form of evidence theory and introduced an association coefficient measure to handle the uncertainty in the process of conflict management. in section 6, we have put forward an algorithm to implement the proposed association coefficient measure on the q-rung orthopair basic probability assignment and also, an application in medical diagnosis is carried out in section 7. section 8 concludes the paper. 2. preliminaries in this section, we have put forward some fundamental concepts of dst and uncertainty measure of evidence. a q-rung orthopair basic probability assignment and its application in medical diagnosis 293 2.1. dempster-shafer theory: basic concept definition 1 (dempster, 1967; shafer, 1976): let 1 2 { , ,..., } n x x x x be a collection of mutually exclusive and exhaustive elements and the collection of all the hypotheses i f is defined by the power set of x such that 1 1 2 2 { ,{ },...,{ },{ , },..., } x n x x x x x . then, x is called as the frame of discernment (fod) such that 1 2 2 2 { , ,..., ,..., }n x i f f f f . definition 2 (dempster, 1967; shafer, 1976): the basic probability assignment (bpa) is a function : 2 [0,1] x m  that satisfies the condition ( ) 0m   and ( ) 1 i i f x m f   (1) where the collection 1 2 2 2 { , ,..., ,..., }n x i f f f f is 2 n possible propositions. definition 3 (dempster, 1967; shafer, 1976): the belief measure of f on x is a function : 2 [0,1] x bel  which satisfies the condition ( ) 0bel   , ( ) ( ) i i f f bel f m f    (2) plausibility measure of f is also a function : 2 [0,1] x pl  which satisfies the condition ( ) 0pl   , ( ) ( ) i i f f pl f m f     (3) definition 4 (dempster, 1967): the dempster’s combination rule for combining two bpas 1 m and 2 m is a joint function 1 2 : 2 [0,1] x m m  defined as     1 2 1 2 1 ( ) ( ) ( ) 1 i j i j f f f m m f m f m f k       (4) in addition,  1 2 ( ) 0m m   and 1 2( ) ( ) i j i j f f k m f m f     represents the conflict coefficient between 1 m and 2 m . two pieces of evidence are said to be in conflict whenever 1k  . the counter-intuitiveness and conflicts of bpas are reduced and managed with the help of various methods of uncertainty measure. 3. some uncertainty measure in dempster shafer theory deng entropy (deng, 2015) is used to measure the uncertainty contained in the bpas, whereas distance measure (jousselme, 2001; cheng, 2019), similarity measure (xiao, 2018) and divergence measure (fei et al. 2018; xiao, 2019; zhou et al. 2020) are the measure to distinguish two belief functions, and is also used to modify the conflicting evidence. definition 5 (deng, 2015): let m be the bpa on the discernment frame x . the deng’s entropy d e of m is defined as 2 2 ( ) ( ) ( ) log 2 1ix i i d i f f m f e m m f           (5) limboo and palash/decis. mak. appl. manag. eng. 5 (1) (2022) 290-308 294 in particular, if 1 i f  , then deng’s belief entropy reduces to  2 2 ( ) ( ) log ( ) x i d i i f e m m f m f     (6) definition 6 (jousselme, 2001): consider 1 m and 2 m be two bpas defined on the fod x . then, the jousselme’s evidence distance between two bpas is defined as 1 2 1 2 1 2 1 ( , ) ( ) ( ) 2 t d m m m m d m m   (7) where 1 m and 2 m be the bpas in the vector form. the matrix d is the jaccard’s matrix of order 2 2 n n  whose elements ( , )i jd f f are the jaccard’s similarity coefficient defined by ( , ) i j i j i j f f d f f f f    , for all , 2 x i j f f  (8) definition 7 (cheng, 2019): cheng’s distance measure between two bpas 1 m and 2 m is defined as 1 2 1 2 1 2 1 ( , ) ( ) ( ) 2 t d m m m m d m m     (9) where d  is the matrix of order 2 2 n n  whose entries ( , )i jd f f are defined as ( , ) i j i j i j i j f f f f d f f f f      , for all , 2 x i j f f  (10) definition 8 (jiang, 2016): the jiang’s correlation coefficient measure between 1 m and 2 m is defined as 1 2 1 2 1 1 2 2 ( , ) ( , ) ( , ) ( , ) c m m r m m c m m c m m  , (11) where 2 1̀ 2 1 1 2 1 2 1 1 | | ( , ) ( ) ( ) | | n n i j i j i j i j f f c m m m f m f f f           (12) definition 9 (pan & deng, 2019): pan & deng’s association coefficient measure between 1 m and 2 m is defined as 1 2 1 2 1 1 2 2 ( , ) ( , ) 1 { ( , ) ( , )} 2 r m m a m m r m m r m m   (13) where | | | |2 1 2 1 1 2 1 2 | | | | 1 1 2 1 2 1 ( , ) ( ) ( ) 2 1 2 1 n n i j i j i j f f f f i j f f i j r m m m f m f                          (14) a q-rung orthopair basic probability assignment and its application in medical diagnosis 295 4. fundamental concepts of fuzzy sets and q-rung fuzzy set this section puts forward some fundamental concepts of fuzzy set theory and fuzzy soft set with related properties and operations. definition 10 (zadeh, 1965): a fuzzy set a in the universal set x is defined as  , ( ) :aa x x x x  , where : [0,1]a x  is the membership function of the fuzzy set a . definition 11 (yager, 2016): a q-rung orthopair fuzzy set m in the universal set x defined as  , ( ), ( ) :m mm x x x x x   , where : [0,1]m x  is the membership function and : [0,1] m x  is the non-membership function that satisfying the condition    0 ( ) ( ) 1 q q m m x x    . the hesitancy degree of the qrung fuzzy set (q-rofs) m is defined as     ( ) 1 ( ) ( ) q q m m m x x x     , 1q  (15) for fixed 1q  , m reduces to intuitionistic fuzzy set (ifs) (atanassov, 1986) and for 2q  , m reduces to pythagorean fuzzy set (pfs) (yager, 2013). definition 12 (yager, 2016; hussain, 2020): let  , -a b q rofs x such that   , ( ), ( ) :a aa x x x x x   and   , ( ), ( ) :b bb x x x x x   , then we have a) a b iff    a bx x  and    a bx x  , x x  . b) a b iff    a bx x  and    a bx x  , x x  . c)      , , :c a aa x x x x x   , where c a is the complement of a . d)   , ( ), ( ) :a b a ba b x x x x x        , where ( ) min{ ( ), ( )} a b a b x x x     and ( ) max{ ( ), ( )} a b a b x x x     e)   , ( ), ( ) :a b a ba b x x x x x        , where ( ) max{ ( ), ( )} a b a b x x x     and ( ) min{ ( ), ( )} a b a b x x x     . f) the score function (peng et al. 2018) of the q-rofss a is given by             ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) { ( )} 21 q q a a q q a a x x q q q a a x x e s a x x x e                    , 1q  definition 13 (maji et al. 2001): let ( )u be the collection of fuzzy sets over the universal set u and e be the set of parameters with a e . a fuzzy soft set over u is a pair  ,f a , where f is a function given by (: )f a u . in general, if ( ) ( )u ifs u , ( ) ( )u pfs u and ( ) ( )u q rofs u , then  ,f a is accordingly called as the intuitionistic fuzzy soft set (ifss) (maji et al. 2001) or pythagorean fuzzy soft set (pfss) (peng et al. 2015) or q-rung orthopair fuzzy soft set (q-rofss) (hussain et al. 2020) over the universal set u respectively. definition 14 (maji et al. 2001): let  1,f x and  2,f x represents the two distinct fsss over the universe u and for all 1 1 x x , 2 2 x x , we have limboo and palash/decis. mak. appl. manag. eng. 5 (1) (2022) 290-308 296 (i)    1 2 1 2, , ,( )h x x f x g x  , where 1 2 1 2( ) ( ) ( )h x , x f x g x     1 2 1 2, , ,( )h x x f x g x  , where 1 2 1 2( ) ( ) ( )h x , x f x g x  5. q-rung evidence set in this section, we have defined a new form of evidence set named q-rung evidence set inspired from the work of li & deng, (2019) and put forward all the related definitions of dempster-shafer theory in this form. definition 15: let x be the frame of discernment. a q-rung basic probability assignment or q-rung orthopair basic probability assignment (q-robpa) on x is defined as the pair ,m m m    in which the function m has the following conditions the support belief degree : 2 [0,1] x m   satisfies the conditions ( ) 1 a x m a    (16) the non-support belief degree : 2 [0,1] x m   satisfies the conditions ( ) 1 a x m a    (17) for all a x , { ( )} { ( )} 1 q q m a m a     , where q tends to  . definition 16: let ,m m m    is a q-robpa on the fod x . then, the belief measure of the focal element a is pair ,bel bel   that satisfies the following conditions: ( ) ( ) b a bel a m b      and ( ) ( ) b a bel a m b      , (18) plausibility measure of is a pair of two function ,pl pl   which satisfies: ( ) ( ) a b pl a m b        and ( ) ( ) a b pl a m b        (19) where , : 2 [0,1] x pl bel      are the support degree whereas , : 2 [0,1] x pl bel      are the non-support degree of belief and plausibility functions respectively. definition 17: let 1 1 1 ,m m m    and 2 2 2 ,m m m    be two distinct q-robpa on the fod x . the combination of 1 m and 2 m is the new mass function defined as: 1 2 ( ) ( ), ( )m m a m a m a     , 2 x a    (20) where 1 2 1 2 ( ) ( ) ( ) 1 ( ) ( ) b c a b c m b m c m a m b m c               and 1 2 1 2 ( ) ( ) ( ) 1 ( ) ( ) b c a b c m b m c m a m b m c               (21) example 1: consider 1 2 { , ,..., } n x x x x be a fod having two q-robpas 1 m and 2 m for the focal elements 1 2 3 , ,f f f are given below 1 1 ( ) 0.6, 0.1m f  , 1 2 ( ) 0.05, 0.8m f  , 1 3 ( ) 0.35, 0.1m f  2 1 ( ) 0.7, 0.2m f  , 2 2 ( ) 0.1, 0.6m f  , 2 3 ( ) 0.3, 0.2m f  a q-rung orthopair basic probability assignment and its application in medical diagnosis 297 using the equations (20) and (21), the combined q-robpa for focal elements are given by 1 ( ) 0.8485, 0.0307m f  , 2 ( ) 0.0101, 0.9831m f  and 3 ( ) 0.1414, 0.0015m f  counter-intuitiveness: if 1 2 ( ) ( ) 1 b c m b m c       and 1 2( ) ( ) 1 b c m b m c       , then the combination rule is conflicting from its membership belief degree of q-robpa. on the other hand, if 1 2 ( ) ( ) 1 b c m b m c       and 1 2( ) ( ) 1 b c m b m c       then the fusion rule is conflicting from its non-membership end. again, if both 1 2 ( ) ( ) 1 b c m b m c       and 1 2 ( ) ( ) 1 b c m b m c       , then the evidence is in conflicting from both the end. the conflict of the above type can be handled by pre-processing the q-robpas via the weighted average technique similar to followed in bpa in the classical evidence theory. the weight of the evidence is obtained by using the various uncertainty measure. we have proposed a new association coefficient measure of belief functions below and shows its efficiency and applicability in conflict resolution in the new qrobpa with example 2. 5.1. association coefficient measure let 1 m and 2 m denotes the bpas on the fod x . the association coefficient measure is defined as 1 2 1 2 1 1 2 2 ( , ) ( , ) 1 { ( , ) ( , )} 2 r m m a m m r m m r m m   (22) where | | 32 1 2 1 1 2 1 2 | | | || | 1 1 (2 1) ( , ) ( ) ( ) (2 1)(2 1)(2 1) n n i j j i ji f f pro i j f f ff i j r m m m f m f              , (23) 1 2 ( , )r m m can also be represented as 1 2 1 2 ( , ) ( ) ( ) i j r m m m f dm f with | | 3 | | | || | (2 1) (2 1)(2 1)(2 1) i j j i ji f f f f ff d        , (24) where d is a positive-definite matrix of order (2 1) (2 1) n n    and it can be expressed as the product of the invertible matrix and its transpose i.e., t d q q . the proposed association coefficient measure 1 2 ( , )a m m will trivially holds the following properties: (i) symmetricity 1 2 2 1 ( , ) ( , )a m m a m m . (ii) boundedness 1 2 0 ( , ) 1a m m  . (iii) 1 2 1 2 ( , ) 1a m m m m   (iv) 1 2 ( , ) 0 i j a m m f f     for all , 2x i j f f  . example 2: consider the {1, 2, 3,...,10}x  be a fod and two q-robpas 1 m and 2 m for the events is given below 1 ({2,3, 4}) 0.05, 0.2375m  , 1 ({7}) 0.05, 0.2375m  , 1 ( ) 0.1, 0.225m x  , 1 ( ) 0.8, 0.05m a  , 1 ({1, 2,3, 4,5}) 0, 0.25m  limboo and palash/decis. mak. appl. manag. eng. 5 (1) (2022) 290-308 298 2 ({2,3, 4}) 0, 0.25m  , 2 ({7}) 0, 0.25m  , 2 ( ) 0, 0.25m x  , 2 ( ) 0, 0.25m a  , 2 ({1, 2,3, 4,5}) 1, 0m  the effectiveness of the new proposed association coefficient measure is shown through the ten cases of different focal elements of a . we have noticed that the conflict degree 1 2 1 ( , )a m m of q-robpa is minimum for the variable event a at {1,2,3,4,5} from both the pair of support as well as non-support belief degree as given in table 1. table 1. conflict coefficient measure of q-robpa sl. no. variable set conflict for support belief degree conflict for non-support belief degree 1 {1} 0.9959 0.2125 2 {1,2} 0.9877 0.2104 3 {1,2,3} 0.9476 0.1997 4 {1,2,3,4} 0.7729 0.1547 5 {1,2,3,4,5} 0.0325 0.0064 6 {1,2,3.…,6} 0.5216 0.1602 7 {1,2, 3, …,7} 0.9393 0.1989 8 {1,2, 3,.…,8} 0.9825 0.2070 9 {1,2, 3, …,9} 0.9934 0.2019 10 {1,2, 3, …, 10} 0.9963 0.1789 0.0000 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 1.0000 1.1000 1.2000 1 2 3 4 5 6 7 8 9 10 conflict in q-robpa figure 1. comparison of conflicts for support and non-support belief degree from the graphical representation, it is noticed that the trends of conflict coefficient of q-robpa first decrease with the addition of elements, attains minimum valued at a equals to {1,2,3,4,5} and increases with the addition of extra elements. the association coefficient measure is effective in handling conflict and modifying the q-robpa. a q-rung orthopair basic probability assignment and its application in medical diagnosis 299 6. application of q-robpa and association coefficient measure in decision-making. in this section, we have proposed a methodology focussing on applying the association coefficient measure on the new q-robpa. the methodology and algorithm are as follows. 6.1. methodology and algorithm let e be the finite set of parameters consisting of disease’s symptoms 1 2 { , ,..., } m a e e e and decision-making parameters 1 2 { , ,..., } n b f f f respectively. suppose ( , )f a be the fuzzy soft set for representing the state of symptoms of disease and ( , )g b be the fuzzy soft set representing the decision-making tools. the steps for the generation of initial q-robpa for the diseases 1 2 { , ,..., } l d x x x are as follows: step i: input two q-rofss ( , )f a and ( , )g b for the assessment of symptoms and decision-making tool relative to the disease as ( ) : 1, 2,..., , i i k i e e x f e i m             and ( ) : 1, 2,..., , j j k j f f x g f j n              , for all k x , the fuzzy soft set can also be written in the matrix form as follows   1 2 1 11 12 1 2 21 22 2 1 2 , m m m l l l lm e e e x a a a x a a a f a x a a a             ;   1 2 1 11 11 1 2 21 22 2 1 2 , n n n l l l ln f f f x b b b x b b b g b x b b b             where   ,, j j e eij i a x   and   , ,j jij i f fb x   are the relative q-rung fuzzy number of i x with respect to the different parameters. step ii: construct t mn number of new parameters 1 1 1 p e f  , 2 1 2 p e f  ,…, t i j p e f  ,..., t m n p e f  based on the “and” operation of the fsss. in matrix form, we have 1 2 1 1 11 12 1 1 1 1 2 21 22 2 2 1 2 1 2 1 ... ... ... ... ... ... i i n i t i i t l l l li li lt x c c c c c x c c c c c m x c c c c c p p p p pt                , where     , min ( ), ( ) , max ( ), ( ) : , [0,1]ij i ij ij ij ij ij ijc x mf a mf b nmf a nmf b a b  represents the membership value of i x with respect to the new parameters. step iii: construct the information structure image matrix m of alternatives by the membership degree as follows limboo and palash/decis. mak. appl. manag. eng. 5 (1) (2022) 290-308 300 1 2 1 1 11 12 1 1 1 1 2 21 22 2 2 1 2 1 2 1 ... ... ... ... ... ... i i t n i t i i t l l l li li lt x c c c c c x c c c c c m x c c c c c p p p p p                 , 1 1 , j j j j p p ij t t p pj j c          where 1 2 { } { , ,..., } ij j j lj c c c c is the information structure image sequence (li et al. 2015). step iv: build t pieces of q-rung belief function i m generated from the new parameters as follows 1 1 11 11 ( ) ,m x c c    , 1 2 21 21( ) ,m x c c    ,…, 1 1 1( ) ,i i im x c c    ,…, 1 1 1( ) ,l l lm x c c    2 1 12 12 ( ) ,m x c c    , 2 2 22 22( ) ,m x c c    ,…, 2 2 2( ) ,i i im x c c    ,…, 2 2 2( ) ,l l lm x c c    ...... ..... .... …. ...... ..... .... …. ...... ..... .... …. 1 1 1 ( ) , t t t m x c c    , 2 2 2( ) ,t t tm x c c    ,…, ( ) ,t i it itm x c c    ,…, ( ) ,t l lt ltm x c c    here 1 1 ( ) 1 ,1 l l ij ijj i i m d c c         , (1 ( ))dij ij ic c e p      , (1 ( ))dij ij ic c e p      and 2 ( ) ( ) ( ) log 2 1i i d i i x i m x e m m x          is the deng’s belief entropy (deng, 2015) of the parameters. step v: the support degree of the q-robpa , 1, 2,..., i m i t are defined as  ( ) ( ), ( )i i isup m sup m sup m    , where ( ) i sup m  and ( ) i sup m  represents the support of membership belief function and non-membership belief function such that 1, ( ) ( , ) t i pro i j j j i sup m a m m        and 1, ( ) ( , ) t i pro i j j j i sup m a m m        step vi: the degree of credibility of q-robpas is given by 1 1 ( ) ( ) ( ) , ( ) ( ) i i i n n i i i i sup m sup m crd m sup m sup m                     , where the credibility ( ) i crd m is the weight of i m such that 1 ( ) 1 n i i crd m   . a q-rung orthopair basic probability assignment and its application in medical diagnosis 301 step vii: the weighted average mass ( )wam m of the evidences m is given by  ( ) ( ), ( )wam m wam m wam m  step viii: the weighted average mass of alternatives is combined separately for the membership and non-membership belief degree up to 1n  by using the dempster’s combination rule. step ix: rank the alternatives based on the score values of its belief measure by using the equation ({ }) ( ) ( ) i i i s x x x   , thus, the patient is suffering from i x if 1 2 max{ ( ), ( ),..., ( )} l s x s x s x with support belief measure { ( ) : 1, 2,..., } i bel x i l   and non-support belief measure { ( ) : 1, 2,.., } i bel x i l   . 7. numerical example in medical diagnosis in this section, we have implemented the various concepts related to uncertainty to execute the case study in medical diagnosis taken from the example 6.2 in basu et al. (2012). consider a patient is under observation have noticed some symptoms among the several symptoms namely fever 1 s , running nose 2 s , weakness 3 s , oro-facial pain 4 s , nausea or vomiting 5 s , swelling 6 s and trismus 7 s respectively. the set 1 2 3 4 { , , , }d d d d d of four possible diseases associated with the set of symptoms, where 1 2 3 4 , , and d d d d stands for the disease acute dental abscess, migraine, acute sinusitis and peritonsillar abscess respectively. let the symptoms and other decisionmaking tools history (h), physical examination (p) and lab investigation (l) together forms the set of parameters e such that 1 2 7 { , ,..., , , , }e s s s h p l associated with possible disease. an expert assessed a patient’s disease possibility based on the responses made by the patient against his symptoms, history, physical examination and laboratory investigation etc. let the parameters 1 2 7 { , ,..., }a s s s and { , , }b h p l forms two qrung orthopair fuzzy soft sets such that 1 2 7 ( , ) { ( ), ( ),..., ( )}f a f s f s f s and ( , ) { ( ), ( ), ( )}g b g h g p g l , where the membership degree is defined by generalised form of fuzzy sets i.e., intuitionistic, pythagorean or q-rung orthopair fuzzy number as 31 2 4 1 ( ) , , , 0.6, 0.4 0.2, 0.8 0.3, 0.7 0.4, 0.6 dd d d f s          , 31 2 4 2 ( ) , , , 0, 0.7 0, 0.7 0.7, 0.4 0, 0.7 dd d d f s          , 1 2 3 4 3 ( ) , , , 0.6,0.4 0.1,0.8 0.3,0.4 0.2,0.7 d d d d f s          , 1 2 3 44( ) , , , 0.9,0.1 0.9,0.1 0.8,0.2 0.7,0.3 d d d d f s          , 1 2 3 4 5 ( ) , , , 0, 0.9 0.8, 0.2 0.3, 0.6 0.1, 0.8 d d d d f s          , 1 2 3 4 6 ( ) , , , 0.7,0.3 0,0.9 0.4,0.6 0.6,0.4 d d d d f s          , limboo and palash/decis. mak. appl. manag. eng. 5 (1) (2022) 290-308 302 1 2 3 4 7 ( ) , , , 0.8, 0.2 0, 0.8 0, 0.8 0.5, 0.4 d d d d f s          ; 1 2 3 4( ) , , , 0.6,0.2 0.8,0.1 0.8,0.2 0.6,0.2 d d d d g h          , 1 2 3 4( ) , , , 0.8, 0.2 0.3, 0.5 0.4, 0.6 0.8, 0.2 d d d d g p          , 1 2 3 4( ) , , , 0.4, 0.6 0.6, 0.4 0.7, 0.4 0.3, 0.7 d d d d g l          since, the patient expressing three symptoms fever 1 s , running nose 2 s , and facial pain 4 s , the nine possible pairs of parameters i p is represented by the pairs 1 ( , )s h , 1 ( , )s p , 1 ( , )s l , 2 ( , )s h , 2 ( , )s p , 2 ( , )s l , 4 ( , )s h , 4 ( , )s p , 4 ( , )s l respectively. we consider the set 1 2 3 4 { , , , }d d d d d as the frame of discernment and each pair is represented as the evidence. the matrix of membership degree of i d relative to the joint parameters obtained from the q-rofsss ( , )f a and ( , )g b is given by 1 2 3 4 5 6 7 8 9 1 2 3 4 0.6, 0.4 0.6, 0.4 0.4, 0.6 0.0,1.0 0.0,1 0.0,1.0 0.6, 0.4 0.8, 0.2 0.4, 0.6 0.2, 0.8 0. p p p p p p p p p d d m d d  2, 0.8 0.2, 0.8 0.0,1.0 0.0,1 0.0,1.0 0.8, 0.2 0.3, 0.7 0.6, 0.4 0.3, 0.7 0.3, 0.7 0.3, 0.7 0.7, 0.3 0.4, 0.6 0.7, 0.3 0.8, 0.2 0.4, 0.6 0.7, 0.3 0.4, 0.6 0.6, 0.4 0.3, 0.7 0.0,1.0 0.0,1.0 0.0,1.0 0.6, 0.4 0.7, 0.3 0.3, 0.7              the fuzzy information structure image matrix m is constructed based on the step i-iii given by 0.40, 0.40, 0.3333, 0.0, 0.0, 0.0, 0.2143, 0.3636, 0.20, 0.16 0.16 0.2143 0.3030 0.2778 0.303 0.3333 0.1111 0.30 0.1333, 0.1333, 0.1667, 0.0, 0.0, 0.0, 0.2857, 0.1364, 0.30, 0.32 0.32 0.2857 0.3030 0.2778 0.303 0.1667 0.3889 0.20 0.2 m  0, 0.20, 0.25, 1.0, 1.0, 1.0, 0.40, 0.1818, 0.35, 0.28 0.28 0.25 0.0909 0.1667 0.0909 0.16 0.3333 0.15 0.2667, 0.2667, 0.25, 0.0, 0.0, 0.0, 0.40, 0.3182, 0.15, 0.24 0.24 0.25 0.3030 0.2778 0.0303 0.16 0.1667 0.35                           now, the generated initial q-robpa i m can be obtained in table 2 and evaluated based on the step i-iv and the initial q-robpas are modified by using the association coefficient measure defined in equation (22)-(24). table 2. q-rung orthopair basic probability assignments of alternatives. 1d 2d 3d 4d d 1 m 0.3346, 0.1419 0.1115, 0.2838 0.1673, 0.2484 0.2231, 0.213 0.1635, 0.1129 2 m 0.3346, 0.1419 0.1115, 0.2838 0.1673, 0.2484 0.2231, 0.213 0.1635, 0.1129 3 m 0.2767, 0.1897 0.1384, 0.2528 0.2076, 0.2213 0.2076, 0.2213 0.1697, 0.1149 4 m 0, 0.2701 0, 0.2701 1, 0.0801 0, 0.2701 0, 0.1087 5 m 0, 0.2462 0, 0.2462 1, 0.1477 0, 0.2462 0, 0.1137 6 m 0, 0.2701 0, 0.2701 1, 0.0801 0, 0.2701 0, 0.1087 7 m 0.1774, 0.2964 0.2366, 0.1482 0.2366, 0.1482 0.1774, 0.2964 0.1720, 0.1108 a q-rung orthopair basic probability assignment and its application in medical diagnosis 303 1d 2d 3d 4d d 8 m 0.3039, 0.0993 0.1140, 0.3476 0.1519, 0.2979 0.266, 0.149 0.1642, 0.1062 9 m 0.1666, 0.2424 0.2499, 0.2424 0.2916, 0.2154 0.125, 0.1885 0.1668, 0.1113 therefore, the association coefficient measure matrix a and a of the belief function for the support belief degree as well as non-support belief degree is given by 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 1 0.986 0.2731 0.2731 0.2731 0.8917 0.9934 0.8387 1 1 0.986 0.2731 0.2731 0.2731 0.8917 0.9 m m m m m m m m m m m m m a m m m m m                     934 0.8387 0.986 0.986 1 0.3437 0.3437 0.3437 0.9488 0.9818 0.9106 0.2731 0.2731 0.3437 1 1 1 0.3938 0.2488 0.4792 0.2731 0.2731 0.3437 1 1 1 0.3938 0.2488 0.4792 0.2731 0.2731 0.3437 1 1 1 0.3938 0.2488 0.4792 0.8917 0.8917 0.9488 0.3938 0.3938 0.3938 1 0.8938 0.9854 0.9934 0.9934 0.9818 0.2488 0.2488 0.2488 0.8938 1 0.8283 0.8387 0.8387 0.9106 0.4792 0.4792 0.4792 0.9854 0.8283 1                             , and 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 1 0.9878 0.8956 0.9462 0.8956 0.8692 0.9735 0.9686 1 1 0.9878 0.8956 0.9462 0.8956 0.8692 0 m m m m m m m m m m m m m a m m m m m                     .9735 0.9686 0.9878 0.9878 1 0.9218 0.9670 0.9218 0.9089 0.9396 0.9783 0.8956 0.8956 0.9218 1 0.9864 1 0.9558 0.8032 0.9419 0.9462 0.9462 0.9670 0.9864 1 0.9864 0.9671 0.8753 0.9814 0.8956 0.8956 0.9218 1 0.9864 1 0.9558 0.8032 0.9419 0.8692 0.8692 0.9089 0.9558 0.9671 0.9558 1 0.7476 0.9372 0.9735 0.9735 0.9396 0.8032 0.8753 0.8032 0.7476 1 0.9147 0.9686 0.9686 0.9783 0.9419 0.9814 0.9419 0.9372 0.9147 1                             the support degree of the q-rung ortho pair basic probability assignment is now obtained as follows 1 ( ) 5.5291, 7.5365sup m  , 2 ( ) 5.5291, 7.5365sup m  , 3 ( ) 5.8443, 7.6130sup m  , 4 ( ) 4.0117, 7.4003sup m  , 5 ( ) 4.0117, 7.4003sup m  , 6 ( ) 4.0117, 7.4003sup m  , 7 ( ) 5.7928, 7.2108sup m  , 8 ( ) 5.4371, 7.0306sup m  , 9 ( ) 5.8393, 7.6326sup m  and the corresponding credibility degree of the belief assignments are obtained as follows 1 ( ) 0.1210, 0.1125crd m  , 2 ( ) 0.1210, 0.1125crd m  , 3 ( ) 0.1270, 0.1136crd m  , 4 ( ) 0.0862, 0.1104crd m  , 5 ( ) 0.0862, 0.1104crd m  , 6 ( ) 0.0862, 0.1104crd m  , 7 ( ) 0.1260, 0.1076crd m  , 8 ( ) 0.1190, 0.1049crd m  ¸ 9 ( ) 0.1276, 0.1139crd m  now, the weighted average mass wam(m) of the alternative are obtained as follows limboo and palash/decis. mak. appl. manag. eng. 5 (1) (2022) 290-308 304 1 ( ) 0.1950, 0.2111m d  , 2 ( ) 0.1194, 0.2604m d  , 3 ( ) 0.4129, 0.1875m d  , 4 ( ) 0.1496, 0.2298m d  , ( ) 0.1231, 0.1112m d  the above wam(m) of alternative is combined for eight times to itself by using the dempster’s combination rule (21) and the final belief measure for i d is shown below 1 ( ) 0.00902, 0.14785m d  , 2 ( ) 0.00078, 0.53130m d  , 3 ( ) 0.98794, 0.07461m d  , 4 ( ) 0.00225, 0.24575m d  , and ( ) 0.000002, 0.00001m d  since, the final belief measures of alternatives are in the form of intuitionistic fuzzy number and this can be ranked based on the score of the intuitionistic bpas, then we have 1 ({ }) 0.1393s d   , 2 ({ }) 0.5305s d   , 3 ( ) 0.9133s d  , 4 ( ) 0.2435s d   finally, from values of the score functions we can conclude that the patient has the possibility of suffering from the disease of 3 d with the support belief degree 0.98794 and non-support belief degree 0.07455 respectively. table 3. comparison of final belief measure different methods type of bpa ranking order 3 ({ })bel d score value of 3 d li et al.’s, method (2015) discrete 3 1 4 2 d d d d   0.8349 na wang et al.’s method (2016) discrete 3 1 4 2 d d d d   0.9906 na xiao’s, method (2018) discrete 3 1 4 2 d d d d   0.99996 na chen et al.’s method (2019) discrete 3 1 4 2 d d d d   0.9989 na the proposed method q-rung fuzzy number 3 1 4 2 d d d d   0.9879, 0.0746 0.9133 from the comparison as shown in table 3, the proposed method suggests the same decision and follows the same ranking order 3 1 4 2 d d d d   as earlier method suggests (li et al. 2015; wang et al. 2016; xiao, 2018; chen et al. 2019). the advantages of the proposed methodology over the others are that rest of the methods is based on the discreate number while our proposed method has more flexible one in the sense that the assessment of alternatives with respect to the parameters can be made based on membership and non-membership belief degree. in addition, the conflicts in the belief if exist will handled by the proposed association coefficient measure. however, the methodology has certain limitation as assessment of alternatives is based on the human subjective expertise, the assessment may give false decision if the decision-maker deliberately provides some false assessment membership belief degree. 8. conclusion in this paper, we have proposed a new q-rung orthopair basic probability assignment consisting of membership and non-membership belief degree to provide a q-rung orthopair basic probability assignment and its application in medical diagnosis 305 decision-maker’s flexibility in assigning his belief degree to a proposition. we have investigated various essential concepts of the classical dempster-shafer theory and related uncertainty measures in the literature of evidence theory. to cope with uncertainty in the q-robpa, we have further implemented our novel association coefficient measure to obtain the pre-process evidence. finally, a methodology is developed to apply the proposed q-robpa and association coefficient measure with a hypothetical case study in medical diagnosis, compared with the existing results. this study reveals that the decision of an alternative from the proposed algorithm follows the same ranking order as earlier did with its support degree of belief and nonsupport degree of belief. from the study, a medical expert can make an action plan for the patient’s treatment, which has high possibility of belief degree based on the appropriate and flexible assessment from his previous experience in the field. the proposed methodology offers comprehensive advantages to the decisionmaker for the assessment of an alternative through the membership as well as nonmembership belief degree. the q-rung orthopair basic probability assignment can easily represent the decision-maker’s views on the alternatives from his experience where he has scope to assign membership degree for favourable case and nonmembership belief degree for non-favoured cases from same type of situation. from the perspective of the limitations of the methodology, as the belief degree assessment is based expert’s knowledge and information, so the intentional false exercise of information sharing may give abrupt results. in this regard, the experienced and loyal decision expert is required for the assessment of these alternatives. in the future, more general basic probability assignment based on the picture fuzzy set and spherical fuzzy sets, etc may be used to get more accurate, precise result and implement the method for a large real-time statistical data set of medical decisionmaking. author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research received no external funding. data 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(2020). a new divergence measure of pythagorean fuzzy sets based on belief function and its application in medical diagnosis. mathematics, 8(1), 142. © 2021 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license 2 soft multiset topology.dvi abstract the aim of this paper is to introduce the notion of soft multi-set topology (sms-topology) defined on a soft multi-set (sms). soft multi-set and soft multi-set topology are fundamental tools in computational intelligence, which have a large number of applications in soft computing, fuzzy modeling and decision-making under uncertainty. the idea of power whole multi-subsets of a sms is defined to explore various rudimentary properties of sms-topology. certain properties of sms-topology like sms-basis, sms-subspace, sms-interior, sms-closure and boundary of sms are explored. furthermore, the multicriteria decision-making (mcdm) algorithms with aggregation operators based on sms-topology are established. algorithm i (i = 1, 2, 3) are developed for the selection of best alternative for biopesticides, for the selection of best textile company, for the award of performance, respectively. some real life applications of the proposed algorithms in mcdm problems are illustrated by numerical examples. the the reliability and feasibility of proposed mcdm techniques is shown by comparison analysis with some existing techniques. keywords: soft multi-sets; soft multi-set topology; aggregation operators, algorithms; mcdm. 1 introduction modeling and handling uncertainties has become an issue of great importance in the solution of sophisticated problems originating in a vast range of various fields such as computational intelligence, artificial intelligence, data analysis, information fusion, image processing, signal processing, environmental sciences and medical sciences. mathematical models like multi-sets (blizard, 1989), fuzzy sets (zadeh, 1965), soft sets (molodtsov, 1999) and rough sets (pawlak, 1982) are fundamental tools for uncertainty, hesitancy and vagueness in the real life circumstances. the researchers have been developed some extension of fuzzy sets like intuitionistic fuzzy sets (atanassov, 1986), bipolar fuzzy sets (zhang, 1994), pythagorean fuzzy sets (yager, 2013; yager and abbasov, 2013) and q-rung orthopair fuzzy sets (yager, 2017) which have a large number of applications 70 decision making: applications in management and engineering vol. 3, issue 2, 2020, pp. 70-96. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame2003070r certain properties of soft multi-set topology with applications in multi-criteria decision making muhammad riaz 1*, naim çagman 2, nabeela wali 3 and amna mushtaq 4 1 department of mathematics, university of the punjab, lahore, pakistan. 2 department of mathematics, tokat gaziosmanpasa university, tokat, turkey. 3 department of mathematics, university of the punjab, lahore, pakistan. 4 department of mathematics, university of the punjab, lahore, pakistan. * corresponding author. e-mail addresses: mriaz.math@pu.edu.pk (m. riaz), naim.cagman@gop.edu.tr (n. çagman), nabeelawali.math@gmail.com (n. wali), amna44mushtaq@gmail.com (a. mushtaq) original scientific paper received: 5 june 2020; accepted: 19 august 2020; available online: 12 september 2020. in computational intelligence, decision making under uncertainty and many other fields of science and engineering. indeed, the real power of these sets are in their ability to handle and manipulate verbally-stated information into mathematical modeling and seeking feasible solutions to complicated real life problems. additionally, fuzzy sets and its extensions are strong mathematical models to solve real world problems which can not be solved by classical mathematical techniques. fuzzy sets, extensions of fuzzy sets, rough sets, soft sets and hybrid structures of these sets have been studied by many researchers like ali, (2009,2011); cagman et al., (2011); chen (2005); feng et al., (2010,2011,2018); garg and rani, (2019); hashmi et al., (2019); karaaslan and hunu, (2020); kumar and garg, (2018), maji et al., (2002,2003); naeem et al., (2019), peng and yang (2015), peng et al., (2017), pie and miao (2005), roy and maji (2007); riaz et al., (2019);, riaz and hashmi (2019); riaz and tehrim, (2019); shabir and naz (2011); zhang and xu (2014); zhan et al., (2015,2019); and zhang (1994). multi-set theory and soft multi-set theory have been studied by many researchers including alkhazaleh et al. (2011); babitha and john (2013); balami and ibrahim (2013); girish and john (2009,2019); kumar and naisal (2016); mukherjee et al. (2014); syropoulos (2001) and tokat and osmanoglu (2011,2013). a large number of mcdm methods have been developed by the researchers under rough sets, fuzzy sets and soft sets. but these methods do not deal with real life situations under the universe of soft multi-sets. due to repetition of objects or objects have multiplicity more than one and variety of attributes under consideration in the universe of soft multi-sets it is necessary to develop novel mcdm approaches. the goal of this article is deal with these challenges and to extend the notion of soft multi-sets and soft multi-set topology towards mcdm problems. the topological and algebraic structures of soft multi-sets have large number of applications in soft computing, decision-making, data analysis, data mining, expert systems, information aggregation and information measures. the remaining article is arranged as follows: in section 2, we use power whole multi-subsets of a sms to introduce some basic concepts of sms-theory. in section 3, we present some new results of sms-topology and certain properties including basis, subspace, interior, closure and boundary of soft multi-sets (smss). in section 4, we present algorithm 1, algorithm 2 and algorithm 3 for the selection of best alternative for biopesticides, for the selection of best textile company, for the award of performance, respectively. we also present applications of sms-topology for mcdm by using proposed algorithms. at the end, the sum up of this research studies is given in the in section 5. 2 preliminaries in this section, we study few primary rudiments of multi-sets (mss) and soft multi-sets (smss). definition 2.1. ”a multi-set (ms) over z is just a pair < z, f >, where f : z → w is a function, z is a crisp set and w is a set of whole numbers. also in order to avoid any confusion we will use square brackets for multi-sets and braces for sets. multiset a is given by a =< z, f >= [k1 z1 , k2 z2 , ..., kn zn ], where z1 occuring k1 times, z2 occuring k2 times and so on (syropoulos, 2001). definition 2.2. let a =< z, f > and b =< z, g > be two multi-sets. multiset a is a submulti-set of b, 71 riaz et al./decis. mak. appl. manag. eng. 3 (2) (2020) 70-96 denoted by a ⊆ b if for all z ∈ a, f(z) ≤ g(z) (syropoulos, 2001). definition 2.3. a submulti-set a =< z, f > of b =< z, g > is a whole submulti-set of b with each element in a having full multiplicity as in b. i.e. f(z) = g(z), for every z in a (babitha and john (2013)) definition 2.4. let [z]n denotes the set of all mss whose elements are in z such that no element in a multi-set appears more than n times. let a ∈ [z]n be a multi-set. the power whole multi-set of a denoted by pw(a) is defined as the set of all whole sub mss of a. the cardinality of pw(a) is 2m, where m is the cardinality of the support set (root set) of a” (babitha and john (2013)). in the sequel, h indicates to universal multi-set, e is a set of attributes or parameters , pw(h) is a power whole multi-set of h and a ⊆ e. example 2.5. let ... m = [2/r, 1/y, 1/k] be a multi-set. then the set of all sub mss of m is pw(a) = { ... m1 = [0/r, 0/y, 0/k], ... m2 = [0/r, 0/y, 1/k], ... m3 = [0/r, 1/y, 0/k], ... m4 = [0/r, 1/y, 1/k], ... m5 = [1/r, 0/y, 0/k], ... m6 = [1/r, 0/y, 1/k], ... m7 = [1/r, 1/y, 0/k], ... m8 = [1/r, 1/y, 1/k], ... m9 = [2/r, 0/y, 0/k], ... m10 = [2/r, 0/y, 1/k], ... m11 = [2/r, 1/y, 0/k], ... m12 = [2/r, 1/y, 1/k] } and card(p(m)) = (2 + 1)(1 + 1)(1 + 1) = 12. furthermore, the power whole multi-set is given by pw(m) = { ... m1, ... m2, ... m3, ... m4, ... m9, ... m10, ... m11, ... m12} and its cardinality is given by card(pw(m)) = 23 = 8. definition 2.6. ”a soft multi-set (sms) ωa on the universal multi-set h is defined by the set of all ordered pairs ωa = {(ν, ωa(ν)) : ν ∈ e, ωa(ν) ∈ pw(h)}, where ωa : e → pw(h) such that ωa(ν) = ∅ if ν /∈ a. throughout this paper, sm(h) denotes the family of all smss over h with attributes from e. now, we elaborate the definition of soft multi-set by the succeeding example” (babitha and john (2013)). example 2.7. let h = [ 2 r1 , 4 r2 , 3 r3 , 5 r4 , 7 r5 , 6 r6 , 9 r7 ] be the universal multi-set of classrooms, e = {comfortable, air conditioned, well decorated, flipped classroom} and a = e. then the sms ωa is given by ωa = {(comfortable, [ 2 r1 , 5 r4 ]), (air conditioned, [ 6 r6 , 9 r7 ]), (well decorated, [ 2 r1 , 4 r2 ]), (flipped classroom, [ 3 r3 , 7 r5 , 9 r7 ])}. definition 2.8. ”let ωa ∈ sm(h). if ωa(ν) = ∅, ∀ ν ∈ e, then ωa is called an empty or null sms, denoted by ωφ (see babitha and john (2013)). 72 certain properties of soft multi-set topology with applications in multi-criteria decision making definition 2.9. let ωa ∈ sm(h). then ωa is said to be a-universal sms, denoted by ωâ, if ωa(ν) = h, ∀ ν ∈ a. if a = e, then a-universal soft multi-set is said to be an universal or absolute sms, denoted by ω ê ” (babitha and john (2013)). definition 2.10. let ωa, ωb ∈ sm(h). then, ωa is a soft multi subset of ωb, denoted by ωa⊆̂ωb, if ωa(ν) ⊆ ωb(ν) for all ν ∈ e” (babitha and john (2013)). definition 2.11. let ωa, ωb ∈ em(h). then, the union ωa∪̂ωb, the intersection ωa∩̂ωb, the difference ωa\̂ωb of ωa and ωb are defined by the approximate functions ωa∪̂b(ν) = ωa(ν) ∪ ωb(ν), ωa∩̂b(ν) = ωa(ν) ∩ ωb(ν), ωa\̂b(ν) = ωa(ν) ⊖ ωb(ν), respectively, and the complement ω c a of ωa is defined ω c a(ν) = h ⊖ ωa(ν), for all ν ∈ e. note that (ω c a) c = ωa and ω c φ = ωê. definition 2.12. a soft multi-set ωa over h is called soft multi-set point (sms-point), if there is exactly one ν ∈ a, such that ωa(ν) 6= ∅ and ωa(µ) = ∅, ∀µ ∈ a \ {ν}. the sms-point ωa is in the sms δa, if for the element ν ∈ a, ωa(ν) ⊆ δa(ν). example 2.13. let h = [ 2 a , 3 b , 4 c ], a = {ν, µ} = e. let ωa = {(ν, [ 2 a ])} and δa = {(ν, [ 2 a , 3 b ]), (µ, [3 b , 4 c ])}. since ωa(ν) = [ 2 a ] ⊆ [ 2 a , 3 b ] = δa(ν) and ωa(µ) = ∅ ∀µ ∈ a \ {ν}. therefore, ωa is a sms-point of sms δa. , where proposition 2.14. let ωa, ωb ∈ sm(h). then (i) (ωa∪̂ωb) c = σca∩̂σ c b, (ii) (ωa∩̂ωb) c = σca∪̂σ c b. 3 soft multi-set topology different approaches have bee studied by the researchers to define soft multi-set topology (sms-topology) (mukherjee et al. (2014), and tokat and osmanoglu (2011,2013)). in this section, we introduce the notion of sms-topology on a soft multi-set and its analogous properties by using the concept of power whole sub multi-sets to use the full multiplicity or zero multiplicity of each objects. definition 3.1. let ωa be a sms over h. the soft power whole multi-set of the sms ωa is denoted by p̃w(ωa) and is defined as p̃w(ωa) = {ωai : ωai⊆̃ωa, i ∈ i} and its cardinality is given by |p̃w(ωa)| = 2 ∑ i∈n |xi|, where |xi| is the cardinality of the support set xi of approximation image multi-set ... mi with respect to parameter ëi, where i ∈ n. example 3.2. let h = [ 5 a , 4 b , 3 c ], e = {ë1, ë2, ë3}, a = {ë1, ë2} ⊆ e and a soft multi-set over h is ωa = {(ë1, [ 5 a , 4 b ]), (ë2, [ 4 b , 3 c ])}. then |p̃w(ωa)| = 2 |x1|+|x2| = 22+2 = 24 = 16, where |x1| = 2, since x1 = {a, b} and |x2| = 2, since x2 = {b, c}. 73 riaz et al./decis. mak. appl. manag. eng. 3 (2) (2020) 70-96 the soft power whole multi-set of the soft multi-set ωa is given by p̃w(ωa) = {ωa1, ωa2, · · ·, ωa16}, where ωa1 = ω∅, ωa2 = {(ë1, [ 5 a ])}, ωa3 = {(ë1, [ 4 b ])}, ωa4 = {(ë1, [ 5 a , 4 b ])}, ωa5 = {(ë2, [ 4 b ])}, ωa6 = {(ë2, [ 3 c ])}, ωa7 = {(ë2, [ 4 b , 3 c ])}, ωa8 = {(ë1, [ 5 a ]), (ë2, [ 4 b ])}, ωa9 = {(ë1, [ 5 a ]), (ë2, [ 3 c ])}, ωa10 = {(ë1, [ 5 a ]), (ë2, [ 4 b , 3 c ])}, ωa11 = {(ë1, [ 4 b ]), (ë2, [ 4 b ])}, ωa12 = {(ë1, [ 4 b ]), (ë2, [ 3 c ])}, ωa13 = {(ë1, [ 4 b ]), (ë2, [ 4 b , 3 c ])}, ωa14 = {(ë1, [ 5 a , 4 b ]), (ë2, [ 4 b ])}, ωa15 = {(ë1, [ 5 a , 4 b ]), (ë2, [ 3 c ])}, ωa16 = ωa. example 3.3. let h = [ 1 2 , 1 3 , 2 4 , 3 5 , 2 6 , 5 7 , 1 8 , 5 9 , 4 10 ] and e = {ë1, ë2, ë3, ë4, ë5, ë6} where ë1 denotes divisibility by 2, ë2 denotes divisibility by 3, ë3 denotes divisibility by 4, ë4 denotes divisibility by 5, ë5 denotes divisibility by 6, ë6 denotes divisibility by prime numbers. let a = {ë3, ë4, ë5} ⊆ e and a soft multi-set over h is ωa = {(ë3, [ 2 4 , 1 8 ]), (ë4, [ 3 5 , 4 10 ]), (ë5, [ 2 6 ])}. then |p̃w(ωa)| = 2 |x1|+|x2|+|x3| = 22+2+1 = 25 = 32, where |x1| = 2, since x1 = {4, 8}, |x2| = 2, since x2 = {5, 10} and |x3| = 1, since x3 = {6}. the soft power whole multi-set of the sms ωa is given by p̃w(ωa) = {ωa1, ωa2, · · ·, ωa32}, where ωa1 = ω∅, ωa2 = {(ë3, [ 2 4 ])}, ωa3 = {(ë3, [ 1 8 ])}, ωa4 = {(ë3, [ 2 4 , 1 8 ])}, ωa5 = {(ë4, [ 3 5 ])}, ωa6 = {(ë4, [ 4 10 ])}, ωa7 = {(ë4, [ 3 5 , 4 10 ])}, ωa8 = {(ë5, [ 2 6 ])}, ωa9 = {(ë3, [ 2 4 ]), (ë4, [ 3 5 ])}, ωa10 = {(ë3, [ 2 4 ]), (ë4, [ 4 10 ])}, ωa11 = {(ë3, [ 2 4 ]), (ë4, [ 3 5 , 4 10 ])}, ωa12 = {(ë3, [ 1 8 ]), (ë4, [ 3 5 ])}, 74 certain properties of soft multi-set topology with applications in multi-criteria decision making ωa13 = {(ë3, [ 1 8 ]), (ë4, [ 4 10 ])}, ωa14 = {(ë3, [ 1 8 ]), (ë4, [ 3 5 , 4 10 ])}, ωa15 = {(ë3, [ 2 4 ]), (ë5, [ 2 6 ])}, ωa16 = {(ë3, [ 1 8 ]), (ë5, [ 2 6 ])}, ωa17 = {(ë3, [ 2 4 , 1 8 ]), (ë5, [ 2 6 ])}, ωa18 = {(ë3, [ 2 4 , 1 8 ]), (ë4, [ 3 5 ])}, ωa19 = {(ë3, [ 2 4 , 1 8 ]), (ë4, [ 4 10 ])}, ωa20 = {(ë3, [ 2 4 , 1 8 ]), (ë4, [ 3 5 , 4 10 ])}, ωa21 = {(ë4, [ 3 5 ]), (ë5, [ 2 6 ])}, ωa22 = {(ë4, [ 4 10 ]), (ë5, [ 2 6 ])}, ωa23 = {(ë4, [ 3 5 , 4 10 ]), (ë5, [ 2 6 ])}, ωa24 = {(ë3, [ 2 4 ]), (ë4, [ 3 5 ]), (ë5, [ 2 6 ])}, ωa25 = {(ë3, [ 2 4 ]), (ë4, [ 4 10 ]), (ë5, [ 2 6 ])}, ωa26 = {(ë3, [ 2 4 ]), (ë4, [ 3 5 , 4 10 ]), (ë5, [ 2 6 ])}, ωa27 = {(ë3, [ 1 8 ]), (ë4, [ 3 5 ]), (ë5, [ 2 6 ])}, ωa28 = {(ë3, [ 1 8 ]), (ë4, [ 4 10 ]), (ë5, [ 2 6 ])}, ωa29 = {(ë3, [ 1 8 ]), (ë4, [ 3 5 , 4 10 ]), (ë5, [ 2 6 ])}, ωa30 = {(ë3, [ 2 4 , 1 8 ]), (ë4, [ 3 5 ]), (ë5, [ 2 6 ])}, ωa31 = {(ë3, [ 2 4 , 1 8 ]), (ë4, [ 4 10 ]), (ë5, [ 2 6 ])}, ωa32 = ωa. definition 3.4. ”let ωa be a soft multi-set over universal multi-set h. a sms-topology on a soft multi-set ωa, denoted by τ̃, is a collection of soft multi subsets of ωa having the following properties: (i) ω∅, ωa ∈ τ̃. (ii) union of any number of members of τ̃ belongs to τ̃ i.e. {ωai⊆̃ωa : i ∈ i ⊆ n}⊆̃τ̃ ⇒ ⋃̃ i∈i ωai ∈ τ̃. (iii) intersection of finite number of members of τ̃ belongs to τ̃ i.e. {ωai⊆̃ωa : 1 ≤ i ≤ n, n ∈ n}⊆̃τ̃ ⇒ ⋂̃ 1≤i≤nωai ∈ τ̃. then a sms topological space is denoted by (ωa, τ̃)” (mukherjee et al. (2014), and tokat and osmanoglu (2011,2013)). example 3.5. let h = [ 5 a , 4 b , 3 c ], e = {ë1, ë2, ë3}, a = {ë1, ë2} ⊆ e and a soft multi-set over h is ωa = {(ë1, [ 5 a , 4 b ]), (ë2, [ 4 b , 3 c ])} as given in example 3.2. then τ̃1 = {ω∅, ωa}, τ̃2 = p̃w(ωa), and τ̃3 = {ω∅, {(ë1, [ 4 b ])}, {(ë1, [ 4 b ]), (ë2, [ 3 c ])}, {(ë1, [ 5 a , 4 b ]), (ë2, [ 4 b ])}, ωa} are three sms topologies on the soft multi-set ωa. likewise τ̃4 = {ω∅, {(ë1, [ 5 a ])}, {(ë1, [ 4 b ])}, ωa} is not a sms-topology on ωa. example 3.6. take soft multi-set (sms) ωa = {(ë3, [ 2 4 , 1 8 ]), (ë4, [ 3 5 , 4 10 ]), (ë5, [ 2 6 ])}. which is same as given in example 3.3. so that τ̃1 = {ω∅, ωa}, τ̃2 = {ω∅, ωa24, ωa26, ωa30, ωa} or 75 riaz et al./decis. mak. appl. manag. eng. 3 (2) (2020) 70-96 τ̃2 = {ω∅, {(ë3, [ 2 4 ]), (ë4, [ 3 5 ]), (ë5, [ 2 6 ])}, {(ë3, [ 2 4 ]), (ë4, [ 3 5 , 4 10 ]), (ë5, [ 2 6 ])}, {(ë3, [ 2 4 , 1 8 ]), (ë4, [ 3 5 ]), (ë5, [ 2 6 ])}, ωa} and τ̃3 = p̃w(ωa) are sms topologies on the sms ωa. throughout this work, we use the following definition of complement in a sms topological space. definition 3.7. the soft multi complement ωc̃b of a soft multi subset ωb in a sms topological space (ωa, τ̃) is defined as ωc̃b = ωa\̃ωb. definition 3.8. let τ̃ be a sms-topology then each of its element is called soft open multi-set (soms) and the complement of each soft open multi-set is called called a soft closed multi-set. example 3.9. let τ̃2 be the sms-topology which considered in example 3.6. since ωa24 = {(ë3, [ 2 4 ]), (ë4, [ 3 5 ]), (ë5, [ 2 6 ])} is a soft open multi-set. then ωc̃a24 = {(ë3, [ 1 8 ]), (ë4, [ 4 10 ])} is a soft closed multi-set. remark. the union of two sms-topologies on a sms ωe may not be a sms-topology on ωe. example 3.10. let h = [ 2 g , 4 h , 6 i ], e = {ë1, ë2}, and τ̃1 = {ω∅, ωẽ, ω1e , ω2e , ω3e , ω4e }, τ̃2 = {ω∅, ωẽ, ω5e , ω6e , ω7e , ω8e } be two sms topologies on ωẽ where ω1e , ω2e , ω3e , ω4e , ω5e , ω6e , ω7e and ω8e are smss over h defined as follows: ω1e = {(ë1, [ 4 h ]), (ë2, [ 2 g ])}, ω2e = {(ë1, [ 4 h , 6 i ], (ë2, [ 2 g , 4 h ])}, ω3e = {(ë1, [ 2 g , 4 h ]), (ë2, x)}, ω4e = {(ë1, [ 2 g , 4 h ]), (ë2, [ 2 g , 6 i ])}, ω5e = {(ë1, [ 4 h ]), (ë2, [ 2 g ])}, ω6e = {(ë1, [ 4 h , 6 i ], (ë2, [ 2 g , 4 h ])}, ω7e = {(ë1, [ 2 g , 4 h ]), (ë2, [ 2 g , 4 h ])}, ω8e = {(ë1, [ 4 h ]), (ë2, [ 2 g , 6 i ])}. now, we define τ̃ = τ̃1∩̃τ̃2 = {ω1e , ω2e , ω3e , ω4e , ω5e , ω6e , ω7e , ω8e }. if we take ω2e ∪̃ω7e = he. then he(ë1) = f2e (ë1) ∪ f7e (ë1) = [ 4 h , 6 i ] ∪ [ 2 g , 4 h ] = h he(ë2) = f2e (ë2) ∪ f7e (ë2) = [ 2 g , 4 h ] ∪ [ 2 g , 4 h ] = [ 2 g , 4 h ] but he /∈ τ̃. thus τ̃ is not a sms-topology on ωẽ. definition 3.11. let (ωa, τ̃) be a sms topological space and b̃⊆̃τ̃. if every element of τ̃ can be written as a union of members of b̃, then b̃ is called a soft multi basis for the sms-topology τ̃. example 3.12. let h = [ 5 a , 4 b , 3 c ], e = {ë1, ë2, ë3}, a = {ë1, ë2} ⊆ e and a sms over h is ωa = {(ë1, [ 5 a , 4 b ]), (ë2, [ 4 b , 3 c ])}. let τ̃2 = p̃w(ωa). then b̃ = {ω∅, ωa2, ωa3, ωa5, ωa6} or b̃ = {ω∅, {(ë1, [ 5 a ])}, {(ë1, [ 4 b ])}, {(ë2, [ 4 b ])}, {(ë2, [ 3 c ])}} is a soft multi basis for the sms-topology τ̃2. 76 certain properties of soft multi-set topology with applications in multi-criteria decision making example 3.13. consider the sms-topology τ̃3 that is given in example 3.6. since τ̃3 = p̃w(ωa). then b̃ = {ω∅, ωa2, ωa3, ωa5, ωa6, ωa8} is a soft multi basis for the sms-topology τ̃3. definition 3.14. let (ωa, τ̃ωa) be a sms topological space and ωb is contained in ωa. let τ̃ωb be the collection of ωbi such that ωbi = ωai∩̃ωb where each ωai are contained in τ̃ωa. then τ̃ωb is called a soft multi subspace topology or soft multi relative topology on ωb. hence (ωb, τ̃ωb ) is soft multi subspace of (ωa, τ̃ωa). theorem 3.15. let (ωa, τ̃ωa) be a sms topological space and ωb⊆̃ωa. then a soft multi subspace topology τ̃ωb on ωb is a sms-topology. proof. (i) since ωb⊆̃ωa and ωφ⊆̃ωa. then clearly ωφ and ωb are contained in τ̃ωb this is so because ωφ∩̃ωb = ωφ and ωa∩̃ωb = ωb, where ωφ, ωa are in τ̃ωa. (ii)-(iii) since τ̃ωa sms topology, then by the given relations ⋂̃n i=1 (ωai∩̃ωb) = ( ⋂̃n i=1 ωai)∩̃ωb, ⋃̃ i∈i (ωai∪̃ωb) = ( ⋂̃ i∈i ωai)∩̃ωb τ̃ωb is the sms topology on ωb. example 3.16. let us consider sms-topology τ̃3 on ωa as given in example 3.5. let ωb = ωa10 = {(ë1, [ 5 a ]), (ë2, [ 4 b , 3 c ])}, and τ̃3 = {ω∅, {(ë1, [ 4 b ])}, {(ë1, [ 4 b ]), (ë2, [ 3 c ])}, {(ë1, [ 5 a , 4 b ]), (ë2, [ 4 b ])}, ωa} then τ̃ωb = {ω∅, ωa6, ωa9, ωa10}. so (ωb, τ̃ωb ) is soft multi subspace of (ωa, τ̃3). example 3.17. let us consider the sms-topology τ̃2 that is given in example 3.6. let ωb = ωa11 = {(ë3, [ 2 4 ]), (ë4, [ 3 5 , 4 10 ])} then τ̃ωb = {ω∅, ωa9, ωa11}. so (ωb, τ̃ωb ) is soft multi subspace of (ωa, τ̃2). definition 3.18. ”let (ωa, τ̃) be a sms topological space and ωb⊆̃ωa. then the soft multi interior of ωb, denoted by int(ωb) or ω ◦ b, is the soft multi union of all soft open multi subsets of ωb”. example 3.19. let us consider the sms-topology τ̃3 given in example 3.5. if ωb = ωa13 = {(ë1, [ 4 b ]), (ë2, [ 4 b , 3 c ])}, then ω◦b = ω∅∪̃ωa3∪̃ωa12 = ωa12. example 3.20. let us consider the sms-topology τ̃2 given in example 3.6. if ωb = ωa17 = {(ë3, [ 2 4 , 1 8 ]), (ë5, [ 2 6 ])}, then ω◦b = ω∅. definition 3.21. let (ωa, τ̃) be a sms topological space and ωb⊆̃ωa. then the soft multi closure of ωb, denoted by cl(ωb) or ωb, is the soft multi intersection of all soft closed super multi-sets of ωb. example 3.22. let us consider the sms-topology τ̃3 given in example 3.5. if ωb = ωa10 = {(ë1, [ 5 a ]), (ë2, [ 4 b , 3 c ])}, then ωc̃a3 = {(ë1, [ 5 a ]), (ë2, [ 4 b , 3 c ])} = ωb and ω c̃ φ = ωa are soft closed super multi-sets of ωb. hence ωb = ωa∩̃ωb = ωb. 77 riaz et al./decis. mak. appl. manag. eng. 3 (2) (2020) 70-96 example 3.23. let us consider the sms-topology τ̃2 given in example 3.6. if ωb = ωa3 = {(ë3, [ 1 8 ])}, then ωc̃a24 = ωa13, ω c̃ a26 = ωa3 and ω c̃ φ = ωa are soft closed super multi-sets of ωb. hence ωb = ωa3∩̃ωa13∩̃ωa = ωa3. theorem 3.24. let (ωa, τ̃) be a sms topological space and ωb, ωc⊆̃ωa. then, (i) (ω◦b) ◦ = ω◦b (ii) ωb⊆̃ωc ⇒ ω ◦ b⊆̃ω ◦ c (iii) ω◦b∩̃ω ◦ c = (ωb∩̃ωc) ◦ (iv) ω◦b∪̃ω ◦ c⊆̃(ωb∪̃ωc) ◦. (v) (ωb) = ωb (vi) ωc⊆̃ωb ⇒ ωc⊆̃ωb (vii) (ωb∩̃ωc)⊆̃ωb∩̃ωc (viii) (ωb∪̃ωc) = ωb∪̃ωc. (ix) ω◦b⊆̃ωb⊆̃ωb proof. the proof follows by definition 3.18 and definition 3.21. example 3.25. let u = [ 2 g , 4 h , 6 i ], e = {ë1, ë2} and τ̃ = {ω∅, ωẽ, ω1e , ω2e , ω3e , ..., ω7e }, where ω1e = {(ë1, [ 2 g , 4 h ]), (ë2, [ 2 g , 4 h ])}, ω2e = {(ë1, [ 4 h ]), (ë2, [ 2 g , 6 i ])}, ω3e = {(ë1, [ 4 h , 6 i ]), (ë2, [ 2 g ])}, ω4e = {(ë1, [ 4 h ]), (ë2, [ 2 g ])}, ω5e = {(ë1, [ 2 g , 4 h ]), (ë2, u)}, ω6e = {(ë1, u), (ë2, [ 2 g , 4 h ])}, ω7e = {(ë1, [ 4 h , 6 i ]), (ë2, [ 2 g , 4 h ])}. then (ω ẽ , τ̃) is a soft multi-set topological space. let ωe and ω̈e are defined as follows: ωe = {(ë1, [ 2 g , 6 i ]), (ë2, ∅)}, ω̈e = {(ë1, [ 4 h , 6 i ]), (ë2, [ 2 g , 4 h ])}. then ωe∩̃ω̈e = {(ë1, [ 6 i ]), (ë2, ∅)}. now, ωe = ωẽ∩̃ω c̃ 2e ∩̃ωc̃4e = ω c̃ 2e and ω̈e = ωẽ. therefore ωe∩̃ω̈e = ωe. also ωe∩̃ω̈e = ∩̃{ωẽ, ω c̃ 1e , ωc̃2e , ω c̃ 4e , ωc̃5e } = ω c̃ 5e . so ωe∩̃ω̈e⊆̃ωe∩̃ω̈e but ωe∩̃ω̈e*̃ωe∩̃ω̈e. hence, ωe∩̃ω̈e 6= ωe∩̃ω̈e. definition 3.26. let (ωa, τ̃) be a soft multi-set topological space and ωb⊆̃ωa. the soft multi interior of soft multi complement of ωb is called the soft multi exterior of ωb and is denoted by ext(ωb) or ω ẽ b. in other words, ωẽb = (ω c̃ b) ◦. example 3.27. from example 3.5, we take sms-topology τ̃3. then for 78 certain properties of soft multi-set topology with applications in multi-criteria decision making ωb = ωa14 = {(ë1, [ 5 a , 4 b ]), (ë2, [ 4 b ])}, then ωc̃a14 = {(ë2, [ 3 c ])} = ωa6. hence ω ẽ b = (ω c̃ b) ◦ = ωφ, (because null soft multi-set is the only soft open multi subset contained in ωc̃b). example 3.28. from example 3.6, we take sms-topology τ̃2. then for ωb = ωa30 = {(ë3, [ 2 4 , 1 8 ]), (ë4, [ 3 5 ]), (ë5, [ 2 6 ])}, then ωc̃b = {(ë4, [ 4 10 ])} = ωa6. hence ω ẽ b = (ω c̃ b) ◦ = ωφ, (because null soft multi-set is the only soft open multi subset contained in ωc̃b). theorem 3.29. let (ωa, τ̃) be a sms topological space and ωb, ωc⊆̃ωa. then, (i) (ωb∪̃ωc) ẽ = (ωb) ẽ∩̃(ωc) ẽ, (ii) (ωb) ẽ∪̃(ωc) ẽ⊆̃(ωb∩̃ωc) ẽ. proof. (i) (ωb∪̃ωc) ẽ = ((ωb∪̃ωc) c̃)◦ = (ωc̃b∩̃ω c̃ c) ◦ = (ωc̃b) ◦∩̃(ωc̃c) ◦ = (ωb) ẽ∩̃(ωc) ẽ (ii) (ωb) ẽ∪̃(ωc) ẽ = (ωc̃b) ◦∪̃(ωc̃c) ◦ ⊆̃(ωc̃b∪̃ω c̃ c) ◦ = (ω◦b∪̃(ω c̃ c) ◦ = (ωb∩̃ωc) ẽ. definition 3.30. ”let (ωa, τ̃) be a sms topological space. a soft multi point α ∈ ωa is said to be a soft multi interior point of the soft multi-set ωa if there is a soft open multi-set ωb such that α ∈ ωb⊆̃ωa. moreover, if α is soft multi interior point of the soft multi-set ωa then ωa is called soft multi neighborhood (or soft multi open neighborhood) of α. thus ν̃(α) = {ωb : ωb ∈ τ̃} is the family of soft multi neighborhoods of α” (mukherjee et al. (2014), and tokat and osmanoglu (2011,2013)). example 3.31. let ωa = {(ë1, [ 5 a , 4 b ]), (ë2, [ 4 b , 3 c ])} be the soft multi-set as given in example 3.5 and τ̃3 = {ω∅, {(ë1, [ 4 b ])}, {(ë1, [ 4 b ]), (ë2, [ 3 c ])}, {(ë1, [ 5 a , 4 b ]), (ë2, [ 4 b ])}, ωa} be a sms-topology on the soft multi-set ωa. let α = (ë1, [ 5 a , 4 b ]) ∈ ωa then α ∈ ωa14⊆̃ωa, where ωa14 = {(ë1, [ 5 a , 4 b ]), (ë2, [ 4 b ])} is the soft multi open neighborhood of α. similarly α ∈ ωa⊆̃ωa this shows that ωa is multi soft neighborhood of α. thus ν̃(α) = {ωa14, ωa} is the family of soft multi neighborhoods of α. example 3.32. let ωa = {(ë3, [ 2 4 , 1 8 ]), (ë4, [ 3 5 , 4 10 ]), (ë5, [ 2 6 ])} be the soft multi-set as given in example 3.6 and τ̃2 = {ω∅, {(ë3, [ 2 4 ]), (ë4, [ 3 5 ]), (ë5, [ 2 6 ])}, {(ë3, [ 2 4 ]), (ë4, [ 3 5 , 4 10 ]), (ë5, [ 2 6 ])}, {(ë3, [ 2 4 , 1 8 ]), (ë4, [ 3 5 ]), (ë5, [ 2 6 ])}, ωa} be a sms-topology on the sms ωa. let α = (ë3, [ 2 4 , 1 8 ]) ∈ ωa then α ∈ ωa30⊆̃ωa, where ωa30 = {(ë3, [ 2 4 , 1 8 ]), (ë4, [ 3 5 ]), (ë5, [ 2 6 ])} is the soft multi open neighborhood of α. similarly α ∈ ωa⊆̃ωa this shows that ωa is soft multi neighborhood of α. thus ν̃(α) = {ωa30, ωa} is the family of soft multi neighborhoods of α. theorem 3.33. let τ̃ be a sms topology on sms ωa. then a subset ωb of ωa is said to be open if and only if it is neighborhood of each of its own soft multi point. proof. let ωb be soft multi open subset of ωa. then for each soft multi point λ in ωb, we have λ∈̃ωb⊆̃ωb. this shows that ωb is a neighborhood of each of its own soft multi point. conversely, suppose that ωb is a neighborhood of each of its own soft multi point. then for each soft multi point λ∈̃ωb there exists soft multi open set ωuλ such that λ∈̃ωuλ⊆̃ωb. this shows that ωb = ∪̃{λ}⊆̃∪̃ωuλ⊆̃ωb. thus we get ωb = ⊆̃∪̃ωuλ. this proves that ωb is soft multi open set. 79 riaz et al./decis. mak. appl. manag. eng. 3 (2) (2020) 70-96 definition 3.34. ”let (ωa, τ̃) be a sms topological space and ωb⊆̃ωa and α ∈ ωa. if every multi soft neighborhood of α soft multi intersects ωb in some soft multi points other than α itself, then α is called a soft multi limit point of ωb. the collection of all soft multi limit points of ωb is denoted by ω ′ b. in other words, if (ωa, τ̃) is a sms topological space and ωb⊆̃ωa and α ∈ ωa, then α ∈ ω ′ b ⇔ ωc∩̃(ωb\̃{α}) 6= ωφ for all ωc ∈ ν̃(α)”. example 3.35. consider example 3.31. if ωb = ωa14 and α = (x1, [ 5 a , 4 b ]) ∈ ωa, then α ∈ ω ′ b, since ωa∩̃(ωb\̃{α}) 6= ωφ and ωa14∩̃(ωb\̃{α}) 6= ωφ. example 3.36. consider example 3.32. if ωb = ωa30 and α = (ë3, [ 2 4 , 1 8 ]) ∈ ωa, then α ∈ ω ′ b, since ωa∩̃(ωb\̃{α}) 6= ωφ and ωa30∩̃(ωb\̃{α}) 6= ωφ. theorem 3.37. let (ωa, τ̃) be a sms topological space and ωb⊆̃ωa. then, ωb∪̃ω ′ b = ωb. proof. if α ∈ ωb∪̃ω ′ b, then α ∈ ωb or α ∈ ω ′ b. in this case, if α ∈ ωb, then α ∈ ωb. if α ∈ ω ′ b, then ωc∩̃(ωb\̃{α}) 6= ωφ for all ωc ∈ ν̃(α), and so ωc∩̃ωb 6= ωφ for all ωc ∈ ν̃(α); hence, α ∈ ωb. conversely, if α ∈ ωb, then α ∈ ωb or α ∈ ω ′ b. in this case, if α ∈ ωb, it is trivial that α ∈ ωb∪̃ω ′ b. if α /∈ ωb, then ωc∩̃(ωb\̃{α}) 6= ωφ for all ωc ∈ ν̃(α). therefore, α ∈ ω ′ b, so α ∈ ωb∪̃ω ′ b. hence ωb∪̃ω ′ b = ωb. theorem 3.38. let (ωa, τ̃) be a sms topological space and ωb⊆̃ωa. then, ωb is soft closed multi-set if and only if ω′b⊆̃ωb. proof. ωb = ωb ⇔ ωb∪̃ω ′ b = ωb ⇔ ω ′ b⊆̃ωb. theorem 3.39. let (ωa, τ̃) be a sms topological space and ωb, ωc⊆̃ωa. then, (i) ω′b⊆̃ωb (ii) ωb⊆̃ωc ⇒ ω ′ b⊆̃ω ′ c (iii) (ωb∩̃ωc) ′⊆̃ω′b∩̃ω ′ c (iv) (ωb∪̃ωc) ′ = ω′b∪̃ω ′ c (v) ωb is a soft closed multi-set ⇔ ω ′ b⊆̃ωb. proof. the proof is straightforward. theorem 3.40. let (ωa, τ̃) be a sms topological space and ωb, ωc⊆̃ωa. then, (i) (ωc̃b) = (ω ◦ b) c̃ (ii) (ωb) c̃ = (ωc̃b) ◦ (iii) ω◦b = ((ω c̃ b)) c̃ (iv) ωb = ((ω c̃ b) ◦)c̃ (v) (ωb\̃ωc) ◦⊆̃ωb ◦\̃ωc ◦ . proof. (i) let α ∈ ωb such that α /∈ ω ◦ b. then, for each soft multi open neighborhood of ωc of α, ωc soft multi intersects ωc̃b. otherwise, for some soft multi open neighborhood ωc of α, ωc∩̃ω c̃ b = ωφ or ωc⊆̃ωb. since ω◦b is the largest soft open multi-set in ωb, therefore α ∈ ωc⊆̃ω ◦ b, which is a contradiction. therefore, α ∈ (ωc̃b). hence, (ω ◦ b) c̃⊆̃(ωc̃b). 80 certain properties of soft multi-set topology with applications in multi-criteria decision making conversely, suppose α ∈ ωc̃b, then by definition 3.34, α ∈ ω c̃ b or α is a soft multi limit point of ω c̃ b. if α ∈ ωc̃b, then α ∈ (ω ◦ b) c̃. in the second case, α /∈ ω◦b. otherwise, by the definition of soft multi limit point, ω◦b∩̃ω c̃ b 6= ωφ, which is false. therefore, (ω c̃ b)⊆̃(ω ◦ b) c̃. combining, we get (i). (ii) clearly (ωb) c̃ = ( ⋂̃ ωai ⊇̃ωb,ω c̃ ai ∈τ̃ ωai) c̃ = ⋃̃ ωc̃ai = (ω c̃ b) ◦. (iii) and (iv) are directly obtained by taking the complements of (i) and (ii), respectively. (v) (ωb\̃ωc) ◦ = (ωb∩̃ω c̃ c) ◦ = ω◦b∩̃(ω c̃ c) ◦ = ω◦b∩̃(ωc) c̃ ⊆̃ω◦b∩̃(ω ◦ c) c̃ = ω◦b\̃ω ◦ c. definition 3.41. let (ωa, τ̃) be a sms topological space and ωb⊆̃ωa. the soft multi frontier or boundary of ωb is denoted by ωr(ωb) or ω b̃ b and is defined as ω b̃ b = ωb∩̃ω c̃ b. stated differently, the soft multi points that do not belong to soft multi interior and exterior of ωb are in ω b̃ b. example 3.42. from example 3.5, we take sms-topology τ̃3, then for ωb = ωa14 = {(ë1, [ 5 a , 4 b ]), (ë2, [ 4 b ])}, then ωc̃a14 = {(ë2, [ 3 c ])} = ωa6. hence ω b̃ b = ωb∩̃ω c̃ b = ωa∩̃ωa6 = ωa6. example 3.43. let us consider the sms-topology τ̃2 given in example 3.6. if ωb = ωa30 = {(ë3, [ 2 4 , 1 8 ]), (ë4, [ 3 5 ]), (ë5, [ 2 6 ])}, then ωc̃a30 = {(ë4, [ 4 10 ])} = ωa6. hence ω b̃ b = ωb∩̃ω c̃ b = ωa∩̃ωa6 = ωa6. theorem 3.44. let (ωa, τ̃) be a sms topological space and ωb, ωc⊆̃ωa. then, (i) ωb̃b⊆̃ωb (ii) ωb̃b = (ω c̃ b) b̃ (iii) ωb̃b = ωb\̃ω ◦ b. proof. (i) the proof is clear by definition of a soft multi boundary. (ii) take as given α ∈ ωb̃b ⇔ ωc∩̃ωb 6= ωφ and ωc∩̃ωb c̃ 6= ωφ for all ωc ∈ ν̃(α) ⇔ ωc∩̃ωb c̃ 6= ωφ and ωc∩̃(ωb c̃)c̃ 6= ωφ for all ωc ∈ ν̃(α). hence ω b̃ b = (ω c̃ b) b̃. (iii) by using the definitions of a soft multi closure and a multi soft interior, we have ωb\̃ω ◦ b = ωb∩̃(ω ◦ b) c̃ = ωb∩̃( ⋃̃ ωbi ⊆̃ωb,ωbi ∈τ̃ ωbi) c̃ = ωb∩̃( ⋂̃ ωbi c̃) = ωb∩̃(ωbi c̃) = ωb̃b. theorem 3.45. let (ωa, τ̃) be a sms topological space and ωb⊆̃ωa. then, (i) (ωb̃b) c̃ = ω◦b∪̃(ω c̃ b) ◦ = ω◦b∪̃ω ẽ b (ii) ωb = ωb∪̃ω b̃ b (iii) ω◦b = ωb\̃ω b̃ b. proof. (i) ω◦b∪̃(ω c̃ b) ◦ = ((ω◦b) c̃)c̃∪̃(((ωc̃b) ◦)c̃)c̃ = [(ω◦b) c̃∩̃((ωc̃b) ◦)c̃ ]c̃ = [ωc̃b∩̃ωb ] c̃ = (ωb̃b) c̃. (ii) ωb∪̃ω b̃ b = ωb∪̃(ωb∩̃ω c̃ b) = [ωb∪̃ωb ]∩̃[ωb∪̃ω c̃ b ] = ωb∩̃[ωb∪̃ω c̃ b ] = ωb∩̃ωa = ωb. (iii) ωb\̃ω b̃ b = ωb∩̃(ω b̃ b) c̃ = ωb∩̃(ω ◦ b∪̃(ω c̃ b) ◦) (by (i)) = [ωb∩̃ω ◦ b]∪̃[ωb∩̃(ω c̃ b) ◦] = ω◦b∪̃ωφ = ω ◦ b. remark. from theorem 3.45, it follows that ωa = ω ◦ b∪̃ω ẽ b∪̃ω b̃ b. theorem 3.46. let (ωa, τ̃) be a sms topological space and ωb⊆̃ωa. then, (i) ωb̃b∩̃ω ◦ b = ωφ (ii) ωb̃b∩̃ω ẽ b = ωφ. 81 riaz et al./decis. mak. appl. manag. eng. 3 (2) (2020) 70-96 proof. (i) ωb̃b∩̃ω ◦ b = (ωb∩̃ω c̃ b)∩̃ω ◦ b = ωb∩̃(ω ◦ b) c̃∩̃ω◦b = ωφ. (ii) ωb̃b∩̃ω ẽ b = (ω c̃ b) ◦∩̃(ωb∩̃ω c̃ b) = (ω c̃ b) ◦∩̃ωb∩̃ω c̃ b = (ωb) c̃∩̃ωb∩̃ω c̃ b = ωφ. theorem 3.47. let (ωa, τ̃) be a sms topological space and ωb⊆̃ωa. then, (i) ωb is soft open multi-set ⇔ ωb∩̃ω b̃ b = ωφ (ii) ωb is soft closed multi-set ⇔ ω b̃ b⊆̃ωb. (iii) ωb is both soft open multi-set and soft closed multi-set ⇔ ω b̃ b = ∅. proof. (i) let ωb is soft open multi-set. then ω ◦ b = ωb. thus ωb∩̃ω b̃ b = ω ◦ b∩̃ω b̃ b = ωφ (by theorem 3.46(i)). conversely, let ωb∩̃ωb = ωφ. then, ωb∩̃[ωb∩̃ω c̃ b] = ωφ, ωb∩̃ω c̃ b = ωφ, or ω c̃ b⊆̃ω c̃ b, which implies that ωc̃b is soft closed multi-set and hence, ωb is soft open multi-set. (ii) let ωb is soft closed multi-set. then ωb = ωb. now, ω b̃ b = ωb∩̃ω c̃ b⊆̃ωb = ωb, or ω b̃ b⊆̃ωb and conversely. (iii) we know that ωb is open ⇔ (ωb) ◦ = ωb and ωb is closed ⇔ ωb = ωb. also by theorem 3.45, we obtain ωb = ωb∪̃ω b̃ b and ω ◦ b = ωb\̃ω b̃ b. this completes the proof. theorem 3.48. let (ωa, τ̃) be a sms topological space and ωb, ωc⊆̃ωa. then, (i) [ωb∪̃ωc] b̃⊆̃[ωb̃b∩̃ω c̃ c]∪̃[ω b̃ c∩̃ω c̃ b] (ii) [ωb∩̃ωc] b̃⊆̃[ωb̃b∩̃ωc ]∪̃[ω b̃ c∩̃ωb ]. proof. proof is obvious. theorem 3.49. let (ωa, τ̃) be a sms topological space and ωb⊆̃ωa. then, ((ωb̃b) b̃)b̃ = (ωb̃b) b̃. proof. (i) ((ωb̃b) b̃)b̃ = (ωb̃b) b̃∩̃((ωb̃b) b̃)c̃ = (ωb̃b) b̃∩̃((ωb̃b) b̃)c̃ (1) now, consider ((ωb̃b) b̃)c̃ = [(ωb̃b)∩̃(ω b̃ b) c̃ ]c̃ = (ωb̃b∩̃(ω b̃ b) c̃)c̃ = (ωb̃b) c̃∪̃((ωb̃b) c̃)c̃. therefore, (((ωb̃b) b̃)c̃) = [(ωb̃b) c̃∪̃((ωb̃b) c̃ )c̃ ] = ((ωb̃b) c̃ )∪̃(((ωb̃b) c̃)c̃) = ωc∪̃((ωc)c̃) = ωa (2) where ωc = ((ω b̃ b) c̃ ). from (1) and (2), we have ((ωb̃b) b̃)b̃ = (ωb̃b) b̃∩̃ωa = (ω b̃ b) b̃. definition 3.50. let (ωa, τ̃) be a sms topological space and ωb⊆̃ωa. then ωb is said to be a soft clopen multi-set if ωb is both soft open and soft closed multi-set. example 3.51. since ωφ and ωa are always present in τ̃, so ωφ and ωa are soft open multi-sets. moreover, ωφ and ωa are also soft closed multi-sets since ω c̃ φ = ωa and ω c̃ a = ωφ. in fact, these two soft multi-sets are soft open and soft closed multi-sets simultaneously. hence, ωφ and ωa are soft clopen multi-sets. example 3.52. let us consider the sms-topology τ̃3 given in example 3.6. let ωb, ωc∈̃τ̃3, where ωb = {(ë3, [ 2 4 , 1 8 ]), (ë4, [ 3 5 ]), (ë5, [ 2 6 ])}, and ωc = {(ë4, [ 4 10 ])}. then ωc̃c = {(ë3, [ 2 4 , 1 8 ]), (ë4, [ 3 5 ]), (ë5, [ 2 6 ])} = ωb. hence ωb is a soft clopen multi-set. 82 certain properties of soft multi-set topology with applications in multi-criteria decision making theorem 3.53. let (ωa, τ̃) be a sms topological space and ωb⊆̃ωa. ω b̃ b = ωφ if and only if ωb is soft clopen multi-set. proof. suppose that ωb̃b = ωφ. first we prove that ωb is a soft closed multi-set. consider ωb̃b = ωφ ⇒ ωb∩̃(ω c̃ b) = ωφ ⇒ ωb⊆̃((ω c̃ b) c̃) = ω◦b⊆̃ωb ⇒ ωb⊆̃ωb ⇒ ωb = ωb. this implies that ωb is a soft closed multi-set. now we now prove that ωb is a soft open multi-set. consider ωb̃b = ωφ ⇒ ωb∩̃(ω c̃ b) = ωφ or ωb∩̃(ω ◦ b) c̃ = ωφ ⇒ ωb⊆̃ω ◦ b ω◦b = ωb. this implies that ωb is a soft open multi-set. conversely, suppose that ωb is a soft clopen multi-set. then, ωb̃b = ωb∩̃(ω c̃ b) = ωb∩̃(ω ◦ b) c̃ = ωb∩̃ω c̃ b = ωφ. 4 mcdm based on sms-topology there are different kinds of decision-making methods for selection of a best alternative. sometimes it is quite difficult to select an appropriate decision-making method with similar situation in our real life problems. however, mcdm method based on sms-topology plays a enthusiastic role in our daily life and this is very helpful in selection of a best alternative. mcdm is the thought process of selecting a logical choice from the available options. the concept of aggregation operators in the framework of soft sets and fuzzy soft sets have been introduced by çağman et al. (2011). we used the notion of aggregation operators to compute aggregate fuzzy soft sets and aggregate multi-sets. 4.1 mcdm for selection of best alternative of biopesticides a big challenge to the agricultural department is to enlarge the production and to meet the demands of the increasing world population without destroying the environment. in modern agricultural exercises, the check of pests is generally completed by means of the extreme usage of agrochemicals, which is source of ambient pollution and the improvement of repellent pests. but biopesticides can proffer a best substitute to synthetic pesticides empowering safer check of pest communities. it is always a challenging task for a farmer to choose a best agrochemicals for biopesticides. every farmer has to face many difficulties to save his fields from pests. for these challenging tasks various components are take into examination by the farmer either searching for agrochemicals in order to provide safety from pests attack, improve the soil quality, increase the quantity of crops, enhance the quality of crops. major components of biopesticides include microbial pesticides, biochemical pesticides and biological control agent. the examples of biopesticides include insects, virus, bacteria, fungi, protozoan, and nematodes. table 1 gives the comparison of merits and demerits of biopesticides and chemicals. 83 riaz et al./decis. mak. appl. manag. eng. 3 (2) (2020) 70-96 biopesticides chemicals pesticides environmentally intelligent farming conflicting to intelligent farming cheaper, affordable costly, expansive warmly to non-target genus dangerous to non-target genus do not cause pollution serious pollution to the environment pests never develop resistance pests eventually become resistance expanding market inclination reduce market inclination fight their intended pests end up affecting non target species derived from living organisms contain non-living organism table 1: comparison analysis of biopesticides and chemicals algorithm 1 the selection of best alternative for biopesticides step 1: input a suitable parameter set s and universal multi-set h. step 2: input smss ωa and ωb over h. step 3: construct sms-topology τ̂ containing ωa and ωb as soft open mss in τ̂. step 4: compute the aggregate fuzzy soft sets by using the formula, γa = {(µi, γa(µi)) : µi ∈ s}, where γa(µi) = { ki/|ωa(µi)| ωi : ki ωi ∈ ωa(µi)}. step 5: find resultant fuzzy soft set γa ∨ γb = γa×b by applying ’or’ operation on γa and γb. step 6: use comparison table of γa ∨ γb to calculate row-sum (ri) and column-sum (ti) for ωi, ∀ i. step 7: calculate the resulting score ri of ωi, ∀ i. step 8: optimal choice is ωj that has max{ri}. step 9: compute the sms boundary of soft open multi-sets. step 10: here non-null sms boundary of sms that contains kj ωj is a decision set. figure 1 shows a brief flow-chart of algorithm 1. assume that a farmer wants to safe his fields from pests by using leading alternative of biopesticides without damaging the sustainability of environment. let h = [ 30 ω1 , 25 ω2 , 28 ω3 , 30 ω4 ] be the universe of some plants, where ω1 = sheesham (dalbergia sissoo), ω2 = safeda (eucalyptus), ω3 = sukh chain (pongamia pinnata), ω4 = neem (azadirachta indica) and the multiplicity of ωi (i = 1, 2, 3, 4) denotes the number of plants corresponding to ωi. consider the set of attributes s = {µ1, µ2, µ3, µ4}, where µ1 = provide safety from pests attack, µ2 = improve the soil quality, µ3 = increase the quantity of crops, 84 certain properties of soft multi-set topology with applications in multi-criteria decision making start input a multi-set h input a parameter set s input smss ωa, ωb constuct sms-topology τ̂ s.t ωa, ωb ∈ τ̂ compute γa = {(µi, γa(µi)) : µi ∈ s}, where γa(µi) = { ki/|ωa(µi)| ωi : ki ωi ∈ ωa(µi)}, find γa ∨ γbconstruct the comparison table of γa ∨ γb calculate score ri = ri − ti choose ωj that has max{ri} if ωa(µ) = ∅, ∀ µ ∈ s, ωa ∈ τ̂ yes no ωb̂a = ωφ ω b̂ a 6= ωφ select that ωb̂a that contains kj ωj stop ∀ ωa ∈ τ̂ figure 1: graphical representation of algorithm 1 µ4 = enhance the quality of crops. we here use the following algorithm to choose the best alternative of agrochemicals for biopesticides without damaging the environment to safe the fields from pests. two decision makers (dms) ω1 and ω2 presented the report to farmer on plant production by using traditional farming system. let the dms ω1 and ω2 select two sets of attribute a = {µ1, µ2, µ3, µ4} and b = {µ1, µ2, , µ3}, respectively. then dms construct two smss named as ωa and ωb over h given by ωa = {(µ1, [ 30 ω1 , 25 ω2 , 30 ω4 ]), (µ2, [ 25 ω2 , 28 ω3 , 30 ω4 ]), (µ3, [ 30 ω4 ]), (µ4, h)} and ωb = {(µ1, [ 30 ω1 , 25 ω2 ]), (µ2, [ 25 ω2 , 28 ω3 ]), (µ3, [ 30 ω4 ])}. 85 riaz et al./decis. mak. appl. manag. eng. 3 (2) (2020) 70-96 the first sms ωa can be written as: ωa µ1 µ2 µ3 µ4 ω1 30 0 0 30 ω2 25 25 0 25 ω3 0 28 0 28 ω4 30 30 30 30 the second sms ωb can be written as: ωb µ1 µ2 µ3 ω1 30 0 0 ω2 25 25 0 ω3 0 28 0 ω4 0 0 30 here we make a sms-topology on ωa as τ̂ = {ωφ, ωa, ωb}, where ωφ is an empty sms. now we find the aggregate fuzzy soft sets γa and γb given by γa = {(µ1, { 0.35 ω1 , 0.29 ω2 , 0.35 ω4 }), (µ2, { 0.30 ω2 , 0.33 ω3 , 0.36 ω4 }), (µ3, { 1 ω4 }), (µ4, { 0.26 ω1 , 0.22 ω2 , 0.24 ω3 , 0.26 ω4 })} and γb = {(µ1, { 0.54 ω1 , 0.45 ω2 }), (µ2, { 0.47 ω2 , 0.52 ω3 }), (µ3, { 1 ω4 })}. the fuzzy soft set γa can be written as: γa µ1 µ2 µ3 µ4 ω1 0.35 0 0 0.26 ω2 0.29 0.30 0 0.22 ω3 0 0.33 0 0.24 ω4 0.35 0.36 1 0.26 the fuzzy soft set γb can be written as: γb µ1 µ2 µ3 ω1 0.54 0 0 ω2 0.45 0.47 0 ω3 0 0.52 0 ω4 0 0 1 we apply here ’or’ operation on γa and γb, then we get 4 ∗ 3 = 12 attributes of the form µij = (µi, µj), ∀ i = 1, 2, 3, 4 and j = 1, 2, 3. we find the fuzzy soft set for the set of attributes a × b = {µ11, µ12, µ13, µ21, µ22, µ23, µ31, µ32, µ33, µ41, µ42, µ43}. after applying ’or’ operation we get fuzzy soft set γa ∨ γb given as: 86 certain properties of soft multi-set topology with applications in multi-criteria decision making γa ∨ γb = {(µ11, { 0.54 ω1 , 0.45 ω2 , 0 ω3 , 0.35 ω4 }), (µ12, { 0.35 ω1 , 0.47 ω2 , 0.52 ω3 , 0.35 ω4 }), (µ13, { 0.35 ω1 , 0.29 ω2 , 0 ω3 , 1 ω4 }), (µ21, { 0.54 ω1 , 0.45 ω2 , 0.33 ω3 , 0.36 ω4 }), (µ22, { 0 ω1 , 0.47 ω2 , 0.52 ω3 , 0.36 ω4 }), (µ23, { 0 ω1 , 0.30 ω2 , 0.33 ω3 , 1 ω4 }), (µ31, { 0.54 ω1 , 0.45 ω2 , 0 ω3 , 1 ω4 }), (µ32, { 0 ω1 , 0.47 ω2 , 0.52 ω3 , 1 ω4 }), (µ33, { 0 ω1 , 0 ω2 , 0 ω3 , 1 ω4 }), (µ41, { 0.54 ω1 , 0.45 ω2 , 0.24 ω3 , 0.26 ω4 }), (µ42, { 0.26 ω1 , 0.47 ω2 , 0.52 ω3 , 0.26 ω4 }), (µ43, { 0.26 ω1 , 0.22 ω2 , 0.24 ω3 , 1 ω4 })}. now the tabular form of γa ∨ γb is written as: γa ∨ γb µ11 µ12 µ13 µ21 µ22 µ23 µ31 µ32 µ33 µ41 µ42 µ43 ω1 0.54 0.35 0.35 0.54 0 0 0.54 0 0 0.54 0.26 0.26 ω2 0.45 0.47 0.29 0.45 0.47 0.30 0.45 0.47 0 0.45 0.47 0.22 ω3 0 0.52 0 0.33 0.52 0.33 0 0.52 0 0.24 0.52 0.24 ω4 0.35 0.35 1 0.36 0.36 1 1 1 1 0.26 0.26 1 now we find the comparison-table of fuzzy soft set γa ∨ γb by using the algorithm which is given by roy and maji in (2007). the comparison-table is given below. ω1 ω2 ω3 ω4 ω1 12 6 6 5 ω2 6 12 6 6 ω3 6 7 12 3 ω4 9 6 9 12 here we calculate the column-sum (ti) and row-sum (ri) after that we calculate the score (ri) for each ωi, i = 1, 2, 3, 4. row-sum (ri) column-sum (ti) score (ri = ri − ti) ω1 29 33 -4 ω2 30 31 -1 ω3 28 33 -5 ω4 36 26 10 table 2: tabular form of score score (ri = ri − ti) from table 2, we see that the topmost score is 10 which is gained by ω4. which shows that neem plant is selected to safe the fields from pests. now problem is that where to grow the neem plants to protect the field from pests. to solve this problem, we find the sms boundary of soft open multi-sets. if the sms boundary of at least one soft open multi-sets is not a null smss and contains 30 ω4 in non-null µ-approximate elements, ∀ µ ∈ s, then neem plants can grow on the corners of the field. if the sms boundary of all soft open multi-sets are null smss, then neem plants cannot grow on the corners of the field. 87 riaz et al./decis. mak. appl. manag. eng. 3 (2) (2020) 70-96 now compute the sms boundary of ωφ, ωa and ωb given as: ωb̂φ = ωφ, ω b̂ a = ωφ and ω b̂ b = ωb∩̂ω c b = ωa∩̂ω c b = ω c b = {(µ1, [ 30 ω4 ]), (µ2, [ 30 ω4 ]), (µ4, h)}. which shows that ωb̂b contains 30 ω4 in non-null µ-approximate elements ∀ µ ∈ s. so farmer decides to grow neem plants on the corners of field. the attention should be given to grow neem plants as a reassuring choice to exchange agrochemicals in agriculture pest control. neem can conduce to acceptable development and the determination of pest control problems in agriculture which can be best alternative to plant fertilizer. the proposed algorithm 1 is used in the environment of smss information for the selection of best alternative of biopesticides and the results are compared as indicated in the table 3. method ranking of alternatives the optimal alternative algorithm 1 (proposed) ω4 ≻ ω2 ≻ ω1 ≻ ω3 ω4 algorithm (çağman et al., 2011) ω4 ≻ ω2 ≻ ω1 ≻ ω3 ω4 algorithm (riaz et al., 2019) ω4 ≻ ω2 ≻ ω1 ≻ ω3 ω4 table 3: comparison of final ranking with existing methods using algorithm 1. 4.2 mcdm by using sms-topology for the selection of best textile company we present two modified algorithms based on sms-topology for a decision-making problem. at the end, we show the comparison of ranking of objects obtained by algorithm 2 and algorithm 3. furthermore we present another interesting application in agriculture for decision-making to find the optimal choice by using sms-topology and boundaries of soft open multi-set. algorithm 2 the selection of best textile company input: step 1: consider a universe of multi-set (ms) u. step 2: a set e of attributes. step 3: construct sms fa and fb. output: step 4: write sms-topology τ̃ in which fa and fb are open smss in τ̃. step 5: write the aggregate multi-sets of all open smss by using the formula, f ∗a = [ f ∗a(ωi) ωi : ωi ∈ x], where f ∗a(ωi) = σjωij. step 6: add f ∗a and f ∗ b to find decision ms. step 7: select the object with greatest multiplicity determined by max f ∗a⊕b(σ). 88 certain properties of soft multi-set topology with applications in multi-criteria decision making start stop input a multi-set u input a parameter set e input soft msets choose that σ that has max f ∗a⊕b(σ) construct sms compute f ∗a = [ f ∗a(ωi) ωi : ωi ∈ x] where f ∗a(ωi) = σjωij , ∀ fa ∈ τ̃ add f ∗a and f ∗ b that is f ∗ a ⊕ f ∗ b fa and fb topology τ̃ s.t fa, fb ∈ τ̃ figure 2: graphical representation of algorithm 2 graphical representation of algorithm 2 is shown in the figure 2. here we introduce another algorithm for sms-topology in decision-making. now we give algorithm 3 and compare the optimal decision obtained by algorithm 2. algorithm 3 the award of performance input: step 1: consider a universe of multi-set u. step 2: a set e of attributes. step 3: construct smss fa and fb. output: step 4: write sms-topology τ̃ containing fa and fb as open smss in τ̃. step 5: find the cardinal mss of all open smss by using the formula, cfa = [ cfa(λi) λi : λi ∈ e], where cfa(λi) = σiωij. step 6: find the aggregate multi-sets by using the formula, ... mf ∗ a = ... mfa ∗ m t cfa , → (1) where ... mfa, ... mcfa and ... mf ∗ a are representation matrices of fa, cfa and f ∗ a, respectively. step 7: adding f ∗a and f ∗ b to find decision mset. step 8: select the object that has greatest multiplicity i.e. max f ∗a⊕b(σ). a brief sketch of algorithm 3 is given in the figure 3. assume that government of a country is interested to give the ”award of performance” to best textile company of country to appreciate the contribution of the company. let u = [ 2 ω1 , 2 ω2 , 1 ω3 , 1 ω4 , 1 ω5 , 1 ω6 , 1 ω7 ] be the multi-set of big textile companies of the state, and the multiplicity of ωi, i = 1, 2, ..., 7 denotes the 89 riaz et al./decis. mak. appl. manag. eng. 3 (2) (2020) 70-96 start stop input a multi-set u input a parameter set e input soft msets choose that σ that has max f ∗a⊕b(σ) construct sms-topology τ̃ compute cfa = [ cfa(λi) λi : λi ∈ e], where cfa(λi) = σiωij., ∀ fa ∈ τ̃ add f ∗a and f ∗ b that is f ∗ a ⊕ f ∗ b fa and fb find f ∗a and f ∗ b s.t fa, fb ∈ τ̃ figure 3: graphical representation of algorithm 3 number of branches of company ωi that are selected for the award. let x = {ω1, ω2, ω3, ω4, ω5, ω6, ω7} be the support set of u. the set of parameters is given as e = {λ1, λ2, λ3, λ4, λ5} where λ1 = best hosiery, λ2 = best export, λ3 = healthy working environment, λ4 = use of modern technology, λ5 = expert workers. we here use the following algorithm 2 to select the best company of the state for the ”award of performance. the dms ω1 and ω2 construct two squads named as squad-ω1 and squad-ω2, respectively. then they choose two sets of attributes a = {λ1, λ2, λ3} and b = {λ1, λ2} and use them to construct soft multi-sets (smss) fa and fb over u given by fa = {(λ1, [ 2 ω1 , 2 ω2 , 1 ω3 , 1 ω4 ]), (λ2, [ 2 ω1 , 2 ω2 , 1 ω6 , 1 ω7 ]), (λ3, [ 2 ω1 , 2 ω2 , 1 ω5 , 1 ω6 , 1 ω7 ])} and fb = {(λ1, [ 2 ω1 , 2 ω2 , 1 ω4 ]), (λ2, [ 2 ω2 , 1 ω6 , 1 ω7 ])}. the 1st sms fa can be written as 90 certain properties of soft multi-set topology with applications in multi-criteria decision making fa λ1 λ2 λ3 ω1 2 2 2 ω2 2 2 2 ω3 1 0 0 ω4 1 0 0 ω5 0 0 1 ω6 0 1 1 ω7 0 1 1 the 2nd sms fb can be written as fb λ1 λ2 ω1 2 0 ω2 2 2 ω3 0 0 ω4 1 0 ω5 0 0 ω6 0 1 ω7 0 1 now we construct a sms-topology as τ̃ = {fφ, fa, fb, fẽ}, where fφ and fẽ are empty soft and absolute soft msets, respectively. write aggregate multi-sets of all open smss given by f ∗a = [ 6 ω1 , 6 ω2 , 1 ω3 , 1 ω4 , 1 ω5 , 2 ω6 , 2 ω7 ], f ∗b = [ 2 ω1 , 4 ω2 , 1 ω4 , 1 ω6 , 1 ω7 ], f ∗φ = [ 0 ω1 , 0 ω2 , 0 ω3 , 0 ω4 , 0 ω5 , 0 ω6 , 0 ω7 ] and f ∗ ẽ = [ 10 ω1 , 10 ω2 , 5 ω3 , 5 ω4 , 5 ω5 , 5 ω6 , 5 ω7 ]. in order to evaluate decision multi-set, the dms added the sets f ∗a and f ∗ b. thus f ∗a⊕b(σ) = f ∗ a(σ) + f ∗ b(σ), ∀ σ ∈ x. thus f ∗a ⊕ f ∗ b = [ 8 ω1 , 10 ω2 , 1 ω3 , 2 ω4 , 1 ω5 , 3 ω6 , 3 ω7 ]. since max f ∗a⊕b(σ) = 10 which shows that ω2 has the highest multiplicity, so ω2 is chosen for the ”award of performance”. next we use algorithm 3 on the same data as above and then compare the optimal results. the dms ω1 and ω2 consider smss (data same as above) fa and fb over u given by fa = {(λ1, [ 2 ω1 , 2 ω2 , 1 ω3 , 1 ω4 ]), (λ2, [ 2 ω1 , 2 ω2 , 1 ω6 , 1 ω7 ]), (λ3, [ 2 ω1 , 2 ω2 , 1 ω5 , 1 ω6 , 1 ω7 ])} and fb = {(λ1, [ 2 ω1 , 2 ω2 , 1 ω4 ]), (λ2, [ 2 ω2 , 1 ω6 , 1 ω7 ])}. again consider first sms fa given as 91 riaz et al./decis. mak. appl. manag. eng. 3 (2) (2020) 70-96 fa λ1 λ2 λ3 ω1 2 2 2 ω2 2 2 2 ω3 1 0 0 ω4 1 0 0 ω5 0 0 1 ω6 0 1 1 ω7 0 1 1 now consider second sms fb given as fb λ1 λ2 ω1 2 0 ω2 2 2 ω3 0 0 ω4 1 0 ω5 0 0 ω6 0 1 ω7 0 1 now we make a sms-topology as τ̃ = {fφ, fa, fb, fẽ}, where fφ and fẽ are empty soft and absolute soft msets, respectively. here we find the cardinal msets of all soft open msets given by cfa = [ 6 λ1 , 6 λ2 , 7 λ3 ], cfb = [ 5 λ1 , 4 λ2 ], cfφ = [ 0 λ1 , 0 λ2 , 0 λ3 , 0 λ4 , 0 λ5 ] and cf ẽ = [ 9 λ1 , 9 λ2 , 9 λ3 , 9 λ4 , 9 λ5 ]. the aggregated multi-set f ∗a is calculated by first decision maker by using (1), ... mf ∗ a =   2 2 2 2 2 2 1 0 0 1 0 0 0 0 1 0 1 1 0 1 1     6 6 7   =   38 38 6 6 7 13 13   that means, f ∗a = [ 38 ω1 , 38 ω2 , 6 ω3 , 6 ω4 , 7 ω5 , 13 ω6 , 13 ω7 ]. furthermore, the aggregate multi-set for fb is calculated by second decision maker, 92 certain properties of soft multi-set topology with applications in multi-criteria decision making ... mf ∗ b =   2 0 2 2 0 0 1 0 0 0 0 1 0 1   [ 5 4 ] =   10 18 0 5 0 4 4   which is, f ∗b = [ 10 ω1 , 18 ω2 , 0 ω3 , 5 ω4 , 0 ω5 , 4 ω6 , 4 ω7 ]. now we find the final decision multi-set by adding f ∗a and f ∗ b only. thus f ∗a⊕b(σ) = f ∗ a(σ) + f ∗ b(σ), ∀ σ ∈ x. thus f ∗a ⊕ f ∗ b = [ 48 ω1 , 56 ω2 , 6 ω3 , 11 ω4 , 7 ω5 , 17 ω6 , 17 ω7 ]. since max f ∗a⊕b(σ) = 56 which shows that ω2 has the greatest multiplicity, so ω2 is chosen for the ”award of performance”. it is interesting to note that algorithm 2 and algorithm 3 provides the same optimal decision. the proposed algorithm 2 and algorithm 3 are used in the environment of soft multi-sets information systems for the award of performance and the results are compared with existing methods as indicated in the table 4. method ranking of alternatives the optimal alternative algorithm 2 (proposed) ω2 ≻ ω1 ≻ ω6 = ω7 ≻ ω4 ≻ ω5 ≻ ω3 ω2 algorithm 3 (proposed) ω2 ≻ ω1 ≻ ω6 = ω7 ≻ ω4 ≻ ω5 ≻ ω3 ω2 algorithm (çağman et al., 2011) ω2 ≻ ω1 ≻ ω6 ≻ ω7 ≻ ω4 ≻ ω5 ≻ ω3 ω2 algorithm (riaz et al., 2011) ω2 ≻ ω1 ≻ ω6 ≻ ω7 ≻ ω4 ≻ ω5 ≻ ω3 ω2 table 4: comparison of final ranking by using algorithm 2 and algorithm 3 the comparison analysis of final ranking determined by algorithm 2, algorithm 3, çağman et al. (2011) and riaz et al. (2011) is also shown by multiple bar chart in the figure 4. 93 riaz et al./decis. mak. appl. manag. eng. 3 (2) (2020) 70-96 figure 4: multiple bar chart view of final ranking 5 conclusion the algebraic and topological structures of soft multi-sets (smss) are quite different from traditional crisp sets. moreover the mcdm methods developed under rough sets, fuzzy sets and soft sets do not deal with real life situations under the universe of soft multi-sets. due to the repetition of objects in the universe of soft multi-sets there is a need to develop novel mcdm methods. the goal of this article is deal with these challenges and to extend the notion of sms-topology towards mcdm problems. we initiated the idea of sms-topology which is defined on soft multi-sets for a fixed set of attributes. we used the idea of power whole multi-subsets of a soft multi-set in the construction of sms-topology. the notions of sms-basis, smssubspace, sms-interior, soft multi-set closure and boundary of soft multi-set are introduced. additionally, the concept of sms-topology is extended to develop novel multi-criteria decision-making (mcdm) methods. to meet these objectives, algorithm 1, algorithm 2 and algorithm 3 are presented for the selection of best alternative for biopesticides, for the selection of best textile company and for the award of performance, respectively. the aggregation operators are used to compute aggregate fuzzy soft sets and aggregate multisets. based on proposed mcdm methods some real life applications are justified by illustrative examples. soft multi-sets and sms-topology have large number of applications in soft computing, decision-making, data analysis, data mining, expert systems, information aggregation and information measures. references ali m.i. 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(1994). bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis, proceedings of the first international joint conference of the north american fuzzy information processing society biannual conference, 305-309. 96 certain properties of soft multi-set topology with applications in multi-criteria decision making © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 2, issue1, 2019, pp. 105-114. issn: 2560-6018 eissn:2620-0104 doi:_ https://doi.org/10.31181/dmame1901105t * corresponding author. e-mail address: jasmina.bunevska@uklo.edu.mk (j. bunevskatalevska), mitegeo@yahoo.com (m. ristov) and marija.malenkovska@uklo.edu.mk (m. malenkovska todorova). development of the methodology for selecting the optimal type of pedestrian crossing jasmina bunevska talevska1*, mite ristov2, and marija malenkovska todorova1 1 faculty of technical sciences bitola, st. kliment ohridski university, bitola, north macedonia 2 high school josip broz bitola, bitola, north macedonia received: 27 october 2018 accepted: 02 february 2019 available online: 03 march 2019 original scientific paper abstract: the world health organization in its agenda on sustainable development 2030 sets a goal to reduce the number of traffic-related accidents by 50%. according to the trend toward reducing the number of traffic-related accidents and the latest statistics report by sia bitola, we have found that this is a great challenge for our city and a very high goal which we could try to reach. namely, we have started a pedestrian safety initiative by trying to provide infrastructural facilities and elements that are planned and designed according to the security principles and which correspond to the projected speed and road function as well as safe infrastructure for pedestrians, the elderly and persons with disabilities. the main objective of this paper is to develop a case study methodology regarding the selection of pedestrian crossing types on the case study location example. namely, the vissim simulation model for the studied location has been introduced, and the general conclusions have been adopted based on the multi-criteria decision-making process analysis. the most important aim is directed towards obtaining pedestrian safety while bearing in mind the role of pedestrian safety within the current safety goals. key words: pedestrian safety; pedestrian crossing, ahp, vissim simulation 1. introduction pedestrians are the most vulnerable road users. in many countries, collisions with pedestrians are a leading cause of death and injury, and over half of all road deaths are caused by collisions between vehicles and pedestrians that occur in a number of situations, especially including walking while trying to cross the road. the process of pedestrian traffic is influenced by a number of factors, of which the urban mailto:jasmina.bunevska@uklo.edu.mk mailto:mitegeo@yahoo.com mailto:marija.malenkovska@uklo.edu.mk talevska et al./decis. mak. appl. manag. eng. 2 (1) (2019) 105-114 106 environment and streetscape are very important (aashto, 2010, 2014). the severity of pedestrian crashes is strongly dependent on traffic speed whereas the risk of pedestrian injuries is increased by a number of factors related to the road environment, including high traffic speed, inadequate crossing facilities, lack of pedestrian crossing opportunities (gaps in passing traffic), number of lanes to cross, complexity and unpredictability of traffic movements, inadequate separation from traffic and poor crossing sight distance as well. in oecd countries, traffic accidents cause 41% of fatalities among 14-year-olds. in spain, nine children aged between 6 and 14 died from a traffic-related accident while five of them were pedestrians (road safety inspection manual for school zones, 2014). moreover, according to principle 2 of the declaration of the rights of the child, "the child shall enjoy special protection, and shall be given opportunities and facilities, by law and by other means, to enable him/her to develop physically, mentally, morally, spiritually and socially in a healthy and normal manner and in conditions of freedom and dignity"(geneva declaration, 1959), the protection of the most vulnerable traffic group has to be one of the priorities of local authorities; hence it is one of the operational objectives of our recently launched pedestrian safety initiative "to provide safe school zones and routes." 2. methodology since there is no unique methodology for selecting an appropriate pedestrian crossing facility, the process for its selection revolves around the question of why it is considered desirable to provide specific assistance for pedestrians at a particular location or what it is that the designer seeks to achieve. the second stage, which follows after the overall need has been identified, is to identify a set of facilities that may have a detrimental impact on the safety of all users. typically, this choice of possible devices is based on the characteristics of the road on which the facility is to be installed and the basic choice sets are outlined in the tables, respectively. making a decision regarding the selection of a pedestrian crossing type is based on several criteria in order to create a solution that is fair for all the participants. the main objective of this paper is to develop a case study methodology regarding the selection of the most suitable pedestrian crossing type for the city of bitola, macedonia. the most important aim is directed towards obtaining pedestrian safety while bearing in mind the role of pedestrian safety within the current safety goals. 2.1. geomorphological and transport position of bitola the city of bitola, the second largest city in macedonia (77,004 inhabitants, census 2002), is located in its southwest part, on the edge of the pelagonija valley. it is located at the foothills of the baba mountain with the peak pelister (2601 m), near the greek border, 13 km away (ristov, 2015). the city stretches from both sides of the river dragor; to the north, it is surrounded by the bairo hills, as part of the cloud-snow mass with the peak kale (890m); to the south, it is surrounded by the hill tumbe cafe (744m) as a branch of neolica, i.e. baba mountain. to the east, bitola is widely open to the pelagonia valley, and towards the west it is open to the floodplains of the river dragor, the gavatian overbearing valley and the peak pelister. bitola is spread on a terrain that is sloped from west to east, from 720 m to 585 m, with an average altitude of 652 m. regarding the traffic situation, it can be said that bitola is relatively poorly development of the methodology for selecting the optimal type of pedestrian crossing 107 connected. this unfavorable traffic connection had its beginnings in the early 20th century, when with the reshaping of the borders, a large number of traffic routes lost their meaning or completely disappeared. hence the bitola gravitational area was reduced and deformed (dimitrov, 1998). 2.2. problem identification according to the standing classification of city street intersections (gupcb, 1999; sia, 2016) vasko karangeleski st. is classified as the main street. after the appropriate analysis, "mobility versus accessibility", it is determined that this street does not meet the criteria for that classification (administrative office of the primary school elpida karamandi bitola, 2018). furthermore, according to the data from the report of sia bitola for 2003-2016, a total of 38 pedestrians were injured on the said street (13 of whom were severely injured while 25 suffered minor injuries). from this data it can be concluded that there were no injured pedestrians only in 2010 and 2013 while in 2003, 2007 and 2016 an increased number of injured pedestrians was noted (administrative service at the pedagogical faculty bitola, 2018).the traffic situation particularly deteriorated after the constriction of the primary school elpida karamandi, the kindergarten majski cvet, the day centre for persons with disabilities and the faculty of pedagogy since a large number of children and students who live east off the street were forced to cross vasko karangeleski st. on a daily basis. today, this street is crossed by 228 pupils (world bank, 2012; who, 2016) who live to the east of the street as well as 380 students who are enrolled as full-time students in the academic year 2017/2018 (adriazola-steil et al., 2015). the crossing of the street is made difficult due to the fact that the children, most of whom are still very young, have to cross four lanes at once and deal with the lack of traffic culture by the drivers who fail to follow the rules of pedestrian crossings. 2.3. pedestrian safety initiative following the identification of the problem and the initiative launched by the parents of the children who attend the school, a campaign was organized, in cooperation with the municipal council on road traffic safety, professors in the traffic and transport department at the faculty of technical sciences bitola and non-government organizations in bitola, to raise awareness among the drivers about the pedestrians' need for safe crossing of streets as well as an initiative to the relevant institutions to help find a suitable solution for the pedestrians on vasko karangeleski st. the proposals of the parents and the ngos are related to the placement of vertical signalization (call buttons), which would be operational during the arrival and departure times of the students in the school; they would enable a relatively fast flow of vehicles on the main street as well as help to avoid any unnecessary stops when there are no pedestrians, as is the case with conventional traffic lights. in order to acquire information regarding citizens' opinion over pedestrian safety on the analyzed location, we used a questionnaire of six questions and an online survey that involved 770 citizens (figures 1-3). namely, gender equality was a key factor since 56.8% of the surveyed were women while 43.2% were men. in terms of age, most of the surveyed were between 31 and 40 (32.6%) and 21 and 30 years of age (25.7%). when asked "how safe is it to cross the street at the pedestrian crossing?” 40.3% answered that it is not safe at all whereas 57.1% answered that it is partially safe. talevska et al./decis. mak. appl. manag. eng. 2 (1) (2019) 105-114 108 figure 1. graphical presentation of pedestrian crossing safety when determining the main cause for the lack of safety at the pedestrian crossing, the lack of traffic culture among drivers and unmarked pedestrian crossings are listed as the two main reasons. these are followed by illegally parked vehicles in front of pedestrian crossings, its illumination, and ultimately the lack of traffic police and traffic signalization. what is quite evident from the answers of citizens is that despite the fact that the number one reason for the lack of pedestrian safety on pedestrian crossings dominates, all of the given causes contribute to some degree to the reduction of pedestrian safety. figure 2. graphical presentation of pedestrian insecurity while crossing when asked "do you think that the four lanes of the street can be safely crossed by a primary school pupil?” a high percentage of 75.8% answered that it would be impossible. figure 3. graphical presentation of meaning regarding pedestrian crossing length development of the methodology for selecting the optimal type of pedestrian crossing 109 when asked "do you think that the placement of light signalization (call button) on vasko karangeleski st. will increase the safety of the students who use the pedestrian crossing?" almost 86% of the surveyed answered yes, which speaks to the need of regulating traffic on this location. figure 3. graphical presentation of meaning regarding signalized pedestrian crossing design 3. multi-criteria decision-making approach we have decided to use a multi-criteria decision-making analysis based on the ahp (analytic hierarchy process) approach which synthesizes the aspects of different opinions by weighing up many subjective factors and which studies the unique common result (saaty& tran, 2007). the level of consistency allows us to form an adjustment of judgments. at the end of the process we have answered how to best make a decision in a complex and subjective situation with more than a few realistic options. namely, for the application of the ahp method we set the goal, i.e. three alternatives, adequate number of criteria and subcriteria for precise ranking of the alternative, as in (table 1). pairwise comparisons are used to determine the relative importance of each alternative in terms of each criterion (figure 4). table 1. competing alternatives and criteria a1 signalized pedestrian crossing a2 pedestrian crossing with refuge median island a3 – pedestrian overpass k1safety criterion subcriterion: driving speed (n,1.1) subcriterion: traffic flow (n,1.2) subcriterion: length of the pedestrian crossing (road width) (n,1.3) k2price criterion subcriterion: cost of design (n,2.1) subcriterion: cost of construction (n,2.2) subcriterion: cost of maintenance (n,2.3) k3environment & comfort criterion subcriterion: noise and environmental impact (n,3.1) subcriterion: comfort(n,3.2) subcriterion: access for the disabled (n,3.3) talevska et al./decis. mak. appl. manag. eng. 2 (1) (2019) 105-114 110 figure 4. adopted ahp excel software tool for pedestrian crossing type selection 3.1. establishment of structural hierarchy 3.1.1. signalized pedestrian crossings as an alternative it consists of signal displays, line markings and lighting. in general, fixed-time signals are the rule in urban areas for reasons of regularity, network organization, predictability, and reducing unnecessary delays. in certain, less-trafficked areas, actuated signals (call buttons, loop detectors) may be appropriate; however, these must be programmed to minimize delay, which will increase compliance. the pedestrian crossing signals at midblock crossing locations are widely used in most developed countries. they can be classified into four types: fixed time pedestrian actuated crossing, pelican crossing, puffin crossing and toucan crossing. fixed time pedestrian actuated crossing (figure 5) is a stand-alone pedestrian actuated (or automatic) signal control. pedestrians can call green phase by pushing the button, though, traffic must be able to see pedestrian crossing points in time to stop for them. advance warning signs should be used if visibility is poor. parking should be removed from near pedestrian crossings to provide adequate sight distance. figure 5. vissim microscopic simulation for location under study development of the methodology for selecting the optimal type of pedestrian crossing 111 figure 6. design of puffin pedestrian crossing for location under study 3.1.2. pedestrian crossing with median island as an alternative crossing a busy road with fast flowing traffic can be very difficult. pedestrian median islands (figure 7) can help pedestrians to cross such roads safely by allowing them to cross in two stages and deal with one direction of traffic flow at a time. they can be used where there is a demand for pedestrians to cross the road but where the number of pedestrians is not high enough to warrant a signalized pedestrian crossing. median islands can be part of no-signalized pedestrian crossing and are usually used on wide, multi-lane roads, with the function of narrowing the lanes for vehicular traffic. they must be clearly visible to traffic both day and night. figure 7. design of pedestrian crossing type with median island for location under study 3.1.3. pedestrian overpass as an alternative one effective way of preventing crashes between vehicles and pedestrians is placing them at different levels, or 'grade separating' them. in urban situations where the pedestrian crossing signals would cause congestion or crashes (due to high traffic speeds), a grade separated pedestrian crossing, such as an overpass or an underpass, talevska et al./decis. mak. appl. manag. eng. 2 (1) (2019) 105-114 112 may be used. outside of urban areas, in situations where there is pedestrian demand in high speed environments, this treatment may also be applied. grade separated pedestrian crossings reduce pedestrian crashes but they also have some disadvantages: they are costly, pedestrians may avoid them if there are a lot of steps to climb up or down. what is more, if they are not well-lit and patrolled, they may pose a personal security risk. pedestrians tend only to use crossing facilities located at, or very near, to where they want to cross the road. where a lot of cycling traffic is present, a pedestrian underpass or overpass can be used by cyclists as well as pedestrians, but this will require shallow approach ramps and therefore additional land. 3.1.4. decision hierarchy criteria and sub-criteria safety (k1) is a condition in which a pedestrian can normally cross at a pedestrian crossing in the process neither disturbed nor degraded due to various threats and dangers, adapted according to (sia, 2016). driving speed (n1.1) has been identified as a key risk factor in road traffic injuries, influencing both the risk of a road crash as well as the severity of the injuries that result from crashes (world bank, 2012). excess speed is defined as exceeding the speed limit. at inappropriate speed, the pedestrian cannot properly estimate the moment at which the vehicle will reach the pedestrian crossing, i.e. the point of intersection between the paths of the vehicle and the pedestrian while the motorist is not able to stop the vehicle on time. the greater the difference in the speed between the pedestrian and the vehicle, the greater the danger to the pedestrian. if traffic flow (n1.2) saturation results in situations in which the time gap between the approaches of two succeeding vehicles is shorter than the time required to cross the road, the method of stopping the vehicle has to be applied in order to perform the crossing. if traffic is of a higher intensity resulting in even scarcer occurrences of suitable intervals to cross the road, the pedestrians lose patience and recklessly step onto the roadway. the consequences of such actions may be catastrophic and in such situations zebra crossings do not usually match the needs and signalized crossing needs to be constructed. should traffic lights cause very long queues of vehicles, and pedestrian waiting time exceeds the limit of patient waiting, then the pedestrian crossings are grade-separated, by constructing overpasses. the length of the pedestrian crossing (n1.3) is in correlation with traffic safety. the crossing time using a longer pedestrian crossing means a longer stay of the pedestrian on the roadway and a higher risk of getting injured. on a multi-lane road the vehicles moving along the right kerb often obscure the view of vehicles that move along the farther lane. this phenomenon is especially noted in cases when small children want to cross the street and the motorists fail to notice them on time. this problem is especially emphasized in the vicinity of schools. price (k2) is the value that is put to a product or service and is the result of a complex set of calculations, research and understanding and risk taking ability. a pricing strategy takes into account segments, ability to pay, market conditions, competitor actions, trade margins and input costs, amongst others. it is targeted at the defined customers and against competitors. in forming the criteria of prices for pedestrian facilities, costs of design (n2.1), construction (n2.2), and maintenance (n2.3), have been taken into consideration and studied as separate subcriteria. bearing in mind that there are no exact numerical indicators the criterion environment & comfort (k3) serves as an additional assistance to the decisionmakers. development of the methodology for selecting the optimal type of pedestrian crossing 113 exposure to noise (n3.1), in everyday urban life is considered to be an environmental stressor. a specific outcome of reactions to environmental stress is a fast pace of life that also includes a faster pedestrian walking speed. on zebra crossings and signalized crossings pedestrians are exposed at a high noise level whereas in underpasses and overpasses they are much better protected. the closer the vehicles are to the pedestrian crossing and the larger their number, the greater the influence of noise. it is most expressed at the peak hours when the greatest number of vehicles and pedestrians is on the road. the sub-criterion aesthetic and environment (n3.2) considers the negative impacts of pedestrian crossing construction on the environment, changes of the streetscape, unpleasant experiences as well as a feeling of personal protection. the concept of accessible design for disabled persons (n3.3) ensures both direct access, i.e. unassisted, and indirect access, that is compatibility with a person's assistive technology (for example, computer screen readers). this intends to make everything accessible to all people regardless of their having any disability or not. 4. conclusions the crossing opportunities available to pedestrians on the studied location are below the desired level of service. historical records of crashes in the vicinity of the location are a serious factor that indicates the need for providing crossing assistance. the methodology of selecting a pedestrian crossing proposed by this research is comprehensive. for this purpose an adopted ahp excel software tool has been developed and a vissim microscopic simulation was used to model the alternatives under a range of likely pedestrian volumes and a range of likely vehicle volumes. the process of alternatives evaluation by calculating the weight values of criteria and alternatives has been performed by comparing the pairs of criteria, based on the questionnaire results for different target groups of citizens, professionals, disabled and healthy persons. namely, professionals give weights to the traffic safety and priority to the overpasses. regarding environment and price the best alternative is median islands. both healthy and persons with disabilities equally value signalized pedestrian crossing. analysis results are in correlation with the design principles showing that on undivided four lane main street (two lanes per direction), without allowed side parking, at a speed limit of 50 km/h, and with the distance of 150-200 meters to the next intersection light signals or median islands are recommended. bearing in mind the traffic culture of drivers, we assume that the introduction of signalized mid-block crossing with pedestrian-actuated control and fixed-time operation is the optimal type for the location under study. the methodology and the results from various research allow tests to be administered on possible scenarios in order to decide on the best type of pedestrian crossing. however, we must take into account all the specific features of the location so as to introduce the best conditions, both in terms of space and maintenance plan, as well as suitable traffic signalization. talevska et al./decis. mak. appl. manag. eng. 2 (1) (2019) 105-114 114 references administrative office of the primary school elpida karamandi bitola. (2018). bitola, north macedonia. administrative service at the pedagogical faculty bitola. (2018).bitola, north macedonia. adriazola-steil, c., wei li & b. welle (2015). designing safer cities for children. journal of transportation engineering, 132(1), 40–51. american association of state highway and transportation officials (aashto). (2010). highway capacity manual, transportation research board, usa. american association of state highway and transportation officials (aashto). (2010). highway safety manual, transportation research board, usa. declaration of the rights of the child. (1959). geneva declaration, proclaimed by general assembly resolution 1386(xiv). dimitrov, n. (1998). bitola, urban-geographical development, society for science and art, bitola. general urban plan of the city of bitola (gupcb). (1999). third major amendments and supplements, book i: final report, bitola. ristov, m. (2016). factors and principles for determining the most suitable housing space in the city of bitola, master thesis, faculty of natural sciences and mathematics, institute of geography, skopje. road safety inspection manual for school zones (2014).foundation mapfree, geneva. saaty, t.l. & tran, l.t. (2007). on the invalidity of fuzzifying numerical judgments in the analytic hierarchy process. mathematical and computer modelling, 46(7), 962-975. safety information agency (sia). (2016). report on accidents accidents 2003-2016. north macedonia, north macedonia safety agency. world bank. (2012).cycling and walking, usa. world health organization (who). (2016). report on sustainable development 2030. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 5, issue 1, 2022, pp. 246-263. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame211221090a * corresponding author. e-mail addresses: hdarora@amity.edu (h. d. arora), anathani@amity.edu (a. naithani) significance of topsis approach to madm in computing exponential divergence measures for pythagorean fuzzy sets h. d. arora1 and anjali naithani1* 1department of mathematics, amity institute of applied sciences, amity university uttar pradesh, noida, india received: 29 march 2022; accepted: 17 december 2021; available online: 21 december 2021. original scientific paper abstract: managers nowadays face challenging decisions on daily basis and must weigh a growing number of factors while making such decisions. one of the most common and popular research domains in decision theory is the multiple attribute decision-making (madm) problem which allows us to consider several factors into consideration. in this paper, the primary goal is to uncover the important aspect of divergence measures based on exponential function under pythagorean fuzzy sets (pfss) and study its application to multi attribute decision making. pfss is a more tensile and powerful approach than intuitionistic fuzzy set (ifs) to depict uncertainty. numerical example has been illustrated and sensitivity analysis has been carried out to validate our proposed measures. moreover, a comparative study of the results for the proposed measures demonstrates the efficacy of the proposed distance measures. key words: intuitionistic fuzzy set, pythagorean fuzzy sets, similarity measure, exponential measure, topsis method, multi attribute decision making 1. introduction evaluations of alternative measures are a challenging and complex task due to several variables which relate to specific decisions in many decision-making problems, such as environmental, social, physical, organizational, and social criteria. researchers have developed various decision-making methods over the last few years to help policymakers to analyze the strategic planning in the industry and to resolve them. furthermore, because of the increasing uncertainty and complexity of attributes and the vagueness of human thinking, the study of madm in an uncertain environment has received much emphasis. for decision-makers, therefore, it is important to significance of topsis approach to madm in computing exponential divergence measures… 247 understand the nature and significance of insecurity to improve their capabilities to take the best decision to decrease risk in their final decisions. multi-attribute decisionmaking, a systematic method, can be a useful tool for fuzzy set has its participation and non-participation values totaled to 1. atanassov (1986, 1989) created the notion of intuitionistic fuzzy set (ifs) to better express uncertainty by easing this constraint. the participation degree and non-participation degree of an ifss are both real numbers in the range [0, 1], and their sum is less than 1. another ifss parameter, the hesitation degree, is derived from the difference between 1 and their summation. ifss theory has been successfully applied to a variety of real-world challenges such as decision-making. remarkable outcomes on ifss have been carried out by many researchers (peng et al., 2017; thao, et al., 2019). pfss is a generalization of ifs that has the prerequisite that the sum of square of perception and non-perception grade ≤ 1. the space of all intuitionistic membership values is also pythagorean membership values, but not the other way around. garg (2017) introduced an improved ranking order interval valued pfss using topsis technique. indeed, the hypothesis of pfss has been widely considered, as demonstrated by various researchers (garg, 2018; liang and xu, 2017). in association with the uses of pfss, rahman et al., (2017, 2018) proposed a few ways to deal with aggregation operators (ao) and mcdm problems. pfss have drawn the attention of researchers and are being applied in decision making (liu et al. 2020; mahanta and panda, 2021; fei and deng, 2020, farhadinia, 2021), medical diagnosis (ejegwa, 2020; zhou et al. 2020), stock portfolio problem (kalifa, 2020), belief function (xiao, 2020). the characteristics and applicability of the measures in pattern recognition, medical diagnosis, multi-criteria decision-making, and clustering analysis were reviewed by singh and ganie (2020). overall, the possibility of pfss has pulled in incredible considerations of numerous researchers, and the idea has been functional to a few applied regions viz., aggregation operators (khan et al., 2019), social network analysis (wang et al. 2020), mcdm (gao and wei, 2018; rahman and abdullah, 2019), information measures and many more (yager, 2014; ejegwa, 2019). pamucar et al. (2017) investigated the sensitivity of madm approaches to changes in criteria weight, as well as the methods' consistency in response to changes in measurement scale and created criteria. a divergence metric for pfss is a tool that reflects how analogous two or more pfss are to each other. indeed, there is a second concept of similarity measurement for pfss. pfs similarity measures have been investigated from several angles in recent times (ejegwa, 2018). to address the shortcomings of existing measures, peng (2019) proposed new pythagorean distance and similarity measurements (adabitabar et al. 2020). modification of zhang and xu’s (2014) distance measure for pfss and its application to pattern recognition was carried out (ejegwa, 2020). some formulae of pythagorean fuzzy information measures on similarity measures and corresponding transformation relationships were also developed (peng et al., 2017, 2019). similarity measures for trigonometric function for fss, ifss and pfss were also proposed (taruna et al., 2021), ifss and pfss (wei and wei, 2018; mohd and abdullah, 2018) were also proposed (maoying, 2013). some complex pfss distance measures have been established, and their features have been investigated (ullah et al. 2020). the similarity measures of the ifss and pfss are broadly used in various disciplines, comparable to the pattern identification (peng and garg, 2019), the clinical finding (son and phong, 2016), decision-making (zhang et al., 2019). however, lu and ye (2018) offered similarity measure of ivfss on log function. agheli et al. 2021 recently proposed a method for calculating pythagorean similar measure for two pythagorean fuzzy value by making use of t-norm and s-norm. arora/decis. mak. appl. manag. eng. 5 (1) (2022) 246-263 248 for supplier evaluation and selection, pamucar et al. (2020) suggested a fuzzy neutrosophic decision-making approach. many researchers analysed madm approach using topsis method (hwang and yoon, 1981). many researchers like adeel et al., 2019; akram and adeel, 2019; akram et al., 2018; balioti et al., 2018; biswas and kumar, 2018; askarifar et al., 2018; wang and chen, 2017; gupta et al., 2018; kumar and garg, 2018 and many more have applied topsis method in various problems of decision making like supplier selection, selection of land, robotics, medical diagnosis, ranking of water quality, human resource selection personnel problem, and many other real life situations flavoured with fss and generalized fss. in this article, we are exploring the resourcefulness of exponential divergence measures of pfss in the application to pattern recognition and multi attribute decision making. this paper is organized as follows: section 2 introduces preliminaries of fss, ifss and the pfss. section 3 comprises of the concept of proposed exponential similarity measures of pfss. we introduce exponential similarity measures and weighted similarity measures of the pfss and its numerical computations to validate our measures. application is also provided in section 4. section 5 deliberates discussion about the methodology discussed and sensitivity analysis of the proposed measures. section 6 compares the new exponential similarity measures with the existing similarity measure by an example. finally, section 7 summarizes the document and delivers directions for future experiments. 2. preliminaries in this segment, we bring in some basic theories related to fss, ifss and pfss applied in the article. definition 2.1. [zadeh, 1965]. let x be a nonempty set. a fuzzy set 𝑃 in 𝐸 = {𝑥1, 𝑥2 … , 𝑥𝑛 } is characterized by a membership function: 𝑃 = {〈𝑥, 𝛿𝑃(𝑥)〉|𝑥 ∈ 𝐸} (1) where 𝛿𝑃(𝑥): 𝐸 → [0,1] is a measure of belongingness of degree of membership of an element 𝑥 ∈ 𝐸 in 𝑃. definition 2.2. [atanassov, 1986]. an ifs p in x is given by 𝑃 = {〈𝑥, 𝛿𝑃(𝑥), 𝜁𝑃 (𝑥)〉|𝑥 ∈ 𝐸} (2) where 𝛿𝑃(𝑥), 𝜁𝑃 (𝑥): 𝐸 → [0,1], 0 ≤ 𝛿𝑃(𝑥) + 𝜁𝑃 (𝑥) ≤ 1, ∀ 𝑥 ∈ 𝐸. the number 𝛿𝑃(𝑥) and 𝜁𝑃 (𝑥) represents, respectively, the membership degree and non-membership degree of the element 𝑥 to the set p. for each ifs 𝑃 in 𝐸, if 𝜂𝑃(𝑥) = 1 − 𝛿𝑃(𝑥) − 𝜁𝑃 (𝑥), ∀ 𝑥 ∈ 𝐸. (3) then 𝜂𝑃(𝑥) is called the degree of indeterminacy of 𝑥 to ã. definition 2.3. [yager, 2013]. an ifs 𝑃 in 𝐸 is given by 𝑃 = {〈𝑥, 𝛿𝑃(𝑥), 𝜁𝑃 (𝑥)〉|𝑥 ∈ 𝐸} where 𝛿𝑃(𝑥), 𝜁𝑃 (𝑥): 𝐸 → [0,1], with the condition that 0 ≤ 𝛿𝑃 2(𝑥) + 𝜁𝑃 2(𝑥) ≤ 1, ∀ 𝑥 ∈ 𝐸 (4) and the degree of indeterminacy for any pfs 𝑃 and 𝑥 ∈ 𝐸 is given by 𝜂𝑃 2 (𝑥) = √1 − 𝛿𝑃 2(𝑥) − 𝜁𝑃 2(𝑥) (5) significance of topsis approach to madm in computing exponential divergence measures… 249 3. exponential divergence measures in this segment, new exponential divergence measures of the pfss are proposed. preposition 1. let 𝛸 be nonempty set and p, q, r ∈ pfs (𝛸). the divergence measure between p and q is a function 𝐷𝑖𝑣: 𝑃𝐹𝑆 × 𝑃𝐹𝑆 → [0,1] satisfies (p1) boundedness: 0 ≤ 𝐷𝑖𝑣(𝑃, 𝑄) ≤ 1 (p2) separability: 𝐷𝑖𝑣(𝑃, 𝑄) = 0 ⇔ 𝑃 = 𝑄. (p3) symmetric: 𝐷𝑖𝑣(𝑃, 𝑄) = 𝐷𝑖𝑣(𝑄, 𝑃) (p4) inequality: if r is a pfs in 𝛸 and 𝑃 ⊆ 𝑄 ⊆ 𝑅, then 𝐷𝑖𝑣(𝑃, 𝑄) ≤ 𝐷𝑖𝑣(𝑃, 𝑅) and 𝐷𝑖𝑣(𝑄, 𝑅) ≤ 𝐷𝑖𝑣(𝑃, 𝑅). in several circumstances, the weight of the elements 𝑥𝑖 ∈ 𝑋 must be considered. for instance, in decision making, the attributes usually have distinct significance, and thus ought to be designated unique weights. as a result, we propose two weighted logarithmic divergence measures between p and q, as follows: let 𝑃, 𝑄 ∈ pfs (𝛸) such that 𝑋 = {𝑥1, 𝑥2 … , 𝑥𝑛 } then 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑄) = 2 𝑛 ∑ [|𝛿𝑃 2(𝑥𝑖 ) − 𝛿𝑄 2(𝑥𝑖 )|. 2 −|𝛿𝑃 2 (𝑥𝑖)−𝛿𝑄 2 (𝑥𝑖)|−1 + |𝜁𝑃 2(𝑥𝑖 ) − 𝑛 𝑖=1 𝜁𝑄 2 (𝑥𝑖 )|. 2 −|𝜁𝑃 2(𝑥𝑖)−𝜁𝑄 2 (𝑥𝑖)|−1] (6) 𝐷𝑊𝑃𝐹𝑆𝐸 (𝑃, 𝑄) = 2 𝑛 ∑ 𝜔𝑖 [|𝛿𝑃 2(𝑥𝑖 ) − 𝛿𝑄 2(𝑥𝑖 )|. 2 −|𝛿𝑃 2 (𝑥𝑖)−𝛿𝑄 2 (𝑥𝑖)|−1 + |𝜁𝑃 2 (𝑥𝑖 ) − 𝑛 𝑖=1 𝜁𝑄 2 (𝑥𝑖 )|. 2 −|𝜁𝑃 2(𝑥𝑖)−𝜁𝑄 2 (𝑥𝑖)|−1] (7) 𝜔 = (𝜔1, 𝜔2, … , 𝜔𝑛 ) 𝑇 is the weight vector of 𝑥𝑖 (𝑖 = 1,2, … , 𝑛), with 𝜔𝑘 ∈ [0,1], 𝑘 = 1,2, … , 𝑛, ∑ 𝜔𝑘 = 1 𝑛 𝑘=1 . if 𝜔 = ( 1 𝑛 , 1 𝑛 , … 1 𝑛 ) 𝑇 , then the weighted exponential divergence measure reduces to proposed measure. if we take 𝜔𝑘 = 1, 𝑘 = 1,2, … , 𝑛, then then 𝐷𝑊𝑃𝐹𝑆𝐸 (𝑃, 𝑄) = 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑄). theorem 3.1. the pythagorean fuzzy divergence measures defined in equation (6) (7) are valid measures of pythagorean fuzzy divergence. proof. all the necessary four conditions to be a divergence measure are satisfied by the new divergence measures as follows: (p1) boundedness: 0 ≤ 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑄) ≤ 1 proof. since the values 0 ≤ 𝛿𝑃(𝑥𝑖 ) ≤ 𝛿𝑄(𝑥𝑖 ) ≤ 1 and 0 ≤ 𝜁𝑃 (𝑥𝑖 ) ≤ 𝜁𝑄 (𝑥𝑖 ) ≤ 1, therefore, 0 ≤ |𝛿𝑃 2(𝑥𝑖 ) − 𝛿𝑄 2(𝑥𝑖 )| ≤ 1 and 0 ≤ |𝜁𝑃 2(𝑥𝑖 ) − 𝜁𝑄 2(𝑥𝑖 )| ≤ 1. since minimum values of all the expression is 0, then the measure 𝐷𝑃𝐹𝑆𝐿 (𝑃, 𝑄) will have value as 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑄) = 2 1 (0 . 2−1 + 0 . 2−1 ) = 0 . also, if the maximum value of the above expressions is 1, then 𝐷𝑃𝐹𝑆𝐿 (𝑃, 𝑄) = 2 1 (1 . 2−2 + 1 . 2−2 ) = 1. thus, 0 ≤ 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑄) ≤ 1. measure 𝐷𝑊𝑃𝐹𝑆𝐸 (𝑃, 𝑄) can be proved similarly. (p2) separability: 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑄) = 0 ⇔ 𝑃 = 𝑄. proof. for two pfss p and q in 𝑋 = {𝑥1, 𝑥2 … , 𝑥𝑛 }, if 𝑃 = 𝑄, then 𝛿𝑃 2(𝑥𝑖 ) = 𝛿𝑄 2(𝑥𝑖 ) and 𝜁𝑃 2(𝑥𝑖 ) = 𝜁𝑄 2 (𝑥𝑖 ). thus, |𝛿𝑃 2(𝑥𝑖 ) − 𝛿𝑄 2(𝑥𝑖 )| = 0 and |𝜁𝑃 2(𝑥𝑖 ) − 𝜁𝑄 2(𝑥𝑖 )| = 0. therefore, 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑄) = 0. if 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑄) = 0, this implies |𝛿𝑃 2(𝑥𝑖 ) − 𝛿𝑄 2(𝑥𝑖 )|. 2 −|𝛿𝑃 2 (𝑥𝑖)−𝛿𝑄 2 (𝑥𝑖)|−1 + |𝜁𝑃 2(𝑥𝑖 ) − 𝜁𝑄 2(𝑥𝑖 )|. 2 −|𝜁𝑃 2 (𝑥𝑖)−𝜁𝑄 2 (𝑥𝑖)|−1 = 0 ⇒ |𝛿𝑃 2(𝑥𝑖 ) − 𝛿𝑄 2 (𝑥𝑖 )| = 0 and |𝜁𝑃 2(𝑥𝑖 ) − 𝜁𝑄 2(𝑥𝑖 )| = 0. arora/decis. mak. appl. manag. eng. 5 (1) (2022) 246-263 250 therefore 𝛿𝑃 2(𝑥𝑖 ) = 𝛿𝑄 2(𝑥𝑖 ) and 𝜁𝑃 2(𝑥𝑖 ) = 𝜁𝑄 2 (𝑥𝑖 ). hence p = q. measure 𝐷𝑊𝑃𝐹𝑆𝐸 (𝑃, 𝑄) can be proved similarly. (p3) symmetric: 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑄) = 𝐷𝑃𝐹𝑆𝐸 (𝑄, 𝑃) proofs are self-explanatory and straight forward. (p4) inequality: if r is a pfs in 𝛸 and 𝑃 ⊆ 𝑄 ⊆ 𝑅, then 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑄) ≤ 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑅) and 𝐷𝑃𝐹𝑆𝐸 (𝑄, 𝑅) ≤ 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑅). proof. if 𝑃 ⊆ 𝑄 ⊆ 𝑅, then for 𝑥𝑖 ∈ 𝛸, we have 0 ≤ 𝛿𝑃(𝑥𝑖 ) ≤ 𝛿𝑄(𝑥𝑖 ) ≤ 𝛿𝑅 (𝑥𝑖 ) ≤ 1 and 1 ≥ 𝜁𝑃 (𝑥𝑖 ) ≥ 𝜁𝑄 (𝑥𝑖 ) ≥ 𝜁𝑅 (𝑥𝑖 ) ≥ 0. this implies that 0 ≤ 𝛿𝑃 2(𝑥𝑖 ) ≤ 𝛿𝑄 2(𝑥𝑖 ) ≤ 𝛿𝑅 2(𝑥𝑖 ) ≤ 1 and 1 ≥ 𝜁𝑃 2(𝑥𝑖 ) ≥ 𝜁𝑄 2(𝑥𝑖 ) ≥ 𝜁𝑅 2(𝑥𝑖 ) ≥ 0. thus, we have, |𝛿𝑃 2(𝑥𝑖 ) − 𝛿𝑄 2(𝑥𝑖 )| ≤ |𝛿𝑃 2(𝑥𝑖 ) − 𝛿𝑅 2(𝑥𝑖 )| ; |𝛿𝑄 2(𝑥𝑖 ) − 𝛿𝑅 2(𝑥𝑖 )| ≤ |𝛿𝑃 2(𝑥𝑖 ) − 𝛿𝑅 2(𝑥𝑖 )| and |𝜁𝑃 2(𝑥𝑖 ) − 𝜁𝑄 2 (𝑥𝑖 )| ≤ |𝜁𝑃 2(𝑥𝑖 ) − 𝜁𝑅 2(𝑥𝑖 )| ; |𝜁𝑄 2 (𝑥𝑖 ) − 𝜁𝑅 2(𝑥𝑖 )| ≤ |𝜁𝑃 2(𝑥𝑖 ) − 𝜁𝑅 2(𝑥𝑖 )| from the above we can write, 2−|𝛿𝑃 2 (𝑥𝑖)−𝛿𝑄 2 (𝑥𝑖)|−1 ≤ 2−|𝛿𝑃 2 (𝑥𝑖)−𝛿𝑅 2 (𝑥𝑖)|−1 ⇒ |𝛿𝑃 2(𝑥𝑖 ) − 𝛿𝑄 2 (𝑥𝑖 )|. 2 −|𝛿𝑃 2 (𝑥𝑖)−𝛿𝑄 2 (𝑥𝑖)|−1 ≤ |𝛿𝑃 2(𝑥𝑖 ) − 𝛿𝑅 2(𝑥𝑖 )|. 2 −|𝛿𝑃 2 (𝑥𝑖)−𝛿𝑅 2 (𝑥𝑖)|−1 ⇒ |𝛿𝑃 2(𝑥𝑖 ) − 𝛿𝑄 2 (𝑥𝑖 )|. 2 −|𝛿𝑃 2 (𝑥𝑖)−𝛿𝑄 2 (𝑥𝑖)|−1 + |𝜁𝑃 2(𝑥𝑖 ) − 𝜁𝑄 2 (𝑥𝑖 )|. 2 −|𝜁𝑃 2(𝑥𝑖)−𝜁𝑄 2 (𝑥𝑖)|−1 ≤ |𝛿𝑃 2(𝑥𝑖 ) − 𝛿𝑅 2(𝑥𝑖 )|. 2 −|𝛿𝑃 2 (𝑥𝑖)−𝛿𝑅 2 (𝑥𝑖)|−1 + |𝜁𝑃 2(𝑥𝑖 ) − 𝜁𝑅 2(𝑥𝑖 )|. 2 −|𝜁𝑃 2(𝑥𝑖)−𝜁𝑅 2 (𝑥𝑖)|−1 ⇒ 2 𝑛 [∑ {|𝛿𝑃 2(𝑥𝑖 ) − 𝛿𝑄 2(𝑥𝑖 )|. 2 −|𝛿𝑃 2 (𝑥𝑖)−𝛿𝑄 2 (𝑥𝑖)|−1 + |𝜁𝑃 2(𝑥𝑖 ) − 𝜁𝑄 2 (𝑥𝑖 )|. 2 −|𝜁𝑃 2(𝑥𝑖)−𝜁𝑄 2 (𝑥𝑖)|−1} 𝑛 𝑖=1 ] ≤ 2 𝑛 [∑ {|𝛿𝑃 2(𝑥𝑖 ) − 𝛿𝑅 2(𝑥𝑖 )|. 2 −|𝛿𝑃 2 (𝑥𝑖)−𝛿𝑅 2 (𝑥𝑖)|−1 + |𝜁𝑃 2(𝑥𝑖 ) − 𝑛 𝑖=1 𝜁𝑅 2(𝑥𝑖 )|. 2 −|𝜁𝑃 2(𝑥𝑖)−𝜁𝑅 2 (𝑥𝑖)|−1}] ⇒ 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑄) ≤ 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑅). similarly, 𝐷𝑃𝐹𝑆𝐸 (𝑄, 𝑅) ≤ 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑅). similar proofs can be made for 𝐷𝑊𝑃𝐹𝑆𝐿 (𝑃, 𝑄) ≤ 𝐷𝑊𝑃𝐹𝑆𝐸 (𝑃, 𝑅) and 𝐷𝑊𝑃𝐹𝑆𝐸 (𝑄, 𝑅) ≤ 𝐷𝑊𝑃𝐹𝑆𝐸 (𝑃, 𝑅). 3.1. numerical verification of the distance measures based on the parameters suggested by wei and wei (2018), we verify whether proposed divergence measures satisfy above four properties: example 1. let 𝑃, 𝑄, 𝑅 ∈ 𝑃𝐹𝑆(𝑋) for 𝑋 = {𝑥1, 𝑥2, 𝑥3}. suppose 𝑃 = {⟨𝑥1, 0.6, 0.2⟩, ⟨𝑥2, 0.4, 0.6⟩, ⟨𝑥3, 0.5, 0.3⟩}, 𝑄 = {⟨𝑥1, 0.8, 0.2⟩, ⟨𝑥2, 0.7, 0.3⟩, ⟨𝑥3, 0.6, 0.3⟩} and 𝑅 = {⟨𝑥1, 0.9, 0.1⟩, ⟨𝑥2, 0.8, 0.2⟩, ⟨𝑥3, 0.7, 0.1⟩} calculating the distance using proposed distance measures are as follows: 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑄) = 2 3 [ {|0.62 − 0.82|. 2−|0.6 2−0.82|−1 + |0.22 − 0.22|. 2−|0.2 2−0.22|−1} + {|0.42 − 0.72|. 2−|0.4 2−0.72|−1 + |0.62 − 0.32|. 2−|0.6 2−0.32|−1} + {|0.52 − 0.62|. 2−|0.5 2−0.62|−1 + |0.32 − 0.32|. 2−|0.3 2−0.32|−1} ] = 2 3 [0.115302742 + 0 + 0.131263519 + 0.111958138 + 0.050962343 + 0] = 0.272991161. 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑅) = 2 3 [ {|0.62 − 0.92|. 2−|0.6 2−0.92|−1 + |0.22 − 0.12|. 2−|0.2 2−0.12|−1} + {|0.42 − 0.82|. 2−|0.4 2−0.82|−1 + |0.62 − 0.22|. 2−|0.6 2−0.22|−1} + {|0.52 − 0.72|. 2−|0.5 2−0.72|−1 + |0.32 − 0.12|. 2−|0.3 2−0.12|−1} ] significance of topsis approach to madm in computing exponential divergence measures… 251 = 2 3 [0.16470964 + 0.014691304 + 0.172074629 + 0.12817118 + 0.101609437 + 0.037842305] = 0.41273233. 𝐷𝑃𝐹𝑆𝐸 (𝑄, 𝑅) = 2 3 [ {|0.82 − 0.92|. 2−|0.8 2−0.92|−1 + |0.22 − 0.12|. 2−|0.2 2−0.12|−1} + {|0.72 − 0.82|. 2−|0.7 2−0.82|−1 + |0.32 − 0.22|. 2−|0.3 2−0.22|−1} + {|0.62 − 0.72|. 2−|0.6 2−0.72|−1 + |0.32 − 0.12|. 2−|0.3 2−0.12|−1} ] = 2 3 [0.075551627 + 0.014691304 + 0.067593784 + 0.024148408 + 0.05939904 + 0.037842305] = 0.186150981. the detailed computation for the proposed measures can be summarized in the table 1: table 1. numerical illustration to validate proposed measures proposed measure 1 numerical values proposed measure 2 numerical values 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑄) 0.272991 𝐷𝑊𝑃𝐹𝑆𝐸 (𝑃, 𝑄) 0.093874 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑅) 0.412732 𝐷𝑊𝑃𝐹𝑆𝐸 (𝑃, 𝑅) 0.138443 𝐷𝑃𝐹𝑆𝐸 (𝑄, 𝑅) 0.186151 𝐷𝑊𝑃𝐹𝑆𝐸 (𝑄, 𝑅) 0.061395 from the above computations, it supports that 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑄) ≤ 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑅) and 𝐷𝑃𝐹𝑆𝐸 (𝑄, 𝑅) ≤ 𝐷𝑃𝐹𝑆𝐸 (𝑃, 𝑅). also, 𝐷𝑊𝑃𝐹𝑆𝐸 (𝑃, 𝑄) ≤ 𝐷𝑊𝑃𝐹𝑆𝐸 (𝑃, 𝑅) and 𝐷𝑊𝑃𝐹𝑆𝐸 (𝑄, 𝑅) ≤ 𝑆𝑊𝑃𝐹𝑆𝐸 (𝑃, 𝑅). 4. applications of exponential divergence measures one of the most crucial aspects of building model is deciding on criteria. as a result, criteria are critical components that allow options to be compared from a particular perspective. users are often satisfied with a product when its characteristics fit their tastes and expectations. the most essential product selection criteria for consumers must be determined to design an effective decision model. pfss has been frequently used to handle multi attribute decision making (madm) problems in pythagorean fuzzy environments due to its excellent skill in representing uncertain information. many pfss based madm algorithms have been proposed. we are presenting a novel pythagorean fuzzy madm approach based on the topsis method in this part. under the pythagorean fuzzy condition, madm problem can be depicted by a flowchart as shown in figure 1. arora/decis. mak. appl. manag. eng. 5 (1) (2022) 246-263 252 figure 1. topsis algorithm approach 4.1. a case study a multinational company wishes to buy smartphones for their white-collar workers. because of the large number of smart phones to be acquired, the process is crucial. with members as procurement manager (ɗ𝟏), human resource manager (ɗ𝟐) and quality manager (ɗ𝟑), a decision committee of three experts is formed by the company with the goal of determining the most appropriate smartphone among five possible options as model 1 (𝒜1), model 2 (𝒜2), model 3 (𝒜3), model 4 (𝒜4) and model 5 (𝒜5). experts assist in the decision-making process and employ smartphone choosing criteria. there are five options available as storage capacity in gigabytes(ℂ𝟏), weight in grams(ℂ𝟐), camera specifications in pixels(ℂ𝟑), screen size in inches(ℂ𝟒), battery life in hours(ℂ𝟓). these models were chosen due to their similar costs in the indian market. organization assigns weights to these criteria as 𝜔 =0.25, 0.35, 0.20, 0.12, and 0.08, respectively. step 1: establish the decision matrix (x) for each criterion, the options are first examined by the decision makers ɗ𝟏, ɗ𝟐 and ɗ𝟑 using pair-wise comparisons. the assessment of the given alternatives in the form of pythagorean fuzzy sets by these three decision makers are examined in table 2-4. significance of topsis approach to madm in computing exponential divergence measures… 253 table 2. data set in the form of a decision matrix (x) of decision maker ɗ1 decision maker alternatives vs criteria ℂ𝟏 ℂ𝟐 ℂ𝟑 ℂ𝟒 ℂ𝟓 ɗ𝟏 𝒜1 <0.8,0.1> <0.1,0.6> <0.2,0.8> <0.6,0.1> <0.1,0.6> 𝒜2 <0.0,0.8> <0.4,0.4> <0.6,0.3> <0.1,0.7> <0.1,0.8> 𝒜3 <0.6,0.1> <0.4,0.5> <0.3,0.0> <0.7,0.2> <0.3,0.4> 𝒜4 <0.7,0.3> <0.3,0.4> <0.7,0.2> <0.8,0.1> <0.2,0.5> 𝒜5 <0.5,0.3> <0.5,0.4> <0.7,0.2> <0.6,0.1> <0.4,0.7> table 3. data set in the form of a decision matrix (x) of decision maker ɗ2 decision maker alternatives vs criteria ℂ𝟏 ℂ𝟐 ℂ𝟑 ℂ𝟒 ℂ𝟓 ɗ𝟐 𝒜1 <0.4,0.0> <0.0,0.7> <0.3,0.3> <0.1,0.8> <0.4,0.0> 𝒜2 <0.3,0.5> <0.6,0.2> <0.6,0.1> <0.2,0.4> <0.3,0.5> 𝒜3 <0.1,0.7> <0.9,0.0> <0.2,0.7> <0.8,0.0> <0.1,0.7> 𝒜4 <0.4,0.3> <0.8,0.1> <0.2,0.6> <0.2,0.7> <0.4,0.3> 𝒜5 <0.4,0.5> <0.5,0.3> <0.8,0.2> <0.7,0.3> <0.3,0.6> table 4. data set in the form of a decision matrix (x) of decision maker ɗ3 decision maker alternatives vs criteria ℂ𝟏 ℂ𝟐 ℂ𝟑 ℂ𝟒 ℂ𝟓 ɗ𝟑 𝒜1 <0.2,0.6> <0.8,0.3> <0.4,0.5> <0.1,0.7> <0.6,0.5> 𝒜2 <0.6,0.3> <0.5,0.3> <0.6,0.3> <0.5,0.2> <0.2,0.6> 𝒜3 <0.7,0.2> <0.6,0.2> <0.7,0.2> <0.6,0.3> <0.3,0.4> 𝒜4 <0.3,0.8> <0.2,0.6> <0.3,0.6> <0.6,0.3> <0.4,0.8> 𝒜5 <0.2,0.6> <0.8,0.3> <0.4,0.5> <0.1,0.7> <0.6,0.5> step 2: calculation of normalized decision matrix (x) in the crisp environment, to avoid the complicated normalization formula used in classical topsis, simpler formulas are used to transform the various criteria scales into a comparable scale. ℂ𝟏, ℂ𝟑, and ℂ𝟒 are benefit criteria, while ℂ𝟐 is cost qualities, according to these experts. however, in case of pythagorean fuzzy environment, normalized matrix can be constructed by replacing membership and non-membership values in cost attributes, whereas there will not be any change in case of benefit attributes. the results are shown in table 5-8. table 5. normalized values of decision maker ɗ1 in terms of pfss (x) decision maker alternatives vs criteria ℂ𝟏 ℂ𝟐 ℂ𝟑 ℂ𝟒 ℂ𝟓 ɗ𝟏 𝒜1 <0.8,0.1> <0.6,0.1> <0.2,0.8> <0.6,0.1> <0.1,0.6> 𝒜2 <0.0,0.8> <0.4,0.4> <0.6,0.3> <0.1,0.7> <0.1,0.8> 𝒜3 <0.6,0.1> <0.5,0.4> <0.3,0.0> <0.7,0.2> <0.3,0.4> 𝒜4 <0.7,0.3> <0.4,0.3> <0.7,0.2> <0.8,0.1> <0.2,0.5> 𝒜5 <0.5,0.3> <0.4,0.5> <0.7,0.2> <0.6,0.1> <0.4,0.7> arora/decis. mak. appl. manag. eng. 5 (1) (2022) 246-263 254 table 6. normalized values of decision maker ɗ2 in terms of pfss (x) decision maker alternatives vs criteria ℂ𝟏 ℂ𝟐 ℂ𝟑 ℂ𝟒 ℂ𝟓 ɗ𝟐 𝒜1 <0.4,0.0> <0.7,0.0> <0.3,0.3> <0.1,0.8> <0.4,0.0> 𝒜2 <0.3,0.5> <0.2,0.6> <0.6,0.1> <0.2,0.4> <0.3,0.5> 𝒜3 <0.1,0.7> <0.0,0.9> <0.2,0.7> <0.8,0.0> <0.1,0.7> 𝒜4 <0.4,0.3> <0.1,0.8> <0.2,0.6> <0.2,0.7> <0.4,0.3> 𝒜5 <0.4,0.5> <0.3,0.5> <0.8,0.2> <0.7,0.3> <0.3,0.6> table 7. normalized values of decision maker ɗ3 in terms of pfss (x) decision maker alternatives vs criteria ℂ𝟏 ℂ𝟐 ℂ𝟑 ℂ𝟒 ℂ𝟓 ɗ𝟑 𝒜1 <0.2,0.6> <0.3,0.8> <0.4,0.5> <0.1,0.7> <0.6,0.5> 𝒜2 <0.6,0.3> <0.3,0.5> <0.6,0.3> <0.5,0.2> <0.2,0.6> 𝒜3 <0.7,0.2> <0.2,0.6> <0.7,0.2> <0.6,0.3> <0.3,0.4> 𝒜4 <0.3,0.8> <0.6,0.2> <0.3,0.6> <0.6,0.3> <0.4,0.8> 𝒜5 <0.2,0.6> <0.3,0.8> <0.4,0.5> <0.1,0.7> <0.6,0.5> step 3: identify the fuzzy positive ideal solution (fpis) and negative ideal solution (fnis) fpis maximizes the benefit and minimizes the cost, whereas the fnis maximizes the cost and minimizes the benefit. for each decision maker, we compute fpis and fnis for the pfss using 𝐴𝑘+ = {𝑟1 𝑘+, 𝑟2 𝑘+, … , 𝑟𝑛 𝑘+} = {(max 𝑖 (𝑟𝑖𝑗 𝑘 ) ∕ 𝑗 ∈ 𝐼) , ((min 𝑖 (𝑟𝑖𝑗 𝑘 ) ∕ 𝑗 ∈ 𝐽))}; (8) 𝐴𝑘− = {𝑟1 𝑘−, 𝑟2 𝑘−, … , 𝑟𝑛 𝑘−} = {(min 𝑖 (𝑟𝑖𝑗 𝑘 ) ∕ 𝑗 ∈ 𝐼) , ((max 𝑖 (𝑟𝑖𝑗 𝑘 ) ∕ 𝑗 ∈ 𝐽))} (9) where i refer to the benefit criteria and j, the cost criteria. the subsequent values are presented in table 8. table 8. fuzzy positive and negative ideals for each decision makers decision maker fpis and fnis ℂ1 ℂ2 ℂ3 ℂ4 ℂ5 ɗ1 𝐴+ 𝐴− <0.8,0.1> <0.0,0.8> <0.6,0.1> <0.4,0.5> <0.7,0.0> <0.2,0.8> <0.8, 0.1 <0.1,0.7> <0.4,0.4> <0.1,0.8> ɗ2 𝐴+ 𝐴− <0.4,0.0> <0.1,0.7> <0.7,0.0> <0.0,0.9> <0.8,0.1> <0.2,0.7> <0.8,0.0> <0.1,0.8> <0.4,0.0> <0.1,0.7> ɗ3 𝐴+ 𝐴− <0.7,0.2> <0.2,0.8> <0.6, 02> <0.2,0.8> <0.7,0.2> <0.3,0.6> <0.7,0.2> <0.1,0.7> <0.6,0.0> <0.2,0.8> step 4: calculate the separation distance of each competitive alternative from the ideal and nonideal solution separation measures 𝐷(𝒜𝑖 , 𝐴 +), 𝐷(𝒜𝑖 , 𝐴 −) and the weighted exponential divergence measure proposed in equation (7) of each alternative from fpis and fnis have been calculated using formulae (10) and (11) and presented in table 9. significance of topsis approach to madm in computing exponential divergence measures… 255 𝐷(𝒜𝑖 , 𝐴 +) = 2 𝑛 ∑ 𝜔𝑖 [|𝛿𝑃 2(𝑥𝑖 ) − 𝛿𝑄 2+(𝑥𝑖 )|. 2 −|𝛿𝑃 2 (𝑥𝑖)−𝛿𝑄 2+(𝑥𝑖)|−1 + |𝜁𝑃 2(𝑥𝑖 ) − 𝑛 𝑖=1 𝜁𝑄 2+(𝑥𝑖 )|. 2 −|𝜁𝑃 2(𝑥𝑖)−𝜁𝑄 2+(𝑥𝑖)|−1] (10) 𝐷(𝒜𝑖 , 𝐴 −) = 2 𝑛 ∑ 𝜔𝑖 [|𝛿𝑃 2(𝑥𝑖 ) − 𝛿𝑄 2−(𝑥𝑖 )|. 2 −|𝛿𝑃 2 (𝑥𝑖)−𝛿𝑄 2−(𝑥𝑖)|−1 + |𝜁𝑃 2(𝑥𝑖 ) − 𝑛 𝑖=1 𝜁𝑄 2−(𝑥𝑖 )|. 2 −|𝜁𝑃 2 (𝑥𝑖)−𝜁𝑄 2−(𝑥𝑖)|−1] (11) table 9. separation measures for ideal solutions w.r.t. each decision maker alternativ es ɗ1 ɗ2 ɗ3 𝐷1 (𝐴𝑖 , 𝐴 +) 𝐷1 (𝐴𝑖 , 𝐴 −) 𝐷2(𝐴𝑖 , 𝐴 +) 𝐷1 (𝐴𝑖 , 𝐴 +) 𝐷1 (𝐴𝑖 , 𝐴 −) 𝐷2(𝐴𝑖 , 𝐴 +) 𝒜1 0.03861 0.08584 0.03659 0.10277 0.06359 0.05815 𝒜2 0.09158 0.03120 0.08431 0.07717 0.10660 0.03200 𝒜3 0.05103 0.08746 0.11772 0.01963 0.05403 0.06523 𝒜4 0.03305 0.09852 0.10238 0.04890 0.04483 0.06909 𝒜5 0.05061 0.07748 0.05728 0.09883 0.06889 0.06916 step 5: measure the relative closeness of each location to the ideal solution and rank the preference order for each competitive alternative the relative closeness of the potential model with respect to the ideal solution is computed. relative closeness coefficient with respect to each decision maker can be found using the formula 𝑅𝑖 = 𝐷(𝐴𝑖,𝐴 −) 𝐷(𝐴𝑖,𝐴 +)+𝐷(𝐴𝑖,𝐴 −) (12) where 0 ≤ 𝑅i ≤ 1, 𝑖 = 1,2, … , 𝑚 the value of 𝑅𝑖 signifies that higher the value of the relative closeness, the higher the ranking order and hence the better the performance of the alternative. ranking of the preference in descending order thus allows relatively better performances to be compared. the ranking results obtained by pythagorean fuzzy topsis approach is demonstrated in table 10. table 10. ranking results obtained from topsis approach alternatives ɗ1 ɗ2 ɗ3 𝑅1 ranking 𝑅2 ranking 𝑅3 ranking 𝒜1 0.6897 2 0.7374 1 0.4776 4 𝒜2 0.2541 5 0.4779 3 0.2308 5 𝒜3 0.6315 3 0.1428 5 0.5469 2 𝒜4 0.74882 1 0.3232 4 0.6064 1 𝒜5 0.6048 4 0.6330 2 0.5009 3 5. discussion table 10 shows the ranking of the 5 considered smart phones according to the three decision makers. for decision maker 1, 𝐷1 , model 4 (𝒜4) is the best smartphone. same is the case for decision makers three, 𝐷3. on the other hand, model 1 (𝒜1) is the best choice for decision maker 𝐷2. further, to validate this result, sensitivity analysis will be carried out in the next section, arora/decis. mak. appl. manag. eng. 5 (1) (2022) 246-263 256 5.1. sensitivity analysis if decision makers find distinct ranking for the alternatives, the overall findings of the best alternatives remain unclear. to overcome the ambiguity about the best alternatives with respect to the decision-makers, we aggregate the ideal distance measurement values of every decision-maker different values of the experts are aggregated by assigning a priority value value 𝜌 = (𝜌1, 𝜌2, … , 𝜌𝑠) 𝑇 to each expert such that 𝜌𝑠 > 0 and ∑ 𝜌𝑘 = 1 𝑠 𝑘=1 . the distance measure of each expert is aggregated by using these weight vectors and the overall measurement values of the alternatives are obtained which can be depicted in table 11 as 𝜗𝑖 + = ∑ 𝜌𝑘 𝐶𝑖𝑗 +𝑠 𝑘=1 (13) 𝜗𝑖 − = ∑ 𝜌𝑘 𝐶𝑖𝑗 −𝑠 𝑘=1 (14) also, ℜ𝑖 = 𝜗𝑖 − 𝜗𝑖 ++𝜗𝑖 − (15) where 0 ≤ ℜ𝑖 ≤ 1, 𝑖 = 1, 2, … , 5 table 11. aggregated closeness coefficient and ranking for each smartphone 𝐶𝑎𝑠𝑒 1: 𝜌1 = 0.45, 𝜌2 = 0.35, 𝜌3 = 0.20 alternatives ℜ𝑖 ranking selected smartphone 𝒜1 0.6677 1 𝒜2 0.3401 5 𝒜3 0.4415 4 𝒜1 𝒜4 0.5577 3 𝒜5 0.5953 2 𝐶𝑎𝑠𝑒 2: 𝜌1 = 0.35, 𝜌2 = 0.27, 𝜌3 = 0.37 alternatives ℜ𝑖 ranking selected smartphone 𝒜1 0.6268 1 𝒜2 0.3154 5 𝒜3 0.4637 4 𝒜1 𝒜4 0.5679 3 𝒜5 0.5743 2 𝐶𝑎𝑠𝑒 3: 𝜌1 = 0.38, 𝜌2 = 0.33, 𝜌3 = 0.29 alternatives ℜ𝑖 ranking selected smartphone 𝒜1 0.6485 1 𝒜2 0.3325 5 𝒜3 0.4423 4 𝒜1 𝒜4 0.5536 3 𝒜5 0.5855 2 𝐶𝑎𝑠𝑒 4: 𝜌1 = 0.40, 𝜌2 = 0.30, 𝜌3 = 0.30 alternatives ℜ𝑖 ranking selected smartphone 𝒜1 0.6448 1 𝒜2 0.3250 5 𝒜3 0.4565 4 𝒜1 𝒜4 0.5659 3 𝒜5 0.5834 2 significance of topsis approach to madm in computing exponential divergence measures… 257 𝐶𝑎𝑠𝑒 5: 𝜌1 = 0.29, 𝜌2 = 0.25, 𝜌3 = 0.46 alternatives ℜ𝑖 ranking selected smartphone 𝒜1 0.6092 1 𝒜2 0.3081 5 𝒜3 0.4659 4 𝒜1 𝒜4 0.5653 3 𝒜5 0.5655 2 𝐶𝑎𝑠𝑒 6: 𝜌1 = 0.29, 𝜌2 = 0.25, 𝜌3 = 0.46 alternatives ℜ𝑖 ranking selected smartphone 𝒜1 0.6626 1 𝒜2 0.3347 5 𝒜3 0.4495 4 𝒜1 𝒜4 0.5639 3 𝒜5 0.5926 2 sensitivity analysis concluded that by assigning different priorities to the opinions of decision makers, the result of the proposed method remains the same, as 𝒜1 came out to be the best-model in all the cases, thereby substantiating the validity and reliability of the proposed method. 6. comparative study to demonstrate the dominance of the proposed divergence measure, a comparison between the proposed weighted exponential divergence measure and the existing measures is conducted based on the numerical cases suggested. table 12 represents a comprehensive evaluation of the divergence measures for pfss. the numerical data in table 12 have been analysed, and it has been discovered that the results produced using our suggested divergence measure given in equation 8 are like those obtained using existing measures. as a result, the accompanying table demonstrates that the proposed divergence measure is consistent across all approaches, as the best alternative remains the same. table 12. comparison of existing with the proposed divergence measures measure ranking peng et al. (2017) 𝒜 1 › 𝒜 4 › 𝒜 5 › 𝒜 3 › 𝒜 2 ejegwa, 2018 measure i 𝒜 1 › 𝒜 5 › 𝒜 4 › 𝒜 3 › 𝒜 2 ejegwa, 2018 measure ii 𝒜 1 › 𝒜 5 › 𝒜 4 › 𝒜 3 › 𝒜 2 ejegwa, 2018 measure iii 𝒜 1 › 𝒜 5 › 𝒜 4 › 𝒜 3 › 𝒜 2 zhang et al., 2019 measure i 𝒜 1 › 𝒜 5 › 𝒜 4 › 𝒜 3 › 𝒜 2 zhang et al., 2019 measure ii 𝒜 1 › 𝒜 5 › 𝒜 4 › 𝒜 3 › 𝒜 2 zhang et al., 2019 measure iii 𝒜 1 › 𝒜 5 › 𝒜 4 › 𝒜 3 › 𝒜 2 zhang et al., 2019 measure iv 𝒜 1 › 𝒜 5 › 𝒜 4 › 𝒜 3 › 𝒜 2 it can be determined by studying the above findings that there are differences in ranks when different scenarios are applied, indicating that the model is sensitive to changes. it is observed that 𝒜1 is the best alternative. in accordance with the findings of bohar et al., (2020), spearman’s rank correlation is used to find the correlation of attributes. when written in mathematical form, spearman’s coefficient of correlation is denoted by ℛ and is defined by arora/decis. mak. appl. manag. eng. 5 (1) (2022) 246-263 258 ℛ = 1 − 6 ∑ ď𝑖 2𝑛 𝑖=1 𝑛(𝑛2−1) (16) where ď𝑖 = difference in ranks of the “ith” element: 𝑛 = number of observations ℛ = value of correlation coefficient the value of ℛ will always lies between -1 and 1. if ℛ = 1, there is a perfectly positive correlation; if ℛ = −1, then the ranks are exactly opposite. however, if ℛ = 0, then the ranks are uncorrelated. spearman’s rank correlation among existing and proposed measures is shown in table 13 as table 13. spearman’s rank correlation for various existing and proposed measures p e n g e t a l. ( 2 0 1 7 ) e je g w a , 2 0 1 8 m e a su re i e je g w a , 2 0 1 8 m e a su re i i e je g w a , 2 0 1 8 m e a su re i ii z h a n g e t a l. , 2 0 1 9 m e a su re i z h a n g e t a l. , 2 0 1 9 m e a su re i i z h a n g e t a l. , 2 0 1 9 m e a su re ii i z h a n g e t a l. , 2 0 1 9 m e a su re ii i z h a n g e t a l. , 2 0 1 9 m e a su re i v p ro p o se d m e a su re peng et al. (2017) 1 ejegwa, 2018 measure i 0.9 1 ejegwa, 2018 measure ii 0.9 1 1 ejegwa, 2018 measure iii 0.9 1 1 1 zhang et al., 2019 measure i 0.9 1 1 1 1 zhang et al., 2019 measure ii 0.9 1 1 1 1 1 zhang et al., 2019 measure iii 0.9 1 1 1 1 1 1 zhang et al., 2019 measure iii 0.9 1 1 1 1 1 1 1 zhang et al., 2019 measure iv 0.9 1 1 1 1 1 1 1 1 proposed measure 0.9 1 1 1 1 1 1 1 1 1 the table of spearman's coefficient of correlation values shows that there is perfectly positive correlation in almost all the cases. the graphical representation (figure 2) is also shown for better understanding of the selection procedure. significance of topsis approach to madm in computing exponential divergence measures… 259 figure 2. comparison of existing with the projected divergence measure 7. conclusion the paper offers new exponential divergence measures which comply with the conventional parameters of pfss. the credibility of the proposed divergence measures through numerical computations has been confirmed as well. further, these divergence measures have been employed to the application of madm problem for the selection of smartphones. this analysis depicts an extension of topsis methodology under pythagorean fuzzy sets (pfss) environment. the technique for order preference by similarity to ideal solutions (topsis) notable and powerful technique for multi attribute decision making (madm) issues. the goal of this investigation is to broaden topsis to handle madm problems under pfss. however, in this topsis approach, sometimes there could be severe loss of data and misleading results in the fuzzy environment. to overcome this, a sensitivity analysis has been done for better reliability and accuracy of the decision. a case study is taken to rank five leading smartphones based on five criteria using the proposed divergence measures. these weighted divergence measures can be applied to complex decision making and risk analysis in the future. furthermore, criteria weights can be chosen using entropy and sensitivity analysis of the obtained results with the results obtained by the basic classical methods topsis, fuzzy topsis methods, and intuitionistic fuzzy topsis can be done. author contributions: h. d. arora made a significant contribution to the article's concept or design. the author is also responsible for methodology, drafting, and investigations. anjali naithani contributed to software validation, formal analysis, data processing, and critical supervision. she also assisted in critically edited the article for essential intellectual content. both authors agreed to be responsible for all aspects of the work, including ensuring that any questions about the work's accuracy or integrity are thoroughly examined and resolved. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 d1(p,q) d2(p,q) d3(p,q) d4(p,q) d5(p,q) d6(p,q) d7(p,q) d8(p,q) proposed measure comparison of existing with the proposed divergence measures r1 r2 r3 r4 r5 arora/decis. mak. appl. manag. eng. 5 (1) (2022) 246-263 260 funding: this research received no external funding. data availability statement: the authors certify that the data supporting the findings of this study are available within the article. acknowledgments: authors thank the anonymous reviewers and editors for their careful reading of our manuscript and their insightful comments and suggestions. conflicts of interest: the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. references adabitabar f.m., agheli b., & baloui je. 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(2020). a new divergence measure of pythagorean fuzzy sets based on belief function and its application in medical diagnosis. mathematics, 8, 142. © 2021 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 5, issue 2, 2022, pp. 329-361. issn: 2560-6018 eissn: 2620-0104 doi:_ https://doi.org/10.31181/dmame181221030d * corresponding author. e-mail addresses: madhushreedascs@gmail.com (m. das), royarindamroy@yahoo.com (a. roy), samirm@iimcal.ac.in (s. maity), samarjit.kar@maths.nitdgp.ac.in (s. kar), shatadru.sengupta@gmail.com (s. sengupta) solving fuzzy dynamic ship routing and scheduling problem through modified genetic algorithm madhushree das1, arindam roy1, samir maity2, samarjit kar3* and shatadru sengupta4 1 department of computer science and application, prabhat kumar college, west bengal, india 2 department of data science, university of kalyani, west bengal, india 3 department of mathematics, national institute of technology, west bengal, india 4 department of computer applications, haldia institute of technology, haldia, india received: 25 june 2020; accepted: 18 december 2021; available online: 29 december 2021. original scientific paper abstract: main motto of ship routing and scheduling is to reduce the total transportation cost of each ship or vessel without interrupting the demand and supply. in this study, we have proposed a ship routing and scheduling model for commercial ships where, to ensure unhindered demand and supply of products at various ports in a fixed time frame, the dynamic demand and supply of each port were considered under a fuzzy environment. additionally, simultaneous loading and unloading and a fixed load factor is used to minimize port time and reduce risks, and this aspect of our work makes it realistically inclined. we also show, in our work, speed optimization to reduce fuel consumption and carbon emission. in practice, cost parameters cannot be always determined, it fluctuates at a certain range from time to time. we have treated the imprecise cost parameters as triangular fuzzy numbers. with a view to working with the developed model, a modified genetic algorithm (mga) with a new selection technique, namely an in-vitro-fertilization-based crossover, and a generation-dependent mutation is proposed. the proposed sustainable ship routing algorithm with dynamic demand and supply in an uncertain environment gives a novelty in the literature. another novelty is incurred through the proposed mga in the heuristic search algorithms. this algorithm has produced numerical results superior to those of other heuristic algorithms. we have also established the efficiency of the proposed algorithm through statistical experiments. mailto:madhushreedascs@gmail.com mailto:royarindamroy@yahoo.com mailto:samirm@iimcal.ac.in mailto:samarjit.kar@maths.nitdgp.ac.in mailto:shatadru.sengupta@gmail.com das et al./decis. mak. appl. manag. eng. 5 (2) (2022) 329-361 330 key words: ship routing and scheduling, risk factor, modified genetic algorithm, bird’s nest building material selection technique, possibility approach. 1. introduction shipping industries play an important role in the growth of the economy of a country, and forms a core part of its trade. in the 20th century, the maritime transportation grew exponentially. bulk cargo is generally moved either by sea, or by road, or, by air. seaborne trade is the cheapest form, and it has spread to 89.6% of total global trade in terms of volume and 70.1% in terms of value in the recent years. in ship routing, each ship starts its journey from initial ports at the beginning of the planning horizon and visits from one port to another port with its containers. a ship visits a set of ports within any planning horizon, and at each of these ports some loading/unloading operations are carried out. to transport the cargos, a heterogeneous fleet of ships are used. there are a fixed capacity and sailing speed for each individual ship. this model aims to design appropriate shipping routes and schedules to minimizing the costs associated with transportation. the present investigation considered heterogeneous ships with a variety of capacities. determining the navigation of ships or vessels in the maritime environment is a ship routing problem that is similar to the vehicle routing problem on land. some of the challenges in ship routing and scheduling are as follows: 1. because a trip may last for numerous days, each ship has a fixed ‘time window’ to operate at the port. a fixed time is allotted to each ship to complete the operation (loading/unloading) at the port. 2. a linear ship route is not always possible because of several factors. 3. sometimes two captains may not agree on the same route for the same container for several reasons. 4. because of variable weather conditions, maritime transport is highly uncertain. calculation of the traveling time and cost is difficult because the travel time is affected by the wind speed, direction, and current. commercial ship operations are categorized as follows: 1) linear, 2) tramp and 3) industrial. similar to bus transport, a linear operation in sea involves visiting all ports in its route, and ending at the destination port. tramp ships are analogous to taxis, and their goal is to maximize profit while ensuring unhindered demand and supply of products. industrial shipment minimizes the cost of shipping products by using the best route. cargo is allotted to ships at the source port, which is then transported to the destination. industrial shipping is of two types, namely cargo routing and inventory routing. the cargo routing problem is specified by the time frame for loading and unloading operations, and the demand and supply at specific ports may not be fixed. by contrast, in the inventory routing problem, the product requirement at a particular port is fixed. in this research, we have focused on the cargo routing problem. in practice, the demand and supply of products at the port are not fixed, it varies from time to time. to satisfy this condition, the loading/unloading operation of some products is determined spontaneously at the port according to the demand/supply. because of uncertain market conditions, dynamic demand/supply is typical. loading and unloading operations are performed simultaneously to optimize the port time as well as port congestion. in this study, we considered those operations simultaneously where there is spontaneous and simultaneous loading/unloading. the traveling costs depend on various factors, such as the geographical areas, the weather conditions solving fuzzy dynamic ship routing and scheduling problem through modified genetic… 331 during the sailing period, and the product being transported, and hence are considered as vague or fuzzy in this model. we have thus worked to minimize the shipping cost and maintain sustainability keeping uncertainty in mind, at the same time ensuring that all cargoes are lifted from the loading port and discharged at the unloading port. this model used container-based cargo to transport goods. risk factors such as, thunderstorms, tsunamis, icebergs, and piracy, which depends on the time period of sailing and choosing the routes, are also considered as fuzzy numbers. the total risk is considered to be less than the maximum risk which is assigned for the entire trip of that ship. it is this fuzzy approach renders the problem realistic. 1.1. motivation in ship routing and scheduling problems, a ship with a fixed capacity starts its voyage and visits a set of ports in a fixed time frame to satisfy the demand and supply of those ports. a port can be visited by only one ship at a particular time. de et al. (2017) considered numerous sub-time window concepts in ship routing and scheduling problems. the travelling time of a ship is the time to travel from one port to another port and its operation time is the time needed mainly for loading and unloading. however, within a fixed time frame, due to lack of proper scheduling of ships, there is a delay in cargo transportation from the origin port to the destination port. a penalty cost is imposed for early/late arrival or loading/unloading delay. if any ship exceeds the fixed time frame, it had to pay a penalty calculated according to the additional hours it operates outside the fixed time frame. in this model, loading and unloading operations of various goods is performed simultaneously but unloading operation from a ship is performed before the loading operation starts. here the main research motive is total cost minimization and optimum path determination for an industrial ship routing and scheduling in a fixed time frame, in which risk is minimum in a sustainable environment but in uncertain market conditions. because of covid-19, social behavior as well as demand and supply fluctuate randomly, affecting the shipping industry. imran et al. (2020) designed a ship routing and scheduling model for static demand and supply, but the model is not robust in uncertain or dynamic situations. our second research motive is the facilitating the transport of cargo whose demand and supply are determined dynamically. yang et al. (2021) designed a tramp ship routing model to address port congestion because of static demand and supply. the result revealed that dynamic demand and supply reduce port congestion and render the system robust. next, we focused on container-based cargo shipping to minimize the cost and path of travel. container-based shipping provides an end-to-end approach to customers and shipping companies. finally, we focused on increasing the efficiency and effectiveness of the solution method. in this study, we develop a novel selection technique and generation-dependent mutation to achieve a better solution. a nature-based heuristic algorithm requires less time and computational cost. considering fuzzy numbers, we introduced uncertainty or dynamic attributes to the model and adopted the possibility and necessity approach by using the graded mean integration value defuzzification method. thus, we can once again the novelties of the proposed study are as follows: we developed a dynamic cargo ship routing and scheduling algorithm under uncertain environments. to maintain sustainability conditions, a novel mathematical function was incorporated to study the fuel consumption corresponding to the total carbon emission and the limitations of the journey considered. das et al./decis. mak. appl. manag. eng. 5 (2) (2022) 329-361 332 we developed an optimal route design with a maximum allowable risk for that trip. a novel ga (new selection, crossover, and mutation strategies introduced) was developed to address the proposed model. we considered traveling cost and risk factors as fuzzy values for uncertain environment. the rest of this paper is organized as follows: section 2 describes studies relevant to the present investigation. section 3 describes mathematical preliminaries. section 4 discusses the proposed dynamic ship routing model. the defuzzification of the proposed model is presented in section 5. in section 6, a modified ga (mga) is presented. section 7 illustrates the outcome of numerical experiments. statistical test significance is discussed in section 8. section 9 provides results and discussions. managerial insights are provided in section 10. finally, conclusions and future scope are presented in section 11. 2. literature study the ship routing and scheduling problem (srsp) is similar to the traveling salesman problem (tsp) for cargo on land. bausch et al. (1998) was the first to develop a short-term ship routing and scheduling model. subsequently, numerous studies have focused on ship routing and scheduling problems. psaraftis (2019) described a few conceptual hypotheses for the srsp. however, the study did not provide any practical implementations derived from these. alfandri et al. (2019) proposed a weekly demand service between a pair of ports for barge containers using a liner service to maximize the profit. however, transport delays may occur in liner shipping for certain routes because of uncertain weather conditions. rabbani et al. (2019) described a model to determine the best route to minimize the shipping costs and carbon emissions and maximize job creation in ships and ports. in this study, variable speed was considered. cost minimization and job creation maximization contrast each other. noshokaty (2021) used information technology in tramp ship routing and scheduling problems to address commodity forecasting. in this model, the method of solution increases the time complexity, when all constraints are considered in commodities forecasting. zhao et al. (2019) described how operation measures and objectives depend on fuel prices and vessel loads. to generate profits, the speed of a vessel is decreased for carrying a higher vessel load. this scenario sometimes results in missing the time window because of the low-speed voyage. homsi et al. (2020) described the generation of elementary routes for tackling routing problems related to industrial and tramp ships. however, load-dependent fuel consumption, sea condition, and emission-related vital points were not considered there. liqan et al. (2020) considered ocean currents for speed optimization to minimize the total fuel consumption. however, this optimization model had been tested for only one ship, and hence the results cannot be exactly generalized. numerous algorithms have been proposed to optimize ship routing problems, and each algorithm has some pros and cons. laura et al. (2016) used algorithms, such as the dijkstra’s algorithm, dynamic programming, and iterative methods, to solve ship route optimization problems. these solution methods are problem-dependent, and each method has advantages and disadvantages. these methods are computationally expensive for np-hard problems. de et al. (2017) described a hybrid particle swarm optimization (pso) algorithm to solve timewindow-based ship routing problems with multidimensional features. in this study, pso marginally increases the computational time. wang et al. (2018) proposed a solving fuzzy dynamic ship routing and scheduling problem through modified genetic… 333 hybrid mutation operator to maintain the population diversity in ship routing solutions. this method requires a long time to converge to the optimal solution. alhamad et al. (2019) developed a tabu search method to solve the ship routing problem and revealed that if the problem size increases, the execution time increases drastically. fan et al. (2019) used a variable neighborhood search algorithm to solve tramp scheduling, which revealed moderate computation time to generate optimal or near-optimal solutions. roy et al. (2020) proposed a real-life in-vitro-fertilization (ivf)-based crossover in a ga to provide a solution to the tsp. in this study, the ivf crossover provided the concepts of having three parents and diversified the solutions. pratap et al. (2019) described their own ship routing and scheduling model without considering the relationship between carbon emissions with fuel consumption in 2019, fan et al. proposed speed optimization in tramp shipping where they have discussed about how to reduce greenhouse gasses. in 2018, according to unctad, sustainable shipping can reduce 50% of the world’s carbon dioxide (co2) within the year 2050 by using slow steaming and alternate fuel. zhang et al. (2021) described fuel consumption as a black-box concept. however, this study did not provide a transparent fuel consumption concept. lan et al. (2020) described various carbon emission policies and presented a comparative analysis of these policies. however, the carbon emission policies sometimes reduce the profit of the shipowner. fan et al. (2019) described the process of multi-type tramp shop scheduling to minimize the total costs of shipping companies. however, real-time ship scheduling was not considered in this study. wang et al. (2018) did not consider wave and wind disturbance, which considerably affects the ship route design. sun et al. (2019) described the uncertain planning stage and demand–supply aspects of customers in real time, but only the logistic network forward flow was considered; reverse flows were not considered. maity et al. (2018) introduced a rough set-based ga, in which an age-dependent selection technique and age-oriented min point crossover was considered. however, this method works with both a high computational time and a high amount of computational resources. dong & bain (2020) described a hybrid a* and ga to solve ship pipe route design. this hybrid algorithm requires considerable execution time. akter et al. (2020) proposed the use of type-2 fuzzy and fuzzy random data to solve a four-dimensional (4d) tsp. in this study, small-size data sets are used. the use of large size data sets becomes computationally expensive. 3. mathematical preliminaries in this study, we presented preliminary fuzzy number concepts and their membership values and the defuzzification technique required for the proposed model. 3.1. triangular fuzzy numbers here, ~ ( , , )a p q r is a normalized triangular fuzzy number (tfn), which is a subset of s (real numbers). it’s membership function is given as follows: das et al./decis. mak. appl. manag. eng. 5 (2) (2022) 329-361 334 ~ 0, , ( ) , 0, a x p x p p x q q p x s x q x s s q x s                  (1) figure 1. graphical representation of triangular fuzzy number 3.2. fuzzy possibility and necessity approach assuming ~ p and ~ q are two fuzzy numbers with membership functions ~ ( ) p x and ( ) q y , respectively ,x y s ~ ~ ~ pos( * ) = sup min ( ), ( ) , , q p p q x y x y s           (2) where * any one of the relations , , , ,     and represents real numbers. ~ ~ ~ ~ nes * 1 pos( * )p q p q        (3) where nes represents necessity. if ~ ~ ~ , ,p q r s and ~ ~ ~ ( , )p f q r where :f s s s  is a binary operation, then the membership function ~ ( ) p x of ~ a is defined as follows: ~ ~( ) = sup min ( ), ( ) , , and z = ( , ) q p p z x y x y s f x y            (4) 1 0 p (q, 1) m (p, 0) n (q, 0) o (s, 0) solving fuzzy dynamic ship routing and scheduling problem through modified genetic… 335 for a tfn ~ 1 2 3 ( , , )b b b b has three parameters 1 2 3 b b b  . by using (1), the membership function ~ ( ) b x is expressed as follows: ~ 1 1 2 2 1 3 2 3 3 2 , ( ) , 0, otherwise b x b for b x b b b b x x for b x b b b                (5) from the aforementioned definitions, the following lemmas can easily be derived: lemma 1. if ~ 1 2 3 ( , , )b b b b is a tfn with 0 < 1 b and d is a crisp number, then ~ 1 pos ( < d)p  , if 1 1 2 1 d b b b     . lemma 2. if ~ 1 2 3 ( , , )b b b b is a tfn with 0 < 1 b and d is a crisp number, then ~ 1 nes ( < d)p  if 3 1 3 2 1 b d b b      . lemma 3. if ~ 1 2 3 ( , , )b b b b and ~ 1 2 3 ( , , )e e e e is tfn with 0 < 1 b and 0 < 1 e then ~ ~ 1 pos ( < )b e  if 3 1 1 3 2 2 1 e b e e b b       . lemma 4. if ~ 1 2 3 ( , , )b b b b and ~ 1 2 3 ( , , )e e e e is tfn with 0 < 1 b and 0 < 1 e , then ~ ~ 1 nes ( < )b e  if 3 1 1 2 1 3 2 1 b e e e b b        . where 1  is a predefined possibility. 3.3. defuzzification of the fuzzy number by using graded mean integration the fuzzy number graded mean integration method with the integral value of graded mean 1  -level of the generalized fuzzy number was used for defuzzification. assuming ~ b is a generalized fuzzy number, then the graded mean integration value (gmiv) of ~ b is denoted by the following equation: 1 ~ 1 1 0 p( )= {(1 ) ( ) ( )} /b x u l x ur x dx x dx     . 1 1 1 0 =2 {(1 ) ( ) ( )}x u l x ur x dx     . (6) das et al./decis. mak. appl. manag. eng. 5 (2) (2022) 329-361 336 where [0,1]u  if, 1u  , an optimistic value. if, 0u  , a pessimistic value. and if, 0.5u  , moderately optimistic value. using the gmiv of tfn ~ 1 2 3 ( , , )b b b b equation (6) becomes a crisp value 1 2 3 [(1 ) 2 ] / 3u b b ub   . when 0.5u  , the expression becomes 1 2 3 1 [ 4 ] 6 b b b  . 4. proposed dynamic ship routing model to develop the proposed model, a set of ships, a group of ports, and a set of products were considered. typically, a ship visits some of the ports to ensure unhindered demand and supply of those ports within a time frame. in this model, a ship starts its journey from a port, visits several ports, and ends its journey at the destination port, which is neither the starting port nor a visited port. for a ship, exceeding the given time window results in a penalty cost. loading and unloading are performed at the port, but the unloading operation is performed before loading. in practice, the cost may not be deterministic, is imprecise, and considered as fuzzy parameters. the model was developed by using various dependent and independent variables and several constraints. the assumption of this model are as follows. figure 2. graphical representation of the proposed model 4.1. assumptions of the model to construct the model mathematically, we assumed the following points: 1. a ship visits a set of ports in its voyage in a fixed time frame. solving fuzzy dynamic ship routing and scheduling problem through modified genetic… 337 2. we consider heterogeneous ships for this model. each ship has a different capacity. 3. a container contains only one product at a time. it has a particular demand port and supply port. in between these two ports, containers cannot be unloaded from the ship. 4. a ship sails at a certain speed in which fuel consumption is minimum and it does not violate the time frame. the total carbon emission does not exceed the maximum carbon emission index. 5. costs and risks are not fixed, but they vary in a certain range. however, the total risks do not exceed the maximum allowable risk for that route. 6. ship bunkering, cleaning, and maintenance are performed at the operation time at the port if required. 7. on a weekly basis, each ship visits the port. demand and supply vary dynamically at ports. 4.2. mathematical model mathematical indices and parameters are stated in table 1. mathematical formulations are given below. the indices obtained from de et al. (2017) work partially. table 1. symbol and indices indices sets symbols descriptions symbols descriptions ,l m time period l time periods set s ships s ships set ,p q ports p ports set r products r products set parameters symbols descriptions pq d distance between port p and port q. a pl t the beginning of the time window at port p in the time period l. b pl t the ending of the time window at port p in the time period l. pr c operational cost (loading/unloading) for a single unit of product r at port p. pr t the loading/unloading time required for single unit of product r at port p. plr w the demand of product r at port p in time period l. pqs c the transportation cost from port p to port q for ship s. sl e storage capacity of ship s in time period l. pr y total storage capacity associated with the depot at port p for the product r. pr u the setup time required for operation (loading/unloading) at port p for product r. das et al./decis. mak. appl. manag. eng. 5 (2) (2022) 329-361 338 pl c penalty cost per hour, associated with operation delay in time l at port p. s h carbon emission per hour for ship s. g constant (relationship constant of speed and fuel consumption). s f fuel consumption rate per hour for ship s. max e maximum carbon emission for that trip. max b maximum bunker consumption for that trip in crushing speed. pr j 1, if product r has a supply at port p. 2, if a demand for product r at port p. 0, if no demand/supply for product r at port p. the following binary variables were assumed: 1, if ship began its operation at port in time and then travel from port to port and initiate its operation at port in time . 0, otherwise; , ; , . plqms s p l p x q q m l m l p q p           1, if product is loaded/unloaded in shi p at port in time period . 0, otherwise; ; ; ; . plsr r s p o l r r l l s s p p          1, if finally ends its travel at port after an operation which is started at time by ship . 0, otherwise; ; ; . pls p z l s l l s s p p         the following continuous variables were assumed: 1 pl t : the starting time of operation at port p in time period l; ,l l p p  . e pl t : the operation ending time at port p in period l; ,l l p p  . pl n : total operating time outside the time frame in time period l at port p; , .l l p p  solving fuzzy dynamic ship routing and scheduling problem through modified genetic… 339 1 plsr q : total quantity of product r loaded/unloaded at port p from ship s in time l; ,l l p p  , r r , s s ; 1 plsr q = 0 if prj = 0. ik cnt : i number of containers of kth type, where i=1,2,3,4,5,6,7; k=1. 1 plsr i : total amount of product r available on ship s after an operation that started in time l while departing from port p; ,l l p p  , r r , s s . 1 plr s : the stock level at port p of product r in time period l; ,l l p p  , r r . pqs v : velocity of ship s while traveling from port p to port q. ,p q p ; s s . pqs v = 0 if 0 plqms x  and p = q. rf pq : risk factor between port p and port q; ,p q p . max r : maximum risk on that trip. objective function: 1 , , pqs plqms pr plsr plsr pl pl p q p l m l s s p p l l s s r r p p l l minimize c x c o q c n                (7) equation (7) minimizes the traveling cost, operation cost, and penalty cost. the constraints are as follows: 1 plqms p p l l s s x     ,q p m l    (8) constraint (8) represents that at least a single ship can operate in port p at the given time l. 1 pls p p l l z    s s  (9) constraint (9) represents that ship s at some port p ended its route at the time l. 1a b pl pl pl t t t  ,p p l l    (10) constraint (10) represents the time frame range. 1 0 pqe pl pm plqms pqs d t t x v           , ; , ;p q p l m l s s      (11) constraint (11) represents that after finishing the operation at port p, ship s travels some distance between port p and port q with a fixed velocity. if 1 0 pqe pl pm plqms pqs d t t x v           , then the ship is not traveling in the sea, that is, 0 plqms x  . 1 1 0 e pl pl pr pr plsr s s r r s s r r t t u t q          ,p p l l    , s s  (12) das et al./decis. mak. appl. manag. eng. 5 (2) (2022) 329-361 340 constraint (12) represents the sum of the setup time, starting time of operation, and that the total loading/unloading time is equal to the ending of each operation. e b pl pl pl t t n  (13) constraint (13) represents penalty time at port p. 1 ( 1)sl plsr plsr ik s l p p r r e o q cnt e      ,p p l l    , s s  , r r  (14) constraint (14) represents that after unloading the product from the ship, the empty space in the ship should be greater than the previous period. products are unloaded container wise. 1 1 ( 1)p l sr pr plsr ik sl p p r r i j q cnt e      ,p p l l    , s s  , r r  (15) constraint (15) represents that the quantity product r in ship s at port p in time period (l-1) is onboard and addition or removal of the quantity of product r loaded/unloaded at port p in time period l should not exceed the empty capacity of ship s in time period l. 1 0 sl plqms plsr q p l l r r e x i       , ; , ;p q p l m l s s      , r r  (16) constraint (16) represents that an upper bound of ship s of product r while sailing does not exceed the ship capacity. 1 1 1 ( 1) 0 p m r pmsr pmr pmr r r s q w s       (17) constraint (17) represents that the demand for each product r is satisfied at time m at port p. 1 max , log( ) ( ) e pq pqs pl pl ps p q p p p l l g d v t t f b        ,p p l l    , s s  (18) constraint (18) represents that the total fuel consumption for ship s for a trip is less than maximum bunker consumption. here, fuel consumption is considered as a logarithmic function of velocity. when ships travel from port p to port q, the fuel consumption is log( ) pqs g v . 1 ( ) 0 e pl pl ps p p l l t t f     a certain fuel consumption occurs at the port when ship s is in port p. 1 0 plr pr s y  ,p p l l    , r r  (19) constraint (19) represents the storage capacity of port p for product r. max , rf pq p q p r   (20) solving fuzzy dynamic ship routing and scheduling problem through modified genetic… 341 constraint (20) represents that the total risk achieved by the port is less than equal to the maximum risk on that trip. 1 max , * ( * log( ) ( ) ) e s pq pqs pl pl ps p q p p p l l h g d v t t f e        p q , s s  (21) constraint (21) represents the total carbon emission at port and sea, which is always less than the maximum carbon emission index for that trip. in the proposed model, we assumed speed to be a logarithmic function, which is appropriate for considering traveling time and carbon emission. {0,1} plqms x  , ; , ;p q p l m l s s      , p q (22) {0,1} pls z  ,p p l l    , s s  (23) {0,1} plsr o  ,p p l l    , s s  , r r  (24) constraints (22) – (24) represent the binary variables. 0 pqs v  , ; ;p q p s s p q     (25) 1 0 plr s  ,p p l l    , r r  (26) 1 1 , plsr plsr q i ,p p l l    , r r  , s s  (27) 1 , , e pl pl pl t t n ,p p l l    (28) constraints (25) – (28) represent the nonnegative constraints. 5. defuzzification technique for the proposed model the traveling costs and the risk factors are fuzzy numbers, which are denoted as ~ ( , )c i j and ~ ( , )r i j respectively, where ~ maxr is the maximum risk level of a particular path. in the proposed model, the total traveling cost and risk are expressed as follows: ~ , 1 ~ ~ max , to minimize z = ( , ) subject to ( , ) m i j i j m i j i j c x x r x x r           (29) where i j x x , i, j= 1, 2, ……, m. m = total number of nodes on that route. 5.1. possibility approaches (optimistic) by using equation (2), we obtain the following objective and constants: das et al./decis. mak. appl. manag. eng. 5 (2) (2022) 329-361 342 minimize f ~ 2 , 1 ~ ~ max 2 , subject to pos ( , ) pos ( , ) m i j i j m i j i j c x x f r x x r                          (30) where i j x x , i, j= 1, 2, ……, m. m = total number of nodes on that route. where 2 2 ,  are levels of possibility, respectively (predefined). traveling fuzzy cost for a path is   ~ 1 2 3 ( , ) ( , ) , ( , ) , ( , )c i j c i j c i j c i j , and corresponding fuzzy risk is   ~ 1 2 3 ( , ) ( , ) , ( , ) , ( , )r i j r i j r i j r i j , and maximum risk   ~ max 1 2 3 , ,r r r r . the aforementioned problem is reduced by using lemma 1 and lemma 3 as follows: to minimize f 1 2 2 1 3 1 2 3 2 2 1 subject to f f f f r r r r r r               (31) where , ( , ) m k i j k i j f c x x  , k =1, 2, 3. and , ( , ) m k i j k i j r r x x  , k =1, 2, 3. where i j x x , i, j= 1, 2, ……, m. m = total number of nodes on that route. the objective function in equation (31) becomes: 1 2 2 1 minimize ( )f f f  subject to 3 1 2 3 2 2 1 r r r r r r       here, 2 2 ,  are possibility levels (predefined). 5.2. necessity approaches (pessimistic) by using equation (3), we obtain the following objective and constants: solving fuzzy dynamic ship routing and scheduling problem through modified genetic… 343 minimize f ~ 3 , 1 ~ ~ max 3 , subject to nes ( , ) nes ( , ) m i j i j m i j i j c x x f r x x r                          (32) where i j x x , i, j= 1, 2, ……, m. m = total number of nodes on that route. traveling fuzzy cost for a path is   ~ 1 2 3 ( , ) ( , ) , ( , ) , ( , )c i j c i j c i j c i j , and corresponding fuzzy risk is   ~ 1 2 3 ( , ) ( , ) , ( , ) , ( , )r i j r i j r i j r i j , and maximum risk   ~ max 1 2 3 , ,r r r r . the aforementioned problem is reduced by using lemma 2 and lemma 4: to minimize f 3 3 3 2 3 1 3 2 1 3 2 subject to 1 1 f f f f r r r r r r                 (33) where , ( , ) m k i j k i j f c x x  , k =1, 2, 3. and , ( , ) m k i j k i j r r x x  , k =1, 2, 3. where i j x x , i, j= 1, 2, ……, m. m = total number of nodes on that route. the objective function in equation (33) becomes: 3 3 3 2 minimize (1 )( )f f f   subject to 3 1 3 2 1 3 2 r r r r r r       here, necessity levels are 3 3 ,  (predefined). 5.3. gmiv approach by applying gmiv for the defuzzification on srsp using equation (29), we obtain the following equation: das et al./decis. mak. appl. manag. eng. 5 (2) (2022) 329-361 344 to minimize 1 2 3 1 [ 4 ] 6 z f f f   subject to 1 2 3 1 2 3 1 1 [ 4 ] [ 4 ] 6 6 r r r r r r     here, we consider w = 0.5, for an optimal solution. 6. mga in 1975, holland was the first to propose natural and artificial system ga, which was subsequently used in natural evolution and optimization problems. adaptive heuristic search algorithms evolved from ga. this algorithm used a random search technique to obtain an optimized result. although random search is used, it is not random, and the previous information is used to direct the search in an appropriate search space to obtain optimum results. ga provides superior optimization when high dimensional search spaces and numerous variables are present. 6.1. proposed bird’s nest building material selection technique to solve the ship routing and scheduling model through mga, we proposed a novel bird’s nest building material selection technique (bbmst) for selection. not all birds excel at making nests. hamerkop (scopus umbretta), ruby-throated hummingbird (archilochus colubirs), sociable weaver (philetairus socius), baya weaver (ploceus philippinus), etc make good nests. among these birds, “sociable weaver” birds select the best material for nest building, and they check the building material with their beak. we selected the best solution from all possible solutions set to obtain the optimum result in ship routing. in the bbmst algorithm, first, the fitness values of all chromosomes are calculated, and the minimum cost path, which is the main objective of this model, is implemented. in each iteration, we selected the minimum cost of the chromosomes, and based on the cost of other chromosomes, we calculated the threshold value. the probability of selection was randomly generated, which is between 0 and 1. if the threshold value is less than the probability of selection, then the current chromosome is selected, otherwise the minimum threshold valued chromosome is considered as fittest for the ivf crossover process. let c[i][j] represent the traveling cost between i th port to j th port, n is the set of ports, and pv is the total number of chromosomes. algorithm1: bbmst selection process require: set of given port n and the size of the population is pv. ensure: a set of fittest chromosomes. 1. begin 2. for ( i = 1 to pv) 3. judge fitness; 4. end for; 5. b = ( minimum-fitness ) chromosome ; 6. ;population ofavgb fitness1  7. for(i= 1 to pv) solving fuzzy dynamic ship routing and scheduling problem through modified genetic… 345 algorithm1: bbmst selection process 8. generate a temporary population, where fitness < b1; 9. end for; 10. for(i= 1 to pv) 11. randomly select three edges sj, j (1, 2, 3) ; 12. generate i r = (cost (sj)/ fitnessi) //cost(sj) represents the traveling cost between two end vertices/nodes of sj . 13. r = random [0, 1]; 14. if (any two i r r ) 15. select the chromosome i; 16. else 17. select the chromosome b; 18. end for 19. end figure 3. ivf crossover 6.2. ivf crossover in the proposed mga, ivf is used for the crossover process. in the ivf process, three parents, namely one father (pr1), one mother (pr2), and a surrogate mother (sr) are selected randomly, depending on their crossover probability (pc) and bring them in mating pool, and two children were created, child1 and child2. first randomly generate a node (for example: ai) and place it at the first position at chid1 and update the parents with ai. next, the least cost valued node is selected from three parents in mating pool. repeat the previous step until all the nodes are not selected and always check whether the same node is selected previously or not. similarly, the same steps are performed for child2. in the matting pool of ga, two new offspring are created with superior information, which is inherited from parents. the ivf is a parallel flow process, which receives the input portion of various parents from the ga population. the individuals are supplied by the ivf process. figure 3 displays the ivf process in a pictorial form. the algorithm steps of ivf crossover are given below: das et al./decis. mak. appl. manag. eng. 5 (2) (2022) 329-361 346 algorithm2: ivf crossover 1. begin 2. choose three parents (pr1, pr2 and sr) randomly from mating pool depending on crossover probability pc. 3. nodes in three parents arepr1: (a1, a2, …, an), pr2: (r1, r2, …., rn) and sr (s1, s2, ….., sn ). 4. randomly choose a node ai. 5. place that ai node at the beginning of child1 and update the three parents. 6. in child1 place ai at the first position. 7. find the minimum cost between (ai, a1), (ai, r1) and (ai, s1). 8. choose the minimum cost node and place it to the next position of child1 and update parents. 9. repeat steps 7 and 8 until all nodes are not selected without repetition of any node. 10. repeat the same steps for child2. 11. end 6.3. generation-dependent mutation after the crossover process, the algorithm goes through the mutation process. in this algorithm, we used generation-dependent mutation. the mutation prevents the solution to be trapped in a local minimum and premature convergence. the mutation maintains genetic diversity in the population. in a population, a random change in some of genes result in one chromosome. this phenomenon produces a novel offspring with a new genetic structure. mutation helps escape from local minima, find the global minima, and maintain the diversity of the population. the probability of generation-dependent mutation ( m p ) is expressed as follows: number of current generation m k p  , [0,1]k  when the number of generations increases, m p decreases normally. 6.3.1. mutation process the proposed ship routing and scheduling model is a node-dependent problem. to mutate the chromosome x = ( 1 2 , ,...., n x x x ), ( 1 2 , ,...., n y y y ), to find the mutated node * m t p n , where n = total number of nodes in a chromosome. if 2 m r p , 2r random [0, 1], then the corresponding chromosome is to be selected for mutation. first, we generate two distinct numbers , i j x x randomly between [1, n]. to obtain mutated solutions, i x and ,j x are interchanged. the same process repeated times to obtain the best offspring. solving fuzzy dynamic ship routing and scheduling problem through modified genetic… 347 algorithm3: generation-dependent mutation 1. begin 2. set g= number of the current generation. 3. ( ) m k p sqrt g  , [0,1]k  4. calculate * m t p n ; // the total number of mutated nodes in the // chromosome. 5. for (i =0 to pop-size) //pop-size = population size. 6. 2r  = rand (0, 1); 7. if( 2 m r p ) 8. select current chromosome; 9. c = rand [1, n]; 10. d = rand [1, n]; 11. if (c == d) 12. goto step 9; 13. end for 14. for (j = 1 to n) // n =total number of nodes of a route . 15. if (x [j ] == c) 16. s = j ; 17. if(x[j] == d) 18. t = j ; 19. x[s] =d ; //replace c by d. 20. x[t]= c ; //replace d by c. 21. end for 22. repeat steps 8 to 20 up to t times. 23. end if 24. end for 25. end because the process is an np-hard problem, the exact approach may cause high computational time. meta-heuristic approaches provide an optimal result in a reasonable time. in this study, we proposed a modified ga-based bbmst, in vitro fertilization, and generation-dependent mutation. 6.4. complexity analysis we considered p as the initial population, n as the number of nodes, g as the number of generations, c p as the probability of crossover, m p as the probability of mutation, and c m p p . therefore, the time complexity of three operations, selection, crossover, and mutation are o( pn ), o( 2 cp p n ), and o( 2 mp p n ), respectively. for the proposed algorithm, complexity is as follows: das et al./decis. mak. appl. manag. eng. 5 (2) (2022) 329-361 348 6.4.1. time complexity because the proposed mga consists of bbmst selection, ivf crossover, and generation-dependent mutation, the time complexity of this algorithm is the maximum time complexity of these three processes. o( pn ), o( 2 pn ), and o( 2 pn ) are the time complexity of three processes. therefore, o( 2 pn ) is the time complexity of the proposed algorithm. 6.4.2. space complexity in the proposed mga, the total space is considered as the population multiplied by the number of nodes, that is, pn . as p > n, the space complexity of the proposed algorithm is o( pn ). figure 4. flowchart of mga solving fuzzy dynamic ship routing and scheduling problem through modified genetic… 349 7. numerical experiments the optimum value is obtained by running this algorithm up to 500 generations considering the crossover probability as 0.61 and considering generation-dependent mutation. we used a 2.40 ghz i7 processor with 12gb of ram with windows 10 for our numerical calculations. the use of fuzzy theory renders the model robust. in this model, we considered 3 ships and 15 ports. we collected data from the haldia port and traveling cost in 10k of each cell value in matrix, which is expressed as follows: table 2. input data: deterministic traveling cost matrix i/j 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 ∞ 25 28 28 30 26 15 37 40 30 41 23 31 4 35 2 17 ∞ 20 28 25 14 30 14 28 4 37 25 32 50 11 3 14 8 ∞ 10 25 15 9 32 40 30 31 47 22 33 45 4 28 30 7 ∞ 20 25 30 35 22 37 11 33 40 21 32 5 37 22 35 30 ∞ 20 25 30 9 28 33 44 15 27 19 6 25 30 25 8 28 ∞ 32 40 32 30 15 34 27 41 11 7 28 25 30 22 37 40 ∞ 10 32 20 36 45 8 25 13 8 20 5 32 40 35 25 40 ∞ 22 37 10 37 29 15 50 9 30 40 35 25 20 22 37 32 ∞ 28 42 31 30 7 33 10 28 30 28 20 11 32 37 40 30 ∞ 36 22 32 23 16 11 12 24 37 29 52 19 37 6 42 31 ∞ 25 14 36 39 12 35 21 13 46 34 29 37 28 19 30 17 ∞ 16 34 29 13 42 36 31 26 25 12 30 24 19 27 36 23 ∞ 7 24 14 38 15 24 42 18 29 46 27 33 19 19 45 25 ∞ 31 15 41 29 11 28 41 27 34 29 9 28 16 45 29 34 ∞ the loading and unloading operation cost for each product and penalty cost for the outside time from timeframe for each port and cost in 1 k of each cell of the matrix is presented in table 3. table 3. operation cost and penalty cost matrix operation cost for each product 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 15 8 11 13 12 7 14 9 10 8 7 14 9 12 12 2 8 8 12 7 9 8 9 9 12 9 9 9 8 9 9 3 7 9 8 8 7 7 8 7 9 10 6 9 13 9 9 4 10 8 15 5 6 10 9 7 10 12 8 7 13 8 7 penalty cost of ports 15 12 10 16 18 12 8 10 9 14 7 8 10 9 11 by using the deterministic traveling cost presented in table 2, operation cost and penalty cost presented in table 3, we obtained the results for the mga and ga to obtain the optimum cost of the proposed algorithm. the final path for the four ships and their optimum cost for both the algorithms are presented in the given table: das et al./decis. mak. appl. manag. eng. 5 (2) (2022) 329-361 350 table 4. result for deterministic model si no ga mga operation cost penalty cost final path traveling cost final path traveling cost 1 2-12-6-4-1-10 146 2-6-4-1-1210 131 5720 22 11-8-7-5-9 109 7-8-11-5-9 97 4080 40 14-15-3-13 83 14-3-13-15 79 2843 41 2 11-1-8-2-1012 156 11-8-1-2-1012 112 5769 22 13-4-5-15-7 144 4-5-13-15-7 138 3269 21 3-6-9-14 105 14-3-6-9 108 3605 60 3 3-2-10-5-8-4 121 10-5-2-8-3-4 117 6293 44 9-14-11-12-6 84 12-9-14-11-6 72 3602 38 13-15-1-7 120 1-7-13-15 74 2748 21 4 4-2-15-13-7 128 2-15-13-4-7 124 4173 9 8-12-11-9-514 178 12-11-8-5-914 112 4751 50 6-1-10-3 115 10-3-6-1 108 3719 44 5 2-13-5-15-3 101 13-5-2-15-3 83 4695 53 12-9-1-8-11-6 140 12-9-8-11-16 124 4856 50 7-14-10-4 91 7-14-10-4 92 3092 0 6 6-1-8-2-10 140 8-2-6-1-10 102 5272 12 12-14-11-3-54 170 12-3-5-1411-4 141 4667 54 7-13-15-9 83 7-13-15-9 83 2704 37 7 11-2-1-9-4 134 9-11-1-2-4 135 4678 40 12-7-8-14-6 116 12-7-8-14-6 116 4208 10 13-15-3-10-5 113 15-3-13-10-5 100 3757 53 8 2-4-11-12-10 106 4-12-11-2-10 95 4421 10 6-15-3-8-9 106 8-3-6-15-9 97 3890 60 13-14-5-1-7 105 1-14-5-13-7 95 4332 33 9 2-10-9-5-7 107 2-10-5-9-7 89 5223 40 12-6-15-4-14 127 12-15-6-4-14 123 2933 10 1-3-8-11-13 126 8-11-1-3-13 114 4487 53 to simulate practical applications, we obtained the fuzzy traveling cost matrix. here, the triangular cost fuzzy matrix is used for the proposed model as follows: solving fuzzy dynamic ship routing and scheduling problem through modified genetic… 351 table 5. input data: fuzzy cost matrix i/j 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 ∞ (22, 24, 27) (25, 27, 29) (24, 28, 30) (31, 33, 35) (24, 27, 29) (13, 16, 18) (32, 34, 36) (39, 41, 42) (29, 31, 33) (40, 43, 45) (20, 22, 24) (30, 32, 34) (15, 16, 18) (31, 34, 38) 2 (16, 18, 20) ∞ (20, 22, 24) (24, 26, 29) (23, 25, 27) (13, 15, 17) (29, 31, 33) (11, 14, 16) (26, 29, 31) (14, 16, 17) (36, 39, 40) (23, 26, 28) (31, 33, 35) (47, 50, 52) (11, 13, 15) 3 (13, 15, 17) (16, 17, 20) ∞ (9, 12, 14) (23, 26, 28) (14, 16, 18) (17, 19, 21) (29, 31, 33) (36, 39, 42) (27, 31, 34) (28, 31, 35) (42, 45, 48) (45, 48, 50) (18, 21, 23) (40, 43, 45) 4 (24, 27, 29) (25, 28, 31) (16, 18, 19) ∞ (19, 21, 23) (24, 26, 27) (29, 31, 33) (30, 33, 35) (18, 21, 23) (34, 36, 37) (9, 12, 14) (29, 32, 34) (38, 41, 43) (21, 23, 25) (30, 33, 35) 5 (33, 35, 37) (21, 23, 25) (32, 35, 37) (28, 30, 33) ∞ (20, 22, 24) (20, 23, 26) (31, 33, 35) (8, 10, 12) (27, 29, 31) (31, 34, 36) (42, 45, 47) (13, 16, 18) (26, 29, 31) (18, 20, 22) 6 (24, 26, 28) (29, 31, 33) (24, 27, 29) (16, 19, 21) (27, 29, 31) ∞ (31, 33, 35) (39, 41, 43) (30, 33, 35) (25, 28, 31) (14, 16, 18) (33, 35, 37) (25, 27, 29) (38, 40, 42) (9, 12, 14) 7 (27, 29, 31) (24, 26, 28) (29, 31, 33) (21, 23, 24) (32, 36, 39) (40, 42, 43) ∞ (9, 11, 12) (28, 31, 33) (19, 21, 23) (33, 35, 38) (42, 46, 48) (6, 9, 11) (20, 23, 25) (11, 13, 15) 8 (20, 21, 23) (15, 19, 21) (31, 35, 37) (37, 39, 41) (33, 36, 38) (23, 25, 27) (38, 40, 43) ∞ (21, 23, 25) (35, 38, 40) (8, 11, 13) (32, 34, 36) (23, 26, 28) (13, 15, 18) (47, 49, 51) 9 (31, 33, 34) (36, 38, 40) (34, 36, 37) (24, 26, 27) (21, 23, 25) (20, 23, 25) (36, 38, 39) (29, 31, 33) ∞ (26, 29, 30) (40, 42, 44) (31, 33, 35) (31, 33, 34) (6, 8, 10) (29, 31, 33) 10 (27, 29, 31) (30, 32, 34) (25, 27, 29) (19, 21, 23) (10, 12, 13) (31, 33, 35) (35, 36, 38) (39, 41, 43) (31, 32, 35) ∞ (36, 37, 39) (19, 21, 23) (29, 31, 35) (22, 24, 26) (16, 18, 20) 11 (10, 13, 15) (23, 25, 27) (38, 39, 41) (25, 28, 32) (51, 53, 55) (19, 21, 23) (34, 37, 38) (17, 19, 21) (42, 43, 45) (29, 30, 32) ∞ (24, 26, 28) (13, 15, 17) (32, 35, 37) (37, 39, 40) 12 (34, 36, 38) (22, 23, 24) (13, 15, 16) (44, 46 ,48) (35, 36, 38) (27, 29, 31) (35, 38, 39) (27, 29, 30) (19, 21, 23) (28, 29, 31) (17, 18, 20) ∞ (14, 16, 18) (32, 33, 35) (27, 29, 31) 13 (37, 39, 41) (36, 38, 40) (32, 33, 35 (23, 25, 27) (24, 26, 28) (11, 13, 15) (29, 31, 33) (20, 23, 25) (19, 22, 24) (25, 27, 29) (32, 35, 37) (20, 22, 24) ∞ (7, 9, 10) (23, 25, 27) 14 (35, 37, 39) (13, 16, 18) (23, 25, 27) (41, 43, 44) (17, 19, 21) (28, 30, 31) (42, 44, 47) (25, 27, 29) (29, 32, 34) (17, 19, 21) (15, 18, 20) (41, 43, 46) (24, 26, 27) ∞ (32, 33, 35) 15 (39, 41, 43) (27, 29, 31) (10, 12, 14) (28, 29, 30) (38, 41, 43) (27, 29, 30) (31, 33, 35) (29, 31, 33) (7, 10, 11) (27, 29, 30) (15, 17, 19) (42, 45, 47) (25, 27, 29) (33, 35, 37) ∞ das et al./decis. mak. appl. manag. eng. 5 (2) (2022) 329-361 352 table 6. result for fuzzy model si no ga mga final path traveling cost final path traveling cost 1 2-4-12-6-1-10 512.83 2-6-4-1-12-10 444.83 5-7-8-11-9 305.83 7-8-11-5-9 305.17 15-3-14-13 184.67 15-3-14-13 184.67 2 11-2-8-1-10-12 502.67 11-1-8-2-10-12 447.17 13-4-5-15-7 380.00 4-5-15-13-7 373.67 3-6-9-14 222.67 3-6-9-14 222.67 3 3-2-10-5-8-4 447.16 3-2-8-10-5-4 434.33 9-14-11-12-6 266.17 12-9-14-11-6 251.33 13-15-1-7 296.50 13-15-1-7 296.50 4 4-2-15-13-7 394.50 2-15-4-13-7 364.00 8-12-11-9-5-14 631.17 12-11-8-5-9-14 436.34 6-1-10-3 282.00 3-6-10-1 239.00 5 5-2-13-15-3 361.00 5-2-15-13-3 329.33 8-12-11-8-1-6 514.00 12-9-8-11-1-6 442.50 7-14-10-4 222.17 14-10-4-7 236.17 6 8-1-2-6-10 392.67 1-2-8-6-10 336.17 3-14-11-12-5-4 535.50 4-11-12-5-3-4 536.67 7-13-15-9 137.67 7-15-9-13 181.50 7 11-1-2-9-4 398.00 11-2-1-9-4 325.83 7-8-14-12-6 362.83 12-7-8-14-6 327.83 13-15-3-10-5 324.17 13-15-3-10-5 324.17 8 2-4-11-12-10 344.50 2-12-4-11-10 313.83 6-15-3-8-9 279.00 6-15-3-8-9 279.00 13-14-1-5-7 366.17 13-14-5-1-7 285.83 9 10-5-2-9-7 352.33 2-5-9-10-7 342.33 12-6-15-4-14 362.33 12-6-15-4-14 362.33 8-11-3-1-13 350.83 3-1-8-11-13 312.50 by using the fuzzy triangular cost matrix presented in table 5, we calculated the final path for three ships and their optimum cost for our mga and ga. the aforementioned table reveals the results for the fuzzy data. table 7 presents the deterministic risk matrix between every two ports. we considered a value between 0 and 1 as the deterministic risk factor. solving fuzzy dynamic ship routing and scheduling problem through modified genetic… 353 table 7. input data: deterministic risk matrix i/j 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 ∞ 0.5 0.8 0.7 0.82 0.59 0.51 0.61 0.38 0.71 0.57 0.61 0.63 0.58 0.42 2 0.78 ∞ 0.6 0.52 0.9 0.32 0.47 0.51 0.48 0.6 0.63 0.58 0.71 0.61 0.5 3 0.56 0.7 ∞ 0.5 0.47 0.49 0.55 0.63 0.7 0.8 0.67 0.6 0.53 0.63 0.39 4 0.61 0.77 0.73 ∞ 0.51 0.54 0.64 0.74 0.48 0.5 0.61 0.62 0.45 0.64 0.8 5 0.48 0.67 0.55 0.68 ∞ 0.55 0.51 0.62 0.7 0.8 0.71 0.72 0.5 0.54 0.49 6 0.39 0.57 0.66 0.68 0.88 ∞ 0.91 0.75 0.76 0.88 0.58 0.71 0.71 0.55 0.72 7 0.73 0.52 0.7 0.53 0.71 0.56 ∞ 0.59 0.67 0.63 0.56 0.66 0.67 0.59 0.6 8 0.78 0.54 0.67 0.69 0.58 0.73 0.55 ∞ 0.77 0.68 0.67 0.77 0.69 0.47 0.54 9 0.61 0.56 0.47 0.66 0.64 0.83 0.66 0.88 ∞ 0.57 0.73 0.74 0.72 0.6 0.69 10 0.49 0.63 0.56 0.59 0.63 0.81 0.72 0.48 0.49 ∞ 0.68 0.61 0.56 0.63 0.63 11 0.53 0.68 057 0.58 0.7 0.49 0.55 0.57 0.58 0.72 ∞ 0.63 0.57 0.43 0.7 12 0.5 0.76 0.62 0.62 0.63 0.66 0.71 0.44 0.57 0.61 0.6 ∞ 059 0.39 0.61 13 0.49 0.48 0.67 0.63 0.59 0.51 0.55 0.47 0.49 0.5 0.7 0.78 ∞ 0.51 0.53 14 0.47 0.54 0.64 0.72 0.76 0.66 0.56 0.73 0.61 0.57 0.45 0.48 0.47 ∞ 0.51 15 0.63 0.7 0.73 0.51 0.69 0.82 0.48 0.6 0.63 0.58 0.49 0.7 0.46 0.51 ∞ because risk factors are dependent on various weather conditions and other factors, it is appropriate to take risk factors as a fuzzy value instead of a crisp value. the table displays the fuzzy triangular risk factor between ports. in our problem, we obtained the tfn for the risk matrix. das et al./decis. mak. appl. manag. eng. 5 (2) (2022) 329-361 354 table 8. fuzzy risk matrix i/j 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 ∞ 0.48, 0.51, 0.61 0.75, 0.81, 0.85 0.67, 0.7, 0.78 0.79, 0.83, 0.87 0.53, 0.57, 0.63 0.47, 0.5, 0.56 0.58, 0.6, 0.65 0.35, 0.37, 0.4 0.69, 0.7, 0.75 0.53, 0.56, 0.61 0.58, 0.6, 0.66 0.6, 0.64, 0.69 0.53, 0.57, 0.61 0.39, 0.41, 0.44 2 0.75, 0.77, 08 ∞ 0.58, 0.59, 0.63 0.49, 0.51, 0.54 0.87, 0.91, 0.92 0.3, 0.33, 0.35 0.45, 0.48, 0.5 0.49, 0.52, 0.55 0.46, 0.49, 0.51 0.58, 0.61, 0.62 0.61, 0.63, 0.65 0.55, 0.57, 0.6 0.69, 0.72, 0.73 0.59, 0.62, 0.63 0.49, 0.51, 0.53 3 0.53, 0.55, 0.59 0.68, 0.69, 0.74 ∞ 0.48, 0.5, 0.53 0.45, 0.48, 0.49 0.47, 0.49, 0.51 0.52, 0.54, 0.57 0.6, 0.64, 0.65 0.68, 0.71, 0.73 0.78, 0.81, 0.83 0.65, 0.66, 0.68 0.58, 0.61, 0.63 0.5, 0.54, 0.55 0.61, 0.64, 0.66 0.79, 0.81, 0.84 4 0.59, 0.62, 0.64 0.73, 0.76, 0.79 0.7, 0.74, 0.77 ∞ 0.48, 0.52, 0.55 0.52, 0.55, 0.57 0.62, 0.65, 0.67 0.72, 0.75, 0.76 0.46, 0.48, 0.5 0.48, 0.51, 0.53 0.59, 0.62, 0.64 0.58, 0.61, 0.63 0.43, 0.44, 0.46 0.6, 0.65, 0.67 0.47, 0.48, 0.51 5 0.45, 0.47, 0.5 0.63, 0.66, 0.7 0.53, 0.55, 0.57 0.64, 0.67, 0.7 ∞ 0.53, 0.56, 0.58 0.49, 0.52, 0.53 0.6, 0.63, 0.65 0.68, 0.71, 0.73 0.79, 0.82, 0.83 0.7, 0.72, 0.74 0.69, 0.71, 0.73 0.48, 0.51, 0.53 0.52, 0.55, 0.57 0.46, 0.48, 0.5 6 0.35, 0.38, 0.41 0.55, 0.57, 0.61 0.62, 0.65, 0.68 0.65, 068, 0.71 0.86, 0.88, 0.9 ∞ 0.89, 0.91, 0.93 0.73, 0.76, 0.78 0.73, 0.75, 0.78 0.85, 0.87, 0.9 0.56, 0.59, 0.6 0.69, 0.72, 0.74 0.69, 0.73, 0.74 0.53, 0.56, 0.58 0.69, 0.72, 0.74 7 0.7, 0.74, 0.78 0.49, 0.53, 0.55 0.68, 0.71, 0.74 0.5, 0.54, 0.56 0.68, 0.72, 0.74 0.53, 0.55, 0.58 ∞ 0.56, 0.58, 0.6 0.65, 0.66, 0.69 0.6, 0.64, 0.65 0.53, 0.55, 0.58 0.63, 0.65, 0.67 0.65, 0.68, 0.69 0.57, 0.59, 0.61 0.58, 0.61, 0.63 8 0.75, 0.77, 0.8 0.51, 0.53, 0.56 0.65, 0.68, 0.7 0.65, 0.68, 0.71 0.56, 0.59, 0.61 0.7, 0.72, 0.75 0.52, 0.56, 0.58 ∞ 0.75, 0.78, 0.79 0.66, 0.69, 0.71 0.65, 0.68, 0.7 0.74, 0.76, 0.79 0.66, 0.68, 0.72 0.44, 0.46, 0.49 0.52, 0.55, 0.57 9 0.58, 0.61, 0.64 0.53, 0.55, 0.58 0.45, 0.48, 0.5 0.63, 0.65, 0.68 0.62, 0.65, 0.67 0.8, 0.82, 0.85 0.63, 0.65, 0.68 0.85, 0.87, 0.89 ∞ 0.55, 0.56, 0.58 0.7, 0.74, 0.76 0.72, 0.73, 0.75 0.7, 0.73, 0.75 0.58, 0.61, 0.63 0.66, 0.68, 0.7 10 0.45, 0.48, 0.52 0.6, 0.64, 0.67 0.53, 0.55, 0.58 0.55, 0.57, 0.61 0.6, 0.62, 0.65 0.77, 0.79, 0.84 0.69, 0.73, 0.75 0.45, 0.47, 0.5 0.47, 0.5, 0.53 ∞ 0.66, 067, 0.7 0.59, 0.62, 0.64 0.53, 055, 0.58 0.6, 0.63, 0.65 0.6, 0.64, 0.66 11 0.51, 0.54, 0.59 0.63, 0.66, 0.71 0.54, 0.56, 0.59 0.56, 0.59, 0.61 0.68, 0.71, 0.73 0.46, 0.48, 0.5 0.51, 0.54, 0.57 0.55, 0.57, 0.59 0.55, 0.57, 0.6 0.7, 0.73, 0.75 ∞ 0.6, 063, 0.65 0.55, 0.56, 0.59, 0.41, 0.42, 0.45 0.68, 0.71, 0.73 12 0.47, 0.51, 0.55 0.71, 0.75, 0.78 0.59, 0.63, 0.65 0.58, 0.6, 0.63 0.6, 0.62, 0.65 0.62, 0.65, 0.67 0.69, 0.72, 0.73 0.42, 0.45, 0.47 0.55, 0.58, 0.6 0.58, 0.62, 0.63 0.57, 0.59, 0.63 ∞ 0.56, 0.58, 0.62 0.36, 0.38, 0.4 0.58, 0.61, 0.63 13 0.45, 0.48, 0.51 0.43, 0.47, 0.5 0.64, 0.67, 0.69 0.6, 0.64, 0.66 0.55, 0.58, 0.6 0.49, 0.52, 0.54 0.53, 0.55, 0.58 0.44, 0.48, 0.51 0.47, 0.48, 0.5 0.47, 0.49, 0.53 0.68, 0.71, 0.73 0.75, 0.79, 0.81 ∞ 0.47, 0.5, 0.53 0.52, 0.4, 0.56 14 0.44, 0.46, 0.5 0.51, 0.54, 0.57 0.61, 0.63, 0.67 0.69, 0.73, 0.75 0.72, 0.75, 0.77 0.62, 0.65, 0.68 0.53, 0.54, 0.57 0.7, 0.74, 0.76 0.59, 0.62, 0.63 0.55, 0.58, 0.6 0.43, 0.46, 0.48 0.45, 0.47, 0.49 0.45, 0.47, 0.49 ∞ 0.47, 0.51, 0.54 15 0.6, 0.64, 0.66 0.68, 0.71, 0.73 0.7, 0.72, 0.75 0.48, 0.52, 0.54 0.65, 0.68, 0.7 0.8, 0.83, 0.84 0.44, 0.47, 0.5 0.56, 0.59, 0.63 0.61, 0.64, 0.66 0.57, 0.6, 0.61 0.47, 0.49, 0.51 0.68, 0.71, 0.73 0.44, 0.77, 048 0.48, 0.52, 0.53 ∞ the table below displays the risk achieved by the deterministic risk factor and fuzzy risk factor from tables 7 and 8, respectively, at a particular path for both the algorithms. solving fuzzy dynamic ship routing and scheduling problem through modified genetic… 355 table 9. result for risk factor si no risk value achieved for deterministic model risk value achieved for fuzzy model ga mga maximum risk (rmax) ga mga maximum risk (rmax) 1 2.78 2.65 7.23 2.30 2.23 6.44 3.14 3.14 9.01 2.09 1.93 5.23 2.09 1.77 5.25 3.01 3.01 8.41 2 2.46 2.30 6.01 2.11 2.12 6.44 2.70 2.62 9.21 1.81 1.74 5.23 2.05 1.89 7.44 3.26 3.26 8.41 3 2.86 2.55 8.88 2.48 2.39 6.44 2.98 2.99 9.00 1.69 1.55 5.23 1.81 1.71 6.68 2.70 2.70 8.41 4 2.46 2.23 8.34 2.34 2.27 6.32 3.11 3.05 7.28 2.99 2.98 8.56 2.22 1.44 6.98 1.47 1.42 5.20 5 2.64 2.49 8.95 2.50 2.47 6.32 3.25 3.21 8.90 3.46 3.23 8.56 1.68 1.75 8.32 2.15 1.75 5.20 6 2.53 2.50 8.59 2.07 1.95 6.32 3.05 3.09 9.17 2.91 2.70 8.56 1.71 1.71 7.64 2.11 1.96 5.20 7 2.41 2.35 8.29 2.93 2.83 8.76 2.85 2.85 8.61 2.04 1.96 6.73 1.93 1.83 6.25 2.15 2.15 7.68 8 2.51 2.50 7.48 3.54 3.53 8.76 3.53 3.49 10.28 2.31 2.29 6.73 1.70 1.64 8.25 2.45 2.23 7.68 9 1.81 1.72 7.32 1.90 1.85 5.20 1.93 1.95 8.17 2.31 2.26 7.66 2.50 2.46 8.54 3.44 3.42 8.32 8. statistical test we compared the statistical test quantitative decision of the proposed algorithm with other those of other conventional algorithms. the analysis of variance (anova) was performed to indicate the statistical significance of the proposed algorithm with the conventional algorithms (rwga and pbga). 8.1. anova for the efficiency test the performance of mga, rw selection-based ga (rwga), and probabilistic selection-based ga (pbga) for solving the standard ship routing problem was compared. we obtained various parametric values, such as the different number of vessels, v3, v4, and v5; different number of containers, c7, c10, c15, c20, and c23, and four instances, namely instance 1 (i1), instance 2 (i2), instance 3 (i3), and instance 4 (i4). various instances obtained various source ports and destination ports for the container, which changed the path of ships. in this model, nine types of data sets were considered for the anova test. these nine data sets were categorized into two, short sea and deep-sea categories. short sea data sets are represented as three ships and seven containers (v3c7), three ships and ten containers (v3c10), three ships and fifteen containers (v3c15), four ships and fifteen containers (v4c15), four ship and 20 containers (v4c20), five ship and 20 containers (v5c20). deep sea data sets are represented as three ships and 15 containers (v3dc15), three ships, and 20 containers (v3dc20), four ships, and 23 containers (v4dc23). we obtained the result das et al./decis. mak. appl. manag. eng. 5 (2) (2022) 329-361 356 for three types of the algorithm by using the standard ship routing library given by hemmati et al., (2014). the results for benchmark data are expressed as follows: table 10. comparative results for benchmark data short sea deep-sea v3c7 v3c10 v3c15 v4c15 v4c20 v5c20 v3dc15 v3dc20 v4dc23 mga i1 876483 1313111 664424 454848 655617 642901 17263074 24286620 46565448 i2 876580 1162171 593090 458165 642039 669206 15369754 24621576 48265056 i3 899058 1320132 650774 418006 667173 548434 15906567 26041704 44681920 i4 897202 1392051 683387 424909 663714 627702 15743047 24509110 48519056 rwga i1 1323232 1813450 861581 461202 800196 798402 22402576 33192620 65667660 i2 1331933 1624214 895387 498499 738716 747174 2143396 32698632 65663988 i3 1185146 1846947 801951 491465 725799 736120 21872220 32146192 60964440 i4 1335549 1779040 861432 437312 727999 788223 21539136 34659412 61995028 pbga i1 1245731 1553214 842612 482627 800231 788321 20982165 33101152 50501933 i2 1013155 1411325 791313 471737 712913 792319 19036152 31987645 52901340 i3 1256324 1402389 782451 500321 773211 699939 18862371 29136691 57631210 i4 1200135 1428362 802314 492001 690023 700291 19116692 29900267 7010005 for all three algorithms, we obtained 500 maximum generation (max-gen) and 100 as the population size (pop-size). the three algorithms were tested by using ship routing benchmark data. the results were obtained 100 win runs for all three algorithms. we compared more than two algorithms for efficiency tests. therefore, the anova is the best choice. we considered 100 runs and used the number of successful runs for three algorithms, namely mga, rwga, and pbga, which are given in the following table: table 11. number of wins run for different algorithms short sea deep-sea v3c7 v3c10 v3c15 v4c15 v4c20 v5c20 v3dc15 v3dc20 v4dc23 mga i1 82 97 80 84 78 87 81 78 87 i2 86 89 87 88 95 91 82 84 88 i3 91 92 82 93 90 94 79 90 81 i4 90 96 88 94 86 91 89 93 86 rwga i1 72 69 64 62 61 62 67 68 70 i2 75 70 71 69 64 63 66 68 69 i3 72 76 68 75 71 74 64 70 72 i4 74 68 68 71 72 70 69 70 68 pbga i1 60 61 54 56 60 59 54 62 55 i2 61 62 58 61 55 54 63 55 60 i3 59 55 61 54 56 61 60 58 56 i4 58 55 63 59 61 55 61 59 58 the anova test presents a comparison of flexibility between the groups and flexibility within the groups. the sum of the square provides a total variation between the groups and within groups. the degree of freedom is calculated between groups as (number of samples of the individual group – 1) and within groups as (sum of all the samples – number of the sample of individual groups). the mean of the square is the sum of the square divided by the degree of freedom (mean of square = sum of square/degree of freedom). f is calculated as the comparison of the mean of the square between groups and the mean of the square within groups. the given table presents the anova test result for the proposed model and two other algorithms: solving fuzzy dynamic ship routing and scheduling problem through modified genetic… 357 table 12. anova test result instance no. source of variation sum of square degree of freedom mean of square f instance 1 between groups 3149.85 2 1574.93 76.62 within groups 493.33 24 20.55 total 3643.18 26 instance 2 between groups 3931.18 2 1965.60 149.71 within groups 315.11 24 13.13 total 4246.30 26 instance 3 between groups 4124.74 2 2062.37 116.31 within groups 425.56 24 17.73 total 4550.30 26 instance 4 between groups 4605.41 2 2302.71 291.98 within groups 189.56 24 7.90 total 4794.96 26 the total sample size is nine for each algorithm, and the number of the algorithm is three. the critical value of v ≈ 3.40 and p is extremely smaller than α = 0.05 for all the cases. because the f critical value is smaller than f, we rejected the null hypothesis, which provides statistical significance to the result. a significant difference was observed between the algorithms. therefore, the proposed modified algorithm is more efficient than the other two algorithms. 9. result discussion we considered various sets of input values for dynamic ship routing and scheduling problems to achieve appropriate numerical outcomes. table 2 provides the crisp data input for the traveling cost. table 3 provides the operation cost of each product and the penalty cost for violating the time frame in case of each port. table 4 displays the path and cost as the output from our ship routing model with a crisp cost matrix obtained by using classical ga and mga. table 5 provides the triangular fuzzy input data for traveling cost. table 6 presents the output of our ship routing model considering fuzzy data obtained using classical ga and mga. in tables 4 and 6, we can see that the proposed algorithm provided the optimum result for crisp data input as well as the fuzzy data input. the use of crisp risk factors is presented in table 7, whereas the use of fuzzy risk factors is exhibited in table 8. table 9 displays the risk achieved for the deterministic as well as the fuzzy models for both ga and our mga. here, rmax for a route represents the maximum risk value on that route. table 9 reveals that the proposed mg exhibited the best result for both the crisp and fuzzy data sets. the comparative results for the ship routing benchmark instances for the mga, rwga, and pbga are displayed in table 10, which reveals that the proposed mga provides the best results for benchmark instances. in table 11, we have compared the winning runs of our algorithm and those of two other algorithms. table 12 displays the anova test results for the three algorithms working on four instances of the data input for the standard ship routing problem. from table 12, it seems we can conclude that the proposed modified algorithm works more efficiently and provides better results than the other two algorithms. das et al./decis. mak. appl. manag. eng. 5 (2) (2022) 329-361 358 10. managerial insights the substitution of the new model gives more space to the manager for making decisions regarding which approach to adopt for incurring maximum more profit for the organization. with the implementation of our model, a manager has the freedom to design the shipping route under an uncertain environment, a situation which can be easily applied to the shipping industry in real life. our proposed model can handle dynamic demand and supply, and also generate optimum routes and minimize the overall costs of the transportation by ship, which is the principal motto of any ship routing and scheduling algorithm. risk factors are considered in calculating optimum routes, which provides a realistic approach to the shipping industry. the use of cost minimization and time window reduces overall costs and maintain shipping discipline. that is why the proposed model can generate more profit than other previous models. the demand and supply of products at the port may vary because of market conditions or social issues and use of this model leads to less congestion at the port. considering the cost and risks parameter as the fuzzy number makes the model realistic because costs and risks factor are changed time to time. this dynamic environment makes ship routing and scheduling problems more suitable for practical use. therefore, management should adopt this realistic model, which is cost effective and manages the risks well. 11. conclusions and future scope in our model, we have assumed that ships start their sailing from starting ports, visit other ports to ensure the unhindered demand and supply of products at those ports, and end their routing in their destination ports. we obtained the optimum paths and traveling costs for various ships, and hence the optimal cost is calculated by adding the cost of traveling, loading–unloading operations, and penalties (levied to ensure port discipline). cost efficiency and environmental friendliness are prioritized by considering a few aspects such as the dynamic demand and supply models followed in ports, container-based shipping, and techniques for saving fuel. fuzzy cost and fuzzy risk factor are introduced to represent the impreciseness of the parameters. therefore, in our model, a real-time ship routing approach is proposed in an uncertain environment. the advantages of this model are its ability to the determined optimal path and cost when various real-life parameters are uncertain. this model maintains the port time as well as the traffic, and handles the container congestion very smoothly. we have used bbmst selection and ivf crossover techniques in our work. a generation-dependent mutation was used in the proposed algorithm in which, when the number of generations increased, the algorithm provided a better result. the advantages of the proposed methodology are that the novel bbmst selects the better solution for further steps, which gives optimal solution in less computational time. the ivf-based crossover method has been used on very few instances of work before. the proposed mga clearly revealed itself as being superior other rwga and pbga in our numerical experiments. the anova test revealed that the proposed algorithm works efficiently. however, the model has some limitations. the maximum risk (rmax) is selected randomly to ensure the total risk does not exceed the maximum risk. dynamic weather routing may define various risks of a route for a ship for a particular time, which is not considered. we did not consider the container type, so our models are based only on standard containers. there also are some limitations in the solving solving fuzzy dynamic ship routing and scheduling problem through modified genetic… 359 method: as the approach is heuristic, a high computational complexity exists. when numerous constraints are considered, the mga falls short in solving the problem. the futures cope of this model is actually quite wide. various types of containers (foldable container and smart container) can be introduced in the future. we can predict the dynamic risks of each route using artificial intelligence and modern technology in the future. various ranking-based fuzzy approaches, including the type2 fuzzy approach, can be used to solve complex uncertain routing problems. we may use rough numbers, fuzzy systems, or even neutrosophic numbers in the future. author contributions: madhushree das: conceptualization, methodology, software, data curation, writing original draft, visualization, investigation. arindam roy: conceptualization, methodology, data curation, writing review & editing, visualization, investigation. samir maity: methodology, software, data curation, editing. samarjit kar: supervision, validation, writing review & editing. shatadru sengupta: supervision, validation, writing review & editing. funding: this research was supported by department of science and technology and bio-technology, west bengal by grant number 1001 (sanc.) /st/p/s&t/16g13/2018 dated 05.08.2019. acknowledgments: the authors would like to thank the reviewers and editor for their constructive comments and suggestions and especially appreciated the editorin-chief for their valuable comments, which improved the quality and presentation of this article. conflicts of interest: the authors declare no conflicts of interest. references aktar, m. s., de, m., maity, s., mazumder, s. k., & maiti, m. 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(2019). bi-objective optimization of vessel speed and route for sustainable coastal shipping under the regulations of emission control areas. sustainability, 11(22), 1-24. © 2021 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 2, issue 1, 2019, pp. 13-34. issn: 2560-6018 eissn: 2620-0104 doi:_ https://doi.org/10.31181/dmame1901013h * corresponding author. e-mail address: malek.hassanpour@yahoo.com (m. hassanpour) evaluation of iranian wood and cellulose industries malek hassanpour 1* 1 department of environmental science, ucs, osmania university, telangana state, india received: 22 october 2018; accepted: 17 january 2019; available online: 7 february 2019. original scientific paper abstract: iranian wood and cellulose industries (iwci) are distinguished via a minimum quantity of wood consumptions with high wastages rates along with favourite products generation. iwci exposed to lots of obstacles in the way of maturation and expansion especially in terms of technologies assigned and overdependence on input materials entered into industries cycle. present cluster study of iwci empirically targeted an assessment of technologies, input and output materials streams, existing facilities in industries individually. spss software along with delphi fuzzy theory and fuzzy technique for order of preference by similarity to ideal solution (topsis) methods were assigned to evaluate the data of industries as findings of iranian evaluator team once before construction of industries. t-test analysis had represented significant differences around (pvalue≤ 0.001, 0.002) among main criteria of iwci such as the number of employees, power, water and fuel exploitations and the land area occupied by each industry. using friedman test the ranks values were obtained about 2.59, 4, 1.53, 1.88 and 5 for the number of employees, power, water, fuel consumed and land area applied in the location of industries. analytical hierarchy process (ahp) via delphi fuzzy set, fuzzy topsis and topsis resulted to a hierarchical classification among iwci. key words: evaluation; iranian wood and cellulose industries; topsis. 1. introduction the use of wood in iranian ancient refers to before the aryan migration from about 4200 bc. wood industry has got an extensive range of applications both as commercial and industrial demands in iran. obviously, population growth aligned with escalated consumption patterns, industries and urbanity developments, have culminated demands for wood and its products in iran. iranian statistics centre has recently reported to around 226 industrial production sites of furniture with approximately 10,000 employees are currently running along with around 46,700 wood industries offices operating 117,000 at the native workshops. the value-added of wood products has been recently reported approximately 30% apart of value-added percentage mailto:malek.hassanpour@yahoo.com malek hassanpour/decis. mak. appl. manag. eng. 2 (1) (2019) 13-34 14 associated with the furniture industry (totally 70%). 5% of the industry’s value-added devoted to both industrial printing (4.6%) plus iranian cellulose industries. the per capita consumption rates for various paper and paperboard types has been estimated at around 23 kg in 2016 with a rise from 12-13 kg to 23 kg in comparison to 10 years ago. this amount has been forecasted in high amounts, (with a factor of 2) for other nations over the world. on the other hand, iranian people stake in various paper and paperboard consumptions are negligible. the prominent stake for both of paper and paperboard productions has devoted to linerboard and fluting applications which comprise approximately 50%; employed in sheets and cardboard boxes generations and their equipment. in the swot analysis, many strength points determined for iwi such as longtime production background, various academic and vocational centres and also well trained and well-experienced labour forces in various fields, creating a high value-added percentage, high-quality products manufacturing in comparison to imported products. however, many drawbacks have also reported for aforementioned industries such as dependency to rare domestic resources, old fashionable equipment and machinery, exhausted devices, bereavement in special tariff proclamations, high transportation outlays and deficiency of investment for requested infrastructure. according to aforementioned advantages and drawbacks, stakeholders need to consider to some opportunities to pave the way for more advancement and development in the field of wood industries. globally, the lumber & wood products are divided to many sections such as (1) hardwood dimension and flouring mills (2) millwork (3) hardwood veneer and plywood (4) softwood veneer and plywood (5) structural wood members (6) nailed and lock-corner wood boxes and shook (7) wood pallets and skids (8) wood containers (9) wood preserving (10) wood products, (11) pulp mills (12) wood kitchen cabinets (13) prefabricated wood buildings and components (14) wood household furniture, except upholstered wood television, radio, and phonograph cabinets (15) wood office furniture (16) sawmills and planing mills (17) special product sawmills (18) particleboard. in iran, there are many cases of wood and cellulose products industries such as cooler bangs (1), carton (2), industrial drying wood (3), hydrophilic cotton (4), sheet rolls and packing (5), wax paper (6), booklet (7), hasp (8), decal (9), multilayer paper bags (10), row board (11), wooden and paper disposable products (12), wooden pencil (13), carbon paper (14), parquet (15) wooden sandpaper (16) (iranian industries organization, 2018). in accordance with the approval of government agencies, any industrial project prior to construction requires the financial, technical and environmental assessments etc. according to the current assessment of the iranian industries organization, in a cluster study, about 16 types of wood and cellulose industries have been identified. in the present study, raw data are generally presented in the framework of a phd thesis with existing methods for evaluating the project and obtaining the best possible decision-making processes. using multi-criteria decision making (mcdm) models to weight and rank the various data will result to generate different values for the same data employed. the madm practices need to each alternative to be evaluated against amounts of rating devoted to the attributes, factors and criteria containing various units of measurement for each of them. to compare obtained results associated with each factor or criterion a normalization process is accomplished and the results will offer its own value in integrating the diverse measurement units. the main reason for the normalization process gets back to shift the various assessed units into a non-dimensional scale. by the way, normalized values follow non-declining amounts in the range of 0 and 1 (gul et al., 2018). applying ahp, for decision-making processes gets back to saaty (1980), evaluation of iranian wood and cellulose industries 15 in an effective and robust practice to model the sophisticated decision difficulties. this practice encompasses complex factors and criteria by deconstructing and dividing them into various easy sub-items so that assign the hierarchical classification, in which the main objective placed in the top level, sub-objectives or accessory options at below clusters and in the following the possible options are embedded in the last level. by the definition, the ahp method is an economic multi-criteria practice of analysis pertaining to a weighting style, in which lots of proper contributions are released based on their relative importance. topsis method, first time acknowledged by hwang & yoon (1981), who employed the basic implication of positive and negative ideal solutions in which the determined factors and criteria should have the shortest distance from the positive ideal solution, and the farthest distance from the negative ideal solution (yazdani-chamzini et al., 2014). in the uncertainty situation, topsis method is assigned to realize and identify the difficulties so it offers a certain solution. an ideal solution includes the best response or alternative amounts for each factor and criterion. in some cases using topsis for identifying ambiguous data brings some other difficulties so in this cases in order to overcome this restriction, the fuzzy set theory can be employed with the traditional topsis approach to permitting decisionmakers to integrate vague data, non-obtainable information, and relatively ignorant facts into the decision model to solve various difficulties and challenges successfully (zare et al., 2016). therefore, according to the objective of paper as evaluation of iwci, the present study included the flow diagrams of running processes, input and output materials flows entered and outsourced from industries along with equipment and facilities used at each industry. the fuzzy delphi logic and fuzzy topsis and topsis (based on real data) were assigned to assess the factors and criteria and in the following industries hierarchically classified, weighted and ranked, values were calculated based on available information. 2. literature review mardani et al. (2016) assessed around 10 biggest iranian hotels via fuzzy set. yazdani-chamzini et al. (2013) assigned fuzzy topsis to assess the difficulties of investment strategy selection. zagorskas et al. (2014) investigated the growth in building refurbishment of new-build projects and historical buildings preservation involvements via topsis technique. nikas et al. (2018) evaluated the gap between climate policy to find a methodological framework to remove existing complex problems using both delphi and topsis methods. cavallaro et al. (2016) employed a prioritization method for factors and criteria of combined heat and power systems via both fuzzy shannon entropy and fuzzy topsis methods. moghimi & anvari (2014) utilized fuzzy mcdm approach among 8 iranian cement companies pertaining to financial statements. 3. methodology 3.1. friedman test present cluster research of iwci was empirically performed to evaluate and assess the data of industries. in order to carry out the research, secondary data were gathered from the iranian industrial organization database along with findings of evaluator team of environment protection agency. then secondary data were processed by the malek hassanpour/decis. mak. appl. manag. eng. 2 (1) (2019) 13-34 16 mcdm methods supported by spss software (ibm spss statistic 20) in order to classify the aforementioned industries hierarchically. data were analyzed using the friedman test and statistic tests for distinguishing initial ranking and realizing significant relations among them. friedman test assumes the data as a matrix with certain columns and rows ([xij] n×k in a matrix with n rows, k columns). actually, to the object, i is added the rank ri, j by judge number j, where it appears in whole n objects and m amount. therefore, taking into account equations 1 to 6, the initial processing is done on the data by software. then, equation 5 is used for a general ranking of any factor having the specified values in the columns. the overall ranking can be checked with the analogous test to friedman test called kendall. kendall's w is a non-parametric statistic test and can be assigned for normalization of the results of friedman test, as well as investigating agreement among values. w in equation 9 is linearly joined to the mean value of the spearman's rank correlation coefficients between all pairs of the available rankings. the symbol of s (in equation 8), is the sum of squared deviations appeared below. therefore, equations 6 to 9 are applied to process total rank given to object i which obtained from the friedman test. the results obtained at this step can be used to investigate friedman test results (wittkowski, 1998). ȓ. j = 1 n ∑ 𝑟𝑖𝑗𝑛𝑖=1 (1) ȓ = 1 nk ∑ ∑ 𝑟𝑖𝑗𝑘𝑗=1 𝑛 𝑖=1 (2) sst = n ∑ (ȓ. 𝑗 − ȓ)2.𝑗=1 (3) sse = 1 n(k−1) ∑ ∑ (𝑟𝑖𝑗 − ȓ)2𝑘𝑗=1 𝑛 𝑖=1 (4) q = sst sse (5) ri = n ∑ (𝑟𝑖, 𝑗, . . )𝑚𝑗=1 (6) rave = 1/n ∑ ri𝑛𝑖=1 (7) s = ∑ (ri − rave)2𝑛𝑖=1 (8) w = 12 s m2(𝑛3−𝑛) (9) 3.2 fuzzy set theory in this section, the equations of 10 to 17 are introduced, which are explained below. the delphi fuzzy system used in this research is displayed as triangular fuzzy numbers according to figure 1. the weighing system complies from a pattern as, ∑ wj𝑛𝑗 , (j=0-1). initially, the factors and criteria used are represented by linguistic words, real and fuzzy numbers according to table 1. evaluation of iranian wood and cellulose industries 17 table 1. delphi fuzzy set linguistic words symbol fuzzy no crisp no very low vl (0.09,0, 0.1) 0.1362 low l (0.2, 0.1, 0.1) 0.2272 slightly low sl (0.3, 0.1, 0.2) 0.3695 medium m (0.5, 0.1, 0.1) 0.5 slightly high sh (0.6, 0.1, 0.2) 0.6304 high h (0.8, 0.1, 0.1) 0.7727 very high vh (0.85, 0.1, 0) 0.8636 current fuzzy values (m, a, b) are able to transform as m2+b to m1-a. by the equations of 10 to 12 (n= m, a, b) fuzzy numbers can be displayed in figure 1. by the way, fuzzy numbers are represented by some symbols and also real numbers which can be converted to fuzzy numbers. in this research, equation 13 was used to prioritize factors. using a data classification system, the actual numbers obtained by the evaluator team were classified in certain intervals. as a result, table 5 was formulated as a criterion/factor versus symbol in the likert scale. the special vector (a vector is defined as a rank value obtained from criteria and factors in columns) was acquired by the results of the friedman test. the weighted sum vector (wsv) is the summation of the weight of each criterion (w) multiply in assigned fuzzy number (d) according to equation 14. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 figure 1. a triangular fuzzy numbers (shiroye, 2013) µr(m) = 1 − 1 1+𝑎 ∗ (1 − 𝑚) (10) µl(m) = 1 − 1 1+𝑏 ∗ (𝑚) (11) a = ∑ (𝑊𝑗. 𝑊𝑖𝑗)𝑗 (12) wsv = ∑ 𝐷 × 𝑊 (13) ci = ƛ max −𝑚 𝑚−1 (14) malek hassanpour/decis. mak. appl. manag. eng. 2 (1) (2019) 13-34 18 ȓ. j = 1 n ∑ 𝑟𝑖𝑗𝑛𝑖=1 (15) using equation 15, the natural attribution of incompatibility can be figured out upon a matrix set for data in which ƛ𝑚𝑎𝑥 is always ≥ m. ƛ𝑚𝑎𝑥 and m are the biggest eigenvalue of the pairwise comparison and criteria number respectively. therefore, ƛ max − 𝑚 represents the incompatibility degree in the matrix. in the equation 16, the symbols of ci and ri are the consistency index and random index which saaty (1980) used them for a matrix holding a set of data from 1 to 10 and recognized a compatibility value as cr ≤ 0.1. the incidence of random inconsistencies suggested by saaty (1980) is according to table 2. table 2. incidence of random inconsistencies (saaty 1980) m 1 2 3 4 5 6 7 8 9 10 ri 0.0 0.0 0.58 0.9 1.12 1.24 1.32 1.41 1.45 1.49 cr = ci ri (16) z x = ƛ max 𝑋 (17) the current research, obtained data were the findings of iranian evaluator team once prior to the implementation of the industries sites. therefore, data are offered as a reference information and there is no possibility of changing data. therefore, the conditions described in equation 16 cannot be applied to the evaluation style of this research. the studies and assumptions mentioned by saaty (1980) are governed by the questionnaire methods and if the results are not met the assumptions and conditions of the formula or any failure to follow the results with the assumptions and conditions needs modifying and changing even rechecking the privileges, scores and marks given by experts. equation 17 is utilized to estimate the priority vectors so z, x and max are the values of pairwise comparison matrix, priority vector or principal eigenvector and maximum or principal eigenvalue of matrix z (shirazi et al., 2017; shiroye, 2013). 3.3 fuzzy topsis procedure using the fuzzy topsis method to extract the final weight of data, is a type of evaluation of matrix containing industries criteria in which aij is the numerical value of each industry i, according to the index j. topsis method is a very strong evaluation method and a technique for prioritizing by analogy to the ideal response. based on the fact that the selected option should be kept in the shortest distance from the ideal response and the furthest distance from the worst response. in this research, the topsis method was selected based on hwang's rule for choosing the best options. equation 18 was used to convert the matrix of industries factors into a non-dimension matrix. nd = aij √∑ (𝑎𝑖𝑗)2 𝑚𝑖=1 (18) the next step was to create a non-dimension matrix with the assumption that the weights (wn.n) are indexed. the non-dimension matrix is obtained by equation 19. therefore, the special vector (obtained from the friedman test) was conducted on a non-dimension matrix to get the values for v. evaluation of iranian wood and cellulose industries 19 v = nd × wn. n (19) the next step was to identify the ideal positive solution (a+) and the ideal negative solution (a-) according to the equations of 20 and 21. to perform this purpose the amounts were extracted based on equations at each column of v. a+= {(max 𝑉𝑖𝑗|𝑗 ∈ 𝐽), (min 𝑉𝑖𝑗|𝑗 ∈ 𝑗′)|𝑖 = 1,2, … , 𝑚} = {v1+, v2+, . . vj+, vn+} (20) a−= {(min i 𝑉𝑖𝑗|𝑗 ∈ 𝐽), (max 𝑉𝑖𝑗|𝑗 ∈ 𝑗′)|𝑖 = 1,2, … , 𝑚} = {v1−, v2−, . . vj−, vn−} (21) then the distance between each option was calculated using euclidean intervals according to equations 22 and 23. the relative proximity to the ideal solution was calculated in accordance with equation 24. on the other hand, equation 24 represents approach coefficient (zagorskas et al. 2014; nikas et al. 2014; mukhametzyanov & pamucar, 2018). di+= {(∑ (𝑉𝑖𝑗 − 𝑉𝑗 +𝑛𝑗=1 ) 2 } 0.5 ; 𝑖, = 1,2,3, … 𝑚 (22) di−= {(∑ (𝑉𝑖𝑗 − 𝑉𝑗 −𝑛𝑗=1 ) 2 } 0.5 ; 𝑖, = 1,2,3, … 𝑚 (23) cli+= di− di(+)+(𝑑𝑖−) (24) 4. results and discussion the wood was, at first, a vital ingredient for the construction of primary tools, homes and boats for moving in the rivers. then, it was employed to make most of the useful things that people relied on for centuries to develop their lives style. part of the technology of wood has left over by the efforts of industrialists, but most of it has been lost and replaced by other materials and methods that are the result of the industrial revolution of mankind. wood is the only natural renewable resource. oil and coal and other mines will eventually end, but a well-maintained forest will indefinitely continue to produce wood. wood has a prominent place in the global economy. the annual production of wood in the world is 2,500 million cubic meters. the physical, chemical and mechanical properties of wood have made it a unique product for lots of applications at this time. wood is one of the most useful materials we have which is sturdy, but it can be easily cut and made in different shapes. the bulk of wood comes from the trunk or body of trees. wood hardness; this is important in the quality of work with those and other uses, such as parquet, which is continuously affected by wear. softwoods are more likely to be consumed in carpentry. the impact resistance of wood is different because of the heterogeneous construction of wood in different directions and sizes. the wood in the direction of the impact has a lot of pressure, but it changes due to the introduction of a lot of force. flexural strength; the wood affected by bending is noticeably deformed. if the force applied is more than flexural, it will break the fibre. as the wet stick is more flexible, its resistance to impact is greater. in general, the more porous the wood is, the less the impact is. wood durability; wood is not a durable object, it is worn out by insects and fungi. of course, thicker wood is more durable and can be increased by some methods. nowadays, there is a lot of consumption for wood in many other industries, including printing, chasing, furniture, malek hassanpour/decis. mak. appl. manag. eng. 2 (1) (2019) 13-34 20 carpentry, shoemaking, coiling, carving and railway wagoning, and many other industries, especially in the motherboard industry. today, many products, such as a variety of compact fibres, bone fragments, chipboard, refractory boards, triplex and five-ply boards, and many others are used in machine systems, building and refurbishment work etc. therefore, we tried to present wood applications and existing technologies to produce and make woody equipment. our data were raw results of iranian evaluator team once before construction of manufacturers in terms of energy consumed, input and output materials injected into generation process along with accessible facilities in each industry. figure 2 shows the iwci and their production processes and running technologies. table 3 includes input materials entered to iwci and table 4 contains iwci, number of staff, land area used and energy consumptions. up to down: cooler bangs (1), carton (2), industrial drying wood (3), hydrophilic cotton (4), sheet rolls and packing (5), wax paper (6), booklet (7), hasp (8), decal (9), multilayer paper bags (10), row board (11), wooden and paper disposable products (12), wooden pencil (13), carbon paper (14), parquet (15), sandpaper (16) figure 2. iwci and their production processes evaluation of iranian wood and cellulose industries 21 table 3. input materials entered to iwci industry initial materials (1) wood (1890t); nylon networks (43260 kg); packaging bags (9700 kg); stapler needles (29120 bundle) (2) three layers paper sheets (1454117 kg); five layers paper sheets (955704 kg); silicate glue (25498 kg); dye (9956 kg); nylon cords (1100 kg) (3) wood pollens (9500 m3) (4) raw cotton (440t); bleach with activity of 11-12 (55t); naoh, 98% (17.6t); washing liquid (4.4t); h2so4 (4.4t); nylon, thickness of 0.02 mm (40t); softener (4.4t); thiosulfate (8.8t) (5) paper, 30 g/m2 (947.5t); three layers packaging cartons in sizes of 75*23*50 cm3 (139000 no); cardboard pipes, l= 23 cm (100t); plastic bags (16.7t) (6) paper rolls having 500 kg (685t); al sheets, thickness of 10 micron (285t); paraffin as rolls of 500 kg (52t); special gum (3.1t); packing paper (3.2t) (7) paper of 60 g (379t); cardboard, 175 g (43t); plastic yarn (312000 g); stapler wires (686 kg); ink (22.8 kg); cartons in sizes of 66*52.5*18 cm3 (17333 no) (8) timber (400 m3); timber layers of 2.5 mm (40000 kg); formaldehyde jum 60% (8000 l); glue (160 kg); axe (60000 no); spool 27-30 (15000 no); brass pieces (15000 no); paper washer (120000 no); bolts and nuts (120000 no); hasp bar (30000 no); prong (30000 no); nuts layout (120000 no); polished oil (600 l); thinner 2000 (200 l); washing soap (400 kg); nail with grade of 4 and 5 (100 kg) (9) velvet and raw papers (6250000 no); resin paste (312500 kg); ink (800 kg); resin glue (15625 kg) (10) craft paper (2232t); crepe paper (84t); paper yarn (84t); filter cords as sweeper (18t); gum, liquid silicate (180t); ink (12t); pp strips, w= 2 cm (400000 m) (11) wood veneer (126000 pieces); urea glue (6300 kg); filler and fixer pastes (8220 kg); sandpaper (1260 m2) (12) dry wood (240000 kg); pe cover (27t); nylon cover (20457 m2); plastic boxes (210000 no); packaging carton (580 no); tape (10000 m) (13) slat in dimensions of 184*71*5.2 cm3 (340200 no); graphite of pencil (46638 no); glue aw (6674.4 kg); black dye (30034.8 kg); other dyes (3337.2 kg); al cellophone (2182 rolls); boxes having 12 empty spaces (687204 rolls); packaging cartons having 288 empty spaces (13772 rolls); tape (1000 rolls) malek hassanpour/decis. mak. appl. manag. eng. 2 (1) (2019) 13-34 22 industry initial materials (14) raw paper with width around 674 mm, length of 3000 m (1285 roll); ink (36t); ink of paper backside (26t); carton with dimension of 100*105*88 cm3 (4500 no); boxes of 10*35*22 cm3 (450000) (15) oak pollen (4934 m3); paper sheet, w= 50 cm (157000 m2); carton in sizes of 49*49 cm2 (25050 no); pp rope (5300 m); glue materials (1500 kg) (16) alo 93-98.5% (133000 kg); formaldehyde urea gum (326000 kg); craft paper (490000 kg); wood ink (10200 kg); gum (10200 kg) w= width, l= length, pp= polypropylene, pe= polyethylene table 4. iwci, numer of staff, land area used and energy consumptions land (m2) fuel (gj) water (m3) power (kw) employees nominal capacity (t) industry 9500 3 10 125 29 1400 (1) 3500 3 5 100 20 1500 (2) 5400 29 12 174 24 7500 (3) 4000 35 17 187 29 400 (4) 5800 10 6 228 30 1000 (5) 2400 3 4 58 16 1000 (6) 2100 29 12 174 30 2600000 (7) 4600 23 10 212 10 120000 (8) 4000 7 7 116 23 6250 (9) 5100 7 8 155 35 12000 (10) 15700 25 20 575 72 12000 (11) 3300 5 13 152 30 7565000 (12) 2100 3 8 99 13 324000 (13) 2100 3 3 30 15 450000 pockets (14) 20600 74 60 359 42 150000m+150000 m2 (15) 7300 31 12 209 20 2000000 m2 (16) 4.1 delphi fuzzy set spss software, ahp and fuzzy topsis methods were assigned to classify around 16 iwci. using friedman test the ranks values were obtained about 2.59, 4, 1.53, 1.88 and 5 for the number of employees, power, water, fuel consumed and land area. tables 5 and 6 show likert spectrum defined for criteria, fuzzy set possessing values and linguistic words respectively. evaluation of iranian wood and cellulose industries 23 narimisa and narimisa (2016) used paired comparisons matrix among main factors of isfahan oil refinery so it resulted to a prioritization style as economic > land use > environmental > social. azizi et al (2009) assigned ahp and expert choice 2000 upon iranian particle board industries among major criteria intensities, so results revealed that the density of the products and its high intensity had the highest priority. azizi (2007) assessed iranian facial tissue industries based on weighing factors via ahp method and expert choice software. it revealed that softness, time of absorption, appearance quality, basis weight and price criteria had high priority respectively. t a b le 5 . c ri te ri a / s y m b o ls v e rs u s fa ct o rs b a se d o n l ik e rt s ca le c ri te ri a / sy m b o ls e m p lo y e e s p o w e r (k w ) w a te r (m 3 ) f u e l (g j) l a n d ( m 2 ) s y m b o l v e ry h ig h 1 2 1 -1 4 0 + 6 0 0 + 6 0 + 2 5 0 1 6 5 0 1 -2 4 0 0 0 v h h ig h 1 0 1 -1 2 0 5 0 1 -6 0 0 5 1 -6 0 2 0 1 -2 5 0 1 2 5 0 1 -1 6 5 0 0 h s li g h tl y h ig h 8 1 -1 0 0 4 0 1 -5 0 0 4 1 -5 0 1 0 1 -2 0 0 1 0 0 0 1 -1 2 5 0 0 s h m e d iu m 6 1 -8 0 3 0 1 -4 0 0 3 1 -4 0 7 6 -1 0 0 7 5 0 1 -1 0 0 0 0 m s li g h tl y l o w 4 1 -6 0 2 0 1 -3 0 0 2 1 -3 0 5 1 -7 5 5 0 0 1 -7 5 0 0 s l l o w 2 1 -4 0 1 0 1 -2 0 0 1 1 -2 0 2 6 -5 0 2 5 0 1 -5 0 0 0 l v e ry l o w 0 -2 0 0 -1 0 0 0 -1 0 0 -2 5 0 -2 5 0 0 v l malek hassanpour/decis. mak. appl. manag. eng. 2 (1) (2019) 13-34 24 t a b le 6 . f u z z y d e ci si o n -m a k in g a p p ro a ch t o p ri o ri ti z e t h e f a ct o rs w e ig h ts l a n d f u e l w a te r p o w e r e m p lo y e e s n o m in a l ca p a ci ty in d u st ry 4 .4 6 m ( 0 .5 ) v l ( 0 .1 3 6 2 ) v l ( 0 .1 3 6 2 ) l ( 0 .2 2 7 2 ) l ( 0 .2 2 7 2 ) 1 4 0 0 (1 ) 2 .4 9 8 l ( 0 .2 2 7 2 ) v l ( 0 .1 3 6 2 ) v l ( 0 .1 3 6 2 ) v l ( 0 .1 3 6 2 ) v l ( 0 .1 3 6 2 ) 1 5 0 0 (2 ) 4 .1 1 s l ( 0 .3 6 9 5 ) l ( 0 .2 2 7 2 ) l ( 0 .2 2 7 2 ) l ( 0 .2 2 7 2 ) l ( 0 .2 2 7 2 ) 7 5 0 0 (3 ) 3 .4 0 8 l ( 0 .2 2 7 2 ) l ( 0 .2 2 7 2 ) l ( 0 .2 2 7 2 ) l ( 0 .2 2 7 2 ) l ( 0 .2 2 7 2 ) 4 0 0 (4 ) 4 .3 7 s l ( 0 .3 6 9 5 ) v l ( 0 .1 3 6 2 ) v l ( 0 .1 3 6 2 ) s l ( 0 .3 6 9 5 ) l ( 0 .2 2 7 2 ) 1 0 0 0 (5 ) 2 .0 4 3 v l ( 0 .1 3 6 2 ) v l ( 0 .1 3 6 2 ) v l ( 0 .1 3 6 2 ) v l ( 0 .1 3 6 2 ) v l ( 0 .1 3 6 2 ) 1 0 0 0 (6 ) 2 .9 5 v l ( 0 .1 3 6 2 ) l ( 0 .2 2 7 2 ) l ( 0 .2 2 7 2 ) l ( 0 .2 2 7 2 ) l ( 0 .2 2 7 2 ) 2 6 0 0 0 0 0 (7 ) 3 .4 3 1 l ( 0 .2 2 7 2 ) v l ( 0 .1 3 6 2 ) v l ( 0 .1 3 6 2 ) s l ( 0 .3 6 9 5 ) v l ( 0 .1 3 6 2 ) 1 2 0 0 0 0 (8 ) 3 .0 9 7 l ( 0 .2 2 7 2 ) v l ( 0 .1 3 6 2 ) v l ( 0 .1 3 6 2 ) l ( 0 .2 2 7 2 ) l ( 0 .2 2 7 2 ) 6 2 5 0 (9 ) 3 .8 s l ( 0 .3 6 9 5 ) v l ( 0 .1 3 6 2 ) v l ( 0 .1 3 6 2 ) l ( 0 .2 2 7 2 ) l ( 0 .2 2 7 2 ) 1 2 0 0 0 (1 0 ) 8 .8 5 h ( 0 .7 7 2 7 ) v l ( 0 .1 3 6 2 ) l ( 0 .2 2 7 2 ) h ( 0 .7 7 2 7 ) m ( 0 .5 ) 1 2 0 0 0 (1 1 ) 3 .0 9 7 l ( 0 .2 2 7 2 ) v l ( 0 .1 3 6 2 ) v l ( 0 .1 3 6 2 ) l ( 0 .2 2 7 2 ) l ( 0 .2 2 7 2 ) 7 5 6 5 0 0 0 (1 2 ) 2 .0 4 3 v l ( 0 .1 3 6 2 ) v l ( 0 .1 3 6 2 ) v l ( 0 .1 3 6 2 ) v l ( 0 .1 3 6 2 ) v l ( 0 .1 3 6 2 ) 3 2 4 0 0 0 (1 3 ) 2 .0 4 3 v l ( 0 .1 3 6 2 ) v l ( 0 .1 3 6 2 ) v l ( 0 .1 3 6 2 ) v l ( 0 .1 3 6 2 ) v l ( 0 .1 3 6 2 ) 4 5 0 0 0 0 p o ck e ts (1 4 ) 9 .1 5 v h ( 0 .8 6 3 6 ) s l ( 0 .3 6 9 5 ) h ( 0 .7 7 2 7 ) m ( 0 .5 ) s l ( 0 .3 6 9 5 ) 1 5 0 0 0 0 m + 1 5 0 0 0 0 m 2 (1 5 ) 4 .3 1 s l ( 0 .3 6 9 5 ) l ( 0 .2 2 7 2 ) v l ( 0 .1 3 6 2 ) s l ( 0 .3 6 9 5 ) v l ( 0 .1 3 6 2 ) 2 0 0 0 0 0 0 m 2 (1 6 ) evaluation of iranian wood and cellulose industries 25 4.2 fuzzy topsis procedure using equation 18 the existing data in table 6 were shifted to present data of table 7. in the following was used equations of 19-24 to obtain fuzzy topsis values and their weights according to table 8. table 7. defuzzification matrix land fuel water power employees nominal capacity (t) industry 0.318 0.184 0.1362 0.174 0.25 1400 (1) 0.144 0.184 0.1362 0.104 0.15 1500 (2) 0.235 0.307 0.2272 0.174 0.25 7500 (3) 0.144 0.307 0.2272 0.174 0.25 400 (4) 0.235 0.184 0.1362 0.284 0.25 1000 (5) 0.086 0.184 0.1362 0.104 0.15 1000 (6) 0.086 0.307 0.2272 0.174 0.25 2600000 (7) 0.144 0.1.0 0.1362 0.284 0.15 120000 (8) 0.144 0.184 0.1362 0.174 0.25 6250 (9) 0.235 0.184 0.1362 0.174 0.25 12000 (10) 0.492 0.184 0.2272 0.174 0.549 12000 (11) 0.144 0.184 0.1362 0.174 0.25 7565000 (12) 0.086 0.184 0.1362 0.104 0.15 324000 (13) 0.086 0.184 0.1362 0.104 0.15 450000 pockets (14) 0.55 0.5 0.7727 0.384 0.4056 150000m+ 150000 m2 (15) 0.235 0.307 0.1362 0.284 0.25 2000000 m2 (16) ideal and anti-ideal solutions in the topsis procedure were complied from the obtained values for a+ and athat in the following has been explained; a+= 1.42, 1.536, 0.347, 0.94, 2.75 and a= 0.388, 0.416, 0.208, 0.345, 0.43. based on ideal and anti-ideal amounts were computed di+ and diand also cli+. in lots of researches, ahp is applied to extract weights for criteria, while fuzzy topsis employed to support the ranking of options. mardani et al (2016) evaluated around 10 biggest iranian hotels via fuzzy set theory in different provinces focusing on prominent key energy-saving technologies and solutions. so, 17 key energy factors were chosen in the first screening among about 40 energy factors classified into 5 groups. findings revealed rank ratios around 0.403, 0.225, 0.151, 0.091 and 0.083 for the equipment efficiency, system efficiency, heating and cooling demands reductions, energy management and renewable energy respectively. the fuzzy ahp among 17 factors presented ranks around 0.662, 0.541 and 0.532 for active space cooling, building insulation and tourist accommodation service respectively. malek hassanpour/decis. mak. appl. manag. eng. 2 (1) (2019) 13-34 26 yazdani-chamzini et al. (2013) used fuzzy topsis to assess the problem of investment strategy selection. the fuzzy topsis methodology applied for prioritizing the existing alternatives. the findings offered that the implemented model has a high potential to evaluate the data. zagorskas et al. (2014) studied the growth in building refurbishment of new-build projects and historical buildings preservation t a b le 8 . f u z z y t o p s is v a lu e s a n d t h e ir w e ig h ts cl i+ d i d i+ l a n d f u e l w a te r p o w e r e m p lo y e e s n o m in a l ca p a ci ty ( t) in d u st ry 0 .0 1 2 1 .2 2 1 0 1 .7 3 6 9 1 .5 9 0 .3 4 5 0 .2 0 8 0 .6 9 6 0 .6 4 7 1 4 0 0 (1 ) 0 .1 0 2 0 .2 9 1 .7 4 4 0 .7 2 0 .3 4 5 0 .2 0 8 0 .4 1 6 0 .3 8 8 1 5 0 0 (2 ) 0 .3 0 8 0 .8 8 1 .9 7 1 .1 7 5 0 .5 7 7 0 .3 4 7 0 .6 9 6 0 .6 4 7 7 5 0 0 (3 ) 0 .1 . 9 0 .5 5 2 .3 5 7 0 . 7 2 0 .5 7 7 0 .3 4 7 0 .6 9 6 0 .6 4 7 4 0 0 (4 ) 0 .3 1 1 2 .1 2 1 .1 7 5 0 .3 4 5 0 .2 0 8 1 .1 3 6 0 .6 4 7 1 0 0 0 (5 ) 0 2 .7 8 0 .4 3 0 . 3 4 5 0 . 2 0 8 0 .4 1 6 0 .3 8 8 1 0 0 0 (6 ) 0 .7 7 0 .2 1 8 2 .6 1 0 0 .4 3 0 . 5 7 7 0 .3 4 7 0 .6 9 6 0 .6 4 7 2 6 0 0 0 0 0 (7 ) 0 .2 0 0 .7 7 2 .3 9 1 0 . 7 2 0. 3 4 5 0 . 2 0 8 1 .1 3 6 0 .3 8 8 1 2 0 0 0 0 (8 ) 0 .1 6 0 .0 . 2 .0 0 . 7 2 0 . 3 4 5 0 . 2 0 8 0 .6 9 6 0 .6 4 7 6 2 5 0 (9 ) 0 .2 5 5 0 .7 2 .0 3 8 1 .1 7 5 0 . 3 4 5 0 . 2 0 8 0 .6 9 6 0 .6 4 7 1 2 0 0 0 (1 0 ) 0 .6 8 3 2 .3 1 .0 6 9 2 .4 6 0 . 3 4 5 0 . 3 4 7 0 .6 9 6 1 .4 2 1 2 0 0 0 (1 1 ) 0 .1 6 6 0 .0 . 2 .0 0 7 0 . 7 2 0 . 3 4 5 0 . 2 0 8 0 .6 9 6 0 .6 4 7 7 5 6 5 0 0 0 (1 2 ) 0 2 .7 8 0 .4 3 0 . 3 4 5 0 . 2 0 8 0 .4 1 6 0 .3 8 8 3 2 4 0 0 0 (1 3 ) 0 2 .8 0 .4 3 0 . 3 4 5 0 . 2 0 8 0 .4 1 6 0 .3 8 8 4 5 0 0 0 0 p o ck e ts (1 4 ) 0 .8 8 2 .7 3 0 .3 7 2 .7 5 0 .9 4 0 . 3 4 7 1 .5 3 6 1 .0 5 0 1 5 0 0 0 0 m + 1 5 0 0 0 0 m 2 (1 5 ) 0 .3 7 1 .0 9 0 1 .8 4 0 1 .1 7 5 0 . 5 7 7 0 . 2 0 8 1 .1 3 6 0 .6 4 7 2 0 0 0 0 0 0 m 2 (1 6 ) evaluation of iranian wood and cellulose industries 27 involvements in terms of practice for assigning best insulation options. according to the research, 5 modern insulation materials had chosen and evaluations revealed that topsis technique with grey numbers was a dominant technique to realize. nikas et al. (2018) evaluated the gap between climate policy to find a methodological framework and remove existing complex problems using both delphi and topsis methods. by the way, they reached to find ranks for factors and criteria and closeness to ideal solutions. cavallaro et al. (2016) studied a prioritization method for factors and criteria of combined heat and power systems via fuzzy shannon entropy and fuzzy topsis. findings represented a classification as turbine > steam turbine > fuel cell > reciprocating engine > micro-turbine. moghimi & anvari (2014) employed fuzzy mcdm approach among 8 iranian cement companies listed in the tehran stock exchange based on financial statements. hence, the ranking of companies has done as sabhan, sarab, sedasht, safar, sekaroun, sakarma, sanir and sahrmoz with priority scores of 0.55, 0.51, 0.50, 0.49, 0.42, 0.37, 0.36 and 0.33 respectively. radfar & ebrahimi (2012) used fuzzy multi-criteria decision making for iranian shipping industries to prioritize the investment methods in technology transfer. obtained results led to introduce joint venture and the subsidiary companies as the highest and lowest priorities, respectively. parsa et al. (2016) utilized fuzzy topsis technique for national iranian gas company to evaluate performance. it was performed a scoring and ranking system among them. sorayaei et al. (2012) used a fuzzy network model for forecasting stock exchange of the automobile industries. so, the results indicated the bubble growth of stock exchange of iran automobile industries. kavousi & salamzadeh (2016) applied topsis technique for national iranian copper industries to identify and prioritize factors influencing the success of a strategic planning process. in the following steps, indicators were weighted and prioritized. ebrahimnejad et al. (2008) asserted his findings by fuzzy build operate transfer + madm in order to evaluate iranian power plant industry in terms of risk identification and management. therefore, a new ranking model was presented based on fuzzy. tash & nasrabadi (2013) exploited fuzzy topsis for ranking of iran's monopolistic industry. behrouzi et al. (2011) investigated 133 automotive industries using fuzzy madm + spss analysis in order to performance measurement. the classifying options, weighting and ranking systems were the prominent findings of this research. zare et al. (2016) employed fuzzy topsis by using the nearest weighted interval approximations for the aluminum waste management system selection problem. by the way, a few scenarios introduced to figure out the solutions, so scenarios were ranked based on their closeness coefficient to the ideal solution. therefore, scenario of s4 was distinguished as the most prominent practice with a weight of 0.723514 and then following scenario of s1 with a value of 0.448137, scenario s5 with a value of 0.354226, scenario s2 with a value of 0.314215 and scenario s3 with a value of 0.204909 were ranked from second to fifth as an overwhelming method to compute and prioritize factors respectively. 4.3 topsis method in this step same procedure was done on data to classify iwci. the difference between this method and the previous one was the use of real data for industries malek hassanpour/decis. mak. appl. manag. eng. 2 (1) (2019) 13-34 28 classification. therefore, the existing data (in table 4) were shifted to table 9 and then to table 10 using the equation of 18-24. t a b le 9 . m a tr ix b a se d o n ( re a l d a ta ) in t a b le 4 l a n d f u e l w a te r p o w e r e m p lo y e e s n o m in a l ca p a ci ty ( t) in d u st ry 0 .3 0 .0 2 9 0 .1 3 6 0 .1 0 0 0 .2 3 5 1 4 0 0 (1 ) 0 .1 1 0 0 .0 2 9 0 .0 6 . 0 .1 1 2 0 .1 6 2 1 5 0 0 (2 ) 0 .1 7 0 0 .2 7 9 0 .1 6 4 0 .1 9 5 0 .1 9 5 7 5 0 0 (3 ) 0 .1 2 6 0 .3 3 7 0 .2 3 2 0 .2 0 9 0 .2 3 5 4 0 0 (4 ) 0 .1 . 3 0 .0 9 6 0 .0 . 2 0 .2 5 5 0 .2 4 3 1 0 0 0 (5 ) 0 .0 7 5 0 .0 2 . 0 .0 5 0 0 .0 6 5 0 .1 3 1 0 0 0 (6 ) 0 .0 6 6 0 .2 7 9 0 .1 6 4 0 .1 9 5 0 .2 4 3 2 6 0 0 0 0 0 (7 ) 0 .1 4 5 0 .2 2 1 0 .1 3 6 0 .2 3 7 0 .0 8 1 2 0 0 0 0 (8 ) 0 .1 2 6 0 .0 6 7 0 .0 9 5 0 .1 3 0 .1 8 6 6 2 5 0 (9 ) 0 .1 6 1 0 .0 6 7 0 .1 0 9 0 .1 7 3 0 .2 8 3 1 2 0 0 0 (1 0 ) 0 .4 9 6 0 .2 4 0 .2 7 3 0 .6 4 4 0 .5 8 3 1 2 0 0 0 (1 1 ) 0 .1 0 4 0 .0 4 8 0 .1 7 7 0 .1 7 0 0 .2 4 3 7 5 6 5 0 0 0 (1 2 ) 0 .0 6 6 0 .0 2 8 0 .1 0 9 0 .1 1 0 0 .1 0 5 3 2 4 0 0 0 (1 3 ) 0 .0 6 6 0 .0 2 8 0 .0 4 1 0 .0 3 3 0 .1 2 1 4 5 0 0 0 0 p o ck e ts (1 4 ) 0 .6 5 1 0 .7 1 3 0 .8 2 0 .4 0 2 0 .3 4 0 1 5 0 0 0 0 m + 1 5 0 0 0 0 m 2 (1 5 ) 0 .2 3 1 0 .2 9 8 0 .1 6 4 0 .2 3 4 0 .1 6 2 2 0 0 0 0 0 0 m 2 (1 6 ) evaluation of iranian wood and cellulose industries 29 t a b le 1 0 . t o p s is v a lu e s cl i+ d i d i+ l a n d f u e l w a te r p o w e r e m p lo y e e s n o m in a l ca p a ci ty ( t) in d u st ry 0 .3 1 0 1 .0 3 .1 1 1 1 .5 0 .5 0 5 2 0 .2 0 . 0 0 .5 6 0 .6 0 . 6 5 1 4 0 0 (1 ) 0 .0 9 9 0 .0 .0 0 0 .0 . 7 0 .5 5 0 .0 5 0 5 2 0 .1 0 0 0 0 0 .0 0 . 0 .0 1 9 5 . 1 5 0 0 (2 ) 0 .2 5 4 1 .1 6 3 .4 1 2 0 .8 5 0 .5 2 4 5 2 0 .2 5 0 9 2 0 .7 8 0 .5 0 5 0 5 7 5 0 0 (3 ) 0 .2 5 2 1 .1 7 3 .0 7 0 .6 3 0 .6 3 3 5 6 0 .3 5 0 9 6 0 .. 3 6 0 .6 0 . 6 5 4 0 0 (4 ) 0 .2 7 1 .2 5 3 .3 6 0 .9 1 5 0 .1 . 0 0 . 0 .1 2 5 0 6 1 .0 2 0 .6 2 9 3 7 1 0 0 0 (5 ) 0 .0 0 0 0 .1 . 9 0 .2 5 0 .3 7 5 0 .0 5 2 6 0 0 .0 . 2 6 2 0 .2 6 0 .3 3 6 7 1 0 0 0 (6 ) 0 .1 9 6 0 .9 2 3 .7 7 0 .3 3 0 .5 2 4 5 2 0 .2 5 0 9 2 0 .7 8 0 .6 2 9 3 7 2 6 0 0 0 0 0 (7 ) 0 .2 1 6 0 .9 8 6 3 .5 6 0 .7 2 5 0 .4 1 5 4 8 0 .2 0 8 0 8 0 .9 0 . 0 .2 0 7 2 1 2 0 0 0 0 (8 ) 0 .1 2 9 0 .5 7 3 .8 5 0 .6 3 0 .1 2 5 9 6 0 .1 4 5 3 5 0 .5 2 0 .4 8 1 7 4 6 2 5 0 (9 ) 0 .2 0 2 0 .9 1 3 .5 7 0 .8 0 5 0 .1 2 5 9 6 0 .1 6 6 7 7 0 .6 9 2 0 .7 3 2 9 7 1 2 0 0 0 (1 0 ) 0 .7 3 .5 4 7 1 .5 2 .4 8 0 .4 5 1 2 0 .4 1 7 6 9 2 .5 7 6 1 .5 1 1 2 0 0 0 (1 1 ) 0 .1 6 5 0 .7 4 7 3 .8 0 .5 2 0 .0 9 0 2 4 0 .2 7 0 8 1 0 .6 8 0 .6 2 9 3 7 7 5 6 5 0 0 0 (1 2 ) 0 .0 9 0 .4 1 2 4 .1 8 2 0 .3 3 0 .0 5 2 6 4 0 .1 6 6 7 7 0 .4 4 0 .2 7 1 9 5 3 2 4 0 0 0 (1 3 ) 0 .1 0 2 0 .4 8 6 4 .2 4 0 .3 3 0 .5 2 6 4 0 .0 6 2 7 3 0 .1 3 2 0 .3 1 3 3 9 4 5 0 0 0 0 p o ck e ts (1 4 ) 0 .7 6 5 3 .7 7 5 1 .1 5 5 3 .2 5 5 1 .3 4 0 4 4 1 .2 5 4 6 1 .6 0 8 0 .8 8 0 6 1 5 0 0 0 0 m + 1 5 0 0 0 0 m 2 (1 5 ) 0 .2 8 7 1 .2 7 3 .1 4 6 1 .1 5 5 0 .5 6 0 2 4 0 .2 5 0 9 2 0 .9 3 6 0 .4 1 9 5 8 2 0 0 0 0 0 0 m 2 (1 6 ) malek hassanpour/decis. mak. appl. manag. eng. 2 (1) (2019) 13-34 30 ideal and anti-ideal solutions in current topsis procedure were complied from the obtained values for a+ and aas; a+= 1.51, 2.576, 1.2546, 1.34044, 3.255 and a= 0.2072, 0.132, 0.06273, 0.05260, 0.33. finally, iwci was classified based on 3 methods of fuzzy set logic, fuzzy topsis, topsis based on real data as below: fuzzy set logic: 15 > 11 > 16 > 5 > 1 > 3 > 10 > 4 > 8 > 7 > 9 = 12 > 2 > 6 > 13 = 14; fuzzy topsis: 15 > 7 > 11 > 1 > 5 > 16 > 3 > 10 > 8 > 4 > 21 > 9 > 2 >; (6=13=14); topsis: 15 > 11 > 1 > 16 > 5 > 3 > 4 > 8 > 10 > 7 > 12 > 9 > 14 > 2 > 13 > 6 further study on the industries of iwci was revealed the statistics and list of facilities and equipment used according to table 11. awareness of the existing facilities in iwci helps stakeholders to understand new developments in utilized facilities. also, the information provided can be compared with the facilities and equipment industries in other countries. table 11. all available facilities of iwci industry facilities (1) saw, 500 kg/h, 15 hp (1 no); bangs producer machine, 260 kg/h, 15 hp (1 no); baling machine, 8 tons/h, 2.5 hp (1 no) (2) lining machines, 10 and 14 m2/min (1 and 1 no); cutting machine, 170 m/h, 4 kw (1 no); dye cast machine (1 no); split machines, 10 m2/min; saw, 3 kw, 30 m/min (3o no); print machine, 3.5 kw (1 no); carton maker machine, 2000 cartons/h, 3 kw (1 no); packaging machine (1 no) (3) motor saw of 590 degree, (1 no); saw with w= 140 cm, 30 kw (1 no); saw 100, 15 kw, 1500 rpm (2 no); cutting machine, 5 kw, 1440 rpm (1 no); grinder, 5 kw (1 no); dryer machines (3 no); wagons, in size of 1.5*3 m2 (48 no); derrick, 5 ton (2 no); compressor, 110 atm, 2000 l, 7 kw, 4 m3/min (1 no) (4) cleaning machine, 130 kg/h, 4 kw (1 no); block machine (1 no); cotton baking pot, 125 kg/h, 35 kw (1 no); feeding tank (1 no); centrifuge, 130 kg/h, 5 kw (1 no); dryer, 300 kg/h, 25 kw (1 no); wraping machine, 150 kg/h, 5 kw (1 no); carding machine, 60 kg/h, 5 kw (1 no) (5) cutting and perforation machine, 5 kw, 10 kg/min (1 no); rolling machine, 8 kw, 4.5 kg/min (8 no); air suction fan, 2 kw (2 no); fitted lab (1 no) (6) roll flattening machine (1 no); gluing machine (1 no); printing machine (1 no); paraffin addition machine (1 no); cutting machine (1 no); derrick, 2 tonss (1 no) (7) cutting machine, 5 kw (1 no); stapler machine, 0.6 kw (2 no); labelling machine, 1.5 kw (1 no) (8) shaver, 5 kw (1 no); saw, 11 kw (1 no); saw sharpener, 1.5 kw (1 no); 5storeys thermal press, 20 kw (2 no); boiler, 0.5 ton, 2 kw (1 no); 5-ways device, 2.5 kw (1 no); perforating machine, 2.5 kw (1 no); fs 1000 machine, w= 1000 mm (1 no); automat sewing machine, 7 kw (1 no); rond sanding, 2 and 3 kw (1 and 1 no); cutting machine, 3 kw (1 no); tape buffing machine, 4 kw (1 no); polishing machine, 4 kw (1 no); drill 1.5 kw (2 no); gum roller and mixer, 5 kw (1 no) evaluation of iranian wood and cellulose industries 31 (9) steel mixing tanks, 1 ton (2 no); printing machine, 2 m/min (1 no); drying and flocking machines, 500 kg (1 no); fluff removal machine, 5 m/s (2 no); screen printing machines, 1 m/min (6 no); sheet dryer machine, 2 m/min (30 no); printing machines, 3 m/min (2 no); flattening machine, 2 m/min (1 no); al frames (500 no); cleaner along with plastic knive (1 no) (10) envelope manufacturing machine, l and w= 5-110 cm and 35-60 cm (1 no); two-sided sewing machine, l= 65-95 cm, capacity of 1500 no/h (2 no); one-sided sewing machine, l= 65-90 cm, capacity 1500 no/h (2 no); packaging machine, in bundles of 100-150, 50 no/h (2 no); gum dough generation device, 1 ton (1 no); feeding roll paper, 50 m/min (1 no); compressor, 7-10 kg/cm2 (1 no); testing and checking equipment (1 no); repair workshop (1 no) (11) derrick, 5 tons (1 no); automatic saw, 48, 38 and 42 inch (1, 1 and 4 no); circular conveyor, l= 3 m (10 no); circular saw, 40 inch (2 no); dryer furnace, model of 10 m bmf-kin (8 no); derrick, 2 tons (1 no); cutting saw (5 no) (12) primary wood cutting machine, 28 inch, 2.5 kw, 5 tons (2 no); secondary wood cutting machine, i 3 model, w= 100 mm, 35 rpm, 30 kw (2 no); low-diameter round timber manufacturing machine, k 20.2, w= 80 mm, d= 80 mm, 20 rpm, 5 kw, weight of packs 550 kg (1 no); wood cutting machine of az-2.5, 3 kw, weigh of packs 50 kg (1 no); wood thickness setting machine, 6 kw, weigh of pack 60 kg (1 no); cutting machine with circular saw, mu-vs 3, 2 kw, weigh of pack, 120 kg (1 no); polishing machine, pot 1000 model, 0.5 kw, 20 rpm (1 no); packaging machine, 10.5 hp, 3 kw, pure weigh of 10 kg (1 no); paper milling machine, ramonas model, 3 tons, 14-18 kw (1 no) (13) complete line of wooden pencil production, 1200 tablet/shift, 28.5 kw (1 no); cyclone along with centrifuge machine, steel carbon, d and h= 68 and 1000 mm (1 no) (14) printing press machine, 100 m/min (1 no); roll flattening machine, 30160 m/min (1 no); gillutine 34 rpm (1 no); lab and repair workshop (1 and 1 no) (15) semi automatic saw, 5.5 and 11 kw (1 and 3 no); saw for cutting dry boards (2 no); multi-saw machine (1 no); automatic grinder, 7.5 kw (2 no); 15-saws machine, 15 kw (1 no); finishing operation line such as buffing and dyeing operations (1 no); wood carving machine, 63 cm, 5.5 kw (1 no); curing machine, 70 cm, 5.5 kw (1 no); saw a80, 6 kw (1 no); automatic packaging machine (1 no); dye drying line (1 no) (16) spray system as electrostatic and gravity (1 no); heating and ventilation as tunnel dryer (1 no); preparation section for resin and gum (1 no); motive power (1 no) w= width, l= length 4.4 statistical analysis results t-test analysis had represented significant differences around (p-value≤ 0.001, 0.002 among the main criteria of iwci such as the number of employees, power, water and fuel consumptions and the land area occupied by each industry. pearson correlation sig. (2-tailed), kendall's correlation coefficient sig. (2-tailed) and malek hassanpour/decis. mak. appl. manag. eng. 2 (1) (2019) 13-34 32 spearman's correlation coefficient analysis had manifested the highest significant differences about 0.886, 0.653 and 0.820 between both factors of fuel and water consumptions respectively. the categories of water, fuel, power consumptions, number of employees and the land area used had shown equal probabilities around 0.982, 0.437 (via one-sample chi-square test), 0.299 (via one-sample kolmogorov smirnov test) and 0.309 and 0.185 (via one-sample kolmogorov smirnov test). therefore, the null hypothesis was retained among factors. kolmogorov – smirnov z was conducted to figure out normal distribution among factors so obtained results revealed values about 0.966, 0.974, 1.243, 0.907 and 1.090 for the number of employees, power, water, fuel consumed and the land area occupied by industries individually. therefore, the obtained findings have supported the presence of a normal distribution trend among factors. hassanpour (2017) investigated 6 different kinds of iranian recycling industries comprising factors of power-water and fuel-land with a result as (p-value ≤.016 and 0.023) via spss analysis respectively. unnisa & hassanpour (2018) came into view a significant difference among factors such as initial feed, employees, power, water, fuel and land (p-value ≤.001) in an assessment upon 0 various kinds of iranian brick manufacturing industries. 5. conclusion by present study was empirically assessed iwci in terms of an inventory of materials, processes and facilities employed. data were evaluated by three methods of delphi logic, fuzzy topsis, topsis along with spss analysis of data. it was found that topsis (based on real data) was more precise than fuzzy topsis and delphi fuzzy set to classify industries. the spss software presented correlations, significant differences and null hypothesis among the data to complete iwci evaluation procedure. some of the main achievements of this study can be cited to awareness of the flow of input materials injected into industries according to the type of materials and their required values, the prediction of the type of pollutants released into the environment and developing researches towards industrial ecology studies, the identification of existing facilities and devices in the industries and as well as technologies employed for the purposes of industry 4.0, getting enough knowledge about the amount of energy consumed in industries and the amount of product produced by each industry, providing economic estimates of industries in the easiest possible way, managing industries regarding the enough information to evaluate efficient industries in studies related to data envelopment analysis etc. references azizi, m., khakifirooz, a., & moghimi, f. 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(2013). selecting the optimal renewable energy using multi criteria decision making. journal of business economics and management, 7(1), 957-978. yazdani-chamzini, a. y., shariati, s., yakhchali, s. h., & zavadskas e. k. (2014). proposing a new methodology for prioritising the investment strategies in the private sector of iran. economic research – ekonomska istzraživanja, 27(1), 320–345. zagorskas, j., zavadskas, e. k., turskis, z., burinskiene, m., blumberga, a., & blumberga, d. (2014). thermal insulation alternatives of historic brick buildings in baltic sea region. energy and buildings, 78, 35–42. zare, r., nouri, j., abdoli, m. a., & atabi, f. (2016). application integrated fuzzy topsis based on lca results and the nearest weighted approximation of fns for industrial waste management-aluminum industry: arak-iran. indian journal of science and technology, 9(2), 2-11. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). https://en.wikipedia.org/w/index.php?title=knut_m._wittkowski&action=edit&redlink=1 https://www.tandfonline.com/author/yazdani-chamzini%2c+abdolreza https://www.tandfonline.com/author/zavadskas%2c+edmundas+kazimieras https://www.tandfonline.com/author/moini%2c+s+hamzeh+haji plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 1, issue 2, 2018, pp. 16-33 issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame1802016b * corresponding author. e-mail addresses: ibrahim.badi@hotmail.com (i. badi), mohamed.ballem@eng.misuratau.edu.ly (a.m. abdulshahed) supplier selection using the rough bwm-mairca model: a case study in pharmaceutical supplying in libya ibrahim badi*1, mohamed ballem1 1 misurata university, faculty of engineering, mechanical engineering department, libya received: 2 january 2018; accepted: 30 june 2018; available online: 3 july 2018. original scientific paper abstract: the quality of health system in libya has witnessed a considerable decline since the revolution in 2011. one of the major problems this sector is facing is the loss of control over supply medicines and pharmaceutical equipments from international suppliers for both public and private sectors. in order to take the right decision and select the best medical suppliers among the available ones, many criteria have to be considered and tested. this paper presents a multiple criteria decision-making analysis using modified bwm (best-worst method) and mairca (multi-attribute ideal-real comparative analysis) methods. in the present case study five criteria and three suppliers are identified for supplier selection. the results of the study show that cost comes first, followed by quality as the second and company profile as the third relevant criterion. the model was tested and validated on a study of the optimal selection of supplier. key words: supplier selection, multi-criteria decision-making, rough numbers, bwm, mairca. 1 introduction selecting and managing medicines and pharmaceutical equipment supplies for primary health care services have a significant impact on the quality of patient care and represent a high proportion of health care costs. in developing countries health services need to choose appropriate supplies, equipment and drugs, in order to meet priority health needs and avoid wasting their limited resources. items can be inappropriate because they are technically unsuitable or incompatible with existing equipment, if spare parts are not available, or, because staff have not been trained to use them (kaur et al., 2001). recently, supplier evaluation and selection have received more attention from various researchers in the literature (mardani et al., mailto:ibrahim.badi@hotmail.com mailto:mohamed.ballem@eng.misuratau.edu.ly supplier selection using rough bwm-mairca model: a case study in pharmaceutical… 17 2016; de boer et al., 2001; govidan et al., 2015; chai et al., 2013; prakash et al., 2015; abdulshahed et al., 2017; badi et al., 2018; stević et al., 2017 a). supplier selection is a multi-criteria problem which includes both quantitative and qualitative factors (liang et al., 2013). generally, the criterion for supplier selection is highly dependent on individual industries and companies. therefore, different companies have different management strategies, enterprise culture and competitiveness. furthermore, company background can make a huge difference and can impact supplier selection. thus, the identification of supplier selection criteria is largely requiring the domain expert’s assessment and judgment. to select the best supplier, it is necessary to make a trade-off between these qualitative and quantitative factors some of which may be in conflict (ghodsypour & o'brien, 1998). the traditional supplier selection methods are often based on the quoted price, which ignores significant direct and indirect costs associated with quality, delivery, and service cost of purchased materials; however, uncertainty is present because the future can never be exactly predicted. the selection of the best supplier is done based on quoted price and considering all the possibilities of the analysis, but there is always uncertainty about indirect costs associated with quality, delivery time, and the like. one of the key problems in the supplier selection is to find the best supplier among several alternatives according to various criteria, such as service, cost, risk, and others. after identifying the criteria, a systematic methodology is required to integrate experts’ assessments in order to find the best supplier. at present, various methods have been used for the supplier selection, such as the analytic network process (anp) and the analytical hierarchy process (ahp) (porras-alvarado et al., 2017). ahp is a common multicriteria decision-making method; it is developed by saaty (saaty, 1979; saaty, 1990) to provide a flexible and easily understood way of analyzing complex problems. the method breaks a complex problem into hierarchy or levels, and then makes comparisons among all possible pairs in a matrix to give a weight for each factor and a consistency ratio. libya began privatizing the pharmaceutical system in 2003. pharmaceutical supplies were previously provided to both public and private sectors by the national company of pharmaceutical industry (ncpi), but drug companies are also permitted to market and supply their products to both public and private health sectors through local agencies. in 2009, over 300 international pharmaceutical manufacturers from europe, asia, and the middle east were registered as permitted drug suppliers for libya (alsageer, 2013). all the drugs consumed in libya are imported except few items, which are manufactured locally. the headquarters of the ncpi until 2003 was responsible for all drug manufacture and imports in libya. its branches are the channels of drugs distribution for governmental hospitals, private pharmacies, and clinics (khalifa et al., 2017). from 2004 till date the libyan secretariat of health, by executing a public tender through medical supply organization (mso), has been responsible for purchasing and distributing drugs to public hospitals and clinics. worth noting is that, on sporadic intervals, the budget has been allocated to the major public hospitals to locally purchase their own general drug demands. however, since 2011 (post-17th february 2011 revolution) mso has lost its control on importing medicines due to receiving many drugs as donations from different international sources without acceptable level of coordination (zhai et al., 2008); this has resulted in the supply of pharmaceuticals and medical equipment growing considerably in recent years. for badi & ballem/decis. mak. appl. manag. eng. 1 (2) (2018) 16-33 18 instance, in misrata (the third-largest city in libya) the number of companies operating in the field of medical supply exceeded 170 companies, and more than 425 companies in tripoli (capital city). the items that are supplied vary but the most common drugs are capsules, injections, ointments, inhalants, solutions, etc.; these drugs and materials are supplied from several countries, including arab (e.g. egypt, morocco, algeria, uae, and jordan), european (e.g. germany, switzerland, and britain), and asian ones (e.g. india, china, and malaysia) as well as america. the suppliers in each of these countries have some special characteristics distinguishing them from others. the closest arab countries have the ability to speed supply and hence the flexibility in providing these drugs more quickly than the rest. on the other hand, products coming from european countries are of better quality, but their prices are higher compared to competitors from other countries. thus, to make informed choices about what to buy and what to select among available suppliers, clear criteria for selection remain important, and efforts should be made to make suitable decision support tools for right decision-making. in this paper, a rough bwm-mairca model for selection of the best supplier is proposed. the presented model is used for the analysis of the supplier selection process in pharmaceutical supplies in libya. in this case study there are three suppliers with high medicine supplies to libya. in order to maintain confidentiality of the supplier, we have denoted the given suppliers as a, b, and c. 2. rough numbers in group decision-making problems, the priorities are defined with respect to multi-expert’s aggregated decision and process subjective evaluation of the expert’s decisions. rough numbers consisting of upper, lower and boundary interval, respectively, determine intervals of their evaluations without requiring additional information by relying only on original data (zhai et al., 2008). hence, the obtained expert decision-makers (dms) perceptions objectively present and improve their decision-making process. according to zhai et al. (2010), the definition of rough number is shown below. let’s u be a universe containing all objects and x be a random object from u . then we assume that there exists a set built with k classes representing dms preferences, 1 2( , ,..., )kr j j j with condition 1 2 ,..., kj j j   . then, , , 1qx u j r q k     lower approximation ( )qapr j , upper approximation ( )qapr j and boundary interval ( )qbnd j are determined, respectively, as follows:  ( ) / ( )q qapr j x u r x j   (1)  ( ) / ( )q qapr j x u r x j   (2)       ( ) / ( ) / ( ) / ( ) q q q q bnd j x u r x j x u r x j x u r x j         (3) the object can be presented with rough number (rn) defined with lower limit ( )qlim j and upper limit ( )qlim j , respectively: supplier selection using rough bwm-mairca model: a case study in pharmaceutical… 19 1 ( ) ( ) ( )q q l lim j r x x apr j m   (4) 1 ( ) ( ) ( )q q u lim j r x x apr j m   (5) where lm and um represent the sum of objects contained in the lower and upper object approximation of qj , respectively. for object qj , rough boundary interval  ( )qirbnd j presents an interval between the lower and the upper limits as: ( ) ( ) ( )q q qirbnd j lim j lim j  (6) the rough boundary interval presents measure of uncertainty. the bigger ( )qirbnd j value shows that variations in the experts’ preferences exist, while smaller values show that the experts have harmonized opinions without major deviations. in ( )qirbnd j are comprised all the objects between lower limit ( )qlim j and upper limit ( )qlim j of rough number ( )qrn j . that means that ( )qrn j can be presented using ( )qlim j and ( )qlim j . ( ) ( ), ( )q q qrn j lim j lim j    (7) since rough numbers belong to the group of interval numbers, arithmetic operations applied in interval numbers are also appropriate for rough numbers (zhu et al., 2015). 3. rough based best-worst method (r-bwm) in order to take into account the subjectivity that appears in group decisionmaking more comprehensively, in this study a modification of the best-worst method (bwm) is carried out using rough numbers (rn). the application of rn eliminates the necessity for additional information when determining uncertain intervals of numbers. in this way, the quality of the existing data is retained in group decisionmaking and the perception of experts is expressed in an objective way in aggregated best-to-others (bo) and others-to-worst (ow) matrices. since the method is very recent, the literature so far only has the traditional (crisp) bwm (rezaei, 2015; rezaei et al., 2015; rezaei, 2016; ren et al., 2017) and modification of the bwm carried out using fuzzy numbers (guo and zhao, 2017). also, stević et al., (stević et al., 2017b) used rough bwm to solve an internal transportation problem of the paper manufacturing company. the approach in this section introduces rn which enables a more objective expert evaluation of criteria in a subjective environment. the proposed modification of the bwm using rn (r-bwm) makes it possible to take into account the doubts that occur during the expert evaluation of criteria. r-bwm makes it possible to bridge the existing gap in the bwm methodology with the application of badi & ballem/decis. mak. appl. manag. eng. 1 (2) (2018) 16-33 20 a novel approach in the treatment of uncertainty based on rn. the following section presents the algorithm for the r-bwm that includes the following steps: step 1 determining a set of evaluation criteria. this starts from the assumption that the process of decision-making involves m experts. in this step, the experts consider a set of evaluation criteria and select the final one  1 2, ,... nc c c c , where n represents the total number of criteria. step 2 determining the most significant (most influential) and worst (least significant) criteria. the experts decide on the best and the worst criteria from the set of criteria  1 2, ,... nc c c c . if the experts decide on two or more criteria as the best, or worst, the best and worst criteria are selected arbitrarily. step 3 determining the preferences of the most significant (most influential) criteria (b) from set c over the remaining criteria from the defined set. under the assumption that there are m experts and n criteria under consideration, each expert should determine the degree of influence of best criterion b on criteria j ( 1, 2,...,j n ). this is how we obtain a comparison between the best criterion and the others. the preference of criterion b compared to the j-th criterion defined by the e-th expert is denoted with e bja ( 1, 2,...,j n ;1 e m  ). the value of each pair e bja takes a value from the predefined scale in interval  1, 9 e bja  . as a result a bestto-others (bo) vector is obtained: 1 2( , ,..., ); 1 e e e e b b b bna a a a e m   (8) where e bja represents the influence (preference) of best criterion b over criterion j, whereby 1 e bba  . this is how we obtain bo matrices 1 ba , 2 ba , …, m ba for each expert. step 4 determining the preferences of the criteria from set c over the worst criterion (w) from the defined set. each expert should determine the degree of influence of criterion j ( 1, 2,...,j n ) in relation to criterion w. the preference of criterion j in relation to criterion w defined by the e-th expert is denoted as e jwa ( 1, 2,...,j n ;1 e m  ). the value of each pair e jwa takes a value from the predefined scale in interval  1, 9ejwa  . as a result an others-to-worst (ow) vector is obtained: 1 2( , ,..., ); 1 e e e e w w w nwa a a a e m   (9) where e jwa represents the influence (preference) of criterion j in relation to criterion w, whereby 1 e wwa  . this is how we obtain ow matrices 1 wa , 2 wa , …, m wa for each expert. step 5 determining the rough bo matrix for the average answers of the experts. based on the bo matrices of the experts’ answers 1 e e b bj n a a      , we form matrices of the aggregated sequences of experts *e ba * 2 1 2 1 2 1 1 1 2 2 2 1 , , , ; , , ,; ; ; ; , e k m b b b b b b b bn bn bn n m m a a a a a aa a a a          (10) supplier selection using rough bwm-mairca model: a case study in pharmaceutical… 21 where  1 2, , ,e mbj bj bj bna a a a  represents sequences by means of which the relative significance of criterion b is described in relation to criterion j. using equations (1)-(7) each sequence e bja is transformed into rough sequence   ( ), ( )e e ebj bj bjr a lim an lim a    , where ( ) e bjlim a represent the lower limits, and ( ) e bjlim a the upper limit of rough sequence  ebjrn a , respectively. so for sequence  ebjrn a we obtain a bo matrix *1 ba , *2 ba , …, *m ba . by applying equation (11), we obtain the average rough sequence of the bo matrix 11 2 1 1 ( ) ( , ,..., ) 1 m l el bj bj ee bj bj bj bj m u eu bj bj e a a m rn a rn a a a a a m               (11) where e represents the e-th expert ( 1, 2,...,e m ), while  ebjrn a represents the rough sequences. we thus obtain the averaged rough bo matrix of average responses ba 1 2 1 , ,...,b b b bn n a a a a      (12) step 6 determining the rough ow matrix of average expert responses. based on the wo matrices of the expert responses 1 e e w jw n a a      , as with the rough bo matrices, for each element e jwa we form matrices of the aggregated sequences of the experts *e wa * 1 2 1 2 1 2 1 1 1 2 2 2 1 , , , ; , , ,; ; ; ; , e m m w w w w w w w nw nw nw m n a a a a aa a a a a          (13) where  1 2, , ,ejw jw jw n m wa a a a  represents sequence with which the relative significance of criterion j is described in relation to criterion w. as in step 5, using (1)-(7), sequences e jwa are transformed into rough sequences   ( ), ( )e e ejw jw jwr a lim an lim a    . thus for each rough sequence of expert e (1 e m  ) a rough bo matrix is formed. equation (14) is used to average the rough sequences of the ow matrix of the experts to obtain an averaged rough ow matrix. 11 2 1 1 ( ) ( , ,..., ) 1 m l el jw jw ee jw jw jw jw m u eu jw jw e a a m rn a rn a a a a a m               (14) badi & ballem/decis. mak. appl. manag. eng. 1 (2) (2018) 16-33 22 where e represents the e-th expert ( 1, 2,...,e m ), while ( )jwrn a represents the rough sequences. thus, we obtain the averaged rough ow matrix of average responses wa 1 2 1 , ,...,w w w nw n a a a a      (15) step 7 calculation of the optimal rough values of the weight coefficients of criteria 1 2[ ( ), ( ),..., ( )]nrn w rn w rn w from set c . the goal is to determine the optimal value of the evaluation criteria, which should satisfy the condition that the difference in the maximum absolute values (16) ( )( ) ( ) ( ) ( ) ( ) jb bj jw j w rn wrn w rn a and rn w rn w rn w   (16) for each value of j is minimized. in order to meet these conditions, the solution that satisfies the maximum differences according to the absolute value ( ) ( ) ( ) b bj j rn w rn a rn w  and ( ) ( ) ( ) j jw w rn w rn w rn w  should be minimized for all values of j. for all values of the interval rough weight coefficients of the criteria ( ) ( ), ( ) [ , ] l u j j j j jrn w lim w lim w w w    the condition is met that 0 1 l u j jw w   for each evaluation criterion jc c . weight coefficient jw belongs to interval [ , ] l u j jw w , that is l u j jw w for each value 1, 2,...,j n . on this basis we can conclude that in the case of the rough of the weight coefficients of the criteria the condition is met that 1 1 n l jj w   and 1 1 n u jj w   . in this way the condition is met that the weight coefficients are found at interval [0,1], ( 1, 2,..., )jw j n  and that 1 1 n jj w   . the previously defined limits will be presented in the following min-max model: 1 1 ( )( ) min max ( ) , ( ) ( ) ( ) . . 1 1; , 1, 2,..., , 0, 1, 2,..., jb bj jw j j w n l jj n u jj l u j j l u j j rn wrn w rn a rn w rn w rn w s t w w w w j n w w j n                             (17) where ( ) ( ), ( ) [ , ] l u j j j j jrn w lim w lim w w w    is the rough weight coefficient of a criterion. model (17) is equivalent to the following model: supplier selection using rough bwm-mairca model: a case study in pharmaceutical… 23 1 1 min . . ; ; ; ; 1; 1; , 1, 2,..., , 0, 1, 2,..., l u u l b b bj bj u l j j l u u lj j jw jw u l w w n l jj n u jj l u j j l u j j s t w w a a w w w w a a w w w w w w j n w w j n                                       (18) where ( ) [ , ] l u j j jrn w w w represents the optimum values of the weight coefficients, ( ) [ , ] l u b b brn w w w and ( ) [ , ] l u w w wrn w w w represents the weight coefficients of the best and worst criterion, respectively, while ( ) , l u jw j jrn a a a      and ( ) , l u bj bj bjrn a a a      , respectively, represent the values from the average rough ow and rough bo matrices (see equations (12) and (15)). by solving model (18) we obtain the optimal values of the weight coefficients of evaluation criteria 1 2[ ( ), ( ),..., ( )]nrn w rn w rn w and *  . the consistency ratio of the rough bwm the consistency ratio is a very important indicator by means of which we check the consistency of the pair wise comparison of the criteria in the rough bo and rough ow matrices. definition 1 comparison of the criteria is consistent when condition ( ) ( ) ( )bj jw bwrn a rn a rn a  is fulfilled for all criteria j, where ( )bjrn a , ( )jwrn a and ( )bwrn a , respectively, represent the preference of the best criterion over criterion j, the preference of criterion j over the worst criterion, and the preference of the best criterion over the worst criterion. however, when comparing the criteria it can happen that some pairs of criteria j are not completely consistent. therefore, the next section defines consistency ratio (cr), which gives us information on the consistency of the comparison between the rough bo and the rough ow matrices. in order to show how cr is determined we start from calculation of the minimum consistency when comparing the criteria, which is explained in the following section. as previously indicated, the pair wise comparison of the criteria is carried out based on a predefined scale in which the highest value is 9 or any other maximum from a scale defined by the decision-maker. the consistency of the comparison badi & ballem/decis. mak. appl. manag. eng. 1 (2) (2018) 16-33 24 decreases when ( ) ( )bj jwrn a rn a is less or greater than ( )bwrn a , that is when ( ) ( ) ( )bj jw bwrn a rn a rn a  . it is clear that the greatest inequality occurs when ( )bjrn a and ( )jwrn a have the maximum values that are equal to ( )bwrn a , which continues to affect the value of  . based on these relationships we can conclude that ( ) ( ) ( ) ( ) ( ) ( )b j j w b wrn w rn w rn w rn w rn w rn w        (19) as the largest inequality occurs when ( )bjrn a and ( )jwrn a have their maximum values, then we need to subtract the value  from ( )bjrn a and ( )jwrn a and add ( )bwrn a . thus we obtain equation (20)  ( ) ( ) ( )bj jw bwrn a rn a rn a             (20) since for the minimum consistency ( ) ( ) ( )bj jw bwrn a rn a rn a  applies, we present equation (20) as        2 2 ( ) ( ) ( ) 1 2 ( ) ( ) ( ) 0 bw bw bw bw bw bw rn a rn a rn a rn a rn a rn a                    (21) since we are using rough numbers, and if there is no consensus between the dm on their preferences of the best criterion over the worst criterion, then ( )bwrn a will not have a crisp value but we will use ( ) , l u bw bw bwrn a a a      . since for rn condition l u bw bwa a applies, we can conclude that the preference of the best criterion over the worst cannot be greater than u bwa . in this case, when we use upper limit u bwa for determining the value of ci, then all the values connected with ( )bwrn a can use the ci obtained for calculating the value of cr. we can conclude this from the fact that the consistency index which corresponds to u bwa has the highest value in interval , l u bw bwa a      . based on this conclusion we can transform equation (21) in the following way:     22 1 2 0 u u u bw bw bwa a a      (22) by solving equation (22) for the different values of u bwa we can determine the maximum possible values of  ,which is the ci for the r-bw method. since we obtain the values of ( )bwrn a , i.e. u bwa on the basis of the aggregated decisions of the dm, and these change the ivfrn interval, it is not possible to predefine the values of  . the values of  depend on uncertainties in the decisions, since uncertainties change the rn interval. as explained in the algorithm for the r-bw method, interval , l u bw bwa a   changes depending on uncertainties in evaluating the criteria. supplier selection using rough bwm-mairca model: a case study in pharmaceutical… 25 if the dm agree on their preference for the best criterion over the worst then bwa represents the crisp value of bwa from the defined scale and then the maximum values of  apply for different values of  1, 2,...,9bwa  , table 1. table1 values of the consistency index (ci) bwa 1 2 3 4 5 6 7 8 9 ci ( max ) 0.00 0.44 1.00 1.63 2.30 3.00 3.73 4.47 5.23 in table 1 values bwa are taken from the scale  1, 2,...,9 which is defined in rezaei (2015). on the basis of ci (table 1) we obtain consistency ratio (cr) * cr ci   (23) cr takes values from interval  0,1 , where the values closer to zero show high consistency while the values of cr closer to one show low consistency. 4. rough mairca method the basic assumption of the mairca method is to determine the gap between ideal and empirical weights. the summation of the gaps for each criterion gives the total gap for every observed alternative. finally, alternatives will be ranked, and the best ranked alternative is the one with the smallest value of the total gap. the mairca method shall be carried out in 6 steps (pamučar et al., 2014; gigović et al., 2016): step 1 formation of the initial decision matrix ( y ). the first step includes evaluation of l alternatives per n criteria. based on response matrices yk=[ykij]l×nby all m experts we obtain matrix * y of aggregated sequences of experts 1 2 1 2 1 2 11 11 11 12 12 12 1 1 1 1 2 1 2 1 2 * 21 21 21 22 22 22 2 2 2 1 2 1 2 2 1 1 1 1 2 2 2 , , , ; , , , , , , , ; , , , , , , , ; , , , ; ; ; ; ; ; ; ; ; , n n n n n n n n n n n n nn nn n m m m m m m m m n m y y y y y y y y y y y y y y y y y y y y y y y y y y y y                           (24) where  1 2, , ,ij ij m ij ijy y y y  denote sequences for describing relative importance of criterion i in relation to alternative j. by applying equations (1) through (7), sequences m ijy are transformed into rough sequences  ij m rn y . consequently, rough matrices y1l, y2l, …,yml will be obtained for rough sequence  ij m rn y , where m denotes the number of experts. therefore, for the group of rough matrices y1, y2, …,ym we obtain rough sequences    1 1 2 2( ), ( ) , ( ), ( ) ,..., ( ), ( )ij ij ij ij jmj ij imir llim y lim y lim im y ln y y y im yl im           . by applying equation (25), we obtain mean rough sequences badi & ballem/decis. mak. appl. manag. eng. 1 (2) (2018) 16-33 26 11 2 1 1 ( ) ( , ,..., ) 1 m l el ij ij ee ij ij ij ij m u eu ij ij e y y m rn y rn y y y y y m               (25) where e denotes e-th expert ( 1, 2,...,e m ), ( )ijrn y denotes rough number ( ) ( ), ( )ijij ijr y yn y lim lim    . in such a way, rough vectors       1 2, ,...,i i i ina rn y rn y rn y of mean initial decision matrix is obtained, where ( ) ( ), ( ) , l u ij ijij ijijyrn y lim lim yy y        denotes value of i -th alternative as per j -th criterion ( 1, 2,..., ;i l 1, 2,...,j n ). 1 2 1 11 12 1 2 21 22 2 1 2 ... ( ) ( ) ... ( ) ( ) ( ) ( ) ... ... ... ... ... ( ) ( ) ... ( ) n n n l l l ln l n c c c a rn y rn y rn y a rn y rn y rn y y a rn y rn y rn y              (26) where l denotes the number of alternatives, and n denotes total sum of criteria. step 2define preferences according to selection of alternatives ia p . when selecting alternative, the decision maker (dm) is neutral, i.e. does not have preferences to any of the proposed alternatives. since any alternative can be chosen with equal probability, preference per selection of one of l possible alternatives is as follows: 1 1 ; 1, 1, 2,..., i i l a a i p p i l l     (27) where l denotes the number of alternatives. step 3 calculate theoretical evaluation matrix elements ( pt ). theoretical evaluation matrix ( pt ) is developed in l x n format (l denotes the number of alternatives, n denotes the number of criteria). theoretical evaluation matrix elements ( ( )pijrn t ) are calculated as the multiplication of the preferences according to alternatives ia p and criteria weights ( ( ), 1, 2,...,irn w i n ) obtained by application of r-bwm. 1 2 1 2 11 12 1 21 22 2 1 2 ( ) ( ) ... ( ) ( ) ( ) ... ( ) ( ) ( ) ( ) ... ... ... ...... ( ) ( ) ... ( ) l n a p p p n a p p p n p pl pl plna l n rn w rn w rn w p rn t rn t rn t p rn t rn t rn t t rn t rn t rn tp               (28) supplier selection using rough bwm-mairca model: a case study in pharmaceutical… 27 where ia p denotes preferences per selection of alternatives, ( )irn w weight coefficients of evaluation criteria, and ( )pijrn t theoretical assessment of alternative for the analyzed evaluation criterion. elements constituting matrix tp will be then defined by applying equation (29) ( ) , l u pij ai i ai i it p rn w p w w      (29) since dm is neutral to the initial selection of alternatives, all preferences ( ia p ) are equal for all alternatives. since preferences ( ia p ) are equal for all alternatives, then matrix (28) will have 1 x n format ( n denotes the number of criteria). 1 2 1 1 2 2 1 ( ) ( ) ... ( ) , , , ... , i n l u l u l u p a p p p p pn pn xn rn w rn w rn w t p t t t t t t              (30) where n denotes the number of criteria, ia p preferences according to selection of alternatives,  irn w weight coefficients of evaluation criteria. step 4 determination of real evaluation ( rt ). calculation of the real evaluation matrix elements ( rt ) is done by multiplying real evaluation matrix elements ( pt ) and elements of initial decision-making matrix ( x ) according to the following equation: ( ) ( ) ( ) , , l u l u rij pij nij pij pij ij ij rn t rn t rn x t t y y           (31) where ( )pijrn t denotes elements of theoretical assessment matrix, and ( ) ij rn y denotes elements of normalized matrix ( ) ij l n y rn y      . normalization of the mean initial decision matrix (25) is done by applying equation (32) and (33) ( ) ( ), ( ) , , l u l u ij ij ij ij ij ij ij ij ij ij ij ij ij y y y y rn y lim y lim y y y y y y y                         (32) b) for the „cost“ type criteria (lower criterion value is preferable) ( ) ( ), ( ) , , u l l u ij ij ij ij ij ij ij ij ij ij ij ij ij y y y y irn y lim y lim y y y y y y y                         (33) where iy  and iy  denote minimum and maximum values of the marked criterion by its alternatives, respectively:  min lij ij j y y   (34)  max uij ij j y y   (35) step 5 calculation of total gap matrix ( g ). elements of g matrix are obtained as difference (gap) between theoretical ( pijt ) and real evaluations ( rijt ), or by actually badi & ballem/decis. mak. appl. manag. eng. 1 (2) (2018) 16-33 28 subtracting the elements of theoretical evaluation matrix ( pt ) with the elements of real evaluation matrix ( rt ) 11 12 1 21 22 2 1 2 ( ) ( ) ... ( ) ( ) ( ) ... ( ) ... ... ... ... ( ) ( ) ... ( ) n n p r l l ln l n rn g rn g rn g rn g rn g rn g g t t rn g rn g rn g                (36) where n denotes the number of criteria, l denotes the number of alternatives, and gij represents the obtained gap of alternative i as per criterion j. gap gij takes values from the interval rough number according to equation (37) ( ) ( ) ( ) , , ij l u l u ij pij r pij pij rij rijrn g rn t rn t t t t t          (37) it is preferable that ( )ijrn g value goes to zero ( ( ) 0ijrn g  ) since the alternative with the smallest difference between theoretical ( ( )pijrn t ) and real evaluation ( ( )rijrn t ) shall be chosen. if alternative ia for criterion ic has a theoretical evaluation value equal to the real evaluation value ( ( ) ( )pij rijrn t rn t ) then the gap for alternative ia for criterion ic is zero, i.e. alternative ia per criterion ic is the best (ideal) alternative. if alternative ia for criterion ic has a theoretical evaluation value ( )pijrn t and the real ponder value is zero, then the gap for alternative ia for criterion ic is ( ) ( )ij pijrn g rn t . this means that alternative ia for criterion ic is the worst (anti-ideal) alternative. step 6 calculation of the final values of criteria functions ( iq ) per alternatives. values of criteria functions are obtained by summing the gaps from matrix (36) for each alternative as per evaluation criteria, i.e. by summing matrix elements ( g ) per columns as shown in equation (38) 1 ( ) ( ), 1, 2,..., n i ij j rn q rn g i m    (38) where n denotes the number of criteria, m denotes the number of the chosen alternatives. ranking of alternatives can be done by applying rules governing ranking of rough numbers described in (stević et al., 2017). 5. calculation part application of the hybrid rough bwm-mairca model is shown using a case study related to the selection of an optimal supplier selection in libya. based on an analysis of the available literature and expert evaluation of suppliers, five criteria were used: price and costs (c1), quality (c2), supplier profile (c3), delivery (c4) and flexibility (c5). four experts took part in the research. the r-bwm was used to determine the weight coefficients of the criteria. after defining the criteria for evaluation, the supplier selection using rough bwm-mairca model: a case study in pharmaceutical… 29 experts also determined the best (b) and worst (w) criteria. on this basis, the experts determined the bo and ow matrices in which the preferences of the b and w over the criteria were considered for the remaining criteria from the defined set. evaluation of the criteria was carried out using a scale  1,9eija  [18]. the bo and ow matrices are presented in table 2. table 2 the bo and ow expert evaluation matrices best: c1 expert evaluation worst: c5 expert evaluation c1 1, 1, 1, 1 c1 8, 7, 8, 7 c2 2, 2, 3, 3 c2 4, 4, 3, 4 c3 2, 3, 3, 2 c3 4, 4, 5, 5 c4 4, 5, 5, 4 c4 2, 3, 2, 3 c5 8, 8, 9, 9 c5 1, 1, 1, 1 using equations (1)-(7) the evaluations in the bo and ow matrices were transformed into rough numbers. after transforming crisp numbers into rough numbers, equations (9)-(15) were used to transform the bo and ow of the expert matrices into aggregated rough bo and rough ow matrices, table 3. table 3 aggregating the rough bo and rough ow matrices best: c1 rn worst: c5 rn c1 [1.00, 1.00] c1 [7.25, 7.75] c2 [2.25, 2.75] c2 [3.56, 3.94] c3 [2.25, 2.75] c3 [4.25, 4.75] c4 [4.25, 4.75] c4 [2.25, 2.75] c5 [8.25, 8.75] c5 [1.00, 1.00] on the basis of the rough bo and rough ow matrices for criteria, the optimal values of the rough weight coefficients of the criteria were calculated. based on model (18) the optimal values of the weight coefficients of the criteria were calculated, table 4. table 4 optimal values of the criteria criterion weights rank c1 [0.4113, 0.4286] 1 c2 [0.2035, 0.2169] 2 c3 [0.1498, 0.1576] 3 c4 [0.1062, 0.1424] 4 c5 [0.0667, 0.0748] 5 by solving the model (18) the value of * is obtained, * 0.8464  . the value of *  is used to determine consistency ratio (cr=0.16), equation (23). since we obtain the value of bwa i.e. u bwa on the basis of the aggregated decisions of the experts, and they affect the interval of the rn, it is not possible to predefine the values of badi & ballem/decis. mak. appl. manag. eng. 1 (2) (2018) 16-33 30 consistency index  . using equation (22), the values of consistency index (  ) is defined (ci=5.04). after calculating the weight coefficients of the criteria, expert evaluation of the alternatives was carried out with the predefined evaluation criteria. once the evaluation process is completed by applying equations from (24) through (26) decisions were aggregated and initial decision-making matrix * y obtained, table 5. evaluation of the alternatives was carried out using a scale  1, 5eijy  . table 5 aggregated initial decision-making matrix criteria/ alternatives c1 c2 c3 c4 c5 a1 [2.05, 2.39] [2.06, 2.43] [2.23, 2.73] [2.25, 3.20] [1.98, 2.86] a2 [2.43, 3.44] [4.58, 4.95] [2.10, 2.77] [4.55, 4.93] [4.00, 4.00] a3 [4.26, 4.76] [4.55, 4.93] [4.54, 4.93] [4.46, 5.00] [4.46, 5.00] after aggregation of evaluated criteria (table 5) preferences were determined as per selection of alternatives pai=1/m=0.33, where m denotes the number of alternatives and pa1=pa2=pa3=0.33. based on preferences pai, and by applying equation (29), theoretical evaluation matrix (tp) rank 1xn, will be obtained. in order to determine real evaluation matrix tr (table 6), elements of the theoretical evaluation matrix will be multiplied with normalized elements of the aggregated initial decision matrix. table 6 real evaluation matrix tr criteria/ alternatives c1 c2 c3 c4 c5 a1 [0.12, 0.14] [0.00, 0.01] [0.00, 0.01] [0.00, 0.02] [0.00, 0.01] a2 [0.07, 0.12] [0.06, 0.07] [0.00, 0.01] [0.03, 0.05] [0.01, 0.02] a3 [0.00, 0.03] [0.06, 0.07] [0.04, 0.05] [0.03, 0.05] [0.02, 0.02] normalization of the initial decision-making aggregated matrix will be done by applying equations (32) and (33). in next step, elements of theoretical evaluation matrix (tp) will be deducted from the elements of real evaluation matrix (tp) to obtain total gap matrix (g). by summing up the rows of the total gap matrix we obtain the total gap for every alternative, equation (37). based on the obtained values of the total gap between theoretical and real evaluations, the initial evaluation of alternatives will be performed, table 7. table 7 values of the total gap of alternatives and their ranking alternatives alternative gap rn(qi) rank a1 [0.13, 0.22] 3 a2 [0.04, 0.17] 1 a3 [0.09, 0.19] 2 supplier selection using rough bwm-mairca model: a case study in pharmaceutical… 31 6 conclusion supplier selection is a very important step in the purchasing process; therefore, to carry out the selection process, it is first important to identify the criteria for selection. this is particularly important for a company operating in the pharmaceutical industry and working mainly with international suppliers. the study addresses the problem of medicine supply from international suppliers for both public and private sectors in libya. five criteria and three suppliers are identified for supplier selection in this problem. this multiple criteria decision-making analysis problem is solved using the rough bwm method. as a result of the presented calculations, it is shown that cost comes first, followed by quality as the second and company profile as the third relevant criterion. references abdulshahed, a. m., badi, i. a. & blaow, m.m.. 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(2017 b). the selection of wagons for the internal transport of a logistics company: a novel approach based on rough bwm and rough saw methods. symmetry, 9(11), 264. stević, ž., pamučar, d., vasiljević, m., stojić, g., & korica, s. (2017 a). novel integrated multi-criteria model for supplier selection: case study construction company. symmetry, 9(11), 279. zhai, l.y., khoo, l.p., &zhong, z.w. (2008). a rough set enhanced fuzzy approach to quality function deployment. international journal of advanced manufacturing technology, 37 (5–6), 613-624. supplier selection using rough bwm-mairca model: a case study in pharmaceutical… 33 zhai, l.y., khoo, l.p., & zhong, z.w. (2010). towards a qfd-based expert system: a novel extension to fuzzy qfd methodology using rough set theory. expert systems with applications, 37(12), 8888-8896. zhu, g.n., hu, j., qi, j., gu, c.c. & peng, j.h. (2015). an integrated ahp and vikor for design concept evaluation based on rough number, advanced engineering informatics, 29, 408–418. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 5, issue 1, 2022, pp. 67-89. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame181221045d * corresponding author. e-mail addresses: ajoykantidas@gmail.com (a.k. das), carlosgranadosortiz@outlook.es (c. granados), fp-intuitionistic multi fuzzy n-soft set and its induced fp-hesitant n-soft set in group decisionmaking ajoy kanti das1* and carlos granados2 1 department of mathematics, bir bikram memorial college, india 2 estudiante de doctorado en matemáticas, magister en ciencias matemáticas, universidad de antioquia, colombia received: 6 june 2021; accepted: 15 november 2021; available online: 9 february 2022. original scientific paper abstract: intuitionistic fuzzy sets (ifss) can effectively represent and simulate the uncertainty and diversity of judgment information offered by decision-makers (dms). in comparison to fuzzy sets (fss), ifss are highly beneficial for expressing vagueness and uncertainty more accurately. as a result, in this research work, we offer an approach for solving group decision-making problems (gdmps) with fuzzy parameterized intuitionistic multi fuzzy n-soft set (briefly, fpimfnss) of dimension q by introducing its induced fuzzy parameterized hesitant n-soft set (fphnss) as an extension of the multi-fuzzy n-soft set (mfnss) based group decision-making method (gdmm). in this study, we use the proposed gdmm to solve a real-life gdmp involving candidate eligibility for a single vacant position advertised by an it firm and compare the ranking performances of the proposed gdmm with the fatimahalcantud method. key words: decision making, fuzzy set, soft set, n-soft set, intuitionistic fuzzy set. 1. introduction soft set theory (sst) was first presented by molodtsov (1999) as a fundamental and useful mathematical method for dealing with complexity, unclear definitions, and unknown objects (elements). since there are no limitations to the description of elements in sst, researchers may choose the type of parameters that they need, significantly simplifying decision making problems (dmps) and making it easier to make decisions in the absence of partial knowledge, it is more effective. while several mathematical tools for modeling uncertainties are available, such as operations analysis, probability theory, game theory, fs theory, rough set theory, and intervalvalued fuzzy set (ivfs), each of these theories has its own set of problems. furthermore, all of these theories lack parameterization of the tools, which means mailto:ajoykantidas@gmail.com mailto:carlosgranadosortiz@outlook.es das and granados/decis. mak. appl. manag. eng. 5 (1) (2022) 67-89 68 they can't be used to solve problems, especially in the economic, environmental, and social realms. in the sense that it is clear of the aforementioned difficulties, sst stands out. the sst is extremely useful in a variety of situations. molodtsov (1999) developed the basic results of sst and successfully applied it to a variety of fields, including the smoothness of functions, operations analysis, game theory, riemann integration, probability, and so on. later, maji et al. (2003) presented several new sst concepts, such as subset, complements, union, and intersection, as well as their implementations in dmps. ali et al. (2009) identified some more operations on sst and demonstrated that de morgan's laws apply to these new operations in sst. to solve the dmps, maji et al. (2002) used sst for the first time. recently, several authors later looked into the more broad properties and applications of sst. fatimah et al. (2018, 2019) studied the concepts of probabilistic sst and dual probabilistic sst in dmps with positive and negative parameters, and alcantud (2020) introduced soft open bases and presented a new construction of soft topology from bases for topology. many academics are interested in hybrid models, as seen by the aforementioned references. many hybridization options for the recently created n-soft sets (nsss) (fatimah et al. 2018) model. this model's primary role is to broaden the scope of sst applications, which deal with qualities that resemble the universe of discourse. because many real-world examples have insisted on their applicability, this paradigm constitutes a practical expansion of sst (fatimah et al., 2018; alcantud et al., 2020; kamachi & petchimuthu, 2020; kushwaha et al., 2020). in addition, it has demonstrated its theoretical flexibility: the model is adaptable to hybridization with alternative theories of ambiguity and uncertainty. akram et al. (2018, 2019, 2019a, 2019b, 2021), chen et al. (2020), liu et al. (2020), and riaz et al. (2020) have built hybrid structures that incorporate other notable properties of approximation knowledge structures. an n-soft structure (riaz et al., 2019) exists as a natural extension of soft topology and is a natural extension of topological studies (alcantud et al., 2020; terepeta, 2019; youssef and webster, 2022). the idea of the fs was started by zadeh (1965), thereafter, many new approaches and ideas have been offered to deal with imprecision and ambiguity, such as the hesitant fuzzy sets (hfss) (torra, 2010), multi-fuzzy sets (mfss) (sebastian & ramakrishnan, 2011), ifss (atanassov, 1986), intuitionistic multi-fuzzy sets (imfss) (shinoj & john, 2012) and so on (abdulkareem et al., 2020, 2021; azam & bouguila, 2019, 2020; mohammed & abdulkareem, 2020). fs has a wide range of applications, including databases, neural systems, pattern recognition, medicine, fuzzy modelling, economics, and multicriteria dmps (alcantud and torra, 2018; al-qudah & hassan, 2017, 2018). maji et al. (2001) described fuzzy soft set (fss), which is a hybrid of fs and sst. fss based decision-making method (dmm) was first proposed by roy and maji (2007). thereafter, the applications of fsss have been gradually concentrated by using these concepts. feng et al. (2010) introducing an adjustable dmm to solve fssbased dmps. thereafter, zhu and zhan (2015) described and presented the t-norm operations on fuzzy parameterized fsss, as well as shown their applications in dmps. çağman et al. (2010, 2011) introduced the concept of fp-fss and its applications in dmps and later on proposed a new idea of fp-sst and shown some applications in dmps. das and kar (2015) presented the concept of hfss and studied its application in dmps. alcantud and mathew (2017) recently defined separable fss with its applications in dmps for both positive and negative qualities. alcantud et al. (2017) developed a methodology for asset assessment using an fss (flexible fss) based dmm. al-qudah and hassan (2018) presented the theory of complex multi-fss fp-intuitionistic multi fuzzy soft n-soft set & its induced fphnss in group decision making 69 as well as studied its entropy and similarity measure. based on revised aggregation operators, peng and li (2019) suggested a dmm using hfss. lathamaheswari et al. (2020) presented the theory of triangular interval type-2 fss and also, shown its applications. petchimuthu et al. (2020) defined the mean operators and generalized products of fuzzy soft matrices and discussed their applications in mcgdm. paik and mondal, (2020) introduced a distance-similarity method to solve fuzzy sets and fsss based dmps. paik and mondal (2021) had shown the representation and application of fsss in a type-2 environment. močkoř and hurtik (2021) used the concept fsss in image processing applications. gao and wu (2021) defined the notion of filter with its applications in topological spaces formed by fsss. dalkılıç and demirtaş (2021) introduced the idea of bipolar fuzzy soft d-metric spaces. dalkiliç (2021) defined topology on virtual fp-fsss. bhardwaj and sharma (2021) described an advanced uncertainty measure using fsss and shown its application in dmps. atanassov (1986) suggested the notion of ifs as a generalization of fs. maji et al. (2001, 2004) defined intuitionistic-fss (ifss) as an important mathematical method for solving dmps in an uncertain situation by combining sst with ifs. das and kar (2014) proposed a gdmm in medical system using ifss and das et al. (2014) suggested a magdm based on interval-valued intuitionistic fuzzy soft matrix. later on, das et al. (2018) proposed a dmm based on intuitionistic trapezoidal fss and krishankumar et al. (2019) presented a framework for magdm using double hierarchy hesitant fuzzy linguistic term set. also, das et al. (2019) presented the concept of correlation measure of hfsss as well as their applications in dmps. the topic of intertemporal fss selection was first raised by alcantud and muoz torrecillas (2017). the algorithms for ivfsss in stochastic mcdm and neutrosophic soft dmm were introduced by peng and others (2017, 2017a). furthermore, based on codas and wdba with novel information measures, peng and garg (2018) suggested unique algorithms for ivfsss in emergency dmps. in the case of sst, zhan and alcantud (2019) provide an updated assessment of the parameter reduction literature. one or more of the following constraints limited the majority of sst researchers (for example (ma et al., 2017) or other updated hybrid model summaries): the evaluations can either be binary integers between 0 and 1, or real values between 0 and 1, such as fsss or separable fsss (maji et al., 2001). both scenarios are discussed by alcantud and santos-garcía (2017), which includes an examination of partial data. in scenarios such as social assessment systems or the presentation of ranking positions in ordinary life, however, we encounter information with a different framework that is not binary. nsss (fatimah et al., 2018) are, nonetheless, the accurate formal expression of the concept of a parameterized description of the universe of objects based on a finite number of ordered grades and the other extended structures of sst that have been linked to social choice were mentioned by fatimah et al. (2018, 2019). the idea of parameter reduction in nsss was recently presented by akram et al. (2020). when the membership degrees of the alternatives are not uniquely defined, such as due to group knowledge or hesitancy (2011, 2015), hfss (2019) are useful. hesitancy is a model that can be combined with other key structures, for contemporary examples see, fatimah and alcantud (2018), liu & zhang (2017, 2017a), peng et al. (2017, 2018). recently, krishankumar et al. (2021, 2021a) presented a decision framework under probabilistic hesitant fuzzy environment with probability estimation for multi-criteria decision making and introduced the idea of interval-valued probabilistic hesitant fuzzy set-based framework for group decision-making with unknown weight information. fatimah https://link.springer.com/article/10.1007/s00521-020-05160-7#auth-raghunathan-krishankumar das and granados/decis. mak. appl. manag. eng. 5 (1) (2022) 67-89 70 and alcantud (2021) introduced the concept of mfnss and developed a gdmm as a generalization of the successful idea of nss and mfs, for solving gdmps. ifss can effectively represent and simulate the uncertainty and diversity of judgment information offered by dms. in comparison to fss, ifss are highly beneficial for expressing vagueness and uncertainty more accurately. as a result, in this paper, we offer an approach for solving gdmps with fpimfnss, by extending the mfnss based gdmm. the new structure combines the advantages of imfs with those of fpsoft sets and nsss, three structures that have received a lot of attention in current years. the constructed method in this paper is very advantageous for solving gdmps. in this study, we use the proposed gdmm to solve a real-life gdmp involving candidate eligibility for a single vacant position advertised by an it firm and compare the ranking performances of the proposed gdmm with the fatimah-alcantud method. the following is the structure of this paper: the essential concepts and conclusions of fs, ifs, soft set, imfs, nss, mfnss, and ifnss are presented in sec. 3, which will be important in later discussions. in sec. 4, we define fpimfnss and its induced fphnss as an extension of the mfnss, along with some fundamental features. in sec. 5, we present an advanced and adjustable gdmm for solving gdmps based on fpimfnsss. in sec. 6, we show the validity of our proposed gdmm with the help of one real-life example, and in sec. 7, we address comparison analysis with the fatimah-alcantud method. finally, in sec. 8, we bring the paper to a conclusion and our future work. 2. literature review torra (2010) was the first to propose the idea of hfss. torra and narukawa (2009) described certain new operations on hfss and used them in dmps. xia and xu (2011) proposed hesitant fuzzy information aggregation in dmps. zhu et al. (2014) suggested a technique for deriving a ranking from hesitant fuzzy preference relations under group dmps. das and kar (2014) proposed a gdmm in medical system using ifss and das et al. (2014) suggested a magdm based on interval-valued intuitionistic fuzzy soft matrix. liu and zhang (2017, 2017a) proposed an mcdm technique with neutrosophic hesitant fuzzy heronian mean aggregation operators and also, developed an extended mcdm technique with the help of neutrosophic hesitant fuzzy information. peng and dai (2017) proposed hesitant fuzzy soft dmms based on copras, mabac, and waspas with combined weights. fatimah and alcantud (2018) initiated the idea of expanded dual hfss and peng and li (2019) proposed a hesitant fuzzy soft dmm with the help of revised aggregation operators, codas and wdba. sebastian and ramakrishnan (2011) defined mfs as an extension of fs. thereafter, shinoj and john (2012) developed the concept of if-multisets and applied it in medical diagnosis. yang et al. (2013) developed the theory of mfss and its application in dmps. dey and pal (2015) developed the concept of generalized mfss and its application in dmps. fatimah et al. (2018) first introduce the idea of nsss and based on nsss they proposed dmms. later on, akram et al. (2018) initiated the theory of a novel model of fuzzy nsss with its applications in dmps. riaz et al. (2019) introduced the theories of n-soft topologies and shown their applications in mcgdm. akram et al. (2019) developed some group dmms based on hnsss. later on, akram et al. (2019) presented a novel structure of hesitant fuzzy nsss with its applications in dmps. akram and adeel (2019) proposed the topsis approach to mcgdm with the help of an interval-valued hesitant fuzzy n-soft environment and akram et al. (2019) developed a new hybrid dmm with if-n-soft rough sets. das et al. (2018) proposed a fp-intuitionistic multi fuzzy soft n-soft set & its induced fphnss in group decision making 71 dmm based on intuitionistic trapezoidal fuzzy soft set and krishankumar et al. (2019) presented a framework for magdm using double hierarchy hesitant fuzzy linguistic term set. also, das et al. (2019) presented the concept of correlation measure of hesitant fuzzy soft sets as well as their applications in dmps. recently, riaz et al. (2020) defined neutrosophic nss and applied it with the topsis method for mcdm. kamacı and petchimuthu (2020) introduced bipolar nss theory with its applications and liu et al. (2020) proposed an mcdm method based on neutrosophic vague nsss. chen et al. (2020) presented a group dmm based on generalized vague nsss. akram et al. (2020) presented the parameter reductions in nsss and shown their applications in dmps and alcantud et al. (2020) presented an nss approach using rough set. recently, krishankumar et al. (2021, 2021a) presented a decision framework under probabilistic hesitant fuzzy environment with probability estimation for multi-criteria decision making and introduced the idea of intervalvalued probabilistic hesitant fuzzy set-based framework for group decision-making with unknown weight information. fatimah and alcantud (2021) defined the theory of mfnss and its applications to dmps. it is clear from the continuing literature analysis that previous researchers built tools for various real-world scenarios using nss, fp-sst, or mfss. previous research studies have not combined nss, fp-sst, and mfss in an if setting. ifss can effectively represent and simulate the uncertainty and diversity of judgment information offered by dms. in comparison to fuzzy sets, ifss are highly beneficial for expressing vagueness and uncertainty more accurately. thus, in this paper, we offer an approach for solving gdmps with fpimfnss by introducing its induced fphnss as an extension of the mfnss based gdmm. mfnss is a fantastic and useful tool to deal with gdmps, but it has some limitations. as a result of combining the three models, this study effort focuses on gdmps in the real world. consequently, the current gap in the real-time practical implementation of combined nss, fp-sst, and mfss in an if setting as a fpimfnss would be filled by this integrated nss, fp-sst, and mfss model. this proposed model would provide the most up-to-date viewpoint and motion for dealing with real-world gdmps. 3. preliminary let us consider ω represents the starting universe and q represents a nonempty set of parameters. let the power set of ω is denoted by p(ω) and pq. let,  , 2,3, 4,5,....q n  and  0,1, 2,3, 4,5,...., 1 .r n  definition 3.1 (zadeh, 1965). an fs z on ω is a set with a structure    o, μ o : ,zz o  where the real-valued function μ : [0, 1] z   is said to be the membership function and  μ oz is called the degree of membership for each object .o  assume that, in this research paper fs(ω) means the collection of all fss on ω. definition 3.2 (atanassov, 1986). an ifs z on ω is a set with a structure z z z={ o, ( o ) , ( o ) : o }   , where the real-valued functions z  :ω[ 0,1] and z  : ω[0, 1] means the membership function and the non-membership function respectively, and    μ o , oz z are called the degree of membership and the degree https://link.springer.com/article/10.1007/s00521-020-05160-7#auth-raghunathan-krishankumar das and granados/decis. mak. appl. manag. eng. 5 (1) (2022) 67-89 72 of non-membership for each object oω, satisfying the condition z z 0 ( o )+ ( o ) 1    for each object oω. assume that, in this research paper ifs(ω) means the collection of all ifss on ω. definition 3.3 (molodtsov, 1999). a soft set over the nonempty universe ω is a pair (, p), where  is a mapping defined by : pp (ω). definition 3.4 (shinoj and john, 2012). an imfs z on ω is a set with a structure  1 1 2 2, ( ( ), ( )), ( ( ), ( )),...., ( ( ), ( )) : ,q qz o o o o o o o o        where the real-valued functions k : [0,1], : [0,1] k       satisfying the condition k 0 ( ) (o) 1 k o    for k =1, 2, 3,...,q and for each oω. in this research paper, ( ) q imfs  means the collection of all imfss on ω. definition 3.5 (fatimah et al., 2018). a triple (ψ, p, n) is said to be an nss on ω, where : p 2 r   is a function, satisfying the condition, for each p ∈ p and o ∈ ω there exists a unique couple ( , ) p o r r such that ( , ) ( ), p o r p p r r . the object o belongs to the collection of p-approximations of the universal set ω with the grade p r , according to the interpretation of the couple ( , ) ( ) p o r p . definition 3.6 (akram et al., 2019). a triple (, p, n) is said to be a hesitant n-soft set (simply, hnss) over ω, where : p 2 r   is a function such that for every p ∈ p and o∈ω there exists at least one couple ( , ) p o r r such that ( , ) ( ), p p o r p r r  . definition 3.7 (akram et al. 2019). the collection h satisfying the condition  0,1, 2,3, 4,5,...., 1h r n     is said to be hesitant n-tuple (simply, hnt). any hnss has a tabular representation consisting of a matrix whose cells are hnts. definition 3.8 (fatimah and alcantud, 2021). let ( , ) ( ) n q mfs  be the set of all qtuples of triples of objects from [0,1]r indexed by ω, i.e., the collection of all objects having the structure  1 1 2 2, ( ( ), ( )), ( ( ), ( )),...., ( ( ), ( )) : ,q qo r o o r o o r o o o    where k : , and : [0,1] k r r     are mappings. an mfnss on ω is a pair (ψ, p), such that ψ is a mapping ( , ) : ( ) n q p mfs   defined by    1 1 2 2, ( ( ), ( )), ( ( ), ( )),...., ( ( ), ( )) : (1)q qp o r o o r o o r o o o      definition 3.9 (fatimah and alcantud, 2021). let  , p be an mfnss on ω, where  1 2, ,..., mp p p p q  and  1 2, ,..., m     , a vector of thresholds [0,1]i  associated with each i p p for i=1,2,..m. then the hnss α-induced by  , p is the triple  , ,h p n where : ( )h p p r    is a mapping, such that   ( ) , ( ) : ( ) , 1, 2,..., : (2)j jj i t i t i j ih p o r o o t q o       definition 3.10 (maji et al., 2001). a pair (, p) is called an ifss over ω, where  is a function given by : p ifs(ω). definition 3.11 (akram et al., 2019) a t r i p l e ( φ , k , n ) i s k n o w n a s i f n s s , i f k = (ψ, p, n) is an nss over ω and : ( )p i f s r    is a mapping, where ( )ifs r means the collection of all ifss on .r fp-intuitionistic multi fuzzy soft n-soft set & its induced fphnss in group decision making 73 example 3.12. let us consider that four candidates, denoted by  1 2 3 4, , ,o o o o  , applied for a single vacant position advertised by an it firm. then, the selection board of the firm has firstly determined a parameter set  1 2 3, ,p p p p , such that p1 = experience, p2 = knowledge of foreign language, and p3 = knowledge of software, which are used to assign grades to candidates. we can calculate a 5-soft set from table 1, where 4 stars mean excellent 3 stars mean very good 2 stars mean good 1 star means regular circle means bad. the set of grades g = {0, 1, 2, 3, 4} can be easily associated with checkmarks as follows: 0 stands for ‘o’ 1 stands for * 2 stands for ** 3 stands for *** 4 stands for ****. then from definition 3.5, the tabular form of 5-soft set k=(ψ, p, 5) can be shown in table 2 and from definition 3.11 the if5ss ( φ , k , 5 ) i s g i v e n b y t a b l e 3 . table 1. information extracted from the related data ω p1 p2 p3 o1 o2 o3 o4 *** * **** ** **** ** *** * * **** ** **** table 2. the 5-soft set k=(ψ, p, 5) ω p1 p2 p3 o1 o2 o3 o4 3 1 4 2 4 2 3 1 1 4 2 3 table 3. the if5ss ( φ , k , 5 ) ω p1 p2 p3 o1 o2 o3 o4 (3,0.3,0.5) (1,0.3,0.4) (4,0.4,0.5) (2,0.6,0.3) (4,0.4,0.5) (2,0.3,0.4) (3,0.5,0.3) (1,0.4,0.5) (1,0.5,0.4) (4,0.5,0.3) (2,0.5,0.4) (3,0.3,0.5) das and granados/decis. mak. appl. manag. eng. 5 (1) (2022) 67-89 74 4. theoretical analysis of fpimfnss in this research paper, we consider ω represents the starting universe and q represents a nonempty set of parameters. let  1 2, ,..., mp p p p q  and  ( ) : ,x px p p p  be an fs over p. let  , 2,3, 4,5,....q n  be two fixed numbers, where q is the dimension of our new structure and n distinguishes how many degrees of satisfaction with the parameters are permitted, allowing us to utilize  0,1, 2,3, 4,5,...., 1r n  as a collection of ordered grades and p(r) means the power set of r. definition 4.1 let us define ( , ) ( ) n q imfs  as the set of all q-tuples of triples of objects from [0,1] [0,1]r  indexed by ω, i.e., the collection of all objects having the structure  1 1 1 2 2 2, ( ( ), ( ), ( )), ( ( ), ( ), ( )),...., ( ( ), ( ), ( )) : ,q q qo r o o o r o o o r o o o o       where k : , : [0,1], : [0,1] k k r r        and satisfying k 0 ( ) (o) 1 k o    for k = 1, 2, 3,....., q. a fpimfnss of dimension q on ω is a pair (ψ, x) such that ψ is a mapping ( , ) : ( ) n q x imfs   defined by ( ) ,x p p x       ( ) 1 1 1, ( ( ), ( ), ( )),...., ( ( ), ( ), ( )) : (3)x p q q q p o r o o o r o o o o         note: simply, we denote the set of all fpimfnsss of dimension q over ω by ( , ) ( , ) n q d p where the parameter set p is fixed. remark 4.2 if n=1, the members in ( , ) ( , ) n q d p can be matched to those in ( ) q imfs  and the element of (1, ) ( , ) q d p is of the form  1 1 2 2, (0, ( ), ( )), (0, ( ), ( )),...., (0, ( ), ( )) : .q qo o o o o o o o       we identify it with  1 1 2 2, ( ( ), ( )), ( ( ), ( )),...., ( ( ), ( )) : ( )qq qo o o o o o o o imfs         in a trivial manner. example 4.3 let  1 2 3, ,o o o  be the universe of candidates and  1 2,p p p is the set of attributes and  0.5 0.71 2,x p p be an fs over p. a fpimf5ss of dimension 2 (ψ, x) on ω is defined by the assignments     0.5 1 1 2 3 0.7 2 1 2 3 ( ) , (3, 0.3, 0.5), (4, 0.5, 0.3) , , (2, 0.3, 0.4), (4, 0.5, 0.3) , , (2, 0.4, 0.5), (3, 0.4, 0.4) , ( ) , (1, 0.4, 0.5), (3, 0.5, 0.4) , , (2, 0.3, 0.4), (4, 0.5, 0.3) , , (3, 0.5, 0.3), (2, 0.5, 0.4) . p o o o p o o o     the tabular representation of the above fpimf5ss of dimension 2 (ψ, x) can be shown in table 4. table 4. the fpimf5ss of dimension 2  , x ω p1 0.5 p2 0.7 o1 o2 o3 (3,0.3,0.5)(4,0.5,0.3) (2,0.3,0.4)(4,0.5,0.3) (2,0.4,0.5)(3,0.4,0.4) (1,0.4,0.5)(3,0.5,0.4) (2,0.3,0.4)(4,0.5,0.3) (3,0.5,0.3)(2,0.5,0.4) fp-intuitionistic multi fuzzy soft n-soft set & its induced fphnss in group decision making 75 definition 4.4 let us consider two fpimfnsss ( , ) ( , ), ( , ) ( , ) n q x y d p    , such that    ( ) ( ) 1 1 1, , ( ( ), ( ), ( )),..., ( ( ), ( ), ( )) : ,x x p p q q q p x p o r o o o r o o o o               ( ) ( ) 1 1 1, , ( ( ), ( ), ( )),..., ( ( ), ( ), ( )) : .y y p p q q q p y p o r o o o r o o o o                 then we say that [1] subset: ( , ) ( , )x y   if ( ). is a fuzzy subset of , . . , ( ) ( ) x y i x y i e p p p p        ( ) ( )( ). , ( ) ( ), ( ) ( ) and ( ) ( ) , and 1, 2,..., x yp p i i i i i i ii p p p p r o r o o o o o o i q                     [2] equal set: ( , ) ( , )x y   if ( ). , ( ) ( ) x y i p p p p        ( ) ( )( ). , ( ) ( ), ( ) ( ) and ( ) ( ) , and 1, 2,..., x yp p i i i i i i ii p p p p r o r o o o o o o i q                     [3] union: ( , ) ( , ) ( , )x y z     , where ,z x y  and  denotes the fuzzy union and ( ) ,z p p z       ( ) 1 1 1, ( ( ), ( ), ( )),..., ( ( ), ( ), ( )) : , where ,z p q q q p o r o o o r o o o o o                1 ( ) max{ ( ), ( )}, ( ) max{ ( ), ( )} and ( ) min{ ( ), ( )}, 1, 2,..., . i i i i i i i i r o r o r o o o o o o o i q               [4] intersection: ( , ) ( , ) ( , )x y z     , where ,z x y  and  denotes the fuzzy intersection and ( ) ,z p p z       ( ) 1 1 1, ( ( ), ( ), ( )),..., ( ( ), ( ), ( )) : , where ,z p q q q p o r o o o r o o o o o                1 ( ) min{ ( ), ( )}, ( ) min{ ( ), ( )} and ( ) max{ ( ), ( )}, 1, 2,..., . i i i i i i i i r o r o r o o o o o o o i q               definition 4.5 we consider a fpimfnss   ( , ), ( , )n qx d p   . its induced fphnss of dimension q is the pair  ,h x , where : ( )h x p r  is a mapping, such that   ( ) 1 2( ) , ( ), ( ),...., ( ) : (4)x p j j jj i i i q i ih p o r o r o r o o   example 4.6 let  1 2 3 4, , ,o o o o  be the universe of candidates and  1 2 3, ,p p p p is the collection of parameters. we consider the fpimf5ss of dimension 3  , x on ω as shown in the table 5. then we have the induced fph5ss of dimension 3  ,h x as in table 6. das and granados/decis. mak. appl. manag. eng. 5 (1) (2022) 67-89 76 table 5. the fpimf5ss of dimension 3  , x ω p1 0.5 p2 0.6 p3 0.7 o1 o2 o3 o4 (3,0.3,0.5) (4,0.5,0.3) (2,0.6,0.3) (2,0.3,0.4) (4,0.5,0.3) (3,0.5,0.3) (2,0.4,0.5) (3,0.4,0.4) (1,0.5,0.4) (1,0.5,0.4) (4,0.6,0.4) (3,0.4,0.5) (1,0.4,0.5) (3,0.5,0.4) (2,0.5,0.3) (2,0.3,0.4) (4,0.5,0.3) (3,0.5,0.3) (3,0.5,0.3) (2,0.5,0.4) (4,0.5,0.4) (1,0.4,0.4) (2,0.6,0.3) (3,0.4,0.3) (2,0.5,0.5) (3,0.3,0.4) (2,0.6,0.3) (3,0.5,0.4) (4,0.4,0.3) (2,0.6,0.3) (1,0.5,0.3) (2,0.3,0.5) (3,0.6,0.3) (4,0.3,0.4) (2,0.6,0.3) (3,0.4,0.3) table 6. the fph5ss of dimension 3  ,h x ω p1 0.5 p2 0.6 p3 0.7 o1 o2 o3 o4 {3, 4, 2} {2, 4, 3} {2, 3, 1} {1, 4, 3} {1, 3, 2} {2, 4, 3} {3, 2, 4} {1, 2, 3} {2, 3, 2} {3, 4, 2} {1, 2, 3} {4, 2, 3} definition 4.7 let us fix   ( , ), ( , ) n qx d p   where  1 2, ,..., mp p p p q  and    1 1 2 2, ( , ), ( , ),..., ( , )m m         , a vector of thresholds , [0,1]i i   associated with each i p p for i=1,2,..m. then the (,)-fphnss induced by  , x is the triple  , , ( , )h x   where : ( )h x p r   is a mapping, such that   ( )( ) , ( ) : ( ) and ( ) , 1, 2,..., : .x p j j jj i t i t i j t i j ih p o r o o o t q o          example 4.8 let  1 2 3 4, , ,o o o o  be the universe of candidates and  1 2 3, ,p p p p is the set of attributes. we consider the fpimf5ss of dimension 3  , x on ω whose tabular information is displayed in table 5 and let (,) = {(0.5, 0.4), (0.5, 0.3), (0.6, 0.4)} be a fixed threshold. then, we obtain (,)-fph5ss whose tabular representation is in table 7. fp-intuitionistic multi fuzzy soft n-soft set & its induced fphnss in group decision making 77 table 7. the (,)-fph5ss  , , ( , )h x   ω p1 0.5 p2 0.6 p3 0.7 o1 o2 o3 o4 {4, 2} {4, 3} {1} {1, 4} {2} {4, 3} {3} {2} {2} {2} {3} {2} 5. gdmm based on fpimfnss now, we present our machine learning algorithm for solving gdmps based on fpimfnss. the steps of our proposed gdmm listed below: algorithm 1 step1: enter a nonempty universe  1 2, ,..., ,no o o  a set of parameters  1 2, ,..., mp p p p , an fs   ( ) :x p x p p p    over p, and a group of dms 1 2 q {m , m ,..., m } . step2: enter the dms observations (ifnsss) 1 2 ( , ), ( , ),..., ( , ), q p p p   as provided by each dm. step3: compute the resultant fpimfnss  , x of dimension q from the ifnsss 1 2 ( , ), ( , ),..., and ( , ) q p p p   step4: enter a threshold (,) =   , [0,1] [0,1], 1, 2,...,j j j m     , where  ,j j  associated with each attribute .jp p step5: obtain the (,)-fphnss  , , ( , )h x   in its tabular form. step6: obtain the scores  ( )j ih o of all the hnts ( )j ih o in  , ,h p n   by taking any operation (say, arithmetic mean, geometric mean, etc.), , and j=1,2,3,....,m i o  step7: compute   1 ( ) ( ) , m i x j j i i j u p h o o      step8: the best optimal choice is to select os if su is maximized. step9: if os has many values, any of os may be selected. remark 5.1 in the 8th-step of our constructed gdmm, one can return to the 4th or 6th steps and change the threshold (α,) or operation respectively that he previously used to adjust the final optimal choice, particularly when there are lots of optimal choices to choose from. das and granados/decis. mak. appl. manag. eng. 5 (1) (2022) 67-89 78 6. result and discussions in this part, we use the proposed gdmm to solve a real-life gdmp involving candidate eligibility for a single vacant position in a job posting. let us consider that four candidates, denoted by  1 2 3 4, , ,o o o o  , applied for a single vacant position advertised by an it firm. the selection of a candidate in this firm is based on star ratings and gradings given by a selection board comprised of three experts: a director, a subject specialist, and a chairman. then, the selection board of the firm has firstly determined a parameter set  1 2 3, ,p p p p , such that p1 = experience, p2 = knowledge of foreign language, and p3 = knowledge of software, which are used to assign grades to candidates. suppose, the three experts observations (if5sss) are in tables 8, 9, and 10 respectively and let  0.5 0.6 0.71 2 3, ,x p p p be an fs over p. then the results of combining the three experts observations, we have the resultant fpimf5ss (ψ, x) of dimension 3 as shown in table 11. let us consider a threshold (, ) = {(0.5, 0.4), (0.5, 0.3), (0.6, 0.3)} associated with the parameter set p. then, we obtain (,)fph5ss as shown in table 12. we use the arithmetic score on hnts. table 13 shows the results of the computations at steps 5 and 6. step 7 suggests that the candidate o4 is the best candidate for the vacant post in the it firm. table 8. director’s observation ω p1 p2 p3 o1 o2 o3 o4 (3,0.4,0.5) (3,0.4,0.4) (2,0.4,0.5) (2,0.5,0.4) (3,0.4,0.5) (3,0.5,0.4) (3,0.5,0.3) (4,0.6,0.2) (4,0.5,0.5) (3,0.6,0.4) (4,0.5,0.3) (4,0.5,0.4) table 9. subject specialist’s observation ω p1 p2 p3 o1 o2 o3 o4 (4,0.6,0.3) (4,0.5,0.3) (4,0.4,0.4) (4,0.6,0.4) (4,0.5,0.4) (4,0.6,0.3) (4,0.5,0.3) (2,0.6,0.3) (3,0.5,0.4) (4,0.5,0.3) (2,0.4,0.5) (2,0.6,0.2) table 10. chairman’s observation ω p1 p2 p3 o1 o2 o3 o4 (2,0.6,0.3) (3,0.6,0.3) (1,0.5,0.4) (3,0.5,0.4) (2,0.6,0.3) (3,0.5,0.3) (4,0.6,0.4) (3,0.6,0.3) (2,0.6,0.3) (2,0.6,0.3) (3,0.6,0.3) (3,0.4,0.3) fp-intuitionistic multi fuzzy soft n-soft set & its induced fphnss in group decision making 79 table 11. the fpimf5ss of dimension 3  , x ω p1 0.5 p2 0.6 p3 0.7 o1 o2 o3 o4 (3,0.4,0.5) (4,0.6,0.3) (2,0.6,0.3) (3,0.4,0.4) (4,0.5,0.3) (3,0.6,0.3) (2,0.4,0.5) (4,0.4,0.4) (1,0.5,0.4) (2,0.5,0.4) (4,0.6,0.4) (3,0.5,0.4) (3,0.4,0.5) (4,0.5,0.4) (2,0.6,0.3) (3,0.5,0.4) (4,0.6,0.3) (3,0.5,0.3) (3,0.5,0.3) (4,0.5,0.3) (4,0.6,0.4) (4,0.6,0.2) (2,0.6,0.3) (3,0.6,0.3) (4,0.5,0.5) (3,0.5,0.4) (2,0.6,0.3) (3,0.6,0.4) (4,0.5,0.3) (2,0.6,0.3) (4,0.5,0.3) (2,0.4,0.5) (3,0.6,0.3) (4,0.5,0.4) (2,0.6,0.2) (3,0.4,0.3) table 12. the (,)-fph5ss  , , ( , )h x   ω p1 0.5 p2 0.6 p3 0.7 o1 o2 o3 o4 {4, 2} {4, 3} {1} {2, 4, 3} {2} {4, 3} {3, 4} {4, 2, 3} {2} {2} {3} {2} table 13. the scores  ( )j ih o with ui ω p1 0.5 p2 0.6 p3 0.7 ui o1 o2 o3 o4 3 3.5 1 4.5 2 3.5 3.5 4.5 2 2 3 2 4.1 5.25 4.7 6.35 7. comparison results in this present sec., we first present the fatimah-alcantud method (fatimah and alcantud, 2021) for solving mfnss based dmps. fatimah and alcantud (2021) proposed the mfnss based approach as follows: algorithm 2 (fatimah and alcantud, 2021): step1: enter a nonempty universe  1 2, ,..., ,no o o  a set of parameters  1 2, ,..., mp p p p , an mfnss  , p of dimension q, a vector 1 2 ( , ,..., ) m     das and granados/decis. mak. appl. manag. eng. 5 (1) (2022) 67-89 80 of threshold, and a weight 1 2 ( , ,..., ) m w w w w , where , [0,1] j j w  associated with each attribute j p p step2: obtain the hnss  , ,h p n in its tabular form. step3: obtain the scores  ( )j ih o of all the hnts ( )j ih o in  , ,h p n   by taking any operation (say, arithmetic mean, geometric mean), , and j=1,2,3,....,m i o  step4: compute   1 ( ) , m i j j i i j u w h o o      step5. the best optimal choice is to select os if su is maximized. step6. if os has many values, any of os may be selected. in this sense, it is impossible to compare the proposed gdmm in sec. 5 to the fatimah-alcantud method because it is the first method suggested in connection to fpimfnss in an intuitionistic fuzzy environment. however, if the simulated problem's uncertainties are reduced, the approach can be compared to the fatimahalcantud method in a substructure mfnss. as a result, we reduce the fpimfnss to mfnss by considering only the membership-values of the objects in the fpimfnss and eliminating their non-membership values. let us consider that four candidates, denoted by  1 2 3 4, , ,o o o o  , applied for a single vacant position advertised by an it firm. the selection of a candidate in this firm is based on star ratings and gradings given by a selection board comprised of three experts: a director, a subject specialist, and a chairman. then, the selection board of the firm has firstly determined a parameter set  1 2 3, ,p p p p , such that p1 = experience, p2 = knowledge of foreign language, and p3 = knowledge of software. suppose, the three experts observations (if5sss) are in tables 14, 15, and 16 respectively and let  0.5 0.6 0.71 2 3, ,x p p p be an fs over p. then the results of combining the three experts observations, we have the resultant fpimf5ss (ψ, x) of dimension 3 as shown in table 17. if we consider the fpimf5ss  , x as shown in table 17, then we may get its reduced fmnss  , p as in shown table 18. we consider weight 1 2 3 ( ( ) 0.5, ( ) 0.6, ( ) 0.7),w w p w p w p    same as our fs  0.5 0.6 0.71 2 3, ,x p p p over p. let (,) = {(0.5, 0.2), (0.5, 0.3), (0.6, 0.3)} be a fixed threshold for our proposed gdmm and  = {0.5, 0.5, 0.6} be a fixed threshold for fatimah-alcantud method. table 14. director’s observation ω p1 p2 p3 o1 o2 o3 o4 (3,0.4,0.5) (3,0.5,0.4) (2,0.5,0.5) (2,0.5,0.4) (3,0.4,0.5) (2,0.5,0.4) (3,0.5,0.3) (4,0.6,0.2) (4,0.5,0.5) (3,0.5,0.3) (4,0.5,0.3) (4,0.5,0.4) fp-intuitionistic multi fuzzy soft n-soft set & its induced fphnss in group decision making 81 table 15. subject specialist’s observation ω p1 p2 p3 o1 o2 o3 o4 (4,0.6,0.2) (4,0.5,0.3) (4,0.4,0.4) (4,0.6,0.4) (4,0.5,0.4) (4,0.6,0.3) (2,0.5,0.3) (2,0.6,0.3) (3,0.5,0.4) (4,0.5,0.3) (2,0.4,0.5) (2,0.6,0.2) table 16. chairman’s observation ω p1 p2 p3 o1 o2 o3 o4 (2,0.6,0.3) (2,0.6,0.2) (1,0.5,0.2) (3,0.5,0.2) (2,0.6,0.3) (3,0.5,0.3) (1,0.6,0.4) (3,0.6,0.3) (2,0.6,0.3) (2,0.6,0.3) (3,0.6,0.3) (3,0.4,0.3) table 17. the fpimf5ss of dimension 3  , x ω p1 0.5 p2 0.6 p3 0.7 o1 o2 o3 o4 (3,0.4,0.5) (4,0.6,0.2) (2,0.6,0.3) (3,0.5,0.4) (4,0.5,0.3) (2,0.6,0.2) (2,0.5,0.5) (4,0.4,0.4) (1,0.5,0.2) (2,0.5,0.4) (4,0.6,0.4) (3,0.5,0.2) (3,0.4,0.5) (4,0.5,0.4) (2,0.6,0.3) (2,0.5,0.4) (4,0.6,0.3) (3,0.5,0.3) (3,0.5,0.3) (2,0.5,0.3) (1,0.6,0.4) (4,0.6,0.2) (2,0.6,0.3) (3,0.6,0.3) (4,0.5,0.5) (3,0.5,0.4) (2,0.6,0.3) (3,0.5,0.3) (4,0.5,0.3) (2,0.6,0.3) (4,0.5,0.3) (2,0.4,0.5) (3,0.6,0.3) (4,0.5,0.4) (2,0.6,0.2) (3,0.4,0.3) table 18. the reduced mfnss  , p of the fpimf5ss  , x ω s1 s2 s3 o1 o2 o3 o4 (3,0.4)(4,0.6)(2,0.6) (3,0.5)(4,0.5)(2,0.6) (2,0.5)(4,0.4)(1,0.5) (2,0.5)(4,0.6)(3,0.5) (3,0.4)(4,0.5)(2,0.6) (2,0.5)(4,0.6)(3,0.5) (3,0.5)(2,0.5)(1,0.6) (4,0.6)(2,0.6)(3,0.6) (4,0.5)(3,0.5)(2,0.6) (3,0.5)(4,0.5)(2,0.6) (4,0.5)(2,0.4)(3,0.6) (4,0.5)(2,0.6)(3,0.4) now, we apply the suggested gdmm as well as the fatimah-alcantud method to the fpimf5ss  , x and its reduced mfnss  , p , as shown in tables 17 and 18, respectively. the decision sets and the ranking orders of the approaches within their das and granados/decis. mak. appl. manag. eng. 5 (1) (2022) 67-89 82 own structures are given in table 19. table 19 shows that according to the fatimahalcantud method three candidates o1, o2, and o4 are eligible for a single vacant position, but according to our suggested gdmm, only one candidate o1 is eligible. in this example, fatimah-alcantud method is unable to determine the best candidate for a single vacant position, whereas we are able to do so. as a result, the proposed strategy has been successfully applied to a problem with additional uncertainty. table 19. the decision sets and ranking orders of the proposed gdmm and fatimahalcantud method dmms (algorithms) decision sets ranking algorithm-1 (proposed gdmm) algorithm-2 (fatimah and alcantud, 2021)  1 2 3 4( , 5.2), ( , 4.5), ( , 4.1), ( , 4.7)o o o o  1 2 3 4( , 4.7), ( , 4.7), ( , 4.05), ( , 4.7)o o o o 1 4 2 3 o o o o 1 2 4 3 o o o o  8. conclusions ifss can effectively represent and simulate the uncertainty and diversity of judgment information offered by dms. in comparison to fss, ifss are highly beneficial for expressing vagueness and uncertainty more accurately. as a result, in this research work, we offer an approach for solving gdmps with fpimfnsss by extending the mfnss (fatimah and alcantud, 2021) based gdmm. in this study, we use the proposed gdmm to solve a real-life gdmp involving candidate eligibility for a single vacant position advertised by an it firm. we also compare the ranking performances of the proposed gdmm with the fatimah-alcantud method and we have shown that the fatimah-alcantud method is unable to determine the best candidate for a single vacant position, whereas we are able to do so. we hope that this proposed model would provide the most up-to-date viewpoint and motion for dealing with real-world gdmps. in a future study, we will extend this proposed gdmm to other real-life applications in the field of pattern recognition and medical diagnostics. fp-intuitionistic multi fuzzy soft n-soft set & its induced fphnss in group decision making 83 abbreviations: dm decision maker dmm decision making method dmp decision making problem fs fuzzy set fss fuzzy soft set gdmm group decision-making method gdmp group decision-making problem hfs hesitant fuzzy set hfss hesitant fuzzy soft set hnfss hesitant n-fuzzy soft set hnt hesitant n tuples if intuitionistic fuzzy ifs intuitionistic fuzzy set ifss intuitionistic fuzzy soft set ifnss intuitionistic fuzzy n-soft set ivfss interval valued fuzzy soft set imfs intuitionistic multi fuzzy set mcdm multi criteria decision making mcgdm multi criteria group decision making mfnss multi-fuzzy nsoft set mfs multi fuzzy set mfss multi fuzzy soft set nss n-soft set sst soft set theory author contributions: research problem, a.k.d. and c.g.; methodology, a.k.d. and c.g.; formal analysis, a.k.d. and c.g.; resources, a.k.d.; writing – original draft preparation, a.k.d. and c.g.; writing – review & editing, a.k.d. and c.g acknowledgement: the authors would like to express their gratitude to the editors and anonymous referees for their informative, helpful remarks and suggestions to improve this paper as well as the 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(2016). fuzzy parameterized fuzzy soft sets and decision making, international journal of machine learning and cybernetics, 7, 1207–1212. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 3, issue 2, 2020, pp. 162-189. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame2003162b * corresponding author. e-mail addresses: sanjibb@acm.org (s. biswas) measuring performance of healthcare supply chains in india: a comparative analysis of multicriteria decision making methods sanjib biswas 1* 1 calcutta business school, west bengal, india received: 2 august 2020; accepted: 1 october 2020; available online: 11 october 2020. original scientific paper abstract: the supply chain forms the backbone of any organization. however, the effectiveness and efficiency of every activity get manifested in the financial outcome. hence, measuring supply chain performance using financial metrics carries significance. the purpose of this paper is to carry out a comparative analysis of supply chain performances of leading healthcare organizations in india. in this regard, this paper presents an integrated multi-criteria decision making (mcdm) framework wherein we derive the weights of the criteria based on experts’ opinions using pivot pairwise relative criteria importance assessment (piprecia) method. we then apply three distinct frameworks such as multi-attributive border approximation area comparison (mabac), combined compromise solution (cocoso) and measurement of alternatives and ranking according to compromise solution (marcos) for ranking purpose. in this context, this paper presents a comparative analysis of the results obtained from these approaches. the results show that large cap firms do not necessarily perform well. further, the results of three mcdm frameworks demonstrates consistency. key words: healthcare supply chain, financial metrics, piprecia, mabac, cocoso, marcos. 1. introduction with rapid development in information technology and communication technology (ict), consumers' nature and requirements have changed to a great extent in order to win the battle at the market place and, more specifically, to survive. organizations are increasingly putting primary emphasis on strengthening the supply chains. the performance of the supply chain stands as a critical deciding factor for ensuring business sustainability. hence, supply chain management (scm) encompasses all related decisions to strike a balance between demand and supply, linked with the financial outcome (huang et al., 2008). in other words, scm addresses the issue of mailto:sanjibb@acm.org measuring performance of healthcare supply chains in india: a comparative analysis... 163 economic sustainability (al-hussaini, 2019). supply chain decision-makers must adequately consider two interdependent objectives such as reduction of cost and improvement of service levels for contributing to the overall profitability of the organization (parasuraman et al., 1991; mentzer et al., 1999; ray et al., 2004; johnson and templar, 2011). the basic premise of the supply chain concept is built on horizontal integration and development, shifting away from the functional brilliance (lester, 1999). hence, all the activities across the supply chain must be performed in sync and directed towards attaining the overall business objectives of meeting the needs and requirements of the customers and fulfilling the stakeholders' expectations. in effect, supremacy in supply chain performance contributes in achieving overall organizational excellence (ellram et al., 2002; d’avanzo et al., 2004; christopher, 2005) which is beyond the local scope of cost optimization (lambert and cooper, 2000; ellram and liu, 2002; farris and hutchison, 2002). supply chain practitioners need to connect operational efficiency with financial investment outcomes (lalonde, 2000). though the explicit linkage of supply chain performance with financial performance is quite complex to realize (frohlich and westbrook, 2001), christopher (1998) mentioned three dimensions of financial performance: profitability, liquidity, and productivity or asset utilization in which the contributions of supply chain performance can be gauged. therefore, it is understood that there is a need to bridge the gap between the supply chain operational framework consisting of performance criteria and the financial metrics for assessing business outcomes. financial metrics help the supply chain decision-makers and executors to understand the impact of the operational decisions and efficiency on the overall profitability of the business unit (tan, 1999; ketzenberg et al., 2006; kremers, 2010; kancharla and hegde, 2016). also, the measurement of supply chain performance in financial terms enables the organizations to get an outlook on future earnings, which would value the shareholders (krause et al., 2009). in this regard, wisner (2011) demonstrated the impact of the supply chain's performance on the organization's financial results. over the years, several researchers have made significant contributions in developing comprehensive performance assessment frameworks for supply chains. one famous framework, such as the scor (supply chain operations reference) model integrates the primary processes (plan, source, make, deliver and return) of supply chain operation with the overall strategy of the organization (kocaoğlu et al., 2013; askariazad and wanous, 2009; parkan and wang, 2007; lockamy and mccormack, 2004). the scor model enables to interconnect the process efficiency with the business effectiveness reflected in, both the financial (e.g., supply chain cost, cost of goods sold or cogs, return on assets, return on working capital) and the operational (e.g., order fulfillment time, supply chain flexibility, supplier relationship, % yield, delivery efficiency, supply chain adaptability, distribution planning, network design) outcomes. in this regard, elgazzar et al. (2012) identified the firm’s financial strategy's priorities and put forth a framework to link the scor model-based supply chain performance measures with financial metrics (identified through du-pont ratio analysis). in tune with this work, in recent times, many researchers have put their efforts into establishing the relationship of supply chain operational performance and financial outcome of the organization across the industry (zhu and sarkis, 2004; li et al., 2006; wagner et al., 2012). innovation is also given due importance for improving supply chain performance (chithambaranathan et al., 2015). as we see that supply chain performance depends on several parameters, mcdm methods have been used by the researchers. in literature we notice applications of various mcdm frameworks related to supply chain performance measurement (for example, bhagwat and sharma, 2007; wong and wong, 2007; varma et al., 2008; yang, 2009; najmi and biswas/decis. mak. appl. manag. eng. 3 (2) (2020) 162-189 164 makui, 2010; pramod and banwet, 2011; elgazzar et al., 2012; bhattacharya et al., 2014; jothimani and sarmah, 2014; rouyendegh et al., 2014; shafiee et al., 2014; tyagi et al., 2014; tseng et al., 2014; dey et al., 2016; uygun and dede, 2016; ghosh and biswas, 2016; moharamkhani et al., 2017; govindan et al., 2017; janaki et al., 2018; sufiyan et al., 2019; grida et al., 2020). in this paper we focus on the healthcare sector in india. in india, healthcare is one of the most talked-about and promising sectors in terms of customers’ attachment and emotion (schneller and smeltzer, 2006), complexity, growth, revenue generation and employment potential. the expected business is around inr 8.6 trillion by 2022. the sector has been emphasized by the govt. of india (goi) as the plan is to spend 2.5 percent of the country’s gdp in public health by 2025 (source: ifbe report). already several initiatives (e.g., ayushman bharat) have been conceptualized and implemented by the goi. a healthcare supply chain is said to be inefficient at utilizing invested capital, which eventually increases operating costs (kwon et al., 2016). hence, as compared to supply chain of the other industries (i.e., commercial supply chains) there is enough scope for improving the performance (de vries and huijsman, 2011) to bring down the operating through effective utilization of resources, provide quality service to the users at an affordable price while maximizing shareholders’ returns. this paper intends to carry out a comparative assessment of supply chain performance of leading organizations belonging to the healthcare sector in india. financial metrics are used as criteria for assessment. from the methodological point of view, in this paper we consider three recently developed mcdm algorithms such as mabac, cocoso, and marcos. we are interested to examine the competitive positions of the sample firms using the lens of these three different algorithms. we aim to compare the results obtained from the applied mcdm methods. we see that most of the past research considered the methods like analytic hierarchy process (ahp), analytic network process (anp), decision making trial and evaluation laboratory (dematel), technique for order of preference by similarity to ideal solution (topsis) for measuring supply chain performance. our work uses a combined framework of both outranking and compromise solution algorithms. this paper adds value to the growing literature in the following way. first, it addresses the issue of performance measurement of healthcare organizations in india. in the literature, there are evidences of linking supply chain performance with financial performance. however, measuring comparative supply chain performances based on financial metrics, particularly for health care supply chains in india seems to be rare. second, in this paper we use a combined subjective and objective methodology. we apply piprecia method to prioritize the criteria based on the opinions of experts in the stated field. we then apply three distinct frameworks such as multi-attributive border approximation area comparison (mabac), combined compromise solution (cocoso) and measurement of alternatives and ranking according to compromise solution (marcos) for for comparing supply chain performance using published financial data. we capture expert opinions for understanding relative priorities of the criteria to infuse practitioners’ views which provides a basis for comparing the results obtained from three distinct algorithms. in this paper, we use a combination of similarity based and compromise solution oriented methods. in order to compare supply chain performance, it is not only required to find closeness to average standard, but also trading off or compromising on performances subject to different attributes assume practical relevance. this is required as financial metrics do not reveal a comprehensive view of operational performance. further, to arrive at the conclusion, we use simple additive weighting (saw) method which uses the score values as calculated by mabac, cocoso and measuring performance of healthcare supply chains in india: a comparative analysis... 165 marcos. to our best knowledge, there has not been any previous work which have attempted to compare the performance of the mcdm algorithms used in this paper. the rest of the paper is organized as follows. in the section 2, the detailed methodology is presented while section 3 encapsulates the findings and includes a brief discussion of the findings. section 4 concludes the paper while pointing out some of the implications of this study and future research agenda. 2. data and methodology in this paper the following steps are followed for carrying out the research work. step 1: selection of sample step 2: identification of the criteria step 3: determination of criteria weights using expert opinion based piprecia method step 4: comparative ranking based on supply chain performance using mabac, cocoso, and marcos algorithms step 5: comparison of ranking results as obtained by using three distinct methods and arrive at a combined final ranking 2.1. sample in this study, leading indian healthcare organizations listed in bse, india are considered. in the selection of sample organizations, the size of the company is taken as the classifier. accordingly, top 20 companies (source: the database of the centre for monitoring indian economy pvt. ltd., cmie prowess iq) are included under the consideration of this study. table 1 provides the list of such companies. table 1. list of companies under study company name code company name code abbott india ltd. a1 glaxosmithkline pharmaceuticals ltd. a11 alembic pharmaceuticals ltd. a2 glenmark pharmaceuticals ltd. a12 alkem laboratories ltd. a3 ipca laboratories ltd. a13 aurobindo pharma ltd. a4 jubilant life sciences ltd. a14 biocon ltd. a5 pfizer ltd. a15 cadila healthcare ltd. a6 piramal enterprises ltd. a16 cipla ltd. a7 sanofi india ltd. a17 divi's laboratories ltd. a8 strides pharma science ltd. a18 dr. reddy's laboratories ltd. a9 sun pharmaceutical inds. ltd. a19 fortis healthcare ltd. a10 torrent pharmaceuticals ltd. a20 2.2. criteria selection in this paper we focus on the following abilities of the supply chains such as customer attractiveness through its products and services, profitable utilization of the working capital, efficient management of the working capital and inventory, and liquidity. accordingly, we select five criteria for comparing relative supply chain performances of the sample organizations. biswas/decis. mak. appl. manag. eng. 3 (2) (2020) 162-189 166 sales growth (sg) is a manifest of acceptance of the firm’s products and services in the marketplace. sg is an indication of improved product and service quality, timeliness in delivery, flexibility, and responsiveness, which increase revenue. hence, sg represents the operational efficiency of the activities carried out across the supply chain and holds a positive linkage with supply chain performance (brewer and speh, 2000). it is measured in terms of an incremental difference in sales value over two consecutive years. return on working capital (rwc) is an important criterion for assessing the performance of supply chain as it entails the asset management efficiency of the firms (okumuş et al., 2019). cash to current liabilities (ccl) or cash ratio reflects the liquidity position of an organization. this ratio is one of the indicators that the creditors look at before taking loan related decision. ccl shows the ability of supply chains to generate cash for meeting short-term requirement such as debt repayment. inventory not only is a cost element for any organization but also it is a part of the total asset (shah and shin, 2006). an effective inventory management adds to the overall profitability of the firm and hence, inventory turnover ratio (itr) needs to be optimized (ganesan et al., 2009). itr indicates the ability of the organization to effectively roll out its inventory. finally, effective management of cash conversion cycle (ccc) increases productivity, revenue generation, and results in a reduction in operating costs (okumuş et al., 2019). gunasekaran et al. (2004) reflected on the significance of converting the materials into cash through sales to ensure the return on investment for the shareholders. ccc stands on three components: cash receivables from the customer end, cash payables to the suppliers, and cash held up in the form of inventories (richards and laughlin, 1980). ccc is, therefore, an indicator of the efficiency of operations and effectiveness of the operational decisions about working capital management (özbayraka and akgün, 2006; bagchi et al., 2012) which significantly impacts the profitability of the firms (jose et al., 1996; padachi, 2006; lazaridis and tryfonidis, 2006; garcia-teruel and martinez-solano, 2007; falope and ajilore, 2009; okumuş et al., 2019). lesser value of ccc signifies better profitability (raheman and nasr, 2007; uyar, 2009) and lesser opportunity cost. researchers (churchill and mullins, 2001; farris and hutchinson, 2002, 2003; bauer, 2007) have pointed out that shorter the period of dso, better it is for the firms to utilize the amount in different activities of the supply chain including sales promotion which has a positive impact on the financial performance. moreover, the higher dso cycle often leads to credit risk. the organizations usually offer discounts against early payments to encourage customers and maintain a mutual relationship (moran, 2011). the nature of dio in this context posits a challenge to the firms. shah and shin (2007) opined that drawing relationships between inventory holding and firm performance. the decisions on inventory management stand a bit complex. on one side, a higher inventory level ensures the timely availability of products. it enables the organizations to combat the effect of surge demand while on the other side, holding an additional inventory shot up the carrying costs and other potential hidden losses and bear a negative impact on the firm’s liquidity. keeping excess inventory results in forecasting error and becomes a potential cause of the bullwhip effect (tangsucheeva and prabhu, 2013). overall lower the dio better is the performance of the supply chain (chen et al., 2005; singhal, 2005; swamidass, 2007; koumanakos, 2008; capkun et al., 2009). on the other hand, more is the value of dpo, better will it be for the firms as the liquidity position is improved (stewart, 1995). however, here lies a situation of tradeoff. extending the payment cycle has a significant negative impact on the relationship between the firm and the suppliers (fawcett et al., 2010). modern scm concepts believe in an end-to-end seamless operation, which demands integration among different chain members and mutual development. higher dpo often generates measuring performance of healthcare supply chains in india: a comparative analysis... 167 a strangulated effect as many suppliers face a liquidity crisis, often results in reduced service level (raghavan and mishra, 2011; timme and wanberg, 2011). it is evident from the literature that the researchers are of double way opinions. for example, farris and hutchison (2002) advocated for longer dpo, while deloof (2003) and garcia-teruel and martinezsolano (2007) observed evidence of better performance with shorter dpo. in general, within a toleration level as set by the nature of the relationship among the suppliers and users, type of supply, and terms and conditions of the service level agreements, it is a common notion to consider higher dpo for better functioning of the organizations within the ccc. some of the recent studies also have reported the use of financial metrics for comparing supply chain performances (avelar-sosa et al., 2019; fekpe and delaporte, 2019; tripathi et al., 2019). table 2 summarizes the criteria considered for this study. table 2. list of criteria criteria code uom definition effect direction sg c1 times (salest-sales t-1)/ sales t-1 (+) rwc c2 times earnings before interest, depreciation, tax and amortization divided by working capital (+) ccl c3 times cash and marketable securities/current liabilities (+) itr c4 times cost of goods sold/average inventory (+) ccc c5 days the average time elapsed between cash disbursement and collection (-) 2.3. methods general notations: ai = no. of alternative options (healthcare companies in this paper); i = 1,2, … . m cj = no. of criteria; j = 1,2, … . n x = [xij]m×n ; decision matrix xij = performance value of ith alternative for jth criterion 𝑋𝑗 𝑚𝑎𝑥 = maximum value for criterion j 𝑋𝑗 𝑚𝑖𝑛 = minimum value for criterion j 𝑊j = weight or importance level for the criterion j r = [rij]m×n ; normalized decision matrix rij = normalized performance value of ith alternative for jth criterion 2.3.1. piprecia method piprecia is an extension of the widely used group decision making approach such as stepwise weight assessment ratio analysis (swara) method as developed by kersuliene et al. (2010). most often in real-life situations, it is very difficult for biswas/decis. mak. appl. manag. eng. 3 (2) (2020) 162-189 168 reaching a consensus while a considerably larger set of decision makers attempts to find out the expected importance of a set of criteria and order them. the computational steps of piprecia are quite similar to swara, but it provides a freedom not to put emphasis on sorting out the criteria based on expected significance in a group decision making environment (biswas and pamucar, 2020; stanujkic et al., 2017; keršulienė and turskis, 2011). the computational steps as described by stanujkic et al. (2017) are as follows: step 1: selection of a set of relevant criteria for evaluating the alternatives. step 2: (optional) sort the criteria according to their expected significances as rated by the decision makers in descending order. for a small number of experts, it works well; however, for a large group of respondents, it is very difficult to arrive at a bias-free group consensus. hence, in that case this step is not required. step 3: determination of the relative importance of the criteria. starting from the second criterion, the relative importance or significance of any criterion cj is given by: 𝑆𝑗 𝑟 = { > 1 when cj ≻ cj−1 1 when cj = cj−1 < 1 when cj ≺ cj−1 (1) here, ‘r’ denotes a particular respondent among all. step 4: find out the coefficient 𝐾𝑗 𝑟 𝐾𝑗 𝑟 = { 1 when j = 1 2 − 𝑆𝑗 𝑟 when j > 1 (2) step 5: determine the recalculated criteria weights 𝑄𝑗 𝑟 = { 1 when j = 1 𝑄𝑗−1 𝑟 𝐾𝑗 𝑟 when j > 1 (3) step 6: calculate the relative criteria weights 𝑊𝑗 𝑟 = 𝑄𝑗 𝑟 ∑ 𝑄𝑗 𝑟𝑛 𝑗=1 (4) in a group decision making environment for each decision maker, the above steps need to be carried out. finally, for obtaining the group weight calculation, in a simple sense, geometric mean (gm) of individual weights is calculated (stanujkic et al., 2017) as given by: 𝑊𝑗 ∗ = (∏ 𝑊𝑗 𝑟𝑅 𝑟=1 ) 1/𝑅 (5) here, ‘r’ is the total number of respondents. final criteria weights are given by: 𝑊𝑗 = 𝑊𝑗 ∗ ∑ 𝑊𝑗 ∗𝑛 𝑗=1 (6) 2.3.2. mabac method mabac uses the distance of the alternatives from the boundary approximation area (upper approximation area or uaa for ideal or desirable solutions and lower approximation area or laa for non-ideal or non-desirable solutions along with border approximation area or baa) based the performance values under the influence of the criteria (pamučar and ćirović, 2015). this method is a widely used approach (debnath et al., 2017) as measuring performance of healthcare supply chains in india: a comparative analysis... 169 it produces a stable solution as compared to topsis and vikor (pamučar and ćirović, 2015) it works with qualitative and quantitative data to classify the best, the worst, and borderline solutions (roy et al., 2018) it is based on a comprehensive, rational and sensible algorithm (xue et al., 2016) it compares the alternatives on relative strength and weakness dimensions under the effect of the criteria (roy et al., 2016). this method has been applied in solving several social science related decision making problems (yu et al., 2017; sharma et al., 2018; vesković et al., 2018; roy et al., 2018; biswas et al., 2019). the methodological steps (in brief) are as under: step 1: formation of the decision matrix x step 2: formation of the normalized decision matrix r normalization: rij = (xij− xj min) (xj max− xj min) ; for beneficial criteria (7) rij = (xij− xj max ) (xj min− xj max) ; for non-beneficial criteria (8) step 3: construction of the weighted normalization matrix y = [yij]m×n where, yij = wj(rij + 1) (9) step 4: determination of the border approximation area (baa) represented as t = [tj]1×n where, tj = (∏ yij m i=1 ) 1/m (10) step 5: derive q matrix related to the separation of the alternatives from baa q = y-t (11) a particular alternative ai is said to be belonging to the upper approximation area (uaa) i.e. t+ if qij > 0 or lower approximation area (laa) i.e. t − if qij < 0 or baa i.e. t if qij = 0. the alternative ai is considered to be the best among the others if more numbers of criteria pertaining to it possibly belong to t+ . step 6: ranking of the alternatives in descending order based on the final appraisal score given by si = ∑ qij n j=1 (12) 2.3.3. cocoso method the primary objective of any mcdm framework is to determine the best feasible solution among the available options under the influence of the set of relevant criteria. now, in real-life situations, many times these criteria are characterized by non-commensurable and conflicting nature. under this circumstance, there is no alternative which can satisfy the requirements of all the criteria to a considerable extent. hence, decision makers need to accept a tradeoff or compromise solution subject to the criteria considered. looking into it, researchers have attempted to develop such models which can deal with multiple criteria (conflicting nature) based compromise solution. the popular techniques such as topsis, copras and vikor have been used extensively in this regard. however, these techniques suffer from following issues biswas/decis. mak. appl. manag. eng. 3 (2) (2020) 162-189 170 topsis and vikor consider the negative ideal solution while calculating the euclidean distance of each alternative the traditional copras and topsis methods do not provide meaningful solutions when work with mixed data and suffer fro bagchi m rank reversal phenomena (aouadni et al., 2017) under this situation, cocoso (yazdani et al., 2018) works with weight aggregation process based on grey relational generation (which enables to cope up with conflicts) and incorporates the following features: it uses the power of weights for aggregation. as a result, it provides relatively stronger distance measurement for modelling purposes. for validation of ranking result (i.e., index) it uses three different aggregation strategies to generate the cumulative score. therefore, it gives a complete ranking index taking compromising and conflicting situations into account. in a nutshell, this method is an integration of simple additive weighting and exponentially weighted product models. the methodological steps can be described as follows (yazdani et al., 2018): step 1: formation of the decision matrix x step 2: derive the normalized decision matrix r cocoso follows a normalization process suggested by zeleny (1973). accordingly, the normalized values are obtained as: rij = xij−xj min xj max− xj min (for beneficial criteria) (13) rij = xj max −xij xj max− xj min (for non-beneficial criteria) (14) step 3: determine the aggregate of the weighted normalized performance values as given by si = ∑ wj n j=1 rij (15) step 4: calculation of the aggregate of the power weight of comparability values pi = ∑ (rij) wjn j=1 (16) step 5: calculations of the relative weights of the alternatives for this step, in cocoso method the relative weights are calculated in three ways such as ki1 = pi+si ∑ (pi+si) m i=1 (17) ki2 = si min 𝑖 si + pi min 𝑖 pi (18) ki3 = 𝛼(𝑆𝑖)+(1−𝛼)pi (𝛼 max 𝑖 si+(1−𝛼) max 𝑖 pi) (19) here, these three strategies consider weighted arithmetic average; relative scores based on sum and product of performance values and allow the decision makers to flexibly select the 𝛼 values which can vary from 0 to 1 (the usual value being 0.5). step 6: find out the final ranking score the final ranking of the alternatives is done depending on the overall ki value (higher value implies more importance) which is given as: ki = (ki1 ki2ki3) 1/3 + 1 3 (ki1 + ki2 + ki3) (20) 2.3.4. marcos method this method is a new addition to the portfolio of compromise solution based mcda (stević et al., 2020). it has been used in solving complex research problems (stević and brković, 2020; stanković et al., 2020). the procedural steps are explained below: measuring performance of healthcare supply chains in india: a comparative analysis... 171 step 1: formation of the extended decision matrix (edm) in the edm the first row is occupied by the anti-ideal solution (ais) values and the last row indicates the ideal solution (is) values. ais indicates the most pessimistic choice whereas is is the most optimistic selection. the values are calculated as follows. 𝐴𝐼𝑆 = min 𝑖 𝑥𝑖𝑗 𝑤ℎ𝑒𝑛 𝑗 ∈ 𝐽 +; max 𝑖 𝑥𝑖𝑗 𝑤ℎ𝑒𝑛 𝑗 ∈ 𝐽 − (21) 𝐼𝑆 = max 𝑖 𝑥𝑖𝑗 𝑤ℎ𝑒𝑛 𝑗 ∈ 𝐽 +; min 𝑖 𝑥𝑖𝑗 𝑤ℎ𝑒𝑛 𝑗 ∈ 𝐽 − (22) here, 𝐽+ represents a set of beneficial criteria (whose effect direction is +ve) and 𝐽− indicates a set of non-beneficial criteria (having –ve effect direction). step 2: normalization the normalized values are given by (using linear normalization rij = xij−ais is− ais (𝑤ℎ𝑒𝑛 𝑗 ∈ 𝐽+) (23) rij = 1 − xij−ais is− ais (𝑤ℎ𝑒𝑛 𝑗 ∈ 𝐽−) (24) step 3: formation of weighted matrix vij = wjrij (25) step 4: calculation of utility degrees alternatives with respect to is and ais 𝐾𝑖 − = 𝑆𝑖 𝑆𝐴𝐼𝑆 (26) 𝐾𝑖 + = 𝑆𝑖 𝑆𝐼𝑆 (27) where, 𝑆𝑖 = ∑ vij 𝑛 𝑗=1 (28) step 5: determination of utility functions with respect to is and ais f(𝐾𝑖 −) = 𝐾𝑖 + 𝐾𝑖 ++ 𝐾𝑖 − (29) f(𝐾𝑖 +) = 𝐾𝑖 − 𝐾𝑖 ++ 𝐾𝑖 − (30) step 6: calculation of the utility function values for the alternatives f(𝐾𝑖 ) = 𝐾𝑖 ++ 𝐾𝑖 − 1+ 1− f(𝐾 𝑖 +) f(𝐾 𝑖 +) + 1− f(𝐾𝑖 −) f(𝐾 𝑖 −) (31) decision rule: the higher is the utility value, better is the alternative 3. findings and discussion table 4-7 present the step by step derivation of the criteria weights using piprecia method. for this purpose, we have approached to three experts who have substantial experience in the stated field. table 3 provides a summary of experts’ profiles. in the tables 4-6, their responses are summarized and subsequently, values of the parameters are calculated using eq. (2), (3), and (4). table 3. experts’ profiles expert 1 2 3 experience 10 years 18 yrs 25 years industry healthcare healthcare, fmcg chemical, healthcare biswas/decis. mak. appl. manag. eng. 3 (2) (2020) 162-189 172 table 4. response of expert 1 & weights of the criteria criteria code sj1 kj1 qj1 wj1 sg c1 1.000 1.000 0.2247 rwc c2 1.15 0.850 1.176 0.2643 ccl c3 0.8 1.200 0.980 0.2203 itr c4 0.4 1.600 0.613 0.1377 nwcc c5 1.1 0.900 0.681 0.1530 table 5. response of expert 2 & weights of the criteria criteria code sj2 kj2 qj2 wj2 sg c1 1.000 1.000 0.2180 rwc c2 1.25 0.750 1.333 0.2907 ccl c3 0.55 1.450 0.920 0.2005 itr c4 0.8 1.200 0.766 0.1671 nwcc c5 0.65 1.350 0.568 0.1238 table 6. response of expert 3 & weights of the criteria criteria code sj3 kj3 qj3 wj3 sg c1 1.000 1.000 0.2787 rwc c2 0.8 1.200 0.833 0.2323 ccl c3 0.9 1.100 0.758 0.2111 itr c4 0.75 1.250 0.606 0.1689 nwcc c5 0.45 1.550 0.391 0.1090 now by applying eq. (5) and (6), we derive the final weights of the criteria (see table 7). table 7. weights of the criteria criteria code wj* wj sg c1 0.239 0.2401 rwc c2 0.261 0.2626 ccl c3 0.210 0.2115 itr c4 0.157 0.1579 nwcc c5 0.127 0.1279 it is important to ensure harmony in a typical group decision making format. for this purpose, we check the consistency of each individual expert’s rating with the aggregated final weight. we calculate spearman’s ρ using ibm spss (version 24) software. spearman’s ρ measures the degree of interrelation in terms of the correlation coefficient among the variables compared. table 8 shows that individual decisions are in sync with the group opinion. measuring performance of healthcare supply chains in india: a comparative analysis... 173 table 8. consistency check i group_weight weight_exp_1 .945* weight_exp_2 .951* weight_exp_3 .910* * correlation is significant at the 0.05 level (2-tailed). now, we move to rank the sample organizations based on their comparative supply chain performance. table 9 exhibits performance values of the alternatives (organizations) under different criteria as considered here. table 9. decision matrix weight 0.2401 0.2626 0.2115 0.1579 0.1279 criteria c1 c2 c3 c4 c5 (+) (+) (+) (+) (-) company a1 0.0973 0.4011 1.96 6.06 90.18 a2 0.2428 2.1991 0.12 4.11 101.15 a3 0.1063 1.2033 0.29 5.76 -5.31 a4 0.1897 0.8828 0.01 2.7 240.85 a5 -0.2638 0.4479 0.54 5.35 144.47 a6 0.1147 2.0538 0.04 4.9 195.12 a7 0.0857 0.5013 0.95 4.31 266.65 a8 0.2713 0.5612 1.69 2.79 242.32 a9 0.1353 0.5313 0.73 4.6 206.03 a10 -0.0084 -0.4166 0.02 15.46 -68.32 weight 0.2401 0.2626 0.2115 0.1579 0.1279 criteria c1 c2 c3 c4 c5 (+) (+) (+) (+) (-) company a11 0.0809 1.5779 0.41 6.43 37.95 a12 0.1372 0.8885 0.12 4.38 66.8 a13 0.1285 0.6983 0.35 3.52 209.58 a14 0.031 -1.8818 0.01 6.43 41.2 a15 0.051 0.4464 2.15 5.38 21.52 a16 0.0703 -0.0758 0.08 14.29 17.57 a17 0.1121 0.8037 0.69 5.74 83 a18 0.0684 1.4684 0.24 3.46 126.97 a19 0.1422 -0.8247 0.04 3.69 233.16 a20 0.3413 4.0801 0.19 3.96 237.09 next, we carry out the comparative analysis of the organizations under study using the mcdm algorithms as used here. first, we apply mabac method. table 10 shows the final rankings based on appraisal scores, obtained using eq. 7-12. proceeding further, we compare the performances of sample organizations using the compromise solution approach cocoso. using the eq. 13-20 we derive the competitive positions of biswas/decis. mak. appl. manag. eng. 3 (2) (2020) 162-189 174 the alternatives (see table 11). finally, table 12 highlights the findings in terms of ranking of the organizations derived as per the procedural steps (eq. 21-31) of the latest compromise solution based mcdm methodology such as marcos. in order to check the consistency among the results obtained from three distinct algorithms, we calculate kendall’s τ and spearman’s ρ using ibm spss (version 24) software. spearman’s ρ measures the degree of interrelation in terms of the correlation coefficient among the variables compared while kendall’s τ measures the probability of concordance and discordance among them (nelsen, 1992). table 10. ranking result (mabac) company sum (si) rank_mabac company sum (si) rank_mabac a1 0.1592 3 a11 0.0758 6 a2 0.0859 5 a12 0.0028 11 a3 0.0658 9 a13 -0.0515 16 a4 -0.0748 17 a14 -0.1371 18 a5 -0.1519 20 a15 0.1794 1 a6 -0.0053 13 a16 0.0713 7 a7 -0.0299 15 a17 0.0561 10 a8 0.1100 4 a18 -0.0215 14 a9 -0.0039 12 a19 -0.1507 19 a10 0.0664 8 a20 0.1610 2 table 11. ranking result (cocoso) company ki rank_cocoso company ki rank_cocoso a1 2.170174 3 a11 2.060398 6 a2 2.067633 4 a12 1.880885 11 a3 2.036865 8 a13 1.699495 15 a4 1.42355 18 a14 1.357961 19 a5 1.270518 20 a15 2.202375 1 a6 1.860046 12 a16 2.062095 5 a7 1.475917 16 a17 1.993071 9 a8 1.922084 10 a18 1.792973 14 a9 1.817654 13 a19 1.44915 17 a10 2.038985 7 a20 2.185628 2 table 12. ranking result (marcos) company kiki+ f(ki-) f(ki+) f(ki) rank a1 4.211465 0.617644 0.12790 0.8721 0.606271 3 a2 3.704441 0.543284 0.12790 0.8721 0.533281 6 a3 2.911298 0.426964 0.12790 0.8721 0.419103 15 a4 3.282138 0.481351 0.12790 0.8721 0.472488 10 a5 2.103364 0.308474 0.12790 0.8721 0.302795 19 a6 3.552041 0.520934 0.12790 0.8721 0.511342 8 a7 3.787117 0.55541 0.12790 0.8721 0.545183 5 a8 4.735647 0.694519 0.12790 0.8721 0.681731 2 a9 3.62839 0.532131 0.12790 0.8721 0.522333 7 a10 2.539582 0.372449 0.12790 0.8721 0.365591 18 a11 3.248171 0.476369 0.12790 0.8721 0.467598 12 a12 2.849102 0.417843 0.12790 0.8721 0.410149 16 measuring performance of healthcare supply chains in india: a comparative analysis... 175 company kiki+ f(ki-) f(ki+) f(ki) rank a13 3.277273 0.480637 0.12790 0.8721 0.471788 11 a14 1.602783 0.23506 0.12790 0.8721 0.230732 20 a15 3.959377 0.580673 0.12790 0.8721 0.569981 4 a16 3.090712 0.453276 0.12790 0.8721 0.444931 13 a17 3.36252 0.493139 0.12790 0.8721 0.484059 9 a18 3.018697 0.442715 0.12790 0.8721 0.434564 14 a19 2.642797 0.387586 0.12790 0.8721 0.38045 17 a20 5.103074 0.748405 0.12790 0.8721 0.734625 1 for identifying top and worst performers, the geometric mean of the year wise ranks is calculated for each organization. in literature, there are instances where researchers (basak and saaty, 1993; barzilai and lootsma, 1997; stanujkic et al., 2015) have mentioned the use of the geometric mean in finding out the synthesized view to reach group consensus in case different opinions or methodologies are adopted. in these cases, the geometric mean is a useful measure for averaging (fleming and wallace, 1986). but, in this paper, for more objective evaluation, we use simple additive weighting (saw) method to compare the results obtained from three algorithms used here. we take the score values of the alternatives obtained from each algorithm and apply saw method (simanaviciene and ustinovichius, 2010) using linear max-min normalization. we assign equal priorities to all three algorithms. table 13 provides the summary of rankings and table 14 shows the result of the consistency test. for further investigation we perform related sample wilcoxon signed rank test (wsrt). table 15 indicates the test result. the null hypothesis for wsrt states that the median of differences between rankings of any two algorithms is equal to zero. table 13. ranking summary company ranking results final rank (saw) mabac cocoso marcos a1 3 3 3 3 a2 5 4 6 5 a3 9 8 15 9 a4 17 18 10 17 a5 20 20 19 19 a6 13 12 8 11 a7 15 16 5 16 a8 4 10 2 4 a9 12 13 7 12 a10 8 7 18 10 a11 6 6 12 6 a12 11 11 16 13 a13 16 15 11 15 a14 18 19 20 20 a15 1 1 4 2 a16 7 5 13 7 a17 10 9 9 8 a18 14 14 14 14 a19 19 17 17 18 a20 2 2 1 1 biswas/decis. mak. appl. manag. eng. 3 (2) (2020) 162-189 176 table 14. consistency test ii rank mabac rank cocoso rank marcos final rank kendall's tau rank_mabac 1 rank_cocoso .884** 1 rank_marcos .463** .368* 1 final_rank .895** .842** .526** 1 spearman's rho rank_mabac 1 rank_cocoso .959** 1 rank_marcos .638** .528* 1 final_rank .980** .952** .690** 1 ** correlation is significant at the 0.01 level (2-tailed). * correlation is significant at the 0.05 level (2-tailed). table 15. result of wsrt pair significance value* decision mabac and cocoso 0.439 the null hypothesis is supported mabac and marcos 0.965 the null hypothesis is supported marcos and cocoso 1.000 the null hypothesis is supported * at the 0.05 level (asymptotic significance, 2-sided) in order to check the stability of the results obtained by using these three algorithms we carry out the sensitivity analysis. sensitivity analysis is useful for achieving a rational and reliable results while reducing subjectivity and bias (mukhametzyanov and pamucar, 2018; pamučar et al., 2016). for carrying out the sensitivity analysis, we perform four experients wherein we replace the weights of the criteria other than that holds the highest weight. it means in each experiement, the weight of a particular criterion (not having the highest weight) gets replaced with the highest value while keeping the priorities of other criteria same. accordingly, we rank the companies under each circumstance applying all three methods as mentioned here. table 16 describes the experiments done for carrying out the sensitivity analysis. table 17-22 demonstrates the results of sensitivity analysis for all the mcdm frameworks. table 16. experimental cases for sensitivity analysis criteria weights actual case 1 case 2 case 3 case 4 c1 0.2401 0.2626 0.2401 0.2401 0.2401 c2 0.2626 0.2401 0.2115 0.1579 0.1279 c3 0.2115 0.2115 0.2626 0.2115 0.2115 c4 0.1579 0.1579 0.1579 0.2626 0.1579 c5 0.1279 0.1279 0.1279 0.1279 0.2626 measuring performance of healthcare supply chains in india: a comparative analysis... 177 table 17. result of sensitivity analysis (mabac) company ranks under different cases actual case 1 case 2 case 3 case 4 a1 3 2 2 2 2 a2 5 5 8 10 9 a3 9 9 10 8 5 a4 17 17 17 18 18 a5 20 20 20 20 19 a6 13 13 14 13 15 a7 15 15 13 14 16 a8 4 4 3 5 8 a9 12 12 11 11 12 a10 8 8 7 3 3 company ranks under different cases actual case 1 case 2 case 3 case 4 a11 6 7 6 7 6 a12 11 11 12 12 11 a13 16 16 16 16 17 a14 18 18 18 17 14 a15 1 1 1 1 1 a16 7 6 5 4 4 a17 10 10 9 9 7 a18 14 14 15 15 13 a19 19 19 19 19 20 a20 2 3 4 6 10 table 18. consistency check iii (among the rankings obtained through sensitivity analysis for mabac) actual case1 case2 case3 case4 kendall's tau actual 1 case1 .979** 1 case2 .895** .916** 1 case3 .821** .842** .884** 1 case4 .705** .726** .726** .842** 1 spearman's rho actual 1 case1 .997** 1 case2 .977** .982** 1 case3 .935** .946** .971** 1 case4 .863** .878** .892** .950** 1 ** correlation is significant at the 0.01 level (2-tailed). biswas/decis. mak. appl. manag. eng. 3 (2) (2020) 162-189 178 table 19. result of sensitivity analysis (cocoso) company ranks under different cases actual case 1 case 2 case 3 case 4 a1 3 3 2 4 3 a2 4 4 7 9 8 a3 8 8 8 6 5 a4 18 18 18 23 19 a5 20 20 20 24 20 a6 12 12 13 13 14 a7 16 16 16 21 18 a8 10 10 10 10 11 a9 13 13 12 12 13 a10 7 7 5 1 2 company ranks under different cases actual case 1 case 2 case 3 case 4 a11 6 6 6 5 6 a12 11 11 11 11 9 a13 15 15 15 19 15 a14 19 19 19 22 16 a15 1 1 1 3 1 a16 5 5 4 2 4 a17 9 9 9 8 7 a18 14 14 14 14 12 a19 17 17 17 20 17 a20 2 2 3 7 10 table 20. consistency check iv (among the rankings obtained through sensitivity analysis for cocoso) actual case1 case2 case3 case4 kendall's tau actual 1 case1 1.000** 1 case2 .937** .937** 1 case3 .800** .800** .863** 1 case4 .737** .737** .800** .874** 1 spearman's rho actual 1 case1 1.000** 1 case2 .986** .986** 1 case3 .916** .916** .956** 1 case4 .890** .890** .926** .970** 1 ** correlation is significant at the 0.01 level (2-tailed). measuring performance of healthcare supply chains in india: a comparative analysis... 179 table 21. result of sensitivity analysis (marcos) company ranks under different cases actual case 1 case 2 case 3 case 4 a1 3 3 3 3 4 a2 6 6 7 8 10 a3 15 15 15 15 17 a4 10 10 11 14 7 a5 19 19 19 19 18 a6 8 8 9 11 9 a7 5 5 5 6 3 a8 2 2 1 1 1 a9 7 7 6 7 5 a10 18 18 18 10 19 company ranks under different cases actual case 1 case 2 case 3 case 4 a11 12 12 12 12 15 a12 16 16 16 17 16 a13 11 11 10 13 8 a14 20 20 20 20 20 a15 4 4 4 4 6 a16 13 13 13 5 13 a17 9 9 8 9 12 a18 14 14 14 16 14 a19 17 17 17 18 11 a20 1 1 2 2 2 table 22. consistency check v (among the rankings obtained through sensitivity analysis for marcos) actual case1 case2 case3 case4 kendall's tau actual 1 case1 1.000** 1 case2 .958** .958** 1 case3 .758** .758** .800** 1 case4 .768** .768** .768** .589** 1 spearman's rho actual 1 case1 1.000** 1 case2 .994** .994** 1 case3 .872** .872** .880** 1 case4 .917** .917** .920** .758** 1 ** correlation is significant at the 0.01 level (2-tailed). this study reveals a number of observations. first, we observe that there is a variation in the comparative positions of the sample organizations as derived by using three mcdm frameworks. top five positions are occupied by a same group of companies with some variations within the group. however, we see the bottom five group shows considerable changes in the positions as we apply three methods. second, if we analyze specifically, mabac based ranking shows highest correlation with the aggregate final result. among the methods, the results obtained from mabac, cocoso, and marcos are statistically consistent with each other as it gets revealed biswas/decis. mak. appl. manag. eng. 3 (2) (2020) 162-189 180 from table 14. here, mabac and cocoso show more consistency between the results obtained by using these algorithms. third, considering overall, we find that the large cap firms have not done well as far as supply chain performance is concerned. fourth, we observe that all methods responds more or less in a similar fashion to the sensitivity analysis. however, looking at the values of correlation coefficients, one can infer that cocoso performs slightly better under different situations. figure 1-3 graphically present the result of sensitivity analysis for all three methods. figure 1. sensitivity analysis (mabac) figure 2. sensitivity analysis (cocoso) 0 5 10 15 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 actual case 1 case 2 case 3 case 4 0 5 10 15 20 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 actual case 1 case 2 case 4 case 3 measuring performance of healthcare supply chains in india: a comparative analysis... 181 figure 3. sensitivity analysis (marcos) 4. conclusion in this study, we attempt to compare a number of leading healthcare companies based on their supply chain performances measured in financial terms. for this purpose, we present a comparative analysis of three distinct algorithms such as mabac, cocoso and marcos. we find that the large cap firms do not perform well. the results obtained from three mcdm frameworks show consistency while cocoso appears to be comparatively better. the present study has a number of managerial and social implications. first, with the effects of the factors like increasing population and pollution level, transformation in the climate, and changes in the lifestyle, healthcare operations have become critical and delicate in nature particularly in the diverse country like india. in addition, as the level of competition has got amplified to a large extent, the pressure of reducing prices for providing requisite service invokes a focused approach by the service providers. scm is one of the key areas which can provide a competitive edge to the organizations. hence, measuring supply chain performances following a multiple criteria based holistic framework linked with financial outcomes enables the organizations to take appropriate strategic and operational decisions. this study provides such a framework. second, understanding relative performances of the focused organization and its competitors help the decision makers to take the appropriate futuristic course of actions. third, most often policy makers need to know the nature of the industry and performances of the key players for formulating policies for the sector. the findings show significant variations across the organizations which might help the policy makers and industry analysts to intervene and formulate contemporary policies. fourth, many a times the price for the offered services is decided from a cost plus margin point of view. heath care is a typical sector where this approach often creates a disconnect between the service provider and the service users (i.e., the patients). this eventually impacts on the long-term business growth and brand value of the organization. measuring supply chain performance and delving into its impact on the profitability of the organization helps the decision makers to come up with innovative and robust service offerings at an affordable price. finally, the comparative analysis of mcdm algorithms. however, in the present study, the opinions of a few experts have been sought for measuring financial performances of the healthcare supply chains. in the future study, a larger 0 5 10 15 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 actual case 1 case 2 case 3 case 4 biswas/decis. mak. appl. manag. eng. 3 (2) (2020) 162-189 182 group of experts and consumers can be approached to identify critical success factors for the healthcare supply chains and based on that a comparative assessment may be carried out. nevertheless, we believe that this limitation does not necessarily dilute the usefulness and relevance of this work. acknowledgment: the author would like to express his sincere gratitude to all anonymous reviewers whose remarks have helped to improve the quality of this work. further, the author extends sincere thanks to ms. priyanka roy and mr. shouvik sarkar, pgdm graduate students at calcutta business school for their immense support in data collection and formatting. author contributions: the author takes public responsibility for appropriate portions of the content. funding: this research received no external funding. conflicts of interest: the author declare no conflicts of interest. references al-hussaini, a. 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(2004). relationships between operational practices and performance among early adopters of green supply chain management practices in chinese manufacturing enterprises. journal of operations management, 22(3), 265289. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 4, issue 1, 2021, pp. 51-84. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame2104051g * corresponding author. e-mail addresses: fri.indra@gmail.com (i. ghosh), tamal5302@yahoo.com (t. datta chaudhuri) feb-stacking and feb-dnn models for stock trend prediction: a performance analysis for pre and post covid-19 periods indranil ghosh1* and tamal datta chaudhuri1 1 calcutta business school, west bengal, india received: 28 october 2020; accepted: 21 january 2021; available online: 6 february 2021. original scientific paper abstract: in this paper, stock price prediction is perceived as a binary classification problem where the goal is to predict whether an increase or decrease in closing prices is going to be observed the next day. the framework will be of use for both investors and traders. in the aftermath of the covid-19 pandemic, global financial markets have seen growing uncertainty and volatility and as a consequence, precise prediction of stock price trend has emerged to be extremely challenging. in this background, we propose two integrated frameworks wherein rigorous feature engineering, methodology to sort out class imbalance, and predictive modeling are clubbed together to perform stock trend prediction during normal and new normal times. a number of technical and macroeconomic indicators are chosen as explanatory variables, which are further refined through dedicated feature engineering process by applying kernel principal component (kpca) analysis. bootstrapping procedure has been used to deal with class imbalance. finally, two separate artificial intelligence models namely, stacking and deep neural network models are deployed separately on feature engineered and bootstrapped samples for estimating trends in prices of underlying stocks during pre and post covid-19 periods. rigorous performance analysis and comparative evaluation with other well-known models justify the effectiveness and superiority of the proposed frameworks. key words: binary classification, kernel principal component (kpca), bootstrapping, stacking, deep neural network. 1. introduction the financial literature is replete with attempts in predicting stock prices. in contrast to the efficient market hypothesis, researchers have identified various factors that can influence stock returns and hence have used them for prediction ghosh et al./ decis. mak. appl. manag. eng. 4 (1) (2021) 51-84 52 purposes. going back to graham and dodd (1934) where they disregarded the fact that “good stocks (or blue chips) were sound investments regardless of the price paid for them”, they distinguished between speculation and investment, and consequently emphasized on factors like management quality, earnings, dividends, capital structure and interest cover. while econometric techniques have been predominantly used to predict stock returns, various machine learning tools like artificial neural network, support vector machine, decision tree, etc. have also been used for the purpose. the literature can be classified according to choice of variables and techniques of estimation and forecasting. to mention a few, the first strand consists of studies using simple regression techniques on cross sectional data. papers by of basu (1977, 1983), jaffe et al. (1989), banz (1981), fama and french (1988, 1992, 1995), strong and xu (1997), and ibbotson and idzorek (1998) fall into this category. the second strand of the literature uses time series models and techniques to forecast stock returns. some papers in this area are by srinivasan and prakasam (2014), babu and reddy (2015) and ahmar and val (2020). econometric tools like autoregressive integrated moving average (arima), autoregressive distributed lag (ardl), generalized autoregressive conditional heteroscedasticity (garch) have been employed to forecast stock prices. papers by mostafa (2010), dutta et al. (2006), shen et al. (2007), chen et al. (2003), wu et al. (2008), perez-rodriguez et al. (2005) and datta chaudhuri et al. (2016, 2017), ghosh et al. (2018) fall in a third category where machine learning tools have been used for prediction of stock returns. majority of these studies applied traditional or variants of artificial intelligence driven (ai) models for prediction of stock returns. sezer et al. (2020) conducted an exhaustive and systematic review of usage of deep learning driven models for financial time series forecasting. their work illustrates the usage of deep neural network (dnn), recurrent neural network (rnn), long shortterm memory network (lstm), convolutional neural network (cnn), restricted boltzmann machine (rbm) method, deep belief network (dbn), auto encoder (ae), and deep reinforcement learning (drl) on plethora of equity market data. work of jiang et al. (2020) also presents a review of applications of deep learning models, features, and deployment text and image data for stock market data. the study outlines effectiveness of additional deep learning models, graph neural network (gnn), gated recurrent unit (gru) and discriminative deep neural network with hierarchical attention (han) for forecasting. usage of technical indicators and feature engineering through principal component analysis (pca) has been reported as well. rundo et al. (2019) thoroughly reviewed frameworks using econometric methods, machine learning, and deep learning methods for predictive modelling of asian, european, and us stock markets. their study also covered the indices commonly used for evaluating models. amongst the machine learning models, support vector machine (svm), decision tree (dt), random forest (rf), boosting, and artificial neural network (ann) have also been successful in modelling financial markets. therefore, in our paper, utilizing deep learning and machine learning frameworks in an integrated framework for predictive analysis, is justified. prediction of stock price movements is critical for stock market traders and portfolio managers as they have to continuously realign their strategies with market volatility. recent times have observed increase in research on stock price prediction based on advanced ai based frameworks. the stock market prediction problem can broadly be categorized into two strands. the first category deals with estimation of closing prices of different stocks, while the second strand attempts to predict the direction of movement i.e. whether stock prices would increase or decrease after a pre-specified time interval. the second category of problem is also referred as feb-stacking and feb-dnn models for stock trend prediction: a performance analysis for… 53 classification problem. the problem is quite challenging as correct estimation of trend can immensely boost investors for trading as compared to buy and hold strategy for long duration. additionally imbalanced distribution of class information of target variable, known as class imbalance often further complicates the task (pirizadeh et al. 2020, bria et al. 2020) which may lead to poor performance in test data cases. predominantly several variations of sampling strategies are used to tackle the problem (shin et al. 2021). our research attempts to develop an integrated research structure capable of modeling class imbalance in order to carry out stock trend classification in indian context. the body of research mentioned above has focused on relatively low volatile and chaotic time horizons. however, the outbreak of covid-19 pandemic has wreaked havoc by disrupting business and global supply chains. to curb infections, nations across the world resorted to strict lockdowns, banned international travels, sealed borders and imposed restriction on movements of goods and people which eventually led to increased uncertainty and stock market volatility. it would be interesting and important to check whether stock price trends can be predicted with some degree of accuracy during the new normal period owing to covid-19 pandemic. it also needs to be seen whether ai driven frameworks can be useful in such situations. one step ahead stock price trend prediction is a process of foretelling whether price of the underlying stock would increase or decrease. an increase would indicate buy signal (up) while decrease would reflect sell signal (down). hence the problem basically takes the form of binary classification. the said problem is often affected by class imbalance, i.e. disproportion between buy and sell ratio. it is highly probable to have class imbalance during the covid-19 period. considering these challenges, it becomes absolutely imperative to design robust frameworks for predictive modeling of stock price trends and test the same in new normal time periods. in this paper, we have considered four indian companies namely, hdfc bank, tata consultancy services (tcs), reliance industries ltd. (reliance), and spice jet limited (spicejet) as examples for predicting their future stock price trends. they belong to four different sectors namely, banking, it, energy and airlines. these companies have been consistently profit making and dividend paying, are leaders in their respective sectors in terms of size and performance and their stocks are extensively traded in the indian stock market. among the four sectors, airlines sector has been a recipient of rapid shock owing to worldwide lockdown due to covid pandemic. thus, our framework would be tested for efficacy on challenging time series data as well. the interested reader can consider other companies and test the efficacy of our framework. this paper considers technical indicators along with macroeconomic variables as explanatory variables for predicting the trend of aforesaid stocks. the exercise has been carried out on different time frames covering pre-covid-19 and covid-19 periods. rigorous feature engineering (fe) process has been evoked using unsupervised feature selection algorithm i.e., kernel principal component analysis (kpca) for better realization and compactness of dataset in high dimensional feature space. the class imbalance obstacle has been resolved through bootstrapping process. both fe and bootstrapping processes are invoked before applying ai algorithms for discovering the association between the explanatory and target variables for precisely predicting the trend. models belonging to two sub-fields of ai, machine learning and deep learning have been exploited for the predicting exercise. stacking, a machine learning framework built upon combination of various other learning algorithms for classification, has also been used for predicting the price trend of the three stocks. the ghosh et al./ decis. mak. appl. manag. eng. 4 (1) (2021) 51-84 54 stacking architecture has been built by combining three ensemble machine learning algorithms namely, random forest (rf), bagging, and gradient boosting (gb). since stacking is driven by both fe and bootstrapping operations, the combined framework has been coined as feb-stacking. deep neural network has been utilized for predicting trends. like the feb-stacking approach, dnn has been deployed in conjunction with fe and bootstrapping processes. hence, the combined framework has been referred as feb-dnn. rigorous classification accuracy measures have been computed to ascertain the predictive accuracy of both feb-stacking and feb-dnn models. profitability of both frameworks has been compared against the profitability of buy and hold strategy. further, comparative study with several benchmark models has been conducted to properly justify the use of the proposed architectures. the major contribution of the present research work lies in designing predictive structures in challenging times like covid-19 where financial markets are highly volatile and when financial markets experience crashes in stock market and worldwide recession. the paper proposes a structured framework for selecting technical and macroeconomic indicators for building the trend prediction frameworks. our approach recognizes the class imbalance problem arising in volatile times and combining such processes with stacking and dnn models and checking the effectiveness in covid-19 pandemic time horizons comprise the novelty of our work. both feb-stacking and feb-dnn are exposed to a battery of performance tests to prove the efficiency. the remaining portion of the article is organized as follows. section 2 outlines the previous related research to comprehend the evolution pattern and identify the existing gaps. subsequently, brief description of the data for accomplishing our research endeavor is provided in section 3. the entire working principle and the research methodology is then elucidated in section 4. next, predictive results are presented in detail and discussed in section 5. section 6 concludes the paper highlighting the key implications and future research potential. 2. previous research stock price predictive modeling has garnered strong focus among researchers and practitioners owing to its practical implications and arduous nature of modeling. as stated earlier, the predictive modeling of financial markets can be categorized in two strands namely, forecasting absolute figures and estimating trend direction. plethora of ai driven models have been reported to be extremely successful in capturing inherent and complex pattern driving stock market dynamics. it should also be noted that research aiming at predictive analysis has not been restricted to stock market time series data only. other financial time series variables viz. volatility, exchange rate and commodity prices too have been explored for forecasting exercises. atsalakis and valavanis (2009) developed a predictive structure based on adaptive neuro fuzzy inference system (anfis) for forecasting returns of stock markets of athens and new york. the model emerged to yield forecasts of supreme accuracy and more profitable than buy and hold (b&h) strategy. zhang et al. (2016) developed a hybrid technical indicator driven stock trend prediction system comprising adaboost, probabilistic support vector machine (psvm) and genetic algorithm. psvm was used as base learner in adaboost while ga assisted in optimal hyper-parameter tuning. rigorous performance inspection demonstrated the classification accuracy and trading benefits. feb-stacking and feb-dnn models for stock trend prediction: a performance analysis for… 55 chatzis et al. (2018) conducted predictive modeling exercises of global stock, bond, and currency markets using a series of machine learning and deep learning models during the time horizons affected by several stock market crash events. they mainly utilized salient fundamental features pertinent to respective market as explanatory features which were evaluated using boruta feature selection algorithm. as predictive modeler, classification trees, support vector machines, random forests, neural networks, extreme gradient boosting, and deep neural networks were used. findings revealed insights of practical relevance. chen and hao (2018) proposed a stock trading signal prediction system incorporating pca and weighted svm. pca was used on raw technical indicators for refinement and feature engineering process. the transformed feature set was used in weighted svm model for prediction performance. efficacy of the proposed model was validated on shanghai and shenzhen stock markets. lei (2018) developed a framework for stock price trend prediction using hybrid framework of rough set (rs) and wavelet neural network (wnn). the framework utilized several technical indicators as explanatory features which were refined through rs based feature selection model. subsequently wnn was trained on selected feature set for performing predictive exercise. efficacy of developed model was validated on trend estimation of sse composite index, csi 300 index, all ordinaries index, nikkei 225 index and dow jones index. bisoi et al. (2019) developed a hybrid granular predictive structure comprising variational mode decomposition (vmd), differential evolution (de), and a robust kernel extreme learning machine (rkelm) technique for forecasting daily prices of bse s&p 500 index (bse), hang seng index (hsi) and financial times stock exchange 100 index (ftse). vmd was deployed to better model the inherent nonlinearity, de was used for optimal parameter tuning while final prediction were drawn using rklm. the framework emerged superior to several well-known algorithms. das et al. (2019) developed an integrated model of feature selection and predictive modeling of bse sensex, nse sensex, s&p 500 index and ftse index. hybrid structure of principal component analysis (pca) and several metaheuristic searching algorithms, firefly optimization (fo) and ga was utilized for feature engineering on a set of technical indicators. subsequently, machine learning algorithms, extreme learning machine (elm), online sequential extreme learning machine (oselm) and recurrent back propagation neural network (rbpnn) were used for estimating forecasts on different time intervals. among these methods, oselm appeared to be superior. zhou et al. (2019) proposed a hybrid predictive framework of empirical mode decomposition (emd) and factorization machine based neural network for daily closing price prediction of shanghai stock exchange composite (ssec) index, the national association of securities dealers automated quotations (nasdaq) index and the standard & poor’s 500 composite stock price index (s&p 500). the predictive performance duly rationalized the efficiency of proposed architecture. ismail et al. (2020) developed a feature engineering structure based on persistent homology to form more meaningful explanatory features from original feature set for trend prediction of kuala lumpur stock exchange. the outcome of persistent homology was fed into logistic regression, artificial neural network, support vector machine and random forest for estimating one day-ahead trend movement. the combination of persistent homology and svm emerged to be the most efficient one. liu and long (2020) proposed a novel deep learning framework for stock market prediction. the framework utilized empirical wavelet transform (ewt) and outlier robust extreme learning machine (orelm) for preprocessing and long short term ghosh et al./ decis. mak. appl. manag. eng. 4 (1) (2021) 51-84 56 memory network (lstm) for forecasting. further fine tuning of lstm was carried out using particle swarm optimization (pso). the framework emerged to be superior to several benchmark models. carta et al. (2021) developed a reinforcement learning framework based on ensemble of deep q learning agents for predictive analysis of stock markets. unlike machine and deep learning models, the reinforcement learning strategy was implemented by training q-learning agent on same training samples. the framework emerged to yield excellent trading performance in comparison to conventional b&h strategy. review of the existing literature clearly indicates extensive usage of machine and deep learning driven models in stock market forecasting and classification. clear trend of hybrid granular models incorporating such models is also apparent. recently, stock market sentiment analysis and reinforcement learning have appeared to significantly contribute to precise modeling of stock market trends and absolute figures too. methodologically, either technical indicators or macro-economic variables have been predominantly used as explanatory features. nevertheless, frameworks built on amalgamation of both types of features to carry out predictive exercises in extreme volatile regimes are absent. on the other hand, behavior, co-movement, causality of various stock markets during the global financial crisis have received serious attention in the literature. characterization of stock market crashes have been elaborated as well. however, development of predictive frameworks to estimate trends during unprecedented or black swan events has seen comparatively less attention. specifically, there is paucity of predictive models to estimate financial market trend during covid-19 pandemic. moreover, the task of trend modeling needs to properly combat class imbalance and proper feature engineering issues. therefore, design of integrated frameworks to yield precise forecasts for severe conditions is of paramount significance. our research attempts to address these challenges and endeavor to design a robust framework which can significantly contribute to the previous literature on stock market prediction. 3. data and variable description 3.1. data to accomplish the research objectives, we have compiled daily closing prices of hdfc bank, tcs, reliance, and spicejet from january, 2014 to july, 2020. for performing stock price trend prediction, the datasets are segregated into two strands reflecting different time horizons. the first set comprises of data ranging from january, 2014 to december, 2019 which has been referred as set a throughout the paper. on the other hand closing price data of underlying stocks from january, 2014 to july, 2020 forms set b. the partitioning has been made in order to assess the classification accuracy of proposed predictive models on relatively less volatile time horizons and on time horizons deeply penetrated by the impact of covid-19 pandemic. thus, any analysis on set a would measure the effectiveness of proposed frameworks in trend estimation during pre-covid time horizons, whereas analysis with set b would measure quality of predictions during post-covid time horizon. figures 1 and 2 exhibit the evolutionary pattern of temporal movements of underlying variables. feb-stacking and feb-dnn models for stock trend prediction: a performance analysis for… 57 figure 1. temporal evolutionary movements of set a dataset during the pre-covid context, i.e. set a dataset, it can be observed that hdfc bank and reliance stock prices more or less exhibit dominance of trend component over short term fluctuations. tcs stock prices on the other hand demonstrate comparatively more fluctuation in addition to trend component. finally, spicejet stock prices exhibit periodic pattern with growth. hence, outcome of visual inspection suggests that banking and energy sector have performed reasonably well, while performance of it sector has undergone certain extent of uncertainty during the said time horizon. the figure of the stock price movement of the airline company reflects seasonality. ghosh et al./ decis. mak. appl. manag. eng. 4 (1) (2021) 51-84 58 figure 2. temporal evolutionary movements of set b dataset visualization of set b dataset, reflecting the impact of covid fear, clearly demonstrates drastic falls in the stock prices of selected companies. of late, stock prices of hdfc bank, tcs and reliance have displayed signs of recovery. the stock prices of spicejet, however, have not recovered from the covid shock as there exist curbs on airline movements to varying extent till now. briefly speaking, the selection of the sectors as well as the segregation of the time horizons, make the forecasting task extremely challenging and arduous. for better understanding of critical properties, descriptive statistics have been computed as well. tables 1 and 2 outline key statistical properties of the datasets. feb-stacking and feb-dnn models for stock trend prediction: a performance analysis for… 59 table 1. descriptive statistics of set a dataset properties hdfc bank tcs reliance spicejet minimum 313.2 1018 400.0 11.25 maximum 1302.4 2278 1610.0 154.00 mean 747.2 1464 757.0 74.15 median 642.5 1280 541.2 71.55 sd 277.630 370.58 336.288 44.219 skewness 0.302 0.938 0.76 0.057 kurtosis -1.284 -0.734 -0.824 -1.312 jarque-bera 123.66*** 249.96*** 184.06*** 10.6.25*** shapiro wilk 0.9216*** 0.797*** 0.841*** 0.914*** frosini test 2.3723*** 4.139*** 3.164*** 1.713*** adf test 2.5072# 1.0334# 1.9645# 0.0794# terasvirta’s nn test 42.92*** 19.459*** 11.885# 9.8803*** hurst exponent 0.8918 0.8844 0.8886 0.8813 ***significant at 1% level of significance, #not significant, sd: standard deviation, adf: augmented dickey fuller, nn: neural network table 2. descriptive statistics of set b dataset properties hdfc bank tcs reliance spicejet minimum 313.2 1018 400.0 11.25 maximum 1302.4 2310 2177.7 154.00 mean 775.2 1515 823.5 72.89 median 732.1 1296 663.9 69.38 sd 282.698 393.257 392.914 42.907 skewness 0.1582 0.670 0.772 0.128 kurtosis -1.3590 -1.218 -0.470 -1.242 jarque-bera test 131.4*** 221.68*** 176.18*** 108.31*** shapiro wilk test 0.9269*** 0.822*** 0.863*** 0.927*** frosini test 2.3531*** 3.985*** 2.963*** 1.5969*** adf test 0.8139# 0.9364# 2.4592# -0.5647# terasvirta’s nn test 25.522*** 52.076*** 32.811*** 6.2511** hurst exponent 0.8936 0.8928 0.8888 0.8828 ***significant at 1% level of significance, #not significant, sd: standard deviation, adf: augmented dickey fuller, nn: neural network it is evident that none of the underlying stocks follow normal distribution while presence of non-stationary evolutionary pattern is also apparent as manifested by outcome of jarque-bera, frosini, and shapiro-wilk tests. results of adf test clearly indicates selected stock prices are non-stationary in nature. outcome of nonlinearity assessment through terasvirta’s neural network test suggests entrenchment of nonlinear traits in all four stocks for set b datasets considering covid-19 period. in set a segment reflecting normal time horizon, tcs, reliance, and spicejet stock prices ghosh et al./ decis. mak. appl. manag. eng. 4 (1) (2021) 51-84 60 have emerged to be nonlinear. on the flipside, estimated hurst exponent figures imply the underlying time series observations of both sets exhibit long memory dependence or persistent pattern as the they are substantially greater than 0.5 (ghosh and datta chaudhuri, 2018). successful usage of technical indicators for predictive modeling of financial time series observations exhibiting persistent pattern has been reported in literature. therefore integration of technical indicators for trend prediction of chosen stocks is justified. since high degree of non-stationary and nonlinear traits with complete nonparametric movements can be observed, deployment of advanced ai models is considered appropriate. 3.2. variables the present work is aimed at stock trend prediction, i.e. to estimate whether oneday ahead closing price would increase or decrease. an increase would indicate an ‘up’ signal while decrease refers ‘down’ signal. thus objective of proposed research methodology is to correctly classify the next day movement. the aforesaid problem is also referred as binary classification the target takes two classes explicitly. mathematically the target (𝑇) can be explained as: 𝑇 = { 0 𝑖𝑓 (𝑃𝑖 − 𝑃𝑖−1) < 0 1 𝑖𝑓 (𝑃𝑖 − 𝑃𝑖−1) ≥ 0 (1) where, 𝑃𝑖 and 𝑃𝑖−1 represent closing prices of two consecutive days of any stock we attempt to develop a robust predictive structure to estimate the future trend direction, i.e. 0 (‘down’) and 1 (‘up’) of hdfc bank, tcs, reliance, and spicejet share prices. as empirical analysis of considered datasets hint at existence of long memory dependence, several technical indicators which are computed by performing simple mathematical operations on closing prices have been selected as explanatory features as outlined in table 3. table 3. list of technical indicators no . feature formulae 1. one day back closing price (lag1) 𝐿𝐴𝐺1 = 𝑃𝑖−1 where 𝑃𝑖−1 denotes closing value at previous day 2. two-day back closing price (lag2) 𝐿𝐴𝐺2 = 𝑃𝑖−2 3. three-day back closing price (lag3) 𝐿𝐴𝐺3 = 𝑃𝑖−3 4. four-day back closing price (lag4) 𝐿𝐴𝐺4 = 𝑃𝑖−4 5. five-day back closing price (lag5) 𝐿𝐴𝐺5 = 𝑃𝑖−5 6. 5-day moving average (ma5) 𝑀𝐴5 = ∑ 𝑃𝑖 𝑗 𝑖=𝑗−4 5 7. 10-day moving average (ma10) 𝑀𝐴10 = ∑ 𝑃𝑖 𝑗 𝑖=𝑗−9 10 8 20-day moving average (ma20) 𝑀𝐴20 = ∑ 𝑃𝑖 𝑗 𝑖=𝑗−19 20 feb-stacking and feb-dnn models for stock trend prediction: a performance analysis for… 61 no . feature formulae 9. 5-day bias (b5) 𝐵5 = 𝑃𝑖−𝑀𝐴5 𝑀𝐴5 10. 10-day bias (b10) 𝐵10 = 𝑃𝑖−𝑀𝐴10 𝑀𝐴10 11. 20-day bias (b20) 𝐵200 = 𝑃𝑖−𝑀𝐴20 𝑀𝐴20 12. 5-day momentum (mtm5) 𝑀𝑇𝑀5 = 𝑃𝑖 − 𝑃𝑖−5 13. 10-day momentum (mtm10) 𝑀𝑇𝑀10 = 𝑃𝑖 − 𝑃𝑖−10 14. 20-day momentum (mtm20) 𝑀𝑇𝑀20 = 𝑃𝑖 − 𝑃𝑖−20 15. 5-day exponential moving average (ema5) 𝐸𝑀𝐴5 = 2 5+1 × 𝑃5 + 5−1 5+1 × 𝐸𝑀𝐴4 , where 𝐸𝑀𝐴1 = 𝑃1 16. 10-day exponential moving average (ema10) 𝐸𝑀𝐴10 = 2 10+1 × 𝑃9 + 10−1 10+1 × 𝐸𝑀𝐴9 17. 20-day exponential moving average (ema10) 𝐸𝑀𝐴20 = 2 20+1 × 𝑃19 + 20−1 20+1 × 𝐸𝑀𝐴19 18. 5-day rate of change (roc5) 𝑅𝑂𝐶5 = 𝑃𝑖−𝑃𝑖−5 𝑃𝑖−5 19. 10-day rate of change (roc10) 𝑅𝑂𝐶10 = 𝑃𝑖−𝑃𝑖−10 𝐸𝐶𝑖−10 20. 20-day rate of change (roc20) 𝑅𝑂𝐶20 = 𝑃𝑖−𝑃𝑖−20 𝐸𝐶𝑖−20 21. upper bollinger band (ub) 𝑈𝐵 = 𝑀𝐴20 + (20 × 𝜎20) where 𝜎20 denotes standard deviation of previous 20 days closing prices 22. lower bollinger band (lb) 𝐿𝐵 = 𝑀𝐴20 − (20 × 𝜎20) 23. difference (diff) 𝐷𝐼𝐹𝐹 = 𝐸𝑀𝐴26 − 𝐸𝑀𝐴12 24. moving average convergence divergence (macd) 𝑀𝐴𝐶𝐷 = 2 × (𝐷𝐼𝐹𝐹 − 𝐷𝐸𝐴); 𝐷𝐸𝐴 = 𝐸𝑀𝐴(𝐷𝐼𝐹𝐹) 25. difference of high and low price (h-l) 𝐻 − 𝐿 = 𝐻𝑃𝑖−1 − 𝐿𝑃𝑖−1 ; 𝐻𝑃𝑖−1 and 𝐿𝑃𝑖−1 denote high and low price of previous day 26. difference of closing and opening price (c-o) 𝐶 − 𝑂 = 𝐶𝑃𝑖−1 − 𝑂𝑃𝑖−1 ; 𝐶𝑃𝑖−1 and 𝑂𝑃𝑖−1 denote closing and opening price of previous day alongside technical indicators, several key macroeconomic features representing sector outlook, raw material prices, market fear, and market sentiment have been added to the explanatory variable list as well. as discussed earlier, majority of past ghosh et al./ decis. mak. appl. manag. eng. 4 (1) (2021) 51-84 62 literature have either relied either on technical features or on macroeconomic constructs. the present paper combines them to achieve better classification accuracy in extreme circumstances. table 4 reports the macroeconomic variables used in the analysis. table 4. macroeconomic variable details stocks macroeconomic indicators components hdfc bank nifty, india vix, nifty bank index market sentiment, market fear, and sectoral outlook tcs nifty, india vix, it sectoral index, rupee-dollar exchange rate market sentiment, market fear, sectoral outlook, and foreign exchange rate reliance nifty, india vix, energy sectoral index, crude oil price market sentiment, market fear, sectoral outlook, and raw material price. spicejet nifty, india vix, crude oil price market sentiment, market fear, and raw material price. technical features remain uniform for all four stocks while the macroeconomic features vary according to the industry segment. the combined set of raw explanatory features will undergo rigorous feature engineering process through kpca technique before being deployed for the prediction process. 4. methodology this section articulates the utilized components of integrated predictive architectures, feb-stacking and feb-dnn chronologically. figure 3 depicts the integrated research framework. feb-stacking and feb-dnn models for stock trend prediction: a performance analysis for… 63 figure 3. flowchart of research framework the structure of the integrated research model shown in figure 3 demonstrates the flow of deployment of the different components in a seamless manner. initially after compilation and segregation of datasets across pre-covid and post-covid regimes, macroeconomic indicators and technical features are arranged as explanatory variables for estimating trend of chosen stocks. subsequently, bootstrapping and kpca have been evoked to sort class imbalance problem and ghosh et al./ decis. mak. appl. manag. eng. 4 (1) (2021) 51-84 64 feature engineering process respectively. stacking and dnn models are then applied on feature engineered and bootstrapped samples for carrying out predictive analysis to automatically estimate trends. a battery of numerical evaluations and statistical tests are utilized to critically assess effectiveness of both forecasting frameworks. we next, briefly expound the principles of utilized research components. 4.1. kernel principal component analysis (kpca) kernel principal component analysis (kpca) is an extension of ordinary pca method (scholkopf et al., 1999), where to tackle non linearity, input data space is mapped into feature space. usually a kernel function is used to carry out the inner products in the feature space without explicitly defining transformation φ. the present research utilizes well known radial basis kernel for accomplishing the task. after the said transformation, orthodox pca is invoked on the transformed dataset. applying kpca transformation, we project the raw set of explanatory features comprising technical and macroeconomic indicators into feature space which would be ideal for precise classification of the futuristic trends of chosen stocks. thus, the objective of fe process through kpca is not to reduce feature set but to obtain retransformation for better predictions. we next explain how class imbalance problem has been tackled in the proposed predictive architectures. 4.2. fixing class imbalance the problem of data classification refers to the imbalance distribution of target variable classes in the dataset. the target variable which we have set in this study is strictly binary in nature. however, anticipation of crash in markets, uneven bearish and bullish phases may lead into severe imbalance in distribution of the target construct. there exists high possibility for models built on such dataset to exhibit overfitting phenomenon, thereby performing poorly in test data segments. thus, it is necessary to balance the ratio of up and down signals of our dataset to be balanced beforehand. literature reports usage of random up and down resampling approaches as bootstrapping driven solution for dealing with class imbalance problem. in this work, we have opted for up-sampling approach to generate artificial data in order to compensate the lagging proportion of a particular class depending on actual count. the ratio of ‘up’ (1) and ‘down’ (0) signals as expressed by equation 1 is estimated beforehand and up-sampling is applied to the lagging signals in order to keep the ratio even. we now proceed to discuss the principles of stacking and dnn used for yielding predictions exhaustively. 4.3. stacking it replicates the working principle of typical ensemble machine learning frameworks where predictions from multiple models are used as inputs to yield the final predictions for developing forecasting framework. in this work, stacking has been applied on predictions obtained through three different ensemble learning models namely, gradient boosting (gb), random forest (rf), and bagging. the final training of stacking is achieved through deploying a separate rf model, with 200 base learners, which acts as final stacking classifier. detailed of constituent models have been elucidated as follows. the stacking framework has been implemented using ensemble utilities of ‘sklearn’ library of python. https://courses.analyticsvidhya.com/courses/ensemble-learning-and-ensemble-learning-techniques?utm_source=blog&utm_medium=comprehensive-guide-for-ensemble-models https://courses.analyticsvidhya.com/courses/ensemble-learning-and-ensemble-learning-techniques?utm_source=blog&utm_medium=comprehensive-guide-for-ensemble-models feb-stacking and feb-dnn models for stock trend prediction: a performance analysis for… 65 4.3.1. gradient boosting (gb) boosting is an ensemble predictive analysis technique where a series of different learning algorithms are applied in a forward-stage wise manner to generate final predictions (schapire and singer, 1999). gradient boosting is a variant of classical boosting algorithm which basically mimics the same principle with an extension of identification of training samples via determination gradient driven error rate computation. decision trees for classification have been used as base learners sequentially in forwards direction. simulation of the method has been carried out using ‘sklearn’ library in python programming environment. in the implementation part of gb algorithm, learning rate (0.05), number of base learners (300), maximum number of feature (7), and maximum depth (5) of decision trees have been considered for hyper parameter tuning which is basically accomplished through ‘gridsearch’ utility of python library. default figures of other parameters have been considered. 4.3.2. random forest (rf) it is an ensemble based machine learning model comprising decision trees as base learners. rf, developed by breiman (2001), is characterized by its high precision, robustness to outliers and effective execution time. since inception, it has garnered tremendous attention among the academic fraternity and practitioners for solving classification and regression tasks (lariviere and van den poel, 2005; liu et al., 2013). since the underlying research problem of the paper is binary classification, decision trees for classification have been chosen as base learners. number of base learners in rf can be arbitrary and depend on complexity of the problem. final assignment of class label information (for classification task) or estimation of continuous outcome (for regression task) on test data set is carried out through majority voting or averaging scheme. three parameters namely, maximum features (8), number of base learners (500), and minimum number of samples for split (2), have been fine-tuned using ‘gridsearch’ utility of python library, while default values of other parameters have been considered. 4.3.3. bagging similar to rf, bagging (bootstrap aggregating) also follows similar ensemble properties for modelling data classification tasks (lemmens and croux, 2006; zheng et al. 2011; simidjievski et al., 2015). it too utilizes decision tree for classification as constituent base learner. majority voting scheme is applied to draw final predictions based on the outcome of individual trees which grow in bootstrapped samples drawn from training samples. outcome of individual clan ensemble based predictive modelling technique however, differs from former in implementation ensemble learning. bagging reduces the variance of unstable learning methods leading to improved prediction. there are differences between bagging and rf. only a subset of features are chosen randomly from set of all features for splitting operations of constituent decision trees in rf, whereas bagging evaluates all features to identify the most suitable for splitting operations. thus, incorporating rf and bagging together in stacking structure would cancel out the effects of over fitting and under fitting. for implementing bagging, number of base learners (350), maximum numbers of features (8), maximum samples (1.0), and feature bootstrapping (false) have been auto-tuned using ‘gridsearch’ utility keeping default values of remaining parameters. gb, rf, and bagging receive technical indicators and macroeconomic indicators of respective stocks outlined in tables 3 and 4 as inputs for predicting the target defined ghosh et al./ decis. mak. appl. manag. eng. 4 (1) (2021) 51-84 66 in equation 1. predictions obtained by the three models are fed as inputs in the stacking framework to obtain the final predictions. the entire modelling, i.e. combination of fe through kpca, bootstrapping via upsampling for sorting class imbalance, and stacking for yielding prediction, has been implemented using python programming language. as stated, stacking combines the outcome of gb, rf, and bagging and treat them as new set of features for explaining the movements of trend. it should be noted that all these methods are dependent on several process hyper-parameters which have been auto tuned invoking ‘gridsearch’ utility of python library. the integrated feb-stacking has been evaluated separately on set a and set b observations for ascertaining performance in pre-covid and postcovid time periods distinctly. for assessing the predictive performance, typical classification measures viz. roc curve, specificity, sensitivity, and various other measures have been used as discussed in sub-sections 4.5 and 4.6 respectively. 4.4. deep neural network (dnn) artificial neural network (ann) models have emerged to be highly effective and successful in modeling complex pattern recognition problems throughout the literature. the ann architecture comprises of three distinct layers, input layer, hidden layer, and output layer. with rapid development and success of deep learning methodologies, a subset of ai field, focus has been put to examine efficacy deep neural network (dnn) structures where multiple hidden layers are incorporated in standard ann architecture for carrying out predictive analysis tasks (liu et al., 2017; qureshi et al., 2017). these hidden layers act as additional feature engineering process in the context of predictive modeling tasks. in this problem, these layers additionally refine the fed input features for performing classification. individual hidden layers of dnn comprise of several neurons connected to neurons of adjacent layers. they receive inputs from the previous layer and estimate output for propagation to next layer. in this work two hidden layers of 50 nodes each, have been deployed. transformation functions are utilized for generation of output through deployment of activation functions. literature reports different activation functions including ‘identity’, ‘sigmoid’, ‘tanh’, and ‘relu’. in this research ‘relu’ (rectified linear unit) function has been used as activation function. the training of dnn is achieved through adjusting connection weights and biases based on the amount of error in the output compared to the expected result encapsulated in the loss function. this learning process is carried out through forwardand back-propagation and solved by the “adam” optimizer, which is an algorithm for optimization of stochastic objective functions, proposed by kingma and ba (2014). all technical and macroeconomic indicators comprise the input layer, which undergoes series of transformations in hidden layer in order to generate the future trend as output. feature engineering and bootstrapping processes are combined with dnn to form feb-dnn model to estimate trend predictions of hdfc bank, tcs, reliance, and spicejet during normal and new-normal time horizons. the model is simulated using keras interface in python programming framework. likewise feb-stacking, set a and set b data samples are used to test predictive ability of feb-dnn at pre-covid and post-covid time frames. to evaluate the classification performance of respective models, visual metric and quantitative indices have been obtained. visual metric in the form of receiver operating characteristic (roc) curve is determined while several quantitative binary classification indices are estimated also. feb-stacking and feb-dnn models for stock trend prediction: a performance analysis for… 67 4.5. receiver operating characteristic (roc) curve it is used for evaluating the predictive performance of a classifier and seldom utilized for model selection. roc curve depicts a visualization of sensitivity represented by vertical axis and 1-specificity represented by horizontal axis. basically, it reflects the probability of correctly specifying a random pair of positive and negative instances. to get quantitative information from roc curve, area under the curve (auc) is estimated. models associated with higher auc values are said to yield better and accurate predictions. it should be close to 1 to indicate superior classification performance. 4.6. quantitative measures to evaluate efficiency of proposed predictive structures, feb-stacking and febdnn, the present research has utilized a series of quantitative indices which are mathematically expressed as: 𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 = 𝑇𝑃 𝑇𝑃+𝐹𝑁 (2) 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐𝑖𝑡𝑦 = 𝑇𝑁 𝑇𝑁+𝐹𝑃 (3) 𝐺 = √𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 ∗ 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐𝑖𝑡𝑦 (4) 𝐿𝑃 = 𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 1−𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐𝑖𝑡𝑦 (5) 𝐿𝑅 = 1−𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐𝑖𝑡𝑦 (6) 𝐷𝑃 = √3 𝜋 [𝑙𝑜𝑔 𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 1−𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐𝑖𝑡𝑦 + 𝑙𝑜𝑔 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐𝑖𝑡𝑦 1−𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 ] (7) 𝛾 = 𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 − (1 − 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐𝑖𝑡𝑦) (8) 𝐵𝐴 = 1 2 (𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 + 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐𝑖𝑡𝑦) (9) 𝑀𝐶𝐶 = (𝑇𝑃×𝑇𝑁)−(𝐹𝑃×𝐹𝑁) √(𝑇𝑃+𝐹𝑃)(𝑇𝑃+𝐹𝑁)(𝑇𝑁+𝐹𝑃)(𝑇𝑁+𝐹𝑁) (10) 𝐹1 = 2𝑇𝑃 2𝑇𝑃+𝐹𝑃+𝐹𝑁 (11) 𝐹𝑀 = √ 𝑇𝑃 𝑇𝑃+𝐹𝑃 × 𝑇𝑃 𝑇𝑃+𝐹𝑁 (12) tp denotes true positive ratio signifying the number of positive cases which are correctly classified as positive. the positive case in this work refers to up signal. tn signifies true negative ratio that accounts for the number of negative cases (i.e. down signal) correctly classified as negative. on the other hand, fn denotes the number of positive cases misclassified as negative while fp implies the number of negative cases predicted as positives. thus, magnitude of tp and tn should ideally be close to 1 for accurate classification whilst fp and fn values should be close to 0. magnitudes of specificity and sensitivity should be close to 1 as well for models to be regarded as supreme. g-mean attempts to measure the balance between the performances of classifying positive and negative classes. poor performance in correctly classifying positive cases would result in low g-mean value in spite of good accuracy in predicting negative cases. lp is positive likelihood ratio measuring the probability of classifying an instance as positive when it is negative actually and probability of classifying an actual positive instance as positive. lr reflects the opposite scenario, i.e. the ratio of probability of classifying an instance as negative when it is actually positive and probability of classifying a negative instance correctly. higher lp and lower lr figures imply precise classification. ghosh et al./ decis. mak. appl. manag. eng. 4 (1) (2021) 51-84 68 dp reflects the discriminant power of underlying classification models. dp values higher than 1 indicates supreme distinguishable capability. youden’s index (𝛾) and balanced accuracy (ba) figures should be close to 1 as well. similarly, mathews correlation coefficient (mcc), f1 score, and fowlkes-mallows (fm) index figures should lie close to 1 to infer high quality predictions. apart from checking the classification accuracy, to measure the practical benefits of deploying feb-stacking and feb-dnn models, trading benefits of both models have been estimated too. 4.7. trading benefits to demonstrate the practical effectiveness of proposed framework a comparison with orthodox buy and hold (b&h) strategy has been conducted. the b&h strategy implies that the investor will invest a quantum of money in a particular stock and hold the same for a predefined time horizons, generally 3 months to 6 months duration. the net profit under this scheme is estimated after the completion of the time horizon. on the contrary, the proposed model suggests to invest for a predicted up (1) signal and to sell for predicted down (0) signal next day. the said process is continued for the entire time horizon. thus the rate of return (𝑅𝑂𝑅) can be calculated as: 𝑅𝑂𝑅 = 𝑛𝑒𝑡 𝑔𝑎𝑖𝑛 𝑖𝑛 𝑠𝑡𝑜𝑐𝑘 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 (13) therefore, based on the estimated 𝑅𝑂𝑅 figures, profitability of b & h strategy, febstacking, feb-dnn respectively can be determined and relative performance can be measured. the said exercise has been performed for time horizon of 3 months at separate time horizons. finally to perform comparative statistical analysis with various other models, diebold-mariano’s pairwise test for equal predictive ability has been evoked. 5. results and discussions executing classification exercise requires designing of training and test data segments systematically. since we have two set of data samples sets a and b, for critically evaluating the performance of feb-stacking and feb-dnn on pre-covid and post-covid contexts respectively, training and test partitions have been formed for both sets in order to ascertain the predictive capabilities during normal and newnormal time horizons. the segmentation is made in forward looking direction which has been reported to be successfully utilized for time series prediction (ghosh et al., 2019). for set a observations ranging from january, 2014 to december, 2018 constitute training data points where as test segment comprises of observations from january, 2019 to december, 2019. the said partitioning evaluates the classification accuracy of feb-stacking and feb-dnn models during the pre-covid time horizons characterized by relatively low volatility and uncertainty. on the other hand, observations of january, 2014 to december, 2019 compose the training samples whilst data points spanning from january, 2020 to july, 2020 make up the test segment for set b. the designed segmentation of set b sample measures the predictive ability of respective models during the time period where the covid-19 pandemic wreaked havoc. as discussed, stacking is implemented by combing output of rf, bagging, gb methods. these methods however are governed by several process parameters. to identify the most competent setting of hyper-parameters, the ‘gridserach’ tool available at keras interface has been evoked. all three constituent ensemble models are highly sensitive to parameters viz. number of base estimators, number of features feb-stacking and feb-dnn models for stock trend prediction: a performance analysis for… 69 for branching operations of base learners, leaf nodes, etc. using ‘gridsearch’ utility these parameters can be varied and combinatorial search operation is performed to select the most prominent combination. on contrary, dnn with 2 hidden layers comprising 30 nodes each have been selected for learning process. rectified linear (relu) activation function has been used at input and hidden layers whilst linear activation function has been applied at output layer. selection of batch size, number of iterations, and optimizer for learning process has been made through performing gridsearch utility of keras. the well-known ‘adam’ optimizer has been found to be the optimal one. 5.1. predictive accuracy the following figures 4-7 exhibit the resultant roc plots alongside auc values for feb-stacking and feb-dnn models on sets a and b. figure 4. roc curve of feb-stacking on test segment of set a observations ghosh et al./ decis. mak. appl. manag. eng. 4 (1) (2021) 51-84 70 figure 5. roc curve of feb-stacking on test segment of set b observations feb-stacking and feb-dnn models for stock trend prediction: a performance analysis for… 71 figure 6. roc curve of feb-dnn on test segment of set a observations ghosh et al./ decis. mak. appl. manag. eng. 4 (1) (2021) 51-84 72 figure 7. roc curve of feb-dnn on test segment of set b observations it can be noticed visually that auc (represented by area in figures) values of resultant roc curves on test data segments of pre-covid and post-covid periods have emerged to be pretty high for both feb-stacking and feb-dnn models which basically implies good trend prediction performance. nevertheless, to validate the inference drawn based on visual metrics, quantitative indices are estimated as well and presented in tables 5-8. at first, table 5 outlines summary of performance of febstacking on set a samples. feb-stacking and feb-dnn models for stock trend prediction: a performance analysis for… 73 table 5. predictive performance of feb-stacking on set a hdfc bank tcs reliance spicejet training data set 𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 0.9870 0.9931 0.9913 0.9870 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐𝑖𝑡𝑦 0.9121 0.9241 0.9226 0.9108 𝐺 0.9488 0.9580 0.9563 0.9481 𝐿𝑃 11.2563 11.6528 11.5796 11.065 𝐿𝑅 0.0143 0.0075 0.0094 0.0143 𝐷𝑃 1.5962 1.7876 1.7266 1.5924 𝛾 0.8991 0.9172 0.9139 0.8978 𝐵𝐴 0.9495 0.9586 0.9569 0.9489 𝑀𝐶𝐶 0.8102 0.8346 0.8328 0.8093 𝐹1 0.8979 0.9186 0.9164 0.8972 𝐹𝑀 0.8935 0.9175 0.9170 0.8928 𝐴𝑈𝐶 0.906 0.934 0.935 0.901 test data set 𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 0.9759 0.9801 0.9790 0.9748 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐𝑖𝑡𝑦 0.9086 0.9154 0.9139 0.9086 𝐺 0.9417 0.9472 0.9458 0.9411 𝐿𝑃 10.6795 10.7046 10.6794 10.665 𝐿𝑅 0.0264 0.0215 0.0219 0.0277 𝐷𝑃 1.4355 1.5027 1.4849 1.4246 𝛾 0.8846 0.8955 0.8929 0.8834 𝐵𝐴 0.9423 0.9477 0.9465 0.9417 𝑀𝐶𝐶 0.7910 0.8247 0.8232 0.7903 𝐹1 0.8823 0.9078 0.9066 0.8807 𝐹𝑀 0.8824 0.9096 0.9073 0.8811 𝐴𝑈𝐶 0.895 0.921 0.923 0.894 it can be noticed that values of performance indicators on both training and test samples clearly lie on the zone which simply indicate remarkable performance of febstacking framework in carrying out directional predictive modeling of stock prices of hdfc bank, tcs, reliance, and spicejet during the pre-covid time horizons. values of sensitivity, specificity, 𝑮, 𝜸, 𝑩𝑨, 𝑴𝑪𝑪, 𝑭𝟏,𝑭𝑴,and 𝑨𝑼𝑪 have emerged to be close to 1. superior capability of the proposed framework in distinctly predicting up and down trend can be inferred. high values of 𝑳𝑷 and low values of 𝑳𝑹 further solidify the claim. therefore, it can be concluded that before the outbreak of covid, i.e. in pre-covid scenario, feb-stacking has accurately predicted future movements of hdfc bank, tcs, reliance, and spicejet stocks. we next, examine the performance of feb-stacking framework on trend prediction of underlying stocks on set b dataset reflecting the scare part of covid-19 pandemic. table 6 summarizes the said findings. ghosh et al./ decis. mak. appl. manag. eng. 4 (1) (2021) 51-84 74 table 6. predictive performance of feb-stacking on set b hdfc bank tcs reliance spicejet training data set 𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 0.9778 0.9437 0.9789 0.9743 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐𝑖𝑡𝑦 0.9091 0.8785 0.9167 0.9069 𝐺 0.9428 0.9105 0.9473 0.9400 𝐿𝑃 10.8248 10.5239 10.6918 10.4651 𝐿𝑅 0.0238 0.0254 0.0214 0.0283 𝐷𝑃 1.4571 1.1482 1.4924 1.4149 𝛾 0.8870 0.8221 0.9473 0.8812 𝐵𝐴 0.9435 0.9111 0.9478 0.9406 𝑀𝐶𝐶 0.7926 0.7711 0.7975 0.7904 𝐹1 0.8936 0.8692 0.9018 0.8917 𝐹𝑀 0.8841 0.8677 0.8924 0.8813 𝐴𝑈𝐶 0.889 0.869 0.908 0.878 test data set 𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 0.9683 0.9357 0.9779 0.9587 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐𝑖𝑡𝑦 0.8987 0.8698 0.9086 0.8914 𝐺 0.9329 0.9021 0.9426 0.9244 𝐿𝑃 9.5587 7.1866 10.6991 8.8278 𝐿𝑅 0.0322 0.0325 0.0243 0.0463 𝐷𝑃 1.3408 1.0954 1.4568 1.2565 𝛾 0.8670 0.8055 0.8865 0.8501 𝐵𝐴 0.9335 0.9028 0.9433 0.9251 𝑀𝐶𝐶 0.7819 0.7625 0.7810 0.7737 𝐹1 0.8847 0.8611 0.8833 0.8788 𝐹𝑀 0.8768 0.8590 0.8857 0.8695 𝐴𝑈𝐶 0.880 0.858 0.899 0.867 like the performance on set a, efficacy of feb-stacking framework in trend modeling is apparent on set b as well as manifested by the figures of chosen performance indicators. however, it must be noted that the classification performance has marginally deteriorated as drop in sensitivity, specificity, 𝑮, 𝑳𝑹, 𝜸, 𝑩𝑨, 𝑴𝑪𝑪, 𝑭𝟏, and 𝑭𝑴 values can be observed whilst an increase in magnitude of 𝑳𝑹 is imminent on both training and test samples. the outcome is expected and logical due to the unprecedented shock induced by covid-19 pandemic. nevertheless, the figures of all these measures indeed indicate predictions of superior quality. therefore, the framework can be regarded to be extremely efficient to yield predictions at extreme events as well. subsequently, we evaluate the predictive capability of feb-dnn on set a and set b datasets. table 7 reports outcome of predictive performance feb-dnn on set a samples. feb-stacking and feb-dnn models for stock trend prediction: a performance analysis for… 75 table 7. predictive performance of feb-dnn on set a hdfc bank tcs reliance spicejet training data set 𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 0.9896 0.9915 0.9904 0.9861 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐𝑖𝑡𝑦 0.9139 0.9227 0.9210 0.9095 𝐺 0.9510 0.9565 0.9551 0.9470 𝐿𝑃 11.2601 11.6509 11.5781 10.8961 𝐿𝑅 0.0174 0.0149 0.0155 0.0153 𝐷𝑃 1.6557 1.7325 1.6975 1.5661 𝛾 0.9035 0.9142 0.9114 0.8956 𝐵𝐴 0.9518 0.9571 0.9557 0.9478 𝑀𝐶𝐶 0.8132 0.8297 0.8328 0.8104 𝐹1 0.9034 0.9159 0.9164 0.8988 𝐹𝑀 0.8976 0.9144 0.9170 0.8943 𝐴𝑈𝐶 0.895 0.933 0.905 0.880 test data set 𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 0.9783 0.9794 0.9753 0.9757 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐𝑖𝑡𝑦 0.9097 0.9138 0.9107 0.9081 𝐺 0.9434 0.9460 0.9424 0.9413 𝐿𝑃 10.8615 10.6780 10.6772 10.6170 𝐿𝑅 0.02489 0.0233 0.0225 0.0268 𝐷𝑃 1.4644 1.4893 1.4356 1.4320 𝛾 0.8880 0.8932 0.8860 0.8838 𝐵𝐴 0.9440 0.9466 0.9430 0.9419 𝑀𝐶𝐶 0.7966 0.8213 0.8209 0.7943 𝐹1 0.8875 0.9044 0.9052 0.8837 𝐹𝑀 0.8849 0.9061 0.9059 0.8811 𝐴𝑈𝐶 0.886 0.922 0.899 0.873 similar to feb-stacking, predictive performance of feb-dnn has emerged to be of supreme quality as manifested by the estimated classification indicators on both training and test data segments. hence, feb-dnn too can be regarded to be an extremely effective tool for trend prediction of chosen stocks during the normal time horizon i.e, pre-covid time frame. table 8 reports quality of performance on set b samples. ghosh et al./ decis. mak. appl. manag. eng. 4 (1) (2021) 51-84 76 table 8. predictive performance of feb-dnn on set b hdfc bank tcs reliance spicejet training data set 𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 0.9769 0.9439 0.9794 0.9548 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐𝑖𝑡𝑦 0.9086 0.8774 0.9186 0.8983 𝐺 0.9428 0.9105 0.9473 0.9261 𝐿𝑃 10.8237 10.5221 10.6927 9.3884 𝐿𝑅 0.0254 0.0639 0.0224 0.0503 𝐷𝑃 1.4459 1.1467 1.5043 1.2515 𝛾 0.8870 0.8221 0.9473 0.8531 𝐵𝐴 0.9435 0.9111 0.9478 0.9402 𝑀𝐶𝐶 0.7914 0.7706 0.7994 0.7876 𝐹1 0.8922 0.8683 0.9031 0.8879 𝐹𝑀 0.8829 0.8668 0.8936 0.8792 𝐴𝑈𝐶 0.884 0.861 0.899 0.871 test data set 𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 0.9657 0.9312 0.9788 0.9489 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐𝑖𝑡𝑦 0.8964 0.8673 0.9101 0.8936 𝐺 0.9329 0.9021 0.9426 0.9208 𝐿𝑃 9.3214 7.0173 10.8877 8.9182 𝐿𝑅 0.0383 0.0793 0.0233 0.0572 𝐷𝑃 1.3153 1.0729 1.4713 1.0965 𝛾 0.8670 0.8055 0.8865 0.8425 𝐵𝐴 0.9335 0.9028 0.9433 0.9213 𝑀𝐶𝐶 0.7804 0.7598 0.7835 0.7746 𝐹1 0.8829 0.8587 0.8856 0.8723 𝐹𝑀 0.8747 0.8573 0.8874 0.8683 𝐴𝑈𝐶 0.877 0.852 0.892 0.855 inspection of classification exercise on dataset carrying impact of covid-19 pandemic reveals similar phenomenon observed in feb-stacking model. classification performance of feb-dnn model in set b has seen a marginal drop in accuracy as compared to set a. however, the overall figures of the indicators on both training and test samples does suggest that feb-dnn has achieved noteworthy performance on highly volatile and uncertain time horizons affected by covid-19 pandemic. 5.2. profitability analysis to evaluate trading benefits of proposed schemes, feb-stacking and feb-dnn, samples of approximately 1 month periods have been selected. during the selected time intervals b&h strategy is invoked to estimate the ror%. finally, ror% based on predictions made by feb-stacking and feb-dnn has been separately computed to perform a buy operation when trend of next day is predicted to be ‘up’ (1) and sell operation if predicted trend of next day is ‘down’ (0). the said exercises have been feb-stacking and feb-dnn models for stock trend prediction: a performance analysis for… 77 repeated on three different time windows to evaluate the trading benefits of respective models. table 9 reports the findings. table 9. outcome of profitability analysis hdfc bank tcs reliance spicejet time period 1/9/2016 – 7/10/2016 ror% of febstacking 13.52% 18.16% 31.25% 11.26% ror% of febdnn 13.41% 18.31% 31.09% 10.98% ror% of b&h strategy 0.067% -5.57% 7.75% -3.46% time period 1/7/20196/8/2019 ror% of febstacking 9.86% 11.20% 8.54% 8.17% ror% of febdnn 9.77% 11.35% 8.69% 7.84% ror% of b&h strategy -11.55% -1.10% -11.05% -12.61% time period 6/5/20208/6/2020 ror% of febstacking 6.59% 11.19% 13.97% 5.32% ror% of febdnn 6.46% 10.97% 13.91% 5.24% ror% of b&h strategy -29.71% -12.05% -9.78% -24.66% time periods have been chosen randomly by critically covering the pre-covid and post-covid time horizons. first two samples assess the trading benefits of proposed models on normal time periods whilst the third sample evaluates profitability during new normal periods. results clearly suggest dominance of both feb-stacking and febdnn models over the orthodox b&h strategy as estimated ror% figures of both models are substantially higher than the latter one on all three occasions. outcome of profitability analysis is of paramount significance for investors as the proposed prediction models have emerged to yield substantial amount of profit even during the time of unprecedented circumstances owing to covid-19 outbreak. performance turned out to be exceptionally superior as compared to b&h strategy for normal time span as well. among the stocks, reliance has emerged to be most profitable which basically implies its superior performance in turbulent time as well. on the flipside, spicejet has turned out to be relatively less profitable, in comparison to the counterparts suggesting low confidence of investors. it must be noted that the proposed frameworks are tailor made for evaluation through ror% to comprehend trading benefits. inspection of risk related performance is beyond the scope of present work. ghosh et al./ decis. mak. appl. manag. eng. 4 (1) (2021) 51-84 78 5.3. comparative performance analysis to ascertain the rationale of development of feb-stacking and feb-dnn model, rf, ann, multiple adaptive regression splines (mars), support vector machine (svm), and recurrent neural network (rnn) models have been applied to perform predictive modeling on set b segment using same set of explanatory features as well. however, exclusive feature engineering through kpca and bootstrapping operations are not attached to these models. ‘gridsearch’ utility nevertheless has been utilized for finding optimal hyper-parameters of competing models. dm pairwise test has been evoked to perform pairwise comparison of underlying models. since the test operates in a pairwise format and the outcome depends on the order of components, competing models are stacked with the index numbers for referring the order in the table for ease of comprehension. a significant positive test statistic figure signifies that the performance of second model is statistically superior to the first model. if test statistic value appears to be significantly negative then opposite scenario prevails, i.e. the superiority of the first model over the second model is implied. tables 10-13 report the outcome of dm test. table 10. comparative performance assessment on hdfc bank models rf (1) ann (1) mars (1) svm (1) rnn (1) febstackin g (1) febdnn (1) rf (2) ann (2) 0.196# mars (2) 0.203# 0.208# svm (2) 0.191# 0.198# 0.221# rnn (2) 0.214# 0.213# 0.202# 0.228# feb-stacking (2) 6.9482*** 6.9678*** 6.9843*** 6.9680*** 6.9396*** feb-dnn (2) 6.9458*** 6.9615*** 6.9856*** 6.9685*** 6.9416*** 0.195# # not significant, *** significant at 1% level of significance table 11. comparative performance assessment on tcs models rf (1) ann (1) mars (1) svm (1) rnn (1) febstacking (1) febdnn (1) rf (2) ann (2) 0.194# mars (2) 0.215# 0.217# svm (2) 0.198# 0.194# 0.229# rnn (2) 0.222# 0.226# 0.234# 0.210# feb-stacking (2) 6.9536*** 6.9614*** 6.9917*** 6.9759*** 6.9421*** feb-dnn (2) 6.9567*** 6.9622*** 6.9895*** 6.9782*** 6.9457*** 0.192# # not significant, *** significant at 1% level of significance feb-stacking and feb-dnn models for stock trend prediction: a performance analysis for… 79 table 12. comparative performance assessment on reliance models rf (1) ann (1) mars (1) svm (1) rnn (1) febstacking (1) febdnn (1) rf (2) ann (2) 0.213# mars (2) 0.197# 0.220# svm (2) 0.223# 0.203# 0.224# rnn (2) 0.235# 0.218# 0.211# 0.241# feb-stacking (2) 6.9488*** 6.9646*** 6.9724*** 6.9693*** 6.9408*** feb-dnn (2) 6.9484*** 6.9679*** 6.9708*** 6.9687*** 6.9443*** 0.189# # not significant, *** significant at 1% level of significance table 13. comparative performance assessment on spicejet models rf (1) ann (1) mars (1) svm (1) rnn (1) febstacking (1) febdnn (1) rf (2) ann (2) 0.207# mars (2) 0.193# 0.204# svm (2) 0.211# 0.189# 0.229# rnn (2) 0.229# 0.213# 0.232# 0.226# feb-stacking (2) 6.9276*** 6.9519*** 6.9631*** 6.9617*** 6.9359*** feb-dnn (2) 6.9327*** 6.9608*** 6.9674*** 6.9622*** 6.938*** 0.196# # not significant, *** significant at 1% level of significance sign and significance levels of dm test statistics clearly imply that feb-stacking and feb-dnn have resulted in statistically superior trend predictions for all four underlying stocks, hdfc bank, tcs, reliance, and spicejet as compared to the five other models. on the other hand, no clear statistical evidence can be found to discriminate the performance of competing models. therefore, outcome of comparative study clearly suggests supremacy of both feb-stacking and feb-dnn over the remaining competing models in precisely estimating the trend of selected stocks in challenging times. therefore, the importance of performing feature engineering bootstrapping apart from using high end stacking and dnn models is also justified. hence, both feb-stacking and feb-dnn frameworks have emerged to be extremely efficient and precise estimation of stock trends in normal and new-normal time horizons. specifically, the performance during the covid-19 pandemic is noteworthy and can immensely benefit traders and investors. our findings reveal that both frameworks, feb-stacking and feb-dnn have emerged to be highly successful in trend classification of hdfc bank, tcs, reliance, and spicejet on both set of exercises. quality of predictions during pre-covid period has emerged to be marginally superior to the predictions obtained in post-covid period. nevertheless, the proposed architectures statistically outperformed several benchmark predictive tools during the said period. the models have appeared to be ghosh et al./ decis. mak. appl. manag. eng. 4 (1) (2021) 51-84 80 highly profitable for trading purposes as well during the covid-19 outbreak. the predictive structures have successfully accomplished the research objectives and can be regarded to be a contribution to the existing trend prediction literature. the strength of both frameworks lies in seamless integration of feature engineering, bootstrapping, and pattern mining process. both frameworks have emerged to be highly successful in generating precise estimates of future across pre-covid and postcovid regimes. as expected, during post-covid time horizons, prediction accuracy has marginally suffered. with availability of future data samples, both feb-stacking and feb-dnn models can be tested for quality of accuracy over a prolonged period affected by covid pandemic. both models require identification of explanatory features beforehand. other advanced deep learning models, gru, lstm, cnn, gnn, etc. have been reported to be extremely successful in stock trend prediction as discussed in the literature. these models are famous for automatic extraction of features for predictive modeling. the present work, nevertheless, relied upon standard dnn model for predictive exercise. since a substantial effort was put to form explanatory features in the form of technical and macroeconomic indicators and subsequent feature refining through kpca, conventional dnn has turned out to be extremely effective in estimating trends with superior precision. however it would be interesting to explore the efficacy of our feature engineering process with aforesaid state-of-the-art deep learning models for stock trend prediction problems. 6. conclusion the present paper addresses a practical research problem of predicting trend of stock prices, particularly in volatile times resulting from the covid-19 pandemic. the developed frameworks have been found to be efficient in estimating future movements of prices of three major indian stocks belonging to three different industry verticals. feb-stacking and feb-dnn frameworks have performed quite well in trend predictions in pre-covid and post-covid periods. although the performance of the proposed architectures marginally deteriorated in post-covid period, quality of predictions still emerged to be statistically superior to several benchmark ones. apart from yielding high quality trend estimations, both frameworks have been found to be profitable as well as compared to orthodox b&h strategy, even at the time of exceedingly high uncertainty and fear in market owing to covid-19 pandemic. the key contributions of the paper are listed below. usage of technical indicators together with carefully chosen macroeconomic variables as proxies for market fear, market sentiment, sector outlook, and raw material availability. transforming the raw independent features comprising technical and macroeconomic indicators through kpca driven fe process to refine and augment the explanatory capabilities of feature set in predicting stock price trends during precovid and post-covid phases. deployment of bootstrapping method for sorting class imbalance problem for strengthening the predictive frameworks. statistically, the contribution of both these steps have been found to be of paramount significance as both feb-stacking and febdnn have outperformed the competitive models. the performance of feb-stacking and feb-dnn has emerged to be better during pre-covid period i.e. normal time horizons than the post-covid period reflecting newnormal time span. nevertheless, the predictive accuracy of proposed models has been feb-stacking and feb-dnn models for stock trend prediction: a performance analysis for… 81 found to be statistically more superior to rf, ann, mars, svm, and rnn in both time periods. increased profitability from use of both frameworks indicate that they can be effectively utilized for trading purposes. the present paper has used stock prices of three companies in the binary trend prediction problem. state-of-the-art deep learning algorithms viz. lstm, cnn, gan, etc. can be explored and compared with presented frameworks on trend predictions of wider variety stocks belonging to different sectors. in future, explainable ai can be added on top of predictive architectures to interpret the positive or negative influence of the 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(2019). emd2fnn: a strategy combining empirical mode decomposition and factorization machine based neural network for stock market trend prediction. expert systems with applications, 115, 136-151. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0317102022r * corresponding author. e-mail addresses: vishnurrr40@gmail.com (v. k. rai), santonabchakraborty@gmail.com (s. chakraborty), s_chakraborty00@yahoo.co.in (s. chakraborty), association rule mining for prediction of covid-19 vishnu kumar rai1, santonab chakraborty2 and shankar chakraborty1* 1 department of production engineering, jadavpur university, kolkata, west bengal, india 2 industrial engineering and management department, maulana abul kalam azad university of technology, west bengal, india received: 13 august 2021; accepted: 26 september 2022; available online: 17 october 2022. original scientific paper abstract: covid-19 is a raging pandemic that has created havoc with its impact ranging from loss of millions of human lives to social and economic disruptions of the entire world. therefore, error-free prediction, quick diagnosis, disease identification, isolation and treatment of a covid patient have become extremely important. nowadays, mining knowledge and providing scientific decision making for diagnosis of diseases from clinical datasets has found wide-ranging applications in healthcare sector. in this direction, among different data mining tools, association rule mining has already emerged out as a popular technique to extract invaluable information and develop important knowledge-base to help in intelligent diagnosis of distinct diseases quickly and automatically. in this paper, based on 5434 records of covid cases collected from a popular data science community and using rapid miner studio software, an attempt is put forward to develop a predictive model based on frequent pattern growth algorithm of association rule mining to determine the likelihood of covid-19 in a patient. it identifies breathing problem, fever, dry cough, sore throat, abroad travel and attended large gathering as the main indicators of covid19. employing the same clinical dataset, a linear regression model is also proposed having a moderately high coefficient of determination of 0.739 in accurately predicting the occurrence of covid-19. a decision support system can also be developed using the association rules to ease out and automate early detection of other diseases. key words: covid-19, association rule mining, frequent pattern growth, prediction, regression. mailto:vishnurrr40@gmail.com mailto:santonabchakraborty@gmail.com mailto:s_chakraborty00@yahoo.co.in rai et al./decis. mak. appl. manag. eng. (2022) 2 1. introduction coronavirus disease 2019 (covid-19) is mainly caused by severe acute respiratory syndrome coronavirus 2 (sars-cov-2) which is contagious in nature. the transmission of covid-19 occurs when a person inhales virus-containing respiratory droplets and airborne particles from an infected patient. the first known case of covid was from wuhan, china. it has now become a raging pandemic creating havoc with its impact ranging from loss of millions of lives to social and economic disruptions around the globe. the impact of covid in india, being the second populous country, is threatening. in india, the first few cases were from kerala resulting in the first wave. total lockdown was imposed from 25th march, 2020 and the number of active cases began to drop from september, 2020. a larger and much powerful second wave hit india on march 2021. presently, india has the largest number of covid cases in asia. as of 12 june 2021, it has the second-highest number of confirmed cases in the world (after the united states) with 29.3 million reported cases of covid-19 infection. it has also the third-highest number of covid19 deaths (after the united states and brazil) with 367,081 deaths. india became the first country to report over 400,000 new cases in a 24 hour period on 30 april 2021. as of 30 june 2021, the total number of confirmed cases in india is 3,03,62,848 with total number of deaths as 3,98,454. the rapid increase of cases in the peak of both first and second waves of covid-19 had put tremendous pressure on the medical infrastructure leading to shortage of hospital beds, oxygen cylinders, vaccines and other medicines in the country. there was also a great chance that the primary health workers were being infected by the same disease while treating the covid patients. the standard diagnostic procedure for covid is to detect the presence of coronavirus’s nucleic acid in human body which is usually performed by real-time reverse transcription polymerase chain reaction (rrt-pcr), transcription-mediated amplification (tma) or by reverse transcription loop-mediated isothermal amplification (rt-lamp) test from a nasopharyngeal swab. each of these testing procedures for covid-19 is time consuming requiring plenty of resources. at the peak of waves, when there are millions of daily cases, it has become crucial for having a more quick and efficient approach to determine whether a person has covid-19 or not. while detecting this disease and treating the infected patients, the concerned healthcare sector is generating huge volume of valuable information which can be effectively deployed for rapid diagnosis, identification and treatment of an individual. in the present day pandemic scenario, mining knowledge and providing scientific decision making for diagnosis of covid-19 from the clinical dataset has turned out to be extremely important. with rapid development of computational facilities, data mining technology has gained increasing attention to discover interesting knowledge in the form of useful patterns, changes, associations, anomalies and structures from large volume of data stored in databases, data warehouses or other data repositories. association rule mining is an effective data mining tool mainly deployed to extract association relationships or correlations/cooccurrences among a given set of items. due to its simplicity in framing rules based on ‘if-then’ statements, association rule mining is now being extensively used to explain patterns from seemingly independent repositories, like transactional databases, relational databases or clinical databases (kaur & madan, 2015; sabthami et al. 2016). the developed rules can assist the physicians in diagnosing patients based on the conditional probability while comparing the symptom relationships in the data from the past cases (hareendran & chandra, 2017; cheng & wang, 2017). in association rule mining for prediction of covid-19 3 this paper, an attempt is put forward to employ association rule mining as an effective predictive tool for diagnosing covid-19 patients based on frequent pattern (fp) growing algorithm. using a large clinical database containing 5434 records of covid cases, breathing problem, fever, dry cough, sore throat, abroad travel and attended large gathering are identified as the most important covid-19 indicators. with the help of rapid miner studio software, the corresponding association rules are framed for prediction of covid-19 disease in a patient. the developed regression model would also aid in covid-19 diagnosis correlating the symptoms and likelihood of this disease. its application would save a lot of time and resources in the case of huge influx of patients. using this predictive model, the patients themselves can envisage whether they have the disease or not and can start taking necessary precautions (stilou et al., 2001). association rule mining, logistic regression, discriminant analysis etc. are different types of machine learning approaches. in association rule mining, simple ‘if…then’ clauses are framed to discover the existent relationships between independent relational databases, transactional databases, and other forms of data repositories, requiring simple counting to observe frequently occurring patterns, and similarities and dissimilarities among different objects. on the other hand, logistic regression employs binary variables where the response records either success or failure for a given event. it can also be extended to combine more than one independent continuous or categorical variable. discriminant analysis develops a set of prediction equations based on independent variables for classifying new items and interpreting the relationship among the considered variables. the application of discriminant analysis assumes that the data is normally distributed and each attribute has the same variance. among these machine learning approaches, association rule mining is the simplest one, requiring no assumption about the underlying distribution of the initial dataset, while framing easy to understand rules. with the help of different parameters, like support, confidence and lift, it can clearly identify the strongest rule penetrating more inside the problem. 2. association rule mining association rule mining is one of the techniques of data mining to extract interesting relations (dependencies) and patterns/links among variables from large seemingly independent datasets in order to draw useful inferences and decisions for practical use. application of association rule mining helps in generating simple ‘ifthen’ statements to analyze frequently occurring patterns in a dataset and/or identify the inherent relationships between independent and dependant variables in the dataset. it can also frame useful rules from qualitative and categorical datasets which are often difficult to interpret (ordonez et al., 2006). an association rule consists of two components, i.e. an antecedent (if) and a consequent (then). an antecedent is an item found within the dataset and a consequent is an item observed in combination with the antecedent (freeda & florence, 2017). thus, in a rule, x→y, x is the antecedent (if) and y is the consequent (then). in a clinical database, the rule {symptom1, symptom2}→{disease1} signifies that a patient having symptom1 and symptom2 would tend to have disease1. for example, there is a set of symptoms a = {a1,a2,,…,an} and b indicates entries of multiple patients in a clinical database b = {b1,b2,…,bn}. each patient contains a subset of elements in a–b  a, and the corresponding association rule is the implication x→y, where x  a, y  a and x∩y = ∅. in a clinical database, an antecedent is a specific symptom or combination of rai et al./decis. mak. appl. manag. eng. (2022) 4 symptoms and a consequent can be a disease caused due to occurrence of the antecedent. the generated rules would thus help the concerned physicians in making faster decisions with correct diagnosis of a disease (kulkarni & mundhe, 2017; lakshmi & vadivu, 2017). the effectiveness of the developed association rules is usually validated using three parameters, i.e. support, confidence and lift. support measures the percentage of items in the given dataset following a particular rule, i.e. how often a rule occurs in the dataset. support(x→y) = p(x∪y) confidence evaluates the probability of inclusion of item x also leading to the inclusion of y. it is the conditional probability of how often a rule is found out to be true. confidence(x→y) = p(y|x) = support(x→y)/support(x) lift finally measures the performance of an association rule. lift(x→y) = confidence(x→y)/support(x→y) higher value of confidence signifies higher strength of a particular rule. on the other hand, higher value of lift symbolizes that having x and y together is not accidental, but due to the presence of a meaningful relationship between them. lift = 1 signifies that the probability of occurrence of antecedent and consequent is not dependant on each other. lift > 1 determines the degree to which antecedent and consequent are dependent to each other. lift < 1 signifies that one item has a negative effect on the other, i.e. one item is a substitute for the other item present in the rule. there are three popular algorithms, i.e. apriori algorithm, eclat (equivalence class clustering and bottom-up lattice transversal) algorithm and frequent pattern (fp) growth algorithm deployed for framing the relevant association rules from a given dataset (prithiviraj & porkodi, 2015). in apriori algorithm (which is an arraybased algorithm), frequent itemsets are used for generation of the association rules. it employs a breadth-first search and hash tree to efficiently identify the frequent itemsets in a transactional database. but, its application is time-consuming and the corresponding runtime may increase exponentially (jain & gautam, 2013; sambasiva rao & uma devi, 2017). the eclat adopts a depth-first search technique to find out the frequent itemsets in a relational database. it has less execution time than apriori algorithm. the fp growth is a tree-based algorithm which employs depth-first search to compress the dataset to form an fp-tree. it is faster than the other two algorithms and its runtime increases linearly. but for large fp-tree, it may not fit in the memory space, thus being expensive to build (thamer, et al., 2020). as in this paper, fp growth algorithm is employed for development of the association rules for effective prediction of covid-19, its procedural steps are detailed out here-in-under. association rule mining using fp growth algorithm mainly consists of two steps, i.e. a) generation of frequent item sets and b) formation of association rules from the frequent item sets. to demonstrate generation of frequent item sets employing fp growth algorithm, let us consider a clinical dataset containing different symptoms for nine infected patients. in this database, p and s represent patient and symptom respectively. p1 = (s1, s2, s5); p2 = (s2, s4); p3 = (s2, s3); p4 = (s1, s2, s4); p5 = (s1, s3); p6 = (s2, s3); p7 = (s1, s3); p8 = (s1, s2, s3, s5) and p9 = (s1, s2, s3) now, the corresponding fp-tree is developed based on the following steps: association rule mining for prediction of covid-19 5 a) scan the dataset to determine the support count of each symptom. remove the less-frequent symptom(s) and sort the frequent symptoms in descending order of their occurrence. b) scan the dataset of one patient at a time resulting in formation of the fptree. for each transaction, i) if it has a set of unique symptoms, form a new path and set the counter for each node to 1. ii) if it shares a set of common symptoms, increase the common symptom itemset node counters and create new nodes, if needed. c) this process needs to be continued until each patient case is mapped into the tree. this algorithm scans the database only twice while directly compressing it into the corresponding fp-tree. in this algorithm, minimum support (basically acts as a cut-off) can be used to classify the frequent and less-frequent item sets in a database. the less-frequent items are ignored while developing the fp-tree. identification of the most appropriate cut-off for subsequent fp-tree generation is a critical task. lower cut-off with minimum support may include many item sets resulting in less significant results. on the other hand, higher cut-off may result in finding out zero item sets with no generation of fp-tree. in this illustrative example, the support count of each symptom is determined as given below (in descending order): s2 = 7, s1 = 6, s3 = 6, s4 = 2 and s5 = 2. now, the patient datasets are rearranged according to the descending order of support count of different symptoms. p1 = (s2, s1, s5); p2 = (s2, s4); p3 = (s2, s3); p4 = (s2, s1, s4); p5 = (s1, s3); p6 = (s2, s3); p7 = (s1, s3); p8 = (s2, s1, s3, s5) and p9 = (s2, s1, s3). based on the dataset with nine patient cases and five symptoms in the illustrative example, the following fp-tree of figure 1 is developed. this fp-tree is generated while considering null as the root node. the count of each symptom for each patient case is highlighted in parenthesis at each node. figure 1. fp-tree for the illustrative example next, the developed fp-tree is mined. the lowest node of the tree is checked first along with its links. the lowest node represents the symptom with minimum support count. from the lowest node, traverse the path in the fp-tree to the null node. each such path is termed as conditional pattern base. the conditional fp-tree is formed rai et al./decis. mak. appl. manag. eng. (2022) 6 while counting the symptoms in the path. the symptoms meeting the minimum support of 2 are considered here for subsequent generation of frequent itemsets, as exhibited in table 1. in this table, six 2-frequent and two 3-frequent itemsets are generated. table 1. generation of frequent itemsets for the illustrative example symptom conditional pattern base conditional fp-tree frequent itemsets s5 {{s2,s1:1},{s2,s1,s3:1}} [s2:2, s1:2] {s2,s5:2},{s1,s5:2},{s2,s1,s5:2} s4 {{s2,s1:1},{s2:1}} [s2:2] {s2,s4:2} s3 {{s2,s1:1},{s2:2},{s1:2}} [s2:4, s1:2], [s1:2] {s2,s3:4},{s1,s3:4},{s2,s1,s3:2} s1 {{s2:4} [s2:4] {s2,s1:4} based on the frequent itemsets in table 1, the corresponding association rules are thereby generated using the following steps: a) generate all non-empty subsets of each frequent itemset u. b) for every non-empty subset f of u, formulate the rule : f→(u–f) if (support_count(u)/support_count(f) ≥ minimum_confidence where minimum_confidence is the threshold confidence level. for example, consider the first frequent itemset u = (s2,s1,s5). generate all the non-empty subsets of u as f: {s1},{s2},{s5},{s1,s2},{s2,s5},{s1,s5},{s1,s2,s5}. for every non-empty subset f of u, the corresponding association rules are framed, as given in table 2. in this table, support_count is the number of occurrences of all the elements in a set (u or f) together in the dataset, confidence calculated = (support_count(u)/support_count(f))×100, support = ((support_count(u))/n)×100, lift = confidence calculated/support and n is the total number of patient cases in the example. here, the threshold confidence value is arbitrarily taken as 80%. it can be noticed from this table that among the generated rules, only rules 3, 5 and 6 are accepted with their confidence levels greater than or equal to the set threshold value. thus, for this example, the following association rules are developed: s5→(s2,s1); (s1,s5)→s2 and (s2,s5)→s1. table 2. association rules for the illustrative example rule no. association rules support_ count(u) support_ count(f) confidence calculated threshold confidence n support lift accepted/ rejected 1 s1→(s2,s5) 2 6 33.33 80 9 22.22 1.50 rejected 2 s2→(s1,s5) 2 7 28.57 80 9 22.22 1.29 rejected 3 s5→(s2,s1) 2 2 100.00 80 9 22.22 4.50 accepted 4 (s1,s2)→s5 2 4 50.00 80 9 22.22 2.25 rejected 5 (s1,s5)→s2 2 2 100.00 80 9 22.22 4.50 accepted 6 (s2,s5)→s1 2 2 100.00 80 9 22.22 4.50 accepted 7 (s1,s2,s5) →(null) rejected it has been noticed that the apriori algorithm of association rule mining has already been successfully deployed for prediction/diagnosis of heart diseases (said et al., 2015; domadiya & rao, 2018; jamsheela, 2021), dengue (jahangir et al., 2018), brain tumor (sengupta et al., 2013), chronic kidney disease (alaiad et al., 2020), infectious diseases (brossette et al., 1998), pandemic diseases (burvin & dhanalakshmi, 2018; aiswarya et al., 2020), covid-19 (çelik, 2020; shawkat et al., 2021; tandan et al., 2021), pediatric primary care (downs & wallace, 2000), association rule mining for prediction of covid-19 7 treatment of patients in an emergency department (sarıyer & taşar, 2020) etc. in this paper, based on a huge dataset of covid-19 patients and using the fp growth algorithm of association rule mining, an attempt is put forward to discover covid-19 symptom patterns and rules which would support the initial identification of severe covid-19 cases for early treatment and isolation. based on the most frequent symptoms, a first-order regression model is also developed to assist prediction of covid-19. 3. data collection in order to predict covid-19 based on development of the corresponding association rules, the related data is collected from kaggle.com which is the world’s largest online data science community. the data consists of the symptoms and other factors responsible for covid-19 infection. they are based on the guidelines provided by the world health organization (who) (www.who.int ) and the ministry of health and family welfare, india (https://main.mohfw.gov.in). the data is in ‘yes’ and ‘no’ format, where ‘yes’ represents the presence of a particular symptom and ‘no’ denotes absence of it. based on the given guidelines, the considered factors for covid-19 infection are as follows: a) breathing problem, b) fever, c) dry cough, d) sore throat, e) running nose, f) asthma, g) chronic lung disease, h) headache, i) heart disease, j) diabetes, k) hyper tension, l) fatigue, m) gastrointestinal, n) abroad travel, o) contact with other covid patient, p) attended large gathering, q) visited public exposed places, r) family working in public exposed places and s) wearing masks at all times. in this database, covid-19 is treated as the decisional (target) variable. the clinical dataset is in tabular form containing 5434 records of infected covid cases. the snapshot of a small portion of the considered dataset is shown in figure 2. figure 2. a portion of the covid-19 dataset 4. rule mining for covid-19 prediction in this paper, using covid-19 symptom dataset and employing fp growth algorithm, the corresponding association rules are extracted for early detection of rai et al./decis. mak. appl. manag. eng. (2022) 8 this disease based on the application of rapid miner studio educational 9.9.000 software. in this software, there are options to select different operators which can perform varying functions ranging from data input to data analysis. each operator has an input node and an output node through which the data is processed. these operators are combined together to perform a specific task. in order to extract the association rules using this software, the following steps are adopted: a) data input: the data is fed into the software through the read csv (commaseparated values) operator. the output node of read csv operator is connected to the input node of fp-growth operator. b) finding the frequent itemsets: the fp-growth operator is utilized to find out the frequent itemsets in the dataset. the output of this operator is then connected to the input node of create association rule operator. c) extraction of the association rules: the create association rule operator extracts the corresponding association rules while considering the input in the form of frequent itemsets from fp-growth operator. d) displaying the results: the create association rule operator has two output nodes, i.e. ite-frequent item sets obtained and rul-association rules extracted. these nodes are finally connected to two result nodes to display both the frequent itemsets and association rules. figure 3 portrays the flow diagram representing the positioning of different operators in a logical way to provide the intended results. the fp growth algorithm thus generates frequent itemsets with size ranging from 1 to 5. the frequent itemsets of sizes 3, 4 and 5 with their corresponding support values for covid-19 prediction are depicted in tables 3-5 respectively. in table 5, there is a frequent itemset of size 5, i.e. {covid-19, dry cough, fever, sore throat, breathing problem} which signifies that frequent occurrence of these four symptoms would lead to covid-19. a support value of 0.374 symbolizes that there are 37.4% patient cases having dry cough, fever, sore throat and breathing problem resulting in covid-19 infection. the corresponding association rules developed by this software are provided in table 6. in this table, ‘premises’ signifies the antecedent and ‘conclusion’ signifies the consequent of the association rule generated. thus, the occurrence of any of these rules would lead to increased likelihood of this disease in a patient. in order to generate these rules, the values of minimum support and confidence are considered as 30% and 1 respectively. the minimum support value of 30% symbolizes that at least 30% of the patient database contains any of these nine rules. on the other hand, lift > 1 indicates the existence of meaningful relationships between the symptoms and covid-19 prediction. the formation of these rules and interrelations among them are pictorially exhibited in figure 4. in this figure, all the variables (symptoms) and rules are shown in different blocks. the block for each rule has the format: rule x (support of the rule x/confidence of rule x) where x is the corresponding association rule number. it provides a visual information of the unique symptoms/factors for covid-19 and inter-connections among the generated rules. now, based on all these association rules, it can be concluded that there are six most important factors, i.e. breathing problem, fever, dry cough, sore throat, abroad travel and attended large gathering responsible for infection of this disease in a person. association rule mining for prediction of covid-19 9 figure 3. flow diagram for extraction of association rules table 3. frequent itemsets of size 3 size support item1 item2 item3 3 0.611 covid-19 dry cough fever 3 0.593 covid-19 dry cough breathing problem 3 0.530 covid-19 dry cough running nose 3 0.392 covid-19 dry cough fatigue 3 0.367 covid-19 dry cough visited public exposed places 3 0.399 covid-19 dry cough headache 3 0.349 covid-19 dry cough contact with covid patient 3 0.415 covid-19 dry cough hyper tension 3 0.376 covid-19 dry cough asthma 3 0.353 covid-19 dry cough attended large gathering 3 0.424 covid-19 dry cough abroad travel 3 0.596 covid-19 fever sore throat 3 0.510 covid-19 fever breathing problem 3 0.384 covid-19 fever running nose 3 0.351 covid-19 fever fatigue 3 0.377 covid-19 fever visited public exposed places 3 0.411 covid-19 fever contact with covid patient 3 0.360 covid-19 fever hyper tension 3 0.378 covid-19 fever attended large gathering 3 0.381 covid-19 fever abroad travel 3 0.526 covid-19 sore throat breathing problem 3 0.370 covid-19 sore throat running nose 3 0.376 covid-19 sore throat visited public exposed places 3 0.401 covid-19 sore throat contact with covid patient 3 0.384 covid-19 sore throat attended large gathering 3 0.374 covid-19 sore throat abroad travel 3 0.376 covid-19 breathing problem contact with covid patient 3 0.355 covid-19 breathing problem attended large gathering table 4. frequent itemsets of size 4 size support item1 item2 item3 item4 4 0.519 covid-19 dry cough fever sore throat 4 0.433 covid-19 dry cough fever breathing problem 4 0.363 covid-19 dry cough fever contact with covid patient 4 0.354 covid-19 dry cough fever abroad travel 4 0.447 covid-19 dry cough sore throat breathing problem 4 0.352 covid-19 dry cough sore throat contact with covid patient 4 0.448 covid-19 fever sore throat breathing problem 4 0.362 covid-19 fever sore throat contact with covid patient rai et al./decis. mak. appl. manag. eng. (2022) 10 table 5. frequent itemsets of size 5 size support item1 item2 item3 item4 item5 5 0.374 covid-19 dry cough fever sore throat breathing problem table 6. association rules generated for covid-19 prediction rule no. premises conclusion support confidence lift 1 abroad travel covid-19 0.451 1 1.24 2 dry cough, attended large gathering covid-19 0.390 1 1.24 3 dry cough, abroad travel covid-19 0.424 1 1.24 4 fever, attended large gathering covid-19 0.378 1 1.24 5 fever, abroad travel covid-19 0.381 1 1.24 6 sore throat, attended large gathering covid-19 0.384 1 1.24 7 sore throat, abroad travel covid-19 0.374 1 1.24 8 breathing problem, attended large gathering covid-19 0.355 1 1.24 9 dr cough, fever, abroad travel 0.354 1 1.24 figure 4. formation of association rules considering the initial dataset, and breathing problem, fever, dry cough, sore throat, abroad travel and attended large gathering as the major factors for covid-19 infection, a linear regression model is developed using the following steps: a) data input: the relevant data is fed into rapid miner software through read csv operator with covid-19 as the decisional (dependent) variable. b) data preprocessing: the replace operator is employed to convert the data from ‘yes’ and ‘no’ to 1 and 0 respectively. the guess types operator is used to transform all the variable data into numerical data. association rule mining for prediction of covid-19 11 c) data processing: the set role operator changes covid-19 variable as a special attribute. it would help in developing the corresponding regression model considering covid-19 as the dependant variable. d) model development: the linear regression operator finally generates the regression equation from the given dataset. the corresponding flow diagram, as exhibited in figure 5, develops the regression equation correlating infection of covid-19 and the main medical factors in the following form: y = 0.030 + (0.208×breathing problem) + (0.175×fever) + (0.243×dry cough) + (0.193×sore throat) + (0.189×abroad travel) + (0.177×attended large gathering) where y is the target variable (presence of covid-19). a value of y less than 0.5 signifies less likelihood of covid-19 in a patient; on the other hand, a value greater than or equal to 0.5 identifies more likelihood of covid-19 infection in a patient. a moderately high coefficient of determination (r2) value as 0.739 provides an indication of acceptable accuracy of the developed predictive model. it indicates that almost 73.9% variation of the dependent variable (presence/absence of covid-19) can be explained by the independent variables (symptoms/factors). figure 5. flow diagram for regression equation 5. conclusion keeping in mind the requirements of early detection of covid-19 for faster isolation and treatment of an infected patient, this paper proposes the application of fp growth algorithm to find out the frequent itemsets and extract the association rues with their corresponding confidence and support values. it is noticed that six factors, i.e. breathing problem, fever, dry cough, sore throat, abroad travel and attended large gathering are mainly responsible for covid-19 infection. a linear regression equation-based predictive model is also developed to correlate those factors and presence of covid-19 in a patient. a moderately high coefficient of determination value suggests that almost 73.9% variation of the dependent variable (presence/absence of covid-19) can be explained by the independent variables (symptoms/factors). it would help in early prediction of this disease, thus saving valuable time and resources. but, if a patient is asymptotic, this model would not rai et al./decis. mak. appl. manag. eng. (2022) 12 provide accurate results. among the existing machine learning techniques, association rule mining has several advantages, like capability of dealing with different forms of data repositories, development of easy to understand clauses, no assumption about the underlying distribution of the data, application of support, confidence and lift parameters for developing the strongest rule etc. as a future scope, it is suggested to develop and integrate association rule mining in a decision support system for early diagnosis of covid-19 and other severe diseases, like kidney related problems, brain tumor etc. with more real-time clinical dataset, those diseases can be diagnosed much faster while evaluating coexistence of the symptoms. author contributions: v.k.r.: data collection, software, analysis; santonab.c.: draft preparation, review, technical writing; shankar c.: technical writing, editing. conflicts of interest: the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper funding: this research received no external funding. data availability statement: the related data is collected from kaggle.com which is the world’s largest online data science community. references aiswarya, p., bhanu sridhar, m., & kavitha, l. 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(2020). a semantic approach for extracting medical association rules. international journal of intelligent engineering & systems, 13(3), 280-292. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). https://ebooks.iospress.nl/bookseries/studies-in-health-technology-and-informatics https://ebooks.iospress.nl/bookseries/studies-in-health-technology-and-informatics https://www.sciencedirect.com/science/article/abs/pii/s0010482521000433?via%3dihub#! plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 2, issue 1, 2019, pp. 115-131. issn: 2560-6018 eissn:2620-0104 doi:https://doi.org/10.31181/dmame1901115n * corresponding author. e-mail addresses: dragana_95_@hotmail.com (d. nenadić) ranking dangerous sections of the road using the mcdm model dragana nenadić1 1university of east sarajevo, faculty of transport and traffic engineering doboj, bosnia and herzegovina received: 13 october 2018; accepted: 25 february 2019; available online: 09 march 2019. original scientific paper abstract: traffic accidents are a matter of great concern in traffic safety since they unexpectedly and sometimes unavoidably cause fatal and non-fatal injuries, or material damage. the causes of traffic accidents can vary but they can always be linked to one of the four basic factors: human, vehicle, road and environment. however, there are some places where traffic accidents happen more frequently than in others. the decision-making process concerning dangerous road sections using the multi-criteria decision-making (mcdm) model involves the definition of quantitative and qualitative traffic safety criteria. the model used in the paper consists of five quantitative and two qualitative traffic safety criteria. based on those criteria the ranking of the prospected sections is carried out. by analyzing the total number of traffic accidents, by their categories and by analyzing the current state of the traffic infrastructure and annual average daily traffic (aadt), seven traffic safety criteria are defined and, in the first phase of the model, are rated and ranked by their importance. by using the full consistency method (fucom), weighted coefficients of the defined criteria are determined followed by ranking of dangerous road sections using the weighted aggregate sum product assessment method (waspas). the obtained results show which of the offered alternatives is best ranked, that is, which section of the road is the safest one. key words: dangerous sections, traffic safety, multi-criteria decisionmaking, fucom, waspas 1. introduction undoubtedly, decision-making is one of the most important aspects of shaping the future of traffic safety, especially in situations where human lives and material goods are endangered, as in the case of traffic accidents. the multi-criteria decision-making (mcdm) methods are gaining importance as potential tools for analyzing and solving nenadić/decis. mak. appl. manag. eng. 2 (1) (2019) 115-131 116 complex real-time problems due to their inherent ability to evaluate different alternatives with respect to various criteria (chakraborty et al., 2015). the multicriteria decision-making methods can be used as an adequate tool for making valid decisions when it comes to traffic safety. approximately 1.35 million people die each year as a result of road traffic accidents. variations in death rates observed across regions and countries also correspond to differences in the types of road users most affected. only in the european union, over 1.1 million traffic accidents have killed more than 30,000 people while 1.5 million people have been injured (world health organization, 2018). based on who data (2018), estimated road traffic death rate in bosnia and herzegovina is 17.7 (per 100 000 population). for the system management, it is necessary (lipovac, 2008):  to know the existing state of affairs,  to define the desired condition, and,  to choose those management measures that would bring the existing situation closer to the desired one. in the field of traffic safety, the concept of management can be defined similarly; so, in order to manage the safety of traffic, it is necessary to know the existing situation, define the desired state of affairs and take measures to bring the existing state to the desired one. in defining the present state of affairs, it is necessary to observe the basic trends in the development of the phenomenon, which includes the prognosis of the occurrence based on the existing condition (for example, the forecast of the number of traffic accident and their consequences, assuming that the current trend continues). this means that nothing is done in terms of solving traffic safety problems, so that the current trend continues. however, this is only the first step in defining the current state. it should be followed by the research based on the definition of the supposed desired state as well as the selection of those management measures that would bring the current state closer to the desired one. because of that, traffic safety includes several models that can be used for defining the current state of affairs (lipovac, 2008):  descriptive model,  prediction model,  risk factor model,  models that show the consequences of traffic accidents, and,  models that rely on monitoring the traffic safety indicators. the descriptive model is trying to describe the state of affairs and traffic safety problems by using three dimensions: exposure, accident risk and consequences of traffic accidents. basic data about road traffic accidents and injuries are collected every day in most countries. police officers write reports on accidents, insurance companies document their clients’ accidents while health workers keep medical records on the road traffic injuries they have treated. the main purpose of documenting this information is usually to assist an agency in carrying out its specific function like investigation, law enforcement and provision of health care. however, this information can also be used for ranking dangerous sections of the road by using the mcdm model. the motorization rate in bosnia and herzegovina has been growing gradually in the last couple of years and so has the number of road accidents. the consequence of such trend is the absence of the road safety system due to the lack of systemic and continual road safety management (lipovac et al., 2015). ranked road sections in terms of risk, together with ranked weights of factors considered as causes of accidents for each section, are highly effectual information for road safety implementing planning (jantakat et al., 2014). that is why it is important to determine ranking dangerous sections of the road using mcdm model 117 the dangerous road sections, which can help us to make a better decision when it comes to improving traffic safety. basic sources about traffic accidents and their consequences are police, hospital and insurance company reports. so, the first step in the descriptive model is to describe the current state of affairs and determine the importance of the traffic safety problems based on that data. in this paper, the description of the current state of affairs is given by data on the total number of traffic accidents by their categories, data on the current state of the traffic infrastructure and aadt for observed locations. in order to determine the significance of the problem, dangerous road sections on the territory of the municipality of derventa will be ranked by using the mcdm model. hence the main goal of this paper is to make a decision about the road section which is estimated as the most dangerous of all the observed ones. that will show the significance of the problem when it comes to traffic safety. the paper is structured in several sections. section 1 (introduction) shows the importance of describing the current state of affairs when it comes to traffic safety. next section (literature review) consists of three parts: a review of the use of the multi-criteria decision-making model, a review of the use of the fucom method in different research fields and a review of the use of the waspas method of evaluating different alternatives. section 3 (methods) consists of two parts; the first one describes in detail three steps of the fucom method while the second one describes the waspas method by its steps. it is on the basis of these steps that the dangerous sections of the road on the territory of the municipality of derventa will be ranked. the main section of the paper is section 4 (case study), which includes forming a multi-criteria model as well as applying the fucom and the waspas methods to the concrete case. the end of this section is reserved for the sensitivity analysis, based on what we can do in the model behavior testing. the last section of the paper (conclusion) represents a brief summary of the things described in the paper as well as an explanation showing us which of the observed locations is ranked best. 2. literature review the research and development of the mcdm methods increased during the 80s and early 90s but it seems that the exponential growth of this process continued (köksalan et al., 2011). the mcdm methods can be applied effectively to determining the value and utility degree of various areas and to establishing the priority order for their implementation (turskis, 2008). triantaphyllou and mann (1995) said that the mcdm plays an important role in real-life problems as there are a large number of everyday decisions to be made which include a huge number of the criteria. according to chen et al. (2015), the mcdm is an effective, systemic and quantitative way of solving vital real-life problems with a large number of alternatives and several (opposing) criteria. according to drezner (1995) the study of location selection has a long and extensive history spanning many general research fields including operations research (or management science), industrial engineering, geography, economics, computer science, mathematics, marketing, electrical engineering, urban planning, etc. according to kahraman et al. (2011) evaluation of specific sites in the selected community is commonly termed microanalysis. many authors (roberts and goodwin, 2002; solymosi and dompi, 1985; cook, 2006; weber and borcherding, 1993) agree that the values of criteria weights are significantly conditioned by the methods of their nenadić/decis. mak. appl. manag. eng. 2 (1) (2019) 115-131 118 determination. but there is no agreement as to which of the methods is the best one for determining criteria weights. according to stević et al. (2018) the main problem of the multi-criteria decision-making (mcdm) is that of choosing an appropriate method for determining criteria weights, as a very important stage, which complicates the decision-making process. if we take the fact that the weights of criteria can significantly influence the outcome of the decision, it is clear that attention must be paid to the objectivity factors of criteria weights. real problems do not usually have the criteria of the same degree of significance. it is, therefore, necessary that the significance factors of particular criteria should be defined by using appropriate weight coefficients for the criteria, so that their sum is one. therefore, the new fucom method for determining the weight coefficients of criteria is proposed (pamučar et al., 2018). the fucom method enables the precise determination of the values of the weight coefficients of all of the criteria at a certain level of the hierarchy. in comparison with similar subjective models (the ahp and the bwm methods) for determining the weight the coefficients of the criteria, the fucom only requires the (n-1) pairwise comparison of the criteria (pamučar et al., 2018). a fucom method is applied to determining the weights of the criteria for the selection of the automatically guided vehicles (agvs) as one important type of material handling equipment in warehouses. the multi-criteria model included several criteria and agvs solutions, based on which the selection of agvsis done. that caused reduction of labor costs, increased reliability and productivity, reduction of the damage of goods, safety improvement, managing and control of the complete system, etc. (zavadskas et al., 2018). solving different problems can be done by using the fucom with some other method. the advantages of the new methodology, delphi-fucom-servqual methodology, are reflected in providing precise treatment of input and output parameters, and obtaining results that are more objective (prentkovskis et al., 2018). according to nunić (2018), solving the problem of the selection of the pvc carpentry manufacturer by using the fucom-mabac model has included all the relevant criteria which are of influence upon the final decision. the objective was to obtain the most suitable offer, that is, the one which involves high quality, which is the lowest possible price, a short time for delivery and montage, a possibility of deferred payment, a longer warranty period with the manufacturer’s reliability but it is not necessary to ignore other relevant facts that may have an impact on the formation of a final decision. according to pamučar et al. (2018) the fucom method was used for evaluation of the level crossing, as a point of the crossing of road and rail traffic in the same level. the presented fucom-mairca model allows consideration of subjectivity in the process of group decision-making through linguistic evaluation of the evaluation criteria. the results obtained using the waspas method show that the use of method and techniques in the field of mcdm can help decision-makers to successfully evaluate defined alternatives (tešić et al 2018). chakraborty and zavadskas (2014) used the waspas approach for solving decision-making problems related to manufacturing, and the findings of this paper were accurate; the proposed method had accurate ranking capability for solving decision-making problems related to manufacturing. according to zavadskas et al. (2012), the waspas method approach enables attaining high accuracy measurement. the use of the waspas technique for assessment and selection of appropriate solutions for occupational safety (dėjus and antuchevičienė, 2012) has revealed that typical solutions for occupational safety are used in the field of road construction; however, they are intended for protecting third persons from accessing dangerous zones next to a construction site rather than for ensuring health and safety of workers. ranking dangerous sections of the road using mcdm model 119 according to stević et al. (2018) the expanded form of waspas method, which includes rough numbers, was used to make decisions that are more precise because an initial matrix has more accurate values, which eliminates subjectivity and reduces uncertainty in a decision-making process. that is why the complete rough bmwrough waspas model is used for the location selection for the construction of a roundabout which is one of the essential factors for increasing mobility in the towns. 3. methods 3.1 full consistency method (fucom) the fucom method was developed by pamučar et al. (2018) for determining the weights of criteria. according to the author, this new method is better than ahp (analytical hierarchy process) and bwm (best worst method). the fucom provides a possibility to validate the model by calculating the error size for obtained weight vectors, by determining the degree of consistency. on the other hand, in other models for determining the weights of criteria, the bwm (rezaei, 2015) and the ahp (saaty, 1980) models, redundancy in pairwise comparison appears which makes them less susceptible to errors in judgment, while the methodological procedure of the fucom eliminates that problem. in the following section, the procedure for obtaining weight coefficients of criteria by applying the fucom is presented: step 1 in this step, the criteria from the predefined set of the evaluation criteria are ranked. the ranking is performed according to the significance of the criteria, i.e. starting from the criterion which is expected to have the highest weight coefficient to the criterion of the least significance: (1) step 2 in this step, comparison of the ranked criteria is carried out and comparative priority , , with k representing the rank of the criteria) of the evaluation criteria, is determined. (2) step 3 in this step, the final values of the weight coefficients of evaluation criteria are calculated. the final values of the weight coefficients should satisfy the following two conditions: (a) the ratio of the weight coefficients is equal to the comparative priority among observed criteria defined in step 2, i.e. the following condition is met: (3) (b) in addition to condition (2), the final values of the weight coefficients should satisfy the condition of mathematical transitivity, i.e. t . then and are obtained.  1 2, ,..., nc c c c (1) (2) ( ) ... j j j k c c c   / ( 1) 1 ( ) k k k k c c     1, 2,...,k n  1/ 2 2/3 /( 1), ,..., k k      1 2, ,..., t n w w w / ( 1) ( ) k k   / ( 1) 1 k k k k w w     / ( 1) ( 1) / ( 2) / ( 2) k k k k k k          / ( 1) 1 k k k k w w     1 ( 1) / ( 2) 2 k k k k w w       1 1 2 2 k k k k k k w w w w w w       nenadić/decis. mak. appl. manag. eng. 2 (1) (2019) 115-131 120 thus, another condition that the final values of the weight coefficients of the evaluation criteria should meet is obtained, namely: (4) based on the defined settings, the final model for determining the final values of the weight coefficients of the evaluation criteria can be defined. ( ) / ( 1) ( 1) ( ) / ( 1) ( 1) / ( 2) ( 2) 1 min . . , , 1, 0, j k k k j k j k k k k k j k n j j j s t w j w w j w w j w j                          (5) by solving model (5), we obtain the final values of evaluation criteria and the degree of consistency (χ) of the results obtained. 3.2. weighted aggregate sum product assessment method (waspas) the weighted aggregate sum product assessment method (waspas) (zavadskaset al., 2012) is one of the best known and often applied multiple criteria decision-making methods for evaluating a number of alternatives in terms of a number given criteria. in general, suppose that a given mcdm problem is defined on m alternatives and n decision criteria. next, suppose that wj denotes the relative significance of the criterion and xij is the performance value of alternative i when it is evaluated in terms of criterion j. waspas methods consist of the following steps: step 1 formatting of initial decision matrix (x). the first step is to evaluate m alternatives by n criteria. alternatives are shown to the vectors: where xij is value of i-th alternatives according to the j-th criterion. (6) / ( 1) ( 1)/ ( 2) 2 k k k k k k w w          1 2, ,..., t n w w w  1 2, ,...,i i i ina x x x  1, 2, 3,..., ; 1, 2, 3,..., .i m j n  1 1 11 1 1 ... ... n n m m mn c c a x x x a x x            ranking dangerous sections of the road using mcdm model 121 step 2 in this step it is necessary to normalize the initial matrix using the following equations: (7) for (8) for step 3 weighing the normalized matrix is done in such a way that the previous (normalized) matrix is multiplied by the weight coefficients: (9) (10) step 4 summarizing all obtained values of the alternatives (summation in rows): (11) (12) step 5 determination of the weighted product model by using the following equations: (13) (14) step 6 determination of the relative values of alternative ai: (15) (16) coefficient λ can be crisp value; and it can be any value from 0, 0.1, 0.2, … , 1.0. step 7 ranking of alternatives. the highest value of the alternative is the best ranked while the smallest value reflects the worst alternative. 4. case study 4.1 forming a multi-criteria model three locations (fig. 1) that are located in the municipality of derventa, of which one is connection between the town of derventa and the town of brod (lužani), one that connects the town of derventa and the town of prnjavor (lug), and one that passes by the town (kninska street), are evaluated based on a total of seven criteria presented in table 1. max ij ij i ij x n x  1, 2 ,..., . n c c c b min i ij ij ij x n x  1, 2 ,..., . n c c c b n ij m n v v      , 1, 2,..., , ij j ij v w n i m j   1 i ij m q q      1 n ij ijj q v   1 i ij m p p      1 ( ) n wj ij ijj p v   1 i ij m a a      (1 ) ij i i a q p      nenadić/decis. mak. appl. manag. eng. 2 (1) (2019) 115-131 122 fig. 1 observed locations in the municipality of derventa the first location is located in the municipality of derventa, and it represents the main road m14.1 the second location is an exit from the town of derventa onto the m16.1 main road towards prnjavor, while the third location is an exit from the town of derventa onto the m14.1 towards brod. table 1 criteria in a multi-criteria model and their interpretation criterion criterion description c1 total number of traffic accidents with killed persons (quantitative data for last 6 years) c2 total number of traffic accidents with seriously injured persons (quantitative data for last 6 years) c3 total number of traffic accidents with slightly injured persons (quantitative data for last 6 years) c4 total number of traffic accidents with property damage only (quantitative data for last 6 years) c5 geometric design of road (qualitative data about curves, road width, upgrade, downgrade, etc.) c6 aadt (besides annual average daily traffic, quantitative data about the structure of traffic flow, car flow) c7 traffic elements (qualitative data about condition of pavement, roadway, road markings (horizontal and vertical signalization) table 1 shows both the criteria and the detailed interpretation of their meaning. the criteria used in this study are traffic safety criteria, commonly used in croatia and serbia (stević et al., 2018). criteria number 1,2,3,4 and 6 represent quantitative data, while criteria number 5 and 7 represent qualitative data. when it comes to the number of traffic accidents, all data are obtained from derventa police station. all data about number of accidents are shown in fig. 2. ranking dangerous sections of the road using mcdm model 123 fig. 2 total number of traffic accidents for last 6 years as the fig. 2 shows, two traffic accidents with killed persons happened in location lug and lužani, while in the kninska street location no traffic accidents with killed persons happened. the highest number of accidents with seriously and slightly injured persons took place at the lug location while the smallest number of traffic accidents with seriously injured persons happened on kninska street location. the smallest number of traffic accidents with property damage only took place at location no. 3, lužani, while the highest number took place at the lug location. aadt data for three locations are shown in fig. 3. fig. 3 aadt data for observed locations criterion number 6, aadt data about observed locations is based on the basis of data from the roads of the republic of srpska’s. aadt for first location is 8753 vehicle/day, for second location is 7875 vehicle/day, and for third location is 4591 vehicle/day. nenadić/decis. mak. appl. manag. eng. 2 (1) (2019) 115-131 124 4.2 determining criteria weights the using fucom method step 1 ranking the criteria based on their importance: 1 2 3 4 6 7 5 c c c c c c c      table 2 the importance of criteria (fucom method) criterion c1 c2 c3 c4 c6 c7 c5 wcj 1 1.8 2 2.5 2.6 3.4 3.4 table 2 presents the importance of criteria used in the multi-criteria decisionmaking model, where we can see that the most important criterion is criterion 1, the total number of traffic accidents with killed persons. after that, the most important criteria are the total number of traffic accidents with seriously and slightly injured persons, separately. then follows criterion 4 referring to the total number of traffic accidents with property damage only. the next criterion by importance is criterion 6, aadt. criteria 5 and 7 have the same importance. step 2 comparison of the ranked criteria is carried out and the comparative priority of the evaluation criteria is determined. comparative priority of the evaluation criteria is obtained by equation (3): 1 2/ 1.8 c c   2 3/ 1.11 c c   3 4/ 1.25 c c   4 6/ 1.04 c c   6 7/ 1.31 c c   7 5/ 1 c c   step 3 the final values of the weight coefficients are calculated by equation (4): 1 3 2 w w  2 4 1.39 w w  3 6 1.30 w w  4 7 1.36 w w  6 5 1.31 w w  regarding the defined limitations, on the basis of expression (5), a finite model for determining the weight coefficients meeting the condition of maximum consistency can be defined. min 3 6 71 2 4 2 3 4 6 7 5 3 61 2 4 3 4 6 7 5 1 1.8 ; 1.11 ; 1.25 ; 1.04 ; 1.31 ; 1 ; . . 2 ; 1.39 ; 1.3 ; 1.36 ; 1.31 ; 1, m j j j j w w ww w w w w w w w w w ww w w s t w w w w w w w                                                 final results for weight coefficients were obtained using the lingo 17 program, and it follows: 1 0.292w  2 0.162w  3 0.146w  4 0.117w  5 0.086w  6 0.112w  7 0.086w  after the completed calculation, it can be concluded that the most important criterion is the total number of traffic accidents with killed persons, whose weight ranking dangerous sections of the road using mcdm model 125 coefficient is 1 0.292w  . deviation from full consistency (dfc) was obtained as 0.00. 4.3 ranking dangerous sections road using waspas method step 1 formatting of initial decision matrix (x). the data shown in figs. 2 and 3 are used to form the initial decision matrix (table 4), in the first step of the waspas method. all criteria have been evaluated by using linguistic scale, presented in table 3. all criteria were evaluated by obtained data, depending on their type, max/min type. table 3 linguistic scale for evaluating qualitative criteria (stević et al., 2017) linguistic scale for criteria max type for criteria min type very poor (vp) 1 9 poor (p) 3 7 medium (m) 5 5 good (g) 7 3 very good (vg) 9 1 criterion 5 (geometric design of road in location number one and location number three) was evaluated by linguistic scale, and it is poor (p). geometric design of road in location number two was evaluated by previous scale, and it is good (g). kninska street, the first location, is the main road m14.1.the beginning of the hall is a crossroad, at an angle of 90 degrees. most of the hall is a straight line, along which there are four mild curves, and one bridge. the main road is characterized by a large number of percussion holes, damaged carriageway and very poor roadblock status, making it the lowest scoring on the scale. the second location, lug, represents the m16.1 main road, from derventa towards prnjavor. this layout of the main road m16.1 is a route along which there are no curves. the condition of the carriageway is rated as good, with curbside and protective equipment alongside the road. the location of lužani is rated at grade 3 because of the fact that on this section of main road m14.1 from derventa towards brod, there is a major damage on road. except for that, the grade “poor” was given because there are two sharp curves on this part of the road, one of which is steep. criterion number seven, traffic elements, was evaluated by linguistic scale, as poor (p) for location one, and medium (m) for location number two and three. location number one is rated poor (p) because of no placed vertical signaling at all the necessary locations along the layout. edge lines as well as dividing lines in certain places are not sufficiently noticeable, especially in night conditions. location number two and three are rated as medium (m) because there are adequate vertical signaling on these sections of the road. all traffic signs are in a good condition. horizontal signaling is well known, visible in day and night conditions. nenadić/decis. mak. appl. manag. eng. 2 (1) (2019) 115-131 126 table 4 initial decision matrix 1 c 2 c 3 c 4 c 5 c 6 c 7 c 1 l 1 5 5 7 3 5 3 2 l 5 7 7 7 7 5 5 3 l 5 5 3 5 3 7 5 min min min min max min max 1 5 3 5 7 5 5 step 2 normalization of the initial matrix using eqs. (7), (8). normalization of the initial matrix (table 5) has been done according to the type of criteria. if it is maximum, we use equation (7), and if it is minimum we use equation (8). the first example represents the minimum criteria, and the second example represents the maximum criteria. 21 1 0.200 5 n   15 3 0.429 7 n   table 5 normalization of the initial decision matrix 1 c 2 c 3 c 4 c 5 c 6 c 7 c 1 l 1.000 1.000 0.600 0.714 0.429 1.000 0.600 2 l 0.200 0.714 0.429 0.714 1.000 1.000 1.000 3 l 0.200 1.000 1.000 1.000 0.429 0.714 1.000 step 3 weighting of the normalized matrix is done in such a way that the previous (normalized) matrix is multiplied by the weight coefficients, by using equation (10). table 6 represents the normalized matrix with weight coefficients which is used to form weighted normalized matrix. table 6 normalized initial matrix with weight coefficients 1 c 2 c 3 c 4 c 5 c 6 c 7 c 1 l 1.000 1.000 0.600 0.714 0.429 1.000 0.600 2 l 0.200 0.714 0.429 0.714 1.000 1.000 1.000 3 l 0.200 1.000 1.000 1.000 0.429 0.714 1.000 cj w 0.292 0.162 0.146 0.117 0.086 0.112 0.086 11 0.292 1.000 0.292v    after using equation (10), like in the previous example, the normalized initial matrix is weighted (table 7). ranking dangerous sections of the road using mcdm model 127 table 7 weighted normalized matrix 1 c 2 c 3 c 4 c 5 c 6 c 7 c 1 l 0.292 0.162 0.088 0.084 0.037 0.112 0.052 2 l 0.058 0.116 0.063 0.084 0.086 0.112 0.086 3 l 0.058 0.162 0.146 0.117 0.037 0.080 0.086 step 4 summarizing all obtained values of the alternatives by using equation (11), (12) (summation in rows): 1 0.292 0.162 0.088 0.084 0.037 0.112 0.052 0.826s         this step implies that every row in table 7 must be summarized. by summing up every row, we form the next table, table 8. table 8 summation in rows 1 c 2 c 3 c 4 c 5 c 6 c 7 c i s 1 l 0.292 0.162 0.088 0.084 0.037 0.112 0.052 0.826 2 l 0.058 0.116 0.063 0.084 0.086 0.112 0.086 0.604 3 l 0.058 0.162 0.146 0.117 0.037 0.080 0.086 0.686 step 5 determination of the weighted product model by using eqs. (13), (14): 0,292 0,162 0,146 0,117 0,086 0,112 0,086 1 1.000 1.000 0.600 0.714 0.429 1.000 0.600 0.794p         weighted product model (table 9) is formed by using eqs. (13) and (14) as in the previous example. the weighted product model is used to determinate the relative values of the alternatives. table 9 weighted product model 1 c 2 c 3 c 4 c 5 c 6 c 7 c i p 1 l 1.000 1.000 0.928 0.961 0.930 1.000 0.957 0.794 2 l 0.625 0.947 0.884 0.961 1.000 1.000 1.000 0.503 3 l 0.625 1.000 1.000 1.000 0.930 0.963 1.000 0.560 step 6 determination of the relative values of alternative i a : 1 0.5 0.828 (1 0.5) 0.8100.794a       2 0.5 0.604 (1 0.5) 0.5540.503a       3 0.5 0.686 (1 0.5) 0.6230.560a       step 7 ranking of alternatives. the final step of the waspas method means ranking of alternatives by their values. by using the fucom method for determining the weight coefficients and the waspas method for ranking the locations, we have obtained that the best alternative is location number 1 (kninska street). after location 1, the best nenadić/decis. mak. appl. manag. eng. 2 (1) (2019) 115-131 128 ranked alternative is location 3; that is why location 2 represents the most dangerous location of the road in terms of traffic safety. all data about values of the alternatives is shown in table 10. table 10 ranking of alternatives i q i p i a 1 l 0.826 0.794 0.810 2 l 0.604 0.503 0.554 3 l 0.686 0.560 0.623 1 3 2 l l l  4.4 sensitivity analysis the results of the multi-criteria models can significantly be influenced by the values of degree of consistency λ. the value of λ goes from 0, 0.1, 0.2, ..., 1. that is why the analysis of the influence of values of λ on the results of the research is done. therefore, in this part of the paper the sensitivity analysis of the ranks of alternatives to changes in value of λ is carried out. the sensitivity analysis is performed through ten situations. in every situation, values of λ is different, starting from 0,0.1,0.2, …,1. the obtained ranges are shown in fig. 4. fig. 4 sensitivity analysis after sensitivity analysis is done, the obtained results show that there is no difference in ranking dangerous sections of the road in the territory of the municipality of derventa. for all changes value of λ the ranking results are the same: 1 3 2 l l l  location number 1 is the best ranked alternative. location number 2 is the most dangerous sections of the road, and it is ranked as the third alternative. ranking dangerous sections of the road using mcdm model 129 5. conclusion this research study presents the use of the multi-criteria fucom-waspas model for ranking dangerous sections of the road in the municipality of derventa. the fucom-waspas model used in the study encompasses seven traffic safety criteria that are evaluated by the linguistic scale presented in the paper. by applying the fucom-waspas model three different sections of the road were ranked. the results obtained were verified through the sensitivity analysis carried out on the basis of different values of degree of consistency λ. in every case, location kninska street was best ranked alternative, while the location lug is ranked as the most dangerous section of the road, from all observed locations at the municipality of derventa. in order to manage the safety of traffic, it is necessary to know the existing situation, which can include ranking dangerous sections of the road. the process of ranking the road sections would help us determine the locations having the priorities when it comes to making decisions about improvements in traffic safety. when we find out which section is the most dangerous section of the road, it is easier to take the management and every other measure to improve safety of traffic starting from the most dangerous section. also, ranking of the road sections gives data to the traffic participants that would serve them as the basis for choosing a safer way to their finish line. in addition, the model presented in this paper introduces new methodological principles for evaluating the dangerous sections of the road, which at the same time contributes to the improvement of theoretical basis of multi-criteria decision making in general. future research related to this paper may imply the improvement of the proposed methodology by defining universal criteria for ranking dangerous sections of the road and the possibility of developing new approaches in the area of multicriteria decision-making. references chakraborty, s., &zavadskas, e. k. 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(2018).http://apps.who.int/gho/data/node.main.a997accesed 12 january 2019. zavadskas, e. k., nunić, z., stjepanović, ž.,&prentkovskis, o. (2018). a novel rough range of value method (r-rov) for selecting automatically guided vehicles (agvs). studies in informatics and control, 27(4), 385-394. zavadskas, e., z. turskis, j. antucheviciene, a. zakarevicius, (2012). optimization of weighted aggregated sum product assessment, elektronikairelektrotechnika, 122, 36. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 2, issue 1, 2019, pp. 132-146. issn: 2560-6018 eissn: 2620-0104 doi:_ https://doi.org/10.31181/dmame1901132b * corresponding author. e-mail addresses: dbozanic@yahoo.com (d. božanić), tesic.dusko@yahoo.com (d. tešić), jovokocic1993@gmail.com (j. kočić). multi-criteria fucom – fuzzy mabac model for the selection of location for construction of single-span bailey bridge darko božanić1*, duško tešić1 and jovo kočić2 1 university of defense in belgrade, military academy, belgrade, serbia 2 ministry of defence, serbian armed forces, pancevo, serbia received: 03 september 2018; accepted: 12 february 2019; available online: 09 march 2019. original scientific paper abstract: selecting the most favorable location for construction of singlespan bailey bridge is ideal for applying multi-criteria decision making. in that regard, it has been developed a model for selecting the most favorable location. the first part of the model is based on the full consistency method (fucom), and it is used for the evaluation of weight coefficients of criteria. the second part of the model presents the fuzzification of the multiattributive border approximation area comparison (mabac) method, which is used in the evaluation of alternatives. additionally, in the paper are presented basic criteria, based on which the selection is to be made key words: fucom, fuzzy mabac method, single-span bailey bridge, selection of location. 1. introduction – problem description the set for launching bailey bridge consists of a number of elements used to make single-span and multi-span bridges (bridges on standing supports) which are designed for overcoming dry and water barriers. these bridges are mounted on the banks, and after mounting their construction are launched over dry or water barrier (slavkovic et al., 2013). they can be easily adapted to different length or capacity requirements. their main disadvantage is large mass of the parts of the set, which can significantly slow down the mounting of the bridge itself. these sets are included in the engineering units of the serbian army. the bridges made of this material can be found throughout serbia, and in some places they represent significant link between the two banks. the selection of location for mounting a single-span bailey bridge is ideal field for the application of multi-criteria decision making methods. potential locations where such bridges could be placed usually have significant differences that more or less mailto:dbozanic@yahoo.com mailto:tesic.dusko@yahoo.com mailto:jovokocic1993@gmail.com multi-criteria fucom – fuzzy mabac model for the selection of location for construction of … 133 affect the speed of assembly and human and material resources necessary during the construction process (gordic et al., 2013). by correct selection of location for such bridge can be prevented potential problems in the process of its construction and later use. in this paper, the selection of location for the construction of a single-span bailey bridge is carried out using the fucom fuzzy mabac method. weight coefficients of criteria are calculated using the fucom method, while for ranking alternatives is used fuzzy mabac method. both methods are very young and have not been largely applied so far. the fucom method was developed in 2018 by pamučar et al. (2018). in the same year prentkovskis et al. (2018) used this method as a part of the model for improving service quality measurement. crisp mabac method was announced for the first time in 2015 by pamučar and ćirović (2015). as a new method, it has been noted by the researchers quickly, and now there are many papers using this method in problem consideration, independently or as a part of a hybrid model (božanić et al., 2016a; peng & yang, 2016; chatterjee et al., 2017; hondro, 2018; majchrzycka & poniszewska, 2018; ji et al., 2018; peng & dai, 2018). in some papers, the method is used in fuzzy environment (roy et al., 2016; xue et al., 2016; sun et al., 2017; hu et al., 2019; yu et al., 2017), and it has also appeared combined with rough numbers (sharma et al., 2018; roy et al. 2017). 2. methods considering that the hybrid fucom – fuzzy mabac model consists of two methods, in the following section of the paper these two methods will be described in detail. 2.1. fucom this method is a new mcdm method proposed in (pamučar et al., 2018). in the following section, the procedure for obtaining the weight coefficients of criteria by using fucom is presented. step 1. in the first step, the criteria from the predefined set of the evaluation criteria  1 2 nc c , c ,..., c are ranked. the ranking is performed according to the significance of the criteria, i.e. starting from the criterion which is expected to have the highest weight coefficient to the criterion of the least significance. thus, the criteria ranked according to the expected values of the weight coefficients are obtained: j(1) j( 2) j( k ) c c ... c   (1) where k represents the rank of the observed criterion. if there is a judgment of the existence of two or more criteria with the same significance, the sign of equality is placed instead of “>” between these criteria in the expression (1) step 2. in the second step, a comparison of the ranked criteria is carried out and the comparative priority ( k / ( k 1)   , k 1, 2,..., n , where k represents the rank of the criteria) of the evaluation criteria is determined. the comparative priority of the evaluation criteria ( k / ( k 1)   ) is an advantage of the criterion of the j( k ) c rank compared to the criterion of the j( k 1)c  rank. thus, the vector of the comparative priorities of the evaluation criteria are obtained, as in the expression: (2) božanić et al./decis. mak. appl. manag. eng. 2 (1) (2019) 132-146 134  1/ 2 2/ 3 k / (k 1), ,...,     (2) where k / ( k 1)   represents the significance (priority) that the criterion of the j( k ) c rank has compared to the criterion of the j( k 1) c  rank. the comparative priority of the criteria is defined in one of the two ways defined in the following part: a) pursuant to their preferences, decision-makers define the comparative priority k / ( k 1)   among the observed criteria. b) based on a predefined scale for the comparison of criteria, decision-makers compare the criteria and thus determine the significance of each individual criterion in the expression (1). the comparison is made with respect to the first-ranked (the most significant) criterion. thus, the significance of the criteria ( j( k )c  ) for all of the criteria ranked in step 1 is obtained. since the first-ranked criterion is compared with itself (its significance is j(1)c 1  ), a conclusion can be drawn that the n-1 comparison of the criteria should be performed. as we can see from the example shown in step 2b, the fucom model allows the pairwise comparison of the criteria by means of using integer, decimal values or the values from the predefined scale for the pairwise comparison of the criteria. step 3. in the third step, the final values of the weight coefficients of the evaluation criteria   t 1 2 n w , w ,..., w are calculated. the final values of the weight coefficients should satisfy the two conditions: (1) that the ratio of the weight coefficients is equal to the comparative priority among the observed criteria ( k / ( k 1)   ) defined in step 2, i.e. that the following condition is met: k k / ( k 1) k 1 w w     (3) (2) in addition to the condition (3), the final values of the weight coefficients should satisfy the condition of mathematical transitivity, i.e. that k / ( k 1) ( k 1) / ( k 2) k / ( k 2)          . since k k / ( k 1) k 1 w w     and k 1 ( k 1) / ( k 2) k 2 w w       , that k k 1 k k 1 k 2 k 2 w w w w w w       is obtained. thus, yet another condition that the final values of the weight coefficients of the evaluation criteria need to meet is obtained, namely: k k / ( k 1) ( k 1) / ( k 2) k 2 w w         (4) full consistency i.e. minimum dfc (  ) is satisfied only if transitivity is fully respected, i.e. when the conditions of k k / ( k 1) k 1 w w     and k k / ( k 1) ( k 1) / ( k 2) k 2 w w         are met. in that way, the requirement for maximum consistency is fulfilled, i.e. dfc is 0  for the obtained values of the weight coefficients. in order for the conditions to be met, it is necessary that the values of the weight coefficients   t 1 2 n w , w ,..., w meet multi-criteria fucom – fuzzy mabac model for the selection of location for construction of … 135 the condition of k k / ( k 1) k 1 w w       and k k / ( k 1) ( k 1) / ( k 2) k 2 w w           , with the minimization of the value  . in that manner the requirement for maximum consistency is satisfied. based on the defined settings, the final model for determining the final values of the weight coefficients of the evaluation criteria can be defined. j( k ) k / ( k 1) j( k 1) j( k ) k / ( k 1) ( k 1) / ( k 2) j( k 2) n j j 1 j min s.t. w , j w w , j w w 1, j w 0, j                          (5) 2. 2. fuzzy мавас method the mabac method is developed by (pamučar & ćirović, 2015). it is developed as the method providing crisp values. in this paper is carried out its fuzzification. the fuzzyfication is performed using triangular fuzzy numbers. a general form of triangular fuzzy number is given in the figure 1. 0 t1 µ(x) x. t2 t3 1 figure 1. triangular fuzzy number triangular fuzzy numbers have the form 1 2 3 t (t , t , t ) . value t1 represents the left distribution of the confidence interval of fuzzy number t, t2 is where the fuzzy number membership function has the maximum value equal to 1, and t3 represents the right distribution of the confidence interval of fuzzy number t (pamučar, 2011). the fuzzyfication of the mabac method is taken from (božanić et al., 2018), and its mathematical formulation is presented in seven steps. božanić et al./decis. mak. appl. manag. eng. 2 (1) (2019) 132-146 136 step 1. forming of the initial decision matrix ( x ). in the first step the evaluation of m alternatives by n criteria is performed. the alternatives are shown by vectors  i i1 i2 ina x , x ..., x , where xij is the value of the i alternative by j criterion (i = 1,2, ... m; j = 1,2, ..., n). 1 2 n 1 11 12 1n 2 11 22 2n m 1m 2m mn c c ... c a x x ... x a x x x x ... ... ... ... ... a x x ... x             (6) where m denotes the number of the alternatives, and n denotes total number of criteria. step 2. normalization of the initial matrix elements ( x ). 1 2 n 1 11 12 1n 2 11 22 2n m 1m 2m mn c c ... c a t t ... t a t t t n ... ... ... ... ... a t t ... t              (7) the elements of the normalized matrix ( n ) are obtained by using the expressions: for benefit-type criteria ij i ij i i x x t x x       (8) for cost-type criteria ij i ij i i x x t x x       (9) where ij x , i x  and i x  represent the elements of the initial decision matrix ( x ), whereby i x  and i x  are defined as follows: i 1r 2r mr x max(x , x ,..., x )   and represent the maximum values of the right distribution of fuzzy numbers of the observed criterion by alternatives. i 1l 2l ml x min(x , x ,..., x )   and represents minimum values of the left distribution of fuzzy numbers of the observed criterion by alternatives step 3. calculation of the weighted matrix ( v ) elements 11 12 1n 21 22 2n m1 m 2 mn v v ... v v v ... v v ... ... ... ... v v ... v             (10) multi-criteria fucom – fuzzy mabac model for the selection of location for construction of … 137 the elements of the weighted matrix ( v ) are calculated on the basis of the expression (11) ij i ij i v w t w  (11) where ij t represent the elements of the normalized matrix ( n ), i w represents the weighted coefficients of the criterion. step 4. determination of the approximate border area matrix ( g ). the border approximate area for every criterion is determined by the expression (12): 1/ m m i ij j 1 g v          (12) where ij v represent the elements of the weighted matrix ( v ), m represents total number of alternatives. after calculating the value of i g by criteria, a matrix of border approximate areas g is developed in the form n x 1 (n represents total number of criteria by which the selection of the offered alternatives is performed).   1 2 n 1 2 n c c ... c g g g ... g (13) step 5. calculation of the matrix elements of alternatives distance from the border approximate area ( q ) 11 12 1n 21 22 2n m1 m 2 mn q q ... q q q q q ... ... ... ... q q ... q             (14) the distance of the alternatives from the border approximate area ( ij q ) is defined as the difference between the weighted matrix elements ( v ) and the values of the border approximate areas ( g ). q v g  (15) the values of alternative i a may belong to the border approximate area ( g ), to the upper approximate area ( g  ), or to the lower approximate area ( g  ), i.e.,  ia g g g      . the upper approximate area ( g  ) represents the area in which the ideal alternative is found ( a  ), while the lower approximate area ( g  ) represents the area where the anti-ideal alternative is found ( a  ), as presented in the figure 2. božanić et al./decis. mak. appl. manag. eng. 2 (1) (2019) 132-146 138 g  g  a  a  1 a 3 a 4 a 2 a 5 a 6 a 7 a g upper approximation area lower approksimation area bordere approksimation area 0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 c ri te ri o n f u n c ti o n s figure 2. display of upper ( g  ), lower ( g  ) and border ( g ) approximate area (pamučar & ćirović, 2015) the membership of alternative i a to the approximate area ( g , g  or g  ) is determined by the expression ij i ij ij g if q 0 a g if q 0 g if q 0          (16) for alternative i a to be chosen as the best from the set, it is necessary for it to belong, by as many as possible criteria, to the upper approximate area ( g  ). the higher the value i q g   indicates that the alternative is closer to the ideal alternative, while the lower the value i q g   indicates that the alternative is closer to the anti-ideal alternative. step 6. ranking of alternatives. the calculation of the values of the criteria functions by alternatives is obtained as the sum of the distance of alternatives from the border approximate areas ( i q ). by summing up the matrix q elements per rows, the final values of the criteria function of alternatives are obtained n i ij j 1 s q , j 1, 2,..., n, i 1, 2,..., m     (17) where n represents the number of criteria, and m is the number of alternatives. step 7. final ranking of alternatives. by defuzzification of the obtained values i s , the final rank of alternatives is obtained. the defuzzification can be performed with the next expressions (seiford, 1996):     13 1 2 1 1defazzy s= t t t t 3 t        (18)   13 2 1defazzy s= t t 1 t 2        (19) multi-criteria fucom – fuzzy mabac model for the selection of location for construction of … 139 3. description of criteria and calculation of weight coefficients the criteria for selecting the most favorable location for a single-span bailey bridge are defined based on the analysis of the available literature. the analysis sets out seven key criteria that have the greatest influence on the selection, and they are the following (kočić, 2017): c1access roads c2scope of work on site arrangement c3properties of banks c4width of water barrier c5masking conditions c6scope of works on joining access roads with the crossing point c7protection of units the concept of access roads (c1) refers to the number and quality of the roads by which the resources are brought to the location for construction and launching of the bridge over the water barrier, or close to it. these are the roads with adequate surface which does not require significant repairs and reconstructions. through this criterion several elements are considered: capacity, number and width of access roads, as well as the position of roads in relation to the barrier (administrative or lateral) (pamučar et al., 2011). the scope of work on site arrangement (c2) represents the workload required for the site arrangement. in other words, it refers to the works necessary for arranging a place of work, where the space for storage of the parts of the set is arranged, parking of motor vehicles, place for stuff operation, space for rest, material disposal, and space for assembly and launching of the bridge (božanić, 2017). properties of the banks (c3) refer to the soil composition of the bank, height of the bank, slope of the bank, forestation, artificial barriers, and the like. the width of water barrier (c4) is defined as the distance from one bank to the other, measured by the surface of water (pifat, 1980). masking conditions (c5) include measures and procedures undertaken to hide the activities and arrangement of the forces, assets and objects from the enemy, in order to lead the enemy to wrong conclusions, to make wrong decisions and apply wrong actions (rkman, 1984). scope of works on joining access roads with the crossing point (c6) refers to the roads that ensure moving the unit from the nearest access road to the crossing point over the water barrier. unit protection (c7) is an integral and essential part of every operation. this criterion includes the assessment of the measures that must be taken to ensure required level of unit protection. the set of criteria from c1 to c7 consists of two subsets: the "c +" is a set of criteria of the benefit type, which means that the higher value of criteria is more favorable (the criteria c1, c3, c5 and c7), and the"c -" is a set of criteria of the cost type, which means that the lower value of criteria is more favorable (the criteria c2, c4 and c6). the criterion c4 is presented as numerical, while the other criteria are presented as linguistic. the weight coefficients of criteria are obtained by applying the fucom method. the evaluation of the weight coefficients is performed by 9 decision makers (dm) – experts in the field of the subject matter. for all decison makers is carried out the evaluation of competence. božanić et al./decis. mak. appl. manag. eng. 2 (1) (2019) 132-146 140 in the first step, the decision makers ranked the criteria. after several rounds of harmonization, three groups of ranks of criteria appeared, which are as follows: dm1, dm2, dm6, dm7, dm8 and dm9: c1>c2>c3>c4>c5>c6,>c7, dm3 and dm5: c1>c2>c5>c4>c3>c6,>c7 dm4: c2>c1>c3>c4>c5>c6,>c7. in the second step, the decision makers compared in pairs the ranked criteria from the step 1. the comparison is made according to the first-ranked criterion, based on the scale  1, 7 . this is how the importance of the criteria is obtained ( ( )j kc  ) for all the criteria ranked in the step 1 (table 1). table 1. importance of criteria dm1 criteria c1 c2 c3 c4 c5 c6 c7 importance ( ( )j kc  ) 1 2 2.5 3 3.1 4 5.5 dm2 criteria c1 c2 c3 c4 c5 c6 c7 importance ( ( )j kc  ) 1 2.5 3 3.5 4 5 5 . . . . . . dm9 criteria c1 c2 c3 c4 c5 c6 c7 importance ( ( )j kc  ) 1 2 2.1 3 4 4.5 6 finally, in the third step and based on the comparison performed by dm, applying the expressions 3-5 are obtained the values presented in the table 2. table 2. weight coefficient of criteria by every dm individually dm1 criteria c1 c2 c3 c4 c5 c6 c7 wj 0.335 0.167 0.134 0.112 0.108 0.084 0.061 dm2 criteria c1 c2 c3 c4 c5 c6 c7 wj 0.375 0.150 0.125 0.107 0.094 0.075 0.075 . . . . . . dm9 criteria c1 c3 c2 c4 c5 c6 c7 wj 0.339 0.170 0.162 0.113 0.085 0.075 0.057 having been obtained the weight coefficients of criteria by every dm, it is performed the calculation of the aggregated weight coefficient. such calculation was carried out by subsequent synthesis of individual decisions by the method of averaging using geometric mean (geometric mean method – gmm) applying the expression (zoranović & srđević, 2003): multi-criteria fucom – fuzzy mabac model for the selection of location for construction of … 141   1     k k bg i i k a a k (20) where: g i a – aggregated value of the weight coefficient,  ia k – value of the weight coefficient for every k-th dm where k=1,...k, k b – additionally normalized competence coefficient of the k-th dm; final, aggregated values of the weight coefficients are presented in the table 3. table 3. final weight coefficient of criteria criteria weight coefficient of criteria c1 0.311 c2 0.198 c3 0.137 c4 0.112 c5 0.098 c6 0.079 c7 0.065 4. model testing the testing of the model, respectively, fuzzy mabac method is performed with six alternatives. before the very beginning of the testing, fuzzy linguistic descriptors had been defined which were used to describe linguistic criteria 1 0.8 0.6 0.4 0.2 1 32 54 0 a b c d e figure 3. graphic display of fuzzy linguistic descriptors (božanić et al., 2016b) every criterion can be described with five values: božanić et al./decis. mak. appl. manag. eng. 2 (1) (2019) 132-146 142 c1, c3, c5 and c7: a=very bad (vb), b=bad (b), c=medium (m), d=good (g) and e=excellent (e). c2 and c6: a=very small (vs), b=small (s), c=medium (m), d=large (l), e=very large (vl). the initial decision making matrix is shown in the table 4. table 4. initial decision making matrix c1 c2 c3 c4 c5 c6 c7 a1 m l e (45, 50, 56) m vs vb a2 g s m (39, 44, 47) vb vl g a3 vb vl g (47, 51, 56) e s m a4 b m vb (46, 48, 51) g vs vb a5 m l b (38, 42, 45) e l g a6 e m g (45, 47, 51) g s b the quantification of linguistic descriptors is shown in the table 5. table 5. quantification of linguistic descriptors c1 c2 c3 c4 c5 c6 c7 a1 (2,3,4) (3,4,5) (4,4,5) (45, 50, 56) (2,3,4) (1,1,2) (1,1,2) a2 (3,4,5) (1,2,3) (2,3,4) (39, 44, 47) (1,1,2) (4,4,5) (3,4,5) a3 (1,1,2) (4,4,5) (3,4,5) (47, 51, 56) (4,4,5) (1,2,3) (2,3,4) a4 (1,2,3) (2,3,4) (1,1,2) (46, 48, 51) (3,4,5) (1,1,2) (1,1,2) a5 (2,3,4) (3,4,5) (1,2,3) (38, 42, 45) (4,4,5) (3,4,5) (3,4,5) a6 (4,4,5) (2,3,4) (3,4,5) (45, 47, 51) (3,4,5) (1,2,3) (1,2,3) applying steps 1 to 7 of the fuzzy mabac method, final values for every alternative are obtained, which allow ranking alternatives and selecting the most favorable location for the construction of a bailey bridge. the table 6 shows final results by alternatives table 6. ranking of alternatives fuzzy mabac method crisp mabac method si rank si rank a1 0.027 3 0.022 4 a2 0.127 2 0.175 2 a3 -0.110 6 -0.174 6 a4 -0.079 5 -0.077 5 a5 0.021 4 0.073 3 a6 0.168 1 0.219 1 as can be noted in the table 6, the rank of criteria slightly differs when applying crisp and fuzzified mabac method. the main difference is in the ranking of alternatives a1 and a5. it is also noted that the obtained values by alternatives are not the same, but that does not have a significant influence to the rank of criteria. multi-criteria fucom – fuzzy mabac model for the selection of location for construction of … 143 5. sensitivity analysis in this section is presented sensitivity analysis, as a logical sequence of the development of the multi-criteria decision-making model.the sensitivity assessment was done by changing the weight coefficients of the criteria, using seven different scenarios, where in each scenario the second criterion was favorable (pamučar et. al. 2017). the display of weight coefficients according to the scenarios is given in table 7. table 7. weight coefficient in different scenario criteria s-0 s-1 s-2 s-3 s-4 s-5 s-6 s-7 c1 0.311 0.4 0.1 0.1 0.1 0.1 0.1 0.1 c2 0.198 0.1 0.4 0.1 0.1 0.1 0.1 0.1 c3 0.137 0.1 0.1 0.4 0.1 0.1 0.1 0.1 c4 0.112 0.1 0.1 0.1 0.4 0.1 0.1 0.1 c5 0.098 0.1 0.1 0.1 0.1 0.4 0.1 0.1 c6 0.079 0.1 0.1 0.1 0.1 0.1 0.4 0.1 c7 0.065 0.1 0.1 0.1 0.1 0.1 0.1 0.4 the values obtained by applying different scenarios are given in table 8. table 8. ranking of alternatives by applying different scenarios alter. index s-1 s-2 s-3 s-4 s-5 s-6 s-7 si r a n k si r a n k si r a n k si r a n k si r a n k si r a n k si r a n k a1 0.024 4 -0.020 5 0.149 1 -0.034 4 -0.021 5 0.131 1 -0.071 5 a2 0.113 2 0.143 1 0.038 4 0.097 2 -0.132 6 -0.105 6 0.143 2 a3 -0.114 6 -0.084 6 0.086 2 -0.064 6 0.091 3 0.067 3 0.040 3 a4 -0.098 5 0.007 3 -0.148 6 -0.048 5 0.007 4 0.084 2 -0.118 6 a5 0.048 3 0.003 4 -0.027 5 0.134 1 0.128 1 -0.046 5 0.152 1 a6 0.189 1 0.094 2 0.064 3 0.051 3 0.094 2 0.046 4 0.019 4 based on sensitivity analysis of the results from the table 8, it can be observed that the model in the midst of change of weight coefficients provides also the change of ranks of the given alternatives. it is interesting to note, though, that the firstranked alternative a6, no matter the scenario, not once was ranked as the fifth or the sixth, and the alternative a3 which was ranked as the last, not in one scenario appeared as the first one. for the mathematical determination of the correlation of ranks, the values of spirman's coefficient were used: n 2 i i 1 2 6 d s 1 n(n 1)     (21) where is:  s the value of the spirman coefficient,  di the difference in the rank of the given element in the vector w and the rank of the correspondent element in the reference vector, božanić et al./decis. mak. appl. manag. eng. 2 (1) (2019) 132-146 144  n number of ranked elements. the rank of each criterion the alternative is determined based on the weight coefficient vector w=(w1, w2, ..., wn). spirman's coefficient takes values from the interval -1,1. when the ranks of the elements completely coincide, the spirman coefficient is 1 ("ideal positive correlation"). when the ranks are completely opposite, the spirman coefficient is -1 ("ideal negative correlation"), that is, when s = 0 the ranks are unregulated. table 9. spirman's coefficient values s-0 s-1 s-2 s-3 s-4 s-5 s-6 s-7 s-0 1 0.964 0.821 0.464 0.750 0.286 0.143 0.429 s-1 1 0.857 0.321 0.857 0.429 0.000 0.571 s-2 1 0.071 0.714 0.214 0.000 0.429 s-3 1 0.214 0.250 0.607 0.607 s-4 1 0.500 -0.071 0.786 s-5 1 0.286 0.696 s-6 1 -0.143 s-7 1 as observed from the table of spearman's coefficient values, it ranges from -0.143 to 0.964. the differences in the ranks of alternatives point out the sensitivity of the model to changes of weight coefficients. on the other hand, low spearman’s coefficient in certain scenarios indicates the necessity of careful evaluation of alternatives by criteria, because potential errors could reflect on the final rank of alternatives. what is important is that the values of spearman’s coefficient, in relation to the s-0 strategy (according to calculated weight coefficients) are fairly high compared to all the other strategies. 6. conclusions the introduction of the model into the decision-making processes has proved to be very useful. in the specific case, deciding based on the application of the model has created the conditions for persons with less experience to make a decision. also, this kind of decision making helps decision makers to perceive complete picture of the impact of all the conditions in which a bailey bridge is constructed. on the other hand, deciding without applying the model creates the possibility of ignoring or neglecting a part of criteria during decision making. the application of the fuzzified mabac 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(in serbian). © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 5, issue 2, 2022, pp. 287-299. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0318062022m * corresponding author. e-mail addresses: toufikmzili95@gmail.com(t. mzili),saidriffi2@gmail.com(m. riffi), dr.mzili.ilyass@gmail.com(i. mzili), gdhiman0001@gmail.com(g.dhiman) a novel discrete rat swarm optimization (drso) algorithm for solving the traveling salesman problem toufik mzili1*, mohammed essaid riffi 1, ilyass mzili1, and gaurav dhiman2 1 department of computer science, laboratory laroseri, faculty of science, chouaib doukkali university, morocco 2 department of computer science, government bikram college of commerce, punjab, india received: 11 february 2022; accepted: 25 may 2022; available online: 18 june 2022. original scientific paper abstract: metaheuristics are often used to find solutions to real and complex problems. these algorithms can solve optimization problems and provide solutions close to the global optimum in an acceptable and reasonable time. in this paper, we will present a new bio-inspired metaheuristic based on the natural chasing and attacking behaviors of rats in nature, called a rat swarm optimizer. which has given good results in solving several continuous optimization problems, and adapted it to solve a discrete, np-hard, and classical optimization problem that is the traveling salesman problem (tsp) while respecting the natural behavior of rats. to test the efficiency of the adaptation of our proposal, we applied the adapted rat swarm optimization (rso) algorithm to some reference instances of tsplib. the obtained results show the performance of the proposed method in solving the traveling salesman problem (tsp). keywords: travelling salesman problem, rat swarm optimization combinatorial optimization, metaheuristic, and nature-inspired. 1. introduction the traveling salesman problem (tsp) (pintea et al., 2017) this is an np-hard (tanaev et al., 1994) problem in combinatorial optimization, in which the traveling salesman wishes to visit a certain number of cities, starting and ending its route with the same city of departure, visiting these cities only once, making the shortest possible route. this is an important problem in theoretical computer science and operations research. mailto:saidriffi2@gmail.com(m.%20riffi) mailto:dr.mzili.ilyass@gmail.com mailto:gdhiman0001@gmail.com mzili et al./decis. mak. appl. manag. eng. 5 (2) (2022) 287-299 288 tsp has several applications, even in its purest formulation, such as astronomy, logistics, transportation, and telecommunications. the resolution of this problem in a reasonable execution time led the researchers to propose an approximation algorithm such as heuristics, such as simulated annealing (sa) (johnson et al., 1989), taboo search (ts) (glover, 1989), local search (korupolu et al., 2000), etc. and metaheuristics such as ant colony( wang et al., 2012), genetic algorithm (ga)( rybickova et al., 2012), particle swarm optimization (pso) (wang et al., 2003), hybrid methods: a novel hybrid penguins search optimization algorithm (mzili et al., 2015), discrete optimization of the search for penguins (mzili et al., 2015), new discrete hybrid pso (bouzidi & riffi, 2014), particle swarm optimization with simulated annealing (fang et al., 2007), discrete cat swarm optimization (bouzidi & riffi ,2013), elephants herding optimization (eho)(hossam et al., 2018), ant colony optimization (aco)(bao, 2015), artificial bee colony (abc)(gündüz et al., 2015), discrete social spıder (baş & ülker, 2021), etc. in general, the most commonly used heuristics are based on nature, animal behavior, and artificial intelligence. these algorithms have several advantages over other heuristics: they can contain information about the entire search space and are very easy to implement. these algorithms have fewer parameters, which means they require less memory than other metaheuristics (dhiman et al., 2022). these algorithms can explore the right balance between search spaces by traversing the entire operation space to find the optimal value. these advantages are a huge motivation to adopt a new bio-inspire-based metaheuristic algorithm (called rso), which has been recently developed to solve continuous optimization problems such as the pressure vessel problem, and the gearbox design problem. welded beam design problem, tension/compression spring design problem, rod support design problem, bearing design problem. the results are much better than the seven best-known meta-heuristics and are robust (dhiman et al., 2020). the objective of this work is to adapt the rat swarm optimization (rso) algorithm, introduced in 2020 by gaurav dhiman, to solve the traveling salesman problem (tsp) a classical discrete optimization problem, very well known for its complexity and very useful in several domains. a new fitness function based on the euclidean distance is proposed to deal with the discontinuity, which has been neglected in other algorithms. this adaptation consists in reconstructing again this method by introducing new mathematical operators and by modifying just the values of the parameters of the method. without touching the definition of the proposed rat behavior. the organization of this research is as follows: in sect. 2, a presentation of the traveling salesman problem; in sect. 3, a presentation of the rso algorithm introduced to solve continued optimization problems. in sect. 4, the adaptation of the rso optimization algorithm to solve the traveling salesman problem. in sect. 5, the results of tests using tsplib instances. finally, comes the conclusion in the last section. 2. the travelling salesman problem the traveling salesman problem (tsp) is one of the oldest and most studied combinatorial optimization problems. this problem aims to find the shortest circuit which allows him to visit a certain number of cities and pass once and only once per city and return to his starting point, at a lower cost, by covering the shortest distance a novel discrete rat swarm optimization (drso) algorithm for solving the traveling… 289 possible. the distances between cities are known. we must find the path that minimizes the distance traveled. 2.1. the importance of resolving tsp: the traveling salesman problem consists of determining whether it is possible to travel through n cities in such a way that the sum of the distances traveled in each city is the least costly. solving the traveling salesman problem is usually very timeconsuming. therefore, new strategies must be found for the solvers. the trade traveler problem (tsp) essentially aims to find the shortest route through a set of points to minimize the cumulative cost of travel overall routes. since a solution must determine the number of cities, it cannot always successfully solve large-scale problems. as such, tsp is one of the np-complete tasks, which means that although there are efficient algorithms, none of them are sure to stop in a finite time. this makes solving tsp more difficult, more important, and more motivating. 3. rso algorithm gaurav dhiman introduced the rat swarm optimizer (rso) (dhiman et al., 2020) in 2020 to solve continuous optimization problems. the rso algorithm is inspired by the chasing and attacking behaviors of rats. rats are long-tailed and medium-sized rodents, socially intelligent by nature and they are territorial animals that live in groups of two males and females, they participate in various activities. such as jumping, running, and tumbling. and boxing. but they are very aggressive, which in many cases results in the death of some animals. this aggressive behavior when hunting and fighting with prey gave rise to this algorithm. the hunting and fighting behaviors of the rats are mathematically modeled to design the rso algorithm and perform the optimization. 3.1. rats behavior modeling this subsection describes the behavior of rats, chasing and fighting. then the proposed rso algorithm is outlined. 3.2. chasing the prey in general, rats are social animals that hunt prey in groups due to their agonistic social behavior. to define this behavior mathematically, we assume that the best researcher knows the location of the prey. other search agents can update their positions against the best search agent obtained so far. to model this mechanism the following equations are proposed: 𝑋 = 𝐴 × 𝑋𝑖 + 𝐶 × (𝑋𝐵𝑒𝑠𝑡 − 𝑋𝑖) (1) where ⃗x represents the positions of rats and x is the best optimal solution. a and c are calculated as follows: 𝐴 = 𝑅 − 𝑥 ( 𝑅 𝑀𝑎𝑥𝐼𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛 ) , 1 ≤ 𝑅 ≤ 5 (2) where x in [0 , 1, 2…max-iteration]. therefore, r and c are random numbers between [1, 5] and [0, 2], respectively. parameters a and c are responsible for better exploration and exploitation during iterations. mzili et al./decis. mak. appl. manag. eng. 5 (2) (2022) 287-299 290 3.3. fighting with prey the process of fighting rats with prey is mathematically defined by the following equation: 𝑋𝑖+1 = |𝑋𝐵𝑒𝑠𝑡 − 𝑋𝑖 | (3) where 𝑋𝑖+1 defines the newly updated position of the rat. it saves the best solution and updates the positions of other search agents against the best search agent. figure 1: shows how rats change their positions according to equations (1) and (3) in a three-dimensional environment. in this figure, the rat (a, b) can actualize its position toward the prey position (a*, b*). by adjusting the parameters, as shown in eq. (2). the different number of positions can be reached around the current position. the adjusted values of parameters a and c guarantee good exploration and exploitation. the rso algorithm will record the optimal solution with the fewest operators. 3.4. rat swarm optimization (rso) algorithm begin: step 1: initialize the rats population pi where i = 1, 2...n. step 2: choose the initial parameters of rso: a, c, and r. step 3: now, calculate the fitness value of each search agent. step 4: the best search agent is then explored in the given search space. step 5: update the positions of search agents using eq(1). step 6: check whether any search agent goes beyond the boundary limit of a search space and then amend it. step 7: again, calculate the updated search agent fitness value and update the vector xbest if there is a better solution than the previous optimal solution. step 8: stop the algorithm if the stopping criteria are satisfied. otherwise, return to step 5. step 9: return the best obtained optimal solution. end figure 1: 3d position vectors of rats a novel discrete rat swarm optimization (drso) algorithm for solving the traveling… 291 4. use rso to solve the tsp this section presents the adaptation of the rso method to solve a tsp. the adaptation of rso consists in redefining the algebraic operators of the algorithm and the structures and stages of this algorithm 4.1. adapted discrete rso to solve the tsp the rso method proposed by dhiman et al. (2020) was defined to solve continuous optimization problems, so it cannot be applied to solve discrete combinatorial optimization problems, since in continuous optimization a number represents the solution, on the other hand in combinatorial optimization, the solution is represented by an order which can be modeled as a vector or a sequence of numbers. to adapt the rso to solve the problems of discrete combinatorial optimization, it is necessary to adapt the operations and the operators while respecting the real behavior of the rats:  the position of a rat represents a solution and it is modeled as a vector of integers where each integer represents a city x = {1 2 3 4 5}.  subtraction between each of the two positions (x y) presents the list of permutations to apply to the y path (vector) to obtain the x path:  x = {1 2 3 4 5} and y = {1 2 4 3 5}  then q=x y  return q= {(2, 3) (4, 5)}..  the operation of multiplication (*) is an operation performed between a real k [0;1] and a set of permutations q,  the result q’ is part of the set q according to the value of k.  the addition operation ⊕ is an effective operation between the solution ⃗x and the set of permutations q, the result is a new solution x’. this operation consists in applying permutations of q’ on x to obtain a new x’ solution.  x= {1 2 3 4 5 6}  q = {(1, 2) (4, 5)}  x’ =x ⊕ q  x’= {2 1 3 5 4 6}. mzili et al./decis. mak. appl. manag. eng. 5 (2) (2022) 287-299 292 4.2. the flowchart of the rso algorithm start generate the initial rats population choose the initial parameters of rso calculate the fitness value of each search agent update the positions of the other search agent using equation (3) calculate the fitness value of the new search agent update the agent of search if is a better solution than the previous one end check stop condition figure 2: the flowchart of the rso algorithm 5. experimental results and comparison 5.1. experimental results: the implementation of the rso optimization algorithm adapted to solve the tsp was carried out on the programming language c ++, and the simulations were carried out on a personal computer equipped with a core i7-3540 m cpu at 3.00 ghz, 8 gb of ram, and windows 10 (64 bits). table 1 shows the results of the executions of this algorithm on several different reference instances of tsplib. the parameters were set as follows: the number of rats was set at 100, the number of iterations varied between 6000 and 8000 depending on each instance, c is a random variable between 0 and 1, and r and x remain the same as in the original algorithm. a novel discrete rat swarm optimization (drso) algorithm for solving the traveling… 293 the table displays the following information:  inst: name of the benchmark instance in the tsplib library.  nb.node: number of nodes.  opt: the best-known solution for the instance.  bestr: the best solution found by the algorithm after ten different executions.  worstr: the worst solution found by the algorithm after ten different executions.  average: the average of ten different executions of the algorithm  time: displays the average time in seconds of ten different executions of the algorithm  err: (%) is the percentage relative error  pdbest(%): the percentage deviation of the best solution length from the optimal table 1: results produced by rso method instance opt bestr average worstr err(%) pdbest time(s) eil51 426 426 432,57 442 1,54 0 6,76 berlin52 7542 7542 7788,79 8111 3,27 0 3,84 st70 675 675 685,46 699 1,55 0 9,39 oliver30 420 420 421,62 426 0,38 0 0,39 eil76 538 549 569,5 591 5,85 2,00 13,35 kroa100 21282 21353 21748,4 21986 2,19 0,33 18,76 krob100 22141 22337 23165,5 23929 2,62 0,87 9,50 eil101 629 653 672,62 696 6,93 3,67 0,39 ch130 6,110 6275 6507 6816 6,49 2,62 13,53 rat99 1211 1229 1274,44 1308 5,23 1,46 0,39 d198 15780 16045 16517,8 16994 4,67 1,65 19,89 5.2. comparison and discussion: in this part, we will try to compare the results of the application of the rso with the results of other more known metaheuristics. the results of aco (ant colony optimization) were taken from (bao, 2015), the results of abc (artificial bee colony) (gündüz et al., 2015), and the results of ha (wang et al., 2011). mzili et al./decis. mak. appl. manag. eng. 5 (2) (2022) 287-299 294 table 2: comparison of experimental results of improved dsro with aco, abc, ha, and dssa instance method best mean worst time(s) oliver30 rso 420 421,62 426 0,39 aco 423.74 424.68 429.36 35.20 abc 439.49 462.55 484.83 1.26 ha 423.74 423.74 423.74 19.63 dssa eil51 rso 426 432,57 442 6,76 aco 450.59 457.86 463.55 112.11 abc 563.75 590.49 619.44 2.16 ha 431.74 443.39 454.97 58.33 dssa 431.87 483.53 berlin52 rso 7542 7788,79 8111 3,84 aco 7548.99 7659.31 7681.75 116.67 abc 9479.11 10,390.26 11,021.99 2.17 ha 7544.37 7544.37 7544.37 60.64 dssa 7659 31432 st70 rso 675 685,46 699 9,39 aco 696.05 709.16 725.26 226.06 abc 1162.12 1230.49 1339.24 3.15 ha 687.24 700.58 716.52 115.65 dssa eil76 rso 549 569,5 591 13,35 aco 554.46 561.98 568.62 271.98 abc 877.28 931.44 971.36 3.49 ha 551.07 557.98 565.51 138.82 dssa 559,31 2720,4 kroa100 rso 21353 21748,4 21986 18,76 aco 22455.89 22,880.12 23,365.46 615.06 abc 49519.51 53,840.03 57,566.05 5.17 ha 22122.75 22,435.31 23,050.81 311.12 dssa 21363 189,380 eil101 rso 653 672,62 696 0,39 aco 678.04 693.42 705.65 527.42 abc 1237.31 1315.95 1392.64 5.17 ha 672.71 683.39 696.04 267.08 dssa a novel discrete rat swarm optimization (drso) algorithm for solving the traveling… 295 figure 3: comparison of experimental results of improved dsro with aco figure 4: comparison of experimental results of improved dsro with dssa 0 100 200 300 400 500 600 700 800 eil51 eil76 st70 eil101 oliver30 rso aco 4 2 6 5 4 9 2 1 3 5 3 7 5 4 2 4 3 1 .8 7 5 5 9 .3 1 2 1 3 6 3 7 6 5 9 e i l 5 1 e i l 7 6 k o a 1 0 0 b e r l i n 5 2 rso dssa mzili et al./decis. mak. appl. manag. eng. 5 (2) (2022) 287-299 296 figure 5: comparison of experimental results of improved dsro with aco, abc, ha, and dssa figure 6: comparison of experimental results of improved dsro with abc, ha eil51 berlin52 eil76 rso 426 7542 549 aco 450.59 7548.99 554.46 abc 563.75 9479.11 877.28 ha 431.87 7544.37 551.07 dssa 431.87 7659 559.31 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 a xi s t it le rso aco abc 426 549 675 635 439.49 877.28 1162.12 1237.31 423.74 551.07 687.24 672.71 0 200 400 600 800 1000 1200 1400 eil51 eil76 st70 eil101 rso abc ha a novel discrete rat swarm optimization (drso) algorithm for solving the traveling… 297 6. discussion the following examples show a comparison of the objective function values of the proposed method with other existing metaheuristics for the oliver30, eil51, berlin52, st70, eil76, kroa100, and eil101 instances of tsplib. the results in table 2 confirm that the rso algorithm can solve multiple tsplib instances in a very reasonable runtime. to prove the robustness of the algorithm, table 1 compares the average solution proposed by the rso algorithm with other methods applied to solve the tsp using tsplib. the table also compares the execution time obtained by the algorithm with the other bio-inspired algorithm that solves tsp. according to the results in table 2 and figures 2, 3, 4, and 5, the results obtained by the rso algorithm are good compared to the other methods and this is evident from the rso curve that is lower than the other methods. this can be justified by the simplicity of rso and its parameters, which can guarantee a much faster convergence than other algorithms. in some cases, this algorithm gave results close to the optimum without reaching it, is it because this algorithm is guided by a single search agent which is the best global search agent, and after several iterations, all the agents and solutions converge towards this optimum. to solve this limitation and to make this optimizer more robust, we will consider adding other improvements or hybridization heuristics. 7. conclusion in this paper, we first presented an adaptation of the rso algorithm proposed by dhiman et al (2020) without hybridization to solve the symmetric psd. this adaptation achieved good performance compared to several metaheuristics. discrete rat swarm optimization (drso) is one of the most intelligent and powerful algorithms. this algorithm can be used to solve any optimization problem, including the traveling salesman problem (tsp). this algorithm has great potential to become a very powerful optimization technique. in drso, each rat is a simple entity capable of moving in one dimension to find the best path in its environment. each rat can visit a point in its neighborhood or search for a new point in a radius area around the current point. the algorithm has been tested on a set of benchmark instances of tsplib. its performance exceeds that of recent methods used to solve tsp, such as aco, abc, ha, and dssa. moreover, the robustness and speed of the rso algorithm encourage its use to solve other combinatorial optimization problems. in the future, we will try to improve this algorithm to obtain better results than the majority of methods and we aim to extend the algorithm to apply it to various application domains and solve any discrete optimization problem such as network optimization, scheduling, transportation problems, vehicle routing problem, electronic manufacturing units, etc. several improvements have been made to the algorithm to solve the quadratic assignment problem, a new problem as important as the tsp, designed to minimize the overall cost of building and operating a facility. author contributions: research problem, m.r., t.m. and i.m.; conceptualization, m.r., t.m. and i.m.; methodology, m.r., t.m., g.d., and i.m.; formal analysis, m.r., t.m.; resources, m.r., i.m.; original drafting, m.r. and t.m.; reviewing and editing, m.r., t.m., i.m; project administration, m.r., t.m., im; supervision, m.r., i.m.; proposal, improvement and ideation, m.r., i.m, g.d. mzili et al./decis. mak. appl. manag. eng. 5 (2) (2022) 287-299 298 all authors have read and approved the published version of the manuscript. funding: this research received no external funding. acknowledgment: the authors would like to express their gratitude to the editors and anonymous referees for their informative, helpful remarks and suggestions to improve this paper as well as the important guiding significance to our research. conflict of interest: the authors declare that they have no conflict of interest. references bao, h. 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(2012). an improved ant colony algorithm for solving tsp problems. mathematics in practice and theory. 4(1), 16-28. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1109/icmlc.2003.1259748 plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 4, issue 2, 2021, pp. 1-25. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame210402001a * corresponding author. e-mail addresses: ahmataytekin@artvin.edu.tr (a. aytekin) comparative analysis of normalization techniques in the context of mcdm problems ahmet aytekin1* 1 department of quantitative methods, artvin çoruh university, turkey received: 30 january 2021; accepted: 3 march 2021; available online: 21 march 2021. original scientific paper abstract: normalization is an essential step in data analysis and for mcdm methods. this study aims to outline the positive and negative features of the normalization techniques that can be used in mcdm problems. in order to compare the different normalization techniques, fourteen sets representing different scenarios of decision problems were used. according to the results, if the decision-maker chooses to take the alternative with the highest value in the criteria and avoid the one with the lowest value, or vice versa, optimization-based normalization techniques should be preferred, whereas the reference-based normalization techniques are considered appropriate for situations where there are ideal values determined by the decision-maker for each criterion. however, if the decision-maker believes that the values in the criteria do not represent the monotonous increasing or decreasing benefit/cost, then non-linear normalization techniques should be used. also, in the event of a change in the conditions mentioned above, the decision maker may opt for mixed normalization techniques. however, some data structures, such as the presence of zero, and negative values in the decision matrix, can prevent the use of some normalization techniques. the choice of the normalization technique may also be affected by the problem of rank reversal, the range of normalized values, obtaining the same optimization aspect for all criteria, and the validity of results. key words: data; mcdm; scaling; normalization. 1. introduction in quantitative research, researchers often try to use methods appropriate for the data structures. to do this, the data is first collected and then compiled. in the compilation process, it is always important to create the data structure required by the relevant method using scaling techniques. in the scaling process, the unit of measurement, the size, and the level of the criteria are changed alongside one or more of the transformations, re-measurement, mailto:ahmataytekin@artvin.edu.tr aytekin/decis. mak. appl. manag. eng. 4 (2) (2021) 1-25 2 normalization, or weighting operations. the differences in various features of the criteria such as measurement levels, size of the range, importance levels and reflecting the decision maker's preferences effectively are prominent reasons for scaling. the other reason for scaling is the need to meet the assumptions of the method used for the research or the decision problem. in this context, the primary purpose of scaling is to provide the appropriate measurements or data structure for the proper method or analysis. normalization is one of the critical processes used in scaling data (jensen, 1984; roberts, 1984; lootsma, 1999; tavşancıl, 2006; kainulainen et al., 2009; sarraf et al., 2013; jahan & edwards, 2015; podviezko & podvezko, 2015; gardziejczyk & zabicki, 2017). there are numerous application areas for normalization including data mining, multivariate statistics and multi-criteria decision making (mcdm) among others. this study will however focus on the effects of the normalization processes on solutions obtained using mcdm methods. normalization is used to obtain criteria that have the same weight, are dimensionless, and are suitable for compensatory processes in mcdm problems. normalization also enables the decision maker to show his preferences regarding the problem to a certain extent. there are many normalization techniques in the literature to achieve this. the choice of the normalization technique depends on the structure of problems and the assumptions of mcdm methods. although not yet sufficient, studies on the comparison of normalization techniques have been increasing in the past decades. these studies, however, usually include a small number of techniques. similarly, studies considering the selection of the normalization technique, and the criteria to be used in this selection process are also limited. another important issue is that every normalization technique cannot be suitable for all decision problems. it is, therefore, necessary to investigate the extent to which the normalization techniques have achieved their purposes of development, their roles in the problem, and the mcdm method, their dimensionlessness, and comparability. in this context, this study will examine the practical comparisons of the normalization techniques, determine the positive and negative features, and outline the selection process of normalization techniques suitable for different data structures. the purpose of the study is thus to provide different perspectives on normalization techniques and a holistic framework for researchers and decision makers. 2. normalization normalization is a scaling process used to make the criteria comparable by eliminating the optimization orientation (benefit, cost), the unit of measurement, and the variation range. through normalization, the data is converted to a specific norm or standard. another term often used interchangeably with normalization is standardization. however, standardization is a normalization process that eliminates unit differences and transforms values to a specific range, such as 0-1 in all criteria. in general, normalization techniques are expected to equalize the effect levels of all criteria (regardless of the weighting process), process the zero and negative values, generate the same normalized value for different units of measure that can be converted into each other (as in the case of g / cm3 – kg / m3), and not cause rank reversal problems while also ensuring symmetry in the cost and benefit optimization orientation. the normalization technique that has these features is considered successful (pavličić, 2001; jahan & edwards, 2015; podviezko & podvezko, 2015). comparative analysis of normalization techniques in the context of mcm problems 3 in the analysis of normalization techniques in the mcdm literature, the decision matrix given in equation (1) will be used. the rows of the decision matrix contain alternatives while the columns carry the criteria. each of its cells/elements shows the quality, feature, or performance value of each alternative in the relevant criterion.                       11 1 1 1, , 1, , n m mn i mx x j nx x x (1) the elements of the decision matrix in equation (1) are expressed as xij. xij is the performance or result value of alternative i in criteria j. the following section of the study outlines the classifications of normalization techniques. 2.1. classification of normalization techniques classification of normalization techniques makes it easier to identify the similarities and differences of the techniques, standardize the concepts in the field, and examine the increasing number of techniques. various approaches can be used in the classification of normalization techniques. however, the most common classifications in the literature are done according to the distance measurements, the linearity of the normalization process or the optimization orientation of the criteria (milani et al., 2005; yoon & kim, 1989; zeng et al., 2013; jahan & edwards, 2015). distance measurements are the most commonly used in the normalization process. the distance-based normalization is the ratio of the distances of the alternatives from the starting point (vector 0) to the sum of distances of all alternatives from the starting point in the relevant criterion. in the distance-based normalization processes, eq. (2) is used in the lp metric. (yoon & kim, 1989, p. 22):   1 1 0 ( ) 0 ij ij m pp ij i x n p x           (2) eq. (2) is used in the benefit criteria. for the cost criteria, the values are converted to benefit with the transformation of 1/xij values. in eq. (2), manhattan distance normalization is performed for p = 1, with euclidean distance normalization for p = 2 and tchebycheff distance normalization for p = ∞ (yoon & kim, 1989). in the literature, manhattan distance normalization is called sum-based linear normalization, while euclidean distance normalization is known as vector normalization. for, normalization processes that are not based on distance, a specific value is used. often, these values are the maximum and the minimum in the criterion. similarly, reference values or large fixed numbers can be used. the linearity of the normalization process is that the utilities or values in the criterion increase or decrease monotonously in a specific direction. in non-monotonic normalization processes, there is no continuous increase or decrease of acceptable performance values within the criterion in a certain direction. for example, in the criterion with a normal distribution, the linear normalization process will not be aytekin/decis. mak. appl. manag. eng. 4 (2) (2021) 1-25 4 appropriate if the desired/ideal values are within three or four standard deviations of the mean. z score normalization, some reference-based normalizations, and nonlinear normalizations are some examples of non-monotonic normalization techniques (zavadskas & turskis, 2008; zeng et al. 2013). we can divide normalization techniques into two fundamental classes based on whether they consider the optimization orientation of the criteria. however, some normalization techniques provide a mixed/integrated normalization process with the idea that the optimization orientation and reference value are vital for different criteria that can be found at the same time in the decision problem. the following section will examine the normalization techniques which depend on the optimization orientation, those that are independent of the optimization orientation, and those that have a mixed structure. 2.2. normalization techniques depending on the optimization orientation most normalization techniques provide normalization according to the optimization orientation of the criteria. the optimization orientation is divided into two benefit and cost. the benefit optimization orientation implies that the increase in the performance values of the alternatives evaluated in criterion j is preferred to the decrease. the cost optimization orientation is that the decline in the performance values of the alternatives in criterion j is preferred to the increase. in general, we can say that the highest (maximum) value in benefitorientation criteria and the lowest (minimum) value in costorientation criteria are preferred. normalization techniques depending on the optimization orientation are given in table 1 (brauers & zavadskas, 2006; zavadskas & turskis, 2008; fayazbakhsh et al., 2009; jahan & edwards, 2015; gardziejczyk & zabicki, 2017). these techniques mainly use performance value totals, the maximum value, and the minimum value in a criterion. table 1. normalization techniques depending on the optimization orientation notation techniques benefit criteria cost criteria references n1 sum-based linear normalization    1 ij ij m ij i x n x    1 1 1 ij ij m i ij x n x gardziejczyk & zabicki (2017) n2 vector normalization    2 1 ij ij m ij i x n x     2 1 1 ij ij m ij i x n x zavadskas & turskis (2008) gardziejczyk & zabicki (2017) n3 logarithmic normalization          1 ln ln ij ij m ij i x n x            1 ln 1 ln 1 ij m ij i ij x x n m zavadskas & turskis (2008) gardziejczyk & zabicki (2017) n4 maximum linear normalization  ij ij ij i x n maks x  1 ij ij ij i x n maks x jahan & edwards (2015) comparative analysis of normalization techniques in the context of mcm problems 5 notation techniques benefit criteria cost criteria references n5 minimum linear normalization   min 1 iji ij ij x n x  min iji ij ij x n x jahan & edwards (2015) n6 0-1 interval normalization using maxmin  ij ij ij i x n maks x  min iji ij ij x n x brauers & zavadskas (2006); jahan & edwards (2015) n7 jüttler-körth normalization    1 ij ij i ij ij i maks x x n maks x    min 1 ij iji ij ij i x x n maks x brauers & zavadskas, (2006); zavadskas & turskis (2008); gardziejczyk & zabicki (2017) n8 stopp normalization  100 ij ij ij i x n maks x  100min iji ij ij x n x brauers & zavadskas (2006); gardziejczyk & zabicki (2017) n9 nonlinear (peldschus) normalization         2 ij ij ij i x n maks x         3 ij ij ij i x n maks x brauers & zavadskas (2006); zavadskas & turskis (2008); gardziejczyk & zabicki (2017) n10 weitendorf’s linear normalization    min min ij iji ij ij ijii x x n maks x x    min ij ij i ij ij ijii maks x x n maks x x brauers & zavadskas (2006) n11 z score normalization depending optimization aspect              1 2 1 m ij i ij ij j ij m j ij j i x xx mn x m      ij j ij j x n fayazbakhsh et al. (2009); gardziejczyk & zabicki (2017) it is possible to further divide the normalization techniques depending on the optimization orientation into four sub-classes: sum-based, a maximum or minimum value-based, range-based, and others. from the techniques in table 1, while n1, n2, and n3 are sum-based, n4-n9 are maximum-minimum value-based, n10 is rangebased, and n11 is evaluated under the other category. in sum-based normalization techniques, the sum of performance values within the criterion is used. it seems that the normalization techniques in this class may lead to the rank reversals problem due to the changes (adding or removing alternatives) in the alternative set. for example, when the alternative performance with the highest performance value is removed from criterion j, which has a benefit optimization orientation, the maximum value used in n4 will change. change of the maximum value in criterion j will require the recalculation of normalized values. also, when the aytekin/decis. mak. appl. manag. eng. 4 (2) (2021) 1-25 6 numerator or denominator values change, all normalized values to be generated by the techniques in table 1 will change in criterion j. another critical issue is the presence of the criterion that has negative values in the decision problem. the presence of negative values in a criterion can prevent effective solutions in n1 and n2 depending on the nature of the problem. the negative and positive values in n1 can be said to be mutually offsetting. also, the negative performance values (xij) in n2 cause negative normalized values. and these situations can prevent the realization of effective solutions. the range of normalized values obtained by the normalization techniques in table 1 are different from each other. the sum of all normalized values is equal to 1 in n1 for benefit-orientation criteria and n3 for all type criteria. also, n1, n2, n3 are expected to generate nij in the range of 0-1. from these techniques, n3 was found to be useful if the values in a criterion are quite different from each other (jahan & edwards, 2015, p. 338). maximum/minimum value-based normalization techniques are said to be less successful than the sum-based normalization techniques in handling the scale effect. also, some of the techniques in this class cannot be effectively applied to cost criteria. in some cases, the normalized values may be higher than 1 in the techniques of this category. this situation is generally undesirable in some mcdm methods (jahan & edwards, 2015, p. 338). among the techniques in this class, it is aimed to provide normalization in the range of 0-1, with the highest value being 1 in n4 and the lowest value being 1 in n5. after normalization using n6, the best performance value depending on the optimization orientation of criteria is expected to equal 1 while all normalized values are expected to fall within the range of 0-1. in n7, the normalized values are expected to be in the range of 0-1 while the best value is equal to 1. since n8 produces large normalized values, it does not seem appropriate for most mcdm methods. the range of normalized values obtained in n9 can be expected to be smaller than the normalized values created by most techniques. n10 is one of the most used techniques in the normalization steps of mcdm methods. providing range-based normalization, n10 is successful in handling the scale effect (jahan & edwards 2015, p. 338). normalized values in n10 are expected to be in the range of 0 and 1. z score normalization is frequently used in the application of multivariate statistical methods. fayazbakhsh et al. (2009) used the z score normalization as dependent of the optimization orientation. optimization orientation dependent z score normalization can generate negative values, but normalized values are usually around 0. this situation restricts n11's usage in mcdm methods (jahan & edwards, 2015). normalization techniques were developed to be used for specific purposes or expectations. we have highlighted these purposes and expectations for the optimization orientation dependent normalization techniques. on the other hand, we will test whether the intended or expected normalized values of these normalization techniques will always be obtained in the application section. 2.3. normalization techniques independent of the optimization orientation for normalization techniques independent of the optimization, a specific reference value/range or a constant is used instead of the optimization orientation of the comparative analysis of normalization techniques in the context of mcm problems 7 criteria. for the techniques in this category, one or more of maximum value, minimum value, mean, standard deviation, reference (ideal/target) value/range, adjustable constant number, and data distributions are used in the normalization process (wu, 2002; shih et al., 2007; jahan et al., 2011; jahan et al., 2012; alpar, 2013; saranya & manikandan, 2013; jahan & edwards, 2015; gardziejczyk & zabicki, 2017; aytekin, 2020). such values as maximum and minimum are also used in the normalization of optimization orientation-dependent techniques. the normalization techniques independent of the optimization orientation have no formula change based on the optimization orientation and use the same equation for all criteria types. normalization techniques independent of the optimization orientation are given in table 2. table 2. normalization techniques independent of the optimization orientation notation technique all criteria references n12 z-score (nonmonotonic) normalization       2 22 ij j j x r ij n e shih et al. (2007) n13 comprehensive normalization              m , min min , 1 ij j ij j ij j ii x r aks maks x r x r ij n e jahan et al. (2011); jahan & edwards (2015) n14 normalization equalizing the average to 1   ij ij j x n alpar (2013) n15 normalization equalizing standard deviation to 1   ij ij j x n alpar (2013) n16 decimal normalization    , 0 10 ij ij x n saranya & manikandan (2013) n17 reference based normalization         1 , min , ij j ij ij j ij jii x r n maks maks x r min x r jahan et al. (2012); jahan & edwards (2015) n18 wu's reference based normalization    ij j ij ij j i x r n maks x r wu (2002); jahan & edwards (2015) n19 aytekin's reference based normalization        1 , 0 10 ij j ij j x r n r aytekin (2020) n20 range normalization between -1 and +1           min 2 min 2 ij ijii ij ij ij ijii maks x x x n maks x x alpar (2013) n21 range normalization between 0 and +1    min min ij iji ij ij ijii x x n maks x x alpar (2013) in the normalization techniques in table 2, rj shows the reference value determined for criterion j. the reference value is the base, source, or guide point that aytekin/decis. mak. appl. manag. eng. 4 (2) (2021) 1-25 8 reflects the decision maker's preferences in the decision problem. the reference value can be identified subjectively by the decision-maker, or it can be determined with the help of scientific tools or techniques (aytekin, 2020). the arithmetic mean is generally used as a reference value in n12. the normalized values obtained in n12 are mainly in the range of 0 to ± 3. jahan et al. (2011) used n13 in the extension of vikor. n14 and n15 allow the decision-maker to adjust the performance values within the criteria according to the average or standard deviation. the ρ parameter in n16 is determined by the digit value of the largest absolute number in the decision matrix. thus, n16 can generate normalized values are between -1 and +1. n17, n18, and n19 are techniques that provide normalization based on reference. jahan et al. (2012), in their study on material selection, proposed n17, which is the extension of weitendorf's linear normalization, based on the reference value. wu (2002) suggested using n18 in the gray relational analysis if the reference value is determined among the maximum and minimum values. aytekin (2020) proposed n19 by integrating reference-based and decimal normalization processes. in this study, n19 is revised to equate the bestnormalized value to 1, and the worst normalized value to 0 according to the reference value. n20 and n21 provide normalization based on range. it is aimed to obtain normalized values between -1 and +1 in n20, and 0-1 range in n21. 2.4. integrated-mixed normalization techniques the different structures of mcdm problems always force researchers to seek new solutions. in mcdm problems, the optimization orientation-based or reference-based approaches are generally adopted for solutions. however, it may be possible for decision-makers to solve the problem using reference values for some criteria and optimization orientations for others. similarly, some of the criteria in the decision matrix may be monotonously increasing or decreasing, while others may not. in these cases, for instance, it is possible to determine the maximum or the minimum value according to the optimization orientation as a reference value or to create a solution by applying transformations to existing techniques. in such a case, there are normalization techniques developed for these integrated/mixed situations (zhou et al, 2006; zeng et al., 2013; jahan & edwards, 2015). the integrated-mixed normalization techniques are given in table 3. table 3. integrated-mixed normalization techniques notation technique all criteria references n22 ideal linear normalization zhou et al (2006); jahan & edwards (2015)     min , for cost criteria , for benefit criteria min , , for refe , ij i ij ij ij ij i ij j ij j x x x n maks x x r maks x r  rence-based criteria            n23 enhanced accuracy normalization zeng et al. (2013); jahan & edwards (2015) comparative analysis of normalization techniques in the context of mcm problems 9 notation technique all criteria references              1 1 1 1 , for benefit criteria min 1 , for cost criteria min 1,96 , for criteria having normal distrubition, if 1,96 1,96 1,96                              ij ij i m ij ij i i ij ij i m ij ij i i ij ij ijm ij i i maks x x maks x x x x x x n x x x x        1 , for criteria having normal distrubition, if 1,96 1,96 1 , for criteria having normal distrubition, if 1,96                                  j ijm ij i ij x x x n22 was proposed by zhou et al. (2006) to determine the ratio of xij to the reference value or maximum/minimum value. zeng et al. (2013) developed n23 for an extension of vikor to provide practical solutions to problems in the health sector. zeng et al. (2013) stated that the criteria having normal distribution are mostly used in the issues in the health sector and that an absolute deviation from the average is acceptable in these criteria. also, they stated that normalization could be achieved with n23, including monotonic increasing or decreasing criteria. apart from the normalization techniques examined in the study, there are still more normalization techniques developed for different purposes like membership functions (rough numbers, triangle, trapezoidal, etc.) which are used in the normalization processes within the framework of fuzzy logic or rough sets (sharma et al., 2018; vafaei et al., 2018a; roy et al., 2019). a comparative review of the normalization techniques given in table 1, table 2, and table 3 will be carried out in the following section of the study. 2.5. comparisons of normalization techniques in the literature normalization techniques have an essential role in the solution of mcdm problems. normalization processes used in the vast majority of mcdm methods enable the criteria of various structures to be dimensionless so that they can be directly compared. however, every normalization technique cannot be suitable for all decision problems. for example, some techniques do not provide eligible normalization for criteria with negative values or 0. other than this, the expected range of normalized values and the rank reversal problem likely to be encountered are among the other determining factors in the selection of the normalization technique. on the other hand, it is not possible to evaluate mcdm methods independently of the normalization techniques they contain. changing the normalization process included in an mcdm method results in the creation of a new extension/derivative of the method. there are many normalization techniques and mcdm methods in the literature. the structure of problems and the assumptions of mcdm methods are prominent factors for choosing the normalization technique. in this context, although not yet sufficient, studies on the comparison of normalization techniques have been increasing in the recent past. these studies, however, usually include a small number aytekin/decis. mak. appl. manag. eng. 4 (2) (2021) 1-25 10 of techniques and exclude most. similarly, those that consider the selection of the normalization technique, and the criteria to be used in this selection process are limited. in one of the prominent studies in comparing normalization techniques, çelen (2014) examined the deposit banks using the fahp-topsis integrated model. the study compared n1, n2, n4, and n10. the consistency and validity of the normalization results were evaluated under four conditions. according to the first of these conditions, the distribution of the normalized values should be similar when compared to other techniques. under this condition, inferences could be made by looking at the mean, standard deviation, smallest-largest values, and kolmogorovsmirnov normality test. in the second condition, the first three and the last three of the values to be obtained by normalization techniques should be the same. the third condition states that the correlations of these rankings should be high while the fourth condition emphasizes the need for the normalization techniques to produce similar scores, and this condition was examined with correlation coefficient values. the pearson correlation coefficient was used for measuring similarities (çelen, 2014, p. 201-203). the use of the kolmogorov-smirnov normality test and pearson correlation coefficient in the study carried out by çelen (2014) can be seen to be unsuitable for mcdm problems. this is because mcdm problems mostly contain data structures that are not normally distributed, and the normal distribution is not generally sought in mcdm problems. normalized values provided by normalization techniques may also have different structures. while some normalization techniques limit the data to values in a particular range, some ensure that obtained values are close to zero, while others provide only positive one-way data. it is thus clear that the structures of the decision problem and the mcdm method have a direct influence on the determination of the normalization process. the primary purpose of the normalization process performed in mcdm problems is not to obtain data with normal distribution, but to get comparable data with equal weight. however, it should be accepted that çelen (2014) gives a different perspective on the comparison of normalization techniques. chakraborty and yeh (2007) suggested rci (ranking consistency index) to compare normalization techniques. in rci, a normalization technique is evaluated based on ranking consistency with other normalization techniques. for this, it is necessary to simulate the decision matrix of at least 4x4 and at most 20x20. finally, the results of the different normalization techniques are analyzed. in another study conducted for the selection of normalization techniques, vafaei et al. (2018a) discussed the appropriate normalization technique for topsis using the reviews previously held in the literature. accordingly, rci, mean and distribution measures of normalized values, kolmogorov-smirnov normality test, ranking consistency of normalization techniques in terms of the first three and last three rows, pearson and spearman correlations were used in the comparison of normalization techniques and order of suitability for topsis. the authors concluded that vector normalization is the best technique for topsis in similarity with the study of chakraborty and yeh (2009). they also stated that comparison based on the normal distribution is questionable (vafaei et al., 2018a). some of the processes mentioned in the comparison of normalization techniques do not always seem to be possible due to the structural features of mcdm problems. it is often not possible for the criteria to have a normal distribution. furthermore, when the number of normalization techniques to be compared is high, and the size of the decision matrix exceeds 20x20, the use of rci may not be effective. another critical problem is that using the pearson correlation coefficient to examine the correlation of comparative analysis of normalization techniques in the context of mcm problems 11 the rankings does not give accurate results. consequently, in the application part of this study, the spearman rank correlation coefficient will be used to analyze the rank correlations of the normalization techniques. the ability to give effective normalization in different criteria structures, differences between normalized values, and their use in mcdm methods will also be examined. normalization is one of the essential process steps in mcdm methods, and it directly affects the solution to the problem. the normalization process used in the mcdm methods should not be considered independent of the method itself. for example, topsis uses euclidean distance in solving the decision problem. in this context, vector normalization, which is the second moment according to the starting vector (0), is used in the normalization process in topsis. like topsis, many other mcdm methods also have normalization procedures for the intended solution. table 4 shows the mcdm methods and the normalization techniques used in the original forms of these methods (the first form, not extension form) and the studies comparing them depending on different normalization techniques. table 4. normalization in mcdm methods methods normalization process in the original method recommended most compatible normalization process normalization techniques compared source ahp n1 n4 + n1 n2, n4, n5, n10, n12 vafaei et al. (2016), copras n1 n1, n2 özdağoğlu (2013) gra n10 n2 n2, n7, n9 chatterjee and chakraborty (2014) topsis n2 n2 n1, n4, n10 chakraborty and yeh (2009), çelen (2014), vafaei et al. (2018a) vikor n10 n13, n23 n13, n23 jahan et al. (2011), zeng et al. (2013) prometheeii n10 n2 n2, n7, n9 chatterjee and chakraborty (2014) maut n1 n1, n2, tchebycheff yoon and kim (1989) moora n2 n1, n2, n4, n5 n1, n4, n5, n10, n12 özdağoğlu (2014) electre-ii n2 n1, n10 pavličić (2001) saw n1, n2, n4 n1, n2, n3, n10, fuzzy trapezoid membership function chakraborty and yeh (2007), vafaei et al. (2018b) as seen in table 4, the most recommended normalization techniques are sumbased linear normalization and vector normalization. the normalization technique should be chosen by considering the nature of the decision problems. in the following section of the study, an applied comparison of the normalization techniques will be given. aytekin/decis. mak. appl. manag. eng. 4 (2) (2021) 1-25 12 3. applied comparative analysis of normalization techniques with different scenarios in this part of the study, a comparison of normalization techniques will be highlighted. the twenty-three normalization techniques in the second section will be compared over a randomly and purposefully generated data set for a valid comparison. these sets were created by considering different scenarios to reflect the general structure of mcdm problems. the literature and scenarios were taken as a basis in determining the number of criteria and alternatives for decision matrices. to this end, studies that conducted the literature review of mcdm methods were examined. the conclusion is that the most observed numbers were four for criteria, and five for alternatives (durucasu et al., 2017). the numbers in this study were considered important as they provide guidance. for this study, however, it was decided that it would be appropriate to have six criteria and six alternatives to reflect the structural differences of scenarios and decision problems to be used in comparing normalization techniques. after determining the number of criteria, and the number of alternatives, the different scenarios to be created were decided. each scenario is named with a different set. to ensure that the criteria in set 1 are different from each other, k1, whose variation range is quite wide compared to other criteria, k2 with a variation range of 0-1, k3 containing 0 and positive values, k4 containing negative values and 0, k5 containing negative, 0 and positive values together, and k6 whose values are all negative were established. while the optimization directions of all criteria in set 1 were determined as benefit (maximum), it was evaluated as cost (minimum) in set 2. in set 3, a scenario was created where all criteria have the benefit optimization orientation and where the ranges do not intersect was created. accordingly, the ranges are 1-10 for k1; 11-100 for k2; 101-1,000 for k3; 1.001-10.000 for k4, 10.001-100.000 for k5 and 100.001-1.000.000 for k6. the optimization orientations of all criteria were set as benefit in set 3 whereas it was determined as the cost in set 4 which has the same decision matrix. in set 5 to set 11, the aim is to investigate the effects of adding and removing alternatives from the decision problem. set 12 was created to examine whether units that can be converted into each other are normalized with the same values. in set 12, the optimization orientations of the criteria are determined as benefit, while they are cost in set 13. set 14 was created based on the values obtained in scaling techniques commonly used in mcdm problems. in this regard, the ten-point direct rating scale for k1, saaty’s fundamental (linear priority) scale for k2, likert type scale for k3, dematel scoring scale for k4, semantic scale for k5, and the hundred-point direct rating scale for k6 were used to determine the values of the alternatives. the ten-point direct rating scale allows alternatives to be evaluated in the range of 1-10. alternatives are evaluated in the range of 1-9 by pairwise comparisons using saaty’s fundamental (linear priority) scale. dematel scoring scale is based on determining the interactions between the alternatives by pairwise comparisons in the range of 0-4 (1-5 in some research). semantic scale generates values in the range of 0-100 using binary comparisons of alternatives in the range of 0-6 (1-7 in some research). the hundred-point direct rating scale allows alternatives to be evaluated in the range of 1-100 (saaty, 1977; e costa & vansnick, 1994; wu, 2008). the fourteen sets in table 5 will be used in the comparison of the normalization techniques. these sets contain decision matrices created for different scenarios. there are six criteria and six alternatives in these matrices. ms excel was used to generate the performance values of the alternatives randomly. for this purpose, the formulas = comparative analysis of normalization techniques in the context of mcm problems 13 randbetween (lower_bound_value; upper_bound_value) and = rand () were used. however, some values were determined purposely; for instance, in set 1, to examine the effects of 0, and in set 3 to check the effects of rank reversals in sets 6-11. the values that could be converted into each other were also purposely assigned in set 12. to examine the rank reversal problem, saw, which has one of the simplest and basic forms of mcdm methods, was used to solve the decision problems. besides, ms excel, sanna, and spss 25.0 were used in the analysis. table 5. the decision matrices for different scenarios set alternatives criteria k1 k2 k3 k4 k5 k6 set 12 a1 750940 0,8675 51 -71 16 -3 a2 200772 0,0687 64 -50 -57 -24 a3 557819 0,9374 0 0 0 -43 a4 827702 0,9138 24 -41 19 -16 a5 26218 0,7912 75 -22 90 -61 a6 401846 0,5273 2 -55 -4 -31 set 3-4 a1 8 29 276 3565 23351 352023 a2 4 79 491 7985 28023 354205 a3 6 68 783 4322 19956 401177 a4 7 32 335 6167 67964 376962 a5 6 21 275 3132 33352 151235 a6 2 98 920 9174 93170 928875 set 5 it was created by removing a6 from set 3. set 6 it was created by removing a5 from set 3. set 7 it was created by removing a5 and a6 from set 3. it was created by adding a7 to set 3. set 8 a7 10 100 1.000 10.000 100.000 1.000.000 it was created by adding a8 to set 3. set 9 a8 1 11 101 1.001 10.001 100.001 set 1011 it was created by adding a7 and a8 to set 3. set 12 13 a1 1 10 100 1.000 10.000 100.000 a2 2 20 200 2.000 20.000 200.000 a3 3 30 300 3.000 30.000 300.000 a4 4 40 400 4.000 40.000 400.000 a5 5 50 500 5.000 50.000 500.000 a6 6 60 600 6.000 60.000 600.000 set 14 a1 2 0,1657 1 0,1719 100 5 a2 7 0,0881 2 0,1442 70 8 a3 8 0,0471 4 0,1821 55 60 a4 4 0,0368 2 0,1793 40 47 a5 9 0,2325 3 0,1355 25 70 a6 7 0,4298 5 0,1869 0 23 in table 5, to show the effect of removing an alternative from the decision matrix in set 5, a6, which ranked first in the solutions obtained with saw in set 3, was removed from the decision problem. in set 6, a5, which took the last place in the solutions obtained with saw in set 3, was removed from the decision problem. in set aytekin/decis. mak. appl. manag. eng. 4 (2) (2021) 1-25 14 7, both a5 and a6 alternatives in set 3 are excluded from the decision matrix. in set 8, a7, a new alternative with the best values in all criteria, was added to the decision matrix specified in set 3. in set 9, a8, a new alternative with the worst values in all criteria, was added to the decision problem in set 3. set 10 was created by simultaneously adding a7 and a8 to set 3. set 11 was created by changing the optimization orientations of set 10. the values in set 12 and set 13 are assigned to represent values that can be converted into each other. in set 14, the values of k2, k4, and k5 should be created by pairwise comparison. values ranging between 0 and 1 were obtained for each alternative as a result of the operations performed using saaty's fundamental scale in k2 and dematel scoring scale in k4. these values were assigned following the structure of these scales; thus, multiple way performance analysis and comparison of normalization techniques can be made with set 1-14. to examine the rank reversals problem, the ranking of the alternatives was obtained via saw using the normalized values. in the ranking process, the criteria were considered to have equal weights. the best values in criteria depending on the optimization orientation are used as reference values for reference-based normalization techniques. the issues determined in the applications of normalization techniques in sets 1-14 are shown in table 6. the table contains information on the techniques regarding the sets in which the normalization process is not completed, the number of rank reversals, the maximum-minimum normalized values observed in the benefit/cost criteria, the ability to cope with the same values expressed in different units, and providing the same optimization orientation. the completion of the normalization process, which is one of the issues in table 6, in all possible data types in a meaningful way (by cleaning all units and making them dimensionless) shows the robustness and usability of the technique. at the end of the normalization process, normalized values between 0 and 1 are preferred while different ranges of normalized values across criteria are undesirable. for example, having normalized values between -1 and +1 in one criterion, and between 0 and 10 in another is not desirable. such a structure will change the effects/weights of the criteria on the decision problem and make the solution of the decision problem to become invalid. the success of the normalization techniques will be examined by looking at the maximum and minimum normalized values. also, normalization techniques are expected to handle unit differences successfully. for example, a distance may be expressed in any of the units of kilometers-hectometers-decametersmeters-decimeters-centimeters. the unit of measurement used does not change the length of the distance. the normalization technique should therefore generate units that can be converted to each other and measure the same thing, with the same normalized values. this standpoint is considered when examining how the techniques cope with different units. the last issue in table 6 is the test of whether the technique gives the same optimization orientation, which is often the benefit orientation, an indication of whether the normalization technique gives one-dimensional values in all criteria. comparative analysis of normalization techniques in the context of mcm problems 15 table 6. issues determined within the scope of set 1set 14 t e ch n iq u e the sets that normalization cannot be completed number of rank reversals benefit criteria cost criteria capability removing unit differences the success of providing the same optimization orientation maximum normalized value minimum normalized value maximum normalized value minimum normalized value n1 set 2 & set 14 11 1,406 -0,891 108 -16 successful successful n2 8 0,853 -0,710 1,710 0,178 successful successful n3 set 1, set 2 & set 14 4 0,696 0 0,200 0,061 fail successful n4 set 1 & set 2 6 20,333 -0,633 1,633 -19,333 successful successful n5 set 1, set 2 & set 14 4 4,563 -19,333 20,333 -3,563 successful successful n6 set 1 & set 2 4 20,333 -0,633 20,333 -3,563 successful successful n7 set 1 & set 2 6 1 -18,333 1 -18,333 successful successful n8 set 1 & set 2 4 2033,333 -63,333 2033,333 -356,250 successful successful n9 set 1 & set 2 6 413,444 0 8406,704 -0,254 successful fail n10 7 1 0 1 0 successful successful n11 16 2,039 -1,831 1,831 -1,917 successful successful n12 19 1,000 0,125 0,999 0,159 successful fail n13 10 1 0,368 1 0,368 successful successful n14 13 8,4375 -5,344 8,438 -5,344 successful fail n15 16 17,435 -2,983 3,711 -2,983 successful fail n16 0 0,980 -0,710 0,980 -0,710 successful fail n17 7 1 0 1 0 successful successful n18 set 1, set 3, set 5-set 10, set 12 & set 14 0 1 0 successful (only in cost criteria) fail n19 0 1 0,226 1 0,064 successful successful n20 7 1 -1 1 -1 successful fail n21 7 1 0 1 0 successful fail n22 set1 & set 2 4 20,333 -0,633 20,333 -3,563 successful successful n23 7 1 0,325 1 0,499 successful successful the first column of table 6 shows the sets in which normalization techniques could not complete all the operations. n2, n10, n11, n12, n13, n14, n15, n16, n17, n19, n20, n21, and n23 completed the normalization process in all criteria in all sets. n3, n4, n5, n6, n7, n8, n9, and n22 could not complete normalization processes in all or some of the criteria because of a zero value in k3; zero and negative values in k4, and negative, zero and positive values in k5. n1, n5, n6, and n22 could not create normalized values for a3 in set 2 due to the error of dividing by zero. the normalization process with n3 could not be completed because of the zero value in k4 in set 14. normalization of the benefit criteria could not be ended with n18. the main reason for this situation in n18 is the selection of reference values depending on the optimization orientation. when the normalization techniques were analyzed in terms of rank reversals for sets 1-14, only n16, n18, and n19 were found not to have rank reversal problems. however, a dramatic change in ρ value in only a specific criterion can cause rank reversals in n16. n18, which achieved normalization in only four sets, is subject to rank reversals due to maximum and minimum values in the decision matrix. in n19, if the reference and ρ values are determined depending on the first decision matrix or independently from the decision matrix, rank reversals are not observed. however, if the reference and ρ values change with changes in the decision matrix, rank reversals should be expected. rank reversal problems were observed in other normalization techniques. among these, n12 had the highest number of rank reversals. aytekin/decis. mak. appl. manag. eng. 4 (2) (2021) 1-25 16 table 6 also shows the range of the values obtained by normalization techniques in benefit criteria can also be observed. accordingly, techniques providing normalization in the range of 0-1 are n3, n10, n12, n13, n17, n19, n21, and n23. n2 and n16 give normalization between -1 and +1. the sum of normalized values from the normalization performed in n1 on the benefit criteria is expected to be 1. on the other hand, n1 generated 1.406 and -0.891 normalized values in k5 in set 1. the following techniques were also seen to have negative normalized values; n1, n2, n4, n5, n6, n7, n8, n9, n11, n14, n15, n16, n20, and n22. in general, normalized values are expected to be within a certain range in all criteria as a result of normalization. at this point, it is more preferred that the normalized values are in the range of 0-1. in set 1set 14, the average of normalized values obtained by normalization techniques was 0.7057, except for n8 and n9, which can produce structurally high normalized values. for example, an examination of the normalized values in benefit criteria reveals that n4, n5, n6, n8, n9, n14, n15, and n22 could produce normalized values larger than five by absolute value. from these techniques, n8 and n9 produced large normalized values; 2033,33 and 413,44, respectively. when the normalized values of the cost criteria were analyzed, n3, n10, n12, n13, n17, n18, n19, n21, and n23 were found to provide normalization in the range of 0-1 whereas n1, n4, n5, n6, n7, n8, n9, n11, n14, n15, n16, n20, and n22 gave negative normalized values. n1, n4, n5, n6, n7, n8, n9, n14, and n22 generated normalized values larger than five by absolute value. among these techniques, n8 and n9 produced four-digit normalized values. set 12 and set 13 were created to examine whether units that could be transformed into each other were normalized in the same way. table 6 shows that technique n3 failed in this regard. another important comparison point is that normalized values can be used as-is in mcdm methods without additional processing. although a significant majority of mcdm methods involve processing steps according to the costbenefit optimization orientation, the normalization technique should shorten this process. in this context, techniques providing one-dimensionality; n1, n2, n3, n4, n5, n6, n7, n8, n10, n11, n13, n19, n22, and n23. the applications performed in sets 1 – 14 provide an opportunity to see the positive and negative features of normalization techniques. in the following section, a general evaluation of each technique will be given. n1 does not provide useful normalization in the cost criteria as well as it does in the benefit criteria. n1 cannot complete the normalization process for cells that have 0 in the cost criteria. besides, n1 produced quite large normalized values, such as 108 and -16, in cost criteria. however, as can be seen in k5 in set 1, if negative, 0, and positive values are included in the benefit criteria, n1 can produce negative normalized values or values greater than 1. also, n1 is prone to rank reversal problems. n1 can give values corresponding to the performances of the alternatives in the range of 0-1 in cases where the optimization of the criterion benefit orientation and all values are positive. thus, the weight of an alternative in the relevant criterion can be easily determined. besides, n1 was also found to have successfully removed unit differences. n2 produced negative values in set 1 in the normalization of k4, k5, and k6. it also generated normalized values larger than 1 in set 2. the main reason for this is the negative values in the decision matrix. n2 is also prone to rank reversal problems as it uses the sum of squares of the decision matrix elements. on the other hand, n2 completed the normalization processes in all criteria. n2 was also found to have comparative analysis of normalization techniques in the context of mcm problems 17 successfully removed unit differences and gave normalized values in the benefit orientation. n3 could not complete the normalization of the criteria having zero and negative values. although n3 provided normalization in the range of 0-1, most of the values tended to be closer to zero. the rank reversal problem in n3 was less than many other techniques. it was determined that n3 could not successfully remove unit differences. on the other hand, n3 can give normalized values in the benefit orientation. another critical point is that n3 provides non-linear normalization. most normalization techniques have a monotonous increasing / decreasing structure. n3 can be said to be an option for decision problems with a different structure. n4 cannot perform normalization in criteria where zero is the maximum value. also, it produces negative normalized values in criteria having zero, negative, and positive values together. n4 gives quite large normalized values only for criteria including negative values. there may be rank reversals with n4. on the other hand, it was seen that n4 successfully removed unit differences and gave normalized values in the benefit orientation. in the cost criteria, n5 cannot complete normalization when 0 is the minimum value whereas it gives normalized value as 0 for all other values in the decision matrix. the use of the minimum value in the decision matrix in the normalization process with n5 can lead to rank reversal problems. n5 can produce quite large normalized values in criteria that have negative and zero values. however, it gives the same normalized values if the same value is expressed in different units. it also gives all normalized values in the benefit orientation. n6 could not complete normalization operations for k4 containing negative values and zero in set 1. similarly, in set 2, n6 could not provide normalization for zero value in k3, k4 and k5 due to cost optimization orientations of these criteria. also, it gave quite large normalized values such as 20,33 in k6, which had negative values in set 1 and set 2. another negative feature of n6 is that it is prone to the rank reversal problem. however, n6 was found to provide normalization in the range of 0-1 in criteria where all values are positive. it is also successful in coping with different units and providing all normalized values in the benefit orientation. at first glance, n7 can be assumed not to produce negative normalized values because it operates with absolute value. on the other hand, negative normalized values for k5 and k6 were observed in set 1 and set 2. normalized values such as 18.33 produced in the mentioned sets are also quite high. n7 provided normalization in the range of 0-1 in other sets. as n7 is dependent on the maximum and minimum values in the decision matrix in the normalization processes, it is prone to the rank reversal problem. on the other hand, n7 is successful in dealing with the unit's differences and providing all normalized values in the same optimization orientation. n8 did not provide normalization for all cells in k4 in set 1, and the cells have zero values in k4, k5, and k6 in set 2. also, while n8 is expected to produce 100 as normalized values for the best value according to the optimization orientations of the criteria, quite large normalized values such as -356.25 and 2033.33 values were observed in set 1 and set 2. the rank reversal problem was also identified in applications done in sets 14. outstanding positive features of n8 include removing the unit's differences and giving all normalized values in the same optimization orientation. another technique that is prone to the problem of rank reversal is n9. n9 could not complete normalization in k4, where zero is the maximum value in set 1 and set 2. however, the normalized values created for set 1 and set 2 had quite high values, such as 413.44 and 8406.70 and negative values. n9 was found to be able to remove the aytekin/decis. mak. appl. manag. eng. 4 (2) (2021) 1-25 18 unit's differences. on the other hand, it was unable to provide the benefit orientation in all criteria. in this context, the same ranks were seen for set 12 and set 13, in which their optimization orientations are different. another critical point is that n9 produces non-linear normalized values. it can therefore be stated that n9 is an option for decision problems with different structures. n10 provided normalization in the range between 0 and 1 in all sets. n10 was also successful in dealing with the unit's differences and providing benefit orientation values. among the normalization techniques, n10 is one of the most robust and successful, but it can cause rank reversals. n11 provided normalization in all sets. the maximum and minimum normalized values observed with n11 are 2.03 in set 9 and -1.91 in set 4. also, the rank reversal problem was observed in normalizations with n11. on the other hand, it was successful in removing unit differences and generating normalized values in benefit criteria. n11 is one of the techniques with a non-monotonous structure. it can be stated that n11 creates an option for decision problems with different structures with these features. techniques n1 to n11 perform the normalization process according to the optimization orientation, while n12 to n21 provide normalization independent of the optimization orientation. in normalization techniques independent of the optimization orientation, specific parameters such as reference, mean, digit value, range, standard deviation, or fixed values are used for normalization. however, this study used the maximum/minimum values as reference values following the optimization orientations of the criteria. furthermore, when the references are determined independently from the decision matrix, the reactions of the techniques will be reported. the arithmetic mean is used as the reference value in normalization processes performed in sets 1 14 with n12. in this case, normalized values ranged from 0-1 were obtained in all sets. in some sets, the same normalized values were generated for criteria with different values. at this point, it should be noted that the values in certain ranges under the normal distribution curve have the same normalized values, and a non-monotonic process is performed. considering that most normalization techniques have a monotonous increasing/decreasing structure, n12 can be said to differ significantly from other techniques. on the other hand, these features, which distinguish n12 from other techniques, prevent it from being successful at giving all values in the benefit orientation. furthermore, n12 was the technique in which the rank reversal problem was common in sets 114. on the other hand, n12 has the capability of removing unit differences. n13 was observed to cause the rank reversal problem. on the other hand, it provides normalization in the range of 0-1 in all sets and successful in coping with unit differences and providing all values in benefit criteria. it gave the minimum normalized value as 0.3678 in all sets. n14 performs the normalization by equating the average value to 1. if a criterion has a large range, normalized values will not be acquired within the desired range using n14. also, normalization based on average value can lead to rank reversal problems. n14 cannot achieve normalization in all benefit criteria. it however has the capability of removing unit differences. n15 shares the same features as n14 except for the normalization by providing the standard deviation equal to 1. in n16, the digit value of the largest absolute number in the decision matrix is taken as the basis. when the maximum or the minimum value changes dramatically in a criterion, n16 can cause the rank reversal problem. however, no rank reversal was observed in comparisons made in sets 114 with n16. n16 cannot provide all values comparative analysis of normalization techniques in the context of mcm problems 19 in the benefit orientation, but it produces normalized values between -1 and +1 in all sets. also, n16 is successful in removing unit differences. although n17 provides normalization based on reference, it also uses the maximum and minimum values in the criterion. this situation can cause the rank reversal problem. n17 produces normalized values in the range of 0-1, succeeds in coping with unit differences, and provides all normalized values in the same optimization orientation. n18 was the technique with the highest number of processing errors in sets 114 since the reference values were determined depending on the optimization orientation of the criteria. revisions were made on set 1 and set 3 to measure the reaction of n18 and other reference-based techniques, n12, n13, n17, and n19. for this purpose, the reference values were different from the best values in the criteria. to do this, new reference values were determined to be 10% better and 10% worse than the best values in the criteria, and normalized values were examined. n18 could not complete the normalization process if the references were 0. also, if the reference was higher than the maximum value in the criterion, n18 produced negative and large normalized values such as -8.85 and -9.37. for the other techniques, no situation other than the issues determined in the context of set 1-14 was encountered. also, the fact that n18 uses the maximum value in the criterion in the normalization process can lead to the rank reversal problem. n18 can remove unit differences. on the other hand, it gives the values in the benefit orientation only if the reference is less than the minimum value in the decision matrix. n19, one of the reference-based normalization techniques, produces normalized values in the range of 0-1 in all sets, but it never creates exactly normalized values as 0. n19 does not cause the rank reversal problem if the reference and ρ values are determined depending on the first decision matrix or independently from the decision matrix. in applications carried out in the context of sets 114, the rank reversal was not observed in n19. besides, the n19 can remove unit differences and provide all values in the benefit orientation. n20 uses the average of the maximum and minimum values and the range in the normalization process. these features, on the other hand, cause the rank reversal. unable to provide all values in the benefit orientation, n20 produces normalized values between -1 and +1. also, n20 succeeded to cope with unit differences. n21 applies the formula used for the benefit orientation of n10 in the same way in all criteria. being independent of the optimization orientation prevents n21 from providing all values in the benefitorientation. n21 provided normalization in the range of 0-1 in all sets and n21 caused the rank reversal problem at the same time, but it managed to cope with unit differences. normalization techniques from n1 to n21 provide normalization either by considering the optimization orientation of the criteria or reference or specific values. the emergence of integrated-mixed normalization techniques in the literature has given researchers different perspectives. in integrated-mixed normalization techniques, the optimization orientations of the criteria or reference/specific values are used under certain conditions. with n22, which is one of the integrated-mixed normalization techniques, normalization could not be achieved in cells that had zero value in set 1 and set 2. also, quite large normalized values, such as 20.33 and -3.56, were observed in normalization depending on the optimization orientation, similar to the techniques using maximum and minimum values in criteria. n22 can cause the rank reversal problem, but it can remove unit differences and gives all normalized values in the benefit orientation. aytekin/decis. mak. appl. manag. eng. 4 (2) (2021) 1-25 20 n23 gives normalized values in the range of 0-1 in all sets and is also successful in dealing with unit differences and providing all normalized values in the benefit orientation. additionally, it produces non-monotonic normalization, but it can cause the rank reversal problem. obtained by normalization techniques. the similarities between these rankings were examined using the spearman rank correlation coefficient, and the results are presented in table 7. an examination of the correlation values in table 7 reveals that the techniques that did not complete all normalization process in criteria having the benefit optimization orientation have a low or meaningless correlation with the other techniques. the same can be said for the techniques not having a linear structure. among the twenty-three techniques, n12 has the lowest correlation with others. n3, n9, n14, n15, n16, n20, and n21 have a low level of correlation with the other techniques. as seen in table 7, the technique that has the most inverse correlation with other techniques is n18. this can be attributed to the low number of normalization processes completed by n18 in sets 1-14. the other techniques were found to provide a high level of correlation (rs> 0.8) with each other in general. 4. conclusions normalization is a scaling process that is frequently used in creating the data structure required for the method to be used in quantitative studies. the normalization process directly affects the results of the analyzes to be performed. however, it wouldn’t be right to use traditional normalization techniques for such a process without question. in mcdm methods, normalization is used to obtain criteria that have the same weight, are dimensionless, and are suitable for compensatory processes. although the vast majority of mcdm methods provide normalization within themselves, the nature of the decision problems makes it necessary to review these processes. on the other hand, changing the normalization process of an mcdm method partially leads to the emergence of a new extension of the relevant method. to this end, this study aimed to outline the positive and negative features of the normalization techniques that are widely used and can be used in mcdm problems, thus giving decision-makers or researchers an insight into the situations in which the considered normalization techniques can work well and where they should not be used. eleven techniques, based on optimization orientations of criteria, ten techniques independent of optimization orientations, and two integrated-mixed techniques were evaluated in fourteen sets, reflecting different criteria and data structures considered for normalization. in the comparison, the emphasis was made on the ranges of normalized values obtained by the techniques, the presence of the rank reversal problem, normalization performance in criteria with different structures, the capacity to remove unit differences, the ability to give all normalized values in the same optimization orientation, and completion of normalization in all sets. the comparisons in sets 1-14 showed the importance of the normalization technique selection process. normalization may not be completed if the normalization technique is not chosen according to the data structure, and should that be the case then the validity of the results may be disputable. the results showed that some normalization techniques are not able to achieve the purposes of development or use in some cases. n1, n3, n4, n5, n6, n7, n8, n9, and n22 could not complete the normalization procedures in all or part of the criteria with zero or negative values in the decision comparative analysis of normalization techniques in the context of mcm problems 21 matrix. n1, n4, n5, n6, n7, n8, n9, n14, n15, and n22 produced quite large normalized values. also, they did not produce normalized values within a certain range, such as 01, in all criteria. these techniques provide very high normalized values due to the criteria having a different structure in set 1 and set 2. the rank reversal problem was not observed in techniques n16 and n19 in sets 1-14. indeed, if the reference and ρ values cannot be changed after being determined at the beginning of the problem, n19 stands out as the only technique that does not cause any rank reversal problems. also, n19 does not use values such as maximum, minimum that will change with the change in the decision matrix. normalization techniques have different structures. however, the nature of the decision matrix, preferences of the decision-maker, and the properties of the mcdm method to be used in the solution of the decision problem should also be considered in the selection of the normalization technique. if the aim is to choose the highest performance value in a criterion and avoid the alternative with the lowest performance value, or vice versa, it would be correct to use techniques that provide normalization depending on the optimization orientation. if the decision-maker has reference/ideal/utopic values determined for each criterion, it will be necessary to use reference-based normalization techniques. if the decision-maker has the opinion that the values in the criteria do not represent the monotonous increasing/decreasing benefit/cost, then the non-monotonic or non-linear techniques should be used. however, some data, that have zero and negative values, may prevent the use of some normalization techniques, as determined in the application section. also, the problem of rank reversal, the ranges of normalized values, the capability of removing unit differences, and the ability to give all normalized values in the same orientation, robustness and validity of the results will affect the choice of the normalization technique. this study presents a comparison of normalization techniques in different criteria structures. it sought to highlight the positive and negative features of the techniques in question thereby guiding decision-makers or researchers on the selection of techniques. the study is also envisioned to provide a perspective on the development of new normalization techniques and the creation of new extensions of mcdm methods by replacing the normalization techniques originally included in the mcdm methods with other techniques. normalization techniques continue to be applied in most areas where data analysis is required. it is, therefore, necessary to conduct comparisons in other fields such as data mining as well. it may also be beneficial for future research to examine the normalization processes associated with fuzzy and rough sets used in mcdm problems. aytekin/decis. mak. appl. manag. eng. 4 (2) (2021) 1-25 22 alternatives in sets 114 have been ranked by saw using the normalized values t a b le 7 . r a n k c o rr e la ti o n s o f n o rm a li z a ti o n t e ch n iq u e s w it h e a ch o th e r in t h e c o n te x t o f sa w r a n k in g n 1 n 2 n 3 n 4 n 5 n 6 n 7 n 8 n 9 n 1 0 n 1 1 n 1 2 n 1 3 n 1 4 n 1 5 n 1 6 n 1 7 n 1 8 n 1 9 n 2 0 n 2 1 n 2 2 n 2 3 n 1 1 ,0 0 0 ,9 6 7 ** ,6 3 8 ** ,8 9 4 ** ,9 0 7 ** ,9 1 1 ** ,8 9 4 ** ,9 1 1 ** ,4 1 0 ** ,8 9 7 ** ,8 9 4 ** -0 ,0 7 2 ,9 0 2 ** ,5 2 2 ** ,4 2 9 ** ,4 0 9 ** ,8 9 7 ** -, 8 7 1 ** ,8 9 8 ** ,4 0 8 ** ,4 0 8 ** ,9 1 1 ** ,8 7 8 ** n 2 ,9 6 7 ** 1 ,0 0 0 ,6 4 4 ** ,9 2 4 ** ,8 9 2 ** ,9 0 4 ** ,9 2 4 ** ,9 0 4 ** ,4 0 2 ** ,9 0 0 ** ,9 0 3 ** -0 ,0 4 6 ,8 9 1 ** ,3 9 5 ** ,3 4 5 ** ,3 4 3 ** ,9 0 0 ** -, 8 4 2 ** ,9 1 4 ** ,3 4 2 ** ,3 4 2 ** ,9 0 4 ** ,8 7 7 ** n 3 ,6 3 8 ** ,6 4 4 ** 1 ,0 0 0 ,6 4 3 ** ,5 5 4 ** ,6 2 9 ** ,6 4 3 ** ,6 2 9 ** ,2 5 0 * ,6 2 1 ** ,6 4 7 ** 0 ,1 8 4 ,6 3 5 ** ,2 4 8 * ,2 4 5 * 0 ,2 0 6 ,6 2 1 ** -, 5 5 0 * ,5 8 3 ** ,2 4 4 * ,2 4 4 * ,6 2 9 ** ,6 2 5 ** n 4 ,8 9 4 ** ,9 2 4 ** ,6 4 3 ** 1 ,0 0 0 ,9 0 7 ** ,9 8 3 ** 1 ,0 0 0 ** ,9 8 3 ** ,4 3 5 ** ,9 4 6 ** ,9 5 2 ** -0 ,0 0 8 ,9 5 9 ** ,3 7 8 ** ,4 2 0 ** ,4 3 3 ** ,9 4 6 ** -, 8 8 5 ** ,9 5 7 ** ,4 1 3 ** ,4 1 3 ** ,9 8 3 ** ,9 2 9 ** n 5 ,9 0 7 ** ,8 9 2 ** ,5 5 4 ** ,9 0 7 ** 1 ,0 0 0 ,9 2 5 ** ,9 0 7 ** ,9 2 5 ** ,3 6 0 ** ,9 4 0 ** ,9 1 2 ** 0 ,0 3 1 ,9 2 1 ** ,3 5 5 ** ,3 7 4 ** ,3 8 9 ** ,9 4 0 ** -, 8 7 1 ** ,9 5 4 ** ,3 7 5 ** ,3 7 5 ** ,9 2 5 ** ,9 1 7 ** n 6 ,9 1 1 ** ,9 0 4 ** ,6 2 9 ** ,9 8 3 ** ,9 2 5 ** 1 ,0 0 0 ,9 8 3 ** 1 ,0 0 0 ** ,4 5 2 ** ,9 4 2 ** ,9 3 6 ** 0 ,0 0 0 ,9 5 5 ** ,3 9 9 ** ,4 3 7 ** ,4 3 7 ** ,9 4 2 ** -, 8 7 1 ** ,9 5 3 ** ,4 1 7 ** ,4 1 7 ** 1 ,0 0 0 ** ,9 2 5 ** n 7 ,8 9 4 ** ,9 2 4 ** ,6 4 3 ** 1 ,0 0 0 ** ,9 0 7 ** ,9 8 3 ** 1 ,0 0 0 ,9 8 3 ** ,4 3 5 ** ,9 4 6 ** ,9 5 2 ** -0 ,0 0 8 ,9 5 9 ** ,3 7 8 ** ,4 2 0 ** ,4 3 3 ** ,9 4 6 ** -, 8 8 5 ** ,9 5 7 ** ,4 1 3 ** ,4 1 3 ** ,9 8 3 ** ,9 2 9 ** n 8 ,9 1 1 ** ,9 0 4 ** ,6 2 9 ** ,9 8 3 ** ,9 2 5 ** 1 ,0 0 0 ** ,9 8 3 ** 1 ,0 0 0 ,4 5 2 ** ,9 4 2 ** ,9 3 6 ** 0 ,0 0 0 ,9 5 5 ** ,3 9 9 ** ,4 3 7 ** ,4 3 7 ** ,9 4 2 ** -, 8 7 1 ** ,9 5 3 ** ,4 1 7 ** ,4 1 7 ** 1 ,0 0 0 ** ,9 2 5 ** n 9 ,4 1 0 ** ,4 0 2 ** ,2 5 0 * ,4 3 5 ** ,3 6 0 ** ,4 5 2 ** ,4 3 5 ** ,4 5 2 ** 1 ,0 0 0 ,4 2 6 ** ,4 2 3 ** -0 ,0 5 2 ,4 3 9 ** ,9 2 6 ** ,9 5 5 ** ,9 4 2 ** ,4 2 6 ** ,9 8 6 ** ,4 1 4 ** ,9 5 9 ** ,9 5 9 ** ,4 5 2 ** ,4 1 0 ** n 1 0 ,8 9 7 ** ,9 0 0 ** ,6 2 1 ** ,9 4 6 ** ,9 4 0 ** ,9 4 2 ** ,9 4 6 ** ,9 4 2 ** ,4 2 6 ** 1 ,0 0 0 ,9 7 5 ** -0 ,0 0 3 ,9 8 2 ** ,3 4 1 ** ,3 9 9 ** ,3 5 3 ** 1 ,0 0 0 ** -, 9 0 9 ** ,9 4 9 ** ,4 0 2 ** ,4 0 2 ** ,9 4 2 ** ,9 4 9 ** n 1 1 ,8 9 4 ** ,9 0 3 ** ,6 4 7 ** ,9 5 2 ** ,9 1 2 ** ,9 3 6 ** ,9 5 2 ** ,9 3 6 ** ,4 2 3 ** ,9 7 5 ** 1 ,0 0 0 0 ,0 0 7 ,9 6 8 ** ,3 3 8 ** ,4 0 2 ** ,3 5 6 ** ,9 7 5 ** -, 8 7 5 ** ,9 2 9 ** ,3 9 9 ** ,3 9 9 ** ,9 3 6 ** ,9 2 4 ** n 1 2 0 ,0 7 2 -0 ,0 4 6 0 ,1 8 4 -0 ,0 0 8 0 ,0 3 1 0 ,0 0 0 -0 ,0 0 8 0 ,0 0 0 -0 ,0 5 2 -0 ,0 0 3 0 ,0 0 7 1 ,0 0 0 -0 ,0 3 1 -0 ,0 8 9 0 ,0 2 7 -0 ,0 5 1 -0 ,0 0 3 0 ,0 0 1 -0 ,0 1 7 -0 ,0 0 6 -0 ,0 0 6 0 ,0 0 0 -0 ,0 1 6 n 1 3 ,9 0 2 ** ,8 9 1 ** ,6 3 5 ** ,9 5 9 ** ,9 2 1 ** ,9 5 5 ** ,9 5 9 ** ,9 5 5 ** ,4 3 9 ** ,9 8 2 ** ,9 6 8 ** -0 ,0 3 1 1 ,0 0 0 ,3 5 7 ** ,4 0 0 ** ,3 5 8 ** ,9 8 2 ** -, 8 9 8 ** ,9 3 5 ** ,3 9 2 ** ,3 9 2 ** ,9 5 5 ** ,9 4 7 ** n 1 4 ,5 2 2 ** ,3 9 5 ** ,2 4 8 * ,3 7 8 ** ,3 5 5 ** ,3 9 9 ** ,3 7 8 ** ,3 9 9 ** ,9 2 6 ** ,3 4 1 ** ,3 3 8 ** -0 ,0 8 9 ,3 5 7 ** 1 ,0 0 0 ,9 0 0 ** ,8 9 3 ** ,3 4 1 ** ,9 1 1 ** ,3 3 9 ** ,8 8 1 ** ,8 8 1 ** ,3 9 9 ** ,3 1 3 ** n 1 5 ,4 2 9 ** ,3 4 5 ** ,2 4 5 * ,4 2 0 ** ,3 7 4 ** ,4 3 7 ** ,4 2 0 ** ,4 3 7 ** ,9 5 5 ** ,3 9 9 ** ,4 0 2 ** 0 ,0 2 7 ,4 0 0 ** ,9 0 0 ** 1 ,0 0 0 ,9 0 9 ** ,3 9 9 ** ,9 6 6 ** ,3 6 0 ** ,9 7 5 ** ,9 7 5 ** ,4 3 7 ** ,3 6 8 ** n 1 6 ,4 0 9 ** ,3 4 3 ** 0 ,2 0 6 ,4 3 3 ** ,3 8 9 ** ,4 3 7 ** ,4 3 3 ** ,4 3 7 ** ,9 4 2 ** ,3 5 3 ** ,3 5 6 ** -0 ,0 5 1 ,3 5 8 ** ,8 9 3 ** ,9 0 9 ** 1 ,0 0 0 ,3 5 3 ** ,9 6 6 ** ,4 0 8 ** ,9 2 9 ** ,9 2 9 ** ,4 3 7 ** ,3 3 4 ** n 1 7 ,8 9 7 ** ,9 0 0 ** ,6 2 1 ** ,9 4 6 ** ,9 4 0 ** ,9 4 2 ** ,9 4 6 ** ,9 4 2 ** ,4 2 6 ** 1 ,0 0 0 ** ,9 7 5 ** -0 ,0 0 3 ,9 8 2 ** ,3 4 1 ** ,3 9 9 ** ,3 5 3 ** 1 ,0 0 0 -, 9 0 9 ** ,9 4 9 ** ,4 0 2 ** ,4 0 2 ** ,9 4 2 ** ,9 4 9 ** n 1 8 -, 8 7 ** -, 8 4 2 ** -, 5 5 0 * -, 8 8 5 ** -, 8 7 1 ** -, 8 7 1 ** -, 8 8 5 ** -, 8 7 1 ** ,9 8 6 ** -, 9 0 9 ** -, 8 7 5 ** 0 ,0 0 1 -, 8 9 8 ** ,9 1 1 ** ,9 6 6 ** ,9 6 6 ** -, 9 0 9 ** 1 ,0 0 0 -, 8 9 8 ** 1 ,0 0 0 ** 1 ,0 0 0 ** -, 8 7 1 ** -, 8 7 5 ** n 1 9 ,8 9 8 ** ,9 1 4 ** ,5 8 3 ** ,9 5 7 ** ,9 5 4 ** ,9 5 3 ** ,9 5 7 ** ,9 5 3 ** ,4 1 4 ** ,9 4 9 ** ,9 2 9 ** -0 ,0 1 7 ,9 3 5 ** ,3 3 9 ** ,3 6 0 ** ,4 0 8 ** ,9 4 9 ** -, 8 9 8 ** 1 ,0 0 0 ,3 5 7 ** ,3 5 7 ** ,9 5 3 ** ,9 2 6 ** n 2 0 ,4 0 8 ** ,3 4 2 ** ,2 4 4 * ,4 1 3 ** ,3 7 5 ** ,4 1 7 ** ,4 1 3 ** ,4 1 7 ** ,9 5 9 ** ,4 0 2 ** ,3 9 9 ** -0 ,0 0 6 ,3 9 2 ** ,8 8 1 ** ,9 7 5 ** ,9 2 9 ** ,4 0 2 ** 1 ,0 0 0 ** ,3 5 7 ** 1 ,0 0 0 1 ,0 0 0 ** ,4 1 7 ** ,3 7 1 ** n 2 1 ,4 0 8 ** ,3 4 2 ** ,2 4 4 * ,4 1 3 ** ,3 7 5 ** ,4 1 7 ** ,4 1 3 ** ,4 1 7 ** ,9 5 9 ** ,4 0 2 ** ,3 9 9 ** -0 ,0 0 6 ,3 9 2 ** ,8 8 1 ** ,9 7 5 ** ,9 2 9 ** ,4 0 2 ** 1 ,0 0 0 ** ,3 5 7 ** 1 ,0 0 0 ** 1 ,0 0 0 ,4 1 7 ** ,3 7 1 ** n 2 2 ,9 1 1 ** ,9 0 4 ** ,6 2 9 ** ,9 8 3 ** ,9 2 5 ** 1 ,0 0 0 ** ,9 8 3 ** 1 ,0 0 0 ** ,4 5 2 ** ,9 4 2 ** ,9 3 6 ** 0 ,0 0 0 ,9 5 5 ** ,3 9 9 ** ,4 3 7 ** ,4 3 7 ** ,9 4 2 ** -, 8 7 1 ** ,9 5 3 ** ,4 1 7 ** ,4 1 7 ** 1 ,0 0 0 ,9 2 5 ** n 2 3 ,8 7 8 ** ,8 7 7 ** ,6 2 5 ** ,9 2 9 ** ,9 1 7 ** ,9 2 5 ** ,9 2 9 ** ,9 2 5 ** ,4 1 0 ** ,9 4 9 ** ,9 2 4 ** -0 ,0 1 6 ,9 4 7 ** ,3 1 3 ** ,3 6 8 ** ,3 3 4 ** ,9 4 9 ** -, 8 7 5 ** ,9 2 6 ** ,3 7 1 ** ,3 7 1 ** ,9 2 5 ** 1 ,0 0 0 *. c o rr e la ti o n i s si g n if ic a n t a t th e 0 .0 5 l e v e l (2 -t a il e d ). ** . c o rr e la ti 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(2006). comparing aggregating methods for constructing the composite environmental index: an objective measure. ecological economics, 59(3), 305-311. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applicatons in management and engineering vol. 2, issue 2, 2019, pp. 19-35. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame1902038b * corresponding author. e-mail addresses: i.badi@eng.misuratau.edu.ly (i. badi), ali.shetwan@eng.misuratau.edu.ly (a. shetwan), a.abdulshahed@eng.misuratau.edu.ly (a. abdulshahed) wisamhomrana93@gmail.com (w. eltayeb) evaluation of solid waste treatment methods in libya by using the analytic hierarchy process ibrahim badi *1, ali shetwan 2, ali abdulshahed 3 and wisam eltayeb 2 1 misurata university, faculty of engineering, mechanical engineering department, libya 2 misurata university, faculty of engineering, industrial engineering department, libya 3 misurata university, faculty of engineering, electrical engineering department, libya received: 13 march 2019; accepted: 9 june 2019; published: 10 june 2019. original scientific paper abstract: evaluation and selection of the appropriate method for solid waste treatment (swt) in libya are a complex problem and require an extensive evaluation process. this is because it is very difficult to develop a selection criterion that can precisely describe the preference of one method over another. waste management is the collection, transport, treatment, recycling or disposal and monitoring waste materials. in this paper, four treatment systems for waste management in libya are evaluated using the analytic hierarchy process (ahp) in respect to four main criteria and twenty-two subcriteria. the treatment systems for waste management are anaerobic digestion, landfilling, incineration and compost. the selected criteria used in the evaluation of four treatment systems are environmental impacts, sociocultural aspects, technical aspects and economic aspects. according to the evaluation, anaerobic digestion ranks the highest in classification in libya. compost ranks higher than landfilling and incineration. furthermore, it should be noted that the rank of waste treatment systems can be changed according to the future technological developments. key words: waste management, multi-criteria evaluation, ahp, libya. 1. introduction during the earliest periods, solid wastes were conveniently and unobtrusively disposed in large open land spaces, as the density of the population was low. as the population and economic growth increases, the solid household waste also increases. however, the population and economic growth not only lead to an increase in volume mailto:i.badi@eng.misuratau.edu.ly mailto:ali.shetwan@eng.misuratau.edu.ly mailto:a.abdulshahed@eng.misuratau.edu.ly mailto:i.wisamhomrana93@gmail.com badi et al./decis. mak. appl. manag. eng. 2 (2) (2019) 19-35 20 of solid household waste but also to great changes in its specification and contents (abduelmajid et al., 2015). the concept of eliminating waste completely is highly unrealistic. however, the best approach is to handle solid waste in such a manner that does not damage the environment, while utilizing methods supported by the denizens of the community who are directly impacted by the solid waste management (swm) program in an area (khan & faisal, 2008). therefore, waste management is a priority issue regarding protection of the environment and conservation of natural resources. libya is a north african country located along the southern coast of the mediterranean basin. its total land area is about 1.8 million km2, most of which (95%) is a desert, whereas the rest is either rangeland (4%), or agricultural land (0.5%), and less than 0.5% is a scattered forested area. due to rapid expansion of industry, urbanization and increasing population, particularly in large cities which are located on the coast, has increased the amount of solid waste generated in libya significantly (badi et al., 2016). in libya issues related to sound municipal solid waste (msw) management including waste reduction and disposal have not been addressed adequately and the collection and the separation treatment of solid waste are still neglected. in this paper, criteria for the assessment of the municipal waste management technologies are analyzed and evaluated. the technology assessment indices calculated with these methods were applied as criterions for multi-criteria analysis, which evaluates individual variants of municipal waste management systems. indices evaluating the performance of the system can be determined with due regard to the technical, environmental, economic, social and other objectives, bearing in mind specific features of the area involved. the aim of this study is to evaluate different waste management methods and their applicability in libya based on multi-criteria decision analysis (mcda). the common methods used in this study are those recommended in the waste management laws and regulations, such as composting, anaerobic digestion, incineration with thermal energy recovery (electricity and heat), and landfill without any form of energy recovery. 2. literature review this literature review studies and investigates various waste management methods and a multi-criteria decision analysis including waste reduction and disposal that is applied to the swm. javaheri et al. (2006) presented study includes multi criteria evaluation method under the name of weighted linear combination by using geographical information technology to evaluate the suitability of the vicinity of giroft city in kerman province of iran for landfill. the major criteria used in the study were geomorphologic, hydrologic, humanistic and land use. the results of the study were afford strategy to the decision makers of giroft city by a variety of options (javaheri et al., 2006). manaf et al. (2009) evaluated the generation, characteristics and management of solid waste in malaysia. it was concluded that the efficiency of solid waste management in malaysia will be increased towards achieving vision 2020 as a developed country (manaf et al., 2009). a case study was conducted by sawalem et al. (2009) to evaluate hospital waste management in libya. the study found that several factors such as the type of healthcare establishment, level of instrumentation and location affect waste generation rates. the results showed that the highest generation evaluation of solid waste treatment methods in libya by using the analytic hierarchy process 21 rates at tripoli medical center are attributed to a larger number of patients due to being in the capital of libya (sawalem et al., 2009). gebril et al. (2010) presented an overview of the current swm practices in benghazi, libya. the objective of the study was to investigate the current practices and challenges that faced msw management in benghazi. it was found that several issues affected in the swm such as lack of suitable facilities and inadequate management and technical skills, improper bin collection and route planning (abdelsalam & gebril, 2010). generowicz et al. (2011) combined the best available techniques, technology quality method and multi-criteria analysis in order to develop indices for evaluating municipal waste management systems. the results showed that incineration of waste is much more beneficial than disposal (generowicz et al., 2011). tabasi and marthandan (2013) presented an overview on the existing researches in the area of clinical waste management. the objective of the study was to investigate different findings regarding associated factors on quantity of waste generation to find, integrate and enhance accessibility to hospital key factors in waste generation forecasting. the results showed that the number of patients, number of beds, bed occupancy rate and type of hospitals were the most important factors in waste generation (tabasi & marthandan, 2013). norkhadijah et al. (2013) investigated the challenges which can be faced to find a suitable place for future landfill in malaysia. based on the fact that limited space is available for landfill development, the conclusion of the study was that, landfill cannot be the ultimate option for much longer (norkhadijah et al., 2013). a study was conducted by gebril (2013) to determine the causes of solid waste pollution in benghazi city, in libya and its surrounding areas. the results showed that solid waste pollution in the city and its surrounding areas is the outcome of poor planning and environmental management, population growth, lack of hardware and equipment for the collection and transport of waste from the city to the landfill site (ali, 2013). hamad et al. (2014) presented an overview on solid waste that can be used as a source of bioenergy in libya including industrial solid waste and health care wastes as biomass sources. the aim of the study was to investigate whether or not solid waste can be used as a source of bioenergy in libya. the results showed that organic matter represents 59% of waste, followed by paper–cardboard 12%, plastic 8%, miscellaneous 8%, metals7%, glass4%, and wood 2%. the technology of incineration is recognized as a renewable source of energy and is playing an increasingly important role in msw management in libya (hamad et al., 2014). najjar et al. (2015) conducted a study to estimate the percentage of total plastic and pvc in particular, in solid household waste in the city of tripoli, libya. the results concluded that the weight percentages of plastic waste and pvc were about 10.52% and 1.36%, respectively. the percentage of pvc from plastic waste was only 12.94% (abduelmajid et al., 2015). babalola (2015) presented a multi-criteria decision analysis to evaluate different waste management options and their applicability in japan. the results showed that anaerobic digestion should be chosen as the best food and biodegradable waste treatment option concerning resource recovery (babalola, 2015). a study was carried out by moftah et al. (2016) to evaluate the generation, composition and density of household solid waste in tripoli city, libya. it was concluded that the total generation quantity, daily generation rate, total volume and density were in tripoli city agreed with those for african and arabic countries. the study showed that tripoli suffers from insufficient msw management and lack of sanitary landfills (moftah et al., 2016). jovanović (2016) presented a method for selection an optimal waste management system in the city of kragujevac, serbia badi et al./decis. mak. appl. manag. eng. 2 (2) (2019) 19-35 22 through an integrated application of the life cycle assessment method and mcdm methods. six different waste management strategies for the territory of the city were formulated and eight parameters were selected (jovanović et al., 2016). omran et al. (2017) conducted a study in the city of al-bayda, libya dealing with solid waste management. the aim of the study was to investigate the major problems facing the city in dealing with swm in terms of generation, collection, handling, transportation, recycling and disposal of msw. the conclusion was, there were major factors impacting the decision-making and operational processes of msw that include lack of resources and services that significantly affect the disposal of waste and inadequate number of waste collection containers. this makes the distance to these containers for many households excessive and thus leading to an increasing likelihood of dumping solid waste in open areas and roadsides (omran et al., 2017). by reviewing the previous studies specifically, the studies that dealt with swm in libya, it can be noted that, vast majority of them focused on the classification of solid waste management rather than the selection of the technology treatment. to fill this research gap, this paper examines the selection of the appropriate method for the solid waste treatment. 3. variants of municipal waste management technology several types of recycling, energy recovery or waste neutralization technologies are used in a system of waste management. each of them shows different technical and environmental characteristics. 3.1. composting composting is a biological process in which the organic matter current in waste is converted into enriched inorganic nutrients. the manure obtained has high nitrogen, phosphorus, and potassium content. composting is often described as nature’s way of recycling is a key ingredient in organic farming. at the simplest level, the process of composting only requires making a heap of wetted organic material (leaves, food waste) and waiting for the materials to breakdown into humus like substance by undergoing biological decomposition after a period of weeks or months (ladan, 2014). the quality of compost depends upon the waste being composted. there are a number of biological or compost related technologies. these are open windrow, vermicomposting, enclosed composting and fermentation (thompson-smeddle, 2011). 3.2. anaerobic digestion anaerobic digestion (ad) is a naturally occurring biological process that uses microorganisms to break down organic material in the absence of oxygen. in other words, ad is a process that makes any organic waste can be biologically transformed into another form, in the absence of oxygen. the production of biogas and other energy-rich organic compounds is mainly produced from the degradation of organic waste by microbial organisms (arshad et al., 2011). a series of metabolic reactions such as hydrolysis, acid genesis, acetogenesis and methanogenesis are involved in the process of anaerobic decomposition (charles et a., 2009). anaerobic digestion can be applicable for a wide range of material including municipal, agricultural and industrial wastes and plant residues (chen et a., 2008). evaluation of solid waste treatment methods in libya by using the analytic hierarchy process 23 3.3. incineration incineration, or thermal oxidation is the process of oxidizing combustible materials by raising the temperature of the material above its auto-ignition point. the process is done in the presence of oxygen, and maintaining it at a high temperature for sufficient time to complete combustion to carbon dioxide and water (epa-cica, 2003). any non-combustible materials (e.g. metals, glass, stones) remain as a solid, known as incinerator bottom ash that always contains a small amount of residual carbon (defr, 2007). the efficiency of the combustion process is affected by the factors such as time, temperature, turbulence (for mixing) and the availability of oxygen. these factors provide the basic design parameters for volatile organic compounds oxidation systems (icac, 1999). 3.4. landfilling landfilling is the ultimate disposal process for the swm. the process is simply dumping the waste in trenches or cells with leveling and compacting by trash compactors to reduce the size and the thickness of the layers, and finally the waste is covered by soil (aljaradin, 2014). the quantity of msw for land disposal can be considerably reduced by setting up waste processing facilities and recycling the waste materials as much as possible. 4. multi-criteria decision making multi-attribute decision making (madm) is the most well-known branch of decision making. madm models deal with decision making problems under the presence of a number of decision criteria. this class of models is very often called multi-criteria decision making (or mcdm). according to many authors mcdm is divided into multi-objective decision making (or modm) and multi-attribute decision making (or madm) (shu et al., 1998; karami, 2011). modm is a mathematical programming problem with multiple objective functions. whereas, the developing of madm models is based on several alternatives according to some criteria are ranked and selected. ranking and selecting will be made among decision alternatives described by some criteria through decision-maker knowledge and experience (karami, 2011; devi et al., 2009; chatterjee et al., 2018; pamučar et al., 2018a). mcdm is approach for finding the optimal alternative from all the feasible alternatives according to some criteria or attributes (stević et al., 2017; pamučar et al., 2018b). 5. analytic hierarchy process analytical hierarchy process is a common mcdm method. it is developed by saaty to provide a flexible and easily understood way of analysing complex problems (saaty, 1979; saaty, 1990). according to (chai et al., 2013) it has been found that ahp method was used more than any other mcdm methods (chai et al., 2013). it breaks a complex problem into hierarchy or levels, and then making comparisons between possible pairs in a matrix to give a weight for each factor and also a consistency ratio. the ahp utilises a tree structure in order to simplify complex decision-making problems resulting in simplified sub problems, which can easily be examined. the ahp method can be distinguished in four main steps: badi et al./decis. mak. appl. manag. eng. 2 (2) (2019) 19-35 24 • creation of a tree structure, which comprises of one goal, the criteria, and alternative solutions. • evaluation of each alternative solution in relation to each criterion. • calculation of the weighting factor of the criteria with subjective evaluation using pairwise comparisons. • synthesis of the results of stages 2 and 3 so as to calculate the overall evaluation of each alternative regarding the degree of achievement of each goal. figure 1 presents the tree structure for the four swt systems. figure 1. tree structure for the four swt systems in the ahp method, pairwise comparisons permit the decision maker to concentrate only on one element at a time. specifically, to explore how strongly important is one criterion related to another with regards to the goal?” the comparisons are the input into a matrix. if the matrix is sufficiently consistent, priorities can then be calculated with formula (1). max aw w (1) where a is the comparison matrix, λmax is the principal eigenvalue and w is the priority vector. the ahp model gives feedback to the decision maker on the consistency of the entered judgments through the measurement of consistency ratio (cr) by using formulas (2) and (3). ri ci cr  (2) 1 max    n ci n (3) where ci is the consistency index, n is the dimension of the comparison matrix, λmax is the principal eigenvalue and ri is the ratio index. the ratio index or random consistency index (ri) is given in table 1. if the consistency ratio is less than 0.1 (<10%) the matrix is regarded as consistent, otherwise the matrix is inconsistent and it is suggested to modify the comparisons in order to reduce the inconsistency (saaty, 1980). if all sub-priorities are available, they are aggregated with a weighted sum in evaluation of solid waste treatment methods in libya by using the analytic hierarchy process 25 order to obtain the overall priorities of the alternatives so as a final judgment can be made based on the ranking (saaty, 1980; saaty & vargas, 2001). table 1. random consistency index (ri) n 1 2 3 4 5 6 7 8 9 ri 0 0 0.58 0.90 1.12 1.24 1.32 1.41 1.45 6. analysis of the results the quantity of msw generated in libya is estimated at 3.2 million tons/year (sawalem et al., 2009; ali, 2013). the treatment of solid waste is done by throwing in open dumps designated by the relevant authorities and in many cases random dumps that are not controlled by the state. lack of suitable facilities, inadequate management and technical skills, improper bin collection and shortages in solid waste plants are among the important issues resulting in poor collection and transportation of municipal solid wastes in libya (abdelsalam & gebril, 2010). however, in libya few msw plants were established in several cities as shown in figure 2. these plants are suffering from many obstacles, because all of them are outdated and need to be updated or replaced. for example, the msw plant in misurata, which was opened in 1982 with a capacity of 120 tons per day, currently the capacity is only 60 tons. figure 2. distribution of solid waste plants in libya for the evaluation of the four treatment systems, with the use of the ahp, 12 cases were carried out. these cases were the base case, equally distributed criteria case, four cases of single-criterion analysis and six cases of multi-criteria analysis. in this paper, qualitative criteria are identified based on questionnaire forms that have been filled badi et al./decis. mak. appl. manag. eng. 2 (2) (2019) 19-35 26 in by environmental experts and academic staff university members. table 2 shows the pairwise comparison matrix of the general and organizational structure of the technology’s sub-criteria. in order to facilitate the solution process for the ahp problem, expert choice software were used to compute the model. table 2. criteria, sub-criteria and their weights criteria weight cr sub-criteria weight environmental 0.581 0.07 air and water pollution 0.286 land use, requirement, and contamination 0.046 material recovery 0.062 waste coverage and elimination 0.061 net energy recovery 0.077 disamenity, such as noise and dust 0.049 socio-cultural 0.204 0.07 acceptance 0.119 usability and compatibility 0.027 policy 0.020 implementation and adoptability 0.023 vulnerability of the area 0.016 technical 0.128 0.08 possibility and robustness 0.054 local labor working experience 0.019 adaptability to existing systems 0.012 handling capacity and continuous process 0.031 prospective future improvement 0.012 economical 0.086 0.08 capital and construction cost 0.038 operating and maintenance cost 0.007 revenue generation and marketability 0.009 financial planning 0.011 employment and job creation 0.016 waste volume and composition 0.005 6.1. base case in the base case, the criteria weights have been calculated using pairwise comparison according to the ahp method. the following weighting factors are used: environmental impacts 58%, socio-cultural aspects 20%, technical aspects 13% and economic aspects 9%. the weighting factors were given to each criterion. these percentages indicated that the environmental impact of each alternative option is the primary concern of this case, while socio-cultural aspects follow. figure 3 presents the rating of alternative options for swt system. as can be seen from figure 2, the anaerobic digestion is the best option when a greater emphasis is given to environmental impact. furthermore, compost and incineration are ranked second and third, respectively. on the contrary, landfill is ranked last. figure 3. overall evaluation of swt system evaluation of solid waste treatment methods in libya by using the analytic hierarchy process 27 the next step of the decision process of the ahp method is the sensitivity analysis, where the input data of criteria weighting are slightly modified in order to observe their impact on the results. if the ranking of treatment systems does not change significantly, the results are said to be robust. bearing in mind that the opinions of experts may vary, a sensitivity study was carried out. the following cases were examined: 6.2. equally distributed criteria (case 1) in case1, the following weighting factors are used: environmental impacts 25%, socio-cultural aspects 25%, technical aspects 25% and economic aspects 25%. figure 4 presents the rating of alternative options for swt system for this case. again, the anaerobic digestion is the best option while landfill is ranked last. 6.3. single-criterion analysis (cases 2–6) in the single-criterion analysis (cases 2–6), the evaluation has been carried out with full emphasis to one criterion while the other four criteria are ignored. 6.3.1. case 2 in case 2, the following weighting factors are used: environmental impacts 100%, socio-cultural aspects 0%, technical aspects 0% and economic aspects 0%. as can be seen from figure (5), the best option is the anaerobic digestion, landfill ranks last, given the fact that it has a high impact on the environment. 6.3.2. case 3 in case 3, the following weighting factors are used: environmental impacts 0%, socio-cultural aspects 100%, technical aspects 0% and economic aspects 0%. figure 6 gives the overall ranking of swt system when emphasis is given to socio-cultural aspects. the anaerobic digestion has the highest ranking while incineration receives the last position. 6.3.3. case 4 in case 4, the following weighting factors are used: environmental impacts 0%, socio-cultural aspects 0%, technical aspects 100% and economic aspects 0%. as can be seen from figure 7, the compost has the highest-ranking while incineration receives the last position. this result was expected since the incineration system requires some technical consideration. figure 6. overall evaluation of swt figure 7. overall evaluation of swt system for case 3 system for case 4 badi et al./decis. mak. appl. manag. eng. 2 (2) (2019) 19-35 28 6.3.4. case 5 in case 5, the weighting factors used are: environmental impacts 0%, socio-cultural aspects 0%, technical aspects 0% and economic aspects 100%. as shown in figure 8, again the anaerobic digestion has the highest-ranking while the landfilling system receives the last position. 6.4. multi-criteria analysis (cases 6–11) according to multi-criteria analysis (cases 6–14), the evaluation of the fourselected swt system has been carried out by giving greater emphasis (a larger weighting factor) to one criterion without ignoring the rest as was carried out in the single-criterion analysis (cases 6–9). in the last two cases, greater emphasis is given to two criteria at the same time (cases 10–11). 6.4.1. case 6 in case 6, the following weighting factors are used: environmental impacts70%, socio-cultural aspects 10%, technical aspects 10% and economic aspects 10%. as can be seen from figure 9, the best swt system is the anaerobic digestion while landfilling system receives the last position. figure 8. overall evaluation of swt figure 9. overall evaluation of swt system for case 5 system for case 6 6.4.2. case 7 in case 7, the following weighting factors are used: environmental impacts 10%, socio-cultural aspects 70%, technical aspects 10% and economic aspects 10%. figure 10 presents the rating of alternative options for swt system for case 7. according to this figure, the best waste treatment system is anaerobic digestion and next is compost while incineration receives the last position. 6.4.3. case 8 in case 8, the following weighting factors are used: environmental impacts 10%, socio-cultural aspects 10%, technical aspects 70% and economic aspects 10%. according to figure 11, the best waste treatment system in case 8 is anaerobic digestion, and next is compost while incineration receives the last position. these outcomes are very similar to the results obtained in case 7. evaluation of solid waste treatment methods in libya by using the analytic hierarchy process 29 figure 10. overall evaluation of swt figure 11. overall evaluation of swt system for case 7 system for case 8. 6.4.4. case 9 in case 9, the following weighting factors are used: environmental impacts 10%, socio-cultural aspects 10%, technical aspects 10% and economic aspects70%. as can be seen from figure 12, the best swt system is the anaerobic digestion while incineration system receives the last position. 6.4.5. case 10 in case 10, the following weighting factors are used: environmental impacts 35%, socio-cultural aspects 35%, technical aspects 15% and economic aspects 15%. as can be seen from figure 13, the best swt system is the anaerobic digestion while incineration system receives the last position. figure 12. overall evaluation of swt figure 13. overall evaluation of swt system for case 9 system for case 10 6.4.6. case 11 in case 11, the following weighting factors are used: environmental impacts 35%, socio-cultural aspects 15%, technical aspects15% and economic aspects 35%. as can be seen from figure 14, the best swt system is the anaerobic digestion while landfilling system receives the last position. badi et al./decis. mak. appl. manag. eng. 2 (2) (2019) 19-35 30 figure 14. overall evaluation of swt system for case 11 table 3 presents the criteria weights for the 12 scenarios conducted to the case study which is described above, and table 4 summarises an overall evaluation and ranking of the four swt systems under examination. the evaluation of the swt systems was carried out using the ahp for 12 cases. these consisted of the base case, the equally distributed criteria, four cases of single-criterion analysis and six cases of multi-criteria analysis. each treatment system presents a solution for the solid waste management system with a certain degree of trade-off between benefit and its consequences related to environmental, social, technical and economic issues. sensitivity analysis is conducted to evaluate the robustness of the selected treatment options. a “what if analysis” figure 15 was performed to see if there were any changes among the selected treatment options. the results display no changes in the ranked results, as the anaerobic alternative remained the most suitable option for the treatment of the swm. as can be seen from figure 15, the majority of cases, the anaerobic digestion is considered to be better than the other systems (landfilling, incineration, and compost) and is higher in ranking. on the contrary, the landfilling and incineration systems rank last in most of the cases. more specifically, in most of the cases (10 out of 12), the first in ranking swt system is considered to be the anaerobic digestion and the worst (7 out of 12) is landfilling. there is a need for improvement in the design of this treatment system, site location, size and management of the disposal sites. existing practices must be improved immediately evaluation of solid waste treatment methods in libya by using the analytic hierarchy process 31 as they create environmental problems. it should be noted that the rank of swt systems can be changed according to the future system development. t a b le 3 . o v e ra ll c ri te ri a w e ig h ts f o r e a ch s ce n a ri o . c ri te ri a c ri te ri a w e ig h ts f o r e a ch c a se ( % ) b a se c a se c a se # 0 1 c a se # 0 2 c a se # 0 3 c a se # 0 4 c a se # 0 5 c a se # 0 6 c a se # 0 7 c a se # 0 8 c a se # 0 9 c a se # 1 0 c a se # 1 1 e n v ir o n m e n ta l im p a ct s 5 8 % 2 5 % 1 0 0 % 0 0 % 0 0 % 0 0 % 7 0 % 1 0 % 1 0 % 1 0 % 3 5 % 3 5 % s o ci o -c u lt u ra l a sp e ct s 2 0 % 2 5 % 0 0 % 1 0 0 % 0 0 % 0 0 % 1 0 % 7 0 % 1 0 % 1 0 % 3 5 % 1 5 % t e ch n ic a l a sp e ct s 1 3 % 2 5 % 0 0 % 0 0 % 1 0 0 % 0 0 % 1 0 % 1 0 % 7 0 % 1 0 % 1 5 % 1 5 % e co n o m ic a sp e ct s 0 9 % 2 5 % 0 0 % 0 0 % 0 0 % 1 0 0 % 1 0 % 1 0 % 1 0 % 7 0 % 1 5 % 3 5 % t a b le 4 . o v e ra ll e v a lu a ti o n a n d r a n k in g s w t s y st e m f o r e a ch c a se s o li d w a st e t re a tm e n t sy st e m b a se c a se c a se # 0 1 c a se # 0 2 c a se # 0 3 c a se # 0 4 c a se # 0 5 s co re (% ) r a n k s co re (% ) r a n k s co re (% ) r a n k s co re ( % ) r a n k s co re (% ) r a n k s co re (% ) r a n k a n a e ro b ic d ig e st io n 4 2 % 1 3 9 .7 % 1 4 6 .2 % 1 3 9 .1 % 1 3 4 .7 % 2 3 8 .4 % 1 l a n d fi ll in g 1 3 % 4 1 5 .7 % 4 9 .9 % 4 1 6 .5 % 3 1 7 .0 % 3 1 9 .3 % 4 in ci n e ra ti o n 1 8 % 3 1 6 .2 % 3 2 1 .7 % 3 8 .8 % 4 1 3 .5 % 4 2 0 .9 % 3 c o m p o st 2 7 % 2 2 8 .5 % 2 2 2 .2 % 2 3 5 .7 % 2 3 4 .8 % 1 2 0 .4 % 2 s o li d w a st e t re a tm e n t sy st e m c a se # 0 6 c a se # 0 7 c a se # 0 8 c a se # 0 9 c a se # 0 1 0 c a se # 0 1 1 s co re (% ) r a n k s co re (% ) r a n k s co re (% ) r a n k s co re ( % ) r a n k s co re (% ) r a n k s co re (% ) r a n k a n a e ro b ic d ig e st io n 4 3 .6 % 1 3 7 .7 % 1 3 5 .2 % 1 3 7 .2 % 1 3 8 .9 % 1 3 8 .8 % 1 l a n d fi ll in g 1 2 .2 % 4 1 7 .2 % 3 1 7 .5 % 3 1 8 .9 % 3 1 5 .9 % 3 1 6 .5 % 4 in ci n e ra ti o n 1 9 .5 % 3 1 0 .6 % 4 1 3 .4 % 4 1 7 .9 % 4 1 4 .5 % 4 1 7 .0 % 3 c o m p o st 2 4 .7 % 2 3 4 .5 % 2 3 3 .9 % 2 2 6 .0 % 2 3 0 .7 % 2 2 7 .7 % 2 badi et al./decis. mak. appl. manag. eng. 2 (2) (2019) 19-35 32 figure 15. sensitivity analysis for the 12 cases of the swm we feel the proposed method plays an important role in ranking of waste treatment systems, especially when it is in a situation where dynamic, and complex real-world problems. one of the most important advantages of the proposed approach is that it is based on a pair-wise comparison. moreover, the method computes the inconsistency index, which is used to determine whether a respondent answered similar items in a consistent manner. 7. conclusion undoubtedly, the waste treatment system in libya is very poor, for instance, more than 97% of the waste is dumped in uncontrolled open areas. as a result, libyan authorities need to take urgent steps in order to address the current situation. in this study, the multi-criteria decision-making approach is identified as a useful means for an integrated evaluation of the appropriate treatment options for the swm. the methodology presented here can be used as a well-organized, strategic decision supporting tool for decision makers, politicians, and planners. it is essential to have consistent goals and objective information about the evaluation process of anaerobic digestion suitability for solid waste treatment based on environmental, sociocultural, technical, and economic aspects. clearly, the anaerobic digestion and composting treatment systems are the two most preferred alternatives. furthermore, a large part of the used fertilizer in the agriculture is imported from abroad, and most of the local fertilizer industries are not competitive in the today market. also, the results show that the incineration alternative is in the last order, due to the inability to compete with current power generation methods in the country (e, g. power generation using fuel oil and natural gas). furthermore, libya is also considered as rich country with renewable energy resources such as solar and wind energy. however, waste incineration is not a competitive alternative renewable energy. the performance of the treatment options based on the criteria mentioned earlier is a robust one similar to the synthesis results. as anaerobic digestion is based on a naturally occurring biological process which produces biogas through anaerobic digestion, this can lead to reduce the main environmental problems of increasing organic waste production and increasing carbon dioxide in the atmosphere. moreover, investments in this waste management facility can be considered to offer another source of revenue generation for waste evaluation of solid waste treatment methods in libya by using the 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(4th ed.). cape town: city of cape town. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/) plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 1, number 1, 2018, pp. 67-81 issn: 2560-6018 doi: https://doi.org/10.31181/dmame180167l * corresponding author. e-mail addresses: veskolukovac@yahoo.com (v. lukovac), milenap@fon.bg.ac.rs (m. popović) fuzzy delphi approach to defining a cycle for assessing the performance of military drivers vesko lukovac*1, milena popović2 1 university of defense in belgrade, military academy, department of logistics, belgrade, serbia 2 university of belgrade, faculty of organizational sciences, belgrade, serbia received: 3 january 2018; accepted: 20 february 2018; published: 15 march 2018. original scientific paper abstract: this paper presents the fuzzy delphi approach to defining a cycle for assessing the performance of military drivers. this approach is based on the delphi decision-making process under uncertainty. these uncertainties are described by linguistic terms modeled with triangular fuzzy numbers. the approach is modeled to take into the account the importance weight of each decision-maker and the homogeneity of their individual fuzzy preferences. the vertex method calculates the distance between the aggregated fuzzy estimation and the triangular fuzzy numbers in which the linguistic terms which experts had chosen are modeled. defuzzification of the fuzzy preference of the experts was carried out by a graded mean integration representation. key words: fuzzy delphi approach, cycle, evaluating performance, military drivers. 1. introduction depending on the work industry, job characteristics and used techniques, estimation can be done on daily, weekly, monthly, quarterly, half yearly or yearly basis (noe et al., 2006; grout, 2008; jovanović et al., 2004; vujić 2008; bogićevićmilikić, 2008). common practice for performance rating in most organizations is on the yearly level. this choice has its advantages and disadvantages. the organization financial reporting dynamics is similar to the performance estimation cycle on the yearly basis; this is one of the advantages for such decision. however, some authors assert that such dynamics does not have to match the time cycle of a certain job, which is why many of the dimensions that are being evaluated stay blurred: in some executive jobs, which are low in the organizational hierarchy, the time cycle can be very short (e.g. seasonal jobs), which leads to a very short period for evaluating the performance of top managers. it is wrong to start with performance evaluation mailto:veskolukovac@yahoo.com mailto:milenap@fon.bg.ac.rs lukovac & popović/decis. mak. appl. manag. eng. 1 (1) (2018) 67-81 68 before it can be measured. in situations when individuals do not work long enough on particular workplace, a premature performance evaluation leads to stimulation of short goals only. on the other side, if you are waiting too long for formal evaluation, estimates can be wrong, whereby a significant loss of the motivation potential is considered. also, it can be considered as a loss of the development potential during evaluation since the employees find out too late what they should improve in their work (bogićevićmilikić, 2008). however, same authors suggest that for some organizations it is completely impossible or either problematic, for practical reasons, to adopt evaluating system in which assessment is carried out in different time periods for different jobs. one of the possible compromises between demands for performance evaluation of all employees on the yearly basis, on one side, and demands for performance evaluation corresponding to the time cycle of a particular job, lies in performance evaluation of employees on the annual basis, except for executives and new employees. for them, the performance evaluation should be done more often. a decision that specifies the start and the end of a single assessment cycle is an important component of the employees’ performance evaluation system. in terms of defining the start and the end of the evaluation period, in literature and practice, there are two basic models (bogićevićmilikić, 2008): 1. model i (anniversary date appraisals)  an appraisal system in which all of an organization's employees are reviewed on the anniversaries of their individual hire dates; 2. model ii (focal point reviews)  also called common date or scheduled reviews, have organizations evaluate all of their employees at one set time, usually at the end of the calendar year. model i has many disadvantages: they are difficult to manage as an employee's role changes from one manager or department to another; the manager is constantly evaluating individual performance rather than that of the department as a whole; it is difficult to complete the process on time. focal performance appraisal strategy can be very helpful if the company is facing changes and must quickly alter its strategy. this model also enables managers to compare the performance of different employees simultaneously, which can result in appraisals that are more accurate and fair. the aim of this paper is to define the cycle for assessing the performance of military drivers. the basic assumption is based on the view that the extended fuzzy delphi model is a convenient tool for achieving this goal. this paper is organized as follows: section 2 describes the basic assumptions of the delphi method; section 3 describes the basic concepts of the fuzzy logic theory; section 4 describes the fuzzy delphi approach to defining a cycle for assessing the performance and results obtained by applying the proposed algorithm; and section 5 provides concluding remarks. 2. delphi method the delphi method is a widely used and accepted method for gathering data from respondents within their domain of expertise. the method is designed as a group communication process which aims at achieving a convergence of opinions on a specific real-world issue. the delphi method is well suited as a means and method for consensus-building by using a series of questionnaires to collect data from a panel of selected subjects (young & jamieson, 2001). the method was implemented fuzzy delphi approach to defining a cycle for assessing the performance of military drivers 69 with a selected set of experts who were anonymous to each other, in as many rounds as necessary for the deviation in the mean limit values of the observed variables to be negligible. after receiving the response from all the participants from the first round, the statistical processing is made, which involves calculating mean values, variance and standard deviation. information about answers given by all experts is put in materials for the second round so that the experts have a chance to change their prognosis. the answers are being collected again and processed in the same way as in the first round. this procedure is repeated until the value of the coefficient of variation is not satisfactory. when an acceptable degree of consensus is obtained the process ends. delphi method is shown in fig. 1. start end the last round ? expert's estimates on the round i distribute questionnare final results yes no statistical analysis responses on the round i figure 1. implementation of the delphi method (lukovac, 2016) fuzzy logic is a very convenient tool for exploiting uncertainties and subjectivity that characterize the delphi method. 3. fuzzy logic theory fuzzy logic theory was introduced by zadeh in 1965 as an extension of the classical notion of set. the fuzzy logic theory is based on fuzzy sets which are a natural extension of the classical set theory. a fuzzy set is determined by a membership function which accepts all intermediate values between 0 and 1. the values of a membership function precisely specify to what extent an element belongs to a fuzzy set, i.e. to the concept it represents. in the fuzzy sets, the decision-maker should determine the form of the membership function. in the literature, the most lukovac & popović/decis. mak. appl. manag. eng. 1 (1) (2018) 67-81 70 common fuzzy numbers are triangular, trapezoidal and bell shape numbers. the use of these fuzzy numbers does not require complex mathematical calculations, and the accuracy of the results obtained is quite satisfactory. according to some authors, the use of higher order fuzzy sets (parabolic shape, logarithmic curve, and itc) has no meaningful application in the uncertainty modeling that exists in real problems (klir & yuan, 1995). in this paper, triangular fuzzy numbers (tfn) were used to model the uncertainty. fig. 2 shows a typical example of a tfn a symbolized by ( , , )a l m r , with peak (or center) m, left width l > 0 and right width r > 0. 0 x 1 l a m r figure 2. fuzzy number a basic operations over tfn are defined in (dubois & prade, 1980). if we consider two tfn  1 1 1, ,a l m r and  2 2 2, ,b l m r , the algebraic rules that apply to these two tfn are:  1 2 1 2 1 2, ,a b l l m m r r     (1)  1 2 1 2 1 2, ,a b l r m m r l     (2)  1 2 1 2 1 2* * , * , *a b l l m m r r (3)  1 2 1 2 1 2: : , : , :a b l r m m r l (4)  1 1 1* * , * , * ,k a k l k m k r k const  (5)   1 1 1 1 1 1 1 1 1 1 1 , , , ,a l m r r m l           (6) defuzzification is the process of producing a quantifiable result in fuzzy logic, given fuzzy sets and corresponding membership degrees. there are many different fuzzy delphi approach to defining a cycle for assessing the performance of military drivers 71 methods of defuzzification available, and which will be used depends on the decision-maker. 4. fuzzy delphi approach to defining a cycle for assessing the performance the need to improve the delphi method by introducing uncertain data was explained in papers (ishikawa et al., 1993; wu, 2011). fuzzy delphi methods (fdm) have been investigated by different researchers. in (chang et al., 2011) author deals with the problem of controlling the quality of services in rail traffic with the fdm. in (tadić et al., 2013) the authors considered the problem of selecting appropriate technologies by following 14 criteria. fdm determines the aggregation of the relative importance of the criteria. in (cheng & lin, 2002), the fdm was developed to determine relative importance of business goals. according to this model, each decision-maker carries out a direct assessment of the importance of business goals on each hierarchical model. then the group’s opinion mean value is calculated, which is also described by the tfn based on the algebra. also, the fuzzy distance between the mean value of a group and fuzzy numbers is calculated, which describes predefined linguistic terms. based on this information, the decision-makers in the first iteration correct their estimations. the consensus is considered to be achieved in the second iteration of the fdm. in the majority of papers, the authors consider that the number of iterations is a criterion according to which the stability of fdm is achieved. in (lukovac, 2016), the authors expose consideration that the difference between the fuzzy numbers of two consecutive iterations for the referred item should not be greater than 0.2. in this paper, the extended fdm (efdm extended fuzzy delphi model) developed in (kashdan, 2004), which takes into account the importance (weights) of decision-makers and the homogeneity of their expressed fuzzy preferences, was used to define the start/end of the military performance assessment cycle. the efdm algorithm consists of six steps: step 1: decision-makers express their opinion by choosing one of the six offered responses described by linguistic terms (analogous to saaty, 1980) via tfn. the domains of these tfn's are defined in the saaty's scale of measurement (chen & tzeng, 2004). value 1 or value 9, indicates the lowest or highest value of the variables. table 1 shows the domains of these tfn's. table 1. linguistic terms of efdm (lukovac, 2016 ; lukovac & popović, 2017) linguistic terms tfn disagree strongly (dst) (1,1,2.5) disagree moderately (dmo) (1.5,3,4.5) disagree a little (dli) (3,4.5,6) agree a little (ali) (4,5.5,7) agree moderately (amo) (5.5,7,8.5) agree strongly (ast) (7.5,9,9) the graphical representation of efdm's linguistic terms from table 1 is shown in figure 3. lukovac & popović/decis. mak. appl. manag. eng. 1 (1) (2018) 67-81 72 0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 µ(x) dst astdmo dli ali amo figure 3. the graphical representation of efdm's linguistic terms (lukovac & popović, 2017) step 2: the aggregation of the decision-makers fuzzy estimations is accessed according to the expression:   1 1 1 , , n a i ei i n a a a a i ei i n a i ei i l l a l m r m m r r                              , i=(1,…,n) (7) where are: a  aggregating experts' fuzzy estimate; a l  the left margin of aggregated fuzzy assessment; a m  the value in which the function of the aggregated fuzzy assessment has the highest value i.e., 1 a m  ; a r  the right margin of aggregated fuzzy assessment; n number of experts; ei   the normalized weight of i expert. step 3: the vertex method calculates the distance (d+) between the aggregated fuzzy estimation and the triangular fuzzy numbers in which the linguistic statements, according to the expression (gigović et al., 2016; pamučar et al., 2011):       2 2 21 * 3 i a i a i a i d l l m m r r           (8) where is: fuzzy delphi approach to defining a cycle for assessing the performance of military drivers 73 i index of linguistic term in table 1, i=(1,...,6); the decision-makers' aggregated opinion can be described by linguistic terms that the least distance value is associated with. step 4: the approach is towards the second iteration of fdm, with the prior knowledge of decision-makers with the results of the first fdm iteration. step 5: the distance between aggregated fuzzy estimations is calculated in two consecutive iterations:     1 1 1, , , , ,a a a a a ad l m r l m r      (9) if the value of the distance between the aggregated stages of the estimation in two consecutive iterations is less than 0.2 (analogously [9]), the decision-makers' consensus has been reached. step 6: the defuzzification of individual fuzzy estimations from the second iteration of the efdm is carried out, and its homogeneity on the saaty's scale, by the mean value, the standard deviation and the coefficient of variation, is investigated: . ; . 30% 3 mean std deviation c variance  (10) if the condition of homogeneity of the individual fuzzy estimation of the decisionmakers is satisfied, it is established that a complete consensus has been reached and the process is therefore completed. otherwise, the process is repeated. in line with the presented efdm algorithm, the definition of the cycle for assessing the performance of military drivers has begun. the expert group was made of 20 decision-makers who conducted the research on the development of a model for the elimination of errors in the system for assessing the performance of drivers of military motor vehicles (lukovac, 2016). experts expressed their preferences about alternatives for the cycle length, as well as its start/end, by choosing one of the linguistic terms in table 1. 4.1. determining the cycle length of assessment experts used linguistic terms from the table 1 to determine the cycle length of assessment. they evaluated the offered alternatives for the time period that the performance of military drivers should be assessed for (3, 6 and 12 months) and their preferences in the second iteration are shown in table 2. based on the distance value between aggregating experts' fuzzy estimate in two consecutive efdm iterations, the first condition for accepting a decision is satisfied in the second iteration. expert weights ( ) e  were obtained by normalizing their coefficients of competence, calculated according to the approach shown in [11]. distances between aggregated fuzzy numbers (table 2) and triangular fuzzy numbers in which the linguistic terms which experts had chosen are modeled, are shown in table 3. lukovac & popović/decis. mak. appl. manag. eng. 1 (1) (2018) 67-81 74 table 2. fuzzy preferences of the experts regarding the time period of the assessment (lukovac, 2016) experts 3 months 6 months 12 months e  1. (1,1,2.5) (1,1,2.5) (7.5,9,9) 0.0540 2. (1,1,2.5) (1.5,3,4.5) (7.5,9,9) 0.0456 3. (1.5,3,4.5) (1,1,2.5) (7.5,9,9) 0.0461 4. (7.5,9,9) (1.5,3,4.5) (1,1,2.5) 0.0476 5. (1,1,2.5) (1.5,3,4.5) (7.5,9,9) 0.0456 6. (1,1,2.5) (1.5,3,4.5) (7.5,9,9) 0.0455 7. (1,1,2.5) (7.5,9,9) (3,4.5,6) 0.0457 8. (1,1,2.5) (1,1,2.5) (7.5,9,9) 0.0561 9. (1,1,2.5) (1,1,2.5) (7.5,9,9) 0.0582 10. (1,1,2.5) (1.5,3,4.5) (7.5,9,9) 0.0463 11. (1,1,2.5) (1.5,3,4.5) (7.5,9,9) 0.0498 12. (1,1,2.5) (1.5,3,4.5) (7.5,9,9) 0.0508 13. (1,1,2.5) (1.5,3,4.5) (7.5,9,9) 0.0557 14. (5.5,7,8.5) (1.5,3,4.5) (1,1,2.5) 0.0547 15. (1,1,2.5) (1.5,3,4.5) (7.5,9,9) 0.0511 16. (1,1,2.5) (1.5,3,4.5) (7.5,9,9) 0.0485 17. (7.5,9,9) (3,4.5,6) (1,1,2.5) 0.0543 18. (1,1,2.5) (1.5,3,4.5) (7.5,9,9) 0.0476 19. (1,1,2.5) (1.5,3,4.5) (7.5,9,9) 0.0505 20. (1,1,2.5) (1.5,3,4.5) (7.5,9,9) 0.0463 aggregation (1.93,2.24,3.58) (1.75,2.93,4.36) (6.28,7.54,7.85) 1 table 3. distance values for the assessment cycle length (lukovac, 2016) linguistic terms 3 months 6 months 12 months disagree strongly 1.090 1.605 5.750 disagree moderately 0.733 0.170 4.267 disagree a little 2.009 1.498 2.792 agree a little 2.978 2.495 1.832 agree moderately 4.458 3.992 0.665 agree strongly 5.947 5.523 1.286 analyzing the results shown in table 3, it can be observed that, for the considered alternatives of 3 and 6 months, the distance value is the smallest for linguistic term "disagree moderately" (0.733 i.e. 0.170); after that term the closest term is "disagree strongly". for alternative of 12 months its closest linguistic term is "agree moderately" (0.665), and then term "agree strongly". distance values between aggregated fuzzy decisions for the assessment cycle length in the first and the second efdm iterations are shown in table 4. fuzzy delphi approach to defining a cycle for assessing the performance of military drivers 75 table 4. distance value between aggregated fuzzy decisions for the assessment cycle length (lukovac, 2016) aggregation 3 months 6 months 12 months iteration i (2.04,2.34,3.61) (1.95,3.01,4.45) (6.18,7.45,7.82) iteration ii (1.93,2.24,3.58) (1.95,2.93,4.36) (6.28,7.54,7.85) distance 0.090 0.136 0.076 since the distance between the aggregated fuzzy estimates of the experts in the first and second iteration of the efdm for selecting the cycle length is less than 0.2, the first condition for accepting a decision is satisfied (according to the proposed efdm algorithm). defuzzification of the fuzzy preference of the experts was carried out by a graded mean integration. representation according to the expression:  1 1 14 / 6defuzzy a l m r   (11) defuzzification of the fuzzy preference of the experts from table 4 carried out by an ibm spss statistics 22.0, is shown in table 5. table 5. indicators of homogeneity of the efdm decision for the assessment cycle length (lukovac, 2016) statistical indicators 3 months 6 months 12 months mean 0.1205 0.1485 0.3691 standard deviation 0.1332 0.0727 0.1444 variance 108% 48% 38% statistical analysis in table 5 indicated the inhomogeneity of the fuzzy preference of the experts, so the efdm process had to be continued with a new (third) iteration. in the third iteration there was no deviation from the fuzzy preference of the experts from the second iteration; therefore, it began to determine the cause of inhomogeneity. in order to determine the cause of inhomogeneity it calculated the distance between the linguistic terms chosen by the experts and a linguistic term that is equivalent to aggregating fuzzy decisions. these results are shown in table 6. table 6 shows that the individual fuzzy preferences of the expert 4, 7, 14 and 17 are far away from the linguistic terms that are equivalent to aggregated fuzzy decisions. in other words, the preferences of these experts are in contrast to the group's preference, which is the cause of inhomogeneity. by eliminating the preference of these four experts, separated aggregated fuzzy decisions were obtained (table 7). lukovac & popović/decis. mak. appl. manag. eng. 1 (1) (2018) 67-81 76 table 6. values of individual distance by experts for the assessment cycle length (lukovac, 2016) experts distances values 3 months (1.5,3,4.5) 6 months (1.5,3,4.5) 12 months (5.5,7,8.5) 1. 1.658 1.658 1.658 2. 1.658 0.000 1.658 3. 0.000 1.658 1.658 4. 5.545 0.000 5.545 5. 1.658 0.000 1.658 6. 1.658 0.000 1.658 7. 1.658 5.545 2.500 8. 1.658 1.658 1.658 9. 1.658 1.658 1.658 10. 1.658 0.000 1.658 11. 1.658 0.000 1.658 12. 1.658 0.000 1.658 13. 1.658 0.000 1.658 14. 4.000 0.000 5.545 15. 1.658 0.000 1.658 16. 1.658 0.000 1.658 17. 5.545 1.500 5.545 18. 1.658 0.000 1.658 19. 1.658 0.000 1.658 20. 1.658 0.000 1.658 table 7. separated aggregated fuzzy decisions (lukovac, 2016) experts 3 months 6 months 12 months e  1. (1,1,2.5) (1,1,2.5) (7.5,9,9) 0.0677 2. (1,1,2.5) (1.5,3,4.5) (7.5,9,9) 0.0571 3. (1.5,3,4.5) (1,1,2.5) (7.5,9,9) 0.0577 5. (1,1,2.5) (1.5,3,4.5) (7.5,9,9) 0.0571 6. (1,1,2.5) (1.5,3,4.5) (7.5,9,9) 0.0570 8. (1,1,2.5) (1,1,2.5) (7.5,9,9) 0.0704 9. (1,1,2.5) (1,1,2.5) (7.5,9,9) 0.0730 10. (1,1,2.5) (1.5,3,4.5) (7.5,9,9) 0.0580 11. (1,1,2.5) (1.5,3,4.5) (7.5,9,9) 0.0625 12. (1,1,2.5) (1.5,3,4.5) (7.5,9,9) 0.0637 13. (1,1,2.5) (1.5,3,4.5) (7.5,9,9) 0.0699 15. (1,1,2.5) (1.5,3,4.5) (7.5,9,9) 0.0640 16. (1,1,2.5) (1.5,3,4.5) (7.5,9,9) 0.0607 18. (1,1,2.5) (1.5,3,4.5) (7.5,9,9) 0.0597 19. (1,1,2.5) (1.5,3,4.5) (7.5,9,9) 0.0634 20. (1,1,2.5) (1.5,3,4.5) (7.5,9,9) 0.0580 aggregation (1.03,1.12,2.62) (1.37,2.46,3.96) (7.5,9,9) 1 fuzzy delphi approach to defining a cycle for assessing the performance of military drivers 77 distances between aggregated fuzzy numbers (table 7) and triangular fuzzy numbers in which the linguistic terms which experts had chosen are modeled, are shown in table 8. table 8. distances values of separated aggregated fuzzy decisions (lukovac, 2016) linguistic terms 3 months 6 months 12 months disagree strongly 0.096 1.213 7.036 disagree moderately 1.563 0.446 5.545 disagree a little 2.989 1.913 4.062 agree a little 3.970 2.909 3.082 agree moderately 5.454 4.407 1.658 agree strongly 6.948 5.937 0.000 based on the results of values of distances (table 8), alternative with cycle length of 3 months is equivalent to linguistic term "disagree strongly" (0.096). alternative with cycle length of 6 months is equivalent to linguistic term "disagree moderately" (0.446). linguistic term "agree strongly" (0.000) is equivalent to an alternative with cycle length of 12 months. distances values of separated aggregated fuzzy decisions in the last two iterations of efdm is 0 (zero) because the experts, in the last (third) iteration of efdm, did not change their fuzzy orientations with respect to the previous ones, and the first condition for accepting a decision is satisfied. the results of statistical analysis of separated defuzzification of fuzzy expert's estimation from table 8, using ibm spss statistics 22.0, are shown in table 9. based on the results (table 11.), alternative a1 has the smallest distance value for linguistic term "disagree strongly" (0.000). alternatives a2 is equivalent to linguistic term "agree strongly", based on the least distance value (0.568). values of distances between aggregated fuzzy numbers iteration i and iteration ii to the time at the start and the end of a cycle for assessing are shown in table 12. table 9. statistical analysis of the homogeneity of the fuzzy preference for the assessment cycle length (lukovac, 2016) statistical indicators 3 months 6 months 12 months mean 0.0844 0.1581 0.5469 standard deviation 0.0245 0.0455 0.0463 variance 28% 28% 8% statistical analysis in table 9 indicates the homogeneity of the fuzzy preference of the experts, which complies with the second condition of the stability of the efdm decision and it can be concluded that the performance of a military driver should be evaluated at 12 months. 4.2. defining to the start/end of the cycle for assessing the efdm algorithm was used to define the start/end of the military performance assessment cycle. an alternative a1 represented an estimation model in which the start and the end of the assessment period relate to the start of employment, while the model of assessment by which all employees are assessed at the same time, i.e., at the end of the calendar year, was an alternative a2. an acceptable consensus was reached in the second iteration of efdm, table 10. lukovac & popović/decis. mak. appl. manag. eng. 1 (1) (2018) 67-81 78 table 10. experts’ fuzzy preference to the start/end of the assessment cycle (lukovac, 2016; lukovac & popović, 2017) experts а1 а2  e 1. (1,1,2.5) (7.5,9,9) 0.0540 2. (1,1,2.5) (7.5,9,9) 0.0456 3. (1,1,2.5) (7.5,9,9) 0.0461 4. (1,1,2.5) (7.5,9,9) 0.0476 5. (1,1,2.5) (5.5,7,8.5) 0.0456 6. (1,1,2.5) (7.5,9,9) 0.0455 7. (1,1,2.5) (7.5,9,9) 0.0457 8. (1,1,2.5) (5.5,7,8.5) 0.0561 9. (1,1,2.5) (7.5,9,9) 0.0582 10. (1,1,2.5) (5.5,7,8.5) 0.0463 11. (1,1,2.5) (5.5,7,8.5) 0.0498 12. (1,1,2.5) (5.5,7,8.5) 0.0508 13. (1,1,2.5) (7.5,9,9) 0.0557 14. (1,1,2.5) (7.5,9,9) 0.0547 15. (1,1,2.5) (7.5,9,9) 0.0511 16. (1,1,2.5) (7.5,9,9) 0.0485 17. (1,1,2.5) (7.5,9,9) 0.0543 18. (1,1,2.5) (5.5,7,8.5) 0.0476 19. (1,1,2.5) (7.5,9,9) 0.0505 20. (1,1,2.5) (5.5,7,8.5) 0.0463 aggregation (1,1,2.5) (6.8,8.3,8.8) 1 the distance values between aggregated fuzzy numbers (table 10) and linguistic terms are shown in table 11. table 11. values of distances to the start/end of the assessment cycle (lukovac, 2010; lukovac & popović, 2017) linguistic terms а1 а2 disagree strongly 0.000 6.516 disagree moderately 1.658 5.008 disagree a little 3.082 3.517 agree a little 4.062 2.529 agree moderately 5.545 1.090 agree strongly 7.036 0.568 based on the results (table 11), alternative a1 has the smallest distance value for linguistic term "disagree strongly" (0.000). alternative a2 is equivalent to linguistic term "agree strongly", based on the least distance value (0.568). values of distances between aggregated fuzzy numbers iteration i and iteration ii to the time at the start and the end of the assessment cycle are shown in table 12. fuzzy delphi approach to defining a cycle for assessing the performance of military drivers 79 table 12. values of distances between aggregated fuzzy numbers to the start/end of the assessment cycle (lukovac, 2016; lukovac & popović, 2017) aggregation a1 a2 iteration i (1.1,1.2,2.7) (6.7,8.2,8.8) iteration ii (1,1,2.5) (6.8,8.3,8.8) distance 0.166 0.080 since the distance values are less than 0.2, the first condition for accepting a decision is satisfied. the mean standard deviation and the variance were performed using statistical software ibm spss 22.0 and are shown in table 13. table 13. the efdm statistical indicators for defining the start/end of the cycle (lukovac, 2016; lukovac & popović, 2017) statistical indicators a1 a2 mean 0.0625 0.4075 standard deviation 0.0052 0.0596 variance 8% 14% statistical analysis in table 13 indicates the homogeneity of the fuzzy preference of the decision-makers, which complies with the second condition of the stability of the efdm decision. 5. conclusion an important component of the performance assessment system for military drivers, which can cause errors in the system for assessing their performance, is a decision concerning the defining the period of their assessment: the duration of one cycle and the determination of the start and the end of one assessment cycle (lukovac, 2010; lukovac et al., 2012, lukovac et al., 2014). the results of the conducted efdm have confirmed that this technique is a suitable tool for correctly defining the cycle for assessing the performance of military drivers and it can be concluded that the performance of a military driver should be evaluated at the same time at the end of a calendar year. the presented efdm enables faster, more complete, more flexible and more realistic modeling of the decision-making process compared to the classic delphi model. developed efdm contributes to the greater stability of the final decision, taking into account the importance of the decision-makers and the homogeneity of their individual fuzzy preferences. also, the proposed efdm is of a general character, and as such, it is applicable to solving similar problems in different areas. in order to upgrade the presented efdm, the direction of further research should focus on linking this technique with one of the multi-criteria decision-making methods under uncertainty conditions, the results of which would be the starting point for the implementation of the presented efdm. lukovac & popović/decis. mak. appl. manag. eng. 1 (1) (2018) 67-81 80 references bogićević milikić, b. (2008). human resource management, faculty of economics belgrade. chang, p-l., hsu, c.w.,& chang, p.c. (2011). fuzzy delphi method for evaluation hydrogen production technologies. international journal of hydrogen energy, 36 (21). chen, m.f., & tzeng, g.h. (2004). combining grey relation and topsis concepts for selecting and expatriate host country. mathematical and computer modeling, 40 (13), 1473-1490. cheng, c.h., & lin, y. (2002) evaluating the best main battle tank using fuzzy decision theory with linguistic criteria evaluation. european journal of operational research, 142. dubois, d., & prade, h. (1980). fuzzy sets and systems: theory and applications, london, uk: academic press, inc (london) ltd. grout, d. (2008). performance evaluation and improvement: questions and answers, manager's handbook, asee, novi sad. ishikawa, a., amagasa, m., shinga, t., tomizawa, g., tatsuta, r., & mieno, h. 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(2010). a model for evaluating the quality of military drivers, magister thesis, faculty of transport and traffic engineering, belgrade. lukovac, v. (2016). a model for evaluating the quality of military drivers, dissertation, university of defense in belgrade, military academy, belgrade. lukovac, v., & popović, m. (2017). fuzzy delphi approach to defining the start and the end of a cycle for assessing the performance of military drivers, the 1st international conference on management, engineering and environment, icmnee 2017, belgrade, september 28-29, 198-209. noe a. r., hollenbeck r. j., gerhart b., & wright m. p. (2006). human resource management, third edition, mate d.o.o, zagreb. http://bg.ac.rs/en/members/faculties/tte.php fuzzy delphi approach to defining a cycle for assessing the performance of military drivers 81 saaty, t.l. (1980). the analytic hierarchy process, mcgraw-hill, new york. tadić, d., pravdić, p., arsovski. z., arsovski, s., & aleksić, a. (2013). ranking and managing business goals of manufacturing organizations by balanced scorecard under uncertainties.technics technologies education management, 8 (3), 13-28. vujić, d. (2008). human resource management and quality, faculty of philosophy, belgrade. wu, k.y. (2011). applying the fuzzy delphi method to analyze the evaluation indexes for service quality after railway re-opening-using the old mountain line railway as an example, proceedings of the 15th wseas international conference on systems, world scientific and engineering academy and society (wseas) stevens point, wisconsin, usa. young, s. j., & jamieson, l. m. (2001). delivery methodology of the delphi: a comparison of two approaches. journal of park and recreation administration, 19 (1), 42-58. plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 1, number 1, 2018, pp. 143-164 issn: 2560-6018 doi: https://doi.org/10.31181/dmame1801143b * corresponding author. e-mail addresses: dbozanic@yahoo.com (d. božanić), tesic.dusko@yahoo.com (d. tešić), jovicamilicevic74@gmail.com (j. milićević). a hybrid fuzzy ahp-mabac model: application in the serbian army – the selection of the location for deep wading as a technique of crossing the river by tanks darko božanić1*, duško tešić1, jovica milićević1 1 university of defense in belgrade, military academy, belgrade, serbia received: 5 january 2018; accepted: 19 february 2018; published: 15 march 2018. original scientific paper abstract: in this paper is presented a hybrid model based on the fuzzified analytical hierarchical process (ahp) method and the fuzzified multiattributive border approximation area comparison (mabac) method. the fahp method is used for defining the weight coefficients of the criteria, while the fmabac method is performed for the ranking of the alternatives. the fuzzification of the ahp method is carried out by defining a variable confidence interval for the values from the saaty’s scale, which is derived from the comparison in pairs and the degree of certainty of the decisionmakers in the comparison they make. the application of the hybrid model is shown on the example of the ranking of the locations for deep wading as a technique of crossing the river by the serbian army tank units. through the paper are elaborated the criteria which condition such choice; also, the application of the method in a particular situation is demonstrated. key words: fuzzy ahp (fahp), fuzzy mabac (fmabac), location for river crossing, deep wading, tank. 1. introduction a modern approach to decision-making increasingly implies the application of several methods with the tendency to exploit positive, i.e. to isolate/reject negative characteristics that different decision-making methods possess. this creates various hybrid models, which differ from case to case. the specificity of the case that is to be solved, and not rarely, the knowledge of the author, influence the choice of the methods which will form a hybrid model. the results of a large number of studies point to the fact that hybrid models provide significantly better results, compared to the application of classic problem solving methods. mailto:dbozanic@yahoo.com mailto:tesic.dusko@%20yahoo.com mailto:jovicamilicevic74@gmail.com božanić et al./decis. mak. appl. manag. eng. 1 (1) (2018) 143-164 144 in this paper a hybrid model composed of two segments is created. firstly, the fuzzification of the saaty’s scale (the ahp method) is carried out in order to obtain weight coefficients of the criteria. the main advantage of this fuzzification consists in its treating the uncertainties that may arise as a result of the uncertainty of the decision-makers about the comparisons they make. secondly, the fuzzification of the mabac method is performed in order to present the values of the alternatives by the criteria in the most realistic form. accordingly, the paper is sectioned in four parts. the first gives the description of the problem to be solved; the second presents the methods determining the hybrid model; in the third, the criteria are considered and their weight coefficients are calculated (using the fahp method), while the fourth shows the application of the fmabac method to a specific example. the main objective of the paper is to improve the decision-making processes in the serbian army, more precisely, to improve the processes of selecting locations for overcoming water barriers, by tanks with a deep wading technique. the process itself can be improved at two levels. firstly, by defining the criteria that the choice to be made is based on, and secondly, by defining the methodology according to which this choice is implemented. 2. problem description the military personnel who command and manage units are liable to come across, in their work, many situations where making significant decisions is needed, especially during combat operations. in these situations, wrong decisions can result in losses of human lives and material resources. therefore, in the military organization, special attention is given to the decision-making process because a human being is in the center of every decision, and, moreover, all the people are not expected to react in the same way in the situations in which they may find themselves (pamučar et al., 2011a). for this reason, the application of a multi-criteria decision-making is an inevitable tool in supporting a decision-making process. in this paper, several methods are applied, fahp and fmabac, to improve and facilitate a decision-making process when selecting a location at the water barrier for deep wading by tanks. crossing water barriers by tanks can be realized in a number of ways: by a wading, by a deep wading, by floating on the water (if a tank possesses amphibious characteristics) and by underwater driving (driving manual for tanks and armored vehicles, 1971). the way of overcoming the obstacle shall depend on the situation and the characteristics of technical resources. for deep wading which is discussed in this paper, special preparation of tanks, stuff and crossing points need to be carried out (slavkovic et al., 2012; gordic et al., 2013). the phrase ‘location for deep wading as a technique of crossing the river by tanks’ implies the location for crossing a water barrier (rivers, canals, lakes and the like) at the maximum water depth of up to 1.80 m and the flow rate of up to 1.5 m/sec, considering that the bottom of the river is suitable (the military lexicon, 1981; tank m-84, description, handling, basic and technical maintenance, 1988). at the river having the width of up to 200 m, the location of crossing is at least 25 m wide, and if the river is over 200 m wide, the width is 40-50 m (tank m-84, description, handling, basic and technical maintenance, 1988). the entrance and exit ramps are set at the crossing point and the control service is formed (tank m-84, description, handling, basic and technical maintenance, 1988). this is organized at special locations which must meet certain conditions. a hybrid fuzzy ahp-mabac model: application in the serbian army – the selection... 145 modern tanks are made in order to perform a deep wading operation when overcoming water barriers, with the basic aim of reducing negative impacts of water obstacles and creating conditions for an uninterrupted operation. the serbian army is equipped with the tank m-84 a crawler vehicle with powerful weapon, strong armor protection and great maneuverability and passability (the military lexicon, 1981). in the literature are outlined some criteria that the crossing points should meet, but without any precise definitions (slavkovic et al., 2013). such an approach makes it inevitable that decision-making about selecting the crossing point relies on the experience and knowledge of decision-makers and their associates in the specific situation. in other words, a situation could also occur in which the decision-makers would not have enough knowledge and experience for choosing such a crossing point. deciding on the selection of a place for organizing deep wading by tanks is performed by ranking the offered locations (alternatives) and selecting the best location for crossing over. 3. description of the methods applied in the following part of the paper triangular fuzzy numbers in the shortest terms are described. the basic principles of fuzzy logic and fuzzy numbers, as well as of the ahp method, are not explained because their description is provided in a large number of papers (saaty, 1980; teodorović & kikuchi, 1994; čupić & suknović, 2010; pamučar et al., 2011a; devetak & terzić, 2011). they also provide a detailed fuzzification of the saaty’s scale, with an overview of different approaches to fuzzification, and the fuzzification of the mabac method. the basic phases with the model steps are shown in fig. 1. p h a s e 1 : f a h p c ri te ri a i d e n ti fi ca ti o n a n d ca lc u la ti o n o f w e ig h t co e ff ic ie n ts o f cr it e ri a step 1: defining comparison matrices /comparison of criteria and the degree of certainty of experts in given claims/comparisons step 2: calculation of the fuzzified matrices of the criteria comparison step 3: calculation of weight fuzzy coefficients of criteria for each expert separately step 4: calculation of aggregated fuzzy weight coefficients of criteria step 5: calculation of final (non-fuzzy) weight coefficients of criteria p h a s e 2 : f m a b a c r a n k in g a lt e rn a ti v e s step 1: forming initial decision making matrix step 2: normalization of initial decision making matrix elements step 3: calculation of weighted matrix (v) elements step 4: determination of border approximate area matrix (g) step 5: calculation of the distance between alternatives and border approximate areas step 6: ranking alternatives figure 1. fahp-fmabac model božanić et al./decis. mak. appl. manag. eng. 1 (1) (2018) 143-164 146 3.1 triangular fuzzy numbers the fuzzification of the mabac method is performed by using triangular fuzzy numbers. a general form of the triangular fuzzy number is given in fig. 2. 0 t1 µ(x) x. t2 t3 1 figure 2. triangular fuzzy number triangular fuzzy numbers have the form 1 2 3t (t , t , t ) . value t1 represents the left distribution of the confidence interval of fuzzy number t, t2 is where the fuzzy number membership function has the maximum value equal to 1, and t3 represents the right distribution of the confidence interval of fuzzy number t (pamučar, 2011). the membership function of fuzzy number t is defined with the following expressions:   1 1 1 2 2 1 2t 3 2 3 3 2 3 0, x t x t , t x t t t x 1, x t t x , t x t t t 0, x t                     (1) for a final, operational role, most often it is necessary to perform defuzzification of the fuzzy number in order to obtain a crisp value. for the defuzzification of fuzzy numbers the following expressions are mostly used (seiford, 1996):     13 1 2 1 1defazzy s= t t t t 3 t        (2)   13 2 1defazzy s= t t 1 t 2        (3) where  represents optimism index 0,1. optimism index () is described as a belief of the decision-makers in a decision-making risk. the most commonly used values are 0, 0.5 and 1 which are used to represent a pessimistic, moderate and optimistic attitude towards risk (milićević, 2014). a hybrid fuzzy ahp-mabac model: application in the serbian army – the selection... 147 3.2. fuzzy ahp method fuzzification of the saaty's scale the analytical hierarchical process is a method often used in multi-criteria decision-making. this method was developed by thomas saaty. it is based on the development of a complex problem into the hierarchy scheme, with the aim at the top, criteria, sub-criteria and alternatives at the levels and sublevels of the hierarchy scheme (saaty, 1980), fig. 3. goal k1 k2 kn-1 kn a1 a2 ... an k1rk11 k12 knmkn1 kn2 level 1 level2 level p-1 level p ... ... ... figure 3. general hierarchical model in ahp to compare the criteria in pairs, the saaty's scale is commonly used (table 1), which is considered a standard for the ahp method. table 1. saaty's scale for a comparison in pairs standard values definition inverse values 1 the same importance 1 3 weak dominance 1/3 5 strong dominance 1/5 7 very strong dominance 1/7 9 absolute dominance 1/9 2, 4, 6, 8 intervalues 1/2, 1/4, 1/6, 1/8 so far, the saaty’s scale has been fuzzified in various ways. the simplest saaty's scale fuzzification is done by using fuzzy numbers with a predetermined confidence interval. in such fuzzification, the confidence intervals of fuzzy numbers are first established followed by the comparison in pairs. this approach to fuzzification can also be called a "sharp" fuzzification (božanić et al., 2015b). unlike "sharp" fuzzification, a "soft" fuzzification assumes that the confidence interval is not predetermined, but it is defined during the decision-making process based on additional parameters (božanić & pamučar, 2016). laarhoven and pedrycz carried out one of the earliest fuzzifications of the saaty's scale in 1983 (john et al., 2014). nowadays, many papers can be found handling this topic. in table 2 are given the examples of the most commonly defined left and right distribution of fuzzy numbers. božanić et al./decis. mak. appl. manag. eng. 1 (1) (2018) 143-164 148 table 2. the saaty's scale for comparison in pairs using fuzzy numbers with a predetermined confidence interval definition standard values fuzzification in kilic et al. (2014), john et al. (2014), li et al. (2009) fuzzification in kamvysi et al. (2014), meng et al. (2014) the same importance 1 (1,1,1) (1,1,1) weak dominance 3 (2,3,4) (2,3,4) strong dominance 5 (4,5,6) (4,5,6) very strong dominance 7 (6,7,8) (6,7,8) absolute dominance 9 (8,9,9) (8,9,10) intervalues 2, 4, 6, 8 (x-1, x, x+1) (x-1, x, x+1) it frequently occurs that, instead of the classic saaty’s scale, a scale based on the same principles as saaty’s is used, but with fewer comparison values (seven, six or five), as presented in martinovic & simon (2014), carnero (2014), bozbura et al. (2007), isaai et al. (2011), deng et al. (2014) junior et al. (2014). regardless of the number of comparisons, they all define the confidence interval in the same way [x-1, x, x + 1], where x presents a standard comparison value. in refs. srđević et al. (2008), gardašević-filipović & šaletić (2010), janacković et al. (2013), janjić et al. (2014), the saaty’s scale is modified so that the differences between t2 and t1, respectively t3 and t2, are not the same for every standard value (table 3), as has happened in most of the previous cases. value "" is obtained from interval 0.5 2  (srđević et al., 2008). table 3. saaty's scale for comparison in pairs with different confidence interval of a fuzzy number (srđević et al., 2008; gardašević-filipović & šaletić, 2010; janacković et al., 2013; janjić et al., 2014) definition standard values fuzzy number the same importance 1 (1, 1, 1) weak dominance 3 (3-, 3, 3+) strong dominance 5 (5-, 5, 5+) very strong dominance 7 (7-, 7, 7+) absolute dominance 9 (9-, 9, 9+) intervalues 2, 4,6,8 (x-, x, x+), x=2,4,6,8 the references cited where fuzzifications of the modified scales are performed represent only a minor part of this topic. the authors often use other types of functions, such as trapezoidal functions, gaussian functions, and in addition to classic, also interval fuzzy numbers (abdullah & najib, 2014; kahraman et al., 2014) etc. the number of values the scale contains for comparison in pairs coincides with the results of psychological experiments which showed that an individual could not simultaneously compare more than 7 ± 2 objects (miler, 1956). a hybrid fuzzy ahp-mabac model: application in the serbian army – the selection... 149 a different ("soft") approach is presented in papers by božanić et al. (2011, 2013), pamučar et al. (2011b, 2012, 2015). in these fuzzifications, the confidence interval of a fuzzy number remains open, or dependent on the person who performs the comparison in pairs. the new parameter the degree of uncertainty ʺ  ʺ is introduced into the calculation of the confidence interval, where under the value ʺ 0  ʺ is described the highest possible uncertainty, while the value ʺ 1  ʺ corresponds to the situation in which with the fullest certainty is known which linguistic expression corresponds to the given comparison. parameter  uses the values from the interval [0, 1]. the presentation of the fuzzified saaty’s scale used in the papers mentioned is given in table 4. table 4. the fuzzification of the saaty’s scale by applying the degree of uncertainty (božanić et al., 2011, 2013; pamučar et al., 2011b, 2012, 2015) definition standard values fuzzy number the same importance 1 (1, 1, 1) weak dominance 3   3 , 3, 2 3  strong dominance 5   5 , 5, 2 5  very strong dominance 7   7 , 7, 2 7  absolute dominance 9  9 ,9,9 intervalues 2, 4, 6, 8   x , x, 2 x ,  x 2, 4, 6, 8 this approach to fuzzification is particularly important in group decision-making since it can be expected that parameter  differs from one to another decisionmaker/analyst/expert (dm/a/e). consequently, the confidence interval of the fuzzy numbers varies from one to another decision-maker/analyst (božanić & pamučar, 2016). in order to determine the weight coefficients in this paper the fuzzification shown in božanić & pamučar (2016), božanić et al. (2015a, 2016b), pamučar et al. (2016) is used. in this fuzzification, several questions are raised, namely, whether dm/a/es are certain about the statements on comparison in pairs, how certain they are about such statements, and whether they are equally certain about every statement. the situation in which a dm/a/e is not sure how to evaluate the relationship between two elements is not rare because the classic saaty’s scale is subjective to some extent. its elements are not precisely explained, which in certain situations can cause some confusion; this, however, does not imply that the saaty's scale is bad, only that there is a wide range of options to upgrade and improve it (božanić & pamučar, 2016). this fuzzification proceeds from two elements: 1) the introduction of fuzzy numbers instead of classic numbers of the saaty’s scale, 2) the introduction of the degree of certainty of decision-makers/analysts in the statements they give during comparison in pairs  (božanić & pamučar, 2016). the basis of the fuzzification is in the assumption that dm/a/es can have a different degree of certainty ji in the accuracy of the comparison in pairs, so it is allowed for the degree of certainty to differ from one to another comparison pair. the value of the degree of certainty is within the interval 0,1. in cases where ji=0, dm/a/es are considered not to have božanić et al./decis. mak. appl. manag. eng. 1 (1) (2018) 143-164 150 any knowledge on the basis of which the comparison can be made, so in such relationship it is defined aji=1. the value of the degree of certainty =1 describes the absolute certainty of dm/a/e in the defined comparison. the overview of a fuzzy number with different degrees of certainty is given in fig. 4. as an example, it is taken a weak-dominance value from the saaty’s scale and the degrees of certainty =1, =0.8 and =0.4. 0 1 0.8  32.4 3.6 b) 0 1 0.4  31.2 5.4 c ) 0 1 1  3 a) figure 4. dependence of the fuzzy number on the degree of certainty (božanić & pamučar, 2016) there are different methods for defining the degree of certainty. this value can be defined in percentages or by using fuzzy linguistic descriptors. in the first case, experts would define the percentage of certainty in comparison in pairs (from 0 to 100%). in the second case, defining of the degree of certainty would be done using fuzzy linguistic descriptors. an example of the fuzzy linguistic descriptor for determining the degree of certainty which is used in this paper is given in fig. 5. 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 very small small medium high very high 0 figure 5. fuzzy linguistic descriptors for evaluating the degree of certainty of experts as can be seen in fig. 5, the degree of certainty of dm/a/es is defined with five linguistic variables: vs very small, s small, m medium, h high and vh very high. the degree of certainty  is used to define the confidence interval of fuzzy numbers when modifying the saaty’s scale, as shown in table 5. table 5. fuzzification of the saaty’s scale using the degree of certainty a hybrid fuzzy ahp-mabac model: application in the serbian army – the selection... 151 definition standard values fuzzy number inverse values fuzzy number the same importance 1 (1, 1, 1) (1, 1, 1) weak dominance 3   ji ji3 , 3, 2 3   ji ji1 (2 )3,1 / 3,1 3  strong dominance 5   ji ji5 , 5, 2 5   ji ji1 (2 )5 ,1 / 5,1 5  very strong dominance 7   ji ji7 , 7, 2 7   ji ji1 (2 )7 ,1 / 7,1 7  absolute dominance 9   ji ji9 , 9, 2 9   ji ji1 (2 )9 ,1 / 9,1 9  intervalues 2, 4, 6, 8   ji jix , x, 2 x  x 2, 4, 6, 8   ji ji1 2 x ,1/ x,1 x  x 2, 4, 6, 8 by defining different values of parameter ji , the left and right distribution of fuzzy numbers change from one comparison to another, according to the expression:           1 2 1 2 1 2 1 2 3 2 2 2 3 2 3 2 2 3 t t , t t , t , t 1 / 9, 9 t t , t , t t t , t 1 / 9, 9 t 2 t , t t , t , t 1 / 9, 9                    (4) t2 value represents the value of the linguistic expression from the classic saaty's scale, in which the fuzzy number has its maximum membership t2 = 1. fuzzy number     1 2 3t t , t , t x , x, 2 x    ,  x 1,9 is defined with expressions: 1 x , 1 x x t x 1, x 1             (5)  2t x, x 1,9   (6)    3t 2 x, x 1,9    (7) inverse fuzzy number      1 3 2 1t 1 / t ,1 / t ,1 / t 1 2 x ,1 / x,1 x      ,  x 1,9 is defined as:          3 1 2 x , 1 2 x 1 1/ t 1 2 x , x 1, 9 1, 1 2 x 1                (8) 21 / t 1 / x, 1 / x 1, 9   (9)  11/ t 1 x , 1/ x 1,9   (10) by using the previously defined scale, the decision makers/analysts enter the values of the criteria compared in pairs into a new, modified matrix: božanić et al./decis. mak. appl. manag. eng. 1 (1) (2018) 143-164 152 1 2 n 1 11 11 12 12 1n 1n 2 21 21 22 22 2n 2n n n1 n1 n 2 n 2 nn nn c c c c a ; a ; a ; a c a ; a ; a ; c a ; a ; a ;                       (11) where ji=ij. in the same way, the alternatives are compared in pairs. the standard steps of the ahp method are further applied. after all the calculations have been completed, the fuzzy values of the criteria functions are obtained by every alternative observed, where defuzzification is performed using expression (2) or (3). the scale shown can be applied in the classic ahp method, where the weight coefficients are first calculated, and then the evaluation of the criteria functions for every observed alternative is made. the scale is also suitable for evaluating the weight of criteria for later application of other methods (topsis, vikor, etc.). the defined scale is also suitable for the process of group decision-making, which has recently shown the tendency of being used more and more. experts’ incorporation significantly improves the quality of decisions made because knowledge and experience are gathered and integrated into one whole. the most commonly used approach in collecting data from experts is the delphi method (mučibabić, 2003). the scale defined in this paper in group decision-making is applied as well as the standard ahp method. 3.3. fuzzy мавас method the mabac method is developed by pamučar & ćirović (2015). the basic setting of the mabac method consists in defining the distance of the criteria function of every observed alternative from the border approximate area. the mabac method was modified with several papers. roy et al. (2017) extended the mabac method using rough numbers. xue et al. (2016) defined an interval-valued intuitionistic fuzzy mabac approach. yu et al. (2017) and roy et al. (2016) modified mabac approach with interval type-2 fuzzy numbers. peng and yang (2016) developed pythagorean fuzzy choquet integral based mabac method. the following text shows the procedure for implementing the fuzzificated mabac method (with triangular fuzzy number) in seven steps, i.e., its mathematical formulation. step 1. forming of the initial decision matrix ( x ). in the first step the evaluation of m alternatives by n criteria is performed. the alternatives are shown by vectors  i i1 i2 ina x , x ..., x , where xij is the value of the i alternative by j criterion (i = 1,2, ... m; j = 1,2, ..., n). 1 2 n 1 11 12 1n 2 11 22 2n m 1m 2m mn c c ... c a x x ... x a x x x x ... ... ... ... ... a x x ... x             (12) where m denotes the number of the alternatives, and n denotes total number of criteria. step 2. normalization of the initial matrix elements ( x ). a hybrid fuzzy ahp-mabac model: application in the serbian army – the selection... 153 1 2 n 1 11 12 1n 2 11 22 2n m 1m 2m mn c c ... c a t t ... t a t t t n ... ... ... ... ... a t t ... t              (13) the elements of the normalized matrix ( n ) are obtained by using the expressions: for benefit-type criteria ij i ij i i x x t x x       (14) for cost-type criteria ij i ij i i x x t x x       (15) where ijx , ix  and ix  represent the elements of the initial decision matrix ( x ), whereby ix  and ix  are defined as follows: i 1r 2r mrx max(x , x ,..., x )   and represent the maximum values of the right distribution of fuzzy numbers of the observed criterion by alternatives. i 1l 2l mlx min(x , x ,..., x )   and represents minimum values of the left distribution of fuzzy numbers of the observed criterion by alternatives step 3. calculation of the weighted matrix ( v ) elements 11 12 1n 21 22 2n m1 m2 mn v v ... v v v ... v v ... ... ... ... v v ... v             (16) the elements of the weighted matrix ( v ) are calculated on the basis of the expression (17) ij i ij iv w t w  (17) where ijt represent the elements of the normalized matrix ( n ), iw represents the weighted coefficients of the criterion. step 4. determination of the approximate border area matrix ( g ). the border approximate area for every criterion is determined by the expression (18): 1/ m m i ij j 1 g v           (18) where ijv represent the elements of the weighted matrix ( v ), m represents total number of alternatives. božanić et al./decis. mak. appl. manag. eng. 1 (1) (2018) 143-164 154 after calculating the value of ig by criteria, a matrix of border approximate areas g is developed in the form n x 1 (n represents total number of criteria by which the selection of the offered alternatives is performed).   1 2 n 1 2 n c c ... c g g g ... g (19) step 5. calculation of the matrix elements of alternatives distance from the border approximate area ( q ) 11 12 1n 21 22 2n m1 m2 mn q q ... q q q q q ... ... ... ... q q ... q             (20) the distance of the alternatives from the border approximate area ( ijq ) is defined as the difference between the weighted matrix elements ( v ) and the values of the border approximate areas ( g ). q v g  (21) the values of alternative ia may belong to the border approximate area ( g ), to the upper approximate area ( g  ), or to the lower approximate area ( g  ), i.e.,  ia g g g    . the upper approximate area ( g  ) represents the area in which the ideal alternative is found ( a  ), while the lower approximate area ( g  ) represents the area where the anti-ideal alternative is found ( a  ), as presented in the fig. 6. g  g  a  a  1 a 3 a 4 a 2 a 5 a 6 a 7 a g upper approximation area lower approksimation area bordere approksimation area 0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 c ri te ri o n f u n c ti o n s figure 6. display of upper ( g  ), lower ( g  ) and border ( g ) approximate area (pamučar & ćirović, 2015) the membership of alternative ia to the approximate area ( g , g  or g  ) is determined by the expression a hybrid fuzzy ahp-mabac model: application in the serbian army – the selection... 155 ij i ij ij g if q 0 a g if q 0 g if q 0           (22) for alternative ia to be chosen as the best from the set, it is necessary for it to belong, by as many as possible criteria, to the upper approximate area ( g  ). the higher the value iq g   indicates that the alternative is closer to the ideal alternative, while the lower the value iq g   indicates that the alternative is closer to the anti-ideal alternative. step 6. ranking of alternatives. the calculation of the values of the criteria functions by alternatives is obtained as the sum of the distance of alternatives from the border approximate areas ( iq ). by summing up the matrix q elements per rows, the final values of the criteria function of alternatives are obtained n i ij j 1 s q , j 1, 2,..., n, i 1, 2,..., m     (23) where n represents the number of criteria, and m is the number of alternatives. step 7 final ranking of alternatives. by defuzzification of the obtained values is , the final rank of alternatives is obtained. the defuzzification can be performed with the expressions (2) or (3). 4. criteria description and definition of weight coefficients the criteria for selecting the most convenient locations for organizing deep wading as a river crossing technique by tanks are defined by an analysis of the available literature. the most detailed description of the conditions that the tanks’ crossing point should meet is provided in (pifat, 1980). applying a detailed analysis, seven key criteria are distinguished, namely: c1 water barrier width represents the distance from one river bank to the other, measured by the surface of the water. when crossing the water barrier, the width affects the speed of crossing over, i.e., the time the unit would be exposed to enemy fire; c2 composition of the bottom-stream bed implies the composition of the river bottom in the geological sense. the type and composition of the bottom has a major or even decisive influence on the possibility of deep wading performance on rivers and canals. a hard, rocky but flat bottom, or the bottom with stable, solid gravel allows the crossing without any prior works, while a soft, muddy or uneven bottom requires greater workloads to reinforce bottom of the river, or it can completely disable crossing over a barrier with this technique; c3 influence of the enemy means that the crossing location should provide the least impact of the enemy on crossing over water barrier. c4 water flow speed refers to water flowing expressed in the unit of time. the speed of the water flow affects drift sideways of the vehicles that cross over the water barrier; božanić et al./decis. mak. appl. manag. eng. 1 (1) (2018) 143-164 156 c5 characteristics of the river bank imply the existence, quality and condition of access roads, composition of the ground on the river bank, height of the bank, slope of the bank, forestation, artificial obstacles, etc. the extent of the work necessary to take on the arrangement of the bank depends on these characteristics; c6 water barrier depth is the distance measured from the water surface to the bottom of the barrier. the maximum water depth at which it is possible to perform river crossing by tanks with a deep wading technique is 1.8 m. c7 masking implies that the site where a deep wading, as a technique of crossing by tanks, will be organized must provide good concealment of the access to the bank and to the water barrier, as well as good masking conditions on the bank in situations where the crossing is stopped (due to the effects of the air force, etc.). the complexity of making a mask also plays an important role. criteria c1, c4 and c6 are numeric, while criteria c2, c3, c5 and c7 are linguistic. the values of the linguistic criteria are described with fuzzy linguistic descriptors, as presented in the fig. 7. 1 0.8 0.6 0.4 0.2 1 32 54 0 vb b m g e figure 7. graphic display of fuzzy linguistic descriptors (božanić et al., 2016a) every criterion can be described with five values: vb – very bad, b – bad, m – medium, g – good and e – excellent. after the key criteria have been defined, the conditions are created for their comparison in pairs. the comparison in pairs is conducted by 11 experts using the saaty's scale. also, the experts define the degree of certainty in the comparisons they make using fuzzy linguistic descriptors shown in fig. 5. the comparison in pairs and the degree of certainty form the initial decision matrix. the first expert defines the following elements of the initial decision matrix: a hybrid fuzzy ahp-mabac model: application in the serbian army – the selection... 157 63 5 71 2 4 1 2 3 4 5 6 7 cc c cc c c c 1; 2; h 2; vh 2; m 3; h 3; h 4;s c 1 / 2; h 1; 1 / 3; m 4; h 2; vh 3; vh 2; h c 1 / 2; vh 3; m 1; 2; vs 1; h 2;s 2; h a c 1 / 2; m 1 / 4; h 1 / 2; vs 1; 1 / 2; vh 2; h 1 / 3; m c 1 / 3; h 1 / 2; vh 1; h 2; vh 1; 3; vh 1 / 3; h c 1 / 3; h 1 / 3; vh 1 / 2;s 1 / 2; h 1 / 3; vh 1; 1 / 2 c        ; vh 1 / 4;s 1 / 2; h 1 / 2; h 3; m 3; h 2; vh 1;                       when fuzzy linguistic descriptors are defuzzified, the following matrix is obtained: 3 5 6 71 2 4 1 2 3 4 5 6 7 c c c cc c c c 1; 2; 0.75 2; 0.933 2; 0.5 3; 0.75 3; 0.75 4; 0.25 c 1 / 2; 0.75 1; 1 / 3; 0.5 4; 0.75 2; 0.933 3; 0.933 2; 0.75 c 1 / 2; 0.933 3; 0.5 1; 2; 0.067 1; 0.75 2; 0.25 2; 0.75 a` c 1 / 2; 0.5 1 / 4; 0.75 1 / 2; 0.067 1; 1 / 2; 0.933 2; 0.75 1 / c c c      3; 0.5 1 / 3; 0.75 1 / 2; 0.75 1; 0.75 2; 0.933 1; 3; 0.933 1 / 3; 0.75 1 / 3; 0.75 1 / 3; 0.933 1 / 2; 0.25 1 / 2; 0.75 1 / 3; 0.933 1; 1 / 2; 0.933 1 / 4; 0.25 1 / 2; 0.75 1 / 2; 0.75 3; 0.5 3; 0.75 2; 0.933 1;                       the next step is the calculation of a fuzzified initial decision-making matrix using the expressions given in table 5: 6 71 2 1 2 3 4 5 6 7 c cc c ... c (1,1,1) (1.5, 2, 2.5) (2.25,3,3.75) (1, 4, 7)... c (0.4, 0.5, 0.67) (1,1,1) (2.8,3,3.2) (1.5, 2, 2.5)... c (0.47, 0.5, 0.54) (1,5,3, 4.5) (1, 2,3.5) (1.5, 2, 2.5)... a` c (0.33, 0.5,1) (0.2, 0.25, 0.33) (1.5,... c c c  2, 2.5) (0.22, 0.33, 0.67) (0.27, 0.33, 0.44) (0.47, 0.5, 0.54) (2.8,3,3.2) (0.27, 0.33, 0.44)... (0.27, 0.33, 0.44) (0.31, 0.33, 0.36) (1,1,1) (0.47, 0.5, 0.54)... (0.14, 0.25,1) (0.4, 0.5, 0.67) (1.87, 2, 2.13) (1,1,1)...                       applying standard steps of the ahp method, the values of the weight coefficients of criteria for the first expert are obtained, and shown in table 6. table 6. weight coefficients of criteria for the first expert criterion fuzzy weight coefficient of criteria weight coefficient of criteria (fahp) weight coefficient of criteria (classic ahp) c1 (0.145,0.271,0.487) 0.269 0.271 c2 (0.108,0.169,0.280) 0.166 0.169 c3 (0.094,0.181,0.344) 0.184 0.181 c4 (0.043,0.077,0.153) 0.081 0.077 c5 (0.081,0.112,0.157) 0.104 0.112 c6 (0.036,0.058,0.099) 0.057 0.058 c7 (0.077,0.133,0.258) 0.139 0.133 božanić et al./decis. mak. appl. manag. eng. 1 (1) (2018) 143-164 158 after the aggregation of the weight coefficients of criteria of all experts, the final weight coefficients of the criteria are obtained, which is shown in table 7. table 7. final weight coefficients of the criteria criterion weight coefficient of criteria (fahp) weight coefficient of criteria (classic ahp) c1 0.243 0.262 c2 0.159 0.169 c3 0.182 0.194 c4 0.097 0.079 c5 0.125 0.109 c6 0.071 0.055 c7 0.123 0.132 the analysis of the results from tables 6 and 7 points to the existence of differences between the application of the standard and the fuzzified saaty’s scale. small differences in values indicate that, when applying the fuzzified scale, the value assigned for comparison in pairs is still a key element. the degree of certainty makes only certain corrections of these comparisons. 5. ranging alternatives applying the fuzzy mabac method the application of the fuzzy mabac method is presented by illustrated alternatives. further in the paper are ranked six alternatives. in the first step, the initial decision matrix ( x ) is defined. 3 5 6 71 2 4 1 2 3 4 5 6 c c c cc c c a (115,120,126) (0.9,1.1,1.3) (1.3,1.5,1.7) (1,1, 2)g e m a (134,140,147) (0.7, 0.9,1.2) (1.1,1.3,1.5) (3, 4, 5)vbe m a (105,110,115) (1.05,1.2,1.4) (1.4,1.6,1.8) (2, 3, 4)ge e x a (120,125,130) (0.8,1vbm a a  ,1.2) (1.3,1.5,1.7) (1,1, 2)g (153,160,170) (0.6, 0.7, 0.8) (1,1.2,1.4) (3, 4, 5)g e e (114,118,126) (1.1,1.15,1.25) (1.3,1.5,1.7) (2, 3, 4)gm m                    then the initial matrix is quantified: 3 5 6 71 2 4 1 2 3 4 5 6 c c c cc c c a (115,120,126) (3, 4, 5) (4, 5, 5) (0.9,1.1,1.3) (2, 3, 4) (1.3,1.5,1.7) (1,1, 2) a (134,140,147) (4, 5, 5) (2, 3, 4) (0.7, 0.9,1.2) (1,1, 2) (1.1,1.3,1.5) (3, 4, 5) a (105,110,115) (4, 5, 5) (3, 4, 5) (1.05,1.2, x a a a  1.4) (4, 5, 5) (1.4,1.6,1.8) (2, 3, 4) (120,125,130) (2, 3, 4) (1,1, 2) (0.8,1,1.2) (3, 4, 5) (1.3,1.5,1.7) (1,1, 2) (153,160,170) (3, 4, 5) (4, 5, 5) (0.6, 0.7, 0.8) (4, 5, 5) (1,1.2,1.4) (3, 4, 5) (114,118,126) (2, 3, 4) (2, 3, 4) (1.1,1.15,1.25) (3, 4, 5) (1.3,1.5,1.7) (2, 3, 4)                    in the second step, the normalization of the initial decision matrix elements is performed. for the normalization the expressions (14) and (15) are used. the results obtained are shown in the normalized matrix ( n ). a hybrid fuzzy ahp-mabac model: application in the serbian army – the selection... 159 2 6 71 1 2 3 4 5 6 c ... c cc a (0.68, 0.77, 0.85) (0.33, 0.67,1) (0.13, 0.38, 0.63) (0, 0, 0.25)... a (0.35, 0.46, 0.55) (0.67,1,1) (0.38, 0.63, 0.88) (0.5, 0.75,1)... a (0.85, 0.92,1) (0.67,1,1) (0, 0.25, 0.5) (0.25, 0.5, 0.75)... n a (0.62, a a  0.69, 0.77) (0, 0.33, 0.67) (0.13, 0.38, 0.63) (0, 0, 0.25)... (0, 0.15, 0.26) (0.33, 0.67,1) (0.5, 0.75,1) (0.5, 0.75,1)... (0.68, 0.8, 0.86) (0, 0.33, 0.67) (0.13, 0.38, 0.63) (0.25, 0.5, 0.75)...                    in the third step, the calculation of the weighted matrix ( v ) is performed by using the expression (17). 2 6 71 1 2 3 4 5 6 c c cc ... a (0.41, 0.43, 0.45) (0.21, 0.27, 0.32) (0.08, 0.1, 0.12) (0.12, 0.12, 0.15)... a (0.33, 0.36, 0.38) (0.27, 0.32, 0.32) (0.10, 0.12, 0.13) (0.18, 0.22, 0.25)... a (0.45, 0.47, 0.49) (0.27, 0.32, 0.32) (0.07,... v a a a  0.09, 0.11) (0.15, 0.18, 0.22) (0.39, 0.41, 0.43) (0.16, 0.21, 0.27) (0.08, 0.1, 0.12) (0.12, 0.12, 0.15)... (0.24, 0.28, 0.31) (0.21, 0.27, 0.32) (0.11, 0.12, 0.14) (0.18, 0.22, 0.25)... (0.41, 0.44, 0.45) (0.16, 0.21, 0.27) (0.08,... 0.1, 0.12) (0.15, 0.18, 0.22)                    in the fourth step, the matrix of the approximate border areas ( g ) is obtained by using the expression (18).   2 6 71 c ... c cc g (0.36, 0.39, 0.41) (0.21, 0.26, 0.30) (0.08, 0.10, 0.12) (0.15, 0.17, 0.20)... the fifth step is the calculation of the matrix elements distance of the alternatives from the border approximate area ( q ). the calculation is made by using the expression (21). 71 2 1 2 3 4 5 6 cc c ... a (0, 0.04, 0.08) ( 0.09, 0, 0.11) ( 0.08, 0.05, 0)... a ( 0.08, 0.04, 0.01) ( 0.03, 0.06, 0.11) ( 0.02, 0.05, 0.09)... a (0.04, 0.08, 0.12) ( 0.03, 0.06, 0.11) ( 0.05, 0.01, 0.06)... q a ( 0.02, 0.02, 0.07) ( 0.14 a a             , 0.05, 0.06) ( 0.08, 0.05, 0)... ( 0.17, 0.11, 0.06) ( 0.09, 0, 0.11) ( 0.02, 0.05, 0.09)... (0, 0.05, 0.09) ( 0.14, 0.05, 0.06) ( 0.05, 0.01, 0.06)...                            by summing up the elements of the matrix q per row, the final values of the criteria functions of alternatives are obtained, as presented in the table 8. božanić et al./decis. mak. appl. manag. eng. 1 (1) (2018) 143-164 160 table 8. fuzzy values of criteria functions of alternatives alternative is a1 (-0.37,-0.10,0.25) a2 (-0.29,0.05,0.34) a3 (-0.20,0.14,0.40) a4 (-0.25,0.06,0.36) a5 (-0.32,-0.02,0.29) а6 (-0.31,0.04,0.36) by defuzzification of the obtained values of the criteria functions of alternatives is obtained the rank of alternatives. in table 9 are shown the results of alternatives ranking after the defuzzification, as well as the application of classic mabac method and the results obtained by the survey of experts in the field of overcoming water barriers. table 9. final values of the criteria function of alternatives alternativе experts classic мавас method defuzzification using the expression (2) si rank si rank a1 6 -0.098 6 -0.071 6 a2 3 0.110 2 0.032 3 a3 1 0.183 1 0.113 1 a4 2 0.031 3 0.053 2 a5 5 -0.024 5 -0.014 5 а6 4 0.027 4 0.029 4 all the methods ranked the a3 alternative at the first place, respectively, the alternatives a5 and a6 are found at the last two positions. significant differences are noted in the ranking of alternatives a2 and a4, which change their rank depending on the method applied or its modification. furthermore, by analyzing the outcome results it is noticed that the differences in the obtained values of the criteria functions of alternatives are less when the fuzzificated model is applied. 6. conclusion the paper presents a successful application of the hybrid model fuzzy ahp fuzzy mabac in the selection of the locations for river crossing by tanks with a deep wading technique . the comparison with the results obtained by an experts’ survey, using the classic and the fuzzified mabac method, leads to the conclusion that the fuzzified mabac method can completely replace expert judgment. on the other hand, the application of the fuzzified ahp method in defining weight coefficients of the criteria takes into account uncertainty during comparison in pairs, which in relation to the classic ahp method, corrects the weight coefficients of the criteria. the significance of the model is also reflected in the fact that the criteria for selecting locations for crossing over water barriers by tanks with a deep a hybrid fuzzy ahp-mabac model: application in the serbian army – the selection... 161 wading technique are defined. also, these criteria are described in basic terms, which provides for a further possibility for their detailed elaboration. the greatest contribution that the model presented in the paper makes lies in the fact that experience is, in a decision-making process, translated into mathematics. this makes the consideration of the problem more comprehensive and at the same time less dependent on the experience of decision-makers. references ***(1971). driving manual for tanks and armored vehicles (only in serbian: priručnik za vožnju tenkova i oklopnih transportera). belgrade: federal secretariat for national defense, general staff of the yugoslav people's army (jna), administration of armored units/ssno, gš jna, uprava oj. ***(1981). the military lexicon (only in serbian: vojni leksikon). belgrade: military publishing institute/vojnoizdavački zavod. ***(1984). tank m-84, description, handling, basic and technical maintenance, book 1 (only in serbian: tenk m-84, opis, rukovanje, osnovno i tehničko održavanje, knjiga 1.) belgrade: federal secretariat for national defense/ssno, technical book. abdullah, l., & najib, l. 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(2017). an interval type-2 fuzzy likelihood-based mabac approach and its aplication in selecting hotels on a tourism website. international journal of fuzzy systems, 19(1), 47-61.s plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 2, issue 1, 2019, pp. 66-85 issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame1901066n * corresponding author. e-mail addresses: mohammad.noureddine@udd.edu (m. noureddine) and milos.ristic@mod.gov.rs (m. ristic) route planning for hazardous materials transportation: multi-criteria decision-making approach mohammad noureddine 1* and milos ristic 2 1 faculty of science and technology, university of djelfa, djelfa, algeria 2 university of defence in belgrade, department of logistics, belgrade, serbia received: 29 august 2018; accepted: 03 february 2019; available online: 28 february 2019. original scientific paper abstract: transport of hazardous material (thm) represents a complex area involving a large number of participants. the imperative of thm is minimization of risks in the entire process of transportation from the aspect of everyone involved in it, which is not an easy task at all. to achieve this, it is necessary in its early phase to carry out adequate evaluation and selection of an optimal transport route. in this paper, optimal route criteria for thm are selected using a new approach in the field of multi-criteria decision-making. weight coefficients of these criteria were determined by applying the full consistency method (fucom). evaluation and selection of hazardous material routes is determined by applying the topsis (technique for order of preference by similarity to ideal solution) and the mabac (multi-attributive border approximation area comparison) methods. in order to establish the stability of models and validate the results obtained from the fucom-topsismabac model, a sensitivity analysis (of ten different scenarios) was performed. the sensitivity analysis implied changes of the weight coefficients criteria with respect to their original value. the proposed route model was tested on the real example of the transport eurodiesel in serbia. key words: hazardous materials routing, fucom, topsis, mabac, multicriteria decision-making. 1. introduction the rapid development of industry, based on the development of techniques and technology increased the usage of substances, materials, elements, which are hazardous to human health and safety as well as environment safety. modern industry, route planning for hazardous materials transportation: multi-criteria decision-making…. 67 especially the one of chemical character, also contributes to the faster development of new materials whose usage can cause huge destruction and damage. the issues of storage and transport, shipping (loading), discharging (unloading) or reloading, or the issues of the activities related to process of transport and storage of these substances are very sensitive. especially during these activities the risk of unwanted consequences is significantly high and each accident can turn into catastrophe. from the aspect of transport, realization of each transport process of hazardous material implies a certain risk of an unwanted(accident) event, caused by scattering (effusion, shedding, etc.) of burden, with the consequences related to the nature of the hazardous material that is being transported. by mentioning all of these risks, the transport safety is a very important and responsible task. in the case of an accident, the consequences can be very large and can cause damage to people and their environment, namely, death, diseases of human beings, plant and animals, pollution of environment, destruction of natural and national resources, damage of industrial buildings, traffic communications and their respective facilities. potential danger, on one hand, and the need for transport of hazardous materials, on the other hand, both lead to the necessity of setting specific requirements related to risk reduction and attempts to increase the safety in the transport of hazardous materials. with the growth of ecological consciousness, there is also a growing demand for reduction of transport risks, but also in handling, in general, hazardous materials. because of these reasons, numerous countries, institutions and organizations have different regulations and other regulatory measures in order to manage the safety of these transport processes. to keep the hazardous materials transport process safe, it is necessary to manage the risk. risk management represents a very complex process, consisting of several steps and elements. certainly one of the most important steps in this process is the selection of routes for the movement of vehicles that carry hazardous load (material). the problem of routing in the transport of hazardous materials, as a problem of multicriteria factors, became popular in the 90s of the last century. approaches to solving this problem are numerous and depend on many factors, such as the methods used to identify risks, the criteria that are considered, the ways in which these criteria are valued, etc. this is necessary because the requirements for transporting hazardous materials are very complicated; this implies a very difficult task for the managers assigned to properly evaluate potential hazardous materials transport routes (thm) that will enable efficient and safe transportation. in order to minimize thm risks, efficient management strategy has become a key risk minimization component (pamucar et al., 2016). when considering the efficiency of the entire thm it is impossible not to notice that it largely depends on adequate route selection because this process represents one of the most important factors that directly affect the overall risk and safety of transport. only by properly evaluating and selecting routes this logistical subsystem can efficiently perform tasks related to end-user supply. in this paper, the choice of optimum route for thm was performed using linear programming and multi-criteria evaluation of the thm route. the weight coefficients of the criteria are determined by linear programming. evaluation and selection of route for thm was performed using topsis and mabac methods. these multi-criteria techniques were chosen because the topsis method is one of the most commonly used multi-criteria techniques (song et al, 2014), while the mabac method is one of the newest methods in this area that has found a wide and efficient application in many areas (yu et al, 2016; xue et al, noureddine and ristic/decis. mak. appl. manag. eng. 2 (1) (2019) 66-85 68 2016; peng and dai, 2016; peng and yang, 2016; roy et al, 2016; gigovic et al, 2017; pamucar et al., 2018). this paper has more goals. the first objective is to improve the methodology for route optimization for the thm. the second goal of this paper is the popularization of the operational research, especially linear programming and multi-criteria techniques, through their application for decision-making in a real business and business system. the third goal of the paper is a proposal of a model that comprehensively addresses the problem of hazmat rutting with respect to both cost aspects and different aspects of risk, as well as a number of uncertainties in the decision-making process. by proposing the new model of the lp-topsis-mabac hybrid model, it is trying to show that academic research models can be more practical and useful for actually planning the routes. the paper is structurally divided into six sections. in the next section, an overview of the literature with an accent on the criteria for selecting the optimal route for thm is given. in addition, the methods used to optimize the thm process have been presented. in the third section, the fucom-topsis-mabac hybrid model algorithms are presented: (1) fucom for defining weight coefficient criteria, (2) topsis model for thm route evaluation and (3) mabac model for thm route evaluation. the fourth section is the application of the above mentioned techniques of operational research to a real problem. in the fifth section, a sensitivity analysis was performed defining different sets with different criteria values based on which the stability of the proposed model was verified. section six is conclusion with the guidelines for future research. 2. literature review multi-criteria decision-making is widely applied in all areas, and when it comes to transport, more precisely the sub-system of transport carried out by thm is often used to select transport routes (pamucar et al., 2016). for the purposes of this paper, the author's works have been analyzed to deal with the problem of choosing the optimal route for the thm and thus the choice of criterion of choice. among them, it was noted that the sources the authors rely on are often similar, so most of the criteria are repeated in the works of different authors. consequently, in this paper are presented and analyzed the characteristic works, which are set out according to the methodology and criteria applied. wijerante et al., (1993) have developed a method for determining undetermined routes in the network when there are multiple, uncertain measures based on which route estimates are made and applied to a transport hazard example in the territory of new york state (united states). in order to evaluate the route options and choose optimal, they based their analysis on three criteria: time of transport, incidence of traffic accidents resulting from hazardous substances and operating costs. the issue of risk modeling in the transport of hazardous materials and the question of the importance of the way of evaluating this risk was addressed by erkut and verter (1998). they presented an overview of the models and methods most commonly used in theory and practice, and their empirical analysis was conducted on the american road network. they concluded that choosing the optimum route for thm depends on the way of risk assessment, i.e. they have shown the impact of different risk assessment models on selecting the optimal route for thm. to consider the thm impact on the environment was of great importance when choosing the route, as shown in monprapussornte al., (2009). the authors also pointed route planning for hazardous materials transportation: multi-criteria decision-making…. 69 out to the possibility of applying a decision support system, such as the multi criteria decision analysis (mcda) and the geographic information system (gis), which make easier the selection of routes while planning thm, while respecting the environmental criteria. in that study monprapussornte al., (2009) the environmental factor has been identified as one of the key factors in addition to those that are economically linked to safety and the ability to react in emergency situations. to establish a network of roads for thm, law and rocchi (2008) conducted research in canada. the goal of this study was to establish a network of thm routes in canada. the authors have analyzed and used current methodological approaches (mcdm, routing algorithms etc.) that used different route agencies when determining routes in some other cities and regions. law and rocchi (2008) have proposed criteria for route evaluation as well as methodology for choosing the optimal route for thm based on the mcdm approach. huang et al., (2004) and huang and fery (2005) dealt with the choice of the thm route in singapore as the third oil refinery in the world. given the increased number of trucks carrying hazardous goods in this city, the authors have pointed out the need to improve the tracking and safety of trucks driving on the city and suburban road network at thm. to select the optimal directions for thm authors proposed risk mapping and gis application in combination with genetic algorithms in this study. samuel (2007) presented a time study, which covers the time period from 1995 to 2007, in which he analyzed 1850 incidents in transport of flammable-liquid substances. focus studies include shipments of hazardous cargo from five us states (california, illinois, iowa, new jersey, and texas), which were selected due to their size and geographic location differences. the main objective of this study was to analyze the frequency of incidents during thm and as a result of the analysis, thirteen criteria for route selection were set out. the importance of safety when transporting hazardous materials was pointed out as well by dilgir, et al., (2005). they consider that thm that run on roads that pass through larger cities are not only a challenge for transporters, but also for city planners and services designed to respond to emergency situations. dilgir, et al., (2005) point out that road safety is a key criterion for efficient route selection for thm and suggest the use of mcdm techniques to solve this problem. sattayaprasert et al., (2008) have proposed multi-criteria models to form an efficient logistic network, with particular reference to the risk inherent in thm. using the analytic hierarchy process (ahp) sattayaprasert et al., (2008) have observed a case of study related to petrol logistics as one of the most frequently transported hazardous cargoes in thailand. the ahp structure of the criteria which they have established is based on the evaluation and opinions of the expert group and the local community. as most of the authors of the previously analyzed papers pointed to the consequences of cargo carrying hazardous cargo, oluwoye (2007) deals with the effects and risks of the environment if accidents occur during this type of transport. oluwoye(2007) states that if an economical and efficient risk management strategy is to be achieved, optimization must be carried out to minimize costs and impact on the environment. milovanovic (2012) also deals with the topic of selecting an adequate route from the aspect of risk management and provides an overview of the risk management process in hazardous materials transport, i.e. the phases of the risk management process, as well as a detailed description of each phase. in order to determine the level of risk milovanovic (2012) defines two types of parameters. the first group of parameters affects the probability of an incident while the other group of parameters affects the consequences of an incident. noureddine and ristic/decis. mak. appl. manag. eng. 2 (1) (2019) 66-85 70 li and leung (2011) also viewed a multi-objective optimization problem as the problem of selecting the transport route hazmat on the urban network. they proposed a compromise programming approach to modify the dijkstring's algorithm while for the attribution of weight coefficient they used the analytic hierarchy process, considering that they will minimize human subjectivity in decision-making. from the previous literature analysis, it can be said that the multi-criteria analysis is used as a tool for achieving the best possible trade – off among different objectives (li & leung, 2011). it should be borne in mind that the optimality of multi-objective solutions in the hazmat routing domain implies the so-called "pareto-optimality”. more about pareto concept can be seen in (das et al., 2012). in the application of the multi-criteria analysis method for selecting the hazmat transport route, hybrid models are often proposed in which these methods combine with the classical shortest path algorithms or geographic information systems gis (ahp method and dijkstra's algorithm) (verma, 2011; li & leung, 2011), ahp method and gis (long &liew, 2003; huang, 2006; sattayaprasert et al., 2008). the application of other multi-criteria analysis methods, such as promethee and topsis, in vehicle routing problems, can be seen in (bandyopadhyay& bhattacharya, 2013; jia et al., 2013; talarico, 2015). the literature review shows that in the literature there are known crisp multicriteria algorithms based on the most common application of gis models with ahp, topsis and promethee algorithms. considering that the topsis method falls into the methods found to be the widest application in solving multi-criteria models (song et al., 2014; stevic et al., 2016; zhang et al., 2017) it is justified to further develop the topsis method algorithm through the application of other approaches. in order to achieve greater objectivity in decision-making over the last several years, numerous multi-criteria models have been developed among which the swords and mabac methods (pamucar&cirovic, 2015).the authors agreed to apply the mabac method due to many advantages it recommends: (1) the mathematical framework of the method remains the same regardless of the number of alternatives and criteria; (2) the possibility of applying in the case of a number of alternatives and criteria; (3) a clearly defined ranking of alternatives is expressed in numerical value, which allows a better understanding of the results; (4) it is applicable to the qualitative and quantitative criterion type and (5) it provides stable solutions regardless of the change in the scale of qualitative criteria and the change in the formulation of the quantitative criteria (pamucar&cirovic, 2015). the original model based on the linear programming (lp) was suggested for determining weight criteria. the main advantages of the lp models are as follows: (1) weight coefficients obtained with the lp model represent fair values since the input data is obtained with a small number comparing to the real criteria; (2) the mathematical framework of the model remains the same regardless of the number of criteria; (3) the lp model provides stable solutions regardless of the type of scale used to represent the expert preferences. taking into account all the advantages of the lp, topsis and mabac models in the decision-making process, the authors have decided in this paper to present the hybrid lp-topsis-mabac model for selecting the optimal route for thm. 3. multi-criteria model for choosing the optimal route for thm the model for optimal route selection for thm is realized through two phases. in the first phase of the hybrid fucom-topsis-mabac model using the linear programming model, the weighting coefficients of the evaluation criterion are route planning for hazardous materials transportation: multi-criteria decision-making…. 71 calculated. in the second phase of the fucom -topsis-mabac model, a thm route evaluation is performed using topsis and mabac models. 3.1. determining weight coefficient criteria fucom model fucom (pamucar et al., 2018) is a new mcdm method for determination of criteria weights. in the following section, fucom algorithm is shown, which implies the following steps: step 1. determining the set of evaluation criteria. this starts from the assumption that the process of decision-making involves m experts. in this step, experts consider the set of evaluation criteria and select the final set of criteria  1 2, ,... nc c c c , where n represents the total number of criteria. step 2. the second step is to rank the criteria according to their significance. the criterion we expect to have the highest weight coefficient gets the first rank, while the least important criterion gets the last rank. the remaining criteria get the rankings between the most important and the least important criterion. the ranks of the criteria are presented by the experts in descending order in accordance with the expected values of weight coefficients ( ) ( ) ( ) (1) ( 2 ) ( ) ... e e e j j j k c c c   , where k represents the rank of the observed criterion, whereas e represents the mark of expert 1 e m  . step 3. the third step is to compare the ranked criteria together and compare the significance of the evaluation criterion. comparative significance of the criterion of evaluation is an advantage that has a higher ranking criterion in relation to the lower rank criterion. the final values of the weight coefficients should meet the following two conditions: (1) the relation of the weight coefficients should be the same as the comparative importance between observed criteria ( ( ) / ( 1) e k k   ), which is defined in step 2, meeting the condition: ( ) ( ) / ( 1)( ) 1 e ek k ke k w w     (1) (2) apart from the condition (1), the final values of the weight coefficients should meet the condition of mathematical transitivity, so that ( ) ( ) ( ) / ( 1) ( 1) / ( 2 ) / ( 2 ) e e e k k k k k k          . taking into consideration the fact that ( ) ( ) / ( 1) ( ) 1 e e k k k e k w w     and ( ) ( ) 1 ( 1) / ( 2) ( ) 2 e e k k k e k w w       , ( ) ( ) ( ) 1 ( ) ( ) ( ) 1 2 2 e e e k k k e e e k k k w w w w w w       is obtained. in that manner, the second condition that the final values of the weight coefficients of the evaluation criteria should meet is: ( ) ( ) ( ) / ( 1) ( 1) / ( 2)( ) 2 e e ek k k k ke k w w         (2) noureddine and ristic/decis. mak. appl. manag. eng. 2 (1) (2019) 66-85 72 step 4. solving the optimization model (3) the final values of the weighting coefficients of the evaluation criteria are calculated  1 2, ,..., t n w w w .the minimum deviation from the maximum consistency (dfc) of the comparison (  ) is only met if transitivity is fully complied with, when the conditions are met, where ( ) ( ) / ( 1)( ) 1 0 e ek k ke k w w      and ( ) ( ) ( ) / ( 1) ( 1) / ( 2)( ) 2 0 e e ek k k k ke k w w          . then, the condition of the maximum consistency is met, respectively, for the obtained values of the weight coefficients, the deviation from the maximum consistency being 0  . in order to meet the mentioned conditions, it is necessary to determine the values of the weight coefficients of evaluation criteria  ( ) ( ) ( )1 2, ,..., t e e e n w w w meeting the condition, where ( ) ( ) / ( 1)( ) 1 e ek k ke k w w       and ( ) ( ) ( ) / ( 1) ( 1) / ( 2)( ) 2 e e ek k k k ke k w w           , while minimizing the values, thus meeting the condition of the maximum consistency. based on the mentioned assumptions, the final model for determining the values of the weight coefficients of the evaluation criteria can be defined as follows: ( ) ( ) / ( 1)( ) 1 ( ) ( ) ( ) / ( 1) ( 1) / ( 2 )( ) 2 ( ) 1 ( ) min . . , , 1, 0, e ek k ke k e e ek k k k ke k n e j j e j s t w j w w j w w j w j                          (3) by solving model (3), the final values of the evaluation criteria  ( ) ( ) ( )1 2, ,..., t e e e n w w w and the dfc ( ( )e  ) for every expert are obtained. 3.2. topsis method the topsis method implies ranking alternatives with respect to the multiple criteria based on distance comparison with an ideal solution and a negative ideal solution (chang et al., 2010). the ideal solution minimizes the cost-type criteria and maximizes the criteria of the benefit type, while the negative ideal solution works the other way around. a simple example is an effort to make (identify) decisions in business decision-making maximizing profit and minimizing the risk. the optimal alternative is the one that is geometrically closest to the ideal solution, that is, the farthest from the ideal negative solution (srdjevic et al., 2002). the ranking of alternatives is based on a "relative connection with an ideal solution", thus avoiding the situation that the alternative simultaneously has the same resemblance to the ideal and the negative ideal solution. the ideal solution is defined by using the best value route planning for hazardous materials transportation: multi-criteria decision-making…. 73 rating alternatives for each individual criterion. a negative ideal solution represents the worst value rating alternative. topsis method consists of 6 steps that are shown in the following section. step 1. normalization of decision matrix values. for the majority of multi-criteria decision-making, the first step is the normalization of the elements of the decision matrix to obtain a matrix in which all elements are non–dimensional in size. the topsis method applies vector normalization that is represented by expressions (4) and (5): 2 1 ij ij n ij i r x r    ,for “benefit“ criteria type, (4) 2 1 1 ij ij m ij i r x r     , fort ”cost” criteria type (5) after normalization, we get a matrix x in which all the elements are standardized and are in the interval [0, 1]. 1 11 12 1 2 21 22 2 3 1 2 ... ... ... ... ... ... m m n n nm a x x x a x x x x a x x x             (6) step 2. multiplication of normalized matrix values x with the weight coefficient criteria ; 1, 2,..., ij ij j v x w j m   (7) using the relation (7) we get elements of weight normalized matrix ( ) ij v v , where everyone is ij v a product of normalized alternate performance and an appropriate weighting coefficient of the criterion. 1 11 12 1 1 1 11 2 12 1 2 21 22 2 2 1 21 2 22 2 3 1 2 3 1 1 2 2 ... ... ... ... ... ... ... ... ... ... ... ... ... ... m m m m m m n n nm n n m nm a v v v a w x w x w x a v v v a w x w x w x v a v v v a w x w x w x                                 (8) step 3. determining ideal solutions. ideal solution a* and negative ideal solution a  are determined by the relation:    1 2(max | ), (min , ), 1,.., , ,...,ij ij ma v j g v j g i n v v v          (9)    ' 1 2(min | ), (max , ), 1,.., , ,...,ij ij ma v j g v j g i n v v v          (10) where :  1, 2,..., |g j m  , for ”benefit”criteria type   ' 1, 2,..., |g j m  ,for ”cost” criteria type step 4. determining the distance of alternatives to ideal solutions. in this step, using the following links: noureddine and ristic/decis. mak. appl. manag. eng. 2 (1) (2019) 66-85 74 2 1 ( ) , 1,..., m i ij j j s v v i n       (11) 2 1 ( ) , 1,..., m i ij j j s v v i n       (12) calculated n dimensional euclidean distances of all the alternatives of an ideal and ideal negative solution. step 5. determining the relative proximity of an alternative to an ideal solution. for each alternative, a relative interval is determined , 1,..., i i i i s q i n s s        (13) where 0 1 i q    . alternative i a is closer to ideal solution if i q  is close to 1, or, which is the same, if i s  is closer to 0. step 6. ranking alternatives. alternatives are ranked by decreasing values i q  . the best alternative is the one whose value * i q is the highest and vice versa. 3.3. mabac method the basic function of the mabac method is to define the distance of the criterion function of each observed alternative from the boundary approximating area. in the following section, the procedure for conducting the mabac method consists of five steps. step 1. normalization of element from initial matrix ( x ): 1 2 1 11 12 1 2 21 22 2 1 2 ... ... ... ... ... ... ... ... n n n m m m mn c c c a t t t a t t t n a t t t             (14) elements of normalized matrix ( n ) are determined using the expression: (a) for the "benefit" type criteria (a higher value criterion is more desirable)) ij i ij i i x x t x x       (15) (b) for “cost “ type criteria (a lower value criterion is more desirable) ij i ij i i x x t x x       (16) where ij x , i x  and i x  represent the elements of initial decision matrix ( x ), whereby i x  and i x  defined as:  1 2max , ,...,i mx x x x   and represents the maximum value of the observed criterion by alternatives and route planning for hazardous materials transportation: multi-criteria decision-making…. 75  1 2min , ,...,i mx x x x   and represents the minimum values of the observed criterion by alternatives. step 2. calculation of weighted matrix elements ( v ).calculation of weighted matrix elements ( v ) are calculated based on expression (17): ij i ij i v w t w   (17) where ij t represent elements of a normalized matrix ( n ), i w represents the weighting criterion coefficients. using expression (17) we get weighted matrix v : 11 12 1 1 11 1 2 12 2 1 21 22 2 1 21 1 2 22 2 2 1 2 1 1 1 2 2 2 ... ... ... ... ... ... ... ... ... ... ... ... ... n n n n n n n n m m mn m m n mn n v v v w t w w t w w t w v v v w t w w t w w t w v v v v w t w w t w w t w                                          where n represents the total number of criteria, m represents the total number of alternatives. step 3. determination of matrix of border approximate domains (g). the boundary approximate area (gao) is determined according to expression (18): 1/ 1 m m i ij j g v          (18) where ij v represent elements of a heavy matrix ( v ), m represents the total number of alternatives. after calculating value i g according to the criteria, a matrix of border approximating areas is formed g (19) formats 1n x ( n represents the total number of criteria by which a choice of alternatives is offered):   1 2 1 2 ... ... n n c c c g g g g (19) step 4. calculation of the matrix elements of the distance of alternatives from boundary approximating area ( q ): 11 12 1 21 22 2 1 2 ... ... ... ... ... ... n n m m mn q q q q q q q q q q             (20) alternative distance from border approximate area ( ij q ) is defined as the difference between the elements of a heavy matrix ( v ) and values of border approximate areas ( g ): 11 1 12 2 1 11 12 1 21 1 22 2 2 21 22 2 1 1 2 2 1 2 ... ... ... ... ... ... ... ... ... ... ... ... ... n n n n n n m m mn n m m mn v g v g v g q q q v g v g v g q q q q v g v g v g v g q q q                                   (21) noureddine and ristic/decis. mak. appl. manag. eng. 2 (1) (2019) 66-85 76 where i g represents a border approximate area for criterion i c , ij v represents the elements of a heavy matrix ( v ), n represents the number of criteria, m represents the number of alternatives. alternative i a may belong to borderline approximate area ( g ), upper approximate area ( g  ) or lower approximate area ( g  ), regarding  ia g g g      . upper approximate area ( g  ) represents the area in which the ideal alternative is ( a  ), lower approximate area ( g  ) represents the area where the anti-ideal alternative is( a  ). step 5. ranking alternatives. calculation of criterion values by alternatives (22) is obtained as a sum of limiting alternatives to border-approximating areas ( i q ). summing up matrix elements q we get the final values of the criterion functional alternatives in a row: 1 , 1, 2,..., , 1, 2,..., n i ij j s q j n i m     (22) 4. selection of thm routes using the fucom-topsis-mabac model the fucom-topsis-mabac model was tested on the example of choosing the optimal route for thm (eurodiesel) in the petroleum industry of serbia. thm is performed on the village leskovac šabac. during thm, the vehicle moves from kragujevac and has a zero drive to the warehouse in the village of leskovac. transport can be carried out over four routes that represent alternatives: a1 route: kragujevac – knic v. leskovac – knic – prelјina – ljig – mionica – valјevo – kocelјeva – vladimirci – šabac a2 route: kragujevac – knic v. leskovac – knic – prelјina – ljig – lajkovac – ub – šabac a3 route: kragujevac – knic – v. leskovac – knic – kragujevac – topola – mladenovac – mali pozarevac – beograd – dobanovci – šimanovci – šabac a4 -route: kragujevac – knic – v. leskovac – knic – kragujevac – batocina – beograd dobanovci – šimanovci – šabac. the analysis of the literature presented in the second section of the work contains five criteria for the evaluation of the thm route: number of rail crossings on the route (c1), existence of traffic jam on the route (c2), number of traffic accidents in the last ten years (c3), reaction of rescue services (emergency aid, fire brigade and police) (c4) and travel line length (c5). the existence of rail crossing of the road route (c1) carrying hazardous goods presents a great danger from the point of view of traffic accidents (incident situations) due to the fact that there is a large stopping distance to braking of locomotives. trails with a greater number of rail crossings have a greater degree of risk than those with fewer or no crossings at all. traffic jams (c2) directly affect the probability of incident situations. increasing the number of vehicles that use a certain part of the route directly affects an increase in probability of incident situations. since traffic accidents with the involvement of individual vehicles are common, traffic jams appear to be an important factor in determining not only the frequency of traffic accidents but also their weights. the route planning for hazardous materials transportation: multi-criteria decision-making…. 77 following relationships were used to estimate the probability of occurrence of incident situations depending on traffic jam:  the ratio of traffic speed and traffic capacity is less than 0.5,  traffic flow velocity and traffic capacity between 0,5 and 0,7 and  traffic speed ratio and road capacity greater than 0.7. in order to estimate the probability of occurrence of an incident situation, depending on the number of traffic accidents (c3) on a particular section, the following scale was used within the route:  1 to 2 traffic accidents per kilometer per year,  from 2 to 7 traffic accidents per kilometer per year and  from 7 to 15 traffic accidents per kilometer per year. emergency response service (c4) represents the time for which city services (fire services, emergency services and police) react in the case of an accident. it is very important to determine the number of properly trained and well-prepared fire brigades and ambulance services as soon as possible from the base to any point along the route. this determines the effects of these services on softening the consequences of an accident involving the participation of vehicles transporting hazardous materials. on the scale from 1 to 9, values are defined that indicate the response time. number one represents a small response time, and the number nine means quite a long response time on a particular route. the minimum distance (c5) between the start and end point of the thm on the route is determined on the basis of available satellite images of the traffic routes. only first and second line roads were considered. this research study involved six road safety experts with a minimum of 10 years of experience in managing the transport of hazardous materials. in the first phase of the fucom-topsis-mabac model, the weighting coefficients of the evaluation criteria are calculated using linear programming. 4.1. fucom: defining the weight of the criteria experiment surveys obtained the ranking criteria and significance of the criteria that was further used in the lp model. table 1 shows the results of surveyed experts. table 1. ranking of criteria and determination of significance experts rank/significance e1 c2 c4 c1 c3 c5 1 2 2.8 3 3.5 e2 c3 c2 c5 c4 c1 1 1.3 1.7 1.5 3 e3 c4 c5 c1 c2 c3 1 1.34 1 1.6 1.45 e4 c5 c4 c2 c1 c3 1 1.28 1.35 1.62 1.07 e5 c5 c4 c2 c3 c1 1 1.2 1.3 1.5 1.6 e6 c5 c4 c1 c2 c3 1 1.2 1.4 1.2 1.3 in the next step, based on the model (3), the weight coefficients of the criteria are estimated. since the research involved six experts, the fucom model, which was noureddine and ristic/decis. mak. appl. manag. eng. 2 (1) (2019) 66-85 78 solved using the lingo 17.0 software, was formed from any expert. fucom models are shown in the next section. 2 4 1 3 52 4 1 3 2 5 4 3 2 4 34 2 5 1 3 1 5 4 1 5 5 1 1 min 2 min 2 , 2.8 , 3 , 1.3 , 1.7 , 1.5 , 3.5 , 2 , 5.6 , 3 , 2.21 , 2.55 10.5 , 1, 0, j j j expert expert w w w w ww w w w w w w w w w ww w w w w w w w w w w w j                                                          5 1 5 1 5 54 1 5 1 2 4 52 4 3 1 2 1 3 5 1 , 4.5 , 1, 0, 3 min 4 min 1.34 , 1 , 1.6 , 1.28 , 1.45 , 1.34 , 1.6 , 2.32 , 1, 0, j j j j j j w w w w j expert expert w ww w w w w w w ww w w w w w w w w j                                                                4 2 2 1 51 4 3 2 1 2 3 5 1 5 4 2 4 2 3 3 5 4 1 2 3 2 1 1.35 , 1.62 , 1.07 , 1.73 , 2.19 , 1.73 , 1, 0, 5 min 1.2 , 1.3 , 1.5 , 1.6 , 1.56 , 1.95 , 2.4 , j j j w w w ww w w w w w w w w j expert w w w w w w w w w w w w w w w                                                         5 4 1 4 1 2 52 4 3 1 2 1 3 5 5 1 1 6 min 1.2 , 1.4 , 1.2 , 1.3 , 1.68 , 1.68 , 1.56 , 1, 0, 1, 0, j j j j j j expert w w w w w w ww w w w w w w w j w w j                                                           by solving the presented linear programming models, the final values of the weight coefficients of each expert are defined, table 2. by evaluating the obtained values, the optimal values of the weight coefficients of the criteria were further determined to be used for the evaluation of the routes using topsis and mabac methods. table 2. calculation of weight coefficients of the criteria experts weight coefficient of criteria c1 c2 c3 c4 c5 e1 0.1017 0.5698 0.0339 0.2849 0.0097 e2 0.0382 0.2932 0.3811 0.1150 0.1724 e3 0.2275 0.1422 0.0981 0.3048 0.2275 e4 0.1171 0.1897 0.1094 0.2561 0.3278 e5 0.0843 0.2023 0.1349 0.2630 0.3156 e6 0.1800 0.1500 0.1154 0.2521 0.3025 average value 0.1248 0.2579 0.1455 0.2460 0.2259 route planning for hazardous materials transportation: multi-criteria decision-making…. 79 4.2. application of the topsis model the topsis method algorithm is applied to initial decision matrix d: 1 2 3 4 5 min min min min min 1 3 0, 65 8 8 232 2 2 0, 50 7 6 233 3 1 0, 45 5 5 250 4 1 0, 20 2 4 280 route f f f f f d                 step 1. normalized matrix (x) is obtained by normalizing the elements of initial decision matrix (d), expression (4) 1 2 3 4 5 min min min min min 0, 2254 0, 3205 0, 3287 0, 3263 0, 5351 0, 4836 0, 4773 0, 4126 0, 4947 0, 5331 0, 7418 0, 5296 0, 5804 0, 5789 0, 499 0, 7418 0,8322 0,8322 0, 6631 0, 4389 f f f f f x                 step 2. by multiplying the normalized matrix and weight coefficients of the criteria, expression (7), a heavier normalized matrix is constructed (8) 1 2 3 4 5 min min min min min 0, 0281 0, 0827 0, 0478 0, 0803 0,1209 0, 0604 0,1231 0, 06 0,1217 0,1204 0, 0926 0,1366 0, 0844 0,1424 0,1127 0, 0926 0, 204 0,1211 0,1631 0, 0991 f f f f f t                 step 3. using expressions (9) and (10) ideal and negative-ideal solutions are calculated: ideal solution: a* = { 0.0926, 0.204, 0.1211,0.1631, 0.12} and negative ideal solution: a‾= {0.0281, 0.0827, 0.0478, 0.0803, 0.0991}. step 4: using expressions (11) and (12) euclidean distance alternatives are calculated from ideal and negative-ideal solutions, table 3. table 3. distance from ideal and negative-ideal solutions alternative si* si‾ a1 0.0218 0.1764 a2 0.0707 0.1141 a3 0.1116 0.0799 a4 0.1764 0.0218 steps 5 and 6. using expression (13) the relative proximity of the alternatives to the ideal solution is calculated and we get the final rank of the alternative: a4> a3> a2> a1. 4.3. application of the mabac model the mabac method algorithm applies to the same initial decision matrix d, as well as the topsis model. noureddine and ristic/decis. mak. appl. manag. eng. 2 (1) (2019) 66-85 80 step1: since all functions of the min type are used to normalize the initial matrix of decision making, we use expression (12). by applying this relation we receive a normalized matrix n: 1 2 3 4 5 min min min min min 1 0 0 0 0 0 2 0, 5 0, 33 0,1667 0, 5 0, 9792 3 1 0, 44 0, 5 0, 75 0, 625 4 1 0, 95 1 1 1 route f f f f f n                 step 2: in step 2, applying expression (13), elements of a heavy normalized matrix are calculated: 1 2 3 4 5 min min min min min 1 0,1248 0, 2579 0,1455 0, 246 0, 2259 2 0,1872 0, 343 0,1698 0, 369 0, 4471 3 0, 2496 0, 3714 0, 2183 0, 4305 0, 3671 4 0, 2496 0, 5029 0, 291 0, 492 0, 4518 route f f f f f v                 step 3: in step 3, boundary approximate domain matrix (g) is calculated. boundary approximation area (gao) for each criterion is determined according to (18). 1 2 3 4 5 0,1954 0, 3585 0,199 0, 3724 0, 2597 c c c c c g        step 4: using expression (21) we calculate the elements of a matrix (20), which represents the distance of an alternative to gao. 1 2 3 4 5 min min min min min 0, 0706 0,1006 0, 0535 0,1264 0,1338 0, 0082 0, 0155 0, 0292 0, 0034 0, 0874 0, 0542 0, 0129 0, 0193 0, 0581 0, 0074 , 0542 0,1444 0, 092 0,1196 0, 0921 f f f f f q                                  step 5: calculation of the value of the criterion functions for each alternative is obtained as the sum of the distance of the alternatives from the boundary approximate fields. by summarizing the elements of the q matrix in rows, we obtain the final values of the criterion functions of the alternative and the final ranking alternative that reads: a4> a3> a2> a1. in table 4 a comparative analysis of the route ranges for thm obtained using topsis and mabac methods is given. table 4. route ranges using topsis and mabac methods route rank topsis mabac a1 4 4 a2 3 3 a3 2 2 a4 1 1 route planning for hazardous materials transportation: multi-criteria decision-making…. 81 4.4. sensitivity analysis of the solution since the results of multi-criteria decision-making depend on the value of the weight coefficient of the evaluation criteria, in the following section the analysis of the sensitivity of the results to the change in the weight of the criteria is presented. sometimes the ranking alternatives vary with very small changes in weight coefficients. therefore, the results of these multi-criteria decision-making methods follow the sensitivity analysis on these changes as a rule. the analysis of the sensitivity of the ranks of alternatives to changes in the weight coefficients of the criteria was carried out through ten scenarios given in table 5. table 5. scenarios of sensitivity analysis scenario weight criteria scenario weight criteria s1 wc1=1.25× wc11(old); wci=0.25× wci(old) s6 wc1=1.55× wc11(old); wci=0.55× wci(old) s2 wc2=1.25× wc11(old); wci=0.55× wci(old) s7 wc2=1.55× wc11(old); wci=0.55× wci(old) s3 wc3=1.25× wc11(old); wci=0.25× wci(old) s8 wc3=1.55× wc11(old); wci=0.55× wci(old) s4 wc4=1.25× wc11(old); wci=0.25× wci(old) s9 wc4=1.55× wc11(old); wci=0.55× wci(old) s5 wc5=1.25× wc11(old); wci=0.25× wci(old) s10 wc5=1.55× wc11(old); wci=0.55× wci(old) the scenarios of the sensitivity analysis are grouped into two phases. within each phase of the sensitivity analysis, the weight coefficients of the criteria were increased by 25% and 55%, respectively. in each of the ten scenarios, only one criterion is favored for which the weight coefficient is increased for the stated values. in the same scenario, with the remaining criteria, weight coefficients were reduced by 25% (s1s5) and 55% (s6-s10). changes in the ranking alternatives during the 10 scenarios in topsis and mabac methods are presented in figure1. 0 0.5 1 1.5 2 2.5 3 3.5 4 s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 a1 a2 a3 a4 a) topsis method b) mabac method 0 0.5 1 1.5 2 2.5 3 3.5 4 s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 a1 a2 a3 a4 figure 1. changes in the ranking alternatives in 10 scenarios the results show that assigning different weight to the criteria through the 10 scenarios shown does not lead to a significant change in the ranking of the alternative. by comparing the first-ranked alternatives (a4 and a3) in scenarios 1-10 with initial noureddine and ristic/decis. mak. appl. manag. eng. 2 (1) (2019) 66-85 82 rankings from topsis and mabac models, we note that the rank of first-ranked alternatives is confirmed. by analyzing the rankings through 10 scenarios, we also notice that the a4 alternative in all 10 scenarios has kept its ranking. based on this, we can conclude that there is a satisfactory closeness of ranks and that the proposed ranking is confirmed and credible. 5. conclusions the new fucom-topsis-mabac model for route evaluation for thm is presented here. verification of the fucom-topsis-mabac model was carried out on a real case from the practice in which the transport of eurodiesel was considered for the needs of the ministry of defense of the republic of serbia. one of the contributions of this paper is the new fucom-topsis-mabac model that provides for an objective aggregation of expert decisions. the second contribution of this paper is the development of the linear programming model for determining the weight coefficients of the evaluation criteria, which contributes to the improvement of the literature that considers the theoretical and practical application of multi-criteria techniques. the third contribution of this study is to improve the methodology of route evaluation for thm through a new approach to determining the weight coefficient of the criteria. using the hybrid fucom-topsis-mabac model, it is possible to solve the problems of multi-criteria decision-making in a simple way and make decisions that have a significant impact on increasing safety and reducing risk in thm. the analysis of the results shows that the ranks of the alternatives using the lp-topsis model are in complete correlation with the obtained ranks of the lp-mabac model. in selecting the most suitable route for thm, both methods (fucom-topsis and fucom-mabac) from the aspect of stability of the obtained results prove to be reliable. this was confirmed by analyzing the sensitivity of multi-criteria techniques, which was done through ten scenarios. further research related to this paper relates to the post analysis of the 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(2016). an interval type-2 fuzzy likelihood-based mabac approach and its application in selecting hotels on a tourism website. international journal of fuzzy systems, 19(1), 47-61. https://www.sciencedirect.com/science/journal/03861112 https://www.sciencedirect.com/science/journal/03861112 route planning for hazardous materials transportation: multi-criteria decision-making…. 85 zhang, y., zhang, y., li, y., liu, s., & yang, j. (2017). a study of rural logistics center location based on intuitionistic fuzzy topsis. mathematical problems in engineering, article id 2323057, https://doi.org/10.1155/2017/2323057. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 5, issue 1, 2022, pp. 113-134. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame030222100y * corresponding author. e-mail addresses: ashraf.labib@port.ac.uk (a. labib), reza@additive-design.com (m.r abdi), sara.hadleigh-dunn@port.ac.uk (s. sara hadleigh-dunn), morteza.yazdani@uam.es (m. yazdani) evidence-based models to support humanitarian operations and crisis management ashraf labib 1, m. reza abdi 2, sara hadleigh-dunn3 and morteza yazdani*4 1 university of portsmouth, pbs, operations & systems management, united kingdom 2 manufacturing systems analyst additive design ltd. broad gate, united kingdom 3 university of portsmouth, pbs, strategy, enterprise, and innovation, united kingdom 4 universidad autónoma de madrid, spain received: 17 october 2021; accepted: 21 january 2022; available online: 7 february 2022. original scientific paper abstract: the term humanitarian operation (ho) is a concept extracted from the need to perform supply chain operations in special, risky, and critical events. understanding and implementing operations under such conditions is a strategic responsibility. due to its importance, we design a framework for organizational learning from major incidents through root cause analysis the case studies contain a purely industrial disaster; bhopal and a mixed industrial-natural disaster; fukushima. an approach is proposed for organizational safety by incorporating techniques related to root cause analysis, by incorporating a hybrid of analytical tools in an innovative dynamic framework and applied to one case study. we also describe the benefits of using such hybrid of techniques. moreover, we employ the analytic hierarchy process, which is applied to the second case study. we incorporate models to analyse data related to the two major disasters. the case studies in two organizations are then compared with respect to their causes and effects along with the models adopted to support ho& crisis management (cm). the main outcome of this work is demonstration of the use of hybrid modelling techniques to analyse disasters in terms of humanitarian operations and crisis management. key words: operations management, analytic hierarchy process, humanitarian operations management, organizational learning, fault tree analysis. mailto:ashraf.labib@port.ac.uk mailto:reza@additive-design.com mailto:sara.hadleigh-dunn@port.ac.uk mailto:morteza.yazdani@uam.es labib et al./decis. mak. appl. manag. eng. 5 (1) (2022) 113-134 114 1. introduction the main theme of the paper is to propose a set of innovative models in a hybrid modelling methodology. in doing so, we propose an integration of tools in a dynamic framework and demonstrate the benefits of such analysis through applying these techniques to case studies. although different models are applied to the two case studies, a comparison is then performed between the two cases in terms of their features and type of analysis performed. studies related to humanitarian operations (ho) and crisis management (cm) have identified the lack of adequate empirical data as a common challenge found in the majority of the research work (starr & van wassenhove, 2014). the final link in the humanitarian supply chain, that is, contact with beneficiaries of ho, was addressed by balcik & ak (2014), and again the data challenge was highlighted in this work. procurement issues in ho were also addressed (eftekhar et al., 2014). there were also other lines of investigation, including one related to the incorporation of itenabled multi-agency ho to enhance mutual benefits in refugee camps (ergun et al., 2014). differentiation of goals and objectives in ho (gralla et al., 2014), and the concept of trading off between two conflicting objectives of equity towards beneficiaries versus cost-efficiency in ho, were also investigated (mccoy & lee, 2014). ho and cm may vary from one disaster to another depending on the type and scale of the disasters. a disaster may be classified with respect to its cause and level of controllability from a purely natural disaster, almost out of causal control, up to a purely man-made disaster with high controllability. the scale could also reflect the effects, ranging from a small local disaster with minor impacts and losses to a large national/international scale with major impacts and casualties. ho&cm could aid both categories with respect to various causes and effects. it has been noted that experience gained from disaster management assists the stakeholders involved to take decisions and promotes the effective establishment of response (gupta et al., 2016). in the structuring process of a decision problem a vague situation is transformed into a structured problem with a set of well-defined elements, relations and operations to represent the informing factors, including the views, opinions and values of multiple decision-makers (ishizaka & labib, 2011). ho&cm problem structuring does not necessarily lead to an optimal solution, but it helps in finding critical elements. for example, structuring ho&cm responses to nuclear incidents and social impacts needs a systematic approach that considers various challenges and issues attached to nuclear incidents. a system index (heng & tao, 2014) was introduced to help policymakers predict the impacts of nuclear accidents in order to reduce risks to public safety. they used the multiple-attribute decision-making method linked to the analytic hierarchy process (ahp) to indicate indices such as public opinion and emergency resources and their weights along with a set of plans against nuclear accidents (alternatives), such as: taking iodine, shelter and evacuation, were set. the factors contributing to disasters and their impacts with the challenges and constraints for ho&cm have also been studied based on experts’ perceptions. for example, challenges such as communication in chemical, biological, radiological and nuclear (cbrn) disasters and finding the best practices for structuring tasks and principles with citizens were studied through a questionnaire survey from the perspective of experts (ruggiero & vos, 2015). communication across ho&cm actors and beneficiaries, including the public, is a key challenge in the pre-disaster stage of operational preparation/prevention while considering ethical issues. the lack of basic knowledge about hazardous materials such as radiation and evidence-based models to support humanitarian operations and crisis management 115 viruses will increase the impacts and potential risks of disasters. one of the best communicating practices in the pre-/post-disaster stage would be using online communication and social media monitoring (assuming electricity is available) with sufficient trained staff and support tools and solutions taking into consideration ethical issues (ruggiero & vos, 2014). for instance, erlandsson et al. (2017) analysed the reasons why donors make charitable decisions towards victims in different conditions. chávez et al. (2017) used bayesian methods in time preference research in intertemporal decision-making in risky choices to estimate parameters in delay discounting to avoid potential abuse. yazdani et al. (2019) proposed selective or methods incorporating a fuzzy anp model and failure mode and effect analysis (fmea) for risk evaluation of several construction projects related to water reservoirs/dams. various internal, external and technical factors were evaluated to acquire the riskiest projects, which would necessitate exceptional attention being given to the dams from pre-construction to post-construction and during their usage. a literature survey undertaken by altay & green (2006) of the papers published to address ho&cm aspects using operational research (or) methods indicates differentiation of the or tools applied for modelling various ho&cm problems at pre/post-disaster stages over a decade. although numerous or methods, such as mathematical programming, decision theory, queuing theory, heuristic methods, probability theory and statistics, and simulations have been used in the area of ho&cm and supply chain management, there is still a lack of adequate applications of or models to highlight critical success/failure factors influencing the performance of ho&cm efforts. according to galindoa & battaa (2013), no major changes or developments in the field of or application in ho&cm have appeared since the work of altay & green (2006). solid research is still required to re-establish the wellintended perception with a system view reflecting all the influencing factors (starr & van wassenhove, 2014; van wassenhove, 2006). the post-disaster analysis must include retrospective analysis via root cause analysis (to learn from failures/successes) followed by a disaster and prospective analysis to find safety measures and plan ho for preparation for/presentation of a new disaster (cacciabue & vella, 2010). this paper focuses on post-disaster analysis, i.e. retrospective and prospective analysis, while considering in-disaster real-time incidents and immediate rescue operations. this paper provides a selection of (hybrid) or models that have been applied to analyse two disaster case studies. the rationale behind the choice of case studies and hybrid tools is provided later on where the two cases are compared using several theoretical lenses. hence, the core research question related to why a comparison of the two case studies is needed is answered in table 1 in the paper, where the analysis in terms of cause and effect, recovery response, and retrospective and prospective analysis are compared. the main research motivation is to investigate selected two major disasters; one man-made and the other combined natural and made-made, and then a set of propose innovative hybrid analytical methods are applied as way of demonstration of their utility. the structure of the paper is as follows. after this introduction and review of the study background, section 2 outlines the research methodology with the rationale behind the choice of case studies. in section 3, we have two subsections. section 3.1 examines the data related to the case study of the bhopal disaster and the developed models for analysis. section 3.2 proposes the data related to the case study of the fukushima disaster and the developed models for analysis, which incorporate a multiple-criteria decision analysis (mcda) approach using the analytic hierarchy labib et al./decis. mak. appl. manag. eng. 5 (1) (2022) 113-134 116 process (ahp) technique. section 4 provides a framework for learning from disasters through comparative analysis of the disasters within the ho&cm context. we outline the main conclusions of the paper in section 5. 2. research methodology the aim of the research is to explore ho&cm influencing elements and structure them through suitable or models for further insight. the paper intends to build knowledge from learning from failures concerning ho&cm that occurred in two major disasters, which have been selected due to the suitability of the available data for using the analytical models. the research consists of three stages: 1) study of major disasters; 2) selecting suitable disasters for the study based on the availability of the data (volume and type); and 3) selecting suitable analytical methods with respect to ho&cm. accordingly, fta, rbd, mcs and ahp are used to structure the disasters’ ho&cm aspects due to exploring elements concerning causes and effects, the relationship/impact of each element with/on the others and the multicriteria nature of ho&cm critical factors while considering alternative strategies for tackling a disaster. the paper promotes the concept of ‘hybrid modelling’ focused on ‘ho&cm cases’. this is in line with shanthikumar & sargent (1983), who classified hybrid approaches into: either a ‘hybrid model’ in the form of procedures, or ‘hybrid modelling’ ie independent types of models. in this paper, we will focus on the hybrid modelling, where different modelling approaches (fta, rbd and mcs from the reliability analysis domain) and ahp (from multiple-criteria decision analysis) are utilized independently in two cases to study the single theme of ho&cm. this is in contrast to stephen & labib (2018), who developed a ‘hybrid model’ approach to one case study where an output of one model acted as an input to the subsequent model. hence, in this paper we demonstrate that the use of independent models, ‘hybrid modelling’, can help us to better understand a certain phenomenon. the paper is an attempt to match the proposed models with possible integration that can connect tangible and intangible aspects of ho&cm for a disaster. the authors initially studied a number of disasters for evaluation and analysis prior to selecting the proposed models and decided to study the two disasters due to data availability, the homogeneity of the ho&cm aspects with the proposed models and the limit on the length of the paper. the paper contributes to finding suitable analytical models according to ease of use of the data in terms of type, scope and volume. the proposed models are shown in an integrated structure, which can enhance the addressing of specific challenges, feasibilities and barriers from different perspectives. 3. the case studies 3.1. the case of the bhopal disaster the case of bhopal was previously studied by labib & champaneri (2012) in terms of the provision of root cause analysis; by ishizaka & labib (2014), through the proposal of a new logical gate for the analysis; by labib (2014), where generic lessons were extracted; and by labib (2015), where it was compared to the fukushima disaster and the unlearning phenomenon was investigated. in this section, we extend the analysis to address the humanitarian operations and crisis evidence-based models to support humanitarian operations and crisis management 117 management aspects of the disaster. we also provide evidence of the involvement of stakeholders in the investigation and the use of or techniques. in our investigation, it was noticed that cut set analysis provides more insight than just using fault tree analysis. this is further enhanced by the systems approach to technical and organizational analysis. 3.1.1. background the following text in this section is a summary adapted from the literature, and more details can be found in labib (2015). the incident occurred at midnight on 2 december 1984, when the tank number 610 (which is one of existing three tanks) containing a lethal toxic gas called methyl isocyanate (mic), was contaminated with water causing exothermal chemical reaction phenomena, which leaked into the atmosphere. the investigation found out that there was two types of failures; first, that the vgs was not sufficiently well designed due to its inability to handle a leak of that magnitude, and second, and to make matters even worse, it was under maintenance during the incident and hence not available at the time of the incident. in other words there were elements of both bad design and bad operation and maintenance (labib, 2015; chouhan, 2005). 3.1.2. modelling the ho&cm aspects of bhopal using fta, rbd and mcs in his account of the disaster, chouhan (2005), who had first-hand experience of the disaster as he was one of the employees of union carbide at bhopal and produced a comprehensive analysis of bhopal, stated that there was no evacuation plan for the neighboring area/communities (chouhan, 2005) the first author conducted a series of workshops. this was attended by experts in the chemical process industry. among the participants, there was an indian engineer who was originally from bhopal and was quite young when it happened. it was interesting to see his account of the accident and especially the fact that several of his young relatives were born suffering from disabilities, which demonstrates the extent of the disaster and the effects not just on the direct casualties but also on the next generation. the fta model in figure 1 shows a revised and extended model provided by the investigation of one of the groups (labib, 2015). however, in this paper we extend the analysis of this case study by developing a systems dynamics approach to analyse failures. 3.1.3. fault tree analysis (fta) a fault tree is a structured method for identification of causal factors and the logical relationships among them. the undesirable event is located at top of the hierarchical model and the different causes failures constitute the basic events further down in the tree. the causes of the top event are ‘connected’ through logic gates such as and gates (all inputs/causes needed for the above failure to occur) and or gates (one of the inputs/causes are needed for the above failure to occur). the authors built a fault tree analysis (fta) model as shown in figure 1, which was then mapped into a reliability block diagram (rbd) as shown in figure 2, and then a cut set analysis was performed on the derived rbd model in order to assess vulnerability of the system, by analyzing combinations of scenarios that can cause a complete failure of eh while system. the rationale behind the logic expression for the top event of bhopal and how his derived can be explained as follows. a major accident in any process industry can be attributed to either due to lack of design integrity or lack of good operation and maintenance. in other words, problems originate from either bad labib et al./decis. mak. appl. manag. eng. 5 (1) (2022) 113-134 118 design or bad use. one can also include a third factor which relates towards the attitude towards safety. hence in our proposed fta model of bhopal, we selected the major causal factors to be attributed to poor design, lack of safety and poor maintenance. the rationale behind the incorporation of a top and gate is that all three factors contributed simultaneously to the realization of the top event of bhopal disaster (bd). figure 1. the developed fta model for the bhopal disaster note that with respect to the idempotent laws (x.x = x and x + x = x), they remove repeated events within cut sets, and repeated cut sets (sinnamon and andrews, 1997). within the law of absorption (x + x.y = x) removes non-minimal cut sets since x.y is redundant. thus, the logic expression for the top event bhopal disaster (bd) derived by labib (2015): bd = (1 + 4).(2 + 5).(3 + 5) bd = (1 + 4).{2.3 + 5.3 + 2.5 + 5.5} bd = (1 + 4).{2.3 + 5.3 + 2.5 + 5} [applying idempotent law: x.x = x] bd = (1 + 4).{2.3 + 5.3 + 5} [applying absorption law: x + x.y = x] bd = (1 + 4).{2.3 + 5} [applying absorption law: x + x.y = x] this is the simplest possible and can be used to redraw an equivalent fault tree. bd = (1 + 4).{2.3 + 5} bd = {1.2.3 + 2.3.4 + 1.5 + 4.5} therefore, minimal cut sets are: 1.2.3; 2.3.4; 1.5; 4.5. note that the minimum number of boxes are 1.5 and 5.5, where 5 is common in both. also note that 5 is equivalent to ‘no adequate training’. evidence-based models to support humanitarian operations and crisis management 119 figure 2. the equivalent rbd of bhopal with/without description of boxes to appreciate the benefit of using an rbd, the following scenario was provided by labib (2014): given a limited amount of budget for improvement, the scenario of ‘no adequate training’ is crucial (a root cause) of the rbd system using minimum cut set analysis. available literature also verifies this argument. see the work of chouhan (2005), who was a technical eyewitness of the incident and in his account, there was major cut back in spending on training across the plant prior to the disaster. this shows the ability of such analysis to capture and extract implicit knowledge. also, in terms of ho, there is evidence, as reported by leveson (2004), that the emergency squad staff at bhopal were not sufficiently qualified and skilled to control such a disaster. this observation coincides with the root cause analysis we carried out that shows the significance of the effect of a lack of training on this disaster. 3.1.4. a systems approach to technical and organizational safety in this section, we extend the analysis of bhopal to a variation of a systems dynamics rather than a reliability approach. such an approach has its advocates, including leveson (2004), hollnagel (2004) and woods & cook (2002). a systems approach can be characterized by three features, according to leveson et al. (2009): “(1) top-down systems thinking that recognizes safety as an emergent system property rather than a bottom-up summation of reliable components and actions; (2) labib et al./decis. mak. appl. manag. eng. 5 (1) (2022) 113-134 120 focus on the integrated sociotechnical system as a whole and the relationships between the technical, organizational and social aspects; and (3) focus on providing ways to model, analyze and design specific organizational safety structures rather than trying to specify general principles that apply to all organizations. the fta and rbd of the bhopal disaster are shown in figures 1 and 2, respectively, and can be represented as a control system as shown in figure 3 in the form of controlling the water level in a storage tank, which is a similar model to that provided by aven (2012). the dynamic modelling here is based on a hydraulic system of a tank that stores an amount of fluid with valves and sensors (limit switches) controlling the flow. figure 3. representing the fta of the bhopal disaster as a storage tank system note that both the fta and the rbd models of the storage tank example are shown below in figure 4. note that from the rbd, one can extract some useful lessons for prevention. for example, if one is given resources to spend on prevention, the most vulnerable box according to the rbd would be box number 5 as its failure affects two boxes (2 and 3), compared to any other box whose failure will only affect less than two boxes. now, box 5 turns out to be the lshh signal, according to the fta model. as an outcome of going back to the tank illustration in figure 3, one can accordingly come up with recommendations such as initiating more preventative maintenance checks on the lshh, or even redesigning the tank to separate the signal into two different signals. another line of thinking concerns minimum cut sets (mcs), which can be done either algebraically as shown above or simply by imagining having a pair of scissors that can cut through the circuit of the rbd model. accordingly, cutting through just two boxes, say 4 and 5, will cause the complete failure of overfilling the tank to occur. boxes 4 and 5 turn out to be related to the two signals of the lsh and the lshh, according to the fta model. consequently, if one needs to evidence-based models to support humanitarian operations and crisis management 121 prevent such a failure at all costs then perhaps, we need to rely on two different electrical power stations to supply the tank with electricity instead of just relying on one source of electrical power. all these analyses demonstrate the power of modelling fta followed by an rbd followed by mcs. therefore, the tank storage example can be considered a simulation, or a mental model, of a control system that represents bhopal as shown in figure 5. figure 4. the equivalent fta and rbd models for the storage tank problem labib et al./decis. mak. appl. manag. eng. 5 (1) (2022) 113-134 122 figure 5. bhopal as a control system please note that in the reliability block diagram (rbd), actions to be taken to increase reliability are not only related to the logical structure of the rbd (i.e. the connections among blocks) but also to the value of the reliability of each block. however, since reliability figures, especially if they relate to qualitative assessments as in the case of a failure such as bhopal, are difficult to measure unless this is done in experimental conditions or if historical data are available, we assume that all blocks are equally weighted. hence, the important factor becomes related to the nature of the configuration of how the blocks are linked to each other, where a series structure implies a weak link as compared to a parallel structure that implies a strong connection, such as in the case of redundancy. in order to understand this mechanism as a mental map, we have three main subsystems, or control systems, as illustrated in figure 6, which are attributed to design control and safety. the first control loop of poor design contains the vgs and water spray, whereas the second control loop of a lack of safety contains switching off the refrigerator and no adequate training. finally, the third control loop of poor maintenance contains the issue of slip blind not installed and no adequate training. the three control loops are highlighted as dotted circles in figure 6. evidence-based models to support humanitarian operations and crisis management 123 figure 6. bhopal as a control mechanism notice that in figure 2, the rbd of the tank is replaced by bhopal’s equivalent system derived from fta. such a system approach enriches our understanding of the fundamental problems that caused the disaster as well as acting as a mental model, so that when deterioration in any of the three control systems is realized, an action can be triggered to respond. such an approach can be considered a variation of systems dynamics, but not exactly the same. it also shows that by using such a dynamic system one is able to capture sociotechnical aspects of a disaster. the reasons and benefits for the selection of these hybrid techniques are illustrated in figure 7 below. the use of fta facilitates problem structuring in terms of modelling the causal factors and their relationships via the use of the logic gates (and and or gates). these relationships are then fed into the rbd, which translates the logic gates into series configurations (for or gates) and parallel configurations (for and gates). such configuration provides the decision maker with an initial overall view about vulnerabilities in the whole system. such vulnerability analysis is then further analyzed using the mcs analysis, where a sensitivity (what if) analysis is carried out to examine various possible combinations of failures (scenarios). such sensitivity analysis can inform the decision maker about safety barriers in terms of either their effectiveness or need for new ones. finally, the use of sda provides a higher-level understanding of how each of the factors affect the performance of the whole system. in such dynamic analysis one is able to simulate factors which tend to be qualitative in nature and hence ideal for understanding socio technical aspects of the disaster. labib et al./decis. mak. appl. manag. eng. 5 (1) (2022) 113-134 124 figure 7. benefits of the hybrid framework 3.2. the case of the fukushima nuclear power plant disaster fukushima is considered, together with chernobyl, the worst disaster to have occurred in the nuclear power industry. the case of fukushima has previously been studied (labib, 2014) and extended (labib & harris, 2015; labib, 2015). in this section, we extend the analysis to address the humanitarian operations and crisis management aspects of the disaster. we also provide evidence in the form of investigation reports recounting observed incidents with real and reliable information. we also describe the use of or techniques carried out for such an investigation. the first author organized a workshop to investigate the fukushima disaster. the participants were from several industries including oil and gas, electric power and nuclear power generation. the workshop was part of a masters-class related to learning from failures. 3.2.1. background of the disaster the disaster is fully described in labib (2014). it is expected that the reactor will take about 50‒60 years to be decommissioned, i.e. for the plant to be accessed and cleared of any radioactive material. 3.2.2. ho&cm aspects of fukushima due to the earthquake and the tsunami, the power was lost around fukushima. the rescue squad and the operators were working in very harsh conditions, trying to cope with three simultaneous disasters: the earthquake, the tsunami and the nuclear meltdown. many of them were trying to resolve the situation at the nuclear plant while they had just lost many of their family members as a consequence of the earthquake linked to the tsunami. in addition, with the lack of electrical power, the operators could not access the gauges in the control room to assess the condition of the reactors. they sometimes had to dismantle car batteries to give them just enough power to glance at the indicators in the control room to discover the reactor’s condition. the disaster is considered a classic example of double jeopardy, and reflects the effects of a multi-hazard combination of an earthquake and a tsunami on the infrastructure system. it is also considered to be a ‘beyond design’ phenomenon, in which safety factors were not taken into account during the initial design stages, which in turn affected the response to the disaster. they are suffering from psychological agony due to the fear of radiation exposure, separation from their family, separation from their community, disruption of communities, loss of work, fault tree analysis (fta) reliability block diagram (rbd) cut sets analysis (csa) system dynamics analysis (sda) problem screening: causal factors problem solving: relationship between factors vulnerability analysis: recommendations & barriers higher level of understanding of relationships among causal factors evidence-based models to support humanitarian operations and crisis management 125 uncertainty about the future, and so on’. the accident has not only deprived people of their homes and lives but also destroyed their communities and caused them to feel the loss of personal pride and dignity. activities such as decontamination of houses, rain gutters, gardens and borders of woods are expected to take place after agreeing with the residents on the methods of decontamination of their houses and gardens and the place for temporary storage of decontamination waste to be generated as a result. it is expected that the storage volume will range between 15 and 28 million square metres. moreover, the economic impact of the disaster at fukushima is enormous as agricultural and fishery businesses are still banned in the neighbourhood of fukushima daiichi. the sales of tourism within the region have been reduced by 90%. the shutdown of nuclear power plants has caused a rise in electricity prices, and a 26% increase in co2 emissions in the electricity generation sector (national academies of sciences, 2016). fukushima had a less severe impact than chernobyl in terms of health-related radiological consequences (national academies of sciences, 2016). the fukushima disaster poses an interesting challenge with respect to evacuation efforts (national academies of sciences, 2016). as mentioned before, a 20 km radius evacuation zone determined by the japanese authorities is considered by many to be quite adequate. questions were rightly posed by the japanese community: does the american government know something that we don’t? this incident prompts the question “what is the appropriate role of foreign authorities in providing recommendations to its travelling or relocated citizens in a nuclear emergency?” 3.2.3. sources of empirical evidence various accident investigation teams published their judgment on the causes of the accident and lessons learned, including aoki & rothwell (2013), who analysed the key related reports. two apparent schools of thought exist. one school of thought held by tepco (the company in charge of fukushima) argues that the fukushima catastrophe was an unavoidable outcome of a natural disaster as it was “beyond the conceivable hypothetical possibilities” (soteigai in japanese), which is a view previously held by those who believe in normal accident theory (nat), a term coined by perrow (1984) that refers to accidents that are so complex by nature that they cannot be foreseen or stopped. a second school of thought argues that there was regulatory oversight and inadequate management and emergency response that allowed the accident to unfold as it did. a particularly strong message came from dr kurokawa, the chairman of the naiic, in his report: “[o]ur report catalogues a multitude of errors and wilful negligence that left the fukushima plant unprepared for the events of 11 march 2011. what this report cannot fully convey is the mindset that supported the negligence behind this disaster. what must be admitted ‒ very painfully ‒ is that this was a disaster ‘made in japan’. its fundamental causes are to be found in the ingrained conventions of japanese culture: our reflexive obedience, our reluctance to question authority, our devotion to ‘sticking with the programme’, our groupism and our insularity” (kurokawa, 2012). the nuclear regulators lacked the knowledge and the responsibility to secure nuclear power safety. the independence of the nuclear watchdog from the ministries and the operators caused the regulatory state, where the industry had a great influence over the regulator. the investigation of a nuclear accident’s impact generally concentrates on the technical tangible elements limited to the disaster region. however, the social intangible impacts of such disasters are rarely studied (heng & tao, 2014; lindell & labib et al./decis. mak. appl. manag. eng. 5 (1) (2022) 113-134 126 perry, 2012). in order to avoid social fear and disorder, macro-analysis of social impacts is crucial and needs to be appropriately studied (slovic, 2012). a holistic analytical method using a system index can be employed to analyse influencing factors concerned with nuclear disaster scenarios and help policymakers to foresee the possible outcomes in order to avoid (or reduce) threats to public safety (heng & tao, 2014). since the fukushima disaster, the debate about the safety of the nuclear industry has been highlighted, particularly in japan, the u.s. and several european countries. in japan, the initiative for embracing nuclear energy as the main source is called the ‘nuclear village’ (genpatsu mura). there are many debates in the press about the viability of this initiative. 3.2.4. or models of the fukushima disaster based on the ahp the ahp was used to support decision-making in terms of selecting the best strategy for of energy based on multiple criteria, in order for a country such as japan to decide in the aftermath of the nuclear disaster. the ahp was chosen due to its ability to deal with both qualitative and quantitative measures, its ability to model a hierarchical structure as a mental model, its ability to provide feedback on the consistency of judgments to the decision-maker and its ability to provide sensitivity (what if) analysis. for information about the ahp, the reader is advised to consult with saaty (1980), ishizaka & labib (2011), abdi & labib (2011), muhammad et al. (2021) and alosta et al. (2021). 3.3. the nuclear safety debate and the humanitarian view inspection of the serious accident at fukushima directs us back to the basic question: ‘what went wrong?’ the humanitarian views of the nuclear authorities usually vary as they might consider the serious accident as offering an opportunity to restrengthen nuclear power, instead of a justification for signaling the lesson that nuclear power is seen as a threat to the public and should be scrapped. the consensus at the world nuclear fuel cycle 2011 conference was in favor of this idea because nuclear energy has been providing utility power globally for a long time, despite the accident at fukushima. this statement was derived from experts’ knowledge with minimum application of the decision support tools available at the time. there are two conflicting views regarding japan’s nuclear power and accordingly two alternative decisions can be derived as follows: option 1 ‒ use alternative sources such as green energy instead of nuclear: this is mainly supported by the environmental agencies and the humanitarian organizations. option 2 ‒ keep the status quo but improve current design of nuclear power stations: this comes from a common opinion derived from nuclear industry professionals. option 3 ‒ no change: in other words, keep the status quo. 3.4. application of mcda the economy criteria refer to economic and financial measures such as cost and value for money. the image criterion is related to the authorities’ opinions. the feasibility criterion refers to technical feasibility. the ahp hierarchy is developed and illustrated in figure 8. evidence-based models to support humanitarian operations and crisis management 127 the goal is to identify appropriate post-disaster hom decisions for alternative energy sources the hierarchy second level is concerned with the important criteria governing the decision making process the last tier of the hierarchy reflects the strategic alternatives to be evaluated structure of the hierarchical model figure 8. the ahp hierarchy for the fukushima disaster 3.4.1. ahp results figure 9 represents the model output in terms of the priorities given to the criteria and alternative decisions. it highlights that the option ‘enhance nuclear safety’ is the most preferred alternative. figure 9. the ahp model outputs showing priorities of criteria (on the left) and alternatives (on the right) 3.4.2. conclusion of the group of participants: this group employed the ahp to evaluate the future options for nuclear power usage in japan following the fukushima accident. their study demonstrates the applicability of the ahp and mcda for ho&cm. the proposed model provides a labib et al./decis. mak. appl. manag. eng. 5 (1) (2022) 113-134 128 framework that encourages a number of key stakeholders to evaluate and resolve the nuclear power debate. the ahp model, coupled with the usage of expert knowledge, could improve the reliability of the model findings and alternative solutions. although there are other mcdm methods compared to ahp such as fucom (pamučar et al, 2018), and bwm (rezaei, 2015), we tend to agree with (pamučar et al, 2018) in that there is no agreement upon the best method of determining criteria weights. however, judging by the current number of papers in using ahp, it is still ranked as one of the most used methods for mcdm. due to its simplicity and offer of feedback on consistency of judgements as well as a facility to carry out a what if (sensitivity analysis) (pamucar & savin, 2020; pamucar & dimitrijevic, 2021). 4. comparative analysis of the two case studies this section overviews the or methods used for analysis of the two large-scale disasters. the decision support tools were selected mainly according to the way in which the decision problem was defined for each disaster along with the type and range of available data, which played a crucial role in finding a matching practicable technique. • a typical detached ahp model for disasters with mixed causes due to diverse views obtained from a number of data sources available for the second case study, i.e. for a mixed (industrial-natural) disaster the data sets were compiled and mcda, through the use of the ahp, was used for selecting an alternative energy source as a post-disaster recovery/reconstruction stage. since there were no reliable and consistent data required for performing unified root cause analysis, no fta model was proposed as being suitable for reflecting both causes and exploring industrial-natural failures. the mixed-cause disaster seemed to be too complex to derive a unified root because the analysis included both industrial and natural causes and effects as in the case of the second case study. • root cause analysis of past incidents through fta for disasters with a uniform cause root cause analysis of past incidents was used through fta for analysis of the technical causes and for exploring pre-disaster failures of the first (industrial) case study. • fta integrated into another analytical tool for disasters with a uniform cause fta was used for the first case study with a uniform cause respectively linked to: 1) an rbd for finding critical failure(s) and a systems approach, as a mental model, for exploring a control mechanism and potential control loops based on the underlying disaster factors that affected the first (industrial) disaster; and 2) the ahp for synthesized ranking of the direct disaster causes and the factors that contributed to the disaster impacts appeared in the second case study. table 1 summarizes comparative analysis of the disasters with respect to various aspects, including the disaster type/degree of cause and effect, controllability, ho&cm and the or models and analysis so that ho&cm communities can benefit. evidence-based models to support humanitarian operations and crisis management 129 table 1. comparative analysis of the case studies criteria bhopal fukushima models fta+rbd ahp neutrality/ objectivity of disaster low medium cause manmade mix (manmade + natural) effect major major pre disaster preparation very low low controllable cause high medium controllable effect high medium dependence on predisaster hom high high dependence on postdisaster hom high high recovery response very low low retrospective analysis high high prospective analysis high medium 5. discussion and conclusion the main contribution of this work is the use of hybrid modelling techniques to analyse disasters in terms of humanitarian operations and crisis management (ho&cm). such tools are often used as ‘mental models’ for both problem solving and problem structuring. problem solving is an efficiency measure (doing things right), for example setting priorities of actions for allocation of resources. problem structuring is about effectiveness (doing the right thing), for example brainstorming of possible scenarios of causal factors that lead to a disaster. unfortunately, most researchers tend to use or models for the former rather than the latter. this is partly due to our original definition of risk and risk assessment. risk is broadly defined, and assessed, based on a combination of severity and occurrence. whilst it is relatively easy to quantify severity, occurrence, as a performance measure, is often difficult to assess and it can be claimed as misleading as a measure in the first place. in this paper it has been demonstrated that the use of or techniques independently can contribute to a better understanding and therefore potentially better management of the ho&cm cases. for example, the hierarchal structures of fta and the ahp can help in brainstorming causal factors. moreover, a systems approach can facilitate mental modelling, which can lead to better problem structuring and decision analysis. in addition, the use of the ahp for setting priorities should be based on ranking for resource allocation to all factors according to their weight rather than utilizing it as a selection exercise and just allocating resources to the top-of-the-list factors and ignoring the rest. in other words, all possibilities need to be considered and resources need to be allocated to all possibilities with varying weights to realize the shift in emphasis from probability to possibility. in doing so, a more robust position is reached, thereby reducing the possibility of error or failure. since data availability and accuracy, along with uncertainty, is the most crucial part of modelling ho using or models, as future research, the authors can develop the proposed models incorporating fuzzy sets for considering vague data, which can be used as the models’ inputs in a fuzzy range in order to facilitate learning ho&cm more realistically through analysis of a range of solutions/outcomes. although the labib et al./decis. mak. appl. manag. eng. 5 (1) (2022) 113-134 130 concept of hybrid modelling has been demonstrated for ho&cm, it can also be applied to cases related to safety science and security studies, where the main difference between safety and security lies in the intention. author contributions: conceptualization, al.; methodology, al, ma, sh, and my; validation, my and sh; formal analysis, al, and ma; investigation, al, ma, my and sh; writing—original draft preparation, al, ma.; writing—review and editing, sh, my; supervision, al.; project administration, my;. all authors have read and agreed to the published version of the manuscript.” authorship must be limited to those who have contributed substantially to the work reported. funding: this research received no external funding data availability statement: the study did not report any data. acknowledgments: the authors are grateful for the anonymous reviewers for their valuable comments and suggestions. conflicts of interest: the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. references abdi, m.r. & labib, a. 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(1994). behind human error: cognitive systems, computers, and hindsight. cseriac state-of-the-art-report, wright-patterson air force base, oh: crew system, ergonomics information analysis centre. yadav, d.k. & barve, a., (2015). analysis of critical success factors of humanitarian supply chain: an application of interpretive structural modeling. international journal of disaster risk reduction, 12, 213-225. yazdani, m., abdi, m.r., kumar, n., keshavarz-ghorabaee, m. & chan, f.t., (2019). improved decision model for evaluating risks in construction projects. journal of construction engineering and management, 145(5), 04019024. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). evaluating the satisfaction of citizens in municipality services by using picture fuzzy vikor method: 2014-2019 period analysis decision making: applications in management and engineering vol. 5, issue 1, 2022, pp. 50-66. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame181221001y * corresponding author. e-mail addresses: bahadirf.yildirim@istanbul.edu.tr (b.f. yıldırım), sultan.kuzu@istanbul.edu.tr (s. k. yıldırım) evaluating the satisfaction level of citizens in municipality services by using picture fuzzy vikor method: 2014-2019 period analysis bahadır fatih yıldırım1 *, sultan kuzu yıldırım2 1 i̇stanbul university, faculty of transportation and logistics, i̇stanbul, turkey 2 i̇stanbul university, school of business, i̇stanbul, turkey received: 19 october 2021; accepted: 7 december 2021; available online: 20 december 2021. original scientific paper abstract: in this study, it is aimed to rank the satisfaction levels of citizens in municipality services. for this purpose, 20 municipal services included in the life satisfaction survey (lss) that the turkish statistical institution regularly applies every year are considered as alternatives. in addition, the satisfaction of citizens was evaluated not only for the last year, but also for the period of 2014-2019, and these years were considered as a set of criteria. lss statistics contains the citizens' responses which involve such opinion as abstain and refusal in addition to yes or no answers. for analyze the effect of all opinion types on decision process, the participant responses constituting the dataset were converted into picture fuzzy numbers (pfns) consisting of 4 parameters (positive, neutral, negative, and refusal). finally, we apply utilize vikor (visekriterijumska optimizacija i kompromisno resenje) method by using pfns arithmetic operators and evaluate the citizens’ satisfaction levels of the municipality services. as a result, it was determined that the municipal services with the highest satisfaction were graveyard (a18) and fire-fighting (a17) activities, while the services with the lowest satisfaction were zoning and city planning (a10) and control of food producing facilities (a20). key words: picture fuzzy sets; vikor; municipal services; satisfaction. 1. introduction the most important duty of local administrations is to provide services that meet the expectations of the citizens. local administrations in turkey is organized in three autonomous types of local government which are special provincial administrations, municipalities and villages (akyıldız, 2012). among them, municipalities are the most suitable local government units to measure the satisfaction of citizens with the execution of public services. it is important for municipal administrations to be evaluating the satisfaction level of citizens in municipality services by using picture fuzzy… 51 sensitive to the needs and wishes of the citizens and to ensure the continuous support, commendation and trust of the citizens. in order for the municipal administrators to be re-elected, it is important that the citizens are satisfied with their duties and the services provided (bostancı & erdem, 2020). in addition, the increasing urban needs with the developing technology reveal the necessity of providing more effective and efficient services for the local governments responsible for meeting these needs in the cities (yıldırım, 2004). the services, duties and authorities offered by the municipalities are spread over a wide area. these duties and authorities are detailed in municipal laws. (laws of municipalities, article 14). tuik evaluates the satisfaction with the municipalities within the scope of the articles in this law with the life satisfaction survey that it conducts regularly every year. in this type of research where criteria and alternatives are numerous, managers prefer numerical decision making techniques rather than emotional decision making. especially in municipal services, multi criteria decision making methods are a very appropriate approach for the level of satisfaction measured by a large number of criteria. consequently, 20 municipal services included in the life satisfaction survey (lss) that the turkish statistical institution regularly applies every year are considered as alternatives and, the period of 2014-2019 years was considered as a set of criteria, we design a decision-making problem. to handle the uncertainty which occurs in many real-life problems, has always been a problem for the researchers and decision makers (mahmood, 2020). however, it is often difficult to exactly assess the level of satisfaction with each service provided in the decision process, because of human judgments which are vague and ambiguous in many circumstances. when there may exist hesitation in the either assessment process or in the preferences of the attributes, picture fuzzy sets are suitable and flexible tool in dealing with fuzziness and uncertainty due to imprecise knowledge or information involving hesitancy. the motivation of this paper is ranking the municipal service alternatives according to citizens' judgements over 2004-2019 time period. in the decision process, it is aimed to make more effective decisions by expressing human judgments with fuzzy numbers. in part of lss, there are 20 questions based on municipal services that measure citizens’ satisfaction levels. items were scored using 5 points likert scale, with additional options for “no idea” or “no service”. for analyzing the effect of all opinion types on decision process, we construct the decision matrix from pfns which are calculated from citizens' responses. 5 point likert options use for calculating pfn’s positive, neutral and negative membership degrees and additional options use for calculating refusal membership degree. as far as we know, in the literature, picture fuzzy vikor method has not been perform to evaluate satisfaction level from municipal service. the originality of this study originating from this point, so we investigate the satisfaction levels of citizens from municipal services using picture fuzzy vikor. since the dataset used in this study contains expressions that represent the neutral view, pfs contains grade of neutral and more suitable for analyzing the satisfaction level. the aim of extending the vikor by using pfs is analyzing the effect of all opinion types obtained from citizens. the rest of the study is organized as follows; in section 2 briefly gives a literature review about evaluating public and municipality satisfaction, paying particular attention to the use of mcdm methods. following section 3. some basic concepts of picture fuzzy sets are given. in the fourth section, the analysis steps of vikor method using picture fuzzy numbers are presented respectively. the application to evaluate satisfaction levels of the municipality services with picture fuzzy vikor method is yıldırım et al./decis. mak. appl. manag. eng. 5 (1) (2022) 50-66 52 proposed in section 5. in the last section, the study is concluded with discuss numerical implementations and future studies are suggested. 2. literature review studies to determine the quality of municipal service in turkey have mostly focused on the evaluation of the surveys with statistical methods. in these studies, satisfaction with municipal services was associated with demographic factors (ince & sahin, 2011; gokus & alpturker, 2011; yucel et al., 2012; sabuncu 2016; bayram & polat, 2021). kelly and swindell (2002) investigated the relationship among them citizen satisfaction level and performance indication in an analysis of municipality services with correlation analysis. folz (2004) carried out a research on comparison of capacity in municipality services. they applicate clustering analysis to category cities into three homogenous classes based on service standards. studies on mcdm techniques and the grade of satisfaction with municipal and public services are given in the table below. table 1. evaluation of public and municipality satisfaction with mcdm author methodology results bostancı (2016) fuzzy ahp according to the neighborhoods, it was determined that the most satisfied neighborhood in kayseri municipality was yenidoğan. ozdogan et al. (2020) fuzzy ahp, fuzzy topsis the most important factor in the ranking of municipal services is green space. social and cultural services take the second. bostancı& erdem (2020) fuzzy dematel, fuzzy topsis when the thematic map determining citizen satisfaction is examined, it is seen that the satisfaction levels are quite high in mimar sinan and adnan menderes regions, and low in fakıuşağı district. ansari et al. (2016) fuzzy ahp, fuzzy topsis it was determined that municipality of district 2 has won the first place in qazvin municipalities. celik et al. (2013) interval type-2 fuzzy sets, gra, topsis according to the results, the public transportation service metrobus with the best customer satisfaction level in istanbul was determined. awasthi (2011) servqual, fuzzy topsis in montreal's subway transportation service, the metro line that provides the highest quality service has been determined as the orange line. bilişik et al. (2013) servqual, fuzzy ahp, fuzzy topsis it has been determined that the public transportation service with the highest satisfaction in istanbul is the metrobus. evaluating the satisfaction level of citizens in municipality services by using picture fuzzy… 53 author methodology results rahimi &najafi (2017) fuzzy anp, fuzzy topsis, fuzzy electre in the research conducted for zanjan, “municipal area 2” was chosen as the most suitable region with the highest score according to the expectations of the citizens. pehlivan & gürsoy (2019) fuzzy topsis, fuzzy multimoora fuzzy aras in turkey, it was found that zonguldak had the highest satisfaction with public services, while van had the lowest satisfaction with public services. nassereddine & eskandari (2017) delphi, gahp, promethee the public transport systems in tehran, in order of increasing importance: van, bus, brt, taxi and metro. li et al. (2020) picture fuzzy multimoora railway lines in shanghai are used to show the effectiveness of the recommended passenger satisfaction assessment technique. gündoğdu et al. (2021) picture fuzzy ahp it has been determined that the most influential factor in the satisfaction of public transport services in budapest is the timetables of the vehicles. it has not been found in the literature that satisfaction with municipal or public services is examined by the vikor method. for this reason, in the last part, studies with vikor and its extended versions are given. kank and park (2014) measured bank customers' satisfaction with mobile services using the vikor method, dincer and hacıoğlu (2013) measured their satisfaction with banking services using the fuzzy vikor method. in the beef industry, meksavang et al. (2019) evaluated and selected a sustainable supplier management with extended picture fuzzy vikor aproach. in parkouhi and ghadikolaei (2017), grey vikor techniques were used for supplier selection. tiwari et al. (2016) applied the product style concept evaluation by using integrated rough vikor method. krishankumar et al. (2020) suggested the intuitionistic fuzzy vikor method to the personnel selection problem. abdel-bassets' et al. (2018) extended vikor method with neutrosophic sets and provided a multicriteria group decision making method, for evaluating e-government websites. gundogdu et al. (2019) investigate the waste management problem using spherical fuzzy vikor method. similarly, gundogdu and kahraman (2019) applied the spherical fuzzy vikor method to the warehouse location selection. zhang et al. (2016) carried out an inpatient admission assessment using the hesitant fuzzy vikor method with linguistic terms at the west china hospital. gül et al. (2019) used the pythagorean fuzzy vikor based decision-making approach in the mining industry for security risk assessment. akram et al. (2019) contributed a novel multiple-attribute group decision-making method which called the trapezoidal bipolar fuzzy vikor method. apart from these studies, qin et al. (2015) and wang et al. (2019) purpose a new approach which vikor method extended with interval type-2 fuzzy for multiattribute decision making. ashraf et al. (2019) evaluated cleaner production in gold mines using novel distance measure method with cubic picture fuzzy numbers. mahmood et al. (2019) used the concept of spherical fuzzy sets for the solution of decision making and medical diagnosis problems. biswas et al. (2021) applied to extend the basic framework of lbwa in the picture fuzzy environment using actual score evaluate of the picture fuzzy numbers. pramanik et al. (2021) presented a comparative analysis of various mcdm methods under asymmetric conditions with varying selection alternative sets and criteria. yıldırım et al./decis. mak. appl. manag. eng. 5 (1) (2022) 50-66 54 ashraf et al. (2019) suggested generalized form of weighted geometric aggregation operator for picture fuzzy information. ali and mahmood (2020) investigated the generalization dice similarity measures based patterns recognition models with picture hesitant fuzzy information. pamučar et al. (2021) applied a new logarithm methodology of additive weights (lmaw) for mcdm. biswas (2020), carry out a comparative analysis of supply chain performances of leading healthcare organizations in india. biswas et al. (2019) have suggested an ensemble approach based on a two-stage framework for portfolio selection. for this purpose, using dea for primary selection of the funds. and then they used mabac approach in the second stage wherein criteria weights have been calculated using the entropy method. 3. picture fuzzy sets (pfs) in this section, we give the definition of pfs and summarize picture fuzzy distance measurement, arithmetic operations, score and accuracy functions. on the basis of intuitionistic fuzzy sets developed by atanassov (1986), the concept of picture fuzzy sets (pfs) was proposed by cuong and kreinovich (2014) to model the complex and uncertain assessments of experts in real decision making problems. because of the grade of a neutral cannot be discussed in intuitionistic fuzzy set, picture fuzzy sets investigated by cuong and kreinovich (2014) which contains positive, abstinence and negative grades (mahmood & ali, 2020). a picture fuzzy set p, on a non-empty set x is defined as,       , , , |p p pp x x x x x x      (1) where  p x represents the positive membership degree of p, the  p x parameter is the neutral membership degree of p and finally the  p x parameter indicates the negative membership degree of p.      , ,p p px x x   parameters,             0 , , 1, 0 1 p p p p p p x x x x x x             (2) provides the conditions. all pfs defined in the x universe have a fourth parameter called the degree of refusal membership, which makes the sum of      , ,p p px x x   parameters equal to 1.        1p p p px x x x       (3) since pfss are developed on the basis of classical fuzzy set and intuitionistic fuzzy set theory, the word "picture" in the title is used to mean "generality" (jovčić et al., 2020). for   0p x  the picture fuzzy set turns into an intuitionistic fuzzy set; it turns into a classical fuzzy set for    , 0p px v x  (jovčić et al., 2020). for convenience, picture fuzzy numbers will be represented by  , ,   triplet consisting of symbols representing parameters and arithmetic operators will be introduced. let  1 1 1 1, , vp   and  2 2 2 2, , vp   be two picture fuzzy numbers. the basic arithmetic operations that can be performed on these two numbers (addition, multiplication, multiplication by constant and exponentiation, respectively) are as follows (si et.al, 2019).  1 2 1 2 1 2 1 2 1 2, ,p v vp         (4) evaluating the satisfaction level of citizens in municipality services by using picture fuzzy… 55  1 2 1 2 1 2 1 2 1 2 1 2, ,p v vp v v          (5)       1 1 1 11 1 , , , 0p v           (6)       1 1 1 1,1 1 ,1 1 , 0p v            (7) the score (s) and accuracy (h) functions can be used for comparing two picture fuzzy numbers (wang et.al, 2017),               1 1 , 0,1 2 , 0,1 s v s v p p p ph h            (8) is calculated with the equation (8). picture fuzzy numbers  1 1 1 1, , vp   and  2 2 2 2, , vp   are sorted according to the following conditions (wei, 2018).                     2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 p p p p p p p p if s s if s s and h h if s s a h p p p p p p pnd h p           (9) distance between  1 1 1 1, , vp   and  2 2 2 2, , vp   picture fuzzy numbers (dutta, 2018),       1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 , 4 1 max , , , 2 d p v v v p v                         (10) is calculated by equation (10). 4. picture fuzzy vikor the vikor method developed by opricovic (1998), as an mcdm approach which determine a compromise solution which is acceptable for all decision makers and solve a discrete multi-criteria decision problems. in vikor method compromise solution takes into account conflict and imponderable criteria due to its potential benefits in compromise solution based ranking, the vikor method has been used in many of areas, singular or hybrid with other mcdm methods and extend with many system theories, in recent years. an extension of vikor to determine the compromise solution on uncertain, imprecise and non-commensurable decision process is picture fuzzy vikor (pf-vikor) approach. pf-vikor basically uses picture fuzzy numbers (pfns) to construct the decision matrix and picture arithmetic operators in the decision process. the principal characteristic of the pf-vikor method is that it calculates the separation measures from the fuzzy positive and the fuzzy negative values with the developed picture fuzzy distance operators, and herewith, the best alternative can be determined by the decision maker according to more precise information. the steps of the purposed pf-vikor approach are given as follows: step 1. determine of the picture fuzzy (pf) decision matrix. for the decision problem, m alternatives and n criteria are determined. the decision matrix is formed by combining the picture fuzzy performance scores of each alternative with  , ,ij ij ij ijr    according to each criteria in the r matrix. yıldırım et al./decis. mak. appl. manag. eng. 5 (1) (2022) 50-66 56 11 12 1 21 22 2 1 2 n n m m mn r r r r r r r r r r              (11) step 2. determination of picture fuzzy positive and negative values according to each criterion, picture fuzzy positive values which is the best  * * * *, ,j j j jf v  and picture fuzzy negative values which is worst  , ,j j j jf v       are determined using equations (12) and (13) according to the optimization direction (benefit or cost) of the criterion. in the equations, 1 j and 2 j represents the set of benefit criteria and cost criteria respectively. 1 * 2 max | , 1, 2, , min | ij i j ij i r j j f j n r j j           (12) 1 2 min | , 1, 2, , max | ij i j ij i r j j f j n r j j            (13) the max and min values in the equations are determined according to the conditional statements in equation (9) by using the pf score function and pf accuracy function are defined in equation (8). step 3. calculation of normalized picture fuzzy differences normalized picture fuzzy differences ij d (i = 1, 2,..., m; j = 1, 2,..., n) are calculated using equation (14).     * * , , j ij ij j j d f r d d f f   (14) the  * ,j ijd f r and   * , j j d f f  values in the equation are calculated with the distance formula shown in equation (10) (dutta, 2018).       * * * * * * * * * 1 4 1 max 2 , j j ij j ij j j i j j ij ij ij j ij ij jj j i d v vr v f v                            (15)       * * * * * * * * * 1 4 1 max 2 , j j j j j j j j j j j j j j j j j j vd vf f v v                                   (16) step 4. obtaining s, r and q values s, r and q values are calculated using the following equations, respectively. the v and (1-v) values in equation (19) are the degree of importance of the strategy to be determined for maximum group benefit and minimum individual regret, and it is generally accepted as 0.50 in studies (zhao et. al, 2017). 1 n c j i j i j s w d    (17)  max ci j ij j r w d (18) evaluating the satisfaction level of citizens in municipality services by using picture fuzzy… 57   * * * * 1i i i s s r r q v v s s r r          (19) * * min max min max i i i i i i i i s s s s r r r r       (20) step 5. the rankings of alternatives by the s, r, and q values three separate rankings are obtained by ordering the s, r and q values of the alternatives from smallest to largest. step 6. propose a compromise solution, the alternative (a(1)), which is the best ranked by the measure min q if the acceptable advantage and acceptable stability conditions are satisfied. in order for the obtained result to be considered valid, the following two conditions must be met. however, in this case, it is stated that the alternative (a(1)) with the minimum q value and in the first place in the ranking is the most ideal alternative.  c1. acceptable advantage:       ( 2) (1) 1 1 q a q a m     c2. acceptable stability: the best alternative a(1) must also be in the first order by s or, and r. the compromise solution is stable within a decision-making process, which could be: “voting by majority rule” (when v > 0.5 is needed), or “by consensus” v ≈ 0.5, or “with veto” (v < 0.5). the following compromise solutions can be proposed if one of the c1 and c2 conditions is not satisfied: alternatives a(1) and a(2) if only condition c2 is not fulfilled or alternatives a(1), a(2),..., a(m) if condition c1 is not fulfilled. a(m) is calculated by using equation         1 1 1 m q a q a m    5. application turkish statistics institute (tuik) has been conducted the life satisfaction survey (lss) since 2003. lss is a key indicator to measure the general happiness perception of citizens, their social value judgments, their general satisfaction in basic living areas and their satisfaction with public services. while lss was carried out on an urbanrural scale until 2013, since 2014 it was carried out throughout turkey. as a result of tuik’s revision -for a homogenous examinationthe period of 2014-2019 was selected in this study. in part of lss, there are 20 questions based on municipal services that measure citizens’ satisfaction level. items were scored using 5 points likert scale, with additional options for “no idea” or “no service”, used as an alternative set in this paper and shown in table 2. table 2. alternative set for municipal services ai service alternative ai service alternative yıldırım et al./decis. mak. appl. manag. eng. 5 (1) (2022) 50-66 58 ai service alternative ai service alternative a1 garbage and environmental cleanliness a11 arrangements for the disabled a2 drainage a12 social aids a3 drinking water a13 cultural activities a4 public transport a14 public education centers a5 municipal police a15 street and road lighting a6 road and pavement construction a16 cleanliness a7 parks and gardens a17 fire-fighting a8 minimization of noise and air pollution a18 graveyard a9 health, fitness center facilities a19 address information systems a10 zoning and city planning a20 control of food producing facilities before giving application steps, so that make it more easily understandable, the flowchart of the pf-vikor method presented in figure 1. determine the alternative set determine the criteria set construct decision matrix determine the picture fuzzy positive and negative values calculate s, r, q values       ( 2) (1) 1 1 q a q a m    is a (1) also be in the first place based on s or/and r set of alternatives is proposed as the best alternatives. both alternatives a (1) and a (2) are proposed as the best alternatives a (1) is the best alternative is yes no yes no calculate picture fuzzy performance scores 20 municipal services 2004-2019 lls datasets figure 1. pf-vikor methodology for evaluating satisfaction level from municipal services step 1. construct the pf decision matrix. for analyzing the effect all opinion types on decision process, we construct the decision matrix from pfns which are calculated from citizens' responses. 5 point likert options use for calculate pfn’s positive, neutral and negative membership degrees and additional options use for calculate refusal membership degree. as an evaluating the satisfaction level of citizens in municipality services by using picture fuzzy… 59 example, the calculation of picture fuzzy performance score of a1 alternative for year 2019 is shown in table 3. table 3. calculation of picture fuzzy performance score of a1 service alternative for year 2019 expressions options count of response total membership degrees positive expression very satisfied 405 6120 0.70 positive membership degree (μ) satisfied 5715 neutral expression neutral 1007 1007 0.12 neutral membership degree (η) negative expression dissatisfied 1149 1476 0.17 negative membership degree (ν) very dissatisfied 327 ineffective expression no idea 38 92 0.01 refusal membership degree (π) no service 54 grand total 8695 1.00 after calculating the picture fuzzy performance scores for overall alternative set according all years, the picture fuzzy decision matrix shown in table 4 was constructed. table 4. pfs decision matrix. 2014 2015 2016 2017 2018 2019 a1 (0.71, 0.08, 0.18, 0.03) (0.73, 0.10, 0.15, 0.02) (0.74, 0.09, 0.15, 0.02) (0.73, 0.09, 0.17, 0.01) (0.72, 0.10, 0.17, 0.01) (0.70, 0.12, 0.17, 0.01) a2 (0.66, 0.08, 0.18, 0.08) (0.67, 0.09, 0.15, 0.08) (0.71, 0.08, 0.15, 0.06) (0.66, 0.09, 0.17, 0.07) (0.65, 0.10, 0.18, 0.07) (0.66, 0.10, 0.18, 0.07) a3 (0.70, 0.08, 0.19, 0.02) (0.72, 0.10, 0.16, 0.02) (0.75, 0.08, 0.15, 0.02) (0.57, 0.11, 0.30, 0.01) (0.57, 0.12, 0.29, 0.01) (0.59, 0.12, 0.27, 0.01) a4 (0.59, 0.09, 0.21, 0.12) (0.59, 0.11, 0.19, 0.11) (0.63, 0.09, 0.17, 0.11) (0.59, 0.11, 0.20, 0.10) (0.60, 0.12, 0.19, 0.09) (0.59, 0.13, 0.20, 0.08) a5 (0.54, 0.09, 0.11, 0.26) (0.52, 0.09, 0.10, 0.29) (0.60, 0.08, 0.11, 0.22) (0.55, 0.09, 0.11, 0.25) (0.53, 0.11, 0.13, 0.24) (0.55, 0.11, 0.11, 0.23) a6 (0.56, 0.10, 0.29, 0.05) (0.56, 0.12, 0.27, 0.05) (0.59, 0.11, 0.26, 0.03) (0.54, 0.12, 0.30, 0.04) (0.54, 0.12, 0.31, 0.03) (0.55, 0.14, 0.28, 0.03) a7 (0.54, 0.10, 0.27, 0.09) (0.55, 0.12, 0.23, 0.09) (0.59, 0.10, 0.24, 0.06) (0.54, 0.12, 0.28, 0.06) (0.54, 0.13, 0.28, 0.05) (0.52, 0.14, 0.29, 0.04) a8 (0.42, 0.09, 0.22, 0.26) (0.42, 0.12, 0.20, 0.26) (0.47, 0.09, 0.20, 0.24) (0.41, 0.12, 0.21, 0.26) (0.40, 0.13, 0.23, 0.25) (0.40, 0.13, 0.22, 0.25) a9 (0.45, 0.09, 0.17, 0.28) (0.46, 0.11, 0.16, 0.27) (0.52, 0.10, 0.15, 0.24) (0.48, 0.11, 0.15, 0.25) (0.48, 0.13, 0.16, 0.23) (0.47, 0.12, 0.17, 0.24) a10 (0.36, 0.08, 0.17, 0.39) (0.34, 0.09, 0.15, 0.43) (0.43, 0.08, 0.14, 0.35) (0.37, 0.09, 0.15, 0.38) (0.38, 0.11, 0.15, 0.36) (0.39, 0.10, 0.15, 0.36) a11 (0.44, 0.09, 0.20, 0.27) (0.46, 0.10, 0.20, 0.24) (0.51, 0.09, 0.18, 0.22) (0.47, 0.11, 0.19, 0.23) (0.48, 0.12, 0.19, 0.20) (0.46, 0.12, 0.21, 0.21) a12 (0.50, 0.09, 0.16, 0.26) (0.52, 0.10, 0.14, 0.23) (0.56, 0.09, 0.13, 0.22) (0.54, 0.11, 0.13, 0.22) (0.53, 0.11, 0.14, 0.22) (0.49, 0.12, 0.15, 0.23) a13 (0.49, 0.10, 0.12, 0.29) (0.50, 0.11, 0.10, 0.29) (0.55, 0.10, 0.10, 0.25) (0.53, 0.11, 0.11, 0.25) (0.53, 0.12, 0.11, 0.24) (0.52, 0.13, 0.11, 0.24) a14 (0.51, 0.07, 0.09, 0.33) (0.52, 0.08, 0.08, 0.32) (0.57, 0.08, 0.08, 0.27) (0.55, 0.09, 0.08, 0.28) (0.55, 0.10, 0.08, 0.27) (0.53, 0.10, 0.09, 0.28) a15 (0.71, 0.10, 0.15, 0.04) (0.72, 0.11, 0.14, 0.03) (0.75, 0.09, 0.13, 0.03) (0.74, 0.10, 0.14, 0.03) (0.74, 0.10, 0.13, 0.03) (0.75, 0.11, 0.12, 0.03) a16 (0.67, 0.11, 0.18, 0.04) (0.69, 0.12, 0.15, 0.04) (0.72, 0.10, 0.14, 0.03) (0.69, 0.12, 0.17, 0.02) (0.68, 0.13, 0.17, 0.02) (0.67, 0.14, 0.17, 0.02) a17 (0.64, 0.07, 0.06, 0.23) (0.66, 0.08, 0.05, 0.22) (0.71, 0.06, 0.05, 0.18) (0.70, 0.06, 0.05, 0.19) (0.69, 0.07, 0.04, 0.20) (0.70, 0.07, 0.05, 0.18) a18 (0.72, 0.06, 0.04, 0.18) (0.74, 0.05, 0.04, 0.17) (0.79, 0.05, 0.04, 0.13) (0.78, 0.05, 0.03, 0.14) (0.79, 0.05, 0.03, 0.13) (0.79, 0.05, 0.04, 0.11) a19 (0.73, 0.08, 0.12, 0.07) (0.72, 0.09, 0.10, 0.09) (0.77, 0.08, 0.09, 0.06) (0.75, 0.08, 0.10, 0.07) (0.73, 0.10, 0.11, 0.07) (0.74, 0.09, 0.10, 0.07) a20 (0.35, 0.08, 0.18, 0.39) (0.33, 0.09, 0.17, 0.42) (0.42, 0.08, 0.15, 0.35) (0.36, 0.10, 0.15, 0.39) (0.35, 0.11, 0.18, 0.36) (0.35, 0.10, 0.17, 0.38) step 2. determination of picture fuzzy positive and negative values yıldırım et al./decis. mak. appl. manag. eng. 5 (1) (2022) 50-66 60 based on table 4, score and accuracy function values are obtained from equation (8). all evaluation criteria belong to beneficial criteria set. the fuzzy positive and the fuzzy negative values are determined according to the conditional statements in equation (9), and listed in table 5. table 5. picture fuzzy positive and negative values 2014 2015 2016 2017 2018 2019 * j f (0.72, 0.06, 0.04, 0.18) (0.74, 0.05, 0.04, 0.17) (0.79, 0.05, 0.04, 0.13) (0.78, 0.05, 0.03, 0.14) (0.79, 0.05, 0.03, 0.13) (0.79, 0.05, 0.04, 0.11) j f  (0.35, 0.08, 0.18, 0.39) (0.33, 0.09, 0.17, 0.42) (0.47, 0.09, 0.20, 0.24) (0.41, 0.12, 0.21, 0.26) (0.40, 0.13, 0.23, 0.25) (0.35, 0.10, 0.17, 0.38) step 3. calculation of normalized picture fuzzy differences the normalized picture fuzzy differences calculated by using equation (14), based on equations (15-16). calculated normalized pf difference values are given in table 6. table 6. calculated normalized pf differences 2014 2015 2016 2017 2018 2019 a1 0.074 0.064 0.071 0.070 0.070 0.059 a2 0.067 0.055 0.068 0.072 0.074 0.059 a3 0.076 0.063 0.069 0.134 0.125 0.100 a4 0.083 0.074 0.089 0.093 0.089 0.081 a5 0.082 0.088 0.099 0.104 0.109 0.090 a6 0.122 0.106 0.136 0.135 0.131 0.106 a7 0.113 0.094 0.121 0.128 0.124 0.114 a8 0.134 0.129 0.167 0.167 0.167 0.146 a9 0.121 0.113 0.142 0.133 0.133 0.121 a10 0.162 0.162 0.188 0.183 0.172 0.153 a11 0.127 0.115 0.148 0.137 0.131 0.125 a12 0.102 0.088 0.120 0.105 0.109 0.112 a13 0.103 0.098 0.123 0.112 0.108 0.103 a14 0.097 0.091 0.115 0.104 0.103 0.100 a15 0.066 0.060 0.061 0.059 0.053 0.040 a16 0.076 0.065 0.070 0.076 0.076 0.065 a17 0.037 0.034 0.042 0.036 0.043 0.035 a18 0.000 0.000 0.000 0.000 0.000 0.000 a19 0.050 0.038 0.040 0.037 0.042 0.030 a20 0.167 0.167 0.191 0.187 0.186 0.167 step 4. obtaining si, ri, and qi values si, ri, and qi values calculated based on equations (17–20), respectively. it was assumed that there was no superiority between the years, so all criteria weights were considered as equal and used equal in equations (17-18). step 5. the rankings of alternatives by the si, ri, and qi values the values of si, ri, and qi for each alternative and the ranking of municipality services based on these values are given in tables 7. table 7. s, r and, q values and municipality services ranking i s rank ir rank iq rank a1 0.406 6 0.074 5 0.384 6 a2 0.396 5 0.074 6 0.380 5 evaluating the satisfaction level of citizens in municipality services by using picture fuzzy… 61 i s rank ir rank iq rank a3 0.568 9 0.134 14 0.618 12 a4 0.509 8 0.093 8 0.484 8 a5 0.572 10 0.109 9 0.555 9 a6 0.737 15 0.136 15 0.703 15 a7 0.693 14 0.128 13 0.660 14 a8 0.909 18 0.167 18 0.863 18 a9 0.764 16 0.142 16 0.732 16 a10 1.020 19 0.188 19 0.970 19 a11 0.783 17 0.148 17 0.754 17 a12 0.635 12 0.120 11 0.613 11 a13 0.648 13 0.123 12 0.625 13 a14 0.610 11 0.115 10 0.588 10 a15 0.340 4 0.066 4 0.333 4 a16 0.428 7 0.076 7 0.401 7 a17 0.226 2 0.043 2 0.218 2 a18 0.000 1 0.000 1 0.000 1 a19 0.237 3 0.050 3 0.243 3 a20 1.063 20 0.191 20 1.000 20 * s 0.000 *r 0.000 s  1.063 r  0.191 step 6. propose a compromise solution based on table 7 and acceptable advantage and stability conditions, “a18 graveyard” alternative determined as the most appreciated municipal service. the variations in the ranking patterns with respect to changes in weights of the strategy of the majority of attributes (v values) are exhibited in table 8. table 8. the degree of possibility of each alternative over others depending on the values of v municipal services v = 0,25 v = 0,5 v = 0,75 rank rank rank garbage and environmental cleanliness 6 6 6 drainage 5 5 5 drinking water 13 12 10 public transport 8 8 8 municipal police 9 9 9 road and pavement construction 15 15 15 parks and gardens 14 14 14 minimization of noise and air pollution 18 18 18 health, fitness center facilities 16 16 16 zoning and city planning 19 19 19 arrangements for the disabled 17 17 17 social aids 11 11 12 cultural activities 12 13 13 public education centers 10 10 11 street and road lighting 4 4 4 yıldırım et al./decis. mak. appl. manag. eng. 5 (1) (2022) 50-66 62 cleanliness 7 7 7 fire-fighting 2 2 2 graveyard 1 1 1 address information systems 3 3 3 control of food producing facilities 20 20 20 based on table 8 we observe that the pf-vikor method is robust and provides rational ranking order. it is clearly seen that the order of the municipal service alternatives in the first and last places has not changed. the rankings contain very minor differences for only a few alternatives depending on the different v values. 6. conclusion and future studies this study introduces an alternative approach for satisfaction level assessment for municipal services and gives a real case study from turkey for the evaluation of twenty municipality services. we assumed that the ratings of municipality service alternatives on the given attributes are expressed using pfns. the importance degrees (weights) were assumed to be equally in this case study. sensitivity analysis is performed over weights of the strategy of the majority of attributes (v values), and from the analysis, as a result, it was found that the pf-vikor method is robust and provides consistent ranking order. in future research, we would like to proceed in the following facets. first, we can determine the importance degrees’ (weights) of years (criteria set) by using a weight assessment model like ahp, anp, bwm etc., or using these methods with fuzzy extensions. second, we can target the decision-making environment where picture fuzzy information is captured by interval valued picture fuzzy numbers. third, an optimization method can develop or other mcdm techniques can be used to determine the importance degrees of criteria objectively. evaluation of municipal services is a strategic decision-making problem for municipal administrations. so, the analysis results can be used by local authorities to benchmark municipal service alternatives. the proposed pf-vikor can be applied to solve many other decision making problems specially in different areas of management science with convenient modifications or hybrid usage with other mcdm methods. in future studies, researchers interested in this field can extend this assessment approach by using different systems theories (spherical, intuitionistic, pythagorean, fermatean, q-rung orthopair fuzzy, neutrosophic or rough sets) and, other mcdm techniques and investigate specific municipality services from selected country's perspective. author contributions: research problem, b.f.y. and s.k.y.; methodology, b.f.y. and s.k.y.; formal analysis, b.f.y. and s.k.y.; resources, s.k.y.; writing – original draft preparation, b.f.y. and s.k.y. writing – review & editing, b.f.y. and s.k.y. funding: this research received no external funding. acknowledgments: the authors are very grateful to the editor and the anonymous reviewers for their constructive suggestions. conflicts of interest: the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. evaluating the satisfaction level of citizens in municipality services by using picture fuzzy… 63 references abdel-basset, m., zhou, y., mohamed, m., & chang, v. 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(2016). inpatient admission assessment in west china hospital based on hesitant fuzzy linguistic vikor method. journal of intelligent & fuzzy systems, 30(6), 3143-3154. © 2021 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applicatons in management and engineering vol. 2, issue 1, 2019, pp. 147-165. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame1901147j * corresponding author. e-mail addresses: aleksandar.jankovic@gmail.com (a. janković), milenap@fon.bg.ac.rs (m. popović) methods for assigning weights to decision makers in group ahp decision-making aleksandar janković 1* and milena popović 2 1 ministry of transport and maritime affairs, directorate for transport, montenegro 2 university of belgrade, faculty of organizational sciences, belgrade, serbia received: 12 september 2018; accepted: 11 january 2019; available online: 9 march 2019. original scientific paper abstract: the method known as the analytical hierarchy process (ahp), a theoretical and methodological concept of multi-criteria analysis, is increasingly used in solving various decision-making problems. ahp is an excellent support to both the individual and group decision-making process, however, the involvement of a greater number of decision makers complicates the process and requires a different approach to when an individual decides alone. the synthesis of individual decisions within a group can be done in various ways, but the problem is how to deal with different levels of consistency when there are a number of decision makers. thus, this paper presents some of the methods for defining the individual weights of decision makers in group ahp decision making. key words: weights; decision makers; analytical hierarchy process; group decision making. 1. introduction decision making is as old as humanity itself. people have always made decisions (without even being aware of it), since decision making is, in fact, an integral part of everyday life. however, as life has become increasingly complex over time, it has also become necessary to master new knowledge in order to make the right decisions. in order to support the work of individual or group decision makers with complex sets of diverse information that cross over at the psychological, technical and other levels during the decision-making process, various mathematical and computer tools have been developed to support the decision-making process. one of these tools is ahp. considering that the ahp is based on individual (subjective) opinion of a decision maker (dm) about decision-making issue, it is always better to make a decision in mailto:milenap@fon.bg.ac.rs methods for assigning weights to decision makers in group ahp decision-making 148 group context, as this reduces the risk of wrong assessment, as the problem is approached from different perspectives based on different knowledge and experience of decision makers, and finally, the decision made has greater legitimacy to be realized. the objective of the research presented in this paper is to indicate the difference in significance of individual decision makers in group ahp synthesis. basic assumption is founded on the attitude that individuality brings participants' subjectivity (education, knowledge, concentration, desire, etc.) into decision-making process, so a quality methodological procedure is necessary that would objectivise final (group) decision. knowing the possibilities to define weights of decision makers directly contributes to the transparency of group decision. the paper with introduction and conclusion consists of four parts. in the second part of the paper titled analytical hierarchical process, mathematical basis of the ahp is presented. in the third section of the paper, a case study is used to present some of the methods for assigning individual weights to decision makers in group ahp. in the conclusion the fourth section of this paper, are pointed out key contributions of the conducted research and the directions for future research. 2. analytical hierarchy process (ahp) the analytical hierarchical process (saaty, 1980) is a method of multi-criteria analysis that is widely used in the world to support individual and group decision making (eskobar et al., 2004; vaidya & kumar, 2006; altuzarra et al., 2007; ho, 2008; arnette et al., 2010; subramanian & ramanathan, 2012; bernasconi et al., 2014). the method is both "analytic" and "hierarchical" because a decision maker decomposes complex problem of decision-making into several decision-making elements between which he establishes hierarchy relation. the word "process" in the name of the method suggests that after the formation of the initial hierarchy of a decision making issue are allowed its iterative modifications (saaty, 1999). the hierarchy of the decision making issue has several levels, with the goal at the top of the hierarchy; the following level contains the criteria, while the alternatives are at the bottom. such hierarchical setting refers to standard decision-making problem, but there are also cases where the hierarchy has four and more levels, respectively, when there are subcriteria between criteria and alternatives. also, there are decision making issues in which the hierarchy has two levels, and then only alternatives are below the goal. after setting the hierarchy, the decision maker compares pairs of elements at a given level of hierarchy with respect to all the elements at the higher level (superiors), in order to determine their mutual importance. in standard ahp, the elements are compared by providing linguistic (semantic) evaluations of mutual importance in relation to the elements at the higher level of the hierarchy using basic scale in the table 1 (saaty, 1980 ). in addition to saaty’s scale, other scales can also be used, such as lootsma’s (lootsma, 1988; lootsma, 1990; lootsma et al., 1990), ma & zheng’s (ma & zheng, 1991), balanced, etc., but the saaty 's scale is used mostly. linear part of the saaty’s scale consists of integers [1,9], and non-linear part of its reciprocal values [1,1/9]. when a dm at the given level of hierarchy evaluates n elements of the decisionmaking process as compared to the superior element according to the scale shown in the table 1, its semantic ratings according to the definitions in the left column are expressed as numerical values from the right column and recorded in a square matrix a. janković & popović/decis. mak. appl. manag. eng. 2 (1) (2019) 147-165 149 table 1. saaty 's relative importance scale definition numerical value absolute dominance of the element i over the element j 9 very strong dominance of the element i over the element j 7 strong dominance of the element i over the element j 5 weak dominance of the element i over the element j 3 the same importance of the elements i and j 1 weak dominance of the element j over the element i 1/3 strong dominance of the element j over the element i 1/5 very strong dominance of the element j over the element i 1/7 absolute dominance of the element j over the element i 1/9 (intervalues) (2,4,6,8) the matrix is positive and reciprocal (symmetrical in relation to the main diagonal). in other words, the elements from the top of triangle of the matrix are reciprocal to the elements from the bottom of triangle, and the elements on the main diagonal are equal to 1 ( ij ij a 1 a , for every i and j; ii a 1 for every i), as shown in the relation 1. 11 12 1n 21 22 2n n1 n 2 nn a a ... a a a ... a a ... ... ... ... a a ... a             (1) if using standard saaty 's scale, then every ij a can have one of 17 values from a discrete interval [1/9,9]. determining weights of the compared elements based on numerical values of the matrix a is called prioritization. prioritization shows a process of determining of priority vectors   t 1 n w w ,..., w from the matrix a, where every i w 0 and it's true n i i 1 w 1.   there are several matrix and optimization methods of prioritization (table 2), but the most commonly used methods are eigenvalue method, logarithmic least square method and the method of additive normalization (blagojević, 2015). table 2. prioritization methods and their authors prioritization methods authors of the method eigenvector method  ev saaty (1980) additive normalization method  an saaty (1980) weighted least squares method  wls chu et al. (1979) logarithmic least squares method  lls crawford & williams (1985) logarithmic goal programming method  lgp bryson (1995) fuzzy preference programming method  fpp mikhailov (2000) due to its simplicity and frequent use, as the example shown in this paper is used the method of additive normalization (an). to obtain a priority vector w it is enough to divide each element from the given column of the matrix a with the sum of the methods for assigning weights to decision makers in group ahp decision-making 150 elements of this column (normalization), then to sum up the elements in each row and finally to divide each resulting sum with the rank of the matrix a. this procedure is described by the relations 2 and 3: n , ij ij i 1 a a , ij 1, 2,..., n    (2) n , ij j 1 i a w , i 1, 2,..., n n     (3) based on the evaluation, by selected prioritization method are determined local weights of decision-making elements, and by synthesis, that is, additive synthesis, at the end are determined weights of alternatives at the lowest level in relation to the element at the highest level (goal), thus completing individual deciding using the ahp. the additive synthesis is presented with the relation 4: i j ij j u w d  (4) where in:  i u  final (global) priority of the alternative i;  j w  weight of the criterion j;  ij d  local weight of the alternative i in relation to the criterion j; in addition to the prioritization method, one of essential characteristics of the ahp is that at all levels of the hierarchy consistency of the decision makers’ evaluation is checked. for testing consistency, saaty ( 1977 ) proposed consistency ratio (cr) used in the an prioritization method. calculating the consistency ratio consists of two steps. in the first step, the consistency index (ci) is calculated using the relation 5: max n ci n 1     (5) where in:  n  the rank of the matrix;  max   the maximum eigenvalue of the comparison matrix; in the second step, the consistency ratio (cr) is calculated as the relationship of the consistency index (ci) and the random index (ri): ci cr ri  (6) the random index (ri) depends on the rank of the matrix and its values are obtained in random generation of 500 matrices (table 3). janković & popović/decis. mak. appl. manag. eng. 2 (1) (2019) 147-165 151 table 3. random index values depending on the matrix rank n 1 2 3 4 5 6 7 8 9 10 ri 0.00 0.00 0.52 0.89 1.11 1.25 1.35 1.40 1.45 1.49 if the consistency ratio (cr) is lower or equal to 0, 10 the result indicates that the decision maker was consistent and there is no need for the re-evaluation (jandrić srđević, 2000). if the consistency ratio (cr) is higher than 0.10, the decision maker should repeat (or modify) his evaluation in order to improve consistency. important feature of the ahp is sensitivity analysis of the final solution. the sensitivity analysis is carried out in order to see the extent to which the changes in the input data reflect the changes in the obtained results (nikolić & borović, 1996). in order to conclude whether the ranking list of the alternatives is sufficiently stable in relation to acceptable changes in input data, it is recommended to check the priority of alternatives for different combinations of input data. this analysis is very easily performed using software packages (softwares) to support decision making. one of the most commonly used is expert choice, which offers five sensitivity analysis options: dynamic, performance, gradient, head to head and 2d . the analysis can be done based on the goal or any other element in the hierarchy. sensitivity analysis based on the goal node shows the sensitivity of alternatives to all elements in the hierarchical tree structure. the stability of the results is performed using dynamic sensitivity analysis (option dynamic). if the rank of the alternatives remains unchanged when alternating the importance of the main criteria by 5% in all combinations, the result is considered to be stable (hot, 2014 ) . the ahp algorithm implementation is shown in the figure 1. start end cr<0.10 set of alternatives, criteria and goals final rank of alternatives yes no model structuring comparison of element in pairs ranking alternatives yes no decision maker and analytic stable solution? figure 1. the ahp algorithm (hot, 2014) methods for assigning weights to decision makers in group ahp decision-making 152 in the ahp there are several ways to consolidate individual decisions in group equivalents (blagojević, 2015):  aggregation of individual priorities aip;  aggregation of individual judgments  aij;  consensus model convergence  ccm;  geometric cardinal consensus model  gccm; however, the synthesis of individual results of the ahp application and making group decision requires prior determination of individual weights of decision makers. this is a specific problem, which is especially difficult if there is no institutional framework defining this issue. therefore, in this paper several possibilities of determining the criteria for defining weights of individual members of the group are presented. 3. possibilities of defining individual weights of group members according to the described methodology for the implementation of the ahp, it is discussed the hierarchy of decision making problems taken from (lukovac, 2016), figure 2, which consists of three levels. figure 2. hierarchy of decision making problems (lukovac, 2016) the goal is to "rank" the persons who can be included as assessors in the process of assessing the performance of drivers, and it is at the top of the hierarchy. the ranking criteria are at the following  intermediate level, which include:  knowledge of the work to be evaluated (k1);  the best possible insight into the work to be evaluated (k2);  objectivity, impartiality, in the evaluation process (k3); alternatives (participants in evaluating assessors) represent the subjects of ranking and they are at the lowest level of the hierarchy, which are:  superior (a1); janković & popović/decis. mak. appl. manag. eng. 2 (1) (2019) 147-165 153  dispatcher (a2);  colleague (a3);  client (a4);  self-assessment (a5); the expert group consisted of twenty decision makers (dms) to which the assessment of the performances of direct executors the drivers, was one of the obligations arising from the functional duty they performed. the decision makers compared the elements presented in a hierarchy in pairs using expert choice 2000 software, which automatically calculates the reciprocal values, so consequently only the elements in the so-called upper triangles of the comparison matrices are evaluated. the comparison matrices of the decision makers are presented in tables 4 to 10. table 4. the comparison matrices of dm 1to dm 3 dm 1 dm 2 dm 3 goal goal goal к1 к2 к3 к1 к2 к3 к1 к2 к3 к1 1 1 1 к1 1 1 1 к1 1 1/2 1/2 к2 1 1 к2 1 1 к2 1 1 к3 1 к3 1 к3 1 k1 k1 k1 a1 a2 a3 a4 a5 a1 a2 a3 a4 a5 a1 a2 a3 a4 a5 a1 1 1 2 5 3 a1 1 1 1 6 3 a1 1 2 1 6 2 a2 1 2 5 3 a2 1 1 6 3 a2 1 2 6 3 a3 1 3 2 a3 1 5 1 a3 1 4 1 a4 1 1 a4 1 1/4 a4 1 1/3 a5 1 a5 1 a5 1 k2 k2 k2 а1 а2 а3 а4 а5 а1 а2 а3 а4 а5 a1 a2 a3 a4 a5 а1 1 1 4 6 4 a1 1 1 4 6 4 a1 1 3 5 6 3 а2 1 4 6 4 a2 1 4 6 4 a2 1 2 5 2 а3 1 5 3 a3 1 5 3 a3 1 5 3 а4 1 1 a4 1 1 a4 1 1/2 а5 1 a5 1 a5 1 k3 k3 k3 а1 а2 а3 а4 а5 а1 а2 а3 а4 а5 a1 a2 a3 a4 a5 а1 1 1 3 5 7 a1 1 1 4 6 9 a1 1 2 5 7 9 а2 1 3 5 7 a2 1 4 6 9 a2 1 2 6 8 а3 1 3 5 a3 1 3 6 a3 1 4 7 а4 1 3 a4 1 6 a4 1 5 а5 1 a5 1 a5 1 methods for assigning weights to decision makers in group ahp decision-making 154 table 5. the comparison matrices of dm 4 to dm 6 dm 4 dm 4 dm 6 goal goal goal к1 к2 к3 к1 к2 к3 к1 к2 к3 к1 1 1 1 к1 1 1 1/2 к1 1 1 1 к2 1 1 к2 1 1/2 к2 1 1 к3 1 к3 1 к3 1 k1 k1 k1 a1 a2 a3 a4 a5 a1 a2 a3 a4 a5 a1 a2 a3 a4 a5 a1 1 2 3 5 3 a1 1 2 2 9 2 a1 1 1 3 7 3 a2 1 2 6 2 a2 1 1 9 1 a2 1 2 8 2 a3 1 4 1 a3 1 9 2 a3 1 6 1 a4 1 ½ a4 1 1/7 a4 1 1/2 a5 1 a5 1 a5 1 k2 k2 k2 а1 а2 а3 а4 а5 а1 а2 а3 а4 а5 a1 a2 a3 a4 a5 а1 1 2 4 6 3 a1 1 3 3 5 3 a1 1 1 2 5 1 а2 1 2 4 2 a2 1 3 5 3 a2 1 2 5 3 а3 1 4 3 a3 1 3 2 a3 1 6 3 а4 1 1 a4 1 1/2 a4 1 1/2 а5 1 a5 1 a5 1 k3 k3 k3 а1 а2 а3 а4 а5 а1 а2 а3 а4 а5 a1 a2 a3 a4 a5 а1 1 1 3 4 7 a1 1 2 4 6 9 a1 1 2 3 5 9 а2 1 3 4 7 a2 1 2 5 7 a2 1 3 5 8 а3 1 4 7 a3 1 3 6 a3 1 5 7 а4 1 4 a4 1 1/2 a4 1 4 а5 1 a5 1 a5 1 table 6. the comparison matrices of dm 7 to dm 9 dm 7 dm 8 dm 9 goal goal goal к1 к2 к3 к1 к2 к3 к1 к2 к3 к1 1 1/2 ½ к1 1 1 1 к1 1 1/2 1/2 к2 1 1 к2 1 1 к2 1 1 к3 1 к3 1 к3 1 k1 k1 k1 a1 a2 a3 a4 a5 a1 a2 a3 a4 a5 a1 a2 a3 a4 a5 a1 1 1 3 7 3 a1 1 1 2 5 2 a1 1 1 1 5 1 a2 1 3 7 3 a2 1 2 5 2 a2 1 1 5 1 a3 1 7 1 a3 1 5 1 a3 1 5 1 a4 1 1/7 a4 1 1/3 a4 1 1/5 a5 1 a5 1 a5 1 janković & popović/decis. mak. appl. manag. eng. 2 (1) (2019) 147-165 155 k2 k2 k2 а1 а 2 а3 а4 а5 а1 а2 а3 а4 а5 a1 a2 a3 a4 a5 а1 1 1/2 2 7 2 a1 1 1 2 6 3 a1 1 1 2 5 2 а2 1 3 7 3 a2 1 2 6 3 a2 1 2 5 2 а3 1 3 2 a3 1 5 2 a3 1 4 1 а4 1 1/3 a4 1 1 a4 1 1 а5 1 a5 1 a5 1 k3 k3 k3 а1 а 2 а3 а4 а5 а1 а2 а3 а4 а5 a1 a2 a3 a4 a5 а1 1 1 4 6 7 a1 1 1 1 5 7 a1 1 1 2 5 9 а2 1 4 6 7 a2 1 1 5 7 a2 1 2 5 9 а3 1 3 7 a3 1 5 7 a3 1 4 7 а4 1 3 a4 1 4 a4 1 4 а5 1 a5 1 a5 1 table 7. the comparison matrices of dm 10 todm 12 dm 10 dm 11 dm 12 goal goal goal к1 к2 к3 к1 к2 к3 к1 к2 к3 к1 1 1 1 к1 1 1 1 к1 1 1 1 к2 1 1 к2 1 1 к2 1 1 к3 1 к3 1 к3 1 k1 k1 k1 a1 a2 a3 a4 a5 a1 a2 a3 a4 a5 a1 a2 a3 a4 a5 a1 1 1 1 4 1 a1 1 1 2 5 3 a1 1 2 2 7 4 a2 1 1 4 1 a2 1 2 5 3 a2 1 2 7 3 a3 1 4 1 a3 1 5 1 a3 1 5 1 a4 1 ½ a4 1 1/4 a4 1 1/3 a5 1 a5 1 a5 1 k2 k2 k2 а1 а2 а3 а4 а5 а1 а2 а3 а4 а5 a1 a2 a3 a4 a5 а1 1 1 2 6 3 a1 1 1 2 5 3 a1 1 1 2 3 3 а2 1 2 6 3 a2 1 2 5 3 a2 1 2 3 3 а3 1 5 1 a3 1 3 1 a3 1 3 1 а4 1 1 a4 1 1/2 a4 1 1/3 а5 1 a5 1 a5 1 k3 k3 k3 а1 а2 а3 а4 а5 а1 а2 а3 а4 а5 a1 a2 a3 a4 a5 а1 1 2 3 4 8 a1 1 1 3 3 7 a1 1 1 2 3 7 а2 1 3 4 8 a2 1 3 3 7 a2 1 2 3 7 а3 1 4 8 a3 1 3 7 a3 1 3 7 а4 1 2 a4 1 3 a4 1 3 а5 1 a5 1 a5 1 methods for assigning weights to decision makers in group ahp decision-making 156 table 8. the comparison matrices of dm 13 to dm 15 dm 13 dm 14 dm 15 goal goal goal к1 к2 к3 к1 к2 к3 к1 к2 к3 к1 1 1 1 к1 1 1 1 к1 1 1 1 к2 1 1 к2 1 1 к2 1 1 к3 1 к3 1 к3 1 k1 k1 k1 a1 a2 a3 a4 a5 a1 a2 a3 a4 a5 a1 a2 a3 a4 a5 a1 1 1 1 4 1 a1 1 1 1 3 2 a1 1 1 2 6 3 a2 1 1 4 1 a2 1 1 3 2 a2 1 2 6 3 a3 1 4 1 a3 1 3 1 a3 1 6 4 a4 1 ½ a4 1 1/3 a4 1 1/3 a5 1 a5 1 a5 1 k2 k2 k2 а1 а2 а3 а4 а5 а1 а2 а3 а4 а5 a1 a2 a3 a4 a5 а1 1 1 2 5 3 a1 1 1 2 4 3 a1 1 1 2 3 3 а2 1 2 5 3 a2 1 2 4 3 a2 1 2 3 3 а3 1 5 3 a3 1 3 1 a3 1 5 2 а4 1 ½ a4 1 1/3 a4 1 1/3 а5 1 a5 1 a5 1 k3 k3 k3 а1 а2 а3 а4 а5 а1 а2 а3 а4 а5 a1 a2 a3 a4 a5 а1 1 1 2 3 5 a1 1 1 2 4 6 a1 1 1 2 4 6 а2 1 2 3 5 a2 1 2 4 6 a2 1 2 4 6 а3 1 3 5 a3 1 3 5 a3 1 3 5 а4 1 3 a4 1 3 a4 1 3 а5 1 a5 1 a5 1 table 9. the comparison matrices of dm 16 to dm 18 dm 16 dm 17 dm 18 goal goal goal к1 к2 к3 к1 к2 к3 к1 к2 к3 к1 1 1 1 к1 1 1/2 1/2 к1 1 1 1 к2 1 1 к2 1 1 к2 1 1 к3 1 к3 1 к3 1 k1 k1 k1 a1 a2 a3 a4 a5 a1 a2 a3 a4 a5 a1 a2 a3 a4 a5 a1 1 1 1 6 2 a1 1 1 1 5 2 a1 1 2 3 7 4 a2 1 1 6 2 a2 1 1 5 2 a2 1 2 6 3 a3 1 6 2 a3 1 5 2 a3 1 5 2 a4 1 ¼ a4 1 1/3 a4 1 1/3 a5 1 a5 1 a5 1 janković & popović/decis. mak. appl. manag. eng. 2 (1) (2019) 147-165 157 k2 k2 k2 а1 а2 а3 а4 а5 а1 а2 а3 а4 а5 a1 a2 a3 a4 a5 а1 1 1 2 4 3 a1 1 1 2 5 4 a1 1 1 2 3 4 а2 1 2 4 3 a2 1 2 5 4 a2 1 2 3 4 а3 1 4 2 a3 1 4 2 a3 1 3 2 а4 1 1/3 a4 1 1/3 a4 1 1/3 а5 1 a5 1 a5 1 k3 k3 k3 а1 а2 а3 а4 а5 а1 а2 а3 а4 а5 a1 a2 a3 a4 a5 а1 1 2 4 6 8 a1 1 1 2 4 8 a1 1 1 3 4 6 а2 1 3 5 7 a2 1 2 4 8 a2 1 3 4 6 а3 1 4 8 a3 1 3 7 a3 1 2 5 а4 1 3 a4 1 4 a4 1 3 а5 1 a5 1 a5 1 table 10. the comparison matrices of dm 19 to dm 20 dm 19 dm 20 goal goal к1 к2 к3 к1 к2 к3 к1 1 1 1 к1 1 1 1 к2 1 1 к2 1 1 к3 1 к3 1 k1 k1 a1 a2 a3 a4 a5 a1 a2 a3 a4 a5 a1 1 1 2 6 3 a1 1 2 2 6 3 a2 1 2 5 3 a2 1 1 4 3 a3 1 4 2 a3 1 4 2 a4 1 1/3 a4 1 1/3 a5 1 a5 1 k2 k2 а 1 а 2 а 3 а 4 а5 а 1 а 2 а 3 а 4 а5 а1 1 2 3 7 4 a1 1 1 3 8 5 а2 1 2 4 3 a2 1 3 8 5 а3 1 3 1 a3 1 7 2 а4 1 1/3 a4 1 1/3 а5 1 a5 1 k3 k3 а 1 а 2 а 3 а 4 а5 а 1 а 2 а 3 а 4 а5 а1 1 3 4 6 7 a1 1 2 3 5 7 а2 1 3 4 6 a2 1 3 4 6 а3 1 3 6 a3 1 3 5 а4 1 3 a4 1 3 а5 1 a5 1 methods for assigning weights to decision makers in group ahp decision-making 158 table 11 presents the vectors of the priority alternatives for each decision maker, obtained by means of relations (1)(6) based on data from the comparison matrices (tables 4 to 10). table 11. vectors of alternatives priorities by decision makers decision maker alternatives cr superior dispatcher colleague client selfassessment dm 1 0.351 0.351 0.165 0.066 0.068 0.02 dm 2 0.341 0.341 0.177 0.056 0.085 0.04 dm 3 0.433 0.257 0.171 0.055 0.083 0.05 dm 4 0.390 0.287 0.172 0.068 0.082 0.03 dm 5 0.424 0.253 0.181 0.045 0.098 0.03 dm 6 0.341 0.313 0.193 0.054 0.099 0.04 dm 7 0.327 0.381 0.150 0.051 0.090 0.03 dm 8 0.309 0.309 0.225 0.063 0.094 0.01 dm 9 0.303 0.303 0.203 0.067 0.127 0.02 dm 10 0.309 0.284 0.206 0.067 0.134 0.03 dm 11 0.330 0.330 0.170 0.068 0.101 0.02 dm 12 0.340 0.307 0.181 0.073 0.099 0.02 dm 13 0.286 0.286 0.224 0.076 0.128 0.01 dm 14 0.301 0.301 0.198 0.076 0.124 0.01 dm 15 0.317 0.317 0.213 0.066 0.088 0.02 dm 16 0.329 0.292 0.218 0.054 0.107 0.02 dm 17 0.319 0.319 0.211 0.067 0.084 0.01 dm 18 0.360 0.316 0.168 0.071 0.085 0.02 dm 19 0.405 0.284 0.16 0.057 0.094 0.02 dm 20 0.388 0.301 0.177 0.054 0.079 0.02 as during the implementation of dynamic alteration in sensitivity analysis of all important criteria by 5% in all combinations (optional dynamic in expert choice software), there was no change in ranking of alternatives, the final results of the conducted individual ahp can be considered stable. since all dms performed all the evaluations, the information base is complete, and thus is fulfilled one of the conditions for starting group syntheses of individual priority vectors from the table 11. however, the synthesis of the individual results of the ahp application into a group decision requires prior defining of individual weights of dms. by means of the case study, five methods for assigning individual weights to decision makers are considered (lukovac, 2016). "the first method" is to assign equal weights to all dms and then synthesize a group decision (srđević et al. 2004). this approach, however, does not treat individual dm consistency and is subject to manipulation and other irregularities. for example, if a dm had personal motive (relative-friendly relations, possibility of corruption, etc.), his ratings could be adjusted and/or inconsistent (to better rank the desired candidates) and would not have suffered any consequences in relation to his inconsistency (weights would remain the same as at other dms). according to this approach, in the specific case the weight ( k  ) of all dm would be 0. 05 ( k 1 / 20  ). "the second method" is to assign to dms the weights based on the values of spearman’s correlation coefficient which shows the compatibility of the individual janković & popović/decis. mak. appl. manag. eng. 2 (1) (2019) 147-165 159 dm with the reference group decision where also his decision was taken into account (srđević et al., 2009). spearman’s correlation coefficient ( s ) is calculated according to the relation 7. n 2 a i 1 2 6 d s 1 n(n 1)     (7) a d is the difference between a u and a v , where a u and a v are the ranks for the alternative a by reference list and by list compared to the reference, and n is the number of alternatives. in a group context, the relation 7 is applied to each of the combinations (group list, the list for k th member of the group, that is, the spearman’s coefficient is calculated according to the number of the members of the group). spearman’s coefficient value may vary between theoretical values of  1 and 1. when the value approaches to 1, the indication is that the ranks are the same or similar, and when the value approaches to zero and  1, the ranks are reverse, or negatively correlated. in this case, the highest weight obtains the dm whose decision was the closest to the group decision (the dm having the highest value of spearman’s coefficient), while the smallest weight obtains the dm whose decision was the furthest from the group decision. all dms are scaled according to the value of spearman’s coefficient. for the purpose of calculating the weights of dms under this possibility, in the considered case, the first thing to be done is making group decision. two basic and most commonly used ways for obtaining group decision in the ahp are the aip and the aij (ramanathan & ganesh, 1994; forman & peniwati, 1998). for consolidating individual decisions into the group one, in this case, the aip method is used, which is characteristic for two aggregations: (a) weight arithmetic mean weight method  wamm. it is provided the alternative i a and its weight value (priority)   k i w for the k-th decision-maker. if all members of the group (g) are assigned appropriate weights k  , the weight arithmetic mean is:     m g k i i k k 1 w w     (8) where in:    g i w final (composite) priority of the alternative i a .  m number of decision makers (group members); assuming, individual weights k  of the members of the group were previously additionally normalized, i.e., m k k 1 1   . (b) geometric mean method  gmm. in this method, the aggregation consists in applying the following expression: methods for assigning weights to decision makers in group ahp decision-making 160      km g k i i k 1 w w     (9) the weights of group members ( k  ) are also previously additionally normalized. in the table 12 are shown the results of the aip synthesis of the individual dm priority vectors from the table 11 in the case where dms are assigned equal weights ( k 0.05  ). table 12. the aip synthesis for k 0.05  aip superior dispatcher colleague client self-assessment wamm 0.345 0.307 0.188 0.063 0.097 gmm 0.345 0.307 0.188 0.063 0.097 from the table 12 it can be seen that the identical values of group vector are obtained of alternatives priorities for both aggregations of the aip synthesis (wamm and gmm). with the synthesis carried out it is fulfilled the condition for calculating spearman’s correlation coefficient for every dm, respectively, for comparing individual dm decisions with a reference, group decision. in the table 13 are shown the weights ( k  ) assigned to the dm based on the obtained spearman’s coefficient of dm based on the correlation 7. table 13. dm weights based on s value decision maker s k dm 1 0.975 0.050 dm 2 0.975 0.050 dm 3 1 0.051 dm 4 1 0.051 dm 5 1 0.051 dm 6 1 0.051 dm 7 0.9 0.046 dm 8 0.975 0.050 dm 9 0.975 0.050 dm 10 1 0.051 dm 11 0.975 0.050 dm 12 1 0.051 dm 13 0.975 0.050 dm 14 0.975 0.050 dm 15 0.975 0.050 dm 16 1 0.051 dm 17 0.975 0.050 dm 18 1 0.051 dm 19 1 0.051 dm 20 1 0.051 "the third method" is to determine the weights of the dms based on their competency for solving given decision making problem (lukovac, 2016). according to this approach, the competency coefficient for each dm is calculated. the obtained competence coefficients are later additionally normalized and assigned as weights of dms. in the table 14 are shown the weights ( k  ) assigned to the dms by the value janković & popović/decis. mak. appl. manag. eng. 2 (1) (2019) 147-165 161 calculated according to their competence ration according to the approach developed in (lukovac, 2016). table 14. dm weights based on k value decision maker k k dm 1 0.6706 0.054 dm 2 0.5624 0.046 dm 3 0.561 0.046 dm 4 0.5833 0.048 dm 5 0.5621 0.046 dm 6 0.5619 0.046 dm 7 0.5598 0.046 dm 8 0.6918 0.056 dm 9 0.7195 0.058 dm 10 0.5686 0.046 dm 11 0.6141 0.050 dm 12 0.6319 0.051 dm 13 0.6946 0.056 dm 14 0.6738 0.055 dm 15 0.6341 0.051 dm 16 0.6018 0.049 dm 17 0.6754 0.054 dm 18 0.5888 0.048 dm 19 0.6242 0.051 dm 20 0.5673 0.046 "the fourth method" is to assign to dms weights obtained by normalizing reciprocal values of their consistency ratios (cr) ( srđević, 2008 ), table 15. table 15. dm weights based on cr value decision maker cr 1/cr k dm 1 0.02 50 0.047 dm 2 0.04 25 0.024 dm 3 0.05 20 0.019 dm 4 0.03 33.3333 0.032 dm 5 0.03 33.3333 0.032 dm 6 0.04 25 0.024 dm 7 0.03 33.3333 0.032 dm 8 0.01 100 0.095 dm 9 0.02 50 0.047 dm 10 0.03 33.3333 0.032 dm 11 0.02 50 0.047 dm 12 0.02 50 0.047 dm 13 0.01 100 0.095 dm 14 0.01 100 0.095 dm 15 0.02 50 0.047 dm 16 0.02 50 0.047 dm 17 0.01 100 0.095 dm 18 0.02 50 0.047 dm 19 0.02 50 0.047 dm 20 0.02 50 0.047 methods for assigning weights to decision makers in group ahp decision-making 162 "the fifth method" is that the weights of the dms are determined by the consistency ratio (cr) and total euclidean distance (ed), explained in (srđević et al., 2009). this possibility was developed in (blagojević et al. , 2010) as a method consisting of the following steps: 1. for every dm, cr and ed are calculated from all comparison matrices; 2. all cr values for every dm are summed separately, and then the same procedure is repeated for ed values; 3. the reciprocal values of the cr and ed values are calculated for every dm; 4. additive normalization is performed (the reciprocal value of a sum for one dm is divided by a sum of reciprocal values of the sums of all dms), especially for cr and ed ; 5. for every dm, the mean value of the normalized values of cr and ed is calculated and it is adopted as its weight in the group ahp decision, respectively, k (normcr normed) / 2   . based on the data from the comparison matrices shown in (lukovac, 2016), for the considered ahp example, in the tables 16  19 the calculation of the weights of dms based on cr and ed is described. table 16. consistency and total euclidean distance dm 1-dm 5 dm 1 dm 2 dm 3 dm 4 dm 5 cr ed cr ed cr ed cr ed cr ed goal 0 0 0 0 0 0 0 0 0 0 k1 0.008 1.109 0.032 2.026 0.044 1.867 0.023 2.896 0.022 4.364 k2 0.056 3.775 0.099 6.483 0.074 4.897 0.054 3.048 0.050 3.226 k3 0.031 3.532 0.079 6.979 0.086 8.032 0.065 4.885 0.063 4.955 ʃ 0.096 8.416 0.210 15.488 0.204 14.796 0.142 10.829 0.135 12.546 1/ʃ 10.43 0.12 4.75 0.06 4.90 0.07 7.02 0.09 7.41 0.08 norm 0.046 0.052 0.021 0.028 0.022 0.030 0.031 0.041 0.033 0.035 k  0.049 0.025 0.026 0.036 0.034 table 17. consistency and total euclidean distance dm 6-dm 10 dm 6 dm 7 dm 8 dm 9 dm 10 cr ed cr ed cr ed cr ed cr ed goal 0 0 0 0 0 0 0 0 0 0 k1 0.029 3.098 0.046 5.800 0.012 2.025 0.000 0.000 0.013 1.514 k2 0.054 2.910 0.021 2.565 0.023 2.206 0.039 2.339 0.042 2.634 k3 0.076 5.836 0.064 5.757 0.032 3.441 0.026 3.638 0.059 4.867 ʃ 0.158 11.844 0.130 14.122 0.067 7.672 0.065 5.977 0.114 9.015 1/ʃ 6.32 0.08 7.71 0.07 14.93 0.13 15.42 0.17 8.78 0.11 norm 0.028 0.037 0.034 0.031 0.066 0.057 0.068 0.074 0.039 0.049 k  0.032 0.033 0.062 0.071 0.044 janković & popović/decis. mak. appl. manag. eng. 2 (1) (2019) 147-165 163 table 18. consistency and total euclidean distance dm 11-dm 15 dm 11 dm 12 dm 13 dm 14 dm 15 cr ed cr ed cr ed cr ed cr ed goal 0 0 0 0 0 0 0 0 0 0 k1 0.026 3.057 0.024 3.049 0.013 1.514 0.018 1.284 0.034 2.836 k2 0.006 0.910 0.034 2.403 0.018 1.985 0.018 1.964 0.052 3.248 k3 0.047 3.900 0.019 2.601 0.027 2.469 0.020 2.482 0.020 2.482 ʃ 0.079 7.866 0.078 8.053 0.059 5.968 0.055 5.730 0.106 8.567 1/ʃ 12.63 0.13 12.88 0.12 16.95 0.17 18.02 0.17 9.43 0.12 norm 0.056 0.056 0.057 0.055 0.075 0.074 0.079 0.077 0.041 0.051 k  0.056 0.056 0.074 0.078 0.046 table 19. consistency and total euclidean distance dm 16-dm 20 dm 16 dm 17 dm 18 dm 19 dm 20 cr ed cr ed cr ed cr ed cr ed goal 0 0 0 0 0 0 0 0 0 0 k1 0.002 0.874 0.001 0.456 0.019 3.260 0.011 1.724 0.017 1.906 k2 0.024 2.291 0.020 2.450 0.051 2.961 0.015 2.050 0.026 4.422 k3 0.069 6.330 0.020 3.294 0.021 3.172 0.070 5.971 0.045 4.003 ʃ 0.096 9.495 0.041 6.200 0.092 9.393 0.096 9.745 0.087 10.331 1/ʃ 10.45 0.11 24.33 0.16 10.91 0.11 10.41 0.10 11.43 0.10 norm 0.046 0.046 0.107 0.071 0.048 0.047 0.046 0.045 0.050 0.043 k  0.046 0.089 0.047 0.046 0.046 4. conclusions decision making, especially at the strategic level, requires more participants in the decision-making process (experts), who have different preferences depending on institutional placements, interests, skills, education and the like. in order to maximally objectify group context, in the procedure of synthesis of individual decisions , in this paper, using specific case, several possibilities for grading individual preferences of decision makers in group ahp synthesis are presented. it is important to emphasize also the difference between the terms "joint" and "group" decision. in the first case it is implied the consensus, and in the second not necessarily. group context treated in this paper fits to another case, no harmonization is performed, no consultation among participants, and the results of individual evaluations are consolidated later. further research should be directed towards analyzing the ahp synthesis results for the shown possibilities of assigning weights to decision makers. the subject of the research should also be directed towards the consensus of decision makers and the so-called joint decision. methods for assigning weights to decision makers in group ahp decision-making 164 references altuzarra a., morenojimenez j.m., & salvador m.a. 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(2006). analytic hierarchy process: an overview of applications. european journal of operational research, 169(1), 1–29. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 2, issue 2, 2019, pp. 81-99. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame1902021a * corresponding author. e-mail addresses: patrick_ay@rediffmail.com (p. anthony), behnoee1354@gmail.com (b. behnoee), malek.hassanpour@yahoo.com (malek hassanpour), dpamucar@gmail.com (d. pamucar) financial performance evaluation of seven indian chemical companies patrick anthony 1*, babak behnoee 2, malek hassanpour 3 and dragan pamucar 4 1 department of commerce and business management, ucc & bm, osmania university, telengana, india 2 department of commerce and business management, ucc & bm, osmania university, telengana, india 3 department of environmental science, ucs, osmania university, telangana, india 4 department of logistics, military academy, university of defence in belgrade, serbia received: 26 february 2019; accepted: 20 may 2019; available online: 22 may 2019. original scientific paper abstract: financial appraises create a prominent media for giving advice in the expansion, development of any society as well as its role in forbearance and stamina in depletion and recession. obviously, manufacturing units have a main role in the development and progress of modern india. indian economic relied on agricultural activities but industries also provide a prominent booster for the economic cycle. the current empirical study investigated the 7 indian chemical companies in terms of financial aspect using ratio analysis, technique for order of preference by similarity to ideal solution (topsis), complex proportional assessment (copras) and data envelopment analysis (dea) along with weighing systems of equal weighing, entropy shannon and friedman test as the objective of research during 2010 to 2018. by the way, present research resulted in weighing and ranking of above-named industries in three classes. the weighing systems of friedman test and entropy shannon were revealed a relatively linear scatter plot with no significant differences between values. dea model had distinguished and classified the efficient companies based on rank values. key words: financial performance, companies, dea, ratio analysis. mailto:patrick_ay@rediffmail.com mailto:behnoee1354@gmail.com mailto:malek.hassanpour@yahoo.com mailto:dpamucar@gmail.com anthony et al./decis. mak. appl. manag. eng. 2 (2) (2019) 81-99 82 1. introduction financial estimations create a prominent media in the expansion, development of any society as well as its role in forbearance and stamina in depression and depletion. at the micro and macro positions, the financial viability of any industrial sector presents the economic achievements and progresses. to figure out the development trend and also any fall or rise parallel with revolution towards sustainability of companies, the financial outcomes are posed as a level of judge. obviously, industries, companies and manufacturing units have a main role in flourish and growth of modern india. indian economic relies on agricultural activities but industries also provide a prominent booster for the economic cycle. the preliminary activities to set up the industries started after british rule in india. the industrial sector encompassed 3 major sectors such as (1) primary sector devoted to the exploitation of raw materials using agricultural activities or mining and aggregate extractions. (2) the second sector included refining, building and construction and manufacturing developments. (3) the third sector is related to distribution, delivery of commodities and marketing purposes (arab et al 2015; kettiramalingam et al 2017). by the 1938 indian chemical council was found in order to further development of companies in this regard. this sector placed the third greatest producer in asia and 12th in the world because of marketing expansion. it has been forecasted the growth rate around 14% per year from $ 160 billion in 2013 to $ 350 billion by 2021. the majority of indian chemical products encompassed based chemicals, which include the petrochemicals, man-made fibres, industrial gases, fertilizers, chlor-alkali, and other organic and inorganic chemicals etc. over 70000 commercial products. also, this sector included 12.5% of the total industrial output and approximately 16.2% of the total exports in india. financial analysis refers to the process of evaluating companies, businesses or projects in terms of budgeting and other financial aspects of these institutions, which is used to determine the suitability of these institutions for investing through financial statements. financial analysis is often used to assess the strength of an institution and its ability to pay debts, as well as its liquidity and profitability. financial analysis often focuses on the profit and loss account, balance sheet and cash flow, which, based on the firm's past, estimates its future performance (kumar and bhatia 2014). many scholars recognize decision making as an essential factor in management. decision-making is the result of a process that ultimately leads to a decision, while those who are not in the decision-making process. only see the result of the decision. in recent years, the attention of academic assemblies has attracted more decision making science in the country and relatively comprehensive research has been done in order to choose the best option in the fields of industry, commerce, trade, mining and so on. among different decision-making methods, depending on the data of this study, copras and topsis methods have been selected as the ranking and weighing systems. weighing systems have been used for data recording. in this study, entropy shannon and friedman have been used for this purpose (bulgurcu 2012; zavadskas et al., 2008). data envelopment analysis (dea) has been empirically declared for evaluation of relative efficiency and inefficiency of various companies and industries etc. the main purpose to figure out the dea in industries refers to the sustainability of industries and companies. dea can be calculated via the ratio of output costs to inputs costs. therefore, financial data of input and output from industries are the main information to investigate the performance of industries. so, in parallel with distinguish input and financial performance evaluation of seven indian chemical companies 83 outputs outlay to estimate dea, we tried to find out both financial items of profit and loss of industries. many kinds of research have completed based on limited criteria of industries and they focused on some single group industries or single industry during a certain period. also, they tried to represent their results based on one methodology either dea or financial analysis (sinha 2015). the current study was conducted to an analysis of financial performance of selected companies with respect to liquidity ratios, turn over ratios, solvency ratio, and profitability ratios along with efficiency classification of companies' based on dea and weighing additive models. the final achievement of the present study includes the sustainability progresses of industries and companies. 2. literature review the financial performance of many companies such as tata steel ltd., jindal steel & power ltd., j s w steel ltd., bhushan steel ltd. and steel authority of india ltd evaluated based on liquidity, solvency, activity and profitability ratios in india (arab et al 2015). kettiramalingam et al (2017) estimated the financial performance using productivity and efficiency relationships as a case study industry in india. the obtained results revealed a rise in the performance of the industry in a period of 20 years. to investigate the interplay between executive compensation and companies performance has been used the ratios analysis as main and important variables by raithatha and komera (2016) in indian companies. 50 listed non-financial companies on pakistani stock market investigated for financial performance via working capital management, inventory turnover, cash conversion cycle, average collection period, and average payment period, return on asset, return on equity and earning per share in a period ranging from 2005 to 2014 (bagh et al 2016). lots of methods have been posed for weighting and ranking systems based on multi-criteria networks and financial ratios analysis such as topsis, vikor, waspas, copras, edas, and aras etc. yalcin et al (2012) set up a weighing system in the hierarchical financial performance system and ranked the criteria in the topsis and vikor models. to compare the financial situation of 13 technology companies has been utilized ratios analysis along with the topsis method in the istanbul stock exchange. the results were used to rank the firm during 2009-2011 (bulgurcu 2012). anderkinda and rakhmetova (2013) surveyed the financial outcomes of industries holding an adverse relationship between them such as liquidity decline, profitability loss, financial instability, raise in expenses and etc. by the way, some economic and financial models have released to further studies. the inventory turnover ratio, debtor turnover ratio, investment turnover ratio, fixed assets turnover ratio and total assets turnover ratio were studied to measure the financial performance of a case study steel industry in india (pinku paul and mukherjee 2013). kumar and bhatia (2014) evaluated the financial performance of tata motors and maruti suzuki using ratios analysis including the liquidity, assets, profitability etc. a study by margineana et al (2015) included ratios analysis and the existing relationship among various kinds of ratios, expenses paid for around 700 staff and raw material flow based on real data during 2006 to 2013. fenyves et al (2015) implemented a benchmarking method to evaluate the performance of companies based on financial analysis. so the study pointed out that the dea procedure was a dominant method to investigate the profit-making trend comparison of companies. rezaee and ghanbarpour (2017) carried out research on the dea model for investigating 59 iranian manufacturing units based on linear multianthony et al./decis. mak. appl. manag. eng. 2 (2) (2019) 81-99 84 group relations. by the way, it was developed a score based on dea performance model for industries individually. rahimi et al (2013) applied a dea model for figuring the performance out for around 22 poultry companies in iran. it was matured the efficiency score in dea solver. dea model has been used for financial performance analysis (liquidity, activities, leverage) profitability (output) to find efficient and inefficient industries for around 36 companies in a period of 5 years. findings paved the way for the classification of companies and figure out the reasons for weakness and strong points among 9 efficient and 27 inefficient units in the group (tehrani et al 2012). some attempts done resulted in figure out the financial analysis of around 85 spanish industries using dea model (rodríguez‐pérez et al 2011). according to discussions outlined the dea model is a dominant method for traditional ratio analysis and it also able to measure a prominent procedure to determine the operational and managerial efficiencies of companies and industries etc (feroz et al 2017). dea model used to measure the efficiency level of 15 insurance companies from 2005 to 2012. so, despite demystifying the efficient companies, it has been reported significant fluctuations between the technical efficiency levels obtained in the distinguished time interval (sinha 2015). saranga and nagpal (2016) used a model of dea to distinguish the efficient and inefficient indian airline companies in terms of operational efficiency of drivers. on the other hands, the efficiency of airline companies was obtained in a high relationship with prices and cost efficiency relied on the technical aspect. a study targeted to evaluate the performance of manufacturing 744 small and medium enterprises based on input and output variables in turkey. by the way, it has been reported to exist around 94 efficient units (bulak and turkyilmaz 2014). a study estimated the efficiency score (relies on value-added amounts) of manufacturing companies of both china and turkey via the dea model. the canonical correlation analysis used to figure out the weight values. the t-test analysis has been selected to compare the significant differences between the efficiency values of two groups of companies. the statistical analysis has been manifested the highest efficiency level to chinese companies (bayyurt and duzu 2008). amini and alinezhad (2016) carried out his research using the dea method for ranking 15 iranian industries. in the following steps, it was found around 8 efficient industries with a score of 1. the research conducted by lu et al (2014) used a similar procedure close to dea to figure out the efficiency of industries. the results appeared with the efficiency scores about 0.905 to 0.973 for 34 chinese life insurance companies from 2006 to 2010. an article devoted to assessing the efficiency and performance of around 40 retail workshops via dea method in the portuguese in the period of 2010 to 2013. it has been reported that the technical efficiency complied from a failure. therefore the authors tried to offer some improvement steps of marketing and selling trends (xavier et al 2015). ahmadi and ahmadi (2012) revealed that dea models can provide efficiency scores scaled to a maximum value of 1 to evaluate efficiency and inefficiency of industries (case study conducted among 23 main industries). so, obtained results revealed amounts of around 0.591, 0.418 and 0.484 for iranian recycling industries at efficiency scale, while values were about 1, 1, and 1 at pure technical efficiency during 2005, 2006 and 2007 respectively. also, results asserted that there are 3 major manufacturing industries and two provinces which are identified as the best performers, namely tobacco, transport equipment and coal coke. among 30 provinces, bushehr and north khorasan provinces have the utmost performance. keramidou et al (2011) evaluated the purely technical and scale efficiency of the greek meat products industry from 1994 to 2007 via dea. the results presented the presence of inefficiencies in firms as well as a waning trend the efficiencies due to mismanagement and wastage of capital. financial performance evaluation of seven indian chemical companies 85 rahmani (2017) used the dea model for estimating the industrial productivity of a country. 3. methodology this study has relied on secondary data obtained from valuable resources (website) and then secondary data came through the following procedures. seven indian large chemical companies were chosen as case studies in a period from 2010 to 2018. companies have been chosen from around the top 10 chemical companies in india. an appropriate performance analysis demands a reliable procedure to measure the availability in the best possible situation. it requires a procedure to conduct the empirical methods and practices such as dea, ratios analysis (turn over ratios, liquidity, profitability and solvency). in order to analyze the collected data, the ibm spss statistics 20 and excel package were used. companies were ranked by the topsis, copras and dea models. 3.1. financial ratio analysis to conduct the financial ratios analysis below equations were used to get the results. below displays the applied equations. (1) (current assets/current liabilities) current ratio (2) (quick assets) / (current liabilities) acid test ratio (3) (absolute liquid assets) / (current liabilities) absolute liquid ratio (4) (net credit sales) / (average trade debtors) debtor turnover ratio (5) (total sales / (total assets) total asset turnover (6) (cost of goods sold) / (average inventory) inventory turnover ratio (7) (shareholder funds) / (total assets) equity ratio (8) outsider funds (total debts)/ (shareholder funds or equity) debt equity ratio (9) (total debts) / (total assets) debt to total capital ratio (10) (fixed assets × 100) / ( net worth) fixed assets to net worth ratio (11) (earnings after tax × 100) / (net sales) net profit margin or ratio (12) (earnings before interest & tax (ebit) × 100) / net capital return on net capital then topsis procedure was assigned for ranking of companies and determining the performance values based on ratio analysis values (bulgurcu 2012). 3.2. friedman test the current empirical study of seven indian chemical industries was accomplished to determine the performance of industries. in the spss software structure, there is a test defined as the friedman test. the friedman test was selected to estimate weight values. this test is used by equations 13 to 17 to estimate the weight of criteria and factors in separate columns. the test structure is formatted so that all values in the columns form a matrix with various rows and columns. the weight of each column is anthony et al./decis. mak. appl. manag. eng. 2 (2) (2019) 81-99 86 then estimated by comparing the values in the columns. in this estimation, higher weights are assigned to columns of higher values and medium weights for average values and vice versa. therefore, the friedman test is used as a highly valid test in estimating the weight of numbers with a variety of values. in the matrix of [rij] n×k the entry rij is the estimated weight of xij within the block of i individually. the test statistic is calculated by equation 17 (eisinga et al 2017). ȓ. j = 1 n ∑ 𝑟𝑖𝑗𝑛𝑖=1 (13) ȓ = 1 nk ∑ ∑ 𝑟𝑖𝑗𝑘𝑗=1 𝑛 𝑖=1 (14) sst = n ∑ (ȓ. 𝑗 − ȓ)2.𝑗=1 (15) sse = 1 n(k−1) ∑ ∑ (𝑟𝑖𝑗 − ȓ)2𝑘𝑗=1 𝑛 𝑖=1 (16) q = sst sse (17) 3.3. topsis method topsis method has been defined pertaining to the smallest distance best possible and ideal solution value and largest distance from the negative on unreliable solution value. so the findings based on the present procedure provide a steady rise and fall in the values. the important stages posed in running the process include (1) set up the matrix of data (2) weight estimation base on hwang's rule (3) set up the non-scale matrix (4) figure out the best solutions values (5) finding the relative proximity and ranking the alternatives. to set up the non-dimension matrix was used the equation 18. in this equation, aij is the numerical value of each industry i, according to the index j. the equal weights were assumed about 0.0715 for 15 criteria individually as they provide the same significance (∑wi=1). the symbol of wi is the weight for each ratio or criterion. then, according to equation 19 the weights assigned to the rows of the matrix as a special vector. the special vector has collected the values in the non-scaled matrix. to find the best ideal values (a+) and (a-) were applied the equations of 20 and 21. the largest and smallest values were assumed as the best ideals solutions in the columns individually. then euclidean distance was employed to find the positive and negative ideal solutions for each company. the distances were calculated regarding the equations of 22 to 24. the higher the cli+, the higher the weighting value will be provided (bulgurcu 2012). nd = aij √∑ (𝑎𝑖𝑗)2 𝑚𝑖=1 (18) v = nd × wn. n (19) a+= {(max 𝑉𝑖𝑗|𝑗 ∈ 𝐽), (min 𝑉𝑖𝑗|𝑗 ∈ 𝑗′)|𝑖 = 1,2, … , 𝑚} = {v1+, v2+,..vj+, vn+} (20) a−= {(min i 𝑉𝑖𝑗|𝑗 ∈ 𝐽), (max 𝑉𝑖𝑗|𝑗 ∈ 𝑗′)|𝑖 = 1,2, … , 𝑚} = {v1-, v2-,..vj-, vn-} (21) di+= {∑ (𝑉𝑖𝑗 − 𝑉𝑗 +𝑛𝑗=1 ) 2 }0.5 ; 𝑖, = 1,2,3, … 𝑚 (22) financial performance evaluation of seven indian chemical companies 87 di−= {∑ (𝑉𝑖𝑗 − 𝑉𝑗 −𝑛𝑗=1 ) 2 }0.5 ; 𝑖, = 1,2,3, … 𝑚 (23) cli+= di− di(+)+(𝑑𝑖−) 𝑖 = 1,2, … , 𝑚 (24) 3.4. entropy shannon weighing system this method like other methods needs to compose a matrix for the existing data. to normalize the existing data was employed equation 25, and 26 and 27 for entropy values. the distance between each of the options was obtained from the entropy value using equation 28. it was used the equation of 29 to release the weight of each indicator by excel 2013. pij = xij ∑ 𝑋𝑖𝑗𝑚𝑖=1 𝑗 = 1, … . , 𝑛 (25) ej = −k ∑ 𝑃𝑖𝑗 × 𝐿𝑛 𝑃𝑖𝑗 𝑖 = 1,2, … , 𝑚𝑚𝑖=1 (26) k = 1 ln 𝑚 (27) dj = 1 − ej (28) wj = dj ∑ 𝑑𝑗 (29) 3.5. dea determining the performance of each company is done using the dea method. in this method, the ranking of each option is done according to the weight assigned to it. in this study, the weight of each column was obtained by the friedman test. then the data was sorted by input and output and according to formulas 30 to 34, and the efficiency of the companies was estimated (xavier et al 2015). dea = 0 ≤ ∑ 𝑈𝑟 𝑌𝑟𝑗𝑆𝑟=1 ∑ 𝑉𝑖 𝑋𝑖𝑗𝑚𝑖=1 ≤ 1 (30) max z = ∑ 𝑈𝑟 𝑌𝑟𝑗𝑆𝑟=1 ∑ 𝑉𝑖 𝑋𝑖𝑗𝑚𝑖=1 ≤ 1 , 𝑗 = 1,2,3, … . 𝑛 (31) ur, vi ≥ 0 (32) 𝐷𝐸𝐴 = output (1)weight (1) + output (2)weight (2) + … + output (s)weight (s) input (1) weight (1) + input (2)weight (2) + … + input (m)weight (m) (33) 3.6. ranking system based on copras copras method is a dominant procedure to rank the alternatives that it was introduced in 1996 firstly. the procedure makes it easy for the decision making processes for multi-criteria options. it follows some steps to complete the ranking operation. equation 35 was employed to normalize the decision matrix. by the way, the xij and w are the values and weighted values respectively. to sum the normalized values, figure out the relative importance of alternatives and the greatest value of relative importance (qmax) were used the equation of 36 to 39 respectively. the smin (minimum value of s-i) and nj (ranking amount), s+j, (maximizing criterion of jth alternative) s-i (minimum value of the sum of minimizing criteria of the j-th option) and s-i (minimizing criteria of the j-th option) were distinguished respectively (zavadskas et al., 2008). anthony et al./decis. mak. appl. manag. eng. 2 (2) (2019) 81-99 88 pij = xij .w ∑ 𝑋𝑖𝑗𝑛𝑖=1 𝑖 = 𝛤, 𝑚; 𝑗 = 𝛤, 𝑛 (34) s + j = ∑ +𝑃𝑖𝑗𝑚𝑖=1 𝑖 = 𝛤, 𝑚; 𝑗 = 𝛤, 𝑛 (35) s − j = ∑ −𝑃𝑖𝑗𝑚𝑖=1 𝑖 = 𝛤, 𝑚; 𝑗 = 𝛤, 𝑛 (36) qj = sj+, + s−min × ∑ 𝑠𝑗−𝑛𝐼=1 sj−,∑ ( 𝑆−𝑚𝑖𝑛 𝑆𝑗− )𝑛𝑖=1 = sj+, + ∑ 𝑆𝑗−𝑛𝐼=1 sj−,∑ ( 1 𝑆𝑗− )𝑛𝑖=1 (37) nj = qj 𝑄𝑚𝑎𝑥 ∗ 100 (38) 4. results and discussion 4.1. financial data analysis financial statements (fs) are summaries of the operating, financing, and investment activities of a business. fs should present useful data to both investors and creditors in making credit, investment, and other business decisions. this usefulness means that investors and creditors can use these statements to predict, compare, and evaluate the amount, timing, and uncertainty of potential cash flows. in other words, fs provides the information needed to assess a company's future earnings and therefore the cash flows expected to result from those earnings. by this study, the financial data of 7 indian industries were collected according to table 1. table 1. financial data of industries during 2010-2018 (profit & loss account in rs, cr) tata chemicals (a) (1) 3,447.99 3,591.36 8,170.30 9,984.39 8,590.23 8,440.93 7,912.63 6,225.27 5,411.70 (2) 3,466.01 3,606.80 8,220.86 10,082.06 8,689.64 8,529.87 7,996.25 6,332.86 5,411.70 (3) 194.49 176.92 164.37 194.75 202.92 365.6 308.57 108.03 88.35 (4) 531.39 479.95 2,041.14 3,778.55 3,194.24 2,988.79 2,864.91 2,198.87 2,724.92 (5) -19.7 39.95 591.34 -850.84 130.19 273.78 -409.36 -10.07 171.17 (6) 258.03 266.66 286.27 330.17 267.05 273.56 239.75 207.38 204.66 (7) 86.51 100.98 215.16 186.78 185.32 203.25 210.19 201.49 189.71 (8) 126.55 129.6 153.5 192.71 158.82 214.29 224.68 204.46 187.19 (9) 1,537.82 1,513.61 2,031.18 3,072.81 2,556.19 2,542.98 2,109.54 1,744.50 717.95 gujarat fluorochemicals (b) (1) 2,044.48 1,417.22 1,319.08 1,309.21 1,134.87 1,504.16 2,065.56 978.97 985.57 (2) 2,050.46 1,421.52 1,338.31 1,320.97 1,140.94 1,596.08 2,069.00 982.85 985.57 (3) 103.02 71.12 52.36 56.19 65.06 56.9 57.64 99.53 49.23 (4) 539.38 374.41 335.54 410.09 320.84 303.47 252.35 212.16 377.57 (5) 38.42 1.19 50.63 -47.05 41.05 -75.08 -94.3 39.66 -9.2 (6) 138.35 120.06 103.04 96.16 80.69 74.53 66.53 55.63 56.97 (7) 47.62 35.18 47.73 51.98 55.28 68.95 57.13 29.87 48.03 (8) 152.14 148.84 144.15 123.85 101.7 96.38 77.82 44.86 57.03 (9) 755.3 615.38 559.59 581.94 507.66 588.8 760.65 350.71 83.35 solar industries india (c) (1) 1,230.54 1,094.29 1,084.25 1,009.18 896.76 884.56 722.62 531.21 480.21 (2) 1,273.27 1,137.31 1,089.50 1,014.75 904.03 886.99 723.75 534.01 480.21 (3) 18.23 13.38 10.19 19.83 17.1 17.64 24.97 24.81 20.09 (4) 750.02 678.57 640.97 599.86 489.22 509.02 393 261.62 218.92 (5) -19.46 -1.79 -2.98 2.37 -3.81 -1.61 -1.87 -0.43 0.19 (6) 69 54.35 43.41 40.42 38.69 32.24 24.15 18.88 16.83 (7) 14.23 13.79 7.92 7.24 14.48 21.91 20.09 11.45 8.27 (8) 26.09 19.28 17.72 17.66 12.57 10.31 8.05 6.64 6.32 (9) 154.81 113.27 164.54 161.13 206.44 162.45 109.63 102.62 95.62 financial performance evaluation of seven indian chemical companies 89 gujarat alkalies & chemicals (d) (1) 2,420.13 2,023.04 1,955.67 1,931.81 1,882.85 1,794.31 1,698.22 1,423.17 1,280.47 (2) 2,454.50 2,070.21 1,995.45 1,948.12 1,896.06 1,814.60 1,710.97 1,434.68 1,280.47 (3) 105.74 55.92 46.23 48.95 30.27 18.7 11.77 12.01 49.26 (4) 1,177.41 1,132.21 1,219.66 675.57 717.22 714.75 720.8 615.79 807.34 (5) 1.47 5.68 -1.95 25.04 -4.47 3.71 -30.57 -4.66 6.2 (6) 201.39 169.76 162.1 167 151.44 118.9 118.13 114.93 119.91 (7) 14.9 12.83 9.93 9.34 6.36 8.34 20.53 21.17 17.48 (8) 127.32 110.92 107.44 98.06 150.65 151.52 138.95 133.12 121.55 (9) 285.78 308.67 268.82 800.46 658.4 481.93 506.43 423.31 64.89 phillips carbon black (e) (1) 2,542.63 1,924.04 1,892.03 2,467.24 2,276.10 2,280.72 2,180.65 1,690.14 1,232.57 (2) 2,546.98 1,926.95 1,894.10 2,470.19 2,277.46 2,284.91 2,186.78 1,695.72 1,232.57 (3) 19.73 18.94 16.76 14.42 20.51 9.27 10.44 20.91 25.62 (4) 1,650.89 1,221.26 1,291.46 1,864.41 1,856.05 1,889.63 1,701.80 1,228.17 937.15 (5) 8.42 15.28 35.61 43.09 -25.54 -26.03 -43.57 11.24 -15.06 (6) 97.18 81.8 72.61 70.16 62.91 58.43 52.35 47.7 36.66 (7) 41.44 51.45 72.1 94.8 80.23 72.13 67.63 43.75 31.21 (8) 60.52 60.62 62.15 57.53 53.74 50.79 48.59 38.58 31.15 (9) 404.44 349.96 311.06 307.34 358.48 289.27 267.01 182.83 100.8 gujarat heavy chemicals (f) (1) 2,905.65 2,780.70 2,532.19 2,361.58 2,210.82 2,106.28 1,868.88 1,469.11 1,215.87 (2) 2,905.65 2,780.70 2,532.19 2,373.61 2,224.21 2,124.95 1,896.73 1,498.17 1,215.87 (3) 35.75 10.78 7.47 11.26 5 2.98 9.63 13.3 14.28 (4) 1,100.08 1,069.91 900.42 903.92 888.6 790.91 770.75 593.61 655.15 (5) 23.62 -43.53 -5.33 -12.77 -10.13 5.58 -24.35 -25.15 11.41 (6) 176.37 158.13 133.24 125.87 121.99 111.03 99.93 95.67 82.98 (7) 124.16 133.77 162.82 163.84 170.53 157.96 184.96 110.43 103.39 (8) 109.53 85.69 81.74 84.45 81.57 81.97 80.85 84.4 76.11 (9) 866.62 790.79 806.23 772.06 762.08 739.55 583.56 479.04 131.13 upl (g) (1) 7,091.00 6,794.00 5,821.76 5,226.20 4,814.85 3,826.27 3,216.99 2,822.46 2,699.10 (2) 7,263.00 6,939.00 5,982.53 5,334.99 4,968.27 3,939.44 3,308.00 2,911.09 2,699.10 (3) 435 325 458.78 240.47 317.84 134.32 151.49 153.59 103.88 (4) 3,517.00 3,029.00 2,833.75 2,438.76 2,014.58 1,838.39 1,557.89 1,270.96 1,415.03 (5) 2 -108 -66.28 -207.37 -153.99 -38.2 -116.85 -51.05 108.57 (6) 486 445 390.41 317.8 257.87 237.46 184.65 153.12 127.36 (7) 135 149 192.61 35.27 243.29 105.99 164.37 293.64 108.34 (8) 666 655 243.94 186.75 169.09 157.76 143.49 114.68 107.91 (9) 1,905.00 1,929.00 1,720.56 1,630.12 1,380.77 1,127.93 876.67 788.52 508.63 revenue from operations [net] (1), total operating revenues (2), other income (3), cost of materials consumed (4), changes in inventories of fg,wip and stock-in trade (5), employee benefit expenses (6), finance costs (7), depreciation and amortization expenses (8), other expenses (9) based on existing data in table 1, one sample t-test had shown a significant difference around 0.001 among criteria such as revenue from operations [net], total operating revenues, other income, cost of materials consumed, changes in inventories of fg, wip and stock-in trade, employee benefits expenses, finance costs, depreciation and amortization expenses and other expenses. it was found the amount of around 0.806 for the cronbach, s alpha reliability test. the distributions of revenue from operations (net), total operation revenues, distribution of other income, distribution of changes in inventories of fg, wip, and stock-in-trade, depreciation amortization expenses and other expenses were obtained normally with mean and standard deviation of 2843.29 and 2273.20, 2877.84 and 2308.48, 2877.84 and 2306.48, 88.06 and 110.02, 64.72 and 140.32, 118.69 and 115.19, 762.99 and 702.47 based on one sample kolmogorov-simonov test. therefore, the null hypothesis was retained for them respectively. the distributions of the cost of materials consumed, employee benefit expenses and finance cost with the mean and standard deviation of 1229.84 and 912.45, 144.00 and 104.08, 88.18 and 74.21 were also achieved normally based on the same test but null hypothesis was rejected for them respectively. chianthony et al./decis. mak. appl. manag. eng. 2 (2) (2019) 81-99 90 square test had revealed a value of 0.000 for all criteria such as revenue from operation (net), total operating revenues, other income, cost of materials consumed, changes in inventories of fg, wip and stock-in-trade, employee benefit expenses, finance costs, depreciation and amortization expenses and other expenses. the friedman test was revealed the mean weights around 8.08, 8.92, 2.68, 6.83, 1.71, 4.37, 2.89, 3.38 and 6.14 for the revenue from operation (net), total operating revenues, other income, cost of materials consumed, changes in inventories of fg, wip and stockin-trade, employee benefit expenses, finance costs, depreciation and amortization expenses and other expenses respectively (with a chi-square value around 446.966). 4.2. performance ranking by topsis based on financial ratio analysis tables 2 and 3 present the data associated with financial ratio analysis from 2010 to 2018 for 7 indian industries and weighted matrix respectively. the columns of tables were composed with the following layout. liquidity ratio (current ratio (1), quick ratio (2), cash ratio (3)); turnover ratio (debt turnover ratio (4), assets turnover ratio (5), inventory turnover ratio (6)); solvency ratio (equity ratio (7), debt-equity ratio (8), debt to total capital ratio (9), (fixed assets/net worth ratio (10))); profitability ratio (net profit margin ratio (11), (return on net worth/equity ratio (12)), return on capital employed ratio (13), return on assets ratio (14), (total debt/equity ratio (15))). the vector of a+=0.036499171, 0.03880029, 0.0314006, 0.03763651, 0.041673431, 0.047213935, 0.042005393, 0.061909557, 0.041915856, 0.047567033, 0.04991575, 0.017316467, 0.042615593, 0.042736267, 0.047403448. the vector of a-= 0.01609707, 0.012867944, 0.002692456, 0.013447954, 0.014037746, 0.014127247, 0.00378427, 0.003621045, 0.008111058, 0.007857528, 0.011726501, 0.007936345, 0.014754619, 0.011453475, 0.004778268. table 4 displays the topsis ranking system results. financial performance evaluation of seven indian chemical companies 91 t a b le 2 . d a ta o f fi n a n ci a l ra ti o a n a ly si s fr o m 2 0 1 0 t o 2 0 1 8 f o r se v e n c o m p a n ie s 1 5 1 4 1 3 1 2 1 1 1 0 9 8 7 6 5 4 3 2 1 c o 0 .4 3 5 .8 4 8 .0 6 1 0 .3 1 3 .2 2 3 2 0 7 .3 6 2 1 .9 3 9 5 3 .2 2 .2 5 7 .8 8 6 1 .7 6 7 3 2 .5 7 .8 8 1 .0 9 1 .3 9 a 0 .2 7 8 .1 2 1 0 .6 9 1 2 .4 2 2 0 .4 3 9 2 5 6 .1 3 1 7 .3 2 5 6 8 7 .7 1 0 .3 4 .5 9 3 8 .9 4 5 2 7 .7 5 8 9 0 .9 3 1 .4 b 0 .4 4 1 0 .9 7 1 6 .0 3 1 9 .5 5 9 .4 7 6 2 0 3 .3 7 2 2 .6 7 9 1 4 .6 9 2 .5 1 1 5 .3 4 1 1 5 .6 7 6 4 .7 7 7 6 .4 2 1 .2 1 .4 9 c 0 .1 2 6 6 .8 6 8 .9 7 1 0 .3 7 1 2 .5 8 6 3 1 .2 5 7 .7 5 3 3 2 .6 7 2 .3 1 0 .5 8 5 5 .8 7 6 0 8 .6 5 9 1 .9 1 .5 8 1 .9 5 d 1 .2 5 2 .9 4 5 .5 5 8 .9 6 3 0 .3 5 1 5 3 6 8 .7 7 3 5 .8 9 2 2 7 5 .9 4 1 .5 9 6 .7 9 9 3 .3 8 4 4 1 .6 9 1 0 .5 8 0 .8 6 e 1 .1 5 6 .4 1 1 .5 2 1 7 .7 6 8 .9 4 1 9 4 1 6 .3 6 4 0 .0 5 1 2 0 4 .1 3 3 .3 3 5 .2 1 7 1 .5 4 8 8 9 .9 8 9 0 .7 8 0 .5 2 4 0 .9 5 4 f 0 .4 5 4 .4 5 6 .5 7 9 .1 6 7 .1 3 5 4 4 5 .9 6 2 0 .5 1 5 2 0 .9 5 1 .3 3 6 6 0 .7 3 1 8 7 0 .7 1 .3 7 1 .7 4 g anthony et al./decis. mak. appl. manag. eng. 2 (2) (2019) 81-99 92 t a b le 3 . w e ig h te d m a tr ix 8 7 6 5 4 3 2 1 c o 0 .0 1 0 3 7 5 3 8 7 0 .0 2 8 3 8 2 0 2 2 0 .0 2 4 2 5 3 3 1 2 0 .0 2 2 2 6 4 2 8 3 0 .0 3 0 9 7 6 8 1 3 0 .0 0 2 6 9 2 4 5 6 0 .0 2 6 7 6 7 2 8 8 0 .0 2 6 0 1 7 3 5 8 a 0 .0 6 1 9 0 9 5 5 7 0 .0 0 3 7 8 4 2 7 0 .0 1 4 1 2 7 2 4 7 0 .0 1 4 0 3 7 7 4 6 0 .0 2 2 3 1 8 1 0 6 0 .0 3 0 4 0 9 7 2 2 0 .0 2 2 8 3 8 1 4 5 0 .0 2 6 2 0 4 5 3 3 b 0 .0 0 9 9 5 6 2 1 3 0 .0 3 1 6 6 1 7 2 3 0 .0 4 7 2 1 3 9 3 5 0 .0 4 1 6 7 3 4 3 1 0 .0 3 2 3 4 1 4 8 4 0 .0 2 6 1 1 1 3 5 9 0 .0 2 9 4 6 8 5 7 4 0 .0 2 7 8 8 9 1 1 c 0 .0 0 3 6 2 1 0 4 5 0 .0 2 9 0 1 2 7 3 4 0 .0 3 2 5 6 3 4 5 7 0 .0 2 0 1 4 0 9 5 7 0 .0 2 5 7 3 9 3 0 .0 3 1 4 0 0 6 0 .0 3 8 8 0 0 2 9 0 .0 3 6 4 9 9 1 7 1 d 0 .0 2 4 7 7 3 1 4 0 .0 2 0 0 5 6 6 2 9 0 .0 2 0 8 9 8 4 7 6 0 .0 3 3 6 6 3 1 9 2 0 .0 1 8 6 7 4 8 9 5 0 .0 3 1 0 9 3 0 8 6 0 .0 1 4 2 4 3 1 4 4 0 .0 1 6 0 9 7 0 7 e 0 .0 1 3 1 0 6 7 0 8 0 .0 4 2 0 0 5 3 9 3 0 .0 1 6 0 3 5 5 0 2 0 .0 2 5 7 8 9 9 4 2 0 .0 3 7 6 3 6 5 1 0 .0 3 1 0 1 7 9 1 6 0 .0 1 2 8 6 7 9 4 4 0 .0 1 7 8 5 6 5 1 8 f 0 .0 1 6 5 5 5 2 2 9 0 .0 1 6 7 7 6 9 2 9 0 .0 1 8 4 6 6 9 8 9 0 .0 2 1 8 8 2 1 5 6 0 .0 1 3 4 4 7 9 5 4 0 .0 2 4 1 5 6 9 3 6 0 .0 3 3 6 4 3 2 8 9 0 .0 3 2 5 6 8 4 9 1 g 1 5 1 4 1 3 1 2 1 1 1 0 9 c o 0 .0 1 6 3 0 6 7 8 6 0 .0 2 2 7 5 1 1 2 1 0 .0 2 1 4 2 7 4 2 8 0 .0 0 9 1 2 3 2 5 3 0 .0 2 1 7 4 2 5 4 4 0 .0 0 7 8 5 7 5 2 8 0 .0 2 2 9 5 1 6 7 9 a 0 .0 1 0 2 3 9 1 4 5 0 .0 3 1 6 3 3 4 0 8 0 .0 2 8 4 1 9 2 5 7 0 .0 1 1 0 0 1 0 4 9 0 .0 3 3 6 0 0 6 1 9 0 .0 2 2 6 7 6 0 6 5 0 .0 1 8 1 2 6 9 0 7 b 0 .0 1 6 6 8 6 0 1 4 0 .0 4 2 7 3 6 2 6 7 0 .0 4 2 6 1 5 5 9 3 0 .0 1 7 3 1 6 4 6 7 0 .0 1 5 5 7 5 0 3 0 .0 1 5 1 9 7 2 8 2 0 .0 2 3 7 2 6 1 5 4 c 0 .0 0 4 7 7 8 2 6 8 0 .0 2 6 7 2 4 7 7 6 0 .0 2 3 8 4 6 6 5 4 0 .0 0 9 1 8 5 2 5 6 0 .0 2 0 5 5 8 3 8 1 0 .0 2 1 1 4 5 2 0 7 0 .0 0 8 1 1 1 0 5 8 d 0 .0 4 7 4 0 3 4 4 8 0 .0 1 1 4 5 3 4 7 5 0 .0 1 4 7 5 4 6 1 9 0 .0 0 7 9 3 6 3 4 5 0 .0 4 9 9 1 5 7 5 0 .0 3 7 6 5 1 0 7 3 0 .0 3 7 5 6 2 0 4 9 e 0 .0 4 3 6 1 1 1 7 2 0 .0 2 4 9 3 2 7 3 6 0 .0 3 0 6 2 5 8 0 3 0 .0 1 5 7 3 0 9 6 9 0 .0 1 4 7 0 3 3 5 4 0 .0 4 7 5 6 7 0 3 3 0 .0 4 1 9 1 5 8 5 6 f 0 .0 1 7 0 6 5 2 4 1 0 .0 1 7 3 3 6 0 4 3 0 .0 1 7 4 6 6 2 7 8 0 .0 0 8 1 1 3 4 9 5 0 .0 1 1 7 2 6 5 0 1 0 .0 1 3 3 4 1 7 4 7 0 .0 2 1 4 5 5 0 5 7 g financial performance evaluation of seven indian chemical companies 93 table 4. topsis ranking system results ranks cli+ (di+)+(di-) didi+ co. 6 0.317233465 0.142612596 0.045241488 0.097371108 a 4 0.476166445 0.161005577 0.076665453 0.084340124 b 3 0.495475692 0.160104742 0.079328008 0.080776734 c 5 0.38164729 0.158292269 0.060411816 0.097880454 d 2 0.520332447 0.160900794 0.083721904 0.07717889 e 1 0.526029684 0.168044337 0.08839631 0.079648027 f 7 0.308841556 0.144121922 0.044510839 0.099611083 g 4.3. performance analysis based on financial data using dea method in many studies the financial performance evaluation ratios have been defined as asset turnover ratio (input/output), inventory turnover ratio (input/output), receivable accounts turnover ratio (input), quick ratio (input), current ratio (input), cash earned from set activities to company earning ratio (input), interest coverage ratio (input), total debt to equity ratio (input), debt ratio (input/output), earning per share ratio (output), return on assets ratio (output), net profit margin ratio (output), economic value added (output), growth rate of sales (output), growth rate of earnings per share (output), sustainable growth rate (output), price to earnings ratio (input/output), tobin q ratio (output). a study determined the universe of input/output parameters of introduced into dea equations including return on equity, return on assets, net profit margin, earnings/share, receivables turnover, inventory turnover, current ratio, quick ratio, debt to equity ratio, leverage ratio, solvency ratios, price to earnings ratio, price to book ratio, revenue growth rate, net income growth rate and eps growth rate (edirisinghe and zhang 2010). dea is a non-statistical method methodology is used to measure performance in a relative manner and each producer unit or decision maker is compared to the best unit in that industry. of course, the higher the number of units, the better the comparison and the more realistic results. simple ratios do not lead to ranking and comparison of companies' performance, and multiple inputs and outputs in this field should be used. also, through the method of dea, there is no need for a definite form of production function as it is in the economy, and this technique can be used with minimal data. according to our knowledge, financial ratios and indicators make an ad hoc and a relative appraise of corporate performance, however, we know dea can be employed to develop very complex investigations (fenyves et al 2015). table 5 shows the dea score for the seven indian chemical companies [this study]. anthony et al./decis. mak. appl. manag. eng. 2 (2) (2019) 81-99 94 t a b le 5 . d e a s co re f o r th e s e v e n i n d ia n c h e m ic a l co m p a n ie s d e a c o . p ro d u ct iv it y w e ig h ts o u tp u t w e ig h ts in p u t 1 a 3 .8 4 1 6 7 7 0 9 8 .0 8 r e v e n u e f ro m o p e ra ti o n s [n e t] 6 .8 3 c o st o f m a te ri a ls c o n su m e d 0 .9 6 3 b 3 .6 9 7 5 7 9 8 1 7 8 .9 2 t o ta l o p e ra ti n g re v e n u e 1 .7 1 c h a n g e s in i n v e n to ri e s o f f g ,w ip a n d s to ck -i n t ra d e 0 .8 6 3 c 3 .3 1 4 7 2 5 3 4 9 2 .6 8 o th e r in co m e 4 .3 7 e m p lo y e e b e n e fi t e x p e n se s 0 .8 1 7 d 3 .1 3 6 2 2 5 5 1 2 2 .8 9 f in a n ce c o st s 0 .7 1 3 e 2 .7 3 7 6 9 9 4 7 7 3 .3 8 d e p re ci a ti o n a n d a m o rt iz a ti o n e x p e n se s 0 .8 6 2 f 3 .3 0 8 2 7 6 2 0 4 6 .1 4 o th e r e x p e n se s 0 .8 1 8 g 3 .1 3 8 6 9 0 4 7 4 w e ig h ts v a lu e s b a se d o n f ri e d m a n t e st financial performance evaluation of seven indian chemical companies 95 4.4. performance analysis based on financial data using copras method the criteria used for weighing by entropy shannon were encompassed; revenue from operations [net] (1), total operating revenues (2), other income (3), cost of materials consumed (4), changes in inventories of fg,wip and stock-in trade (5), employee benefit expenses (6), finance costs (7), depreciation and amortization expenses (8), other expenses (9). there are negative and positive relations among 9 aforementioned criteria. therefore, the weighting and ranking systems were selected entropy shannon and copras. table 6 includes weighted values based on entropy shannon procedure. table 6. weighted values based on entropy shannon procedure criteria e dj=1-ej wj ∑ 𝑑𝑗 k 1 1.995278628 -0.99527863 0.133719351 7.443041120.5139 2 1.994522361 -0.99452236 0.133617744 3 1.817204902 -0.8172049 0.10979449 4 2.001968116 -1.00196812 0.134618108 5 0.776434672 0.223565328 -0.03003683 6 2.008943625 -1.00894363 0.135555294 7 1.946715084 -0.94671508 0.12719466 8 1.959818549 -0.95981855 0.128955159 9 1.942155183 -0.94215518 0.12658202 table 7. the ranking system developed in copras method co. total revenue total expenses rank based on revenue score rank based on expenses score a 35.44 131.94 3 2 b 13.75 44.2 6 4 c 57.04 291.766 2 1 d 14.3 29.67 5 7 e 15.12 38.7 4 5 f 11.44 31.23 7 6 g 84.089 53.883 1 3 it was found a significant difference about 0.012 between total revenue and total expenses values (between seven industries) in table 7 according to the t-test analysis. 4.5. the relationship between the weights values obtained from the friedman test and entropy shannon it was conducted a scatter plot for the data of weights values obtained from the friedman test and entropy shannon base on the results of profit & loss accounts according to figure 1. anthony et al./decis. mak. appl. manag. eng. 2 (2) (2019) 81-99 96 figure 1. scatter plot developed for the weights values obtained from the friedman test and entropy shannon according to the t-teat analysis, there is no significant difference between the weights values obtained from the friedman test and entropy shannon. moreover, the scatter plot is representing that there is a relatively linear relationship between both weight values obtained from friedman test and entropy shannon with receding the weight values associated to a criterion of changes in inventories of fg, wip and stockin-trade. 5. conclusion by the present study, we tried to figure out the efficiency of seven indian industries. the obtained results for the efficiency of industries were approached to full efficiency of industries in most cases. the statistical analysis revealed significant differences among the data of industries. the friedman test has provided valuable weights for raw values. the entropy shannon weighting system has provided the positive and negative weights for existing values and also sought the highest consistency with the copras ranking system. by the way, the copras ranking system had classified industries based on negative and positive criteria (expenses and revenues). the topsis procedure ranked the industries based on the available ratio analysis and it has emerged a good agreement among the industries ratio values. the profit and loss analysis made clear the output incomes and input expenses. also, it resulted in output and input criteria for introducing into the dea model. the findings based on the copras model predict the situation of industries for the further financial statement concept. with regard to a rise in the expenses, the ranking system for the income will be taken lots of fluctuations. acknowledgement: this research was conducted as corresponding author phd research work. i would like to extend my thanks to the managers and colleagues in -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.002.004.006.008.0010.00 financial performance evaluation of seven indian chemical companies 97 osmania university because of their support in offering and collecting the data and resources in running the program. references ahmadi, v. & ahmadi a. 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(2008). a new logarithmic normalization method in games theory. informatica, 19(2), 303-314. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 2, issue 2, 2019, pp. 1-18. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame1902001v * corresponding author. e-mail addresses: m.j.vilela-ibarra@rgu.ac.uk (m. vilela), g.f.oluyemi@rgu.c.uk (g. oluyemi), a.petrovski@rgu.ac.uk (a. petrovski). a fuzzy inference system applied to value of information assessment for oil and gas industry martin vilela*1, gbenga oluyemi 1 and andrei petrovski 2 1 school of engineering, robert gordon university, aberdeen, united kingdom 2 school of computing, robert gordon university, aberdeen, united kingdom received: 20 march 2019; accepted: 21 may 2019; available online: 23 may 2019. original scientific paper abstract: value of information is a widely accepted methodology for evaluating the need to acquire new data in the oil and gas industry. in the conventional approach to estimating the value of information, the outcomes of a project assessment relate to the decision reached following boolean logic. however, human thinking logic is more complex and include the ability to process uncertainty. in addition, in value of information assessment, it is often desirable to make decisions based on multiple economic criteria, which, independently evaluated, may suggest opposite decisions. artificial intelligence has been used successfully in several areas of knowledge, increasing and enhancing analytical capabilities. this paper aims to enrich the value of information methodology by integrating fuzzy logic into the decisionmaking process; this integration makes it possible to develop a human thinking assessment and coherently combine several economic criteria. to the authors’ knowledge, this is the first use of a fuzzy inference system in the domain of value of information. the methodology is successfully applied to a case study of an oil and gas subsurface assessment where the results of the standard and fuzzy methodologies are compared, leading to a more robust and complete evaluation. key words: value of information, fuzzy logic, fuzzy inference system, oil and gas industry, uncertainty. mailto:m.j.vilela-ibarra@rgu.ac.uk mailto:g.f.oluyemi@rgu.c.uk mailto:a.petrovski@rgu.ac.uk vilela et al./decis. mak. appl. manag. eng. 2 (2) (2019) 1-18 2 1. introduction 1.1. review of value of information value of information (voi) is a prescriptive methodology embedded in the discipline of decision analysis that has the aim of assessing the value assocted with gathering information. to that end, the methodology maximizes an objective function, which defines the value of a project. grayson (1960), raiffa and schlaifer (1961) and newendorp (1967) were the pioneers in the field of decision making for data acquisition in the oil and gas industry. subsequently, more research and applications, such as those of warren (1983), lohrenz (1988), demirmen (1996), newendorp and schuyler (2000) and koninx (2000), among others, expanded the scope of the subject, adding more robustness to the methodology. recently, more applications have emerged—like those of clemen (1996), coopersmith and cunningham (2002), suslick and schiozer (2004)—which enrich the process of assessing the voi decision problem from a methodological perspective. several papers, such as walls (2005) and vilela et al (2017), have discussed the use of utility theory in voi assessment in the oil and gas industry; similarly, santos and schiozer (2017) discussed the impact of the risk attitude of the decision makers in voi assessments; kullawan et al (2017) developed a discretized-programming approach, based on value of information, to optimize stochastic-dynamic the geosteering operations; steineder et al (2018) discussed the maximization of the voi on a horizontal polymer pilot project; all these researchers used one or more crisp decision criteria to make decisions. in the oil and gas industry, the scope of a project varies from the complex exploitation of hydrocarbon fields to theoretical reservoir studies or laboratory tests; project’s economic benefits are calculated based on the estimated figures of hydrocarbons’ production and price, operating cost, taxes, royalties, and investments. all these figures carry uncertainties because it is not possible to predict their future fluctuations accurately—in particular, future hydrocarbon production is uncertain due to a combination of: (a) the uncertainties associated with the reservoir parameters (permeability, thickness, top reservoir, well producibility, aquifer support, etc.); (b) the uncertainties associated with the methods used to estimate future production based on the reservoir parameters (dynamic reservoir models, decline curve analysis, etc.) on occasion, additional data can be acquired to change the uncertainty in the reservoir parameters; however, acquiring data involves a cost that could be greater than the benefits of the data. changes in the reservoir parameters’ uncertainties translate into changes in the value of the project. in general, acquiring additional data makes sense in cases in which the outcome from the data acquisition can change the decisions being made. for a project with uncertain outcomes, the voi is the difference between the expected value (ev) of the project with and without the newly acquired data (clemen, 1996): with information without informationvoi ev ev  (1) where both values, 𝐸𝑉𝑤𝑖𝑡ℎ 𝑖𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 and 𝐸𝑉𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝑖𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 , assess the outcome of the project in these circumstances and refer to a future situation. a fuzzy inference system applied to value of information assessment for oil and gas industry 3 in the ‘without-information’ case, for 𝜅 possible scenarios (which include endorsing the project with the current knowledge and uncertainties and the alternative of relinquishing it), the ev of the project corresponding to the 𝑗𝑡ℎ scenario is defined as:     1 n j ji i i v a u p s   (2) where 𝑢𝑗𝑖 is the value of the state of nature 𝑠𝑖 for the scenario 𝑎𝑗 and 𝑝(𝑠𝑖 ) is the prior probability of the state of nature 𝑠𝑖 . the most often used decision criterion is to select the alternative that maximizes the ev:    * max j j ev a ev a (3) equation (3) is the 𝐸𝑉𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝑖𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 term shown in equation (1). similarly, in the ‘with-information’ alternative, for 𝜅 possible scenarios and for each possible data outcome, 𝑥𝑘 , the ev for the 𝑗 𝑡ℎ alternative is:   1 | ( | ) n j k ji i k i ev a x u p s x   (4) where 𝑢𝑗𝑖 is the value of the state of nature 𝑠𝑖 for the scenario 𝑎𝑗 ; 𝑝(𝑠𝑖 |𝑥𝑘 ) is the posterior probability of the state 𝑠𝑖 given the outcome 𝑥𝑘 ; and the term 𝐸𝑉(𝑎𝑗 |𝑥𝑘 ) is the expected project value for the 𝑗𝑡ℎ alternative given the outcome 𝑥𝑘 . similarly, as in the ‘without-information’ case, the optimum alternative in the ‘withinformation’ case for a given data outcome 𝑥𝑘 (ev conditioned on the outcome 𝑥𝑘 ) is the one that maximizes the ev:  *| max ( | )k j k j ev a x ev a x (5) the unconditional maximum ev (which is the ev of the project considering the data acquisition outcomes) is the sum of the conditional ev weighted with the corresponding marginal probabilities:  * * 1 ( | ) ( ) n k k k ev a ev a x p x   (6) the voi is the difference between the estimates of ev in equation (6) and equation (3). so far, the discussion has focused on the classical methodology to assess the voi in a decision problem in which the output values (hydrocarbon production, total benefits, etc.) are uncertain due to uncertainties in the input variables; these uncertainties have been included using probabilistic measures. in the next section, we will include the imprecision in the input variables by making use of fuzzy logic. 1.2. review fuzzy logic fuzzy logic, pioneered by zadeh (1965), is one of the most prolific areas of artificial intelligence, which has enriched the analysis of challenging and complex problems. vilela et al./decis. mak. appl. manag. eng. 2 (2) (2019) 1-18 4 bellman and zadeh (1970) introduced an important distinction between randomness and fuzziness: while randomness relates the uncertainty concerning membership or non-membership of an object or event to a non-fuzzy set (a crisp set), fuzziness deals with classes in which there may be degrees of membership (between the fulland the no-membership relationship). these distinct sources of uncertainties are managed during different phases of the voi assessment: phase 1) assessing: assesses, using one or more decision criteria, whether the new data add value to the project or not. phase 2) categorization: relates the values obtained during the assessing phase to the options available for the decision problem. during the phase 1, the uncertain nature of the input variables (e.g. reservoir parameters) and outcome values (e.g. financial benefits or economic parameter values) is captured using probabilistic analysis based on ev calculations. in the standard voi approach, phase 2 is implemented using crisp criteria to make decisions that do not correspond with human fuzzy thinking for making decisions. following bellman and zadeh (1970), the uncertainty related to fuzziness is a major source of uncertainty in many decision processes. in classical set theory, the events (values) belong (or not) to sets in a crisp manner that is represented by the “characteristic function”, defined by equation (7), which is a mapping from the input variables to the boolean set {0,1}: 1 , 0, k m x m otherwise     (7) thus, an event (e.g. x) belongs totally or not at all (1 or 0) to a set; these kinds of relationships follow boolean logic. as a practical example, in subsurface reservoirs, the characteristic function allows establishing a boolean relationship (1 or 0, i.e. totally belongs or totally excluded) between quantitative input variables (e.g. reservoir depth of 5000 ft) and descriptive terms (e.g. deep reservoir). fuzzy logic extends the mapping between events and sets using the membership function (mf) to include all the values between 0 and 1, [0,1]; mathematically, the mf is a mapping from a given universe of discourse “x” to the continuous unit intervals that are the membership values. equation (8) shows the mathematical expression for the mf:      / 0,1m x y y   (8) which shows that the values of the mf belong to the interval [0,1]. the membership values measure the degree of belonging of each event to a given set, representing the “degree of membership” of the mentioned event to that set. in this logic, an event (e.g. reservoir depth of 5000 ft) belongs partially (with a value between 0 and 1) to a set (e.g. deep reservoir). in the standard voi, the results of the assessment are a set of crisp values that measure the project benefits or losses of the different alternatives under evaluation. comparing those values with a set of threshold values, a decision is made regarding the project fate; however, a decision made in this manner is limited because it does not follow the human thinking for decision making which works with fuzzy categories like “the project is viable to endorse”, “the project is unviable to endorse” or “the project needs reframing”. a fuzzy inference system applied to value of information assessment for oil and gas industry 5 1.3. review fuzzy inference systems in practice, fuzzy logic is implemented through a process called a “fuzzy inference system” (fis). a fis is a non-linear procedure that derives its output based on fuzzy reasoning and a set of if-then rules. the fis performs approximate reasoning like the human brain, albeit in a much more straightforward manner. the fis is one of the most prolific applications of fuzzy logic. it has been used recently in very different areas and within various problem domains, such as: the assessment of water quality in rivers (ocampo, 2008), the improvement of image expansion quality (sakalli et al, 1999), the differential diagnosis of non-toxic thyropathy (guo and ling, 2008), the development of a fuzzy logic controller for a traffic junction (pappis and mamdani, 1997), maintenance scheduling of smart grid systems (malakhov et al, 2012), the design of fire monitoring sensors in coal mines using fuzzy logic (muduli et al, 2017), the estimation of the impact of tax legislation reforms on the tax potential (musayev et al, 2016), pipeline risk assessment (jamshidi et al., 2013), depression diagnosis (chattopadhyay, 2014), river discharge prediction assessments (jayawardena et al., 2014), geological strength index calculation and slope stability assessments (sonmez et al, 2004), the regulation of industrial reactors (ghasem, 2006) and the use of a fuzzy logic approach for file management and organization (gupta, 2011). in the domain of the oil and gas industry several applications of fis have been reported such as the streamline based fuzzy logic workflow to redistribute water injection by accounting for operational constraints and number of supported producers in a pattern (bukhamseen et al, 2017), the identification of horizontal well placement (popa, 2013), for estimating strength of rock using fis (sari, 2016), for predicting the rate of penetration in shale formations (ahmed et al, 2019). fuzzy logic has been used in combination with others artificial intelligence techniques such as adaptative neuro-fuzzy inference system (anfis) on practical applications, e.g. for predicting the inflow performance of vertical wells producing two-phase flow (basfar et al, 2018) or to predict geomechanical failure parameters (alloush et al, 2017); fis has also been used in conjunction with analytical hierarchical processes to evaluate the water injection performance in heterogeneous reservoirs (oluwajuwon and olugbenga, 2018). from the point of view of applications, there are two kinds of fis (guillaume, 2001): 1) fuzzy expert systems or fuzzy controllers: fuzzy rules built on expert knowledge. this kind of fis uses the ability of fuzzy logic to model natural language. 2) automatic learning from data: neural networks have become the most popular tool using a numerical performance index, typically based on the mean square error. these kinds of development are distinguished by their accuracy, and their main drawback is their “black-box” approach. for the current application, we will focus on the first kind of fis. from a methodological perspective, the fis can be understood as a general procedure that transforms a set of input variables into a set of outputs, following the workflow shown in figure 1. vilela et al./decis. mak. appl. manag. eng. 2 (2) (2019) 1-18 6 figure 1. fuzzy inference system as shown in figure 1, fis as a procedure entailing five blocks in which the inputs and outputs are in crisp form. for a mandani fis, shown in figure 1, the outcome is a crisp number, independently of the number of crisp parameters used the asses the value of the project (e.g. npv, dpi, irr, etc.); this is fis aggregation process; in general, higher fis values means higher value of the project and vice versa. 1.4. objective of this research work the objective of this work is to investigate whether considering the fuzzy nature of human thinking can impact the decision’s assessment in voi problems, especially in oil and gas projects; to reach this objective we integrate fuzzy inference system into the voi assessment. 2. application 2.1. reservoir description an exploration campaign conducted in algeria discovered a medium-sized oil field located at 5200 ft. tvd ss. four wells were drilled—the discovery well and three appraisal wells. the range of original oil in place (minimum and maximum figures) has been assessed; the fluid characteristics are known based on samples taken from the appraisal wells. the operator’s technical team has estimated, based on the available information, technical experience, and analogue fields, that the main source of subsurface uncertainty is the well productivity. the four wells drilled were tested for six hours; however, considerable uncertainty remains regarding well productivity due to reasons described in table 1. a fuzzy inference system applied to value of information assessment for oil and gas industry 7 table 1. causes of well productivity uncertainty reason for uncertainty comment quality and reliability of the well test possible calibration issues on well testing equipment duration of the tests well test period too short, no enough to reach stabilized flow based on the information gathered during the exploration phase and from similar fields in the same basin, a material balance model is built to represent the forecast oil rate for the high-, mediumand low-development scenarios, as shown in figure 2. figure 2. high, medium and low cases of the oil production rate the difference between the profiles is the well model used in each case. the full development of the field includes twelve vertical wells, four of which have already been drilled—at present these are “suspended”, to be used in the development phase of the project if a decision is taken to move the project forwards; otherwise, those wells should be abandoned entirely. the rig will be available in four months, and each well can be drilled in two months; the duration of the campaign to drill and complete the remaining eight wells included in the development plan is sixteen months. the first period of oil production was planned to have a fixed plateau rate followed by a period of oil rate decline (figure 2). 2.2. decision problem at this stage, the operator company must decide whether acquiring additional information would increase the project’s value. alternative a: without-information. the decision on project development is made based on the current information using the expected value (ev) of the net present value (npv) and the discounted profit to investment ratio (dpi), which is discussed further in section 3.3. prior probabilities are assigned according to the technical team members’ judgment on the subjective probabilities of realizing the different states of nature; the economic parameters are estimated based on the assumptions and assessments included in the high-, mediumand low-production scenarios. if this vilela et al./decis. mak. appl. manag. eng. 2 (2) (2019) 1-18 8 option is chosen, the first oil can be reached in two years’ time, including facilities and wells. alternative b: with-information. additional information is acquired regarding the uncertain parameters of the reservoir and, subsequently, based on the outcomes of the data obtained, a decision is made on the future development of the project. the operator’s technical team has estimated, based on the reservoir and fluid properties, that, to obtain meaningful well test results, the minimum well test duration per well should be four months. it was decided that two of the appraisal wells could be used to perform an extended well test (ewt) in each one. following these assumptions, there will be a delay of one year (four months rig move + eight months ewt) compared with the without-information alternative. after the test results have been gathered, the technical team expects to have a more certain criterion to assign well deliverability, although uncertainty will still be present because the data are not perfect. the cost associated with the well test in these two wells is us$20 million. it should be considered that, if the project is relinquished now, the us$90 million already spent on exploration and appraisal will be lost; additionally, the abandonment cost for the 4 drilled wells, us$4 million, and the facilities’ abandonment cost, us$10 million, should be added to the economic evaluation. if new data are acquired and afterwards the decision is made to abandon the project, the cost of the data acquisition must be added to the previously mentioned costs. the outcome of the assessment of the alternatives without-information and withinformation will result in one of the following options: 1) the project is viable to endorse: it will proceed to the development phase, which necessitates a large investment; 2) the project is not viable to endorse : it will be relinquished, carrying the losses associated with the exploration costs; 3) the project needs additional analysis before deciding: it will be reframed. 2.3. economic parameters for decision making two economic parameters are used to make the decision: the net present value (npv) and the discounted profit to investment ratio (dpi). the npv is the yearly net cash flow discounted to the weighted average cost of capital (wacc—the average rate of return with which a company expects to compensate all its different investors, in which the weights are the fraction of each financing source in the company’s target capital structure), which in this case is 10.5%; the dpi is the result of dividing the discounted net cash flow by the discounted sum of the investment using the wacc. the values of npv and dpi are shown in section 2.4.2. 2.4. classical voi assessment as discussed in section 1.1, the voi is described by equation (1); in this section, the classical approach to the voi is discussed. 2.4.1. decision rules based on the operator’s portfolio of projects, the criterion for making decisions on projects with a financial investment higher than us$500 million is: a) a project with npv lower than us$100 million is unviable to endorse, which means that it is relinquished, b) a project with npv higher than us$500 million is viable to endorse and, c) a project with npv between us$100 million and us$500 million is reframed to find alternative development options. a fuzzy inference system applied to value of information assessment for oil and gas industry 9 regarding the dpi: a) a project with dpi higher than 0.5 is viable to endorse, b) a project with dpi lower than 0.0 is unviable to endorse and, c) a project with dpi between 0.0 and 0.5 should be reframed. 2.4.2. voi assessment for the without-information and with-information alternatives for the without-information alternative, table 2 shows the prior probabilities, the calculated npv, and dpi of each state of nature and the ev of the without-information alternative. table 2. prior probabilities, npv, dpi and expected values for the withoutinformation alternative state of nature prior probabilities (%) npv (us$ million) dpi (fract.) s1=high 25 2,139 2.27 s2=medium 40 414 0.42 s3=low 35 -631 -0.61 evnpv-a1 (us$ million) 479 evnpv-a2 (us$ million) -102 evnpv (us$ million) 479 evdpi-a1 (fraction) 0.52 evdpi-a2 (fraction) -1.00 evdpi (fraction) 0.52 for the with-information alternative, the technical team members estimated the reliability probabilities for the well test. it is acknowledged that, in a developed field, wells perform differently depending on their location and well test results are representative of a specific location; additionally, the duration of the test, although designed to capture the well performance, might not be long enough to assess the longrange well operation. table 3 shows the reliability probabilities of the well test estimated by the technical team members. table 3. reliability probability of the well test reliability probability x1=high productivity x2=medium productivity x3=low productivity 𝑝(𝑥𝑘 |𝑠1) 0.9 0.1 0.0 𝑝(𝑥𝑘 |𝑠2) 0.1 0.8 0.1 𝑝(𝑥𝑘 |𝑠3) 0.0 0.1 0.9 reliability probabilities are used together with prior probabilities to obtain posterior probabilities, which are combined with the project values to generate the expected value of the net present value (evnpv) and the ev of the discounted profit to investment ratio (evdpi). the results of these assessments are shown in table 4. vilela et al./decis. mak. appl. manag. eng. 2 (2) (2019) 1-18 10 table 4. posterior probabilities, residual probabilities and expected values for the with-information alternative x1=high productivity x2=medium productivity x3=low productivity 𝑝(𝑠1|𝑥𝑘 ) 0.85 0.07 0.00 𝑝(𝑠2|𝑥𝑘 ) 0.15 0.84 0.11 𝑝(𝑠3|𝑥𝑘 ) 0.00 0.09 0.89 𝑝(𝑥𝑘 ) 0.27 0.38 0.36 𝐸𝑉𝑁𝑃𝑉(𝐴1|𝑥𝑘 ) 1,667 357 -497 𝐸𝑉𝑁𝑃𝑉(𝐴1|𝑥𝑘 ) -114 -114 -114 𝐸𝑉𝑁𝑃𝑉(𝐴1|𝑥𝑘 ) 1.667 357 -114 𝐸𝑉𝑁𝑃𝑉(𝑈𝑆$ 𝑚𝑖𝑙𝑙𝑖𝑜𝑛) 537 𝐸𝑉𝐷𝑃𝐼(𝐴1|𝑥𝑘 ) 1.76 0.37 -0.48 𝐸𝑉𝐷𝑃𝐼(𝐴1|𝑥𝑘 ) -1.00 -1.00 -1.00 𝐸𝑉𝐷𝑃𝐼(𝐴1|𝑥𝑘 ) 1.76 0.37 -0.48 𝐸𝑉𝐷𝑃𝐼( 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛) 0.44 2.4.3. results of the voi using the classical approach decision based on npv: based on the results obtained in section 2.4.2, and using the decision rules (section 2.4.1), it can be concluded that, the without information project should be reframed and the with information project should be endorsed; in this manner, acquiring data increase the project’s value; summarizing, according to the nvp figures, the preferred alternative is to perform the well test (the withinformation alternative) and, depending on the data outcomes, decide whether the project should be endorsed or not. decision based on dpi: using dpi as the decision criterion, the without-information alternative suggests endorsing the project, while the with-information alternative suggests reframing the project; summarizing, the alternative that maximizes the dpi is the without-information one, which means that it is not recommended to perform the well test. at this stage, using two financial criteria, we have two contrasting recommendations about the future of the project. making a decision using these crisp criteria does not include the sophisticated elements used by human thinking in which, the criteria may overlap. in addition, from the independent npv and dpi results, it is not clear which is the optimum alternative unless some form of weighted valuation is made by combining the two economic parameters into one. 2.5. fis voi assessment up to this stage, the criterion to decide the future of the project has been based on linguistic variables like “not endorse”, “endorse”, “viable”, “unviable”, “high”, “medium” and “low”. indeed, a crisp relationship is established between the npv and dpi and the linguistic variables: if the npv is less than us$ 𝑿 million, the project is “unviable to endorse”, if the npv is higher than us$ 𝒀 million, the project is “viable to endorse” and if the nvp is higher than 𝑿 but lower than 𝒀, the project should be reframed; similar relationships apply to the dpi criterion. a fuzzy inference system applied to value of information assessment for oil and gas industry 11 however, it is worth recognizing that these criteria are fuzzy and not always aligned. the fuzziness occurs because, if the project npv is us$ 𝑿 − 𝜺 million, where 𝜺 is a given amount, the crisp logic decision criterion catalogues the project as “unviable to endorse”, although 𝜺 could be “small” compared with 𝑿. in the same manner, if the project value is us$ 𝒀 + 𝜺, in a crisp decision, the project is catalogued as “viable to endorse”, although 𝜺 could be “small” compared with 𝜺. the no alignment between the criteria happens because very often the two indices, npv and dpi, can produce a contradictory assessment of the same problem; for example, it could be the case that, using the npv, the project is “viable to endorse” but, using the dpi, the project is “unviable to endorse” or vice versa, as has been witnessed in this case study. these cases suggest that fuzzy logic can be used advantageously to make voi decisions by providing a more versatile tool to assess these decision problems; fuzzy logic is implemented through the fis. 2.5.1. fis building and application the fis used in this work was developed using matlab. the input parameters in the fis are the npv and dpi; for each input parameter, six membership functions are built, representing the linguistic variables high nvp or nvp viable to endorse, low nvp or nvp unviable to endorse, mid nvp or nvp for reframing, high dpi or dpi viable to endorse, low dpi or dpi unviable to endorse and mid dpi or dpi for reframing; the corresponding mfs are: nvp high, nvp mid, nvp low, dpi high, dpi mid and dpi low. in matlab, a set of predefined mfs—triangular-shaped functions—are selected. these mfs are chosen because they capture the technical team members’ interpretation of the degree to which the npv and dpi figures belong to the three categories into which the range of potential values are divided. equation (9) shows the mathematical form of the triangular-shaped mf:   0 ; , , 0 x a x a a x b b a f x a b c c x b x c c b c x                  (9) the comparison between the output of the fis for the with-information and that for the without-information alternative indicates which alternative has more value (the better decision). a mamdani fis with the centroid defuzzification method was used in this assessment. figure 3 shows the design of the fis using matlab. in section 1.3 we show the figure 2 which describe the fis process; that figure is shown below but now numbering the steps, in order, we are following in this work vilela et al./decis. mak. appl. manag. eng. 2 (2) (2019) 1-18 12 figure 3. fis implementation step1: the crisp data is generated, in this case, the project value, npv and dpi step 2: data is fuzzified using the membership function located in the data base; those mf describe the degree of belonging of different input values which is defined according with the analyst belief. in this work, the mf used for npv and dpi are shown in figure 4 and figure 5. figure 4. membership function for npv (input) a fuzzy inference system applied to value of information assessment for oil and gas industry 13 figure 5. membership function for dpi (input) step 3: once input variables are fuzzified, the decision rules, which are part of the knowledge base, are applied to the membership functions in the decision-making unit; in the mandani type inference, the decision rules are a mapping from the input membership function to the output membership functions, which are also part of the knowledge base; the rules aggregation process generate the fuzzy outcome. the output membership functions used in this work, describing the different decision options are show in figure 6. figure 6. membership function for the decision (output) vilela et al./decis. mak. appl. manag. eng. 2 (2) (2019) 1-18 14 the decision rules indicate the manner in which the two fuzzy financial parameters combine to result in a fuzzy decision. in this work we define the rules shown in table 5 to include the cases of interest, table 5. fuzzy rules rules if then rule 1 (npv is npv_high) & (dpi is dpi_high) endorse rule 2 (npv is npv_high) & (dpi is dpi_mid) endorse rule 3 (npv is npv_high) & (dpi is dpi_low) reframing rule 4 (npv is npv_mid) & (dpi is dpi_low) reframing rule 5 (npv is npv_low) & (dpi is dpi_low) no endorse rule 6 (npv is npv_low) & (dpi is dpi_high) reframing rule 7 (npv is npv_mid) & (dpi is dpi_mid) endorse step 4: fuzzy output gets into the defuzzification interface to generate crisp output. step 5: the value of the project is the crisp out; different crisp outputs are compared and, the one with the higher value is the optimum decision. the mfs of the nvp are chosen in accordance with past decisions taken by the decision maker, as discussed in section 1.2. the rationale for the selection of these mfs is that, for very high or very low npv values, the npv belongs to only one set, the npv_high or the npv_low, with a membership value of 1; for the intermediate npv value, the npv belongs partially to the three fuzzy sets. this fuzzy representation of the criteria for categorizing the project is based on past decisions made by the field operator company. the selection of the mfs needs to be updated once more decisions have been taken. the mfs for the dpi are chosen following the same procedure used for the npv the authors define a set of seven rules that determine the logic of this decision; these rules (if-then rules) are made by pairs of npv and dpi figures and a consequential sentence (then). the rules do not pretend to be exhaustive but must be coherent. all the rules were built using the and connector; although, in general, they can be defined equally well with or. 2.5.2. fis applied to the without-information and with-information alternatives; voi assessment referred to section 1.3, the outcome of fis (a crisp number) is the value of the project resulting from aggregating the project’s values in terms of npv and dpi; in addition, the fis assessment includes the imprecision in the terms used to decide whether a project worth or not to endorse (section 2.2). for evaluating the project using the fis developed in section 2.5.1, the crisp values for npv and dpi estimated in section 2.4.2 table 2 (us$479 million and 0.52), are input in the fis; the outcome of the assessment made by the fis indicates that the value associated with the without-information alternative is 7.2. similarly, in the with-information alternative, the npv and dpi figures (us$537 million and 0.44), contained in table 3, are input in the fis; as a result, the fis assessment for the with-information alternative is 6.97. due to the fact that, the value of the fis for the without-information alternative is higher than the value of the fis for the with-information case, the best alternative for the decision problem discussed is to endorse the project now and move it forward to the development phase without acquiring additional data. a fuzzy inference system applied to value of information assessment for oil and gas industry 15 this result is explained by the fact that, although the data acquisition reduces the uncertainty regarding well deliverability, the cost of this data acquisition, in terms of the additional investment and oil production delay, is higher than the increased project value due to uncertainty reduction. 3. conclusions in this study, an fis has been successfully implemented with the aim of assessing the voi of an oil and gas project. in the discussed case study, the use of the fis was able to introduce the fuzzy thinking of the decision maker into a subsurface voi assessment while removing ambiguity coming from the use of more than one economic parameter for decision making. the proposed methodology for voi assessment using fis has improved the conventional approach because: 1) instead of using a boolean relationship between project valuation and project decision, the fis uses a fuzzy human thinking approach to make decisions; 2) the fis uses a coherent method to integrate more than one criterion into the assessment, while, in the conventional voi approach, when more than one criterion is used, they can reach contradictory outcomes which conduct to inconclusive assessment. in addition to the aspects discussed above, the fis provides a tool for “selflearning” in which the quality of the voi assessments can be improved through continuous updating of the decision-making unit, knowledge base, and fuzzification and defuzzification interfaces with actual decisions, progressively generating a more robust fis and making the system act closer and closer to the way in which humans make decisions. the fis brings the voi methodology closer to the decision maker’s reasoning. these are important advantages of the fuzzy compared with the classical voi assessment. the fuzzy approach for voi assessment requires a longer and more complex analysis of the data to be acquired and their outcomes. however, this additional effort worth due to the impact it has in the decision. as a summary, the use of the fis makes it possible to have a system that can integrate the linguistic variables that are part of human language, reasoning, and understanding, but not necessarily part of the boolean logic used in the standard voi, into the prescriptive voi assessment. voi assessment using the fis brings the decision-making process one step forward with respect to the classical voi approach. to have tools and methods that replicate the human reasoning process for assessing voi increases the confidence of the decision maker in those procedures, thereby increasing their use and making the tools more reliable. decisions are made by human, and because human thinking is approximated more accurately by imprecise logic than by crisp logic, this research work successfully develop a methodology that integrate the human logic in the voi assessment, in special to problems in the oil and gas industry; 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(1965). fuzzy sets. information and control, 8, 338-352. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 2, issue 2, 2019, pp. 36-53. issn: 2560-6018 eissn:2620-0104 doi: https://doi.org/10.31181/dmame1902040b * corresponding author. e-mail addresses: papunbiswas@ieee.org (p. biswas), bbpal18@hotmail.com (b.b. pal) a fuzzy goal programming method to solve congestion management problem using genetic algorithm papun biswas 1 and bijay baran pal 2* 1 department of electrical engineering, jis college of engineering, west bengal, india 2 department of mathematics, university of kalyani, west bengal, india received: 25 january 2019; accepted: 25 april 2019; available online: 27 april 2019. original scientific paper abstract: the objective of this work is to present a priority-based fuzzy goal programming (fgp) method for solving the congestion management (cm) problem in electric power transmission lines by employing genetic algorithm (ga). to formulate the model for this problem, membership functions which are associated with the fuzzy model goals are converted into membership goals by assigning highest membership value (unity) as goal level and adding underand over-deviational variables to each of them. in solution process, a ga computational scheme is addressed within the framework of fgp model to achieve aspired goal levels of goals according to their priorities in imprecise environment. the standard ieee 30-bus 6-generator test system is taken as a case example to show the effectiveness of the approach. a comparison of model solution is also compared with solution of another approach studied previously. keywords: congestion management; fuzzy goal programming; genetic algorithm; membership function; overload alleviation; particle swarm optimization 1. introduction congestion in thermal power supply system in bhattacharya et al. (2001) refers to overloading situation in transmission lines when thermal bounds and line capacities of the power supply system are violated in chung et al. (2015). congestion actually occurs when power flow in a transmission line is higher than the flow allowed by operating reliability limits in bachtiar nappu & arief (2016). as such, congestion in power system would have to be rectified as and when needed to ensure system security. further, a lack of paying proper attention to congestion of the system may lead to widespread blackouts which give birth to negative impact to social and economic perspectives. therefore, congestion management in emami & sadri (2012) mailto:papunbiswas@ieee.org mailto:bbpal18@hotmail.com a fuzzy goal programming method to solve congestion management problem using genetic… 37 appears as one the key issues to maintain security and reliability of transmission network. the mathematical programming method for estimating voltage dropping and line loading for out of service of each network element was first introduced in abiad & stagg (1963) in 1963. then, different classical optimization methods based on load flow were studied for cm in mamandur & berg (1978), and medicherla et al. (1979) in the past century. the decomposition of spot prices to reveal congestion cost component in a pool model was presented in finney et al. (1997). the dc-optimal power flow (dc-opf) based approach to compute congestion cost was also propounded by singh et al. (1998). the real-time operational environment based cm was studied in fang & david (1999) and wang & song (2000) in last two decades. an optimal dispatch with the consideration of dynamic security constraints for cm was discussed in singh & david (2000). rau (2000) presented the ac-opf driven approach to cm along with congestion cost allocation. then, an effective model to location of unified power flow controller (upfc) for cm was deeply studied in verma et al. (2001). the use of thyristor-controlled series compensation (tcsc) to reduce congestion cost is also presented in lee (2002). to manage congestion, a minimum load curtailment problem was proposed in rodrigues & dasilva (2003). an opf model with multiplicity of objectives and a set of voltage security constraints was also discussed in milano et al. (2003) with regard to avoiding congestion through the use of location marginal price. the use of rescheduling of generation and load with voltage security constraints for cm was also discussed in yamin & shahidepour (2003). an efficient cm approach using real and reactive power rescheduling via optimal allocation of reactive power resources was proposed in kumar et al. (2004). a simple cost effective model for generation rescheduling and load shedding was also studied in talukdar et al. (2005) in the past. the heuristic methods in hazra & sinha (2007); dutta & singh (2008); balaraman & kamaraj (2010) for global optimizations have been made successfully to solve cm problems in the recent past. hazra & sinha (2009) put forth an efficient approach based on fuzzy estimation for identifying collapse sequences to reach the optimal solution of a cm problem. fuzzily described adaptive bacterial foraging algorithm and gravitational search method have also been studied in venkaiah & vinod kumar (2011) and vijaya kumar et al. (2013) previously. to overcome the various drawbacks associated with the previous approaches concerning cm in thermal power supply system, a priority-based fgp method for multiobjective decision making (modm) is addressed in this paper to model cm problem and a ga computational scheme is adapted to reach decision in imprecise premises. in model formulation, fuzzy representations of different objectives are considered for minimization of overload alleviation and operation cost subject to various constraints associated with the problem. the experimental test on standard ieee 6-generator 30-bus system is made to expound the effective use of the method. the solution is also compared with solution achieved by using particle swarm optimization (pso) technique in hazra & sinha (2007) is performed to present superiority of the proposed method. now, fgp model formulation of a modm problem is discussed in the section 2. biswas and pal/decis. mak. appl. manag. eng. 2 (2) (2019) 36-53 38 2. fgp problem formulation in fuzzy environment, objectives are generally described fuzzily, whereas structural resource constraints may be fuzzy or crisp and that depends on how the model parameters are involved there in the decision situation. in line with the work of dubois (1987), the generic form of a fuzzy programming (fp) problem can be exhibited as follows. find x for:   ( ) ; : , ; ; 0 ~ k k n t m l u f x g subject to x s b x r b r x x x x ~                          (1) where x is a vector of decision variables, gk be the imprecise goal level of kth objective ( )kf x , k = 1,2,...., k, ≳ and ≲ indicate fuzziness of ≥ and ≤ restrictions, respectively, and where a is a real matrix and b is a constant vector and t means transposition, lx and u x denote the vectors of lowerand upper-limits, respectively, of the vector x , and where l and u indicate lower and upper, respectively. also, it is assumed that the feasible region ( )s  is bounded. now, characterization of fuzzy goals is by associated membership functions concerned with measuring degree of achievement of each of them in a decision making horizon. 2.1. characterization of membership function let kt and tuk be lowerand upper-tolerance ranges, respectively, regarding achievement of aspired level gk of kth fuzzy goal. then, membership function, say ( )k x , associated with ( )kf x can be characterized as follows. for ~ type of constraint, ( )k x appear as in zimmermann (1987): 1 ( ) ³ , ( ) ( ) ( ) ( ) , 0 ( ) , k k k k k k k k k k k k k k when f x g f x g t x if g t f x g t when f x g t              (2) where ( )k kg t denotes the lower-tolerance limit to achieve the stated fuzzy goal. further, for ~ type of constraint, ( )k x can be presented as: a fuzzy goal programming method to solve congestion management problem using genetic… 39 1 ( ) , ( ) ( ) ( ) ( ) , 0 ( ) , k k k uk k k k k k uk uk k k uk if f x g g t f x x if g f x g t t if f x g t               (3) where (gk + tuk) denotes the upper-tolerance limit to achieve the stated fuzzy goal. the membership functions in (2) and (3) can be graphically depicted as in figure 1 and figure 2, respectively. ( )k x 1 k t  gk )(k xf figure 1. represented graph of the membership function in (2) ( )k x 1 gk tuk ( )kf x figure 2. graph of the membership function in (3) the formulation of an fgp model under a pre-emptive priority structure by defining membership goals is described in section 2.2. 2.2. fgp model since in a modm context, various conflicting goals are dealt for achieving the aspired levels, priority-based fgp is adopted by pal & chakraborti (2013) for formulating the model of the problem. in priority-based fgp, priorities are assigned to goals according to importance of achieving goal levels, where a set of goals which seems equally important for their goal achievements are included at a same priority level and numerical weights are introduced there according to relative weights of importance to achieve goal levels. the generic form of a priority-based fgp model can be presented as follows. biswas and pal/decis. mak. appl. manag. eng. 2 (2) (2019) 36-53 40 1 2 1 2 1 f , ind so ( , ,..., ) [ ( ), ( ) ..., as to : mi .nimize and satisfy ( ) ,. ., ( ) ] ( ) ( ) 1 ) ( ) ( n r r k k k k k k k uk k k k uk x x x x z p d p d p d p d f x g t d d t g t f x d d t                    (4) subject to the given constraints set as described in (1). here k kd ,d   ≥0, 20, 1, ,..., k k k kd d    , are under and over-deviational variables introduced to kth goal, and where z represents the vector of r priority achievement function. ( )rp d  is a linear function of vector of weighted under-deviational variables, and ( )rp d  is of the form:  ; 1, 2, .( ) . , . r rk rk k k p kd kd w       (5) where drk is renamed for d  k to represent it at rth priority level, wrk ( > 0) is the numerical weight associated with drk and it is the weight of importance of achieving kth goal level relative to others which are grouped together at rth priority level and where wrk values are determined in pal et al. (2003) as: 1 ( )1 ( ) lk r uk r t rk t w      (6) for ( )k x in (2) and (3), respectively, where ( )k rt and ( )uk rt are used to present kt and ukt , respectively, at rth priority level. also, the relationship among the priorities is 1 2 . . . . . .r rp p p p     , where “>>>” implies “much greater than”. in the formulated model, the notion of using pre-emptive priorities is that the goals which are at rth priority level rp are preferred most to achieve the corresponding aspired levels before taking the achievement problem of goals included at next lower priority level 1rp  . now, to design the model of a cm problem, it is worth noting that objectives and some system constraints are with nonlinear characteristics. to avoid computational complexity in awerbach et al. (1976) with nonlinearity in model goals and constraints as well as to overcome the burden of hand calculations for linearization of them using approximation technique in pal et al. (2009), ga as a goal satisfier in deb (2002) for multiobjective decision analysis is considered for searching solution of the problem. the ga computational scheme is presented in the section 3. 3. ga computational scheme for cm problem the three probabilistically defined operators in goldberg (1989): selection, crossover and mutation are used to generate new population (i.e., new solution candidates) in the ga scheme to search solution. the real-value coded chromosomes are considered to perform operations with ga in random fashion. to evaluate a function, say ( )veval e , the fitness score of a chromosome, say v, according to a fuzzy goal programming method to solve congestion management problem using genetic… 41 maximization or minimization of an objective function defined by decision maker (dm) in the decision making context. in the proposed modm model, since ( )veval e is a single-objective linear program, roulette-wheel selection, arithmetic crossover and uniform mutation are adapted to search decision of the problem. the algorithmic steps of ga computational process are described in the following section 3.1. 3.1. ga algorithm step 1. representation and initialization. let e denote the double vector representation of chromosome in a population as 1 2( , ,..., )ne x x x . the population size is defined by pop_size, and pop_size chromosomes are randomly initialized in the domain of searching solution. . step 2. fitness function. the fitness value of each chromosome is judged by the value of an objective function. the fitness function is defined as: 1 ( ) ( ) ( ) , 1, 2, 3, ..., k r v r v rk rk v k eval e z w d v pop size               (7) where v( )rz is achievement function (z) in (4) for measuring the fitness value of vth chromosome, when attainments of goals included at rth priority level pr is considered. the best value of a chromosome is determined as   * min 1, 2,..., | ve eval e v pop size  (8) in course of searching minimum value of achievement function. step 3. selection stage. the simple roulette-wheel scheme is employed for selection of two parents for mating purpose in solution search process. step 4. crossover stage. the probability of crossover is defined by parameter pc. the single-point crossover in goldberg (1989) is applied here with a view to obtaining offspring that always satisfy linear constraints set. dom number [0,1], cr r p  is satisfied. for example, if two parents 1 2,e e s are selected, then the arithmetic crossover is defined as: 1 1 1 1 1 2 2 2 2 1 1 2, ,e e e e e e       for generating two offspring 1 1e and 1 2e , where 1 2, 0   with 1 2 1,   1 1 1 2, .e e s step 5. mutation. a parameter pm is defined as the probability of mutation. the mutation operation is made uniformly, where for a random number [0,1]r  , a chromosome is selected for mutation provided that .mr p step 6. termination. the solution search process terminates when best decision for a chromosome is received at a certain generation number in decision making premises. the pseudo code of the ga is as follows: biswas and pal/decis. mak. appl. manag. eng. 2 (2) (2019) 36-53 42           : 1 1 1 1 initialize population of chromosomes e x evaluate the initialized population by computing its fitness measure while not termination criteria do x x select e x from e x crossover e x mutate e x evaluat        1 e e x end while  now, formulation of fgp model of cm problem is discussed in the section 4. 4. cm problem formulation the various objectives that are inherently associated with a cm problem are defined as follows. 4.1 defining the objective functions (a) “overload alleviation” function. in decision premises, the alleviation of overload on a transmission line is essentially needed to ensure security and stability of system, and thereby taking preventing measure against happening of system outage. here, transmission line overload can be alleviated by line switching, generation rescheduling and load shedding. the alleviation of overload in the system takes the form: max 2 1 1 ( ) nl i i i f s s    (9) where, f1 represents cumulative overload, nl is number of overloaded lines, and where max i is and s be the mva flow and mva capacity of line i in power supply system, respectively. also, square form of objective is made to avoid masking effect. (b) operational cost function. in this context, the total incurring cost for thermal power plant operation and which is associated with cm problem can be expressed as sum of the fuel cost and cost of load shedding. the total operational cost function is expressed as: )()( 2 , 1 1 ' , ''2 2 kshd ng i pl k kkshdkkgiigiii lclbapcpbaf      (10) where f2 denotes total operating cost, ng be the number of participating generators, pl is used to represent number of associated loads, pgi is generation of power from ith generator, lshd,k is amount of load shedding at bus k, and where i i ia ,b ,c are cost coefficients of objective associated with generation of power from generator gi, and ' ' ' k k ka ,b ,c are cost coefficients of objective associated with load shedding at bus k. (c) power-loss function. a certain function called real power-loss function which is inherent to a power transmission line and directly affect the ability to transfer power. the mathematical a fuzzy goal programming method to solve congestion management problem using genetic… 43 expression of real power-loss function, f3 (mw) can be defined as in talukdar et al. (2005):    tl ji, jiji 2 j 2 il3 δδvvvvgf 1 )]cos(2[ (11) where tl represents total transmission lines, gl be the conductance of lth line, vi and vj are voltage magnitudes, i and j are voltage phase angles at the end buses i and j of lth line, respectively, of the system, where ‘cos’ designates cosine function. 4.2. definitions of system constraints the constraints on the power generation system r are as follows: a) power balance constraints. the power balance constraints appear as: 1 1 [ cos ( ) sin ( )] 0 [ sin ( ) cos ( )] 0 h gi di i j ij i j ij i j j h gi di i j ij i j ij i j j p p v v g b q q v v g b                         (12) where h be the number of buses, pgi and qgi are realand reactive-power of the generator connected to ith bus, respectively, and where pdi and qdi be realand reactive-power of the load connected to ith bus, respectively, gij and bij indicate transfer conductance and susceptance between bus i and bus j, respectively, δi and δj are bus voltage angles of buses i and j, respectively. b) determining the generation capacity & voltage constraint. similar to conventional power generation and dispatch system, constraints on power generation and voltage appear as: min max min max min max , , ; 1, 2,..., . i i i i i i i i i g g g g g g p p p q q q v v v i n        (13) now, to show the effective use of the proposed approach, an example is considered in the section 5. 5. case example the ieee 30-bus 6-generator test system talukdar et al. (2005) is addressed to present the effectiveness of the method. the diagram of the system depicted in figure 3 below. the diagram shows that the system is with 6 generators, 41 lines and 30 buses. the total demand on 21 load buses is 283.4 mw. biswas and pal/decis. mak. appl. manag. eng. 2 (2) (2019) 36-53 44 figure 3. diagram of ieee 30-bus test system the model data were collected from the studies (talukdar et al., 2005; hazra & sinha, 2007) made previously. the cost-coefficients of power generation and that of load shedding are presented in the table 1 and table 2, respectively. table 1. power generation cost –coefficient data generator type (ti) maximum gen capacity (mw) a b c t1 < 25 0.0 2025.00 1.500 t2 50 0.0 1875.00 1.425 t3 100 0.0 1800.00 1.350 t4 200 0.0 1650.00 1.250 t5 250 0.0 1575.00 1.500 t6 300 0.0 1575.00 1.250 t7 350 0.0 1500.00 1.350 t8 400 0.0 1500.00 1.250 t9 500 0.0 1200.00 1.500 t10 > 500 0.0 1200.00 1.000 a fuzzy goal programming method to solve congestion management problem using genetic… 45 table 2. load shedding cost-coefficient data load in a bus (mw) ' ka ' kb ' kc <=10 0.0 1200 1.00 <=20 0.0 1200 1.50 <=30 0.0 1500 1.25 <=40 0.0 1500 1.35 <=50 0.0 1575 1.25 <=60 0.0 1575 1.5 <=75 0.0 1650 1.25 <=100 0.0 1800 1.35 <=125 0.0 1875 1.425 >125 0.0 2025 1.5 the data associated with transmission lines and loads at buses are presented in the table 3 and table 4, respectively. table 3. transmission-line data line no. from bus no. to bus no. line impedance line no. from bus no. to bus no. line impedance r(p.u.) x(p.u.) r(p.u.) x(p.u.) 1 1 2 0.0192 0.0575 22 15 18 0.1070 0.2185 2 1 3 0.0452 0.1852 23 18 19 0.0639 0.1292 3 2 4 0.0570 0.1737 24 19 20 0.0340 0.0680 4 3 4 0.0132 0.0379 25 10 20 0.0936 0.2090 5 2 5 0.0472 0.1983 26 10 17 0.0324 0.0845 6 2 6 0.0581 0.1763 27 10 21 0.0348 0.0749 7 4 6 0.0119 0.0414 28 10 22 0.0727 0.1499 8 5 7 0.0460 0.1160 29 21 22 0.0116 0.0236 9 6 7 0.0267 0.0820 30 15 23 0.1000 0.2020 10 6 8 0.0120 0.0420 31 22 24 0.1150 0.1790 11 6 9 0.0000 0.2080 32 23 24 0.1320 0.2700 12 6 10 0.0000 0.5560 33 24 25 0.1885 0.3292 13 9 11 0.0000 0.2080 34 25 26 0.2544 0.3800 14 9 10 0.0000 0.1100 35 25 27 0.1093 0.2087 15 4 12 0.0000 0.2560 36 28 27 0.000 0.3960 16 12 13 0.0000 0.1400 37 27 29 0.2198 0.4153 17 12 14 0.1231 0.2559 38 27 30 0.3202 0.6027 18 12 15 0.0662 0.1304 39 29 30 0.2399 0.4533 19 12 16 0.0945 0.1987 40 8 28 0.6360 0.2000 20 14 15 0.2210 0.1997 41 6 28 0.0169 0.0599 21 16 17 0.0824 0.1932 biswas and pal/decis. mak. appl. manag. eng. 2 (2) (2019) 36-53 46 table 4. bus-load data bus no. load bus no. load p(p.u.) q(p.u.) p(p.u.) q(p.u.) 1 0.000 0.000 16 0.035 0.018 2 0.217 0.127 17 0.090 0.058 3 0.024 0.012 18 0.032 0.009 4 0.076 0.016 19 0.095 0.034 5 0.942 0.190 20 0.022 0.007 6 0.000 0.000 21 0.175 0.112 7 0.228 0.109 22 0.000 0.000 8 0.300 0.300 23 0.032 0.016 9 0.000 0.000 24 0.087 0.016 10 0.058 0.020 25 0.000 0.000 11 0.000 0.000 26 0.035 0.023 12 0.112 0.075 27 0.000 0.000 13 0.000 0.000 28 0.000 0.000 14 0.062 0.016 29 0.024 0.009 15 0.082 0.025 30 0.106 0.019 table 5 exhibits various simulation runs which were carried out in the test system. table 5. simulation runs run different simulation cases 1 overload simulation with reduction of capacity of line 1-2 from 130 mw to 50 mw. 2 overload simulation with reduction of capacity of line 1-3 and 2-4 from 130 mw to 50 mw and 65 mw to 15 mw. 3 overload simulation for outage of unit 3 at bus 5 and with reduction of capacity of line 2-5 from 130 mw to 50 mw. in this case, the optimization toolbox under matlab (matlab r2010a) has been employed to conduct the experiments by employing ga at different stages for program evaluation. the computational environment is intel pentium iv with 2.66 ghz. clockpulse and 3 gb ram. in the solution search process, initial population= 50; roulettewheel selection; single-point crossover with probability= 0.8; mutation probability= 0.07 and maximum generation number= 100 are taken into account for exploration and exploitation of search space in the domain of interest. then, following the procedure and fitting the data presented in tables 1 table 5, the membership goals can be obtained by addressing the second goal expression in (4). the executable fgp models for individual three simulation runs under a priority structure considered for the system are presented as follows. run-1: simulation of system under overload by reducing capacity of line 1-2 from 130 mw to 50 mw the model appears as find 1-2( , ) { 1, 2,5,8,11,13}ig s p i  so as to: a fuzzy goal programming method to solve congestion management problem using genetic… 47 1 1 2 2 3 1 1 1 , 2.50 27942 2 minimize z p d p d d                            and satisfy 1 2 3 2 1 2 1 1 2 2 2 2 1 1 2 2 5 5 8 8 2 2 11 11 13 13 2 2 3 : [{3.0 ( 50) } / (3.0 0.5)] 1, : [{547942 (1650 1.25 1875 1.425 1875 1.425 2025 1.5 2025 1.5 2025 1.5 )} / (537942 530000)}] 1, : [(5.50 ) / (5.50 f f f s d d p p p p p p p p p p p p d d f                                3 33.5)] 1,d d       subject to 1 1 2 1 2 1 2 3 1 3 1 3 [ {5.2246 cos ( ) 15.6467 sin ( )} {1.2437 cos ( ) 5.0960 sin ( )] 0 gp v v v                  (14) 2 2 1 2 1 2 1 4 2 4 2 4 5 2 5 2 5 6 2 6 2 6 0.217 [ {5.2246 cos ( ) 15.6467 sin ( )} {1.7055 cos ( ) 5.1974 sin ( ) {1.1360 cos ( ) 4.7725sin ( )} {1.6861cos ( ) 5.1165sin ( )}] 0 gp v v v v v                                   (15) 5 5 2 5 2 5 2 7 5 7 5 7 0.942 [ {1.1360 cos ( ) 4.7725 sin ( )} {2.9540 cos ( ) 7.4493sin ( )] 0 gp v v δ δ δ δ v δ δ δ δ           (16) 8 8 6 8 6 8 6 28 8 28 8 28 0.3 [ {6.2893 cos ( ) 22.0126 sin ( )} {1.4308 cos ( ) 0.4499 sin ( )] 0 gp v v v                   (17) 13 13 12 13 12[ { 7.1429sin ( )}] 0gp v v      (18) 11 11 9 11 9[ { 4.8077 sin ( )}] 0gp v v      (19) 1 1 2 1 2 1 2 3 1 3 1 3 [ {5.2246 sin ( ) 15.6467 cos ( )} {1.2437 sin ( ) 5.0960 cos ( )] 0 gq v v v                  (20) 2 2 1 2 1 2 1 4 2 4 2 4 5 2 5 2 5 6 2 6 2 6 0.217 [ {5.2246 sin ( ) 15.6467 cos ( )} {1.7055sin ( ) 5.1974 cos ( ) {1.1360 sin ( ) 4.7725 cos ( )} {1.6861sin ( ) 5.1165 cos ( )}] 0 gq v v v v v                                   (21) 5 5 2 5 2 5 2 7 5 7 5 7 0.942 [ {1.1360 sin ( ) 4.7725 cos ( )} {2.9540 sin ( ) 7.4493 cos ( )] 0 gq v v v                   (22) 8 8 6 8 6 8 6 28 8 28 8 28 0.3 [ {6.2893sin ( ) 22.0126 cos ( )} {1.4308 sin ( ) 0.4499 cos ( )] 0 gq v v v                   (23) 11 11 9 11 9[ { 4.8077 cos ( )}] 0gq v v      (24) 13 13 12 13 12[ { 7.1429 cos ( )}] 0gq v v      (25) (equality constraints) 4012,300,350 ,5015,800,000 13118 521   ggg ggg pp1p1 pp22p5 (26) biswas and pal/decis. mak. appl. manag. eng. 2 (2) (2019) 36-53 48 20 100 , 15 80 , 15 60 , 10 50 , 15 60, , 1, 2, 5,8,11,13 2 5 8 11 13 i g g g g g g q q q q q 0.95 v 1.1 i              (27) (generator constraints) 6,29,3021,23,24,2,18,19,20,4,15,16,1710,12,13,13,4,5,7,8,i1v0 il 2,,05.85.  (load-bus voltage constraint) run-2: simulation of system under overload by reducing capacity of line 1-3 and 24 from 130 mw to 50 mw and 65 mw to 15 mw in this case, the executable model is found as: find ),( gii ps so as to: 1 1 2 2 3 1 1 1 , 15 21942 2 minimize z p d p d d                            and satisfy 1 2 3 2 2 1 3 2 4 1 1 2 2 2 2 1 1 2 2 5 5 8 8 2 2 11 11 13 13 2 2 :[(25 {( 50) ( 15) }) / (25 10)] 1, :[{(551942 (1650 1.25 1875 1.425 1875 1.425 2025 1.5 2025 1.5 2025 1.5 } / (551942 530000)] 1, : f f f s s d d p p p p p p p p p p p p d d                                  3 3 3[(6.50 ) / (6.50 4.00)] 1,f d d        subject to the constraints in (14)-(27). run-3: simulation of system under overload with outage of unit 3 at bus 5 and by reducing capacity of line 2-5 from 130 mw to 50 mw the executable model is obtained as follows. find ( , )i gis p so as to: 1 1 2 2 3 1 1 1 , 3 21942 1.50 minimize z p d p d d                            and satisfy 1 2 3 2 2 5 1 1 2 2 2 1 1 2 2 8 8 11 2 2 11 13 13 2 2 3 3 3 : [{5 ( 50) } / (5.00 2.00)] 1, : {(551942 (1650 1.25 1875 1.425 2025 1.5 2025 1.5 2025 1.5 ) / (551942 530000)} 1, : [(10.00 ) / (10.00 8.50)] f f f s d d p p p p p p p p p p d d f d d                                    1, subject to the problem constraints in (14) (27). the goal achievement function ( z ) defined for the three runs actually describes the evaluation function in ga search process for solving the problem. the evaluation function for determining the fitness of a chromosome is given as: 3 1 ( ) ( ) ( ) , 1, 2, 3,..., 50; 1, 2r v r v rk rk v k eval e z w d v r                a fuzzy goal programming method to solve congestion management problem using genetic… 49 the best value of objective )( *z for the fittest chromosome is determined as: * min{ ( ) 1, 2,..., 50 }.ve eval e v  the solutions obtained from the three runs of the test system are presented in the table 6. table 6. solution achievements under different runs run overload-condition solution line/ unit mva flow mva capacity mva flow powerloss (mw) png (ng=1,2,5,8,11,13) cost (rs/ hr) 1 1-2 61.25 50 49.55 3.78 (82.27, 58.59, 50.00, 35.00, 30.00, 31.32) 536633.75 2 1-3 and 2-4 35.44 20.30 50 15 35.50 14.95 4.50 (112.44, 20.46, 50.00, 35.00, 30.00, 40.00) 538219.96 3 2-5 and out of unit 3 40.89 50 48.12 8.76 (122.22, 31.24, 0.00, 35.00, 30.00, 40.00) (31.2 mw loadshaded) 550064.38 it is clear from the results that the decision is a satisfactory one from the view point of proper management of mva flow with incurring of minimum operational cost of the power plant in imprecise environment. to show the effective use of the approach, a performance comparison is made in the section 6. 6. performance comparison the pso technique in hazra & sinha (2007) is considered for a solution comparison. the resulting decision is presented in the table 7. table 7. results of three simulation cases under pso technique case overloaded condition solution line/unit mva capacity mva flow cost (rs/ hr) 1 line 1-2 50 49.16 541171 2 line 1-3 and line 2-4 50 15 12.31 14.99 542465 3 line 2-5 and unit 3 out 50 49.88 565979 the mva flow and total incurring cost of the cm problem under the proposed model and pso technique are diagrammatically presented in figure 4 and figure 5, respectively. biswas and pal/decis. mak. appl. manag. eng. 2 (2) (2019) 36-53 50 figure 4. graphical representation of mva flow comparison figure 5. graphical representation of cost comparison the result comparisons show that the proposed approach is superior over the pso to arrive at appropriate decision in imprecise environment. 7. conclusion the main merit of the method presented here is that the fuzzy characteristics regarding attainment of objectives values are preserved there in all possible instances of executing the model of the cm problem. again, computational complexity arising out of the nonlinearity in the goals and constraints associated with the model can easily be avoided here with the use of ga based solution search approach for solving problems in imprecise environment. the proposed method is also advantageous in the 0 10 20 30 40 50 case 1 case2 case3 comparison of mva flow: different simulation cases mva base-capacity proposed ga based fgp approach pso based approach 520000 530000 540000 550000 560000 570000 case 1 case 2 case3 comparison of cost of cm: different simulation cases proposed ga based fgp approach pso based approach a fuzzy goal programming method to solve congestion management problem using genetic… 51 sense that here a multi-objective optimization problem can be converted into a goal oriented single objective optimization problem for achieving a compromise solution in the decision making horizon. further, the proposed approach is flexible enough to accommodate different other restrictions as and when needed for cm in electric power transmission system. however, the use of interval data in pal (2018), instead of considering fuzziness of model parameters, towards promoting cm performances and thereby improving quality of solution is an interesting alley of research for optimization of a power supply problem. acknowledgments: the authors would like to thank anonymous reviewers for useful comments and suggestions to improve the quality and clarity of presentation of the paper. references abiad, a.h.e., & stagg, g.s. 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(1987). fuzzy sets, decision making and expert systems. kluwer academic publisher, boston, dordrecht, lancaster. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). http://www.sciencedirect.com/science/article/pii/s1568494612005340 http://www.sciencedirect.com/science/article/pii/s1568494612005340 http://www.sciencedirect.com/science/article/pii/s1568494612005340 http://www.sciencedirect.com/science/journal/15684946/13/5 plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 2, issue 2, 2019, pp. 54-64. issn: 2560-6018 eissn:2620-0104 doi: https://doi.org/10.31181/dmame1902049s * corresponding author. e-mail addresses: amalendu.si@gmail.com (a. si), sujit.das@nitw.ac.in (s. das), kar_s_k@yahoo.com (s. kar) an approach to rank picture fuzzy numbers for decision making problems amalendu si 1, sujit das 2* and samarjit kar 3 1 department of computer science & engineering, mallabhum institute of technology, india 2 department of computer science and engineering, national institute of technology warangal, warangal, india 3 department of mathematics, national institute of technology durgapur, durgapur, india received: 25 january 2019; accepted: 27 april 2019; available online: 29 april 2019. original scientific paper abstract: comparison of picture fuzzy numbers (pfns) are performed using score and accuracy values. but when both of the score and accuracy values are equal, those pfns are said to be identical. this article presents a novel method to compare the pfns even when the score and accuracy values of those pfns are equal. the proposed ranking method is based on positive ideal solution, positive and negative goal differences, and score and accuracy degrees of the picture fuzzy numbers. a new score function is proposed to calculate the actual score value which depends on the positive and negative goal differences and the neutral degree. finally, a real-life example has been used to illustrate the efficiency of the proposed method. key words: picture fuzzy set, picture fuzzy number, positive ideal solution, positive goal difference, negative goal difference. 1. introduction picture fuzzy set (pfs) (cong, 2014) is an extension of fuzzy set (fs) (zadeh, 1965) and intuitionistic fuzzy set (ifs) (li, 2008, das et al., 2018, das et al., 2017), and it can easily manage the uncertain nature of human thoughts by introducing the neutral and refusal membership degrees. in pfs, the authors have divided the hesitation margin of ifs into two parts which are neutral membership degree and refusal membership degree. when both of the neutral and refusal membership degrees are zero, i.e., hesitation margin becomes zero, then the pfs returns to ifs. sometimes, fs and ifs find it difficult to express the situations when human thoughts involve more options like ‘yes’, ‘abstain’, ‘no’ and ‘refusal’. pfs is preferable to handle mailto:amalendu.si@gmail.com mailto:sujit.das@nitw.ac.in mailto:kar_s_k@yahoo.com an approach to rank picture fuzzy numbers for decision making problems 55 this type of situations using the positive, neutral, negative, and refusal membership degrees. the general election of a country is a good example of such situation where the voters can give their opinions like ‘vote for’, ‘abstain’, ‘vote against’ and the ‘refusal’ of the election (cuong and kreinovich, 2013, 2014). suppose one candidate and 1000 voters are participating in an election process. among them, 400 voters vote for the candidate, 100 voters are not interested in casting their vote, i.e., they remain to abstain from the voting process, 300 voters are giving their vote against the candidate, and 200 voters refuse to cast their vote for the candidate, i.e., they vote for nota. this kind of situation may happen in reality and it is outside the scope of ifs and fs, since fs and ifs don’t support neutrality. as another example, suppose an expert takes the opinion of a person regarding some object. now the person may say that 0.4 is the possibility that the object is good, 0.3 is the possibility that the object is not good, 0.2 is the possibility that the object is good and as well as not good, and 0.1 is the possibility that (s)he does not know about the object. this issue is also not handled by the fss or ifss. due to having the capability of accepting more opinions, pfs has become an important tool to deal with imprecise and ambiguous information and been applied in many real-life problems by some researchers (zhang and xu, 2012, cuong and kreinovich, 2013, cuong, 2013). son (2016) investigated the application of pfs in clustering algorithms to exploit the hidden knowledge from a mass of data sets by proposing a hierarchical picture clustering (hpc) method. motivated by the application of pfs in decision making, garg (2017) proposed a series of aggregation operators in the context of pfs and presented a decision-making approach using the proposed aggregation operators. wang et al. (2017) proposed picture fuzzy set based geometric aggregation operator and compared two picture fuzzy numbers (pfns) using score and accuracy functions. guiwu (2016) used pfs in decision-making problem and proposed cross entropy of pfss. using the idea of cross entropy, the author introduced a new ranking method in pfs environment. singh (2015) defined correlation coefficient of pfs and applied it to clustering analysis problem. many authors have contributed to rank the corresponding numbers in the framework of fuzzy sets (atanassov and georgiev, 1993, das et al., 2015, 2017, 2018), intuitionistic fuzzy sets (atanassov, 1989,bhatiaand kumar, 2013, kumar and kaur, 2012) and intuitionistic multi fuzzy sets (li, 2005, li, 2008, liu, 2007, si and das, 2017). most of these ranking methods are based on the comparative analysis of a pair of sets, measurement of the distance between the sets, and measurement of the distance of a set from a central point. in the comparative and distance measurement methods, the membership, nonmembership and neutral degrees are considered to have similar importance. another important concern is that the neutral membership degree is considered just like a positive or negative membership degree. no focus is given even when the neutral degree increases or decreases. but in our real life, some situations are totally different and the membership, non-membership and neutral degrees play different roles and have various types of functionality in the decision making process or ranking among them. in this article, we propose a new method to calculate the score to rank the pfns using positive ideal solution, negative ideal solution and average neutral value of the alternatives. neutral degree of pfs has an active role in our proposal. we consider the average value of the neutral degree as a pivot point concerning the all other neutral degrees. then, we provide a practical example to analyze the proposed method for ranking to take the decision. remaining of the article organized is as follows. some relevant ideas of picture fuzzy sets are recalled in section 2. we propose the new ranking method in section 3 si et al./decis. mak. appl. manag. eng. 2 (2) (2019) 54-64 56 followed by a real-life example in section 4. a comparative study is given in section 5. finally, we conclude in section 6. 2. preliminaries this section briefly presents the relevant ideas of picture fuzzy set and some of its operations. 2.1 picture fuzzy set a picture fuzzy set (pfs) a on the universe x is an object in the form of        , , ,a a aa x x x x x x    (1) where    0,1a x  be the degree of positive membership of x in a, similarly    0,1a x  and    0,1a x  are respectively called the degrees of neutral and negative membership of x in a. these three parameters       ,a a ax x and x   of the picture fuzzy set a satisfy the following condition      , 0 1a a ax x x x x        . then, the degree of refusal membership  a x of x in a can be estimated accordingly,         , 1a a a ax x x x x x         (2) the neutral membership   a x of x in a can be consideredasdegree of positive membership as well as degree of negative membership whereas refusal membership   a x can be explained as not to take care of the system. when,  , 0ax x x   , then the pfs reduces into ifs. for a fixed ,x a         , , ,a a a ax x x x    is called picture fuzzy number (pfn),where    0,1a x  ,    0,1a x  ,    0,1a x  ,    0,1a x  and         1a a a ax x x x       (3) simply, pfn is representedas       , ,a a ax x x   . 2.2 operations on pfs for two pfss  , ,a a aa    and  , , ,b b bb    cong (2014) defined some operations as given below.                 , max , , min , , min ,a b a b a ba b x x x x x x x x x        (4)                 , min , , min , , max ,a b a b a ba b x x x x x x x x x        (5)        , , ,a a aa x x x x x x    (6) an approach to rank picture fuzzy numbers for decision making problems 57 cuong and kreinovich (2013) and cuong (2013) defined some properties on pfss as given below. a b if             , , ,a b a b a bx x x x x x x x          (7) a b if  a b and b a  (8) if a b and b c then a c (9) a a (10) 2.3 distance between picture fuzzy sets distances between the two pfss are defined in (cuong and son, 2015, son, 2016). the distance between two pfss  , ,a a aa    and  , ,b b bb    in  1 2, ,..., nx x x x is calculated as follows. normalized hamming distance                1 1 , n h a i b i a i b i a i b i i d a b x x x x x x n              (11) normalized euclidean distance                   2 2 2 1 1 , n e a i b i a i b i a i b i i d a b x x x x x x n              (12) example 1: let a ={(0.7,0.2,0.1),(0.8,0.1,0.1),(0.7,0.1,0.2)} and b={(0.6,0.2,0.2),(0.8,0.2,0.0), (0.9,0.0,0.1)} are two picture fuzzy sets of dimensions 3. then                 0.7 0.6 0.2 0.2 0.1 0.2 1 , 0.8 0.8 0.1 0.2 0.1 0.0 3 0.7 0.9 0.1 0.0 0.2 0.1 0.1 0.0 0.1 1 1 0.8 0 6.0 0.1 0.1 0.2 0.2 0.4 3 3 3 0.2 0.1 0 0 . .2 1 hd a b                                                 wang et al. (2017) defined some special operations of picture fuzzy set. they proposed the following operations on pfns  , ,a a aa    and  , ,b b bb    .      . , , 1 1 1a a b b a b a b a ba b                 (13)    , , 1 1 , 0a a a a aa                (14) example 2: let a =(0.7,0.2,0.1) and b=(0.6,0.2,0.2) are two picture fuzzy sets and λ=5. a.b=(0.7+0.2)*(0.6+0.2)-0.2*0.2, 0.2*0.2, 1-(1-0.1)*(1-0.1)=(0.68,0.04,0.19). aλ=a5=(0.7+0.2)5-(0.2)5, (0.2)5, 1-(1-0.1)5= (0.16807-0.00032), 0.00032, 1-0.59 si et al./decis. mak. appl. manag. eng. 2 (2) (2019) 54-64 58 =(0.16,0.00032,0.41) 2.4 comparison of picture fuzzy sets wang et al. (2017) used the score function and accuracy function to compare the pfss. let  , , ,c c c cc     be a picture fuzzy number, then a score function s(c) is being defined as   c cs c    and the accuracy function h(c) is given by   c c ch c      where    1,1s c   and    0,1h c  . then, for two picture fuzzy numbers c and d i. if    s c s d , then c is higher than d, denoted by c>d; ii. if    s c s d , then a.    h c h d , implies that c is equivalent to d, denoted by c=d; b.    h c h d , implies that c is higher than d, denoted by c>d. example 3: let c = (0.7,0.2,0.1) and d=(0.6,0.2,0.2) are two picture fuzzy sets. now, s(c)=0.7-0.1=0.6, s(d)=0.6-0.2=0.4, h(c)=0.7+0.2+0.1=0.9, h(d)=0.6+0.2+0.2=1. since s(c)>s(d), therefore c>d. 3. proposed method for ranking pfns it is known that the ranking of the fuzzy sets depends on the membership value of the elements. fuzzy numbers with higher membership value are ranked first. in an intuitionistic fuzzy set (ifs), the rank of intuitionistic fuzzy numbers (ifns) depend on membership values as well as non-membership values. the ifs which has the highest membership value and smaller non-membership value will have the first rank (zhang and xu, 2012). below, some situations are given, which appears during the ranking of ifns. let  ,a aa   and  ,b bb   be two ifns, then i. if a b  and a b  then a b ii. if a b  and a b  then a b iii. if a b  and a b  then a b here, situation 1 and 2 clearly defines the rank between the ifss a and b, but unable to provide the rank as given in situation 3, when both the membership and non-membership values are equal. motivated by the ranking procedure in ifs, wang et al. (2017) proposed the comparison technique between two pfss with the help of score and accuracy function. but their proposed method cannot discriminate the pfns when the score and accuracy values are same. we observed that the neutral membership grade could be considered to contribute in positive membership grade as well as negative membership grade. this motivated us to propose a new ranking method for the pfns even when the score and accuracy values are equal. the proposed approach is given below in a stepwise manner. let  , , , 1, 2,...,i i i if i n    be the set of pfns, where ,i i iand   respectively denote the positive, neutral and negative membership degree. an approach to rank picture fuzzy numbers for decision making problems 59 step 1: the positive ideal solution (pis)  , ,f       of the pfns  , ,i i i if    ,  1, 2.......i n is determined, where    , , max , min , mini i i i ii f             (15) step 2: the positive goal difference (pgd) * i and negative goal difference (ngd) * i of each of the pfns  , ,i i i if    ,  1, 2,...,i n are computed by * i i      and * i i      . step 3: absolute score of each pfn is calculated as  * *1i i ip     (16) the absolute score of a pfn is computed using the membership and nonmembership grade only. it completely ignores the neutral membership grade. however the neutral membership grade has an important contribution in finding the score which is narrated in the following steps. step 4: next the average neutral degree is computed, where 1 / . n i i               . step 5: estimate the actual score is (given below) of the pfns  , ,i i i if    ,  1, 2,...,i n using the average neutral degree. when actual scores of the two pfns if and jf are the same, then go to step 6.  1 i i i p s      (17) here the actual score will be always a finite value because the difference between average neutral degree and individual neutral degree of an pfn is never equal to 1, i.e.,   1.i   step 6: i) if i j i jand     then i js s . ii) if i j i jand     then iii) if i j  then i js s otherwise i js s . remark 3.1. absolute score pi =1 if i    and i    , i.e., when membership degree highest and non-membership degree is lowest, then the absolute score will be at most. remark 3.2. the actual score basically depends on the neutral degree. if the neutral degree of all pfns are same then actual score equal to absolute score. similarly, the actual score increases if the neutral degree decreases alternatively actual score decrease when the neutral degree increases. si et al./decis. mak. appl. manag. eng. 2 (2) (2019) 54-64 60 4. practical example in this section, we present a practical example to demonstrate the evaluation of the students and their ranking concerned with the multiple-choice questions (mcqs) based examination system with picture fuzzy information to illustrate the proposed method. suppose n be the number of students who are appearing in a competitive examination where the question paper is composed of multiple-choice questions. during the evaluation process, normally this kind of exams assign some positive marks opting for the correct choice and negative marks for opting the wrong choice. this exam system does not consider the non-attempted questions in the evaluation process. in the exam, some students may attempt all the questions while some other students may not attempt all the questions. now, among the attempted questions, two cases may arise. in the first case, all answers may be correct and in the second case, some answers may be correct while the rest are wrong. let’s consider, i , i and i be the percentage of correct answers, wrong answers, and not attempted questions respectively for the ith candidate. in the examination system, we consider that there are no wrong questions or out of syllabus questions. so, 1.i i i     we assume that there is no refusal membership value of the students in this examination system. the result of the ith student can be presented by a pfn  , ,i i i if    where i=1,2,..,n. table 1 shows the results of seven students (t1, t2, t3, t4, t5, t6, t7 ) for a particular mcq based exam using pfns. table 1. students’ results using pfns students t1 t2 t3 t4 t5 t6 t7 result (0.64, 0.22, 0.14) (0.74, 0.15, 0.11) (0.72, 0.19, 0.09) (0.82,0.1, 0.08) (0.82,0.1 4,0.04) (0.9,0.05, 0.05) (0.68,0.1, 0.22) to find out the ranking of the students, we illustrate the proposed approach as given below. step1: calculate the pis  0.9, 0.22, 0.03f   is calculated using eq. (15) and table 1. step 2: the positive and negative goal differences of individual students are given in table 2. table 2. pgd and ngd of students students t1 t2 t3 t4 t5 t6 t7 pgd 0.26 0.16 0.18 0.08 0.08 0.0 0.22 ngd 0.11 0.08 0.06 0.05 0.00 0.02 0.19 step 3: table 3 shows the absolute score of each student using eq. (16). table 3. absolute score of each student students t1 t2 t3 t4 t5 t6 t7 absolute score 0.63 0.76 0.76 0.87 0.92 0.98 0.59 an approach to rank picture fuzzy numbers for decision making problems 61 step 4: table 4 shows an actual score of each student using eq. (17). table 4. actual score of each student students t1 t2 t3 t4 t5 t6 t7 actual score 0.68 0.77 0.80 0.84 0.92 0.98 0.59 now the ranking list of students is found as 6 5 4 3 2 1 7 .t t t t t t t      5. comparative study to present the comparative study, we have compared the proposed method with the right marks (rm) method (lesage et al., 2013), multiple mark question (mmq) method (tarasowaand auer, 2013) and score and accuracy function based method (wang et al., 2017) with the same example stated above in section 4. as presented in (lesage et al., 2013), the degree of correctness ( i ) considers the score of the students which is presented in table 5. according to the rm method, the more be the score, the higher be the rank. this method does not consider the negative marking for the wrong answers. so the students are privileged to guess the answers to the unknown questions and attempt those questions. at this moment the students utilize the ambiguity of this technique. this technique does not measure the actual knowledge of the students. in table 5, both the students t4 and t5 score 0.82 along with 0.08 and 0.04 incorrect answers respectively. this method does not give any penalty for incorrect answers, and there is no proper method to handle when scores are the same. table 5. score of students according to the right marks method students t1 t2 t3 t4 t5 t6 t7 actual score 0.64 0.74 0.72 0.82 0.82 0.9 0.68 to overcome the drawbacks of right mark method, the authors introduced negative marking (nm) method (lesage et al., 2013) which measures the score of students as the difference between the degree of correctness ( i ) and degree of incorrectness ( i ), which is presented in table 6. this technique incorporates some penalty for the wrong answers. so, the students try to attempt their known questions to maximize their scores. but the categories of students, who attempt as many questions as possible without adequate knowledge can get the advantage of this technique. this method attempt to minimize the guessing tendency of the students with a penalty of the wrong answers but to guess advantage remains. in this negative marking method, there’s no provision for handling the situation when the score of two or more students are the same. in the following table (table 6), one can find that the score of two students t2 and t3 are similar. as a result, we are unable to get the proper rank of the students using the methods presented in (lesage et al., 2013). but the proposed method is capable of ranking the students even when the score values are equal. the example described in section 4 illustrates that ranking of the student (t3) is higher than the rank of the student (t2) since the actual score (0.80) of the student (t3) is more than that of the student (t2) which is (0.77). si et al./decis. mak. appl. manag. eng. 2 (2) (2019) 54-64 62 table 6. score of students according to negative marks method students t1 t2 t3 t4 t5 t6 t7 actual score 0.50 0.63 0.63 0.74 0.78 0.85 0.46 next multiple mark question (mmq) (tarasowa and auer, 2013) method has been compared with the proposed method. in mmq method, there are options to mark more than one right choices for a particular question for the checking the depth of knowledge of the students. but in case of mathematical problems, there’s generally one correct answer. so, mmq method is suitable for medical entrance examination but not appropriate for all types of examinations. in our proposed method with a single right answer for every question, we mainly consider the guessing tendency of the examinees and calculate the actual score based on the positive and negative answers and not attempted questions. in this method, if an examinee cleverly attempts more questions based on his assumptions and if these are found wrong, he will lag behind regarding score. table 7 shows the score value and accuracy value of the individual pfns associated with the seven students using the score and accuracy function (wang et al., 2017). comparison method of two ifns using score and accuracy functions is mentioned in section 2.4, where the alternatives are ranked based on their score and accuracy values. but there is no proper clarification when the alternatives have the same score value. in table 7, student t2 and t3 have the same score and accuracy values. therefore we can’t compare those two students. but our proposed method can compare two pfns even if the score and accuracy values are equal. as per the proposed method, the rank of t3 is higher than that of t2, i.e., t3>t2 since the actual scores of the students t3 and t2 are 0.80 and 0.77 respectively. table 7. score and accuracy value according to score and accuracy function students t1 t2 t3 t4 t5 t6 t7 score value( i i ) 0.50 0.63 0.63 0.74 0.78 0.85 0.46 accuracy value ( )i i i    1 1 1 1 1 1 1 6. conclusion in this article, a new approach is presented to rank the pfns. the new approach is different and improved 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(20107). some geometric aggregaton operators based on picture fuzzy sets and their application in multiple attribute decision making. italian journal of pure and applied mathematics, 37, 477–492. © 2018 by the authors.submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 6, issue 1, 2023, pp. 34-56. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0601051229022s * corresponding author. e-mail addresses: saloua.said.5@gmail.com (s. said), h.bouloiz@uiz.ac.ma (h. bouloiz), meryam09@gmail.com (m. gallab) new model for making resilient decisions in an uncertain context: the rational resiliencebased decision-making model (r2dm) saloua said1*, hafida bouloiz1 and maryam gallab 2 1 systems engineering and decision support laboratory, ibn zohr university, ensa agadir, morocco 2 mis-listd laboratory, computer science department, mines-rabat school (ensmr), rabat, morocco received: 22.10.2021; accepted: 12.11.2022; available online: 06.12.2022. original scientific paper abstract: the present paper came into existence with the specific purpose of providing an optimized process that enables making resilient decisions in an uncertain context, and here our interest is particularly focused on the activity of new venture creation and on the entrepreneurial decision-making logic, in particular, effectuation theory. within this framework, the rational resilience-based decision-making model (r2dm) is introduced. the relevant steps of this model are: (1) the identification of the problem and the available options. in this instance, the studied situation is the effectual customer cocreation case, and the available alternatives are planning, visionary, adaptative and transforming approaches, (2) the definition of the selection criteria that should be used to evaluate the available alternatives. in our case, these criteria are the six principles of entrepreneurial resilience, which are set out in detail, (3) the choice of the methodology to be followed in assessing the available options. to that end, three interconnected methods, based mainly on logical thinking and reasoning, are proposed. they are respectively devoted to entrepreneurial resilience (er) calculation, options classification using logistic regression algorithm, and the determination of the most resilient route to reach objectives employing graph theory. the obtained results are compared to what is advocated in the literature and conclusions are made. key words: resilience, decision-making, uncertainty, entrepreneurship, effectuation. new model for making resilient decisions in an uncertain context: the rational resilience… 35 1. introduction to make a profit, entrepreneurs must secure capital investment, implement highly specialized competencies, mobilize outside resources, identify opportunities, and take numerous risks (de winnaar and scholtz, 2018). in the face of this complexity of the entrepreneurship process, traditional management decision theory has shown its limitations (long et al., 2021). companies have developed the habit of being aware of their environment and adopting serial decision-making, where current decisions are influenced by previous ones (abzug, 2017), in contrast to new ventures that operate in a highly uncertain context with a genuine shortage of information. therefore, effectuation theory has emerged to resolve this issue. it is a decision-making process designed for expert entrepreneurs to help them create ventures in an environment marked by high uncertainty and resource scarcity (ghorbel et al., 2021). this theory provides benefits ranging from knightian uncertainty management to supporting control logic and sustainability approaches (sarasvathy and kotha, 2001). this theory is founded on five principles: (1) bird in hand, which alludes to the broad notion of the adage "a bird in the hand is worth two in the bush." in practice, an entrepreneur should begin by implementing the resources at hand rather than waiting for the perfect opportunity (sarasvathy, 2014), (2) affordable loss, which can be used as a substitute for the npv (net present value) traditional approach. according to this principle, it is necessary to make decisions within an acceptable level of risk rather than depending on uncontrollable predictions (silberzahn, 2016), (3) crazy-quilt, a principle inspired by patchwork. its key feature is that entrepreneurship is a social process that grows through the commitment of stakeholders (masango & lassalle, 2020), (4) lemonade, a term derived from the phrase "when life throws you lemons, make lemonade." it displays the capacity to turn adversity into an opportunity (pacho & mushi, 2021), (5) pilot in the plane. this concept incorporates the deep philosophy of effectuation theory. it argues the shifting from a prediction to a control logic since, as abraham lincoln once said, the only way to forecast the future is to mold it (sarasvathy et al., 2014). these principles highlight key personality traits and personal qualities that entrepreneurs should possess to overcome the numerous difficulties related to entrepreneurship. proactivity, stress tolerance, self-efficacy, a need for autonomy, innovation, and creativity are examples of this (branicki et al., 2017). it is important to note that these personality traits should associate with a system's ability to perform well under challenging circumstances. we are referring here to entrepreneurial resilience. this ability to sustain business in the face of toughness is denoted by access to material resources, development of an attractive personal identity, the experience of power and control, adherence to cultural traditions, the experience of social justice, and cohesion with others (hedner et al., 2011). the present paper proposes a decision-making process that guarantees to reinforce the entrepreneurial resilience in new businesses while making decisions in uncertain situations. the rational resilience-based decisionmaking model (r2dm) represents an attempt to establish a link between resilience and effectuation. in scientific literature, the studies with the same purpose remain small in number. examples include a qualitative study that links emotional resilience and effectual logic, on the one hand, and cognitive resilience and causal logic, on the other, and highlights the role played by these two resilience dimensions in supporting entrepreneurs in uncertain environment (d’andria et al., 2018), an analysis that demonstrates two types of coping strategies as a result of juxtaposing effectuation theory and resilience: effectual coping and causal coping (liu, 2019), a discussion of the resilience of family businesses by mentioning the significance of effectuation logic to these kinds of organizations (chrisman et al., 2011), as well as a contribution to said et al./decis. mak. appl. manag. eng. 6 (1) (2023) 34-56 36 fostering a better understanding of the role that effectuation and causation logics play in post-disaster entrepreneurial decision-making (akinboye & morrish, 2022). r2dm constitutes a qualitative and quantitative study, which proposes six pillars of entrepreneurial resilience (er) by taking into consideration psychological resilience related to the entrepreneur, and organizational resilience associated with the new venture. furthermore, it introduces two new methods allowing to calculate the value of entrepreneurial resilience for each available option, and then classify alternatives according to their er values into two main categories: resilient and non-resilient decisions. the next step is to identify the most resilient route to achieve objectives, a method to be used especially when dealing with interdependent choices (options and sub-options), and this is the ultimate result of our model. to achieve this, we made use of machine learning and operations research. it is also worth mentioning that the r2dm model is rational with clearly defined steps and logical reasoning applied at some point. this style has been selected because it has proven to be the most relevant and advanced compared with other decision-making types (intuitive, dependent, and avoidant) (uzonwanne, 2018). in order to give an exhaustive presentation of our model, the remainder of this article is organized in three parts. in the section 2, the r2dm model will be described in detail. first and foremost, the problem is identified. as an example of application, we opt for the effectual customer co-creation case. the objective is to compare the solution recommended by effectual logic in this situation, based on resources acquisition, stakeholders engagement and so on, with the result obtained through the use of the r2dm model, and then, if the same result is produced by the two approaches, discuss the reliability of our model and how it can make effectual logic more effective since it can provide the same results by taking into account a more global criterion, which is resilience. secondly, the six principles of entrepreneurial resilience, considered as selection criteria, are described in greater detail. thereafter, the three quantitative methods serving to determine the optimal decision from a resilience point of view are presented. section 3 is dedicated to the presentation of results and discussion. finally, conclusions are drawn and opportunities for future research are discussed. 2. rational resilience-based decision-making model (r2dm) 2.1. problem identification: the effectual customer co-creation case the effectuation process states that a stakeholder's commitment serves as the lifeblood of any new company. early customer acquisition opens up new avenues and inspires new objectives; otherwise, the small business is put on hold (sarasvathy, 2014). there are more interactions with clients, but there are also more constraints. for instance, the entrepreneur might learn after meeting a client that the factors governing market demand have considerably changed and that interest in the product in question is conditional upon making specific alterations. effectuation theory offers the entrepreneur four alternatives to deal with this predicament (silberzahn & enrico, 2016). the first among them is planning. this consists of making consistent attempts to position the business as accurately as possible. entrepreneurs who do not want to change their products look for other suitable customers using the market segmentation strategy. the second paradigm is visionary. it describes the attitude of entrepreneurs who cling to their optimistic vision of the future. in our context, this refers to passionately believing in the product and considering it avant-garde. in this case, the entrepreneur promotes his vision through various communication channels. a high level of prediction characterizes new model for making resilient decisions in an uncertain context: the rational resilience… 37 these two approaches. as for planning, the process can take a lot of time without any assurance that the entrepreneur can find his market niche. by adhering to the second approach, visionary, financial resources may be consumed over a long period without prospering in convincing customers of the importance of the product. the third alternative is adaptative. the entrepreneur concurs in making the requested changes by mobilizing adequate resources and time to satisfy the customer and win his commitment. lastly, the fourth possibility is transforming. this revolves around setting co-created goals by obtaining customers’ commitment. adaptative and transforming approaches agree that the entrepreneur should adjust the product. the transforming strategy ensures that an actual profit will be generated in exchange for devoting time and resources. however, in the adaptative approach, resources and time are at risk because there is no guarantee that the customer will acquire the transformed product without a prior commitment. the effectuation theory supports the transforming approach because it favors control with minimal prediction. the entrepreneur needs to make a final decision by selecting one of the options previously mentioned. 2.2. selection criteria identification: entrepreneurial resilience principles 2.2.1. first er principle: adapt or pivot where required one of the main causes of startups and new small businesses failing is the inability to pivot (mccarthy, 2017). the entrepreneur has a problem over whether to stick with the previously chosen strategy or alter it and even adopt a new strategy, if required, in the case of any unexpected change (khurana et al., 2020). adaptation involves a level of flexibility, which can be defined as the ability to change easily according to the situation (george-weinstein, 2020). this ability can be measured by the workload that has to be managed within an acceptable amount of time, the response time that designates the time needed to interact, the generated costs, and the quality achieved (gong & janssen, 2010). 2.2.2. second er principle: weigh the options despite their extensive knowledge of the subject, decision-makers may still find it difficult to weigh the possibilities. this can be far more challenging for someone starting a small business since, in addition to their lack of expertise, many other factors are at play and must be carefully considered before taking any action. the impact on internal stakeholders or on the development of human resources (cardon & stevens, 2004), communication quality (khoshnodifar et al., 2016), customer satisfaction (russell-bennett et al., 2007), defined objectives, financial and investment issues, possession of the necessary knowledge and expertise, application of the work plan, dynamism, and enthusiasm are a few examples (kiritz, 2015). there are arguments that this method might slow down the business because a lot of time is spent examining the many options. however, making thoughtful decisions results from carefully balancing the available alternatives and minimizing the risk through reducing costs, implementation time, workload, and enhancing quality. 2.2.3. third er principle: turn adversity into opportunity the ability to recognize hidden opportunities in unfavorable circumstances is a tremendous advantage. every negative experience is perceived as an engaging task rather than a challenge to overcome. for instance, reaching consumers who are not pleased with the competitors' services, or hiring skilled but demanding employees. said et al./decis. mak. appl. manag. eng. 6 (1) (2023) 34-56 38 key lessons can be identified in corporate history, e.g., ryanair’s recession survival in 2008/2009 (curran, 2010). the ability to recognize opportunities in times of crisis can be called opportunity agility (stephan et al., 2022), and is based on four metrics, which are costs, time, robustness, and scope (yauch, 2011). 2.2.4. fourth er principle: do more than is required the extra mile and customer satisfaction are closely related concepts in entrepreneurial jargon. consumer demands and expectations should get constant special attention. additional efforts have to be made to demonstrate a sincere interest in the customer. this may also present a chance to raise the inherent quality of the finished good or service (yi & gong, 2008). customer satisfaction can be measured by quality that refers to the ability of businesses to constantly improve their products and services to meet the customer expectations, loyalty that can be built via flexibility and agility, and trust that can lead to customer retention, and therefore, to a minimization of risks (saad et al., 2022). 2.2.5. fifth er principle: feel self-confident entrepreneurship may expose, by its very nature, the new business founders to reiterated failures. displaying a confident attitude would empower them to embrace and surmount these defeats straightforwardly. over and above that, others can feel the entrepreneur’s confidence, trust him, and be prepared to conduct business with him (gelaidan & abdullateef, 2017). entrepreneurs’ self-confidence is denoted by believing in their own abilities, acting independently in making decisions, having a positive self-concept, and daring to express opinions (febrianto et al., 2022). 2.2.6. sixth er principle: bounce back one of the best ways to build resilience is to bounce back (hoegl & hartmann, 2020). unwanted occurrences may devastate a person's life or business. after that, one can suffer significant losses and feel forced to start again. da capo, a musical word that denotes starting a piece of music from the beginning, can be used to figuratively refer to this situation. giving up is not an option for an entrepreneur who wants to be successful (aldianto et al., 2021). the bouncebackability can be characterized by adaptive capacity and recovery potential (rector et al., 2019). the question now being asked is whether a quantitative measure can be ascribed to entrepreneurial resilience. in the interest of shedding some light on this issue, the following paragraphs will illustrate our principal findings of the present research. 2.3. optimal decision identification: methodology 2.3.1. method n° 1: er calculation using logical thinking and reasoning we start by evaluating how much each er principle is involved in the given alternatives. the options are divided into two categories: resisting the change by declining to make the requested modifications and keeping the product in its original state or complying with the demand by deciding to make the necessary modifications. if a change is refused, planning or visionary techniques are envisaged, and if it is accepted, adaptative or transforming strategies can be used. according to the formulated judgement, a mastery level is assigned to the relevant principle as described in the following table: new model for making resilient decisions in an uncertain context: the rational resilience… 39 table 1. levels of mastery of er principles principle’s value level of mastery pi ϵ [0.8, 1] the principle is very well mastered pi ϵ [0.6, 0.8[ the principle is well mastered pi ϵ [0.4, 0.6[ the principle is moderately mastered pi ϵ [0.2, 0.4[ the principle is weakly mastered pi ϵ [0, 0.2[ the principle is unmastered the value of entrepreneurial resilience is the sum of the length of each interval (0.2 for all of them) divided by 2 and weighted as follows: unmastered principle: 10% (multiplied with 0.1), weakly mastered principle: 15% (multiplied with 0.15), moderately mastered: 20% (multiplied with 0.2), well mastered principle: 25% (multiplied with 0.25), very well mastered principle: 30% (multiplied with 0.3). table 2. calculation of the er value for the option: opposing the changes er principles train of thought principle value er value adapt or pivot where required the entrepreneur does not adjust to the situation since he decides to keep intact his initial product. verdict: in this situation, this principle is unmastered. pi ϵ [0, 0.2[ 0.1 weigh the options the entrepreneur's approach does not take into consideration the minimization of risks. on the contrary, he chooses a situation that is far from being free of ambiguity and uncertainty. verdict: in this situation, this principle is unmastered. pi ϵ [0, 0.2[ turn adversity into opportunity resistance to this change has ruins opportunities that might be arising from working with the customer who asked for the product modification. verdict: in this situation, this principle is unmastered. pi ϵ [0, 0.2[ do more than is required the entrepreneur is unable to understand and accept the customer's needs. verdict: in this situation, this principle is unmastered. pi ϵ [0, 0.2[ feel selfconfident the insistence on maintaining the product in its original state shows the entrepreneur's great confidence in himself and in his product. verdict: in this situation, this principle is very well mastered. pi ϵ [0.8, 1] bounce back the entrepreneur does not find it difficult to start from scratch and to look for new customers. pi ϵ [0.8, 1] said et al./decis. mak. appl. manag. eng. 6 (1) (2023) 34-56 40 verdict: in this situation, this principle is very well mastered. with regard to the option ‘opposing the change’, and by reference to table 2, two er principles represent the strength of this action. in fact, this choice reflects a strong confidence about the capacity of the product to reach the consumer in its present state, and about the aptitude of the entrepreneur for identifying potential clients in the best possible way. for this reason, the level of self-confidence in this situation is thought to be between 0.8 and 1. moreover, by taking this approach, the commercialization process does not advance with the target client, and the entrepreneur must undergo the whole process from the beginning by seeking for new customers. consequently, the capacity of bouncing back is also estimated at between 0.8 and 1. however, four er principles are identified as unmastered principles. the first one is adaptation. by declining the client’s request, the entrepreneur refuses to accept this situation that forces him to change his initial strategy and vision. therefore, this principle is estimated to be between 0 and 0.2. concerning the principle related to minimizing risks (weigh the options), it is not fulfilled since the entrepreneur runs, in this situation, a substantial risk with respect to finding a suitable client within a short time frame while mobilizing the minimum possible resources. in consequence, this principle is also between 0 and 0.2. the third unmastered principle is turning adversity into opportunity. indeed, the entrepreneur misses the opportunity that could possibly be hidden behind the modification demand, which might seem, at first glance, complicated and time-consuming. the value of this principle is thereupon between 0 and 0.2. the last principle described as unmastered is associated with the acceptation and comprehension of the consumer’s needs, which are completely absent in this situation. as a result, the value attributed to this principle is between 0 and 0.2. the er value is: er1 = (((0.2-0)/2) × 0.1) + (((0.2-0)/2) × 0.1) + (((0.2-0)/2) × 0.1) + (((0.2-0)/2) × 0.1) + (((1-0.8)/2) × 0.3) + (((1-0.8)/2) × 0.3) = 0.1. (1) that means that the option ‘opposing the change’ is 10% resilient. for proper interpretation of this result, we propose an application of the resilience scale introduced by (said et al., 2019), and devoted to classifying processes according to their resilience level (called echelon) (figure 1). figure 1. resilience scale (said et al., 2019) new model for making resilient decisions in an uncertain context: the rational resilience… 41 this scale will be used, in this context, to define an available option as per its resilience (table 3). in this case, the option ‘opposing the change’ is unconscious. table 3. classification of available options on resilience scale using er values er value option’s echelon er ϵ ]0.25, 0.3] the option is expert er ϵ ]0.2, 0.25] the option is progressing er ϵ ]0.15, 0.2] the option is aspiring er ϵ ]0.1, 0.15] the option is informed er ϵ ]0, 0.1] the option is unconscious regarding the option ‘planning approach’, which is a sub-option of the alternative ‘opposing the change’, it is considered as an informed option since the corresponding er value is 0.125 (12.5%). er2 = (((0.6-0.4)/2) × 0.2) + (((0.2-0)/2) × 0.1) + (((0.6-0.4)/2) × 0.2) + (((0.80.6)/2) × 0.25) + (((0.8-0.6)/2) × 0.25) + (((0.8-0.6)/2) × 0.25) = 0.125. (2) by referring to table 4, it can be noticed that three er principles are well mastered. this indicates that, by adopting this option, the self-confidence of the entrepreneur is between 0.6 and 0.8 since it is admitted that the product may only be suitable for a specific category of consumers. on a different note, it is true that the entrepreneur has not been able to respond to the needs of the client by modifying the product. nonetheless, he seems keen to reach and retain his target customers through segmentation, even though this strategy is accompanied with risks. hence, the values of the two principles ‘do more than is required’ and ‘bounce back’ are between 0.4 and 0.6. in addition, adaptation and turning adversity into opportunity are moderately mastered with a value between 0.4 and 0.6. lastly, weighing the options is unmastered because there is no limitation of risks in this case. consequently, the associated principle’s value is 0 and 0.2. table 4. calculation of the er value for the planning approach er principles train of thought principle value er value adapt or pivot where required the entrepreneur tries to adapt to the situation by changing his strategy (selection of target customers). however, the risk of not finding potential customers is very high. verdict: in this situation, this principle is moderately mastered. pi ϵ [0.4, 0.6[ 0.125 weigh the options the risk is always there and is not limited in this case. verdict: in this situation, this principle is unmastered. pi ϵ [0, 0.2[ turn adversity into opportunity the entrepreneur tries to use the situation to his advantage by reducing his room for manoeuvre, thus saving time and resources. verdict: in this situation, this pi ϵ [0.4, 0.6[ said et al./decis. mak. appl. manag. eng. 6 (1) (2023) 34-56 42 er principles train of thought principle value er value principle is moderately mastered. do more than is required the segmentation paradigm allows for a better understanding of customers and their expectations. verdict: in this situation, this principle is well mastered. pi ϵ [0.6, 0.8[ feel selfconfident the entrepreneur is convinced that his product can be interesting for a certain category of customers. verdict: in this situation, this principle is well mastered. pi ϵ [0.6, 0.8[ bounce back the entrepreneur was able to pursue a different strategy than the one originally envisioned. however, this change is associated with several risks. verdict: in this situation, this principle is well mastered. pi ϵ [0.6, 0.8[ the visionary approach, detailed in table 5, requires a huge self-confidence that is estimated, in the present instance, between 0.8 and 1. moreover, this strategy revolves around the communication and promotion of the product until the right customers are found, while calling up important resources. that is why it was deemed that the principles ‘do more than is required’ and ‘bounce back’ are moderately mastered with values between 0.4 and 0.6. on the other hand, the principles relating to adaptation, weighing the options, and turning adversity into opportunity are unmastered since the visionary approach constitutes a risk-taking experience par excellence. as a result, this approach is 10% resilient (er value = 0.10) and, through referring to the resilience scale, this option is unconscious. er3 = (((0.2-0)/2) × 0.1) + (((0.2-0)/2) × 0.1) + (((0.2-0)/2) × 0.1) + (((0.6-0.4)/2) × 0.2) + (((1-0.8)/2) × 0.3) + (((0.6-0.4)/2) × 0.2) = 0.1. (3) table 5. calculation of the er value for the visionary approach er principles train of thought principle value er value adapt or pivot where required the entrepreneur refuses to adapt to current customer needs. verdict: in this situation, this principle is unmastered. pi ϵ [0, 0.2[ 0.1 weigh the options the entrepreneur uses extra resources to persuade customers with his product instead of trying to meet their needs. the risk is high in this situation. verdict: in this situation, this principle is unmastered. pi ϵ [0, 0.2[ turn adversity into opportunity there is no immediate opportunity. the entrepreneur follows a logic of prediction and targets opportunities pi ϵ [0, 0.2[ new model for making resilient decisions in an uncertain context: the rational resilience… 43 er principles train of thought principle value er value whose probability of occurrence in the future is unknown verdict: in this situation, this principle is unmastered. do more than is required the entrepreneur tries to better understand the profile of customers who might be interested in and satisfied with his product. verdict: in this situation, this principle is moderately mastered. pi ϵ [0.4, 0.6[ feel selfconfident the entrepreneur has full confidence in his product and is willing to use all means to make it a success. verdict: in this situation, this principle is very well mastered. pi ϵ [0.8, 1] bounce back the entrepreneur has not given up on the customer, but rather has increased communication around his product in order to be able to convince him in addition to the other customers. verdict: in this situation, this principle is moderately mastered. pi ϵ [0.4, 0.6[ for the option ‘complying with the change’, five principles out of a total of six are very well mastered. in fact, this option is proof of the perfect capacity of adaptation while not fearing to go by a road different to that originally envisaged, capturing the opportunity presenting itself while taking in mind the necessity of minimizing risks, bending towards the specific needs of customers, and restarting the process if necessary. the remaining principle ‘feel self-confident’ is well mastered since the entrepreneur has doubts about the completeness of his product in its initial state. thus, the option ‘complying with the change’ is aspiring (er value = 0.175 (17.5%)) (table 6). er4 = (((1-0.8)/2) × 0.3) + (((1-0.8)/2) × 0.3) + (((1-0.8)/2) × 0.3) + (((1-0.8)/2) × 0.3) + (((0.8-0.6)/2) × 0.25) + (((1-0.8)/2) × 0.3) = 0.175. (4) table 6. calculation of the er value for the option: complying with the changes er principles train of thought principle value er value adapt or pivot where required the entrepreneur adapts perfectly to the situation. verdict: in this situation, this principle is very well mastered. pi ϵ [0.8, 1] 0.175 weigh the options the entrepreneur minimizes the risk of losing a potential customer and not finding a new one. verdict: in this situation, this principle is very well mastered. pi ϵ [0.8, 1] said et al./decis. mak. appl. manag. eng. 6 (1) (2023) 34-56 44 er principles train of thought principle value er value turn adversity into opportunity the entrepreneur seizes the opportunity to sell his product and improve its quality at the same time. verdict: in this situation, this principle is very well mastered. pi ϵ [0.8, 1] do more than is required the entrepreneur has perfectly understood and accepted the needs of his customer. verdict: in this situation, this principle is very well mastered. pi ϵ [0.8, 1] feel selfconfident the entrepreneur believes that his product should be improved. verdict: in this situation, this principle is well mastered. pi ϵ [0.6, 0.8[ bounce back the entrepreneur is able to start from scratch if necessary. verdict: in this situation, this principle is very well mastered. pi ϵ [0.8, 1] the option ‘adaptative approach’, presented in table 7, is a sub-option of ‘complying with the changes’ alternative. that said, in the absence of a commitment from the client, the principle ‘feel self-confident’ is considered as weakly mastered, weighing the options is moderately mastered as the risks are increased in this situation, adaptation and turning adversity into opportunity are well mastered since the results are not guaranteed in the present case, and finally, the principles ‘do more than is required’ and ‘bounce back’ are very well mastered because the interest expressed to the customer’s needs is perfect, in this situation, and the entrepreneur has no objection to relaunch the process of product conception. the er value of the adaptative approach is 0.145. this implies that the latter is informed. er5 = (((0.8-0.6)/2) × 0.25) + (((0.6-0.4)/2) × 0.2) + (((0.8-0.6)/2) × 0.25) + (((10.8)/2) × 0.3) + (((0.4-0.2)/2) × 0.15) + (((1-0.8)/2) × 0.3) = 0.145. (5) table 7. calculation of the er value for the adaptive approach er principles train of thought principle value er value adapt or pivot where required the entrepreneur adapts to the situation, but he puts himself under pressure if the customer does not make a commitment. verdict: in this situation, this principle is well mastered. pi ϵ [0.6, 0.8[ 0.145 weigh the options the entrepreneur minimizes the risk of losing a potential customer and not finding a new one. however, without a commitment from the customer, there is a risk that the latter changes his mind. verdict: in this situation, this principle is moderately mastered. pi ϵ [0.4, 0.6[ turn the entrepreneur has taken the pi ϵ [0.6, 0.8[ new model for making resilient decisions in an uncertain context: the rational resilience… 45 er principles train of thought principle value er value adversity into opportunity opportunity to sell his product while improving its quality. in this situation, however, these results are not guaranteed. verdict: in this situation, this principle is well mastered. do more than is required the entrepreneur has perfectly understood and accepted the needs of his customer. verdict: in this situation, this principle is very well mastered. pi ϵ [0.8, 1] feel selfconfident the entrepreneur believes that his product should be improved, and he has not succeeded in obtaining a commitment from his customer. verdict: in this situation, this principle is weekly mastered. pi ϵ [0.2, 0.4[ bounce back the entrepreneur is able to start from scratch if necessary. verdict: in this situation, this principle is very well mastered. pi ϵ [0.8, 1] the last option is the ‘transforming approach’, which requires a commitment on the part of the client before proceeding with any changes, and this has a very positive impact on er principles values that fluctuates between very well mastered (five principles) and well mastered (one principle). this option is aspiring (er value = 0.175). er6 = (((1-0.8)/2) × 0.3) + (((1-0.8)/2) × 0.3) + (((1-0.8)/2) × 0.3) + (((1-0.8)/2) × 0.3) + (((0.8-0.6)/2) × 0.25) + (((1-0.8)/2) × 0.3) = 0.175. (6) table 8. calculation of the er value for the transforming approach er principles train of thought principle value er value adapt or pivot where required the entrepreneur adapts perfectly to the situation. verdict: in this situation, this principle is a strength. pi ϵ [0.8, 1] 0.175 weigh the options the entrepreneur minimizes the risk of losing that customer, but also of wasting resources and time unnecessarily. verdict: in this situation, this principle is a strength. pi ϵ [0.8, 1] turn adversity into opportunity the entrepreneur seized on the opportunity to sell his product while improving its quality. verdict: in this situation, this principle is a strength. pi ϵ [0.8, 1] do more than the entrepreneur perfectly understands pi ϵ [0.8, 1] said et al./decis. mak. appl. manag. eng. 6 (1) (2023) 34-56 46 er principles train of thought principle value er value is required and accepts the needs of his client. verdict: in this situation, this principle is a strength. feel selfconfident the entrepreneur admits the fact that his product needs to be improved, but in return he manages to keep his customer through obtaining a commitment from him. verdict: in this situation, this principle is developing well. pi ϵ [0.6, 0.8[ bounce back the entrepreneur is able to restart again from the beginning if necessary. verdict: in this situation, this principle is a strength. pi ϵ [0.8, 1] the optimal solution that can be selected is simply the one with the highest value of entrepreneurial resilience, which is, in this case, complying with the change by adopting the transforming approach (er = 0.175). this first method suggests the calculation of the er value for each available option based on the values of the six er principles, and, in the context of our studied case, the best solution can be easily detected since we are dealing with very few options. however, in complex situations, this task will become much more difficult. considering this, we propose, in the next paragraph, a second method aimed at classifying available alternatives into resilient and non-resilient options. the part of logical reasoning introduced under the umbrella of method n°1 and serving to determine the mastery levels of er principles for each option, will be used in method n°2 in order to build the dataset. then, the option class (1: resilient, 0: non-resilient) will be identified through the use of a logistic regression model. 2.3.2. method n° 2: options classification using logical thinking and reasoning and logistic regression algorithm the issue addressed here can be regarded as a binary classification problem since we have two classes, namely resilient options belonging to class 1 and non-resilient options, which are in class 0. in order to bring about a resolution to this problem, we decide to use the logistic regression algorithm, a statistical model, which is widely used in machine learning (rymarczyk et al., 2019) for studying the relationships between a variable y to be predicted, and a set of explanatory variables xi. in the present instance, y stands for entrepreneurial resilience (er) and {x1, x2, x3, x4, x5, x6} represent the six principles of er. the environment chosen to write and run our model is google colaboratory or colab. we also used pyspark, an interface to apache spark in python. this is considered as one of the most optimized data structures in machine learning since it enables high-performance computations (el bouchefry & de souza, 2020). after installing pyspark and creating a sparksession as an entry point, we import the dataset, which is, for this case, a csv file named “entrepreneurial_resilience.csv”. the latter consists of seven columns. the first six columns correspond to the er principles and the last column to the class to which the option belongs, 1 for resilient options and 0 for non-resilient options). the sheet contains 1200 lines as well. they new model for making resilient decisions in an uncertain context: the rational resilience… 47 are populated with er principles’ values for several options available to deal with different situations related to managing cash flow issues, launching new products, hiring suitable candidates, building consumer loyalty, stepping out from the comfort zone, coping with cyber security issues, and so forth. the er principles’ values are calculated while following the same process previously explained within the framework of the rational resilience-based decision-making model. as for deciding on the class to which a given option belongs (resilient or non-resilient), this can be achieved through the application of a few rules. in effect, an option can be deemed as resilient on condition that one of the following scenarios apply: (1) if no principle among the six er principles is very well mastered, we must have at least four principles that are well mastered, (2) if there is only one principle that is very well mastered, we must have at least three well mastered principles, (3) if two principles are considered as very well mastered, we must have at least two principles that are well mastered, (4) if three principles are considered as very well mastered, only one principle has to be well mastered, (5) if four or more principles are found to be very well mastered, the option is adjudged as resilient. in applying such a method to the options described in tables (2, 4, 5, 6, 7, and 8), one observes that only the three alternatives: complying to the change, adaptative approach, and transforming approach can be taken into account. as regards with our logistic regression model, table 9 illustrates the first five lines of the imported dataset. table 9. data sample visualization s p a k j d result 0 0.2 to 0.4 0.6 to 0.8 0.6 to 0.8 0.4 to 0.6 0.4 to 0.6 0.8 to 1 0 1 0.6 to 0.8 0.6 to 0.8 0 to 0.2 0.6 to 0.8 0.6 to 0.8 0.2 to 0.4 1 2 0 to 0.2 0.2 to 0.4 0.8 to 1 0 to 0.2 0 to 0.2 0.8 to 1 0 3 0.4 to 0.6 0 to 0.2 0 to 0.2 0 to 0.2 0.2 to 0.4 0.6 to 0.8 0 4 0.6 to 0.8 0.8 to 1 0 to 0.2 0.8 to 1 0.6 to 0.8 0.2 to 0.4 1 in the first line, we have one principle that is considered as very well mastered, namely ‘bounce back’ symbolized by the letter d, and only two principles, which are well mastered. we refer to ‘weigh the options’ (p), and ‘turn adversity into opportunity’ (a). consequently, the option can be qualified as non-resilient (result = 0). in the second row, for example, we have no principle, which is very well mastered. however, four principles are deemed as well mastered, which are ‘adapt or pivot where required’ (s), ‘weigh the options’ (p), ‘do more than is required’ (k), and ‘feel self-confident’ (j), so the decision is considered as resilient (result = 1). some summary statistics, such as the mean value, the standard deviation (stddev), the minimum and the maximum, were also calculated. these values are represented in table 10. table 10. dataset’s summary statistics summary s p a k j d result count 1200 1200 1200 1200 1200 1200 1200 mean 0.4984 0.5038 0.5055 0.4882 0.4992 0.5030 0.2491 stddev 0.2856 0.2903 0.3005 0.2896 0.2928 0.2853 0.4327 min 0.0 0.0 0.0 0.0 0.0 0.0 0.0 max 1.0 1.0 1.0 1.0 1.0 1.0 1.0 said et al./decis. mak. appl. manag. eng. 6 (1) (2023) 34-56 48 according to the mean value, the typical mastery level of er and its principles is ‘moderately mastered’ since they run around 50%. as per the standard deviation, its low value indicates that the columns’ values are closed to the mean. the correlation between each principle and the entrepreneurial resilience was determined as well. as illustrated in table 11, the positive correlation coefficients indicate that the increase of er principles values results in a rise in the result, which stands for the er value. in addition, the fact that the correlation coefficients range between 0.24 and 0.3 states that there is a weak uphill linear relationship between the six principles and the er. table 11. correlations between er and its principles s p a k j d result correlation to results 0.2413 0.2519 0.2953 0.2735 0.2779 0.2767 1.0 then, a subset of our database (723 of 1200 elements) is selected in order to build and train the logistic regression model. table 12 shows a summary of the resulting model. table 12. summary statistics of the model summary result prediction count 723 723 mean 0.2351 0.2102 stddev 0.4243 0.4077 min 0.0 0.0 max 1.0 1.0 after that, the model is evaluated by comparing the generated predictions with the actual data. by taking into consideration the two columns, displayed in table 13, namely ‘result’ and ‘rawprediction’, which reflects the direct confidence calculation, we can proceed to the identification of the model’s accuracy. in our case, it is about 95%. table 13. summary statistics of the model features result rawprediction probability prediction [0 to 0.2, 0 to 0.2, 0.8 to 1, … 0 [5.2503… [0.9947… 0.0 [0 to 0.2, 0 to 0.2, 0.8 to 1, … 1 [-0.1942… [0.4515… 1.0 the last step is to save the model and then upload it back to the environment. lastly, the model is subjected to a test using a file of 10 lines containing only the values of the er principles as represented by table 14, and the model calculates the value of the class to which the option belongs. new model for making resilient decisions in an uncertain context: the rational resilience… 49 table 14. test sample s p a k j d 0.6 to 0.8 0.8 to 1 0.8 to 1 0.6 to 0.8 0.6 to 0.8 0.6 to 0.8 0.4 to 0.6 0.2 to 0.4 0 to 0.2 0.8 to 1 0.2 to 0.4 0 to 0.2 0.4 to 0.6 0.4 to 0.6 0.6 to 0.8 0.8 to 1 0 to 0.2 0.8 to 1 0.8 to 1 0.8 to 1 0.2 to 0.4 0.8 to 1 0.4 to 0.6 0.4 to 0.6 0.2 to 0.4 0.4 to 0.6 0.6 to 0.8 0.4 to 0.6 0.8 to 1 0.4 to 0.6 0.2 to 0.4 0.8 to 1 0.4 to 0.6 0.8 to 1 0 to 0.2 0.8 to 1 0.6 to 0.8 0.4 to 0.6 0 to 0.2 0.6 to 0.8 0.8 to 1 0.8 to 1 0.2 to 0.4 0.8 to 1 0.4 to 0.6 0.4 to 0.6 0.6 to 0.8 0.4 to 0.6 0.2 to 0.4 0 to 0.2 0.4 to 0.6 0.6 to 0.8 0 to 0.2 0.2 to 0.4 0.2 to 0.4 0.6 to 0.8 0.6 to 0.8 0.8 to 1 0.4 to 0.6 0.8 to 1 the results are shown in table 15. according to the predictions of the model, only three of the ten options can be classified as resilient. table 15. test results features prediction [0.6, 0.8[, [0.8, 1], [0.8, 1], [0.6, 0.8[, [0.6, 0.8[, [0.6, 0.8[ 1.0 [0.4, 0.6[, [0.2, 0.4[, [0.8, 1], [0.8, 1], [0.2, 0.4[, [0, 0.2[ 0.0 [0.4, 0.6[, [0.4, 0.6[, [0.6, 0.8[, [0.8, 1], [0, 0.2[, [0.8, 1] 0.0 [0.8, 1], [0.8, 1], [0.2, 0.4[, [0.8, 1], [0.4, 0.6[, [0.4, 0.6[ 0.0 [0.2, 0.4[, [0.4, 0.6[, [0.6, 0.8[, [0.4, 0.6[, [0.8, 1], [0.4, 0.6[ 0.0 [0.2, 0.4[, [0.8, 1], [0.4, 0.6[, [0.8, 1], [0, 0.2[, [0.8, 1] 0.0 [0.6, 0.8[, [0.4, 0.6[, [0, 0.2[, [0.6, 0.8[, [0.8, 1], [0.8, 1] 1.0 [0, 0.2[, [0.8, 1], [0.4, 0.6[, [0.4, 0.6[, [0.6, 0.8[, [0.4, 0.6[ 0.0 [0, 0.2[, [0, 0.2[, [0.4, 0.6[, [0.6, 0.8[, [0, 0.2[, [0.2, 0.4[ 0.0 [0.4, 0.6[, [0.6, 0.8[, [0.6, 0.8[, [0.8, 1], [0.4, 0.6[, [0.8, 1] 1.0 it must be emphasized that this method can be ideally used during brainstorming sessions, for instance, in order to filter out instantaneously resilient and non-resilient options. nevertheless, the identification of the optimal decision is not attainable under this approach, which is dedicated exclusively to classification. therefore, an additional method, supporting this objective, needs to be provided. however, we noted that, in certain cases, the optimal decision is not reduced to a single option to select, but it can include a series of alternatives that should be applied one after the other for purposes of achieving objectives in the most resilient way. in light of this, the next proposed method sets out to determine the most resilient way to reach objectives by employing logical thinking and operations research, more precisely, graph theory. 3.3.3. method n° 3: determination of the most resilient route to reach objectives using logical thinking and reasoning and graph theory as already explained in method 1, when dealing with independent options, the optimal solution that can be selected is the one with the highest value of entrepreneurial resilience. still and all, our studied situation implies interdependent alternatives and thus should be handled differently. for this purpose, a method, aiming at identifying the most resilient path leading to objectives achievement, is said et al./decis. mak. appl. manag. eng. 6 (1) (2023) 34-56 50 introduced. it is conceived from the inspiration of the dijkstra's algorithm for solving the shortest path problem (enayattabar et al., 2018). first off, a weighted graph representing the information gathered in table 2, table 4, table 5, table 6, table 7, and table 8, is created using python, and more specifically, the networkx library, which is designed for the study of graphs and networks (modarresi & symons, 2019). in this graph, the available options are the nodes, and the er values are the weights. the obtained graph is shown by the following figure (figure 2). figure 2. weighted graph for determining the most resilient path the node ‘a’ corresponds to the identified problem or the triggering event. in this case, it is about the product change demand by the customer and whether or not approving this request. regarding the nodes from ‘b’ to ‘h’, they are referring to the available options, ‘b’ is relative to the option “following the change”, ‘c’ stands for the option “opposing the change”, ‘d’ represents the option of adaptative approach, as for ‘e’, it designates the transforming option, on the other side stands ‘f’ for the planning approach as alternative, and ‘h’ denotes the option for visionary paradigm. the last node ‘i’ means goal attainment. in fact, the weight of each edge connecting points ‘d’, ‘e’, ‘f’, ‘h’ to the node ‘i’ is the average of the other weights of the edges forming the same chain. to give an example, the weight of the edge between the two vertices ‘d’ and ‘i’ (weight=0.16, cf. figure 2) constitutes the arithmetic mean of the weights of the edges connecting ‘a’ and ‘b’ (weight=0.175, cf. figure 2) and the nodes ‘b’ and ‘d’ (weight=0.145, cf. figure 2). this means that the approach followed between ‘a’ and ‘d’ to achieve the objective ‘i’, which is, in this case, reaching product commercialization, is 16% resilient. in order to detect the most resilient path on our graph, we commence by drawing a table with lines and columns corresponding to the nodes of the graph. in each cell of the table, we enter the weight of the edge connecting two consecutive nodes. if it is not applicable, the field is greyed out. this step is illustrated with the table below (table 16). table 16. how to identify the most resilient path? a b c d e f h i a 0.175 0.1 b 0.145 0.175 c 0.125 0.1 d 0.16 e 0.175 f 0.1125 h 0.1 i new model for making resilient decisions in an uncertain context: the rational resilience… 51 thereafter, we select the largest value for each line. in our case, we take the value 0.175 for the first row. for the second one, we choose the value 0.175. to find the most optimal path, we apply the following rule: if the highest value of a selected row is greater than or equal to the minimum of the values selected in the previous rows, we take that value, otherwise we go to the next row. regarding the line 2, we have 0.175 which is equal to the value 0.175 selected from the first row, that is why we retain this value. in the third line, we have 0.125 and 0.1, which are less than 0.175. therefore, none of these values are adopted, and thence we move to the fourth row, which contains a single value that does not satisfy the conditions (0.16 < 0.175), so this value is not considered. in the fifth row, we have 0.175. this value is included in our list. in the last two rows, no value (0.1125, 0.1) is valid. to sum up, the selected values are the fields in green. by replacing these values with the corresponding nodes, we obtain the most resilient path with respect to the situation under study: mostresilientpath = [a, b, e, i]. this path corresponds to the option of transforming. it is the most reliable strategy for achieving goals from a resilience perspective, and the obtained result for the present situation, applying our rational resilience-based decision-making model, is in line with the recommendations of the effectuation theory. 3. results and discussions in this study, the following methods were proposed. firstly, the calculation of the entrepreneurial resilience (er) value for a given available option through a logical analysis of the six corresponding er principles by responding to these questions: regarding the studied option, what are the estimated rates of adaptation and pivot, wisely weighing the options to minimize risks, turning adversity into opportunity and uncovering hidden opportunities, paying particular attention to the consumers' needs, feeling confident about one’s capacities and products, and the ability to bounce back? once these rates are obtained, the er value is calculated. the er values help us classify the available options on the resilience scale and identify the most suitable solution from a resilience perspective, but only in straightforward situations. this method reveals its limitations when there are numerous connected alternatives or with options of the same rank. the second method is introduced to address this issue. it is conducted in two stages: the first consists in classifying the available options into resilient and non-resilient alternatives using a logistic regression model. the output is a shortened list containing only the resilient options. the second and last step is identifying the most resilient path to achieve objectives. this goes through applying the first method to the shortlist of resilient options obtained thanks to the binary classification and determining the most resilient path using a weighted graph with the table implemented to interpret this graph. to demonstrate the trustworthiness of these methods, we have decided to apply them in a situation that falls under the effectual logic. in the result, the findings obtained are on the same wavelength as the recommendations of the effectual logic. this opens the discussion about the need to further incorporate resilience into the logic of entrepreneurial decision-making, particularly the effectual logic that emphasizes control over prediction (goel and carry, 2006). it has been established in the scientific literature that coupling control, a strategy for ensuring system performance, with resilience to deal with change under uncertainty allows the creation of an optimized system (hoekstra et al., 2018). in our earlier works, we have addressed a variety of topics, including the relationship between resilience and response capability in the context of unfavorable occurrences, as well as the benefits of enhancing resilience on resources (said et al., 2019) and said et al./decis. mak. appl. manag. eng. 6 (1) (2023) 34-56 52 process functioning optimization (said et al., 2020). all of which is to say that resilience may be thought of as a universal and all-encompassing indicator, which can embrace and satisfy all the crucial features of any organization (new venture or already existing system). 4. conclusions during the drafting of this manuscript, it was observed that making decisions based on their resilience is not widespread despite what is at stake, especially for economic activities performed in an uncertain context, such as new venture creation. to make up for this shortfall, we proposed a novel process designed to help small business owners who need to improve and optimize their decisions. we are talking about a rational resilience-based decision-making model (r2dm). to better explain the practical use of this model, we have chosen effectual customer co-creation as the studied situation. this gives rise to four alternatives (planning, visionary, adaptative, and transforming) that can be considered as input in the decision-making process. once the problem is identified, the second step is to select the criteria based on which the decision is taken. in our case, entrepreneurial resilience (er) principles (adapt or pivot where required, weigh the options, turn adversity into opportunity, do more than is required, feel self-confident, and bounce back) are the parameters against which the available options are assessed. afterward, we laid out in detail three methods that can be used, optimally, in conjunction, or separately, if necessary, to assess the available options and then select the most suitable one. the first method involves calculating the entrepreneurial resilience (er) value through a logical appraisal of the selection criteria for each option. the second proposed approach suggests a classification of the available alternatives through the use of a newly developed logistic regression model, which is aimed at pushing down the list of the eventual options by distinguishing the resilient alternatives from the non-resilient ones. the last method was introduced to determine the most resilient path to achieve objectives, more specifically, when dealing with interconnected options, by implementing a weighted graph. the results obtained after carrying out this study using the proposed methods are closely aligned with the recommendations of the effectuation theory. this can be interpreted as a sign that resilient decisions are informed and enlightened decisions that guarantee, first and foremost, long-term small-business continuity and success. on the other hand, by examining the three methods detailed in this essay, we notice that they are mainly predicated upon logical thinking and reasoning. nevertheless, cognitive scientists and neuroscientists have pointed out that humans are mentally predisposed to making erroneous judgments because of some unconscious mechanisms, such as the confirmation bias that leads to the neglect of strategic data or other possible readings and scenarios, and the groupthink, which represents the existence of insidious pressure to conform to the dominant opinion even if this latter is manifestly wide of the mark. in our future work, natural language processing (nlp) can be employed to increase the accuracy of the rational resilience-based decision-making model. the intention is to obtain a model allowing the identification of the mastery level of each principle by relying directly on the textual description of the given option. furthermore, we can also examine the most common and occurring reasons why startups and new ventures fail, and review, based on this study, the er principles (add, amend, or reposition them), and eventually, suggest additional methods devoted to decreasing the probability of failure of the new ventures through making resilient decisions. as a further matter, the table used to identify the most resilient path based on the new model for making resilient decisions in an uncertain context: the rational resilience… 53 weighted graph should be converted into an automated tool to ensure effective running regardless of the importance of the size of the options list. author contributions: conceptualization, s.s.; methodology, s.s.; software, s.s.; validation, s.s., h.b. and m.g.; formal analysis, s.s.; investigation, s.s.; resources, s.s., h.b. and m.g.; data curation, s.s., h.b. and m.g.; writing—original draft preparation, s.s.; writing—review and editing, h.b. and m.g.; visualization, h.b. and m.g.; supervision, h.b. and m.g.; project administration, s.s., h.b. and m.g.; funding acquisition, n.a. all authors have read and agreed to the published version of the manuscript. funding: this research received no external funding. conflicts of interest: the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. references abzug, z. m. 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(2008). if employees “go the extra mile,” do customers reciprocate with similar behavior? psychology & marketing, 25(10), 961–986. https://doi.org/10.1002/mar.20248 © 2023 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1111/etap.12088 https://doi.org/10.3917/entin.028.0076 https://hal.archives-ouvertes.fr/hal-01892750 https://hal.archives-ouvertes.fr/hal-01892750 https://doi.org/10.1177/10422587221104820 https://doi.org/10.1007/978-3-319-20928-9_2474 https://doi.org/10.1108/17410381111112738 https://doi.org/10.1002/mar.20248 plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 5, issue 1, 2022, pp. 90-112. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0310022022n * corresponding author. e-mail addresses: monikanarang5@gamil.com (monika narang), mcjoshi69@gmail.com (mahesh chandra joshi), kiranbisht96@gmail.com (kiran bisht), arun_pal1969@yahoo.co.in (arun pal). stock portfolio selection using a new decisionmaking approach based on the integration of fuzzy cocoso with heronian mean operator monika narang1*, mahesh chandra joshi1, kiran bisht2 and arun pal2 1 department of mathematics, kumaun university, india 2 department of mathematics, statistics and computer science, cbsh, gbpua&t, pantnagar, india received: 29 july 2021; accepted: 21 january 2022; available online: 10 february 2022. original scientific paper abstract: the main objective of stock portfolio selection is to distribute capital to selected stocks to get the most profitable returns at a lower risk. the performance of a stock depends on a number of criteria based on the riskreturn measures. therefore, the selection of shares is subject to fulfilling a number of criteria. in this paper, we have adopted an integrated approach based on the two-stage framework. first, the heronian mean operator (improved generalized weighted heronian mean and improved generalized geometric weighted heronian mean) is combined with the traditional combined compromise solution (cocoso) method to present a new decisionmaking model for dealing with stock selection problem. second, base-criterion method is used to calculate the relative optimal weights of the specified decision criteria. despite the uncertainties, the advanced cocoso-h model eliminates the efficacy of anomalous data and make complex-decisions more flexible. a case study of stock selection for portfolio under national stock exchange (nse) is discussed to validate the applicability of the proposed model. different portfolio (𝑃1,𝑃2& 𝑃3) have been constructed using particle swarm optimization (pso). the outcome shows the prominence and stability of the proposed model when compare to previous studies. key words: multi-criteria decision-making (mcdm), heronian mean (hm), combined compromise solution (cocoso), pso, portfolio analysis. mailto:monikanarang5@gamil.com mailto:mcjoshi69@gmail.com mailto:kiranbisht96@gmail.com mailto:arun_pal1969@yahoo.co.in stock portfolio selection using a new decision-making approach based on the integration of… 91 1. introduction investing in the stock market over the past few decades has procreated increasing interest among investors as it offers an opportunity for flexible and transparent options of money to diversify risk with the potential for returns. the stock market is influenced by many direct and indirect factors and is full of opacity. therefore, an investor has to examine a stock scrupulously before investing in it. in today's global economy, it is very difficult to analyse large amounts of information to arrive at an investment decision. stock selection process include complex decision-making with many and often conflicting objectives. in the process of stock portfolio selection, there are broadly two stages: (a) some suitable shares are chosen; (b) the percentage of total investment for each share is obtained through different weighing schemes or through optimization techniques. the central problem is how to rank a group of stocks by evaluating them in terms of several criteria. multi-criteria decision making (mcdm) is a controlled decision tool for calculating the weight of the evaluation criterion as well as ranking the alternatives present in problems with quantitative and qualitative criteria (zeleny & cochrane, 1973). mcdm methods have recently been getting phenomenal popularity and widespread applications (durmic et al., 2020; pamucar et al., 2020; gorcun et al., 2021; narang et al., 2021; bozanic et al., 2021). mcdm problems could be categorized into two classes: modm and madm. modm (multiple-objective decision-making methods) refer to handling continuous problems with infinite number of options. on the other hand, madm (multiple-attribute decision-making methods) refer to discrete representations of a problem with many conflicting criteria and a limited number of alternatives. there are two main goals for solving practical problems by mcdm methods: calculating the optimum weight of the criterion and setting the rank of the alternatives. scientists and researchers gave a new insight on how to mend the quality of decision making over the past decades and cited several methods for weight processing of criteria and alternatives as well as how to rank alternatives (fontela & gabus , 1972; saaty, 1980; saaty, 1996; rezaei, 2015; haseli et al., 2019; benayoun et al., 1996; hwang & yoon, 1981; zavadskas, 1994; pamucar et al., 2021; zavadskas et al., 2001; yazdani et al., 2018). base-criterion method (haseli et al., 2019) is the latest mcdm method that calculate the weights of criteria. in this method, one criterion is chosen as the basecriterion out of all the specified decision criteria. then the relative importance of basecriterion to other criteria is determined on the numerical scale of 1/9 to 9. the concept of fuzziness which includes uncertainty and inaccuracy is first formalized by zadeh (1965). so, the fuzzy extension of bcm is proposed (haseli et al., 2020). the cocoso method (yazdani et al., 2018) is new as well as a unique structure among several mcdm methods. it is highly capable of working with incomplete and uncertain data. currently, the utility of cocoso method is increasing in many fields (pamucar et al., 2018; yazdani et al., 2019; wen et al., 2019; peng et al., 2020, deveci et al., 2021). in the proposed study, the cocoso method is modified by integrating it with the heronian mean operator (cocoso-h) under fuzzy environment to rank the stocks. a fuzzy base-criterion method (f-bcm) is used to find the relative importance of criteria in a stock selection process. the derived weights of criteria are then used in the cocoso-h to obtain the most suitable stock. illustration is done using stocks under nse. historical data is used to apply f-cocoso-hbcm model to rank the stocks. various portfolio has been then constructed by using particle swarm optimization based on the ranking obtained by proposed model. the return of the portfolio has been found to be satisfactory as compared to the previous studies. this hybrid approach (fnarang et al./decis. mak. appl. manag. eng. 5 (1) (2022) 90-112 92 cocoso-h-bcm) for solving stock selection problem has been discussed for the first time. the aggregation of information provided by decision makers is a fundamental need of an information processing system such as decision making. aggregation functions play a key role in mcdms to slacken the dimensions of the criterion (detyniekie, 2001). according to the conventional aggregation operators, criteria are independent, and the efficacy of the criterion is additive. whereas, in real-world decision-making problems there are always different types of interrelationships between decision criteria, so this independent hypothesis cannot ordinarily be satisfied. the most fundamental and plausible aggregation operators are the choquet integral (ci) & sugeno integral (si), power average (pa), bonferroni mean (bm) and heronian mean. the hm operator (beliakov et al., 2007); sykora, 2009b; sykora, 2009a) has the peculiarity of catching the correlations of the aggregated arguments. the operators based on hm describes the interrelationships between different variables, explain the interrelationship between a decision criterion and itself and also differ the interrelationship between 𝐶𝑗 and 𝐶𝑖 from the interrelationship between criteria 𝐶𝑖 and 𝐶𝑗. in the cocoso method, criteria values procured by applying the weighted product method (wpm) and the weighted sum method (wsm) employ a significant impact on the final ranking of alternatives. the character of the wsm function is a simple linear one. wpm and wsm ignore the mutual impact of criteria. the traditional method can distort the result of aggregation and leads to incorrect prioritization of alternatives. hence the application of the heronian mean operator (igwhmimproved generalized weighted heronian mean and iggwhmimproved generalized geometric weighted heronian mean) is introduced to overcome this drawback in the traditional cocoso method. the reasons for the combination of f-bcm and cocoso-h are as follows:  despite uncertainties it can make complex stock selection processes much easier and efficient in multi-dimensional decision analysis systems.  in the course of the selection process, f-bcm make use of subjective information that reflects the judgment and conduct of humans. unlike ahp and f-bwm, the f-bcm method permits fully consistent pairwise comparison of evaluation criteria.  reforming the traditional cocoso method using new aggregation operators which has the ability of capturing the correlations of the aggregated arguments and removes the effect of the anomalous data. the remaining paper is organized as follows: section 2 focuses on the existing literature of stock portfolio selection. section 3 decodes the proposed model. section 4 gives a case study, results and discussions. section 5 concludes the paper. 2. related work in the literature, there are number of approaches to handle and construct a portfolio. portfolio optimization is based on the modern portfolio theory (mpt) (markowitz, 1952) established fifty years ago. the mpt is based on the principle that investors want the highest return with the lowest risk. markowitz introduced the mean-variance method for the stock portfolio decision problem. this method works on the concept that “when the risk of stock portfolio is constant, we should try to maximize the return rate of stock portfolio and when the return rate of stock portfolio is constant, we should try to minimize the risk of stock portfolio”. this theory was stock portfolio selection using a new decision-making approach based on the integration of… 93 widely accepted and adopted by various researchers. but market efficiency is considered a core assumption in mpt, getting information about the markets every time is expensive and time-consuming (grossman & stiglitz, 1980). the capital asset pricing model (capm) was developed in the early 1960s (sharpe, 1964; treynor, 2015; lintner, 1965a; lintner, 1965b). the capm is based on the concept that all risks should not affect asset prices. it establishes the relationship between expected return and systematic risk. despite some shortcomings the capm formula is still widely used and is capable of easy comparison of investment options. however, it is difficult to evaluate the ability of firms with different inputs and outputs. the data envelopment analysis (dea) models (charnes et al., 1978; banker et al., 1984) can purposefully combine multiple inputs and outputs of a unit into a single measure of overall organizational capacity. the dea methodology is very efficient in a financial application such as measuring managerial efficiency using a company's financial statements. analytical hierarchy process (ahp) (saaty, 1980) has been proposed to cope with the stock portfolio decision problem by evaluating the performance of each company in various levels of criteria. dea, a nonparametric method, has been used (edirisinghe & zhang, 2008) for selecting and screening of stocks. huang (2008) gave a new definition of risk and employed a genetic algorithm to deal with the stock portfolio decision making problem. generally, in a portfolio selection problem the decision maker simultaneously considers conflicting objectives like rate of return, liquidity, and risk. so, the multi-objective programming (such as goal programming, compromise programming) are used to select the portfolio (abdelaziz et al., 2007; ballestero, 2001; aouni et al., 2005). the need to model the portfolio selection within the mcdm frame has been proposed (hurson & zopounidis, 1995) in 1995. after analysing the relevance of multi-criteria decision systems for financial decisions, a detailed discussion and review on portfolio selection (zopounidis & doumpos, 2013) is provided. an integrated and innovative method for building and selecting equity portfolios have been developed (xinodas et al., 2010), which takes into account the inherent multidimensional nature of the problem while allowing dms to incorporate their priorities in the decision process. due to accurate information and subjective opinions of experts that often appear in the stock portfolio decision making process, crisp values are insufficient to solve problems. the use of linguistic assessment in place of numerical values may be a more appropriate approach. a new decision-making method (chen & hung, 2009) is introduced for stock portfolio selection using linguistic valuation along with computing. in the proposed approach, they have used linguistic assessment to express the opinion of the experts and combined the linguistic topsis (technique for order preference by similarity to an ideal solution) and linguistic electre (elimination and choice expressing reality) methods for dealing with stock selection problem. recently a fuzzy-anp (analytical network process) approach (galankashi et al., 2020) is developed to rank various tehran stock exchange (tse) portfolios. keeping in mind the uncertain nature of the portfolio selection problems, ammar and khalifa (2003). moved a formulation of fuzzy portfolio optimization. a method of group decision making has been developed (tirayaki & ahlatcioglu, 2005) in an ambiguous environment. in this method, tirayaki shifted the linguistic value of the experts to a triangular fuzzy number and used a new fuzzy ranking and weighted algorithm to derive the investment ratio of each stock. ranking is one of the strategies to derive the concept of converting raw information into relevant information for decision making. a hybrid multi-criteria model (fazli & jafari, 2012) for investment in stock exchange is developed. in this methodology, narang et al./decis. mak. appl. manag. eng. 5 (1) (2022) 90-112 94 demetal (decision-making trial and evaluation laboratory) method is used to build a relations-structure between criteria while vikor (vlsekriterijumska optimizacija i kompromisno resenje) method is used to select the most preferrable alternative for investment. a fuzzy cross-entropy-mean–variance–skewness models for portfolio optimization under several constraints under bombay stock exchange (bse) has been proposed (bhattacharya et al., 2014). a new hybrid mcdm approach is introduced (poklepovic & babic, 2014) to rank the stocks based on spearman’s rank correlation coefficient. an investor wants to create a balanced portfolio with shares representing different sectors. an attempt has been made to provide a dea-topsis based framework (mansouri et al., 2014) in the context of tse. a new ranking methodology (dedania et al., 2015) that is based on the principle of comparing a company with companies in the same field have been proposed because the attributes that affect the growth of the company vary for different sectors. a popular mcdm method, ahp, is applied to achieve the rank of portfolio (hota et al., 2015) for further decision-making process. dincher (2015) has proposed a profit-based stock selection approach in the banking sector using fuzzy ahp and moora (the multi-objective optimization ratio analysis) method. a new technique has been proposed (boonjing & boongasame, 2017) for portfolio selection with two significant financial ratios (dividend yield and net profit margin) using the electre iii method to enable small investors to make trading decisions easily. a portfolio selection model (thakur et al., 2018) have been developed that prioritizes high-ranked stocks. in this approach, they have identified critical factors using the fuzzy delphi method and used the dempster – shaffer evidence theory to rank the stocks under nse. ant colony optimization (aco) is used to optimize (or construct) the portfolio. to predict the best performing company in the it industry, a new best-worst method (bwm) has been implemented (krishna et al., 2018). a hybrid approach dea-copras (complex proportional assessment) (gupta et al., 2019) has been applied for portfolio selection at risk-return interface based on nse. in which, dea is applied to calculate the efficiency of the stock and copras is used to rank the stocks. recently, a hybrid multi-criteria decision-making approach have been presented (mills et al., 2020) under grey environment incorporating an integrated anp and demetal that provides both ranking and weighting information for optimal portfolio selection. ahp-topsis (gupta et al., 2020), a hybrid multi-criteria decision-making technique, has been developed. using which they have ranked the financial performance of selected indian private banks. a hybrid fuzzy copras base criterion method (narang et al., 2021) has been implemented to rank the stocks under nse. in this methodology, ranked asset algorithm has been used for the capital distribution among the stocks according to their rank. 3. data and methodology to tackle the hesitant nature of the information of human mind, fuzzy sets are used. it is necessary to use linguistic variables for modeling the decisions of the decision maker’s that can be expressed by trapezoidal fuzzy numbers. 3.1. preliminaries definition (trapezoidal fuzzy number) (nourianfar & montazer, 2013; savitha & george, 2017) a trapezoidal fuzzy number (trfn) represented by 𝑇 is defined as (𝑎,𝑏,𝑐,𝑑) where the membership function is expressed by stock portfolio selection using a new decision-making approach based on the integration of… 95 𝜇𝑇(𝑥) = { 0, 𝑥 ≤ 𝑎 𝑥−𝑎 𝑏−𝑎 , 𝑎 ≤ 𝑥 ≤ 𝑏 1, 𝑏 ≤ 𝑥 ≤ 𝑐 𝑑−𝑥 𝑑−𝑐 , 𝑐 ≤ 𝑥 ≤ 𝑑 0, 𝑥 ≥ 𝑑 (1) definition (statistical beta distribution) (rahmani et al., 2016) the crisp value 𝜇𝑇 corresponding to the trapezoidal fuzzy number (𝑎,𝑏,𝑐,𝑑) based on statistical beta distribution method (sbdm) can be obtained as follows 𝜇𝑇 = 2𝑎+7𝑏+7𝑐+2𝑑 18 (2) the readers can refer (nourianfar & montazer, 2013) to learn about basic operations of trfns. definition (heronian mean) (liu & zhang, 2017) a hm operator of a set of nonnegative values 𝑋 = {𝑥1,𝑥2,…𝑥𝑛} is: 𝐻𝑀(𝑥1,𝑥2,…𝑥𝑛) = 2 𝑛(𝑛+1) ∑ ∑ √𝑥𝑖𝑥𝑗 𝑛 𝑗=𝑖 𝑛 𝑖=1 (3) 3.2. the proposed f-cocoso-h-bcm method 3.2.1. deriving weight of criteria through f-bcm fuzzy base-criterion method has been moved (haseli et al., 2020) in which the decision maker's opinions are expressed linguistically as human decisions are fraught with uncertainty and ambiguity. f-bcm is capable to obtain fully consistent results and calculate crisp weights using less pairwise comparisons than the existing mcdm methods such as ahp (saaty, 1980), and bwm (rezaei, 2015). this method is more accurate and at less time consuming because the execution of secondary comparisons is not necessary. although, bcm and fucom (full consistency method) (pamucar et al; 2018) both the method performs n-1 pairwise comparisons to calculate optimal weights of criteria. but in bcm, complexity is low in terms of selecting a particular criterion as a base criterion as compared to fucom. so, the fuzzy bcm method is preferred to calculate the weights of the criteria in this paper. the fuzzy pairwise comparison matrix is as follows: �̃� = [ (1,1,1,1) �̃�12 �̃�13 … �̃�1𝑛 �̃�21 (1,1,1,1) �̃�23 … �̃�2𝑛 ⋮ ⋮ ⋮ ⋱ ⋮ �̃�𝑛1 �̃�𝑛2 �̃�𝑛3 … (1,1,1,1)] (4) where �̃�𝑖𝑗 represent the relative fuzzy importance of criteria 𝑖 to 𝑗. similarly, 𝑇𝑗𝑖 represent the relative fuzzy importance of criteria 𝑗 to 𝑖. 𝑇𝑖𝑗 is a trapezoidal fuzzy number and when 𝑖 = 𝑗, �̃�𝑖𝑗 = (1,1,1,1). table 1 represents the corresponding trfns for pairwise comparison of criteria. table 1. trfns of f-bcm for pairwise comparison linguistic set trfns equally important (1,1,1,1) moderately important (1,1,2,3) strongly important (2,3,4,5) very strongly important (4,5,6,7) extremely important (6,7,8,9) absolutely important (8,9,9,9) narang et al./decis. mak. appl. manag. eng. 5 (1) (2022) 90-112 96 the step-wise procedure of f-bcm is as follows. step-1 determine and evaluates a set of criteria {𝐶1,𝐶2,𝐶3, ….,𝐶𝑛} accordance with the opinion of decision maker. step-2 identify one of the criteria as a base-criteria from a set of criteria. step-3 pairwise comparisons are performed in this step. the relative fuzzy preference of the base criteria over the remaining criteria is derived using table 1. the resulting vector of fuzzy base comparisons as follows. �̃�𝐵 = (�̃�𝐵1, �̃�𝐵2, �̃�𝐵3,… . . , �̃�𝐵𝑛) �̌�𝐵 represents a fuzzy base-criteria over the rest of the criteria vector. �̌�𝐵𝑗 represents the fuzzy importance of the base-criteria over the rest of the criteria and it is obvious that �̌�𝐵𝐵 = (1,1,1,1). step-4 for identification of optimal fuzzy weights, a non-linear programming model on the basis of components derived from �̌�𝐵 vector is as follows. min𝜉 such that { | �̃�𝐵 �̃�𝑗 − �̃�𝐵𝑗| ≤ 𝜉 ∑ 𝑅(�̃�𝑗) = 1 𝑛 𝑗=1 𝑎𝑗 𝑤 ≤ 𝑏𝑗 𝑤 ≤ 𝑐𝑗 𝑤 ≤ 𝑑𝑗 𝑤 𝑎𝑗 𝑤 ≥ 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑗 (5) where �̃�𝐵 = (𝑎𝐵 𝑤,𝑏𝐵 𝑤, 𝑐𝐵 𝑤,𝑑𝐵 𝑤), �̃�𝑗 = (𝑎𝑗 𝑤,𝑏𝑗 𝑤, 𝑐𝑗 𝑤,𝑑𝑗 𝑤), 𝜉 = (𝑎𝜉,𝑏𝜉, 𝑐𝜉,𝑑𝜉). the equation 5 can be rewritten as min𝜉 such that { | (𝑎𝐵 𝑤,𝑏𝐵 𝑤,𝑐𝐵 𝑤,𝑑𝐵 𝑤) (𝑎𝑗 𝑤,𝑏𝑗 𝑤,𝑐𝑗 𝑤,𝑑𝑗 𝑤) − (𝑎𝐵𝑗,𝑏𝐵𝑗, 𝑐𝐵𝑗,𝑑𝐵𝑗)| ≤ (𝑘 ∗,𝑘∗,𝑘∗,𝑘∗) ∑ 𝑅(�̃�𝑗) = 1 𝑛 𝑗=1 𝑎𝑗 𝑤 ≤ 𝑏𝑗 𝑤 ≤ 𝑐𝑗 𝑤 ≤ 𝑑𝑗 𝑤 𝑎𝑗 𝑤 ≥ 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑗 (6) after solving the equation 6, the optimal fuzzy weights can be transformed to crisp values by make use of sbdm presented in equation 2. the pairwise comparison of elements in base-criterion method done under the principle that 𝑎𝐵𝑎𝑠𝑒,𝑖 ∗ 𝑎𝑖,𝑗 = 𝑎𝐵𝑎𝑠𝑒,𝑗 and trfns satisfy the constraint ( 1 9 , 1 9 , 1 9 , 1 8 ) ≤ (𝑎𝑖𝑗,𝑏𝑖𝑗, 𝑐𝑖𝑗,𝑑𝑖𝑗) ≤ (8,9,9,9) or ( 1 9 , 1 9 , 1 9 , 1 8 ) ≤ (𝑎𝐵𝑗,𝑏𝐵𝑗,𝑐𝐵𝑗,𝑑𝐵𝑗) (𝑎𝐵𝑖,𝑏𝐵𝑖,𝑐𝐵𝑖,𝑑𝐵𝑖) ≤ (8,9,9,9). this ensures the fully consistent solution of the optimization problem with 𝜉 = 0. 3.2.2. ranking of alternatives through fuzzy cocoso-h the traditional cocoso (combined compromise solution) method for mcdm is primarily developed by yazdani et al. (2018). foundation of the cocoso method is based on the idea of saw (simple additive weighting), mew (multiplicative exponential weighting) and waspas (the weighted aggregates sum product assessment) methods. stock portfolio selection using a new decision-making approach based on the integration of… 97 after defining a set of alternatives and criteria, the stepwise procedure to solve the cocoso-h mcdm model is as follows. step-1 decision matrix: the assessment of 𝑚 alternatives 𝐴 = {𝐴1,𝐴2, . . ,𝐴𝑚} with respect to 𝑛 criteria 𝐶 = {𝐶1,𝐶2, . . ,𝐶𝑛} is performed in the matrix 𝑇.̌ decision maker give their opinions from a linguistic point of view. table 2 unfolds the linguistic terms and their corresponding trfns. �̃� = [ (𝑇11 (𝑎) ,𝑇11 (𝑏) ,𝑇11 (𝑐) ,𝑇11 (𝑑) ) (𝑇12 (𝑎) ,𝑇12 (𝑏) ,𝑇12 (𝑐) ,𝑇12 (𝑑) ) … (𝑇1𝑛 (𝑎) ,𝑇1𝑛 (𝑏) ,𝑇1𝑛 (𝑐) ,𝑇1𝑛 (𝑑) ) (𝑇21 (𝑎) ,𝑇21 (𝑏) ,𝑇21 (𝑐) ,𝑇21 (𝑑) ) (𝑇22 (𝑎) ,𝑇22 (𝑏) ,𝑇22 (𝑐) ,𝑇22 (𝑑) ) … (𝑇2𝑛 (𝑎) ,𝑇2𝑛 (𝑏) ,𝑇2𝑛 (𝑐) ,𝑇2𝑛 (𝑑) ) ⋮ ⋮ ⋱ ⋮ (𝑇𝑚1 (𝑎) ,𝑇𝑚1 (𝑏) ,𝑇𝑚1 (𝑐) ,𝑇𝑚1 (𝑑) ) (𝑇𝑚2 (𝑎) ,𝑇𝑚2 (𝑏) ,𝑇𝑚2 (𝑐) ,𝑇𝑚2 (𝑑) ) … (𝑇𝑚𝑛 (𝑎) ,𝑇𝑚𝑛 (𝑏) ,𝑇𝑚𝑛 (𝑐) ,𝑇𝑚𝑛 (𝑑) )] (7) table 2. trfns for cocoso-h linguistic sets trfns very low (1,1,2,3) low (1,2,3,4) medium low (2,3,4,5) medium (3,4,5,6) medium high (4,5,6,7) high (5,6,7,8) very high (6,7,8,9) very very high (7,8,9,9) extremely high (8,9,9,9) step-2 normalized decision matrix: 𝑁 = [ (𝛾11 (𝑎) ,𝛾11 (𝑏) ,𝛾11 (𝑐) ,𝛾11 (𝑑) ) (𝛾12 (𝑎) ,𝛾12 (𝑏) ,𝛾12 (𝑐) ,𝛾12 (𝑑) ) … (𝛾1𝑛 (𝑎) ,𝛾1𝑛 (𝑏) ,𝛾1𝑛 (𝑐) ,𝛾1𝑛 (𝑑) ) (𝛾21 (𝑎) ,𝛾21 (𝑏) ,𝛾21 (𝑐) ,𝛾21 (𝑑) ) (𝛾22 (𝑎) ,𝛾22 (𝑏) ,𝛾22 (𝑐) ,𝛾22 (𝑑) ) … (𝛾2𝑛 (𝑎) ,𝛾2𝑛 (𝑏) ,𝛾2𝑛 (𝑐) ,𝛾2𝑛 (𝑑) ) ⋮ ⋮ ⋱ ⋮ (𝛾𝑚1 (𝑎) ,𝛾𝑚1 (𝑏) ,𝛾𝑚1 (𝑐) ,𝛾𝑚1 (𝑑) ) (𝛾𝑚2 (𝑎) ,𝛾𝑚2 (𝑏) ,𝛾𝑚2 (𝑐) ,𝛾𝑚2 (𝑑) ) … (𝛾𝑚𝑛 (𝑎) ,𝛾𝑚𝑛 (𝑏) ,𝛾𝑚𝑛 (𝑐) ,𝛾𝑚𝑛 (𝑑) )] (8) where normalized values of decision matrix 𝛾𝑖𝑗 = {𝛾𝑖𝑗 (𝑎) ,𝛾𝑖𝑗 (𝑏) ,𝛾𝑖𝑗 (𝑐) ,𝛾𝑖𝑗 (𝑑) } = {𝛾𝑖𝑗 (𝑎) = 𝛾 𝑖𝑗 (𝑎) 𝛾𝑗 + ;𝛾𝑖𝑗 (𝑏) = 𝛾 𝑖𝑗 (𝑏) 𝛾𝑗 + ;𝛾𝑖𝑗 (𝑐) = 𝛾 𝑖𝑗 (𝑐) 𝛾𝑗 + ;𝛾𝑖𝑗 (𝑑) = 𝛾 𝑖𝑗 (𝑑) 𝛾𝑗 + ; (9) where 𝛾𝑗 + = max 𝑖 (𝑇𝑖𝑗 (𝑑) ) step-3 weighted sequences of alternatives: weighted sequences are enumerated by make use of fuzzy igwhm function and fuzzy iggwhm function (liu and zhang, 2017). iwghm is defined as 𝑆𝐻𝑖 𝑝,𝑞= (∑ ∑ (𝑤𝑖�̌�𝑖) 𝑝(𝑤𝑗�̌�𝑗) 𝑞𝑛 𝑗=𝑖 𝑛 𝑖=1 ) 1 𝑝+𝑞 (∑ ∑ (𝑤𝑖) 𝑝(𝑤𝑗) 𝑞𝑛 𝑗=𝑖 𝑛 𝑖=1 ) 1 𝑝+𝑞 narang et al./decis. mak. appl. manag. eng. 5 (1) (2022) 90-112 98 = ( ( (∑ ∑ (𝑤𝑖�̌�𝑖 (𝑎) ) 𝑝 (𝑤𝑗�̌�𝑗 (𝑎) ) 𝑞 𝑛 𝑗=𝑖 𝑛 𝑖=1 ) 1 𝑝+𝑞 (∑ ∑ (𝑤𝑖) 𝑝(𝑤𝑗) 𝑞𝑛 𝑗=𝑖 𝑛 𝑖=1 ) 1 𝑝+𝑞 ),( (∑ ∑ (𝑤𝑖�̌�𝑖 (𝑏) ) 𝑝 (𝑤𝑗�̌�𝑗 (𝑏) ) 𝑞 𝑛 𝑗=𝑖 𝑛 𝑖=1 ) 1 𝑝+𝑞 (∑ ∑ (𝑤𝑖) 𝑝(𝑤𝑗) 𝑞𝑛 𝑗=𝑖 𝑛 𝑖=1 ) 1 𝑝+𝑞 ), ( (∑ ∑ (𝑤𝑖�̌�𝑖 (𝑐) ) 𝑝 (𝑤𝑗�̌�𝑗 (𝑐) ) 𝑞 𝑛 𝑗=𝑖 𝑛 𝑖=1 ) 1 𝑝+𝑞 (∑ ∑ (𝑤𝑖) 𝑝(𝑤𝑗) 𝑞𝑛 𝑗=𝑖 𝑛 𝑖=1 ) 1 𝑝+𝑞 ),( (∑ ∑ (𝑤𝑖�̌�𝑖 (𝑑) ) 𝑝 (𝑤𝑗�̌�𝑗 (𝑑) ) 𝑞 𝑛 𝑗=𝑖 𝑛 𝑖=1 ) 1 𝑝+𝑞 (∑ ∑ (𝑤𝑖) 𝑝(𝑤𝑗) 𝑞𝑛 𝑗=𝑖 𝑛 𝑖=1 ) 1 𝑝+𝑞 ) ) (10) where 𝑝,𝑞 ≥ 0, 𝑤𝑗 unfold the relative weights of criteria, 𝑤𝑗 > 0 and ∑ 𝑤𝑗 = 1. 𝑛 𝑗=1 iggwhm is defined as 𝑃𝐻𝑖 𝑝,𝑞 = 1 𝑝+𝑞 ∏ ∏ (𝑝𝛾𝑖 + 𝑞𝛾𝑗) 2(𝑛+1−𝑖)𝑤𝑗 𝑛(𝑛+1)∑ 𝑤𝑘 𝑛 𝑘=𝑖𝑛 𝑗=𝑖 𝑛 𝑖=1 =(( 1 𝑝+𝑞 ∏ ∏ (𝑝𝛾𝑖 (𝑎) + 𝑞𝛾𝑗 (𝑎) ) 2(𝑛+1−𝑖)𝑤𝑗 𝑛(𝑛+1)∑ 𝑤𝑘 𝑛 𝑘=𝑖𝑛 𝑗=𝑖 𝑛 𝑖=1 ),( 1 𝑝+𝑞 ∏ ∏ (𝑝𝛾𝑖 (𝑏) +𝑛𝑗=𝑖 𝑛 𝑖=1 𝑞𝛾𝑗 (𝑏) ) 2(𝑛+1−𝑖)𝑤𝑗 𝑛(𝑛+1)∑ 𝑤𝑘 𝑛 𝑘=𝑖 ),( 1 𝑝+𝑞 ∏ ∏ (𝑝𝛾𝑖 (𝑐) +𝑛𝑗=𝑖 𝑛 𝑖=1 𝑞𝛾𝑗 (𝑐) ) 2(𝑛+1−𝑖)𝑤𝑗 𝑛(𝑛+1)∑ 𝑤𝑘 𝑛 𝑘=𝑖 ),( 1 𝑝+𝑞 ∏ ∏ (𝑝𝛾𝑖 (𝑑) + 𝑞𝛾𝑗 (𝑑) ) 2(𝑛+1−𝑖)𝑤𝑗 𝑛(𝑛+1)∑ 𝑤𝑘 𝑛 𝑘=𝑖𝑛 𝑗=𝑖 𝑛 𝑖=1 )) (11) where 𝑝,𝑞 ≥ 0, 𝑤𝑗 unfold the relative weights of criteria, 𝑤𝑗 > 0 and ∑ 𝑤𝑗 = 1. 𝑛 𝑗=1 parameters 𝑝 and 𝑞 represent the stability parameters. step-4 relative significance: three pooling strategies are enumerated for each option. the concernment of alternatives within the strategies could be procreated by make use of the following equations. 𝐾𝑖𝐻𝑎 = 𝑆𝐻𝑖+𝑃𝐻𝑖 ∑ (𝑆𝐻𝑖+𝑃𝐻𝑖) 𝑚 𝑖=1 (12) 𝐾𝑖𝐻𝑏 = 𝑆𝐻𝑖 min 𝑖 (𝑆𝐻𝑖) + 𝑃𝐻𝑖 min 𝑖 (𝑃𝐻𝑖) (13) 𝐾𝑖𝐻𝑐 = 𝜆𝑆𝐻𝑖+(1−𝜆)𝑃𝐻𝑖 𝜆max 𝑖 (𝑆𝐻𝑖)+(1−𝜆)max 𝑖 (𝑃𝐻𝑖) , 0 ≤ 𝜆 ≤ 1 (14) the coefficient 𝜆 unfolds the stagnation and ductility of the proposed fuzzy cocosoh model. step-5 final table for the ranking of alternatives: the rank of an alternative is defined on the basis of the value of 𝐾𝑖ℎ. the higher the value of 𝐾𝑖ℎ, the higher the priority of the alternative. 𝐾𝑖𝐻 = (𝐾𝑖𝐻𝑎+𝐾𝑖𝐻𝑏+𝐾𝑖𝐻𝑐) 3 + (𝐾𝑖𝐻𝑎.𝐾𝑖𝐻𝑏.𝐾𝑖𝐻𝑐) 1 3⁄ (15) stock portfolio selection using a new decision-making approach based on the integration of… 99 start collect the information collect the information select the alternatives from the set of opted alternatives determine the base-criteria perform base-comparisons solve the bcm model for the weight of criteria collect the numeric data of the selected alternatives determine the decisionmatrix normalize the decisionmatrix enumerate the weighted sequences by using igwhm & iggwhm execute the cocoso-h model to find the table for the ranking of alternatives figure 1. flowchart of the f-cocoso-h-bcm model 3.3. a case study stock market plays a significant role in economic growth of a country. it is very challenging to choose an investor’s stock because the stock market is full of uncertainties and very unpredictable. a profitable decision could be made by an investor based on some fundamental and technical analysis of stocks. 3.3.1. structure of alternatives and criteria investors believe that there is no one right way to examine stocks because of the multidimensional uncertainties. knowing the financial data and loss information of any company can go a long way in helping investors in the selection of stocks. there are many techniques to select the stocks for the investment like fundamental analysis and technical analysis. fundamental approach sifts the key ratio of a market to shape its financial health and takes several factors into account like earnings ratios and risk, for future forecast. several important fundamental factors are used by topmost investors and portfolio managers to evaluate and select stocks. with the help of extensive literature survey and based on the opinion of experts, we have opted 4 fundamental criteria. the criteria are as follows: 1. revenue: revenue reflects growth of the company. the increasing rate of revenue indicates the increasing demand of company’s product in the market. 2. return on equity (roe): return on equity measures the return or profit earned per share by equity holder. company having high roe consider good for investment. narang et al./decis. mak. appl. manag. eng. 5 (1) (2022) 90-112 100 3. debt equity ratio (der): debt equity ratio of a company is calculated by dividing the debt of the company to the share holders’ fund. it reflects the ability of a company to outstand of all the debts if the market downturn eventually. this ratio is used for selecting a safe investment. 4. price to earnings ratio (p/e): price to earnings ratio depicts the price per share corresponding to the earning per share. it tells the stocks of the company are overvalued or not. so, the two criteria revenues (𝐶1) and roe (𝐶2) are considered as beneficial criteria. der (𝐶3) and p/e (𝐶4) are taken as non-beneficial or cost criteria. 15 stocks from nse have been selected as alternatives to investigate them over four above mentioned criteria. the 11 years (from jan 2010 to dec 2020) historical data of the stocks is collected from http://www.investello.com and http://www.ratestar.in which serves as a collection of evidence to support or disprove the notion that the stock concerned is going to perform well in the future. quarterly data has been used for this study. the single numeric value shown in table 3 is taken as the evaluation value of each alternative to each criterion. 1. tata consultancy services limited (tcs) (𝑠𝑡1) 2. hdfc bank limited (hdfcbank) (𝑠𝑡2) 3. titan company limited (titan) (𝑠𝑡3) 4. oil & natural gas corporation limited (ongc) (𝑠𝑡4) 5. hindustan uniliver limited (hindunilvr) (𝑠𝑡5) 6. divi’s laboratories limited (divislab) (𝑠𝑡6) 7. reliance industries limited (reliance) (𝑠𝑡7) 8. pidilite industries limited (pidilitind) (𝑠𝑡8) 9. jsw steel limited (jswsteel) (𝑠𝑡9) 10. aurobindo pharma limited (auropharma) (𝑠𝑡10) 11. bajaj finance limited (bajfinance) (𝑠𝑡11) 12. dr. reddy’s laboratories limited (drreddy) (𝑠𝑡12) 13. kotak mahindra bank limited (kotakbank) (𝑠𝑡13) 14. asian paints limited (asianpaint) (𝑠𝑡14) 15. jubilant foodworks limited (jublfood) (𝑠𝑡15) table 3. ema (exponential moving average) of actual numerical values of each criterion stocks/criteria 𝐶1(𝑅𝑒𝑣𝑒𝑛𝑢𝑒) 𝐶2(𝑅𝑂𝐸) 𝐶3(𝐷𝐸𝑅) 𝐶4(𝑃/𝐸) 𝑠𝑡1 141259 35.43 0.0002 5.28 𝑠𝑡2 118167.2 15.78 0.3572 26.45 𝑠𝑡3 18139.04 22.64 0.2457 77.33 𝑠𝑡4 390249 9.567 0.481 9.729 𝑠𝑡5 37694.8 78.64 0.0079 70.78 𝑠𝑡6 4783.9 19.31 0.0085 40.61 𝑠𝑡7 511663.6 10.25 0.7372 22.35 𝑠𝑡8 66991.0 24.87 0.039 64.41 𝑠𝑡9 707483.7 14.70 1.60 14.52 𝑠𝑡10 19118.55 19.32 0.453 15.85 𝑠𝑡11 18341.78 17.55 4.52 43.23 𝑠𝑡12 16556.73 13.05 0.264 30.28 𝑠𝑡13 42500.38 12.64 0.8391 34.39 http://www.investello.com/ http://www.ratestar.in/ stock portfolio selection using a new decision-making approach based on the integration of… 101 stocks/criteria 𝐶1(𝑅𝑒𝑣𝑒𝑛𝑢𝑒) 𝐶2(𝑅𝑂𝐸) 𝐶3(𝐷𝐸𝑅) 𝐶4(𝑃/𝐸) 𝑠𝑡14 18434.92 17.55 0.0569 64.99 𝑠𝑡15 3346.76 22.09 0.0004 83.68 4. results and discussions in this part, the application of our proposed methodology to rank the performance of different stocks by investigating them under some selected criteria in the financial trading is discussed. 4.1. enumeration of relative weights of criteria in the first step, relative optimal weights of the criteria are calculated that directly affect the ranking of the alternatives in further work. step-1: at first, decision maker selects 𝐶2:roe as a base-criteria from a set of criteria {𝐶1:revenue, 𝐶2:roe, 𝐶3:der, 𝐶4:p/e}. step-2: the fuzzy base-comparisons are performed based on linguistic terms shown in table 1. step-3: the fuzzy base-comparison vector based on trfns revealed by the decision maker for pairwise comparisons of the fuzzy base-criterion with other criteria as follows. �̃�𝐵 = {(4,5,6,7),(1,1,1,1),(6,7,8,9),(2,3,4,5)}; step-4: a non-linear constrained optimization problem is created based on the equation 6 as follows. min𝜉∗̃ such that { | (𝑎2,𝑏2,𝑐2,𝑑2) (𝑎1,𝑏1,𝑐1,𝑑1) − (4,5,6,7)| ≤ (𝑘∗,𝑘∗,𝑘∗,𝑘∗) | (𝑎2,𝑏2,𝑐2,𝑑2) (𝑎3,𝑏3,𝑐3,𝑑3) − (6,7,8,9)| ≤ (𝑘∗,𝑘∗,𝑘∗,𝑘∗) | (𝑎2,𝑏2,𝑐2,𝑑2) (𝑎4,𝑏4,𝑐4,𝑑4) − (2,3,4,5)| ≤ (𝑘∗,𝑘∗,𝑘∗,𝑘∗) 1 18 ( 2𝑎1 + 7𝑏1 + 7𝑐1+2𝑑1 + 2𝑎2 + 7𝑏2 + 7𝑐2 + 2𝑑2 + 2𝑎3 +7𝑏3 + 7𝑐3+2𝑑3 + 2𝑎4 + 7𝑏4 + 7𝑐4 + 2𝑑4 ) = 1 𝑎1 ≤ 𝑏1 ≤ 𝑐1 ≤ 𝑑1 𝑎2 ≤ 𝑏2 ≤ 𝑐2 ≤ 𝑑2 𝑎3 ≤ 𝑏3 ≤ 𝑐3 ≤ 𝑑3 𝑎4 ≤ 𝑏4 ≤ 𝑐4 ≤ 𝑑4 𝑎1,𝑎2,𝑎3,𝑎4 > 0 𝑘∗ ≥ 0 (16) by solving the equation16, the optimal fuzzy weights are derived, which are: �̌�1 ∗ = (0.095,0.106,0.121,0.144); �̌�2 ∗ = (0.576,0.608,0.638,0.665); �̌�3 ∗ = (0.073,0.079,0.086,0.096); �̌�4 ∗ = (0.133,0.159,0.202,0.288); by applying equation 2, the optimal fuzzy weights of criteria are transformed into crisp values as: �̌�1 = 0.11, �̌�2 = 0.62, �̌�3 = 0.08, �̌�4 = 0.19. the relative weights will be used in the fuzzy cocoso-h model. 4.2. selection of the extremely abiding stock through fuzzy cocoso-h the second phase involves ranking the selected stocks based on the specified decision-criteria. narang et al./decis. mak. appl. manag. eng. 5 (1) (2022) 90-112 102 step-1: the stocks on each criterion are rated by the decision maker using the actual numerical values of each criterion in table 3 and fuzzy linguistic variables in table 2, based on which initial decision matrix (table 4) is derived. step-2: using equation 8 and 9, the normalized fuzzy decisional matrix (table 5) is obtained. step-3: the weighted sequences 𝑆𝐻𝑖 and 𝑃𝐻𝑖 (table 6) have been obtained using elements of the normalized decisional matrix and the relative weights calculated by fbcm for the values of the parameters 𝑝 = 𝑞 = 1. we have validated 𝑝 = 𝑞 = 1 because they not only make calculation easier, but also fully capture the correlations among the specified decision criteria. fuzzy weighted sequences are transformed into crisp sequences using equation 2. table 4. initial decision matrix of stocks. stocks decision-criteria 𝐶1 𝐶2 𝐶3 𝐶4 𝑠𝑡1 (6,7,8,9) (5,6,7,8) (7,8,9,9) (6,7,8,9) 𝑠𝑡2 (6,7,8,9) (1,2,3,4) (4,5,6,7) (6,7,8,9) 𝑠𝑡3 (1,2,3,4) (3,4,5,6) (5,6,7,8) (1,2,3,4) 𝑠𝑡4 (7,8,9,9) (1,1,2,3) (4,5,6,7) (8,9,9,9) 𝑠𝑡5 (2,3,4,5) (8,9,9,9) (7,8,9,9) (1,2,3,4) 𝑠𝑡6 (1,1,2,3) (2,3,4,5) (7,8,9,9) (4,5,6,7) 𝑠𝑡7 (8,9,9,9,) (1,1,2,3) (2,3,4,5) (6,7,8,9) 𝑠𝑡8 (5,6,7,8) (3,4,5,6) (6,7,8,9) (2,3,4,5) 𝑠𝑡9 (5,6,7,8) (1,2,3,4) (2,3,4,5) (7,8,9,9) 𝑠𝑡10 (1,2,3,4) (2,3,4,5) (4,5,6,7) (7,8,9,9) 𝑠𝑡11 (1,2,3,4) (2,3,4,5) (1,1,2,3) (4,5,6,7) 𝑠𝑡12 (1,2,3,4) (1,2,3,4) (5,6,7,8) (5,6,7,8) 𝑠𝑡13 (3,4,5,6) (1,2,3,4) (2,3,4,5) (5,6,7,8) 𝑠𝑡14 (1,2,3,4) (2,3,4,5) (6,7,8,9) (2,3,4,5) 𝑠𝑡15 (1,1,2,3) (3,4,5,6) (8,9,9,9) (1,1,2,3) example 1: fuzzy 𝑆𝐻𝑖 of 𝑠𝑡1for i=1, j=1 is derived by using equation 10 is as follows: =((0.11*0.67)*(0.11*0.67)+(0.11*0.67)*(0.62*0.56)+…….+(0.62*0.56)*(0.62*0.56 )+(0.62*0.56)*(0.08*0.78)+…………….+(0.19*0.67)*(0.19*0.67))^0.5/(0.11*0.11+0.11 *0.62+0.11*0.08+0.11*0.19+0.62*0.62+0.62*0.08+0.62*0.19+0.08*0.08+0.08*0.19+ 0.19*0.19)^0.5 = 0.58 example 2: fuzzy 𝑃𝐻𝑖 of 𝑠𝑡1 for i=1, j=1is calculated by using equation 11 is as follows: =0.5*((0.67+0.67)^((2*4*0.11)/(20*(0.11+0.62+0.08+0.19))))*((0.11+0.62)^((2* 4*0.62)/(20*(0.11+0.62+0.08+0.19))))*((0.11+0.08)^((2*4*0.08)/(20*(0.11+0.62+0 .08+0.19) )) )*……………*(0.19*0.19)^((2*1*0.19)/(20*(0.19))) = 0.66 example 3: crisp value sh1 of 𝑠𝑡1 is calculated by make use of equation 2 is as follows: = (2*0.58+7*0.69+7*0.80+2*0.91)/18 =0.75 step4: by using equations 12,13 and 14 the relative significance of the stocks is procured. the coefficient 𝜆 is assumed to be 1 2⁄ under the third pooling strategy. this stock portfolio selection using a new decision-making approach based on the integration of… 103 value of 𝜆 is the middle most value between 0 and 1 and decision makers generally consider this value. step-5: table 7 presents the final ranking of stocks based on equation 15. narang et al./decis. mak. appl. manag. eng. 5 (1) (2022) 90-112 104 table 6. the weighted fuzzy sequences and their corresponding crisp values of the cocoso-h model stocks 𝑃𝐻𝑖 fuzzy 𝑆𝐻𝑖 fuzzy 𝑃𝐻𝑖 crisp 𝑆𝐻𝑖 crisp 𝑠𝑡1 (0.58,0.69,0.80,0.91) (0.66,0.84,0.92,0.99) 0.75 0.87 𝑠𝑡2 (0.29,0.39,0.50,0.61) (0.37,0.53,0.69,0.85) 0.45 0.61 𝑠𝑡3 (0.30,0.41,0.52,0.63) (0.22,0.35,0.49,0.64) 0.46 0.42 𝑠𝑡4 (0.34,0.38,0.47,0.54) (0.44,0.51,0.66,0.78) 0.43 0.59 𝑠𝑡5 (0.67,0.78,0.89,0.92) (0.54,0.69,0.85,0.94) 0.83 0.76 𝑠𝑡6 (0.29,0.39,0.50,0.60) (0.31,0.41,0.57,0.71) 0.44 0.50 𝑠𝑡7 (0.30,0.40,0.50,0.60) (0.35,0.50,0.64,0.78) 0.45 0.57 𝑠𝑡8 (0.37,0.48,0.59,0.69) (0.38,0.52,0.67,0.83) 0.52 0.60 𝑠𝑡9 (0.29,0.39,0.50,0.58) (0.32,0.48,0.63,0.75) 0.44 0.52 𝑠𝑡10 (0.32,0.43,0.54,0.63) (0.33,0.48,0.63,0.76) 0.48 0.55 𝑠𝑡11 (0.24,0.35,0.46,0.57) (0.18,0.27,0.41,0.55) 0.40 0.35 𝑠𝑡12 (0.22,0.33,0.44,0.55) (0.23,0.38,0.54,0.69) 0.38 0.46 𝑠𝑡13 (0.22,0.33,0.43,0.54) (0.23,0.37,0.52,0.67) 0.38 0.45 𝑠𝑡14 (0.25,0.36,0.47,0.57) (0.23,0.37,0.51,0.66) 0.41 0.44 𝑠𝑡15 (0.32,0.40,0.51,0.61) (0.27,0.32,0.46,0.58) 0.46 0.40 table 7. relative significance and the final ranking of stocks. stocks 𝐾𝑖𝐻𝑎 𝐾𝑖𝐻𝑏 𝐾𝑖𝐻𝑐 𝐾𝑖𝐻 rank 𝑠𝑡1 0.105 4.45 0.952 2.601 1 𝑠𝑡2 0.069 2.92 0.623 1.707 4 𝑠𝑡3 0.057 2.42 0.521 1.420 10 𝑠𝑡4 0.066 2.83 0.603 1.65 6 𝑠𝑡5 0.103 4.36 0.939 2.341 2 𝑠𝑡6 0.061 2.58 0.553 1.513 9 𝑠𝑡7 0.066 2.82 0.602 1.53 7 𝑠𝑡8 0.073 3.10 0.663 1.813 3 𝑠𝑡9 0.064 2.74 0.585 1.603 8 𝑠𝑡10 0.067 2.86 0.613 1.675 5 𝑠𝑡11 0.049 2.06 0.443 1.207 15 𝑠𝑡12 0.055 2.33 0.497 1.361 13 𝑠𝑡13 0.054 2.28 0.448 1.334 14 𝑠𝑡14 0.055 2.34 0.502 1.371 12 𝑠𝑡15 0.056 2.34 0.505 1.375 11 on the basis of the obtained values, it is concluded that the final ranking of the stocks using the f-cocoso-h-bcm model is 𝑠𝑡1 > 𝑠𝑡5 > 𝑠𝑡8 > 𝑠𝑡2 > 𝑠𝑡10 > 𝑠𝑡4 > 𝑠𝑡7 > 𝑠𝑡9 > 𝑠𝑡6 > 𝑠𝑡3 > 𝑠𝑡15 > 𝑠𝑡14 > 𝑠𝑡12 > 𝑠𝑡13 > 𝑠𝑡11 4.3. sensitivity analysis the sensitivity analysis helps to examine the effects of changing criterion weightings on the ranking performance of alternatives. in this case study, the criterion stock portfolio selection using a new decision-making approach based on the integration of… 105 c2 is determined as the most influential criterion because it has the highest weight value. when evaluating the effects of modification of criterion weighting on the preference rating of stocks, 10 scenarios have been developed. table 8 represents the changes in weighting of criteria for different scenarios. while changing the weights, it has been kept in mind that the sum of the weights should be 1. almost similar ranking result has been obtained for minor changes (<=5%) in criteria weighting. whereas more changes (>5%) in the value of criteria weights change the ranking of alternatives. figure 2 shows the ranking obtained by the original weighting and the variation in ranking upon change in the criterion weighting. the outcomes shows that the proposed model is sensitive to changes in the weighting of the criterion. it is also important to emphasize that st1, which represent the best solution, do not change the ranking in either scenario, meaning this is insensitive to changes in the significance of the criterion. in the same order, the spearman's correlation coefficient (scc) between the prime ranking and variation in ranking based on change in weights has been calculated. table 8 shows that the scc [0.884,0.996], which is quite high. table 8. changes in weightings of criteria for different scenarios changes in weights of criteria different scenarios scc between prime ranking and different scenarios increase in w1, w3, w4 decrease in w2 1 % 1 % scenario 1 0.996 2 % 2 % scenario 2 0.985 3 % 3 % scenario 3 0.985 4 % 4 % scenario 4 0.985 5 % 5 % scenario 5 0.985 10 % 7 % scenario 6 0.975 15 % 10 % scenario 7 0.967 20 % 12 % scenario 8 0.964 30 % 15 % scenario 9 0.939 50 % 30 % scenario 10 0.884 figure 2. effects of changing criterion weightings on the ranking performance of stocks narang et al./decis. mak. appl. manag. eng. 5 (1) (2022) 90-112 106 4.4. portfolio analysis in this section, a percentage of the total investment is obtained for each stock. the two main objectives of investors aremaximize the return and minimize the risk. an optimization method produces a more appropriate portfolio with respect to the risk and return of the portfolio. sortino ratio of the portfolio is considered as the objective function to optimize the total return under controllable risk. the sortino ratio is calculated by the following formula: 𝑆.𝑅.𝑜𝑓 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 = 𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑟𝑒𝑡𝑢𝑟𝑛−𝑟𝑖𝑠𝑘 𝑓𝑟𝑒𝑒 𝑟𝑒𝑡𝑢𝑛 𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑑𝑜𝑤𝑛𝑠𝑖𝑑𝑒 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 = ∑ 𝑥𝑖𝑟𝑖−𝑟𝑓𝑖 ∑ 𝑥𝑖𝑑𝑖𝑖 (17) where, 𝑟𝑖 = return of the 𝑖 𝑡ℎ ranked asset of the portfolio 𝑥𝑖 = weight of the the 𝑖 𝑡ℎ ranked asset of the portfolio 𝑟𝑓 = risk free rate 𝑑𝑖 = downside deviation of the 𝑖 𝑡ℎ ranked asset of the portfolio sortino ratio penalize only negative volatility or downside deviation from the mean return as the upside deviation are beneficial for the investor. high value of sortino ratio is considered as a good investment. hence, the optimization function for assigning the weights to the stocks is formulated as follows: max ∑ 𝑥𝑖𝑟𝑖−𝑟𝑓𝑖 ∑ 𝑥𝑖𝑑𝑖𝑖 (18) such that ∑ 𝑥𝑖 =𝑖 1,∀𝑥𝑖 > 0,∀𝑥𝑖 < 𝑚 4.3.1. optimization using pso particle swarm optimization technique is used to solve the aforementioned optimization problem. pso is a meta-heuristic optimization technique inspired by the behavior flock of birds, proposed by kennedy & eberhart (2010). suppose, a swarm consist of m particle in n dimensional search space and an optimization problem considering n candidate solutions such as {𝑋1,𝑋2 …𝑋𝑁}. at 𝑡 iteration, the position and the velocity of the 𝑖𝑡ℎ particle is denoted as 𝑥𝑖(𝑡) = (𝑥𝑖1,𝑥𝑖2 …𝑥𝑖𝑛) and 𝑣𝑖(𝑡) = (𝑣𝑖1,𝑣𝑖2 …𝑣𝑖𝑛), respectively. the best position 𝑃𝑏𝑒𝑠𝑡 visited by the 𝑖 𝑡ℎ particle is denoted as 𝑝𝑖(𝑡) = (𝑝𝑖1,𝑝𝑖2 …𝑝𝑖𝑛) and the particle that attained the best position in the previous iteration is denoted by 𝑔𝑏𝑒𝑠𝑡(𝑡) = (𝑔1,𝑔2 …𝑔𝑛). at next iteration the new position 𝑥𝑖(𝑡 + 1) and velocity 𝑣𝑖(𝑡 + 1) of the 𝑖 𝑡ℎ particle is calculated by the following equation: 𝑣𝑖𝑗(𝑡 + 1) = 𝜔 ∗ 𝑣𝑖𝑗(𝑡) + 𝑐1 ∗ 𝑟1 ∗ (𝑝𝑖𝑗(𝑡) − 𝑥𝑖𝑗(𝑡)) + 𝑐2 ∗ 𝑟2 ∗ (𝑔𝑗(𝑡) − 𝑥𝑖𝑗(𝑡)) (19) 𝑥𝑖𝑗(𝑡 + 1) = 𝑥𝑖𝑗(𝑡) + 𝑣𝑖𝑗(𝑡 + 1) (20) where 𝜔 is the inertia weight and 𝑐1, 𝑐2 are cognitive and social learning parameters, 𝑟1 and 𝑟2 are random numbers such that 𝑟1,𝑟2 ∈ (0,1). the 𝑃𝑏𝑒𝑠𝑡 and 𝑔𝑏𝑒𝑠𝑡 values are evaluated until the given number of iterations. we used three years data 2018-2020 to construct three portfolios 𝑃1,𝑃2 and 𝑃3 consisting top 5, 7 and 10 ranked stocks respectively. the return and the downside deviation of each stock are calculated to implement the optimization problem using pso. here we consider 5 particles, 𝜔 = 0.5 , 𝑐1 = 1, 𝑐2 = 2 and optimize up to 50 iterations. table 9 shows the weights of the stocks obtained by pso. stock portfolio selection using a new decision-making approach based on the integration of… 107 table 9. weights of the stocks obtained by pso stock s 𝑠𝑡1 𝑠𝑡5 𝑠𝑡8 𝑠𝑡2 𝑠𝑡10 𝑠𝑡4 𝑠𝑡7 𝑠𝑡9 𝑠𝑡6 𝑠𝑡3 for 𝑃1 0.26 0.22 0.20 0.17 0.15 for 𝑃2 0.22 0.18 0.15 0.13 0.11 0.10 0.09 for 𝑃3 0.20 0.17 0.15 0.14 0.10 0.08 0.06 0.04 0.03 0.01 sortino ratio greater than one is considered as a good investment. from the table 10, we observe for two portfolios 𝑃2 and 𝑃3 , sortino ratio is greater than one. the investor prefers the portfolio with high sortino ratio as it depicts more return per unit for the downside risk. hence, the portfolio 𝑃2 attain maximum sortino ratio among all portfolios. it also gained maximum return of 16.72%. the downside risk is high for 𝑃1 with sortino ratio less than one. from the analysis, we can conclude that although all three portfolios gave similar return but 𝑃2 and 𝑃3 are profitable and save portfolios with less downside risk. we have also compared the results obtained by our proposed model with earlier study in table 11. this verifies the robustness and rootedness of the proposed ranking system in multi-dimensional decision analysis systems. table 10. performance of different portfolio optimized by pso portfolio portfolio return portfolio downside deviation sortino ratio 𝑃1 0.1654 0.20 0.82 𝑃2 0.1672 0.14 1.194 𝑃3 0.1636 0.15 1.091 table 11. comparison of proposed model with earlier study model (thakur et al., 2018) proposed model year 2016 2021 ex. return 0.1301 0.1672 5. conclusions in this paper, a new integrated f-cocoso-h-bcm strategy is proposed to solve the decision-making problem through some specific modifications to the main structure. the purpose of this study is to demonstrate that the hm aggregation operator can be used for fusion of criterion functions in the decision matrix of the cocoso model as it has ability to detect correlations in specified decision criteria. the improved cocosoh model removes the impact of awkward data in stock selection. furthermore, features of igwhm and iggwhm make decisions much more flexible. f-bcm forges the cocoso-h model more powerful by defining the optimal values of relative weights. since it is an integrated model therefore there are some limitations for finding the criteria’s weight and ranking of alternatives. a hybrid approach (f-cocoso-h-bcm) based on a fusion of three strategies (the cocoso model, the heronian mean and the bcm) is proposed, which has major novelties and contributions as follows: narang et al./decis. mak. appl. manag. eng. 5 (1) (2022) 90-112 108  this study is a presentation of the novel fuzzy cocoso-h-bcm model that serves the purpose of evaluating stocks in fuzzy environment.  however, despite the uncertainty in the decision-making process and the lack of quantitative information, the presented methodology makes it possible to evaluate a set of alternatives.  the f-cocoso-h-bcm approach enables the flexible decision-making with less computation. it also represents a new reference for researchers in the field of stock portfolio selection.  the outcomes (expected return of 0.1672 or 16.72%) of the portfolios validate the rationality, stability, effectiveness and rootedness of the presented methodology. the effectiveness and flexibility of the proposed f-cocoso-h-bcm are properties that make it recommended for use in management, supplier selection and engineering applications. in future, the pythagorean, type-2, neutrosophic, intuitionistic and hesitant concepts which are the generalizations of fuzzy set can be used in decision making process under uncertainty with the cocoso-h model. author contributions: the authors confirm contribution to the paper as follows: conceptualization, monika narang; methodology, monika narang; software, kiran bisht; validation, arun kumar pal; investigation, mahesh chandra joshi; writing— original draft preparation, monika narang; writing—review and editing, arun kumar pal; visualization, arun kumar pal; supervision; project administration, mahesh chandra joshi, arun kumar pal; all authors have reviewed the results and approved the final version of the manuscript. funding: this research received no external funding. data availability statement: preliminary data has been collected https://www.ratestar.in/, https://www.investello.com/ and https://in.finance.yahoo.com/. conflict of interest: the authors declare that they have no known competing 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(2013). multicriteria decision systems for financial problems. top. 21(2), 241–261. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0318062021g * corresponding author. e-mail addresses: sayan.103030@gmail.com (s. gupta), gautam.bandyopadhyay@dms.nitdgp.ac.in (g. bandyopadhyay), arupmitra2001@gmail.com (a. mitra), sb.16ms1302@phd.nitdgp.ac.in (s. biswas) an integrated framework for classification and selection of stocks for portfolio construction: evidence from nse, india sayan gupta1*, gautam bandyopadhyay1, sanjib biswas1, and arup mitra2 1 department of management studies, nit, durgapur, india 2 department of management studies, makaut, haringhata, india received: 15 november 2021; accepted: 16 february 2022; available online: 18 june 2022. original scientific paper abstract: investment decision making is a complex process, influenced by a number of conflicting objectives. investors want to maximize their wealth through investing in the stock market while offsetting the risk to the extent possible. to a common investor, risk is an important aspect to be minimized. in this paper we present a distant framework of stock selection for portfolio construction combining bayesian classifier and a widely used multi-criteria decision making (mcdm) technique such as the technique for order of performance by similarity to ideal solution (topsis) along with entropy method. the study period is 2013 to 2020. we formulate our research design by considering risk adjusted ratios like sharpe ratio, treynor ratio, information ratio, jensen ratio, and calmar ratio to compare the nse 100 listed stocks. using dp omnibus test, the desired sample of companies following the non-normal distribution was achieved. using financial beta, we have selected the outcome based on the nature of their ‘return’ and ‘risk'. the entropy-topsis framework has been used to study the profitability of stocks, rank wise for each year, and finally, the bayes portfolio model help to select the overall profitability associate with low risk for the construction of the portfolio. we notice year wise inconsistency among the performance of the stocks. key words: portfolio selection, equity stocks, bayesian method, dp omnibus test, risk adjusted return ratios, mcdm, entropy, topsis. 1. introduction stock market (sm), more specifically the equity market has been an area of interest to the researchers, practitioners and common investors over many decades. there has mailto:sayan.103030@gmail.com mailto:arupmitra2001@gmail.com mailto:sb.16ms1302@phd.nitdgp.ac.in gupta et al./decis. mak. appl. manag. eng. 2 been a plethora of research work conducted on formulation of investment and/or trading strategy to optimize the risk and return at a given level of invested amount. the selection of appropriate stocks and prudent allocation of the total funds among them lead to an effective portfolio management which stands as a cornerstone of successful investment strategy (ren et al., 2017). portfolio construction is a complicated task for the common investors considering the up and downtrend of the market. there are a number of considerations of the common investors while selecting the stocks such as high return, low risk, and appropriate time to enter and exit the market, period of holding the stocks, and selection of the sectors among others. the extant literature is rife in significant contributions in the stated field of security analysis and portfolio management (sapm) by various scholars in the modern era started with the two seminal work such as concepts and guidelines for security analysis and value investing (graham et al., 1934) and mean-variance analysis based portfolio selection (markowitz, 1952). in subsequent years, the growing field of sapm was notably contributed and expanded by sharpe (1964), lintner (1965), mossin (1966), and black (1993) (capital asset price model and market equilibrium); fama (1970) (efficiency and equilibrium of capital markets); stattman (1980), banz (1981), reinganum (1981), basu (1983), rosenberg et al. (1985), bhandari (1988), chan et al. (1991) (impact of firm characteristics on average stock returns); fama and french (1992, 1993) (three factor asset pricing model for stock selection); jegadeesh and titman (1993), grinblatt et al. (1995), cooper et al. (2004) (momentum and contranian based analysis for stock investment strategy); carhart (1997) (four factor asset pricing model); huang et al. (2011) (behavioural bias in selection of stocks); chong and phillips (2012), hsu and li (2013) (volatility assessment for stock selection); peachavanish (2016) (integrated fundamental and technical analysis based investment decision making) and fama and french (2017, 2018) (multifactor model). one generalized view from these research is evident that investors consider multiple perspectives such as market performance indicators like price to earnings ratios, price to book value ratio, beta, return, and volatility, fundamental attributes like return on investment, return on net worth, asset size etc., and technical indicators while formulating their portfolios. there have been efforts in applying classification models for selection of stocks to invest. cluster analysis in various forms have been used in several research (for instance, da costa et al., 2005; dose and cincotti, 2005; brida and risso, 2010; tabak et al., 2010; silva and marques, 2010; nanda et al., 2010; baser and saini, 2015; peachavanish, 2016; iorio et al., 2018) wherein the analysts considered fundamental and technical attributes for assessing comparative efficiencies and classify the stocks in different categories in the context of global markets (e.g., india, thailland, brazil). the advantage of using clustering stems from efficiency based classification of the stocks of varying characteristics that helps in understanding the interplay among the stocks, construction of portfolios with diverse stocks to reduce systematic risk considerably and effective utilization of the funds. in some work (for example, baks et al., 2001; cabrera et al., 2018; jammalamadaka et al., 2019; hoseini ebrahimabad et al., 2019; de rossi et al., 2020; ampomah et al., 2021; platanakis et al., 2021) the authors have used bayesian approach in determining the suitability of the stocks in terms of their market performances and predicted returns vis-à-vis investment decision making. from the above discussions, it may be inferred that stock selection depends on multiple perspectives that are complex and conflicting in nature. hence, the extant literature has garnered attentions of the researchers (for instance, poklepović and babić, 2014; vezmelai et al., 2015; mashayekhi and omrani, 2016; hatami-marbini and an integrated framework for classification and selection of stocks for portfolio… 3 kangi, 2017; aouni et al., 2018; alali and tolga, 2019; yildiz, 2020; peng et al., 2021; nguyen et al., 2022) and convinced them to apply multi-criteria decision making (mcdm) frameworks in formulation of the investment strategies. therefore, it is amply evident from the literature that stock selection for constructing portfolio using the models of predictive analytics inspired by probabilistic and ai/ml concepts, mcdm techniques and statistical analysis are quite common. however, a combined two stage approach based on classification and mcdm models are quite rare in the literature. further, most of the early work concentrated on market performance indicators, technical analysis and fundamental ratios. the risk adjusted ratios like sharpe ratio (sr), treynor ratio (tr), and information ratio (ir) as used in the present paper have been noticed in use in the literature related to mutual funds. the present study aims to identify the stocks having low-risk propensities and associated with average to high return to construct a fruitful portfolio allocation for the common investors. we have considered the non-normal stocks from the nse 100 using some filtering process while disregarding highly volatile stocks. we consider the stocks and its applicability, to investigate portfolio allocation and estimate the potential performance. a topsis based scheme mcdm has been used to classify and select stocks subject to the influence of the financial risk adjusted performance factors and finally using posterior bayesian optimization for risk less optimal returns. the research questions (rq) that the present study endeavours to enquire are rq1. do all stocks (over the study period) follow same type of distribution? rq2. what are stocks that follow non-normal distributions? rq3. what are the stocks that show low risk propensity associated with average to high return? rq4. to what extent do the stocks perform differently on yearly basis over the study period? in the present study we intend to find answer of the above-mentioned rqs and thereby to suggest a suitable portfolio for the common investors. this paper fills the gap in the literature and contributes in the following ways.  firstly, it provides an integrated model for classification and multi-attribute based ranking. in the present study we use the probabilistic bayesian model in conjunction with mcdm algorithm which seems to be rare in the extant literature.  secondly, we use risk adjusted return ratios such as sr, tr, and ir for comparing stock performance.  thirdly, in indian context, the kind of study similar to our work is not available in the literature as we found in our limited search. the reminder of this paper is presented in the following way. in section 2, we present some of the related work. section 3 discusses the research methodology while in section 4 the summary of findings is included along with discussions. in section 5 the validation test and sensitivity analysis are included while in section 6 we mention some of the research implications and concluding remarks. section 7 concludes the paper while highlighting some of the future scope. 2. related work the mcdm algorithms were developed and introduced in the financial market by several researchers xidonas et al. (2009) reported that mcdm can solve any financial decision, either institutional or private, for investment opportunities. hurson (1997) performed a comparative analysis among multi-criteria methods such as measuring gupta et al./decis. mak. appl. manag. eng. 4 attractiveness by categorical based evaluation techniques (macbeth) and multiutility theory (mut) for portfolio selection and optimization. in the croatian stock market a combined framework of copras, linear assignment, promethee, saw and topsis was used by poklepović and babić (2014). in another study (vezmelai et al., 2015), the authors considered the criteria like economic value added (eva), return on equities (roe), return on assets (roa), q-tobin, earnings per share (eps) and price/earnings per share (p/e) for conducting a comparative assessment of selected stocks in tehran stock market using electre-iii method. dincer and hacioglu (2015) used financial stress and conflict risk as the basis for stock selection and applied a combined framework of ahp-topsis-vikor. mashayekhi and omrani (2016) put forth a trapezoidal fuzzy number based framework of data envelopment analysis (dea) using the fundamental mean variance model of markowitz at risk-return interface to derive the efficient portfolio. bayramoglu and hamzacebi (2016) carried a fundamental analysis of the stock performance using grey relational analysis (gra) in the borsa istanbul stock exchange, turkey. hatami-marbini and kangi (2017) contributed in selecting stocks in untapped sections of tehran stock market with future expectation of appreciation of return using new fuzzy distance measures and extension of classical topsis method. a use of multi-objective optimizations is noticed for multi-criteria based stock selection and portfolio optimization following the mean-variance framework in aouni et al. (2018). some authors (for instance, pätäri et al., 2018) have attempted to contribute a comparative framework of several mcdm models to provide the investors best possible way to select the stocks for investing. the work of makui and mohammadi (2019) considered behavioural aspects and carried out a comparative analysis of relative utilities for stock selection using utastar method on the basis of risk, return and liquidity. alali and tolga (2019) experimented with equally weighted portfolio formulation vis-à-vis the mean-variance one using todim method and reported an insignificant benefit of their proposed portfolio. gupta et al. (2019a) used deacopras combination for portfolio strategy. yildiz (2020) applied topsis method for performance analysis of the turkish stock market indices. there are some other studies in recent past that have used mcdm algorithms for stock selection purpose. for example, cheng et al. (2021) focused on the sports and leisure industry and used a multi-criteria based decision tree method considering fundamental attributes to propose a stock selection framework. peng et al. (2021) applied electre i method in conjunction with z-numbers for portfolio formulation. the work on indian it sector by ghosh (2021) used a combined framework of grey correlational analysis-ahp-topsis. in the context of vietnamese market, nguyen et al. (2022) experimented with critic-dematel method for exploring the impact of covid-19 on commercial banks. a fuzzy base criterion method and copras was utilized for portfolio selection in the research of narang et al. (2021). vásquez et al. (2022) considered an integrated framework of ahp-topsis for portfolio formulation with equity stocks after analysing the performance of colombian market during the period 2012-2017. in another work (gupta et al., 2021), a comparative analysis of the financial performance of public sector banks of india has been carried out using a framework of critic-topsis approach. 3. materials and method in this paper we followed a two steps approach. in the first step we classified the stock through a series of filtering. in this process of classification we adopted a an integrated framework for classification and selection of stocks for portfolio… 5 probability based approach and applied bayesian method at the final filtration stage. in initial stage we select financial stocks data from nse-100 (national stock exchange) from march 2013 to march 2020 as per criteria of diversification and applying the filtering process. we consider the stocks which are having non-normal distributions in the next step, among the final list of selected stocks, we carried a comparative analysis for deriving performance based preferential order based on market perception. the market perception is captured in terms of risk returns based attributed, calculated using closing prices. therefore in the second stage we applied a widely used multi-attribute decision makes process such as topsis. finally bayes portfolio model explains the overall risk on the basis of prior information collected from the outcome of topsis model to construct a fruitful portfolio. performing the methods step by step we find out portfolio bucket with desire returns with low risk aversion, which may help investor to take decision in portfolio selection. we introduce a probabilistic approach to estimate the posterior distribution of the target rank conditionally to the predictors. two desirable properties of a prior distribution for nonparametric problems. (i) the support of the prior distribution should be large--with respect to some suitable topology on the space of probability distributions on the sample space. (ii) posterior distributions given a sample of observations from the true probability distribution should be manageable analytically. the work flow diagram describing the research methodology is given in the figure 1. 3.1. sample out of the 100 selected stocks 14 stocks were discarded because of incomplete data. table 1 shows all the 86 companies those were ultimately considered initially from the list of nse 100 companies for this study. as is evident from the figure 1, we remove 50 stocks from our analysis in the first stage of the filtration process and only 36 stocks having non-normal distribution enters the second stage of the filtration. we then consider financial beta values and the stocks having higher beta values have been discarded as from the perspectives of the common investors we only consider low risk. we get 15 stocks and finally through perceptual mapping we derive our final sample of 6 stocks (having low risk and considerably higher return) for mcdm based comparative analysis. table 1: initial list of 86 companies from nse 100 s/l name s/l name s/l name 1 acc 31 bpcl 61 havells 2 adaniports 32 britannia 62 hcltech 3 ambujacem 33 cadilahc 63 hdfc 4 ashokley 34 cipla 64 hdfcbank 5 asianpaint 35 coalindia 65 heromotoc o 6 auropharma 36 colpal 66 hindalco 7 axiabank 37 concor 67 hindpetro 8 bajajfinsv 38 dabur 68 hindunilvr 9 bajajhldng 39 divislab 69 hindzinc 10 bafinance 40 dlf 70 icicibank 11 bankbaroda 41 drreddy 71 idea 12 bergepaint 42 eichermot 72 indusindbk 13 bhartiartl 43 gail 73 infratel 14 biocon 44 godrejcp 74 infy 15 boschltd 45 grasim 75 ioc 16 itc 46 pageind 76 tatasteel 17 jswsteel 47 pel 77 tcs gupta et al./decis. mak. appl. manag. eng. 6 s/l name s/l name s/l name 18 kotakbank 48 petronet 78 techm 19 l&tfh 49 pfc 79 titan 20 lupin 50 pghh 80 ubl 21 m&m 51 pidiltind 81 ultracemco 22 marico 52 pnb 82 upl 23 maruti 53 powergrid 83 vedl 24 mothersumi 54 reliance 84 wipro 25 nestleind 55 sbin 85 yesbank 26 nhpc 56 shreecem 86 zeel 27 nmdc 57 siemens 28 ntpc 58 srtransfin 29 ofss 59 sunpharma 30 ongc 60 tatamotors figure 1. work flow diagram of the research methodology the data are downloaded from nse website and cmie prowess iq database and company annual reports. statistical calculations have been done using jamovi (version 2.2.5) and r & excel. an integrated framework for classification and selection of stocks for portfolio… 7 3.2. definitions a) financial beta beta is a measure of systematic risk. a beta value of more than 1 indicates that the stock is more unpredictable than the more extensive market and a value under 1 demonstrates that a stock with lower impulsiveness, it is derived from the capital asset pricing model. beta is presumably a superior pointer of present moment instead of long term risk. traditionally beta coefficient is defined as 𝑅𝑖𝑡 = 𝛼 + 𝛽𝑖 𝑅𝑚𝑡 + 𝑒𝑖𝑡 (1) where, 𝑅𝑖𝑡 is the return on asset 𝑖 at time 𝑡 𝑅𝑚𝑡 is the return of the market at time 𝑡 𝛼𝑖 and 𝛽𝑖 are the intercept and slope (beta)coefficient the market model is commonly estimated using ordinary least squares regression (ols). in this instance the ols estimate of beta is simply: 𝛽𝑖 = 𝐶𝑜𝑣(𝑅𝑖𝑡;𝑅𝑚𝑡) 𝑉𝑎𝑟(𝑅𝑚𝑡) (2) b) financial ratios financial ratios are the vital indicators helping to find out performance in terms of profitability, liquidity, growth prospect, and stability of a company from its financial reports. financial ratio can give a blueprint, how an association is performing vis-àvis its competitors and industry at large. while financial ratios offer useful information about an organization, they should be coordinated with various estimations, to get a broader picture of the company’s financial wealth. in this paper we consider market performance of the stocks under study. the ratios used for this paper are briefly described in the following table (see table 2). table 2: definitions of the ratios used in the paper ratio formula explanation sharpe ratio(sr) (sharpe 1966) 𝑆𝑅 = [𝑅𝑎−𝑅𝑏] 𝜎𝑎 𝑆𝑅 = 𝑆ℎ𝑎𝑟𝑝𝑒 𝑅𝑎𝑡𝑖𝑜, 𝑅𝑎 = 𝐴𝑠𝑠𝑒𝑡𝑠 𝑅𝑒𝑡𝑢𝑟𝑛, 𝑅𝑏 = 𝑅𝑖𝑠𝑘 𝑓𝑟𝑒𝑒 𝑅𝑒𝑡𝑢𝑟𝑛, 𝜎𝑎 = 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑠𝑠𝑒𝑡 𝑒𝑥𝑐𝑒𝑠𝑠 𝑟𝑒𝑡𝑢𝑟𝑛 treynor ratio (tr) (treynor, 1965) 𝑇𝑅 = (𝑅𝑎 − 𝑅𝑏 ) 𝛽𝑎⁄ 𝑇𝑅 = 𝑇𝑟𝑒𝑦𝑛𝑜𝑟 𝑅𝑎𝑡𝑖𝑜, 𝑅𝑎 = 𝐴𝑠𝑠𝑒𝑡𝑠 𝑅𝑒t𝑢𝑟𝑛, 𝑅𝑏 = 𝑅𝑖𝑠𝑘 𝑓𝑟𝑒𝑒 𝑅𝑒𝑡𝑢𝑟𝑛, 𝛽𝑎 = 𝐴𝑠𝑠𝑒𝑡𝑠 𝐵𝑒𝑡𝑎 jensen alpha(ja) (jensen, 1968) �̅� = (𝑅𝑎 − 𝑅𝑏 − 𝛽𝑎 × (𝑅𝑎 − 𝑅𝑏 )) 𝛼 = 𝐽𝑒𝑛𝑠𝑒𝑛 𝐴𝑙𝑝ℎ𝑎 information ratio (goodwin, 1998) 𝐼𝑅 = (𝑅𝑎 − 𝑅𝑐 ) 𝜎𝑏 ⁄ 𝑅𝑎 = 𝐴𝑠𝑠𝑒𝑡𝑠 𝑅𝑒𝑡𝑢𝑟𝑛 , 𝑅𝑐 = 𝐼𝑛𝑑𝑒𝑥 𝑅𝑒𝑡𝑢𝑟𝑛, 𝜎𝑏 = 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑑𝑖𝑓𝑓e𝑟𝑒𝑛𝑐𝑒𝑠 gupta et al./decis. mak. appl. manag. eng. 8 ratio formula explanation sortino ratio(sor) (sortino and van der meer, 1991) 𝑆𝑜𝑅 = (𝑅𝑎 − 𝑅𝑏 ) 𝜎𝑑⁄ 𝜎𝑑 = 𝐷𝑜𝑤𝑛𝑠𝑖𝑑𝑒 𝑟𝑖𝑠𝑘 calmar ratio(cr) (young, 1991) 𝐶𝑅 = (𝑅𝑎 − 𝑅𝑏 ) 𝐷𝑚𝑎𝑥⁄ 𝐷𝑚𝑎𝑥 = 𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝐷𝑟𝑎𝑤 𝐷𝑜𝑤𝑛 in the present study we use risk adjusted ratios such as sr that can also be used to determine if a portfolio's excess returns are the consequence of sound investment selections or excessive risk. the standard deviation is a measurement of the square root of the variance and measures the dispersion of a dataset relative to its mean, and its shows how far a portfolio's return deviates from its expected return. the standard deviation also reveals the volatility of the portfolio. when compared to similar portfolios with a lower level of diversification, adding diversification should improve the sharpe ratio. the sharpe ratio of a portfolio determines its risk-adjustedperformance. for capturing the market perception, we have followed the risk-return based attribute the ratios which are considered on this paper have mostly being followed in mutual fund assessment. in this respect the present paper at value to the growing literature. since these ratios consider stocks return, risk-free return, bench mark return, risk parameters. 3.3. methods in this sub-section we present the methods used in this paper briefly. a) dp omnibus test the normal distribution is the most commonly used distribution when performing statistical procedures and applications, especially for parametric methods, because it is the most widely accepted way to verify normality assumptions. dp omnibus test is best suited for sample sizes between 20 and 1000. the test uses skewness and kurtosis √b1 and b2, respectively, and tests for normality of a random sample of the population (pearson et al., 1977; wyłomańska et al., 2020). dp omnibus test used to find out the stocks are follows normal distribution or not. here in this paper we have selected the stocks, those were non normal in nature, so that we can apply the bayesian classifier to classify the stocks based on the prior information. the equation (yap and sim, 2011) is shown below. dp = z2 (√b1) + z2 (b2) (3) b) 3×3 investor perceptual map it’s a graphical representation of an objects to check the position of the items with respect to other items in a two dimensional space, which divides into 9 quadrants. the 9 quadrant are as follows: hl= high low , hm= high medium , hh= high high , ml= medium low , mm= medium medium , mh= medium high , ll= low low , lm= low medium and lh= low high. in this study we have check the position of our stocks on the basis of risk-return interface shown in the figure 2. an integrated framework for classification and selection of stocks for portfolio… 9 range= maximum –minimum. (4) figure 2: representation of 3×3 investor perceptual map c) topsis method the technique for order of preference by similarity to ideal solution (topsis) is a widely popular mcdm algorithm that considers two extreme solution points such as positive ideal solution (pis) or most optimistic solution and negative ideal solution (nis) or most pessimistic solution as references (hwang and yoon, 1981). the euclidean distances of ‘m’ number of alternatives under the influences of ‘n’ number of criteria are calculated with respect to pis and nis. subsequently, the alternative closest to the pis (i.e., furthest to nis) is considered to be the best choice while trading off the impacts of the conflicting criteria. topsis has been used in investment decision making quite frequently (for instance, vásquez et al., 2022; hassanzadeh and valmohammadi, 2021; atukalp, 2021; biswas et al., 2019; gupta et al., 2019; karmakar et al., 2018). the algorithmic steps for topsis method is given in table 3. d) entropy method the entropy method is one of the widely used approaches to determine the weights of the criteria using objective information (biswas et al., 2021; pramanik et al., 2021; biswas et al., 2019; laha and biswas, 2019; karmakar et al., 2018). entropy is essentially a measure of disorder. according to the seminal work of shannon (1948) on information theory, the entropy method assigns higher weights to the criteria that carry substantial information. the steps are given in the table 4. e) bayes model probability is the degree of the prospect that an occasion will occur. probability is quantified as 0 to 1 (wherein 0 suggests impossibility and 1 suggests certainty). bayes theorems entails in the pattern space, here occur an event b for which p (b)>0 and the analytics intention is to computes a conditional probability of p (ak/b). thomas bayes (1702-1761) indicates the relation among one conditional probability and its inverse and offer a mathematical rule of revising an estimate for forecast in mild of revel in and observation. in chance idea and facts bayes theorem (opportunity bayes regulation and bayes rule) describes the chance of an occasion primarily based totally on situations that is probably associated with the occasion. bayesian inference is a technique of statistical inference wherein bayes theorem is used to replace the chance for a speculation of evidence, it worried with 1) prior probability that is preliminary chance primarily based totally on the existing degree of data and 2) posterior chance that is revised chance primarily based totally on extra data, for an unknown parameter θ, its posterior π(θ | x) is a conditional distribution of θ below sampler x and it includes all of the data this is available (avramov, 2002). in this study, collecting the prior information from the outcome of topsis model, we calculate the posterior probability gupta et al./decis. mak. appl. manag. eng. 10 of each stocks considering seven years and find out the overall expected variance of each stocks using bayes model. the posterior probabilities is the parameter 𝜃 given the evidence 𝑋 : 𝑝(𝜃|𝑋) wherein the probability of the evidence is given by the parameter: 𝑝(𝑋|𝜃). the probability distribution function is 𝑝(𝜃) and the observations 𝑥 have likelihood 𝑝(𝑥|𝜃). the equation is: 𝑝(𝜃|𝑥) = 𝑝(𝑥|𝜃) 𝑝(𝑥) 𝑝(𝜃) (5) where 𝑝(𝑥) is the normalizing constant and it’s calculated as 𝑝(𝑥) = ∫ 𝑝(𝑥|𝜃)𝑝(𝜃)𝑑𝜃 (6) for continuous 𝜃 or by summing 𝑝(𝑥|𝜃)𝑝(𝜃), the overall possible values of θ for disctete θ.(see; avramov, 2002) in this paper we followed a two steps approach. in the first step we classified the stock through a series of filtering. in this process of classification, we adopted a probability based approach and applied bayesian method at the final filtration stage. in the next step, among the final list of selected stocks, we carried a comparative analysis for deriving performance based preferential order based on market perception. the market perception is captured in terms of risk returns based attributed, calculated using closing prices. therefore, in the second stage we applied a widely used multi-attribute decision makes process such as topsis. we introduce a probabilistic approach to estimate the posterior distribution of the target rank conditionally to the predictors. two desirable properties of a prior distribution for nonparametric problems. (i) the support of the prior distribution should be large-with respect to some suitable topology on the space of probability distributions on the sample space. (ii) posterior distributions given a sample of observations from the true probability distribution should be manageable analytically. table 4. computational steps of entropy method steps of the entropy method formula step1: creation of decision matrix 𝐴 = [ 𝑎11 ⋯ 𝑎1𝑛 ⋮ ⋱ ⋮ 𝑎𝑚1 ⋯ 𝑎𝑚𝑛 ] step 2: calculation of the normalized matrix 𝑝𝑖𝑗 = 𝑎𝑖𝑗 ∑ 𝑎𝑖𝑗 𝑛 𝑖=1 𝑖 = 1,2 … … . 𝑛 ; 𝑗 = 1,2 … . 𝑘 step 3: calculation of entropy value 𝐸𝑗 = −𝑒 ∑ 𝑝𝑖𝑗 , 𝑛 𝑖=1 𝑒 = 1 𝑙𝑜𝑔 𝑛 ; 𝑗 = 1,2 … … 𝑘 step 4: determination of entropy weights 𝑤𝑖 = 𝑑𝑗 ∑ 𝑑𝑗 𝑛 𝑖=1 𝑑𝑗 = 1 − 𝐸𝑗 ; 𝑗 = 1,2 … . . 𝑘 an integrated framework for classification and selection of stocks for portfolio… 11 table 3. computational steps of topsis method steps calculation step1: decision matrix 𝑌 = [ 𝑦11 ⋯ 𝑦1𝑛 ⋮ ⋱ ⋮ 𝑦𝑚1 ⋯ 𝑦𝑚𝑛 ] m: alternative, n: criteria step2: normalized matrix 𝑌𝑖𝑗̅̅ ̅ = 𝑌𝑖𝑗 √∑ 𝑌𝑖𝑗 2𝑛 𝑗=1 step3: calculate weighted normalized matrix 𝑉𝑖𝑗 = 𝑌𝑖𝑗̅̅ ̅ × 𝑊𝑗 step 4: find out the pis and nis pis: 𝑉𝑗 + = {𝑀𝑎𝑥 𝑉𝑖𝑗 ; 𝑗 ∈ 𝐽 +; 𝑀𝑖𝑛 𝑉𝑖𝑗 ; 𝑗 ∈ 𝐽 − } nis 𝑉𝑗 − = {𝑀𝑖𝑛 𝑉𝑖𝑗 ; 𝑗 ∈ 𝐽 +; 𝑀𝑎𝑥 𝑉𝑖𝑗 ; 𝑗 ∈ 𝐽 − } step 5: calculation euclidean distance from the ideal worst and best 𝑑𝑖 − = [∑(𝑉𝑖𝑗 − 𝑉𝑗 −) 2 𝑛 𝑗=1 ] 0.5 𝑑𝑖 + = [∑ (𝑉𝑖𝑗 − 𝑉𝑗 +) 2𝑛 𝑗=1 ] 0.5 step 6. calculation of the closeness coefficient 𝑆𝑖 = 𝑑𝑖 − 𝑑𝑖 ++ 𝑑𝑖 − decision rule higher the value of 𝑆𝑖 , better is the alternative 4. findings and discussions in this section we exhibit step by step data analysis and the findings. first, we calculate the rate of return (ror) of the stocks pertaining to the initial sample of 86 companies. the rate of return (ror) has been calculated from the stocks using the expression followed in guha et al. (2016). return (rs) = ln ( 𝑙i li−1 ) . 100% (7) where li the closing price of the current month and li-1 is that of the immediately preceding month. then the average rors (aror) of all 86 stocks have been calculated by considering the average of the returns of each stock over a period of 84 months as considered in the study (kindly refer table 5). gupta et al./decis. mak. appl. manag. eng. 12 table 5. aror for the stocks of the initial sample of 86 stocks company ror company ror company ror company ror acc 0.002 14 drreddy 0.006 775 m&m 0.0049 2 tataste el 0.001 19 adanipo rts 0.007 075 eicherm ot 0.007 99 marico 0.0114 02 tcs 0.012 77 ambujac em 0.001 34 gail 0.001 86 maruti 0.0143 82 techm 0.009 029 ashokle y 0.008 073 godrejc p 0.008 302 mother sumi 0.0055 35 titan 0.015 386 asianpai nt 0.014 53 grasim 0.001 052 nestlei nd 0.0150 92 ubl 0.003 245 auroph arma 0.020 639 havells 0.015 676 nhpc 0.0000 5982 ultrace mco 0.006 568 axiaban k 0.004 48 hcltech 0.009 363 nmdc 0.0064 1 upl 0.017 004 bajajfin sv 0.021 232 hdfc 0.008 111 ntpc 0.0040 5 vedl 0.010 44 bajajhld ng 0.008 082 hdfcban k 0.012 071 ofss -0.0028 wipro 0.002 169 bafinan ce 0.035 239 heromo toco 0.000 413 ongc 0.0132 3 yesbank 0.015 96 bankbar oda 0.011 01 hindalc o 0.000 521 pageind 0.0193 8 zeel 0.006 3 bergepa int 0.023 403 hindpet ro 0.013 077 pel 0.0054 73 bhartia rtl 0.005 932 hinduni lvr 0.020 381 petron et 0.0128 69 biocon 0.021 195 hindzin c 0.002 949 pfc 0.0002 09 boschlt d 0.000 533 iciciban k 0.006 342 pghh 0.0167 08 bpcl 0.010 989 idea 0.036 88 pidiltin d 0.0194 93 britann ia 0.027 699 indusin dbk 0.001 68 pnb 0.0177 4 cadilah c 0.006 991 infrate l 0.001 33 powerg rid 0.0048 57 cipla 0.001 28 infy 0.006 839 relianc e 0.0126 01 an integrated framework for classification and selection of stocks for portfolio… 13 company ror company ror company ror company ror coalindi a 0.009 42 ioc 0.001 765 sbin 0.0006 1 colpal 0.008 357 itc 0.002 19 shreece m 0.0174 73 concor 0.004 894 jswstee l 0.009 251 siemens 0.0084 25 dabur 0.014 164 kotakba nk 0.016 412 srtrans fin 0.0005 9 divislab 0.016 588 l&tfh 0.004 33 sunpha rma 0.0017 9 dlf 0.006 37 lupin 0.000 76 tatamo tors 0.0157 3 as seen from the table 5 some stocks (highlighted in light blue shed) have generated negative aror. we discard those stocks for the next step. it is noticed that the stocks having -ve arors exhibited more negative returns during the previous period. from the investors’ point of view, a stock generating more number of negative monthly returns given a study period is not promising (gupta et al., 2019b; guha et al., 2016). therefore, we filter out these 28 stocks that lead to a sample of 58 stocks for the next level of the filtration process. in the next step, we run the normality test using the dp omnibus test and select only the stocks which are not having the normal distribution shown (see table 6). table 6. results of the normalization test stock no. 1 2 3 4 5 6 7 8 9 10 normality_ test adan iport s asho kley asia npai nt auro phar ma axiab ank bajajf insv bajaj hldng bafina nce berge paint bharti artl omnibus: 14.2 02 12.35 4 0.32 95 0.755 4 58.60 75 78.25 22 105.2 009 27.947 3 26.54 78 0.495 p value: 0.00 0824 3 0.002 077 0.84 81 0.685 4 1.88e -13 2.20e -16 2.20e -16 8.54e07 1.72e -06 0.7807 stock no. 11 12 13 14 15 16 17 18 19 20 normality_ test bioc on bosch ltd bpcl brita nnia cadila hc cipla colpa l concor dabu r divisla b omnibus: 2.73 71 22.30 58 3.60 55 40.81 93 6.379 1 0.240 1 4.056 6 56.982 8 2.395 7 36.635 9 p value: 0.25 45 1.43e -05 0.16 48 1.37e -09 0.041 19 0.886 9 0.131 6 4.23e13 0.301 8 1.11e08 stock no. 21 22 23 24 25 26 27 28 29 30 normality_ test drre ddy godre jcp grasi m havel ls hclte ch hdfc hdfcb ank herom otoco hinda lco hindpe tro omnibus: 22.4 798 7.045 5 47.8 567 5.300 9 2.209 4 22.10 51 39.13 45 3.7109 49.46 38 54.379 4 p value: 1.31 e-05 0.029 52 4.06 e-11 0.070 62 0.331 3 1.59e -05 3.18e -09 0.1564 1.82e -11 1.56e12 gupta et al./decis. mak. appl. manag. eng. 14 stock no. 31 32 33 34 35 36 37 38 39 40 normality_ test hind unilv r hindz inc icici bank infy ioc jswst eel kotak bank marico marut i mothe rsumi omnibus: 15.1 563 0.298 3 24.6 786 13.07 86 6.126 7 29.32 8 9.605 7 54.459 8 31.77 87 15.135 8 p value 0.00 0511 5 0.861 5 4.38 e-06 0.001 446 0.046 73 4.28e -07 0.008 206 1.49e12 1.26e -07 0.0005 168 stock no. 41 42 43 44 45 46 47 48 49 50 normality_ test nestl eind nhpc page ind pel petro net pfc pghh pidilti nd powe rgrid relian ce omnibus: 2.00 1 2.543 3 2.93 96 9.574 2 16.61 96 16.86 33 1.592 1 1.0069 0.548 7 32.094 6 p value: 0.36 77 0.280 4 0.23 0.008 337 0.000 2461 0.000 2179 0.451 1 0.6044 0.760 1 1.07e07 stock no. 51 52 53 54 55 56 57 58 normality_ test shre ece m sieme ns tech m titan ubl ultrac emco upl wipro omnibus: 3.90 65 0.864 3 5.17 05 11.17 55 8.913 7 1.821 4 13.66 24 6.0068 p value: 0.14 18 0.649 1 0.07 538 0.003 743 0.011 6 0.402 2 0.001 08 0.0496 2 findings from table 6 suggest that 36 stocks (in bold font) out of the 58 do not follow the normal distribution pattern and are thus non-parametric in nature. in this study we consider the stocks, those are deviated from the normal distribution as we adopt a non-parametric method for comparative ranking and use the bayesian classifier. further we find the financial beta value of each 36 stocks (see table 7). table 7. calculations of beta values stocks beta stocks beta stocks beta adaniports 1.61253 drreddy 0.08840 kotakbank 1.09241 ashokley 1.88335 godrejcp 1.04473 marico 0.24641 axiabank 1.88717 grasim 0.68448 maruti 1.75989 bajajfinsv 1.33048 hdfc 1.15587 mothersumi 1.58371 bajajhldng 0.93469 hdfcbank 0.82681 pel 1.06787 bafinance 1.20713 hindalco 1.56231 petronet 0.71448 bergepaint 1.1147 hindpetro 0.94296 pfc 1.32451 boschltd 1.41777 hindunilvr 0.5529 reliance 0.49102 britannia 0.49815 icicibank 1.66739 titan 1.24017 cadilahc 0.64286 infy 0.18335 ubl 0.90054 concor 1.16298 ioc 1.02975 upl 1.30440 divislab 0.06692 jswsteel 1.16933 wipro 0.17133 in this stage of filtration, we further consider the stocks having beta values ranging less than 1 as higher the beta value, higher is the systematic risk i.e., vulnerability to changes in the macro environment. therefore, after filtration we get 15 such stocks (highlighted in bold font) having beta values ranging from 0 to 1. in the final stage of the filtration we draw a 3×3 perceptual map (see figure 2). an integrated framework for classification and selection of stocks for portfolio… 15 figure 3. 3×3 investor perceptual map from the graphical representation of 15 stocks (figure 3) it is evident that 6 stocks are fallen under the quadrants of high return associated with low risk and medium return associated with low risk are good to invest for the common investor, as the propensity of the risk is low with respect to the other stocks. the six stocks are pointed bold in the table 8 along with their risk-return shown below. table 8. formation of the final sample of 6 stocks – risk and return values the 6 stocks (highlighted in bold fonts) are selected for the final sample for which we apply the integrated framework of entropy-topsis for year wise comparative assessment. we use the entropy method to calculate year wise weights of the criteria considered for comparing the stocks (kindly refer table 9). table 10 shows year wise ranking of the stocks using topsis method. sl.no. stocks sd mean 1 bajajhldng 0.09322 0.00808 2 britannia 0.1212 0.02769 3 cadilahc 0.08112 0.00699 4 divislab 0.0882 0.01658 5 drreddy 0.07601 0.00677 6 grasim 0.29701 0.00105 7 hdfcbank 0.11791 0.01207 8 hindpetro 0.25536 0.01307 9 hindunilvr 0.05798 0.02038 10 infy 0.07224 0.00683 11 marico 0.11756 0.0114 12 petronet 0.07332 0.01286 13 reliance 0.16348 0.0126 14 ubl 0.08744 0.00324 15 wipro 0.0707 0.00216 gupta et al./decis. mak. appl. manag. eng. 16 table 9. year wise criteria weights (entropy method) criteria entropy_weights 20132014 20142015 20152016 20162017 20172018 20182019 20192020 return 0.068 47 0.015 3 0.099 31 0.097 12 0.009 38 0.036 59 0.065 82 sharp ratio 0.185 26 0.032 22 0.204 63 0.161 62 0.027 61 0.234 21 0.119 28 treynor ratio 0.199 59 0.108 13 0.107 12 0.182 41 0.409 53 0.235 55 0.276 52 informatio n ratio 0.174 49 0.251 03 0.087 01 0.181 68 0.249 89 0.036 76 0.285 91 jensen ratio 0.179 79 0.523 42 0.146 76 0.203 11 0.205 39 0.228 05 0.161 97 calmar ratio 0.192 36 0.069 87 0.355 15 0.174 03 0.098 25 0.228 82 0.090 48 table 10. year wise ranking of the stocks (topsis method) stocks topsis_rank_year_on_year 20132014 20142015 20152016 20162017 20172018 20182019 20192020 britan nia 1 6 6 4 6 2 5 divisla b 5 5 5 6 5 1 1 hdfcba nk 2 2 1 1 2 5 2 hindun ilvr 3 4 3 5 1 3 4 marico 4 3 4 3 4 4 6 petron et 6 1 2 2 3 6 3 in the final stage of the study we find the overall expected rank and expected standard deviation of the stocks based on the prior outcome of topsis method as shown in the table 10.let 𝑌𝑡 be the discrete random variable of i th stock, where i consist with 1 to 6 ie, 𝑌𝑖1 = 𝐵𝑅𝐼𝑇𝐴𝑁𝑁𝐼𝐴, ……𝑌𝑖6 = 𝑃𝐸𝑇𝑅𝑂𝑁𝐸𝑇, and pt is the posterior probability of tth years, where t= 1 to 7, which is 𝑷𝒕 = (𝑷(𝑬𝒊) ∗ 𝑷(𝑨/𝑬𝒊)/𝑺𝒖𝒎(𝑷(𝑬𝒊) ∗ 𝑷(𝑨/𝑬𝒊))and a be an event the rank obtain using by topsis methods for each stocks in every year table 9. the probability of each stocks p(ei) = 1/7, and 𝑃(𝐴/𝐸𝑖 )the event where p is random probability of each stocks , a be the rank which is obtain from topsis and ei is the sum of the rank of stocks for each year i.e. 21, shown in the (table 11) for the 1st stock britannia. now we calculate the expected rank for each stocks as e(x)=∑ 𝑌𝑡 𝑃𝑡 , where, e(x) is the expectation of rank, 𝑌𝑡 is outcome from topsis of each stocks and 𝑃𝑡 is cross-ponding posterior probability of each stocks for 7 years, the rank shows (table 12) the minimum expectation possible when posterior probability is minimum, and selecting stocks as per the minimum expectation of rank, which investigates for portfolio selection, in this study. an integrated framework for classification and selection of stocks for portfolio… 17 table 11: posterior probability for britannia using bayes model britan nia brita nnia brit ann ia brit anni a brit anni a britann ia britannia p(ei) ran k p(a/ ei) p(ei)*p( a/ei) pi-> (p(ei)*p(a/ei))/su m(p(ei)*p(a/ei)) 0.14285 year 1 1 p1 0.05 0.00681 0.03333 0.14285 year 2 6 p2 0.29 0.04081 0.2 0.14285 year 3 6 p3 0.29 0.04081 0.2 0.14285 year 4 4 p4 0.19 0.02721 0.13333 0.14285 year 5 6 p5 0.29 0.04081 0.2 0.14285 year 6 2 p6 0.10 0.01360 0.06666 0.14285 year 7 5 p7 0.24 0.03401 0.16666 sum(p(ei )*p(a/ei) ) sum 0.20408 1 similarly we calculate the posterior probability of other 5 stocks (shown in the annexure 1). table 12: overall rank estimation using bayes model stocks expected_rank variance rank britannia 5.13333 11.54046 6 divislab 4.92857 10.55235 5 hdfcbank 2.86666 6.15214 1 hindunilvr 3.69565 6.8107 2 marico 4.21428 8.41918 3 petronet 4.30434 10.13146 4 the year on year performance i.e. from 2013 to 2020 of the stocks for constructing portfolio depends on topsis model (table 10), and finally the table 12 depicts the overall performance of stocks for portfolio construction by using bayes model, which will help the common investor to invest their capital with minimum and maximum portfolio weightages as rank-wise for short span (year wise) and long span (overall) on the basis of their perception. we check the year to year consistency of ranking of the stocks and notice that ranking order varies which is a common phenomenon in the stock market given the changes in the macroeconomic factors. therefore, we select the ranking order to table 12 as final to formulate the portfolio. gupta et al./decis. mak. appl. manag. eng. 18 5. validation and sensitivity analysis the results obtained using the mcdm models are vulnerable to changes in the given conditions such as criteria selection, inclusion and exclusion of the alternatives, change in the criteria weights, change in the alternative’s performance value among others (biswas, 2020; biswas et al., 2021a). hence, it is required to validate the result and perform the sensitivity analysis for examining the stability in result subject to changes in the given conditions (stević et al., 2020; mukhametzyanov and pamucar, 2018; pamucar et al., 2017).in this paper, for validation purpose, we carry out comparative ranking of the final six stocks using copras method (zavadskas et al., 2007) and compare the results with that obtained using topsis method for all the years under study as used in pamucar et al. (2021); sahu et al. (2021); varatharajulu et al. (2021); dehdasht et al. (2020), si et al. (2020) and biswas and anand (2020). table 13 indicates that ranking orders (among topsis and copras) are considerably consistent year wise that implies the validity of the results. table 13. comparison of topsis and copras year wise ranking (validation purpose) fy 2013-2014 2014-2015 2015-2016 2016-2017 method/ stock topsis copras topsis copras topsis copras topsis cop ras britannia 1 1 6 6 6 6 4 4 divislab 5 5 5 5 5 4 6 6 hdfcbank 2 2 2 2 1 1 1 1 hindunilvr 3 4 4 4 3 3 5 5 marico 4 3 3 3 4 5 3 3 petronet 6 6 1 1 2 2 2 2 fy 2017-2018 2018-2019 2019-2020 method/ stock topsis copras topsis copras topsis copras britannia 6 6 2 2 5 4 divislab 5 5 1 1 1 1 hdfcbank 2 2 5 5 2 2 hindunilvr 1 1 3 3 4 5 marico 4 4 4 4 6 6 petronet 3 3 6 6 3 3 we now move forward to carry out the sensitivity analysis. we follow the approach of biswas and anand (2020). table 14 shows the exchange of weights for 2013-2014 for the six criteria and other year’s calculations shown in annexure file and table 15 shows the result of the experimentation with exchange of weights among the pair of criteria in five occasions for each fy and figure 4 shows the graphical representation of sensitivity analysis. it is evident from the table 14 and subsequently from the figure 3 that the results are quite stable. an integrated framework for classification and selection of stocks for portfolio… 19 gupta et al./decis. mak. appl. manag. eng. 20 figure 4. graphical representation of sensitivity analysis table 14. exchange of weight for 2013-2014 for the six criteria weight_exchange ret sharp ratio treynor ratio inforamntion ratio jensen ratio clamar origin al_wei ght 0.06847 0.18526 0.19959 0.17449 0.17979 0.19236 t1 0.19959 0.18526 0.06847 0.17449 0.17979 0.19236 t2 0.06847 0.19959 0.18526 0.17449 0.17979 0.19236 t3 0.06847 0.18526 0.17449 0.19959 0.17979 0.19236 t4 0.06847 0.18526 0.17979 0.17449 0.19959 0.19236 t5 0.06847 0.18526 0.19236 0.17449 0.17979 0.19959 other year’s calculations shown in annexure file. an integrated framework for classification and selection of stocks for portfolio… 21 table 15. result of sensitivity analysis fy 2013-2014 2014-2015 2015-2016 stock origin al t 1 t2 t3 t4 t5 origi nal t 1 t2 t3 t4 t5 origi nal t 1 t2 t3 t4 t5 britan nia 1 1 1 1 1 1 6 6 2 6 2 5 6 6 4 5 4 6 divisl ab 5 2 4 4 4 5 5 5 6 5 6 4 5 4 2 4 2 2 hdfcb ank 2 4 2 2 2 2 2 1 3 3 3 2 1 2 5 3 3 3 hindu nilvr 3 3 3 3 3 3 4 3 5 4 5 3 3 3 6 6 6 4 maric o 4 5 5 5 5 4 3 2 4 2 4 1 4 5 3 2 5 5 petro net 6 6 6 6 6 6 1 4 1 1 1 6 2 1 1 1 1 1 fy 2016-2017 2017-2018 2018-2019 stock origin al t 1 t2 t3 t4 t5 origi nal t 1 t2 t3 t4 t5 origi nal t 1 t2 t3 t4 t5 britan nia 4 4 4 4 4 4 6 3 2 5 5 4 2 1 2 1 2 2 divisl ab 6 6 6 6 6 6 5 6 6 6 6 5 1 6 1 4 1 1 hdfcb ank 1 1 1 1 1 1 2 2 3 2 2 6 5 4 5 5 5 5 hindu nilvr 5 5 5 5 5 5 1 1 1 1 1 1 3 2 3 2 3 3 maric o 3 3 3 3 3 3 4 4 5 4 3 3 4 3 4 3 4 4 petro net 2 2 2 2 2 2 3 5 4 3 4 2 6 5 6 6 6 6 fy 2019-2020 stock origin al t 1 t2 t3 t4 t5 britan nia 5 5 5 5 5 4 divisl ab 1 1 1 1 2 1 hdfcb ank 2 2 2 2 1 3 hindu nilvr 4 3 4 4 4 2 maric o 6 6 6 6 6 6 petro net 3 4 3 3 3 5 6. research implications and conclusion this article focuses on the simple theorem of homogeneous beliefs of risk-free assets and normal returns for all investors, including those with influence functions, choose investment portfolios, and the combination of risk-free assets. the present work considers non-normally distributed returns and risk-free assets. this objective examines whether investors in influence, companies will choose an effective gupta et al./decis. mak. appl. manag. eng. 22 investment portfolio when returns are incompletely non-normally distributed and borrow or borrow at risk-free interest rates. also shows that the unlevered investment portfolio of investors is very close to the beta coefficient, regardless of how the portfolio is constructed; the degree of inter-temporal changes in the portfolio's beta coefficient will decrease as the number of securities in the portfolio increases. however, regardless of whether the investment portfolio is highly concentrated or widely diversified, there are significant differences in the portfolio’s speculation capital. after the filtration process on the basis of low risk and potential return, the framework implement topsis method aims to reshape the reality of the portfolio construction process. it is a flexible combined with various data analysis techniques to evaluate financial indicators in the form and check the best possible outcome type for stocks as a ranking wise for decision-making and construction in a multidimensional context. the bayesian portfolio model (one of the most widely used portfolio construction tools) and prior information about improving the efficiency of risk estimation expenditure and managing the uncertainty of the portfolio under the risk conditions of the prior assumptions of the portfolio optimization problem the integration of the potential model for the test dataset. the advantage of our model is as follows.  the mean and variance are only the first and second central moments of a random variable and are not sufficient to evaluate the entire distribution of the variable. however, the mean and variance do capture the most important information. therefore, in order to avoid complexity in calculating higher moments of the variable’s distribution, the mean and variance are the only parameters considered in forming the portfolios.  in finance point of view we have taken only the risk assessment parameter ie. sharpe ratio, treynor ratio, jensen alpha, information ratio, sortino ratio and calmar ratio, which are taken as a criteria in decision making model.  for the bayesian approach, we need the prior distribution of the stock returns and an updated data set. in this paper we are also derived from the returns of all stocks, consider as a prior information. the prior distribution could be estimated to apply the bayesian approach, the priors could be categorized into two cases: informative and uninformative. as we notice, in the uninformative case, i.e. not a lot of information is known about the prior distribution. some hidden parameters can also the affects the constructed model. 7. future scope people always want to make the optimal financial decision. however, many investors ignore the uncertainties of the parameters and models, which lead to a suboptimal portfolio at last. from this point of view, these models may be of some practical significance and enlightenment. besides, in the future work, we can try to take other informative priors information into consideration, try to expand the models to the multi-stage situation, or even try other frameworks instead of mean-variance framework, such as the utility function, safety-first framework, and so on. in addition, one of our limitation of our study is that as we are only concentrated on the technical parameter of the stocks and exclude the fundamental parameter like return on equity, return on capital employed, eva etc. this gap can be address in the future study. further work to extend and improve the methodology proposed in this paper should focus on four points: (a) methodologies in web-based decision-making information an integrated framework for classification and selection of stocks for portfolio… 23 systems to support investment decisions in real time ( b) choosing decision making weights from entropy to ahp c) taking into account a decision-making parameters such as quality of management decision and the company’s fundamental position in the market, set as a criteria in a qualitative direction, and (d) expand the focus of the methodology to include additional asset classes. further, topsis model sometimes may be suffering from rank reversal problem. hence, more checking is required. nevertheless, this paper shows a considerably unique approach of classification and ranking of stocks for portfolio selection which we hope to be of use to the individual investors and policy makers. author contributions: author contribution: conceptualization, s.g, g.b, s.b and a.m.; methodology, s.g and g.b.; software, s.g and a.m; validation, s.g., g.b. and s.b; formal analysis, s.g and g.b; investigation, s.g, s.b; resources, s.g, g.b.; data curation, s.b. and a.m.; writing—original draft preparation, s.b, s.g. and a.m.; writing—review and editing, s.g., g.b and s.b; visualization, s.g and a.m.; supervision, g.b; all authors have read and agreed to the published version of the manuscript. funding: this research received no external funding. conflicts of interest: the authors declare no conflicts of interest. references alali, f., & tolga, a. c. 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(2007). multi-attribute assessment of road design solutions by using the copras method. the baltic journal of road and bridge engineering, 2(4), 195-203. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 5, issue 1, 2022, pp. 208-224. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0315052022u * corresponding author. e-mail addresses: anilutku@munzur.edu.tr (a. utku), semakayapinar@munzur.edu.tr (s.k. kaya) multi-layer perceptron based transfer passenger flow prediction in istanbul transportation system anıl utku1 and sema kayapınar kaya2* 1 munzur university, department of computer engineering, tunceli, turkey 2 munzur university, department of industrial engineering, tunceli, turkey received: 1 april 2022; accepted: 15 may 2022; available online: 15 may 2022. original scientific paper abstract: predicting passenger movement in transportation networks is a critical aspect of public transportation systems. it allows for a greater understanding of traffic patterns, as well as efficient system evaluation and monitoring. it could also help with precise timing to emergencies or important events, as well as the improvement of urban transport system weaknesses and service quality. the number of transfer passengers demand in istanbul, turkey's biggest and most developed metropolis, was used to construct a realworld forecasting model in this study. the number of transfer passengers has been forecasted using popular machine learning methods such as knn (knearest neighbours), lr (linear regression), rf (random forest), svm (support vector machine), xgboost and mlp. the dataset utilized is made up of hourly passenger transfer counts gathered at two public transportation transfer stations in istanbul in january 2020. using mse, rmse, mae and r2 parameters, each model's experimental data have been thoroughly evaluated. mlp has more successfully other machine learning algorithms in the majority of transportation lines, according to the experimental results. key words: machine learning, passenger flow management, transfer data. 1. introduction in 2020, the city, which straddles the bosporus and is located in both europe and asia, have a population of over 15 million people, contributing for 20 percent of turkey's total population. (tuik, 2021). in according to world demographics data, istanbul is the most crowded city in europe and the world's fifteenth most densely populated metropolis (statista, 2020). the number of passengers utilizing public transportation is significantly higher as a result of the high density of the population (pamucar et al., 2020). while nearly 11 million 500 thousand people use public multi-layer perceptron based transfer passenger flow prediction in istanbul transportation… 209 transportation in istanbul every day, passengers who prefer highway transportation (metrobus, public urban transportation, private bus, etc.) account for nearly 84 %, followed by railway transportation (metro, light metro, tram, etc.) at around %14, and sealine transportation at just under 2% (iett, 2021). despite the constant increase in the number of people and vehicles, the fact that the proportion of cars per thousand people is also constantly rising is important in terms of showing the increased traffic density in istanbul. according to the traffic index published by tomtom, europe's largest navigation systems company, istanbul is the fifth city with the highest traffic density of 51% in the world (tomtom traffic index, 2021). with the increase of urban transportation challenges, forecasting the number of people entering and departing istanbul's transit terminals has become more challenging. passenger flow forecasting provides a better understanding of travel patterns, efficient monitoring and evaluation of the system status of istanbul transportation system. it may also help in the prompt response to crises or special events, as well as the correction of defects and enhancement of public transportation service quality. several predicting methodologies have been proposed to enhance the effectiveness of passenger forecasting models, encompassing mathematical modelling methods, statistical methods, and non-parametric methodology. the machine learning-based (ml) framework is one of the most well-known non-parametric approaches today (boukerche and wang, 2020). it's a subset of ai that integrates the problem of learning from data samples with the concept of reasoning in generally (boukerche & wa ng, 2020). it's a subfield of ai that relates the difficulty of learning from sample data with the concept of reasoning in overall (tom mitchell, 2006). there are two stages to any learning process: (i) particular a given dataset, calculation of unknown relationships in a system particular a given dataset, calculation of unknown relationships in a system (ii) predicted connections are used to forecast new platform outputs. machine learning has also been shown to be an interesting topic of study in passenger demand prediction, with several applications (liu et al., 2020, zheng et al., 2021, wang et al., 2021, hayadi et al., 2021; gummadi and edara (2018); ye et al. (2019); messinis and vosniakos, (2020); liu et al. (2020); hayadi et al. (2021); guo et al. (2021); wang et al. (2021); bozanic et al. (2021); yang et al.( 2021); ge et al. (2021); kamandanipour et al. (2022); müller-hannemann et al. (2022). the ability to predict passenger traffic in transportation networks is critical to public transportation management. it helps to improve transportation services, provide early warnings for unusual traffic situations, and make cities smarter and safer. furthermore, transfer passenger flow prediction can improve the transfer operation efficiency reduce the transfer waiting time and enhancing passengers’ satisfaction. to address this problem, the transfer passenger flow transferring a various modes transportation (metro & tram, bus & metrobus, rail, and ferries & sea-bus) in istanbul has been developed for the first time in the literature. the followings are some of the study's contributions: i. this paper offers a clear theoretical foundation and decision support for the practical work of using intelligent technologies to optimize the predictive performance of the number of passengers moving between different modes, including "metro and tram," "bus and metrobus," "rail," and "ferries and sea-bus." ii. as istanbul has a very heavy traffic; the number of lines can be increased or decreased according to the number of passengers. accurate transfer passenger volume is the fundamental of transportation scheduling in istanbul. iii. this enhances the service standards of an urban public transportation system and exposes passengers with real-time transfer passenger demand information across several routes, allowing them to make greater decision to travel. utku and kaya /decis. mak. appl. manag. eng. 5 (1) (2022) 208-224 210 iv. prediction transfer passenger flow assists istanbul iett authorities and management in increasing public transit reliability of the system, improving passenger experience, and maximizing routing plans. the motivation of this paper is the prediction of the number passengers transferring in various lines in istanbul recorded at 1-hour intervals between 1-31 january 2020. istanbul is turkey's biggest and most developed metropolis therefore, the dataset utilized comprises of passenger transfer numbers on several transportation lines in istanbul. the goal of this research is to anticipate the number of transfer passengers in istanbul. the number of transfer passengers was determined based on passenger data gathered during one-hour intervals. the istanbul public urban transport company (iett), private public bus (oho), motor/boat, and the iett tunnel will have been subjected to empirical investigations. time information such as hourly, daily, and weekly has been revealed in this fashion on certain lines. the goal of this research is to use machine learning techniques to predict the amount of transfer passengers on transportation lines using a different knn, lr, rf, svm, xgboost and mlp methods. 2. literature review with the development of big data technology, using machine learning algorithms to detect the principles of urban passenger movement has become one of the research hotspots in the field of public transportation. in recent decades, there has been a huge amount of work on passenger flow and forecasting using statistical methodological approaches notably machine learning. xie et al. (2014) employed a combination of seasonal decomposition (sd) and least squares support vector regression (lssvr) methods to forecast air passenger volume for a short amount of time. sun et al. (2015) proposed a hybrid wavelet and support vector machine (svm) methods that consist of three significant levels to predict the number of people entering and leaving the subway in beijing. roos et al. (2017) proposed a predicting technique, which is based on dynamic bayesian network (bn) built to function even passenger flow data is missing or uncertain. ni et al. (2017) created a combination time series model based on seasonal arima and loss function (lf), using data from the twitter social media platform to monitor subway passengers. toqué et al. (2017) addressed a passenger flow predicting in multimodal transport using ml methods such as random forest (rf) and long-short term memory (lstm) neural networks. zhang et al. (2017) predicted the short-term passenger data taken from gps device and smart card system in favour of the two-step real time prediction (2rtp) approach based on the extended kalman filtering (kf) method. milenkovıć et al. (2018) estimated an arima analysis to simulate the monthly number of train passengers while considering seasonal variations into consideration. gummadi and edara (2018) employed the arima and seasonal arima to estimate bus passenger flow in india's transport industry over a short period of time. ye et al. (2019) aimed to predict the daily bus passenger traffic amount using the arima method and examined the outcomes of predictions in the case of complete weekday non-peak data collected from january to march 2018. li et al. (2020) predicted shared passenger demand in various locations with a hybrid algorithm based on wt-fcbflstm (wavelet transform, fast correlation-based filter, and long short-term memory). liu et al. (2020) focused on a short-term estimation model for local bus passenger flow using svm. hayadi et al. (2021) proposed a random forest (rf) using multi-layer perceptron based transfer passenger flow prediction in istanbul transportation… 211 the location data from the gps devices in the buses, the location of the bus stop used for operation management, and the volume of traffic estimated by an image processing method. li et al. (2021) adopted the seasonal arima and svm to predict the periodic flow of railway passenger. guo et al. (2021) proposed a regression tree combined with copula-based simulations employing passenger level data to generate real-time distributional estimates of travels in an airport. rajendran et al. (2021) developed a logistic regression (lr), artificial neural networks (ann), rf, and gradient boosting (gb) for assessing air taxi demand considering various factors such as temperature, weather conditions and visibility. zheng et al. (2021) designed an integrated lr, a fully connected neural network (nn) and lstm model for anticipating a metro station’s abnormally substantial passenger movement. rodríguez-sanz et al. (2021) presented two rf algorithms that allow for the integration of flight data and passenger judgement for predicting the duration of queues at check-in counters and the security control area at parma de mallorca airport in spain. wang et al. (2021) established a lightgbm method to estimate railway high passenger parameters like railway specifications, past weather trends, and public transport time sequence. the lightgbm methodology outscored the xgboost, rf, and arima algorithms, as according their findings. yang et al.( 2021) proposed a prediction model based on transit passenger flow using the wavelet analysis (wa) and lstm combination model for the short-term period. abeyrathna et al. (2021) investigated the relationship between the regression tsetlin (rt) machine algorithm and pandemic events such as daily covid-19 cases and deaths, pandemic control measures to estimate the number of transport passengers under different scenarios. jackson et al. (2021) benefited from various bayesian network (bn) models for predicting bus schedule time. ge et al. (2021) implemented a combination of differentially arima and svm to achieve a highly predictive model for passenger flow in shanghai-guangzhou railway station. kamandanipour et al. (2022) presented a multi-layer ann system to forecast the strength of demand caused by seasonal conditions using train ticket service data. müller-hannemann et al. (2022) investigated a new technique of approximating scenario-based resilience employing xgboost, catboost, svr and ann models which are based on carefully selected important aspects of public transport systems. wood et al. (2022) analysed its use of traditional lr analysis and a rf model to unveil future passenger occupancies on a bus when it reaches at next stops using real-time data from bus operating and meteorological data. reitmann and schultz (2022) developed the gradient boosting (xgboost) algorithm and the point-of-interest (poi) model, helping in the reduction of the passenger flow forecast model's total training time, to forecast bus passenger flow in beijing. comparisons of these models are listed in table 1 in detail. table 1. literature review of passenger flow prediction author (year) models passenger type period xie et al. (2014) sd, lssvr air short sun et al. (2015) wavelet, svm subway short roos et al. (2017) bn metro short ni et al. (2017) seasonal arima, lf subway short toqué et al. (2017) rf, lstm multi model long zhang et al. (2017) 2rtp, kf bus short utku and kaya /decis. mak. appl. manag. eng. 5 (1) (2022) 208-224 212 author (year) models passenger type period milenkovıć et al. (2018) arima railway short li et al. (2020) wt-fcbf-lstm railway long liu et al. (2020b) svm bus short li et al. (2021) seasonal arima, svm urban short guo et al. (2021) rt urban short rajendran et al. (2021) lr, ann, rf, gb taxi urban short zheng et al. (2021) nn, lstm, lr metro short rodríguez-sanz et al. (2021) rf airport long wang et al. (2021) lightgbm, xgboost, rf, arima railway long yang et al. (2021) wa, lstm transit short abeyrathna et al. (2021) rt public transport short jackson et al. (2021) bn bus short ge et al. (2021) arima, svm railway long kamandanipour et al. (2022) ann railway short müller-hannemann et al. (2022) xgboost, catboost, svr and ann public transport wood et al. (2022) lr, rf bus short reitmann and schultz (2022) xgboost, poi bus short 3. machine learning-based passenger flow prediction the amount of immediate data produced by urban transportation systems is also expanding, thanks to the growth of big data, internet of things, sensor networks, and cloud computing applications. in topics like safety management, emergency response efficiency, and urban traffic management, passenger flow forecast in urban transportation networks is critical. passenger flow planning is important for concerns including scheduling, traffic planning, and passenger flow control. the goal of this research is to anticipate the number of transfer passengers in istanbul, turkey's largest and most developed metropolis, using passenger flow data. the dataset utilized comprises of passenger transfer numbers on various transportation lines in istanbul, such as transfers and normal boarding, recorded for one month between january 1, 2020 and january 31, 2020. the objective of this research is to use machine learning algorithms to forecast the amount of transfer passengers on transportation lines. in practice, knn (k-nearest neighbors), lr (linear regression), rf (random forest), svm (support vector machine), xgboost (extreme gradient boosting), and mlp (multi-layer perceptron) have been examined then, each model's experimental findings have been thoroughly examined using mse, rmse, mae, and r2 metrics. 3.1. original data analysis in this study, a dataset consisting of the transfer numbers of passengers such as transfer and normal boarding in different transportation lines in istanbul recorded at 1-hour intervals between 1-31 january 2020 by istanbul metropolitan municipality has been used. the dataset used consists of 23163 rows of transportation data. the multi-layer perceptron based transfer passenger flow prediction in istanbul transportation… 213 dataset contains id, date_time, transport_type_id, transport_type_desc, line, transfer_type_id, transfer_type, number_of_passenger parameters. in this study, iett, öho, motor/boat and iett tunnel transfer lines have been selected for prediction because they have the highest number of transfer passengers. iett transfer line refers to all bus lines offered by the istanbul metropolitan municipality. öho transfer line refers to all bus lines offered by private public bus companies. motor/boat, on the other hand, refers to all sea transportation made by marine vehicles. iett tunnel refers to all transfers made using the underground metro. table 2 shows the first 10 rows of the dataset used as an example. table 2. a sample from the dataset date_time line transfer_type number_of_passenger 1.01.2020 00:00 motor_tekne normal 1393 1.01.2020 00:00 kabataş_bağcılar normal 4310 1.01.2020 00:00 aksaray_airport normal 2936 1.01.2020 00:00 kabataş_bağcılar transfer 1586 1.01.2020 00:00 kadıköy-kartal metro transfer 677 1.01.2020 00:00 kirazlı-olimpiyatköy transfer 10 1.01.2020 00:00 edirnekapı-sultançiftliği normal 793 1.01.2020 00:00 şehir hatları transfer 59 1.01.2020 00:00 taksim-4.levent normal 8119 3.2. methodology in this study, popular machine learning algorithms commonly used in the literature such as knn, lr, rf, svm, xgboost and mlp have been applied. the dataset has been pre-processed before applying to the models. possible blank or incorrect fields in the data have been checked. after the data pre-processing step, training, validation, and test datasets have been selected. 80% of the dataset is split into training and 20% testing. 10% of the training data have been split for validation. validation data has been used for the optimization of model parameters. time series data refers to series of numbers ordered according to a time index. time series data refers to series of numbers ordered according to a time index. in supervised learning problems, it is aimed to estimate the output from the inputs by using a function like y=f(x). time series data can be transformed into supervised learning problem for use in time series analysis. the time series data can be transformed into a supervised learning problem by using the values from the previous time step to predict the value in the next time step as seen in figure 1. utku and kaya /decis. mak. appl. manag. eng. 5 (1) (2022) 208-224 214 figure 1. converting time series data to supervised learning problem in this study, time series data has been converted to supervised learning problem by using the sliding window method as seen in figure 1. the number of previous timestamps determines the size of the sliding window. in this study, the size of the sliding window has been determined as 3 as a result of the experimental studies. in order to optimize the parameters of the machine learning algorithms used, 10% of the training data has been used for validation. by using the optimized parameters, algorithms have been applied and prediction values have been obtained. the pseudo code of the developed system is presented below: input: passenger transfer data on iett, oho, motor/boat and iett tunnel lines output: predicted passenger numbers 1: start. 2: checking the missing and incorrect areas in the data (data pre-processing). 3: splitting training, validation and test sets and normalizing the data. 4: optimizing model parameters using validation data. 5: walk forward validation. 6: have the parameters with the lowest mse value been selected? if yes go to step 7, if no go to step 4. 7: creation of the model. 8: making predictions using the created model. 9: calculation of mse, rmse, mae and r2 values according to the prediction results. 10: finish. 3.3. developed model in this study, a comparative analysis of the passenger number estimation problem of the mlp-based model developed with popular machine learning algorithms is presented. mlp is a neural network model inspired by the neuron structure in the brain. mlp is a combination of perceptron’s that bind in different ways and operate in different activation functions. it consists of input nodes, hidden nodes and output multi-layer perceptron based transfer passenger flow prediction in istanbul transportation… 215 nodes. input nodes provide input information to the network. no computation is performed on any of the input nodes. this only relay information to hidden nodes. hidden nodes are structures that are not directly connected to the outside world, perform calculations and transmit information from input nodes to output nodes. a hidden layer is created with a collection of hidden nodes. while a network has only a single input layer and a single output layer, it can have zero or multiple hidden layers. mlp has one or more hidden layers. output nodes, on the other hand, are responsible for information processing and transferring information from the network to the outside world. the developed mlp model takes the passenger flow data in the training dataset as input and predicts the passenger numbers in the test dataset. according to the obtained result, the training process has been continued. the architecture of the developed model is shown in figure 2. figure 2. the architecture of the developed model in the developed mlp-based model, there are an input layer, three hidden layers and an output layer as seen in figure 6. hidden layers represent an intermediate processing step that is combined using weighted sums to obtain the classification result. the developed model is a sequential model with linear layers. there is a dropout layer between the input layer and the hidden layer. in the output layer, there are two output units that return the prediction of the probability of customer loss. relu activation function is used in the input layer and hidden layers, and the sigmoid activation function is used in the output layer. 3.4. experimental results in this study, a dataset consisting of the transfer numbers of passengers such as 1month transfer and normal boarding in different transportation lines in istanbul recorded at 1-hour intervals for 2020 has been used. iett, öho, motor/boat and iett tunnel transfer lines with the highest transfer numbers have been selected for prediction. knn, lr, rf, svm, xgboost and mlp algorithms, which are widely used in the literature, have been applied to the dataset. for each algorithm, the experimental results obtained using mse, rmse, mae and r2 metrics have been compared. utku and kaya /decis. mak. appl. manag. eng. 5 (1) (2022) 208-224 216 the iett transfer line covers all bus lines offered by the istanbul metropolitan municipality. iett transfer line consists of passenger flow information transferring in 687 different time zones. 80% of this data is split for training and 20% for testing. after the train/test split, 6070 rows of data have been used in the training and 1518 rows of data have been used in the testing. figure 3 shows the change in the number of transfer passengers on the iett line over time. table 3 show the average mse, rmse, mae and r2 results obtained for each algorithm for iett line. figure 3. change over time in the number of transfer passengers on the iett line table 3. experimental results for each model according to the mse, rmse, mae and r2 for iett line model mse rmse mae r2 knn 259682.920 509.590 323.066 0.958 lr 635367.832 797.091 653.525 0.906 rf 365886.694 604.883 354.582 0.944 svm 237559.150 487.400 317.741 0.946 xgboost 392332.530 626.364 411.415 0.942 mlp 227419.633 476.885 315.104 0.961 the experimental results show that the mse values of knn, lr, rf, svm, xgboost and mlp are 259682.920, 635367.832, 365886.694, 237559.150, 392332.530, 227419.633, respectively. the rmse values of knn, lr, rf and svm are 509.590, 797.091, 604.883, 487.400, 626.364, 476.885, respectively. the mae values of knn, lr, rf and svm are 323.066, 653.525, 354.582, 317.741, 411.415, 315.104, respectively. the r2 values of knn, lr, rf and svm are 0.958, 0.906, 0.944, 0.946, 0.942, 0.961, respectively. the öho line covers all passenger transfers offered by private public bus companies. öho transfer line consists of passenger flow information transferring in 716 different timestamps. 80% of this data is split for training and 20% for testing. multi-layer perceptron based transfer passenger flow prediction in istanbul transportation… 217 after the train/test split, 572 rows of data have been used in the training and 144 rows of data have been used in the testing. figure 4 shows the change in the number of transfer passengers on the öho line over time. table 4 show the average mse, rmse, mae and r2 results obtained for each algorithm for öho line. figure 4. change over time in the number of transfer passengers on the öho line table 4. experimental results for each model according to the mse, rmse, mae and r2 for öho line model mse rmse mae r2 knn 1335463.741 1155.624 712.375 0.965 lr 2050366.100 1431.909 1117.355 0.949 rf 1332583.670 1154.375 710.408 0.967 svm 1640037.728 1280.639 943.779 0.959 xgboost 2236156.800 1495.378 931.525 0.944 mlp 1252185.815 1119.011 692.050 0.969 the experimental results show that the mse values of knn, lr, rf and svm are 1335463.741, 2050366.100, 1332583.670, 1640037.728, 2236156.800, 1252185.815, respectively. the rmse values of knn, lr, rf and svm are 1155.624, 1431.909, 1154.375, 1280.639, 1495.378, 1119.011, respectively. the mae values of knn, lr, rf and svm are 712.375, 1117.355, 710.408, 943.779, 931.525, 692.050, respectively. the r2 values of knn, lr, rf and svm are 0.965, 0.949, 0.967, 0.959, 0.944, 0.969, respectively. motor/boat transfer line refers to all transfers made by sea vehicles that provide sea transportation. motor/boat transfer line consists of passenger flow information transferring in 618 different timestamps. 80% of this data is split for training and 20% for testing. after the train/test split, 494 rows of data have been used in the training and 124 rows of data have been used in the testing. figure 5 shows the change in the number of transfer passengers on the motor/boat line over time. table 5 show the utku and kaya /decis. mak. appl. manag. eng. 5 (1) (2022) 208-224 218 average mse, rmse, mae and r2 results obtained for each algorithm for motor/boat line. figure 5. change over time in the number of transfer passengers on the motor/boat line table 5. experimental results for each model according to the mse, rmse, mae and r2 for motor/boat line model mse rmse mae r2 knn 57453.940 239.695 160.711 0.884 lr 48366.810 219.924 168.390 0.903 rf 55962.547 236.565 159.844 0.887 svm 34556.885 185.894 136.343 0.93 xgboost 45494.010 213.293 144.571 0.907 mlp 30629.115 175.012 125.101 0.938 the experimental results show that the mse values of knn, lr, rf and svm are 57453.940, 48366.810, 55962.547, 34556.885, 45494.010, 30629.115, respectively. the rmse values of knn, lr, rf and svm are 239.695, 219.924, 236.565, 185.894, 213.293, 175.012, respectively. the mae values of knn, lr, rf and svm are 160.711, 168.390, 159.844, 136.343, 144.571, 125.101, respectively. the r2 values of knn, lr, rf and svm are 0.884, 0.903, 0.887, 0.930, 0.907, 0.938, respectively. iett tunnel transfer line refers to all transfers made using the underground metro. iett tunnel transfer line consists of passenger flow information transferring in 502 different timestamps. 80% of this data is split for training and 20% for testing. after the train/test split, 401 rows of data have been used in the training and 101 rows of data have been used in the testing. figure 6 shows the change in the number of transfer passengers on the iett tunnel line over time. multi-layer perceptron based transfer passenger flow prediction in istanbul transportation… 219 figure 6. change over time in the number of transfer passengers on the iett tunnel line table 6 show the average mse, rmse, mae and r2 results obtained for each algorithm for iett tunnel line. table 6. experimental results for each model for iett tunnel line model mse rmse mae r2 knn 1909.902 43.702 34.096 0.879 lr 2619.355 51.179 40.426 0.835 rf 2070.691 45.504 34.879 0.869 svm 2336.181 48.334 36.738 0.852 xgboost 2832.245 53.218 40.686 0.836 mlp 1904.229 43.637 32.560 0.88 the experimental results show that the mse values of knn, lr, rf and svm are 1909.902, 2619.355, 2070.691, 2336.181, 2832.245, 1904.229, respectively. the rmse values of knn, lr, rf and svm are 43.702, 51.179, 45.504, 48.334, 53.218, 43.637, respectively. the mae values of knn, lr, rf and svm are 34.096, 40.426, 34.879, 36.738, 40.686, 32.560, respectively. the r2 values of knn, lr, rf and svm are 0.879, 0.835, 0.869, 0.852, 0.836, 0.880, respectively. the prediction results of the developed mlp-based model are shown in figure 7. utku and kaya /decis. mak. appl. manag. eng. 5 (1) (2022) 208-224 220 figure 7. prediction results of developed mlp-based model the prediction results of the developed model for the iett line in figure 7.a, the öho line in figure 7.b, the motor/boat line in figure 7.c and the iett tunnel line in figure 7.d are shown. as can be seen in the figure 7, the mlp-based model successfully predicted the patterns in the training and test data. 4. conclusions and future studies in this study, a comparative analysis of popular machine learning algorithms such as knn, lr, rf, svm, xgboost and mlp for passenger flow prediction is presented. the experimental results for iett, öho, motor/boat and iett tunnel lines have been extensively tested using mse, rmse, mae and r2. for the iett line, the mlp has more successful than the other models compared. after mlp, svm, knn, rf, xgboost and lr have been successful, respectively. for the öho line, the mlp has more successful than the other models compared. after mlp, rf, knn, svm, lr and xgboost have been successful, respectively. for the motor/boat line, the mlp has more successful than the other models compared. after mlp, svm, xgboost, lr, rf and knn have been successful, respectively. for the iett tunnel line, the mlp has more successful than the other models compared. after mlp, knn, rf, svm, xgboost and lr have been successful, respectively. experimental results show that these machine learning methods can be used in passenger flow prediction problems. among the compared algorithms, mlp achieved successful results in all of the transportation lines. mlp is a neural network model developed based on biological neural network structures. the mlp consists of interconnected processing units, similar to the functioning of neurons. mlp's ability to detect non-linear, linear or non-linear distributed data makes it perform well on most datasets. xgboost is a machine learning model that uses a gradient boosting multi-layer perceptron based transfer passenger flow prediction in istanbul transportation… 221 framework. xgboost is a decision-tree and gradient-boosting based machine learning model. it works successfully on non-structured data such as images, text and audio. knn may be inefficient in terms of performance on small datasets. svm is successful when having a limited set of points. svm is good at outliers as it will only use the most relevant points to find support vectors. for this reason, svm have successful results in this study. lr is expected to be successful when the dataset is truly linear, especially when there are many features with a very low signal-to-noise ratio. however, rf may fail to model linear combinations of many features. all methods compared in this study had successful results. all methods had r 2 values above 0.90 for the iett line, above 0.94 for the öho line, above 0.88 for the motor/boat line, and above 0.84 for the iett tunnel line. experimental results showed that the developed mlp-based model gives better results than the compared models for all transfer lines used in the prediction of the number of passengers. the prediction of the number of passengers is an important factor for the urbanization phenomenon and city management. transportation planning is also important in terms of avoiding disruptions in transportation and reducing the traffic load. the developed model can be applied to real-world problems by using effective passenger predicting in the field of transportation planning. in future studies, longer-term predictions can be made using passenger data over a larger time period. in addition, the results can be evaluated by applying different models such as deep learning. in this study, traditional machine learning methods and mlp, which is a neural network-based model, are compared in practice. here, it is aimed to benefit from the prominent features of neural networks in the time series prediction problem. the ability of a neural network to process data in detail stems from its ability to reveal hidden patterns between input and output data. an important advantage of neural networks is that they have the ability to learn and generalize information. mlp is tolerant of missing values and can model complex relationships such as nonlinear trends. it can also support multiple inputs. one of the important limitations of this study is that it only considers the number of transfer passenger volume prediction. for this reason, different external factors such as transfer time, rush hours and holiday days could be examined for passenger prediction model in the future. secondly, ml algorithms such as knn, lr, rf, svm, xgboost and mlp methods was employed during the short-term prediction process. in the further study, a state of art deep neural network algorithm could be developed to improve the prediction result for the number of transferring passengers. author contributions: conceptualization, software, methodology, validation, writing, visualization, editing, a.u.; review, writing, original draft preparation, resources, editing, s.k.k. all authors have read and agreed to the published version of the manuscript. funding: this research received no external funding. acknowledgement: the authors would like to express their gratitude to the editors and anonymous referees for their informative, helpful remarks and suggestions to improve this paper as well as the important guiding significance to us researches. data availability statement: in this section, please provide details regarding where data supporting reported results can be found, including links to publicly archived datasets analysed or generated during the study. you might choose to exclude this statement if the study did not report any data. utku and kaya /decis. mak. appl. manag. eng. 5 (1) (2022) 208-224 222 conflicts of interest: the authors declare no conflicts of interest. references abeyrathna, k. d., rasca, s., markvica, k., & granmo, o.c. 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(2021). hybrid model for predicting anomalous large passenger flow in urban metros. iet intelligent transport systems, 14(14), 1987–1996. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). https://data.tuik.gov.tr/bulten/index?p=adrese-dayal%c4%b1-n%c3%bcfus-kay%c4%b1t-sistemi-sonu%c3%a7lar%c4%b1-2020-37210&dil=1 https://data.tuik.gov.tr/bulten/index?p=adrese-dayal%c4%b1-n%c3%bcfus-kay%c4%b1t-sistemi-sonu%c3%a7lar%c4%b1-2020-37210&dil=1 https://data.tuik.gov.tr/bulten/index?p=adrese-dayal%c4%b1-n%c3%bcfus-kay%c4%b1t-sistemi-sonu%c3%a7lar%c4%b1-2020-37210&dil=1 https://doi.org/10.1155/2021/9963394 plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 5, issue 2, 2022, pp. 316-328. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0318062022c * corresponding author. e-mail address: sudiptohaki@hotmail.com (s. chaki), dbose@nitttrkol.ac.in (d. bose) optimisation of spot-welding process using taguchi based cuckoo search algorithm sudipto chaki1* and dipankar bose2 1automobile engineering department, mckv institute of engineering, india, 2 mechanical engineering department, national institute of technical teachers training and research, india received: 2 february 2022; accepted: 8 june 2022; available online: 18 june 2022. original scientific paper abstract: the present work evaluated the efficiency of a taguchi-based cuckoo search (cs) algorithm for optimizing the spot-welding process. the l9 orthogonal array of the taguchi method is used for the conduction of required experiments. during the study input parameters are welding current (ka), hold time (cycles), welding time (cycles), and electrode pressure (kpa) while peak load, kn has been considered as an output parameter. the required objective function is developed through regression model formulation. initially, cs operating parameters such as maximum number of iterations, number of nests, and probability to discover a cuckoo egg by host bird is optimized through the taguchi method and are found as 40, 20, 0.5 respectively. that optimized cs further optimizes the spot-welding process. the maximum peak load of 34.5 kn is obtained if the welding current is 30.7 ka, welding time is 32 cycles, hold time is 20 cycles, and electrode pressure is 480 kpa respectively. experimental validation yields a very low % error of 1.93% during optimization with an optimized cs method and while the error is substantially high if optimization is conducted using a non-optimized cs method key words: spot welding, cuckoo search method, taguchi method, optimization, peak load. 1. introduction spot welding is a popular electric resistance welding process in which workpieces are held together under pressure exerted by electrodes and metal surface points in contact are get joined by the heat obtained from resistance to electric current. it is a mailto:sudiptohaki@hotmail.com mailto:dbose@nitttrkol.ac.in optimization of spot-welding process using taguchi based cuckoo search algorithm 317 quick process and enables the easier joining of even dissimilar metals. the process is widely applied in the automotive sector. the controllable parameters of spot welding such as electrode tip diameter, weld nugget, weld current, holding time, cycle time, etc, bear a very complex relationship between them. in the recent past, researchers employed the conventional statistical design of experiment (doe) based techniques like the taguchi method, response surface methods (rsm), grey relational analysis (gra), etc to optimize spot-welding process parameters. attempts to optimize tensile strength for spot-welded joints of titanium alloy [bozkurt and çakır (2021), fatmahardi et al (2021)], aluminum alloys [raheem, (2021)], dissimilar materials combining steels and composites [neystani et al. (2019)], steels [tyagi et al.(2022), kumar et al. (2021), ebrahimpour et al (2021)], have been computed through the taguchi method [bozkurt and çakır (2021), fatmahardi et al (2021), neystani et al. (2019), tyagi et al.(2022), kumar et al. (2021)] and rsm [raheem, (2021), ebrahimpour et al. (2021)]. but such processes generally involve high mathematical complexity among operating parameters and possible experimental noise. therefore, the application of those traditional statistical optimization methods is often found inadequate for their inability to reach global minima for the multimodal problem. soft computing techniques involving genetic algorithm (ga), simulated annealing (sa), particle swarm optimization (pso), etc can be efficiently employed for optimizing such highly non-linear problems with improved accuracy. the same has been employed already for electric discharge machining (edm) [chinmaya et al. (2017)], laser material processing [chaki et al. (2012)], submerged arc welding, etc [choudhary et. al. (2020)]. however, limited applications of such algorithms are found in the spot-welding domain. recently, the tensile strength of spot-welded joints for mild steel was optimized by pattanaik et al. (2018) using l25 taguchi-based gra and ga. cao et al. (2021) optimized spot-welded joint strength of aluminum to steel using a ga and rsm. dhawale et al. (2019) optimized the strength of spot-welded joints using pso and experimental validation incurred negligible error with 0.57% error. zhao et al. (2014) optimized failure energy of spot-welded titanium alloy joints using box–behnken l17 experimental design of rsm and artificial fish swarm algorithm. the literature survey indicates applications of several conventional statistical methods including taguchi, rsm, etc to optimize the resistance spot welding processes [bozkurt and çakır (2021), fatmahardi et al (2021), neystani et al. (2019), tyagi et al. (2021), kumar et al. (2021)] raheem, (2021), ebrahimpour et al. (2021)]. however, those optimization processes have inherent limitations to stuck into local minima often resulting in faulty optimized output for multimodal problems. however, further nature-inspired algorithms like ga, pso, sa etc. are used for solving spot welding [pattanaik et al. (2018), cao et al. (2021), dhawale et al. (2019), zhao et al. (2014)] and similar problems [chinmaya et al. (2017), chaki et al. (2012), choudhury et al. (2020)] with greater accuracy where process parameters bear complex mathematical relationships among them. but, the optimization performance of those algorithms is governed by several operating parameters, whose selection is generally user-dependent. its wrong selection often leads to inadequate optimization output. therefore, prior optimization of those controlling parameters is necessary and if operated through an optimized parameter setting then only the best-optimized output from the algorithm may be obtained. however, such optimized nature-inspired algorithms are not employed so far for the spot-welding process. cuckoo search (cs) [yang & deb (2010)], a biologically inspired method for optimization is controlled by certain parameters like the number of nests, iterations, the probability with which the host bird discovers the cuckoo egg in nests, etc. cs is employed in the field of time chaki and bose /decis. mak. appl. manag. eng. 5 (2) (2022) 316-328 318 series and forecasting [jiang et al. (2016), xiao et al. (2017)], data mining [swathypriyadharsini & premalatha (2021)], non-traditional machining [saravanan et al. (2020), shastri & mohanty (2021)], scheduling [gu et al. (2021)] etc. present work employed an optimized cs algorithm to optimize the spot-welding process. the taguchi method optimized the cs algorithm during the study. such taguchi optimized cs is not employed earlier for optimizing the spot-welding process. further, the performance of that optimized cs is compared with a non-optimized cs to understand the usefulness of the proposed method. 2. methodology a taguchi integrated cuckoo search (cs) optimization method is employed for optimizing the rsw process using an optimized cs. the controllable input parameters in the rsw process are welding current, ka (i) 2) welding time, cycles (wt), 3) hold time, cycles (ht), and 4) electrode pressure, kpa (ep) while tensile strength is the output parameter. further optimization of spot welding is computed through the following steps: i) conduction rsw experiments using taguchi l9 experimental design matrix to generate an input-output dataset. ii) objective function formulation through developing suitable regression equation iii) optimization of cs parameters through taguchi method for determining an optimized cs algorithm iv) optimizing the spot-welding process using that optimized cs algorithm and determining the model efficiency through experimental validation. the outline of the process has been given in fig.1. detailed steps for taguchi integrated cuckoo search (cs) optimization method are given as follows: step a: development of the experimental dataset i. generate taguchi l9 design matrix ii. conduct experiments following experimental design iii. measure output characteristics iv. generate the input-output dataset step b: formulation of the objective function i. normalize input-output dataset between 0 and 1. ii. develop regression model using the normalized dataset 𝑌 = 𝑓(𝑥1, 𝑥2, . . , 𝑥𝑛 ) + 𝜀, where, (x1,x2,…,xn) are n number of independent input variables, y produces a certain response, 𝜀 represents noise or error in the response y. iii. consider it as the objective function for the cuckoo search (cs) algorithm. step c: optimization of cuckoo search (cs) parameters i. generate taguchi l9 design matrix for a) maximum iteration numbers (imax) , b) number of nests (n), c) probability to discover a cuckoo egg by host bird (pa) ii. for (i=1 to 9) { select imax, n, and pa as cs input parameters call cs subroutine store optimum fitness function value as output for ith set of input parameters } optimization of spot-welding process using taguchi based cuckoo search algorithm 319 iii. dataset is generated with imax, n and pa as inputs (xi’s)and fitness function as output parameter (yi) iv. compute taguchi quality loss function and normalise for all yi’s v. compute signal to noise ratio (s/n ratio) vi. compute level wise mean of s/n ratio for each factor vii. determine optimized imax, n and pa step d: process optimization with optimized cs algorithm i. set, imax, n and pa, at optimized value as determined in step c ii. call cs subroutine iii. determine optimized fitness function value (# such as maximum tensile strength or peak load of rsw joint in present work) cuckoo search (cs) subroutine begin set, the objective function using the regression equation developed in step b maximum iteration numbers (imax) number of nests (n) probability to discover a cuckoo egg by host bird (pa) iterate: generate randomly a cuckoo (i) from lev՜y flight evaluate fitness function value (fi) randomly select nest j and determine fitness function value (fj) if (fi < fj) replace j using the new solution end abandon worst nests by a fraction pa randomly generate new solution from all cuckoos fitness evaluated keep the better solutions current best is found through ranking continue (until iteration > imax) figure 1. outline of the taguchi based cuckoo search algorithm cuckoo search (cs) algorithm objective function formulation regression model development taguchi method for cs parameter setting quality loss function computation experimental dataset generation optimized parameter setting for cs spot welding process optimisation with optimised cs chaki and bose /decis. mak. appl. manag. eng. 5 (2) (2022) 316-328 320 2.1. experiment a resistance spot welding setup (manufactured by electro weld industries) operated by ac (50 hz) 200 kva rating (@ 50% duty cycle) is used for conducting experiments using 3 levels, 4-factor experimental design on mild steel (is 2062, gr. b) having block dimensions 100mm x 25mm x3mm. the experimental setup used is given in fig.2. figure 2. experimental setup table 1 specifics the values of control factors at different levels. the possible variations of control factors for table1 have been determined through pilot experiments. altogether nine experiments are conducted using taguchi l9 experimental design. the input parameters considered during experimentation are, 1) welding current, ka (i) 2) welding time, cycles (wt), 3) hold time, cycles (ht), and 4) electrode pressure, kpa (ef). for welded joints, the output parameter is the ultimate tensile strength which is the maximum force required to tear the spot joints from each other. a universal testing machine measured it through peak load, kn (l). table 2 furnishes the experimental dataset. input and output parameters are as follows: x = [i wt ht ep], 𝑌 = [𝑃𝐿] optimization of spot-welding process using taguchi based cuckoo search algorithm 321 (a) before (a) after figure.3. job specimen before and after resistance spot welding table 1. control factors with their levels control factors symbol level of control factors level 1 level 2 level 3 welding current, ka i 29.5 30 30.7 welding time, cycles wt 25 28 32 welding hold time, cycles ht 20 25 30 electrode pressure, kpa ep 480 550 620 2.2. development of regression model a first-order regression model with interaction effects is developed to determine the approximate relationship between four input variables and the output variable. the experimental dataset is normalized prior development of the model to ensure better optimization performance. the input and output variables are normalized as follows before modeling: xnor = x − xmin xmax − xmin ynor = y ymax ⁄ (2) where xmax and xmin represent the maximum and minimum values of the input parameters and ymax represents the maximum value of the output parameter. the model is developed using minitab16 software has been given below: 𝑃𝐿 = 0.592122 + 0.387324 × i + 0.307468 × wt + 0.521117 × ht − 0.117952 × ep − 0.0806291 × i × wt − 0.430832 × i × ht − 0.556653 × wt × ht (3) the model is evaluated by the regression coefficient (r-square) value. r2 and r2 (adjusted) values obtained are 0.9715 and 0.7731 respectively. near unity r2 and r2 (adjusted) value indicates the developed model is adequate. chaki and bose /decis. mak. appl. manag. eng. 5 (2) (2022) 316-328 322 table 2. input design matrix and experimental details trial no. input parameters output parameter welding current, ka welding time, cycles welding hold time, cycles electrode pressure, kpa peak load kn i wt ht ep pl 1 29.5 25 20 480 17.15 2 29.5 28 25 550 21.55 3 29.5 32 30 620 21.5 4 30 25 25 620 23.8 5 30 28 30 480 28.5 6 30 32 20 550 28.25 7 30.7 25 30 550 28.6 8 30.7 28 20 620 27.15 9 30.7 32 25 480 27.6 maximum value 30.7 32 20 480 28.6 minimum value 29.5 25 30 620 2.3. formulation of objective function it is intended to maximize the tensile strength of the joint in terms of peak load. the objective functions along with constraints are given below: the objective functions: maximize pl (i, wl, ht, ep) subject to constraints: 29.5 ≤ i ≤ 30.7 25 ≤ wt ≤ 32 20 ≤ ht ≤ 30 480 ≤ ep ≤ 620 (4) however, as the cs algorithm is inherently developed for minimization problems, the –ve sign is incorporated before the output variable to convert it into a maximization problem. along with, as regression model is formed with normalized dataset the objective function and constraints have been reduced into the following form: minimize: 𝑃𝐿 = −(0.592122 + 0.387324 × i + 0.307468 × wt + 0.521117 × ht − 0.117952 × ep − 0.0806291 × i × wt − 0.430832 × i × ht − 0.556653 × wt × ht) subject to constraints: 0 ≤ i, wt, ht, ep ≤ 1 (5) optimization of spot-welding process using taguchi based cuckoo search algorithm 323 2.4. cuckoo search algorithm cuckoo search (cs) [yang & deb (2010)] is a nature-inspired metaheuristic algorithm based on the brood parasitism characteristics observed in cuckoo species by which it can lay its eggs in the nests of other host birds (often other species). upon discovering the eggs do not belong to them, the host bird will remove those eggs from nests, or move to another nest leaving that nest. the probability of discovering the egg laid by a cuckoo by the host bird is pa ∈ [0, 1]. here, each nest refers to one egg that also refers to one cuckoo. during optimization, n numbers of host nests (xi, i=1,….,n) are randomly selected as the initial population and fitness value (fi) of eggs of host nests are computed. further, the new nest is randomly generated by a cuckoo(i) through levy flight using the equation, 𝑥𝑖 𝑡+1 = 𝑥𝑖 𝑡 + 𝛼 ⊕ 𝐿�́�𝑣𝑦(𝜆) where, where α > 0 indicates the step size of steps, t represents iteration number, and random walk-through lévy flight (λ) is accomplished by a lévy distribution of 𝐿�́�𝑣𝑦~𝑢 = 𝑡−𝜆, (1 < λ ≤ 3). further, corresponding to updated solution points (nests) fitness value (fj) of eggs is computed. upon comparing with the initial fitness value (fi), if improvement in fitness value is observed, the updated solution is accepted. during computation, the pa fraction of the worst nests is abandoned. repetition of the above process is continued till stopping criteria are achieved. here, controllable parameters are the maximum number of iterations, number of nests, and probability to discover a cuckoo egg by host bird (pa). 2.5. taguchi method for optimization of cs parameters taguchi’s method studies the entire factor space through a limited set of experiments based on an orthogonal array. the method designates the desirable value of output as ‘signal’ and the undesirable value as ‘noise’. the term ‘signal-to-noise ratio’ or s/n ratio is an indicator of the deviation of output from the desired value. l 9 orthogonal array, based on three levels three factors experimental design is used. factors are the maximum iteration numbers (i), the number of nests (n), and the value of pa respectively. the numeric value of variables at different levels is furnished in table 3. for a specific set of factor values, cs computes optimized objective function value pl and employs it for computing loss function. for maximization of pl higherthe-better quality characteristic is required and the loss function (f i) can be given as: 𝑓𝑖 = 1 𝑛 ∑ 1 𝑦𝑖 2 𝑛 𝑖=1 (6) where, yi is the observed output at the ith trial, and n is the number of trials. quality loss functions obtained are normalized and signal to noise ratio or s/n ratio (snr) is calculated corresponding to ith trial condition as: snr𝑖 = −10 log10(𝑓i) (7) 3. results and discussion 3.1. optimization of cs parameters using taguchi method the taguchi design matrix with control parameter settings of cs has been shown in table 3. a program code for cs is developed in matlab2010 using the objective chaki and bose /decis. mak. appl. manag. eng. 5 (2) (2022) 316-328 324 function developed (eq.5). further, the program is run for each parameter setting of cs given in table 4 to find the corresponding objective function value (pl). it is further evaluated for determining the quality loss function, its normalization, and signal-tonoise ratio using eq. (6) eq. (7) and shown in table 4. table 3. cs parameters with different levels symbol cs parameters level 1 level 2 level 3 imax maximum iteration numbers 20 30 40 n number of nests 10 20 30 pa probability to discover a cuckoo egg by host bird 0.1 0.3 0.5 table 4. quality loss values with s/n ratio using l9 orthogonal array exp no. coded input parameters output parameter quality loss normalized quality loss snr (db) imax n pa objective fn (pl) 1 1 1 1 -1.1848 0.7124 1.0000 0.0000 2 1 2 2 -1.1892 0.7071 0.9926 0.0322 3 1 3 3 -1.2042 0.6896 0.9680 0.1411 4 2 1 2 -1.1922 0.7036 0.9876 0.0541 5 2 2 3 -1.2053 0.6884 0.9663 0.1490 6 2 3 1 -1.1951 0.7002 0.9828 0.0752 7 3 1 3 -1.2029 0.6911 0.9701 0.1317 8 3 2 1 -1.2021 0.6920 0.9714 0.1259 9 3 3 2 -1.1885 0.7079 0.9938 0.0271 further, from the influence of controlling parameters of cs on the s/n ratio (table 5) the optimized level of cs parameters is determined as follows: maximum number of iterations (level 3): 40 number of nests (level 2): 20 probability to discover a cuckoo egg by host bird, pa (level 3): 0.5 comparison between optimized cs parameter setting with initial parameter setting indicates reasonable improvement in snr (i.e. 0.1562) and provided in table 6. 3.2. optimization of spot-welding process parameters finally, the spot-welding process is optimized through an optimized cs with the above-mentioned controlling parameter setting. fig.2 shows the nature of variation of the best function value with iterations leading to convergence. optimization of spot-welding process using taguchi based cuckoo search algorithm 325 table 5. effect of factors on s/n ratio factor mean of s/n ratio (db) maximum minimum rank level 1 level 2 level 3 maximum iteration numbers (imax) 0.1733 0.2783 0.2847a 0.1114 2 number of nests (n) 0.1858 0.3071a 0.2433 0.1213 3 probability to discover a cuckoo egg by host bird (pa) 0.2011 0.1134 0.4218a 0.3084 1 a optimum level table 6. improvement in performance at the optimum parameter level figure 4. performance of optimized cs during optimisation the optimized (maximum) value of pl is found as 1.2063. the optimized value peak load (pl) after denormalization is 34.5 kn with corresponding operational input parameter welding current (i), welding time (wt), hold time (ht), and electrode pressure (ep) values as, 30.7 ka, 32 cycles, 20 cycles, and 480 kpa respectively. upon comparing the operational input parameter setting with corresponding level values provided in table 1 it has been found that the maximum peak load indicator of ultimate tensile strength of the joint will be obtained at high welding current, high welding time, low hold time, and low electrode pressure. a validation experiment is conducted with optimized parameter setting and compared with optimized output for evaluating the accuracy of the optimized cs algorithm and furnished in table 7. the 0 10 20 30 40 -1.25 -1.2 -1.15 -1.1 -1.05 -1 number of iterations o b je c ti v e f u n c ti o n v a lu e best function value : -1.2063 initial parameter setting optimal cs parameters level i1n1pa1 i3n2pa3 pl -1.1848 -1.2063 msnr (db) 0 0.1562 improvement of snr (db) 0.1562 chaki and bose /decis. mak. appl. manag. eng. 5 (2) (2022) 316-328 326 optimized cs method can predict optimized output with an absolute % error of 1.93%. computation has been conducted using a pentium iv, 3 ghz, and 512 mb pc. table 7. results of spot-welding optimization using optimized cs and experimental validation operational input parameters output parameter welding current welding time welding hold time electrode pressure peak load ka cycles cycles kpa kn normalised value 1 1 0 0 1.2063 denormalised value 30.7 32 20 480 34.50 experimental value 30.7 32 20 480 35.18 absolute % error 1.93 4. conclusion present work aimed to develop a taguchi-based cs algorithm for enhanced optimization performance and employed to optimize spot welding process. the process is intended to maximize the peak load leading to a maximum tensile strength of the welded joint. the conclusions are as follows: (i) taguchi method determined the optimum value of controllable parameters of cs algorithm such as the number of iterations (imax), number of nests (n), and probability to discover a cuckoo egg by host bird (pa). an optimum combination of cs parameters with imax, n, and pa values as 40, 20, and 0.5 respectively will yield the best optimization result for the spot-welding problem under study. (ii) the performance of the cs algorithm is improved considerably with the optimized cs parameter combinations. on comparing that performance with a nonoptimized initial parameter setting, an improvement of 0.1562 db is measured in terms of snr. (iii) finally, optimization of spot welding with the optimized cs yields a maximum peak load of 34.5 kn for 30.7 ka welding current, 32 cycles welding time, 20 cycles hold time and 480 kpa electrode pressure. (iv) the maximum peak load for the welded joint is found at high welding current, high welding time, low hold time, and low electrode pressure. (v) experimental validation of the optimized result indicates a low error of 1.93% which signifies the accuracy of the model. (vi) the taguchi-based cs algorithm may be used effectively for any other process. the process may be further extended to optimize operating parameters of other popular algorithms like the water cycle algorithm, krill herd algorithm, mine blast algorithm, interior search algorithm, etc. author contributions: conceptualization, s.c. and d.b.; methodology, s.c..; software, s.c.; validation, s.c. and d.b.; formal analysis, s.c. and d.b.; investigation, d.b.; resources, s.c..; writing—original draft preparation, s.c.; writing—review and editing, s.c. and d.b.; all authors have read and agreed to the published version of the manuscript. optimization of spot-welding process using taguchi based cuckoo search algorithm 327 funding: this research received no external funding. acknowledgments: authors would like to thank the national institute of technical teachers’ training and research, kolkata for providing necessary technical support. conflicts of interest: the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. references bozkurt, f., & çakır, f.h. 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accepted: 4 may 2019; available online: 5 may 2019. original scientific paper abstract: in the present study, a novel memetic genetic algorithm (nmga) is developed to solve the traveling salesman problem (tsp). the proposed nmga is the combination of boltzmann probability selection and a multi-parent crossover technique with known random mutation. in the proposed multi-parent crossover parents and common crossing point are selected randomly. after comparing the cost/distance with the adjacent nodes (genes) of participated parents, two offspring’s are produced. to establish the efficiency of the developed algorithm standard benchmarks are solved from tsplib against classical genetic algorithm (ga) and the fruitfulness of the proposed algorithm is recognized. some statistical test has been done and the parameters are studied. key words: tsp, memetic ga, multi-parent crossover. 1. introduction one best example of a well known intensively studied the combinatorial optimization problem is tsp. tsp is also too much related to different type of transportation problem (kundu, 2017; kar, 2018) with vehicle convenience. it is also an example of np-hard problem (lawler & lenstra, 1985; das et al., 2010; das et al., 2011). many researchers are trying to solve tsp with reasonable time and space. but still, there have lacunae to solve such kind of np-hard problems. presently two ways are implemented such as direct method (lin-kernighan helsgaun, scant-method, sant-cycle method) and indirect such as heuristic or metaheuristics. the classical tsp involves finding the shortest path/minimum cost with the visit of all cities exactly ones except the starting one. the probe on the efficient algorithm for tsp is an open problem. mailto:royarindamroy@yahoo.com mailto:pkcollege.contai@gmail.com mailto:sumanta@iimcal.ac.in optimal decisions on pricing and greening policies of multiple manufacturers under… 101 moscato (1989) introduced the word memetic algorithm (ma) as a combination of population-depended global search and the heuristic local search based on every of the individuals. from the different context of view, mas are recently used with different names like hybrid evolutionary algorithms (martinez-estudillo, 2005), many kinds of literature found in (jampani, 2010; li & feng, 2013; silberholz, 2013; hiremath & hill, 2013; nesmachnow, 2014; skinner, 2015), a different lamarckian evolutionary algorithms are studied in (omran, 2016), very recently a cultural algorithms developed found in (reynolds & peng, 2004). in the case of colonial optimization, the various number of instances of ma have been notifiable across a broad scope of application realm, generally merging to better-tone solutions much expeditious than the established affected counterparts. real-world complex problems have been successfully satisfied with memetic algorithms. although various researcher avail procedures nearly bound up to ma, through other names like hybrid genetic algorithms are also exploited. nowadays mas are applied in different research areas included pattern recognition (aguilar & colmenares, 1998), artificial neural networks (ichimura & kuriyama, 1998), circuit design (harris & ifeachor, 1998), robotic motion planning, beam orientation (haas et al., 1998), electric service restoration (kumar et al., 2006), medical expert systems (wehrens, 1993), single machine scheduling (chyu & chang, 2010), etc. a study on multi-parent ma is found in (wang et al., 2010) and they concluded that different combination got better results from others. ye et al., (2014) developed a multi-parent recombination operator for solving linear ordering problem but they do not restrict to chosen the parent because it increases the computational complexity. at the present investigation, only four parents are selected from the mating pool and randomly a common crossing point is chosen. genetic algorithms (gas) were first proposed by holland (1992) whose ideas were applied and expanded on by goldberg (1998). the classical ga has three operators, such as selection, crossover and mutation. different kinds of selection operators (rw, ranking, tournament, etc.) and cyclic, partial-map, ordered based, etc. crossovers are available with a random mutation to solve the discrete optimization problem by ga. in our proposed method (nmga) are the combinations of probabilistic selection and adaptive fourparents crossover with the classical ergodic mutation. now the crossover is taken from the realistic social observations. we see that some child born with legal parents but they adapted by other parents and grown up under them. in third world countries, it is very common. in the proposed crossover methods four parents are used and modified them finally comparing the costs/distance, the offspring is created. the proposed algorithm has the following key features:  boltzmann probabilistic selection,  multi-parent adaptive crossover,  four parents,  random crossing point,  comparing the cost/distance genes are selected,  test on standard tsplib problems. the remaining part of this paper is presented as follows: section 1, a short introduction is presented. in section 2, we describe mathematical pre-requisite. in section 3, the proposed modified memetic algorithm is presented. in section 4, some numerical experiments are done. again in section 5, a brief discussion is given. finally, in section 6, a conclusion with future scope is studied. roy et a., /decis. mak. appl. manag. eng. 2 (2) (2019) 100-111 102 2. classical definition of tsp the goal of a traveling salesman problem (tsp) is that a salesperson would create a path. this path should be an ideal path. ideal path means path would be the shortest while salesman completes his visit across a finite number of cities, visiting each city only once and finished at the starting city. let g=(v, a) is a graph. g has n vertices and v is a set of this n vertices. a is also a set of arcs or edges of this g. then g=(v, a). let c = c(i,j) be a distance ( or cost) matrix associated with a. the intent of tsp is determining a minimum distance or cost circuit passing through each and every vertex only once except starting node. this type of circuit is familiar as a tour or hamiltonian circuit or cycle. in case of symmetric tsp c(i,j) = c(j,i) for all i,j∈ v. (n1)! path will be generated for symmetric tsp and (n-1)!/2 path will be generated for asymmetric tsp. now mathematically tsp defined as below. minimize ( , ) ij i j z c i j x   subject to 1 1 n ij i x   for 1, 2,...,j n ; 1 1 n ij j x   for , 1, 2,...,i n ; | | 1, n n ij i s j s x s s q       ; where {0,1}, , 1, 2..,ijx i j n  . now, q  {1, 2, 3,.., n } set of nodes ij x = the decision variable, 1 ij x  if thesalesman visits from city-i to city-j, 0 ij x  otherwise. then the above tsp reduces todetermine a complete tour 1 2 1 ( , ,..., , ) n x x x x ; 1 1 1 1 ( , ) ( , ) n i i n i z c x x c x x      where, , , 1, 2..., . i j x x i j n  along with sub tour elimination criteria optimal decisions on pricing and greening policies of multiple manufacturers under… 103 | | 1, n n ij i s j s x s s q       , where, {0,1}, , 1, 2..,ijx i j n  . 3. proposed memetic genetic algorithm here we propose a probabilistic selection, multi-parent crossover with the simple random mutation for solving the tsp. 3.1. representation considering n cities available to make a complete tour which stands for a solution. say an integer vector xi of n-dimensional. where xi =(xi1,xi2,··· ,xin) is used as cities, and xi1, xi2, ···, xnstand for n successive cities in a tour. at the beginning need a group of paths (tours) for the salesman. these paths are randomly generated for ga. these initial paths is a group of possible solutions for the ga part of this algorithm. 3.2 selection 3.2.1. probabilistic selection the main objective of tsp that minimizing the path cost/distance. so here minimum fitness value (say fmin) of the choromosome play a vital role. matting pool is formed using the boltzmann-probability (roy et al., 2018) of all chromosome from the initial population. now (( ( ))/ ) ;min j f f x t b p e   t=t0(1-a)k, k=(1+100*(g/g)), g=ongoing generation number, g= maximum generation, t0= rand[10,150], f(xi) corresponding fitness/objectives of chromosome corresponding to xi, a=rand[0,1], i=chromosome number. 3.3. multi-parent crossover nowadays in our society adaptation is very common matter from the different practical situation. here except original parents (father, mother), there are one more parents (father, mother) taken as a part. inspiring this realistic happening here a new approach with four parents (first two are original parents and the other two are adoptive parents) are used to produce offspring. this urged crossover method choose four individuals or parents in an ergodic manner to create offspring. to collect optimum results of a tsp, we make a journey from one node to next node maintaining minimum traveling cost based on tsp condition. following the above conception, we make the crossover procedure in the following condition. at first select (randomly) four individuals ( parents ) from the mating pool. give an example here. pr1, pr2, pr3, pr4 are the parents and ch1, ch2 are the offspring. pr1: pr2: pr3: pr4: 1 2 0 3 4 0 2 4 3 1 4 0 3 1 2 3 4 1 2 0 roy et a., /decis. mak. appl. manag. eng. 2 (2) (2019) 100-111 104 generate random number between 0 and 4. suppose it is 2. then according to our proposed algorithm it would be the starting node of a new offspring (ch1). ch1: now we have to comparing minimum traveling cost between 2 node··· (1) (1st node of parent1 [pr1] ) 2 node··· (0) (1st node of parent2 [pr2] ) 2 node··· (4) (1st node of parent3 [pr3] ) 2 node··· (3) (1st node of parent4 [pr4] ) and if say the traveling cost node (2) to node (4) is minimum from rest three paths, then next node of the new offspring (ch1) would be 4. so it should be like as ch1: the above process will continue until the new offspring (ch1) gets its all nodes maintaining the condition of tsp. similarly, generate the second offspring but in reverse order than another. here r1 and r2 are two randomly generated variable two nodes from the given set of nodes. 3.4. random mutation an ergodic number r is created for every solution of p(t). here r is generated between a range [0,1] and r < pm is a condition, if the condition is true, then a solution is selected for mutation. two nodes are selected ergodic manner from each chromosome and they interchange their positions and replace it in the offspring set. 3.5. proposed novel memetic algorithm ( nmga ) nmga algorithm: 1. input: for crossover procedure (pc), maximumgen(s0), (pop−size) and for mutation procedure (pm). 2. output: the best solution. 3. begin 4. approve generation t = 0. 5. (initialize) ergodic manner and generate approve population p(t), here f(xi),i= 1,2,··· ,(pop − size) state the all chromosomes. 6. all solution will be judged it’s efficiency one by one from the approve population p(t) 7. repeat up to (18) till (t < s0) 8. modify the current generation such as t = t + 1 9. decide (pb) for all chromosome over p(t) to subsection (3.2.1) 10. construct the mating pool on the basis psand pb 11. for crossover parents will be chosen based on pc over mating pool 12. according to subsection (3.3) the crossover operation will be conducted based on exclusive chromosomes/solutions 13. produce offspring and the parents will be replaced 14. repeat (9) to (11) based on pc 15. followed by the subsection (3.4) mutation process will be executed 16. offspring will be selected for mutation depend on pm 17. interchange the position between selected nodes 18. replace offspring 19. determine the effectiveness and save the local best and near best solutions 2 4 2 optimal decisions on pricing and greening policies of multiple manufacturers under… 105 20. repeat (5) to (18) 21. (for best result) store the global best and near best results 22. stop proposed nmga pseudo code: begin generation t = 0; for (i=1 to pop-size) produce chromosome ergodic manner; end for; for (i=1 to pop-size) judge fitness; end for; for (gen=1 to max-gen) { for (j=1 to pop size) r=rand[0,1]; t0= rand[5,100]; a=rand[0,1]; k=(1+100*(i/g)); t=t0(1-a)k; (( ( ))/ ) ;min j f f x t b p e   if ( s r p ) select the current chromosome; j++; else if (r b p ) select f(x j ) ; j++; else select the corresponding chromosome of f mi n ; j++; end for; end for; for (s=1 to (noc ∗ pc)) r1=rand[0, n-1]; r2=rand[0, n-1]; pr1=rand[0, pv-1]; pr2=rand[0, pv-1]; pr3=rand[0, pv-1]; pr4=rand[0, pv-1]; ch1[0]=r1; i=1; do{ ch1[i] = min {c(r1,pr1[0]),c(r1,pr2[0]),c(r1,pr3[0]),c(r1,pr4[0])}; i=i+1;} while(ch1[ n-1]); ch2[n-1]=r2; i = n − 2 ; do{ ch2[i] = min {(r2,pr1[n−1]),c(r2,pr2[n−1]),c(r2,pr3[n−1]),c(r2,pr4[n− 1])}; i = i − 1 ; } while(ch2[0]) ; end for for(a=0 to noc) { roy et a., /decis. mak. appl. manag. eng. 2 (2) (2019) 100-111 106 if (rand[0,1] < pm) mutate; } for (i=1 to noc) evaluate fitness; end for } stop 3.6. termination criteria the proposed algorithm is terminated if it finishes the user-defined maximum number of generations or iterations, or the difference between consecutive iterations less than some predefined values which are earlier. 4. numerical experiments 4.1. test results of nmga we have taken benchmarks from tsplib (reinelt, 1991) and select 53 standard instances form 7 city to 318 cities to test the performance of our proposed algorithm nmga. table 1 shows the comparison of performance between proposed nmga and standard classical ga through the results presented in the form of percentage error. the total comparison held the basis of total cost. we have taken the best, average and the worst outcome of both nmga and classical ga under 100 independent runs and the best solution is presented with relative percentage error. the parameters of the nmga given in table 2 for the same nodes of the benchmarks instance kora100 with 100 cities problem. we have increases pop-size, maxgen, pc and pm of an instance as a parameter. 4.1.1. sensitivity of cpu-time w.r.t. pc and pm sensitivity analysis is performed for cpu-time on the basis of concerned values of the key parameters pm and pc and outcomes are projected in figure1 (three dimensions linear graph using statistica). it is observed that for fixed value of pc, as pm increases, cpu-time increases. also, it is observed that for a fixed value of pm, as pc increases, cpu-time also increases. figure 1. sensitivity analysis optimal decisions on pricing and greening policies of multiple manufacturers under… 107 5. discussion the superiority of the developed algorithm is established by solving standard benchmarks from tsplib which given in table 1. this proposed algorithm nmga is coded in c++ based on few keys like the maximum number of chromosomes (100) and a maximum number of iterations (5000). table 1 is used to show the differences between nmga and ga for few benchmark tsp references in tsplib. it is observed that the percentage of error is lesser in nmga than the classical standard ga. here, 53 standard instances from 7 to 318 cities are studied and most of the cases nmga produced better results. a parameter analysis is done which is given in table 2 for the standard benchmark kora100 with 100 cities problem. table 1. performance (relative error) of benchmarks from instances nmga classical ga best worst average best worst average sh-07 0 0 0 0 0 0 sp11 0 0 0 0 0.15 0.07 uk12 0 0 0 0.04 0.22 0.15 lau15 0 0 0 0.29 0.55 0.44 gr17 0 0 0 0.22 0.52 0.39 wg22 0 0.02 0.01 0.65 1.07 0.94 fri26 0 0.02 0.01 0.76 1.09 0.98 bay29 0 0.06 0.01 1.01 1.25 1.12 bayg29 -0.02 0.04 0.02 0.92 1.22 1.10 wi29 -0.00 0.06 0.02 1.29 1.77 1.59 ha30 0 0.08 0.02 1.14 1.53 1.37 dj38 0.00 0.04 0.01 1.90 2.24 2.12 dantzig42 0.00 0.11 0.03 1.98 2.45 2.28 swiss42 0.00 0.11 0.03 1.75 2.04 1.91 att48 2.16 2.40 2.25 9.07 10.60 10.11 eil51 0.00 0.06 0.03 1.94 2.22 2.11 berlin52 0.00 0.14 0.05 1.90 2.24 2.14 wg59 0.00 0.18 0.07 2.73 3.17 2.99 st70 0.00 0.14 0.04 3.18 3.60 3.48 eil76 0.01 0.09 0.06 3.04 3.59 3.39 pr76 0.01 0.26 0.11 3.20 3.59 3.45 rat99 0.01 0.12 0.06 14.68 16.28 15.80 kroa100 0.00 0.32 0.12 5.25 6.00 5.76 krob100 0.02 0.27 0.10 5.05 5.64 5.44 kroc100 0.02 0.22 0.10 5.19 6.14 5.90 krod100 0.03 0.21 0.10 5.08 5.70 5.50 kroe100 0.02 0.25 0.09 5.24 5.87 5.63 rd100 0.01 0.20 0.11 4.65 5.25 5.07 eil101 0.04 0.15 0.09 3.51 3.85 3.71 lin105 0.01 0.26 0.13 5.95 6.50 6.28 pr107 0.01 0.20 0.09 9.28 10.36 9.95 pr124 0.01 0.41 0.14 8.74 9.59 9.26 bier127 0.07 0.39 0.20 3.60 3.83 3.74 ch130 0.03 0.20 0.12 5.37 5.85 5.70 pr136 0.09 0.36 0.20 6.18 6.70 6.49 pr144 0.01 0.39 0.15 10.71 11.60 11.24 roy et a., /decis. mak. appl. manag. eng. 2 (2) (2019) 100-111 108 in table 2, it is used to calculate the goodness of parameter of selection (ps) in nmga. it shows that to get the optimal solution of the standard tsp kroa100, psindicates the given space better for ps= 0.34. table 2. parameters for nmga of kroa100 instance instance pc pm popsize gen result cpu-time(sec) 0.34 0.01 4673 21417 5526 0.02 4065 21344 5733 0.001 3407 21322 5405 0.003 4957 21298 5373 0.005 2427 21294 5380 0.007 2173 21316 4810 0.008 3376 21285 4470 0.009 3068 21384 4214 kroa100 0.2 0.01 70 2384 21322 3331 0.25 2787 21386 3232 0.30 4958 21333 3916 0.35 2612 21285 5574 0.40 4868 21294 6164 0.45 3883 21412 6256 0.50 3708 21349 6149 0.55 3505 21365 4407 0.60 4406 21316 4438 0.70 4467 21535 6640 0.75 4320 21334 6874 0.80 4975 21831 9982 0.85 3905 21335 9840 0.34 0.01 50 3438 21390 3175 60 4638 21474 8382 kroa150 0.06 0.37 0.18 6.91 7.80 7.52 krob150 0.08 0.34 0.20 6.92 7.88 7.58 pr152 0.14 21.88 13.26 11.31 11.94 11.67 u159 0.03 0.27 0.16 8.13 8.80 8.45 qa194 0.09 0.27 0.17 2.28 3.20 2.23 rat195 0.06 0.25 0.16 32.79 35.82 34.81 d198 0.10 0.29 0.20 9.20 10.05 9.66 kroa200 0.13 0.45 0.26 8.95 9.73 9.37 krob200 0.16 0.42 0.28 8.73 9.43 9.14 ts225 0.08 0.35 0.19 10.13 10.75 10.49 tsp225 0.07 0.27 0.17 8.30 8.87 8.61 pr226 0.11 0.84 0.23 16.82 18.62 18.12 gil262 0.12 0.37 0.24 9.03 9.58 9.33 pr264 0.20 0.41 0.30 17.90 20.19 19.37 a280 0.15 0.37 0.27 10.61 11.33 11.00 pr299 0.12 0.37 0.24 12.85 13.74 13.35 lin318 0.17 0.40 0.29 11.57 12.20 11.94 optimal decisions on pricing and greening policies of multiple manufacturers under… 109 instance pc pm popsize gen result cpu-time(sec) 72 4453 21322 4470 85 3460 21322 4952 110 2706 21285 7664 150 4700 21294 8974 it is evident from table 2 that, for standard three parameters pc, pm and pop − size, our proposed algorithm nmga give us optimum or near optimum result easily. thus the importance of the parameters is discussed in table 2. 6. conclusion in this paper, a novelty introduced in ga regarding selection and crossover, called novel memetic genetic algorithm (nmga). nmga is tested with few test references from tsplib and examined with classical ga. in nmga, boltzmann probabilistic selection and a new four parents crossover are worked with ergodic mutation. the concept of ma is not new in the area of tsps, but the idea of multi-parent(four) crossover on the basis of the memetic concept is new, establishes our proposed algorithm as highly np-hard combinatorial optimization problem. realistically, it is true that multi-parent crossover especially four parent crossover may not be so much truthful than two parent crossover for a specific problem or the complexity may be high than other. but the numerical analysis proves the efficiency of our proposed algorithm. the improvement of developed nmga is in natural form and it is also applicable to solve another discrete problem like network optimization, wellknown graph theory, popular standard transportation problems, vehicle routing problem, and electronics manufacturing units, etc. although we have to get the much superior results by nmga, there is also have a huge scope for improvement 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(2014). a multi-parent memetic algorithm for the linear ordering problem. arxiv preprint arxiv:1405.4507. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 5, issue 1, 2022, pp. 194-207. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0314052022b * corresponding author. e-mail address: bipradasbairagi79@gmail.com (b. bairagi) a novel mcdm model for warehouse location selection in supply chain management bipradas bairagi*1 1 department of mechanical engineering, haldia institute of technology, india received: 1 april 2022; accepted: 4 may 2022; available online: 13 may 2022. original scientific paper abstract: the present investigation reveals a novel method for the evaluation of warehouse location for leagile supply chain entailing fuzzy multi criteria analysis (fmca). an attempt has been made to apply the concept of decision theory for selecting the warehouse under contradictory criteria. aggregate modified weighted value (mwv) of normalized score of alternative is determined to evaluate benefit cost ratio (bcr) which is considered as the warehouse selection index. the proposed algorithm is illustrated with a suitable numerical example to adjudge its desirable importance in capability and practicability. it also ensured that the achieved result clearly matches with those of previous research works. finally, sensitivity analysis has been carried out that justify and supports the application of proposed algorithm in finding the most favorable outcome to the selection problem of warehouse location. key words: leagile supply chain, warehouse location selection, fmcdm, benefit cost ratio. 1. introduction supply chain is a system of association concerned with upstream, mid-stream and downstream link in diverse procedure and actions so as to create value in terms of services and products for customer satisfaction (lee & billington, 1992). the center of attention of the lean manufacturing has basically been on the diminution or abolition of waste. lean is concerning doing extra with a smaller amount. the concept of lean work well if demand is predictable and stable and if variety is small (agarwal et al. (2006). on the other hand for unpredictable demand and additional varieties of customer’s requisite, a higher level of agility is necessary (lee, 2002). the objective of a supply chain is to exploit the value produced and robustly interrelated through effectiveness of supply chain. the ultimate achievement and profitability of a business depends on proper planning, appropriate design and suitable operation of a supply chain. the effectiveness of a usual supply chain largely a novel mcdm model for warehouse location selection in supply chain management 195 is affected by the transportation, inventory, information and facilities (chopra & meindl, 2001), proximity to customers and markets, suppliers’ availability, and even social issues as the potential stability for the warehouse location selection (heizer & render, 2004; stevenson, 2006). ocampo et al. (2020) applied group topsis for selection of warehouse location considering diverse allocations of expert priority. ulutas et al. (2021) introduced a novel combined grey based mcdm approach in selection of ware house location. micale et al. (2021) advocated an integrated topsis with interval-valued electre tri method for appropriate decision making in storage location problem. kabak and keskin (2018) used multi-criteria decision making and gis approach for selection of warehouse for storing hazardous materials. dey et al. (2017) introduced and applied an mcdm model with group heterogeneity for selection of the best warehouse location. pang and chan (2017) employed a data mining based new step by step algorithm foe assignment of storage location in warehouse. emec and akkaya (2018) applied stochastic analytical hierarchy process combining with fuzzy vikor method for the purpose of right decision making in appropriate warehouse location selection under mcdm environment. the analysis of the gap of the above literature survey exposes that previous researchers have applied mcdm techniques for selection of warehouse location. but this endeavor is not enough for exhaustive and wide decision making regarding proper selection of warehouse location. thus an attempt in the current investigation is made to suggest a novel method for the evaluation of warehouse location for leagile supply chain entailing fuzzy multi criteria analysis (fmca). the current paper has objective of improving the warehouse location selection techniques by using the concept of decision theory. the proposed algorithm has been employed to choose the top warehouse location amongst a set of realistic alternatives. the result of this novel algorithm is compared with works of past researchers on the identical problem. lastly, an appropriate example is solved to illustrate the proposed algorithm. this investigation proves that the proposed algorithm is compatible for selection of multidimensional warehouse location problem. the current study also enhances the models used for the same. the paper is organized as follows. section 2 presents a revision on he fuzzy set theory. section 3 is dedicated for the proposed algorithm. in the section 4, an appropriate example is cited and solved. sensitivity analysis is conducted in section 5. section 6 explores the significance of the results of the work and section 7 presents some significant concluding remarks on the proposed model. 2. the fuzzy set theory decision makers usually have a preference subjective to objective assessment of fuzzy information. theory of fuzzy set is used to convert these subjective data into numerical (objective) values (chiou, 2005). a number of important definitions on fuzzy set are presented in the following subsection 2.1. 2.1. some important fuzzy definitions definition 1: a fuzzy set a ~ is defined in a universe of discourse denoted by x specified by  x a ~ , called membership function, which connects every member x (a bairagi/decis. mak. appl. manag. eng. 5 (1) (2022) 194-207 196 real number) in x in a interval where x belong to [0, 1]. (zadeh, 1965, dubois and prade, 1978). definition 2: a triangular fuzzy number (tfn) q is defined as a triplet  1 2 3, ,q q q . membership function is characterized as follows (chu, 2002; keufmann & gupta, 1991).   1 1 1 2 2 1 3 2 3 2 3 3 0, , , 0, . q x q x q q x q q q x x q q x q q q x q                  (1) membership function of a tfn  1 2 3, ,q q q q is graphically shown in fig. 1. figure1. membership function of a tfn  1 2 3, ,q q q q definition 3: let  1 2 3, ,q q q q and  1 2 3, ,r r r r be two tfns, then the distance between the two fuzzy numbers can be calculated as        1 2 22 1 2 2 3 3 1 , 3 d q r q r q r q r            this method of calculating distance between two fuzzy numbers is termed as vertex method (klir & yuan, 1995). 2.2. fuzzy operations let   321 ,, ~ qqqq  and  1 2 3, ,r r r r are two triangular fuzzy numbers. (a) addition:  1 1 2 2 3 3, ,q r q r q r q r     (b) multiplication of a fuzzy number  1 2 3, ,q q q q with a real number k  1 2 3, ,k q kq kq kq  where rkk  &0 (c) multiplication commutative property k q q k   where rkk  &0 . (d) division of a fuzzy number   321 ,, ~ qqqq  with a real number k 1 0 q1 q3 q2 x a novel mcdm model for warehouse location selection in supply chain management 197 31 2( ) , , qq q q k k k k         where rkk  &0 (dubois, & prade, 1978). 3. proposed algorithm a multi-criteria decision-making procedure has been applied for the evaluation of the most excellent warehouse location among various suitable alternatives. a quantitative approach of decision theory has been utilized in order to improve the selection procedure of warehouse location. the steps of the process of the newly proposed algorithm have been furnished below. step 1: formation of decision making committee: the committee unanimously chooses effective criteria and selects the alternatives preliminarily. let, d1, d2...dp are the decision-makers; c1, c2… cn are the selected criteria; where number of benefit criteria is ‘b’ and that of cost criteria is ‘c’such that   .ncb  and a1, a2… am are initially selected warehouse locations. step 2: formation of decision matrix: the committee makes a short list of alternative warehouses for further assessment on the basis of the selection criteria. each alternative warehouse is given a score by the committee (or each member of the committee) with respect to each attribute; this score is termed as performance rating or simply rating. performance ratings under objective criteria are expressed in crisp (specific) values and under subjective criteria are expressed in linguistic terms due to vagueness, imprecision, and ambiguity. the words or phrases like ‘good’, ‘very good’, ‘medium’, ‘poor’, ‘very poor’ etc. are called linguistic variables which are measured by human perception, feelings, experience etc. a decision matrix with m number of alternatives, 1 ... ... t i ma a a   , 1 j n(c c c ) where n is criteria number.   1 11 1 11 1i m 1 ... ... a a a j n k jk nk m n i k ijk ink m k mjk mnk c c c x x x dm x x x x x x                    (2) ijk x is performance rating in linguistic variable which is converted into fuzzy number ijk x where,  , ,ijk ijk ijk ijkx a b c or  , , ,ijk ijk ijk ijk ijkx a b c d is the performance rating of ith candidate (alternative) for j th factor (criterion) given by kth decision maker. step 3a: formation of weight matrix: the committee members give diverse importance to the criteria in linguistic variables according to their knowledge and experience. each linguistic variable is changed into corresponding tfn. the weight matrix is presented in eq. (3). bairagi/decis. mak. appl. manag. eng. 5 (1) (2022) 194-207 198 1 11 1 11 1 1 ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... j n k p j j jk jp n n nk np c c c w w wc w c w w w c w w w                   (3) here  jkjkjkjkw  ,, or  jkjkjkjkjkw  ,,, is the importance weight of jth (factor) criterion awarded by the k th (experts) decision maker in fuzzy numbers (triangular or trapezoidal) respectively step 3b: defuzzification and average weight: for decision-making committee, the defuzzified average weight of the criteria is considered. defuzzified average weight of each criterion is calculated by using eq. 4(a) for triangular fuzzy number or eq. 4 (b) for trapezoidal fuzzy number. )( 3 1 1 jkjk p k jkj p w     4(a) )( 4 1 1 jkjkjk p k jkj p w     , j = 1, 2… n. 4(b) step 3c: normalization of weight: normalization process of defuzzified average importance weight of each criterion is accomplished by eq. 4(c).    n j j jn j w w w 1 , j = 1, 2, 3,… ,n. 4(c) step 4a: defuzzification and average rating: determination of defuzzified average performance rating of every alternative for every subjective criterion by following eq. (5a) for triangular fuzzy number or eq. (5b) for trapezoidal fuzzy number. average ratings for the objective criteria are calculated by the eq. 5(c).     p k ijkijkijkij cba p x 1 3 1 5(a)     p k ijkijkijkijkij dcba p x 1 4 1 5(b)     p k ijkij x p x 1 1 5(c) step 4b: normalization of rating: average rating of each alternative is normalized using eq. (6).    m i ij ijn ij x x x 1 (6) step 5: calculation of modified weighted values: in this method, modified weighted normalized rating is advocated for the assessment of alternative for each benefit and non-benefit criterion. performance weight replaces both interest factor as well as time a novel mcdm model for warehouse location selection in supply chain management 199 period under consideration. the benefit of using importance weight in its place of both interest rate and time period is that the computed coefficient of normalized rating provides a modified weight which depends upon corresponding weight. modifiedweighted benefit and non benefit criteria value is calculated using eq. (7) and eq. (8) respectively.   n jwn j n ij b ij wxmwv  1/ (7) n jw n j n ij c ij wxmwv        1/ (8)  b ijmwv modified weighted value of normalized rating of alternative i under benefit criterion j, cijmwv modified weighted value of normalized rating of alternative i under cost criterion j, nijx corresponding normalized rating of alternative i under criterion j, njw corresponding importance weight of jth criterion. weight of criteria has been modified using the eq. (7) and eq. (8) respectively. here importance weight ( njw ) is simultaneously an analogue to the interest rate and number of periods of cash flows in finding the future value in engineering economy. so, the weight of criteria successfully replaces discounted rate (or interest rate) and time period. the value so obtained will be termed as modified weighted value throughout the paper. this modified weighted value is multiplied with normalized rating in order to obtain the modified normalized rating which gives the measurement of benefit or cost. step 6: aggregate modified weighted value: modified weighted values of ratings under benefit criteria and cost criteria are separately added for calculating aggregate modified-weighted value. aggregate modified-weighted value under benefit criteria and cost criteria reflect the assessment of total beneficial scores and cost scores respectively. aggregate modified-weighted values are calculated using the following simple eq. (9) and eq.(10) respectively.   1 / 1 n j b w b n n i ij j j amwv x w    (9)   1 / 1 n j c w c n n i ij j j amwv x w    (10) b iamwv  aggregate modified weighted value of alternative i for all benefit criteria. ciamwv  aggregate modified weighted value of alternative ai for all cost criteria. step 7: benefit-cost ratio (bcr): in this paper, benefit-cost ratio is proposed to consider warehouse selection index. benefit-cost ratio of an alternative is expressed by the ratio of aggregate modified weighted value of ratings under benefit criterion to that of the cost criterion. the higher value of benefit cost ratio is desirable. the following eq. (11) is used for calculating benefit cost ratio (bcr) of ith alternative.  ibcr =  cibi mwvmwv / (11) the higher the benefit cost ratio is, the better the alternative is. step 8: selection: the alternative warehouses are arranged in order of decreasing benefit cost ratio. higher benefit-ratio is desirable. the best warehouse is one which bairagi/decis. mak. appl. manag. eng. 5 (1) (2022) 194-207 200 has the highest benefit cost ratio. similarly the worst warehouse is one which is which attains the least benefit cost ratio. the above algorithm is applied to solve a warehouse location selection problem and to demonstrate implementation of the paradigm. 4. illustrative example demonstrating proposed algorithm the proposed algorithm has been illustrated with an example. this has been presented dividing it into two sub-sections: warehouse location selection problem definition, calculation and discussions. 4.1 warehouse location selection problem definition the proposed algorithm has been demonstrated with an example on warehouse location selection. the objective is to develop a process for the combination of different criteria pertinent to selection of warehouse location with a view to obtain a comprehensive ranking order of the alternative warehouses. the example on warehouse location selection has been cited from chen et al. (2006). in the present example, a homogeneous decision-making committee is composed of three decisionmakers or experts namely d1, d2 and d3. each members of the homogeneous decision making committee has equal importance weight. through a screening test, the committee preliminarily takes three alternatives warehouse locations a1, a2 and a3 under consideration of further assessment. the committee also considers five subjective decision criteria viz. cost (c1), possibility of expansion (c2), availability of required material (c3), human resource (c4) and proximity to market (c5). 4.2 calculation and discussions owing to subjective, vague and imprecise, the performance ratings of the alternatives are with respect to all the five criteria are estimated by linguistic variables. linguistic variables are easy use and understand. linguistic variables requires less efforts and less time with compared to other mode of expression. that is why decision makers prefer linguistic variables to objective measurement. in the current problems decision makers uses seven degrees of linguistic variables to express their assessment and perception regarding the alternative warehouse locations towards the criteria. the seven degrees of linguistic variables used for expressing performance ratings, their respective acronyms and the corresponding triangular fuzzy numbers have been accommodated in table 1. the degree of importance weights of various criteria in decision making on proper selection of warehouse location varies from criterion to criterion and decision maker to decision makers. the decision makers involved in the decision making process are inspired to use a common set of seven degrees of linguistic variables for measuring importance weights of the criteria. a novel mcdm model for warehouse location selection in supply chain management 201 the seven degrees of linguistic variables used for expressing importance weights, their respective acronyms and the corresponding triangular fuzzy numbers have been arranged in table 2. the decision makers estimate the weights of criteria in linguistic variables and are represented in table 3. since the criteria have different dimensions, they are normalized in order to convert into dimensionless quantity so as to compare one another. defuzzified average values of weight are calculated using eq. 4(a). normalized weights are calculated using eq. 4(c). defuzzified and normalized values of weights are shown in table 4. performance rating of each warehouse is estimated by the each decision makers with respect to each criterion which are estimated by the knowledgeable decision makers. cost criteria are expressed in numerical values. and the remaining four table 1. linguistic variables with fuzzy numbers for ratings linguistic variables acronym fuzzy numbers extremely poor v p ( 0, 0, 1 ) poor p ( 0, 1, 3 ) slightly poor m p ( 1, 3, 5 ) fair f ( 3, 5, 7 ) medium good m g ( 5, 7, 9 ) good g ( 7, 9, 10 ) extremely good v g ( 9, 10, 10 ) (source: chen et al. 2006) table 2. linguistic variables, acronyms and tfn for the estimation of weight linguistic variables acronym fuzzy numbers very low vl ( 0, 0, 0.1 ) low l ( 0, 0.1, 0.3 ) medium low ml (0.1, 0.3, 0.5) medium m ( 0.3, 0.5, 0.7 ) medium high mh (0.5, 0.7, 0.9 ) high h ( 0.7, 0.9, 1.0 ) very high vh ( 0.9, 1.0, 1.0 ) (source: chen et al. 2006) table 3. the linguistic weight of the criteria. criteria d1 d2 d3 c1 h vh vh c2 h h h c3 mh h mh c4 mh mh mh c5 h h h (source: chen et al. 2006) table 4. defuzzified and normalized weight of the criteria values c1 c2 c3 c4 c5 defuzzified values 0.93 0.87 0.75 0.70 0.87 normalized values 0.226 0.21 0.182 0.17 0.211 bairagi/decis. mak. appl. manag. eng. 5 (1) (2022) 194-207 202 criteria are evaluated by prescribed linguistic variables in specified degrees. it can be easily observed that the performance ratings of the warehouse a1, a2, and a3 with respect to criterion c1 by the decision maker d1 are 6, 3, 4 million respectively. the linguistic variables for the alter natives a1, a2, and a3 estimated by decision maker d1 with respect to criterion c2 are expressed as g, eg, and mg respectively. the linguistic variables for the alter natives a1, a2, and a3 estimated by decision maker d2 with respect to criterion c2 are expressed as eg, eg, and g respectively. the remaining performance rating of the three alternative warehouses with respect to the criteria as assessed by the decision makers are provided in the following table 5. average and normalized values of rating of objective criteria as well as defuzzified and normalized values of rating of alternatives under subjective criteria are shown in table 6. the normalized decision matrix is represented in table 7. modified weighted normalized rating is calculated. for example, modified weighted value of alternative a1 for benefit criterion c2 (expansion possibility) is computed as   3206.021.01/3052.0 21.012  b mwv . modified weighted value of alternative a1 for cost criterion c1 (investment cost) is computed as   4636.0226.01/4375.0 226.011  c mwv . aggregate modified weighted ratings are determined. for alternative a1 for benefit criteria (c2, c3, c4, c5) , the calculation process is    1 12 13 14 15 0.3206 0.3145 0.3442 0.2491 1.228b b b b bamwv mwv mwv mwv mwv         table 5. performance ratings of the warehouses by the decision makers criteria alternative d1 d2 d3 c1 a1 a2 a3 6 (million) 3 (million) 4 (million) 8 (million) 4 (million) 5 (million) 7 (million) 5 (million) 6 (million) c2 a1 a2 a3 g e g m g e g e g g f e g e g c3 a1 a2 a3 f g g g g s g g g e g c4 a1 a2 a3 eg g g g g e g g g e g c5 a1 a2 a3 f g g f f g f g g (source: chen et al. 2006) table 6. average normalized rating of objective criteria and defuzzified normalized rating of subjective criteria c1 c2 c3 c4 c5 alternative av nv d v n v d v n v d v n v d v nv a1 7 0.4375 8.0 0.3052 7.44 0.3032 9.00 0.3335 5.00 0.237 a2 4 0.25 9.66 0.3686 8.66 0.3529 8.66 0.3209 7.44 0.3526 a3 5 0.3125 8.55 0.3262 8.44 0.3439 9.33 0.3457 8.66 0.4104 av= average value, dv =defuzzified values, nv =normalized values a novel mcdm model for warehouse location selection in supply chain management 203 the modified weighted values and aggregate modified weighted values are determined and presented in table 8. benefit-cost ratio (bcr) is calculated by using eq. (11)., bcr for alternative a1 is calculated as  1bcr =  1 1/b camwv amwv =  4636.0/228.1 .6488.2 benefit-cost ratios and the ranking order of the alternative warehouse locations are accommodated in table 9. the final ranking of the alternatives is decided on the basis of benefit cost ratio vs. alternatives. the ranking orders of the warehouse locations under consideration are graphically depicted in figure 2, for achieving higher visibility and clarity. figure 2. benefit cost ratio of alternatives warehouse locations the graphical representation clearly shows that the ranking order of the warehouse locations as per the proposed method is a2>a3>a1. the best warehouse location is a2. a comparison of the results obtained by the proposed approach with those of past researches available in the open journals has been made and shown in table 10. 0 1 2 3 4 5 6 b e n e fi tc o s t r a ti o a1 a2 a3 alternatives table 7. normalized decision matrix c1 c2 c3 c4 c5 weight 0.226 0.21 0.182 0.17 0.211 a1 0.4375 0.3052 0.3032 0.3335 0.237 a2 0.25 0.3686 0.3529 0.3209 0.3526 a3 0.3125 0.3262 0.3439 0.3457 0.4104 table 8. modified weighted values, aggregate modified weighted values modified weighted values aggregate modified weighted values alternatives c imwv 1 b imwv 2 b imwv 3 b imwv 4 b imwv 5 c iamwv b iamwv a1 0.4636 0.3206 0.3145 0.3442 0.2491 0.4636 1.228 a2 0.2649 0.3873 0.3660 0.3312 0.3706 0.2649 1.455 a3 0.3311 0.3427 0.3671 0.3568 0.4314 0.3311 1.498 table 9. benefit–cost ratio and ranking order of the alternatives alternatives benefit-cost ratio ranking order a1 2.6488 3 a2 5.4926 1 a3 4.5243 2 bairagi/decis. mak. appl. manag. eng. 5 (1) (2022) 194-207 204 5. sensitivity analysis a sensitivity analysis of the problem of warehouse location selection has been carried out and the result has been depicted in figure 3. the sensitivity analysis is the graphical representation of benefit cost ratio of warehouses with respect to variable decision making attitude. the following eq. (12) governs the warehouse selection index in sensitivity analysis.   ** 1 ci b ii mwvmwvwsi   (12) figure 3. sensitivity analysis of selection index with respect to coefficient of attitude *b imwv and *c imwv are aggregate normalized value and modified weighted value of alternative i for benefit and cost criteria respectively.  = coefficient of decision making attitude in the range 10  .the sensitivity plot clearly shows that warehouse location 2 has the maximum bcr in the range of 87.00   .the result of the sensitivity plot in the range of 87.00   is also in line with the ranking order of warehouse locations. but warehouse location 3 has the highest benefit cost ratio while 187.0  which is shown in table 11. the sensitivity analysis gives a readymade and prompt solution to the current problem under unpredictable coefficient of decision attitude. 0.5714 0.8197 1 0.9712 0.8 1 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 coef f icient of attitude w a re h o u s e s e le c ti o n i n d e x warehouse a1 warehouse a2 warehouse a3 table 10. comparison of results (ranking) among various papers alternatives chen t.c.(2006) proposed method a 1 3 3 a 2 1 1 a 3 2 2 (α) a novel mcdm model for warehouse location selection in supply chain management 205 6. discussions the concept of lean is well applicable where demand is comparatively steady and predictable with low variety. on the other hand, for unstable demand and varieties of customer’s need, to a large extent of higher level “agility” is necessary. the leagile supply chain integrates the lean and agile paradigms contained by a total supply chain approach by placing the decoupling point for the best suit of the need. the decision making in the leagile supply chain is very intricate in nature. the approach proposed in the paper is an aid to the managers for correct decision for the leagile supply chain. the proposed algorithm recommends the benefit cost ratios as the selection criteria of warehouse location. accordingly, the warehouse locations are ranked as follows: a2 > a3 > a1 it is observed that a2 is the best warehouse location. it has also been revealed that the ranks of warehouse locations found by applying the proposed algorithm that employs the proposed methodology produces the similar result obtained by past researchers using different method as shown in table 9. the method validates the judgment behind the approach which complies with the technique adopted by the researchers as demonstrated in literature survey. the proposed algorithm has a number of significant features as follows: a. the modification of weight in the proposed algorithm is exclusive in nature. b. it is able of managing fuzziness of the decision making environment. c. the method is capable of considering subjective and objective criteria to select the best warehouse location. d. the algorithm is simple, easier and straightforward. 7. conclusions lean supply chain is capable of maximizing profit through waste reduction whereas agile supply chain maximizes profit by delighting the customers. leagility in the supply chain makes the upstream cost effective whereas the downstream becomes more service oriented. in this paper, the algorithm has incorporated the concept of modified weighted value into the decision theory for the selection of warehouse in a supply chain which may be very handy for the decision makers. this paper gives a revised version of weight avoiding direct use by employing engineering economy. this modification of weight has not yet been reported in any research work. the consistency of results of the cited problem with those of other works strongly justifies the concept of modification of weight. benefit cost ratio is considered as the key selection parameter for warehouse location selection. the importance weight of criteria play a great role in the evaluation process as it simultaneously act as the interest rate and number of periods of cash flows. table 11. sensitivity analysis option range of coefficient of attitude(α) selection of warehouse 1. 0 ≤ α < 0.87 select 2a 2. 0.87 < α ≤ 1 select 3a 3. α = 0.87 indifferent towards 2 a & 3a bairagi/decis. mak. appl. manag. eng. 5 (1) (2022) 194-207 206 this algorithm is simple, easier and capable of considering fuzziness. this new method may be thought of an extended version of simple additive weighting (saw) method with modified weight applying engineering economy or financial management. this approach is appropriate for implementation in other managerial decision making problems. by transforming it into computerized algorithm, a large number of criteria, alternatives, and decision makers’ view can be considered. author contributions: conceptualization, b.b.; methodology, b.b.; validation, b.b.; formal analysis, b.b.; investigation, b.b.; resources, b.b.; writing—original draft preparation, b.b.; writing—review and editing, b.b.; visualization, b.b.; supervision, b.b.; the author has read and agreed to the published version of the manuscript. funding: this research received no external funding. conflicts of interest: the author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. references agarwal, a., shankar, r. & tiwari, m.k. 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(1965). fuzzy sets. information and control, 338–353. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). https://www.researchgate.net/deref/https%3a%2f%2forcid.org%2f0000-0002-9383-3399 https://www.researchgate.net/deref/https%3a%2f%2forcid.org%2f0000-0002-9383-3399 https://www.researchgate.net/journal/engineering-management-in-production-and-services-2543-912x https://www.researchgate.net/journal/engineering-management-in-production-and-services-2543-912x plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 5, issue 1, 2022, pp. 264-289. issn: 2560-6018 eissn: 2620-0104 doi:_https://doi.org/10.31181/dmame0311022022t * corresponding author. e-mail addresses: belkistorgul@gmail.com (b. torğul), edemiralay@ktun.edu.tr (e. demiralay), tpaksoy@yahoo.com (t. paksoy) training aircraft selection for department of flight training in fuzzy environment belkız torğul1*, enes demiralay1 and turan paksoy2 1 konya technical university, faculty of engineering and natural sciences, department of industrial engineering, turkey 2 necmettin erbakan university, faculty of aviation and space sciences, department of aviation management, turkey received: 1 september 2021; accepted: 8 january 2022; available online: 11 february 2022. original scientific paper abstract: the last two decades have seen a growing trend towards the use of aircraft as transportation tools. however, there is a lack of routes because of the insufficient number of pilots. therefore, the increase in usage of aircraft has been limited. to respond to this increase in turkey, it indicates a rise in the number of flight academies. flight academies have emerged as powerful and expensive platforms for flight training. in the new global economy, the aircraft selection problem has become a central issue for flight training departments, which is planned to open in government universities. in this study, an approach based on the fuzzy bwm method is proposed to select more suitable training aircraft in government universities. criterion weights and alternative aircraft rankings were determined using the fuzzy bwm method. afterward, a mathematical model was developed to calculate how many aircraft we need to buy under certain constraints. necmettin erbakan university, which wants to train new and qualified pilots, needs training aircraft and trainers that can provide pilot training. a case study of training aircraft selection was conducted for the necmettin erbakan university department of flight training. as a result, it can be said that 13 aircraft will be sufficient for the flight training department to start education. key words: training aircraft selection, flight training, bwm, fuzzy sets, linear programming model. 1. introduction when asked what the term aviation means, the first answer has been usually to travel by aircraft. however, the design, manufacture, and maintenance operations of aircraft required to travel are also included in the aviation term. it is unknown what mailto:belkistorgul@gmail.com mailto:edemiralay@ktun.edu.tr mailto:tpaksoy@yahoo.com training aircraft selection for department of flight training in fuzzy environment 265 will bring about the combination of advancing time and constantly developing technologies for aeronautics. today, aircraft have been used for human and cargo transportation, agricultural spraying, and military purposes. for those types of aviation to be used actively, personnel who will design and manufacture aircraft; know-hows; new technologies; pilots to use aircraft; and technicians to undertake maintenance and repair of aircraft are required. the beginning of aviation history dates back to the 9th century when abbas ibn firnas made the first flying glider (lienhard, 2019). the chinese book poo phu tau had been claiming the existence of rotary-wing aircraft in the 4th century. leonardo da vinci's glider design, which has survived to the present day, remained only a design in the 15th century but was produced in the 19th century with the materials used in the 15th century. hezarfen ahmet çelebi had traveled 3 km from galata tower to the anatolian side in 1638 with wings he designed inspired by birds. the modern era of aviation history has begun with the hot air balloon designed by the montgolfier brothers (kılıç, 2015). modern aviation history has been continued the development with alphonse pénaud's first structurally balanced aircraft model, the first successful flight in history by felix du temple, and the first motorized aircraft flight of orville and wilbur wright’s brothers. airports have begun to be built in many cities during world war ii. after world war ii, with the pilots' demobilization and the introduction of the aircraft used by the soldiers but surplus to the civilians, there was a great increase in the use of private and commercial aviation, especially in north america. today, the increase in airlines' use in different transportation types such as passenger transportation, cargo transportation, and dangerous goods transportation continues at an accelerated rate. even in the coming years, an increase in using space tourism will be observed with the development of services, increased security, and reliability, as in airline tourism (webber, 2013). airline transportation, which is still not widely used in turkey, is developing rapidly in the world. transportation with aircraft is a more reliable transportation choice in a shorter time compared to other transportation methods. although technological developments are of great importance for airline transportation to take superiority over other transportation methods, the number of well-trained pilots is also crucial. universities working towards the more widespread use of aviation in turkey have started studies to open flight training departments. necmettin erbakan university intends to purchase training aircraft to train pilots in its department of flight training by progressing towards this aim. however, the increased fuel costs and aircraft costs which have increased due to the economic difficulties experienced in the last few years and the changes in exchange rates, have adversely affected the aviation market. while the aviation market has been affected so much, the number of criteria to be considered for selecting aircraft has increased. for this reason, determining the criteria for the selection of training aircraft is of great importance. thus, the aircraft selection process has turned into a multicriteria decision-making (mcdm) problem. there are different options for the solution of mcdm problems. in this study, an approach based on the fuzzy bwm method is proposed for purchasing training aircraft. a case study of the necmettin erbakan university department of flight training has been conducted to show the approach's applicability. the nine crucial criteria affecting aircraft selection problem (max cruise speed, max range, takeoff and landing roll, max climb rate, power output, weight, price, useful fuel capacity, and time before overhaul) has been determined from the surveys conducted with decision-makers who are the academic members of necmettin erbakan university aviation and space sciences faculty. in the next step, torğul et al./decis. mak. appl. manag. eng. 5 (1) (2022) 264-289 266 according to the experts' evaluations, criterion weights that affect the aircraft selection problem have been determined with the fuzzy bwm method. the most important feature that distinguishes this study from other studies in the literature is developing a linear programming model that determines how many aircraft should be bought under certain constraints (budget, minimum flying before overhaul, and fuel consumption). an original approach consisting of fuzzy bwm and a mathematical model are proposed. a case study is made on which criteria should be considered in the training aircraft selection process to set an example for developing flight departments in turkey. a literature study is conducted in the field of aircraft selection, and as a result, a literature matrix is created that relates the studies and the methods used, and it is ensured that the gap in the literature can be seen in future studies. unlike previous studies, more technical features of training aircraft are discussed. the rest of this paper is arranged as follows. in section 2, a literature review is presented. in section 3, a detailed methodology is presented. section 4 provides the relevant problem definition and developed mathematical model. in section 5, a case study of aircraft selection is presented and demonstrated how the proposed approach works. finally, in section 6, the conclusion of this paper and suggestions for future work are presented. 2. literature review this section presents a comparative discussion of the former studies on aircraft selection to highlight the proposed study's contributions. this study differs from other studies because the criteria that affect the training aircraft selection problem are carefully determined, determined which type of aircraft should purchase by flight academies, and the number of aircraft required for the flight academy. the literature review is divided into three paragraphs to avoid complexity. in the paragraphs, aircraft selection studies using mcdm methods in crisp, fuzzy, and both crisp and fuzzy environments are given, respectively. table 1 presents the previous studies and their' criteria and methods used. see et al. (2004) firstly have demonstrated the strengths and weaknesses of mcdm methods which own theoretical and practical flaws commonly employed, using the speed, max cruise range, and the number of passengers criteria for the airline aircraft selection problem. then, a method based on hypothetical equivalent has been proposed and expanded to include hypothetical inequivalent. in this study, criteria affecting the problem have not expressed the aircraft selection process sufficiently was observed. liu & wu (2010) have proposed an evaluation model based on information entropy and data envelopment analysis methods to analyze suitable alternatives for aircraft fleet selection in local transportation airlines. the applicability of the proposed approach has been shown with a numerical example. six alternative aircraft were evaluated under five main criteria, with ten sub-criteria. the approach has been developed in a fuzzy environment to avoid uncertainties during the decisionmaking process. sun et al. (2011) have proposed a new approach for the hypothetical airline aircraft selection problem using electre, saw, and topsis methods. uncertainties may arise in every decision-making problem. the uncertainties have been eliminated using taguchi loss functions to ensure robustness instead of developing the approach in a fuzzy environment. in this study, robustness has also been added as a criterion. investigating the effect of robustness as a criterion on the ranking of alternatives has been conducted by sensitivity analysis. dožić & kalić training aircraft selection for department of flight training in fuzzy environment 267 (2015a) have proposed a new approach using two different mcdm methods, ahp and esm. a hypothetical airline aircraft selection study has demonstrated the applicability of the approach. sensitivity analysis was performed to show the differences between ahp and esm results. dožić & kalić (2015b) have developed a new fleet planning model for airlines operating on short and medium-haul routes. the fleet planning model consists of three stages: fleet composition, fleet sizing, and aircraft selection. the applicability of the model was demonstrated by a hypothetical airline case study located at the belgrade airport. five criteria evaluated seven alternative aircraft. it is inadequate in reaching the appropriate solution in the problem of aircraft selection with selected criteria. paul et al. (2017) have proposed a topsis method-based approach for fighter aircraft selection. the applicability of the proposed approach is shown with a numerical example. four alternative fighter aircraft were evaluated with six criteria. the criteria chosen are not sufficient for alternative fighter aircraft selection. ali et al. (2017) have created a scenario to select new and better aircraft for their existing fleets to develop pakistan air force and pakistan air defense capabilities. they have proposed an approach using the ahp method for the aircraft selection problem. cost-benefit analysis has been carried out for the selected alternative to be compatible with pakistan's financial budget. six alternative aircraft have been evaluated with ten criteria. the criteria chosen are not sufficient for alternative fighter aircraft selection. kiracı & bakır (2018b) have proposed a topsis method-based approach for choosing the most suitable aircraft for airlines with different flight networks. the applicability of the proposed approach is shown with an example of commercial aircraft selection. eight decision-makers have evaluated four alternative aircraft types, most demanded by airline companies with five criteria. it is inadequate in reaching the appropriate solution in the problem of aircraft selection with selected criteria. kiracı & bakır (2018a) have proposed an approach based on ahp, copras, and moora methods, considering cost, performance, and environmental factors for the commercial aircraft most demanded by airline companies. the proposed approach has been shown with a numerical example, and it has been observed that decision-making methods give consistent results. the decision-makers evaluated the four alternative aircraft most demanded by airlines, with seven criteria affecting the selection. no action has been taken to prevent uncertainties that may occur during the decision-making process. petrovic & kankaraš (2018) have proposed an approach based on the hybrid dematel and ahp methods to select air traffic protection aircraft. the applicability of the proposed approach is shown with a numerical example. forty-five decision-makers calculated the weight of 52 sub-criteria under nine main criteria. no precautions have been taken for uncertainties that may arise during the criterion weighting process. ilgın (2019) has proposed a new approach based on the linear physical programming method to remove the disadvantage of the fact that criteria obtained by different decisionmaking methods take physically meaningless and subjective values. the applicability of the proposed approach is shown with a numerical example. five criteria evaluated six alternative aircraft. (yilmaz et al. (2020) have proposed an approach based on ahp and topsis methods for aircraft selection. sixteen decision-makers who are teachers in eskişehir technical university flight school evaluated six alternative training aircraft with four main criteria (strategic criteria, financial criteria, operational criteria, and maintenance criteria). the criteria for aircraft selection are not explicitly indicated. for this reason, the study is insufficient on an appropriate decision-making aspect. hoan & ha (2021) have proposed a novel decision-making approach with the integration of fucom and aras methods for aircraft selection. a case study of suitable fighter jets selection for the vietnam people's air force has demonstrated the torğul et al./decis. mak. appl. manag. eng. 5 (1) (2022) 264-289 268 proposed approach's applicability. three alternative aircraft (su-35, mig-35, and f16) were examined under 13 criteria. however, the selected criteria do not provide proper outcomes for fighter aircraft selection. for instance, constraints such as the maximum range and the useful fuel capacity have been ignored. sensitivity analyses have been conducted, and the results are compared with the weighted product method to prove the method's robustness. do nascimento maêda et al. (2021) have proposed a hybrid approach based on ahp and topsis methods and the dual normalization procedure for the selection of helicopters to be purchased by the brazilian navy, which provides more logistics and combat capacity in naval operations. a real military case study was conducted to improve the performance of the brazilian armed forces. among the six helicopters evaluated by considering the attack helicopters used by developed countries during the selection process, the ah-64e apache was the most suitable helicopter for the brazilian armed forces. wang & chang (2007) have proposed an approach based on the topsis method in a triangular fuzzy environment in initial training aircraft selection for taiwan air force. in the case study conducted to demonstrate the approach's applicability, 15 decision-makers have evaluated seven aircraft under 16 criteria. the selection process has been insufficient due to the usage of criteria alike and is not helpful enough to make the right decision for the training aircraft selection. also, criteria weight has been calculated by taking the average of decision-makers' evaluations, not by any decision-making method. yeh & chang (2009) have proposed a new group method for mcdm problems in a fuzzy environment. a case study of taiwan's domestic airline's empirical aircraft selection has demonstrated the model's applicability. five alternative aircraft have been evaluated under three main criteria with 11 sub-criteria. the specified criterion has been insufficient to reach an appropriate judgment for the aircraft selection problem. ozdemir & basligil (2016) have proposed an approach for the aircraft selection problem using fuzzy anp and choquet integral method. an aircraft purchase case study has been conducted for a turkish airline company. three aircraft were evaluated with ten criteria. the criteria are not conducive to proper aircraft selection. the proposed approach results have been compared with fuzzy ahp, and the same results on f ahp have been obtained in all three methods. (dožić et al. (2018) have proposed a new methodology to assist in selecting aircraft types that best meet market conditions and airline requirements for estimated travel demand based on known route networks and routes. an ahp-logarithmic fuzzy preference programming method-based approach has been developed in the fuzzy environment to eliminate human uncertainty. the pairwise comparison matrix was created from interviews with experts from different airlines and universities. the applicability of the methodology has been demonstrated by the case study of regional airline aircraft selection. according to interviews conducted with experts from different airlines and universities, seven alternative aircraft have been evaluated with ten criteria. the selected criteria are not sufficient to reach the most suitable alternative aircraft. kartika & hanani (2019) have proposed a new approach based on fgd and ahp methods for aircraft selection. the approach's applicability is proved with a case study of indonesia's national flag carrier airline company's aircraft selection to be used on new routes. decision-makers evaluated four alternative aircraft with six criteria. in this study, it is insufficient to decide on the most suitable aircraft with the determined criteria. ahmed et al. (2020) have proposed a new approach using the ahp and efficiency method in a fuzzy environment to eliminate human uncertainty in the regional aircraft selection problem, considering the environmental design and cost impact. inspired by canadian airlines, the framework of the approach was created. four alternative aircraft were evaluated with 15 sub-criteria under five main criteria. training aircraft selection for department of flight training in fuzzy environment 269 the consistency of the results of the proposed approach was checked using sensitivity analysis. the study is insufficient in an appropriate decision-making aspect since more emphasis on environmental criteria, and technical criteria essential for aircraft selection remain in the background. kiracı & akan (2020) have proposed a new hybrid ahp and topsis approach in the interval type 2 fuzzy environment. the applicability of the proposed approach is shown with a numerical example. four alternative commercial aircraft were evaluated under three main criteria (economic performance, technical performance, and environmental impact) with eight sub-criteria. the technical criteria required for aircraft selection do not fully reflect a real-life choice. sánchez-lozano & rodríguez (2020) have proposed a new hybrid ahp and frim approach in a fuzzy environment to the aircraft selection problem. the applicability of the approach has been demonstrated by the spanish air force aircraft selection case study. the necessary evaluations of the proposed approach's application were obtained from the questionnaires conducted with the flight instructors in the 23rd fighter and attack training wing. four alternative aircraft have been evaluated under 13 criteria. the criteria required for the training aircraft selection have been selected, but some crucial criteria like the time before maintenance or usable fuel capacity seem to have been overlooked. karamaşa et al. (2021) have proposed an approach based on neutrosophic ahp and multimoora methods for training aircraft selection for flight training organizations. the aircraft has been evaluated with the help of questionnaires. in order to check the accuracy of the developed approach, a comparative analysis with existing approaches has been made. in line with the comparative analysis, it is observed that the approach produces productive results. bakır et al. (2021) have proposed an approach based on hybrid piprecia and marcos methods in the fuzzy environment for regional aircraft selection. in the case study to demonstrate the approach's feasibility, five decision-makers evaluated six aircraft alternatives under 14 criteria. in addition, a three-stage sensitivity analysis was conducted to demonstrate the accuracy of the approach. mello et al. (2012) have proposed a novel approach based on the naide method for the aircraft selection problem. the applicability of the proposed approach has been proved with a numerical illustration of a turboprop aircraft selection. eight alternative aircraft have been evaluated under 11 criteria which can be stochastic, fuzzy, or crisp measurements. the criteria determined for aircraft selection are not sufficient to lead to the appropriate judgment. the criteria should be elected more specifically. the authors observed a lack of simplicity when approaching diverse types' variables as a weak point of the method. gomes et al. (2014) have proposed a new approach based on the naide method to aircraft selection. an aircraft selection case study of an airline company investing in regional charter flights in brazil has demonstrated the applicability of the approach. eight alternative aircraft have been evaluated under 11 different criteria: crisp, stochastic, and fuzzy. schwening et al. (2014) have proposed a hybrid ahp and topsis method-based approach for agricultural aircraft selection. the applicability of the model has been demonstrated with a case study. the approach has been developed in a fuzzy environment to avoid uncertainties during the decisionmaking process. four alternative agricultural aircraft were evaluated under nine criteria. torğul et al./decis. mak. appl. manag. eng. 5 (1) (2022) 264-289 270 table 1. summary of previous researches on aircraft selection author m c s m r t l g r m c r p o w p u f c t b o t a c f method (see et al., 2004) ✓ ✓ ✓ (max takeoff) ✓ multi-attribute method (wang & chang, 2007) ✓ ✓ ✓ ✓ ✓ ✓ topsis (yeh & chang, 2009) ✓ ✓ ✓ new fuzzy group mcdm (liu & wu, 2010) ✓ ✓ ✓ (max takeoff) ✓ ✓ information entropy and data envelopment analysis (sun et al., 2011) ✓ ✓ (max takeoff) ✓ ✓ electre, saw, and topsis (mello et al., 2012) ✓ ✓ ✓ ✓ ✓ ✓ naiade (gomes et al., 2014) ✓ ✓ ✓ ✓ ✓ ✓ naiade (schwening et al., 2014) ✓ ✓ ✓ ✓ ✓ ✓ ahp and topsis (dožić & kalić, 2015a) ✓ (max takeoff) ✓ ✓ ahp and even swaps method (dožić & kalić, 2015b) ✓ (max takeoff) ✓ ✓ even swaps method (ozdemir & basligil, 2016) ✓ ✓ anp and choquet integral method (ali et al., 2017) ✓ ✓ (max takeoff) ✓ ✓ ahp and cost benefit analysis (paul et al., 2017) ✓ ✓ ✓ (payload) ✓ ✓ topsis (dožić et al., 2018) ✓ ✓ (max takeoff) ✓ ✓ ahp and logarithmic fuzzy preference programming method (kiracı & bakır, 2018b) ✓ ✓ ✓ ✓ ✓ topsis (kiracı & bakır, 2018a) ✓ ✓ ✓ (payload) ✓ ✓ ✓ ahp, copras and moora (petrovic & kankaraš, 2018) ✓ ✓ dematel and ahp (ilgın, 2019) ✓ ✓ ✓ ✓ linear physical programming (kartika & hanani, 2019) ✓ (max takeoff) ✓ ahp and topsis (yilmaz et al., 2020) ✓ (max takeoff) ✓ ✓ ahp and topsis (ahmed et al., 2020) ✓ ✓ ✓ (payload) ✓ ✓ ✓ ✓ ahp and efficacy method (kiracı & akan, 2020) ✓ ✓ ✓ ✓ (max takeoff) ✓ ✓ ✓ ahp and topsis (sánchez-lozano & rodríguez, 2020) ✓ ✓ ✓ (max takeoff) ✓ ✓ (military ) ✓ ahp and the reference ideal method (hoan & ha, 2021) ✓ ✓ ✓ ✓ (max takeoff) ✓ ✓ fucom and aras (do nascimento maêda et al., 2021) ✓ ✓ ✓ (payload) ✓ ahp, topsis, and 2n (karamaşa et al., 2021) ✓ ✓ ✓ ahp and multimoora (bakır et al.,2021) ✓ ✓ ✓ piprecia and marcos this study ✓ ✓ ✓ ✓ ✓ ✓ (empty) ✓ ✓ ✓ ✓ ✓ bwm and linear model * mcs max cruise speed; mr max range; tlgr takeoff and landing ground roll; mcr max climb rate; po power output; w – weight; p – price; ufc useful fuel capacity; tbo time before overhaul; ta training aircraft; c – crisp; f fuzzy training aircraft selection for department of flight training in fuzzy environment 271 3. methodologies the solution to real-world mcdm problems is too complex to be described with quantitative numbers. this complexity is due to uncertain and conflicting qualitative factors. in mcdm problems, criteria or alternatives are evaluated with qualitative judgments. human qualitative judgments often contain uncertainty and abstraction. the fuzzy set theory simulates human logic using a mathematical model, and a solution to real-world problems can be provided according to the human thinking style. for this reason, fuzzy sets have been used to provide a more flexible, convenient, and effective solution for the decision-makers and to obtain results more compatible with real situations in the training aircraft selection problem. in this section, fuzzy set theory, triangular fuzzy numbers, and graded mean integration representation (gmir) of triangular fuzzy numbers used in our study are briefly mentioned. moreover, fuzzy bwm suggested by dong et al. (2021) based upon triangular fuzzy numbers for mcdm is presented in detail. 3.1. fuzzy set theory in 1965, l. a. zadeh noticed that human thinking is primarily fuzzy and interpreted fuzzy sets. fuzzy set theory has been utilized for modeling decision-making processes based upon vague and uncertain information such as decision-makers' judgments (kumar et al., 2017; lima junior et al., 2014). 3.2. triangular fuzzy numbers a fuzzy set is described with a membership function, and all elements of a fuzzy set have membership degrees that range from 0 to 1 (zadeh, 1965). a triangular fuzzy number is indicated in figure 1. a triangular fuzzy number is indicated as (l, m, u) with l < m< u (kargı, 2016; lima junior et al., 2014; alosta et al., 2021). figure 1. triangular membership function. a triangular membership function and its elements are represented as follows (muhammad et al., 2021): 𝜇�̃�(𝑥) = { 0 𝑓𝑜𝑟 𝑥 < 𝑙, 𝑥−𝑙 𝑚−𝑙 𝑓𝑜𝑟 𝑙 ≤ 𝑥 ≤ 𝑚, 𝑢−𝑥 𝑢−𝑚 𝑓𝑜𝑟 𝑚 ≤ 𝑥 ≤ 𝑢, 0 𝑓𝑜𝑟 𝑥 > 𝑢, (1) torğul et al./decis. mak. appl. manag. eng. 5 (1) (2022) 264-289 272 3.3. graded mean integration representation method gmir method proposed by chen & hsieh (2000). gmir 𝑅(�̃�𝑖) of a triangular fuzzy number �̃�𝑖 = (𝑙𝑖,𝑚𝑖,𝑢𝑖) can be calculated by 𝑅(�̃�𝑖) = 𝑙𝑖+4𝑚𝑖+𝑢𝑖 6 (2) the smaller value of 𝑅(�̃�𝑖), the smaller the triangular fuzzy number �̃�𝑖. if w̃𝑗 = (𝑤𝑗 𝑙,𝑤𝑗 𝑚,𝑤𝑗 𝑢) is a triangular fuzzy number (j=1, 2, …, n). a triangular fuzzy weight vector w̃ = [�̃�1, �̃�2,… ,�̃�𝑛] is called a normalized fuzzy weight vector if for every j∈ {1, 2, …, n}, the following holds (bas et al., 2019; dong et al., 2021; guo & zhao, 2017; liao et al., 2013): ∑ 𝑤𝑗 𝑚𝑛 𝑗=1 = 1, 𝑤𝑗 𝑢 + ∑ 𝑤𝑖 𝑙𝑛 𝑖=1,𝑖≠𝑗 ≤ 1, 𝑤𝑗 𝑙 + ∑ 𝑤𝑖 𝑢𝑛 𝑖=1,𝑖≠𝑗 ≥ 1 (3) 3.4. fuzzy best worst method in this section, we present fuzzy bwm suggested by dong et al. (2021) based upon triangular fuzzy numbers for mcdm. fuzzy bwm was previously developed by guo and zhao (2017) and used in many studies. however, according to dong et al. (2021), the multiplication and subtraction operations of triangular fuzzy numbers in fuzzy bwm developed by guo and zhao (2017) are not in accordance with the operation laws of triangular fuzzy numbers. moreover, their final model is a non-linear programming model, whose global optimal solution may not exist. whereas the fuzzy bwm model proposed by dong et al. (2021) is linear in contrast to other papers, it is more reasonable to construct a linear programming model to obtain the optimal fuzzy weights of criteria. we agree with them and therefore used the fuzzy bwm method suggested by dong et al. (2021) for our study. 3.4.1. constructing of the mathematical programming model the optimal weight of all criteria is one where, each pair of 𝑤𝐵/𝑤𝑗 and 𝑤𝑗/𝑤𝑤 have 𝑤𝐵/𝑤𝑗= 𝑎𝐵𝑗 and 𝑤𝑗/𝑤𝑊 =𝑎𝑗𝑊 (rezaei, 2015). however, it is hard to achieve 𝑤𝐵/𝑤𝑗= 𝑎𝐵𝑗 and 𝑤𝑗/𝑤𝑊 =𝑎𝑗𝑊 for all j. because these formulas are equivalent to 𝑤𝐵 = 𝑤𝑗𝑎𝐵𝑗 and 𝑤𝑗 =𝑎𝑗𝑊𝑤𝑊 respectively, it is anticipated to find the fuzzy weights to ensure 𝑤𝐵 = 𝑤𝑗𝑎𝐵𝑗 and 𝑤𝑗 =𝑎𝑗𝑊𝑤𝑊 as much as possible. that is, (𝑤𝐵 𝑙 ,𝑤𝐵 𝑚,𝑤𝐵 𝑢) = (𝑤𝑗 𝑙,𝑤𝑗 𝑚,𝑤𝑗 𝑢)(𝑎𝐵𝑗 𝑙 ,𝑎𝐵𝑗 𝑚 ,𝑎𝐵𝑗 𝑢 ), (4) (𝑤𝑗 𝑙,𝑤𝑗 𝑚,𝑤𝑗 𝑢) = (𝑎𝑗𝑊 𝑙 ,𝑎𝑗𝑊 𝑚 ,𝑎𝑗𝑊 𝑢 )(𝑤𝑊 𝑙 ,𝑤𝑊 𝑚,𝑤𝑊 𝑢 ), (5) eq. (4) and eq. (5) are regarded as fuzzy equations, that is; (𝑤𝐵 𝑙 ,𝑤𝐵 𝑚,𝑤𝐵 𝑢) ≅ (𝑤𝑗 𝑙𝑎𝐵𝑗 𝑙 ,𝑤𝑗 𝑚𝑎𝐵𝑗 𝑚 ,𝑤𝑗 𝑢𝑎𝐵𝑗 𝑢 ), (6) (𝑤𝑗 𝑙,𝑤𝑗 𝑚,𝑤𝑗 𝑢) ≅ (𝑎𝑗𝑊 𝑙 𝑤𝑊 𝑙 ,𝑎𝑗𝑊 𝑚 𝑤𝑊 𝑚,𝑎𝑗𝑊 𝑢 𝑤𝑊 𝑢 ), (7) where ∼ denotes the fuzzy number and so, the symbol ‘‘≅” is a fuzzy version of ‘‘=” for a real number set, and it has the linguistic explanation ‘‘fuzzy equal to”. then, eqs. (6) and (7) are equal fuzzy equations as follows: training aircraft selection for department of flight training in fuzzy environment 273 𝑤𝐵 𝑙 − 𝑤𝑗 𝑙𝑎𝐵𝑗 𝑙 ≅ 0, 𝑤𝐵 𝑚 − 𝑤𝑗 𝑚𝑎𝐵𝑗 𝑚 ≅ 0, 𝑤𝐵 𝑢 − 𝑤𝑗 𝑢𝑎𝐵𝑗 𝑢 ≅ 0, (8) 𝑤𝑗 𝑙 − 𝑎𝑗𝑊 𝑙 𝑤𝑊 𝑙 ≅ 0, 𝑤𝑗 𝑚 − 𝑎𝑗𝑊 𝑚 𝑤𝑊 𝑚 ≅ 0, 𝑤𝑗 𝑢 − 𝑎𝑗𝑊 𝑢 𝑤𝑊 𝑢 ≅ 0, (9) for convenience; 𝑅(𝑤𝑗 𝑙) = 𝑤𝐵 𝑙 − 𝑤𝑗 𝑙𝑎𝐵𝑗 𝑙 ≅ 0, 𝑅(𝑤𝑗 𝑚) = 𝑤𝐵 𝑚 − 𝑤𝑗 𝑚𝑎𝐵𝑗 𝑚 ≅ 0, 𝑅(𝑤𝑗 𝑢) = 𝑤𝐵 𝑢 − 𝑤𝑗 𝑢𝑎𝐵𝑗 𝑢 ≅ 0, (10) 𝑄(𝑤𝑗 𝑙) = 𝑤𝑗 𝑙 − 𝑎𝑗𝑊 𝑙 𝑤𝑊 𝑙 ≅ 0, 𝑄(𝑤𝑗 𝑚) = 𝑤𝑗 𝑚 − 𝑎𝑗𝑊 𝑚 𝑤𝑊 𝑚 ≅ 0, 𝑄(𝑤𝑗 𝑢) = 𝑤𝑗 𝑢 − 𝑎𝑗𝑊 𝑢 𝑤𝑊 𝑢 ≅ 0, (11) the membership functions are constructed below for eq. (10) and eq. (11) respectively; 𝜇(𝑅(𝑤𝑗 𝑡)) = { 1, 𝑖𝑓 𝑅(𝑤𝑗 𝑡) = 0 1 − 𝑅(𝑤𝑗 𝑡 ) 𝑑𝑗 𝑡 , 𝑖𝑓 0 ≤ 𝑅(𝑤𝑗 𝑡) ≤ 𝑑𝑗 𝑡 1 + 𝑅(𝑤𝑗 𝑡 ) 𝑑𝑗 𝑡 , 𝑖𝑓 − 𝑑𝑗 𝑡 ≤ 𝑅(𝑤𝑗 𝑡) < 0 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (12) 𝜇(𝑄(𝑤𝑗 𝑡)) = { 1, 𝑖𝑓 𝑄(𝑤𝑗 𝑡) = 0 1 − 𝑄(𝑤𝑗 𝑡 ) 𝑞𝑗 𝑡 , 𝑖𝑓 0 ≤ 𝑄(𝑤𝑗 𝑡) ≤ 𝑞𝑗 𝑡 1 + 𝑄(𝑤𝑗 𝑡 ) 𝑞𝑗 𝑡 , 𝑖𝑓 − 𝑞𝑗 𝑡 ≤ 𝑄(𝑤𝑗 𝑡) < 0 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (13) where the tolerance parameters 𝑑𝑗 𝑡 > 0 and 𝑞𝑗 𝑡 > 0 (𝑗 = 1,2,…,𝑛;𝑡 = 𝑙,𝑚,𝑢). the membership function of the fuzzy equation 𝑅(𝑤𝑗 𝑡) = 𝑤𝐵 𝑡 − 𝑤𝑗 𝑡𝑎𝐵𝑗 𝑡 ≅ 0 is shown in figure 2. figure 2. membership function of the fuzzy equation 𝑅(𝑤𝑗 𝑡) = 𝑤𝐵 𝑡 − 𝑤𝑗 𝑡𝑎𝐵𝑗 𝑡 ≅ 0. a fuzzy decision 𝑆 could be considered as a fuzzy set, 𝑆 = {(w̃,𝜇𝑆(w̃))|w̃ ∈ 𝑊}, where 𝜇𝑆(w̃) = 𝛽 = 𝑚𝑖𝑛{𝜇(𝑅(𝑤𝑗 𝑡)),𝜇(𝑄(𝑤𝑗 𝑡))|𝑗 = 1,2,…,𝑛;𝑡 = 𝑙,𝑚,𝑢}, (14) torğul et al./decis. mak. appl. manag. eng. 5 (1) (2022) 264-289 274 then, eq. (14) are transformed into: { 𝜇(𝑅(𝑤𝑗 𝑡)) ≥ 𝛽 (𝑗 = 1,2,…,𝑛;𝑡 = 𝑙,𝑚,𝑢) 𝜇(𝑄(𝑤𝑗 𝑡)) ≥ 𝛽 (𝑗 = 1,2,…,𝑛;𝑡 = 𝑙,𝑚,𝑢) 0 ≤ 𝛽 ≤ 1 (15) where 𝛽 indicates the minimum satisfaction degree of the fuzzy constraints. for obtaining the optimal fuzzy weight vector w̃∗ = [�̃�1 ∗, �̃�2 ∗,…,�̃�𝑛 ∗], the following mathematical programming model, which maximizes the minimum 𝛽 is proposed (dong et al., 2021). 𝑀𝑎𝑥 𝛽 𝑠.𝑡.{ 𝜇(𝑅(𝑤𝑗 𝑡)) ≥ 𝛽 (𝑗 = 1,2,…,𝑛;𝑡 = 𝑙,𝑚,𝑢) 𝜇(𝑄(𝑤𝑗 𝑡)) ≥ 𝛽 (𝑗 = 1,2,…,𝑛;𝑡 = 𝑙,𝑚,𝑢) 0 ≤ 𝛽 ≤ 1 (16) 3.4.2. solution of the constructed mathematical programming model since eq. (12) and eq. (13) are piecewise functions, the solution of eq. (16) depends on the risk attitude of the decision-maker. therefore, four approaches are proposed to solve eq. (16).  𝐶𝐵 is the best criterion, and its weight �̃�𝐵 should be the maximum. for 𝑅(𝑤𝑗 𝑡) = 𝑤𝐵 𝑡 − 𝑤𝑗 𝑡𝑎𝐵𝑗 𝑡 ≅ 0, an optimistic decision-maker might believes 𝑅(𝑤𝑗 𝑡) = 𝑤𝐵 𝑡 − 𝑤𝑗 𝑡𝑎𝐵𝑗 𝑡 > 0 and for this selects 𝜇(𝑅(𝑤𝑗 𝑡)) = 1 − 𝑅(𝑤𝑗 𝑡 ) 𝑑𝑗 𝑡 as the membership function, i.e., the right side of fig. 2 and a pessimistic decision-maker might believes 𝑅(𝑤𝑗 𝑡) = 𝑤𝐵 𝑡 − 𝑤𝑗 𝑡𝑎𝐵𝑗 𝑡 < 0 and for this chooses 𝜇(𝑅(𝑤𝑗 𝑡)) = 1 + 𝑅(𝑤𝑗 𝑡 ) 𝑑𝑗 𝑡 as the membership function, i.e., the left side of fig. 2.  𝐶𝑊 is the worst criterion, and its weight �̃�𝑊 should be the minimum. for 𝑄(𝑤𝑗 𝑡) = 𝑤𝑗 𝑡 − 𝑎𝑗𝑊 𝑡 𝑤𝑊 𝑡 ≅ 0, an optimistic decision-maker might believes 𝑄(𝑤𝑗 𝑡) = 𝑤𝑗 𝑡 − 𝑎𝑗𝑊 𝑡 𝑤𝑊 𝑡 > 0 and for this chooses 𝜇(𝑄(𝑤𝑗 𝑡)) = 1 − 𝑄(𝑤𝑗 𝑡 ) 𝑞𝑗 𝑡 as the membership function and a pessimistic decision-maker might believes 𝑄(𝑤𝑗 𝑡) = 𝑤𝑗 𝑡 − 𝑎𝑗𝑊 𝑡 𝑤𝑊 𝑡 < 0 and for this selects 𝜇(𝑄(𝑤𝑗 𝑡)) = 1 + 𝑄(𝑤𝑗 𝑡 ) 𝑞𝑗 𝑡 as the membership function. there are also two cases for a neutral decision-maker.  case 1the neutral decision-maker chooses 𝜇(𝑅(𝑤𝑗 𝑡)) = 1 − 𝑅(𝑤𝑗 𝑡 ) 𝑑𝑗 𝑡 and 𝜇(𝑄(𝑤𝑗 𝑡)) = 1 + 𝑄(𝑤𝑗 𝑡 ) 𝑞𝑗 𝑡 as the membership functions for the fuzzy equations 𝑅(𝑤𝑗 𝑡) = 𝑤𝐵 𝑡 − 𝑤𝑗 𝑡𝑎𝐵𝑗 𝑡 ≅ 0 and 𝑄(𝑤𝑗 𝑡) = 𝑤𝑗 𝑡 − 𝑎𝑗𝑊 𝑡 𝑤𝑊 𝑡 ≅ 0 respectively. training aircraft selection for department of flight training in fuzzy environment 275  case 2the neutral decision-maker chooses 𝜇(𝑅(𝑤𝑗 𝑡)) = 1 + 𝑅(𝑤𝑗 𝑡 ) 𝑑𝑗 𝑡 and 𝜇(𝑄(𝑤𝑗 𝑡)) = 1 − 𝑄(𝑤𝑗 𝑡 ) 𝑞𝑗 𝑡 as the membership functions of the fuzzy equations 𝑅(𝑤𝑗 𝑡) = 𝑤𝐵 𝑡 − 𝑤𝑗 𝑡𝑎𝐵𝑗 𝑡 ≅ 0 and 𝑄(𝑤𝑗 𝑡) = 𝑤𝑗 𝑡 − 𝑎𝑗𝑊 𝑡 𝑤𝑊 𝑡 ≅ 0 respectively. for all decision-makers, by plugging 𝜇(𝑅(𝑤𝑗 𝑡)) of eq. (12) and 𝜇(𝑄(𝑤𝑗 𝑡)) of eq. (13), which they have chosen according to their stances above, into eq. (16), and eq. (16) is converted into the following linear programming models. (1) optimistic approach: 𝑀𝑎𝑥 𝛽 𝑠.𝑡. { 1 − 𝑤𝐵 𝑡 −𝑤𝑗 𝑡𝑎𝐵𝑗 𝑡 𝑑𝑗 𝑡 ≥ 𝛽, 0 ≤ 𝑤𝐵 𝑡 − 𝑤𝑗 𝑡𝑎𝐵𝑗 𝑡 ≤ 𝑑𝑗 𝑡 (𝑗 = 1,2,…,𝑛;𝑡 = 𝑙,𝑚,𝑢) 1 − 𝑤𝑗 𝑡−𝑎𝑗𝑊 𝑡 𝑤𝑊 𝑡 𝑞𝑗 𝑡 ≥ 𝛽, 0 ≤ 𝑤𝑗 𝑡 − 𝑎𝑗𝑊 𝑡 𝑤𝑊 𝑡 ≤ 𝑞𝑗 𝑡 (𝑗 = 1,2,…,𝑛;𝑡 = 𝑙,𝑚,𝑢) 0 ≤ 𝛽 ≤ 1 ∑ 𝑤𝑗 𝑚𝑛 𝑗=1 = 1, 𝑤𝑗 𝑢 + ∑ 𝑤𝑖 𝑙𝑛 𝑖=1,𝑖≠𝑗 ≤ 1,𝑤𝑗 𝑙 + ∑ 𝑤𝑖 𝑢𝑛 𝑖=1,𝑖≠𝑗 ≥ 1 (𝑖, 𝑗 = 1,2,…,𝑛) (17) (2) pessimistic approach: 𝑀𝑎𝑥 𝛽 𝑠.𝑡. { 1 + 𝑤𝐵 𝑡 −𝑤𝑗 𝑡𝑎𝐵𝑗 𝑡 𝑑𝑗 𝑡 ≥ 𝛽, −𝑑𝑗 𝑡 ≤ 𝑤𝐵 𝑡 − 𝑤𝑗 𝑡𝑎𝐵𝑗 𝑡 ≤ 0 (𝑗 = 1,2,…,𝑛;𝑡 = 𝑙,𝑚,𝑢) 1 + 𝑤𝑗 𝑡−𝑎𝑗𝑊 𝑡 𝑤𝑊 𝑡 𝑞𝑗 𝑡 ≥ 𝛽, −𝑞𝑗 𝑡 ≤ 𝑤𝑗 𝑡 − 𝑎𝑗𝑊 𝑡 𝑤𝑊 𝑡 ≤ 0 (𝑗 = 1,2,…,𝑛;𝑡 = 𝑙,𝑚,𝑢) 0 ≤ 𝛽 ≤ 1 ∑ 𝑤𝑗 𝑚𝑛 𝑗=1 = 1, 𝑤𝑗 𝑢 + ∑ 𝑤𝑖 𝑙𝑛 𝑖=1,𝑖≠𝑗 ≤ 1,𝑤𝑗 𝑙 + ∑ 𝑤𝑖 𝑢𝑛 𝑖=1,𝑖≠𝑗 ≥ 1 (𝑖, 𝑗 = 1,2,…,𝑛) (18) (3) mixed approach i: 𝑀𝑎𝑥 𝛽 𝑠.𝑡. { 1 − 𝑤𝐵 𝑡 −𝑤𝑗 𝑡𝑎𝐵𝑗 𝑡 𝑑𝑗 𝑡 ≥ 𝛽, 0 ≤ 𝑤𝐵 𝑡 − 𝑤𝑗 𝑡𝑎𝐵𝑗 𝑡 ≤ 𝑑𝑗 𝑡 (𝑗 = 1,2,…,𝑛;𝑡 = 𝑙,𝑚,𝑢) 1 + 𝑤𝑗 𝑡−𝑎𝑗𝑊 𝑡 𝑤𝑊 𝑡 𝑞𝑗 𝑡 ≥ 𝛽, −𝑞𝑗 𝑡 ≤ 𝑤𝑗 𝑡 − 𝑎𝑗𝑊 𝑡 𝑤𝑊 𝑡 ≤ 0 (𝑗 = 1,2,…,𝑛;𝑡 = 𝑙,𝑚,𝑢) 0 ≤ 𝛽 ≤ 1 ∑ 𝑤𝑗 𝑚𝑛 𝑗=1 = 1, 𝑤𝑗 𝑢 + ∑ 𝑤𝑖 𝑙𝑛 𝑖=1,𝑖≠𝑗 ≤ 1,𝑤𝑗 𝑙 + ∑ 𝑤𝑖 𝑢𝑛 𝑖=1,𝑖≠𝑗 ≥ 1 (𝑖, 𝑗 = 1,2,…,𝑛) (19) (4) mixed approach ii: 𝑀𝑎𝑥 𝛽 𝑠.𝑡. { 1 + 𝑤𝐵 𝑡 −𝑤𝑗 𝑡𝑎𝐵𝑗 𝑡 𝑑𝑗 𝑡 ≥ 𝛽, −𝑑𝑗 𝑡 ≤ 𝑤𝐵 𝑡 − 𝑤𝑗 𝑡𝑎𝐵𝑗 𝑡 ≤ 0 (𝑗 = 1,2,…,𝑛;𝑡 = 𝑙,𝑚,𝑢) 1 − 𝑤𝑗 𝑡−𝑎𝑗𝑊 𝑡 𝑤𝑊 𝑡 𝑞𝑗 𝑡 ≥ 𝛽, 0 ≤ 𝑤𝑗 𝑡 − 𝑎𝑗𝑊 𝑡 𝑤𝑊 𝑡 ≤ 𝑞𝑗 𝑡 (𝑗 = 1,2,…,𝑛;𝑡 = 𝑙,𝑚,𝑢) 0 ≤ 𝛽 ≤ 1 ∑ 𝑤𝑗 𝑚𝑛 𝑗=1 = 1, 𝑤𝑗 𝑢 + ∑ 𝑤𝑖 𝑙𝑛 𝑖=1,𝑖≠𝑗 ≤ 1,𝑤𝑗 𝑙 + ∑ 𝑤𝑖 𝑢𝑛 𝑖=1,𝑖≠𝑗 ≥ 1 (𝑖, 𝑗 = 1,2,…,𝑛) (20) the optimal weight vector w̃∗ can be attained by solving eqs. (17) -(20) separately for all decision-makers. each of eqs. (17)–(20) should have a unique optimal solution torğul et al./decis. mak. appl. manag. eng. 5 (1) (2022) 264-289 276 if the tolerance parameters values 𝑑𝑗 𝑡 and 𝑞𝑗 𝑡 (𝑗 = 1,2,…,𝑛;𝑡 = 𝑙,𝑚,𝑢) are big enough, and the bigger values of 𝑑𝑗 𝑡 and 𝑞𝑗 𝑡, provide the bigger value for the optimal objective value 𝛽∗. as per eq. (14), if the attained optimal objective value 𝛽∗= 1, then all of the criteria comparisons are fully consistent, and thus, 𝛽∗can be utilized to measure the consistency level of the criteria comparisons (dong et al., 2021; pamucar & savin, 2020; pamucar & dimitrijevic, 2021). 3.4.3. fuzzy consistency index a comparison is fully consistent when �̃�𝐵𝑗 × �̃�𝑗𝑊 = �̃�𝐵𝑊 for all 𝑗 = 1,2,…,𝑛. on the other hand, it is possible for some 𝑗 which lead to not fully consistent, that is, the following inequality applies (guo & zhao, 2017): �̃�𝐵𝑗 × �̃�𝑗𝑊 ≠ �̃�𝐵𝑊 (21) let ζ̃ = (ζ𝑙, ζ𝑚, ζ𝑢) be a triangular fuzzy number that must be subtracted from �̃�𝐵𝑗 = (𝑎𝐵𝑗 𝑙 ,𝑎𝐵𝑗 𝑚 ,𝑎𝐵𝑗 𝑢 ) and �̃�𝑗𝑊 = (𝑎𝑗𝑊 𝑙 ,𝑎𝑗𝑊 𝑚 ,𝑎𝑗𝑊 𝑢 ) of eq. (21) and added to �̃�𝐵𝑤 = (𝑎𝐵𝑊 𝑙 ,𝑎𝐵𝑊 𝑚 ,𝑎𝐵𝑊 𝑢 ) of eq. (21) to obtain the highest inequality of eq. (21) (dong et al., 2021). that is; ((𝑎𝐵𝑗 𝑙 ,𝑎𝐵𝑗 𝑚 ,𝑎𝐵𝑗 𝑢 ) − (ζ𝑙, ζ𝑚, ζ𝑢)) × ((𝑎𝑗𝑊 𝑙 ,𝑎𝑗𝑊 𝑚 ,𝑎𝑗𝑊 𝑢 ) − (ζ𝑙, ζ𝑚, ζ𝑢)) = (𝑎𝐵𝑊 𝑙 ,𝑎𝐵𝑊 𝑚 ,𝑎𝐵𝑊 𝑢 ) + (ζ𝑙, ζ𝑚, ζ𝑢) (22) as for the minimum consistency �̃�𝐵𝑗 = �̃�𝑗𝑊 = �̃�𝐵𝑊 ; ((𝑎𝐵𝑊 𝑙 ,𝑎𝐵𝑊 𝑚 ,𝑎𝐵𝑊 𝑢 ) − (ζ𝑙, ζ𝑚, ζ𝑢)) × ((𝑎𝐵𝑊 𝑙 ,𝑎𝐵𝑊 𝑚 ,𝑎𝐵𝑊 𝑢 ) − (ζ𝑙, ζ𝑚, ζ𝑢)) = (𝑎𝐵𝑊 𝑙 ,𝑎𝐵𝑊 𝑚 ,𝑎𝐵𝑊 𝑢 ) + (ζ𝑙, ζ𝑚, ζ𝑢) (23) with regards to the operation rules of triangular fuzzy numbers, eq. (23) could be rewritten as follows (dong et al., 2021): (𝑎𝐵𝑊 𝑙 − ζ𝑢,𝑎𝐵𝑊 𝑚 − ζ𝑚,𝑎𝐵𝑊 𝑢 − ζ𝑙) × (𝑎𝐵𝑊 𝑙 − ζ𝑢,𝑎𝐵𝑊 𝑚 − ζ𝑚,𝑎𝐵𝑊 𝑢 − ζ𝑙) = (𝑎𝐵𝑊 𝑙 + ζ𝑙,𝑎𝐵𝑊 𝑚 + ζ𝑚,𝑎𝐵𝑊 𝑢 + ζ𝑢) (24) → ((𝑎𝐵𝑊 𝑙 − ζ𝑢)2, (𝑎𝐵𝑊 𝑚 − ζ𝑚)2, (𝑎𝐵𝑊 𝑢 − ζ𝑙)2) = (𝑎𝐵𝑊 𝑙 + ζ𝑙,𝑎𝐵𝑊 𝑚 + ζ𝑚,𝑎𝐵𝑊 𝑢 + ζ𝑢) (25) thus, the following equations can be derived (dong et al., 2021): { (𝑎𝐵𝑊 𝑙 − ζ𝑢)2 = 𝑎𝐵𝑊 𝑙 + ζ𝑙 (𝑎𝐵𝑊 𝑚 − ζ𝑚)2 = 𝑎𝐵𝑊 𝑚 + ζ𝑚 (𝑎𝐵𝑊 𝑢 − ζ𝑙)2 = 𝑎𝐵𝑊 𝑢 + ζ𝑢 (26) after solving eq. (26), the fci ζ̃ = (ζ𝑙, ζ𝑚, ζ𝑢) is attained for different �̃�𝐵𝑊 values, as displayed in table 2. table 2. fci for fuzzy bwm. �̃�𝑩𝑾 (1, 1, 1) (2/3, 1, 3/2) (3/2, 2, 5/2) (5/2, 3, 7/2) (7/2, 4, 9/2) fci ( 𝜁) (0, 0, 0) (0, 0, 1.36) (0.34, 0.44, 2.16) (0.71, 1, 4.29) (1.31, 1.63, 5.69) �̃�𝑩𝑾 (9/2, 5, 11/2) (11/2, 6, 13/2) (13/2, 7, 15/2) (15/2, 8, 17/2) (17/2, 9, 19/2) fci ( �̃�) (1.96, 2.30, 7.04) (2.65, 3, 8.35) (3.36, 3.73, 9.64) (4.09, 4.47, 10.91) (4.85, 5.23, 12.15) training aircraft selection for department of flight training in fuzzy environment 277 3.4.4. fuzzy consistency ratio to determine the fuzzy consistency ratio (fcr), we require to minimize the maximum among the deviation between �̃�𝐵 ∗ and �̃�𝑗 ∗�̃�𝐵𝑗and the deviation between �̃�𝑗 ∗ and �̃�𝑗𝑊�̃�𝑊 ∗ for all 𝑗 = 1,2,…,𝑛. that is, to calculate 𝑚𝑖𝑛𝑚𝑎𝑥{|�̃�𝐵 ∗ − �̃�𝑗 ∗�̃�𝐵𝑗|,|�̃�𝑗 ∗ − �̃�𝑗𝑊�̃�𝑊 ∗ |} that is represented by 𝜉∗ = (𝜉∗𝑙,𝜉∗𝑚,𝜉∗𝑢). since subtraction, multiplication, and absolute operations of triangular fuzzy numbers are approximate operations, to attain the exact fuzzy deviation 𝜉∗ = (𝜉∗𝑙,𝜉∗𝑚,𝜉∗𝑢) is difficult. therefore, the approach below is proposed to determine the fuzzy deviation. 𝜉′𝑙 = 1 2𝑛 ∑ (|𝑤𝐵 ∗𝑙 − 𝑤𝑗 ∗𝑙𝑎𝐵𝑗 𝑙 | + |𝑤𝑗 ∗𝑙 − 𝑎𝑗𝑊 𝑙 𝑤𝑊 ∗𝑙|),𝑛𝑗=1 (27) 𝜉′𝑚 = 1 2𝑛 ∑ (|𝑤𝐵 ∗𝑚 − 𝑤𝑗 ∗𝑚𝑎𝐵𝑗 𝑚 | + |𝑤𝑗 ∗𝑚 − 𝑎𝑗𝑊 𝑚 𝑤𝑊 ∗𝑚|),𝑛𝑗=1 (28) 𝜉′𝑢 = 1 2𝑛 ∑ (|𝑤𝐵 ∗𝑢 − 𝑤𝑗 ∗𝑢𝑎𝐵𝑗 𝑢 | + |𝑤𝑗 ∗𝑢 − 𝑎𝑗𝑊 𝑢 𝑤𝑊 ∗𝑢|),𝑛𝑗=1 (29) where 𝜉′𝑙, 𝜉′𝑚 and 𝜉′𝑢 describe the possible lower bound, mode and upper bound of the fuzzy deviation 𝜉∗ = (𝜉∗𝑙,𝜉∗𝑚,𝜉∗𝑢), respectively. to ensure 𝜉∗𝑙 ≤ 𝜉∗𝑚 ≤ 𝜉∗𝑢, i.e., the attained 𝜉∗ = (𝜉∗𝑙,𝜉∗𝑚,𝜉∗𝑢) is a triangular fuzzy number, it is taken; 𝜉∗𝑙 = min{𝜉′𝑙,𝜉′𝑚,𝜉′𝑢}, 𝜉∗𝑚 = median{𝜉′𝑙,𝜉′𝑚,v′𝑢} , ζ̃∗𝑢 = max{𝜉′𝑙,𝜉′𝑚,𝜉′𝑢}, (30) the aim of eq. (30) is to assure 𝜉∗𝑙 ≤ 𝜉∗𝑚 ≤ 𝜉∗𝑢, such that the fuzzy deviation 𝜉∗ = (𝜉∗𝑙,𝜉∗𝑚,𝜉∗𝑢) is a triangular fuzzy number. fcr is identified as 𝐹𝐶𝑅 = �̃�∗ ζ̃ (31) where is demonstrated in table 2 and 𝜉∗ = (𝜉∗𝑙,𝜉∗𝑚,𝜉∗𝑢) is attained by eq. (30). by the operation rules of triangular fuzzy numbers; 𝐹𝐶𝑅 = �̃�∗ ζ̃ = (𝜉∗𝑙,𝜉∗𝑚,𝜉∗𝑢) (ζ𝑙,ζ𝑚,ζ𝑢) = ( 𝜉∗𝑙 ζ𝑢 , 𝜉∗𝑚 ζ𝑚 , 𝜉∗𝑢 ζ𝑙 ) (32) then, based on eq. (2), we could calculate gmir r(fcr) of the fcr; 𝑅(𝐹𝐶𝑅) = 1 6 ( 𝜉∗𝑙 ζ𝑢 ,4 𝜉∗𝑚 ζ𝑚 , 𝜉∗𝑢 ζ𝑙 ) (33)  if 𝑅(𝐹𝐶𝑅) ≤ 0.1, then the comparisons are acceptable consistent.  if 𝑅(𝐹𝐶𝑅) = 0, then all comparisons are fully consistent.  if 𝑅(𝐹𝐶𝑅) > 0.1, then the comparisons are unacceptable consistent and some of the comparisons must be identified to be adjusted until 𝑅(𝐹𝐶𝑅) ≤ 0.1 (dong et al., 2021). for this, the identification and adjustment processes are described in detail in dong et al. (2021). 3.4.5. steps of fuzzy bwm step 1: determine a set 𝐶 = (𝐶1,𝐶2,… ,𝐶𝑛) of decision criteria. torğul et al./decis. mak. appl. manag. eng. 5 (1) (2022) 264-289 278 step 2: determine the best (e.g., the most important or the most desirable) criterion 𝐶𝐵 and the worst (e.g., the least important or the least desirable) criterion 𝐶𝑊. step 3: determine the fuzzy preference of the best criterion overall the other criteria using the linguistic terms and triangular fuzzy numbers listed in table 3. the resulting best-to-others vector would be �̃�𝐵 = [�̃�𝐵1, �̃�𝐵2,…, �̃�𝐵𝑛] where �̃�𝐵𝑗 demonstrates the fuzzy preference of the best criterion 𝐶𝐵 over criterion 𝐶𝑗. �̃�𝐵𝑗 = (𝑎𝐵𝑗 𝑙 ,𝑎𝐵𝑗 𝑚 ,𝑎𝐵𝑗 𝑢 ), 𝑗 = 1,2,…,𝑛 and �̃�𝐵𝐵 = (1,1,1). step 4: determine the fuzzy preference of all the other criteria over the worst criterion using the linguistic terms and triangular fuzzy numbers listed in table 3. the resulting others-to-worst vector would be �̃�𝑊 = [�̃�1𝑊, �̃�2𝑊,…, �̃�𝑛𝑊] where �̃�𝑗𝑊 indicates the fuzzy preference of criterion 𝐶𝑗 over the worst criterion 𝐶𝑊. �̃�𝑗𝑊 = (𝑎𝑗𝑊 𝑙 ,𝑎𝑗𝑊 𝑚 ,𝑎𝑗𝑊 𝑢 ), 𝑗 = 1,2,…,𝑛 and �̃�𝑊𝑊 = (1,1,1) (dong et al., 2021; guo & zhao, 2017; rezaei, 2015). step 5: determine the suitable tolerance parameters’ values 𝑑𝑗 𝑡 and 𝑞𝑗 𝑡 (𝑗 = 1,2,…,𝑛;𝑡 = 𝑙,𝑚,𝑢) for eqs. (17)– (20) according to your preference and the decision-making problems’ characteristics. in general, 𝑑𝑗 𝑡 and 𝑞𝑗 𝑡 can take any values from the interval [1, 9]. step 6: solve one of eqs. (17)– (20) according to the risk attitude (i.e., pessimistic, optimistic or neutral) of the decision-maker to get the optimal fuzzy weight vector w̃∗ = [�̃�1 ∗, �̃�2 ∗,… ,�̃�𝑛 ∗] and the optimal objective value 𝛽∗ by using a mathematical software. step 7: compute 𝜉∗ = (𝜉∗𝑙,𝜉∗𝑚,𝜉∗𝑢) by eq. (30). step 8: attain the fci by table 2 and calculate fcr by eq. (32). step 9: calculate gmir r(fcr) of the attained fcr by eq. (33). step 10: check the consistency (dong et al., 2021). in the step of final ranking of alternatives, with optimal defuzzified weights of the criteria and the normalized scores of the alternatives on the different criteria, x𝑖𝑗; the final aggregate score per alternative, 𝑉𝑖; could be calculated using eq. (34) (rezaei et al., 2016); 𝑉𝑖 = ∑ 𝑊𝑗 x𝑖𝑗𝑗 (34) x𝑖𝑗 = { x𝑖𝑗 max { x𝑖𝑗} , 𝑖𝑓 𝑥 𝑖𝑠 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒(𝑠𝑢𝑐ℎ 𝑎𝑠 𝑞𝑢𝑎𝑙𝑖𝑡𝑦), 1 − x𝑖𝑗 max{ x𝑖𝑗} , 𝑖𝑓 𝑥 𝑖𝑠 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒(𝑠𝑢𝑐ℎ 𝑎𝑠 𝑝𝑟𝑖𝑐𝑒). (35) 4. problem definition an aircraft selection model was designed to provide training aircraft for a flight training department that will start education. for this, in the study, potential training aircraft will be evaluated in terms of important criteria for decision-makers and will decide the number of aircraft to be purchased, taking into account the current constraints and the aircraft weights obtained. the flight training department is a 4-year undergraduate department that trains professional pilots with the necessary skills, competence, theoretical and practical experience to meet national and international airline companies' needs. in addition to training aircraft selection for department of flight training in fuzzy environment 279 laboratories, students carry out their practical training using training aircraft. pilots who graduate from this department can find the opportunity to work in many general aviation sectors, especially in domestic and foreign airlines. to deal with this decision problem, that is essential to consider the candidate aircraft and analyze the criteria requirements representing the technical specification that the aircraft should have indepth. this study suggests a two-stage solution approach for aircraft evolution. in the first stage, fuzzy bwm is used to get the weights of criteria and then aircraft ranking. then, the linear programming formulation of aircraft selection is constructed. in the second stage, aircraft’ weights (priority scores) are combined into the linear programming model with some resource constraints to determine the optimal order number of aircraft. the aircraft’ weights are utilized as coefficients in the objective function to increase purchasing value and how much will be ordered from which plane is determined. all information such as budget, total flying time, and total fuel consumption was assumed fixed and already known. 4.1. mathematical model index: i: set of aircraft (𝑖 = 1,2,…,𝑁) parameters: 𝑃𝑖: unit purchasing cost of aircraft i 𝑇𝑖 : actual flying time before overhaul of aircraft i 𝐹𝑖 : fuel cost per mile of aircraft i 𝑊𝑖: the weight (priority value) of the aircraft i b: total budget allocated to aircraft m: total flying time before overhaul y: total fuel consumption cost per mile decision variable: 𝑋𝑖 : number of aircraft i objective function: 𝑀𝑎𝑥 𝑍= ∑ 𝑊𝑖 𝑋𝑖 𝑁 𝑖 (36) constraints: ∑ 𝑃𝑖𝑋𝑖 𝑁 𝑖 ≤ b (37) ∑ 𝑇𝑖 𝑋𝑖 𝑁 𝑖 ≥ m (38) ∑ 𝐹𝑖 𝑋𝑖 𝑁 𝑖 ≤ y (39) 𝑋𝑖 ≥ 0 𝑎𝑛𝑑 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 , ∀𝑖 (40) the objective function (36) maximizes the total purchasing value. in other words, it allows purchasing aircraft with a higher weight, which means that the best aircraft, according to the criterion evaluation, will be purchased more. constraint (37) is the budget constraint implies that the total purchasing cost of aircraft cannot exceed the allocated budget amount. constraint (38) is the minimum flying time before overhaul constraint means that no aircraft need overhaul until the specified maintenance time, in other words, the flying time of the aircraft should be at least the minimum flying torğul et al./decis. mak. appl. manag. eng. 5 (1) (2022) 264-289 280 time. constraint (39) is the fuel consumption constraint means that the amount of fuel burned by aircraft per mile should not exceed the budget allocated for it. constraint (40) is the non-negativity and integrity constraint. 5. case study: necmettin erbakan university, faculty of aviation and space sciences necmettin erbakan university, faculty of aviation and space sciences, was established in 2010. flight training undergraduate program is a department of the faculty of aviation and space sciences. the faculty also includes aircraft engineering, space and satellite engineering, and aviation management departments. departments other than the flight training department actively give education. the flight training department has not started education yet, as it does not have sufficient infrastructure. the university wants to activate its flight training department and so should be meet needs such as runway, training aircraft, instructors, maintenance technicians. therefore, for this reason, the university should first determine examining suitable criteria and alternatives, and expert decision-makers should evaluate training aircraft alternatives, and this is the subject of our study. the criteria and alternatives for the problem were examined in line with the fuzzy bwm, and the evaluation process was initiated by the instructors’ committee consisting of experts. after the preliminary screening, the group of experts (instructors of the flight training department) identified nine specification criteria (max cruise speed, max range, take-off and landing ground roll, max climb rate, power output, empty weight, price, useful fuel capacity, time before overhand) and eight training aircraft (cessna skyhawk sp (172s), cessna skylane (182t), cessna turbo stationair hd (206), cirrus sr22, cirrus sr20, diamond da62, diamond da40 ng, diamond da42) for the further evaluation process. 5.1. implementing the fuzzy bwm the importance weights for criteria are described with linguistic variables by the decision-makers. the linguistic expressions and their corresponding triangular fuzzy numbers indicating the importance ratings of criteria are given in table 3. table 3. linguistic variables and fuzzy numbers used in the evaluation of criteria (dong et al., 2021; gan et al., 2019; guo & zhao, 2017). linguistic variables for the importance weight of each criterion linguistic variables triangular fuzzy numbers equally important (ei) (1, 1, 1) weakly important (wi) (2/3, 1, 3/2) intermediate-weakly to moderately important(wm) (3/2, 2, 5/2) moderately important (mi) (5/2, 3, 7/2) intermediate-moderately to strongly important(ms) (7/2, 4, 9/2) strongly important (si) (9/2, 5, 11/2) intermediate-strongly to very important(sv) (11/2, 6, 13/2) very important (vi) (13/2, 7, 15/2) intermediate-very to extremely important(ve) (15/2, 8, 17/2) extremely important (eei) (17/2, 9, 19/2) training aircraft selection for department of flight training in fuzzy environment 281 the best criterion is the most important one, while the worst criterion is the least important one in aircraft selection based on the opinion of an expert/decision-maker. as a result of interviews with experts, the best criterion was determined as price, and the worst criterion was determined as empty weight for aircraft selection. next, they performed the criteria comparison by filling out a survey based on the application of the fuzzy bwm, as shown in table 4. table 4. pairwise comparison vectors for best and worst criteria bo max cruise speed max range take-off and landing ground roll max climb rate power output empty weight price useful fuel capacity time before overhand best objective functions: price vi ms wm vi ms eei ei sv si ow worst objective functions: empty weight max cruise speed mi max range sv take-off and landing ground roll ve max climb rate ms power output sv empty weight ei price eei useful fuel capacity mi time before overhand ms then, the fuzzy best-to-others vector, �̃�𝐵 = [�̃�𝐵1, �̃�𝐵2,… , �̃�𝐵9] where �̃�𝐵1= (13/2, 7, 15/2), �̃�𝐵2= (7/2, 4, 9/2), �̃�𝐵3= (3/2, 2, 5/2), �̃�𝐵4= (13/2, 7, 15/2), �̃�𝐵5= (7/2, 4, 9/2), �̃�𝐵6= (17/2, 9, 19/2), �̃�𝐵7= (1, 1, 1), �̃�𝐵8= (11/2, 6, 13/2) and �̃�𝐵9= (9/2, 5, 11/2), and the fuzzy others-to-worst vector, �̃�𝑊 = [�̃�1𝑊, �̃�2𝑊,…, �̃�9𝑊] where �̃�1𝑊= (5/2, 3, 7/2), �̃�2𝑊= (11/2, 6, 13/2), �̃�3𝑊= (15/2, 8, 17/2), �̃�4𝑊= (7/2, 4, 9/2), �̃�5𝑊= (11/2, 6, 13/2), �̃�6𝑊= (1, 1, 1), �̃�7𝑊= (17/2, 9, 19/2), �̃�8𝑊= (5/2, 3, 7/2) and �̃�9𝑊= (7/2, 4, 9/2) are obtained according to table 4. the values of all tolerance parameters 𝑑𝑗 𝑡 and 𝑞𝑗 𝑡 (𝑗 = 1,2,…,𝑛;𝑡 = 𝑙,𝑚,𝑢) are taken as 1. four separate linear programming models are constructed for all decision-makers by putting fuzzy preferences �̃�𝐵j, �̃�𝑗w and tolerance parameters 𝑑𝑗 𝑡 , 𝑞𝑗 𝑡 given above into eqs. (17)-(20), and the optimal fuzzy weights w̃𝑗 ∗ and minimal satisfaction degrees β are obtained by using the gams/cplex 24.0 software separately for all approaches as seen in table 5. then the values of the fuzzy deviations 𝜉∗, fcr and r(fcr) for all approaches are calculated separately based on eqs. (30), (32), and (33) respectively and presented in table 5. since all r(fcr) < 0.1, obtained according to the four approaches, it can be seen that the comparisons for all decision makers' approaches are acceptably consistent. in our study, considering that the decision-maker has the optimistic approach with the best consistency, these weights obtained by this approach will be used in the ranking. in this context, the optimistic approach's fuzzy weights are defuzzified using eq. (2) as follows, and thus they are ready for use in the next step. �̃�1 ∗= 0.052, �̃�2 ∗= 0.087, �̃�3 ∗= 0.161, �̃�4 ∗= 0.052, �̃�5 ∗= 0.083, �̃�6 ∗= 0.012, �̃�7 ∗= 0.396, �̃�8 ∗= 0.060 and �̃�9 ∗= 0.087. torğul et al./decis. mak. appl. manag. eng. 5 (1) (2022) 264-289 282 table 5. criteria weights (�̃�𝑗 ∗) and minimal satisfaction degrees (β) according to the attitude of the decision-maker criteria the attitude of the decision-maker optimistic approach pessimistic approach mixed approach i mixed approach ii max cruise speed (0.042, 0.054, 0.054) (0.043, 0.063, 063) (0.025, 0.033, 0.054) (0.060, 0.069, 0.077) max range (0.077, 0.089, 0.089) (0.081, 0.085, 0.105) (0.048, 0.091, 0.091) (0.111, 0.129, 0.129) take-off ground roll (0.161, 0. 161, 0. 161) (0.188, 0.188, 0.188) (0.120, 0.163, 0.163) (0.214, 0. 231, 0. 231) max climb rate (0.042, 0.054, 0.054) (0.043, 0.063, 0.063) (0.025, 0.054, 0.054) (0.060, 0.077, 0.077) power output (0.077, 0.083, 0.089) (0.081, 0.085, 0.105) (0.048, 0.091, 0.091) (0.111, 0.129, 0.129) empty weight (0.012, 0.012, 0.012) (0.033, 0.036, 0.036) (0.043, 0.043, 0.043) (0.017, 0.017, 0.017) price (0.390, 0. 396, 0. 402) (0.283, 0.322, 0.342) (0.366, 0. 387, 0. 409) (0.146, 0. 154, 0. 163) useful fuel capacity (0.050, 0.062, 0.062) (0.051, 0.072, 0.072) (0.039, 0.063, 0.063) (0.072, 0.089, 0.089) time before overhand (0.077, 0.089, 0.089) (0.063, 0.086, 0.086) (0.074, 0.074, 0.074) (0.105, 0.105, 0.105) β 0.711 0.871 0.798 0.584 ξ̃∗ (0.084, 0.057, 0.050) (0.026, 0.063, 0.088) (0.012, 0.063, 0.078) (0.108, 0.157, 0.181) fcr (0.007, 0.011, 0.010) (0.002, 0.012, 0.018) (0.010, 0.012, 0.016) (0.009, 0.030, 0.037) r(fcr) 0.010 0.011 0.012 0.028 fci(ζ̃) (4.85, 5.23, 12.15) from table 2, because abw=a76= (17/2, 9, 19/2) the following steps have been implemented to establish the aircraft selection framework; first, the aircraft performance on the different criteria is determined by its service centers and decision-makers of the relevant case institution using the technical specification data for aircraft. the scores of aircraft alternatives are shown in table 6. the data of the training aircraft were obtained in light of the information shared on the web pages of the manufacturing companies (cessna aircraft, 2021; circus aircraft, 2021; diamond aircraft, 2021). table 6. decision matrix of aircraft performance specification criteria 1 2 3 4 5 6 7 8 9 aircraft alternatives max cruise speed (ktas) max range (nm) take-off and landing ground roll (ft) max climb rate (fpm) power output (hp) empty weight (lbs) price ($) useful fuel capacity (gal) time before overhaul (hours) 1 cessna skyhawk sp (172s) 124 640 960 730 180 1690 415000 53 2000 2 cessna skylane (182t) 145 915 795 924 230 2000 530000 87 2000 3 cessna turbo stationair hd (206) 161 703 1060 960 310 2365 745000 87 2000 4 cirrus sr22 183 1169 1082 1270 310 2272 654900 92 2000 5 cirrus sr20 155 709 1685 781 215 2122 474900 56 2000 6 diamond da62 192 1283 1574 1029 180 3505 1290000 86 1800 7 diamond da40 ng 154 940 1214 1690 180 1984 535000 48 2000 8 diamond da42 197 1215 919 1550 168 3109 869000 76,4 1800 then, the aircraft scores are normalized using eq. (35). the normalized scores are summarized in table 7. training aircraft selection for department of flight training in fuzzy environment 283 table 7. normalized decision matrix specification criteria 1 2 3 4 5 6 7 8 9 aircraft alternatives max cruise speed (ktas) max range (nm) take-off and landing ground roll (ft) max climb rate (fpm) power output (hp) empty weight (lbs) price ($) useful fuel capacity (gal) time before overhaul (hours) 1 cessna skyhawk sp (172s) 0.629 0.499 0.430 0.432 0.581 0.518 0.678 0.576 1.000 2 cessna skylane (182t) 0.736 0.713 0.528 0.547 0.742 0.429 0.589 0.946 1.000 3 cessna turbo stationair hd (206) 0.817 0.548 0.371 0.568 1.000 0.325 0.422 0.946 1.000 4 cirrus sr22 0.929 0.911 0.358 0.751 1.000 0.352 0.492 1.000 1.000 5 cirrus sr20 0.787 0.553 0.000 0.462 0.694 0.395 0.632 0.609 1.000 6 diamond da62 0.975 1.000 0.066 0.609 0.581 0.000 0.000 0.935 0.900 7 diamond da40 ng 0.782 0.733 0.280 1.000 0.581 0.434 0.585 0.522 1.000 8 diamond da42 1.000 0.947 0.455 0.917 0.542 0.113 0.326 0.830 0.900 weights of criteria (optimistic) 0.052 0.087 0.161 0.052 0.083 0.012 0.396 0.060 0.087 finally, weighted normalized scores (table 8) and then the overall scores of the alternatives are found using eq. (34) and, the result is summarized in table 9. table 8. weighted normalized decision matrix specification criteria 1 2 3 4 5 6 7 8 9 aircraft alternatives max cruise speed (ktas) max range (nm) take-off and landing ground roll (ft) max climb rate (fpm) power output (hp) empty weight (lbs) price ($) useful fuel capacity (gal) time before overhaul (hours) 1 cessna skyhawk sp (172s) 0.033 0.043 0.069 0.022 0.048 0.006 0.269 0.035 0.087 2 cessna skylane (182t) 0.038 0.062 0.085 0.028 0.062 0.005 0.233 0.057 0.087 3 cessna turbo stationair hd (206) 0.042 0.048 0.060 0.030 0.083 0.004 0.167 0.057 0.087 4 cirrus sr22 0.048 0.079 0.058 0.039 0.083 0.004 0.195 0.060 0.087 5 cirrus sr20 0.041 0.048 0.000 0.024 0.058 0.005 0.250 0.037 0.087 6 diamond da62 0.051 0.087 0.011 0.032 0.048 0.000 0.000 0.056 0.078 7 diamond da40 ng 0.041 0.064 0.045 0.052 0.048 0.005 0.232 0.031 0.087 8 diamond da42 0.052 0.082 0.073 0.048 0.045 0.001 0.129 0.050 0.078 table 9. outranking of alternative aircraft aircraft scores normal weights ranks 1 cessna skyhawk sp (172s) 0.605 0.134 3 2 cessna skylane (182t) 0.658 0.144 1 3 cessna turbo stationair hd (206) 0.577 0.126 5 4 cirrus sr22 0.653 0.143 2 5 cirrus sr20 0.549 0.120 7 6 diamond da62 0.363 0.079 8 7 diamond da40 ng 0.605 0.132 4 8 diamond da42 0.559 0.122 6 the ranking of the aircraft in the order are cessna skylane (182t), cirrus sr22, cessna skyhawk sp (172s), diamond da40 ng, cessna turbo stationair hd (206), diamond da42, cirrus sr20 and, diamond da62. the result of the proposed method, cessna skylane (182t) should be preferred if it is considered to purchase only one aircraft type. torğul et al./decis. mak. appl. manag. eng. 5 (1) (2022) 264-289 284 in the next stage, to perform optimal order numbers, the aircraft’s weights based on the results of fuzzy bwm will be used as coefficients in the proposed linear programming model. 5.2. implementing the proposed model aircraft quantitative information (𝑷𝒊 = price($), 𝑻𝒊 = time before overhaul(hours), 𝑭𝒊 = useful fuel capacity max range ∗ 6,44($/𝑛𝑚)) is given in table 6 (1 gallon fuel fee is calculated as 6,44$), and the weights of aircraft have been obtained as a result of the fuzzy bwm. additionally, the maximum acceptable total budget value (b) is taken 6000000 $ and the total flying time before overhaul value (m) is taken 25000 h, and the maximum acceptable total fuel consumption cost per mile value (y) is taken 1 $/nm in the model. the linear formulation of the case problem is presented as: z𝑚𝑎𝑥 = 0.134x1 + 0.144x2 + 0.126x3 + 0.143x4 + 0.120x5+ 0.079x6 + 0.132x7 + 0.122x8 subject to; 415000x1 + 530000x2 + 745000x3 + 654900x4 + 474900x5 + 1290000x6 + 535000x7 + 869000x8 ≤ 6000000 2000𝑥1 + 2000𝑥2 + 2000𝑥3 + 2000𝑥4 + 2000𝑥5 + 1800x6 + 2000x7 + 1800x8 ≥ 25000 0.535x1 + 0.612x2 + 0.799x3 + 0.509x4 + 0.509x5 + 0.431x6 + 0.328x7 + 0.406x8 ≤ 1 x1 ≥ 0,x2 ≥ 0,x3 ≥ 0,x4 ≥ 0,x5 ≥ 0, x6 ≥ 0,x7 ≥ 0,x8 ≥ 0 x1, x2, x3, x4, 𝑥5, 𝑥6,𝑥7,𝑥8 are integer. the linear programming model was solved by gams/cplex 24.0 software package in accordance with these data, and the following results were obtained in table 10. table 9. the optimal solution. z𝑚𝑎𝑥 x1 x2 x3 x4 x5 x6 x7 x8 1.746 9 1 0 0 0 0 3 0 according to the solution results, it has been decided to purchase 13 aircraft, nine from cessna skyhawk sp (172s), one from cessna skylane (182t), and three from diamond da40 ng aircraft. the proposed model decided to purchase the cessna skylane (182t) aircraft with the largest weight, cessna skyhawk sp (172s) aircraft with 3rd weight, and diamond da40 ng aircraft with 4th weight to maximize the total value of purchasing. on the other hand, the reason for not buying the cirrus sr22 aircraft, the 2nd in the weight ranking, is that its price value is higher than our budget constraint. instead of the cirrus sr22 aircraft, the reason for choosing the diamond da40 ng aircraft is that it has a low price and low fuel consumption. cessna turbo stationair hd (206), mainly because of its high fuel consumption, diamond da62 and diamond da42 because of high purchase prices, were not preferred. training aircraft selection for department of flight training in fuzzy environment 285 as a result, it can be said that 13 aircraft will be sufficient for the flight training department to start education, and the university is in a position to cover all expenses of the aircraft to be purchased, such as purchasing, maintenance, and fuel with its existing resources. 6. conclusions aircraft selection is a complex process and an important mcdm problem that considers various fundamental issues. in this context, an appropriate solution method should help top management efficiently evaluate various aircraft alternatives based on consistent criteria (yilmaz et al., 2020). in this paper, we offered a linear programming model for the training aircraft selection problem. we specified the important specification and considered them as aircraft selection criteria. the problem maximizes the number of best aircraft (resulting from the criterion evaluation) to be purchased, while satisfying the budget requirement, fuel consumption limit, and flying time performance constraints. firstly, we used fuzzy bwm to get the weights for criteria and then used them to evaluate the possible aircraft. next, we improved a linear programming model to attain the optimum solution for the problem. finally, we validated the model with a case study by solving it via gams/cplex 24.0 software. we believe that the proposed model framework is sufficiently valid and strong and could be easily applied in applications for a wide variety of decision-making problems. in the literature, the bwm method has proven useful in various problems, but it has been applied for the first time in training aircraft selection. this study can be evolved into a commercial aircraft selection study for airline companies by differentiating the criteria in the case study of the proposed approach. the limitation of the study is that it is limited to nine criteria and eight aircraft alternatives. in future studies, different rankings can be obtained for alternatives by increasing the number of criteria, alternatives, and experts. in particular, all criteria cover the technical features of the aircraft. for example, qualifications such as flight training, the experience of decision-makers, and ergonomics can also be considered as criteria. moreover, unlike the purchase cost, after the aircraft is purchased, costs such as operation and maintenance occur to the purchasing institution or person. however, this information is not provided by the manufacturers as open source. in addition, there are not enough studies in the literature on the selection of trainer aircraft. for this reason, accessible manufacturer data were considered in the training aircraft purchase process. the limitation of the proposed approach is that the input data expressed in linguistic terms is based on decision-makers' opinions and experiences and therefore includes subjectivity. decision-makers practically might not have complete and certain information about objectives and constraints; 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(1965). fuzzy sets. information and control, 8, 338–353. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 2, issue 2, 2019, pp. 112-125. issn: 2560-6018 eissn:2620-0104 doi: https://doi.org/10.31181/dmame1902089b * corresponding author. e-mail addresses:partha4187@gmail.com (p. s. barma), joy77deep@yahoo.co.in (j. dutta), mukherjee.anupam.bnk@gmail.com (a. mukherjee) a 2-opt guided discrete antlion optimization algorithm for multi-depot vehicle routing problem partha sarathi barma 1*, joydeep dutta 1 and anupam mukherjee 2 1 department of computer science and engineering, nshm knowledge campus durgapur, india 2 department of mathematics, national institute of technology durgapur, india received: 25 january 2019; accepted: 30 april 2019; available online: 10 may 2019. original scientific paper abstract: the multi-depot vehicle routing problem (mdvrp) is a real-world variant of the vehicle routing problem (vrp) where the customers are getting service from some depots. the main target of mdvrp is to find the route plan of each vehicle for all the depots to fulfill the demands of all the customers, as well as that, needs the least distance to travel. here all the vehicles start from different depots and return to the same after serving the customers in its route. in mdvrp each customer node must be served by only one vehicle which starts from any of the depots. in this paper, we have considered a homogeneous fleet of vehicles. here a bio-inspired metaheuristic method named discrete antli-on optimization algorithm (dalo) followed by the 2-opt algorithm for local searching is used to minimize the total routing distance of the mdvrp. the comparison with the genetic algorithm, ant colony optimization, and known best solutions is also discussed and analyzed. key words: multi depot vehicle routing problem, antlion optimization (alo), bio-inspired algorithm, combinatorial optimization. 1. introduction supply of goods from source to destination is a challenging operational process in the logistic distribution system. the products can be delivered either directly from the production center or from the stock points located nearby the production site or via distribution warehouses. such kind of problems can be mathematically modeled as a particular type of vrp which belongs to the set of np-hard problems. it consists of a single depot or warehouse to service the demands of different cities, but most of the cases the different company has more than one warehouse to serve the demands. mailto:joy77deep@yahoo.co.in mailto:mukherjee.anupam.bnk@gmail.com a 2-opt guided discrete antlion optimization algorithm for multi-depot vehicle routing problem 113 in such a scenario the problem can be formulated using more than one depot that is called multi-depot vehicle routing problem, in short mdvrp. mdvrp deals with the delivery of items to all the customers with minimum cost or distance. vrp can be used to manage such kind of scenario efficiently. the main task of the basic form of vehicle routing problem is to search the collection of paths to serve customers with some similar vehicles. in the classic form of vrp, a set of customer node is present, the demands of each node and other primary information such as the distance between all pair of nodes, the distance between nodes and depots, number of vehicles and vehicle capacity are known a priory. the vrp can be closed or open. in closed vrp (laporte et al. 1987) vehicles move from a central point called depot, serves each customer and back to the central position such that the total demand served by one conveyance is less than the vehicle capacity. whereas in the case of open vrp (li et al. 2007) after serving the customer the vehicle does not return to the depot. there are many variants of vrp found in the literature; some of them are capacitated vrp (cvrp), vrp with time window (vrptw), vrp that includes pickup and delivery, multi-depot vrp, stochastic vrp, etc. in this paper, we have focused on multi-depot vrp (mdvrp). the pictorial representation of mdvrp is presented in figure 1. in mdvrp, there will be more than one depot. for solving mdvrp, the following two steps can be used: i clustering: allocation of cities to a depot. ii routing: finding the optimum routes for each depot. this sub-problem is similar to vrp. figure1. pictorial representation of mdvrp mdvrp can be solved in two ways considering the two sub-problems, one is route first cluster second, and another is cluster first route second. here we have discretized the ant lion optimization (alo) algorithm to solve the mdvrp. for local searching of routes, the 3-opt algorithm is used. the main contribution of this article is as follows: (1) an improved discrete alo has been proposed to fit the mdvrp; (2) a new encoding scheme to form a solution (ant or antlion) and (3) a hybridization of alo and 2 opt algorithm. barma et al./decis. mak. appl. manag. eng. 2 (2) (2019) 112-125 114 the paper is arranged as per the below sections. the literature review presents in section 2. the motivation behind this work is explained in section 3. section 4 describes the mathematical model for the mdvrp. section 5 deals with the proposed discrete alo. the result and discussion are presented in section 6. the conclusion is in section 7. 2. literature review some of the solving techniques for single depot vrp are exact algorithms like brunch and bound, branch and cut proposed by fisher et al., (1994) and lada et al. in 2001. many heuristic algorithms like cluster first route second (taillard, 1993), savings algorithm (clarke & wright, 1964) also found in the literature. metaheuristic like ga (berger & barkaoui, 2003), pso (chen et al., 2006), aco (reimann et al., 2004) are also used by many researchers to solve single depot vrp. laporte et al., (1984, 1988) formulated the integer linear programs for mdvrp containing degree constraints, sub-tour elimination constraints, chain-barring constraints, and integrality constraints and presented an exact solution. renaud et al., (1996) presents a tabu search heuristic for mdvrp. chao et al. solved the mdvrp using a multi-phase heuristic approach. ombuki-berman & hanshar (2009) applied a genetic algorithm to mdvrp. vianna et al., (1999) proposed an evolutionary algorithm coupled with local search heuristic to minimize the total cost. matos and oliveira (2004) have to use ant colony optimization (aco) to solve mdvrp. guimarães et al., (2019) have published a paper on the multi-depot inventoryrouting problem with the application on a two-echelon (2e) supply chain. it is also showing a stricter policy for inventory management. in 2017, a different version of mdvrp was developed that deals with hazardous materials by yuan et al., (2017). it was solved using a two-stage heuristic method. in the same year, rabbouch et al., (2017) have published a survey paper on mdvrp for heterogeneous vehicles. it also considered the time windows concept. very recently lalla-ruiz & vob (2019) have developed multi-depot cumulative capacitated vrp. it also designed a meta-heuristic approach (popmusic) to solve it. in 2018, one more paper has also been published on mdvrp, and it has been solved using general variable neighborhood search metaheuristic (bezerra et al., 2018). it uses a local search method named randomized variable neighborhood descent. li et al., (2018) have presented a paper on mdvrp with fuel consumption to make the benefits analysis. it finds the factors that affect the benefit ratio. in the same year, one more paper on mdvrp has also been published that deals with multi-compartment vehicles. it uses the hybrid adaptive large neighborhood search (alinaghian & shokouhi, 2018) to solve the problem. one more new variety of mdvrp has been proposed by zhou et al., (2018). they have developed two –echelon mdvrp that introduces the last mile distribution in the city logistics problem. it has been solved using a hybrid multi-population genetic algorithm. silva et al., (2018) have presented a paper on multi-depot online vehicle routing with a soft boundary. recently zhang et al., (2019) have published an article on mdvrp for routing alternate fuel vehicles. they have used the ant colony method. very recently dutta et al., (2019) have designed a modified version of kruskal's algorithm over the ga to solve ovrp for a single depot problem. mukherjee et al., (2019) have developed a special version of the tsp problem that can be mapped on several real-life scenarios. a 2-opt guided discrete antlion optimization algorithm for multi-depot vehicle routing problem 115 3. motivation there are several works that have already been published in the field of vrp using the exact method and meta-heuristics algorithms. but most of the real-life problems fit with the mdvrp. e.g., newspaper distribution, courier services, emergency services, taxi services, and refuse-collection management, etc. in literature, there are some works on mdvrp but in most of the cases they used metaheuristic algorithms, and in few cases, exact algorithms were used. exact algorithms give better result but take longer computational time. meta-heuristic algorithms take less computational time but will not provide the best solution always. so finding good meta-heuristic to address the real-life problem which will give better result in reasonable computational time is a tough job. so here we try to find a hybrid algorithm which will combine an exact algorithm and one meta-heuristic algorithm to address mdvrp. two competitive firms produce two substitute products and sell their products separately in the market. 4. mathematical model the mdvrp can be represented using a graph g = (v, e) where v is the union of two subsets namely, vc = {v1, . . . ,vn} the set of city or customer and vd = {vn+1,..., vn+m} the set of depots, and e is the edge set. a cost or distance matrix c= {cij} is the cost of traveling from city i to city j. each city vi has a demand qi. in this paper symmetric cost or distance matrix is considered and triangular inequality also satisfied in c. here all depots have a finite set of homogeneous vehicles with capacity q. the solution to an mdvrp consists of a set of vehicle routes each starts and ends at the same depot, and each customer node is visited exactly once by only one vehicle. the total demand of customers in each route must not exceed the vehicle capacity q. here the goal is to minimize the total routing cost. in this problem, n nodes are grouped into m cluster where each cluster contain ni: i = 1,2,…,m number of node and each ni clusters are again group by kj groups depending on the vehicle capacity. the mathematical model for mdvrp proposed by lang is given below. 𝑀𝑖𝑛 𝑍 = ∑ ∑ ∑ ∑ 𝑐𝑖𝑗𝑥𝑖𝑗𝑝𝑞 𝑛 𝑗=1 𝑛 𝑖=1 𝑘𝑝 𝑞=1 (1) 𝑚 𝑝=1 subject to ∑ 𝑞𝑖𝑦𝑖𝑞𝑝 𝑛 𝑗=1 ≤ 𝑄 (2) 0 ≤ 𝑛𝑗𝑞 ≤ 𝑛𝑗 (3) ∑ 𝑛𝑗𝑞 = 𝑛𝑗 ∀ 𝑗 = 1 𝑡𝑜𝑚 𝑘𝑝 𝑞=1 (4) a∑ 𝑛𝑗 = n 𝑚 𝑗=1 (5) barma et al./decis. mak. appl. manag. eng. 2 (2) (2019) 112-125 116 ∑ ∑ 𝑦𝑖𝑞𝑝 = 1 𝑘𝑝 𝑞=1 𝑚 𝑝=1 (6) 𝑥𝑖𝑗𝑞𝑝 = { 1 if vehicle p in depot q travels from customer i to customer j 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (7) 𝑦𝑖𝑘𝑚 = { 1 if vehicle k of depot m serves customer i 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (8) the equation (1) is the objective function, minimizes the total traveling distance or cost. equation (2) ensures the capacity constraint of a vehicle. equation (3) guarantees that the vehicles serving the number of customers must not exceed the number of customers in a depot. equation (4) shows that the total number of customers served by the entire route must be equal to the sum of customers served by depot m. each customer must be served from a single depot is ensured in equation (5). equation (6) shows that each customer is serviced not more than once. equation (7) and (8) represents that the decision variables are binary. 5. proposed discrete alo algorithm in this paper, we have used the ant lion optimization algorithm proposed by mirjalili (2015). alo is a bio-inspired algorithm that mimics the foraging behavior of antlion. the steps of alo are given below:  initialization of ant and antlions  random walk of ants  building traps by antlions  entrapment of ants in traps prepared by the antlion  catching preys by antlion  re-building traps.  elitism  here 2-opt algorithm is used to optimize each route covered by one vehicle. 5.1. encoding scheme an mdvrp contains n cities and m depots. we have used cluster first, route second approach. so to represent an ant or ant lion one integer array a of size n is considered, and the array elements will be ranging from 1 to m. an element a[i] represents that ith city will be served from depot a[i]. as an example consider n as 10 and m as 3 then an ant or an antlion will be as in figure 2. figure 2. encoding of an ant from figure 2 it is clear that depot 1 will serve city 2, city 3 and city 8, depot 2 will serve city 1, city 6 and city 7 and depot 3 will serve city 4, city 5 and city 9. a 2-opt guided discrete antlion optimization algorithm for multi-depot vehicle routing problem 117 5.2. fitness evaluation in this paper, the fitness function is considered as same as the objective function. now to evaluate the value of fitness function we have to find the depot corresponding to each city and the vehicle which will serve the city. from the encoding scheme stated above, it is clear that which city will be served from which depot. then we have to find the vehicle routes starting from each depot. here we have applied a very well-known 2-opt algorithm to find the shortest path starts and end in the same depot after serving all the cities in the route. therefore, total fitness value = the total distances traveled by all the vehicles from all depots. consider an ant a as follows. 3 1 3 1 2 1 2 3 2 2 then depot 1 will serve city 2, 4, 6; depot 2 will serve city 5, 7, 9, 10 and depot 3 will serve city 1, 3, 8. now according to the vehicle capacity routes are to be decided from each vehicle from the depot. assume one vehicle is required for depot 1. then the initial route will be as {0, 2, 4, 6, 0} for depot 1. now, this is very similar to the traveling salesman problem. here we have used a 2-opt algorithm for local search to optimize the route length. a similar approach is taken for all the routes from the different depot, and finally, all the route lengths are added to get the fitness value. 5.3. operators of alo the antlion optimizer does a mimic of the relationship of antlions and ants. the ants will move on the search space, and the antlions are building traps to hunt ants. after capturing an ant, the position of the antlion is updated if it becomes fitter. the movement of ant for searching food is stochastic therefore a random walk is as follows 𝑥(𝑡) = [0, 𝑐𝑢𝑚𝑠𝑢𝑚(2𝑟(𝑡1) − 1), 𝑐𝑢𝑚𝑠𝑢𝑚(2𝑟(𝑡2) − 1), … , 𝑐𝑢𝑚𝑠𝑢𝑚(2𝑟(𝑡𝑛) − 1)] (9) where cumsum represents the cumulative sum where n represents the maximum iteration number and t, gives the step of random walk and r(t) is a random function given by: 𝑟(𝑡) = { 1 𝑖𝑓𝑟𝑎𝑛𝑑 > 0.5 0 𝑖𝑓𝑟𝑎𝑛𝑑 ≤ 0.5 (10) the position of ant and antlions are stored in the following matrix respectively 𝑀𝐴𝑛𝑡 = [ 𝐴1,1 ⋯ 𝐴1,𝑑 ⋮ ⋱ ⋮ 𝐴𝑛,1 ⋯ 𝐴𝑛,𝑑 ] (11) 𝑀𝐴𝑛𝑡𝑙𝑖𝑜𝑛 = [ 𝐴𝑙1,1 ⋯ 𝐴𝑙1,𝑑 ⋮ ⋱ ⋮ 𝐴𝑙𝑛,1 ⋯ 𝐴𝑙𝑛,𝑑 ] (12) a fitness function is used to identify the quality of ant and antlion during the optimization process. two different matrices moa and moal are used to store the fitness of all ant and antlion respectively. the matrices are as follows. barma et al./decis. mak. appl. manag. eng. 2 (2) (2019) 112-125 118 𝑀𝑂𝐴 = [ 𝑓([𝐴1,1, 𝐴1,2 , … … , 𝐴1,d]) 𝑓([𝐴2,1, 𝐴2,2 , … … , 𝐴2,d]) ⋮ ⋮ 𝑓([𝐴n,1, 𝐴n,2 , … … , 𝐴n,d])] (13) 𝑀𝑂𝐴𝐿 = [ 𝑓([𝐴𝑙1,1, 𝐴𝑙1,2 , … … , 𝐴𝑙1,d]) 𝑓([𝐴𝑙2,1, 𝐴2,2 , … … , 𝐴𝑙2,d]) ⋮ ⋮ 𝑓([𝐴𝑙n,1, 𝐴𝑙n,2 , … … , 𝐴𝑙n,d])] (14) where f is the objective function. ai,j gives the value of the jth dimension of ith ant, n represents the total number of ants and is similar for antlions. the alo (mirjalili, 2015) was designed to solve continuous problems. in this paper, we are focused on solving mdvrp which is one combinatorial optimization problem. so the operators used in original alo may not work as desired hence we have customized the operators according to our requirement. initialization in this step, two populations of size n for ant and antlion are formed randomly. let us assume n number of customers and m number of depots is present. assume (al1, al2,……, aln) and (a1, a2,……, an) are the populations of antlion and ant respectively. then each alj and aj represents the jth antlion and ant respectively. both alj and aj are a one-dimensional array of size n, and the array elements will range from 1 to m. random walks of ants in case of discrete problem random walk of an ant is implemented by inverting the entities of a randomly selected part of the string. the operation is demonstrated in figure 3. figure 3. random walk of an ant building traps by antlion in alo, each antlion builds a trap to catch one ant. to implement this mechanism, we have used the roulette-wheel selection mechanism to select antlion. roulette wheel selection chooses the fitter antlions for catching ants with higher probability. entrapment of ants in traps ants are moving randomly in search of food while antlions build traps. the higher the fitness, the bigger the trap is. when an ant falls in the trap antlion shoot sand on it; as a result, the ant slides down towards the trap. to realize this scenario crossover operator of ga is used. in this step crossover between one selected antlion and one ant is performed. the operation is pictorially represented in figure 4. one sub-string of an ant is selected randomly, and that substring is copied into the corresponding antlion. a 2-opt guided discrete antlion optimization algorithm for multi-depot vehicle routing problem 119 figure 4. representation of crossover operation catching of prey, re-construction of pit the final step of alo reaches after an antlion catches the prey. to mimic the step, it is considered that catching of ant happens when prey is going to be fitter than the corresponding antlion. then the antlion will change the location to the corresponding ant to increase the chance of catching a new pre. the above scenario is mathematically represented by the equation (15). 𝐴𝑛𝑡𝑙𝑖𝑜𝑛𝑗 𝑡 = 𝐴𝑛𝑡𝑖 𝑡𝑖𝑓𝑓(𝐴𝑛𝑡𝑖 𝑡) > 𝑓(𝐴𝑛𝑡𝑙𝑖𝑜𝑛𝑗 𝑡) (15) where t shows the current iteration, antlion jt shows the position of selected jth antlion at tth iteration, and anttj indicates the position of ith ant at tth iteration. elitism is one of the most important properties of evolutionary algorithms. elitism allows preservation of one or more good solution(s) in one generation for the next generation. in continuous alo it is assumed that the elite solution will influence random walk of every ant. in this paper, we have chosen 5% solutions from the population of antlion as elite, and they replace the worst antlions after the selection for the next generation. 5.4. pseudo codes the 2-opt algorithm croes et al., (1958) have developed the 2-opt technique to solve the tsp. it is a local search algorithm. the pseudo code for the 2-opt is given below. input: cost matrix c, number of city nc do { minchange = 0; for (i = 0; i< nc-2; i++) { for (j = i+2; j <nc; j++) { change= c(i,j)+c(i+1,j+1)-c(i,i+1)-c(j,j+1); if (minchange> change) { minchange = change; mini = i; minj = j; } } } barma et al./decis. mak. appl. manag. eng. 2 (2) (2019) 112-125 120 } while (minchange< 0); 5.5. pseudo codes the discrete alo algorithm input: number of city n, number of depot m, cost matrix c, number of vehicles available in each depot, vehicle capacity q.  perform a random initialization of ant’s population and antlions’ population.  find the ant’s fitness and the antlions’ fitness  search the best antlion to make it elite  while the termination condition is not satisfied  for every ant in the population  select an antlion using roulette wheel selection  perform a random walk  update the position of the ant  end for  calculate the fitness of all ants  replace an antlion with its corresponding ant if it becomes fitter using equation 15.  update elite if an antlion becomes fitter than the elite  end while  return elite 6. result and discussion the discrete alo is implemented in c language on intel core i5 cpu (2.30 ghz), 4gb ram. the performance of the mdvrp is evaluated using some of the benchmark problems proposed by creviera et al., (2007) taken from http://neo.lcc.uma.es/vrp/vrp-instances/multiple-depot-vrp-instances/ online resource of university of malaga, spain. the specifica-tion of some of the benchmark problems is given in table 1. table 1. specification of benchmark instances instance p01 p02 p03 p04 p06 total number of customer 50 50 75 100 100 total number of depots 4 4 5 2 3 number of the vehicle in each depot 8 5 7 12 10 vehicle capacity 80 100 140 100 100 the parameters for the proposed discrete alo are given in table 2. table 2. parameters of discrete alo parameter value population size 70 if total customer<50 else 100 iteration 2500 to 4000 selection roulette wheel elitism 5% of total population size, i.e., 5 the solutions of instance p1 are given in table 3. a 2-opt guided discrete antlion optimization algorithm for multi-depot vehicle routing problem 121 table 3. the solution of instance p01 depot routes 1 vehicle 1: 0 25 18 4 0 vehicle 2: 0 44 45 33 15 37 17 0 vehicle 3: 0 42 19 40 41 13 0 2 vehicle 1: 0 48 8 26 31 28 22 0 vehicle 2: 0 6 27 1 32 11 46 0 vehicle 3: 0 12 47 0 vehicle 4: 0 23 7 43 24 14 0 3 vehicle 1: 0 49 5 38 0 vehicle 2: 0 9 34 30 39 10 0 4 vehicle 1: 0 21 50 16 2 29 0 vehicle 2: 0 35 36 3 20 0 the results of mdvrp instances using discrete alo guided with 2-opt are compared with the exact solution, solution using discrete alo, ga and aco are presented in table 4. table 4. comparison of solutions of mdvrp using discrete alo with ga, aco and exact solution instance exact solution discrete alo guided with 2-opt discrete alo ga aco p01 576.87 576.87 591.45 598.45 576.87 p02 473.53 473.53 483.15 473.53 473.53 p03 641.15 641.15 694.49 641.18 645.15 p04 1001.04 1003.86 1011.36 1006.66 1001.04 p05 750.03 750.03 750.72 752.39 750.11 p06 876.5 876.5 882.48 877.84 876.5 p07 885.8 885.8 907.55 893.36 888.41 p08 4437.68 4449.65 4450.37 4474.23 4437.68 p09 3895.7 3895.7 4085.51 3900.22 3904.92 p10 3663.02 3663.02 3825.73 3680.02 3666.35 p11 3554.18 3554.18 3732.36 3593.37 3569.68 p12 1318.95 1318.95 1318.95 1318.95 1318.95 p13 1318.95 1318.95 1318.95 1318.95 1318.95 p14 1360.12 1360.12 1365.69 1365.69 1360.12 p15 2505.42 2505.42 2554.12 2549.65 2526.06 p16 2572.23 2572.23 2606.22 2606.22 2572.23 p17 2709.09 2709.09 2733.8 2733.8 2709.09 p18 3702.85 3702.85 3871.01 3781.66 3771.35 p19 3827.06 3827.06 3884.81 3884.81 3827.06 p20 4058.07 4058.07 4058.07 4094.86 4058.07 p21 5474.84 5474.84 5824.58 5668.97 5608.26 p22 5702.16 5702.16 5873.41 5873.41 5708.78 p23 6095.46 6095.46 6129.99 6159.9 6124.67 the percentage of the gap in the result found in the proposed method with the other method in the literature is given in table 5. the gap is calculated using the following formula. barma et al./decis. mak. appl. manag. eng. 2 (2) (2019) 112-125 122 𝐺𝑎𝑝 = (𝑍𝑙 − 𝑍𝑝) 𝑍𝑝 ∗ 100 (26) where 𝑍𝑝 represents the objective value obtained by the proposed method, and 𝑍𝑙 is the objective value of the problem by the others method. therefore, the posi tive gap represents the better performance of the proposed algorithm compared to others. whereas negative gap represents the opposite fact. table 5. the percentage of gap in the result in comparison with other methods instance exact solution discrete alo ga aco p01 0 2.527433 3.740877 0 p02 0 2.03155 0 0 p03 0 8.319426 0.004679 0.623879 p04 -0.28092 0.747116 0.278923 -0.28092 p05 0 0.091996 0.314654 0.010666 p06 0 0.682259 0.152881 0 p07 0 2.455408 0.853466 0.294649 p08 -0.26901 0.016181 0.552403 -0.26901 p09 0 4.872295 0.116025 0.236671 p10 0 4.441963 0.464098 0.090909 p11 0 5.013252 1.102645 0.436106 p12 0 0 0 0 p13 0 0 0 0 p14 0 0.409523 0.409523 0 p15 0 1.943786 1.765373 0.823814 p16 0 1.321421 1.321421 0 p17 0 0.912114 0.912114 0 p18 0 4.541367 2.128361 1.849926 p19 0 1.508991 1.508991 0 p20 0 0 0.906589 0 p21 0 6.388132 3.545857 2.436966 p22 0 3.003248 3.003248 0.116096 p23 0 0.566487 1.05718 0.479209 average gap % -0.02391 2.251911 1.049535 0.297781 from the above table, we observe that 2-opt guided discrete alo gives a better result than discrete alo, ga, and aco in most of the case. it is also found that the proposed algorithm fails to yield the exact solution always. the aco gives a better result than discrete alo guided with the 2-opt technique in case of instance p04, p08. 7. conclusion in distribution logistics, two main decision problems are routing and scheduling. the cost of delivering an item from source to the destination is optimized only by efficient routing. single depot vrp often fails to solve real-life scenario because there exists more than one depot. as an np-hard problem, mdvrp is very difficult to solve a 2-opt guided discrete antlion optimization algorithm for multi-depot vehicle routing problem 123 and to find exact solutions by exact methods. in this paper, we proposed a 2-opt 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(2018). a multi-depot two-echelon vehicle routing problem with delivery options arising in the last mile distribution. european journal of operational research, 265(2), 765-778. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). https://www.researchgate.net/profile/harinandan_tunga/publication/320583375_a_method_for_solving_bi-objective_green_vehicle_routing_problem_g-vrp_through_genetic_algorithm/links/59eec9da4585154350e82150/a-method-for-solving-bi-objective-green-vehicle-routing-problem-g-vrp-through-genetic-algorithm.pdf https://www.researchgate.net/profile/harinandan_tunga/publication/320583375_a_method_for_solving_bi-objective_green_vehicle_routing_problem_g-vrp_through_genetic_algorithm/links/59eec9da4585154350e82150/a-method-for-solving-bi-objective-green-vehicle-routing-problem-g-vrp-through-genetic-algorithm.pdf plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 5, issue 2, 2022, pp. 300-315. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0304052022b * corresponding author. e-mail addresses: bipradasbairagi79@gmail.com (b. bairagi) a homogeneous group decision making for selection of robotic systems using extended topsis under subjective and objective factors bipradas bairagi*1 1 department of mechanical engineering, haldia institute of technology, india received: 7 april 2022; accepted: 11 may 2022; available online: 13 may 2022. original scientific paper abstract: selection of the best robotic system considering subjective and objective factors is very imperative decision making procedure. this paper presents an extended topsis based homogeneous group decision making algorithm for the selection of the best industrial robotic systems under fuzzy multiple criteria decision making (fmcdm) analysis. fpis, fnis, positive and negative separation measures, subjective factor measure, and objective factor measure and robot selection index are computed. a case study has been conducted and illustrated for better clarification and verification of proposed algorithm. key words: fmcdm, robotic system selection, homogeneous group decision making, subjective factor measure, objective factor measure. 1. introduction multi criteria decision making (mcdm) is an analysis dealing with the evaluation of alternatives and identifying the best alternative out of a finite number of available alternatives. mcdm procedure can be categorized into classical multi criteria decision making (mcdm) (feng & wang, 2000; wang & lee, 2007) and fuzzy multiple criteria decision making (fmcdm) (wang & lee, 2003). the selection criteria on the basis of which all these decisions are made are objective, subjective and critical in nature. objective criteria can be measured and quantified. subjective criteria are qualitative but neither measurable nor quantifiable. subjective criteria are associated to ambiguity, imprecision, vagueness and uncertainty and realized by human perception and feelings (zadeh, 1965; zimmermann, 1991). critical criteria are those which decide the requirement of further evaluation of data of an alternative. critical criteria of an alternative must be satisfied before further assessment for final selection. while decision making is based on objective criteria with certainty, it is classical mcdm. the mcdm problems are generally solved using a variety of techniques that include topsis method, the total sum (ts), the ahp, saw, dea, electre and a homogeneous group decision making for selection of robotic systems using extended… 301 promethee (wang & lee, 2003; hwang & yoon, 1981). the fuzzy set theory is applied while assessment of alternative and importance of criteria are not possible to determine exactly. the concept of fuzzy is integrated with mcdm and the concerned technique is termed as fmcdm approach. in fmcdm, linguistic term is used to measure performance assessment of alternative and importance of criteria. linguistic term is converted into fuzzy number. in reality where objective measurement is unsatisfactory or insufficient, fuzzy sets considering subjective factors are applied for the evaluation of alternatives. a crisp sets can be defined to express an element is either member or not member in a universe of discourse. a fuzzy set is defined by assigning a value to each individual belonging in its universe of discourse. in the fuzzy sets this value represents its grades of membership (gorge & klir, 2008; majumder et al., 2004). chodha et al. applied entropy based topsis for ranking of robots for industrial purpose (chodha et al. 2021). narayanamoorthy et al. (2019) implemented intuitionstic hesitant fvikor approach and entropy for selection of industrial robots . fu et al (2019) advocated industrial robot selection technique using group stochastic multiple criteria acceptability analysis. nasrollahi et al. (2020) applied promethee method based on fbwm for ranking and selection of industrial robot[13]. ali and rashid (2020) applied best–worst method for appropriate robots selection in performing definite task in industry. yalcin and uncu (2020) used edas approach for proper decision making in selection of industrial robots . shih (2008) proposes an algorithm to explain the procedure of robot selection. the author first divided the criteria into two categories: benefit and cost. the evaluation of alternatives was done using incremental benefit-cost ratio. group topsis was used to find rank of the candidate. selection of robot is based on the incremental benefitcost ratio. though the proposed algorithm is suitable for more than one decision maker, it is too complex for one decision maker. the algorithm is not only complex but also tedious while the alternatives are required to be ranked. chu et al. (2003) proposed a ftopsis method. the purpose is to make sure the matching amid linguistic rating and related objective values. internal arithmetic was used to rank the robots and to defuzzy of rating into crisp values and closeness coefficient. parkan and wu (1999) suggested a technique that illustrates and judges against a number of madm as well as assessment procedure using a robot selection process. these papers are incapable for handling both objective and subjective factors together. bhattacharyya et al. (2002) suggested a technique for selection of material handling equipment under mcdm environment. a topsis based fuzzy hierarchical algorithm was employed for selection of robotic systems for industrial application (kahraman et al. 2007). the gap analysis of the above literature review exposes that previous researchers have attempted to apply mcdm techniques for selection of robots. still, this endeavor is not enough for extensive decision making regarding evaluation and selection of appropriate robots from several available alternatives under mcdm. in the current study, qualitative (subjective) criteria have been considered for performance evaluation of robotic systems. due to existence of ambiguity and imprecision, decision criteria are expressed in terms of linguistic variables which are then converted into suitable fuzzy numbers for quantification. hence the solution procedure of the present study deserves the implementation of fuzzy set theory. in selection of robotic systems, multiple criteria are generally considered. therefore, mcdm technique is appropriate for solving such a problem on robotic system selection. topsis is one of the most well-known mcdm techniques that past researches have used successfully in similar decision making environment. so in the https://www.sciencedirect.com/science/article/pii/s221478532103412x#! bairagi/decis. mak. appl. manag. eng. 5 (2) (2022) 300-315 302 decision making process of the current study, topsis is applied in combination with fuzzy set theory to ensure better applicability of the approach towards the right solution of the problem. the objective of the paper is to aid decision makers by providing a decision making framework that can considers both objective factors and subjective factors with homogeneous group decision making strategy. the remaining part of the paper is arranged in the following manner. section 2 describes the proposed algorithm. section 3 elaborates the case study and furnishes the calculation and discussion in details. section 4 is dedicated for some essential concluding remarks with the direction of future research. 2. proposed algorithm let ‘m’ alternatives to be ranked based on assessment of ‘n’ number of criteria among those ‘p’ number of criteria are subjective (qualitative) and remaining ‘q’ number of criteria are objective (quantitative), where p + q = n; ‘o’ is the (15th letter of the english alphabet) number of homogeneous decision makers of a committee employed in the selection procedure. step1. (a): form a decision matrix with fuzzy performance ratings expressed with linguistic variables offered by every expert to every alternative for every qualitative factor. 1 11 1 1 1 1 1 ... ... ... ... ... ... ... ... ...... ... ... ... ... ... ... ... ... ... ... j n k k k j n k k i i ij in k k km m mj mn c c c x x x a ad x x x a x x x                     (1) here, k ij x represents rating of ith alternative (ai) for jth criterion (cj) offered by decision maker kd . for subjective criteria, performance ratings of alternatives will be expressed with seven degree of linguistic terms depicted in table 1. each linguistic term is converted into a corresponding tfn as per table 1. for objective criteria, performance ratings are represented in crisp value. here, ,i n i is less than or equal to m; ,j n j is less than or equal to n; ,k n k is less than or equal to o. n is the set of natural number. every decision maker forms such a decision matrix. step 1.b: form of fuzzy weight matrix by the decision makers by assigning linguistic variables to each subjective (qualitative) criterion. a homogeneous group decision making for selection of robotic systems using extended… 303 1 1 1 1 1 1 1 1 ... ... ... ... ... ... ... ... ...... ... ... ... ... ... ... ... ... ... ... j n j n k k k k j n o o oo j n c c c w w w d w d w w w d w w w                     (2) k j w denotes importance for criterion j, estimated by dm k. where, k j w is fuzzy and is represented by trapezoidal number for its simplicity. if the criterion is objective then its weight expressed in fuzzy number is transformed into crisp value by defuzzification. step 2: convert linguistic variable into triangular fuzzy number. form average decision matrix in fuzzy numbers (afdm) and average weight matrix in fuzzy numbers (afwm). element of average fuzzy decision matrix is     o k k ijij x k r 1 1 (3) element of average fuzzy weight matrix is     o k k jij w k w 1 1 (4) where here, ,i n i is less than or equal to m; ,j n i is less than or equal to n; ,k n k is less than or equal to o, n is the set of natural number. in the case of objective criteria, operation of finding average performance rating can be debarred. as the weight of objective criteria is fuzzy and qualitative in nature, the operation of finding average weight must be determined. here,   ijijijij r  ,, is a triangular fuzzy number. step 3: determine normalized average fuzzy decision matrix using the eq. (5a) and eq. (5b)            *** *** ,,,,        ijijij ijijijij r , j b (5a)  * * * * * *, , 1 ,1 ,1 ij ij ij ij ij ij ij r                     , j nb (5b) where   ji ij , max*   step 4: determine weighted normalized average fuzzy decision matrix using the following eq. (6).  , ,ij ij ij ij ij ijr r w           (6) step 5: find fuzzy positive ideal solution (fpis) as )1,1,1(   and fuzzy negative ideal solution (fnis) as )0,0,0(   bairagi/decis. mak. appl. manag. eng. 5 (2) (2022) 300-315 304 step 6: find the euclidean distances from fpis and fnis for every alternative using following eq. (7) and eq. (8).       2 2 2 1 1 1 1 1 3 p i ij ij ij j s                   (7)       2 2 2 1 1 3 p i ij ij ij j s                (8) where i is natural number less than or equal to m; and j is natural number less than equal to n. step 7: determine relative closeness (rci) or subjective factor measure (sfmi) for each alternative using eq. (9). i rc i i i s s s      = sfmi (9) where, i is natural number less than or equal to m. in the paper relative closeness (rci) is considered as subjective factor measure (sfmi) owing to rci is the performance measure of ith alternative on the basis of subjective criteria. step 8: determine objective factor measure (ofm) from objective factor cost (ofc). ofm and ofc are inter-related by the following well known mathematical eq. (10) (feng & wang, 2000). 1 1 1                  m i iii ofcofcofm (10) i ofc = ofc for ith alternative, i ofm = ofm for ith alternative; i is natural number less than or equal to m. step 9: evaluate overall robot selection index i rsi using following eq. (11).   iii ofmsfmrsi   1 (11)  is coefficient of attitude having the value in the range 10   . step 10: organize the alternatives in decreasing order of the robot selection indices and select the alternative with maximum rsi value as the best one. 3. case study the above algorithm is illustrated for solving the following case study. the illustration is presented by dividing it into two subsections such as problem definition, calculation and discussion. 3.1 problem definition an eastern indian based automotive manufacturing organization decides to install robotic systems for its new plant. keeping the ever increasing global market competitiveness in view, the management of the organization is searching the way of making correct decision with scientific basis in every step associated with financial investment and future impact. the management also would like to involve its experts (decision makers) and incorporate their knowledge, experience, and opinion in the a homogeneous group decision making for selection of robotic systems using extended… 305 decision making procedure. the top managerial authority forms a decision making committee with three experts, one from marketing department, one from department of financial management and the remaining expert is from the manufacturing unit. each of the experts has experience more than ten years in the respective department. due to having almost equal experience, same age and organizational positions, the competent authority of top management decides to put equal importance to the decision makers. since there are multiple decision makers with equal importance, hence it may be termed as homogeneous group decision making process. the decision makers are reluctant to reveal their introduction and they are comfortable to be mentioned by d1, d2, and d3 respectively. the three homogeneous personnel of the committee bear the responsibility of making decision regarding the selection of decision criteria and estimation of their respective importance weights. through discussions and exchanging personal opinions, the decision making committee identifies and lists five significant subjective decision criteria for assessment and selection of industrial robotic systems. the listed five criteria are programming flexibility (c1), vendor’s service quality (c2), user friendliness (c3), reputation of manufacturer (c4) and cost (c5). out of the five significant criteria, programming flexibility (c1), vendor’s service quality (c2), user friendliness (c3), reputation of manufacturer (c4) are subjective and the remaining criterion cost (c5) is objective in nature. provisional the decision making committee executes a rigorous market survey for a feasible set of industrial robotic systems. based on the minimum requisite fulfillment of the considered criteria a screening test is conducted and a set of five industrial robotic systems is provisionally identified by the decision making committee. they designate the set of five robots by robot1 (r1), robot2 (r2), robot3 (r3), robot4 (r4) and robot5 (r5) which are to be ranked and the best robotic system is to be selected under fmcdm atmosphere for performing specific function in the automatic manufacturing organization. 3.2 calculation and discussions due to vagueness, imprecision and ambiguity associated with the four criteria viz. programming flexibility, vendor’s service quality, user friendliness and reputation of manufacturer seven grades of linguistic variables have been used for assessment of alternatives with respect to the above mentioned criteria. since the linguistic assessment of the alternatives is inappropriate in decision making, the linguistic variables used for assessing performance rating are required to transform into suitable fuzzy numbers for quantification. this investigation suggests triangular fuzzy numbers (tfns) due to its ease of application, simple calculation and proven capability of conveying information. the linguistic variables along with the acronyms and corresponding tfns for performance rating are presented in table 1. table 1. linguistic variables for assessment of performance rating linguistic terms acronym tfns extremely poor ep (0, 0, 1) poor p (0, 1, 3) slightly poor sp (1, 3, 5) medium m (3, 5, 7) slightly good sg (5, 7, 9) good g (7, 9, 10) extremely good eg (9, 10, 10) bairagi/decis. mak. appl. manag. eng. 5 (2) (2022) 300-315 306 while selecting a robotic system the different criteria in general have a varying impact on the selection and decision making. therefore it is very important to estimate appropriate importance weights for the criteria under consideration. in this paper, five grades of different linguistic variables have been used for the assessment of criteria weights by the decision makers. the linguistic variables to be utilized for assessing criteria weights along with the associated acronyms and the corresponding triangular fuzzy numbers (tfns) are presented in table 2. table 2. linguistic variables for assessment of criteria weight linguistic variable acronyms tfns extremely low el (0, 0, 0.1) low l (0, 0.1, 0.3) slightly low sl (0.1, 0.3, 0.5) medium m (0.3, 0.5, 0.7) slightly high sh (0.5, 0.7, 0.9) high h (0.7, 0.9, 0.1) very high eh (0.9, 0.1, 0.1) the decision making committee consists of three experts with diverse decision making attitude towards assessing performance ratings of the alternative robotic systems due to ambiguous nature of subjective decision criteria. this is equally true for estimation of the importance weights of the criteria with subjective nature. each decision maker assess each alternative robotic system with respect to every subjective criterion using one of the seven degrees of prescribed linguistic variables which are collectively arranged in a matrix form known as decision matrix consisting of performance ratings of the alternatives. there are five alternatives r1, r2, r3, r4 and r5 to be assessed with respect to four criteria c1, c2, c3 and c4. the assessment is to be accomplished by three decision makers d1, d2 and d3. therefore the decision matrix consists of 5  4 3=60 entries or performance ratings. for example, decision maker d1 assesses alternative robotic system r1 with sg, g, eg and g with respect to criteria c1, c2, c3 and c4 respectively. decision maker d2 evaluates alternative robotic system r1 with g, g, g and sg with respect to criteria c1, c2, c3 and c4 respectively. similarly decision maker d3 evaluates alternative robotic system r5 with p, g, eg and sg with respect to criteria c1, c2, c3 and c4 respectively. thus each alternative robotic system is assessed by each decision makers with respect to each criterion with performance ratings which are accommodated in the decision matrix presented in table 3. a homogeneous group decision making for selection of robotic systems using extended… 307 table 3. fuzzy decision matrix alternatives criteria decision makers d1 d2 d3 r1 c 1 sg g sg c 2 g g sg c 3 eg g p c 4 g sg g r2 c 1 eg eg eg c 2 sg g eg c 3 p g g c 4 eg eg g r3 c 1 g sg eg c 2 eg g eg c 3 eg eg eg c 4 g eg sg r4 c 1 p p p c 2 eg sg g c 3 g g sg c4 sg g sg r5 c1 g sg p c2 p g g c3 sg g eg c4 g g sg the criteria weights in linguistic variables estimated by the members of the experts of the decision making committee is presented in a criteria versus decision makers in a matrix form known as weight matrix and shown in table 4. table 4. fuzzy weight matrix in linguistic variable decision makers criteria d1 d2 d3 c1 h vh sh c2 vh vh vh c3 vh h h c4 vh vh h c5 h vh vh cost is an objective criterion. in the current problem on industrial robotic system evaluation and selection, the cost criterion is composed of five components viz. cost of acquisition, cost of installation, cost of operation, cost of maintenance and cost of transportation expressed in the unit of $ × 105. the cost of acquisition for the alternative robotic system selections are 2, 1, 0.9, 0.8, 0.9 unit respectively. the total costs with the five components for each of the five alternative robotic systems are shown in table 5. bairagi/decis. mak. appl. manag. eng. 5 (2) (2022) 300-315 308 the above data is originally taken from bhattacharya et al. (2002). the solution and result of the given example with step by step by illustration have been furnished below. decision makers like to assess alternatives using linguistic variables because of ease of expression with linguistic variables and unavailability of accurate information. however, linguistic variable is not suitable for correct decision making. that is why linguistic variables are then converted into suitable fuzzy numbers. the current algorithm suggests triangular fuzzy numbers as the medium for the quantification of linguistic variables used for the assessment of alternative robotic systems. conversion of linguistic variable to triangular fuzzy number is accomplished as per the suggested scale in the paper. the average fuzzy performance rating is calculated from the assessment individual decision makers. for example robotic system r1 is assessed against the criterion c1 (programming flexibility) with sg, g, and sg by the three decision makers d1, d2 and d3 respectively. now following the conversion scales sg is converted into (5, 7, 9), g into (7, 9, 10) and g into (5, 7, 9). the average fuzzy performance rating is calculated as follows. 5 7 5 7 9 7 9 7 9 , , (5.7, 7.7, 9.3) 3 3 3            the average fuzzy performance ratings of other alternatives with respect to each criterion is calculated in the similar way and average fuzzy decision matrix is constructed as shown in table 6. the weights of criterion c1 in linguistic variables assessed by the decision makers d1, d2, and d3 are h, vh and sh respectively. these linguistic weights are converted into the tfns (0.7, 0.9, 0.1), (0.9, 0.1, 0.1) and (0.5, 0.7, 0.9) respectively as per the prescribed conversion school. the average fuzzy weight for criterion c1is computed as follows. 0.7 0.9 0.5 0.9 1 0.7 1 1 0.9 , , (0.7, 0.87, 0.97) 3 3 3            the average fuzzy weights (afw) for other subjective criteria c2, c3 and c4 are calculated as (0.90, 1.0,1), (0.83, 0.97,1) and (0.83, 0.97,1) respectively using same procedure. the average fuzzy weight performance ratings are inserted in table 7. table 5. cost of alternatives in details (reproduced with permission from a. bhattacharya, industrial engineering journal, 2002.) robots r1 r2 r3 r4 r5 cost of acquisition ($ × 105) 2.00 1.00 0.90 0.80 0.90 cost of installation ($ × 105) 0.40 0.30 0.25 0.20 0.45 cost of operation ($ × 105) 0.30 0.20 0.35 0.30 0.35 cost of maintenance ($ × 105) 0.80 0.60 0.25 0.25 0.50 cost of transportation($×105) 0.20 0.10 0.05 0.05 0.14 total costs ($ × 105) 3.8 2.2 1.8 1.6 2.34 table 6. average fuzzy decision matrix ri c1 c2 c3 c4 r1 (5.7 7.7 9.3) (6.3 8.3 9.7) (6.7 8.0 9.0) (6.3 8.3 9.7) r2 (9.0 10.0 10) (7.0 8.7 9.7) (5.7 7.7 9) (8.3 9.7 10) r3 (6.7 8.7 9.7) (8.3 9.7 10) (9 10 10) (7 8.7 9.7) r4 (3.0 5.0 7.0) (7 8.7 9.7) (6.3 8.3 9.7) (5.7 7.7 9.3) r5 (5.0 7.0 8.7) (5.7 7.7 9) (8.3 9.7 10) (6.3 8.3 9.7) a homogeneous group decision making for selection of robotic systems using extended… 309 average fuzzy performance rating of each alternative with respect to ach criteria is normalized to ensure the range of lower, middle and upper values of all average fuzzy performance ratings from 0 (zero) to 1 (one). to accomplish the operation, every points of each average fuzzy performance rating is divided by the greatest point of all which is 10. the fuzzy performance ratings of robotic system r1 under subjective criteria c1, c2, c3 and c4 are (5.7, 7.7, 9.3), (6.3, 8.3, 9.7), (6.7, 8.0, 9.0) and (6.3, 8.3, 9.7) respectively. when each lower, middle and upper point is divided by the greatest point 10, the normalized fuzzy performance ratings for the same are obtained as (0.57, 0.77, 0.93), (0.63, 0.83, 0.97), (0.67, 0.80, 0.90) and (0.63, 0.83, 0.97) respectively. in this way the normalized average performance rating of all alternatives with respect to every criterion is calculated and the related values of normalized average performance ratings are arranged in table 8. table 8. normalized average fuzzy decision matrix ri c1 c2 c3 c4 r1 (0.55 0.77 0.93) (0.63 0.83 0.97) (0.67 0.80 0.9.0) (0.63 0.83 0.97) r2 (0.90 1.0 0.1) (0.70 0.87 0.97) (0.57 0.77 0.9) (0.83 0.97 1.0) r3 (0.67 0.87 0.97) (0.83 0.97 1.0) (0.9 1.0 1.0) (0.7 0.87 0.97) r4 (0.30 0.50 0.70) (0.7 0.87 0.97) (0.63 0.83 0.97) (0.57 0.77 0.93) r5 (0.50 0.70 0.87) (0.57 0.77 0.9) (0.83 0.97 1.0) (0.63 0.83 0.97) normalized average performance rating of each alternative robotic system is integrated with the respective criteria as per the algorithm and the weighted normalized average performance rating for the same is calculated for each alternative versus each criterion. for example, normalized average performance rating of the alternative robotic systems are (0.57, 0.77, 0.93), (0.90, 1, 0.1), (0.67, 0.87, 0.97), (0.30, 0.50, 0.70) and (0.50, 0.70, 0.87) respectively. while these normalized average performance ratings are integrated with importance fuzzy weight (0.7, 0.87, 0.97), for criterion c1, then the resultant weighted normalized average performance ratings (wnapr) of the alternatives with respect to criteria c1 are (0.39, 0.67, 0.90), (0.63, 0.87, 0.97) ,(0.47, 0.87, 0.94), (0.21, 0.75, 0.68) and (0.35, 0.43, 0.84) respectively. the calculation of the weighted normalized average performance rating for alternative robotic system r1 with respect to criterion c1 is as follows. (0.57 0.7, 0.77 0.87, 0.93 0.97) = (0.39, 0.67, 0.90) (0.90 0.7, 1.0 0.87, 1.0 0.97) = (0.63, 0.87, 0.97) (0.67 0.7, 0.87 0.87, 0.97 0.97) = (0.47, 0.87, 0.94) (0.30 0.7, 0.50 0.87, 0.70 0.97) = (0.21, 0.75, 0.68) (0.50 0.7, 0.70 0.87, 0.93 0.97) = (0.35, 0.43, 0.84) it is noted that all non-benefit subjective criteria are normalized in such a way that it is converted into benefit category. therefore we choose fuzzy positive ideal solution (fpis) as (1, 1, 1) and fuzzy negative ideal solution (fnis) as (0, 0, 0) for all subjective table 7. average fuzzy weight matrix c1 c2 c3 c4 wa (0.7 0.87 0.97) (0.90 1.00 1) (0.83 0.97 1) (0.83 0.97 1) bairagi/decis. mak. appl. manag. eng. 5 (2) (2022) 300-315 310 criteria. weighted normalized average performance ratings (wnapr), fpis and fnis value for each subjective criterion in terms of tfn are shown in table 9. table 9. weighted normalized average fuzzy decision matrix ri c1 c2 c3 c4 r1 (0.39 0.67 0.90) (0.57 0.83 0.97) (0.55 0.77 0.90) (0.52 0.80 0.97) r2 (0.63 0.87 0.97) (0.63 0.87 0.97) (0.47 0.75 0.90) (0.69 0.94 1.0) r3 (0.47 0.87 0.94) (0.74 0.97 1.0) (0.75 097 1.0) (0.58 0.84 0.97) r4 (0.21 0.75 0.68) (0.63 0.87 0.97) (0.52 0.80 0.97) (0.47 0.75 0.93) r5 (0.35 0.43 0.84) (0.51 0.77 0.9) (0.69 0.94 1.0) (0.40 0.80 0.97) fpis (1, 1, 1) (1, 1, 1) (1, 1, 1) (1, 1, 1) fnis (0, 0, 0) (0, 0, 0) (0, 0, 0) (0, 0, 0) positive separation measure (psm) denoted by s+ and negative separation measure (nsm) denoted by s+ for each alternative robotic system are calculated by the euclidian distance of each alternative from the fpis and fnis respectively. positive separation measure ( 1 s  ) for alternative robotic system r1 is computed as follows.                             2 2 2 2 2 2 1 2 2 2 2 2 2 1 1 1 1 0.39 1 0.67 1 0.90 1 0.57 1 0.83 1 0.97 3 3 1 1 1 0.55 1 0.77 1 0.90 1 0.52 1 0.80 1 0.97 3 3 1.2550 s s                            similarly positive separation measures for the alternative robotic systems r2, r3, r4 and r5 are computed as 2 0.9795s   , 3 0.8915s   , 4 1.4572s   and 5 1.3126s   respectively. it is noted that positive separation measures are determined by crisp values. negative separation measure ( 1 s  ) for robotic system r1 is computed as follows.                             2 2 2 2 2 2 1 2 2 2 2 2 2 1 1 1 0 0.39 0 0.67 0 0.90 0 0.57 0 0.83 0 0.97 3 3 1 1 0 0.55 0 0.77 0 0.90 0 0.52 0 0.80 0 0.97 3 3 3.0192 s s                            similarly, negative separation measures for the alternative robotic systems r2, r3, r4 and r5 are computed as 2 3.2056s   , 3 3.3880s   , 4 2.7216s   and 5 3.0831s   respectively. it is noted that negative separation measures are determined and expressed in terms of crisp number. positive separation measures and negative separation measures are combined to determine relative closeness (rc) or subjective factor measure (sfm) for each alternative robotic system. the calculation procedure is illustrated as follows. 1 1 3.0912 0.7094 1.2550 3.0912 rc sfm    2 2 3.2056 0.7660 1.2550 3.2056 rc sfm    a homogeneous group decision making for selection of robotic systems using extended… 311 3 3 3.3880 0.7917 0.8915 3.3880 rc sfm    4 4 2.7216 0.6513 1.3126 2.7216 rc sfm    5 5 3.0831 0.7014 1.3126 3.0831 rc sfm    the calculated positive separation measure, negative separation measures, relative closeness (rci) are presented in table 10. table 10. positive and negative separation measures, relative closeness (rci) robots  i s  i s objective factor measure r1 1.2550 3.0912 0.7094 r2 0.9795 3.2056 0.7660 r3 0.8915 3.3880 0.7917 r4 1.4572 2.7216 0.6513 r5 1.3126 3.0831 0.7014 rc1, rc2, rc3, rc4 and rc5 denote the relative closeness of the robotic systems r1, r2, r3, r4 and r5 respectively. sfm1, sfm2, sfm3 sfm4 and sfm5 denote the subjective factor measure of the robotic systems r1, r2, r3, r4 and r5 respectively. in this paper subjective factor measure is defined as the relative closeness determined from subjective criteria. relative closeness of robotic systems is depicted in figure 1. figure 1. relative closeness of robotic systems objective factor measure (ofm) for each alternative robotic system is calculated from the quantitative assessment of objective criterion which is in this case costs. there are five costs components for each alternative robot. the total cost or objective factor cost (ofc) is calculated for each robotic system. this objective factor cost (ofc) is used to compute objective factor measure (ofm) as follows. 1 1 1 1 1 1 1 3.8 0.1132 3.8 2.2 1.8 1.6 2.7 ofm                0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 r e la ti v e c lo s e n e s s robot 1 robot 2 robot 3 robot 4 robot 5 robots representation of relative closeness of the robotics systems bairagi/decis. mak. appl. manag. eng. 5 (2) (2022) 300-315 312 1 2 1 1 1 1 1 2.2 0.1954 3.8 2.2 1.8 1.6 2.7 ofm                1 3 1 1 1 1 1 1.8 0.2389 3.8 2.2 1.8 1.6 2.7 ofm                1 4 1 1 1 1 1 1.6 0.2687 3.8 2.2 1.8 1.6 2.7 ofm                1 5 1 1 1 1 1 2.7 0.1838 3.8 2.2 1.8 1.6 2.7 ofm                the difference between objective factor cost and objective factor measure is that objective factor cost is of cost category and objective factor measure of benefit category. objective factor measures of robotic systems are presented in figure 2. figure 2. objective factor measure of robotic systems at last, subjective factor measure (sfm) and objective factor measure (ofm) are combined together to compute robot selection index for each alternative robot. based on the importance of the subjective factor and number of subjective factor a coefficient of decision making attitude 0.67  is assigned towards the subjective factor measure and  1 0.33  towards objective factor measures. the subjective factor measure and the objective measure of the robotic system r1 are sfm1=0.7094, and ofm1= 0.1132 respectively. the corresponding robot selection index for r1 is calculated as follows. 1 0.7094 0.67 0.1132 0.33 0.5127rsi      similarly the robot selection indices for the other robotic systems r2, r3, r4 and r5 are also calculated below. 2 0.7660 0.67 0.1954 0.33 0.5777rsi      3 0.7917 0.67 0.2389 0.33 0.6093rsi      4 0.6513 0.67 0.2687 0.33 0.5250rsi      5 0.7014 0.67 0.1838 0.33 0.5306rsi      0 0.05 0.1 0.15 0.2 0.25 0.3 o b je c ti v e f a c to r m e a s u re s robot 1 robot 2 robot 3 robot 4 robot 5 robots representation of objective factor measures of robotic systems a homogeneous group decision making for selection of robotic systems using extended… 313 relative closeness (subjective factor measure), objective factor measure, robot selection index are presented in table 11. it is observed that robot selection indices for the robotic systems are in the following order. 14523 rsirsirsirsirsi  higher robot selection index is better and desirable. robot selection indices of the robotic systems are shown in figure 3. figure 3. robot selection indices of robotic systems therefore decision makers of the committee can rank the robots as 14523 rrrrr  and they can make the conclusion that 3r is the best robotic system of the five. 4. conclusions proper selection of robotic system under subjective and objective factors is very hard. in fuzzy environment, due to existence of imprecision, vagueness and ambiguity in information regarding performance of alternatives and weight of criteria the decision making procedure becomes more complex. in the present work, an effort has been made to turn the complexity into simplicity. to assess the performance of the robot an fmcdm method has been proposed, which can tackle subjective criteria, objective criteria as well as group decision. this model considers subjective criteria of benefit category only and objective criteria of cost category only. the proposed algorithm can help decision makers to select robots and similar alternatives with subjective and objective criteria under 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 r o b o t s e le c ti o n i n d e x robot 1 robot 2 robot 3 robot 4 robot 5 robots representation of robot selection index table 11. relative closeness, objective factor measure, robot selection index robot (ri) subjective factor measure (sfmi) objective factor measure ( i ofm ) robot selection index )( i rsi r1 0.7094 0.1132 0.5127 r2 0.7660 0.1954 0.5777 r3 0.7917 0.2389 0.6093* r4 0.6513 0.2687 0.5250 r5 0.7014 0.1838 0.5306 bairagi/decis. mak. appl. manag. eng. 5 (2) (2022) 300-315 314 fuzzy mcdm environment. manual calculation of the data may make the present model tedious and time consuming. the present methodology can be easily be implemented into computer program by the application of visual basic, visual c++ 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(1991). fuzzy set theory – and its application (2nd ed.) boston: kluwer. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0301072022m * corresponding author. email addresses: saeedshahimoridi@gmail.com (s. s. moridi), s.h.moosavirad@uk.ac.ir (s. h. moosavirad), m.mirhosseini@uk.ac.ir (m. mirhosseini), hosseinnikpour@yahoo.com (h. nikpour), r.min.mkh@gmail.com (a. mokhtari) prioritizing power outages causes in different scenarios of the global business network matrix by using bwm and topsis saeed shahi moridi1, seyed hamed moosavirad1*, mitra mirhosseini2, hossein nikpour3, armin mokhtari1 1 department of industrial engineering, faculty of engineering, shahid bahonar university of kerman, kerman, iran 2 energy and environment research center, shahid bahonar university of kerman, kerman, iran 3 south kerman electricity distribution company, kerman, iran received: 14 march 2022; accepted: 24 june 2022; available online: 1 july 2022. original scientific paper abstract. power outage is one of the significant problems for electricity distribution companies. power outages cause customer dissatisfaction and reduce distribution companies' profits and revenues. therefore, the electricity distribution companies are trying to moderate the leading causes of the outage. however, the dynamics of environmental conditions create uncertainties that require prioritizing the solutions to outages causes in different situations. therefore, this study presents a scenario-based approach to prioritizing power outage causes. four case studies have been conducted in four cities of kerman province in iran. first, the prioritization criteria and causes of the outage were identified using literature and interviews with experts in this field. then, the global business network matrix was used to create four possible scenarios. then, the best-worst method and topsis method were applied to weight the prioritizing criteria and prioritize the causes of the outages in different scenarios. the results showed that working in the power network limit zone, as one of the causes of outage in sirjan and jiroft cities, has the most priority. also, the collision of external objects, birds, and annoying trees should be considered by managers as the leading causes of outages in bam and kahnuj cities. keywords: power distribution networks, power outage, scenario planning, topsis, bwm mailto:saeedshahimoridi@gmail.com mailto:s.h.moosavirad@uk.ac.ir mailto:m.mirhosseini@uk.ac.ir mailto:hosseinnikpour@yahoo.com mailto:r.min.mkh@gmail.com moosavirad et al/decis. mak. appl. manag. eng. (2022) 2 1. introduction power outage is referred to a failure in the components and elements of distribution networks that cause the power to be disconnected (lakrevi & holmes, 2014). the outages are generally divided into two categories, including planned and unplanned ones. the planned outage means the outage of subscribers' power with a previous intention. an unplanned outage is any unforeseen outage of facilities and equipment that leads to subscribers' power outages. today, households' need for electricity is increasing, indicating the increasing loss due to outages (carlsson et al., 2021). although households are highly flexible in adapting to outages, this does not mean that they do not valorize reliable electricity sources (wethal, 2020). instead, societies are highly dependent on power systems to supply their energy needs (castillo, 2014). thanks to improved batteries and backup systems, the frequency and length of outages have decreased in many countries (carlsson et al., 2021). on the other hand, the interest in strengthening the power system aiming to withstand outages has increased compared to the past; however, the outage risk can not be reduced entirely (castillo, 2014). power grids are resistant to outages to maintain the safety and security of society and citizens (landegren et al., 2016). these outages occur for various reasons, including weather events, the primary source of power grid outages (shield et al., 2021). therefore, reducing the events, outages, and undistributed energies is essential for electricity distribution companies (ghasemian fard & mousavirad, 2017). focusing on increasing the strength of power systems while reducing the carbon effect, sepúlveda mora and hegedus developed a way to design a flexible, environmentally friendly microgrid. they chose the microgrid configuration based on economic, environmental, and resilience criteria. the proposed microgrid consisted of photovoltaics (pv), battery, natural gas generator, and electric charge of an office building with an average consumption of two megawatt-hours per day. the results of this study showed that the installation of a microgrid with a 600 kw pv array and a 2.8 mwh lithium-ion battery in an office building prevents the release of a maximum of 287 tons of co2 per year; in addition, to endure two days of outage during peak demand (sepúlveda mora & hegedus, 2021). in a study using data from three typhoons, including rammasun, kalmaegi, and mujigae, yuan et al. developed two models, one for the frequency of customers' power outages and another for the poles damaged in xuwen, guangdong, china. their validation showed that the developed models accurately estimate the frequency of customers' power outages and poles damaged by the typhoon. the mean relative errors in the definite customer and pole damage models were 5.1% and 9.6%, respectively. their models, which were used to support better decision-making, had two critical improvements compared to previous works. first, these models used various static and dynamic variables to provide more accurate predictions describing typhoon risk and local environmental conditions. second, they showed that it is possible to predict outages at a resolution of one kilometer skillfully. this spatial resolution was much higher than in previous models (yuan et al., 2020). in a paper, zhai et al. presented an algorithm that generated an artificial power grid scheme solely based on public data for each us city and then simulated outages at the surface of each building under loading hazard using fragility functions. their method provided estimates of the probability of outages due to natural hazards that were more local and at the building level. zhai et al. validated their model by comparing the grid features and outage events based on their method and actual data from the ohio power system. they found that if accurate fragility curves are available, their model prioritizing power outages causes in different scenarios of the global business network matrix 3 will rely less on input data than statistical learning methods, making accurate predictions (zhai et al., 2021). because current analysis and state estimation require accurate information about the electrical system, he and cheng, in an article, developed a practical analysis that did not fit perfectly to these items. they focused on the identification of power lines. they used a machine learning framework to locate the outages and predict single-line and multi-line outages. they examined a wide range of machine learning algorithms and feature extraction methods. their proposed methods used only the phasor angles obtained from the continuous monitoring of the basses. the algorithms were designed to get the necessary dynamic properties of the power system during sudden topological changes. he and cheng tested the prediction efficiency of their proposed plans under different levels of noise and absence. they showed that their proposed plans were more tolerant of missing and noisy data than previous works involving solving power flow or state estimation equations (he & cheng, 2021). in a study using multiple datasets, shield et al. examined the occurrence and variety of major weather-related power outages. they showed that the weather is the cause of 50% of all events, and 83% of customers are affected. lightning is the cause of 47% of weather events, while winter storms and tropical tornadoes result in 31.5% and 19.5% of events, respectively. the average repair cost for installations during major storms was 12 million pounds, with an average repair time of 117.5 hours (about five days). tropical tornados had the highest average number of affected customers, followed by lightning and winter storms. joint rescue teams were used for lightning less than tropical or winter storms. case studies of hurricane harvey, texas, in 2017, hurricane atlantic in 2011, and severe thunderstorms in alabama in 2011 were highlighted as the unique challenges faced by electrical installation. by showing the difference in the number of outage events (r¼ 0.79) and the affected customers (r ¼ 0.65), they concluded that the number of storms primarily influences these values. despite investing in storm-related outage prediction models and system retrofitting, weather-related outages are still a fundamental cause of power outages (shield et al., 2021). in a paper comparing the results of two studies on the willingness to pay (wtp) of swedish households to prevent outages in 2004 and 2017, carlsson et al. examined whether the wtp had changed or not. they found three main differences: (1) from 2004 to 2017, the proportion of households that were reluctant to pay to avoid outage decreased highly, (2) in 2017, the wtp was significantly higher than in 2004; however, (3) wtp has been reduced for the duration of an outage. the results of their study have consequences for encouraging and regulating electricity suppliers because a reliable source of electricity is more important than what has been shown by previous studies (carlsson et al., 2021). in an article, cerrai et al. described the development of two outage prediction models (opm) for snow and ice storm-related outages in power distribution networks and their performance evaluation in the northeastern united states. the first model was based on machine learning (ml) to predict outages in a typical four-kilometer network. the second model was a generalized linear model (glm) to predict complete outages throughout the city. their inputs to both models included (1) numerical weather prediction outputs (nwp), (2) leaf limit zone index (lai) obtained from satellite, (3) land cover data, (4) useful infrastructure data, and (5) the historical data of outage. the most critical variables for both models were the number of assets on the ground, lai, snow density, and the amount of frozen rain for the ice model. the results of cross-validation experiments based on 54 outage events in three order sizes showed a median absolute error percentage of about 70% for both models. according moosavirad et al/decis. mak. appl. manag. eng. (2022) 4 to the results, glm is better than ml models for predicting severe and destructive events. in contrast, ml models have better performance for low-impact events and better describe the spatial distribution of power outages (cerrai et al., 2020). wethal, who aimed to use household perspectives to understand the consequences of the outage and to show the impact of the outage on relationships between infrastructure, methods, customers, and suppliers, used qualitative interviews with norwegian rural households in an article examining how simple life changes during an outage, and how households experience the consequences of such outages. using the three elements of materials, experts, and meanings, wethal showed how outages temporarily cut off the relationship between elements in electricity-dependent methods. he also showed how households made connections between other elements and technologies to continue their daily lives. the practical reassembly of the elements illustrated the complexity of the outage consequences and explained how norwegian rural households could get on well with long-term outages. wethtal's analysis highlighted the difficulties of reducing the effects of the outage to financial issues and acknowledged that families focus on maintaining their daily lives rather than worrying about the economic costs of an outage. adapting during outages indicates high flexibility of households, but it does not mean that they do not valorize reliable electricity sources (wethal, 2020). in a study, ghasemian fard and mousavirad used the theory of system dynamics to identify and prioritize the factors affecting the reduction of outages in the electricity distribution network of the north of kerman province. their research showed a reduction in redistributed energy in the case of implementing the improvement policies. network standardization, staff training, and justification of instructions were some of the policies they proposed to reduce unscheduled outages. in this paper, some factors such as power outages during low-precipitated hours and seasons, in the shortest range, and non-frequent power outages in a specific limit zone were proposed for scheduled outages (ghasemian fard & mousavirad, 2017). al-shaalan examined the practical measures that reduce severe outages and cut off access to energy. he did this in two steps: first, to explore the effects of power outages from consumers' perspectives in riyadh, and second, to propose, analyze, and implement cost-effective strategies that reduce energy consumption and costs. his study showed that in case of an outage at certain times and seasons and for a long time, some customers suffer tangible and intangible losses. finally, practical measures and desirable solutions were proposed in this study to reduce the outage and, consequently, its undesired effects and consequences while not overshadowing the needs, consumer satisfaction, and comfort (al-shaalan, 2017). acknowledging that the margin and sensitivity of power grids are fundamental aspects of the lesser-considered concept of resilience, landegren et al. proposed a simulation-based method. this method considered repair teams and materials required to repair the network components, specifically the power grid. this method was demonstrated in a municipal electricity distribution system in sweden with outages with a severity of 12 independent damage. the overall result of his research was a quantitative assessment of the margin and power grids' sensitivity concerning the repair system resources. the information in this study could play a key role in deciding on the appropriate amount of resources (landegren et al., 2016). in a study using the analytic hierarchy process (ahp) and geographical information system (gis) decision-making methods, yari et al. identified the essential criteria and their weight to identify critical points for installing remote control terminals. they finally ranked all possible options and implemented their proposed prioritizing power outages causes in different scenarios of the global business network matrix 5 method on a medium air pressure feeder in the tehran power distribution network (yari et al., 2017). in an article, hosein abadi et al. located the power switches optimally to smarten the power system using ahp. they conducted a case study on the medium pressure feeders of kowsar alley in arak city and presented the results while considering the economic parameters. the main result of his research was that installing the optimal number of power switches with remote control units in optimal locations has many advantages, including technical and economic benefits (hosein abadi et al., 2016). focusing on self-healing as one of the critical features of smart power distribution networks, hafezi and eslami, in a study, examined the reduction of outages in selfhealing smart grids and presented three different scenarios. the first scenario looked at the traditional network. the second and third ones increased reliability and reduced outage by smartening and adding distributed resources. they used the genetic algorithm to optimize and select distributed generation sources' size, location, and number. the research results showed the smart grids' efficiency and distributed products as part of self-healing (hafezi & eslami, 2016). numerous algorithms and models are available to predict the risks of customers losing access to energy services. many quantitative and qualitative approaches have been used to determine restoration strategies in the event of an outage, and few studies have evaluated restoration strategies regarding anticipated events risk. following the storms that damaged critical infrastructure in the united states, this became a growing limit zone for research. castillo presented various previously completed studies in the literature review and discussed potential limit zones for future research (castillo, 2014). when the power grid shuts down in the case of uncertain data, there is no single way to assess risk, which iešmantas and alzbutas discussed in an article. they used the bayesian method for the statistical analysis of outage data. the bayesian method enabled a more coherent plan to express data uncertainty and obtain network reliability-related metrics. their method showed how to properly manage the statistical inference process with actual north american power grid outage data. various cases of power lines' unreliability and sudden damage at different levels of complexity, ranging from a simple bayesian evaluation to building a more general bayesian hierarchical model, were investigated in that study. iešmantas and alzbutas concluded that diversity has a significant effect on a geographical and environmental level and suggested analyzing the network line uncertainties by considering the characteristics of each line. they showed that this diversity significantly impacts the reliability of the whole network or any network. taking into account cascade outages, they obtained a hierarchical model based on the borel-tanner distribution. they demonstrated the ability to simulate significant outages, the occurrence of which is unlikely according to records in recent decades (iešmantas & alzbutas, 2014). in a case study, jha et al. analyzed the importance of assessing the services' reliability for electricity customers. the study assessed power outages for residential and commercial customers in ten regions of the northern indian state of rajasthan. to obtain feedback from customers, they prepared a questionnaire containing information about the quality of the power supply, outage losses, the customer's desire for an uninterruptible power supply, compensation for increased outages, and alternative measures to neutralize the effects of the outage. this study also calculated the outage cost with two approaches: (1) preparatory action approach and (2) the contingent value approach. the results of this research were significant from the companies' perspective to ensure the continuity of adequate services and customer satisfaction (jha et al., 2012). moosavirad et al/decis. mak. appl. manag. eng. (2022) 6 believing that the former technical methods were ineffective in reducing the outage following the advancement of technology and power grids, rahimkhani and jafartabar identified the factors affecting outages and analyzed them scientifically to reduce outages. using the hierarchical analysis decision-making method and the opinion of experts, they identified and prioritized the factors affecting the reduction of power outages in khuzestan. their studied factors included training, motivational factors, time management, privatization, staff training, maintenance of preventive systems, and management of unauthorized electricity (rahimkhani & jafartabar, 2012). the concept of a specialized microgrid called the independent distribution power system (idaps) was presented in an article by rahman et al. this microgrid managed the power distribution resources of the customers so that these assets could be distributed in an independent network under normal and outage conditions. they predicted that their proposed concept would be helpful in emergencies and in creating a new market for electricity transactions between customers (rahman et al., 2007). so far, numerous and different studies have been done in the field of power outages. for instance, an article screened and ranked fault risks of distribution network (dongli et al., 2018). another study used outage data to rank grid components (nalini ramakrishna et al., 2021). a study, which aimed to identify and asses power systems vulnerabilities, ranked the most vital branches in the transmission grid (gjorgiev & sansavini, 2022). these studies show the importance of research in the area of outages. although prioritization was done in some fields like human error roots (tavakoli & nafar, 2021) and smart cities (hajduk & jelonek, 2021), prioritization of outage causes and uncertainty of upcoming scenarios has been rarely considered in precious studies. previous studies often utilized classic methods such as ahp, which has apparent drawbacks. therefore, the novelty of this study is prioritizing outage causes and addressing challenges that should be considered and the way to get over them. uncertainty is a vital challenge in this area; the main contribution of this paper is to get over uncertainty by considering upcoming scenarios and providing a scenariobased approach to prioritize power outage causes. besides scenario planning, novel beneficial methods, with fewer comparison data and more consistent comparisons, have addressed some drawbacks of the ahp method, which was popular in previous studies. since factors such as tropical or cold regions and environmental pollution are effective in the outage, limit zones with the same characteristics were studied. sirjan, jiroft, kahnuj, and bam were selected respectively as cities of cold, tropical, and high environmental polluted regions. the power outages in these cities in 2019 were 248, 333, 324, and 420 minutes respectively. kerman electricity distribution company needs to review its developed plans to reduce the outages arising from the constantly changing environmental conditions for achieving its planned goals. therefore, this study aims to provide a scenario-based approach to prioritize the power outages causes. the current article is presented in five sections to achieve this goal. the first section shows the necessity of outages prioritization in the introduction. the second section explains the research background in the field of outages. the step-by-step method of the research is described in the second section. the third section presents the research results in the developed scenarios. the fourth section presents the sensitivity analysis. finally, this article's fifth section provides conclusions and suggestions for managers and researchers. prioritizing power outages causes in different scenarios of the global business network matrix 7 2. research methods this study aims to provide a scenario-based approach to prioritize the power outages causes. for this purpose, a case study has been conducted in sirjan city in kerman province, covered by the power distribution company in the south of kerman province. in this research, the literature review and interviews with kerman electricity distribution company experts are used to identify the causes of outages and decision-making criteria regarding the prioritization of outages. the global business network (gbn) matrix approach is also used for scripting the scenario. 2.1. gbn matrix the gbn matrix, introduced in 1991 (schwartz, 1991), is known as the default scenario planning technique. this matrix consists of two dimensions of uncertainty. therefore, this technique is also called the bivariate method or 2×2 matrix. these two dimensions, which form the two axes of a coordinate system, define four regions that produce four scenarios based on the logic of possible futures. this method is used to find two factors causing most of the uncertainties in the studied system. then, four scenarios are formed based on these two factors. an example of this matrix is shown in figure 1. figure 1. gbn matrix next, the best-worst method (bwm) method introduced in 2015 is used to weight the criteria for prioritizing the outage causes. bwm has some similarities with the ahp method, which has been used in this field. for instance, both questionnaires are based on pairwise comparisons, and both of them are set with a nine degrees spectrum. of course, rezaei has mentioned that the 10-degree scale can be used in bwm. however, according to total deviation, minimum violation, conformity, and consistency ratio, bwm significantly outperforms the ahp technique. in a bwm questionnaire, only the best and worst elements are compared instead of all criteria. the ahp technique calculates the number of comparisons with the n (n-1)/2 connection, but the bwm technique uses the (n-2)×2 connection. thus, bwm, with fewer comparison data, has more consistent comparisons, which leads to reliable results. in addition, in bwm, the respondent does not get bored. (rezaei, 2015): moosavirad et al/decis. mak. appl. manag. eng. (2022) 8 the steps of the bwm are described below (rezaei, 2015): step 1 – determining a set of decision-making criteria (c). c={c1,c2,…,cn} (1) step 2 determining the best, most desirable, or most crucial criterion (b) and the worst, most undesirable, or least important criterion (w). step 3 determining the superiority of the best criterion among all criteria using numbers between 1 and 9 and forming the vector ab. ab = (ab1. ab2. … . abn) (2) where abj indicates the superiority of the best criterion over the criterion j. also, abb = 1. step 4 determining the priority of all criteria over the worst criterion using numbers between 1 and 9 and forming the aw vector. aw = (a1w. a2w. … . anw) t (3) where ajw indicates the superiority of criterion j over the worst criterion. also, aww = 1. step 5 finding the optimal weights (w). w = {w1 ∗, w2 ∗ , … , wn ∗ } (4) the optimal weight for the criteria is where wb wj = abj⁄ , wj ww = ajw⁄ for eachwb wj⁄ and wj ww⁄ pair. to establish this condition for all js, the answer which minimizes the maximum| wb wj − abj| and | wj ww − ajw| for all js need to be obtained. given that the conditions for the weights is not harmful and the sum of the weights is equal to one, the result is the following problem: min max j {| wb wj − abj| . | wj ww − ajw|} (5) s, t, ∑ wjj = 1 wj ≥ 0. for all j the previous problem can be changed to the following one. min ξ (6) s, t, | wb wj − abj| ≤ ξ . for all j | wj ww − ajw| ≤ ξ . for all j ∑ wjj = 1 wj ≥ 0. for all j prioritizing power outages causes in different scenarios of the global business network matrix 9 compatibility rate (cr) the compatibility rate is obtained using the optimal value of ξ in the above problem and the compatibility index (ci) in table 1. cr = ξ∗ ci⁄ (7) table 1. compatibility index abw ci 1 0 2 0.44 3 1 4 1.63 5 2.3 6 3 7 3.73 8 4.47 9 5.23 in this research, the technique for preference by similarity to the ideal solution (topsis) method is used to prioritize outage causes in different scenarios. this method has been utilized in similar studies. for instance, an article used a topsisbased decision making technique to evaluate smart cities and rank them based on energy performance (hajduk & jelonek, 2021). another study utilized a combination of topsis, shanon entropy, and mathematical expectation to rank human error roots in the maintenance of power grid (tavakoli & nafar, 2021). topsis method is a favorable option for prioritization due to the following advantages (zavadskas et al., 2016):  effortless decision making by utilizing both positive and negative criteria  the number of alternatives has little effect on the performance.  increasing the number of alternatives and criteria will amplify rank differences lesser.  it can be used for both quantitative and qualitative data.  it is comparatively fast, comprehensible, and easy to implement.  it provides a systematic analytical procedure that is well-structured.  it is possible to apply the number of criteria simultaneous with the decision operation.  in determining the choice set, it is flexible to a great extent.  scalar values that provide a superior understanding of similarities and differences among alternatives lead to a preferential prioritization of the possible options.  in the rank reversal problem, which means change in the alternatives rank when introducing a non-optimal alternative, topsis is known as one of the best techniques. in the topsis method, the distance of an option from the positive ideal point to the negative ideal one is considered. the selected option has the shortest distance from the positive ideal solution and the farthest distance from the negative ideal solution. the steps of this method are as follows (azimifard et al., 2018): moosavirad et al/decis. mak. appl. manag. eng. (2022) 10 step 1 unscaling the decision matrix (d) and creating a normal decision matrix (nd): d = (rij) ; i = 1. … . m. j = 1. … . n (8) nd = (nij) ; i = 1. … . m. j = 1. … . n (9) nij = rij √∑ rij 2m i=1 (10) step 2 creating a normal weighted matrix (v) using the weight vector (w): w = (wj) ; j = 1. … . n (11) mw = [ w1 0 0 0 ⋱ 0 0 0 wn ] (12) v = (vij) = nd × mw ; i = 1. … . m. j = 1. … . n (13) step 3 determining the ideal solution (a +) and the negative ideal solution (a-): a+ = (vj +) = {(max i vij |j ∈ j) . (min i vij |j ∈ j ′′)} ; (14) i = 1. … . m a− = (vj −) = {(min i vij |j ∈ j) . (max i vij |j ∈ j ′′)} ; (15) i = 1. … . m set j contains positive criteria, and j  contains negative criteria. step 4 calculating the distance of each option from the ideal solution (+di ) and the negative solution (-di) by the euclidean method: di+ = {∑ (vij − vj +) 2n j=1 } 0.5 ; i = 1. … . m (16) di− = {∑ (vij − vj −) 2n j=1 } 0.5 ; i = 1. … . m (17) step 5 calculating the proximity of each option with the ideal solution (cli +): cli+ = di− di++di− ; 0 ≤ cli+ ≤ 1 ; i = 1. … . m (18) step 6 ranking the options based on the descending order of cli +; the better option has more cli +. prioritizing power outages causes in different scenarios of the global business network matrix 11 3. results in this study, the following criteria were identified to prioritize the causes of outages through interviewing the experts of the electricity distribution company and reviewing the literature:  frequency of failure  saidi (the system average interruption duration index) in minutes (average outage duration for each customer per year)  undistributed energy  the cost of troubleshooting solutions  the severity of the failure affects reducing losses then, to identify the main factors influencing the outage, the results of investigating 755 cases of outages that occurred in the electricity distribution network of southern kerman province and the interviews with the experts of the southern electricity distribution company of kerman province (dispatching, monitoring, and operation departments and engineering office) were used. consequently, eight factors influencing outages were identified as follows. 1fault in cut-out fuse 2work in network limit zone 3modification and optimization of the network 4unfavorable weather conditions 5fault in the transformer 6collision of external objects, birds, and annoying trees 7fault in the jumper 8other causes. a comparison of the best outage criterion with other criteria in the first scenario is presented in table 2. table 2. comparison of the best outage criterion with other criteria in the first scenario saidi in minutes other options 2 frequency of failure 1 saidi in minutes 3 undistributed energy 9 the cost of troubleshooting solutions 5 the severity of failure affects the reduction of losses in this study, among the criteria related to outages and macro-level operational plans, two key factors affect their prioritization with high uncertainty: 1) the economic conditions of the company and society, 2) weather and environmental conditions. therefore, the gbn matrix was plotted with the following figure in four scenarios. the results of prioritization of outage causes in each scenario are given below: scenario 1: favorable economic, weather, and environmental conditions according to the scenario, it is assumed that the economic condition will improve, and more financial resources will be available. thus, the importance of the criterion of cost of troubleshooting solutions will be reduced. therefore, according to the bwm method, the criterion of "cost of troubleshooting solutions" is given low priority, and consequently, the worst criterion is considered in this method. then, during a meeting with experts in the company, comparisons were performed related to the bwm method, which can be seen in the following tables. now by giving the scores mentioned, the new weights of the criteria are obtained. the inconsistency rate obtained in this weighting equals 0.073, which indicates the validity of the results because it is close to zero. moosavirad et al/decis. mak. appl. manag. eng. (2022) 12 according to scenario 1, the cost of troubleshooting solutions was low. according to this scenario, the weather condition is considered appropriate to prioritize the causes of outages. therefore, the importance of the cause of unfavorable weather conditions is considered to be less than the other seven causes of outages. based on the weighting of the bwm method, according to the above scenario, the prioritization of outage causes using the topsis method is expressed in table 5. scenario 2: unfavorable economic conditions and favorable weather and environmental conditions according to this scenario, we assume that the economic condition will worsen. therefore, the criterion of cost of troubleshooting solutions would be crucial. according to the approach we had in scenario 1, we also used the same approach to weigh the criteria and score the causes. based on the weighting by the bwm method, according to the above scenario, the topsis method is used for prioritization, which is presented in table 6. scenario 3: favorable economic conditions and unfavorable weather and environmental conditions according to scenario 3, we assume that the economic condition will be improved, and more financial resources will be available. thus, the importance of the cost of troubleshooting solutions criterion will be reduced. therefore, the cost of this criterion takes low priority, and consequently, it is considered the worst criterion in this method. according to the approach we had in scenario 1, we also used the same approach to weigh the criteria and score the causes in this scenario. based on weighting by the bwm method, according to the above scenario in unfavorable weather and environmental conditions, the topsis method was used for prioritization. the results are expressed in table 7. scenario 4: unfavorable economic conditions and unfavorable weather and environmental conditions according to scenario 4, we assume that the economic condition will worsen, and the cost of troubleshooting solutions will be crucial. therefore, the criterion of the cost of troubleshooting solutions is given high priority, and as a result, it is considered the best criterion in this method. according to the approach we had in scenario 1, we also used the same approach to weigh the criteria and score the causes. the unfavorable weather condition is also considered for prioritizing the factors according to the scenario. based on the weighting by the bwm method, according to the above scenario, the topsis method was used for prioritization. the results are presented in table 8. the first result obtained with the table ix is to determine the priority of plans to reduce outages. also, by managing low-cost factors such as working in the network limit zone, the network outages can be reduced. table 3. comparison of the worst outage criteria with the worst criterion in the first scenario the cost of troubleshooting solutions other options 8 frequency of failure 9 saidi in minutes 6 undistributed energy 1 the cost of troubleshooting solutions 3 the severity of failure affects the reduction of losses prioritizing power outages causes in different scenarios of the global business network matrix 13 table 4. weight of outage criteria in bwm method in the first scenario weight criterion 0.254 frequency of failure 0.435 saidi in minutes 0.169 undistributed energy 0.040 the cost of troubleshooting solutions 0.102 the severity of failure affects the reduction of losses table 5. prioritization of outage causes in the first scenario in cities by bwm and topsis methods sirjan bam jiroft kahnuj causes 1 2 5 6 fault in cut-out fuse 2 3 3 5 work in network limit zone 4 4 4 4 modification and optimization of the network 7 7 7 7 unfavorable weather conditions 6 6 6 3 fault in transformer 3 1 1 1 collision of external objects, birds, and annoying trees 5 5 2 2 fault in the jumper table 6. prioritization of outage causes in the second scenario in cities by bwm and topsis methods sirjan bam jiroft kahnuj causes 1 1 1 6 fault in cut-out fuse 2 3 3 4 work in network limit zone 6 6 6 5 modification and optimization of the network 4 4 4 7 unfavorable weather conditions 7 7 7 3 fault in transformer 3 2 5 1 collision of external objects, birds, and annoying trees 5 5 2 2 fault in the jumper table 7. prioritization of outage causes in the third scenario in cities by bwm and topsis methods sirjan bam jiroft kahnuj causes 1 2 6 7 fault in cut-out fuse 2 4 4 6 work in network limit zone 5 5 5 5 modification and optimization of the network 4 3 1 2 unfavourable weather conditions 7 7 7 4 fault in transformer 3 1 2 1 collision of external objects, birds, and annoying trees 6 6 3 3 fault in the jumper moosavirad et al/decis. mak. appl. manag. eng. (2022) 14 table 8. prioritization of outage causes in the fourth scenario in cities by bwm and topsis methods sirjan bam jiroft kahnuj causes 1 2 1 6 fault in cut-out fuse 2 3 3 5 work in network limit zone 6 6 6 7 modification and optimization of the network 5 5 5 2 unfavourable weather conditions 7 7 7 4 fault in transformer 3 1 4 1 collision of external objects, birds, and annoying trees 4 4 2 3 fault in the jumper table 9. comparison of the priority of outage causes in target cities in current conditions sirjan bam jiroft kahnuj causes 3 5 1 7 fault in cut-out fuse 4 3 2 6 work in network limit zone 6 6 5 5 modification and optimization of the network 2 1 4 3 unfavourable weather conditions 7 7 7 1 fault in transformer 1 2 3 2 collision of external objects, birds, and annoying trees 5 4 6 6 fault in the jumper figure 2 depicts how ranks of criteria change in different scenarios. it can also be seen that those cost items such as transformer faults, and to some extent, fault in jumpers have low priority in the current condition. according to the previous season statistics, we will see that the network improvement and optimization in sirjan city have received more attention than the others, and these results confirm the accuracy of the results by prioritizing this case in sirjan city lower than in other cities. another vital analysis obtained from the results is that the issues of unplanned outages such as transformer faults, cut-out fuse faults, and jumper faults were not among the highest priorities. instead, the causes of the planned outages, such as working in the network limit zone and network modification and optimization, were among the highest priorities for allocating facilities. this shows that reducing the planned outages should be the priority of distribution companies. prioritizing power outages causes in different scenarios of the global business network matrix 15 figure 2. ranks of criteria in different scenarios 4. sensitivity analysis a sensitivity analysis has been applied for the first scenario to evaluate the sensitivity of the priority of outage causes in target cities due to the changes in criteria's importance. in this sensitivity analysis, the undistributed energy criterion is assumed to be more important than saidi in minutes and frequency of failure criteria, respectively (please see table 10 and table 11). the results of these changes have been demonstrated in table 12 and table 13. moosavirad et al/decis. mak. appl. manag. eng. (2022) 16 table 10. comparison of the best outage criterion with other criteria in the first scenario for sensitivity analysis undistributed energy other options 3 frequency of failure 2 saidi in minutes 1 undistributed energy 9 the cost of troubleshooting solutions 5 the severity of failure affects the reduction of losses table 11. comparison of the worst outage criteria with the worst criterion in the first scenario for sensitivity analysis the cost of troubleshooting solutions other options 7 frequency of failure 8 saidi in minutes 9 undistributed energy 1 the cost of troubleshooting solutions 3 the severity of failure affects the reduction of losses table 12. comparison of outage criteria weights in the first scenario for sensitivity analysis weight after changes weight before changes criterion 0.173 0.254 frequency of failure 0.259 0.435 saidi in minutes 0.428 0.169 undistributed energy 0.038 0.040 the cost of troubleshooting solutions 0.104 0.102 the severity of failure affects the reduction of losses prioritizing power outages causes in different scenarios of the global business network matrix 17 table 13. prioritization of outage causes in the first scenario for sensitivity analysis sirjan bam jiroft kahnuj causes a fte r ch a n g e s b e fo re ch a n g e s a fte r ch a n g e s b e fo re ch a n g e s a fte r ch a n g e s b e fo re ch a n g e s a fte r ch a n g e s b e fo re ch a n g e s 3 1 3 2 6 5 6 6 fault in cut-out fuse 1 2 4 3 1 3 5 5 work in network limit zone 4 4 2 4 3 4 4 4 modification and optimization of the network 7 7 7 7 7 7 7 7 unfavorable weather conditions 6 6 6 6 5 6 3 3 fault in transformer 2 3 1 1 4 1 1 1 collision of external objects, birds, and annoying trees 5 5 5 5 2 2 2 2 fault in the jumper 5. discussion and conclusion the following criteria were identified, prioritizing the causes of outages in this study by interviewing the experts of the electricity distribution company and reviewing the literature. • frequency of defects • saidi in minutes • undistributed energy • the cost of troubleshooting solutions • the severity of the effects of the defect on reducing losses then, to identify the leading causes of the outage using the results of 755 cases of outages in the power distribution network in the south of kerman province, the following eight causes were identified and verified by the experts of the distribution company (dispatching, supervision, and interest departments and the engineering office of the southern distribution company of kerman province) would be selected as influential causes.  fault in cut-out fuse  work in network limit zone  network modification and optimization  unfavorable weather condition  fault in transformer  collisions of external objects and birds and annoying trees  fault in jumper  other causes then, four scenarios were developed based on the gbn scenario method. in general, the results showed that working in the network limit zone is one of the causes of outage in sirjan and jiroft cities has the priority of investigation. also, managers' moosavirad et al/decis. mak. appl. manag. eng. (2022) 18 collision of external objects, birds, and annoying trees should be considered the leading causes of outages in bam and kahnooj cities. therefore, electricity distribution companies can allocate and plan the necessary resources, including budget and time, to reduce outages according to their priorities. other results that can be stated include reducing outages by managing low-cost causes such as working in network limit zone. another vital analysis obtained from the results is that unplanned outages such as transformer faults, cut-out fuse faults, and jumper faults were not prioritized. instead, the planned outages such as work in the network limit zone and modification optimization prioritized allocating facilities. this indicates that reducing the planned outages should be a priority for distribution companies. in future research, other sustainability criteria such as environmental, social, and economic ones can prioritize the causes of outages. also, other outage causes can be added to the list of causes identified in this research. furthermore, other multi-criteria decision-making methods such as vikor or electra can prioritize outage causes and compare the results with the findings of this study. using the approach mentioned in this research to prioritize outage causes in other cities can also play an essential role in the optimal use of resources of electricity distribution companies. author contributions: all authors contributed to the study's conception and design. saeed shahi moridi performed data collection and prepared the initial draft. dr. seyed hamed moosavirad supervised the research and performed data analysis. dr. mitra mirhosseini, who was the first advisor, contributed to the data analysis. hossein nikpour was the research project advisor and contributed to data gathering and analysis results. armin mokhtari wrote and revised the final draft of the manuscript, and all authors commented on this manuscript. all authors read and approved the final manuscript. funding: this research has been carried out with the support of the south kerman electricity distribution company under contract number 0153-97. data availability statement: factual data gathered from south kerman electricity distribution company can be provided based on the request of each reader. acknowledgments: we thank all the company employees who participated in this project. conflicts of interest: the authors have no further financial or non-financial interests to disclose. references al-shaalan, a. m. 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(2021). power outage prediction for natural hazards using synthetic power distribution systems. reliability engineering and system safety, 208, 107348. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 5, issue 1, 2022, pp. 154-168. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0313052022d * corresponding author. e-mail addresses: ana.diniz@edu.ufes.br (a. p. m. diniz), patrick.ciarelli@ufes.br (p. m. ciarelli), evandro@ufes.br (e. o. t. salles), klaus.coco@ufes.br (k. f. coco) long short-term memory neural networks for clogging detection in the submerged entry nozzle ana p. m. diniz1, patrick m. ciarelli1, evandro o. t. salles1, and klaus f. coco1 1 universidade federal do espírito santo, vitória, espírito santo, brazil received: 2 april 2022; accepted: 8 may 2022; available online: 13 may 2022. original scientific paper abstract: the clogging in the submerged entry nozzle (sen), responsible for controlling the steel flow in continuous casting, is one of the main problems faced by steelmaking process, since it can increase the frequency of interruptions in the operation for the maintenance and/or exchange of its equipment. although it is a problem inherent to the process, not identifying the clogging can result in losses associated with the process yield, as well as compromising the product quality. in order to detect the occurrences of clogging in a real steel industry from historical data of process variables, in this paper, different models of long short-term memory (lstm) neural networks were tested and discussed. the overall performance of the classifiers developed here showed very promising results in real data with class imbalance. key words: continuous casting, submerged entry nozzle, clogging, lstm, deep learning. 1. introduction one of the problems faced by the industry concerning the continuous casting process is the accumulation of steel impurities that forms on submerged entry nozzle (sen) of the tundish, causing their obstruction, known as clogging (ikaheimonen, 2002). as evidenced by ikaheimonen (2002), clogging can be formed by several factors, including metallurgical, hydrodynamic and thermodynamic factors, as well as nozzle material, unpredictable disturbances and operational failures. according to rackers (1995), the consequences of clogging include reduced productivity, increased production costs and decreased product quality. clogging events increase the frequency of interruptions in the operation for the exchange and/or maintenance of nozzles and tundishes, which can reduce the useful lifetime by long short-term memory neural networks for clogging detection in the submerged entry nozzle 155 up to half (schmidt, russo & bederka, 1991). in addition, small solid inclusions can break off and enter the steel flow, causing unacceptable defects in the product (bessho et al., 1991, wang et al., 2021, wu et al., 2021). scientific studies show that the clogging phenomenon lasts around 250 seconds, being perceptible only after the first 80 seconds, which leaves a margin of 170 seconds between possible detection and total obstruction (barati et al., 2018). thus, the detection model must act quickly to allow corrective action in less than 2 minutes. although the detection of the beginning of clogging is of fundamental importance so that control actions can be applied and the system operation is prolonged, in practice, the steel industry still does not have effective tools for the detection (rout et al., 2013, pellegrini et al., 2019, wang et al., 2021). due to hostile working conditions, variations in production, process and sensors failures, the data are generally noisy, with outliers and missing values (wang, mao & huang, 2018). even with the different adversities, researches have been developed and, usually, they correlate the clogging occurrences with the gate opening and the casting speed. there is not a large number of studies found in the bibliography, mainly due to the difficulty in obtaining the data due to their confidentiality. in addition, these researches apply techniques associated with the physical parameterization of the plant, which restricts their applicability (ikäheimonen et al., 2002, vannucci & colla, 2011, rout et al., 2013, pellegrini et al., 2019). regarding the models developed in the literature, in addition to being heavily influenced by physical equations, the small number of clogging occurrences against normal operating data end up compromising the classifiers' ability during the learning phase. as a result, the predictive accuracy desired by the industry can hardly be guaranteed (vannucci & colla, 2011, rout et al., 2013, pellegrini et al., 2019). several works in this area were evaluated and the best success rates for clogging prediction were found by vannucci & colla (2011) and pellegrini et al. (2019). the first authors associated neural networks with fuzzy logic to classify clogging events, achieving 76.9% of recall and 80.2% of accuracy. in pellegrini et al. (2019) were applied an online predictive estimation model for the probability of clogging using about 50 process variables. although the authors suggest that the model presented a good classification performance in series identified as possibly subject to clogging, with an overall area under the curve (auc) equal to 0.8, when tested in series with lower probability of clogging incidence, the model presented an accuracy of 75% and a precision of 62%. in ikäheimonen et al. (2002), neural networks were applied to data from a real plant in a problem similar to the one addressed in this study, however, satisfactory results were not obtained in the interest of the industry. another aspect is related to the amount of noise inherent to the input signals, so that a multiplayer perceptron neural network did not behave so well in the initial clogging prediction task. the idea is to apply as few pre-processing techniques as possible, in order to be able to use real data in an online application. as discussed by goodfellow, bengio & courville (2016), the use of deep learning is motivated by the difficulty of traditional algorithms in generalizing problems involving, above all, high-dimensional and highly complex data. deep learning, then, provides a very powerful framework for supervised learning (wang et al., 2021, wu et al., 2021). in order to identify the clogging in the sen, this article evaluates the general performance of classifiers using long short-term memory (lstm) neural networks. this type of algorithm has been used as an important tool in several researches for diniz et al./decis. mak. appl. manag. eng. (2022) 5 (1) (2022) 154-168 156 extracting temporal resources from sequential data (yildirim et al., 2019, essien & giannetti, 2020, wang et al., 2021, wu et al., 2021). these factors motivated us to apply deep neural networks, such as lstm, which is capable of extracting information relevant to clogging detection even with a high-noise signal. therefore, this study is motivated by the contribution in the application of techniques exclusively based on data for the detection of the initial occurrence of obstruction, since the recent researches are based on idealized systems and lack sufficient precision in complex tasks or dynamic environments. in general, the performance of the classifiers developed here showed very promising results in real data applications, obtaining precision and recall levels above 85%. the correct classification of clogging occurrences can contribute to reducing process interruptions and costs associated with production, as well as improving the quality of the final product (vannucci & colla, 2011, pellegrini et al., 2019, wang et al., 2021). this article is divided as follows: section 2 discusses the causes and effects of the occurrence of clogging in the continuous casting process. section 3 presents lstm neural network. in section 4, the dataset is presented together with the step-by-step of the proposed methodology, as well as the performance metrics used in the classification task. in section 5, the results are presented, comparing the performance of the classifiers, followed by section 6 that provides the final considerations and conclusions. 2. clogging in the continuous casting process the continuous casting process is based on the vertical casting of liquid steel from a ladle positioned on a tundish. in figure 1, a typical schematic of the steel flow from the tundish to the mold is presented. the steel flows through the tundish nozzle, being regulated by the slide gate and introduced into the copper mold cavity through the sen and nozzle port. thus, its flow process begins to solidify (rackers, 1995, mourão et al., 2011). the sen has a fundamental role in the stability of the process and quality of the final product, being fundamental in the production of special steels (rackers, 1995). however, throughout the process, an accumulation of impurities from the steel builds up on the nozzle wall, developing the clog. as the obstruction increases, the slide gate must be opened in order to maintain the desired flow. however, when its opening reaches 100%, production must stop and the sen or even the tundish set (composed of the tundish, slide gate and sen) must be replaced in advance (thomas & bai, 2001). from the prototype model used to simulate the casting speed variation by solid deposition over time, barati et al. (2018) established three stages for the formation of clogging. throughout the process, deposition of particles occurs in the sen. when the clogging event starts, during the first 80 seconds, some regions of the middle section of the sen are covered by a smooth layer of clogging (coverage stage). then, there is the bulging phase in which the deposition of particles occurs more quickly, emerging visible particles. this phase occurs up to about 150 seconds and is followed by the branching step where there is the development of a branched structure that grows continuously until the sen cross-section is completely blocked, around 250 seconds. in general, as this phenomenon occurs only after some heats, it is not necessary the sen to be fixed or cleaned at such a high frequency. long short-term memory neural networks for clogging detection in the submerged entry nozzle 157 figure 1. schematic summary of the flow of steel from the tundish to the mold with flow control performed by the slide gate in their research, mourão et al. (2011) found that clogging can be formed not only by solidified steel and the transport of oxides present in it, but also by the aspiration of air in the sen and the chemical reactions. however, they emphasize that the exact causes of clogging, specifically, can be difficult to identify. 3. long short-term memory (lstm) the lstm has a set of recurrently connected memory blocks. each block contains one or more interconnected cells and three multiplicative units, also called forget gate f(t), input gate i(t) and output gate o(t) (haykin, 2011, goodfellow, bengio & courville, 2016, buduma & locascio, 2017). the basic architecture of an lstm cell is shown in figure 2, where x(t) corresponds to the input signal, c(t) and c(t-1) are, respectively, the current state of the memory cell and its previous instant and h(t) and h(t-1) represent its current and previous hidden state, respectively. the signals are sent to the three gates that control the information. the function of forget gate f(t) is to control which parts of the long-term states should be forgotten. the input gate i(t), in turn, has the function of controlling which parts should be added to the long-term states. the output gate o(t) is responsible for controlling the output information h(t) in the current state of time. the gates outputs are calculated using: diniz et al./decis. mak. appl. manag. eng. (2022) 5 (1) (2022) 154-168 158 figure 2. the structure of an lstm cell 𝑓(𝑡) = 𝜑(𝑤𝑓𝑥 . 𝑥(𝑡) + 𝑤𝑓ℎ . ℎ(𝑡 − 1) + 𝑏𝑓 ) (1) 𝑖(𝑡) = 𝜑(𝑤𝑖𝑥 . 𝑥(𝑡) + 𝑤𝑖ℎ . ℎ(𝑡 − 1) + 𝑏𝑖 ) (2) 𝑜(𝑡) = 𝜑(𝑤𝑜𝑥 . 𝑥(𝑡) + 𝑤𝑜ℎ . ℎ(𝑡 − 1) + 𝑏𝑜) (3) where 𝜑(∙) is a nonlinear activation function that, in general, uses the sigmoid function. thus, the updates of the state of the memory cell c(t) and of the hidden state h(t) are generated, respectively, by 𝐶(𝑡) = 𝑓(𝑡)⨀𝐶(𝑡 − 1) + 𝑖(𝑡)⨀ tanh(𝑤𝑐𝑥 . 𝑥(𝑡) + 𝑤𝑐ℎ . ℎ(𝑡 − 1) + 𝑏𝑐 ) (4) ℎ(𝑡) = 𝑜(𝑡)⨀ tanh(𝑐(𝑡)) (5) where tanh(.) represents the hyperbolic tangent activation function and ⨀ denotes the point multiplication operation between two vectors. the terms wfx, wix, wox and wcx correspond to the input weights of each gate, while wfh, wih, woh and wch refer to their respective recurrent weights and the terms bf, bi, bo and bc represent the bias. lstm avoids the disappearance of the gradient through the switch of its gates, which develop a kind of temporal memory. during the training phase, samples from each batch are passed into cells iteratively through states. the hidden state represents short-term memory, while the cell state is long-term memory. it is in this unit that information is propagated through the network, interacting with the cell through the ability to remove or add information through gates. as a result, they are able to identify which temporal information should be transmitted or discarded by the network. after processing each batch, the internal states of each cell are reset (goodfellow, bengio & courville, 2016, buduma & locascio, 2017). the great complexity of networks based on deep learning can lead to a problem known as overfitting, thus, it is common to use a regularization technique called dropout. in it, at each training iteration, there is a random removal of a pre-established percentage of neurons from a given layer, adding them again in the next iteration. considering that a given neuron will not depend on the specific presence of the others, dropout enables the learning of the network to deal with more robust attributes (goodfellow, bengio & courville, 2016, buduma & locascio, 2017). long short-term memory neural networks for clogging detection in the submerged entry nozzle 159 4. dataset and methodology 4.1. dataset in this paper, historical data from 6 months of measurements made in a continuous casting steelmaking process were used. the variables were collected from two tundishes operating on a mold at a rate of one sample per second. it is important to clarify that in this article does not present data, nor specific characteristics of the industrial process, as well as the company name due to the data confidentiality protocol. ideally, the process specialists classify as clogging the corresponding event to the gate opening without increase the mold level, with or without a variation in casting speed. in most cases, the gate opening occurs after increasing oscillations, which may or may not reflect oscillations in level. the nature of the data, however, does not allow the use of simple rules to classify clogging occurrences. for example, there are clogging events where the casting speed is increasing while the gate opening has a much higher rate than expected. figure 3 illustrates an example of the clogging event through the process. starting from sample 10.601, there is a gradual increase in the gate opening, at constant speed casting, without a significant increase in the mold level. however, due to the reduction of the sen section caused by the obstruction, a small increase in level is observed just before the casting speed is reduced. the reduction of casting speed provides a gate closure and, when the casting speed reaches a level below 0.6 m/s, the level starts to rise. it is also observed that during this period there is no exchange of tundishes, since the indicative variable of tundish in operation remains constant. figure 3. example of clogging occurrence observed from selected process variables thus, based on experiments performed in the literature (ikaheimonen, 2002, vannucci & colla, 2011, pellegrini et al., 2019) four process variables were selected: the percentage of the slide gate's total opening (gate opening), mold level, casting speed and the tundish that is operating (tundish operating). researchers also suggest diniz et al./decis. mak. appl. manag. eng. (2022) 5 (1) (2022) 154-168 160 the use of variables related to temperature, pressure and argon flow; however, we did not observe in our dataset a significant correlation between these variables and the studied phenomenon. this may have happened because of the high number of outliers in the dataset. furthermore, it was observed that these variables were not effectively measured in the period considered, possibly due to sensor failures. after selecting the variables, pre-processing was applied for the treatment of outliers and missing data, ensuring the preservation of the relationships between the attributes. the outliers’ occurrences were seen as measurement errors because they are specific cases and, therefore, were treated in order to make them consistent. for this purpose, the maximum and minimum theoretical values assumed by each of the variables during an operation without anomalies were specified. thus, samples with values outside the theoretical range were considered outliers: if a sample had a value below the theoretical minimum, then it was adjusted to the theoretical minimum value assumed by the variable. analogously, if the assumed value was greater than the theoretical maximum, then it was set to its theoretical maximum value. in relation to missing data, no substantial occurrences were identified in the period and variables analyzed. still, the few absence periods longer than a sample were treated by a moving median filter. furthermore, a certain unbalance between the clogging and non-clogging classes was verified, which was expected for this type of problem, since clogging is a failed event. it was observed that a little more than 7% of the samples indicated the occurrence of the event. given the number of available samples, the data were selected in order to promote the balance between the classes of the training data only, without losing great importance information for the modeling. in barati et al. (2018) were observed that clogging is only truly perceived in the sen section between approximately 100 and 150 seconds. therefore, as it is an autoregressive system, the composition of the input matrix was made through a sliding window covering 120 samples and with a step of 1 sample per iteration. in this way, the past behavior of each variable will be used to classify whether in the present there is clogging or not. figure 4 illustrates a time-based sliding window process. following this analysis, given an output, y(t), corresponding to one of the binary classes at time t, it will be related to the input matrix, x(120×4)={x1, x2, x3, x4}t, containing the 4 selected variables with samples delayed 120-time steps, i.e., from (t-119) to t. thus, each matrix formed will be labeled with one of the classes, that is, the event (clogging or non-clogging) that is happening at instant t. the sliding window moves along the series so that, at each step, a new input matrix and its respective class are generated. the dataset after pre-processing is composed of 422,026 matrices of dimensions 4x120, i.e., 120 samples of each of the four input variables (gate opening, speed casting, mold level and tundish operating). the dataset was divided into training, validation and testing, with 70% of the data applied in training (ntr = 292,170 matrices), 17% for validation (nval = 73,044 matrices) and 13% reserved for the testing stage (ntst = 56,812 matrices). long short-term memory neural networks for clogging detection in the submerged entry nozzle 161 figure 4. sliding window process as shown in table 1, each training matrix has a corresponding class, which is distributed in 50% representing the clogging event and the others representing the non-clogging class. on the other hand, the validation and test datasets remain with the old proportion between classes. data standardization (z-score) was performed for each variable of each set according to eq. (6) so that each variable has zero mean and unitary standard deviation. in the equation, xr(t) represents the variable to be standardized, r (where r ∈ {1,2,3,4}) is the index identifying each variable, 𝜇𝑟 is the mean of xr(t) and 𝜎𝑟 is the respective standard deviation (both 𝜇𝑟 and 𝜎𝑟 were computed using training data). standardization was chosen because it better handles possible outliers present in the series (skiena, 2017). 𝑥𝑧−𝑠𝑐𝑜𝑟𝑒 (𝑡) = 𝑥𝑟(𝑡)−𝜇𝑟 𝜎𝑟 (6) table 1. dataset split and proportion of each class. dataset number of matrices split proportion (%) clogging class (%) normal class (%) training 292,170 70 50 50 validation 73,044 17 93 7 tests 56,812 13 93 7 4.2. methodology in structural terms, different parameter configurations, defined by the trial-anderror method, were applied to the lstm classifiers. for this purpose, only the number of cells in the two lstm layers was varied in increments of eight, using all their states. in turn, a fully connected (fc) layer with 200 neurons was always maintained at the output of the last lstm layer. the four input variables, lagged by 120 samples each, were applied to the network using a batch size of 1,200 samples. a limit of 200 epochs for training was chosen, diniz et al./decis. mak. appl. manag. eng. (2022) 5 (1) (2022) 154-168 162 which can be interrupted by early stopping with patience of 5. the weights were updated using adam algorithm with learning rate of 0.001 and cross-entropy as loss function (kingma & ba, 2014, goodfellow, bengio & courville, 2016). due to the stochastic nature of the models used here, the k-fold was applied, with k = 5, for validation and comparison of the classifiers. from then on, eight classifier configurations presented the best performances in the training phase from the validation set. table 2 shows the configurations of each classifier and also the number of trained parameters. table 2. the best-found lstm models and their main parameters associated with their architectures. lstm model lstm layer 1 cells lstm layer 2 cells number of parameters 1 256 128 490.586 2 256 64 362.842 3 256 32 311.258 4 256 16 288.538 5 256 8 277.946 6 128 64 130.906 7 128 32 95.706 8 64 32 37.082 figure 5 represents a schematic diagram of the structure of the networks used in this paper. the lstm networks were implemented with two layers, each containing 32 to 256 cells, with hyperbolic tangent function. it is interesting to mention that, although network configurations containing 32 cells in the first lstm layer were tested, relevant results were not obtained. the dropout was applied to the second lstm layer with a value of 0.3. in classification tasks, the number of output layer neurons will be equal to the number of classes to be predicted. in this layer, it is common to use the softmax activation function. the softmax function helps to map the output of a neural network to a set of categories, by transforming these responses into probabilities that add up to 1. the softmax function is calculated as follows, where j is the number of classes: 𝑠𝑜𝑓𝑡𝑚𝑎𝑥(𝑦𝑗 ) = 𝑒𝑥𝑝 (𝑦𝑗) ∑ 𝑒𝑥𝑝(𝑦𝑖) 𝐽 𝑖 =1 (7) where yi is the i-th neural network output (i = 1, …, j) and yj is the class whose categorical probability is calculated in the equation. in this way, in all models the signal was transferred to a fc layer containing 200 neurons, activation function rectified linear unit (relu) and dropout of 0.5, concatenated to a softmax layer with 2 neurons to generate the probability of classifying the input time series as clogging or non-clogging. long short-term memory neural networks for clogging detection in the submerged entry nozzle 163 figure 5. generic structure of the lstm classifier used in this article the simulations were implemented on a computer in an environment python 3.6, 64 bit operating system, 16 gb of ram, intel(r) core(tm) i7-9750h cpu @ 2.60ghz 2.59 ghz with gpu nvidia geforce titan xp. 4.3. performance criteria in classification tasks the confusion matrix is usually used in the analysis of performance in classification tasks. as seen in figure 6, it is the result of comparing the correct class of each sample in the test set and the class obtained by the classifier. figure 6. confusion matrix diniz et al./decis. mak. appl. manag. eng. (2022) 5 (1) (2022) 154-168 164 in binary classification tasks, the confusion matrix is composed of positive and negative class observations (fawcett, 2016). in this work, the occurrence of clogging will be associated with the positive class and its absence with the negative class. thus, after classification, the values can belong to four possible categories:  true positive (tp): samples that belong to the positive class that were correctly classified;  false positive (fp): samples that belong to the negative class, but they were incorrectly classified as belonging to the positive class;  true negative (tn): samples that belong to the negative class that were correctly classified;  false negative (fn): samples that belong to the positive class, but they were incorrectly classified as belonging to the negative class. from these categories, some performance measures can be calculated, such as accuracy, precision and recall, respectively, according to eq. (8), eq. (9) and eq. (10). 𝑎𝑐𝑐𝑢𝑟𝑎𝑐𝑦 = 𝑇𝑃+𝑇𝑁 (𝑇𝑃+𝐹𝑁+𝑇𝑁+𝐹𝑃) (8) 𝑝𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 = 𝑇𝑃 (𝑇𝑃+𝐹𝑃) (9) 𝑟𝑒𝑐𝑎𝑙𝑙 = 𝑇𝑃 (𝑇𝑃+𝐹𝑁) (10) accuracy measures the overall performance of the model, considering both the proportion of correct classifications of positive and negative cases. in contrast, precision measures the rate of positive examples classified correctly among all those predicted as positive. recall corresponds to the rate of classification of true positives in relation to the total number of positive examples (fawcett, 2016). from the combination of accuracy and recall, it is possible to obtain an indicator of the overall quality of the model, called f1-score. the f1-score calculation is represented in eq. (11). 𝐹1 = 2 ×𝑝𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 ×𝑟𝑒𝑐𝑎𝑙𝑙 (𝑝𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛+𝑟𝑒𝑐𝑎𝑙𝑙) (11) in some cases, it is also interesting to evaluate the matthews correlation coefficient (mcc). the mcc is a more reliable and balanced measure of quality, as it only produces a high score if the classifier correctly predicted most of the positive and negative grades. its equation is presented in eq. (12) (boughorbel, jarray & el-anbari, 2017). 𝑀𝐶𝐶 = (𝑇𝑃×𝑇𝑁)−(𝐹𝑃×𝐹𝑁) √(𝑇𝑃+𝐹𝑃)×(𝑇𝑃+𝐹𝑁)×(𝑇𝑁+𝐹𝑃)×(𝑇𝑁+𝐹𝑁) (12) the mcc varies in the range between -1 and 1, where the extreme values +1 and 1 indicate, a perfect classification and a totally incorrect classification, respectively, while the value 0 indicates a classification equivalent to what would be done randomly (boughorbel, jarray & el-anbari, 2017). long short-term memory neural networks for clogging detection in the submerged entry nozzle 165 5. results the performances of the eight lstm models are shown in table 3 in terms of mean and standard deviation of accuracy, precision, recall, f1-score and mcc. as we can see, the lstm 1, lstm 2 and lstm 6 models reached levels of accuracy and precision above 80%. however, these were also the models that showed the greatest standard deviations. in particular, the lstm 1 model stands out for its 85.30% recall, that is, around 5% more than the other two models. the same can be observed for the mcc metric, where the lstm 1 model also stands out, reaching an average of 0.723 with a standard deviation around 0.145. the expected mcc metric for a dummy classifier, which is based on the majority class, is close to zero. therefore, the obtained results are superior to a dummy classifier. table 3. comparison among the performances of different lstm models.. lstm model accuracy (%) precision (%) recall (%) f1-score (%) mcc 1 86.10±3.87 85.61±3.42 85.30±5.51 85.45±4.22 0.723±0.145 2 82.40±5.11 83.09±5.52 79.71±5.15 81.36±5.33 0.631±0.161 3 75.81±0.31 79.55±0.65 69.49±1.08 74.18±0.81 0.520±0.006 4 74.31±0.68 77.69±2.67 68.52±3.22 72.82±2.92 0.491±0.016 5 75.20±0.41 81.84±1.02 64.79±0.56 72.32±0.72 0.515±0.010 6 82.02±4.33 82.54±7.41 80.89±7.47 81.32±7.44 0.612±0.085 7 75.83±0.19 79.79±0.80 69.22±1.47 74.13±1.04 0.521±0.013 8 75.76±0.38 79.15±0.81 69.95±0.87 74.27±0.84 0.518±0.080 due to the nature of the process, noise and, mainly, control actions by operators may be present in measurements, making it difficult to use simple rules assertively. furthermore, due to the unbalance between classes, although this strategy can result in high rates of global accuracy, the dummy classifier would hinder the identification of examples belonging to rare classes that, in the problem in question, represent the interest class. although precision and recall had values close to each other in the three main models (lstm 1, lstm 2 and lstm 6), it is interesting to note that the average recall did not exceed precision. in accordance with eq. (9) and eq. (10) this result indicates a number of fn higher than the number of fp. in general, it is possible to observe that the reduction in the number of parameters of the second layer compromised the generalizability of the models, mainly in terms of recall, and consequently, f1-score. for example, the lstm 1 and lstm 2 models differ by the number of cells in the second layer and, as shown in table 3, the lstm 1 model, which has the highest degree of complexity among the models, showed greater generalization capacity in the classification task. in practice, models analysis must be based on a trade-off between the values of the performance metrics and the number of parameters involved. comparing the parameters used by lstm 1 and lstm 2 models with those of the lstm 6 model, which also achieved promising results, it appears that there are about 3.74 more parameters in the first model and 2.77 in the second model. in particular, it is observed that the overall performance of the lstm 2 and lstm 6 models does not change considerably, which makes the lstm 6 model more attractive. diniz et al./decis. mak. appl. manag. eng. (2022) 5 (1) (2022) 154-168 166 furthermore, although a small reduction in training time was observed with the decrease in the number of parameters, no significant differences in processing time were observed during the tests of the models. since this is a problem involving real process data, methodologies capable of correctly detecting the presence of clogging with the lowest possible error rate are sought. in this context, the lstm 1 model appears to be the best choice, since it presents the highest performance averages and the smallest deviations from these averages (among the lstm 1, lstm 2, and lstm 6 models). even taking into account the complexities of the classifiers, the models with the highest number of parameters still seem to be the most attractive, since the training and processing time of these models did not show a significant increase compared to the others. in addition, the lstm 1 model exceeded the 75% accuracy and 62% precision of pellegrini et al. (2019), as well as the 76.9% recall of vannucci & colla (2011). although the significant differences in methodologies and datasets, it can be said that the lstm 1 model achieved promising levels of performance criteria in the clogging classification task compared to those found in the bibliography. 6. conclusion the steel industry does not have effective tools that can correctly detect the occurrences of clogging in the sen. clogging can increase the frequency of interruptions in the operation, resulting in an increase in operating costs, decreased productivity, and adversely affect product quality. in order to treat this problem, in this paper, the general performance of classifiers using lstm neural networks in detecting the clogging in the sen of a steel production process were evaluated. therefore, based on the evidences observed in the literature and analyzes made in this work, four variables were selected to compose the input dataset: gate opening, mold level, casting speed and tundish operating. the dataset was then pre-processed in order to deal with the presence of outliers and missing values, ensuring the relationships between the attributes. furthermore, due to the inherent unbalance of the classes, a careful balancing step was necessary for the training, so that the information of great importance for the modeling was not disregarded. with balanced classes, accuracy can be considered in the evaluation of models, taking into account also other metrics: precision, recall, f1-score and mcc. industrial problems require high hit rates and, within the task of classifying clogging occurrences, the model with the highest number of parameters obtained a remarkably superior performance in relation to the other evaluated models, presenting the highest performance averages. in particular, the best model achieved precision and recall averages above 85%. however, even with higher values of mean precision and recall, a higher number of false negatives was found in relation to the number of false positives, considering that the recall did not exceed precision. this characteristic demonstrates that the best model found is more probable to classify a sample referring to the occurrence of clogging (positive) as normal operation (negative) than the reverse. nevertheless, given the significant differences in methodologies and datasets, it can be said that the best model achieved promising levels of performance criteria in the task of classifying clogging compared to those found in the literature. through the proposed methodology, the possibility of applying the models in a real system was verified. however, the model presented is limited in terms of recall and precision. the recall values did not exceed the precision values, which indicates the long short-term memory neural networks for clogging detection in the submerged entry nozzle 167 need to develop other techniques to reduce false negative rates. for future work, it is desirable to implement other types of classifiers with deep learning architectures, such as convlstm, so that even higher hit rates can be achieved and guaranteeing a reduction in fp rates compared to fn rates. it is also intended to apply the models online in a real plant. the fruits of this work may favor productivity gains and reduction of production costs, due to the increase in sen's useful life, for example. it is estimated that the application of a system that allows the identification of the initial occurrence of clogging can contribute to the increase in the useful life of the sen, which may result in a savings of over us$ 15 million per year. author contributions: all authors actively participated in all stages of this research. funding: this study was financed in part by the coordenação de aperfeiçoamento de pessoal de nível superior brasil (capes) finance code 001. data availability statement: this article does not present data, nor specific characteristics of the industrial process, as well as the company name due to the data confidentiality protocol. acknowledgments: the authors would like to thank the financial support provided by capes, as well as the support of the programa de pós-graduação em engenharia elétrica (ppgee). the authors would also like to thank nvidia corporation for the donation of a titan xp gpu used for this research. conflicts of interest: the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. references barati, h.,wu, m., kharicha, a. & ludwig, a. 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(2021). multiscale convolutional and recurrent neural network for quality prediction of continuous casting slabs. processes, 9(1), 33. yildirim, o., baloglu, u. b., tan, r. s., ciaccio, e. j., acharya, u. r. (2019) a new approach for arrhythmia classification using deep coded features and lstm networks. computer methods and programs in biomedicine, 176, 121-133. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 3, issue 1, 2020, pp. 79-91. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame2003001b *corresponding author email adresses: malay.bhattacharjee100@gmail.com (m. bhattacharjee), gautam.bandyopadhyay@dms.nitdgp.ac.in (g. bandyopadhyay), banhi.guha@gmail.com (banhi guha), sanjibb@acm.org (s. biswas). determination and validation of the contributing factors towards the selection of a b-school an indian perspective malay bhattacharjee 1*, gautam bandyopadhyay 1, banhi guha 2, sanjib biswas 3 1 department of management studies, national institute of technology durgapur, durgapur, india 2 amity university, kolkata, india 3 calcutta business school, west bengal, india received: 6 february 2019; accepted: 2 june 2019; available online: 5 june 2019. original scientific paper abstract: this study has tried to see what factors do play a significant role in the b-school selection process of an aspirant. the study also would like to see the constituent elements of the factors which come to the mind of the prospective takers, herein, the prospective students. the study pursued here has significant relevance in its purpose as it is one of its kind in this country, and that the rising market demands a study which would be helpful to the pursuant as well as the service providers. the existing management institutions in the country are challenged for their survival. this study intends to make their work easier by identifying the most important factors which the ‘prospective candidates’ look for while selecting their b-school. for this purpose, we have conducted an exploratory factor analysis (efa) on the responses obtained from a sample of 594 respondents through questionnaire based survey and interviews. further, in order to ascertain the results, we have done the confirmatory factor analysis (cfa) using structural equation modeling (sem) approach of the first order type. the result of sem conforms to that of efa. keywords: management education, b-school selection, exploratory factor analysis (efa), structural equation modeling (sem) mailto:malay.bhattacharjee100@gmail.com mailto:gautam.bandyopadhyay@dms.nitdgp.ac.in mailto:banhi.guha@gmail.com mailto:sanjibb@acm.org bhattacharjee et. al/decis. mak. appl. manag. eng. 3 (1) (2020) 79-91. 80 1. introduction higher education in the management discipline has become very popular in india in the last twenty – five years or so. while, there were 9 (nine) management institutions in india in the year 1958, the number grew to 2450 in the year 2012. a closer look in this respect will show, that, in the year 1988 india had only 87 management institutions. in the year 1998, that is, just over ten years the number rose to 682, a growth of about 783 percent in a single decade. in the next decade the number went to 1523 (223 percent), and in 2018, the number of aicte approved management institutions stand at 3233, and the total intake strength of all these institutions presently happens to be 3,94,843, (aicte official site, 2017-18). as reported in the same source (aicte), the present number of management pursuant in this country as of 2018 is 2,37,889 which accounts for a total 1,55,154 number of seats (i.e., 39.47 percent of the total permissible enrolment) remain not filled up because of no potential takers. the figure reflects that there is huge competition in the management vertical of higher education in india. number of seats being more than the number of aspirants, management institutions are perceived to be faced with a very difficult situation and really needs to fight it out in the market to convince the aspirants to choose their particular institute over others. the authors (melewar & akel, 2005) suggests that, in an environment where soon-to-be prospective students are being considered as the ‘potential target market’ for being the ‘ultimate consumer’ of the service in the offing (pg management education), higher education institutions very well need to employ articulated and definitive strategies to maintain and intensify their competitive strength in the marketplace. therefore, as a consequence higher education institutions are now relentlessly focusing on wooing higher and better quality students (simões & soares, 2010). it can well be perceived that the same must be the case applicable for the majority of the management institutions (other than the reputed iim’s and the big league ones) in india, engaged in post-graduate management education, especially, when the numbers of seats offered are more than the number of ‘takers’. the evolved scenario thus makes it clear that ‘sustainability’ of a pg management institute in india is a big challengewith 30 management institutions closing down in the year 2017-18 primarily because of dearth of students (aicte official site, 2017-18). earlier literatures in the context of higher education institutions (hei’s) not specific to management education suggest that, various universities based stalwarts have conveyed their anguish with regard to the sedate pace of attaining sustainability of the hei’s (boyle, 1999; leal filho, 2000; roome, 1998). velazquez et al. (2005) reports that for attaining sustainability, hei’s world over strategically resort to focusing on education, research, outreach and partnership and sustainability. in the given context, it is all but clear that management education in the post graduation level has turned into a ‘product’ in the marketplace like all products. the prospective studentwho in the actual practice happens to be the consumer of the service does reserve a huge power of choosing his/her b-school. goff et al. (2004) suggests, it is understood that advertisements, promotions and other marketing activities are very likely to increase in the higher education sector. in this regards it is felt that, a proper evaluation agenda needs to be taken up to ascertain which choice factors students contemplate while making a decision on which institution they intend to attend (wiese et al., 2009). considering the issue of choice factors involved concerning hei’s, it has been found that a stream of investigations has focused on the common student – choice models, e.g. punj & staelin (1978) and vrontis et al. (2007). it is primarily felt that the end consumer who also happens to determination and validation of the contributing factors towards the selection of a b-school… 81 be the ultimate decision maker needs to be informed about the efficacies of a particular offering, as, the consumer can only arrive at a self-satisfying decision when he/she can process a host of information in a meaningful and wholesome way (bhattacharjee & bandyopadhyay, 2017). it may so happen that many of the aspiring students have the potential to get selected in multiple institutions at the time of admission. in such situations it is very important for them to know which institution would give them the highest value, which is still not explored by researchers (debnath & shankar, 2009). with this preamble, in this study, we have made an attempt to understand what factors do influence the students while they select their b-school. further, we also put an effort to analyze the constituent elements those make up the factor(s). the rest of the paper proceeds as follows. section 2 summerizes related work while in section 3, we have elaborated upon the methodology. section 4 highlights the results and includes discussions on the findings, where, section 5 puts forth some implications for industries. section 6 finally concludes the paper including some of the limitations and future scope of the study. 2. related work institutions of higher education strategically play a pivotal role in the economic upliftment of a country (yong et al., 2009). evaluating the performance of education is tough to ascertain, it is being a ‘service’, and governed by all the service attributes of intangibility, inseparability, heterogeneity and perishability (lupo, 2013). it is already an established fact that prospective students do take cognizance of their selected choice factors while they contemplate about enrolling in an hei (espinoza et al., 2002; hoyt & brown, 2003; gray & daugherty, 2004; punnarach, 2004). in addition to this, previous literatures also propose that certain choice factors appear to be of higher importance than the others (sevier, 1993; martin, 1994; geraghty, 1997; davis, 1998; freeman, 1999; bers & galowich, 2002; price et al., 2003; shin & milton, 2006). wan endut et al. (2000) have worked on the subject ofbenchmarking of the institutions engaged in higher education. in the introductory part it has already been discussed that we have failed to identify any work on this very subject concerning the determination of factors which influence a prospective pg management aspirant to choose his/her b-school. in this relation it needs to be mentioned that debnath & shankar (2009) has worked on the subject of ranking of b-schools based on tangible and intangible parameters. studies related to the ranking of pg programs of management have been the subject of the works of kedia & harveston (1998), acito et al. (2008) and köksalan et al. (2010). evidences show that ‘teaching’ is also an important parameter in the ranking of b-schools (ar et al., 2013). the choice factors which are being considered for the purpose of this study have been majorly taken from the existing literatures on hei’s. for example, the researchers (beerli palacio et al., 2002; arpan et al., 2003; pabich, 2003) probed the usefulness of ‘image of the institute’ in winning over students to select a particular brand of hei over its competitors. word-of-mouth mostly propagated by the alumni(earmarked here as past students' feedback),plays a significant role in influencing prospective students (espinoza et al., 2002; arpan et al., 2003; seymour, 2002). a campus visit is one of the most important information sources for a prospective student which has a big say in the final decision making process (seymour, 2002). faculty research (mathew, 2014), international links (wiese et al., bhattacharjee et. al/decis. mak. appl. manag. eng. 3 (1) (2020) 79-91. 82 2009), corporate reputation, earmarked here as brand recognition (coetzee & liebenberg, 2004), employment prospects (here, placement-opportunity), academic reputation or track record, entry requirements, affordable fees, location or near to home (wiese et al., 2009) are some of the other dominant factors that affect selection of a b-school. 20 such choice factors could be found (absher & crawford, 1996; jonas & popovics, 1990)though the studies were from different fields of higher education and none of pg management stream. our whole-hearted efforts could not find anything worth mentioning in regard to the area of our work. as already mentioned ‘choice factors’ have been the subject of earlier researchers (though, not in the indian context). major part of the research fraternity has considered the hei’s and not specifically ‘management studies’ as investigations reveal. furthermore, none have pondered the issue of ‘likely factors which influence (purchase) decision making when it comes with a b-school selection by the (consumers) young aspirants. given this as a pretext, the following is understood to be the gap which this study would address in due course: identify what factors the young aspirants of ‘business education’ are considered the most in their selection process of a b-school. 3. research methodology a standard questionnaire was developed for the purpose of primary data collection from the existing 1st year pg management pursuant. the ‘choice elements’ considered has been mostly taken from the existing literatures on hei’s, which resulted in 21 such elements to be examined. a pilot study was conducted among 100 1st year (1st semester) students. the cronbach’s alpha statistic stood at 0.628, which is satisfactory but not good. therefore, the ‘reliability analysis’ indicated some of the elements were not seriously considered by the respondents while answering. exploratory factor analysis (efa) was then carried out taking all the 21 elements into consideration. 7 different factors resulted in with a kmo score of 0.783, but the irony of the fact, four of them comprised of single elements and thus had to be excluded as conceptually a factor should comprise of more than one element. it is empirically accepted that the elements which comprise a particular factor has strong intra-correlation among themselves. the pilot study strongly indicated the reliability and validity of 13 elements, hence, we had to drop the rest of the element initially considered. to carry out the study we have approached 1000 prospective students of mba 1st year. out of this 594 students agreed to take part as a respondent hence the response rate was 59.4 percent. the respondents happen to be 1st year mba students of different b-schools of eastern india. the profile of the respondents families is given in the table 1. the responses of the respondents constituting the sample under study were captured during the period of july to december 2017. the final sample considered for this study was 594 where the cronbach’s alpha statistic went up remarkably with 13 elements and was recorded as 0.857 which is good and indicates ‘high reliability’. the kmo score stood at 0.797 (considerably higher than earlier) and the bafrtlett’s test of sphericity was significant at 0.000 exploratory factor analysis (efa) has been used to derive the number of factors as the authors (bornstedt, 1977; rattray & jones, 2007) were of the view that the construct validity of a research questionnaire can be verified and validated using factor analysis. determination and validation of the contributing factors towards the selection of a b-school… 83 table 1. profile distribution of the families (respondents) income group (rs. per annum) below 3 lakhs 1.2 percent (7) 3 to < 5 lakhs 10.9 percent (65) 5 to < 7 lakhs 32.5 percent (193) 7 to 10 lakhs 38.7 percent (230) above 16.7 percent (99) occupation (principal source of income) service 68.9 percent (409) business 25.8 percent (153) professional/ self-employed 5.4 percent (32) cfa is carried out to ascertain the factor structure derived from a set of observed variables through pca. essentially, it verifies the hypothesis which examines the relationship between observed variables and their underlying latent structure (anderson & gerbing, 1984; curran, et al., 1996; marsh, et al., 1988; brown, 2014), hence, cfa was employed. in order to understand the structural relationship between observed variables and the concerned latent constructs, structural equational modelling (sem) is carried out (schreiber et al., 2006). for understanding the applicability of carrying out the cfa first a ‘pattern matrix’ is derived. the coefficients of the pattern matrix are understood to be the distinctive or unique loads of the resultant factor into variables. as the sample being considered is a quite big interpretation would be relatively accurate. the model thus derived has been further subject to validation test using sem (structural equation modeling). sem has been used to validate the results derived through the application of amos or other techniques (mardani et al., 2017). the contention of the study is a twofold one. firstly, the study would like to see if all the factors are equally important to the would-be management aspirants in their decision making process towards selecting their b-school of choice andsecondly, if all the factors are represented by the same number of constituent elements? based on the above arguments the hypotheses to be tested are as follows: h0a: all the factors are not equally important to the management aspirants while selecting their b-school of choice. h0b: all the factors do not comprise of the same number of constituent elements. 4. findings and discussion table 2 represents the result of efa. in this case we have followed the principle component method (pca) for extraction.the eigenvalues derived from the efa stands at 4.592 towards “institute reputation (f1)”, 2.291 towards “global engagement and stability (f2)” and 1.214 towards “affordability” (f3). at the same time it is evident that ‘institute reputation’ constitutes 7 elements, that is, when somebody considers ‘institute reputation’ actually the individual in a known or unknown way is considering all these seven elements. in the same light we can find that ‘global engagement and stability’ consists of 3 elements while ‘affordability’ consists of 2 elements. the efa results indicate that 3 factors have been created in accordance with the relationship being shared among the elements or variables in this case. therefore, the results indicate that the null hypotheses considered, have been supported in both the cases. in order to proceed further, we have checked the bhattacharjee et. al/decis. mak. appl. manag. eng. 3 (1) (2020) 79-91. 84 results from the pattern matrix as given below (table 3). in this case we have adopted the maximum likelihood method. table 2. efa component matrix components items institutes’ reputation global engagement & stability affordability 2d_faculty_qualification 0.820 2n_easy_of_entry 0.816 2t_brand_recognition 0.758 2e_placement 0.740 2k_past_stu_feed 0.683 2g_average_salary 0.634 2h_corporate_visit 0.590 2j_hostel_facility 2s_foreign_tours 0.810 2f_foreign_placement 0.776 2c_faculty_engagement 0.687 2q_affordable_fee_structure 0.768 2o_edu_loan 0.765 table 3. pattern matrix items factors institutes’ reputation global engagement & stability affordability 2d_faculty_qualification 0.834 2n_easy_of_entry 0.789 2t_brand_recognition 0.735 2e_placement 0.733 2k_past_stu_feed 0.638 2g_average_salary 0.556 2h_corporate_visit 0.52 2j_hostel_facility 2s_foreign_tours 0.803 2f_foreign_placement 0.666 2c_faculty_engagement 0.578 2o_edu_loan 0.873 2q_affordable_fee_structure 0.674 determination and validation of the contributing factors towards the selection of a b-school… 85 the eigenvalues derived from the efa stand at 4.552 towards “institute reputation (f1)”, 2.280 towards “global engagement and stability (f2)” and 1.209 towards “affordability” (f3). at the same time it is evident that ‘institute reputation’ constitutes 7 elements, that is, when somebody considers ‘institute reputation’ actually the individual in a known or unknown way is considering all these seven elements. in the same light we can find that ‘global engagement and stability’ consists of 3 elements while ‘affordability’ consists of 2 elements. the efa results indicate that 3 factors have been created in accordance with the relationship being shared among the elements or variables in this case. thus, the result of cfa perfectly corroborates the results of the efa and efa also indicates strong support towards both the null hypothesesconsidered. for further validation of the model sem has been taken help of. table 4 represents the individual cornbach’s alpha results factor-wise. institute reputation (f1) does bear a alpha score of 0.857, global engagement and stability (f2) 0.662, and affordability (f3) 0.636. this clearly signifies that all the factors derived are all either good or in their acceptable level, thus, proving the reliability of them. table 5 indicates that the model is fit taking all the indices into consideration. the chi-square value of 1.348 is well below the recommended level. p-value of .085, goodness of fit index (gfi) value of 0.988, adjusted goodness of fit index (agfi) value of 0.969, comparitive fit index (cfi) value of 0.996, tucker lewis index (tli) value of 0.991 are well above the recommended level. root mean square of approximation (rmsea) value of 0.024 is found to be well below the recommended level of <0.10. the hoelter level of significance value stood at 629 at 5 percent significance level and 726 at 1 percent significance level. all the figures indicate towards the validity of the model thus created. the sem result perfectly matches with that of the efa and cfa. the elements constituting the factors (f1, f2 and f3 represented in the diagram as f1, f3 and f4) are found to be the same in all the methods. this might be indicative of the robustness of the model and the accuracy of the methods taken help of.elements like faculty qualification, brand recognition, and past students’ feedback, does have a positive bearing on placements. on the other hand placement (campus) does have a positive bearing on the affordable fee structure. in the same light average salary (students) does have a close bearing on ‘ease of entry’. brand recognition of an institute has a close connection with the number of corporate visiting the campus (in a given period for placement), hostel facility offered, the number of foreign placements and number of foreign tours arranged for the students. affordable fee structure does have a positive bearing on foreign tours. brand recognition, and past students’ feedback do have a direct bearing on faculty engagement (faculty retention). that is, when an aspiring candidate considers the issue of placement (campus), automatically intrinsic issues like affordable fee structure, brand recognition, past students’ feedback as well as faculty qualification are also the elements of his consideration. interestingly, elements like brand recognition, past students’ feedback, placement, and faculty qualification being members of the same factor share a huge intra-group correlation, while, affordable fee structure share a high inter-group correlation among themselves. in the same light it can also be suggested that the average salary (students) shares a high intra-group correlation with ease of entry, while, a high inter-group correlation with foreign tours. many more intra-group correlations as well as inter-group correlations has also come up from the study which suggests that all the three factors have unique intrinsic properties (elements) bhattacharjee et. al/decis. mak. appl. manag. eng. 3 (1) (2020) 79-91. 86 in them which has enabled them to share inter-group relationships. that is, though it can be empirically established that, institute reputation (f1) is the dominant factor out of the three and therefore the most important one which young management aspirants consider, the rest are also within the evoked set of the young aspirant during their decision making process. table 4. factor wise reliability and factor analysis items reliability analysis factor analysis cornbach's alpha (α) α if item deleted factor loading > 0.6 less than factor α ≥ 0.50 2d_faculty_qualification 0.857 0.824 0.82 2t_brand_recognition 0.839 0.758 2k_past_stu_feed 0.84 0.74 2e_placement 0.837 0.683 2g_average_salary 0.844 0.634 2h_corporate_visit 0.847 0.59 2n_easy_of_entry 0.818 0.816 2j_hostel_facility 0.86 0.49 2s_foreign_tours 0.662 0.37 0.81 2f_foreign_placement 0.446 0.776 2c_faculty_engagement 0.736 0.687 2o_edu_loan 0.636 na 0.765 2q_affordable_fee_structure na 0.768 table 5. model fit summary model fit indices recommended value obtained value chi-square/df <3.00 1.348 p-value >0.05 0.085 gfi >0.90 0.988 agfi >0.90 0.969 cfi >0.80 0.996 tli >0.95 0.991 rmsea <0.10 0.024 hoelter > sample size (594) 629 at (0.05) & 726 at (0.01) determination and validation of the contributing factors towards the selection of a b-school… 87 figure 1. structural model 5. industry implications in the course of the study the most important learning’s are enumerated below: the elements (essentially choice factors) considered for the study are not equally important to the ultimate consumer (aspiring pg management student) while they go for deciding their b-school. choice factors single-handedly don’t make a factor, rather, more than one choice factors make a factor and as a consequence of that, when a would-be consumer (aspiring management students in this case) considers one or more factors, in the actual practice there is a high probability that the individual may be silently and subconsciously considering all the different elements (here, choice factors) of the concerned factor. the elements of a given factor are deemed to be highly correlated among each other. but, at the same time it has been understood that inter-correlations among different subjects (choice factors here) do exist making the process complex. the b-schools (service providers) which are fighting intensely among themselves to convince the ultimate takers of their product would be extremely benefitted when they are armed with the findings of this study. bhattacharjee et. al/decis. mak. appl. manag. eng. 3 (1) (2020) 79-91. 88 the service providers must give due diligence to all the three factors derived with a higher emphasis on f1 as it happens to be the dominant factor of the three. at the same time they need to give importance to the elements of the other two factors which are having a high bearing on f1. 6. conclusion in this study, our main focus has been on the choice of factors while selecting a bschool from the perspectives of the users. we administered a questionnaire survey on 594 respondents and then subsequently performed efa for understanding dominating factors. we then performed a cfa in order to confirm our results and to understand the basic structural model behind the same. our results show conformity. however, this study being confined to the city of kolkata only is riddled with area (geographical) limitation. in addition to that the study has tried to identify the most important factor(s) which influences the consumer to go for his choice of institution. in the process the demographic factors have not been given due importance, though, in all probabilities, factors like family income if considered might present a very different picture. we would like to go for a detailed investigation in the future taking all the limitations into consideration. acknowledgement: the authors would like to express sincere gratitude to all reviewers for providing valuable comments for further improvement. the first draft of this paper has been presented during the first international conference on “frontiers of operations research & business studies (forbs 2018)” organized by calcutta business school in collaboration with operational research society of india (orsi), kolkata chapter (oct 11-13, 2018) and subsequently some modifications have been incorporated in response to the reviewers’ comments. author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research received no external funding. conflicts of interest: the authors declare no conflicts of interest. references absher, k., & crawford, c. 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(2009). evaluation of dynamic competitiveness of the university of listed companies based on grey relational analysis. international conference on grey systems and intelligent services (pp. 124-128). © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0323062022s * sujit das. e-mail addresses: mukherjerjee.anupam.bnk@gmail.com (a. mukherjee), partha4187@gmail.com (p.s. barma), joy77deep@yahoo.co.in (j. dutta), sujit.das@nitw.ac.in (s. das), dpamucar@gmail.com (d. pamucar) imprecise covering ring star problem anupam mukherjee1, partha sarathi barma2, joydeep dutta3, sujit das4,*, and dragan pamucar5 1 school of applied science and humanities, haldia institute of technology, haldia, india 2 center for distance and online education, university of burdwan, burdwan, india 3 department of computer science kazi nazrul university, asansol, india 4 national institute of technology warangal, india 5 department of logistics, university of defence in belgrade, belgrade, serbia received: 11 march 2022; accepted: 3 june 2022; available online: 23 june 2022. original scientific paper abstract: in this paper, we formulate and solve an imprecise covering ring star problem (icrsp), a generalization of the ring star problem (rsp). for a given network, the objective of this problem is to find a subset of nodes that forms a cycle, and other nodes are left-out nodes. this problem minimizes the total routing cost due to the cycle itself and assignment costs from the left-out nodes assigned to their nearest concentrators (i.e., nodes on the cycle). no assigned node exceeds a predetermined covering distance from its concentrator. the covering distance and the routing and assignment costs are considered fuzzy in the proposed icrsp. a modified genetic algorithm (mga) has been developed and used to solve this model for different confidence levels depending on the corresponding imprecise parameters, reducing it to a deterministic form with fuzzy possibility and necessity approaches. some comparisons with tsp benchmark problems are presented to justify the algorithm's performance. as individual cases, some practical icrsps are also solved and presented numerically. key words: covering salesman problem, genetic algorithm, median cycle problem, ring star problem. 1. introduction 1.1. literature review labbé et al. (1999) proposed a branch-and-cut technique based on some polyhedral properties to solve the model. kedad-sidhoum (2010) gave a chain-based formulation of rsp and proposed a branch-and-cut algorithm to obtain its results. an mukherjee et al./decis. mak. appl. manag. eng. (2022) 2 integer programming approach to the same problem was formulated by simonetti et al. (2011) and solved it by their branch-and-cut method. a greedy randomized adaptive search procedure for rsp was developed by dias et al. (2006). calvete et al. (2013) proposed an evolutionary method based on selection, crossover, and mutation for the same problem considering the installation cost. all the above authors examined their algorithms for rsp over the benchmark instances of tsplib for travelling salesman problem (tsp). labbé et al. (2004) studied a similar problem called the median cycle problem, where the objective is to find a cycle with minimum routing cost from a subset of vertices in a network subject to an upper bound of assignment cost of the unvisited nodes. a variable neighborhood tabu-search method was proposed by moreno p´erez et al. (2003) for the median cycle problem. capacitated m-ring star problem was introduced by baldacci et al. (2007) where the goal is to find m rings with bounded capacity, which pass through a common depot and some other nodes, and all unvisited nodes are assigned to the visited nodes. the authors used a branch-and-cut algorithm for the model. later, najiazimi et al. (2012) proposed a heuristic method for the same model which consists of an integer linear programming based improvement on a variable neighborhood search. introducing the multiple depot concept in the same model, baldacci et al. (2010) further generalized it. sundar et al. (2017) studied a polyhedral analysis and proposed an exact algorithm to solve the multiple-depot ring-star problem. later, baldacci et al. (2017) discussed pricing strategies of capacitated rsps using dynamic programming approaches. covering salesman problem (csp) is similar to rsp, which minimizes the total traveling cost of a cycle through a subset of nodes from a network. the nodes out of the tour are within a predetermined distance from the visited nodes. this model was first introduced by current & schilling (1989) and solved with their proposed simple heuristics covtour. later, golden et al. (2012) defined some variants of csp and proposed two local search algorithms ls1 and ls2, which he used on the data sets initiated from tsplib. salari et al. (2012) introduced an integer programming-based local search process for csp. recently, a hybrid algorithm consisting of dynamic programming and ant colony optimization (aco) was developed by salari et al. (2015). mukherjee et al. (2019, 2021) have worked on ring star and ring tree problems considering an extra layer of rings. this second layer consists of secondary sub-depots. the authors have used their single and multi-objective discrete versions of the antlion optimizer. barma et al. (2021) proposed a multi-objective ring-star vehicle routing problem with perishable items. the authors have considered an m-ring star network to speed up the routing process. this problem has been solved using a non-dominated sorting-based ga and strength pareto evolutionary algorithm. 1.2. the proposed problem this paper introduces the concept of an imprecise covering ring star problem (icrsp). for a given network, a subset of nodes is to be selected to minimize the routing cost of the cycle through the selected nodes, added by the assignment cost of the unvisited nodes to their nearest concentrators. also, the distance of each unvisited node to its nearest concentrator does not exceed a predetermined covering distance (figure 1). here, the routing cost, assignment costs and the covering distance are considered fuzzy and the shortest distance of one node to another node is considered crisp. a smooth formulation is given with possibility and necessity approaches for this model, which is reduced to deterministic forms in both cases. we have developed and used a modified genetic algorithm (mga) where each binary chromosome implies a imprecise covering ring star problem 3 subset of nodes of the given network. we assign the ith position of the chromosome as 1 if it is on the cycle and 0 otherwise. the mga consists of probabilistic selection (mukherjee et al., 2017), random crossover with proposed bi-part mating pool strategy, 4 types of mutations, i.e., exchange, inverse, reduction and augmentation mutations. the fitness function of each chromosome is the sum of the routing and assignment costs of the network. first, the mga is compared with some of the bestknown rsp results (calvete et al., 2013) to justify the performance of the algorithm. after comparing the rsp results with the most renowned for some instances, a numerical experiment is performed to illustrate our icrsp model. for the proposed icrsp model, results with different confidence levels of routing cost, assignment costs and covering distance are presented in this paper. as individual cases, some practical icrsps are presented numerically. 1.3. motivation in the case of disaster management, the routing costs from one node to another and assignment costs at nodes vary depending upon the type of the used conveyance, road condition, and climate condition at the time of journey, etc. similarly, in the case of telecommunication, also, these costs are not permanently fixed. instead, these costs may be vague in a practical sense. again, in the abovementioned practical problems, the covering distance of left out nodes from the corresponding connecting nodes is not fixed, like fixed ‘l’ distance units. instead, the covering distances are considered “near about ‘l’ distance units”. it can be little more than ‘l’ or slightly less than ‘l.’ the practical considerations motivated us to take up the present investigation. fig 1. the proposed icrsp 1.4. novelty thus, the new features/concepts on rsp introduced in the present paper are: • the impreciseness of routing and assignment costs are considered. • the covering distance at each nearest concentrator from the left-out nodes is imprecise. • the proposed fuzzy optimization problem is transformed into a deterministic one using possibility and necessity measures. • a modified genetic algorithm (mga) with a bi-part mating pool crossover strategy is proposed to solve np-hard problems like icrsp. mukherjee et al./decis. mak. appl. manag. eng. (2022) 4 the rest of the paper has been organized as follows. section 2 recalls some mathematical preliminaries. section 3 describes the mathematical formulation of the proposed problem. the solution procedure is presented in section 4. sections 5 and 6 include the numerical illustration and the discussion based on it, respectively. some particular icrsp is projected in section 7. lastly, section 8 concludes the paper. 2. mathematical preliminaries 2.1. fuzzy possibility and necessity approach let �̃� and �̃� be two fuzzy numbers 𝜇�̃�(𝑥) and 𝜇�̃�(𝑥) are their membership functions respectively. then according to zadeh (1994), pos(ã* b̃) = sup{min(μã(x),μb̃(y)),x,y ∈ r,x*y} (1) where pos stands for possibility, and 𝑛𝑒𝑠(𝑎∗ �̃�) = 1 − 𝑝𝑜𝑠(𝑎∗ �̃�̅̅̅̅̅̅̅) (2) where nes stands for necessity. if �̃�, �̃� ∈ ℜ and �̃� = 𝑓(�̃�, �̃�) where 𝑓:ℜ×ℜ → ℜ, then 𝜇𝑐̃ of c̃ is defined as for each 𝑧 ∈ ℜ, 𝜇𝑐̃(𝑧) = sup {min (𝜇�̃�(𝑥),𝜇�̃�(𝑦))} (3) 𝑥,𝑦 ∈ ℜ,𝑧 = 𝑓(𝑥,𝑦) the following lemmas can easily be derived (zadeh et al., 1994) from the above given definitions. lemma 1 if �̃� = (𝑎1,𝑎2,𝑎3) be a tfn with 0 < 𝑎1 and 𝑏 is a crisp number then 𝑃𝑜𝑠(�̃� < 𝑏) ≥ 𝛽 iff 𝑏−𝑎1 𝑎2−𝑎1 ≥ 𝛽 lemma 2 if �̃� = (𝑎1,𝑎2,𝑎3) be a tfn with 0 < 𝑎1 and 𝑏 is a crisp number then 𝑛𝑒𝑠(�̃� < 𝑏) ≥ 𝛽 iff 𝑎3−𝑏 𝑎3−𝑎2 ≤ 1− 𝛽 lemma 3 if �̃� be a tfn with 0 < 𝑎1 and 𝑏 be a crisp number then 𝑝𝑜𝑠(�̃� > 𝑏) ≥ 𝜂 iff 𝑎3−𝑏 𝑎3−𝑎2 ≥ 𝜂 lemma 4 if �̃� be a tfn with 0 < 𝑎1 and 𝑏 be a crisp number then 𝑛𝑒𝑠(�̃� > 𝑏) ≥ 𝜂 iff 𝑏−𝑎1 𝑎2−𝑎1 ≤ 1− 𝜂 where β and η are predefined confidence levels. 3. model formulation 3.1. ring star problem 𝑁 = {𝑥1,𝑥2,𝑥3,…,𝑥𝑛} being a set of nodes which defines the complete network, find a complete cycle (𝑥1,𝑥𝛼1,𝑥𝛼2,𝑥𝛼3,…,𝑥𝛼𝑚},𝑥1), where {𝑥1}⋃𝑥𝛼𝑖 𝑚 𝑖=1 = 𝑁′ ⊆ 𝑁,𝑚 ≤ 𝑛 −1 which minimizes the sum of the routing cost of the cycle through 𝑁′ and the imprecise covering ring star problem 5 assignment costs of the vertices out of the cycle to their nearest concentrators on the cycle, where the node 𝑥1 is considered as depot. hence the objective is to minimize 𝑐(𝑥1,𝑥𝛼1)+∑ 𝑐(𝑥𝛼𝑖,𝑥𝛼𝑖+1)+𝑐(𝑥𝛼𝑚,𝑥1) 𝑚−1 𝑖=1 +∑ 𝑑(𝑥𝑗,𝑦𝑘)𝑥𝑗∈ 𝑁′ 𝑦𝑘∈ 𝑁∖ 𝑁′ 𝑑𝑒𝑔(𝑦𝑘)=1 (4) where 𝑥1 ≠ 𝑥𝛼𝑖 and 𝑥𝛼𝑖 ≠ 𝑥𝛼𝑗 for 𝑖 ≠ 𝑗;∀𝑖, 𝑗 = 1,2,3,…,𝑚, 𝑐(𝑖, 𝑗) = routing cost from node 𝑖 to node 𝑗, 𝑑(𝑖,𝑗) = assignment cost of node 𝑗 to node 𝑖. 3.2. imprecise covering ring star problem in this case, the objective is to minimize �̃�(𝑥1,𝑥𝛼1)+∑ �̃�(𝑥𝛼𝑖,𝑥𝛼𝑖+1)+ �̃�(𝑥𝛼𝑚,𝑥1) 𝑚−1 𝑖=1 +∑ �̃�(𝑥𝑗,𝑦𝑘)𝑥𝑗∈ 𝑁′ 𝑦𝑘∈ 𝑁∖ 𝑁′ 𝑑𝑒𝑔(𝑦𝑘)=1 (5) subject to 𝑥𝑗 ∈ �̅� (𝑥1,δ1̃)∪�̅�(𝑥𝛼𝑖,δ𝛼𝑖 ̃) ∀ 𝑗 ∈ 𝑁 and for some 𝑖 (6) where 𝑥1 ≠ 𝑥𝛼𝑖 and 𝑥𝛼𝑖 ≠ 𝑥𝛼𝑗 for 𝑖 ≠ 𝑗;∀𝑖, 𝑗 = 1,2,3,…,𝑚, �̃�(𝑖, 𝑗) = fuzzy routing cost from node 𝑖 to node 𝑗, �̃�(𝑖, 𝑗) = fuzzy assignment cost of node 𝑗 to node 𝑖, �̅�(𝑎,𝑟) = closed disc with centre 𝑎 and radius 𝑟, δ�̃� = fuzzy covering distance at node 𝑖. considering the aforesaid costs and the covering distance as triangular fuzzy number (tfn), the problem can be rewritten as: 3.2.1. possibility approach (optimistic) determine a complete cycle (𝑥1,𝑥𝛼1,𝑥𝛼2,𝑥𝛼3,…,𝑥𝛼𝑚,𝑥1) to minimize 𝐹1 +𝐹2 (7) subject to 𝑃𝑜𝑠(�̃�(𝑥1,𝑥𝛼1)+∑ �̃�(𝑥𝛼𝑖,𝑥𝛼𝑖+1) 𝑚−1 𝑖=1 +�̃�(𝑥𝛼𝑚,𝑥1) < 𝐹1) ≥ 𝛽1 (8) 𝑃𝑜𝑠 ( ∑ �̃�(𝑥𝑗,𝑦𝑘)𝑥𝑗∈ 𝑁′ 𝑦𝑘∈ 𝑁∖ 𝑁′ 𝑑𝑒𝑔(𝑦𝑘)=1 < 𝐹2 ) ≥ 𝛾1 (9) 𝑃𝑜𝑠(𝛿(𝑥𝑗,𝑥1) ≤ δ1̃) ≥ 𝜂1 and 𝑃𝑜𝑠(𝛿(𝑥𝑗,𝑥𝛼𝑖) ≤ δ𝛼𝑖 ̃) ≥ 𝜂1 (10) ∀𝑗 ∈ 𝑁 and for some 𝛼𝑖, where 𝛽1,𝛾1 and 𝜂1 are confidence levels for routing cost, assignment costs and covering distance and 𝛿(𝑥𝑖,𝑥𝑗) is the shortest distance between 𝑥𝑖 and 𝑥𝑗. the above equations (7)-(10) can be rewritten as: determine a complete cycle (𝑥1,𝑥𝛼1,𝑥𝛼2,𝑥𝛼3,…,𝑥𝛼𝑚,𝑥1) to minimize 𝐹1 +𝐹2 (11) subject to 𝑃𝑜𝑠(𝐶 ̃ < 𝐹1) ≥ 𝛽1 (12) where, mukherjee et al./decis. mak. appl. manag. eng. (2022) 6 𝐶 ̃ = �̃�(𝑥1,𝑥𝛼1)+ ∑ �̃�(𝑥𝛼𝑖,𝑥𝛼𝑖+1) 𝑚−1 𝑖=1 +�̃�(𝑥𝛼𝑚,𝑥1), 𝑃𝑜𝑠(𝐷 ̃ < 𝐹2) ≥ 𝛾1 (13) and, �̃� = ∑ �̃�(𝑥𝑗,𝑦𝑘) 𝑥𝑗∈ 𝑁′ 𝑦𝑘∈ 𝑁∖ 𝑁′ 𝑑𝑒𝑔(𝑦𝑘)=1 𝑃𝑜𝑠(𝛿(𝑥𝑗,𝑥1) ≤ δ1̃) ≥ 𝜂1 and 𝑃𝑜𝑠(𝛿(𝑥𝑗,𝑥𝛼𝑖) ≤ δ𝛼𝑖 ̃) ≥ 𝜂1 (14) ∀𝑗 ∈ 𝑁 and for some 𝛼𝑖. considering all imprecise costs and covering distances as triangular fuzzy numbers, namely, 𝐶 ̃ = (𝐶1,𝐶2,𝐶3),�̃� = (𝐷1,𝐷2,𝐷3), δ̃𝑘 = ((δ𝛼𝑘)1 ,(δ𝛼𝑘)2 ,(δ𝛼𝑘)3 ) and following lemmas 1 and 3, the equations (11) to (14) can be transformed to: determine a complete cycle (𝑥1,𝑥𝛼1,𝑥𝛼2,𝑥𝛼3,𝑥𝛼𝑚,𝑥1) to minimize 𝐹1 +𝐹2 (15) subject to 𝐹1−𝐶1 𝐶2−𝐶1 ≥ 𝛽1 (16) 𝐹2−𝐷1 𝐷2−𝐷1 ≥ 𝛾1 (17) (δ𝛼𝑖 ) 3 −𝛿(𝑥𝑗,𝑥𝛼𝑖 ) (δ𝛼𝑖 ) 3 −(δ𝛼𝑖 ) 2 ≥ 𝜂1 (18) ∀𝑗 ∈ 𝑁 and for some 𝛼𝑖. the above equations (15)-(18) can be rewritten as: determine a complete cycle (𝑥1,𝑥𝛼1,𝑥𝛼2,𝑥𝛼3,𝑥𝛼𝑚,𝑥1) to minimize 𝐶1 +𝛽1(𝐶2 −𝐶1)+𝐷1 +𝛾1(𝐷2 −𝐷1) (19) subject to (δ𝛼𝑖)3 −𝜂1 {(δ𝛼𝑖)3 −(δ𝛼𝑖)2 } ≥ 𝛿(𝑥𝑗,𝑥𝛼𝑖) (20) ∀𝑗 ∈ 𝑁 and for some 𝛼𝑖. 3.2.2. necessity approach (pessimistic) for necessity approach, following the 2 and 4, the equations (5), (6) can be transformed to: determine a complete cycle (𝑥1,𝑥𝛼1,𝑥𝛼2,𝑥𝛼3,𝑥𝛼𝑚,𝑥1) to minimize 𝐹1 +𝐹2 (21) subject to imprecise covering ring star problem 7 𝐶3−𝐹1 𝐶3−𝐶2 < 1−𝛽1 (22) 𝐷3−𝐹2 𝐷3−𝐷2 < 1−𝛾1 (23) 𝛿(𝑥𝑗,𝑥𝛼𝑖 )−(δ𝛼𝑖 ) 1 (δ𝛼𝑖 ) 2 −(δ𝛼𝑖 ) 1 < 1 −𝜂1 (24) ∀𝑗 ∈ 𝑁 and for some 𝛼𝑖, where 𝛽2, 𝛾2 and 𝜂2 are confidence levels for cycle cost, assignment costs and covering distance. the above equations (21)-(24) can be rewritten as: determine a complete cycle (𝑥1,𝑥𝛼1,𝑥𝛼2,𝑥𝛼3,𝑥𝛼𝑚,𝑥1) to minimize 𝐶2 +𝛽2(𝐶3 −𝐶2)+𝐷2 +𝛾2(𝐷3 −𝐷2) (25) subject to (δ𝛼𝑖)2 −𝜂1 {(δ𝛼𝑖)2 −(δ𝛼𝑖)1 } ≥ 𝛿(𝑥𝑗,𝑥𝛼𝑖) (26) ∀𝑗 ∈ 𝑁 and for some 𝛼𝑖. 4. solution procedure using mga 4.1. description in the mga for icrsp, each chromosome is represented as a binary string (calvete et al., 2013) equal to the total number of nodes contained in the network. the 𝑖𝑡ℎ element of the array is 1 if the corresponding node is on the cycle and it is 0 otherwise. as we fix the first node considering as the depot, so the first element of the string is always 1. for the other strings, we randomly generate 0’s and 1’s and fill each of the chromosomes of the total populations. this process is called the initialization procedure. probabilistic selection (mukherjee et al., 2017) is an efficient selection procedure which selects the chromosomes in each generation on the basis of the fitness function fitness(chromosome) = tsp(chromosome) +assignment(chromosome). it is tested by mukherjee et al. (2017) that the probabilistic selection technique is more efficient than ordinary roulette-wheel selection in most cases. after performing the selection procedure, the new population is archived to a set s1 with the considered population size m. mukherjee et al./decis. mak. appl. manag. eng. (2022) 8 fig 2. representation of a chromosome, random crossover and inverse mutation we propose a bi-part strategic-based mating pool selection procedure to obtain offspring. at first, mating pools are selected based on the probability of crossover pc = 0.7 and sorted according to their fitness in descending order. let the selected number of parents be 2q. after sorting, the parents are divided into two groups, each of which contains q chromosomes. hence, the random crossover technique is performed between ith and (q +i)th parents, where 1 ≤ i ≤ q,, i.e., each corresponding parent between two groups. the proposed bi-part mating pool strategy is represented in figure 3. in the next step, four types of mutations have been used to maintain diversity and have a better chance of finding near optimality. these are inverse mutation (mukherjee et al., 2017) (with probability of mutation pm = 0.2) along with exchange mutation, reduction and augmentation local searches (calvete et al., 2013) (with pm = 0.5). a mutated chromosome can replace the corresponding previous one if it has a better fitness value. for tsp evaluation in the fitness function, we use the 2-opt procedure keeping in mind the total time-consuming factor due to the impreciseness of the model, and the cost is calculated depending on confidence level β. another cost, i.e., assignment cost in the fitness function, can easily be evaluated considering the corresponding confidence level γ of impreciseness. after crossover and mutations, the new population is archived in another set s2 with size m. among the total 2m chromosomes of s1∪ s2, the best m chromosomes are selected without repetition and sent to the next generation. each chromosome is considered feasible at each generation if it satisfies the covering distance constraint depending on the confidence level η of covering distance. the representation of a chromosome, random crossover, and inverse mutation are shown in figure 2. imprecise covering ring star problem 9 4.2. algorithm in details 4.2.1. initialization input: number of nodes n, population size (pop-size) output: a set of pop-size chromosomes each having n bits 1. 𝑓𝑜𝑟(𝑖 = 1 to 𝑝𝑜𝑝-𝑠𝑖𝑧𝑒){ 2. 𝑐ℎ𝑟𝑜𝑚𝑒[𝑖][1] = 1; 3. 𝑓𝑜𝑟(𝑗 = 2 𝑡𝑜 𝑛){ 𝑡 = 𝑟𝑎𝑛𝑑( )%2; 𝑖𝑓 (𝑡 = 1){ 𝑐ℎ𝑟𝑜𝑚𝑒[𝑖][𝑗] = 1; } 𝑒𝑙𝑠𝑒 𝑐ℎ𝑟𝑜𝑚𝑒[𝑖][𝑗] = 0; } } 4.2.2. probabilistic selection this is a more efficient selection procedure used previously by the authors mukherjee et al. (2017). this technique was first tested on some tsp benchmark problems and justified its better efficiency and hence used in mga. 4.2.3. random crossover input: total number of nodes 𝑛, 𝑝𝑎𝑟𝑒𝑛𝑡1, 𝑝𝑎𝑟𝑒𝑛𝑡2, fitness function 𝑓 output: 𝑐ℎ𝑖𝑙𝑑1, 𝑐ℎ𝑖𝑙𝑑2 1. 𝑐ℎ𝑖𝑙𝑑1[1] = 𝑐ℎ𝑖𝑙𝑑2[1] = 1; 2. 𝑓𝑜𝑟 (𝑖 = 2 𝑡𝑜 𝑛){ 𝑠 = 𝑟𝑎𝑛𝑑( )%2; 𝑖𝑓(𝑠 = 1){ 𝑐ℎ𝑖𝑙𝑑1[𝑖] = 𝑝𝑎𝑟𝑒𝑛𝑡1[𝑖]; } 𝑒𝑙𝑠𝑒{ 𝑐ℎ𝑖𝑙𝑑1[𝑖] = 𝑝𝑎𝑟𝑒𝑛𝑡2[𝑖]; } } 3. . 𝑓𝑜𝑟 (𝑖 = 2 𝑡𝑜 𝑛){ 𝑡 = 𝑟𝑎𝑛𝑑( )%2; 𝑖𝑓(𝑡 = 1){ 𝑐ℎ𝑖𝑙𝑑2[𝑖] = 𝑝𝑎𝑟𝑒𝑛𝑡2[𝑖]; } 𝑒𝑙𝑠𝑒{ 𝑐ℎ𝑖𝑙𝑑2[𝑖] = 𝑝𝑎𝑟𝑒𝑛𝑡1[𝑖]; } } mukherjee et al./decis. mak. appl. manag. eng. (2022) 10 fig 3. graphical representation of bi-part mating pool strategic crossover 4.2.4. inverse mutation inverse mutation (mukherjee et al. 2017) is a well-known mutation technique that works faster than conventional random mutation (mukherjee et al. 2017). exchange, reduction (vertex removal) and augmentation (vertex addition) mutations are three local search techniques that have been implemented inspired by the authors calvete et al. (2013) and salari et al. (2015). 4.2.5. procedure mga input: maximum number of generation (𝑚𝑎𝑥-𝑔𝑒𝑛), 𝑝𝑜𝑝-𝑠𝑖𝑧𝑒, number of nodes 𝑛, cost matrix, distance matrix crossover probability 𝑝𝑐, mutation probability 𝑝𝑚, probability of crossover 𝑝𝑠 output: minimum icrsp cost 1. procedure_initialization imprecise covering ring star problem 11 2. set gen← 1, 𝑔𝑙𝑜𝑏-𝑏𝑒𝑠𝑡 = 𝑙𝑜𝑐-𝑏𝑒𝑠𝑡 = 𝑀𝐴𝑋-𝐼𝑁𝑇 3. procedure_probabilistic selection based on 𝑝𝑠 4. archive the population to the set 𝑆1 (size 𝑚) 5. for(i=1 to 𝑝𝑜𝑝-𝑠𝑖𝑧𝑒){ if(rand[0,1] < pc) { 𝑖𝑡ℎ is selected for crossover } } 6. procedure_random crossover with bi-part strategy among the mating pools 7. 𝑓𝑜𝑟(𝑖 = 1 to 𝑝𝑜𝑝-𝑠𝑖𝑧𝑒){ 𝑖𝑓(𝑟𝑎𝑛𝑑[0,1] < 𝑝𝑚){ 𝑠𝑒𝑙𝑒𝑐𝑡 𝑖𝑡ℎ chromosome for mutation } } 8. procedure_inverse, exchange, augmentation and reduction mutations according to their 𝑝𝑚 9. archive the new population to another set 𝑆2 (𝑠𝑖𝑧𝑒 𝑚) 10. select best 𝑚 chromosomes from the set 𝑆1 ∪ 𝑆2 11.𝑓𝑜𝑟(𝑖 = 1 to 𝑝𝑜𝑝-𝑠𝑖𝑧𝑒){ 𝑖𝑓(𝑓[𝑖] < 𝑙𝑜𝑐-𝑏𝑒𝑠𝑡 and covering constraint satisfied){ 𝑙𝑜𝑐-𝑏𝑒𝑠𝑡 = 𝑐𝑜𝑠𝑡[𝑖] } } 12. 𝑔𝑒𝑛 ← 𝑔𝑒𝑛 +1 13. 𝑖𝑓(𝑙𝑜𝑐-𝑏𝑒𝑠𝑡 < 𝑔𝑙𝑜𝑏 −𝑏𝑒𝑠𝑡){ 𝑔𝑙𝑜𝑏-𝑏𝑒𝑠𝑡 ← 𝑙𝑜𝑐-𝑏𝑒𝑠𝑡 } 14. 𝑖𝑓(𝑔𝑒𝑛 < 𝑚𝑎𝑥𝑔𝑒𝑛){ 𝑔𝑜𝑡𝑜 𝑠𝑡𝑒𝑝 3 𝑒𝑙𝑠𝑒 goto step 15 } 15. end the mga program is operated until the termination condition is reached. we terminate the program if it yields the same solution in consecutive 50 generations. 5. numerical experiments 5.1. verification with earlier rsp results to justify the performance of the developed mga, we developed it in c programming language and tested it on a pc with intel core i3. we choose some tsp benchmark problems and compare them with the rsp results obtained by calvete et al.'s bbea (calvete et al., 2013) which has been the most efficient algorithm to outperform all other rsp algorithms, are given in table 1. the routing costs 𝑐𝑖𝑗 and the assignment cost 𝑑𝑖𝑗 in the objective function being as follows: 𝑐𝑖𝑗 = 𝛼 𝐼𝑖𝑗,𝑑𝑖𝑗 = (10−𝛼)𝐼𝑖𝑗 mukherjee et al./decis. mak. appl. manag. eng. (2022) 12 where, 𝐼𝑖𝑗 is the travelling costs from node 𝑖 to node 𝑗 in tsp instances and 𝛼 = 3,5,7,9. the different values of 𝛼 consider the different distribution of weights on rings and stars. more precisely, when 𝛼 = 3, the optimal solution should include all the nodes in the ring, as the star nodes assume (10 − 3) = 7 weightage. similarly, when 𝛼 = 9, most vertices should be out of the ring. here, for the tsp part in the fitness function, we use the same tsp solver algorithms as used by calvete et al. (2013) and salari et al. (2015) in their work on csp, i.e., 2-𝑜𝑝𝑡, 3-𝑜𝑝𝑡, and the lin-kernighan method (helsgaun, 2000) in the required cases. 5.2. experiment with proposed models to illustrate our proposed model, we consider 100 nodes and randomly generate a 100 ×100 fuzzy cost matrix whose elements are tfns in the form (𝑎1,𝑎2,𝑎3) with 30 ≤ 𝑎1 ≤ 120 and 𝑎2 = 𝑎1 +1 + 𝑟𝑎𝑛𝑑()%3, 𝑎3 = 𝑎2 + 1+ 𝑟𝑎𝑛𝑑()%3 and a 100 × 100 distance matrix with lower bound 22.5 and upper bound 90. for these fuzzy costs, the proposed imprecise model is reduced to deterministic ones (19,20) and (25,26) using the possibility and the necessity approaches respectively and solved by proposed mga. in table 2, some results for different parametric values of 𝛽 and 𝛾 along with no covering distance restrictions and with a crisp covering a distance of 70 distance units are presented using both the proposed mga and bbea (calvete et al., 2013). the 𝛽 and 𝛾 are the confidence levels of routing and related assignment costs, respectively. we have compared the results obtained by the proposed mga of our icrsp model only with bbea (calvete et al., 2013) since it is the best among all the existing rsp methods. to apply bbea in our icrsp model, we developed it into an equivalent imprecise form and then compared it with our proposed mga. in table 3, we present some results using mga for different values of 𝛽, 𝛾, and 𝜂 with a fuzzy covering distance (65,70,76), where 𝜂 is the confidence level of the covering distance. in the results with uncertain data, the fitness function is simply the sum of two costs-the routing and assignment costs. also, the 2-𝑜𝑝𝑡 procedure for solving tsps is used in the program, as the program gets more intricate than the crisp case. table 1 verification of rsp results: proposed mga and bbea (calvete et al. 2013) instance α best known cost using bbea best cost by mga among 25 different runs time (s) eil51 3 1278 1278 0.76 eil51 5 1995 1995 1.92 eil51 7 2113 2113 2.01 eil51 9 1244 1244 2.27 berlin52 3 22,626 22,626 0.68 berlin52 5 36,115 36,115 1.91 berlin52 7 37,376 37,376 2.04 berlin52 9 20,361 20,361 2.15 brazil58 3 76,185 76,185 1.94 brazil58 5 115,045 115,045 2.54 brazil58 7 126,807 126,807 1.86 brazil58 9 83,690 83,690 1.77 st70 3 2025 2025 1.22 st70 5 3110 3110 2.41 st70 7 3402 3402 2.64 st70 9 2610 2610 2.88 imprecise covering ring star problem 13 instance α best known cost using bbea best cost by mga among 25 different runs time (s) eil76 3 1614 1614 2.72 eil76 5 2460 2460 3.21 eil76 7 2504 2504 2.87 eil76 9 1710 1710 2.46 pr76 3 324,477 324,477 2.82 pr76 5 500,395 500,395 3.56 pr76 7 555,845 555,845 3.74 pr76 9 424,359 424,359 3.96 rat99 3 3633 3633 4.11 rat99 5 5885 5885 4.27 rat99 7 6436 6436 4.82 rat99 9 5150 5150 4.88 kroa100 3 63,846 63,846 4.74 kroa100 5 100,785 100,785 4.28 kroa100 7 115,388 115,388 5.07 kroa100 9 94,265 94,265 4.62 table 2 numerical results of proposed icrsp using proposed mga and bbea (calvete et al. 2013) with crisp covering distance appro ach β γ optimal cost without covering distance optimal cost with covering distance = 70 distance units proposed mga time (s) bbea time proposed mga time (s) bbea time (s) pos 0.6 0.6 3416.39 6.16 = 7.31 3427.1 7.07 = 7.24 0.7 3402.6 5.41 = 6.87 3432.1 6.18 = 7.47 0.8 3504.59 5.22 = 6.43 3509.8 6.94 = 7.85 0.9 3423.1 4.82 = 5.79 3491.29 6.15 = 6.27 0.95 3383.09 5.27 3381.6* 7.92 3512.04 5.08 = 6.11 0.7 0.6 3399 5.59 = 5.74 3484.40 5.87 = 5.44 0.7 3399.89 4.23 = 6.08 3563.5 7.31 = 8.18 0.8 3443 5.52 = 5.97 3477.20* 6.80 3486.29 7.12 0.9 3435.34 7.98 = 7.76 3486.19 7.82 = 8.32 0.95 3428.65 5.69 = 7.11 3471.09 7.54 = 7.92 0.8 0.6 3497.20 6.76 = 7.83 3672.8 6.94 = 8.11 0.7 3506* 5.09 3517.89 6.87 3597.5 7.19 = 6.93 0.8 3511.39* 6.82 3519.1 5.38 3622.39 5.88 = 6.23 0.9 3442 5.19 = 7.86 3489.19 8.14 = 7.91 0.95 3424.1* 6.61 3430.29 7.11 3481.84* 7.56 3493 8.14 0.9 0.6 3454.69 7.25 = 6.57 3608.40 8.39 = 7.83 0.7 3434.90* 6.67 3452.6 7.18 3605.19* 8.14 3611.5 7.66 0.8 3446.6 5.17 = 7.22 3606.3 7.85 = 8.11 0.9 3464.59* 7.08 3467.2 5.47 3633.9* 8.17 3637.29 8.42 0.95 3456.94* 4.29 3462.1 6.22 3632.54* 5.49 3643.5 7.55 0.95 0.6 3445.94 5.28 = 7.45 3606.35 7.74 3604.49* 9.22 0.7 3497.04 6.94 = 7.88 3707.5* 8.21 3711.1 6.16 mukherjee et al./decis. mak. appl. manag. eng. (2022) 14 appro ach β γ optimal cost without covering distance optimal cost with covering distance = 70 distance units 0.8 3463.8* 6.67 3472.59 8.04 3686 5.86 = 6.22 0.9 3482.04 7.11 = 7.74 3720.20 7.66 = 7.89 0.95 3457.44 5.82 = 6.25 3628.84* 5.84 3634.5 5.17 nec 0.6 0.6 3664.20 7.26 = 8.14 3693.20 7.84 = 9.24 0.7 3634.6 6.51 = 5.22 3762.40 7.08 = 8.28 0.8 3637.8 7.12 = 9.37 3730.20* 8.02 3736.10 8.10 0.9 3630.70* 4.22 3642.20 6.28 3690.20 7.53 = 9.14 0.95 3547 5.72 = 7.29 3793.35 8.96 = 10.22 0.7 0.6 3750.6 5.65 = 6.24 3845 7.37 = 9.72 0.7 3606.8 6.58 = 7.55 3798.39 6.49 = 7.62 0.8 3579.89* 7.11 3586.6 6.86 3729.39 8.84 = 11.46 0.9 3721.5 6.62 = 8.11 4056.5 7.48 = 8.44 0.95 3674.69* 11.41 3686.8 6.73 4047.14 8.82 = 9.22 0.8 0.6 3653 6.71 = 8.43 3753.39 7.66 = 8.92 0.7 3501.6 5.16 = 6.74 3664.70 8.28 3658.1* 9.24 0.8 3581.8 7.62 = 6.22 3692.39 6.65 = 8.36 0.9 3633.1 11.85 = 9.52 3788.70 10.24 = 13.45 0.95 3622.85* 6.33 3645.6 7.42 3771.8 8.21 = 8.48 0.9 0.6 3822.69 7.33 = 6.46 3878.59 7.96 = 9.24 0.7 3786.19 8.94 = 10.41 3877.19 9.85 = 11.21 0.8 3785.19 7.16 = 8.64 3880.19 8.34 = 9.72 0.9 3797.20* 6.82 3801.19 8.52 3879.59 7.36 3878.19* 10.08 0.95 3711.1 5.89 = 6.74 3793.44 8.58 = 9.55 0.95 0.6 3812.20 7.65 = 9.52 3853* 9.86 3862.59 11.73 0.7 3555.3 6.82 = 7.58 3842.79 8.65 = 7.52 0.8 3667.20* 8.57 3674.49 6.28 3882.39 9.11 = 11.80 0.9 3648.8 11.15 = 14.53 3843.6 9.64 = 12.54 0.95 3642.6 9.25 = 10.55 3719.54 13.96 = 12.74 table 3 numerical results of proposed icrsp using proposed mga and bbea (calvete et al. 2013) with fuzzy covering distance η β γ optimal cost with covering distance (65, 70, 76) possibility-approach time (s) optimal cost with covering distance (65, 70, 76) necessity-approach time (s) 0.7 0.8 0.8 3449.5998 7.61 4018.6 8.51 0.9 3471.6999 7.40 4133.3999 10.02 0.95 3468.75 6.45 4188.1 9.29 0.9 0.8 3559.6 8.27 4110.3999 11.68 0.9 3587.1 7.19 4114.3999 7.26 0.95 3556.6999 8.22 4139.2998 9.12 0.95 0.8 3642.2001 9.67 4137 10.34 0.9 3665.6501 6.28 4116.2 10.11 0.95 3572.6499 9.03 4003.0541 8.41 0.8 0.8 0.8 3562.6 7.59 4063 9.82 0.9 3548.6 8.95 4133.3999 8.76 0.95 3453.75 8.27 4188.1 8.14 0.9 0.8 3585.2001 9.17 4110.3999 10.27 0.9 3608 7.72 4114.3999 8.46 0.95 3537.5998 8.18 4139.2998 9.82 imprecise covering ring star problem 15 η β γ optimal cost with covering distance (65, 70, 76) possibility-approach time (s) optimal cost with covering distance (65, 70, 76) necessity-approach time (s) 0.95 0.8 3582.0998 7.08 4112.2001 6.94 0.9 3582.0998 9.58 4116.2001 7.49 0.95 3533.6 9.77 4141.1 7.73 0.9 0.8 0.8 3562.6 8.65 4070.3999 8.02 0.9 3655.1999 9.19 4101.1 9.87 0.95 3484.75 8.74 4188.1 7.15 0.9 0.8 3563.1 7.67 4032 7.42 0.9 3624.5 9.91 4137.2001 9.37 0.95 3533.5998 8.71 4137.2998 9.91 0.95 0.8 3601.1499 9.45 3712.3999 9.29 0.9 3728.3498 9.43 3842.9501 8.77 0.95 3693.3 11.52 4188.9501 12.82 0.95 0.8 0.8 3589.4501 8.38 3970.8 8.89 0.9 3691.1999 9.16 4137.2001 8.74 0.95 3453.75 9.78 4146.2998 9.26 0.9 0.8 3589 8.82 3974.7001 9.71 0.9 3565.1 10.22 3782.3999 11.33 0.95 3537.3999 9.11 4188.1 5.86 0.95 0.8 3582.0998 7.18 3834.35 7.54 0.9 3605.6 9.74 3714.3999 8.71 0.95 3613.25 9.52 4141.1 10.96 table 4 some numerical results of particular icrsp using mga where the nodes 4,21,44,67,82 and 91 are always present on the route η β γ optimal cost with covering distance (65, 70, 76) possibility-approach time (s) optimal cost with covering distance (65, 70, 76) necessity-approach time (s) 0.95 0.8 0.8 3704.6999 8.74 4029.3999 8.48 0.9 3780.1 10.85 4157.25 9.82 0.95 3577.6 9.11 4164.35 8.37 0.9 0.8 3704.1999 9.28 4187.2001 11.41 0.9 3691.25 9.71 3994.1 8.58 0.95 3676.2998 8.91 4266.25 9.92 0.95 0.8 3714.1999 7.55 3952.3999 11.87 0.9 3689.1 8.93 3811.1999 8.52 0.95 3722.3999 8.46 4173.1 9.89 6. discussion in table 1, the best result of the considered benchmark problems among 25 different runs of the proposed mga is the same as the results given by calvete et al. (2013), where the parameter α and the objective function are considered the same as mentioned by the above authors. table 2 presents near-optimal costs without and with a crisp covering distance separately, both in the possibility and the necessity approaches, using bbea and proposed mga. the route's cost with a covering distance is higher than that without the same restriction, which is as per our expectation. in most cases, the optimal solution is best at β=0.7 when γ is fixed. from the whole table, we notice that mga performs better than bbea in many results for different confidence levels, though sometimes it takes slightly more time than bbea to result in a better solution. from table 3, it is clear that the increase in the magnitude of η (confidence level of imprecise covering distance) increases the optimal cost for any fixed β and γ. in all cases, the possibility approach gives better optimal solutions than the necessity approach, which is as per expectations. mukherjee et al./decis. mak. appl. manag. eng. (2022) 16 7. conclusion an imprecise covering ring star problem (icrsp) is proposed and solved by a proposed modified genetic algorithm (mga). the routing costs and the assignment costs are considered imprecise values. also, an imprecise covering distance constraint is introduced, which is very significant in real-life cases and has not been discussed in the ring star problems by earlier researchers. each type of cost is considered a fuzzy variable, and the icrsp model is reduced to a deterministic form in both fuzzy possibility and necessity approaches. in our proposed mga, we use probabilistic selection, which selects the best chromosomes from children and offspring sets at each generation. bi-part mating pool strategy-based crossover keeps better diversity and good fitness values when making offsprings than the ordinary parent selection. at each iteration, the best population is collected from the union of populations before crossover and after mutation and sent to the next iteration, which speeds up the procedure. in most cases, our proposed mga algorithm takes lesser time than bbea to solve icrsps. also, in many cases, mga yields better results than bbea, though sometimes it may take slightly more runtime, resulting in a better solution. this problem can also be formulated in different imprecise environments such as rough, random, fuzzy-random, fuzzy-rough, etc. it also can be extended to a multiobjective icrsp by considering other objective functions, such as maximizing flow rate or minimizing the congestion rate in the case of telecommunication networks. moreover, the proposed mga can be used for optimization problems in other areas such as inventory control, transportation, supply chain, etc. the mga can also be extended to the multi-objective mga with better strategies. author contributions: conceptualization, a. mukherjee; methodology, a. mukherjee; validation, a. mukherjee; formal analysis, p. s. barma; investigation, s. das; resources, j. dutta; writing—original draft preparation, a. mukherjee; writing— review and editing, s. das; visualization, s. das; supervision, d. pamučar; the author has read and agreed to the published version of the manuscript. funding: this research received no external funding. conflicts of interest: the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. references baldacci. r., dell’amico. m. & gonzález. j.s., (2007) the capacitated m-ring star problem, operations research 55 (6) 1147-1162. baldacci. r. & dell’amico. 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(2017) multiple depot ring star problem: a polyhedral study and an exact algorithm, j glob optim 67: 527. https://doi.org/10.1007/s10898-0160431-7 tsplib : http://elib.zib.de/pub/mp-testdata/tsp/tsplib/tsp zadeh, l.a., (1994) fuzzy logic and soft computing: issues, contentions and perspectives, in proceedings of iizuka94 third international conference on fuzzy logic, neural nets and soft computing (vols. 1-2). iizuka, japan zhao, f., sun, j., li, s., & liu., w, (2009) a hybrid genetic algorithm for the traveling salesman problem with pickup and delivery, international journal of automation and computing, 06(1), 97-102. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1007/s10898-016-0431-7 https://doi.org/10.1007/s10898-016-0431-7 http://elib.zib.de/pub/mp-testdata/tsp/tsplib/tsp plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 5, issue 2, 2022, pp. 260-286. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0318062022p * corresponding author. e-mail addresses: osmanpala@kmu.edu.tr (o. pala). a mixed-integer linear programming model for aggregating multi–criteria decision making methods osman pala1* 1 department of econometrics, karamanoğlu mehmetbey university, turkey received: 2 february 2022; accepted: 18 june 2022; available online: 18 june 2022. original scientific paper abstract: selecting an mcdm method to use in any decision-making problem is always a difficult issue regarding that there is no agreement generally on which method is the most appropriate one. this paper addressed a proposal of a hybrid approach for this problem. under the assumption that there is no superiority among well-established and accepted mcdm methods, we defined a minimax strategy based on the fact that the highest total rank deviation between mcdms and the proposed hybrid approach in terms of alternative rankings should be as low as possible. even though mcdm methods often rank the alternatives differently, many methods perform similar ranking due to sharing alike mathematical operations. to avoid positive bias towards these methods in an integrated approach, we focused on a prioritizing scheme that supports differentiated rankings from others. this prioritizing scheme also contributed to hindering the problem of selecting mcdms with constraining the compound effect of similar rankings. we developed a hybrid decisionmaking model combining different mcdm methods with prioritizing them by using a mixed-integer linear programming model. we compared the proposed approach with some well-known prioritizing methods and the results revealed that the proposed approach produced better outcomes in obtaining the desired outputs. key words: multiple criteria analysis; aggregating mcdms; comparative analysis; minimax strategy. 1. introduction decision-making has always been an important part of everyone’s life. while some decisions need to be made daily, some others should be taken with long-term strategic considerations. if someone needs to address the problem through decision-making, then it is called a decision problem. these problems become multi-criteria decisiona mixed-integer linear programming model for aggregating multi–criteria decision making… 261 making (mcdm) problems when it is necessary to evaluate them according to more than one criterion. solving mcdm problems require a process that involves determining the most appropriate alternative among several options with considering the perspective of decision-makers and all criteria. to use in this process, the mcdm methods that have their mathematical basis were developed. some of the most common mcdm methods are weighted sum model (wsm), weighted product model (wpm), elimination et choice translating reality (electre) by roy (1968), decision-making trial and evaluation laboratory (dematel) by gabus & fontela (1972), analytic hierarchy process (ahp) by saaty (1980), technique for order of preference by similarity to ideal solution (topsis) by hwang & yoon (1981), preference ranking organization method for enrichment evaluation (promethee) by brans (1982), compromise ranking method (vikor) by opricovic (1998). numerous mcdm methods have been proposed to assist decision-makers in solving mcdm problems. however, as baydas et al. (2022) stated it is very complicated and difficult to choose an mcdm method for practitioners in their decision-making problems. these mcdm methods differ from each other in different aspects, such as using continuous or discrete data, qualitative or quantitative criteria, and in purpose like choosing, ranking, sorting the alternatives (zavadskas et al. 2014). moreover, mcdm methods employ different techniques to normalize decision matrix and use those outcomes to calculate utility scores of alternatives by varied mathematical operations, such as addition, multiplication, exponentiation, or logarithm. therefore, distinct mcdm methods often yield conflicting results. for example, pamucar et al. (2021) used six different types of mcdm methods in their study and found six different rankings. overcoming this problem is not an easy task because there is no theoretical superiority between any two mcdm methods. baydas & elma (2021) stated that the elements in every mcdm problem can vary and these changes affect the outcomes of an mcdm in different ways, and added that there can be no absolute superiority among the mcdm methods. kou et al. (2012) offered using various mcdm methods instead of one, to get more trustful results. if we agree with this opinion, then a new question arises. how can we aggregate a group of mcdm methods? this question has attracted quite the attention of researchers. to answer this question, some combining methods have been proposed in which each of them used dissimilar perspectives, such as weighting mcdm methods with spearman’s rank correlation coefficient introduced by kou et al. (2012) and peng (2015). using alternatives’ utility scores, which were obtained by different mcdm methods, as input of response surface methodology to produce final rankings introduced by wang et al. (2016). employing borda count method, in which rankings obtained by mcdms are summed, by barak & mokfi (2019). ranking alternatives with multimoora method that uses delphi method and dominance theory to reach an agreement on the final ranking introduced by brauers & zavadskas (2010). biswas (2020) used wsm to obtain a synthesized ranking of three equally weighted mcdm methods. mohammadi & rezaei (2020) used half-quadratic theory and consider an mcdm method that has a different ranking from the others as an exception and rated with a lower weight in their optimization model which evaluates the mcdm importance in the overall ranking. pramanik et al. (2021) employed wsm to aggregate five mcdm methods that have been assessed of identical importance in the overall ranking. although many approaches have been developed to address the combining mcdm methods, these techniques have some issues to discuss. first, using only ranking without utility scores like in borda count and dominance theory, the final rankings' precision and accuracy rates would be significantly decreased. second, it is obvious pala/decis. mak. appl. manag. eng. 5 (2) (2022) 260-286 262 that mcdm methods with similar algorithms tend to produce more alike rankings. with that in mind, even assigning equal weights to all methods may prioritize the correlated ones and underrate the uncorrelated ones while achieving the final aggregate ranking. under the assumption that there is no superiority among mcdm methods in terms of decision theory, we cannot let similar methods gain upper hand in the combined final rankings. therefore, it will be a challenging issue to the determination of mcdm methods that will be chosen to form the appropriate combination that avoids the dominance of similar methods. at this point, the aggregation problem can be seen as an mcdm problem in which the best mixing ratio of the methods is the decision-making problem, and also the methods are the criteria. in this case, it is necessary to answer the question of how should the mcdm methods, which are expressed as criteria in the aggregation problem, be prioritized. in mcdm theory, we can categorize the criteria weighting approaches into three groups: subjective, objective, or combination of both of them. subjective approaches such as point allocation by doyle et al. (1997), direct rating by bottomley & doyle (2001), swing by von winterfeldt & edwards (1986), a ranking method by kirkwood & corner (1993), pairwise comparison by saaty (1980) has been widely popular in the real-world applications. subjective approaches assume that the criteria weights can be predetermined by the decision-maker’s judgments which are based on their knowledge and expertise on the mcdm problem. however, if we cannot predetermine superiority among mcdm methods, then both the subjective approaches and the mixed ones that are all greatly based on predetermination and decision-makers preferences will have important shortcomings to use in the aggregation problem. as mentioned before, we defined the criteria of the aggregation problem as mcdm methods. so, the utility scores of alternatives obtained by each mcdm method in the actual problem can be put together into a whole to form the decision matrix of the aggregation problem. in this context, weighting mcdm methods according to utility scores would be specific to the dataset of the actual problem, and the general or individual judgments would not play a role in the process. hence, these subjective approaches are inappropriate to use, as they rely heavily upon the decision-maker’s judgments. meanwhile, there are significant numbers of objective methods which prioritize the criteria using only the decision matrix. objective methods, such as the entropy method by deng et al. (2000), the standard deviation (sd) and the criteria importance through intercriteria correlation (critic) methods by diakoulaki et al. (1995), the correlation coefficient and standard deviation (ccsd) by wang & luo (2010), the integrated determination of objective criteria weights (idocriw) by zavadskas & podvezko (2016) a modified entropy used by biswas et al. (2019), and the entropy and correlation coefficients (ewm-corr) by mukhametzyanov (2021) stand out as the most notable ones. although objective weighting methods are considered appropriate to prioritize mcdm methods, in this study, we proposed a new mixed-integer linear programming model that produces the importance level of each method for the aggregation problem in a better approach, intending to minimize the maximum total rank reversals from each ranking of mcdm methods in the final ranking. this would also maximize the lowest rank correlation between the final ranking and the rankings of the methods used in the final ranking. the main motivation of this study is the lack of an appropriate approach that eliminates the necessity of excessive pre-examination for choosing a group of mcdm methods that would be used to solve mcdm problems. so, the proposed direct prioritization scheme will consider the final ranking in a better way with behaving fairly and equally for each mcdm method whether they have similar properties or not. a mixed-integer linear programming model for aggregating multi–criteria decision making… 263 the main purpose of this study is to aggregate the scores of mcdm methods with a unique perspective aiming to minimize the maximum total rank deviations between any single mcdm ranking and the final aggregated ranking. so, the proposed approach which is called aggregation with minimax total rank deviation (amtrd) was compared with some well-known objective weighting methods to reveal its performance on attaining the goals in two different mcdm problems. we selected a group of distinguished mcdm methods to illustrate the performance of the amtrd approach in acquiring an aggregated ranking. there is no limitation to selecting another group of mcdm methods. therefore the practitioners can form different groups of mcdm with the amtrd approach. the remainder of this paper is organized as follows. in section 2 a brief explanation of the novel reconciling process of the amtrd and some mcdm methods that were used to illustrate the proposed approach. two different illustrative examples are presented in section 3, followed by the comparative analysis and further analysis in sections 4 and 5 and the conclusion is the final part of the paper. 2. methodology mcdm methods are the specifically produced tools aiming to assess mcdm problems. before using these methods, we should form a decision matrix. in this matrix, we generally present alternatives on rows and criteria on columns. first, all methods deal with the normalization of the decision matrix according to the type of criteria. each method has its normalization procedure for the decision matrix and also handles cost and benefit criteria differently. then, the methods assess the alternatives according to criteria. finally, we obtain ranking/utility values/scores of the alternatives. in this study, we emphasize the amtrd approach to aggregate a group of mcdm methods. the point to be underlined here is the proposed method focuses on the final scores of the mcdm methods and does not relies on specific mcdms. however, to elucidate the amtrd we used some well-known mcdm methods as an example. these methods and our proposed amtrd approach are mentioned as follows. 2.1. critic method in decision-making problems where there is more than one criterion, the importance of the criteria should be determined. a significant number of techniques have been developed to make this assessment, although most of them are subjective approaches. however, by using these subjective techniques, different criteria weights can be obtained from the same decision-maker. in addition, different decision-makers can make different evaluations with the same method (diakoulaki et al. 1995). addressing these shortcomings of subjective methods, diakoulaki et al. (1995) proposed an objective approach known as critic. this method depends on both correlation coefficients between criteria and standard deviations of criteria. critic can be applied with the following steps (jahan et al. 2012): first, a decision matrix ( ) ij nxm x x should be obtained for ( 1,..., ) i a i n alternatives and ( 1,..., ) j c j m criteria, and then we have to normalize criteria using eqs. (1) and (2) where min j x and max j x are minimum and maximum elements of criterion j . second, eqs. (3) and (4) provided correlation coefficients and standard deviations, pala/decis. mak. appl. manag. eng. 5 (2) (2022) 260-286 264 respectively. finally, with eqs. (5) and (6), the importance levels of criteria j w are obtained. min max min for benefit criteria ij j ij j j x x z x x    (1) max max min for cost criteria j ij ij j j x x z x x    (2) 1 2 2 1 1 ( )( ) , , 1,..., ( ) . ( ) n j kij ik i jk n n j kij ik i i z z z z j k m z z z z                  (3) 2 1 1 ( ) , 1,..., . n jj ij i z z j m n       (4) 1 (1 ), 1,..., . m j j jk k c j m      (5) 1 , 1,..., . j j m k k c w j m c     (6) 2.2. aras method aras (a new additive ratio assessment) depends on comparing values of alternatives to optimum values which are added by decision-makers. the mcdm approach of aras was proposed by zavadskas & turskis (2010). they first formed the decision matrix x in eq. (8) using eq. (7) as follows; max min benefit criteria and cost criteria odj j j x x if j x if j   (7) 1 2 11 12 1 21 22 2 1 2 ... ... ... . . 1,..., 0,..., . . . . ... od od odm m m n n nm x x x x x x x x x x i n j m x x x                             (8) afterward, they normalized the decision matrix using eqs. (9) and (10) for benefit and cost criteria, respectively. they obtained a criteria weighted matrix by using j w and eq. (11). lastly, the optimality function values and the utility degree of alternatives are calculated using eqs. (12) and (13). the higher i k scores represent more favorable results. a mixed-integer linear programming model for aggregating multi–criteria decision making… 265 0 for benefit criteria ij ij n ij i x r x    (9) * * * 0 1 ; for cost criteria ij ij ij n ij ij i x x r x x     (10) ij ij j v r w (11) 1 , 0,1,..., m i ij j s v i n    (12) 0 i i s k s  (13) 2.3. copras method the mcdm approach copras (complex proportional assessment method) was firstly proposed by zavadskas & kaklauskas (1996). later, the method was undergone some changes by podvezko (2011). the decision matrix ( ) ij nxm x x can be normalized and weighted simultaneously by eq. (14). the score values for the alternatives are obtained by using eqs. (15) or (16) depends on whether the criterion is benefit (j=1,..,k) or cost (j=g+1,..,m) type, respectively. the relative importance levels of alternatives can be calculated by eq. (17) and the performance index values of alternatives can be computed by eq. (18), as well. the alternative with the highest performance index is determined as the best alternative. 0 ij j ij n ij i x w d x    (14) 1 , 0,1,..., k i ij j s d i n     (15) 1 , 0,1,..., m i ij j k s d i n      (16) 1 1 1 n i i i i n i i i s q s s s           (17) .100% max{ } i i i q p q  (18) pala/decis. mak. appl. manag. eng. 5 (2) (2022) 260-286 266 2.4. edas method keshavarz ghorabaee et al. (2015) proposed the edas (evaluation based on distance from average solution), which depends on the average values of criteria. they first calculate the mean values for each criterion using the decision matrix ( ) ij nxm x x in eq. (19). then the positive and negative distance from average matrices pda and nda can be calculated for benefit criteria using eqs. (20) and (21), for cost criteria employing eqs. (22) and (23). 1 1,... n ij i j x av j m n     (19) max(0, ( )) ij j ij j x av pda av   (20) max(0, ( )) j ij ij j av x nda av   (21) max(0, ( )) j ij ij j av x pda av   (22) max(0, ( )) ij j ij j x av nda av   (23) positive sums and negative sums for the alternatives are obtained by using criteria weight j w in eqs. (24) and (25), respectively. 1 , 1,..., m i j ij j sp w pda i n    (24) 1 , 1,..., m i j ij j sn w nda i n    (25) thereafter, normalized sp and sn values can be acquired by utilizing eqs. (26) and (27), respectively. lastly, the assessment values are obtained by employing eq. (28) and edas ends with ranking alternatives in a decrescent manner. , 1,..., max( ) i i i sp nsp i n sp   (26) 1 , 1,..., max( ) i i i sn nsn i n sn    (27)   1 , 1,..., 2 i i i as nsp nsn i n   (28) 2.5. moosra method moosra (multiobjective optimization based on simple ratio analysis) was developed by das et al. (2012) to overcome the problems found in other mcdm techniques. das et al. (2015) used moosra for evaluating performance in the a mixed-integer linear programming model for aggregating multi–criteria decision making… 267 education system. they initially formed the ( ) ij nxm x x decision matrix and normalized it using eq. (29). subsequently, they weighted the normalized matrix with criteria weight j w and acquired the performance scores of alternatives in eq. (30) according to benefit (j=1,..,g) and cost (j=g+1,..,m) criteria. finally, the decision-making process ends with ranking alternatives in descending order. 2 1 ij ij n ij i x r x    (29) 1 1 g j ij j i m j ij j g w r ps w r       (30) 2.6. waspas method zavadskas et al. (2012) proposed waspas (the weighted aggregated sum product assessment) which is consists of both weighted sum and weighted product model. the decision matrix ( ) ij nxm x x can be normalized by employing eqs. (31) and (32). weighted sum model and weighted product model can be acquired by using criteria weight j w and calculating eqs. (33) and (34) as follows: max for benefit criteria ij ij j x z x  (31) min for cost criteria j ij ij x z x  (32) 1 , 1,..., m i j ij j wsm w z i n    (33)   1 1,..., j m w i ij j wpm z i n    (34) lastly, the alternatives are ranked in descending order according to performance scores acquired by eq. (35).   1 1,..., 2 i i i q wsm wpm i n   (35) 2.7. the proposed amtrd method suppose there is a decision matrix ( ) ij nxm x x that includes mcdm methods as criteria 1 ,..., m c c and also alternatives as 1 ,..., n a a where ij x denotes the scores of ia in terms of j c . the decision matrix that has only benefit criteria can be normalized by using eq. (1). afterward, the aggregated scores can be calculated from the normalized pala/decis. mak. appl. manag. eng. 5 (2) (2022) 260-286 268 score matrix ( ) ij nxm z z . however, the weights of methods also have to be defined before obtaining the final scores of alternatives. since the proposed amtrd approach aimed to minimize the maximum rank reversals from each ranking of mcdm methods, we suggested using the most common and basic mcdm technique as wsm to acquire the final ranking. the wsm obtain the i s , the overall scores of alternative i , as follows (fishburn, 1967): 1 , 1,..., m i j ij j s w z i n    (36) we can use a mixed-integer linear programming model to obtain criteria weights that would be used in eq. (36) to minimize the deviations of final ranking from the ranking of each method as follows; minimize dev ;subject to 1 1 ( , 1,..., . 1,..., . ) m m t t j ij j kj ikt j j w z w z hy n i k n t m for i k           (37) 1 1 ( , 1,..., . 1,..., . ) m m t t j ij j kj ikt j j w z w z hv n i k n t m for i k           (38) 1 1 ( 1,..., ) n n ikt ikt i k dev y v t m       (39) 1 1 m j j w   (40)  0,1 ( , 1,..., . 1,..., . )ikty i k n t m for i k     (41)  0,1 ( , 1,..., . 1,..., . )iktv i k n t m for i k     (42) 0 ( 1,..., ) j w j m  (43) the decision variable called the dev is used to ensure that the maximum sum of deviations between the ranking of any mcdm method and the aggregated amtrd ranking is minimized. suppose that t is also an mcdm method as j . we sorted rows of ( ) ij nxm z z according to scores of alternatives for each mcdm method ( 1,...,t m ) in a descended manner as 1 2 3 ... t t t t n score score score score    before the optimization process and obtained m different ij t z matrices. a mixed-integer linear programming model for aggregating multi–criteria decision making… 269 in eq. (37), where if the condition that 1 1 m m t t j ij j kj j j w z w z     is met, which refers to newly acquired scores are in order of amtrd amtrd i k score score as with the method t that has also k t t i score score . so there is no deviation for the rank of alternatives i and k between the mcdm method t and the amtrd approach. but if the condition is not met then the binary variable ikt y which refers to whether there is a downward rank deviation or not for alternatives i and k according to mcdm method t would be equal to 1. here j w are the mcdms weights that would be obtained in the optimization process. so, basically i amtrd score would be equal to 1 m t j ij j w z   according to wsm that we used to obtain aggregated rankings. in eq. (38), where if the condition that 1 1 m m t t j ij j kj j j w z w z     is met, which refers to newly acquired scores are in order of amtrd amtrd i k score score as with the method t that has also k t t i score score . so there is no deviation for the rank of alternatives i and k between the method t and the amtrd approach. but if the condition is not met then the binary variable ikt v which refers to whether there is an upward rank deviation or not for alternatives i and k according to method t would be equal to 1. in eq. (37) and eq. (38), we expected both 1 n ikt k y   and 1 n ikt k v   are equal to 0 when the amtrd approach and the method t rank the alternative i in the same rank. for example, if amtrd ranks alternative i 3 levels lower than the method t then 1 3 n ikt k y   and a downward deviation of 3 points occurs for only alternative i . conversely, if amtrd ranks alternative i 2 levels upper than the method t then 1 2 n ikt k v   and an upward deviation of 2 points occurs for only alternative i . in eq. (37) and eq. (38), using greater or equal and lesser or equal types of constraints can lead both ikt y and ikt v are equal to 0, when amtrd amtrd i k score score and t t i k score score . to overcome this issue we can use a very small but also meaningful constant number n with the value of 0.0001 and turn the greater or equal and lesser or equal types of constraints into greater and lesser types, respectively. here h , which is a sufficiently large number and used as 1000 to guarantee that the constraints hold. since in some cases, the value of 1 for ikt y or ikt v cannot be adequate to run the necessary restrictions correctly. eq. (39) would constraints that dev will be greater than the each number of rank deviations between methods and the amtrd approach. since, the sum of rank deviation between the method t and the amtrd approach is equal to 1 1 n n ikt ikt i k y v    . while eq. (40) limits the sum of j w equal to 1, eqs. (41), (42), and (43) are binary and nonnegative constraints of ikt y , ikt v , and j w , respectively. pala/decis. mak. appl. manag. eng. 5 (2) (2022) 260-286 270 let us consider an example of n=4 alternatives and m=3 mcdm methods. then we have three different mcdm results for the same four alternatives. assume that (4 3) ij x x x is given as follows; 12 0.9 100 10 0.5 80 5 1 45 9 0.7 15 x             we can obtain the normalized score matrix (4 3) ij x z z using eq. (1) as follows; 1.000 0.800 1.000 0.714 0.000 0.765 0.000 1.000 0.353 0.571 0.400 0.000 z             after that, we can sort rows of ( ) ij nxm z z according to scores of alternatives for each mcdm method ( 1, 2, 3t  ) in a descended manner and obtained 3 different ij t z matrices as follows, 1 1.000 0.800 1.000 0.714 0.000 0.765 0.571 0.400 0.000 0.000 1.000 0.353 ij z             2 0.000 1.000 0.353 1.000 0.800 1.000 0.571 0.400 0.000 0.714 0.000 0.765 ij z             3 1.000 0.800 1.000 0.714 0.000 0.765 0.000 1.000 0.353 0.571 0.400 0.000 ij z             so, the eq. (37) in the amtrd model can be defined with 18 constraints as follows; a mixed-integer linear programming model for aggregating multi–criteria decision making… 271 1 1 1 1 1 1 1 11 2 12 3 13 1 21 2 22 3 23 121 2 2 2 2 2 2 1 11 2 12 3 13 1 21 2 22 3 23 122 3 3 3 3 3 3 1 11 2 12 3 13 1 21 2 22 3 23 123 1 1 1 1 1 11 2 12 3 13 1 31 2 * * * * * * * * * * * * * * * * * * * * * * * w z w z w z w z w z w z hy n w z w z w z w z w z w z hy n w z w z w z w z w z w z hy n w z w z w z w z w                          1 1 32 3 33 131 2 2 2 2 2 2 1 11 2 12 3 13 1 31 2 32 3 33 132 3 3 3 3 3 3 1 11 2 12 3 13 1 31 2 32 3 33 133 1 1 1 1 1 1 1 11 2 12 3 13 1 41 2 42 3 43 141 2 2 1 11 2 12 * * * * * * * * * * * * * * * * * * * * * z w z hy n w z w z w z w z w z w z hy n w z w z w z w z w z w z hy n w z w z w z w z w z w z hy n w z w z                          2 2 2 2 3 13 1 41 2 42 3 43 142 3 3 3 3 3 3 1 11 2 12 3 13 1 41 2 42 3 43 143 1 1 1 1 1 1 1 21 2 22 3 23 1 31 2 32 3 33 231 2 2 2 2 2 2 1 21 2 22 3 23 1 31 2 32 3 33 * * * * * * * * * * * * * * * * * * * * * * w z w z w z w z hy n w z w z w z w z w z w z hy n w z w z w z w z w z w z hy n w z w z w z w z w z w z hy                           232 3 3 3 3 3 3 1 21 2 22 3 23 1 31 2 32 3 33 233 1 1 1 1 1 1 1 21 2 22 3 23 1 41 2 42 3 43 241 2 2 2 2 2 2 1 21 2 22 3 23 1 41 2 42 3 43 242 3 3 3 1 21 2 22 3 23 1 41 * * * * * * * * * * * * * * * * * * * * * * n w z w z w z w z w z w z hy n w z w z w z w z w z w z hy n w z w z w z w z w z w z hy n w z w z w z w z                          3 3 3 2 42 3 43 243 1 1 1 1 1 1 1 31 2 32 3 33 1 41 2 42 3 43 341 2 2 2 2 2 2 1 31 2 32 3 33 1 41 2 42 3 43 342 3 3 3 3 3 3 1 31 2 32 3 33 1 41 2 42 3 43 343 * * * * * * * * * * * * * * * * * * * * w z w z hy n w z w z w z w z w z w z hy n w z w z w z w z w z w z hy n w z w z w z w z w z w z hy n                          the eq. (38) would also be defined with 18 constraints as follows; pala/decis. mak. appl. manag. eng. 5 (2) (2022) 260-286 272 1 1 1 1 1 1 1 21 2 22 3 23 1 11 2 12 3 13 211 2 2 2 2 2 2 1 21 2 22 3 23 1 11 2 12 3 13 212 3 3 3 3 3 3 1 21 2 22 3 23 1 11 2 12 3 13 213 1 1 1 1 1 31 2 32 3 33 1 11 2 * * * * * * * * * * * * * * * * * * * * * * * w z w z w z w z w z w z hv n w z w z w z w z w z w z hv n w z w z w z w z w z w z hv n w z w z w z w z w                          1 1 12 3 13 311 2 2 2 2 2 2 1 31 2 32 3 33 1 11 2 12 3 13 312 3 3 3 3 3 3 1 31 2 32 3 33 1 11 2 12 3 13 313 1 1 1 1 1 1 1 31 2 32 3 33 1 21 2 22 3 23 321 2 2 1 31 2 32 * * * * * * * * * * * * * * * * * * * * * z w z hv n w z w z w z w z w z w z hv n w z w z w z w z w z w z hv n w z w z w z w z w z w z hv n w z w z                          2 2 2 2 3 33 1 21 2 22 3 23 322 3 3 3 3 3 3 1 31 2 32 3 33 1 21 2 22 3 23 323 1 1 1 1 1 1 1 41 2 42 3 43 1 11 2 12 3 13 411 2 2 2 2 2 2 1 41 2 42 3 43 1 11 2 12 3 13 * * * * * * * * * * * * * * * * * * * * * * w z w z w z w z hv n w z w z w z w z w z w z hv n w z w z w z w z w z w z hv n w z w z w z w z w z w z hv                           412 3 3 3 3 3 3 1 41 2 42 3 43 1 11 2 12 3 13 413 1 1 1 1 1 1 1 41 2 42 3 43 1 21 2 22 3 23 421 2 2 2 2 2 2 1 41 2 42 3 43 1 21 2 22 3 23 422 3 3 3 1 41 2 42 3 43 1 21 * * * * * * * * * * * * * * * * * * * * * * n w z w z w z w z w z w z hv n w z w z w z w z w z w z hv n w z w z w z w z w z w z hv n w z w z w z w z                          3 3 3 2 22 3 23 423 1 1 1 1 1 1 1 41 2 42 3 43 1 31 2 32 3 33 431 2 2 2 2 2 2 1 41 2 42 3 43 1 31 2 32 3 33 432 3 3 3 3 3 3 1 41 2 42 3 43 1 31 2 32 3 33 433 * * * * * * * * * * * * * * * * * * * * w z w z hv n w z w z w z w z w z w z hv n w z w z w z w z w z w z hv n w z w z w z w z w z w z hv n                          so, the eq. (39) would be defined with 3 constraints as follows; 121 131 141 231 241 341 211 311 411 321 421 431 122 132 142 232 242 342 212 312 412 322 422 432 123 133 143 233 243 343 213 313 413 323 423 433 dev y y y y y y v v v v v v dev y y y y y y v v v v v v dev y y y y y y v v v v v v                                     the mixed-integer linear programming model of amtrd is coded in the matlab2021 environment and is presented in appendix 1. after solving the optimization model and obtaining j w , the final ranking would be acquired by using eq. (36). the proposed amtrd method is neither dependent on any specific mcdm method nor relies on a certain number of methods. a decision-maker can consider and group different mcdm methods that are reliable on a specific problem in amtrd. the amtrd only needs scores of alternatives obtained by each mcdm method which they have conflicts about the alternative ranks between them, to aggregate the mcdms. considering and using the scores, the magnitude of the difference in scores in any mcdm method will also be included in the model. in this way, both ranking and ratings will collectively affect the final score and rankings. a mixed-integer linear programming model for aggregating multi–criteria decision making… 273 3. experimental results in this section, we used two different cases to analyze amtrd in different circumstances. in case 1, we consider five mcdm methods, and for each method, the criteria were weighted with the critic method. to demonstrate the proposed method can be effective in any circumstances we formed a group of seven mcdms for case 2 and the criteria were equally weighted for all the methods in case 2. 3.1 case 1 in case 1, better life index (bli), a social index to compare well-being across the countries and being carried out by organization for economic co-operation and development (oecd), was used to demonstrate the capabilities of the proposed hybrid approach with aggregating five mcdms namely aras, copras, edas, moosra, and waspas. in case 1, the same data, which were examined by depren & kalkan (2018), were used. bli data for the year 2017 with eleven main criteria and twenty-four sub-criteria were taken into consideration (oecd, 2017). information about these criteria and also sub-criteria weights which were obtained by critic were given in table 1. in this study, these sub-criteria weights were evaluated under the main criteria due to the independence of the main criteria from each other, and the critic method was used for obtaining these weights, while depren & kalkan (2018) considered all the sub-criteria at the same level and used entropy method for weighting them. with these two disparate perspectives and also using different weighting methods such as critic and entropy, the results of criteria weights differ. for example, according to depren & kalkan (2018), the “personal earnings” subcriterion was found as the most important factor on the “jobs” main criterion while this sub-criterion was followed by “employment rate”, “long-term unemployment rate” and “labour market insecurity”, respectively. in comparison with depren & kalkan (2018) and our results, while the most important factor remained the same for the “jobs” main criterion, the second most important factor was changed as “labour market insecurity”, in our study. table 1. sub criteria weights of bli with critic main criteria main criteria no sub criteria sub criteria weight unit criteria type housing c1 dwellings without basic facilities 0.25256 percentage min housing expenditure 0.42091 percentage min rooms per person 0.32652 ratio max income c2 household net adjusted disposable income 0.52068 us dollar max household net financial wealth 0.47932 us dollar max jobs c3 labour market insecurity 0.17577 percentage min employment rate 0.16890 percentage max long-term unemployment rate 0.22764 percentage min personal earnings 0.42770 us dollar max community c4 quality of support network 1.00000 percentage max education c5 educational attainment 0.36937 percentage max student skills 0.25738 average score max years in education 0.37325 years max environment c6 air pollution 0.49417 micrograms /cubic metre min water quality 0.50583 percentage max civic engagement c7 stakeholder engagement for developing regulations 0.48124 average score max voter turnout 0.51876 percentage max pala/decis. mak. appl. manag. eng. 5 (2) (2022) 260-286 274 main criteria main criteria no sub criteria sub criteria weight unit criteria type health c8 life expectancy 0.41054 years max self-reported health 0.58946 percentage max life satisfaction c9 life satisfaction 1.00000 average score max safety c10 feeling safe walking alone at night 0.56169 percentage max homicide rate 0.43831 ratio min work-life balance c11 employees working very long hours 0.53648 percentage min time devoted to leisure and personal care 0.46352 hours max after obtaining sub-criteria weights in case 1, we used these weights and the same data with depren & kalkan (2018) in five different mcdms to acquire scores of alternatives for all eleven main criteria. so, at this stage, we had a total of five different decision matrices, each with 11 criteria and 38 alternatives. these decision matrices are directly associated with the mcdm methods that were used to obtain them. after that, using these decision matrices, main criteria weights were evaluated by using critic for each related mcdm method. as expected the main criteria weights differ for each mcdm method since the evaluation process of mcdms and its outcomes as the importance values of alternatives for main criteria differs among the mcdm methods in decision matrices. the only alternative importance values for the c4 and c9 main criteria have not changed according to each mcdm method. due to having only one criterion as a sub-criterion, it was not necessary to make any calculations with mcdm methods and so oecd original survey results were used as alternative importance values for those criteria. contrastingly, the importance values of alternatives for the main criteria other than c4 and c9 vary according to the mcdm approaches used. next, alternative final scores were obtained by each mcdm method using five different decision matrices and different criteria weights. the final rankings of the alternatives differed according to the mcdm methods. according to the aras method, the countries in the top three were listed as the netherlands, russia, and sweden, respectively. in the copras rankings’, russia, netherlands, and sweden were the first three countries with the highest scores. according to edas rankings’, the top three were the united states, australia, and canada. iceland, norway, and united kingdom were listed as the top three countries while rankings’ were made with moosra. according to waspas rankings’, norway, netherlands, and sweden were the three leading countries, respectively. due to the differences in the results of alternative ranking obtained with the mcdm methods, there was not any common ranking or consensus among the methods. this situation, which poses a problem in decision-making, could be seen not only in the upper ranks but also in the lower ranks and overall mcdm rankings’. as it is known, although there is no superiority among mcdm methods, some methods may resemble each other more than others. accordingly, it will be possible to overcome the problem of compromising these methods properly only by reconciling these methods and taking the methods’ affinity issue into account. table 2 presents the normalized scores of alternatives for each mcdm method used in the optimization process of amtrd to obtain weights of five mcdms and the final scores and ranking with the proposed amtrd approach. the weights of five individual mcdm in amtrd were calculated according to their order in table 2 as 0.00000, 0.53755, 0.45271, 0.00000, and 0.00974. so, the three mcdm namely copras, edas, and waspas were adequate to cover all five mcdm and addressed the amtrd rank in this problem. a mixed-integer linear programming model for aggregating multi–criteria decision making… 275 depending on the scores of amtrd, the united states, canada, and russia are the top three countries, respectively, for the bli. while the united states was the best and canada was the third for edas, russia was the best for copras. it seems that copras and edas were the methods pulling the wire for amtrd rankings’, however, the amtrd also established links with the other remaining methods via copras and edas. table 2. ranking countries with hybrid amtrd for bli (case 1) aras copras edas moosra waspas amtrd australia 0.59734 0.52227 0.69634 0.76715 0.85289 0.60429(4) austria 0.46633 0.42622 0.05885 0.62707 0.66520 0.26223(21) belgium 0.51433 0.44537 0.29698 0.55582 0.72645 0.38093(14) canada 0.60660 0.54226 0.69187 0.75721 0.87328 0.61321(2) chile 0.14864 0.13189 0.00202 0.16510 0.23048 0.07405(37) czech rep. 0.32313 0.31931 0.12586 0.40897 0.50713 0.23356(23) denmark 0.59260 0.54494 0.28247 0.76389 0.85193 0.42911(10) estonia 0.31503 0.28732 0.21735 0.35984 0.44045 0.25714(22) finland 0.48906 0.44424 0.29299 0.60795 0.72868 0.37854(15) france 0.42203 0.38498 0.02009 0.49676 0.63121 0.22219(26) germany 0.54477 0.48736 0.15773 0.71426 0.79012 0.34108(17) greece 0.16962 0.18855 0.02762 0.20638 0.28895 0.11667(33) hungary 0.16738 0.18090 0.03669 0.20707 0.25660 0.11636(34) iceland 0.84244 0.58351 0.23304 1.00000 0.97091 0.42862(11) ireland 0.46890 0.40209 0.28109 0.56431 0.66030 0.34982(16) israel 0.30185 0.29721 0.22060 0.40888 0.44792 0.26400(20) italy 0.29474 0.29176 0.03698 0.33026 0.44098 0.17787(30) japan 0.43250 0.38012 0.26562 0.62355 0.51778 0.32962(18) korea 0.58025 0.30875 0.09216 0.46512 0.49997 0.21256(28) latvia 0.21258 0.20980 0.09867 0.22301 0.29009 0.16027(31) luxembourg 0.73999 0.49146 0.27524 0.67705 0.88233 0.39738(13) mexico 0.24120 0.17245 0.27349 0.21144 0.21911 0.21865(27) netherlands 1.00000 0.77352 0.27740 0.92919 0.99724 0.55110(5) new zealand 0.50919 0.45504 0.41408 0.65853 0.74625 0.43933(9) norway 0.82264 0.57891 0.29301 0.93433 1.00000 0.45358(8) poland 0.24120 0.24634 0.11963 0.30360 0.38357 0.19030129) portugal 0.18319 0.19539 0.02954 0.25636 0.28247 0.12116(32) slovak rep. 0.25922 0.26098 0.19052 0.30120 0.40333 0.23047(24) slovenia 0.37037 0.35216 0.20659 0.46306 0.55809 0.28827(19) spain 0.37551 0.34023 0.07677 0.44454 0.56330 0.22313(25) sweden 0.86184 0.61733 0.45088 0.78653 0.99458 0.54565(7) switzerland 0.83133 0.55118 0.54173 0.78636 0.97193 0.55100(6) turkey 0.12706 0.12936 0.02241 0.15579 0.20709 0.08170(36) united king. 0.63882 0.56885 0.23512 0.93236 0.82157 0.42023(12) united states 0.68253 0.59228 1.00000 0.80599 0.89962 0.77985(1) brazil 0.14587 0.15131 0.00871 0.18589 0.20164 0.08724(35) russia 0.89040 1.00000 0.16019 0.90347 0.31252 0.61311(3) south africa 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000(38) http://stats.oecd.org/oecdstat_metadata/showmetadata.ashx?dataset=bli2017&coords=%5blocation%5d.%5bisr%5d&showonweb=true&lang=en pala/decis. mak. appl. manag. eng. 5 (2) (2022) 260-286 276 3.2. case 2 in this study, we borrowed a material selection case from shanian and savadogo (2006). the decision matrix of the problem with eight alternatives and twelve criteria in which the first, fourth, fifth, and twelfth are the cost and the rest are beneficial criteria are presented in table 3. table 3. decision matrix of case 2 c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12 a1 8.25 560 940 0.78 15183 2916 380 560 138 465 105 18.64 a2 8.65 460 600 0.71 12472 2395 220 460 125 465 205 13.99 a3 8.94 50 210 0.08 1355 260 45 50 122 460 398 3 a4 8.95 340 380 0.48 9218 177 115 340 135 460 390 3.46 a5 2.67 190 295 0.25 20317 1966 87 191 73.59 741 152 2.81 a6 8.06 690 1030 1.55 5909 2174 350 800 190 189 17 5.99 a7 8.63 95 270 0.17 2711 520 63 100 116 174 185 3.32 a8 7.08 267 355 0.48 1957 720 110 265 205 329 50 1.04 source: shanian and savadogo (2006). as mentioned before, all criteria weights in case 2 considered in each mcdm as equal with the value of 0.08333. first, we obtained alternatives’ ranking for seven mcdms, namely aras, copras, edas, moosra, waspas, wsm, and wpm. the eqs. (33) and (34) present the calculations of the wsm and wpm which are basic ranking methods, respectively. table 4 represents the scores and ranks of alternatives for each mcdm and also amtrd. the weights of seven individual mcdm in amtrd were obtained according to their order in table 4 as 0.24290, 0.06798, 0, 0.37440, 0, 0, and 0.31473. so, the three mcdm namely aras, copras, moosra, and wpm were adequate to cover all seven mcdm and addressed the amtrd rank in this problem. however, using amtrd we found out the best alternative is a8 and second best is a6 while also it was considered as the top according to aras, copras, edas, and wsm which two of them were not included in the final phase of amtrd. table 4. ranking alternatives with mcdms and hybrid amtrd for case 2 aras copras edas moosra waspas wsm wpm amtrd scores of alternatives a1 0.50271 97.03645 0.46362 1.80552 0.46416 0.54666 0.38165 0.59815 a2 0.42847 84.69563 0.42089 1.74793 0.41581 0.46510 0.36653 0.41990 a3 0.47157 84.63413 0.31088 2.67523 0.37096 0.45064 0.29127 0.59805 a4 0.41977 86.87727 0.57101 2.45927 0.41881 0.44734 0.39028 0.73403 a5 0.41959 72.21532 0.29427 1.99619 0.39069 0.43345 0.34793 0.41980 a6 0.51769 100.00000 0.60762 2.28269 0.45815 0.55642 0.35988 0.73414 a7 0.31515 69.23272 0.17624 1.80032 0.29286 0.31062 0.27509 0.01877 a8 0.44897 84.07422 0.39773 2.79296 0.41257 0.45042 0.37472 0.83989 rank of alternatives a1 2 2 3 6 1 2 2 4 a2 5 4 4 8 4 3 4 6 a3 3 5 6 2 7 4 7 5 a4 6 3 2 3 3 6 1 3 a5 7 7 7 5 6 7 6 7 a6 1 1 1 4 2 1 5 2 a7 8 8 8 7 8 8 8 8 a8 4 6 5 1 5 5 3 1 a mixed-integer linear programming model for aggregating multi–criteria decision making… 277 since the results of weighting with the proposed approach reveal that some methods can represent others with greater importance in the proposed amtrd approach, it can be expected that the goal of the amtrd which is the minimization of the maximum rank deviation from the mcdm methods achieved. with this regard, in order to demonstrate the effectiveness of the proposed method, the correlation between the methods in terms of ranking results was analyzed and also compared with different well-known methods and finally, the proposed approach was analyzed in terms of its validity. 4. comparative analysis to analyze and compare the effectiveness of the amtrd regarding to the minimax total rank deviation strategy, the critic method which favors the uncorrelated criteria in an mcdm problem was chosen. additionally, with extracting the standard deviations from eq. (5) the greatly enhanced version of critic for focusing on correlation coefficients (cc) also compared with amtrd. the results of the equal mean (em) approach, which is the basic method in criteria weighting also represented to draw a wider perspective. all these models were formed using final scores which were obtained by five different mcdm methods as a decision matrix for case 1. appendix 2 presents the results of each weighting method and the rankings for the bli problem with them obtained by using wsm to make fair comparisons. according to the results, both cc and critic weigh the edas method most. however, amtrd differs from them with prioritizing copras as the highest of all mcdm methods. apart from this; it is observed that the obtained results with cc were much closer to critic than amtrd's. so, amtrd generally parted from them in the weighting scheme. while the united states was the leading alternative for the bli in each approach including em, the runner-up alternatives for bli using amtrd, cc, and critic were canada, switzerland, and the netherlands, respectively. em approach and critic ranked the same alternatives at the top four with a different order which can be seen as a sign of closer ranking results for them. to compare the aggregating methods for case 2, we used scores of single mcdms in table 4, we obtained weights of the seven method with the order in table 4 and for cc as; 0.11148, 0.10853, 0.11246, 0.31352, 0.09357, 0.10049, 0.15995 and for critic as; 0.09760, 0.10560, 0.10627, 0.35267, 0.08368, 0.08744, 0.16673. while cc and critic have significantly close weight values they also ranked the alternatives identically with the amtrd, the only exception is the ranking between a6 and a8 were reversed. amtrd which differs a lot from both cc and critic in terms of weightings revealed similar rankings with them. the em presents the rank of alternatives as 2-56-3-7-1-8-4 which differs from all of them. to identify the total rank deviation among both rankings with weighting methods and single mcdm rankings more approximately, the total rank deviation (trd) was defined as follows; suppose that, there is n number of alternatives that have to be ranked. while also there are two different rankings available as ( )i n r r and ( )i n t t then trd between them would be; 1 n rt i i i trd r t    (44) pala/decis. mak. appl. manag. eng. 5 (2) (2022) 260-286 278 using eq. (43) in tables 5 and 6, the trd values between individual mcdms and each aggregated rankings are presented for case 1 and 2, respectively. table 5. trd values between individual mcdms and aggregated rankings’ for case 1 cc critic em amtrd aras 82 72 58 122 copras 84 74 52 110 edas 170 178 192 132 moosra 90 86 70 124 waspas 102 88 76 146 max trd 170 178 192 146 table 6. trd values between individual mcdms and aggregated rankings’ for case 2 cc critic em amtrd aras 10 10 6 12 copras 8 8 4 10 edas 8 8 4 10 moosra 14 14 20 12 waspas 12 12 6 12 wsm 12 12 8 14 wpm 14 14 10 14 max trd 14 14 20 14 according to max trd values in table 5 and 6, for case 1 the amtrd has the minimum max trd with 146, while cc, critic, and em follow it with 170, 178, and 192, respectively. in case 2 the three approaches amtrd, cc, and critic outperform em in terms of max trd. the results indicated that the amtrd is the better option for obtaining minimum max trd which is the main purpose of the amtrd. 5. further analysis to analyze further effectiveness of amtrd and also to demonstrate that it is not dependent on specific mcdm methods, case 1 and 2 were investigated using different groups of mcdms. in table 7 the max trd between individual mcdms and aggregated rankings’ were presented for case 1. we sequentially discarded an mcdm from the original model for case 1. so, this approach provides five different models with four mcdms in each for case 1. using all these models with different weighting schemes we acquired final rankings via wsm, and then we calculated the trd and max trd for each model to reveal which weighting schemes provide better results in line with the minimax strategy. while amtrd outperformed the other weighting methods in all models, the runner-up was cc for four models and critic only surpassed cc only in one model. em could not compete with any of them concerning minimax strategy. a mixed-integer linear programming model for aggregating multi–criteria decision making… 279 table 7. max trd values between group of mcdms and aggregated rankings’ for case 1 the groups of mcdms in the model cc critic em amtrd copras, edas, moosra, waspas 148 154 170 124 aras, edas, moosra, waspas 154 174 186 136 aras, copras, moosra, waspas 70 62 78 58 aras, copras, edas, waspas 168 170 180 130 aras, copras, edas, moosra 148 150 194 128 considering different models for case 2 we focused on different combinations of the mcdms and obtained seven models and in table 8 the max trd values between individual mcdms and aggregated rankings were listed for case 2. in terms of minimax strategy, amtrd and em outperformed the other two models, while the proposed method slightly has a better mean of max trd than em. but more importantly, amtrd had the minimum max trd values in all models. additionally, an important issue to be considered was the em had better results than both cc and critic in line with the minimax strategy for the first time. this consequence falsified our foresight that cc is the superior version of critic for minimax strategy. since cc even failed to dominate over the em, it cannot be a valid option anymore. similar to all other models, amtrd has maintained its best performance in terms of max trd values in these models as well. table 8. max trd values between group of mcdms and aggregated rankings’ for case 2 the groups of mcdms in the model cc critic em amtrd aras, wsm, wpm 12 12 12 12 copras, wsm, wpm 12 12 10 10 edas, wsm, wpm 8 8 8 8 moosra, wsm, wpm 18 18 14 14 waspas, wsm, wpm 12 12 10 10 aras, waspas, wsm, wpm 12 12 12 10 aras, copras, waspas, wsm, wpm 12 12 10 10 considering the inferences that can be made from all the results, with the amtrd method, the best result was always obtained according to max trd. while the efficiency of amtrd is more evident when the number of alternatives in the problem is higher, the approach also maintains its performance in ranking fewer alternatives. meanwhile, we cannot have a conclusion on which the other compared methods have an edge over others. as mentioned earlier in this study, the amtrd approach should consider the individual mcdm rankings’ that constitute it without any difference between them and the similarity rate in the rankings should be as high as possible in all of them. in this context, it is desirable to minimize the max trd between individual mcdms and this would be a minimax strategy to perform. so, amtrd outperformed cc, critic, and em in the context of minimax strategy in all cases and models. so, these results indicate that removing any mcdm did not crucially affect the performance of the amtrd and also reveal the robustness of the proposed method. pala/decis. mak. appl. manag. eng. 5 (2) (2022) 260-286 280 5. results and discussions this paper aimed to develop an aggregating method that has its original point of view and appropriately aggregates the mcdms in line with its assumptions. some aggregation methods have been proposed in the past. however, they mostly relied on the thought that mcdms which have similar rankings should be more important than the others in the aggregated rankings. contrary to this idea, we proposed a new approach that does not underestimate any mcdm method, assuming that there is no superiority between mcdm methods, which is one of the important assumptions in decision theory, in the evaluation process. to prevent positive bias towards the methods with similar rankings, the basis of the proposed amtrd method is structured to minimize the max trd of final rankings from any rankings of mcdm methods. owing to this approach, overall, the importance of the methods remains relatively the same. because the methods with similar rankings have interaction and so their compound significance highly affects the final rankings. however, the methods with different rankings offset this high effect with their higher individual weights that were acquired by amtrd. in the amtrd approach, the decision-maker can select a group of mcdm methods that he/she considers reliable and uses their scores to obtain aggregated rankings in line with the minimax strategy. the two cases were used in the study to demonstrate the effectiveness of the proposed approach in both mcdm problems with a big and small number of alternatives. different models were handled with amtrd and as indicated by the overall results the amtrd aggregated groups of mcdm methods reliably in line with the minimax strategy, whether the number of methods in the group is small or big, or which methods compromise the group. while the proposed approach had significantly better results from any weighting scheme in case 1 with 38 alternatives, the amtrd maintained its assignments even the number of alternatives decreased to 8 as in case 2. furthermore, even when the number of mcdms used in amtrd was reduced from five to four in case 1, it can produce similar ranking results with the model that has five mcdm ratings and these outcomes display the robustness of the amtrd. in addition, the amtrd outperformed the other well-known approaches in minimizing the max trd of aggregated rankings between any individual mcdm methods. besides, a balance, rather than a net correlation, was established between the correlation of amtrd and mcdm ranking results and the weight of mcdms in the amtrd. so there is no such guarantee that if the weight of the mcdm is high or low then its rank correlation with amtrd also would be high or low. these overall results display that the desired reconciliation with amtrd is achieved in the aggregation of mcdm methods. in future studies, there are sufficient opportunities for reconciling different mcdms with different weighting approaches in the decision-making process. author contributions: research problem, o.p.; methodology, o.p.; formal analysis, o.p.; resources, o.p.; writing – original draft preparation, o.p.; writing – review & editing, o.p. funding: this research received no external funding. conflicts of 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(2018). comprehensive performance evaluation of electricity grid corporations employing a novel mcdm model. sustainability, 10(7), 2130. appendix 1. matlab script of amtrd optimization model. clc; clear; %%%% amtrd method %%% %r decision matrix(nxm) (n=number of alternatives, m=number of mcdms) %example case 2 with seven mcdms r=[0.50271 97.03645 0.46362 1.80552 0.46416 0.54666 0.38165 0.42847 84.69563 0.42089 1.74793 0.41581 0.46510 0.36653 0.47157 84.63413 0.31088 2.67523 0.37096 0.45064 0.29127 0.41977 86.87727 0.57101 2.45927 0.41881 0.44734 0.39028 0.41959 72.21532 0.29427 1.99619 0.39069 0.43345 0.34793 0.51769 100.00000 0.60762 2.28269 0.45815 0.55642 0.35988 0.31515 69.23272 0.17624 1.80032 0.29286 0.31062 0.27509 0.44897 84.07422 0.39773 2.79296 0.41257 0.45042 0.37472]; %r(:,[3,4])=[]; you can extract any mcdms. %but also do not forget make necessary changes in constraint functions n=size(r,1); %number of alternatives m=size(r,2); %number of criteria % min-max normalization [0,1] for i=1:n for j=1:m r1(i,j)= (r(i,j)-min(r(:,j)))/(max(r(:,j))-min(r(:,j))); end end rn=r1; % standard deviation pala/decis. mak. appl. manag. eng. 5 (2) (2022) 260-286 284 for j =1:m b=0; for i=1:n b=b+(rn(i,j)-mean(rn(:,j)))^2; end sigma(j,1)=sqrt((1/(n))*b); end % correlation coefficients for j=1:m for k =1:m aw=0; bw=0; cw=0; for i=1:n aw=aw+(rn(i,j)-mean(rn(:,j)))*(rn(i,k)-mean(rn(:,k))); bw=bw+(rn(i,j)-mean(rn(:,j)))^2; cw=cw+(rn(i,k)-mean(rn(:,k)))^2; end rnc(j,k) = aw/(sqrt(bw*cw)); end eksirnc=(1-rnc).^(1/1); cj(j)=sum(eksirnc(j,:)); crj(j)=sigma(j)*sum(eksirnc(j,:)); end for j=1:m cc(j)=cj(j)/sum(cj); %cc cr(j)=crj(j)/sum(crj); %critic end %sorting rows for each mcdms scores rn1=sortrows(rn,1,'descend'); rn2=sortrows(rn,2,'descend'); rn3=sortrows(rn,3,'descend'); rn4=sortrows(rn,4,'descend'); rn5=sortrows(rn,5,'descend'); rn6=sortrows(rn,6,'descend'); rn7=sortrows(rn,7,'descend'); h=1000; % a sufficient large number n=0.0001; % a sufficient small number % variable definitions wprob = optimproblem; % weights wler = optimvar('wler',m,'lowerbound',0,'upperbound',1); % variables y y1 = optimvar('y1',(n-1)*n/2,'type','integer','lowerbound',0,'upperbound',1); y2 = optimvar('y2',(n-1)*n/2,'type','integer','lowerbound',0,'upperbound',1); y3 = optimvar('y3',(n-1)*n/2,'type','integer','lowerbound',0,'upperbound',1); y4 = optimvar('y4',(n-1)*n/2,'type','integer','lowerbound',0,'upperbound',1); y5 = optimvar('y5',(n-1)*n/2,'type','integer','lowerbound',0,'upperbound',1); y6 = optimvar('y6',(n-1)*n/2,'type','integer','lowerbound',0,'upperbound',1); y7 = optimvar('y7',(n-1)*n/2,'type','integer','lowerbound',0,'upperbound',1); % variables yy yy1 = optimvar('yy1',(n-1)*n/2,'type','integer','lowerbound',0,'upperbound',1); yy2 = optimvar('yy2',(n-1)*n/2,'type','integer','lowerbound',0,'upperbound',1); yy3 = optimvar('yy3',(n-1)*n/2,'type','integer','lowerbound',0,'upperbound',1); yy4 = optimvar('yy4',(n-1)*n/2,'type','integer','lowerbound',0,'upperbound',1); yy5 = optimvar('yy5',(n-1)*n/2,'type','integer','lowerbound',0,'upperbound',1); yy6 = optimvar('yy6',(n-1)*n/2,'type','integer','lowerbound',0,'upperbound',1); yy7 = optimvar('yy7',(n-1)*n/2,'type','integer','lowerbound',0,'upperbound',1); % objective variable zler= optimvar('zler','lowerbound',0); % constraint predefinitions a1 = optimconstr((n-1)*n/2); a2 = optimconstr((n-1)*n/2); a3 = optimconstr((n-1)*n/2); a4 = optimconstr((n-1)*n/2); a5 = optimconstr((n-1)*n/2); a6 = optimconstr((n-1)*n/2); a7 = optimconstr((n-1)*n/2); c1 = optimconstr((n-1)*n/2); c2 = optimconstr((n-1)*n/2); c3 = optimconstr((n-1)*n/2); c4 = optimconstr((n-1)*n/2); c5 = optimconstr((n-1)*n/2); a mixed-integer linear programming model for aggregating multi–criteria decision making… 285 c6 = optimconstr((n-1)*n/2); c7 = optimconstr((n-1)*n/2); b1 = optimconstr(1); b2 = optimconstr(1); b3 = optimconstr(1); b4 = optimconstr(1); b5 = optimconstr(1); b6 = optimconstr(1); b7 = optimconstr(1); a11 = optimconstr(1); % creating constraint functions iter=0; itera=0; for i=1:n for j=1:n if i<j iter=iter+1; a1(iter)=(rn1(i,:)*wler)>=(rn1(j,:)*wler)-h*y1(iter)+n; a2(iter)=(rn2(i,:)*wler)>=(rn2(j,:)*wler)-h*y2(iter)+n; a3(iter)=(rn3(i,:)*wler)>=(rn3(j,:)*wler)-h*y3(iter)+n; a4(iter)=(rn4(i,:)*wler)>=(rn4(j,:)*wler)-h*y4(iter)+n; a5(iter)=(rn5(i,:)*wler)>=(rn5(j,:)*wler)-h*y5(iter)+n; a6(iter)=(rn6(i,:)*wler)>=(rn6(j,:)*wler)-h*y6(iter)+n; a7(iter)=(rn7(i,:)*wler)>=(rn7(j,:)*wler)-h*y7(iter)+n; elseif i>j itera=itera+1; c1(itera)=(rn1(i,:)*wler)<=(rn1(j,:)*wler)+h*yy1(itera)-n; c2(itera)=(rn2(i,:)*wler)<=(rn2(j,:)*wler)+h*yy2(itera)-n; c3(itera)=(rn3(i,:)*wler)<=(rn3(j,:)*wler)+h*yy3(itera)-n; c4(itera)=(rn4(i,:)*wler)<=(rn4(j,:)*wler)+h*yy4(itera)-n; c5(itera)=(rn5(i,:)*wler)<=(rn5(j,:)*wler)+h*yy5(itera)-n; c6(itera)=(rn6(i,:)*wler)<=(rn6(j,:)*wler)+h*yy6(itera)-n; c7(itera)=(rn7(i,:)*wler)<=(rn7(j,:)*wler)+h*yy7(itera)-n; end end end b1(1)=zler(1)>=sum(y1)+sum(yy1); b2(1)=zler(1)>=sum(y2)+sum(yy2); b3(1)=zler(1)>=sum(y3)+sum(yy3); b4(1)=zler(1)>=sum(y4)+sum(yy4); b5(1)=zler(1)>=sum(y5)+sum(yy5); b6(1)=zler(1)>=sum(y6)+sum(yy6); b7(1)=zler(1)>=sum(y7)+sum(yy7); a11(1)=sum(wler)==1; % constraint definitions wprob.constraints.a1 = a1; wprob.constraints.a2 = a2; wprob.constraints.a3 = a3; wprob.constraints.a4 = a4; wprob.constraints.a5 = a5; wprob.constraints.a6 = a6; wprob.constraints.a7 = a7; wprob.constraints.c1 = c1; wprob.constraints.c2 = c2; wprob.constraints.c3 = c3; wprob.constraints.c4 = c4; wprob.constraints.c5 = c5; wprob.constraints.c6 = c6; wprob.constraints.c7 = c7; wprob.constraints.b1 = b1; wprob.constraints.b2 = b2; wprob.constraints.b3 = b3; wprob.constraints.b4 = b4; wprob.constraints.b5 = b5; wprob.constraints.b6 = b6; wprob.constraints.b7 = b7; wprob.constraints.a11 = a11; % objective function definition wprob.objective=zler; % solving mixed integer linear programming model by branch and bound algorithm opts = optimoptions('intlinprog','maxnodes',100000); [sol,fval]=solve(wprob,'options',opts); pala/decis. mak. appl. manag. eng. 5 (2) (2022) 260-286 286 appendix 2. comparison between different weighting schemes for case 1 mcdm weights cc critic em weights weights weights aras 0.14867 0.15838 0.20000 copras 0.18256 0.15295 0.20000 edas 0.35370 0.32373 0.20000 moosra 0.13523 0.15328 0.20000 waspas 0.17985 0.21166 0.20000 final scores rank final scores rank final scores rank australia 0.68758 5 0.69803 6 0.68720 8 austria 0.37239 19 0.39501 19 0.44873 18 belgium 0.46863 15 0.48468 16 0.50779 16 canada 0.69335 3 0.70389 5 0.69424 7 chile 0.11067 36 0.11846 36 0.13562 36 czech republic 0.29736 26 0.31079 26 0.33688 24 denmark 0.54402 12 0.56606 11 0.60717 12 estonia 0.30404 25 0.31258 25 0.32400 26 finland 0.47071 14 0.48767 15 0.51259 15 france 0.32083 22 0.34197 22 0.39101 20 germany 0.46444 16 0.48860 14 0.53885 14 greece 0.14928 33 0.15744 33 0.17622 33 hungary 0.14504 34 0.15211 34 0.16973 34 iceland 0.62404 8 0.65690 8 0.72598 5 ireland 0.43761 17 0.45302 17 0.47534 17 israel 0.31301 23 0.32216 24 0.33529 25 italy 0.23413 28 0.24724 28 0.27894 28 japan 0.40509 18 0.41780 18 0.44392 19 korea 0.32805 21 0.34607 21 0.38925 22 latvia 0.18713 31 0.19328 31 0.20683 31 luxembourg 0.54733 11 0.57200 10 0.61321 11 mexico 0.23208 30 0.23190 30 0.22354 30 netherlands 0.69301 4 0.71999 2 0.79547 2 new zealand 0.52850 13 0.54319 13 0.55662 13 norway 0.63783 7 0.66856 7 0.72578 6 poland 0.23318 29 0.24233 29 0.25887 29 portugal 0.15882 32 0.16754 32 0.18939 32 slovak republic 0.26684 27 0.27419 27 0.28305 27 slovenia 0.35542 20 0.36851 20 0.39006 21 spain 0.30652 24 0.32373 23 0.36007 23 sweden 0.68554 6 0.70795 4 0.74223 3 switzerland 0.69697 2 0.71760 3 0.73651 4 turkey 0.10874 37 0.11488 37 0.12834 37 united kingdom 0.55583 9 0.58110 9 0.63934 10 united states 0.83409 1 0.83637 1 0.79608 1 brazil 0.11379 35 0.12024 35 0.13868 35 russia 0.54998 10 0.55046 12 0.65332 9 south africa 0.00000 38 0.00000 38 0.00000 38 © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering issn: 2560-6018 eissn: 2620-0104 doi:_https://doi.org/10.31181/dmame0319102022d * corresponding author. e-mail address: anamika1994.dash@gmail.com (a. dash), bcgiri.jumath@gmail.com (b. c. giri), aksarkar.jumath@gmail.com (a. k. sarkar) coordination of a single-manufacturer multiretailer supply chain with price and green sensitive demand under stochastic lead time anamika dash1*, bibhas c. giri1 and ashis kumar sarkar1 1 department of mathematics, jadavpur university, kolkata 700032, india received: 5 december 2021; accepted: 1 september 2022; available online: 17 october 2022. original scientific paper abstract: when dealing with uncertainties in supply chain and ensuring customer satisfaction, efficient management of lead time plays a significant role. likewise, besides managing inventory and pricing strategies adeptly in multi-retailer supply chain, it has become inevitable to the firms to embrace green and sustainable business practices. in this context, this paper considers a two-level supply chain consisting of a single manufacturer and multiple retailers in which the manufacturer produces a single product and delivers it to the retailers in some equal-sized batches. each retailer faces a price and green sensitive market demand. the lead time is assumed to be a random variable which follows a normal distribution. shortages for retailer inventory are allowed to occur and are completely backlogged. the centralized model and a decentralized model based on leader-follower stackelberg gaming approach are developed. a price discount mechanism between the manufacturer and retailers is proposed. for the acceptance of this contract, the upper and lower limits of the price discount rate are established. numerical outcomes exhibit that the price discount mechanism effectively coordinates the supply chain and enhances both environmental and economical performances. a sensitivity analysis with respect to some key parameters is performed, and certain managerial insights are emphasized. key words: two-level supply chain, multiple retailers, stochastic lead time, price and green sensitive demand, price discount mechanism. 1. introduction the growing importance of environmental protection and pollution reduction has been felt all over the world in recent years. green supply chain management aims to prevent pollution while also producing environmentally friendly products. it involves many activities including green manufacturing, green packaging, green distribution, mailto:anamika1994.dash@gmail.com mailto:bcgiri.jumath@gmail.com mailto:aksarkar.jumath@gmail.com dash et al./decis. mak. appl. manag. eng. (2022) 2 remanufacturing and waste management. many industries (walmart, coca-cola, nike, adidas, and others) are showing great interest in environmentally friendly supply chains. they are successfully influencing consumers' attitudes toward green products by emphasizing the benefits and necessity of a green supply chain. lg india has pioneered the creation of eco-friendly electronic gadgets. they have strictly used halogen or mercury, trying to reduce the use of dangerous substances in their products. tcs has already earned the title of newsweek’s top greenest company in the world, with a global green score of 80.4% due to its worldwide recognized sustainability practices. dell has promoted an efficient and effective safe disposal system by allowing their customers to return their product to the company for free. as consumer awareness grows, more people are willing to buy environmentally friendly products and are willing to pay more for those products. the government is also trying to make people aware of eco-friendly products through various guidelines and legislation. researchers and practitioners are focusing on integrating environ-mental concerns into supply chain management. lead time plays a vital role in supply chain management. the assumption of deterministic lead time is not valid in most real world situations because of various reasons such as delays in production process, transit time, inspection, loading and unloading, and so on. therefore, dealing with stochastic lead time is very fascinating and challenging. to avoid a planned shortage at the buyer’s end and to efficiently manage the phenomena of early arrival, researchers are developing supply chain models with stochastic lead time (he et al., 2005; lieckens and vandaele, 2007; barman et al., 2021b). price is another important factor that influences the customer demand. in this context, a good quality product with relatively lower price always attracts customers. in traditional supply chain management, the manufacturer determines the quality of the product and the retailers set their selling prices independently. therefore, it has become an important managerial concern to implement an effective coordination between the manufacturer and the retailers for balancing the social and economical issues equitably. suitable coordination schemes can improve the efficiency of the entire supply chain by creating incentives for all members to adopt it. through such coordination mechanisms, the members of the supply chain develop a collaborative relationship between themselves. researchers have performed a significant amount of work to coordinate the supply chain with an appropriate contract such as revenue sharing contract (zhang and feng, 2014; mondal and giri, 2021), cost sharing contract (saha and goyal, 2015; zhu et al., 2018), delay in payments (ebrahimi et al., 2019; duary et al., 2022), etc. in today’s competitive market, manufacturers do not rely on a single retailer to sell their produced goods; instead, they deal with multiple retailers. in this study, we consider a two-level supply chain which is comprised of a single manufacturer and multiple retailers trading for a single product. the manufacturer delivers the retailers' order quantities in equal-sized batches and invests in green technologies to produce eco-friendly products. the product's greening level and selling price influence customer demand at each retailer. replenishment lead time is assumed to be random. both the centralized and decentralized models are considered. we demonstrate cooperation between the manufacturer and the retailers by a price discount mechanism. our primary goal is to fulfill the research gap and find answers to the following research questions: • what will be the optimal strategies of the manufacturer and retailers when the market demand is price and green sensitive? coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 3 • what is the impact of a price discount contract on the optimal decisions of the supply chain? • is the price discount mechanism capable of coordinating the supply chain? • what is the effect of greening investment on the profitability of the supply chain? the contributions of this study are as follows: firstly, we incorporate a price discount mechanism with green initiatives in a single-manufacturer multi-retailer supply chain model under stochastic lead time. secondly, we examine whether the proposed price discount contract is able to coordinate the supply chain or not. finally, we look at the influence of the price discount contract on supply chain members' profitability and determine the conditions under which they accept the price discount contract. the rest of this paper is structured as follows: section 2 contains a brief review of the existing literature relevant to this work. section 3 introduces notations and assumptions that are used throughout the paper. the problem description is given in section 4. in section 5, mathematical models are formulated. section 6 is devoted to numerical analysis. a sensitivity analysis of some key parameters is performed in section 7. section 8 discusses some managerial implications of this study. finally, section 9 concludes the paper with some limitations and future research directions. 2. literature review in this section, we review some of the existing literatures which are related to our current work across four research streams: priceand green-sensitive demand, stochastic lead time, single-manufacturer multi-retailer supply chain model and price discount contract. 2.1 priceand green-sensitive demand price is one of the important factors that influence market demand. a preliminary work focusing on price dependent demand was carried out by whitin and thomson (1955). later, many researchers and practitioners (ho et al., 2008; yang et al., 2009; lin and ho, 2011; atamer et al., 2013; rad et al., 2014; jaggi et al.,2015; alfares and ghaithan, 2016) have done numerous works on price dependent demand. researchers and practitioners are currently focused on issues including the reduction of harmful effects of production on the environment. swami and shah (2013) studied a vertical supply chain consisting of a single manufacturer and a single retailer where the members put an effort for greening their operations, and the customer demand at the retailer’s end is price and green sensitive. zanoni et al. (2014) investigated a two-level joint economic lot size model with customer demand sensitive to price and environmental quality, and concluded that investing in improving a product's environmental performance is more beneficial, and implementing an integrated policy can increase both environmental and economic performance. ghosh and shah (2015) explored the positive impact of a cost sharing contract on the optimal decisions of a green supply chain to enhance the profit level and produce items with higher greening quality. li et al. (2016) initiated e-commerce in green supply chain management and proposed a coordination mechanism for decentralized dual channel green supply chain. basiri and heydari (2017) investigated coordination issues in a green supply chain with a non-green traditional product and a substitutable green product under price, greening level and sales effort dependent demand. giri et al. (2018) analyzed a two-level closed-loop supply chain model where the customer demand is affected by dash et al./decis. mak. appl. manag. eng. (2022) 4 selling price, warranty period and greening level of the product. they proposed a revenue sharing contract in order to develop both social and economic performances. heydari et al. (2019) developed a three-tier dual channel supply chain model with price and green sensitive demand that is not only economically beneficial but also reduces the selling price in both channels. heydari et al. (2021) proposed a hybrid coordination scheme of cost sharing contract and revenue sharing contract in a twolevel green supply chain with price and green sensitive demand. in a two-level supply chain model with imperfect production system, price, advertisement, and green sensitive customer demand, giri and dash (2022) established a cost-sharing contract between the manufacturer and the retailer. sepehri and gholamian (2022) investigated the impacts of shortages in a sustainable inventory model with price and emission sensitive demand considering quality improvement and inspection process concurrently. 2.2 stochastic lead time to address the shortcomings of deterministic lead time, researchers devised supply chain models that take into account the stochastic nature of lead time. sajadieh et al. (2009) developed a single vendor single buyer supply chain model with stochastic lead time following exponential distribution and deterministic demand, and exhibited a significant cost reduction in integrated system than decentralized ones. hoque (2013) presented an integrated inventory model with stochastic lead time following normal distribution under combined equal and unequal batch shipment policy. lin (2016) considered an integrated vendor-buyer model with stochastic lead time, and demonstrated that further investment can reduce lead time variability and achieve enough savings for the entire system. giri and masanta (2019) derived optimal production and shipment policy for a closed-loop supply chain model with stochastic lead time, and observed that learning in production and remanufacturing leads to a significant cost reduction for the supply chain. giri and masanta (2020) developed a closed-loop supply chain model with learning in production, price and quality sensitive demand under stochastic lead time, and elaborated the positive impact of learning in production process on the optimal decisions. sarkar et al. (2020a) investigated an integrated vendor-buyer model considering time value of money with partially backlogged shortage under stochastic lead time where the lead time is variable but dependent on the order size of the buyer and production rate at the vendor. safarnezhad et al. (2021) derived optimal ordering, pricing and inspection policies in a vendor-buyer supply chain model with price dependent demand and stochastic lead time. hoque (2021) developed a single-manufacturer multi-retailer supply chain model under stochastic lead time where the manufacturer delivers the lots to the retailers either only with equal batch sizes or only with unequal batch shipments. 2.3 single-manufacturer multi-retailer supply chain model to come closer to the reality, focusing on multi-retailer models has become a great topic of interest for the researchers. recently, giri and roy (2016) considered a supply chain model consisting of a single manufacturer and multiple retailers with price sensitive customer demand. they found that lead time reduction by paying extra crashing cost does not affect the retail price significantly but enhances the entire system profit. chen and sarker (2017) investigated a single-manufacturer multiretailer production-inventory model for deteriorating items with price sensitive demand under just-in-time delivery environment. they solved the model using coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 5 particle swarm optimization (pso) and quantum-behaved pso (qbpso) techniques. majumder et al. (2018) studied a single-vendor multi-buyer supply chain model with variable production rate and controllable lead time reduction where the production cost at the vendor is a function of the production rate. chan et al. (2018) proposed a coordination mechanism in a single-vendor multi-buyer supply chain model with stochastic demand, and synchronized the manufacturer’s production cycle and retailers’ ordering cycle. ben-daya et al. (2019) developed a single manufacturer multi-retailer closed-loop supply chain model with an environment-friendly approach of remanufacturing the used products under consignment stock policy. giri et al. (2020b) developed a single-manufacturer multi-retailer inventory model with stochastic lead time and price sensitive demand. esmaeili and nasrabadi (2021) presented a single-vendor multi-retailer supply chain model for deteriorating items with trade credit and inflationary conditions, where the demand is price sensitive. najafnejhad et al. (2021) used an imperialist competitive algorithm to solve a singlevendor multi-retailer inventory model under vendor managed inventory policy considering upper limits of inventories as decision variables. nandra et al. (2021b) studied a single-vendor multi-buyer model that took into account variable production cost, imperfect items and environmental factors. malleeswaran and uthayakumar (2022) introduced a discrete investment for ordering cost reduction in a singlemanufacturer multi-retailer epq model with green and environmental sensitive consumer demand and reworking system under carbon emissions policies. 2.4 price discount contract coordination between manufacturers and retailers has received a lot of attention as a means of improving inventory control, and researchers have done a lot of work to coordinate the supply chain with the appropriate contract. as we consider a price discount coordination scheme in our study, we cover some literatures which address similar issues. viswanathan and piplani (2001) analyzed a single-vendor multi-buyer model with a coordination mechanism in which the vendor specifies the replenishment period and all the buyers agree to order at the same time in exchange for a price discount. li et al. (2011) investigated the impact of a price discount mechanism in a single-vendor single-buyer supply chain model with service level constraint and controllable lead time. aljazzar et al. (2017) dealt with a three-level supply chain with two types of trade credit mechanism, and concluded that implementing both delay in payment and price discount coordination mechanisms at a time lead more profit for the entire supply chain rather adopting these contracts individually. nouri et al. (2018) proposed a compensation-based wholesale price contract between the manufacturer and the retailer where the customer demand is stochastic and dependent on innovation and promotional efforts. furthermore, they devised a profit-sharing strategy on the basis of bargaining power of the members. xu et al. (2018) investigated the role of a price discount contract in coordinating a dualchannel supply chain under carbon emission capacity regulation, with consumer demand in both online and offline channels influenced by the product's selling price. they provided the necessary conditions for which the price discount contract coordinates the dual-supply chain in both online and offline modes. sarkar et al. (2020b) suggested a price discount coordination mechanism in a two-level supply chain with price sensitive customer demand to encourage the supply chain players to take part in joint decision-making strategy. yang et al. (2021) explored the optimal cooperation strategy between an upstream supplier and two competing manufacturers considering a wholesale price contract and manufacturers' technology investment. in order to reduce products’ carbon emissions. zu et al. (2021) analyzed dash et al./decis. mak. appl. manag. eng. (2022) 6 a single-manufacturer single-retailer supply chain model under two different mechanisms viz. wholesale price contract and consignment contract. zhang et al. (2022) performed a comparative analysis between wholesale price contract and costsharing contract in a two-level green supply chain model. they looked at which contract is more effective in improving the product's greenness and promoting demand, taking into account the consumer reference pricing effect. 2.5 research gaps in the existing literature table 1 summarizes the research gaps in the existing literature as follows:  although there are numerous research papers available that explore stochastic lead time and single-manufacturer multi-retailer supply chain models, no attempt has been made to maximize individual profits of supply chain stakeholders. the majority of these research focused on maximizing (or minimizing) overall supply chain profit (or cost).  most of these studies considered deterministic customer demand. they overlooked some crucial factors such as the selling price, greening level, promotional effort, advertising and product quality, all of which have an impact on market demand.  no one has incorporated environmental awareness into a single-manufacturer multi-retailer supply chain model with stochastic lead time, and none of these studies looked at the influence of greening investment on both the supply chain's economic and environmental performance.  almost no study has ever suggested a channel coordination mechanism. the above literature review reveals a significant research gap and indicates that no attempt has been made in implementing price discount coordination mechanism in a single-manufacturer multi-retailer supply chain model with price and green sensitive demand under stochastic lead time. it would be interesting and contributory to consider all the genuine issues like the stochastic nature of lead time, the impact of retail price and environmental awareness on market demand, single-manufacturer multi-retailer business situations and so on under one umbrella. although, hoque (2021) extended the model of hoque (2013) in multi-retailer scenario, but he considered the demand of each retailer as deterministic and minimizes the total cost of the supply chain. in this paper, our aim is to fulfill this research gap and implement an appropriate coordination scheme which efficiently improves each supply chain member’s profitability as well as environmental performance. a comparison of the present work with the relevant existing literature is presented in table 1. coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 7 table 1. a comparison of the present model with some existing literature authors retailer batch shipment demand lead time coordination sajadieh et al. (2009) single equal deterministic stochastic no li et al. (2011) single equal deterministic controllable price discount hoque (2013) single equal & unequal deterministic stochastic no sarkar et al. (2017) single equal deterministic no no giri et al. (2018) single no price, green and warranty period sensitive no revenue sharing sarkar et al. (2018) multiple equal deterministic variable no giri and masanta (2019) single equal deterministic stochastic no giri et al. (2020a) single equal & unequal price and green dependent no cost sharing sarkar et al. (2020b) single equal price dependent no price discount agrawal and yadav (2020) multiple equal price dependent constant profit sharing esmaeili and nasrabadi (2021). multiple no price dependent no no nandra et al. (2021a) multiple equal deterministic controllable no sarkar et al. (2021) single equal online & offline price dependent distribution free approach & normal no safarnezhad et al. (2021) single no price dependent stochastic no hoque (2021) multiple equal or unequal deterministic stochastic no this paper multiple equal price and green dependent stochastic price discount 3. notations and assumptions the following notations are used for developing the proposed model: parameters: 𝑅 production rate (units/ year) 𝐴𝑣 set-up cost per set-up ($/set-up) ℎ𝑣 manufacturer’s holding cost per item per unit time ($/unit /year) 𝐹 transportation cost per batch shipment($/shipment) 𝑤 unit wholesale price($/unit) 𝐼 greening investment parameter ($) 𝑁 number of retailers (positive integer) dash et al./decis. mak. appl. manag. eng. (2022) 8 𝑄 total order quantity [= ∑ 𝑄𝑖 𝑁 𝑖=1 ](units) 𝐷 total market demand [= ∑ 𝐷𝑖 𝑁 𝑖=1 ](units /year) 𝐿 lead time, a random variable with p.d.f. 𝑓𝐿(.) 𝑖-th retailer: 𝐴𝑖 ordering cost per order ($/order) ℎ𝑖 holding cost per item per unit time ($/unit /year) 𝐷𝑖 demand rate [𝑅 > ∑ 𝐷𝑖 𝑁 𝑖=1 ](units /year) 𝑎𝑖 basic market demand (units /year) 𝛼𝑖 consumer sensitivity coefficient to greening level 𝛽𝑖 consumer sensitivity coefficient to retail price 𝑄𝑖 order quantity (units) 𝑐𝑖 shortage cost per item per unit time ($/unit /year) 𝑟𝑖 reorder point(units) 𝜎𝑖 standard deviation of the lead time decision variables: 𝑛 number of batches delivered to each retailer (positive integer) 𝜃 greening improvement level 𝑞𝑖 batch size of the 𝑖-th retailer (units) 𝑝𝑖 unit retail price of the 𝑖-th retailer ($/unit) 𝜙 price discount ratio, 𝜙 ∈ [0,1] (.)^𝑑 decision variable in decentralized policy (.)^𝑐 decision variable in centralized policy (.)^𝑐𝑜 decision variable in coordinated mechanism profit functions: 𝐴𝐸𝑃𝑚 average expected profit of the manufacturer($/year) 𝐴𝐸𝑃𝑖 average expected profit of the 𝑖-th retailer($/year) 𝐴𝐸𝑃𝑠 average expected profit of the supply chain ($/year) the basic assumptions for developing the proposed model are as follows: 1. a single manufacturer produces a single item and meets the demand of multiple retailers (sarkar et al., 2018). 2. the manufacturer transfers the products to the retailers in a number of equal sized batches (sarkar et al., 2020b). 3. the retailers face a consumer demand dependent on the selling price and greenness of the product (ghosh and shah, 2015). we assume that the demand rate of the 𝑖-th retailer is a linear function of retail price and greening level of the product given by 𝐷𝑖(𝑝𝑖,𝜃) = 𝑎𝑖 − 𝛽𝑖𝑝𝑖 + 𝛼𝑖𝜃, where 𝑎𝑖 is the basic market demand, 𝛽𝑖 and 𝛼𝑖 are positive integers such that 𝑎𝑖 + 𝛼𝑖𝜃 > 𝛽𝑖𝑝𝑖 for all 𝑖 = 1,2, . . . . ,𝑁. 4. the manufacturer produces the product at a constant production rate 𝑅 in one set-up and the production rate is greater than the sum of demands of all retailers i.e., r > ∑ di n i=1 (hoque, 2021). 5. shortages are allowed and are assumed to be completely backlogged (sarkar et al., 2018). coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 9 6. the 𝑖-th retailer places his next order when his inventory stock level reaches to a certain reorder level 𝑟𝑖 (hoque, 2013). 7. the lead time to meet the retailer’s demand is a random variable which follows a normal distribution and the lead time for each shipment is independent of the others (hoque, 2013). 8. annual greening investment for the product is taken as 𝐼𝜃2, which is increasing and convex in the greening improvement level 𝜃 (ghosh and shah, 2015). 4. problem definition this study develops a green supply chain model where the single manufacturer deals with multiple retailers for a single product. figure 1 exhibits the schematic diagram of the proposed model. figure 1. logistics diagram of the proposed single-manufacturer multiretailer green supply chain model the manufacturer produces the items at a fixed production rate in a single set-up and delivers the order quantities of the retailers with an equal sized batch shipment policy. due to various unavoidable circumstances such as late start in production, varying transportation time, loading, unloading, etc., the batches may arrive early or late at the retailers. to deal with this type of delivery uncertainty, lead time is treated as a stochastic random variable which follows a normal distribution. customer demand is assumed to be affected by the retail price and environmental performance of the product. the manufacturer adopts a green investment strategy to maintain his environmental responsibility as well as stimulate the customer demand in an ecoconscious market. in both decentralized and centralized settings, the manufacturer's and all retailers' optimal pricing and inventory strategies are derived. following that, a wholesale price discount contract is implemented between the manufacturer and the retailers to coordinate the supply chain. 5. model formulation we suppose that the manufacturer sells the produced items to 𝑁 retailers. the manufacturer transfers the ordering quantity 𝑄𝑖 of the 𝑖-th retailer in 𝑛 equal batches of size 𝑞𝑖. total order quantity of 𝑁 retailers is 𝑄. therefore, 𝑄𝑖 = 𝑛𝑞𝑖 and 𝑄 = ∑ 𝑄𝑖 𝑁 𝑖=1 . dash et al./decis. mak. appl. manag. eng. (2022) 10 the 𝑖-th retailer places the next order when the inventory stock reaches to a level 𝑟𝑖. the shipment is expected to arrive to the retailer’s end at or before the time of selling this 𝑟𝑖 quantity. the mean lead time is 𝑟𝑖 𝐷𝑖 . due to various reasons, the batches may reach early or late. we assume that the lead time follows a normal distribution. depending on the length of the lead time, three cases may arise: case (i) when the batch 𝑞𝑖 reaches to the retailer earlier i.e., 0 < 𝑙𝑖 < 𝑟𝑖 𝐷𝑖 . in this case, similar to hoque (2013), the inventory holding area of the 𝑖-th retailer can be determined from figure 2(a) as = 𝐴𝑟𝑒𝑎 ( abcd + efg + ghde) = 1 2 (𝑟𝑖 − 𝐷𝑖𝑙𝑖 +𝑟𝑖)𝑙𝑖 + 1 2 (𝑞𝑖 − 𝐷𝑖𝑙𝑖) (𝑞𝑖 −𝐷𝑖𝑙𝑖) 𝐷𝑖 + 𝑟𝑖(𝑞𝑖 − 𝐷𝑖𝑙𝑖) 𝐷𝑖 = 1 2 [ 𝑞𝑖 2 𝐷𝑖 + 2𝑞𝑖 ( 𝑟𝑖 𝐷𝑖 − 𝑙𝑖)] where 𝑟𝑖 = 𝑞𝑖𝐷𝑖 𝑅 then the order quantity 𝑄𝑖 of the 𝑖-th retailer is given by 𝑛 2 [ 𝑞𝑖 2 𝐷𝑖 + 2𝑞𝑖 ( 𝑟𝑖 𝐷𝑖 −𝑙𝑖)] the holding cost refers to the investment in storing the unsold products. the expected inventory holding cost for the order quantity 𝑄𝑖 of the 𝑖-th retailer is ℎ𝑖 ∫ 𝑛 2 [ 𝑞𝑖 2 𝐷𝑖 + 2𝑞𝑖 ( 𝑟𝑖 𝐷𝑖 −𝑙𝑖)]𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 𝑟𝑖 𝐷𝑖 0 figure 2. inventory of 𝑖th retailer under stochastic lead time coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 11 case (ii) when the batch 𝑞𝑖 reaches late to the 𝑖-th retailer and the lead time 𝑙𝑖 lies in the range 𝑟𝑖 𝐷𝑖 ≤ 𝑙𝑖 ≤ 𝑟𝑖+𝑞𝑖 𝐷𝑖 . in this case, shortages occur at the retailer’s end. from figure 2(b), the shortage area at the 𝑖-th retailer is obtained as = 𝐴𝑟𝑒𝑎 ( cde) = 1 2𝐷𝑖 (𝐷𝑖𝑙𝑖 − 𝑟𝑖) 2. so, the expected shortage cost of the 𝑖-th retailer for 𝑛 batches is given by 𝑛𝑐𝑖 2 ∫ (𝐷𝑖𝑙𝑖 − 𝑟𝑖) 2 𝐷𝑖 𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑟𝑖 𝐷𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 inventory holding area of the 𝑖-th retailer for the batch qi is = 𝐴𝑟𝑒𝑎 ( abc + fgh+ efhj) = ri 2 2di + (qi-dili) 2 2di + ri(qi-dili) di = (𝑞𝑖 − 𝐷𝑖𝑙𝑖+𝑟𝑖) 2 2𝐷𝑖 hence the expected inventory holding cost of the 𝑖-th retailer for 𝑛 shipments is obtained as 𝑛ℎ𝑖 ∫ (𝑞𝑖 − 𝐷𝑖𝑙𝑖 + 𝑟𝑖) 2 2𝐷𝑖 𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑟𝑖 𝐷𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 it is assumed that, during this delay period, the batches remain in the manufacturer’s stockhouse. so, it causes an extra holding cost to the manufacturer. the extra inventory for this delayed delivery is ∑𝑁𝑖=1 𝑛𝑞𝑖(𝐷𝑖𝑙𝑖−𝑟𝑖) 𝐷𝑖 . so, in this case, the additional inventory holding cost for the manufacturer is ℎ𝑣 ∑∫ 𝑛𝑞𝑖(𝐷𝑖𝑙𝑖 −𝑟𝑖) 𝐷𝑖 𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑟𝑖 𝐷𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 𝑁 𝑖=1 . case(iii) when the batch 𝑞𝑖 arrives late to the retailer with lead time in the range 𝑟𝑖+𝑞𝑖 𝐷𝑖 ≤ 𝑙𝑖 < ∞. in this case, shortages occur at the retailer’s end and from figure 2(c), the shortage area for the batch 𝑞𝑖 is obtained as = 𝐴𝑟𝑒𝑎 ( cdef ) = 𝑞𝑖 2 2𝐷𝑖 + 𝐴𝑟𝑒𝑎( defg ) so, the expected shortage cost of the 𝑖-th retailer for all batch shipments is 𝑛𝑐𝑖 ∫ [ 𝑞𝑖 2 2𝐷𝑖 + 𝑞𝑖 ( 𝐷𝑖𝑙𝑖 − 𝑞𝑖 − 𝑟𝑖 𝐷𝑖 )] ∞ 𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 similar to case(ii), the additional expected inventory holding cost for the manufacturer is ℎ𝑣 ∑∫ 𝑛𝑞𝑖(𝐷𝑖𝑙𝑖 − 𝑟𝑖) 𝐷𝑖 ∞ 𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 𝑁 𝑖=1 . dash et al./decis. mak. appl. manag. eng. (2022) 12 combining all three cases, the expected holding cost of the 𝑖-th retailer for all batches is given by 𝑛ℎ𝑖 ∫ 1 2 [ 𝑞𝑖 2 𝐷𝑖 + 2𝑞𝑖 ( 𝑟𝑖 𝐷𝑖 − 𝑙𝑖)] 𝑟𝑖 𝐷𝑖 0 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 + 𝑛ℎ𝑖 ∫ (𝑞𝑖 − 𝐷𝑖𝑙𝑖 + 𝑟𝑖) 2 2𝐷𝑖 𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑟𝑖 𝐷𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 and the expected shortage cost for all batch shipments is 𝑛𝑐𝑖 ∫ (𝐷𝑖𝑙𝑖 − 𝑟𝑖) 2 2𝐷𝑖 𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑟𝑖 𝐷𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 + 𝑛𝑐𝑖 ∫ [ 𝑞𝑖 2 2𝐷𝑖 + 𝑞𝑖 ( 𝐷𝑖𝑙𝑖 −𝑞𝑖 − 𝑟𝑖 𝐷𝑖 )] ∞ 𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 5.1. decentralized model (dm) in the decentralized model, the manufacturer and the retailers independently take their decisions in order to maximize their own profits. here we consider the retailers to be the stackelberg leader and the manufacturer as the follower. the manufacturer sets the number of shipments and greening level of the products. then taking these responses into consideration, the retailers decide their optimal retail price and batch sizes. average expected profit of the manufacturer the manufacturer's total extra holding cost from cases(ii) and (iii) is ℎ𝑣 ∑∫ 𝑛𝑞𝑖(𝐷𝑖𝑙𝑖 − 𝑟𝑖) 𝐷𝑖 ∞ 𝑟𝑖 𝐷𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 𝑁 𝑖=1 figure 3. joint inventory of the manufacturer and the retailers in figure 3, the trapezium abcd represents the joint inventory of the manufacturer and retailers. the average inventory of the manufacturer-retailer system is = 𝐴𝑟𝑒𝑎 ( abcd)× 𝐷 𝑄 coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 13 = 1 2 ×(𝐴𝐵 + 𝐶𝐷) × 𝑄 × 𝐷 𝑄 = 1 2 [ ∑ 𝑞𝑖 𝑁 𝑖=1 𝑅 + ( 𝑄 𝐷 + ∑ 𝑞𝑖 𝑁 𝑖=1 𝑅 − 𝑄 𝑅 )]𝐷 = 𝐷∑ 𝑞𝑖 𝑁 𝑖=1 𝑅 + 𝑄 2 (1− 𝐷 𝑅 ) average inventory holding of 𝑁 retailers is ∑ ( 𝑞𝑖 2 2𝐷𝑖 )( 𝐷𝑖 𝑄𝑖 )𝑁𝑖=1 = ∑ 𝑞𝑖 2 2𝑄𝑖 𝑁 𝑖=1 therefore, the average inventory holding of the manufacturer is 𝐷∑ 𝑞𝑖 𝑁 𝑖=1 𝑅 + 𝑄 2 (1 − 𝐷 𝑅 ) −∑ 𝑞𝑖 2 2𝑄𝑖 𝑁 𝑖=1 the set-up cost incorporates the costs of materials and labours to get ready the machinery system for processing the new production lot of goods. it plays an important role in start-up of a new business and smooth running of it. the 𝑖-th retailer places an order of quantity 𝑄𝑖. the manufacturer produces the total order quantity 𝑄 = ∑ 𝑄𝑖 𝑁 𝑖=1 . the cycle length of the manufacturer is 𝑄 𝐷 . therefore, the average set up cost is 𝐴𝑣𝐷 q . investment for greening supports the environmentally-conscious business practices. in this case, the manufacturer's average greening investment is 𝐼𝜃2. the average expected profit of the manufacturer is 𝐴𝐸𝑃𝑚(𝑛,𝜃) = 𝑤𝐷 − ℎ𝑣 [ 𝐷∑  𝑁𝑖=1𝑞𝑖 𝑅 + 𝑄 2 (1− 𝐷 𝑅 ) − ∑  𝑁𝑖=1 𝑞𝑖 2 2𝑄𝑖 ]− ℎ𝑣 ∑   𝑁 𝑖=1 ∫   ∞ 𝑟𝑖 𝐷𝑖 (𝐷𝑖𝑙𝑖 −𝑟𝑖)𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 − 𝐴𝑣𝐷 𝑄 − 𝐼𝜃2 = 𝑤𝐷 − 𝐴𝑣𝐷 𝑄 − ℎ𝑣 [ 𝐷∑  𝑁𝑖=1𝑞𝑖 𝑅 + 𝑄 2 (1− 𝐷 𝑅 ) −∑  𝑁𝑖=1 𝑞𝑖 2 2𝑄𝑖 ]− ∑  𝑁𝑖=1 ℎ𝑣𝐷𝑖𝜎𝑖 √2𝜋 − 𝐼𝜃2 (1) from (1), we have 𝜕𝐴𝐸𝑃𝑚 𝜕𝑛 = 𝐴𝑣𝐷 𝑛2𝑠 − ℎ𝑣𝑠 2 + ℎ𝑣𝑠𝐷 2𝑅 − ℎ𝑣𝑠 2𝑛2 (2) 𝜕𝐴𝐸𝑃𝑚 𝜕𝜃 = [𝑤 − 𝐴𝑣 𝑄 − ℎ𝑣𝑠 𝑅 + ℎ𝑣𝑄 2𝑅 − (∑𝑁𝑖=1 ℎ𝑣𝜎𝑖 √2𝜋 )]𝑢 − 2𝐼𝜃 (3) 𝜕2𝐴𝐸𝑃𝑚 𝜕𝑛2 = − 2𝐴𝑣𝐷 𝑛3𝑠 + ℎ𝑣𝑠 𝑛3 (4) 𝜕2𝐴𝐸𝑃𝑚 𝜕𝜃2 = −2𝐼 (5) 𝜕2𝐴𝐸𝑃𝑚 𝜕𝑛𝜕𝜃 = 𝐴𝑣𝑢 𝑛2𝑠 + ℎ𝑣𝑠𝑢 2𝑅 (6) 𝜕2𝐴𝐸𝑃𝑚 𝜕𝜃𝜕𝑛 = 𝐴𝑣𝑢 𝑛2𝑠 + ℎ𝑣𝑠𝑢 2𝑅 , where 𝑠 = ∑𝑁𝑖=1 𝑞𝑖 and 𝑢 = ∑ 𝑁 𝑖=1 𝛼𝑖 (7) proposition 1. the average expected profit function of the manufacturer is jointly concave in 𝑛 and 𝜃 if 8𝐼𝑅2𝑛𝑠(2𝐴𝑣𝐷𝑠 − ℎ𝑣𝑠 2) > (2𝐴𝑣𝑅𝑢 + ℎ𝑣𝑢𝑛 2𝑠2)2. proof. considering 𝑛 as real, the hessian matrix is 𝐻 = ( 𝜕2𝐴𝐸𝑃𝑚 𝜕𝜃2 𝜕2𝐴𝐸𝑃𝑚 𝜕𝜃𝜕𝑛 𝜕2𝐴𝐸𝑃𝑚 𝜕𝑛𝜕𝜃 𝜕2𝐴𝐸𝑃𝑚 𝜕𝑛2 ) = ( −2𝐼 𝐴𝑣𝑢 𝑛2𝑠 + ℎ𝑣𝑠𝑢 2𝑅 𝐴𝑣𝑢 𝑛2𝑠 + ℎ𝑣𝑠𝑢 2𝑅 − 2𝐴𝑣𝐷 𝑛3𝑠 + ℎ𝑣𝑠 𝑛3 ) (8) (1) dash et al./decis. mak. appl. manag. eng. (2022) 14 here, 𝜕2𝐴𝐸𝑃𝑚 𝜕𝜃2 = −2𝐼 < 0 . so, the expected average profit function of the manufacturer will be concave in 𝜃 and 𝑛 if |𝐻| > 0. substituting the values of the partial derivatives from the above and using the condition |𝐻| > 0, we get after simplification, 8𝐼𝑅2𝑛𝑠(2𝐴𝑣𝐷𝑠 − ℎ𝑣𝑠 2) > (2𝐴𝑣𝑅𝑢 + ℎ𝑣𝑢𝑛 2𝑠2)2. proposition 2. at the equilibrium, the optimal number of shipments to each retailer, and the optimal greening level of the product are as follows: 𝑛∗ = √ 𝑅(2𝐴𝑣𝐷−ℎ𝑣𝑠 2) ℎ𝑣𝑠 2(𝑅−𝐷) (9) 𝜃∗ = [𝑤− 𝐴𝑣 𝑄 − ℎ𝑣𝑠 𝑅 + ℎ𝑣𝑄 2𝑅 −(∑𝑁𝑖=1 ℎ𝑣𝜎𝑖 √2𝜋 )]𝑢 2𝐼 (10) proof. at the equilibrium, we have 𝜕𝐴𝐸𝑃𝑚 𝜕𝑛 = 𝐴𝑣𝐷 𝑛2𝑠 − ℎ𝑣𝑠 2 + ℎ𝑣𝑠𝐷 2𝑅 − ℎ𝑣𝑠 2𝑛2 = 0 (11) and 𝜕𝐴𝐸𝑃𝑚 𝜕𝜃 = [𝑤 − 𝐴𝑣 𝑄 − ℎ𝑣𝑠 𝑅 + ℎ𝑣𝑄 2𝑅 − (∑𝑁𝑖=1 ℎ𝑣𝜎𝑖 √2𝜋 )]𝑢 − 2𝐼𝜃 = 0 (12) solving equations (11) and (12), we get the optimal values of 𝑛 and 𝜃 as given in equations (9) and (10) above. for integer optimal value of 𝑛, 𝑛𝑜𝑝𝑡 = { ⌊𝑛∗⌋, 𝑖𝑓 𝐴𝐸𝑃𝑚(⌊𝑛 ∗⌋,𝜃) ≥ 𝐴𝐸𝑃𝑚(⌈𝑛 ∗⌉,𝜃) ⌈𝑛∗⌉, 𝑖𝑓 𝐴𝐸𝑃𝑚(⌊𝑛 ∗⌋,𝜃) ≤ 𝐴𝐸𝑃𝑚(⌈𝑛 ∗⌉,𝜃) taking these response functions of the manufacturer, the retailers then set their batch sizes and retail prices. average expected profit of the 𝑖-th retailer since the expected cycle length for the 𝑖-th retailer is qi di , therefore, the average ordering cost of the 𝑖-th retailer is given by 𝐴𝑖𝐷𝑖 𝑄𝑖 . from manufacturing to delivery to the end customer and even returns, transportation is essential to the entire production process. it is practically impossible for a logistics firm to conduct business efficiently without transportation. as the number of shipments increases, the transportation cost increases. since the manufacturer delivers order quantity to the 𝑖-th retailer in 𝑛 shipments and the expected cycle length for the 𝑖-th retailer is 𝑄𝑖 𝐷𝑖 , therefore, the average variable transportation cost is 𝑛𝐹𝐷𝑖 𝑄𝑖 . the expected total profit of the 𝑖-th retailer is 𝑝𝑖𝑄𝑖 − 𝑤𝑄𝑖 − 𝐴𝑖 − 𝑛𝐹 − 𝑛ℎ𝑖 2 [∫   𝑟𝑖 𝐷𝑖 0 [ 𝑞𝑖 2 𝐷𝑖 +2𝑞𝑖 ( 𝑟𝑖 𝐷𝑖 − 𝑙𝑖)]𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 +∫   𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑟𝑖 𝐷𝑖 (𝑞𝑖 − 𝐷𝑖𝑙𝑖 + 𝑟𝑖) 2 𝐷𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖] − 𝑛𝑐𝑖 2 [∫   𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑟𝑖 𝐷𝑖 (𝐷𝑖𝑙𝑖 − 𝑟𝑖) 2 𝐷𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 +∫   ∞ 𝑟𝑖+𝑞𝑖 𝐷𝑖 [ 𝑞𝑖 2 𝐷𝑖 + 2𝑞𝑖 ( 𝐷𝑖𝑙𝑖 −𝑞𝑖 − 𝑟𝑖 𝐷𝑖 )]𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖] coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 15 therefore, the average expected profit of the 𝑖-th retailer is obtained as 𝐴𝐸𝑃𝑖(𝑞𝑖,𝑝𝑖) = 𝑝𝑖𝐷𝑖 − 𝑤𝐷𝑖 − (𝐴𝑖+𝑛𝐹)𝐷𝑖 𝑄𝑖 − ℎ𝑖 2 [∫   𝑟𝑖 𝐷𝑖 0 [𝑞𝑖 + 2(𝑟𝑖 − 𝐷𝑖𝑙𝑖)]𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 +∫   𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑟𝑖 𝐷𝑖 (𝑞𝑖−𝐷𝑖𝑙𝑖+𝑟𝑖) 2 𝑞𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖] − 𝑐𝑖 2 [∫   𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑟𝑖 𝐷𝑖 (𝐷𝑖𝑙𝑖−𝑟𝑖) 2 𝑞𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 +∫   ∞ 𝑟𝑖+𝑞𝑖 𝐷𝑖 [𝑞𝑖 + 2(𝐷𝑖𝑙𝑖 − 𝑞𝑖 −𝑟𝑖)]𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖] (13) proposition 3. the average expected profit of the 𝑖-th retailer is concave in 𝑞𝑖 for given 𝑝𝑖 if 2(𝐴𝑖+𝑛𝐹)𝐷𝑖 𝑛 + (ℎ𝑖 + 𝑐𝑖)∫ 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝐷𝑖 2𝑙𝑖 2 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 > 0. proof. differentiating (13) twice with respect to 𝑞𝑖, we obtain ∂𝐴𝐸𝑃𝑖 ∂𝑞𝑖 = (𝐴𝑖+𝑛𝐹)𝐷𝑖 𝑛𝑞𝑖 2 − ℎ𝑖 2 ∫   𝑞𝑖 𝑅 0 ( 𝑅+2𝐷𝑖 𝑅 )𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 − ℎ𝑖 2 ∫   𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 ( 𝑞𝑖 2(𝑅+𝐷𝑖) 2−𝑅2𝐷𝑖 2𝑙𝑖 2 𝑞𝑖 2𝑅2 )𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 + 𝑐𝑖 2 ∫   𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 ( 𝑅2𝐷𝑖 2𝑙𝑖 2−𝑞𝑖 2𝐷𝑖 2 𝑞𝑖 2𝑅2 )𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 + 𝑐𝑖 2 ∫   ∞ 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 ( 𝑅+2𝐷𝑖 𝑅 )𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 (14) ∂2𝐴𝐸𝑃𝑖 ∂𝑞𝑖 2 = − 2(𝐴𝑖+𝑛𝐹)𝐷𝑖 𝑛𝑞𝑖 3 − ℎ𝑖 ∫   𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝐷𝑖 2𝑙𝑖 2 𝑞𝑖 3 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 − 𝑐𝑖 ∫   𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝐷𝑖 2𝑙𝑖 2 𝑞𝑖 3 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 (15) for given 𝑝𝑖, the average expected profit function of the 𝑖-th retailer is concave in 𝑞𝑖 if 𝜕2𝐴𝐸𝑃𝑖 𝜕𝑞𝑖 2 is negative. this implies the condition 2(𝐴𝑖+𝑛𝐹)𝐷𝑖 𝑛 + (ℎ𝑖 + 𝑐𝑖)∫ 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝐷𝑖 2𝑙𝑖 2 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 > 0. proposition 4. the average expected profit of the 𝑖-th retailer is concave in 𝑝𝑖 for given 𝑞𝑖 if 2𝛽𝑖 + (ℎ𝑖 + 𝑐𝑖)∫ 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝛽𝑖 2(𝑞𝑖−𝑅𝑙𝑖) 2 𝑞𝑖𝑅 2 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 > 0. proof. differentiating (13) with respect to 𝑝𝑖, we obtain ∂𝐴𝐸𝑃𝑖 ∂𝑝𝑖 = 𝐷𝑖 − 𝛽𝑖𝑝𝑖 + ℎ𝑖 ∫   𝑞𝑖 𝑅 0 𝛽𝑖(𝑞𝑖 − 𝑅𝑙𝑖) 𝑅 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 + ℎ𝑖 ∫   𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝛽𝑖(𝑞𝑖 − 𝑅𝑙𝑖)[𝑞𝑖𝑅 + 𝐷𝑖(𝑞𝑖 −𝑅𝑙𝑖)] 𝑞𝑖𝑅 2 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 +𝑤𝛽𝑖 + (𝐴𝑖 + 𝑛𝐹)𝛽𝑖 𝑄𝑖 + 𝑐𝑖 ∫   𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝛽𝑖𝐷𝑖(𝑞𝑖 −𝑅𝑙𝑖) 2 𝑞𝑖𝑅 2 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 − 𝑐𝑖 ∫   ∞ 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝛽𝑖(𝑞𝑖 − 𝑅𝑙𝑖) 𝑅 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 (16) (12) (13) (14) (15) (16) (16) dash et al./decis. mak. appl. manag. eng. (2022) 16 ∂2𝐴𝐸𝑃𝑖 ∂𝑝𝑖 2 = −2𝛽𝑖 − (ℎ𝑖 + 𝑐𝑖)∫   𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝛽𝑖 2(𝑞𝑖−𝑅𝑙𝑖) 2 𝑞𝑖𝑅 2 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 (17) since 𝛽𝑖,ℎ𝑖 and 𝑐𝑖 all are positive, therefore, it implies that 2βi + (hi +ci)∫ qi(r+di) rdi qi r βi 2(qi-rli) 2 qir 2 fl(li)dli > 0. therefore, the average expected profit function 𝐴𝐸𝑃𝑖 is concave in 𝑝𝑖 for given 𝑞𝑖 if the above condition satisfies. solution algorithm taking the best response from the manufacturer, the average expected profit of the 𝑖-th retailer can be optimized using the following solution algorithm. to optimize the expected average profit of the 𝑖-th retailer, we consider initial guess values to the decision variables of the remaining (𝑁 − 1) retailers. step 1: set 𝑘 = 1. step 2: set 𝑖 = 1 and 𝑞𝑗 = 𝑞𝑗 (𝑘−1) , 𝑝𝑗 = 𝑝𝑗 (𝑘−1) for all 𝑗 = 𝑖 + 1,𝑖 + 2, . . . . . ,𝑁. step 3: optimize 𝐴𝐸𝑃𝑖 taking 𝑛 and 𝜃 from the response functions of the manufacturer and 𝑞𝑗 = 𝑞𝑗 (𝑘−1) ,𝑝𝑗 = 𝑝𝑗 (𝑘−1) for all 𝑗 = 𝑖 +1,𝑖 + 2, . . . . . ,𝑁. set the optimal results as 𝑞𝑖 = 𝑞𝑖 (𝑘) and 𝑝𝑖 = 𝑝𝑖 (𝑘) . step 4: set 𝑖 = 𝑖 +1. step 5: optimize 𝐴𝐸𝑃𝑖 taking 𝑛 and 𝜃 from the manufacturer’s response functions and 𝑞𝑗 = 𝑞𝑗 (𝑘) , 𝑝𝑗 = 𝑝𝑗 (𝑘) for 𝑗 = 1,2, . . . , 𝑖 − 1 and 𝑞𝑗 = 𝑞𝑗 (𝑘−1) ,𝑝𝑗 = 𝑝𝑗 (𝑘−1) for 𝑗 = 𝑖 + 1, 𝑖 + 2, . . . . ,𝑁. set the optimal results as 𝑞𝑖 = 𝑞𝑖 (𝑘) and 𝑝𝑖 = 𝑝𝑖 (𝑘) . step 6: repeat steps 4 and 5 until 𝑖 = 𝑁. step 7: stop if 𝑞𝑗 (𝑘) = 𝑞𝑗 (𝑘−1) and 𝑝𝑗 (𝑘) = 𝑝𝑗 (𝑘−1) for all 𝑗 = 2,3, . . . . . ,𝑁 and consider 𝑞𝑗 (∗) = 𝑞𝑗 (𝑘) and 𝑝𝑗 (∗) = 𝑝𝑗 (𝑘) for all 𝑗 = 1,2,3, . . . . . ,𝑁. otherwise, set 𝑘 = 𝑘 +1 and repeat steps 2 to 6. step 8: evaluate the optimal values of 𝑛∗ and 𝜃∗ taking 𝑞𝑗 ∗ and 𝑝𝑗 ∗ for all 𝑗 = 1,2,3, . . . . . ,𝑁 . step 9: using these results, calculate optimal values of 𝐴𝐸𝑃𝑚 and 𝐴𝐸𝑃𝑠. 5.2. centralized model (cm) in this scenario, the manufacturer and all the retailers of the supply chain act jointly as a single decision maker. they determine the optimal selling prices of the product, greening improvement level, number of shipments and batch sizes in order to maximize the entire system profit rather than focusing on their individual profits. the average expected profit of the supply chain is 𝐴𝐸𝑃𝑠(𝑛,𝜃, 𝑞𝑖, 𝑝𝑖) = ∑ 𝑝𝑖𝐷𝑖 − 𝐴𝑣𝐷 𝑄 − ℎ𝑣 [ 𝐷∑  𝑁𝑖=1 𝑞𝑖 𝑅 + 𝑄 2 (1 − 𝐷 𝑅 )− ∑  𝑁𝑖=1 𝑞𝑖 2 2𝑄𝑖 ]𝑁𝑖=1 −∑  𝑁𝑖=1 ℎ𝑣𝐷𝑖𝜎𝑖 √2𝜋 − 𝐼𝜃2 − ∑ [ (𝐴𝑖+𝑛𝐹)𝐷𝑖 𝑄𝑖 + ℎ𝑖 2 [∫   𝑟𝑖 𝐷𝑖 0 [𝑞𝑖 + 2(𝑟𝑖 − 𝑁 𝑖=1 coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 17 𝐷𝑖𝑙𝑖)]𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 +∫   𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑟𝑖 𝐷𝑖 (𝑞𝑖−𝐷𝑖𝑙𝑖+𝑟𝑖) 2 𝑞𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖] + 𝑐𝑖 2 [∫   𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑟𝑖 𝐷𝑖 (𝐷𝑖𝑙𝑖−𝑟𝑖) 2 𝑞𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 +∫   ∞ 𝑟𝑖+𝑞𝑖 𝐷𝑖 [𝑞𝑖 + 2(𝐷𝑖𝑙𝑖 − 𝑞𝑖 −𝑟𝑖)]𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖]] (18) proposition 5. the average expected profit of the system is concave in 𝑛 for given θ, 𝑞𝑖 and 𝑝𝑖 if ℎ𝑣𝑠 2 < 2(𝐴𝑣𝐷 +𝑠𝑔), where 𝑔 = ∑ 𝑁 𝑖=1 𝐴𝑖𝐷𝑖 𝑞𝑖 and the optimal number of shipments is given by 𝑛∗ = √ 𝑅(2𝐴𝑉𝐷−ℎ𝑣𝑠 2+2𝑠𝑔) ℎ𝑣𝑠 2(𝑅−𝐷) (19) proof. considering 𝑛 as real, from equation (18), we derive the following partial derivatives: 𝜕𝐴𝐸𝑃𝑠 𝜕𝑛 = 𝐴𝑣𝐷 𝑛2𝑠 − ℎ𝑣𝑠 2 + ℎ𝑣𝑠𝐷 2𝑅 − ℎ𝑣𝑠 2𝑛2 + 𝑔 𝑛2 (20) 𝜕2𝐴𝐸𝑃𝑠 𝜕𝑛2 = − 2𝐴𝑣𝐷 𝑛3𝑠 + ℎ𝑣𝑠 𝑛3 − 2𝑔 𝑛3 where 𝑠 = ∑ 𝑞𝑖 𝑁 𝑖=1 (21) the average expected profit of the system will be concave in 𝑛, for given θ, 𝑞𝑖 and 𝑝𝑖, if ∂2aeps ∂n2 < 0, which implies that ℎ𝑣𝑠 2 < 2(𝐴𝑣𝐷 + 𝑠𝑔). if the above condition holds then the system profit function attains the maximum value with respect to 𝑛, and the optimal value of 𝑛 can be obtained by using the first order optimality condition i.e., 𝜕𝐴𝐸𝑃𝑠 𝜕𝑛 = 0. solving it for 𝑛, one can get the optimal number of shipments as 𝑛∗ = √ 𝑅(2𝐴𝑉𝐷−ℎ𝑣𝑠 2+2𝑠𝑔) ℎ𝑣𝑠 2(𝑅−𝐷) . for integer optimal value of 𝑛, 𝑛𝑜𝑝𝑡 = { ⌊𝑛∗⌋, 𝑖𝑓 𝐴𝐸𝑃𝑠(⌊𝑛 ∗⌋,𝜃,𝑞𝑖,𝑝𝑖 ) ≥ 𝐴𝐸𝑃𝑠(⌈𝑛 ∗⌉,𝜃,𝑞𝑖,𝑝𝑖) ⌈𝑛∗⌉, 𝑖𝑓 𝐴𝐸𝑃𝑠(⌊𝑛 ∗⌋,𝜃,𝑞𝑖,𝑝𝑖) ≤ 𝐴𝐸𝑃𝑠(⌈𝑛 ∗⌉,𝜃,𝑞𝑖,𝑝𝑖) proposition 6. for given values of 𝑛, 𝑞𝑖 and 𝑝𝑖, the average expected profit function of the supply chain is concave in 𝜃 if 2𝐼 + ∑𝑁𝑖=1 (ℎ𝑖 + 𝑐𝑖)∫ 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝛼𝑖 2(𝑞𝑖−𝑅𝑙𝑖) 2 𝑞𝑖𝑅 2 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 > 0. proof. from equation (18), we get 𝜕𝐴𝐸𝑃𝑠 𝜕𝜃 = −[ 𝐴𝑣 𝑄 + ℎ𝑣𝑠 𝑅 − ℎ𝑣𝑄 2𝑅 + (∑ ℎ𝑣𝜎𝑖 √2𝜋 𝑁 𝑖=1 )]𝑢 − 2𝐼𝜃 − ∑[−𝑝𝑖𝛼𝑖 + (𝐴𝑖 + 𝑛𝐹)𝛼𝑖 𝑄𝑖 𝑁 𝑖=1 +ℎ𝑖 ∫ 𝛼𝑖(𝑞𝑖−𝑅𝑙𝑖) 𝑅 𝑞𝑖 𝑅 0 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 +ℎ𝑖 ∫ 𝛼𝑖(𝑞𝑖−𝑅𝑙𝑖)[𝑞𝑖𝑅+𝐷𝑖(𝑞𝑖−𝑅𝑙𝑖)] 𝑞𝑖𝑅 2 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 +𝑐𝑖 ∫ 𝛼𝑖𝐷𝑖(𝑞𝑖−𝑅𝑙𝑖) 2 𝑞𝑖𝑅 2 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 − 𝑐𝑖 ∫ 𝛼𝑖(𝑞𝑖−𝑅𝑙𝑖) 𝑅 ∞ 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖] (22) (21) (22) dash et al./decis. mak. appl. manag. eng. (2022) 18 𝜕2𝐴𝐸𝑃𝑠 𝜕𝜃2 = −2𝐼 − ∑ (ℎ𝑖 + 𝑐𝑖)∫ 𝛼𝑖 2(𝑞𝑖−𝑅𝑙𝑖) 2 𝑞𝑖𝑅 2 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝑁 𝑖=1 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 (23) it can be easily seen that 𝜕2𝐴𝐸𝑃𝑠 𝜕𝜃2 < 0 if 2𝐼 + ∑𝑁𝑖=1 (ℎ𝑖 + 𝑐𝑖)∫ 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝛼𝑖 2(𝑞𝑖−𝑅𝑙𝑖) 2 𝑞𝑖𝑅 2 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 > 0. therefore, we can conclude that the average expected profit function of the supply chain system is concave in θ for given 𝑛, 𝑞𝑖 and 𝑝𝑖, if this condition holds. proposition 7. the average expected profit function of the supply chain system is concave in 𝑞𝑖 for given 𝑛, θ and 𝑝𝑖 if 2𝐴𝑣𝐷 𝑠3 + 2(𝐴𝑖 + 𝑛𝐹)𝐷𝑖 𝑞𝑖 3 + 𝑛(ℎ𝑖 + 𝑐𝑖) ∫ 𝐷𝑖 2𝑙𝑖 2 𝑞𝑖 3 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 > 0 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 proof. differentiating (18) with respect to 𝑞𝑖, we get 𝜕𝐴𝐸𝑃𝑠 𝜕𝑞𝑖 = 𝐴𝑣𝐷 𝑛𝑠2 − ℎ𝑣𝐷 𝑅 − 𝑛ℎ𝑣 2 (1 − 𝐷 𝑅 )+ ℎ𝑣 2𝑛 + (𝐴𝑖+𝑛𝐹)𝐷𝑖 𝑛𝑞𝑖 2 − ℎ𝑖 2 ∫ ( 𝑅+2𝐷𝑖 𝑅 ) 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 𝑞𝑖 𝑅 0 − ℎ𝑖 2 ∫ ( 𝑞𝑖 2(𝑅 + 𝐷𝑖) 2 − 𝑅2𝐷𝑖 2𝑙𝑖 2 𝑞𝑖 2𝑅2 ) 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 + 𝑐𝑖 2 ∫ ( 𝑅2𝐷𝑖 2𝑙𝑖 2 − 𝑞𝑖 2𝐷𝑖 2 𝑞𝑖 2𝑅2 ) 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 + 𝑐𝑖 2 ∫ ( 𝑅+2𝐷𝑖 𝑅 ) 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 ∞ 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 (24) 𝜕2𝐴𝐸𝑃𝑠 𝜕𝑞𝑖 2 = − 2𝐴𝑣𝐷 𝑛𝑠3 − 2(𝐴𝑖 + 𝑛𝐹)𝐷𝑖 𝑛𝑞𝑖 3 − ℎ𝑖 ∫ 𝐷𝑖 2𝑙𝑖 2 𝑞𝑖 3 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 −𝑐𝑖 ∫ 𝐷𝑖 2𝑙𝑖 2 𝑞𝑖 3 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 (25) now, the average expected profit of the entire supply chain is concave in 𝑞𝑖 for given 𝑛, θ and 𝑝𝑖, if 𝜕2𝐴𝐸𝑃𝑠 𝜕𝑞𝑖 2 < 0, which gives 2𝐴𝑣𝐷 𝑠3 + 2(𝐴𝑖+𝑛𝐹)𝐷𝑖 𝑞𝑖 3 + 𝑛(ℎ𝑖 + 𝑐𝑖)∫ 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝐷𝑖 2𝑙𝑖 2 𝑞𝑖 3 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 > 0. proposition 8. the average expected profit function of the supply chain system is concave in 𝑝𝑖 for given 𝑛, 𝜃 and 𝑞𝑖 if 2𝛽𝑖 +(ℎ𝑖 +𝑐𝑖)∫ 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝛽𝑖 2(𝑞𝑖−𝑅𝑙𝑖) 2 𝑞𝑖𝑅 2 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 > 0. proof. differentiating (18) with respect to 𝑝𝑖, we get (23) (24) (25) (26) coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 19 𝜕𝐴𝐸𝑃𝑠 𝜕𝑝𝑖 = 𝐷𝑖 −𝛽𝑖𝑝𝑖 + 𝐴𝑣𝛽𝑖 𝑄 + ℎ𝑣𝛽𝑖𝑠 𝑅 − ℎ𝑣𝑄𝛽𝑖 2𝑅 + ℎ𝑣𝛽𝑖𝜎𝑖 √2𝜋 + (𝐴𝑖 + 𝑛𝐹)𝛽𝑖 𝑄𝑖 + ℎ𝑖 ∫ 𝛽𝑖(𝑞𝑖 −𝑅𝑙𝑖) 𝑅 𝑞𝑖 𝑅 0 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 +ℎ𝑖 ∫ 𝛽𝑖(𝑞𝑖 − 𝑅𝑙𝑖)[𝑞𝑖𝑅 + 𝐷𝑖(𝑞𝑖 − 𝑅𝑙𝑖)] 𝑞𝑖𝑅 2 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 + 𝑐𝑖 ∫ 𝛽𝑖𝐷𝑖(𝑞𝑖−𝑅𝑙𝑖) 2 𝑞𝑖𝑅 2 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 (26) −𝑐𝑖 ∫ 𝛽𝑖(𝑞𝑖−𝑅𝑙𝑖) 𝑅 ∞ 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 𝜕2𝐴𝐸𝑃𝑠 𝜕𝑝𝑖 2 = −2𝛽𝑖 − (ℎ𝑖 + 𝑐𝑖)∫ 𝛽𝑖 2(𝑞𝑖−𝑅𝑙𝑖) 2 𝑞𝑖𝑅 2 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 (27) the average expected profit function of the entire supply chain is concave in 𝑝𝑖 for given 𝑛, 𝜃 and 𝑞𝑖, if 𝜕2𝐴𝐸𝑃𝑠 𝜕𝑝𝑖 2 < 0 i.e., if 2𝛽𝑖 + (ℎ𝑖 + 𝑐𝑖)∫ 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝛽𝑖 2(𝑞𝑖−𝑅𝑙𝑖) 2 𝑞𝑖𝑅 2 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 > 0. 5.3. coordinated model (com) supply chain players agree to accept the joint-decision making policy only if it provides a better profit than the decentralized model scenario. to motivate supply chain members to make integrated decisions, an incentive strategy is required. in this section, we propose a coordination mechanism between the manufacturer and the retailers, which motivates the members to accept the integrated decision-making policy. in this coordination mechanism, the manufacturer requests the retailers to decide their optimal batch sizes (𝑞𝑖) and retail prices (𝑝𝑖) according to the centralized policy and, in return, the manufacturer decreases his wholesale price (𝑤). suppose that the manufacturer offers the price discount scheme to the 𝑖-th retailer as follows: 𝑤 = { 𝑤, 𝑖𝑓 𝑞𝑖 < 𝑞𝑖 𝑐∗ 𝑤(1 −𝜙𝑖), 𝑖𝑓 𝑞𝑖 ≥ 𝑞𝑖 𝑐∗ (28) for this price discount scheme, the average expected profit of the 𝑖-th retailer becomes 𝐴𝐸𝑃𝑖 𝑐𝑜(𝑞𝑖 𝑐,𝑝𝑖 𝑐,𝜙𝑖) = 𝑝𝑖 𝑐𝐷𝑖 𝑐 −(1 −𝜙𝑖)𝑤𝐷𝑖 𝑐 − (𝐴𝑖+𝑛 𝑐𝐹)𝐷𝑖 𝑐 𝑄𝑖 𝑐 − [ ℎ𝑖 2 ∫ [𝑞𝑖 𝑐 + 2(𝑟𝑖 − 𝑟𝑖 𝐷 𝑖 𝑐 0 𝐷𝑖 𝑐𝑙𝑖)]𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 +ℎ𝑖 ∫ (𝑞𝑖 𝑐 −𝐷𝑖 𝑐 𝑙𝑖+𝑟𝑖) 2 2𝑞𝑖 𝑐 𝑟𝑖+𝑞𝑖 𝑐 𝐷 𝑖 𝑐 𝑟𝑖 𝐷 𝑖 𝑐 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖]− [ 𝑐𝑖 2 ∫ (𝐷𝑖 𝑐 𝑙𝑖−𝑟𝑖) 2 𝑞𝑖 𝑐 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 + 𝑟𝑖+𝑞𝑖 𝑐 𝐷 𝑖 𝑐 𝑟𝑖 𝐷 𝑖 𝑐 𝑐𝑖 2 ∫ [𝑞𝑖 𝑐 + 2(𝐷𝑖 𝑐𝑙𝑖 − 𝑞𝑖 𝑐 − 𝑟𝑖)] ∞ 𝑟𝑖+𝑞𝑖 𝑐 𝐷 𝑖 𝑐 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖] (29) and the average expected profit of the manufacturer becomes (27) (28) (29) dash et al./decis. mak. appl. manag. eng. (2022) 20 𝐴𝐸𝑃𝑚 𝑐𝑜(𝑛𝑐,𝜃𝑐,𝜙𝑖) = ∑(1 −𝜙𝑖)𝑤𝐷𝑖 𝑐 − 𝐴𝑣𝐷 𝑐 𝑄𝑐 − ℎ𝑣[ 𝐷𝑐 ∑ 𝑞𝑖 𝑐𝑁 𝑖=1 𝑅 + 𝑄𝑐 2 (1− 𝐷𝑐 𝑅 ) 𝑁 𝑖=1 −∑ 𝑞𝑖 𝑐 2𝑛𝑐 ] −∑ ℎ𝑣𝐷𝑖 𝑐𝜎𝑖 √2𝜋 𝑁 𝑖=1 𝑁 𝑖=1 − 𝐼𝜃2 (30) proposition 9. the minimum value of 𝜙𝑖 for which the 𝑖-th retailer accepts the coordination mechanism is 𝜙𝑖 𝑚𝑖𝑛 = (𝑝𝑖 𝑑 𝐷𝑖 𝑑 −𝑝𝑖 𝑐 𝐷𝑖 𝑐 )−𝑤(𝐷𝑖 𝑑 −𝐷𝑖 𝑐 )−∆𝑑+∆𝑐 𝑤𝐷𝑖 𝑐 where, ∆= (𝐴𝑖 +𝑛𝐹)𝐷𝑖 𝑄𝑖 + ℎ𝑖 2 [∫ [𝑞𝑖 + 2(𝑟𝑖 − 𝐷𝑖𝑙𝑖)] 𝑟𝑖 𝐷𝑖 0 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 +∫ (𝑞𝑖 − 𝐷𝑖𝑙𝑖 + 𝑟𝑖) 2 𝑞𝑖 𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑟𝑖 𝐷𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖] +[ 𝑐𝑖 2 ∫ (𝐷𝑖𝑙𝑖 − 𝑟𝑖) 2 𝑞𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 + 𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑟𝑖 𝐷𝑖 𝑐𝑖 2 ∫ [𝑞𝑖 + 2(𝐷𝑖𝑙𝑖 − 𝑞𝑖 − 𝑟𝑖)]𝑓𝐿 ∞ 𝑟𝑖+𝑞𝑖 𝐷𝑖 (𝑙𝑖)𝑑𝑙𝑖] proof. the retailer's goal in engaging in the coordination is to find the minimum discount level so that his profit is more or equal to the profit in the decentralized situation. so, 𝐴𝐸𝑃𝑖 𝑐𝑜(𝑞𝑖 𝑐,𝑝𝑖 𝑐,𝜙𝑖) ≥ 𝐴𝐸𝑃𝑖 𝑑(𝑞𝑖 𝑑,𝑝𝑖 𝑑) (31) solving the inequality (31), we get 𝜙𝑖 ≥ (𝑝𝑖 𝑑 𝐷𝑖 𝑑 −𝑝𝑖 𝑐 𝐷𝑖 𝑐 )−𝑤(𝐷𝑖 𝑑 −𝐷𝑖 𝑐 )−∆𝑑+∆𝑐 𝑤𝐷𝑖 𝑐 (32) therefore, if the wholesale price discount offered by the manufacturer does not satisfy the above condition, the 𝑖-th retailer will not accept the contract. so, to motivate the 𝑖-th retailer, the manufacturer should give at least 𝜙𝑖 discount level given by 𝜙𝑖 𝑚𝑖𝑛 = (𝑝𝑖 𝑑 𝐷𝑖 𝑑 −𝑝𝑖 𝑐 𝐷𝑖 𝑐 )−𝑤(𝐷𝑖 𝑑 −𝐷𝑖 𝑐 )−∆𝑑+∆𝑐 𝑤𝐷𝑖 𝑐 (33) proposition 10. the maximum discount level offered by the manufacturer to the 𝑖-th retailer is given by 𝜙𝑖 𝑚𝑎𝑥 = 𝑤(𝐷𝑖 𝑐 −𝐷𝑑)−∇𝑐+∇𝑑+∑ (1−𝜙𝑗)𝑤𝐷𝑗 𝑐𝑁 𝑗=1 𝑗≠𝑖 𝑤𝐷𝑖 𝑐 (34) coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 21 where,∇= 𝐴𝑣𝐷 𝑄 + ℎ𝑣 [ 𝐷∑ 𝑞𝑖 𝑁 𝑖=1 𝑅 + 𝑄 2 (1 − 𝐷 𝑅 ) − ∑ 𝑞𝑖 2𝑛 𝑁 𝑖=1 ]+ ∑ ℎ𝑣𝐷𝑖𝜎𝑖 √2𝜋 + 𝐼𝜃2 𝑁 𝑖=1 proof. the manufacturer will offer the price discount scheme if he/she gains more profit after giving price discount to all the retailers in this coordination than the decentralized scenario. so, if the manufacturer provides a 𝜙𝑖 discount level to the 𝑖-th retailer, then 𝐴𝐸𝑃𝑚 𝑐𝑜(𝑛𝑐,𝜃𝑐,𝜙𝑖) ≥ 𝐴𝐸𝑃𝑚 𝑑(𝑛𝑑,𝜃𝑑) (35) simplifying (35), we get, 𝜙𝑖 ≤ 𝑤(𝐷𝑖 𝑐 −𝐷𝑑)−∇𝑐+∇𝑑+∑ (1−𝜙𝑗)𝑤𝐷𝑗 𝑐𝑁 𝑗=1 𝑗≠𝑖 𝑤 𝐷𝑖 𝑐 (36) so, if the manufacturer gives 𝜙𝑖% price discount to the 𝑖-th retailer, then the maximum allowable discount level for the manufacturer will be 𝜙𝑖 𝑚𝑎𝑥 = 𝑤(𝐷𝑖 𝑐 −𝐷𝑑)−∇𝑐+∇𝑑+∑ (1−𝜙𝑗)𝑤𝐷𝑗 𝑐𝑁 𝑗=1 𝑗≠𝑖 𝑤 𝐷𝑖 𝑐 from propositions (9) and (10), it can be observed that, the 𝑖-th retailer will accept the discount offer for all 𝜙 ≥ 𝜙𝑖 𝑚𝑖𝑛 . therefore, all the 𝑁 retailers will accept the discount scheme (28) if 𝜙 ≥ 𝜙𝑚𝑖𝑛, where 𝜙𝑚𝑖𝑛 = 𝑚𝑎𝑥{𝜙1 𝑚𝑖𝑛 ,𝜙2 𝑚𝑖𝑛 ,𝜙3 𝑚𝑖𝑛 , . . . . . . . . . . ,𝜙𝑁 𝑚𝑖𝑛 } and the manufacturer will provide this price discount only if 𝜙 ≤ 𝜙𝑚𝑎𝑥 = min {𝜙1 𝑚𝑎𝑥,𝜙2 𝑚𝑎𝑥,𝜙3 𝑚𝑎𝑥,……. ,𝜙𝑁 𝑚𝑎𝑥} hence, for all 𝜙 in [𝜙𝑚𝑖𝑛,𝜙𝑚𝑎𝑥] the coordination through the price discount scheme (28) will result better profit level for both the manufacturer and the retailers than the decentralized scenario. since the manufacturer sells the product to all the retailers at the same wholesale price, therefore, we assume that he/she offers the same price discount ratio 𝜙 to each retailer. 6. numerical analysis in this section, we consider three numerical examples to analyze the behaviour of our proposed model and its applicability. here we focus on the scenario where one manufacturer is trading with two retailers. example 1: the following set of parameter-values presented in table 2 are considered to demonstrate the proposed model numerically. as it is difficult to get access to the actual industrial data, some of the parameter-values are taken from hoque (2013) and the rest are hypothetical. the p.d.f. of lead time (𝑙𝑖) of the 𝑖-th retailer is 𝑓𝐿(𝑙𝑖) = 1 √2π𝜎𝑖 𝑒 − 1 2𝜎 𝑖 2(𝑙𝑖− 𝑟𝑖 𝐷𝑖 ) 2 . dash et al./decis. mak. appl. manag. eng. (2022) 22 table 2. set of parameter-values for example 1 parameter value parameter value 𝑅 3000 units/ year 𝑎1 1500 units / year 𝐴𝑣 $400 /set up 𝑎2 1500 units / year 𝐴1 $40 /order 𝛽1 4 𝐴2 $45 / order 𝛽2 4.5 𝑤 $100 / unit 𝛼1 2 𝐹 $10 /shipment 𝛼2 1.5 ℎ𝑣 $3.5/ unit / year 𝜎1 0.12 ℎ1 $5.8 / unit / year 𝜎2 0.13 ℎ2 $5 / unit / year 𝐼 $40 𝑐1 $7 / unit / year 𝑁 2 𝑐2 $7 / unit / year figure 4. concavity of average expected profit function of the first retailer figure 5. concavity of average expected profit function of the second retailer as shown in figures 4 and 5, for given parameter-values, the average expected profit functions of both the retailers are found to be concave with respect to the batch sizes and retail prices of the product. the optimal results are obtained using the computational software mathematica 10.0 with the command findmaximum. from the numerical results given in table 3, we observe that the optimal order quantity, retail price, greening level of the product and number of shipments decided in the centralized scenario gives more system profit than that obtained in the decentralized scenario. in the centralized scenario, both the retailers can sell the product to the end customers at a cheaper price than the decentralized case. since the customer demand is assumed to be price sensitive, the lower priced product attracts more customers. coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 23 table 3. optimal results for different models models 𝑛∗ θ* 𝑞1 ∗ (unit) 𝑞2 ∗ (unit) 𝑝1 ∗ ($/unit) 𝑝2 ∗ ($/unit) ϕmin 𝜙𝑚𝑎𝑥 ϕ 𝐴𝐸𝑃1 ∗ ($/year) aep2 * ($/year) 𝐴𝐸𝑃𝑚 ∗ ($/year) 𝐴𝐸𝑃𝑠 ∗ ($/year) dm com cm 4 6 6 4.36 7.87 7.87 91.98 71.77 71.77 111.79 79.96 79.96 238.74 189.82 189.82 217.53 168.34 168.34 0.14 0.27 0.205 76577 83132 61799 66707 105584 115684 243960 265523 265523 also, the manufacturer can produce more greener product by making the optimal decisions jointly. as the product's greening level has a positive impact on customer demand, customer demand in the centralized case is considerably higher than in the decentralized case, and all the retailers increase their order quantities. as a result, joint-decision making generates a higher system profit than the separate profit optimization. figure 6. price discount rate vs. average expected profit it is also observed from table 3 that the channel coordination can be achieved through the price discount mechanism between the manufacturer and the retailers. the optimal results of the models reflect that embracing the price discount coordination mechanism boosts not only the total system profit but also the profits of individual supply chain members. for the first retailer, the minimum discount ratio to undertake the coordination mechanism is obtained as 0.12 and for the second retailer, it is 0.14. it is clear that if the manufacturer offers 12% discount, the first retailer will accept the offer but the second retailer will not, as it will cause a loss to him. therefore, to motivate both the retailers for participating in the coordination, the manufacturer has to give at least 14% price discount. again, the maximum discount ratio for which the manufacturer does not face any loss is obtained as 0.27. so, the manufacturer can dash et al./decis. mak. appl. manag. eng. (2022) 24 offer each retailer a maximum discount of 27%. therefore, the win-win situation which occurs in the interval [𝜙𝑚𝑖𝑛,𝜙𝑚𝑎𝑥] is appeared as [0.14,0.27]. for any value of 𝜙 in this interval, the price discount mechanism becomes profitable for the manufacturer and both the retailers than the decentralized scenario. for 𝜙 lying in the interval [0.14,0.27], the average expected profits of the first and second retailers vary within the intervals [$78215,$88049] and [$61799,$71610], respectively and the average expected profit of the manufacturer lies within the interval [$105584,$125503]. in all cases, the average expected system profit remains $265523 i.e., our suggested coordination method effectively achieves channel coordination and results in the supply chain members sharing extra profit that occurs in the centralized scenario. naturally, whenever the value of 𝜙 increases from 𝜙𝑚𝑖𝑛, the retailers profitability increases gradually and attains their maximum profits at 𝜙𝑚𝑎𝑥 while the manufacturer’s profit decreases, and at 𝜙𝑚𝑎𝑥, the manufacturer attains the same profitability as that of the decentralized case. this fact is plotted in figure 6. the supply chain members can fix the value of 𝜙 through bargaining. here we take the value of 𝜙 as the mean of the feasible interval [0.14,0.27] i.e., 0.205. example 2: we consider the set of parameter values given in table 4 to demonstrate the model, and the optimal results thus obtained are provided in table 5. table 4. set of parameter-values for example 2 parameter value parameter value r 4000 units/ year 𝑎1 2000 units / year 𝐴𝑣 $500 /set up 𝑎2 1800units / year 𝐴1 $50 /order 𝛽1 4.2 𝐴2 $50/ order 𝛽2 5 𝑤 $90/ unit 𝛼1 3 𝐹 $15/shipment 𝛼2 2.5 ℎ𝑣 $3/ unit / year 𝜎1 0.12 ℎ1 $6/ unit / year 𝜎2 0.13 ℎ2 $5.5/ unit / year 𝐼 $30 𝑐1 $7.4 / unit / year 𝑁 2 𝑐2 $7.4 / unit / year table 5 shows that, when compared to a decentralized system, integrated decision making provides higher supply chain profit. both the order quantity of each retailer as well as the product's greening improvement level increase in the centralized scenario compared to the decentralized scenario. in addition, the product's retail price falls at both the retailers. as a consequence, customers are enticed by a greener product at a lesser cost, which significantly increases market demand. in the coordinated model, the minimum wholesale price discount ratios for the two retailers are obtained as 2% and 7%, respectively, while the maximum allowable price discount ratio for the manufacturer is 17%. therefore, for any price discount lying in the interval [7%,17%], a win-win situation arises, i.e., the wholesale price contract benefits every member of the supply chain. the value of 𝜙 is taken as the mean of this feasible interval [7%,17%] i.e., 12%. coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 25 table 5. optimal results for different models models 𝑛∗ 𝜃∗ 𝑞1 ∗ (unit) 𝑞2 ∗ (unit) 𝑝1 ∗ ($/unit) 𝑝2 ∗ ($/unit) ϕmin 𝜙𝑚𝑎𝑥 ϕ 𝐴𝐸𝑃1 ∗ ($/year) 𝐴𝐸𝑃2 ∗ ($/year) 𝐴𝐸𝑃𝑚 ∗ ($/year) 𝐴𝐸𝑃𝑠 ∗ ($/year) dm com cm 4 5 5 8.22 19.94 19.94 132.42 112.12 112.12 161.82 124.16 124.16 286.18 245.58 245.58 227.19 185.35 185.35 0.07 0.17 0.12 161024 170783 93638 97731 131443 140097 386105 408611 408611 from table 5, it can be noticed that, by accepting the wholesale price discount contract, the profits of the two retailers are increased by 6% and 4%, respectively. furthermore, the manufacturer earns about 7% more profit from this contract. 6.1. a comparative study with existing literature in this section, we attempt to compare the findings of our study to some previous research. sarkar et al. (2020b) developed a single-vendor single-buyer model with equal-sized batch shipment policy and price-dependent demand in this direction. they did, however, take into account variable backorder and the inspection process, that are not considered in this study. furthermore, their model didn’t take into account for stochastic lead time and greening investment. to compare the proposed model to sarkar et al. (2020b), common parameter values from sarkar et al. (2020b) are used, while the remaining parameter values are chosen at random. the proposed model is compared to sarkar et al.'s (2020b) model in two different situations: without greening investment and with greening investment. the parameter values considered are given in table 6. table 6. set of parameter-values for comparative study parameter value parameter value 𝑎1 11,000 units / year 𝛽1 320 𝐴𝑣 $200/set up 𝛼1 3 𝐴1 $20/order 𝜎1 0.02 𝑤 $10/ unit 𝐼 $80 𝐹 $5/shipment 𝑁 1 ℎ𝑣 $2/ unit / year 𝐷 𝑅 0.4 ℎ1 $5/ unit / year 𝑐1 $7.5/ unit / year for the case of without green sensitivity of the customer demand, we set 𝛼1 = 0, 𝐼 = 0. figure 7 shows a comparative graphical representation of the average expected supply chain profit. their centralized model obtains optimal batch size as 279 units, optimal number of shipments as 5, optimal retail price as $17 and optimal profit of the entire supply chain as $92021. whereas our proposed model without green investment results the optimal batch size as 155.2 units, optimal number of shipments as 9, optimal retail price as $17.30, and the average expected supply chain profit as $92287. furthermore, the proposed model with stochastic lead time and greening investment provides the optimal batch size as 223.26 units, optimal number of shipments as 8, optimal retail price as $28.24, optimal green level as 35.07 and the dash et al./decis. mak. appl. manag. eng. (2022) 26 average expected supply chain profit as $152186. as a conclusion of the above numerical results, it is apparent that adding the stochastic lead time and greening investment strategy makes the supply chain significantly more profitable. figure 7. comparison with existing literature 7. sensitivity analysis in order to explore the impact of model parameters on the optimal decisions as well as the average expected system profit, in this section we vary one parametervalue at a time while keeping other parameter-values unchanged in example 1. the results are shown in table 7 from which the following conclusions can be drawn: from table 7 and figure 8, a significant change in overall profit of the system under the price discount coordination mechanism is observed for higher basic market demand. the first retailer can charge a higher price for the product whenever the customer demand increases at his side. this is because the first retailer compensates the effect of higher price by the higher market demand. he places order for more quantity from the manufacturer. consequently, the profit of the first retailer as well as the manufacturer increases significantly. the changes in the order quantity and retail price of the product for the second retailer are almost negligible. as a result, the overall profit of the system increases. similar scenario occurs whenever the market demand increases at the second retailer’s side. coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 27 figure 8. average expected profit vs. 𝑎2 table 7. sensitivity analysis of the parameters 𝑎1, 𝛽1, 𝛼1 and 𝐹 parameters values n* θ* 𝑞1 ∗ (unit) q2 * (unit) 𝑝1 ∗ ($/unit) 𝑝2 ∗ ($/unit) 𝐴𝐸𝑃1 𝑐𝑜 ($/year) 𝐴𝐸𝑃2 𝑐𝑜 ($/year) 𝐴𝐸𝑃𝑚 𝑐𝑜 ($/year) 𝐴𝐸𝑃𝑠 𝑐𝑜 ($/year) 𝑎1 1300 1400 1500 1600 1700 6 6 6 6 6 7.24 7.55 7.87 8.19 8.50 65.20 68.50 71.77 75.10 78.20 78.88 79.41 79.96 80.55 81.17 164.687 177.253 189.82 202.387 214.955 168.26 168.301 168.34 168.383 168.424 55569 68719 83132 98812 115761 66592 66649 66707 66765 66823 108057 111879 115684 119486 123285 230219 247242 265523 285065 305865 𝛽1 3 3.5 4 4.5 5 6 6 6 6 6 9.47 8.55 7.87 7.34 6.92 71.90 71.83 71.77 71.58 71.48 80.07 80.01 79.96 79.57 79.90 253.508 217.082 189.82 168.652 151.736 168.607 168.455 168.34 168.253 168.183 131649 103873 83132 67049 54229 67014 66838 66707 66605 66524 114825 115343 115684 115917 116088 313488 286055 265523 249571 236841 α1 1.5 2 2.5 3 3.5 6 6 6 6 6 6.67 7.87 9.08 10.31 11.56 71.58 71.77 72.01 72.27 72.58 79.87 79.96 80.07 80.18 80.30 189.104 189.82 190.69 191.718 192.908 168.143 168.34 168.543 168.747 168.954 82276 83132 84177 85419 86868 66478 66707 66940 67177 67418 116083 115684 115210 114654 114012 264837 265523 266327 267250 268298 f 0 10 20 30 40 11 6 4 4 3 7.87 7.87 7.87 7.87 7.87 38.99 71.77 97.95 103.46 124.43 43.67 79.96 111.89 116.70 144.46 189.752 189.82 189.89 189.911 189.98 168.282 168.34 168.41 168.421 168.48 83199 83132 83084 83022 82983 66751 66707 66680 66626 66605 115816 115684 115588 115563 115496 265766 265523 265352 265211 265084 figure 9 depicts how the price sensitivity of the consumer demand affects the decision variables and the system profit. the figure shows that when the price sensitivity of consumer demand increases, the greater price of the product influences the customers' choice of alternatives. as a result, if customer demand becomes more price sensitive, the corresponding retailers lower their product prices to meet market demand, reducing the product's greenness. figure 9 and table 7 show that, under the coordination scheme, the average expected profit of both retailers and the total system profit decrease at a decreasing rate as price elasticity increases. dash et al./decis. mak. appl. manag. eng. (2022) 28 figure 9. average expected profit vs. 𝛽2 table 7 shows the effect of customers' environmental awareness on optimal decisions and supply chain members’ profitability. when the values of 𝛼1 and 𝛼2 increase, customers are more concerned about the environmental performance of the product and they are willing to spend more for environmentally friendly products. in such a scenario, to satisfy the customers requirement, the manufacturer increases the greening level of the product. this fact is presented in figure 10. figure 10. product’s greening level vs. 𝛼2 however, the higher greening level increases the expense of the manufacturer. so, the average expected profit of the manufacturer gradually decreases. on the other hand, the retailers can enhance the retail price of the product and achieve higher profitability with higher greener product. it is further observed that the average expected system profit increases for greater values of 𝛼1 and 𝛼2. coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 29 figure 11. product’s greening level vs. 𝐼 table 7 illustrates the effect of the greening investment on the coordinated profit of the supply chain members and the profit of the entire supply chain. figure 11 shows that the product's greening level falls rapidly while 𝐼 increases. when 𝐼 increases, the manufacturer produces lower greener product in order to curb his expenditure but it makes a negative impact on customer demand. so, for higher 𝐼, the profitability of the retailers decreases and the average expected profit of the entire supply chain also decreases gradually. the effect of the transportation cost is found to be negligible on the supply chain’s profitability. if we ignore the transportation cost then the optimal number of shipments is obtained as 11. as the transportation cost increases, the optimal number of shipments declines from 11. figure 12 reflects that the system profit decreases at a diminishing rate as f increases. figure 12. average expected profit vs. 𝐹 dash et al./decis. mak. appl. manag. eng. (2022) 30 8. managerial insights from the numerical study and sensitivity analysis of our proposed model, the following key managerial insights are derived: (1) business managers can improve the sales volume and economic efficiency by adopting green manufacturing technologies and suitable coordination scheme. though, from the point of view of social welfare, it is always desirable to produce green products, the firms should estimate the profitable growth before adopting green manufacturing. from the outcomes of sensitivity analysis, it is evident that environmental awareness of the consumers and greening investment play a crucial role in the profitability of the supply chain members. by participating in the proposed price discount coordination mechanism, the business managers can improve the greening quality of the product to a remarkable higher level. it not only increases their profits but also maintains their social responsibility and increases their reputation in the business market for adopting such green initiative. (2) the proposed price discount scheme is capable of coordinating the supply chain. under this mechanism, the manufacturer reduces his wholesale price and increases greening level of the product and encourages the retailers to set their prices and ordering quantities according to the centralized model. this improves the economic level of all members. moreover, by participating in such coordination, the end customers get more eco-friendly product at a cheaper price than if they used an individual optimization strategy. retailers should also remember that when consumers' price sensitivity is too high, they should lower their sales price to retain profitability. (3) the business managers may not always agree to adopt joint decision-making process even though it yields higher profit for the entire supply chain but it may not be profitable for all the chain members. to convince the members to make coordination, such price discount scheme is very effective as in this scheme increment of each member’s profitability is guaranteed. all the members could enjoy the coordination agreement as it is beneficial both socially and economically. (4) the delivery of the order quantity may not reach to the retailer’s end in time due to various reasons such as variation in transportation time, inspection time, loading and unloading times, etc. therefore, the business managers should understand the stochastic nature of the lead time and account for all possibilities of early arrival, on time arrival, and late arrival to conduct the business efficiently. 9. concluding remarks in this study, we have designed a two-level supply chain model consisting of a single manufacturer and multiple retailers. to develop a realistic model, the lead time between placing an order and receiving its delivery is taken to be stochastic in nature. the retailers face a price sensitive demand from the end customers. the customer demand is also affected by the greening improvement level of the product as determined by the manufacturer. we have studied the decentralized model where supply chain members optimize their own profits without worrying about the profit of others. stackelberg gaming approach is used where the retailers are assumed to act as the leader and the manufacturer as the follower. a solution algorithm is suggested to find the optimal solution of the proposed model. the performance of the whole supply chain is also investigated under integrated decision-making model. though the entire supply chain experiences a better economical and environmental performances in the centralized scenario but it may not be beneficial for all the members coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 31 individually. since the retail price of the product is decided by the retailers and the manufacturer determines the greening level of the product, and both these factors influence the customers demand, it is therefore essential to make these decisions in an efficient and coordinated manner which enriches the profit levels of each members. a price discount mechanism has been proposed to convince the supply chain members to make decisions in a coordinated manner. the maximum and minimum satisfactory discount rates are found so that all the members become interested for participating in this price discount coordination. this coordination mechanism is effective in both cases whether the market demand is high or low. there are some limitations of this study and the present model can be extended in many directions to further enhance the scope of our study. it is widely adopted in the literature but the policy of equal sized batch shipment is very limited in nature, and it may not be always possible to supply the order quantities of all the retailers in some integer number of equal sized batches. so, it would be more realistic to consider a combined equal and unequal sized batch shipment policy (hoque, 2013). another limitation of this study is that it is based on a single product being traded between the manufacturer and the retailers. to simulate a real-world scenario, it can be expanded to include many items (barman et al., 2021a) and multiple manufacturers. another shortcoming of our study is the consideration of complete backlogging strategy. it is desirable to consider partially backlogging of shortages for a more realistic approach (duary et al., 2022). in our study, we have considered constant production rate, perfect production system at the manufacturer. one can enrich the study by taking into account variable production rate (sarkar et al., 2018) and/or imperfect production system (sepehri and gholamian, 2022). the competition between the retailers will be another interesting research idea (mondal and giri, 2020). our developed model can be modified by considering bargaining between manufacturer and retailers to share the profits among all the members (nouri et al., 2018). in our study, we have proposed a price discount coordination scheme. it would be interesting to employ other contracts such as greening cost sharing contract between the manufacturer and the retailers (giri and dash, 2022). consideration of set up cost reduction investment (sarkar et al., 2017), and promotional effort (ebrahimi et al., 2019) would also be fruitful extensions of this model. author contributions: anamika dash conceptualization, methodology, software, validation, formal analysis, investigation, resources, data curation, writing—original draft preparation, funding acquisition. bibhas c. giri conceptualization, methodology, formal analysis, investigation, writing—review and editing, visualization, supervision. ashis kumar sarkar conceptualization, investigation, supervision. all authors have read and agreed to this version of the manuscript. funding: this research was funded by the department of science and technology, government of india (grant number if170698). data availability statement: the authors confirm that the data supporting the findings of this study are available within the article. acknowledgments: the authors would like to thank the editor and the reviewers for their comments which led to considerable improvement in this article. dash et al./decis. mak. appl. manag. eng. 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(2021). optimal control of carbon emission reduction strategies in supply chain with wholesale price and consignment contract. environmental science and pollution research, 28(43), 61707-61722. https://doi.org/10.1007/s11356-021-15080-1 © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 5, issue 2, 2022, pp. 241-259. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0324062022e e-mail addresses: mehrdad.ra@gmail.com (m. rasoulzadeh), s.a.edalatpanah@aihe.ac.ir (s. a. edalatpanah), mohammad.fallah43@yahoo.com (m. fallah), najafi1515@yahoo.com (s. e. najafi) a multi-objective approach based on markowitz and dea cross-efficiency models for the intuitionistic fuzzy portfolio selection problem mehrdad rasoulzadeh1, seyyed ahmad edalatpanah2, mohammad fallah1, and seyyed esmaeil najafi3 1 department of industrial engineering, central tehran branch, islamic azad university, tehran, iran. 2 department of applied mathematics, ayandegan institute of higher education, tonekabon, iran 3 department of industrial engineering, science and research branch, islamic azad university, tehran, iran. received: 11 april 2022; accepted: 23 june 2022; available online: 3 july 2022. original scientific paper abstract: nowadays, investors' main concerns are choosing the best portfolio so that the highest possible investment return can be achieved by accepting the least risk. in this regard, the classical markowitz model is one of the most widely used models which helps investors get closer to their goals. data envelopment analysis (dea) is also a practical technique that can analyze the efficiency of firms. few models can address companies' internal performance simultaneously in addition to considering the goals of markowitz models. also, we study the return and price fluctuations of assets in the market with the intuitionistic fuzzy numbers for the first time. therefore, in this paper, we combine all these tools with returns of intuitionistic fuzzy numbers, proposing a new combined markowitz and the cross dea models. furthermore, to get the best portfolio of assets, this model obtains the efficiency of all companies and, simultaneously, fully covers all constraints of the markowitz model. to show the model's practicality, we studied a case study based on information from 50 active enterprises on tehran stock exchange. we solved the proposed model using the non-dominated sorting genetic algorithm ii (nsga-ii). the obtained results and the comparisons made with the existing approaches show the effectiveness of the proposed model. * corresponding author mailto:mehrdad.ra@gmail.com mailto:s.a.edalatpanah@aihe.ac.ir mailto:mohammad.fallah43@yahoo.com mailto:najafi1515@yahoo.com rasoulzadeh et al./decis. mak. appl. manag. eng. 5 (2) (2022) 241-259 242 key words: markowitz mean-variance model, data envelopment analysis (dea), intuitionistic fuzzy set, cross-efficiency. 1. introduction an increment in wealth is always the most imperative goal for every investment. increasing wealth in any investment is entwined with the two concepts of risk and returns. risk and returns have a direct correlation; the higher the returns expectations, the risk of investment will also be accompanied by a more elevated risk. investors have always sought to utilize varied available models to select investments with maximum returns and the lowest risk. several types of research have been performed so far in this respect to seek the optimum combination of assets by using mathematical models. in a multi-objective optimization model, markowitz (1952) was capable of simultaneously selecting the highest returns and lowest risk in the mathematical approach, through which the maximum stock investment portfolio came into effect. in this model, the investor chooses from amidst the responses present on the efficient frontier, where there are responses in the most desirable conditions possible, from the viewpoint of risk and returns, thus, selecting the answer under consideration. the markowitz model, which has been introduced, takes the historical data relative to the corporate returns and investigates the variance and their mean (averages) as a basis of calculation. after presenting the markowitz model, numerous supplementary researches were performed on it. konno and yamazaki (1991) employed the risk criterion of absolute deviation from the mean instead of risk. kellerer et al. (2000) developed it by incrementing new constraints: fixed cost and the minimum amount of transactions to this model. chang et al. (2000) added the maximum number of constraints to the shares present in the portfolio (cardinality). by adding the boundary constraints and cardinality fernandez and gomez (2007) used the neural network algorithm and solved the model. soleimani et al. (2009) presented a selection of the markowitz stock portfolio model, with constraints consisting of the minimum number of transactions, minimizing cardinality. similarly, in another study, chang et al. (2009) applied diverse risk conditions, such as the semi-variance, absolute deviation variance, and variance with skewness, imposed in the model and solved them using the genetic algorithm. in a paper in 2014, aouni and colapinto rendered appropriate management approaches for the stock portfolio by utilizing the concepts of goal programming (gp). algarvio et al. (2017) dealt with and managed the portfolio in relevance with risk and the optimization of retailers operating in the electrical market and presented a model for optimizing portfolios created by the final consumers using the markowitz hypothesis. zhao (2018) utilized the markowitz model to select options for the stock market. hunjra et al. (2020) compared the performance of risk models (mean-variance, semi-variance, mean absolute deviation, and conditional value-at-risk) in different economic scenarios, namely crisis, recovery, and growth. they implemented their investigations on the stock exchange of pakistan, bombay, and dhaka. the results indicated that conditional value-at-risk presented better results for each scenario in each country and portfolio performance was inconsistent in different methods. alongside these issues, dea was also introduced by charnes and cooper (1978) and was used for optimization and attaining the optimum combination for the portfolio concerning assets (edirisinghe and zhang 2015a). huang (2015) offered an integrated method for optimizing the stock portfolio, which comprises decisionmaking in relevance to the screening of stocks, stock selection, and the allocation of a multi-objective approach based on markowitz and dea cross-efficiency models for … 243 investment. edirsinghe and zhang (2015b) discussed using fiscal ratios to survey the efficiencies of companies. they investigated the datum of budgetary statements and employed dea techniques to determine rfs. this index indicates the competitive acceptance of a company in comparison with other companies. by compiling dea and the multi-criteria decision-making (mcdm) method, goodarzi et al. (2017) dealt with optimizing the stock portfolio. in the initial step, they utilized ratios as inputs and outputs in dea; then, they computed the cross-efficiency for each unit by using the optimal weights. hoe et al. (2017) employed dea and financial ratios to evaluate and compare technology companies listed in malaysia. subsequently, the inefficient companies attained alleviation or improvement criteria by taking advantage of the efficient companies. puri et al. (2017) presented a multi-component data envelopment analysis (mc-dea) model with inaccurate data. they proposed a new standard weights approach by applying interval arithmetic and unified production frontier to find unique weights for measuring these interval efficiencies. jin et al. (2020) proposed a decision-making model based on dea and the concept of probabilistic hesitant fuzzy numbers to construct a decision-making approach with probabilistic hesitant fuzzy preference relations (phfprs) to determine optimal selection among alternatives. chang et al. (2021) used the envelopment analysis of nested dynamic network data to evaluate the portfolio. the effect of the alternative optimized solutions on the dea cross-efficiency for portfolio selection was studied by amin and hajjami (2021). they revealed that these optimal solutions produce cross-efficiency matrices and portfolios of low risk with a higher expected return than the conventional cross-efficiency matrix for the portfolios. a few researchers also utilized uncertain parameters for the problem. huang (2008) described and rendered a new description of risk for selecting a stock portfolio in the fuzzy environment; on the basis of this, a new model was proposed, and a combined intelligent (smart) algorithm was proposed to solve it. the genetic algorithm solved an optimized portfolio model with cardinality constraints and uncertain data by sadjadi et al. (2012). guo (2012) used the fuzzy set theory to solve the mean-variance markowitz model and expand it to a fuzzy portfolio selection model. carlsson (2017) suggested an approved and mixed fuzzy programming by considering the future cash flows, in the form of a single trapezoidal fuzzy number, to select the prime research and development profile. in an attempt to choose a stock portfolio, zhou et al. (2018) took advantage of the qualitative data presented by experts and investors, which are contemplated as uncertain elements for stock selection. they chose two groups of investors, namely, the public and those prone to risk acceptance. two models were selected for portfolios, and scores were proposed based on maximum ranking and the deviation norm. in 2018, chen et al. evaluated and dealt with the efficiency of a stock portfolio in a fuzzy environment with several risk criteria (probable variance, probable semi-variance, and a probable absolute semideviation). they demonstrated that, in fuzzy conditions, the portfolio efficiency is more precise and offers better responses. lamb et al. (2012) unveiled the uncertainty in estimating the efficiency of dea. they employed bootstrap to develop random dea models for funds, the extraction of confidence intervals, and the development of techniques for comparing and ranking funds and indicating ratings. in research, lim et al. (2014) utilized the perception of dea cross-efficiency to select a stock portfolio. in addition to using the mean cross efficiency scores, they also took advantage of the modifications in the (variance) crossefficiency scores. the achieved model was implemented on stock from the stock market of korea, illustrating that the proposed model can be an optimistic tool for stock portfolio selection. in another research (2016), in developing the previous rasoulzadeh et al./decis. mak. appl. manag. eng. 5 (2) (2022) 241-259 244 model, they accounted for the returns of assets in the form of a single trapezoidal fuzzy number, thence proceeded to solve the model with the nsga ii algorithm and viewpoint and compared the responses. omrani and mashayekhi (2017) introduced a hybrid model based on the markowitz mean-variance model to select the stock portfolio. in addition to the risk and returns, the portfolio's efficiency is also taken into account by them simultaneously. their model was a four-objective model, which synchronously maximizes the mean stock returns and efficiency; it simultaneously minimizes the risk of the stock portfolio. to appraise the efficiency, they employed the dea cross-efficiency approach. next, the model was applied to the multi-objective genetic algorithm with a non-dominating sorting (nsga ii). they implemented this model on 52 companies on the tehran stock exchange and compared the responses with those of the markowitz model. chen et al. (2020) rendered a multi-objective model in a fuzzy environment by combining the semi-variance, variance, and crossefficiency dea models. in their model, the cross-efficiency was on the basis and debated upon the 'sharp ratio.' they solved the proposed model with the firefly algorithm. likewise, in another paper, chen (2021) implicated and selected the optimal portfolio; with a hybrid of models, such as the semi-variance, variance, and the cross-efficiency dea model, with non-dominating fuzzy inputs and outputs .for more details regarding the portfolio optimization with dea, see rasoulzade and fallah (2020). although there are numerous investigations in the field of portfolio optimization using fuzzy data envelopment analysis, however, there are only a few studies on this topic using the fuzzy set extensions, such as intuitionistic fuzzy sets, neutrosophic sets, etc; (yang et al. 2020, mao et al. 2020, edalatpanah 2018 and edalatpanah 2020). in recent years, data with uncertainty have been considered in numerous researches with varied formats. one of these types of uncertain data is the intuitionistic fuzzy, which is employed in various dea studies; and has been contemplated upon by researchers for portfolio selection. hajiagha et al. (2013) have presented a dea model with intuitionistic fuzzy inputs and outputs. it was for the first time that puri et al. (2015) analyzed the efficiencies of the optimistic and pessimistic inputs and outputs data from the intuitionistic fuzzy outlook. edalatpanah (2019) rendered a developed dea model in a triangular intuitionistic fuzzy environment. in this study, he proposed a new ranking function that considers the interaction between the membership and non-membership function in the diverse intuitionistic fuzzy sets. javaherian et al. (2021) proposed a new dea model to evaluate the efficiency of decision-making units by using two structures and triangular intuitionistic fuzzy data. yu et al. (2021) developed a unified intuitionistic fuzzy multi-objective linear programming (ifmolp) model for such portfolio selection problems. the nonmembership functions were made by the pessimistic, optimistic, and mixed approaches to perfect the traditional intuitionistic fuzzy (if) inequalities and if theory. in the current paper, we seek to attain a model by combining the markowitz model with returns of the trapezoidal intuitionistic fuzzy type, including the cross-efficiency dea model, to achieve a model where the optimum portfolio of assets comes to hand. by being attentive to the nonlinear structure of the model attained, the nsga ii algorithm will be utilized to solve it, and the responses shall be compared with the fuzzy and classical models. the nsga ii algorithm was introduced by deb (2002). these algorithms are based upon two cross efficiencies performed like the traditional and ordinary genetic algorithms. however, its arrangement and sorting of responses are not on the fundaments of a lower amount or more significant amount of the objective function. a multi-objective approach based on markowitz and dea cross-efficiency models for … 245 in this algorithm, sorting is performed based on non-dominance; responses that do not dominate each other are grouped in the same class. in recent years, the utilization of the nsga ii algorithm to solve problems has been such that the utmost portfolio is sought after and has been used in numerous ways; this can be illustrated in (kaucic et al. 2019, pal et al. 2021, karimi 2021, eftekharian et al. 2017). the categorizing of information in the present paper is as given. in the next section, we shall concisely present some of the required concepts. next, the developed model and the intuitionistic fuzzy returns shall be introduced. then, a numerical example will be solved based on the available data for the companies actively operating on the tehran stock exchange. eventually, we shall render the results that have come to hand. 2. methodology 2.1. the markowitz mean-variance model the matter of selecting an efficient portfolio is one of the concepts that markowitz discussed. an efficient portfolio signifies a portfolio's selection is from assets, of steady returns, the minimum of risk, or in a given risk, the maximum of returns (kazemi 2012). model 1: min σp 2 = ∑ ∑ wiwj cov(ri. rj) n j=1 n i=1 (1) s.t. r̅p = e(rp) = ∑ wir̅i ≥ r n i=1 (2) ∑ wi = 1 n i=1 (3) wi ≥ 0 i = 1.2.3. … . n (4) in the abovementioned model, we have the following definitions: ͞r͞i : mean or average returns on the ith assets cov(ri .rj) : the covariance of the ith and jth asset returns n: number of assets having the capacity to be invested r: minimum return expected by investors in the investment under consideration wi: the ith asset weight in the investment portfolio in the model mentioned above, constraint (2) imposes the minimum of returns cases expected by the investor as a constraint in the model. similarly, constraint (3) sets the total of the portfolio weights to be equivalent to 1, where the relative constraint is a total budget constraint. this model can be in the form of a dual-objective form and the form of a simultaneous increase of returns and a decrease in the risk portfolio as well, and taken into consideration as given below: model 2: rasoulzadeh et al./decis. mak. appl. manag. eng. 5 (2) (2022) 241-259 246 max ∑ wir̅i n i=1 (5) min σp 2 = ∑ ∑ wiwjcov(ri. rj) n j=1 n i=1 (6) s.t. ∑ wi = 1 n i=1 (7) wi ≥ 0 i = 1.2.3. … . n (8) 2.2. data envelopment analysis model (dea) with cross-efficiency to calculate the efficiency of each dmu, charnes and cooper (1978) rendered the following: model 3: max ∑ uryro s r=1 ∑ vixio m i=1 (9) s.t. ∑ uryrj s r=1 ∑ vixij m i=1 ≤ 1 j = 1. … . n (10) ur. vi ≥ 0 ; r = 1. … . s i = 1. … . m (11) in the abovementioned model xij and yrj are the inputs and outputs of the jth dmu respectively. ur and vi are the weights of the inputs and outputs respectively, which the model is seeking to bring to hand. likewise, the objective function is to maximize the weighted sum of the outputs to the weighted sum of the inputs. the standard input-axis model of return-to-scale (crs) data envelopment analysis is as follows: model 4: max ∑ uryro s r=1 (12) s.t. ∑ uryrj s r=1 − ∑ vixij m i=1 ≤ 0 j = 1. … . n (13) ∑ vixio m i=1 ≤ 1 (14) xi ,yr ≥ ε ∀i. r (15) where, n: the number of decision-making units (dmus), m and s: the number of inputs and outputs, respectively, xij and yrj: the amounts for the ith inputs and rth outputs, respectively, for the jth dmus. vi and ur: the weights allocated to the ith inputs and rth outputs, respectively, which the model computes. the efficiency scores of each dmu are obtained by solving model (3) and bringing the optimum responses to hand. let us assume that, 𝑢𝑟 ∗ and 𝑣𝑖 ∗ are the optimum responses, which have been achieved from the model (3) for the kth unit. through equation (16), the cross-efficiency of the other dmus, which are evaluated by the kth unit, can be calculated. ekl = ∑ ur ∗yrl s r=1 ∑ vi ∗xil m i=1 (16) a multi-objective approach based on markowitz and dea cross-efficiency models for … 247 now, by computing the cross-efficiency for the entire units of dmus; and then placing them in a n* n matrix, which is reputedly known (and brought to hand) as the (𝑒𝑖𝑗 )𝑖. 𝑗 = 1. … . 𝑛, cross -efficiency matrix. in this matrix, each element of (𝑒𝑖𝑗 ) is the cross-efficiency of the jth unit, which has been evaluated by the ith unit. in other words, in column (l) of the abovementioned matrix, (𝑒.𝑙) is the cross-efficiency vector of unit l . now, for every dmu like l, the mean is taken from every column (such as l), from which we can obtain and conclude the cross-efficiency ranking of that unit which is represented as 𝑒�̅� (lim et al., 2014): (17) el̅ = 1 n ∑ ekl n k=1 for each dmu to achieve the highest efficiency score in dea, it designates the highest weights to its points of strength; and the lowest weights to its points of weakness. in other words, each dmu in dea does not require considering the selected set of weights for the other dmus, and only utilizes its weights. though, when dea is used to assign an asset portfolio option within a multi-criteria decision-making (mcdm) framework, the weights to measure efficiency are determined externally and may modify over time. so this mechanism is no longer appropriate for use, although it proves valid for evaluating efficiency. to eliminate the aspect of risk due to weight changes, the utilization of dea cross-efficiency is one of the modes to confront the problem. based on evaluating cross-efficiency, all the dmus are ranked at a desirable level, and their efficiency for all the indexes is relatively good. they are resilient to weight change, and the variance of their cross-efficiency is relatively small. however, units that have performed well in a series of criteria have a lower score, are exposed to modifications in weight, and have a high cross-efficiency performance. due to the reasons mentioned, the cross-efficiency evaluation helps to select a stock portfolio in which the dmus are stable (lim et al. 2014). 2.3. mean and variance cross-efficiency models in selecting an asset portfolio, the risk of weight modifications consists of two parts: the sole and single risk of each dmu unit; and the risk between the units of dmus. the risk of individual dmus could be illustrated by the variance of the crossefficiency, of each distinctive unit of dmu, in the stock portfolio, and the risk between the units of dmus, can be demonstrated by the covariance between each pair of units. an uncomplicated application of evaluating cross-efficiency successfully reduces risks for each of the units, though it is incapacitated in considering the risk between the dmus (lim et al. 2014). to conduct this matter, returns and risk for each dmu are regarded with a cross-efficiency and variance (performance) score, respectively. similarly, for the stock portfolio 𝛺; whereas, with the unique dmus, the returns and risk are described as 𝑉𝛺 = 𝑤 𝑇 ∑𝑤 and 𝐸𝛺 = 𝑤 𝑇 �̅�, where, ∑ is the matrix of the crossefficiency variance; and (k,1) is the consistent element of the covariance between the cross-efficiency of the kth unit and the ith unit. likewise, the w's are considered a weight vector, which is contemplated as a sum of 1. in this case, an optimum stock portfolio is achieved by solving the quadratic optimization model, given hereunder, that comes to hand (schaerf 2002). model 5: min vω (18) s.t. (19) rasoulzadeh et al./decis. mak. appl. manag. eng. 5 (2) (2022) 241-259 248 eω ≥ (1 − γ)eω b etw = 1 (20) w≥ 0 (21) in which γ is the return risk swap-over parameter. 𝐸𝛺 𝑏 is the maximum of returns attainable of the stock portfolio and e is an appropriate return vector, the elements of which are similar. in models where the inputs and outputs can take adverse or negative values, the utilization of radial dea/crs models is inappropriate. collective vrs models offer the scopes to accommodate negative data as inputs and outputs (pastor 2007). amidst the varied collective dea models, we take advantage of the vrs collective model compared to other models, as there are numerous features, such as comprehensiveness and stability, in relevance with transfer and the change of interest rate. the dea -vrs cumulative model with inefficient criteria is as below (cooper et al. 1999): model 6: min 1 (m+s) (r−s− + r+s+) (22) s.t. xλ + s− = xk (23) yλ − s+ = yk (24) etλ = 1 (25) λ. s−. s+ ≥ 0 (26) that, 𝑋 = 𝑥𝑖𝑗 ∈ 𝑅 𝑛∗𝑚 and 𝑌 = 𝑦𝑟𝑗 ∈ 𝑅 𝑛∗𝑠 demonstrates the inputs and outputs, data matrix respectively. here, each column represents one of the units, and each row displays a level of one of the aspects of the factors of the relative dmu. 𝑅− and 𝑅+ is also described in the following form: r− = ( 1 r1 − . 1 r2 − . 1 r3 − . … . 1 rm − ) (27) r+ = ( 1 r1 + . 1 r2 + . 1 r3 + . … . 1 rs +) (28) ri − = max j=1……n {xij} − min j=1….n {xij} i = 1,...,m (29) ri + = max j=1…..n {yij} − min j=1….n {xyij} r= 1,…,s (30) the dual model (6) is as given hereunder: model 7: max ek d = pyk − qxk + ξ s.t. (31) py − qx + ξe < 0 (32) p ≥ 1 m+s r+ (33) q ≥ 1 m+s r− (34) in the abovementioned model, vectors p and q are weights of the inputs and outputs. in the case that, for the unit (1), the optimum response is illustrated by * the cross-efficiency for the lth unit, which is appraised by the kth unit, signifies that, 𝑒𝑘𝑙 ∗ is computed as follows: ekl ∗ = pk ∗ yk − qk ∗ xk + ξk (35) a multi-objective approach based on markowitz and dea cross-efficiency models for … 249 2.4. a combined markowitz and a cross-efficiency dea model to evaluate the risk, returns, and efficiency of the model rendered below, omrani and mashayekhi (2017) proposed the said model. model 8: 𝑀𝑎𝑥 ∑ wi n i=1 r̅i (36) min ∑ ∑ wiwj n j=1 cov(ri. rj) n i=1 (37) max ∑ wi n i=1 e̅i (38) min ∑ ∑ wiwj n j=1 cov(ei. ej) n i=1 (39) s.t. ∑ zi ≤ h n i=1 (40) lizi ≤ wi ≤ uizi i=1,…,n (41) ∑ wi = 1 n i=1 (42) wi ≥ 0 i = 1. … . n (43) in the abovementioned model, 𝑧𝑖 is a binary variable; when the i th asset takes place in the stock portfolio, it sums it as (1); if this is not the case, it is (0). variables li and ui are respectively in relevance with the minimum and maximum percentage of investments of the total budget, which pertains to the ith variable in the stock portfolio. in contrast, h is the maximum number of stocks selected in the stock portfolio. in model (8), the objective functions (36 and 37) are the same ones present in the markowitz model. the mentioned is employed to maximize returns and minimize risk returns. the objective functions (38 and 39) optimize portfolio efficiency and reduce portfolio risk. constraint (40) restricts the number of stocks present in the portfolio. constraint (41) demonstrates the maximum and minimum of the total budget deficit, which is liable for allotment to each share. the values h, 𝑙𝑖 and 𝑢𝑖 can vary and be following the investor's opinion. 3. a combined markowitz model with intuitionistic fuzzy returns and a cross-efficiency model as a development of the model rendered in the prior section (model 8), we seek to contemplate the absence of certainty in the form of trapezoidal intuitionistic fuzzy numbers for returns. the new model combines the markowitz and cross-efficiency dea models with intuitionistic fuzzy numbers. in this relevance, permit us to initially present and denote the definitions we will require in this section. �̃�𝐼 which is a fuzzy intuitionistic trapezoidal number, can be contemplated upon as 𝐴 = (𝑎1. 𝑎2. 𝑎3. 𝑎4. 𝑏1. 𝑏2. 𝑏3. 𝑏4), where the membership and non-membership functions 𝜇𝐴𝐼 and 𝜈𝐴𝐼 , respectively, are denoted as given below (puri and yadav 2015) μãi = { fa(x) a1 ≤ x < a2 1 a2 ≤ x ≤ a3 ga(x) a3 < x ≤ a4 0 otherwise (44) rasoulzadeh et al./decis. mak. appl. manag. eng. 5 (2) (2022) 241-259 250 νãi = { ha(x) b1 ≤ x < b2 0 b2 ≤ x ≤ b3 ka(x) b3 < x ≤ b4 1 otherwise (45) such that, 0 ≤ μãi (x) + νãi (x) ≤ 1 and 𝑎1. 𝑎2. 𝑎3. 𝑎4. 𝑏1. 𝑏2. 𝑏3 . 𝑏4 ϵ r in a way that 𝑏1 ≤ 𝑎1 ≤ 𝑏2 ≤ 𝑎2 ≤ 𝑎3 ≤ 𝑏3 ≤ 𝑎4 ≤ 𝑏4. functions 𝑓𝐴 and 𝑘𝐴 are continuous piecewise, non-descending functions, respectively, in the intervals of [𝑎1. 𝑎2) and (𝑏3. 𝑏4] and functions ga and ha are continuous piecewise non-descending functions, respectively, in the intervals of (𝑎3. 𝑎4] and [𝑏1. 𝑏2). the expected distance of a fuzzy intuitionistic number �̃�𝐼 , in the above form, is a precise 𝐸𝐼(�̃�𝐼 ) interval, which has been shown as 𝐸𝐼(�̃�𝐼 ) = [𝐸𝐿 (�̃� 𝐼 ). 𝐸𝑅 (�̃� 𝐼 )] and can be computed as given hereunder (grzegorzewski 2003). el(ã i) = b1+a2 2 + 1 2 ∫ ha(x)dx b2 b1 − 1 2 ∫ fa(x)dx a2 a1 (46) er(ã i) = a3+b4 2 + 1 2 ∫ ga(x)dx a4 a3 − 1 2 ∫ ka(x)dx b4 b3 (47) on the fundaments of which, the expected value is calculated as follows: ev(ãi) = el(ã i)+er(ã i) 2 (48) assuming: fa(x) = x−a1 a2−a1 (49) ga(x) = x−a4 a3−a4 (50) ha(x) = x−b2 b1−b2 (51) ka(x) = x−b3 b4−b3 (52) in this case, we have: ev(a) = 1 8 (∑ ai + 4 i=1 ∑ bi) 4 i=1 (53) for calculating the variance, we have: var(x) = e(x2) − (e(x))2 (54) el(x 2) = b1+a2 2 + 1 2 ∫ ha(x 2)dx b2 b1 − 1 2 ∫ fa(x 2)dx a2 a1 (55) eu(x 2) = a3+b4 2 + 1 2 ∫ ga(x 2)dx a4 a3 − 1 2 ∫ ka(x 2)dx b4 b3 (56) el(x 2) = − 1 3 (a1 2 + a2 2 + b1 2 + b2 2 + a1a2 + b1b2) + a1 + a2 + b1 + b2 (57) eu(x 2) = − 1 3 (a3 2 + a4 2 + b3 2 + b4 2 + a3a4 + b3b4) + a3 + a4 + b3 + b4 (58) e(x2) = el(x 2)+eu(x 2) 2 (59) by inserting the variance formula and a simplification, we shall have: var(aĩ ) = 1 4 (∑ ai + ∑ bi) − 1 12 (a1a2 4 i=1 + a3a4 + b1b2 + b3b4 4 i=1 + ∑ ai 2 + ∑ bi 2 4 i=1 ) − 1 64 (∑ ai 4 i=1 + ∑ bi 4 i=1 )2 4 i=1 (60) a multi-objective approach based on markowitz and dea cross-efficiency models for … 251 by inserting computations in the model, we shall gain access to the model rendered below: model 9: max e (∑ r̃i i wi n i=1 ) = 1 8 (∑(ai1 + ai2 + ai3 + ai4 + bi1 + bi2 + bi3 + bi4)wi n i=1 ) (61) min σ2(∑ r̃i i wi n i=1 ) = 1 4 ∑ ((∑ aij + ∑ bij) − 1 12 (ai1ai2 4 j=1 + ai3ai4 + bi1bi2 + 4 j=1 n i=1 bi3bi4 + ∑ aij 2 + ∑ bij 24 j=1 ) − 1 64 (∑ aij 4 j=1 + ∑ bij 4 j=1 ) 2 wi 4 j=1 ) (62) max ∑ wie̅i n i=1 (63) min ∑ ∑ wiwjcov(ei. ej) n j=1 n i=1 (64) s.t. ∑ zi ≤ h n i=1 (65) lizi ≤ wi ≤ uizi i = 1. … . n (66) ∑ wi = 1 n i=1 (67) wi ≥ 0 i = 1. … . n (68) in the abovementioned model,(𝑎1𝑗 . 𝑎2𝑗 . 𝑎3𝑗 . 𝑎4𝑗 . 𝑏1𝑗 . 𝑏2𝑗 . 𝑏3𝑗 . 𝑏4𝑗 ) illustrates the j th asset returns, which has been considered a trapezoidal fuzzy number and the other parameters equate to the parameters defined in the fuzzy form. with due attention to the fact that in the abovementioned model, such as the upper and lower limits of investments per share; and likewise, the number of maximum shares present; and have come to hand in the portfolio, is determined by the investor. we shall eliminate the maximum constraints of the number of shares in this model; it will be abolished. thereby, we shall deal with solving the following model in this paper as follows: model 10: max e (∑ r̃i i wi n i=1 ) = 1 8 (∑(ai1 + ai2 + ai3 + ai4 + bi1 + bi2 + bi3 + bi4)wi n i=1 ) (69) min σ2(∑ r̃i i wi n i=1 ) = 1 4 ∑ ((∑ aij + ∑ bij) − 1 12 (ai1ai2 4 j=1 + ai3ai4 + bi1bi2 + 4 j=1 n i=1 bi3bi4 + ∑ aij 2 + ∑ bij 24 j=1 ) − 1 64 (∑ aij 4 j=1 + ∑ bij 4 j=1 ) 2 wi 4 j=1 ) (70) max ∑ wie̅i n i=1 (71) min ∑ ∑ wiwjcov(ei. ej) n j=1 n i=1 (72) s.t. lizi ≤ wi ≤ uizi i = 1. … . n (73) rasoulzadeh et al./decis. mak. appl. manag. eng. 5 (2) (2022) 241-259 252 ∑ wi = 1 n i=1 (74) wi ≥ 0 i = 1. … . n (75) 4. solving a numerical example so to solve a numerical example and compare the responses that have come to hand, we have utilized data and information relative to the companies of the tehran stock exchange. for this objective, we have selected 50 companies from those present in the tehran stock exchange from 2018 to 2021 in the tableau of the tehran stock exchange. by paying heed to the fact that banks, insurance enterprises, and investment companies have a diverse fiscal structure from that of other companies, these categories of companies are not incorporated in the list of companies. similarly, the information published in the fiscal statements protracting to 19/03/2021 has been used to survey companies' efficiency. 4.1. value returns the returns are considered a trapezoidal intuitionistic fuzzy number by considering the company's efficiency for three years (2016 to 2019). these returns, which are being considered, are for 50 companies; and have been rendered in table (1). table 1. trapezoidal intuistic-fuzzy returns asset id intuitionistic fuzzy returns asset id intuitionistic fuzzy returns 1 (0.04,0.09,0.14,0.19,-0.13,0.08,0.15,0.61) 26 (0.05,0.1,0.16,0.21,-0.21,0.09,0.17,0.71) 2 (0.03,0.1,0.16,0.23,-0.21,0.08,0.18,0.63) 27 (0.05,0.08,0.11,0.13,-0.14,0.07,0.11,0.4) 3 (0.05,0.09,0.12,0.15,-0.17,0.08,0.13,0.52) 28 (0.04,0.07,0.09,0.12,-0.16,0.06,0.1,0.38) 4 (0.01,0.05,0.08,0.12,-0.21,0.04,0.09,0.44) 29 (0.09,0.11,0.14,0.16,-0.11,0.11,0.14,0.43) 5 (0.07,0.1,0.12,0.15,-0.13,0.09,0.13,0.56) 30 (0.03,0.08,0.13,0.18,-0.28,0.07,0.14,0.58) 6 (0.04,0.06,0.09,0.11,-0.18,0.06,0.09,0.35) 31 (0.03,0.09,0.15,0.22,-0.22,0.08,0.17,0.71) 7 (0.05,0.12,0.18,0.25,-0.23,0.1,0.2,0.74) 32 (0.08,0.13,0.17,0.22,-0.16,0.12,0.18,0.64) 8 (0.04,0.09,0.14,0.19,-0.23,0.08,0.15,0.66) 33 (0.07,0.09,0.1,0.11,-0.04,0.08,0.1,0.4) 9 (0.04,0.1,0.16,0.22,-0.22,0.09,0.17,0.64) 34 (0.02,0.05,0.08,0.11,-0.14,0.04,0.09,0.34) 10 (0.07,0.1,0.13,0.16,-0.16,0.09,0.13,0.56) 35 (0.03,0.08,0.13,0.17,-0.24,0.07,0.14,0.54) 11 (0.04,0.08,0.11,0.14,-0.18,0.07,0.12,0.51) 36 (-0.04,0.08,0.2,0.31,-0.2,0.05,0.22,0.72) 12 (0.08,0.11,0.14,0.17,-0.13,0.11,0.15,0.46) 37 (0.03,0.1,0.18,0.26,-0.18,0.08,0.2,0.89) 13 (0.04,0.07,0.11,0.14,-0.2,0.07,0.12,0.52) 38 (-0.01,0.09,0.18,0.27,-0.21,0.06,0.2,0.72) 14 (0.08,0.11,0.14,0.16,-0.16,0.1,0.14,0.43) 39 (0.06,0.11,0.16,0.22,-0.18,0.1,0.18,0.67) 15 (-0.01,0.03,0.07,0.11,-0.2,0.02,0.08,0.32) 40 (0.04,0.07,0.11,0.14,-0.18,0.07,0.12,0.54) 16 (0.07,0.1,0.14,0.17,-0.13,0.09,0.15,0.53) 41 (0.07,0.1,0.12,0.14,-0.12,0.09,0.13,0.48) 17 (0.05,0.1,0.14,0.19,-0.22,0.08,0.15,0.56) 42 (0.06,0.13,0.21,0.28,-0.12,0.12,0.22,0.84) 18 (0.07,0.1,0.14,0.17,-0.22,0.09,0.14,0.55) 43 (0.05,0.07,0.09,0.11,-0.17,0.07,0.1,0.42) 19 (0.04,0.08,0.12,0.16,-0.15,0.07,0.13,0.59) 44 (-0.02,0.08,0.17,0.26,-0.24,0.05,0.19,0.78) 20 (0.06,0.11,0.17,0.22,-0.22,0.1,0.18,0.7) 45 (0.1,0.12,0.13,0.14,-0.03,0.11,0.13,0.37) 21 (0.04,0.07,0.09,0.12,-0.12,0.06,0.1,0.33) 46 (0.06,0.09,0.12,0.15,-0.16,0.08,0.13,0.35) 22 (0.05,0.12,0.19,0.26,-0.19,0.1,0.21,0.7) 47 (0.05,0.08,0.11,0.14,-0.14,0.07,0.12,0.57) 23 (0.06,0.1,0.13,0.16,-0.17,0.09,0.14,0.44) 48 (0.03,0.08,0.12,0.17,-0.19,0.06,0.14,0.5) 24 (0.04,0.09,0.15,0.2,-0.26,0.08,0.16,0.71) 49 (0.13,0.16,0.19,0.22,-0.13,0.15,0.2,0.52) 25 (0,0.06,0.12,0.18,-0.45,0.04,0.13,0.47) 50 (0.04,0.08,0.11,0.15,-0.24,0.07,0.12,0.49) a multi-objective approach based on markowitz and dea cross-efficiency models for … 253 we have finally solved the model attained by the nsga ii algorithm and the matlab 2014 software. 4.2. the input and output values seven criteria from the input and 7 criterions from the output have been used to survey the efficiency. the list of criteria has been given in table (2). the information relative to each issue inserted in the audited financial statements, prolonging to 19/03/2020, has been extracted and computed. table 2. inputs and outputs 4.3. model parameters to solve the model with the nsga ii algorithm is as follows: the population has been contemplated as(𝑁𝑝𝑜𝑝), 100, 𝑝𝑚 = 0.1, 𝑝𝑐 = 0.8; the maximum number of iterations is equivalent to 200, 𝜇 = 2; the model has been coded by the matlab 2014 (mechanism). similarly, the minimum investment per the ( 𝑙𝑖 ) share by the investor is 10 percent, and the maximum investment for each )𝑢𝑖 ) share has been considered as and equates to 30 percent. type parameter description perspective input turnover of accounts receivable period income divided by accounts receivable productivity inventory of materials and goods period income divided by inventories asset turnover period income divided by assets current ratio current assets over current debts liquidity instantaneous ratio quick assets as to the current debts ratio of debt to the shareholders' equity total debt divided by shareholders' equity debt ratio total debts over total assets leverage ratio output return on the shareholders' equity net profit on the shareholders' equity profitability return on assets net profit on assets net profit margin net profit on sales earnings per share net profit on the number of shares income growth rate current period income divided by the previous period income minus one growth net profit growth rate net profit for the current period divided by the net profit for the prior period minus one growth rate of earnings per share current period eps divided by previous period eps minus one rasoulzadeh et al./decis. mak. appl. manag. eng. 5 (2) (2022) 241-259 254 4.4. results two specifications, such as those given hereunder, are considered to select the stock portfolio. concerning each of the two, we have computed the optimal portfolio, based on the information of the companies, including the returns, from the existing shares; eventually, we have compared the responses. portfolio-1: a combination of the markowitz and the cross-efficiency model with fuzzy returns (mashayekhi and omrani 2016). portfolio-2: a combination of the markowitz and the cross-efficiency model, with intuitionistic fuzzy returns (model 11). the model results based on the weights assigned to each share are given in table (3). the conceptions of the expected rate of returns, portfolio efficiency, and portfolio risk on the fundaments of the rate of returns, as well as the portfolio risk in terms of efficiency, are considered, respectively, following equations (76 to 79) (mashayekhi and omrani 2016). e(∑ r̃i i wi n i=1 ) (76) ∑ wie̅i n i=1 (77) σ2(∑ r̃i i wi n i=1 ) (78) ∑ ∑ wiwjcov(ei. ej) n j=1 n i=1 (79) in implementing the model, a set of optimum pareto responses come to hand. the investor can select one of these pareto responses, as an investment portfolio, based on criteria. various criteria can be contemplated for choosing the desired portfolio. we have considered the highest returns, and in each model, the response with the highest returns is considered in this paper. the reactions achieved following these criteria are shown in table (3) and tabel (4). table 3. weights of the assets in the selected portfolio for the intuitionistic fuzzy returns model asset id 7 37 42 44 weights of the assets 0.23 0.23 0.23 0.31 table 4. weights of the assets in the selected portfolio for the fuzzy returns model asset id 18 32 42 49 weights of the assets 0.20 0.20 0.32 0.28 likewise, the responses in relevance to the objective function in each model have been rendered in table (5). table 5. optimum values of the objective function function function type objective fuzzy intuitionistic fuzzy z1 maximum expected rate of returns 0.16 0.18 z2 minimum portfolio risk, based on the rate of returns 0.003 0.24 z3 maximum portfolio efficiency 0.41 0.72 z4 minimum portfolio risk, based on efficiency 0.16 0.07 a multi-objective approach based on markowitz and dea cross-efficiency models for … 255 as can be observed in the abovementioned table, the intuitionistic fuzzy model has a higher expected rate of returns than the fuzzy model. the same also exhibits a better efficiency than the fuzzy model. regardless of the direct correlation between risk and returns, based on the fundaments of the rate of returns, the risk portfolio and similarly, in accordance with efficiency, has also shown increment. some other pareto responses relevant to any model are presented in table (6). table 6. some pareto solutions obtained from the proposed model type of model serial number expected rate of returns portfolio risk, based on the rate of returns portfolio efficiency portfolio risk, based on efficiency intuitionistic fuzzy 1 0.06 0.11 0.23 0.04 2 0.12 0.18 0.57 0.02 3 0.16 0.22 0.83 0.03 fuzzy 1 0.08 0.0009 0.55 0.04 2 0.11 0.0002 0.53 0.16 3 0.14 0.005 0.80 0.05 5. conclusion this paper combined markowitz's mean-variance model and a cross-efficiency model to introduce a four-objective model that increased efficiency and decreased the covariance of cross-efficiencies besides increasing returns and reducing the portfolio risk. although many studies have addressed portfolio optimization using markowitz's and cross-efficiency models independently, a few studies have combined these models and benefitted from the advantages of both models. correspondingly, the returns are assumed as intuitionistic trapezoidal fuzzy numbers to represent return uncertainty. a non-dominated sorting genetic algorithm (nsga-ii) was also used to solve the new model. moreover, the proposed model was implemented on 50 firms listed on the tehran stock exchange. the results were compared in two cases intuitionistic trapezoidal fuzzy numbers and trapezoidal fuzzy numbers. the results indicated that, despite significant improvement in portfolio efficiency in the case of intuitionistic trapezoidal fuzzy returns, the portfolio risk was increased substantially in response to a slight increase in portfolio returns. some extensions can be considered for future studies, such as adding constraints on transaction costs and the number of stocks within the portfolio. also, other cases of return or efficiency uncertainty can be treated. furthermore, it is suggested to find new models by developing efficiency measurement structures, such as cross-efficiency in network dea. author contributions: research problem, m.r. and s.a.e.; conceptualization, m.r. and s.a.e.; methodology, s.a.e.; software, m.r.; validation, s.a.e. and m.f. and s.e.n.; formal analysis, m.r. and s.a.e.; investigation, m.r.; resources, m.r and s.a.e.; data curation, m.r. and s.a.e.; writing, m.r. and s.a.e. and s.e.n.; reviewing and editing, m.f.; visualization, m.r. and s.a.e.; supervision, s.a.e. and m.f.; all authors have read and approved the published version of the manuscript. acknowledgments: the authors would like to express their gratitude to the editors and anonymous referees for their informative, helpful remarks and suggestions to improve this paper as well as the important guiding significance to our researches. rasoulzadeh et al./decis. mak. appl. manag. eng. 5 (2) (2022) 241-259 256 funding: this research received no external funding. conflict of interest: the authors declare that they have no conflict of interest. references afshar, k. m., khaliliaraghi, m., & sadat, k. a. 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(2018). portfolio selection and risk investment under the hesitant fuzzy environment. knowledge-based systems, 144, 21-31. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 5, issue 1, 2022, pp. 225-245. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0313052022b * corresponding author. e-mail addresses: partha.bhattacharya1@gmail.com (p. p. bhattacharya), kousikbhattacharya12@gmail.com (k. bhattacharya), skdemamo2008.com@gmail.com (s. kumar de)demamo2008.com@gmail.com (s. k. de) a study on pollution sensitive sponge iron based production transportation model under fuzzy environment partha pratim bhattacharya1, kousik bhattacharya2 and sujit kumar de2* 1 department of computer science and engineering, makaut, west bengal, india 2 department of mathematics, midnapore college (autonomous), west bengal, india received: 29 march 2022; accepted: 13 may 2022; available online: 13 may 2022. original scientific paper abstract: over the last few years, sponge iron-based production transportation and pollution problems for major sponge iron producing countries are triggering a critical issue. the excess of marginal pollution from production industries and their disintegration takes drives towards the change of policymaking. the sustainable development of any country signifies the reduction of biohazards, which in turn improves the health index and livelihood status of people across the world. keeping this in mind, a cost depreciation problem for the bi-layer integrated supply chain model has been built up. we consider the functional dependencies among all considerable decision variables like production rate, consumption rate which leads to the pollution rate of different countries exclusively. in this study, we have shown how production and rail freight transport relates to pollution. to draw several graphs and numerical computations we use matlab software and c programming via solution algorithm respectively. the comparative study has been presented using general fuzzy as well as cloudy fuzzy systems. lastly, we have justified our proposed model using sensitivity analysis along with graphical interpretation. key words: production, pollution, transportation, cloudy fuzzy, modeling, optimization. 1. introduction this section has been splitted into two subsections namely general overview and motivation and specific study. bhattacharya et al./decis. mak. appl. manag. eng. 5 (1) (2022) 225-245 226 1.1. general overview due to globalization, the manufacturing of iron as well as steel and other materials, such as aluminum and materials for chemical-based products are playing a substantial role in the competitive market. we know, the total energy use and different levels of pollution come from various stages in the production process. the ferroalloys production is minor in contrast with base materials such as steel and aluminum. the major portion of complete man-made pollution has been incorporated by the severe environmental effects of silicon and ferroalloys. taking consideration of the by products of the production of sponge iron, the daily increment in use of general assets like electrical as well as electronic products, chemical materials, iron, and steel items is alarming contents. also, the large content of daily exhausted baby food products, lapsed drugs are the burning issues for our environment. the manufacturers have extensively marketed for updated items to remain in competition in the global game. with the inclusion of upgraded commodities, old stocks become useless for consumption and thus generate various liabilities causing a severe impact on our surroundings. to realize, control as well as minimizing the surrounding pollution linked with the manufacturing process, their products and actions, the insight of life-cycle concept is booming method for any industry. it is important to consider the environmental issues in sponge iron production from a broad view. for the superior perception of the environmental problem the understanding of drawbacks with a production chain from "cradle to grave" using life cycle assessment (lca) is in dire necessity. the inclusion of the lca study dishes out an integrated analysis of resources, substantial, and health effects on the system. also, it paves the way for environmental advancements by carrying out significant opportunities. in a complete lca, the total environmental cost consists of all material sources as well as energy resources inducted in the process from raw materials up to production and transportation. 1.2. motivation and specific study in the literature, several research articles are available in which most of them are associated to cost benefits and controlling carbon emissions from the vehicles used in the transportation itself. sarkar et al. (2015) investigated the outcome of an uneven lot size model for changeable establishment cost and carbon discharge cost in an sc problem. madadi et al. (2010) came about a multi-tiered inventory management settlement with shipment cost emolument. recently, the consequence of changing shipment and outpouring of carbon in the three-echelon sc model has been reported by sarkar et al. (2016). depending on electrical energy on railway shipment, bryan et al. (2008) proposed a model. a detailed review with an introduction to controlling novel mechanization for carbon combustion had been presented by sithole et al. (2018). in the case of ferromanganese and steel, sjoqvist et al. (2001) reported the outcome of carbon excretion during cleaning. it has to be mentioned, the correlation among manufacturing, shipment, carbon release, and environmental contamination is also being included in the ferro industry-related conveying problem. recent works suggest the severe knock-on our surrounding by the transport sector and this setback forces us to reconsider the environmental effect due to the transport organizing and operations. the main culprits from transportation are consist of different oxides of carbon and nitrogen, as well as different organic chemicals. the rising environmental consciousness among people, enormous competition as well as strict policies from the government enforce the manufacturer for minimization of this severe pollution for the sake of mankind [nouira et al. (2016)]. the important model by benjaafar et al. (2010) a study on pollution sensitive sponge iron based production transportation model … 227 describes the way of controlling carbon footprint in supply chains. aarthi (2017), chen et al. (2013), mancini et al. (2016), akten & akyol (2018) suggest a different model (like the eoq model) and methods to combat the rising carbon footprint. grzywiński et al. (2019a, 2019b) analysed various optimization techniques using metaheuristic algorithm and jaya algorithm. eirgash et al. (2019) described a multi-objective inventory model with trade credit. bera et al. (2020) studied the impacts of air pollution in covid situation in the urban areas. a risk assessment of bankrupt cases in european countries was done by misu and madaleno (2020). abualigah et al. (2021) developed an arithmetic optimization algorithm to solve supply chain management problems. it has been observed that all earlier authors have focused to measure the carbon emission due to production, although along with it, transportation plays a major role. the fuzzy system is utilized when there is the existence of some non-random uncertain parameters in the system. after zadeh (1965) developed the fuzzy set theory, there is multiple reports by renowned scientists across the globe [kumar et al. (2012), de and sana (2013), de et al. (2014)]. along with this, the production mechanism has been considerably investigated using cloudy fuzzy set [de and mahata (2016)), karmakar et al. (2017, 2018)] and triangular dense fuzzy set [de and beg (2017)]. after the invention of triangular dense lock fuzzy sets by de (2017), de and mahata (2019) developed a supply chain backordering model under triangular lock fuzzy environment bhattacharya et al. (2020a, 2020b) developed pollution sensitive inventory models with the effect of corruption as well as global warming and solved these via fuzzy system. giri et al. (2021) solved a price dependent multi-item inventory model using intuitionistic fuzzy number. the above-reported literature suggests no one has investigated the industrial supply chain (sc) problem that includes the pollution due to production as well as transportation. indeed, methodology over fuzzy learning theory was not popularly utilized yet. hence in our study, we present out an article that includes cost minimization two-layer sc problem having two-way pollution channel under learning fuzzy environment. we solve the specific inventory management problem into three sub cases: one by crisp approach, another by general fuzzy approach and the other by cloudy fuzzy approach. we have also developed a solution algorithm to solve the problem in each case. we also include a sensitivity analysis table to show the stability of the parameters involving in the model. the organization of this article is developed as follows: section one is introduction followed by motivation and specific study. section 2 includes preliminaries that focuses definition of general and cloudy fuzzy sets and their defuzzification techniques. section 3 describes notations, assumptions and a case study. section 4 indicates formulation of crisp inventory model. section 5 includes the general fuzzy mathematical model and its defuzzification method; section 6 develops cloudy fuzzy mathematical model and its defuzzification method with a solution algorithm, pseudo code of c programming; section 7 and 8 indicates numerical illustration and sensitivity analysis respectively. sections 9 develops graphical illustrations; section 10 represents the merits and demerits of the article and finally section 11 keeps a conclusion followed by scope of future work. 2. preliminaries in this section, we shall give some definitions and basic formulae that are used to formulate and solve the proposed model. bhattacharya et al./decis. mak. appl. manag. eng. 5 (1) (2022) 225-245 228 2.1. pollution function from the report by karmakar et al. (2017), we have taken the differential equation which governed the production-pollution rate is: { �̇� = 𝑎𝑋 − 𝑟𝑋2 − 𝛼𝑋𝑌, 𝑎,𝑟,𝛼 > 0 �̇� = −𝑐𝑌 + 𝛾𝑋𝑌, 𝑐,𝛾 > 0 (1) from above, the pollution y (%) with production rate (p) is governed by 𝑦 = 0.45 + 0.01𝑝 − 0.25𝑙𝑜𝑔(𝑝) (2) 2.2. normalized general triangular fuzzy number (ngtfn) let a be an ngtfn having the form �̃� = 〈𝑎1,𝑎2,𝑎3〉. then the membership function of the fuzzy set �̃� is defined by 𝜇(�̃�) = { 0, 𝑖𝑓 𝑎 < 𝑎1 𝑎𝑛𝑑 𝑎 > 𝑎3 𝑎−𝑎1 𝑎2−𝑎1 , 𝑖𝑓 𝑎1 ≤ 𝑎 ≤ 𝑎2 𝑎3−𝑎 𝑎3−𝑎2 , 𝑖𝑓 𝑎2 ≤ 𝑎 ≤ 𝑎3 (3) now, the index value of 𝜇(�̃�) [due to yager (1981)] is obtained as 𝐼(�̃�) = 1 2 ∫ [𝐿(𝛼) + 𝑅(𝛼)]𝑑𝛼 1 0 = (𝑎1+2𝑎2+𝑎3) 4 (4) for the left and right α-cuts 𝐿(𝛼) = 𝑎1 + (𝑎2 − 𝑎1)𝛼 and 𝑅(𝛼) = 𝑎3 − (𝑎3 − 𝑎2)𝛼 respectively. 2.3. cloudy normalized triangular fuzzy number (cntfn) (de and mahata, 2016) a fuzzy number �̃� = 〈𝑎1,𝑎2,𝑎3〉 is called cloudy normalized fuzzy number if, after an infinite time, the set converges to a singleton crisp set. that is, if the time 𝑡 → ∞, the set �̃� becomes 𝐴 = {𝑎2}. for example, we consider the fuzzy number �̃� = 〈𝑎2 (1 − 𝜌 1+𝑡 ),𝑎2,𝑎2 (1 + 𝜎 1+𝑡 )〉, for 0 < 𝜌,𝜎 < 1 (5) here we see that both lim 𝑡→∞ 𝑎2 (1 − 𝜌 1+𝑡 ) and lim 𝑡→∞ 𝑎2 (1 + 𝜎 1+𝑡 ) converges to 𝑎2. then its membership function for 𝑡 ≥ 0 is given by 𝜇(𝑥,𝑡) = { 0 𝑖𝑓 𝑥 < 𝑎2 (1 − 𝜌 1+𝑡 ) 𝑎𝑛𝑑 𝑥 > 𝑎2 (1 + 𝜎 1+𝑡 ) 𝑥−𝑎2(1− 𝜌 1+𝑡 ) 𝜌𝑎2 1+𝑡 𝑖𝑓 𝑎2 (1 − 𝜌 1+𝑡 ) ≤ 𝑥 ≤ 𝑎2 𝑎2(1+ 𝜎 1+𝑡 )−𝑥 𝜎𝑎2 1+𝑡 𝑖𝑓 𝑎2 ≤ 𝑥 ≤ 𝑎2 (1 + 𝜎 1+𝑡 ) (6) now the index value of �̃� is given by 𝐼( �̃�) = 1 2𝑇 ∬ {𝐿−1(𝛼,𝑡) + 𝑅−1(𝛼,𝑡)} 𝛼=1,𝑡=𝑇 𝛼=0,𝑡=0 𝑑𝛼𝑑𝑡 = 𝑎2 [1 + 𝜎−𝜌 4 𝑙𝑜𝑔(1+𝑇) 𝑇 ] (7) for the left and right 𝛼 -cuts 𝐿−1(𝛼,𝑡) = 𝑎2 (1 − 𝜌 1+𝑡 + 𝜌𝛼 1+𝑡 ) and 𝑅−1(𝛼,𝑡) = 𝑎2 (1 + 𝜎 1+𝑡 − 𝜎𝛼 1+𝑡 ) respectively. a study on pollution sensitive sponge iron based production transportation model … 229 3. notations and assumptions in this section we shall discuss the notations and assumptions that are used throughout the proposed model. notations 𝑝 : production rate per cycle (mt/year) (decision variable) 𝑦 : pollution index (%) 𝜏1 : production run time (decision variable) (year) 𝜏2 : transportation time (year) 𝜏3 : inventory exhaust time (year) 𝑑 : rate of demand for each cycle (mt/year) 𝑞 : total quantity in order (mt) 𝛿 : deterioration rate per unit time 𝑙 : transportation distance (mile) 𝐶𝑝 : manufacturing expenditure for each item ($) ℎ𝑝 : carrying expenditure for each item for each interval of time at maker’s plant($) ℎ𝑟 : carrying expenditure for each item for each interval of time at dealer’s shop ($) 𝐶𝑝𝑜𝑙 : pollution expenditure ($) (per one item) 𝐶𝑡 : transportation cost ($) (per unit mt per mile) 𝐶𝑑 : deterioration price ($) (for each item for each interval of time) 𝐶𝑐 : global social expenditure of carbon ($) 𝑘1 : setup cost at production plant ($) 𝑘2 : setup cost at retailer side ($) assumptions 1. replenishments are instantaneous. 2. shortages are not allowed. 3. lead time is zero. 4. a producer has the sole responsibility to transport the items to a retailer. 5. deterioration occurs and deteriorated items cannot be recoverable. 6. a producer has a separate transportation facility. 7. pollution during production is controlled by the inbuilt technology of the production process but 100% pollution reduction is not possible. 8. no deterioration is viewed in the final product during transportation. 3.1. case study let us extend the case study performed by karmakar et al. (2017, 2018). these studies were involved in the manufacturing and pollution of a sponge iron industry. our focus of interest is to measure pollution due to the transportation of products by a freight train. also, through managerial insights as well as learning experiences, we try to cut down the standard inventory expenditure. with a diameter of 1200 km (estimated), in this single managerial controlled industry, the different orders are put bhattacharya et al./decis. mak. appl. manag. eng. 5 (1) (2022) 225-245 230 down immediately using different shipment systems. the various expenditure information obtained from the industry is presented in table 1. table 1. data information for the concerned industry establishment expenditure per cycle $20000 carrying expenditure for each cycle in manufacturing plant per mt $5 degradation value in each cycle in manufacturing plant per each thing $10 degradation fraction 0.1 contamination expenditure for a manufacturer for one mt $43.89 carrying expenditure for each cycle in dealer plant for one mt $10 manufacturing price for each mt $327.56 social expenditure for carbon per mt $417 shipment expenditure for unit gallon fuel $3.5 distance crossed in a freight train 600 miles the research problem is i) is it possible to control the contamination and reach the least annual average expenditure in our proposed si production? ii) what is the ideal quantity of order numbers which results in a minimum inventory cost? iii) whether our cloudy fuzzy system is more effective to reduce the pollution of the supply chain as well as average inventory cost than the crisp and general fuzzy system. 4. formulation of crisp mathematical model we consider the above assumptions and notations for developing an imperfect production process by bhattacharya et al. (2021). the proposed mathematical model for average inventory cost minimization is governed by 𝑧 = 1 𝜏1 [𝐻𝐶 + 𝑃𝐶 + 𝐷𝐶 + 𝑇𝐶 + 𝑇𝑃𝐶 + 𝑆𝐶 + 𝑃𝑃𝐶] 𝑧 = ℎ𝑝𝑝 𝛿 (1 + 𝑒−𝛿𝜏1−1 𝛿𝜏1 ) + 𝐶𝑑𝑝(1 + 𝑒−𝛿𝜏1−1 𝛿𝜏1 ) + 𝐶𝑝𝑝 𝛿 (1 + 𝑒−𝛿𝜏1−1 𝛿𝜏1 ) + 𝐶𝑝𝑜𝑙𝑝 + 𝐾1 𝜏1 + 𝐶𝑡 × 0.00424628𝑙𝑑 + 𝐶𝑐 × 0.0000431445𝑙𝑑 + ℎ𝑟𝑑𝜏1 2 + 𝐾2 𝜏1 (8) this sc model is represented by a study on pollution sensitive sponge iron based production transportation model … 231 { 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑧 = ℎ𝑝𝑝 𝛿 (1 + 𝑒−𝛿𝜏1−1 𝛿𝜏1 ) + 𝐶𝑑𝑝(1 + 𝑒−𝛿𝜏1−1 𝛿𝜏1 ) + 𝐶𝑝𝑝 𝛿 (1 + 𝑒−𝛿𝜏1−1 𝛿𝜏1 ) + 𝐶𝑝𝑜𝑙𝑝 + 𝐾1 𝜏1 + 𝐶𝑡 × 0.00424628𝑙𝑑 + 𝐶𝑐 × 0.0000431445𝑙𝑑 + ℎ𝑟𝑑𝜏1 2 + 𝐾2 𝜏1 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜,𝜏2 = 3 2 𝜏1, 𝜏3 = 5 2 𝜏1 𝑞 = 𝑝 𝛿 (1 − 𝑒−𝛿𝜏1) = 𝑑𝜏1 𝑦 = 0.45 + 0.01𝑝 − 0.25𝑙𝑜𝑔(𝑝) (9) 5. construction of sc model under general fuzzy system all our expenditure parameters (𝐶̅) with demand rate (d) in our prescribed sc model also obeys tfn is organized as 𝐶�̃� = 〈𝐶𝑖1,𝐶𝑖2,𝐶𝑖3〉, 𝑖 = 1,2, . . ,9, = (ℎ𝑝,𝐶𝑑,𝐶𝑝,𝐶𝑝𝑜𝑙,𝐾1,𝐾2,ℎ𝑟, 𝐶𝑡,𝐶𝑐 ), and �̃� =< 𝑑1,𝑑2,𝑑3 >. also, due to fuzzification of parameters like demand rate, the order quantity, production rate, and pollution level will assume values in the following: { q̃ =< q1,q2,q3 ≥< d1τ1,d2τ1,d3τ1 > p̃ =< p1,p2,p3 ≥< d1τ1δ(1 − e −δτ1),d2τ1δ(1 − e −δτ1),d3τ1δ(1 − e −δτ1) > ỹ =< y1,y2,y3 > =< 0.45 + 0.01p1 − 0.25logp3 ,0.45 + 0.01p1 − 0.25logp3 ,0.45 + 0.01p1 − 0.25logp3 > (10) then the corresponding fuzzy problem of the crisp problem (9) can be written as { minz̃ =̃ p̃∑ cĩ fi 4 i=1 + c5̃ f5 + c6̃ f6 + d̃ ∑ cĩ fi 9 i=7 subject to,τ2 = 3 2 τ1,τ3 = 5 2 τ1, q̃ =̃ p̃ δ (1 − e−δτ1) =̃ d̃τ1 ỹ =̃ 0.45 + 0.01p̃ − 0.25logp̃ (11) where { 𝒇𝟏 = 𝟏 𝜹 (𝟏 + 𝒆−𝜹𝝉𝟏−𝟏 𝜹𝝉𝟏 ),𝒇𝟐 = (𝟏 + 𝒆−𝜹𝝉𝟏−𝟏 𝜹𝝉𝟏 ), 𝒇𝟑 = 𝟏 𝜹 (𝟏 + 𝒆−𝜹𝝉𝟏−𝟏 𝜹𝝉𝟏 ),𝒇𝟒 = 𝟏, 𝒇𝟓 = 𝟏 𝝉𝟏 ,𝒇𝟔 = 𝟏 𝝉𝟏 , 𝒇𝟕 = 𝝉𝟏 𝟐 , 𝒇𝟖 = 𝟎.𝟎𝟎𝟒𝟐𝟒𝟔𝟐𝟖𝒍,𝒇𝟗 = 𝟎.𝟎𝟎𝟎𝟎𝟒𝟑𝟏𝟒𝟒𝟓𝒍 (12) 5.1. defuzzification under general fuzzy system obeying tfn, all of our fuzzy objectives can be written as �̃� =< 𝑧1,𝑧2,𝑧3 > and the components are represented as: { z1 = p1 ∑ ci1 fi 4 i=1 + c51 f5 + c61 f6 + d1 ∑ ci1 fi 9 i=7 z2 = p2 ∑ ci2 fi 4 i=1 + c52 f5 + c62 f6 + d2 ∑ ci2 fi 9 i=7 z3 = p3 ∑ ci3 fi 4 i=1 + c53 f5 + c63 f6 + d3 ∑ ci3 fi 9 i=7 (13) using equation (4), our fuzzy problem (11) is converted to crisp cost minimization problem by replacing the respective index parameter with mentioned constraints are written as minimize i(z̃) = 1 4 (z1 + 2z2 + z3) (14) bhattacharya et al./decis. mak. appl. manag. eng. 5 (1) (2022) 225-245 232 subject to { 𝜏2 = 3 2 𝜏1,𝜏3 = 5 2 𝜏1, 𝐼(�̃�) = (𝑑1+ 2𝑑2+ 𝑑3) 4 , 𝐼(�̃�) = 𝐼(�̃�)𝜏1, 𝐼(�̃�) = 𝐼(𝑝) 𝛿 (1 − 𝑒−𝛿𝜏1), 𝐼(�̃�) = 0.45 + 0.01 𝐼(𝑝) − 0.25𝑙𝑜𝑔[𝐼(𝑝)] (15) and the values of 𝑧𝑖 , 𝑖 = 1,2,3 are found from (13) 6. formulation of cloudy fuzzy model with cloud type flexibility, all our cost parameters (𝐶̅) with indent rate( 𝑑) connected with the model represented as: { 𝐶̅ 𝑖 ̃ = < 𝐶𝑖1,𝐶𝑖2 ,𝐶𝑖3 >=< 𝐶𝑖2 (1 − 𝜌𝑐 1+𝑡 ),𝐶𝑖2 ,𝐶𝑖2 (1 + 𝜎𝑐 1+𝑡 ) > �̃� =< 𝑑1,𝑑2,𝑑3 >=< 𝑑2 (1 − 𝜌𝑑 1+𝑡 ),𝑑2,𝑑2 (1 + 𝜎𝑑 1+𝑡 ) > (16) where 𝜌𝑐 ,𝜎𝑐 ,𝜌𝑑 ,𝜎𝑑 are fuzzy system deviation parameters for cost vector and demand rate respectively. then the cloudy fuzzy problem will be of the form (11) whose fuzzy cost parameters (𝐶̅) and fuzzy demand rate ( �̃�) follow the membership function as per subsection 2.3. simultaneously, the fuzzy order quantity, fuzzy production rate, and fuzzy pollution level are of the form given in (17). { �̃� =< 𝑞1,𝑞2,𝑞3 >=< 𝑑1𝜏1,𝑑2𝜏1,𝑑3𝜏1 > 𝑝 =< 𝑝1,𝑝2,𝑝3 >=< 𝑑1𝜏1𝛿(1 − 𝑒 −𝛿𝜏1),𝑑2𝜏1𝛿(1 − 𝑒 −𝛿𝜏1),𝑑3𝜏1𝛿(1 − 𝑒 −𝛿𝜏1) > �̃� =< 𝑦1,𝑦2,𝑦3 > =< 0.45 + 0.01𝑝1 − 0.25𝑙𝑜𝑔𝑝3 ,0.45 + 0.01𝑝1 − 0.25𝑙𝑜𝑔𝑝3 ,0.45 + 0.01𝑝1 − 0.25𝑙𝑜𝑔𝑝3 > (17) 6.1. defuzzification of cloudy fuzzy model from equation (11), our fuzzy problem has been transformed into a similar crisp problem using equation (7). all our fuzzy components are represented in (16-17). we have replaced the respective index parameter with mentioned constraints and might be presented as 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝐼(�̃�) = 1 2𝑡1 ∫ (𝑧1 + 2𝑧2 + 𝑧3) 𝑡1 0 (18) subject to the constraints a study on pollution sensitive sponge iron based production transportation model … 233 { 𝜏2 = 3 2 𝜏1, 𝜏3 = 5 2 𝜏1, 𝐼(d̃) = 𝑑2 + 𝑑2 4 (𝜎𝑑 − 𝜌𝑑) log(1+𝑡1) 𝑡1 , 𝐼(�̃�) = 𝑑2𝑡1 + 𝑑2 4 (𝜎𝑑 − 𝜌𝑑) log(1 + 𝑡1) , 𝐼(𝑝) = 𝛿 1−𝑒−𝛿𝑡1 [𝑑2𝑡1 + 𝑑2 4 (𝜎𝑑 − 𝜌𝑑) log(1 + 𝑡1)] 𝐼(�̃�) = 0.45 + 0.01[ 𝛿 1−𝑒−𝛿𝑡1 {𝑑2𝑡1 + 𝑑2 4 (𝜎𝑑 − 𝜌𝑑)log(1 + 𝑡1)}] −0.25log[ 𝛿 1−𝑒−𝛿𝑡1 [𝑑2𝑡1 + 𝑑2 4 (𝜎𝑑 − 𝜌𝑑) log(1 + 𝑡1)]] (19) (for details see appendix 2-3) and the values of new 𝑧𝑖 , 𝑖 = 1,2,3 can be obtained with the replacement of fuzzy components given in (16-17) into the relations (13). 6.2. solution algorithm here, we shall develop a solution algorithm for solving the model under crisp, general fuzzy and cloudy fuzzy environment. step 0: start. step 1: set the main nonlinear equality constrained crisp arithmetic optimization problem 𝑍(𝑋) stated in equation (9). step 2: optimize the crisp problem and store the results at 𝑋0 = (𝑍0 ,𝑌0). step 3: using the results of step 2, formulate the non-linear problem via general fuzzy system in (11) and (12) and solve the defuzzified problem (14) and (15) via yager’index method. step 4: store the results obtained from step 3 at 𝑋1 = (𝑍1 ,𝑌1). step 5: formulate the problem (9) in cloudy fuzzy system in (17) and solve the defuzzified cloudy system at (18) subject to the constraints (19). step 6: store the results obtained from step 5 at 𝑋2 = (𝑍2 ,𝑌2). step 7: compare the solutions by computing the inequalities 𝑋0 < 𝑋1 < 𝑋2 or 𝑋0 > 𝑋1 > 𝑋2 or 𝑋0 > 𝑋1 < 𝑋2 etc. step 8: take optimum solution 𝑋2 when 𝑋0 > 𝑋1 > 𝑋2. step 9: end. the pseudo code of c programming is given below. _______________________________________________________________ #include <stdio.h> #include <math.h> #include <stdlib.h> #include <time.h> void main() { int i,lower =100, upper=999, count=10; float t1[count],hp=5.0,d=600.0,lambda[count],p[count],z[count],del=0.01, cd=10.0, cp=327.56, cpol=43.89,k1=10000,ct=3.5,l=600.0,cc=417.0,k2=10000, hr=4,f1,f2,f3,f4,f5,f6,f7,f8,f9; for(i=0;i<count;i++) { t1[i]=(float)((rand()%(upper-lower+1))+lower)/1000; } for(i=0;i<count;i++) { bhattacharya et al./decis. mak. appl. manag. eng. 5 (1) (2022) 225-245 234 lambda[i]=del*t1[i]/(1-(exp(-del*t1[i]))); } for(i=0;i<count;i++) { p[i]=d*lambda[i]; } for(i=0;i<count;i++) { f1=(1+((exp(-del*t1[i]))-1)/(del*t1[i]))/del; f2=del*f1; f3 = f1; f4 =1; f5 =1/t1[i]; f6 = f5; f7 = 0.5*t1[i]; f8=0.00424628*l; f9=0.0000431445*l; z[i]=p[i]*(hp*f1+cd*f2+cp*f3+cpol*f4)+k1*f5+k2*f6+d*(hr*f7+ct*f8+cc*f9); } float min_z = z[0]; int min_index = 0; for(i=1;i<count;i++) { if(min_z>z[i]) { min_z=z[i]; min_index=i; } } float min_t1=t1[min_index]; float min_p = p[min_index]; float min_lambda[min_index]; float t2=1.5*min_t1; float t3 = 2.5*min_t1; float q = d*min_t1; float y = 0.45 + (0.01*min_p)-(0.25*log(min_p)); printf("z\t\t%f\n",min_z); printf("t1\t\t%f\n",min_t1); printf("t2\t\t%f\n",t2); printf ("t3\t\t%f\n",t3); printf ("p\t\t%f\n",min_p); printf ("q\t\t%f\n",q); printf ("y\t\t%f\n",y); printf ("d\t\t%f\n",d); } ________________________________________________________________________ 6.3. schematic diagram here we include a schematic diagram of the proposed study which shows the novelty of the article (given in figure 1). a study on pollution sensitive sponge iron based production transportation model … 235 input all cost components and demand production sector transportation inventory management model solve the model via crisp technique and store results in (x1, z1 ) solve the model via general fuzzy technique and store results in ( x2 , z2 ) solve the model via cloudy fuzzy technique and store results in ( x 3 , z3 ) take min and store it in z along with optimum design variable x get the optimal result ( x , z) pollution generation consider non-random uncertainty of the parameters consider non-random uncertainty and learning experiences figure 1. schematic diagram of the study 7. numerical illustration from the data set mentioned in table 1 (subsection 3.1) and using the pollution function, the obtained minimized results have been displayed in table 2. also, the computed results using general fuzzy and cloudy fuzzy of the problem related to sc cost have been recorded in table 2. we have considered fuzzy system parameters (𝜌𝑐 ,𝜎𝑐 ,𝜌𝑑 ,𝜎𝑑) = (0.3,0.1,0.2,0.1) for our numerical computations. table 2. minimized solutions of sc model under different environments model 𝑝∗ (mt) 𝑦∗ (%) 𝜏1 ∗ (year) 𝜏2 ∗ (year) 𝜏3 ∗ (year) 𝑞∗ (mt) 𝑧∗ ($) 𝑍∗ − 𝑍∗ 𝑍∗ × 100% crisp 611.37 4.96 0.3768 0.5652 0.9620 226.07 117542.80 0 general fuzzy 596.12 4.82 0.3779 0.5668 0.9447 221.06 114160.20 -2.88 cloudy fuzzy 598.34 4.83 0.3751 0.5626 0.9377 220.27 111893.40 -4.81 bhattacharya et al./decis. mak. appl. manag. eng. 5 (1) (2022) 225-245 236 table 2 represents the optimal sc expenditure for minimal order quantity, cycle time, contamination level, and manufacturing rate for three separate cases which are crisp, general as well as cloudy fuzzy systems. cloudy fuzzy results minimum sc expenditure $ 111893.40 for contamination share of 4.83% for the manufacturing time 0.3751 year with 220.27 mt of customer order. the sc expenditure grows to $ 114160.20 while negligible minimization of the pollution level reaches to 4.82% for 221.06 mt customer order using a normal fuzzy system. the crisp model of sc is remarkably costly ($ 117542.80) when air contamination level reaches to 4.96% for the customer order of 226.07 mt. comparing with the crisp optimal solution, the sc cost-benefit for the cloudy fuzzy model becomes 4.81% which is superior to the 2.88% that was found for the general fuzzy system. 8. sensitivity analysis after obtaining the best efficient cost in cloudy fuzzy, the sensitivity dependence for the same has been investigated. for observing the variation of sc cost with contamination level, customer order, manufacturing rate with several expenditure parts, all fuzzy variables (𝜌𝑐,𝜎𝑐, 𝜌𝑑,𝜎𝑑) have been changed with (+50%, +30%, -30%, -50%) accordingly and the outcomes have represented in table 3. table 3. sensitivity study with % variation of (𝜌𝑐,𝜌𝑑,𝜎𝑐,𝜎𝑑) fuzz y para mete rs % chan ge 𝜏1 ∗ (year) 𝜏2 ∗ (year) 𝜏3 ∗ (year) 𝑦∗ (%) 𝑞∗ (mt) 𝑧∗ ($) 𝑝∗ (mt) 𝑍∗ − 𝑍∗ 𝑍∗ × 100% 𝑌∗ − 𝑌∗ 𝑌∗ × 100% 𝜌𝑐 0.3 +50 0.372 0.558 0.930 4.83 218.42 108570.4 598.24 -7.63 -2.56 +30 0.373 0.560 0.933 4.83 219.13 109900.3 598.28 -6.5 -2.54 -30 0.377 0.565 0.942 4.84 221.38 113884.5 598.41 -3.11 -2.5 -50 0.378 0.567 0.946 4.84 222.10 115210.8 598.45 -1.98 -2.5 𝜎𝑐 0.1 +50 0.376 0.564 0.940 4.84 220.75 113211.4 598.37 -3.68 -2.52 +30 0.376 0.563 0.939 4.84 220.56 112684.2 598.36 -4.13 -2.52 -30 0.375 0.562 0.937 4.84 219.98 111102.4 598.33 -5.48 -2.52 -50 0.374 0.561 0.936 4.84 219.79 110575.0 598.32 -5.93 -2.52 𝜌𝑑 0.2 +50 0.561 0.842 1.403 4.78 323.30 116773.3 592.50 -0.65 -3.65 +30 0.499 0.749 1.248 4.80 289.86 114235.8 595.15 -2.81 -3.15 -30 0.374 0.560 0.934 4.91 222.26 112626.8 606.08 -4.18 -1.03 -50 0.373 0.559 0.932 4.96 223.59 113115.2 611.25 -3.77 -0.02 𝜎𝑑 0.1 +50 0.374 0.560 0.934 4.90 221.74 112786.4 604.78 -4.05 -1.27 +30 0.374 0.561 0.935 4.87 221.15 112429.4 602.21 -4.35 -1.77 -30 0.442 0.663 1.105 4.82 258.01 112359.9 596.85 -4.41 -2.8 -50 0.481 0.722 1.203 4.81 279.99 113321.5 595.74 -3.59 -3.02 table 3 shows that all the cloudy fuzzy system parameters are moderately sensitive relative to the crisp optimal solution. our range of manufacturing run time, cycle time, air contamination share, the customer order, the manufacturing rate, and the average inventory cost assume value between (0.3719-0.5611) year, (0.9298-1.4028) year, (4.779-4.959) %, (218.42-323.30) mt, (592.50-611.25) mt and $ (108570.40116773.30) respectively. the overall cost-benefit lies within (0.65, 7.63) % and the contamination change within (0.02, 3.65) %. by this study, we also notice that the a study on pollution sensitive sponge iron based production transportation model … 237 maximum cost-benefit occurs from the 50% increment of all left fuzzy deviation parameters of all unit costs and the maximum contamination reduction occurs due to the 50% increment of left fuzzy deviation parameter of demand rate explicitly. 9. graphical illustrations different types of figures depend on the obtained outputs on several optimized solutions represented in table 2 and table 3 has been drawn. figure 2. different inventory cost for separate cases figure 2 shows minimum sc cost occurs due to cloudy fuzzy systems rather than crisp and general fuzzy systems. in the cloudy fuzzy system, the average sc cost takes a value around $112000 whereas it takes jumps to around $114000 and $117000 corresponding to the general fuzzy and crisp model respectively. figure 3. average sc expenditure with different pollution figure 3 represents the dependence of average inventory expenditure with different contamination levels while other parameters remain independent. with contamination level reaches 4.833%, the expenditure goes to a minimum. but, at the smaller contamination level of 4.8 %, the average inventory expenditure reaches its maximum value. so, in place of achieving a cleaner environment, we are bound to pay more. the strong dependence of sc expenditure on contamination level is represented in the above graph. 108000 110000 112000 114000 116000 118000 crisp general fuzzy cloudy fuzzy a v e ra g e s c c o st ( $ ) different methodology 108000 110000 112000 114000 116000 a v e ra g e s c c o st ( $ ) pollution (%) bhattacharya et al./decis. mak. appl. manag. eng. 5 (1) (2022) 225-245 238 figure 4. variation of average sc cost due to production run time expenditure of inventory with manufacturing time has represented in figure 4. the least value for sc cost has been obtained with 0.3719 years of manufacturing time. similarly, with an increase in production time, the greatest average sc cost has been obtained. figure 5. variation of pollution rate with respect to production rate naturally, the contamination level reaches its minimum with a minimum production rate. we see in figure 5 that the pollution level remains almost stable within the manufacturing limit (598.24-598.45) mt. the pollution level increases slowly during the production range (592.5-598.24) mt. but when the production rate is increased (more than 598.45 mt) then the level of pollution is also increased almost exponentially. 108000 109000 110000 111000 112000 113000 114000 115000 116000 117000 a v e ra g e s c c o st ( $ ) production run time (year) 4.75 4.78 4.81 4.84 4.87 4.9 4.93 4.96 p o ll u ti o n ( % ) production rate (mt) a study on pollution sensitive sponge iron based production transportation model … 239 figure 6. variation of order quantity with respect to production run time figure 6 reveals the dependence of customer order (mt) with manufacturing time (year). we see that the order quantity curve has increased with the maximum slope with the rise of production run time of more than 0.54 years. when the production run time lies within 0.37-0.38 years then the order quantity curve takes a horizontal line by taking value near 220 mt. but interestingly, the order quantity curve gets an arc upwards (concave) having the range 220-290 mt with manufacturing time 0.38 ~ 0.5 year exclusively. figure 7 shows the surface-like structure of sc cost due to changes in production run time and the pollution level. we see that the cost is minimum at the pollution level 4.94% and production run time 0.55 years approximately. also, the average sc cost reaches its maximum at the minimum pollution level 4.78% and the production run time 0.55 years approximately. figure 7. dependence of manufacturing cost with production run time and pollution 210 260 310 360 0.37 0.38 0.44 0.48 0.5 0.56 o rd e r q u a n ti ty ( m t ) production run time (year) a v e ra g e s c c o st ( $ ) bhattacharya et al./decis. mak. appl. manag. eng. 5 (1) (2022) 225-245 240 the surface curve gets a bend at the middle of the graph the optimum of which parametric values are production run time near 0.45 years, pollution level near 4.84 %, and average sc cost near $1.12 x 105. figure 7. change of mean sc cost with customer order and manufacturing time figure 8 represents the dependence of mean sc expenditure on customer order and manufacturing time. with approximately 0.98 years of manufacturing time and a customer order of 220 mt the average expenditure reaches a minimum, but it attains the greatest value for 0.56-year manufacturing time and 320 mt customer order. 10. merits and demerits in this section we shall discuss the merits and demerits of the proposed approach. merits: i) pollution sensitive sc model has been analysed intelligently with the help of cloudy fuzzy number that gives the measure of learning experiences over time. ii) we incorporated a sensitivity analysis table to show the limitations and stability of the parameters involving in the model. iii) a comparative study with solution algorithm has been done to show the superiority of the optimum results in cloudy fuzzy system. iv) our real case study data supports the learning model with new operational method. v) any decision maker can easily use this method before going to final decision. demerits: since, this model is solely devoted to learning theory so, lack of information gathering can harm the model. however, the numerical study is not checked by some other learning theory to draw an absolute global decision. a v e ra g e s c c o st ( $ ) figure 8. variation of system cost with respect to production time and order quantity a study on pollution sensitive sponge iron based production transportation model … 241 11. conclusions in this study, we have developed a two-channel pollution level on a two-layer supply chain deteriorated sponge iron manufacturing model under a cloudy fuzzy environment. for channel 1 of pollution corresponds to the number of items produced per time and that for the second channel it depends upon the distances travelled and the items transported. the production run time and the demand quantity may matter over the minimization of sc cost and pollution control. the basic novelty of the study is the incorporation of non-random uncertainty of the parameters of the model. we wish to know the average inventory cost, cycle time, pollution level (%) when the demand rate and all cost parameters are getting nonrandomly uncertain. this problem has been solved by using general fuzzy set theory. another important contribution of this article is to study the effect of learning experience in the inventory system. we want to study at what extent the decision maker could be able to reduce the average inventory cost and the pollution reduction for sustainable production within specific cycle time via learning experiences in the inventory process. to incorporate learning experience, we have introduced cloudy fuzzy number for the system parameters. we have also developed a solution algorithm and pseudo code of c programming to solve the mathematical model in different approaches like crisp, general fuzzy and cloudy fuzzy system. moreover, we have incorporated a sensitivity analysis of the parameters to study the limitations and their stability for model validation. the table shows that we can control 3.65 % pollution by controlling the demand cut of 50 %. the restoration of sc cost may rise to 7.63 % by the control of all unit cost components cut by 50% each. we see that the minimum average inventory cost $117542.80 occurs in cycle time 0.962 year due to production 611.37 mt (metric ton), order quantity 226.07 mt with pollution contribution 4.96% in crisp problem. in general, fuzzy situation, the minimum average inventory cost $114160.20 occurs in cycle time 0.945 year due to production 596.12 mt (metric ton), order quantity 221.06 mt with pollution contribution 4.82% exclusively. in cloudy fuzzy approach, the minimum average inventory cost $111893.40 occurs in cycle time 0.938 year due to production 598.34 mt (metric ton), order quantity 220.27 mt with pollution contribution 4.83% alone. so, the decision maker could be able to minimize the average inventory cost up to 2.88 % in general fuzzy approach and 4.81 % by applying cloudy fuzzy technique respectively. also, for sustainability, a situation has come to balance production-demand-pollution-production run time altogether and this is only possible when the dm opts cloudy fuzzy system. however, some common managerial insights from this study can be drawn as follows: i) cloudy fuzzy approach is better than crisp and general fuzzy approach. ii) the decision maker/manager could not ignore the issue of environmental pollution that are being deposited into the environment day by day from both the production process and transportation process. rather, it should have to accept before going to furnish final decision over production-based inventory management problems. iii) increase of production (order quantity) carries some pollution in environment. so, it can be reduced through sustainable production. scope of future work various model can be studied using different fuzzy systems related to learning theory like monsoon fuzzy theory, fuzzy approximate reasoning, doubt fuzzy approach etc. in near future. taking different types of fuzzy set like neutrosophic fuzzy sets, some bhattacharya et al./decis. mak. appl. manag. eng. 5 (1) (2022) 225-245 242 other complex and more realistic models can be developed in future. also, several pollution function may be developed in the way of developing new research. author contributions: conceptualization, s. k. de and k. bhattacharya; methodology, p. p. bhattacharya; software, p. p. bhattacharya; validation, p. p. bhattacharya, k. bhattacharya and s. k. de; formal analysis, p. p. bhattacharya; investigation, p. p. bhattacharya; resources, p. p. bhattacharya; data curation, p. p. bhattacharya; writing—original draft preparation, p. p. bhattacharya; writing—review and editing, s. k. de; visualization, k. bhattacharya; supervision, s. k. de; project administration, s. k. de; funding acquisition, not applicable. all authors have read and agreed to the published version of the manuscript. funding: this research received no external funding. acknowledgments: the authors are thankful to the honourable editor-in chief, associate editors and the respected anonymous reviewers for their constructive comments to improve the quality of the article. conflicts of interest: the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. appendix a a.1: 𝜏3 − 𝜏2 = 2(𝜏2 − 𝜏1), 𝜏1 = 2(𝜏2 − 𝜏1) this implies 𝜏2 = 3 2 𝜏1, 𝜏3 = 5 2 𝜏1 a.2: the cloudy fuzzy objective value �̃� = 〈𝑧1,𝑧2,𝑧3〉 can be reduced as follows: { 𝑧1 = d2τ1δ (1−e−δτ1) {( ℎ𝑝 𝛿 + 𝐶𝑑 + 𝐶𝑝 𝛿 )(1 + 𝑒−𝛿𝜏1−1 𝛿𝜏1 ) + 𝐶𝑝𝑜𝑙}(1 − 𝜌𝑑 1+t )(1 − 𝜌𝑐 1+t ) + ( 𝐾1 𝜏1 + 𝐾2 𝜏1 )(1 − 𝜌𝑐 1+t ) +𝑑2 { ℎ𝑟𝜏1 2 + 𝐶𝑡0.00424628𝑙 + 𝐶𝑐0.0000431445𝑙}(1 − 𝜌𝑑 1+t )(1 − 𝜌𝑐 1+t ) 𝑧2 = d2τ1δ (1−e−δτ1) {( ℎ𝑝 𝛿 + 𝐶𝑑 + 𝐶𝑝 𝛿 )(1 + 𝑒−𝛿𝜏1−1 𝛿𝜏1 ) + cpol} + k1 τ1 + k2 τ1 +d2 { hrτ1 2 + ct0.00424628l + cc0.0000431445l} z3 = d2τ1δ (1−e−δτ1) {( ℎ𝑝 𝛿 + 𝐶𝑑 + 𝐶𝑝 𝛿 )(1 + 𝑒−𝛿𝜏1−1 𝛿𝜏1 ) + 𝐶𝑝𝑜𝑙}(1 + 𝜎𝑑 1+t )(1 + 𝜎𝑐 1+t ) + ( 𝐾1 𝜏1 + 𝐾2 𝜏1 )(1 + 𝜎𝑐 1+t ) +𝑑2 { ℎ𝑟𝜏1 2 + 𝐶𝑡0.00424628𝑙 + 𝐶𝑐0.0000431445𝑙}(1 + 𝜎𝑑 1+t )(1 + 𝜎𝑐 1+t ) now, 1 2 ∫ (𝑍1 + 2𝑍2 + 𝑍3)𝑑𝑡 𝑡1 0 = 1 2 ∫ [ 𝑑2𝜏1𝛿 (1−𝑒−𝛿𝜏1) {( ℎ𝑝 𝛿 + 𝐶𝑑 + 𝐶𝑝 𝛿 )(1 + 𝑒−𝛿𝜏1−1 𝛿𝜏1 ) + 𝐶𝑝𝑜𝑙} × {2 + (1 − 𝜌𝑑 1+𝑡 )(1 − 𝜌𝑐 1+𝑡 ) + 𝑡1 0 (1 + 𝜎𝑑 1+𝑡 )(1 + 𝜎𝑐 1+𝑡 )} + ( 𝐾1 𝜏1 + 𝐾2 𝜏1 )(4 + 𝜎𝑐−𝜌𝑐 1+𝑡 ) + 𝑑2 { ℎ𝑟𝜏1 2 + 𝐶𝑡0.00424628𝑙 + 𝐶𝑐0.0000431445𝑙} × {2 + (1 − 𝜌𝑑 1+𝑡 )(1 − 𝜌𝑐 1+𝑡 ) + (1 + 𝜎𝑑 1+𝑡 )(1 + 𝜎𝑐 1+𝑡 )}]𝑑𝑡 a study on pollution sensitive sponge iron based production transportation model … 243 = 1 2 ∫ [[( 𝑑2𝜏1𝛿 (1−𝑒−𝛿𝜏1) ){( ℎ𝑝 𝛿 + 𝐶𝑑 + 𝐶𝑝 𝛿 )(1 + 𝑒−𝛿𝜏1−1 𝛿𝜏1 ) + 𝐶𝑝𝑜𝑙} + 𝑑2 { ℎ𝑟𝜏1 2 + 𝑡1 0 𝐶𝑡0.00424628𝑙 + 𝐶𝑐0.0000431445𝑙}] × {4 + (𝜎𝑑+𝜎𝑐)−(𝜌𝑑+𝜌𝑐) 1+𝑡 + (𝜎𝑑𝜎𝑐+𝜌𝑑𝜌𝑐) (1+𝑡)2 } + ( 𝐾1 𝜏1 + 𝐾2 𝜏1 )(4 + 𝜎𝑐−𝜌𝑐 1+𝑡 )]𝑑𝑡 = [[( 𝑑2𝜏1𝛿 (1−𝑒−𝛿𝜏1) ){( ℎ𝑝 𝛿 + 𝐶𝑑 + 𝐶𝑝 𝛿 )(1 + 𝑒−𝛿𝜏1−1 𝛿𝜏1 ) + 𝐶𝑝𝑜𝑙} + 𝑑2 { ℎ𝑟𝜏1 2 + 𝐶𝑡0.00424628𝑙 + 𝐶𝑐0.0000431445𝑙}] × 1 2 {4𝑡1 + ((𝜎𝑑 + 𝜎𝑐) − (𝜌𝑑 + 𝜌𝑐))𝑙𝑜𝑔(1 + 𝑡1) + 𝑡1 1+𝑡1 (𝜎𝑑𝜎𝑐 + 𝜌𝑑𝜌𝑐)} + 1 2 ( 𝐾1+𝐾2 𝜏1 ){4𝑡1 + (𝜎𝑐 − 𝜌𝑐)𝑙𝑜𝑔(1 + 𝑡1)}] a.5 adding left and right 𝛼cuts of membership function of cloudy fuzzy demand �̃� we get, 𝐿−1(𝛼,𝑡) + 𝑅−1(𝛼,𝑡) = 𝑑1 + 𝑑3 + 𝛼(−𝑑1 + 2𝑑2 − 𝑑3) now, 𝐼(�̃�) = 1 2𝑡1 ∬ {𝑑1 + 𝑑3 + 𝛼(−𝑑1 + 2𝑑2 − 𝑑3)} 𝛼=1 𝛼=0 𝑑𝛼𝑑𝑡 = 1 2𝑡1 ∫ [{(𝑑1 + 𝑑3)𝛼} 1 0 − {(𝑑1 − 2𝑑2 + 𝑑3) 𝛼2 2 } 1 0 ]𝑑𝑡 𝑡1 𝑡=0 = 1 2𝑡1 ∫ [(𝑑1 + 𝑑3) − 1 2 (𝑑1 − 2𝑑2 + 𝑑3)]𝑑𝑡 𝑡1 𝑡=0 = 1 2𝑡1 ∫ 1 2 [(𝑑1 + 𝑑3 + 2𝑑2)]𝑑𝑡 𝑡1 𝑡=0 = 1 2𝑡1 ∫ 1 2 [𝑑2 {(1 − 𝜌 1+𝑡 ) + (1 + 𝜎 1+𝑡 )} + 2𝑑2]𝑑𝑡 𝑡1 𝑡=0 = 1 2𝑡1 ∫ 𝑑2 2 [4 + 𝜎−𝜌 1+𝑡 ]𝑑𝑡 𝑡1 𝑡=0 = 1 2𝑡1 [ 𝑑2 2 {4𝑡1 + (𝜎 − 𝜌)𝑙𝑜𝑔(1 + 𝑡1)}] = ( 1 2𝑡1 × 𝑑2 2 × 4𝑡1) + 1 2𝑡1 × 𝑑2 2 × (𝜎 − 𝜌)𝑙𝑜𝑔(1 + 𝑡1) = 𝑑2 + 𝑑2 4 (𝜎 − 𝜌) 𝑙𝑜𝑔(1+𝑡1) 𝑡1 = 𝑑2 [1 + 𝜎 − 𝜌 4 𝑙𝑜𝑔(1 + 𝑡1) 𝑡1 ] references aarthi, h. t. 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(1965). fuzzy sets. information control, 8, 338-356. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0313052021b * corresponding author. e-mail address: bipradasbairagi79@gmail.com (b. bairagi) a new framework for green selection of material handling equipment under fuzzy environment bipradas bairagi*1 1 department of mechanical engineering, haldia institute of technology, india received: 1 april 2022; accepted: 13 may 2022; available online: 13 may 2022. original scientific paper abstract. in the rapidly changing global circumstances, managements of industrial organizations are making decisions for their survival in business atmosphere in future. decision makers in industries are steering their respective organizations towards for appropriate decision making satisfying the condition of ‘go green’. appropriate decision making in fuzzy environment is always a hard task. the current investigation explores a new multi criteria decision making approach for green selection of material handling equipment under fuzzy environment. the proposed technique has the capability of capturing effects of economical, environmental and social factors of benefit, non-benefit and target based criteria under uncertainty and vague information. the proposed method is illustrated with a suitable example on material handling equipment selection under fuzzy environment. the result clearly shows that the proposed technique is useful and effective in the decision making process regarding green material handling selection under fuzzy environment. key words: mcdm, green material handling equipment selection, decision making, fuzzy sets. 1. introduction in today’s highly competitive global market scenario industrial organizations worldwide are facing difficulty and their existence as well as survival is under uncertainty. in this circumstance the industrial organizations are persistently seeking means of resolving the difficulties and trying to attain satisfactory condition of going green. selection of material handling equipment under multi criteria decision making environment plays a crucial role for the sustainable development especially for manufacturing organizations. substantial development needs consideration of economical, environmental and social factors. green material handling equipment selection procedure gives emphasis on meeting requirements of present without compromising the capability of future generations to meet their needs. therefore for bairagi/decis. mak. appl. manag. eng. (2022) 2 gaining sustainable development the decision makers consciously incorporates social economical or environmental criteria in material handling selection. a wide range of literature on material handling equipment selection shows that the previous search works in this regard have not given enough attention. new research with proper decision making attitude is still essential to show top level management the ray of hope for their survival and existence in the tough competitive business world. selection of material handling equipment for manufacturing application is essential. previous researchers’ attention is inadequate to select material handling equipment considering green factors. a broad literature survey on material handling equipment selection explores this deficit of research work on the field. goswami and behera (2021) made an investigation for the capability and applicability of two well-known mcdm approaches aras and copras for the evaluation and selection of conveyors, agv and robots as material handling equipment. soufi et al. (2021) introduced an ahp based mcdm methodology for the evaluation and selection of material handling equipment to be utilized in manufacturing systems. satyam et al. (2021) applied a multi attribute decision making approach for evaluation and selection of conveyors as material handling equipment. nguyen et al. (2016) advocated a combined multi criteria decision making model for the evaluation and selection of conveyor as the material handling equipment based on fuzzy analytical hierarchy process and fuzzy aras with vague and imprecision information. mathewa and sahua (2018) made a comparison among the novel multi-criteria decision making approaches by solving a problem on material handling equipment selection. none of the above researchers considered green issues in their investigations. fonseca et al. (2004) developed a prototype expert system for the purpose industrial conveyor selection. poon et al. (2011) addressed selection and allocation of mhe for stocha production material demand problems by using genetic algorithm. lashkari et al. (2004) proposed an integrated model of operation allocation and material handling equipment selection in cellular manufacturing systems. sujono and lashkari (2007) presented a method for determining the operation allocation and material handling equipment. chan et al. (2001) advocate an intelligent material handling equipment selection advisor (mhesa) composed of three modules. kulak (2005) proposed a decision support system named fuzzy multi-attribute material handling equipment selection (fumahes). tuzkaya et al. (2010) developed an integrated fuzzy multi-criteria decision making methodology combining fuzzy sets, analytic network process and (promethee) for mhe selection. chatterjee et al. (2010) gave solution the robot selection problem using two suitable multi-criteria decision-making methods and compared the relative performances. shih (2008) presented a group topsis to select robots by incremental benefit–cost ratio. the gap analysis of the above literature review exposes that, despite the fact that previous researchers have attempted to apply mcdm techniques for selection of material handling equipment, still, this effort is inadequate for exhaustive and extensive decision making regarding green selection of proper material handling devices (mhd) from several available alternatives under multiple criteria decision making. the literature survey clearly shows that the previous researchers did not address the problem of green selection of material handling at a satisfactory level. the objective of the present study is to develop a decision making framework that can be an essential tool in solving problem regarding green selection of material handling equipment for industrial organizations. a new framework for green selection of material handling equipment under fuzzy… 3 the motivation of the research work is our environment. our planet is changing constantly. due to rapid global industrialization, our surrounding environment is being constantly harmed from pollutant substance emitted from industries. and its adverse effect is falling on all living creature including human society. we the human being can steer the direction of change towards the welfare of the beautiful planet only by making careful and wise decision in every step of our practical life. we can reduce the harmful effect on the earth caused by the use of different material handling equipment in industry. the lack of suitable decision making frame work in the open literature has ignited the motivation to accomplish the current research work. the rest of the paper is organized as follows. section 2 presents the proposed algorithm. section 3 describes an empirical example on green material handling equipment selection. section 4 is dedicated for essential concluding remarks. 2. proposed algorithm this research work proposes a novel algorithm consists of 8-steps as follows. step 1: construction of performance rating matrix: a matrix consisting of performance rating of alternatives with respect to criteria is estimated by the experts or decision makers. since the decision is to make under uncertainty, linguistic variables are recommended for estimation of performance of alternatives. seven degrees of linguistic variables with abbreviation and corresponding triangular fuzzy number (tfn) are suggested for performance estimation as presented in table 1. the performance rating matrix is denoted by   nmijpr lvm   (1) lvij denotes performance rating of alternative ai with respect to criterion cj in linguistic variable. here, m and n represents number of alternatives and criteria respectively. step 2: construction of weight matrix: in a decision making process different criteria may have different importance. this importance weight is estimated by the decision makers on the basis of their unanimous decision making attitude through discussion. seven degrees of linguistic variables with abbreviation and corresponding triangular fuzzy number (tfn) suggested for weight estimation are presented in table 2. the weight matrix for measuring criteria importance is formed as follows.   njw lwm   11 (2) lw1j denotes the linguistic weight for criterion cj. here n is number of criteria. table1. seven degrees of linguistic variables for assessing performance description abbreviation triangular fuzzy numbers extremely high eh (1,1,1) very high vh (0.8,0.9,1) high h (0.6,0.7,0.8) medium m (0.4,0.5,0.6) low l (0.2,0.3,0.4) very low vl (0,0.1,0.2) extremely low el (0,0,0) bairagi/decis. mak. appl. manag. eng. (2022) 4 step 3: conversion of rating from linguistic terms to tfn: linguistic rating is not suitable for decision making under multiple conflicting criteria. that is why they are required to be transformed into fuzzy numbers. the current approach suggests triangular fuzzy numbers for making calculation simple and straightforward. table 1 shows the rating and equivalent fuzzy numbers. performance rating matrix in fuzzy numbers is presented as follows.     nm u ij m ij l ijnmijprf rrrrm   ,,~ (3) , l ijr , m ijr u ijr stand for lower , middle and upper value of respective tfn respectively. step 4: conversion of weight from linguistic terms to tfn: linguistic weight is not suitable for decision making under multiple conflicting criteria. that is why these are required to be changed into fuzzy numbers. the present approach suggests triangular fuzzy numbers for making calculation simple and straightforward. table 2 shows the rating and equivalent fuzzy numbers. weight matrix in fuzzy numbers is presented as follows.     nm u j m j l jnmjwf wwwwm   1111 ,, ~ (4) , l ijw , m ijw u ijw stand for lower , middle and upper value of respective tfn respectively. step 5: normalization of fuzzy performance rating: based on the desired value, criteria can be divided in 3 different categories, viz. benefit criteria (higher value is desired), non-benefit criteria (lower value is desired) and target based criteria (neither higher nor lower value is desired instead a certain target value is desired). normalization of fuzzy rating is implemented using appropriate formula depending on the nature of criteria. following formulae are recommended for executing normalization process. for benefit criteria:             u ij ji u ij u ij ji m ij u ij ji l ij ij r r r r r r r (max , )(max , )(max ,,, (5) for non-benefit criteria             u ij ji l ij u ij ji m ij u ij ji u ij ij r r r r r r r (max 1, )(max 1, )(max 1 ,,, (6) table 2. seven degrees of recommended linguistic weights of criteria description abbreviation triangular fuzzy number absolutely important ai (1,1,1) extremely important ei (0.8,0.9,1) very important vi (0.6,0.7,0.8) medium important mi (0.4,0.5,0.6) ordinarily important oi (0.2,0.3,0.4) slightly unimportant s (0,0.1,0.2) unimportant ui (0,0,0) a new framework for green selection of material handling equipment under fuzzy… 5 for target based criteria             )(max)(max 1, )(max)(max 1, )(max)(max 1 ,,,,,, u ij ji l ij u ij ji l t u ij ji m ij u ij ji m t u ij ji u ij u ij ji u t ij r r r r r r r r r r r r r (7) the normalization process is carried out in so as to convert all the fuzzy performance ratings in benefit sense irrespective of nature of criteria. step 6: weighted normalized rating: weighted normalized rating is the rating modified by respective criteria weight. this paper recommends following nonlinear integration of an alternative rating with each criterion weight. for benefit criteria eq. (8) is applied.       1 1 0 0 0 , , , , , max max max l m u j j ij w w wl m u ij ij ijwn ij l u u ij ij ij i j i j i j r r r r dx dx dx r r r                   (8) for non-benefit criteria eq. (9) is applied.       1 1 0 0 0, , , 1 , 1 , 1 max max max u m l j j ijw w wu m l ij ij ijwn ij u u u ij ij ij i j i j i j r r r r dx dx dx r r r                                                    (9) for target based criteria eq. (10) is used.             1 1 0 0 0 , , , , , , 1 , 1 , 1 max max max max max max u m l j j ij w w wu m lu m l ij ij ijwn t t t ij u u u u u u ij ij ij ij ij ij i j i j i j i j i j i j r r rr r r r dx dx dx r r r r r r                         (10) step 7: defuzzification of performance rating is accomplished using the following eq. (11)-eq. (13) for benefit criteria used eq. (11) is employed. (11) for non-benefit criteria       1 1 0 0 0 , , , 1 1 4 1 1 6 max max max u m l j j ij w w wu m l ij ij ijnw ij l u u ij ij ij i j i j i j r r r r dx dx dx r r r                                                     (12) for target based criteria             1 1 0 0 0 , , , , , , 1 1 4 1 1 6 max max max max max max u m l j j ij w w wu m lu m l ij ij ijt t t u u u u u u ij ij ij ij ij ij i j i j i j i j i j i j r r rr r r dx dx dx r r r r r r                             (13)       1 1 0 0 0 , , , 1 4 6 max max max l m u j j ij w w wl m u ij ij ijwn ij u u u ij ij ij i j i j i j r r r r dx dx dx r r r                    bairagi/decis. mak. appl. manag. eng. (2022) 6 step 8: performance index: performance index indicates responsible for measuring performance, ranking and selection. the current investigation advocates the following technique in calculation performance indicator (pi) by integrating individual contribution of each criterion towards the evaluation of alternatives.                                                      nbcj l ij w u ij ji l ij m j w u ij ji m ij u j w l ij ji u ij dx r r dx r r dx r r 0 , 1 0 , 1 0 , max 1 max 14 max 1 6 1 ,                            tbcj l ij w u ij ji l ij u ij ji l t m j w u ij ji m ij u ij ji m t u j w u ij ji u ij u ij ji u t dx r r r r dx r r r r dx r r r r 0 ,, 1 0 ,, 1 0 ,, maxmax 1 maxmax 14 maxmax 1 6 1 (14) hence pii denotes performance index of ith alternative. performance index has its beneficial sense, that is higher value is desirable. thus alternatives are arranged in the descending order of their respective pi value. the alternative with the highest value of performance index is considered the best alternative. the alternative with least value of performance index is considered as the worst alternative and so on. 3. illustrative example a south-east asian manufacturing organization plans to select material handling equipment including the factors related to green. a selection committee comprising of manager, experts and decision makers is formed. the committee chooses seven selection criteria keeping green selection in view under fuzzy environment. safety (c1), operating friendliness (c2), environment friendliness (c3), robustness (c4), cost (c5), repeatability (c6), and human-interaction (c7) are the six selection criteria for material handling equipment. safety, operating friendliness are social factors, cost is economical and environment friendliness is environmental factors. these factors are selected keeping green factors in view. safety, operating friendliness, environment friendliness, and robustness are benefit criteria; cost and repeatability are non-benefit or cost criteria; whereas human-interaction is target based criteria. four alternative material handling equipment designated by a1, a2, a3 and a4 are preliminarily selected after initial screening for further evaluation. 3.1 estimation of linguistic rating a detailed question-answer interview is conducted for the decision making committee to construct a performance rating matrix in terms of linguistic variable as furnished in table 3. it is observed that criteria c1, c2, c3 and c4 are benefit criteria, c5 and c6 are non-benefit criteria and c7 is target based criteria (tbc). table 4 represents performance rating matrix in terms of linguistic variable. the alternative material handling equipment a1 is awarded linguistic performance rating viz. eh, vh, m, eh, m, vh, and l with respect to criteria c1, c2, c3, c4, c5, c6 and c7. eh (extremely high rating is better and desirable) under benefit criteria. m and vh are provided under non-benefit or cost criteria. comparatively, m (medium) is better than vh (very high). for c7, l (low) is expected value.                      bcj u ij w u ij ji u ij m j w u ij ji m ij l j w l ij ji l ij i dx r r dx r r dx r r pi 0 , 1 0 , 1 0 , maxmax 4 max6 1 a new framework for green selection of material handling equipment under fuzzy… 7 table 3. performance rating matrix in terms of linguistic variables estimated by decision making committee criteria benefit criteria non-benefit criteria tbc* a lt e rn a ti v e ai c1 c2 c3 c4 c5 c6 c7 a1 eh vh m eh m vh l a2 vh h eh m vh m eh a3 m vl eh vh h l vh a4 eh m vh mi vl eh m *tbc stands for target based criteria; target value for tbc is set to m (medium). the importance weights of criteria, c1, c2, c3, c4, c5, c6 and c7 are estimated as ai, ei, mi, oi, mi, ei, and ai in linguistic abbreviation. it implies that criteria c2 and criteria c6 are jointly given the highest importance. criterion robustness (c4) is assumed relatively less important. table 4. weight matrix in terms of linguistic variables estimated by decision making committee criteria benefit criteria non-benefit criteria tbc* c1 c2 c3 c4 c5 c6 c7 weight ai ei mi oi mi ei ai *tbc stands for target based criteria; target value for tbc is set to m (medium). 3.2 calculation and illustration since the decision is to make in fuzzy environment, therefore performance rating of alternative and the importance weights of selection criteria have been extracted in the form of linguistic variables. these linguistic variables are not directly suitable for decision making. so conversion of linguistic variables into corresponding fuzzy number is an essential step. the conversion scale for the performance rating are as follows: extremely high (1,1,1), very high (0.8,0.9,1), high (0.6,0.7,0.8), medium (0.4,0.5,0.6), low (0.2,0.3,0.4), very low(0,0.1,0.2) and extremely low (0,0,0). conversion of linguistic terms to tfn is performed and shown in table 5. table 5. fuzzy performance rating in terms of tfn benefit criteria non-benefit criteria tbc* ai c1 c2 c3 c4 c5 c6 c7 a1 (1,1,1) (0.8,0.9,1) (0.4,0.5,0.6) (0.2,0.3,0.4) (0.4,0.5,0.6) (0.8,0.9,1) (0.2,0.3,0.4) a2 (0.8,0.9,1) (0.6,0.7,0.8) (0.2,0.3,0.4) (0.4,0.5,0.6) (0.8,0.9,1) (0.4,0.5,0.6) (1,1,1) a3 (0.4,0.5,0.6) (0,0.1,0.2) (1,1,1) (0.8,0.9,1) (0.6,0.7,0.8) (0.2,0.3,0.4) (0.8,0.9,1) a4 (0.2,0.3,0.4) (0.4,0.5,0.6) (0.8,0.9,1) 0.4,0.5,0.6) (0,0.1,0.2) (1,1,1) (0.4,0.5,0.6) *tbc stands for target based criteria the conversion scale for the linguistic weight of criteria into triangular fuzzy number are as follows: absolutely important (1,1,1), extremely important(0.8,0.9,1), very important(0.6,0.7,0.8), medium important (0.4,0.5,0.6), ordinarily important (0.2,0.3,0.4), slightly unimportant, (0,0.1,0.2) and unimportant (0,0,0). conversion of weight from linguistic terms to triangular fuzzy number is presented in table 6. bairagi/decis. mak. appl. manag. eng. (2022) 8 table 6. weight matrix in terms of linguistic variables criteria benefit criteria non-benefit criteria tbc* c1 c2 c3 c4 c5 c6 c7 weight (1,1,1) (.8,0.9,1) (.4,0.5,0.6) (.2,0.3,0.4) (.4,0.5,0.6) (.8,0.9,1) (1,1,1) *tbc stands for target based criteria normalization of fuzzy performance rating is accomplished and the normalized performance fuzzy ratings are calculated by using respective eq.( 5)eq.(7). for example, normalized fuzzy performance rating for the alternative a1, with respect to criteria c1 is carried out as follows. the criterion safety (c1) is benefit criterion. therefore, eq. (5) has been used. the highest grade under the criterion c1 is (1,1,1). the upper point of the tfn is 1. then the normalized performance rating for this case is obtained by dividing the each of the point, lower, middle, upper with 1(maximum upper point under the criterion). the computed normalized performance rating is (1,1,1). similarly the rest of the normalized performance ratings are calculated. to take the account of the variation, different importance weights have been integrated with the already calculated normalized performance ratings. for the purpose, a new equation is introduced in the current proposed algorithm. weighted normalized rating is computed by using eq. (8) -eq. (10) and shown in table 7. table 7. weighted normalized fuzzy performance ratings benefit criteria non-benefit criteria tbc* ai c1 c2 c3 c4 c5 c6 c7 a1 (1,1,1) (.64,0.81, 1) (0.16,0.25,.36) (0.04,.09,0.16) (0.16,0.25,0.36) (0,0.09,0.2) (0.6,0.8, 1) a2 (0.8,0.9,1) (.48,0.63,.8) (0.08, .15,.24) (0.08,0.15,.24) (0.0,0.05,0.12) (0.32,0.45,0.6) (0.4,0.5,0.6) a3 (0.4,0.5,0.6) (0,0.09,0.2) (0.16, 0.5 ,0.6) (0.16,0.27,0.4) (0.08,0.15,0.24) (0.48,0.63,0.8) (0.6,0.6,0.6) a4 (0.2,0.3,0.4) (.32,0.45,.6) (0.32,.45, 0.6) (0.08,.15,0.24) (0.16,0.45,0.6) (0,0,0) (1,1,1) *tbc stands for target based criteria weighted normalized performance ratings have been expressed in terms of triangular fuzzy number. tfn is not suitable for making final decision regarding the performance evaluation of the alternative material handling equipment. therefore, it is essential to convert the fuzzy number into quantified corresponding crisp number. the process of converting the fuzzy number into crisp number is termed as defuzzification. conversely, the conversion process of crisp number into corresponding fuzzy number is termed as fuzzification. in the current study the defuzzification process is completed through the application of three different newly introduced appropriate equations. eq. (11) has been implemented for benefit criteria, eq. (12) has been used for cost or non-benefit criteria, eq. (13) has been applied for target oriented criterion c7. table 8 shows defuzzified weighted normalized ratings as calculated by using eq. (11)eq.(13). performance index is the index of an alternative material handling equipment that express extent of benefit over the cost as well as non beneficial performance. for the purpose of estimation of performance index of each of the alternative material handling equipment a new, integration oriented technique has been proposed in the investigation as presented in eq.(14). the performance index for each individual alternative has been shown in table 8. a new framework for green selection of material handling equipment under fuzzy… 9 table 8. defuzzified weighted normalized rating, performance index and rank defuzzified weighted normalized rating performance index benefit criteria non-benefit criteria tbc* rank ai c1 c2 c3 c4 c5 c6 c7 pi a1 1 0.816 0.256 0.096 0.256 0.096 0.8 3.32 1 a2 0.9, 0.636 0.156 0.156 0.056 0.456 0.5 1.96 4 a3 0.5 0.096 0.560 0.276 0.156 0.636 0.6 2.28 3 a4 0.3 0.456 0.456 0.156 0.456 0 1 2.82 2 *tbc stands for target based criteria theoretically, the lower limit and the upper limit of performance index associated with any alternative can be 0 (zero) and n (where n is the number of criteria) respectively. in other words, the range of the performance index may vary from 0 to n, that is zero is the lower limit and n (number of criteria) is the upper limit of the result (performance index). figure 1. performance indices of alternatives the performance indices of the alternative material handling equipment are diagrammatically represented in figure1, for improved visualization, enhanced clarification and better comparison. different alternatives are plotted along the horizontal axis, and the respective performance indices are depicted along the vertical axis. it is clearly observed from the figure1 that alternative material handling equipment a1 attains the highest performance index. a4 has the second highest performance index, a3 posses the next higher performance index, lastly a2 has the lowest performance index. the alternatives of material handling equipment are ranked accordingly. the performance indices of the alternatives al, a2, a3 and a4 are 3.32, 1.96, 2.28, and 2.82 respectively. the alternatives are arranged as per the descending order of their performance indices. therefore the ranking order of the alternatives are arranged as a1>a4>a3>a2. it is seen that the alternative a1 has the highest performance index. hence a1 is considered the best alternative and a2 is the worst one and so on. the ranking order is depicted in figure 2. bairagi/decis. mak. appl. manag. eng. (2022) 10 figure 2. ranking order of the alternatives 3.2 comparison of the results the results obtained by the proposed method have been compared with and validated by two well-known tools viz. technique of order preference by similarity to ideal solutions (topsis) and multi objective optimization on the basis of ratio analysis (moora). for the purpose of making comparison and validation, the same ranking problem on material handling equipment selection is once again solved by the methods separately. in topsis method, the closeness coefficient (cc) for each of the alternative is calculated. in moora method, net score is computed for each alternative material handling equipment. the relevant information is furnished in the table 9. it is observed that the closeness coefficient or relative closeness of the alternative a1, a2, a3 and a4 are 0.60, 0.42, 0.41 and 0.45 respectively. therefore the ranking orders of the alternatives are a1>a4>a2>a3. it is obvious that topsis method selects alternative a1 as the best alternative, a4 as the second best alternatives as the proposed methods. though, the ranking orders of the a2 and a3 do not match. in moora method, alternative a1 is ranked 1 and regarded as the best method. the graphical representation of the ranking orders obtained by the proposed method and the two well-known existing methods topsis and moora is portrayed in figure 3. the alternatives are plotted along horizontal axis and the ranking orders are plotted along vertical axis. it is found that alternative a1 has been selected as the best alternative by all the three multi-criteria decision making techniques. these results verify and validate the novel framework in decision making under fuzzy environment. table 9: comparison and validation of the results proposed method topsis moora ai performance index rank cc rank net score rank a1 3.32 1 0.60 1 0.79 1 a2 1.96 4 0.42 3 -0.07 2 a3 2.28 3 0.41 4 -0.16 4 a4 2.82 2 0.45 2 -0.1 3 a new framework for green selection of material handling equipment under fuzzy… 11 figure 3. comparision of ranking orders 4. conclusions the current investigation explores a new fuzzy mcdm framework in green selection of material handling equipment for industrial purpose considering multiple conflicting criteria. the novel framework is able to integrate economical, environment and social factors with vague information to meet the necessary requirements for green selection. the application of the proposed framework in solving the material handling evaluation problem clearly shows the way of implementation and the ability of the framework to make proper decision considering multiple criteria under fuzzy environment. the framework clearly selects the alternative material handling equipment a1 as the best one with the highest performance index 3.32. therefore it can be concluded that the proposed technique introduced in the current research work might be a useful and essential aid to managerial decision makers of manufacturing industry in solving problem on green selection of material handling equipment. the result of the proposed method is supported and validated by two exiting well-known multi-criteria methods. the novelty of the current study can be highlighted as follows:  introduction of a novel mathematical model for fuzzy decision making.  incorporation of green factors in evaluation of material handling equipment.  consideration of benefit, non-benefit and target value based criteria.  application of integration in fuzzy decision making environment. there are some limitations of the present approach though it has many advantages. this method cannot be used for objective factors or a mixture of objective and subjective factors. it has not addressed the consideration of heterogeneous group decision making. the current investigation is limited to independent criteria and decision making under fuzzy environment. an insight of the research exposes that a new paradigm has been explored in the current investigation. in this research work, performance rating and weights of criteria are estimated in the prescribed degrees of linguistic form. these are efficiently integrated for calculating the performance indices for proper decision making in the specific domain. the proposed model can be used for solving similar multi criteria decision making problems where decision is to make under fuzzy environment. the bairagi/decis. mak. appl. manag. eng. (2022) 12 appropriate implementation of the technique can be a useful managerial tool in solving industrial decision making problem. consideration of interdependent factors, incorporation of heterogeneous group decision making process and decision making with both subjective and objective factors, in green selection of material handling equipment for industrial purpose might be some important directions of future research. author contributions: conceptualization, b.b.; methodology, b.b.; validation, b.b.; formal analysis, b.b.; investigation, b.b.; resources, b.b.; writing—original draft preparation, b.b.; writing—review and editing, b.b.; visualization, b.b.; supervision, b.b.; the author has read and agreed to the published version of the manuscript. funding: this research received no external funding. conflicts of interest: the author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. reference chan, f.t.s., ip, r.w.l. & lau, h. 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(2010). an integrated fuzzy multicriteria decision making methodology for material handling equipment selection problem and an application. expert systems with applications, 37, 2853–2863. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license decision making: applications in management and engineering vol. 3, issue 1, 2020, 43-59. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame2003034d * corresponding author. e-mail addresses: manoranjande.1987@gmail.com (m. de), bcgiri.jumath@gmail.com (b.c. giri). optimal decisions on pricing and greening policies of multiple manufacturers under governmental regulations on greening and carbon emission manoranjan de 1 and bibhas chandra giri 1* 1 department of mathematics, jadavpur university, kolkata, west bengal, india received: 25 january 2019; accepted: 16 june 2019; available online: 27 june 2019. original scientific paper abstract: this paper determines the optimal decisions on pricing and greening strategies of substitutable products which are manufactured by duopoly competitor firms in a consumer sensitive market where a consumer can choose a particular product by its retail price and greening level. the firms simultaneously produce these substitute products under carbon emission regulations enacted by government administration. government’s carbon emission regulation like carbon tax or cap and trade may not be enough to direct optimization of social greening welfare but may force to mandate the firms to satisfy a standard greening level to handle it. a penalty or subsidy is levied per unit difference in greening standards as well as with the cape and trade regulation on carbon emission. the contesting firm managers face the problem of fixing the conflicts on carbon penalty and greening investment to decide the optimum policies. for numerical examples, the optimal decisions of the firm managers are obtained by maximizing the profit following government’s mandatory regulations. some managerial insights are outlined and sensitivity analyses on key parameters of the model are graphically presented. key words: green product, carbon emission, substitute product, duopoly firm, government regulations. 1. introduction the commercial ecological conflict has taken a new dimension especially when sustainability standards of operations are part of practice. generally, in a production firm, products are made by means of continuing responsibility to ecological standards in their affirmation and operations because those impact on the global or local mailto:manoranjande.1987@gmail.com mailto:bcgiri.jumath@gmail.com de and giri/decis. mak. appl. manag. eng. 3 (1) (2020) 43-59 44 environment, community, society, or economy. thus, constitution of a state obviously narrates that it is the commitment of the state to anchor and improve the nature and to conserve the green environment and untamed existence of the country. it powers a commitment on every local to guarantee and upgrade the normal territory for regular life and to deal with violators. it accommodates creation of vitality through advancement of sustainable power source assets as per the states of condition, full scale financial conditions and atmosphere to decrease the carbon emissions and in addition to diminish reliance on the non-renewable energy sources. regarding issues on government impositions of green cess, a tax earmarked for special purposes, across different states is also under-lining green activism towards achieving goals of greener environment. inconvenience of green cess is one such choice to influence citizens and stakeholders to take part towards green activism for protecting environment although many of the cases are challenges to the stake-holders who are affected commercially by such imposition. in this article, we would base on the law of issues relating to green cess. we investigate for optimal decisions on prices and green levels for two competing cofirms which produce substitute products with different greening levels. 2. literature review in the past two decades, various government directions were established to ensure the earth and check carbon flows into the atmosphere. for instance, the canadian central government prohibited brilliant lights from being made or imported into canada; however, a couple of claims to fame radiant lights were exempted. the arrangement was planned to diminish vitality utilization. therefore, the approach prompted the expanded utilization of conservative bright lights (cfls) and light emitting diodes (led) lights, which are more vitality productive (blackwell 2015). another case of how the administration assumes a job in ensuring the earth is exhibited through the boycott of plastic froth holders in zimbabwe because of the thing discharging harmful synthetic substances when warmed. because of these worries, zimbabwe's environment management agency requested eateries to utilize recyclable or biodegradable bundles (mhofu 2017). the agency recommends restaurants to use paper packaging or encourage patrons to partake of their food on site. the change in consumer demand for green products and processes, non-official pressures on government legislation and business, make it challenging for organizations to process their supply discipline and re-orient the products. ghosh and shah (2012), swami and shah (2013) and ghosh et al. (2018) studied a channel for the manufacturer and the retailer who invested in greening efforts, and this phenomenon is reflected in the expression of the demand. the study of product replacement (maity and maiti (2009), krommyda et al. (2015), de et al. (2016), de et al. (2018)) has gained significant attention of the researchers, because it contributes to the success of the company's decisions regarding material/product planning, price and inventory control. recently, substitute products dependent on the stock-level displayed to each other was investigated by pan et al. (2018). the decision of the production firms to produce more eco-friendly products stems from their desire to raise profit through improving customer satisfaction. hafezi and zolfagharinia (2018) investigated a green product development model where product types, market price(s) and quality dimensions were considered as decisions of the optimal decisions on pricing and greening policies of multiple manufacturers under… 45 production firm. the main objective was to derive how governmental regulations can be set as a driver of green production and for the benefit of the environment. krass et al.(2013) investigated the technology selection and production decisions model of a profit-maximizing firm under carbon tax. the increase of the tax rate does not necessarily induce the firm to adopt cleaner technology. zhang and xu (2013) discussed a firm’s optimal production quantities under cap-and-trade regulation. they found that cap-and-trade regulation can induce the firm to produce more carbon efficient products. after that, he et al. (2015) focused on a production lot-sizing problem for a carbon-intensive firm under cap-and-trade regulation. the most relevant studies in a competitive environment were carried out by hua et al. (2011) and sun (2012) who solved their problems through game theory. a few studies provided a comprehensive analytical investigation on the role of government control (e.g., chen 2001, zhang et al. 2012, gouda et al. 2016). in this paper, we formulate a model where two competitive firms manufacture two substitute products and sell these products separately in a consumer sensitive market. the model is solved by simultaneous nash game theory. in addition to being confined to a competitive environment, the current study evaluates the role of government regulations on green level and carbon emission and meets some important gaps in literature, where different optimal strategies on prices and green levels are found out for the firms and associated green development costs are numerically investigated and presented graphically. 3. formulation of model 3.1 assumptions a) two competitive firms produce two substitute products and sell their products separately in the market. b) products are substitutable on the basis of their prices and green levels. c) demand of each firm is considered to be linearly decreasing in price and increasing in greening level. d) the effect of one firm’s own price on its market demand is greater than that of its competitor. e) the effect of product greening level on quantity demanded for a firm is more than that of its competitor. f) total potential demand of the market is divided into two parts provided by the significant loyalty of product to the customers. g) the firms incur a cost of greening which is a quadratic function of green level of the product. h) government mandates to the firms to keep a standard greening level and limiting amount of carbon emission. a penalty or subsidy is levied per unit difference in greening standards and carbon emission per unit product produced. 3.2. notations i s : selling price per unit product (dollar/unit). i c : production cost per unit product (dollar/unit). gi : greening level of each product (per unit). g0 : government mandated standard greening level (per unit). de and giri/decis. mak. appl. manag. eng. 3 (1) (2020) 43-59 46 rg : penalty or subsidy levied per unit difference in greening standards per unit produced (dollar/unit). c0 : government mandated limiting amount of carbon emission per unit product (ton/unit). rc : tax or reward levied per unit difference in amount of carbon emission per unit produced (dollar/unit). di : demand of each product in the market (unit). cei : amount of carbon emission to produce a product (ton/unit). ii : investment parameter of greening (dollar-unit 2 ). a : potential demand in the market (unit).  : degree of customer loyalty to a product, 0 1  . α, β : demand sensitivity to price of the product, 𝛼 > 𝛽 > 0, (unit 2 /dollar). δ : demand sensitivity to product greening level,  > 𝛿 > 0, (unit 2 ). 3.3. model development in our model, two manufacturers act as competitors by producing substitute product with different quality in a duopoly green sensitive market where customers choose one type of product on the basis of its retail price and greening level. therefore, demands confronted by the manufacturers are linear functions of selling prices and green levels of the substitute products. both demands to the manufacturers are considered downward sloping in own selling price and upward sloping in own green degree. this consideration is similar to ghosh and shah (2012). the impact of a manufacturer’s own price on its market demand is greater than that of its competitor i.e., 𝛼 > 𝛽 and also the impact of greening level on demand for a manufacturer is more than that of its competitor (δ) [cf. chen (2001)]. further, according to li et al. (2016) the potential market demand a of the green products is assumed to be constant and it is fractionally divided into two parts a and (1-)a on the basis of the degree of customer loyalty  to a desire product. thus, the demand functions for two competitive manufacturers can be written as follows: 1 1 1 2 2 1 2 1 2 2 1 1 2 2 2 1 2 1 ( , , , ) ( , , , ) (1 ) d s g s g a s s g g d s g s g a s s g g                      (1) moreover, the manufacturers acquire an expense 2 i i i g of greening which is a quadratic function of green level i g of the product [cf. savaskan et al. (2004), ghosh and shah (2012), swami and shah (2013)]. further, government regulates the manufacturing firms to keep a standard greening level g0 and constraint measure of carbon emission c0 per unit production. the firm is penalized if it does not acquire the standard greening level g0 or it excesses greater carbon than the permitted cap c0. while in a case where the firm provides more greening level and less carbon emission than mandated standards, a reward or subsidy proportional to the difference is awarded to the firm by the government. following yang and xiao (2017) and ghosh et al. (2018), penalties or rewards (rg and rc) are imposed per unit difference in greening benchmarks (g0 gi) and carbon emission (cei-c0) per unit production. thus, the profit functions of each firm under these government regulation schemes are: about:blank about:blank about:blank about:blank about:blank about:blank about:blank about:blank about:blank optimal decisions on pricing and greening policies of multiple manufacturers under… 47 2 1 1 1 1 1 1 1 1 0 1 1 1 0 1 2 2 2 2 2 2 2 2 2 0 2 2 2 0 2 1 1 2 2 ( , ) ( ) ( ) ( ) ( , ) ( ) ( ) ( ) , , , 0 g c g c s g s c d i g r g g d r ce c d s g s c d i g r g g d r ce c d s g s g                  (2) here, the objective is to maximize (2) with respect to key decision variables selling price and greening level of a product under government mandated regulations. table 1. different indices term expression a 2 1 0 1 02 2 2 0 2 0 1 [{2 (1 ) } 2 { ( )} 4 { ( )}] g c g c a c r g r ce c c r g r ce c                   b 2 2 0 2 02 2 1 0 1 0 1 [{2(1 ) } 2 { ( )} 4 { ( )}] g c g c a c r g r ce c c r g r ce c                   1 a 2 2 1 [2 ( ) ] 4 g r         2 a 2 2 1 [2 ( )] 4 g r         3 a 1 1 1 2 2 2( )( ) g i a r a a      4 a 1 2 1 2 1 2 ( )( ) ( ) g a r a a a a a           5 a 2 1 1 2 2 2( )( ) g i a r a a      e 1 2 1 0 1 0 1 ( ){ ( )} ( )( ) g c g a a a c r g r ce c a r a a b               f 1 2 2 0 2 0 1 ( ){ ( )} ( ){(1 ) } g c g a a b c r g r ce c a r a b a                proposition 1. (a) the optimal greening levels for maximum profit achieved by the manufacturers under competition are * 5 4 1 1 1 1 22 4 3 5 * 4 3 2 2 1 1 22 4 3 5 = , ( )( ) > 0 = , ( )( ) > 0 g g a e a f g when i a r a a a a a a e a f g when i a r a a a a a                   (3) (b) the optimal selling prices for maximum profit achieved by the manufacturers under competition are * * * 1 1 1 2 2 * * * 2 1 2 2 1 = = s a a g a g s b a g a g     (4) de and giri/decis. mak. appl. manag. eng. 3 (1) (2020) 43-59 48 where * 1 g and * 2 g are given in equation (3). all the helping symbols are described in table 1. proof. we utilize in reverse induction technique to tackle the objectives. firstly, optimum selling prices i s ’s are obtained by given greening levels i g ’s. we determine the objective functions in the following form: 1 1 1 1 1 0 1 1 0 1 2 2 1 2 1 1 2 2 2 2 2 0 2 2 0 2 2 1 1 2 2 1 1 2 2 ( , ) = { ( ) ( )}( ) ( , ) = { ( ) ( )}{(1 ) } , , , 0 g c g c g s g s c r g g r ce c a s s g g i g s g s c r g g r ce c a s s g g i g s g s g                                   (5) the first order partial derivatives of 1 and 2 with respect to 𝑠𝑖 are 1 1 1 1 2 1 2 1 1 0 1 0 2 2 2 2 1 2 1 2 2 0 2 0 ( , ) = 2 ( ) { ( )} ( , ) = (1 ) 2 ( ) { ( )} g g c g g c s g a s s g r g s c r g r ce c s g a s s g r g s c r g r ce c                                      (6) also, we have 2 2 = 2 < 0, = 1,2 i i for i s      thus, the profit function i is strictly concave in is . equating to zero the first order partial derivatives 1 1 s   and 2 2 s   given in (6), we get 1 2 1 2 1 0 1 0 1 2 2 1 2 0 2 0 2 = ( ) { ( )} 2 = (1 ) ( ) { ( )} g g c g g c s s a g r g c r g r ce c s s a g r g c r g r ce c                                 (7) solving the equations (7) for 1 s and 2 s simultaneously, we obtain the equilibrium price for the manufacturers as 1 1 1 2 2 2 1 2 2 1 = = s a a g a g s b a g a g     (8) where a, b, a1 and a2 are given in table 1. the corresponding profits of the manufacturers at the equilibrium prices are: 1 1 2 1 0 1 0 1 1 2 2 2 1 2 1 2 1 2 1 1 2 1 2 2 0 2 0 1 2 2 1 2 1 2 2 2 1 1 2 2 ( , ) = { ( ) ( ) }{( ) ( ) ( ) } ( , ) = { ( ) ( ) }{(1 ) ( ) ( ) } g c g g c g g g a c r g r ce c a r g a g a a b a a g a a g i g g g b c r g r ce c a r g a g a b a a a g a a g i g                                                      (9) optimal decisions on pricing and greening policies of multiple manufacturers under… 49 to find the optimal green level, we differentiate the profit function i  given in (9) partially with respect to i g and equating it to zero. the best activities for the manufacturers in equilibrium are given by 3 1 4 2 4 1 5 2 = = a g a g e a g a g f    (10) where a3, a4, a5, e and f are given in table 1. the solution of the above equation (10) for 1 g and 2 g are 5 4 1 2 4 3 5 4 3 2 2 4 3 5 = = a e a f g a a a a e a f g a a a     (11) the second partial derivatives of the profit functions with respect to green level are 2 1 32 1 = a g     and 2 2 52 2 = a g     . this shows that the profit functions are strictly concave in the green level when 3 > 0a and 5 > 0a . proposition 2. the impact of own greening level on its optimal selling price decision is greater or less than that of its competitor according as . g r or       proof: the difference between the coefficients of * 1 g and * 2 g in equation (4) is 1 2 a a and which is given by 1 2 2 2 2 2 2 2 1 1 [2 ( ) ] [2 ( )] 1 (2 )( )] 4 4 4 a a r rg g r g                                  as >  and >  by our assumption, we have a1 – a2 > or < 0 according as . g r or       hence, the proof is complete. 4. numerical results in the preceding section, we have obtained the optimal values of different decision variables and objective functions. we choose the following parameter-values for our numerical analysis: let the base demand in the market be a=1500 units, degree of customer loyalty to order a product λ=0.50, demand response to own price α=0.750 unit2/dollar, demand response to competitor’s price β=0.50 unit2/dollar, demand response to own greening level γ=5.00 unit2, demand response to competitor’s greening level δ=2.75 unit2, government standard greening level is mandated by g0 = 8.0/unit, penalty or subsidy levied per unit difference in greening standards per unit product rg=1.5 dollar/unit, production cost for both products c1=5 dollar/unit, c2=4 de and giri/decis. mak. appl. manag. eng. 3 (1) (2020) 43-59 50 dollar/unit, amount of carbon emission to produce both products ce1=40 ton/unit and ce2=50 ton/unit, government offered limiting amount of carbon emission c0=10 ton/unit, tax or reward levied per unit difference in amount of carbon emission rc=30 (dollar/unit), investment parameters of greening for both products i1=700 dollarunit2 and i2=600 dollar-unit2. in this set-up, the optimal solution is obtained as follows: optimal selling prices 𝑠1 ∗ = 1468.25 dollar/unit, 𝑠2 ∗ = 1523.90 dollar/unit, optimal greening levels g1 ∗ = 2.20957, g2 ∗ = 2.14343 and the optimal profits 𝜋1 ∗ = 227235 dollar, 𝜋2 ∗ = 156715 dollar. the concavity property and contour plot of profit function is graphically shown in figure 1. figure 1. concavity property and contour plot of profit function. 4.1 impacts of greening investment cost observation 1: a) given greening investment cost i i of its own, the optimal pricing * i s of the product is increasing in competitor’s cost of greening investment j i . refer to figure 2(a). figure1(a): concavity of manufacturer’s profit function against selling price and green level. figure 1(b): contour plot of profit function against selling price and green level. optimal decisions on pricing and greening policies of multiple manufacturers under… 51 b) given greening investment cost j i of its competitor, the optimal pricing * i s of the product is decreasing in its own cost of greening investment i i . refer to figure 2(b). figure 2. optimal selling price against greening investment. observation 2: a) given manufacturer’s own greening investment cost, the level of greening of each manufacturer is increasing in its competitor’s greening investment cost. refer to figure 3(a). b) given greening investment cost of its competitor, greening level of a product is decreasing in its own greening investment cost. refer to figure 3b. figure 3: optimal greening level against greening investment. observation 3 a) given greening investment cost i i of its own, the optimal profit * i  of the manufacturer is increasing in competitor’s cost of greening investment j i . refer to figure 4(a). b) * i s vs. ii a) * i s vs. j i a) * i g vs. j i b) * i g vs. ii de and giri/decis. mak. appl. manag. eng. 3 (1) (2020) 43-59 52 b) given greening investment cost j i of its competitor, the optimal profit * i  of the manufacturer is decreasing in its own cost of greening investment i i . refer to figure 4(b). figure 4: optimal profit against greening investment. 4.2. impacts of government regulations on greening and carbon emission observation 4 a) for given one manufacturer’s selling price, it competitor’s selling price is increasing with the government mandated greening level 0 g . refer to figure 5(a). b) the relative optimal selling price difference between the two manufacturers is decreasing in mandated greening level 0g . refer to figure 5(b). figure 5. selling price vs. govt. standard greening level observation 5 a) * i  vs. j i b) * i  vs. ii a) manufacturer’s selling price vs. 0 g b) relative selling price vs. vs. 0 g optimal decisions on pricing and greening policies of multiple manufacturers under… 53 a) for given one manufacturer’s greening level, it competitor’s greening level is decreasing with the government mandated greening level 0 g . refer to figure 6(a). b) the relative optimal greening level difference between the two manufacturers is increasing in mandated greening level 0 g . refer to figure 6(b). figure 6. manufacturer’s greening level vs. govt. standard greening level. observation 6 a) each manufacturer’s optimal profit is decreasing with the government mandated greening level 0 g . refer to figure 7(a). b) the relative optimal profit difference between the two manufacturers is decreasing in mandated greening level 0 g . refer to figure 7(b). figure 7. manufacturer’s profit vs. govt. standard greening level. observation 7 a) product’s greening level vs. 0 g b) relative greening level vs. 0 g a) manufacturer’s profit vs. 0 g b) relative profit vs. 0 g de and giri/decis. mak. appl. manag. eng. 3 (1) (2020) 43-59 54 a) the relative optimal selling price difference between the two manufacturers is increasing in government penalty or subsidy on greening. refer to figure 8(a). b) the relative optimal greening level difference between the two manufacturers is increasing in government penalty or subsidy on greening. refer to figure 8(b). c) the relative optimal profit difference between the two manufacturers is decreasing in government penalty or subsidy on greening. refer to figure 8(c). figure 8: relative price, greening and profit vs. penalty or subsidy on greening. observation 8 a) the relative optimal selling price difference between the two manufacturers is increasing in government penalty or subsidy on carbon emission. refer to figure 9(a). b) the relative optimal greening level difference between the two manufacturers is decreasing first and then increasing in government penalty or subsidy on carbon emission. refer to figure 9(b). a) relative price vs. g r b) relative green vs. g r c) relative profit vs. g r optimal decisions on pricing and greening policies of multiple manufacturers under… 55 c) the relative optimal profit difference between the two manufacturers is increasing in government penalty or subsidy on carbon emission. refer to figure 9(c). figure 9. relative price, greening and profit vs. penalty or subsidy on carbon emission. 4.3. impacts of customer loyalty λ on price, green level, demand and profit of manufacturer the degree of customer loyalty λ to the competitors in the market is very important. a reasonable number of customer’s loyal to a product may affect the manufacturer’s decision for both competitors. varying the value of λ in the interval [0,1] the following observation are made: observation 9 manufacturer’s optimal price, greening level, demand and profit are having opposite characteristics compared to those of its competitor for changing customer loyalty parameter λ. refer to figures 10(a)-10(d). a) relative price vs. penalty or subsidy on c r b) relative green level vs. penalty or subsidy on c r c) relative profit vs. penalty or subsidy c r de and giri/decis. mak. appl. manag. eng. 3 (1) (2020) 43-59 56 figure 10. manufacturer’s prices, greening levels, demands and profits vs. customer loyalty parameter. 4.4. impacts of demand keys on optimum decisions and profit table 2 presents some sensitivity analysis on demand keys for optimum decision variables and profits of the model by changing the key values -30%, -15%, 15% and 30%, respectively, taking one at a time and keeping the other parameters unchanged. from table 2, the following observations are made: observation 10 a) as α increases, the selling price and the greening level decrease. as a result, demands and profits decrease more sensitively. b) β plays negative role compared to α. it increases the market demands and profits of the manufacturers. c) increasing in γ marginally increases the values of optimal decisions and profit. d) similar to comparing between price keys α and β, δ follows negative sense of γ because it decreases optimum results of the model. a) manufacturer’s prices vs. λ b) manufacturer’s greening levels vs. λ c) manufacturer’s demands vs. λ. d) manufacturer’s profits vs. λ optimal decisions on pricing and greening policies of multiple manufacturers under… 57 table 2. sensitivity analysis of demand keys (∆𝑋denotes % change in x) keys changes in % * 1 s * 2 s * 1 g * 2 g * 1 d * 2 d * 1  * 2  -30.00 56.55 54.28 139.36 168.02 75.38 96.38 337.41 447.99 α -15.00 20.00 19.16 48.81 59.00 30.26 39.17 99.30 127.45 15.00 -12.63 -12.06 -31.41 -38.15 -23.66 -31.16 -49.27 -58.73 30.00 -21.33 -20.36 -53.85 -65.54 -43.82 -58.05 -75.68 -86.43 -30.00 -13.49 -12.72 -34.86 -41.39 -35.91 -42.34 -58.95 -66.77 β -15.00 -7.21 -6.82 -18.65 -22.16 -19.19 -22.68 -34.71 -40.23 15.00 8.37 7.94 21.81 25.94 22.28 26.43 49.54 59.87 30.00 18.21 17.31 47.82 56.92 48.47 57.61 120.47 148.47 -30.00 -0.20 -0.20 -30.77 -30.85 -0.74 -0.86 -0.70 -0.82 γ -15.00 -0.12 -0.11 -15.48 -15.54 -0.41 -0.49 -0.41 -0.48 15.00 0.16 0.14 15.71 15.81 0.51 0.60 0.53 0.63 30.00 0.35 0.33 31.70 31.96 1.12 1.31 1.18 1.39 -30.00 0.14 0.14 5.97 6.06 0.40 0.49 0.64 0.78 δ -15.00 0.07 0.07 2.97 3.02 0.20 0.24 0.31 0.38 15.00 -0.06 -0.06 -2.96 -3.00 -0.19 -0.23 -0.29 -0.35 30.00 -0.12 -0.12 -5.89 -5.96 -0.37 -0.44 -0.56 -0.69 5. managerial insights from the analysis of the results given in the previous section, the following managerial insights can be derived: a) in competition, more greening investment parameter leads to a lower greening level of own product under the government regulation. therefore, it is less beneficial to the firms as well as environment. b) decreasing of standard greening level mandated by government attracts the firms to increase their products’ greening levels. c) increasing of penalty or subsidy on unit difference of greening level increases the relative price and green level of the product. as a result, the relative profits of the firm decrease. this phenomenon indicates a hard competition between the contesting firms. d) increasing of penalty or subsidy on unit difference of carbon emission increases the relative price in all over range but relative green level decreases first and then it is increases. thus, resulting profit follows a concave nature with increasing of penalty or subsidy on carbon emission. so, for increasing in penalty or subsidy on carbon emission, soft competition is found up to a certain growth of it and after that, competition becomes tightened to the manufacturer. 6. conclusion the commitment of investigation lies in multi-manufacturer models including competitive greening costs and standard government regulations in a duopoly price and green sensitive market. the models are promptly connected to different ventures of demand and evaluated numerically in a deterministic setting of parameter. the formulated models are solved through simultaneous move game between the contesting firms. several observations are made on the basis of numerical results and graphical presentations. some managerial insights are also derived for the competing firms. de and giri/decis. mak. appl. manag. eng. 3 (1) (2020) 43-59 58 one can extend this model to the case with competing manufacturers-retailers supply chain model. it can prompt extra bits of knowledge of sequential game theory. this model may be considered with uncertain demand like stochastic or fuzzy due to asymmetry information about consumer’s loyalty. acknowledgement: the authors are sincerely thankful to the esteemed reviewers for their comments and suggestions based on which the manuscript has been improved. the second author gratefully acknowledges the financial support from csir, govt. of india (grant ref. no. 25(0282)/18/emr-ii). author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research received no external funding. conflicts of interest: the authors declare no conflicts of interest. references blackwell, r. 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(2017). pricing and green level decisions of a green supply chain with governmental interventions under fuzzy uncertainties. journal of cleaner production, 149, 1174–1187. zhang, b., & xu, l. (2013). multi-item production planning with carbon cap and trade mechanism. international journal of production economics, 144(1), 118–127. zhang, x., xu, x., & he, p. (2012). new product design strategies with subsidy policies. journal of systems science and systems engineering, 21(3), 356–371. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). https://www.voanews.com/a/zimbabwe-ban-plasticfoam/3945349.html plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 5, issue 2, 2022, pp. 372-395 issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0318062022t * corresponding author. e-mail addresses: stuzcu@politics.ankara.edu.tr (s. e. tuzcu), spturkoglu@ybu.edu.tr (s. p. türkoğlu) how vulnerable are high-income countries to the covid-19 pandemic? an mcdm approach sevgi eda tuzcu1* and serap pelin türkoğlu2 1ankara university, faculty of political sciences, department of business administration, ankara, turkey 2 ankara yıldırım beyazıt university, şereflikoçhisar berat cömertoğlu vocational school, department of management and organization, ankara, turkey received: 4 march 2022; accepted: 13 may 2022; available online: 18 june 2022. original scientific paper abstract: this paper tries to determine the most vulnerable points of high– income countries during the covid-19 pandemic in an mcdm setting. for this aim, we use the entropy method to obtain criteria weights and the piv method for the comparisons. we employ a wide range of criteria that account for political, demographic, capacity, and covid-19 indicators including vaccination. our sample consists of 40 hics. the results reveal that countries with less equitable healthcare systems and with more vaccine hesitancy are more vulnerable to covid-19. hospital bed capacity, a strict government policy, and a lower percentage of the population who smoke add to the success of countries in this combat. we compare our findings with saw and maut techniques as well and obtain very similar rankings. therefore, we conclude that the piv method can be used for national performance evaluations with a reduced rank reversal problem and computational simplicity. key words: high–income countries, mcdm, entropy method, piv method, covid-19 pandemic. 1. introduction the unexpected start of the covid-19 pandemic in 2020 has exacerbated debates over how to respond to the spread of infectious disease nationwide. at the beginning of the pandemic, all countries applied similar strategies. within time, it turns out that the same strategies have not provided parallel results for everybody. the national performance, in other words, the success in this pandemic is highly related to the countries’ own dynamics. during the pandemic, it is put forth clearly that low– and middle–income countries (lmics) are more open to the spread of this disease and its negative consequences due to lack of social distancing, crowded households, extreme https://tureng.com/tr/turkce-ingilizce/vocational%20high%20school https://tureng.com/tr/turkce-ingilizce/vocational%20high%20school how vulnerable are high-income countries to the covid-19 pandemic? an mcdm approach 373 poverty, lack of hygiene and medical capacity. the high–income countries (hics), however, have more resources to fight against the pandemic. yet, some of them have not shown a performance as good as prior expectations. this study examines the areas and the reasons that hics are also fragile against the pandemic. we contribute to the covid-19 literature by assessing the vulnerabilities of 40 hics in the combat against this disease within a multicriteria decision-making framework. for this framework, we identify an exhaustive set of criteria including vaccination, population characteristics, per capita income status, covid-19 indicators, healthcare capacity, and governmental policy indicators. criteria weights are defined based on the entropy method. then, we evaluate the national performances of hics by using a novel mcdm technique, namely proximity indexed value (piv) method. more specifically, the aims of this paper are threefold: i. evaluating the national performance of 40 hics in the combat against the covid-19 pandemic with an mcdm framework; ii. comparing the relative importance of criteria and defining in what areas hics are more vulnerable to such a disease, iii. determining the contribution of vaccination policies to pandemic management. in this paper, we have several motivations to limit our sample only to hics rather than lmics. the first reason is the reports and rankings by global health security index (ghs index). ghs index report (2019) noted that infectious diseases can be a significant risk for the international economy and security as much as climate change or political instabilities1. ghs index evaluates the relative capabilities and provides a benchmark for 195 countries in 6 different categories. although this report emphasizes that none of the countries is perfectly ready for future pandemics, the hics have a much higher ghs index score compared to the other countries, so they are the best-prepared ones. no doubt, wealth is an important weapon to manage the spread and effects of the pandemic. in lmics, however, the resources and the availability of measures that can be taken are limited. this is one of the reasons that previous literature mostly concerns the fight that is going on (for example, türkoğlu & tuzcu, 2021). we, on the other hand, focus on the possible flaws in the pandemic management of the hics, the least vulnerable and most wealthy countries. ghs index (2019) indicates that the usa and the uk share the top first and second places. despite their high rankings, during the covid-19 pandemic, these countries have shown a relatively bad performance in terms of the number of new cases and deaths. in fact, most hics are criticized for their delayed responses to the covid-19 pandemic. some low-ranked countries like vietnam and china, on the other hand, have a relatively good performance in the battle against covid-19. this conflicting ghs index score and performance situation leads us to the question: why is it not possible for some hics to obtain the best results against covid-19 despite the existence of all resources? in what areas are they more vulnerable? the second reason to investigate the current situation of hics is the availability of vaccines. chen (2021) notes that the vaccination rate can only be an important determinant in the progress of the current pandemic in high and upper-middleincome countries. by february 2022, every 2 of 3 people in hics have been vaccinated against covid-19, while this rate is every 1 in 8 people in low–income countries.2 this situation widens the gap between hics and other countries, but also the inequity inside the hics. marti and puertas (2021) state that vulnerability shows the degree of a society’s potential to protect and make vulnerable populations more resilient to 1 https://www.ghsindex.org/wp-content/uploads/2019/10/2019-global-health-securityindex.pdf accessed 21 february 2019. 2 https://data.undp.org/vaccine-equity/accessibility/ accessed 21 february 2022. https://www.ghsindex.org/wp-content/uploads/2019/10/2019-global-health-security-index.pdf https://www.ghsindex.org/wp-content/uploads/2019/10/2019-global-health-security-index.pdf https://data.undp.org/vaccine-equity/accessibility/ tuzcu and türkoğlu/decis. mak. appl. manag. eng. 5 (2) (2022) 372-395 374 disasters. vaccination is another determinant of vulnerability. therefore, vaccination status must be taken into account while discussing the national performance against covid-19. however, at the time of this research, taking vaccination into account as a determinant is only possible when the sample is restricted with the hics due to the global availability of covid-19 vaccines. therefore, vaccine availability becomes another motivation to concentrate only on the hics. hodgins and saad (2020) indicate that now the aim in the fight against covid-19 is not to keep the country completely virus–free. fisher et al., (2020) indicate that after two years passed with covid-19, “flatting the curve” by implementing severe restrictions is not a viable option anymore. now, the aim has become the ability to live with this disease, that is to say, to maximize benefit while minimizing the harm caused by covid-19 precautions. therefore, we need to determine the most vulnerable points of countries even among the wealthiest ones. however, the aim of “maximizing benefit while minimizing harm” is conflicting in nature. fisher et al., (2020) also argue that there is no one reliable indicator of performance against covid-19. many criteria are required to assess the ability to live with covid-19. considering the need for many performance indicators and the conflicting nature of covid-19, we believe that the best setting to assess countries is to employ an mcdm framework. with this setting, we believe that some dimensions that the ghs index overlook will be taken into account, and a better country ranking will be provided. this study also contributes to the scarce literature on the application of mcdm techniques to covid-19-related problems. the application of mcdm provides reliable solutions only when the compared units are similar. previous studies often cluster their sample. for example, aydın and yurdakul (2020) apply k–means clustering first, then compare the country performances with a data envelopment analysis. some consider oecd (i.e. yiğit, 2020; çalış boyacı, 2021) or the european countries (i.e. marti & puertas, 2021; markowicz & rudawska, 2021) with an mcdm setting. as noted by hodgins and saad (2020), these countries are very different in terms of demographic characteristics, geography, and economies. comparing non–homogenous groups will not provide a correct picture of the real performance of these countries. we believe that examining a less analyzed and rather homogenous sample with hics will contribute to the literature in order to show their vulnerabilities and sample-specific policy suggestions will be possible. in this study, we employ an entropy-based proximity indexed value (piv) approach to evaluate the national performance rankings of hics. the entropy method is a well– known objective criteria weighting technique that has been used in several mcdm problems. the weights, in this method, are free from the decision–makers’ judgments and determined based on the differences among criteria. hence, it is consistent with the nature of covid-19. the piv method, on the other hand, calculates the distance from the ideal solution and ranks alternatives accordingly. previous studies, such as khan et al., (2019) and zamiela et al., (2021: 8) find that this method provides robust rankings when compared to other well–established methods, such as topsis. to test the robustness of our results, we also crosscheck the rankings from the piv method with two other approaches, namely simple additive weighting (saw) and multiattribute utility theory (maut). the spearman rank correlations and wilcoxon rank tests show that rankings from these two approaches are highly correlated with those obtained from the piv technique and there is no statistically significant difference between the means. therefore, we conclude that our findings are robust. the results of this study reveal the hics’ most vulnerable areas against the current pandemic which is different than lmics. in the lmics, türkoğlu, and tuzcu (2021) show that the weakest point in this struggle is the level of extreme poverty which determines the ability of social distancing and achieving hygiene standards. in the how vulnerable are high-income countries to the covid-19 pandemic? an mcdm approach 375 hics, income has a different role. its importance in the analysis is much less than lmics. the allocation of the available resources, rather than their absolute amount, becomes important for the success of the covid-19 pandemic. this is why some strong and high ghs index scored countries, like the usa or the uk, obtain lower rankings in this analysis. the socioeconomic disadvantage within the hics still determines the rate of success and makes the population of these countries vulnerable to covid-19. vaccination policy is also another determinant of the success of hics against the pandemic. the common point of the countries with the lowest rankings is the low vaccination rates. in fact, it is a worrisome finding despite the fact that most of the vaccines produced so far are held by hics. the reluctance towards the vaccine is mostly due to misinformation about the vaccines and the lack of trust in governments. it is urgent that these governments adopt specific policies to strengthen trust and combat misinformation. public figures and leadership might play a role to weaken vaccine hesitancy. otherwise, especially the elders and the disadvantaged shares of the population in terms of sociodemographic status will continue to be open to this disease. the novelties of this study are as follows: first, we concentrate on a less analyzed rather homogenous group of countries and determine their national performance against the covid-19 pandemic. second, due to the conflicting nature of the disease itself, and the pandemic management aims, we employ an mcdm setting, namely the entropy-based piv method. in this way, we are able to apply a very new mcdm technique to a current problem. next, since these countries are the wealthiest ones in the world, we also show in which aspects they cannot manage the pandemic efficiently despite the availability of resources. in other words, we put forth the weakest points of the hics so that we can make policy implications for future similar diseases. last, by discussing the role of vaccinations, we also contribute to this line of research and show possible reasons for vaccine hesitancy towards the covid-19 vaccine. based on the findings, this study highlights the importance of the following policy implications: even in the wealthiest countries in the world, socioeconomically disadvantaged groups are more open to contagious diseases, such as covid-19. the equity in the access to the healthcare systems as well as in the distribution of other available resources is the key to the success in current and future pandemics. countries must adopt policies that protect disadvantaged groups by providing free or at least more affordable healthcare. vaccine hesitancy has also a vital role in the determination of country performance and vulnerability. therefore, governments must deal with vaccine hesitancy by having transparency about vaccination policy, providing correct information, and persuading especially the disadvantaged and elder population for the necessity of the vaccines. as correctly stated by the who, “no one is safe until everyone is safe.”3, not even in the wealthiest countries. the rest of this study is organized as follows: the following section describes the current literature about the national performance against covid-19 and the usage of several mcdm techniques in this issue. the third section explains our sample and choice of mcdm technique, namely the entropy-based proximity indexed value method. we present our findings in the next section. we also apply a sensitivity analysis and compare our results with other well–established mcdm techniques in the next section, and the last section concludes with managerial implications. 3 https://www.who.int/news-room/commentaries/detail/a-global-pandemic-requires-aworld-effort-to-end-it-none-of-us-will-be-safe-until-everyone-is-safe accessed 04 march 2022. https://www.who.int/news-room/commentaries/detail/a-global-pandemic-requires-a-world-effort-to-end-it-none-of-us-will-be-safe-until-everyone-is-safe https://www.who.int/news-room/commentaries/detail/a-global-pandemic-requires-a-world-effort-to-end-it-none-of-us-will-be-safe-until-everyone-is-safe tuzcu and türkoğlu/decis. mak. appl. manag. eng. 5 (2) (2022) 372-395 376 2. literature review stern consequences of covid-19 on every aspect of life have been discussed heavily in the past two years. the performance of countries against covid-19 has caught the special interests of academics as well. in this study, we try to determine the best performing hics and understand what are their vulnerabilities through a combination of mcdm techniques. for this aim, we can examine the current covid-19 literature from two aspects: the studies that compare country performances with methodologies other than mcdm techniques constitute the first group. the application of mcdm methods to performance evaluations establishes the second group of papers. in the first line of studies, country performances and success factors are discussed. fisher, et al. (2020) argue that in the early times of the pandemic, countries took precautions without considering the situation in their own society. now, success can be defined as the ability to live with covid-19. therefore, the vulnerabilities of countries must be understood and the national performance must be assessed accordingly. jamison et al. (2020) assess the performance of 35 countries based on the doubling times of new cases and deaths attributed to covid-19. their aim is to show the impacts of government policy choices both on health and economic outcomes. aydin and yurdakul (2020) cluster and evaluate 142 countries according to several indicators with a novel dea approach. cartaxo et al., (2021) develop a vulnerability assessment model and cluster countries based on their exposed risk due to covid-19. they adopt an entropy-based model to determine the similarity between covid-19 exposure of countries according to 49 indicators from the social, economy, population, and health categories. they show that contrary to expectations, covid-19 has not only hit the most vulnerable countries with low resource capacity hard but also put developed and wealthiest countries at risk. their results indicate that the usa and japan are among the countries at highest risk due to exposure to covid-19. in fact, the usa has the same similarities to covid-19 spread with india and brazil, two highly affected countries by covid-19. markowicz and rudawska (2021) note that improvements in the healthcare system and rational decision-making are only possible through a thorough assessment of the country's performance during the covid-19 pandemic. for this aim, they suggest an evaluation framework based on a set of demographic, epidemiological, healthrelated quality of life, financial resources, and access to healthcare system indicators. they develop a standardized distance measure using data from 28 european union countries and the usa. higher values of this measure show a better situation for the underlying country. the second field of literature that is related to our study is the application of mcdm techniques to national performance assessments. in this area, pal et al. (2020) forecast long term country-specific risks by using artificial intelligence to predict and cluster them as high-risk, low-risk, and recovering countries. samanlioglu and kaya (2020) evaluate precautions other than the healthcare system that governments apply, such as mobility restrictions, full lockdowns, school closures, and declaration of a state of emergency, by using a hesitant fuzzy ahp method. kayapinar kaya (2020) compares the sustainable development performance of oecd countries before and during the covid-19 pandemic by employing the mairca method. the results of this study indicate that developing countries are more vulnerable to the covid-19 pandemic than their developed counterparts in terms of their sustainable development levels. khan, ali, and pamucar (2021) offer a new fuzzy fucom-qfd approach for the assessment how vulnerable are high-income countries to the covid-19 pandemic? an mcdm approach 377 of healthcare systems during the covid-19 pandemic that can be applied to the systems in different countries. alkan and kahraman (2021) evaluate different governmental strategies and restrictions applied during the covid-19 pandemic by using q–run orthopair fuzzy topsis specification. in this model, government strategies constitute alternatives while different costs of these strategies, future loss, time, and the effects of implementing these restrictions on human rights are considered as the criteria. their findings indicate mandatory quarantine that limits social interaction and a strict isolation strategy as the best strategy that a government can adopt in this fight. yiğit (2020) assesses the national performance of 36 oecd countries with the topsis methodology where the countries are the alternatives and healthcare indicators are the criteria. the results from this study highlight that contrary to expectations, some countries with high healthcare expenditures and long life expectancy, are not classified among the best countries in the struggle against covid19. similar to yiğit (2020), çalış boyacı (2021) investigates the performances of oecd countries by employing topsis, copras, and aras methods while the criteria weights are obtained through the swara technique. in this study, the criteria selection also depends on healthcare statistics. türkoğlu and tuzcu (2021) evaluate 22 middle–high–income countries with an extensive set of indicators, including healthcare capacity, socio–demographic situation, and covid-19 indicators with an sdv-based rov method. their findings confirm that poverty levels are as important as hospital capacity. they also show that demographic characteristics, like the average population age, are a significant determinant of country performance in this battle. marti and puertas (2021) examine the vulnerability of the european countries to the covid-19 health crisis by using topsis. they assess the vulnerability from a multi– viewpoint, namely from society, work, and health. hence, the criteria selected for the mcdm framework are indicators reflecting the situation in these categories, and the countries constitute the alternatives. these criteria reflect the most vulnerable groups to the disease itself and also the economic consequences due to the restrictive measures. previous literature applying mcdm techniques for national performance assessment is rather scarce. among them, some compare national performances of non–homogenous country groups such as markowicz & rudawska (2021), yiğit (2020), and çalış boyacı (2020). although covid–19 does not differentiate countries, continents, or borders, it is clear that a stronger economic situation is a powerful weapon against its fight as put forth clearly by hodgins and saad (2020). using a heterogeneous country group casts doubt on the obtained results. some of the studies, such as çalış boyacı (2020), zizovic et al. (2021), and khan, ali, and pamucar (2021), mostly consider healthcare indicators. assessing national performance, however, requires a multidimensional perspective, including socio– demographic indicators, governmental attitudes, and economic situation as well as healthcare indicators (cartaxo et al. 2021; marti & puertas 2021). last, literature often focuses on lmics (for example, türkoğlu & tuzcu, 2021) and their performance against covid-19. this is a natural choice because they are in a more difficult position due to inefficiency and scarcity of available resources. in this study, we contribute to the existing literature by concentrating on the wealthiest and rather homogenous group of countries, namely hics with an mcdm framework. hodgins and saad (2020) state clearly that the strategies applied by hics may not always be appropriate for lmics. the reverse is also true. besides the availability of resources, the demographic characteristics of lmics are considerably tuzcu and türkoğlu/decis. mak. appl. manag. eng. 5 (2) (2022) 372-395 378 different than hics. therefore, the situation against covid-19 in lmics may not always be representative of the current state in hics. despite the availability of vaccines and more resources, income inequalities and bad management affect considerably their performances in this combat. by employing a wide selection of criteria with a multidimensional perspective, including the vaccination status, we believe that our study reflects the strongest and most vulnerable countries against covid-19 more clearly for hics. based on the findings of this paper, it is possible to make policy suggestions to remedy the vulnerabilities. 3. data and methodology 3.1. sample description this study aims to rank the performance of hics to identify their possible weak points in the struggle against covid-19. according to the world bank income groups, there are 80 countries described as hics. we identify 40 countries that have a population of over 1,000,000 people and which have complete vaccination data for the analyses. these countries are australia, austria, bahrain, chile, croatia, cyprus, czech republic, denmark, estonia, finland, france, germany, greece, hong kong sar, china, ireland, israel, italy, japan, kuwait, latvia, lithuania, netherlands, new zealand, norway, oman, poland, portugal, qatar, saudi arabia, slovak republic, slovenia, spain, sweden, switzerland, taiwan, china, trinidad and tobago, united arab emirates, united kingdom, united states, and uruguay. as of the date of the analysis, this sample covers 50.7% of the total cases worldwide. we use a rather homogenous sample for the analyses in terms of available resources during the pandemic. as it is well known, ghs index ranked most hics as the more prepared countries in case of the existence of a contagious disease. the sample’s average ghs index score is 55.36 which is well above the 2021 world’s overall average score of 38.9. table 1 panel a and b give descriptive statistics and pearson correlations for total cases, total deaths, and gdp per capita for the sample to provide a general outline. table 1. descriptive statistics and correlations for selected variables panel a. descriptive statistics mean median min max std dev total cases 211027.6 231729.5 824.397 413076 117330.7 total deaths 1541.569 1542.045 10.339 3573.06 1096.42 gdp per capita 35478.41 29388.71 10434.78 87097.04 19711.37 panel b. correlations among selected variables total cases total deaths gdp per capita total cases 1 total deaths 0.5476* 1 0.0003 gdp per capita 0.1235 -0.3385* 1 0.4478 0.0326 the values in parentheses show p–values. * represents significance at 5% level. table 1 panel a reveals that the current pandemic situation varies considerably. both the number of total cases and total deaths show high standard deviations. the how vulnerable are high-income countries to the covid-19 pandemic? an mcdm approach 379 correlations presented in panel b indicate no significant association between income and the number of total cases. total deaths, however, are significantly and negatively correlated with gdp per capita. it seems that countries with higher income experience lower deaths which highlights the importance of wealth in this combat. 3.2. methodology many studies in the literature, such as cartaxo et al. (2021) and marti & puertas (2021) state that the struggle against covid-19 has many aspects that the countries deal with at the same time not only healthcare status. instead, studies must adopt a multi–viewpoint to describe country performances. mcdm, in this sense, is a very useful tool to evaluate many and often conflicting criteria and determine the best to worst performing alternatives. we employ the entropy-based piv method as an mcdm approach in this paper to rank the performance of hics while considering policy, healthcare capacity, and demographic criteria as well as covid-19 related indicators. the findings are also compared with two other mcdm approaches, namely saw and maut as sensitivity analysis. the general framework of this study is summarized in figure 1 for clarity. figure 1. the framework of this study tuzcu and türkoğlu/decis. mak. appl. manag. eng. 5 (2) (2022) 372-395 380 following cartaxo et al. (2021), we identify 14 criteria including political, demographic, capacity, and covid-19 indicators and vaccination policy, in order to provide a general view for each country against the pandemic. the selection of criteria is based on the previous studies that evaluate different groups of countries during the covid-19 pandemic, i.e. aydin & yurdakul, 2020; yiğit, 2020; arsu, 2021; cartaxo et al., 2021; george et al., 2020; nguyen et al., 2021; fisher, teo & nabarro, 2020. as in markowicz and rudawska (2021), the criteria selection is based on a set of questions. demographic indicators are chosen to answer whether the population is more open to catching covid-19 and its negative consequences. capacity indicators try to measure whether the healthcare system can deal with covid-19 in terms of human resources and infrastructure. policy and covid-19 indicators aim to assess whether the governmental response is effective in terms of preventing the spread of the disease and deaths. these criteria, their definition, and the sources that the data comes from can be seen in table 2. table 2. the criteria employed in the analyses, and data sources indicators criteria definition data source policy indicators government response government response stringency index: a composite measure based on 9 response indicators value from 0 to 100. https://www.bsg.ox.ac. uk/research/researchprojects/covid-19government-responsetracker (access date: 04.01.2022) capacity indicators hospital beds per thousand the number of hospital beds per 1000 people in a given country. https://ourworldindata. org (access date: 04.01.2022) current health expenditure the amount of health expenditures as a percentage of the gdp of a given country. the world bank database (2018) total vaccinations per hundred the number of vaccines applied per 1,000 people in a given country. https://ourworldindata. org (access date: 17.02.2022) income gdp per capita (current us$) the world bank database (2020) (access date: 04.01.2022) demographic indicators cardiovascul ar death rate the annual number of deaths due to cardiovascular diseases per 100,000 people in a given country. https://ourworldindata. org (access date: 04.01.2022) diabetes prevalence the rate of people aged between 20 and 79 with type 1 and type 2 diabetes https://ourworldindata. org (access date: 06.01.2022) share of adults who smoke the share of adults, aged 15 years and older, who smoke any tobacco https://ourworldindata. org (access date: 06.01.2022) https://www.bsg.ox.ac/ https://ourworldindata.org/grapher/adults-smoking-2007-2018 https://ourworldindata.org/grapher/adults-smoking-2007-2018 https://ourworldindata.org/grapher/adults-smoking-2007-2018 how vulnerable are high-income countries to the covid-19 pandemic? an mcdm approach 381 product on a daily or non-daily basis. population ages 65 and above the rate of people age 65 and above to the total population in a given country. the world bank database (2020) (access date: 04.01.2022) population density population density (people per sq. km of land area) the world bank database (2020) (access date: 04.01.2022) covid-19 indicators total cases the number of total confirmed covid-19 cases per 1,000,000 people https://ourworldindata. org (access date: 17.02.2022) total deaths total number of deaths due to covid-19 per 1,000,000 people https://ourworldindata. org (access date: 17.02.2022) total recovered the number of patients recovered from covid-19 infection per 1,000,000 people. https://www.worldomet ers.info/coronavirus (access date: 17.02.2022) total tests the number of total tests to diagnose covid-19 infections per 1,000 people. https://ourworldindata. org (access date: 17.02.2022) 3.2.1. entropy method the entropy approach is a widely used method to weight criteria in mcdm studies. it is first developed by shannon (1948) and employed in many areas including the areas of bank diversification (çınar et al. 2018), stock markets (baydaş & elma 2021), robotics (chodha et al., 2022), power generation problems (emovon & samuel, 2017) and informatics (kannan & thiyagarajan, 2021). it is a reliable weighting method and provides high reliability in the objective criteria determination (dashore et al., 2013: 2183; işık and adalı, 2017: 85; gupta and kumar, 2022: 78). it is an easily applicable objective weighting method in which decision-makers are not included in the process of establishing the relative importance of criteria. instead, this method relies on the contrasts among criteria, and the weights are determined accordingly (mukhametzyanov, 2021). in other words, the information in the decision matrix and the relation between alternatives and criteria become highlighted in the entropy method (žižovic, miljkovic & marinkovic 2020), which is consistent with the nature of the covid-19 pandemic. to obtain criteria weights with the entropy method, the decision matrix is normalized through eq. (1). 𝑦𝑖𝑗 = 𝑥𝑖𝑗 ∑ 𝑥𝑖𝑗 𝑚 𝑗=1 (1) there are m alternatives and n criteria, i indicates the total number of criteria and j indicates the number of alternatives, where i=1,2,…,n and j=1,2,…,m. “xij” are the elements of the decision matrix, while, yij reflects the normalized matrix. in the second step, the entropy value, ei, is computed for each criterion as shown in eq. (2). tuzcu and türkoğlu/decis. mak. appl. manag. eng. 5 (2) (2022) 372-395 382 𝑒𝑖 = −𝑘 ∑ 𝑦𝑖𝑗ln⁡(𝑦𝑖𝑗)⁡ 𝑚 𝑗=1 ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ (2) where is k a constant term stated as k=1/ln(m) in order to assure that ei lies between 0 and 1. finally, weights for each criterion are computed as stated in eq. (3). wi = 1−ei ∑ 1−ei n i=1 ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡ (3) 3.2.2. proximity indexed value (piv) method the piv method depends on the idea that the selected alternative must be closest to the ideal solution/ the best value. to do so, a proximity index is calculated by the linear distance of each normalized alternative from the best alternative’s value (mufazzal & muzakkir, 2018: 430). piv method provides the advantages of reduced rank reversal problem observed in many mcdm techniques such as topsis as well as computational simplicity. it also offers robust rankings when the results are compared to many other mcdm techniques (khan et al., 2019; zamiela et al., 2021: 8). in fact, goswami et al., (2021: 1154) indicate that the piv method provides more reliable solutions when its rankings are compared to traditional techniques, such as ahp, topsis, copras, and vikor. the steps of the piv method can be described below (mufazzal and muzakkir, (2018: 430) and khan et al. (2019)): first of all, the decision matrix is set and normalized as follows: ri = 𝑌𝑖 ∑ 𝑌𝑖 2𝑚 𝑖=1 i=1,…,m; (4) yi is the decision value of the ith alternative. the next step is to compute the weighted normalized decision matrix as shown in eq. (5): vi = 𝑤𝑖 ∗ ri (5) where wi, is the criterion weight. this step is followed by the computation of the weighted proximity index (wpi), which is shown by ui, by using eq. (6) and eq. (7): 𝑢𝑖 = 𝑣𝑚𝑎𝑥 − 𝑣𝑖 (for beneficial criteria) (6) 𝑢𝑖 = 𝑣𝑖 − 𝑣𝑚𝑖𝑛 (for non–beneficial criteria) (7) the final step is to obtain the overall proximity value, di. 𝑑𝑖 = ∑ 𝑢𝑖 𝑛 𝑗=1 (8) the alternatives are ranked based on the di value obtained from eq. (8). the lowest di score represents the minimum deviation from the best solution, therefore becomes the top alternative. 4. findings as explained below, the mcdm analysis begins with the normalization of the decision matrix by using eq. (1). second, entropy values are obtained by employing eq. (2) and criteria weights are found through eq. (3). the criteria cardiovascular death rate, diabetes prevalence, the share of adults who smoke, the population aged 65 and above, population density, total cases, and total deaths, are non–beneficial, how vulnerable are high-income countries to the covid-19 pandemic? an mcdm approach 383 whereas the criteria government response, hospital beds per thousand, current health expenditure, total vaccinations per hundred, income, total recovered, and total tests are beneficial. the criteria weights are provided in table 3. table 3. criterion weights c ri te ri a c a rd io v a sc u la r d e a th r a te d ia b e te s p re v a le n ce s h a re o f a d u lt s w h o s m o k e p o p u la ti o n a g e d 6 5 + p o p u la ti o n d e n si ty t o ta l c a se s t o ta l d e a th s t o ta l v a cc in a ti o n s g o v e rn m e n t r e sp o n se h o sp it a l b e d s c u rr e n t h e a lt h e x p e n d it u re in co m e t o ta l r e co v e re d t o ta l t e st s w j 0 .0 6 9 1 0 .0 5 6 5 0 .0 8 2 2 0 .0 8 4 4 0 .0 6 3 7 0 .0 7 0 5 0 .0 5 4 1 0 .0 7 7 3 0 .0 8 1 4 0 .0 9 1 3 0 .0 7 1 0 0 .0 6 7 7 0 .0 6 9 1 0 .0 6 1 6 table 3 reveals that the entropy method puts the highest weight on hospital bed capacity, followed by demographic characteristics and government response. the entropy method determines criteria weights based on the contrasts of the selected attributes of the alternatives. this means that the countries in the sample vary the most in terms of hospital capacity, population age, smoking habits, and government response. total vaccinations per hundred people have the 5th position. the sample has similar characteristics in terms of total covid-19 tests and deaths. cartaxo et al. (2021) argue that to understand the true nature of pandemic management, risk factors, their effects, and interactions must be determined, and governmental policies must be made accordingly. based on the weights shown in table 3, it is also possible to make such a relative assessment for pairs of criteria. following de nardo et al. (2020), we create the matrix of criteria weights comparison to demonstrate the interactions between criteria more clearly as suggested in cartaxo et al. (2021). this relative assessment can be observed in table 4. tuzcu and türkoğlu/decis. mak. appl. manag. eng. 5 (2) (2022) 372-395 384 t a b le 4 . m a tr ix o f c o m p a ri so n f o r c ri te ri a w e ig h ts cardiov. death rate diabetes prev. share of smoking pop. ages 65+ pop. density total cases total deaths total vaccination s gov. response hospital beds current health exp. income total recovered total tests w j 0 .0 6 9 0 .0 5 7 0 .0 8 2 0 .0 8 4 0 .0 6 4 0 .0 7 0 0 .0 5 4 0 .0 7 7 0 .0 8 1 0 .0 9 1 0 .0 7 1 0 .0 6 8 0 .0 6 9 0 .0 6 2 c a rd io v a sc u la r d e a th r a te 0 .0 6 9 1 .0 0 0 1 .2 2 3 0 .8 4 1 0 .8 1 8 1 .0 8 5 0 .9 8 0 1 .2 7 8 0 .8 9 4 0 .8 4 9 0 .7 5 7 0 .9 7 3 1 .0 2 0 1 .0 0 0 1 .1 2 1 d ia b e te s p re v a le n c e 0 .0 5 7 0 .8 1 8 1 .0 0 0 0 .6 8 8 0 .6 6 9 0 .8 8 8 0 .8 0 2 1 .0 4 5 0 .7 3 1 0 .6 9 4 0 .6 1 9 0 .7 9 6 0 .8 3 4 0 .8 1 8 0 .9 1 7 s h a re o f a d u lt s w h o sm o k e 0 .0 8 2 1 .1 8 9 1 .4 5 4 1 .0 0 0 0 .9 7 3 1 .2 9 0 1 .1 6 6 1 .5 1 9 1 .0 6 3 1 .0 0 9 0 .9 0 0 1 .1 5 7 1 .2 1 3 1 .1 8 9 1 .3 3 3 p o p u la ti o n a g e s 6 5 + 0 .0 8 4 1 .2 2 2 1 .4 9 4 1 .0 2 8 1 .0 0 0 1 .3 2 6 1 .1 9 8 1 .5 6 1 1 .0 9 2 1 .0 3 7 0 .9 2 5 1 .1 8 9 1 .2 4 6 1 .2 2 2 1 .3 7 0 p o p u la ti o n d e n si ty 0 .0 6 4 0 .9 2 2 1 .1 2 7 0 .7 7 5 0 .7 5 4 1 .0 0 0 0 .9 0 4 1 .1 7 7 0 .8 2 4 0 .7 8 2 0 .6 9 7 0 .8 9 7 0 .9 4 0 0 .9 2 2 1 .0 3 3 t o ta l c a se s 0 .0 7 0 1 .0 2 0 1 .2 4 7 0 .8 5 8 0 .8 3 5 1 .1 0 7 1 .0 0 0 1 .3 0 3 0 .9 1 2 0 .8 6 5 0 .7 7 2 0 .9 9 2 1 .0 4 0 1 .0 2 0 1 .1 4 3 t o ta l d e a th s 0 .0 5 4 0 .7 8 3 0 .9 5 7 0 .6 5 8 0 .6 4 1 0 .8 4 9 0 .7 6 7 1 .0 0 0 0 .7 0 0 0 .6 6 4 0 .5 9 2 0 .7 6 2 0 .7 9 8 0 .7 8 3 0 .8 7 7 t o ta l v a cc in a ti o n s 0 .0 7 7 1 .1 1 9 1 .3 6 8 0 .9 4 1 0 .9 1 6 1 .2 1 4 1 .0 9 7 1 .4 2 9 1 .0 0 0 0 .9 4 9 0 .8 4 7 1 .0 8 9 1 .1 4 1 1 .1 1 9 1 .2 5 4 g o v e rn m e n t r e sp o n se 0 .0 8 1 1 .1 7 8 1 .4 4 1 0 .9 9 1 0 .9 6 5 1 .2 7 9 1 .1 5 5 1 .5 0 6 1 .0 5 3 1 .0 0 0 0 .8 9 2 1 .1 4 7 1 .2 0 2 1 .1 7 9 1 .3 2 1 h o sp it a l b e d s 0 .0 9 1 1 .3 2 1 1 .6 1 5 1 .1 1 1 1 .0 8 1 1 .4 3 4 1 .2 9 5 1 .6 8 8 1 .1 8 1 1 .1 2 1 1 .0 0 0 1 .2 8 6 1 .3 4 8 1 .3 2 1 1 .4 8 1 c u rr e n t h e a lt h e x p e n d it u re 0 .0 7 1 1 .0 2 8 1 .2 5 7 0 .8 6 4 0 .8 4 1 1 .1 1 5 1 .0 0 8 1 .3 1 3 0 .9 1 9 0 .8 7 2 0 .7 7 8 1 .0 0 0 1 .0 4 8 1 .0 2 8 1 .1 5 2 in co m e 0 .0 6 8 0 .9 8 0 1 .1 9 9 0 .8 2 5 0 .8 0 2 1 .0 6 4 0 .9 6 1 1 .2 5 3 0 .8 7 6 0 .8 3 2 0 .7 4 2 0 .9 5 4 1 .0 0 0 0 .9 8 0 1 .0 9 9 t o ta l r e co v e re d 0 .0 6 9 1 .0 0 0 1 .2 2 3 0 .8 4 1 0 .8 1 8 1 .0 8 5 0 .9 8 0 1 .2 7 8 0 .8 9 4 0 .8 4 9 0 .7 5 7 0 .9 7 3 1 .0 2 0 1 .0 0 0 1 .1 2 1 t o ta l t e st s 0 .0 6 2 0 .8 9 2 1 .0 9 1 0 .7 5 0 0 .7 3 0 0 .9 6 8 0 .8 7 5 1 .1 4 0 0 .7 9 7 0 .7 5 7 0 .6 7 5 0 .8 6 8 0 .9 1 0 0 .8 9 2 1 .0 0 0 w j is t h e w e ig h t fo r e a ch c ri te ri o n how vulnerable are high-income countries to the covid-19 pandemic? an mcdm approach 385 the matrix shown in table 4 can be read from left to right and it is not symmetric. although hospital capacity has the highest weight in table 3, the findings from this matrix indicate that it is slightly more important than the share of the population that has 65 years and older (1.08) and the share of the population who smokes (1.11). however, it is 1.615 times more important than diabetes prevalence and 1.688 times more important than total deaths attributed to covid-19. vaccination policy is more important than the number of deaths (1.429), diabetes prevalence in the country (1.368), total tests (1.254), and population density (1.214). it is also as important as current health expenditures. gdp per capita is one of the least important criteria in this analysis. these findings confirm that demographic and socioeconomic factors are as important as medical capacity when it comes to preventing cases and deaths as stated in wildman (2021). once the criteria weights are decided using the entropy method, the rankings of hics are computed using the steps of the piv technique. the rankings presented in table 5 are based on the normalized decision matrix that is obtained by employing eq. (4) 4. table 5. the rankings obtained from the piv technique countries di ranking austria 0.1180 1 japan 0.1258 2 cyprus 0.1351 3 ireland 0.1354 4 switzerland 0.1361 5 australia 0.1364 6 denmark 0.1389 7 france 0.1408 8 norway 0.1412 9 germany 0.1430 10 new zealand 0.1444 11 israel 0.1453 12 united arab emirates 0.1487 13 united kingdom 0.1521 14 finland 0.1536 15 netherlands 0.1544 16 italy 0.1546 17 united states 0.1554 18 qatar 0.1557 19 taiwan, china 0.1565 20 portugal 0.1621 21 kuwait 0.1626 22 greece 0.1632 23 oman 0.1637 24 spain 0.1653 25 saudi arabia 0.1661 26 czech republic 0.1669 27 uruguay 0.1669 28 4 in the papers using mcdm techniques, it is common to provide initial decision matrix and normalized matrix. however, having 40 countries and 14 different criteria makes these matrices rather hard to examine. to conserve space, we do not provide them inside the main text of this paper, but they are available on request. tuzcu and türkoğlu/decis. mak. appl. manag. eng. 5 (2) (2022) 372-395 386 countries di ranking slovenia 0.1671 29 sweden 0.1673 30 slovak republic 0.1704 31 chile 0.1721 32 lithuania 0.1741 33 poland 0.1744 34 estonia 0.1755 35 bahrain 0.1787 36 trinidad and tobago 0.1839 37 croatia 0.1844 38 latvia 0.1848 39 hong kong sar, china 0.2080 40 average 0.1582 the findings in table 5 reveal that the best-performing countries in the sample only consisting of hics are austria, japan, and cyprus. among these three countries, japan has the highest ghs index score in 2021 (60.5), while austria has the average score (56.9) and cyprus is below the average (41.9). the worst performing countries, however, are croatia, latvia, and hong kong, china. the usa has the 18th ranking, while the uk has the 14th place alongside the criticisms toward the ghs index. the average proximity indexed value is 0.1582. when the best performing countries are closely examined, it is seen that austria has one of the highest hospital bed capacities per thousand people and applied strict lockdown policies. although the share of adults who smoke is high in this country, its population is relatively younger than the hics considered in this analysis. japan has an elder population, but the percentage of people with a smoking habit is relatively lower. it is the country with the highest hospital capacity per thousand people. the percentage of health expenditure in japan is also very high. since the beginning of the pandemic, australia and new zealand have applied very strict lockdown policies with speed testing5. their unique geography and traveling restrictions also help to control the spread of new cases. although they have hospital capacity and health expenditures on the sample average, australia and new zealand obtain relatively better rankings in the sample. in this success, very small numbers of total covid-19 cases and covid-19 caused deaths, as well as very low population density, play a role as well. wilson (2021) also states the importance of the strong leadership that new zealand exhibits during the pandemic as one of the keys to success in pandemic management. germany has the largest population in europe but obtains a high ranking in our analysis. as noted in cartaxo et al., (2021), its mass testing policy provides the country with a lower incidence rate. comparing the uk, germany has been more successful in terms of cases and deaths attributed to covid-19 despite the uk’s clear advantage of population density and population age. when closely examined, it is seen that germany is one of the countries with the highest hospital capacity. its health expenditures are also above the sample average. markowicz and rudawska (2021) find that the usa, alongside ireland, sweden, and luxembourg, is the country with the best healthcare system capacity. this is confirmed by our sample as well: the usa has the highest health expenditure among the hics accounted for in this sample. the share of health expenditures of the uk is 5 https://www.who.int/westernpacific/news-room/feature-stories/item/new-zealand-takesearly-and-hard-action-to-tackle-covid-19 accessed 28 february 2022. https://www.who.int/westernpacific/news-room/feature-stories/item/new-zealand-takes-early-and-hard-action-to-tackle-covid-19 https://www.who.int/westernpacific/news-room/feature-stories/item/new-zealand-takes-early-and-hard-action-to-tackle-covid-19 how vulnerable are high-income countries to the covid-19 pandemic? an mcdm approach 387 also above the sample average. yet, these two countries obtain average rankings in table 5. it is interesting to note that the number of total deaths attributed to covid-19 is well above the sample average for these countries. despite these high levels of health expenditures, their hospital capacity per thousand people is one of the lowest among the hics considered in this analysis. these countries are known for their low capacity of government response and control particularly at the beginning of the pandemic. cartaxo et al., (2021) attribute the very high rate of mortality observed in the usa and the uk to the burden on the hospital system. they also criticize especially the usa for the lack of equity in the healthcare system. our findings confirm the overall result put forth by cartaxo et al., (2021) that it is not possible to efficiently manage the available healthcare resources, no matter how vast they are, without providing equity in the system. our findings are also in line with yiğit (2021) that high healthcare expenditures and long life expectancy are not enough to describe the best countries in the struggle against covid-19. wildman (2021) indicates that, in developed countries like the uk and the usa, health outcomes are a factor of income inequality, which also strongly affects the socioeconomic vulnerability inside societies. particularly during the covid-19 pandemic, it is observed that if the percentage of people with low income and insecure jobs increases in a country, their vulnerabilities to disease increases, so the country's performance decreases. this situation describes the mid-ranking of the usa. for the uk, the poorest areas are most affected which contributed to the worsening of the country's rankings. as in the uk, the netherlands also demonstrates a lower performance and obtain the 16th ranking despite its high preparedness level and the strong healthcare system. marti and puertas (2021) emphasize that the new covid-19 cases in this country rose very quickly at the beginning of the pandemic despite their very small household size. the initial herd immunity strategy that the netherlands applied alongside the uk played a significant role in this low performance. the nordic countries in our sample, namely norway, denmark, finland, and sweden, have obtained different performance rankings. among these countries, denmark and norway have the 7th and 9th positions respectively. finland has the 15th ranking, while sweden demonstrates a low performance with 30th place among 40 countries in the sample. they have similar demographic characteristics in terms of age and chronic disease prevalence. all these countries implement less than average strict restrictions, but the smoothest government response belongs to sweden. since the first cases of covid-19, sweden has claimed that the pandemic might prolong for an indefinite period, and very strict lockdowns would be hard to continue (gordon, grafton & steinshamn 2021). our data also reveals that the share of adults who smoke is much higher in sweden than in other nordic countries and the sample average. therefore, compared to other nordic countries, sweden has experienced more deaths due to covid-19 which contributed to its low ranking. among these countries, denmark has applied more social distancing measures than others since the beginning of the pandemic (gordon, grafton & steinshamn 2021). hong kong adopted a strict policy toward covid-19, applied mass testing, and has experienced a much smaller number of cases than most western countries. yet, it still shares the lowest ranking with latvia and croatia. its low scorecard can be attributed to the distrust in government applications and the high burden on the healthcare system (silver, 2022). when the worst performing countries are investigated, one can observe the following common points: croatia, hong kong, and latvia have an elder population tuzcu and türkoğlu/decis. mak. appl. manag. eng. 5 (2) (2022) 372-395 388 with a relatively high percentage of people who smoke. their hospital capacity and the percentage of health expenditures are among the last in the sample. the hics considered in this analysis do not show a very high dispersion in terms of the number of vaccinations applied against covid-19. yet, the countries with the worst performance during the pandemic have lower vaccination rates, particularly croatia, than the rest. misinformation and lack of trust in the government caused an important delay in vaccination rates in latvia, while the delta variant hit the country hard6. croatia shares similar vaccination rates with latvia as well. despite its high level of development and available resources, hong kong exhibits a very low scorecard in the pandemic. the country is criticized due to the delayed vaccination policy7 as well. our findings regarding vaccination confirm the prior evidence put forth by vaccine hesitancy literature. for instance, aw et al. (2021) show that vaccine hesitancy toward covid-19 is the highest in hong kong and the usa. when compared to lmics, they state that this hesitancy has much more severe consequences on hics, since vaccine hesitancy affects less the vaccine uptake decision in lmics. aw et al. (2021) also indicate that vaccine hesitancy in hics is more common among non–white, younger and female populations with low socioeconomic status. that is, even in hics, vaccine hesitancy makes the already vulnerable population more open to the negative consequences of the covid-19 pandemic. our findings, in line with previous literature such as cartaxo et al., (2021) and wildman (2021) once more emphasize that factors only related to the healthcare system and socioeconomic status are not enough to designate the true course of events when it comes to covid-19 pandemic. political and demographic indicators must also be considered to determine vulnerabilities. we also show that the most pronounced factors in the determination of success for hics and lmics are different. türkoğlu and tuzcu (2021) demonstrate that for uppermiddle-income societies, population density and extreme poverty play vital roles in pandemic management. in countries where extreme poverty is a serious case, it is not easy to access basic hygiene materials and apply social distancing. for hics, however, income has a relatively low significance compared to extreme poverty. instead, for the hics, the distribution of the available resources becomes more important than the number of resources. here, the success in pandemic management is determined mostly by the equity in the healthcare system and the relative income inside the country. in contrast to the lmics, population density does not become prominent as a factor of success in the pandemic management of hics since these countries are mostly less dense in population even in the major cities. the ratio of the elder population especially comes to the forefront in the hics. as the population gets older, the rate of comorbidities and the burden on the overall healthcare system. interestingly, the prevalence of chronic diseases, like diabetes or cardiovascular diseases, in societies is not a determinant of hics performance. this finding can be attributed to the good healthcare services in these countries. yet, our findings reveal that hospital bed capacity in hics is as important as for lmics. based on the results presented in this study, we can conclude that the lack of equity in access to the healthcare system makes hics vulnerable to the pandemic as well as 6 https://www.euronews.com/my-europe/2021/08/30/how-distrust-and-disinformationhave-left-latvia-lagging-on-vaccine-rollout accessed 27 february 2022. 7 https://www.bloomberg.com/news/articles/2022-02-24/why-hong-kong-is-now-one-ofthe-worst-places-to-be-in-covid-era accessed 27 february 2022. https://www.euronews.com/my-europe/2021/08/30/how-distrust-and-disinformation-have-left-latvia-lagging-on-vaccine-rollout https://www.euronews.com/my-europe/2021/08/30/how-distrust-and-disinformation-have-left-latvia-lagging-on-vaccine-rollout https://www.bloomberg.com/news/articles/2022-02-24/why-hong-kong-is-now-one-of-the-worst-places-to-be-in-covid-era https://www.bloomberg.com/news/articles/2022-02-24/why-hong-kong-is-now-one-of-the-worst-places-to-be-in-covid-era how vulnerable are high-income countries to the covid-19 pandemic? an mcdm approach 389 the national healthcare capacity. it is seen that the already vulnerable share of society is very open to the risks of covid-19 as well. population characteristics, particularly the prevalence of smoking and age, are weak points of hics as well. although hics have access to the covid-19 vaccinations more than any other societies, countries with low performance have the lowest vaccination rates in the sample. it is more worrisome in hics than lmics. this resistance to the vaccine is mostly due to the lack of confidence in government policies and misinformation about covid-19. vaccine hesitation makes hics more vulnerable to a preventable disease. 5. sensitivity analysis in this section, we compare the rankings of hics that are obtained with the piv method with two other well-known mcdm techniques, namely simple additive weighting (saw) and multiattribute utility theory (maut). saw method is also known as the weighted sum model in the literature. the advantage of this model lies in the logic that depends on the proportional linear transformation of raw data based on the weighted average so that the relative ranking of standardized scores does not change. (afshari et al., 2010). maut, on the other hand, is a systematic method that defines and analyzes more than one variable to ensure a common platform for the decision-making process. the key factor in the maut techniques is to obtain a utility function that depends on single utility functions and their respective weightings (kim & song, 2019). the applications of saw and maut depend on the weights obtained from the entropy method as in the piv technique. the country rankings provided by these three methods are compared in table 6. table 6. the comparison of rankings from the piv, saw, and maut techniques the visual comparison in table 6 demonstrates very parallel rankings for all three methods. all three methods identify the same countries as the best performing and worst-performing ones and the rankings are consistent with each other. this finding is also confirmed with two non–parametric tests, namely spearman rank correlations and wilcoxon rank tests. with spearman rank correlations, we are able to observe the 0 5 10 15 20 25 30 35 40 45 piv saw maut tuzcu and türkoğlu/decis. mak. appl. manag. eng. 5 (2) (2022) 372-395 390 association between rankings, while the wilcoxon rank test analyzes the equality of mean ranks. the results are demonstrated in table 7. table 7. spearman rank correlations and wilcoxon rank tests for piv, saw, and maut piv saw maut piv 1 saw 0.8473* 1 0.0000 maut 0.9360* 0.8809* 1 0.0000 0.0000 h0 piv=saw piv=maut z 0.492 0.027 p-value 0.6228 0.9784 the upper part of table 7 demonstrates spearman rank correlations. the findings point out that correlations among piv, saw, and maut are significant at 0.01 level and positive. the lowest correlation coefficient is 0.8473. in other words, a strong association in the same direction exists between piv and saw and between piv and maut techniques. the bottom part of the table tests the null hypothesis of the equality of mean ranks for the piv and saw and the piv and maut, respectively. the p–values are well above any acceptable significance level meaning that the mean rankings of piv are not different from than saw or maut rankings. the sensitivity analysis confirms that piv provides similar rankings to other mcdm techniques while offering a reduced rank reversal problem and computational advantages. 6. conclusion this paper focuses on the vulnerable parts of the hics during the covid-19 pandemic in an mcdm setting while considering a wide set of demographic, policy, capacity, and covid-19 indicators, including vaccination policy. our results reveal significant differences between hics and lmics in terms of success against this disease. the most pronounced difference between the two groups of countries lies in the effects of income. in lmics, extreme poverty is as critical as healthcare capacity since it determines the ability to apply social distancing measures and to achieve even simple hygiene standards. in hics, however, income has a different function. its role is less important than the lmics. the distribution of available resources, no matter how vast they are, becomes the determinant factor of the success against covid-19. this is the reason behind the low rankings of countries with high ghs index scores like the usa in our analysis. without an equitable healthcare system, countries, particularly the socially disadvantaged portions of societies, are very vulnerable to infectious diseases like the current pandemic. prevalence of diabetes or cardiovascular diseases does not increase the vulnerability of the population of hics very much in contrast to other countries. this result can be attributed to the good health systems of hics. however, hospital capacity still plays an essential role in the determination of success against covid-19. how vulnerable are high-income countries to the covid-19 pandemic? an mcdm approach 391 one of the most important conclusions of our study is about the vaccination policy as a factor in country performance. we show that the lowest ranking countries also have the lowest vaccination rates. this finding cannot be attributed to the lack of vaccines in these countries because a greater portion of worldwide shots has gone to the hics. it can be explained by the lack of trust in governments and the misinformation about the disease and the contents of vaccines. in fact, the reluctance toward the covid-19 vaccine is more worrisome in hics than lmics. this hesitancy makes particularly the elder population of hics more vulnerable to the disease and increases the rates of hospitalization and deaths. these countries must adopt policies to increase the trust in vaccinations and combat misinformation. here, strong leadership and public figures might have an important role to increase the vaccination rates. transparency in vaccination policies is also important to raise confidence and persuade especially the disadvantaged and elder population for the necessity of the vaccines. the protection of the vulnerable population is crucial to have a completely safe environment for all. we also demonstrate that the disadvantaged groups, even inside the hics, are more fragile to covid-19 and similar diseases. the equity in the distribution of available resources and access to the healthcare system is as important as the quantity of the resources. governments must adopt policies that facilitate access to the healthcare system, especially by making them more affordable. last, our sensitivity analysis shows that the piv method provides reliable results in comparison of national performances against covid-19 with a reduced reversal problem and provides a computational advantage. some limitations of this study should also be mentioned. one of the most important limitations of this study is the different vaccination types applied by different countries. in most countries, more than one type of covid-19 vaccine is being applied. the efficiency of these different vaccines against covid-19 may be different, but discussing it is beyond the scope of this paper. second, the definition of death attributed to covid-19 was not unique at the beginning of the pandemic. in time, a convergence to some has been achieved. yet, this variation in definitions might generate a limitation when comparing the findings with other studies. the dynamic nature of the covid-19 pandemic might limit the generalizability of our findings to some extent. however, we believe that our study is important to demonstrate the ongoing state of the pandemic in hics. last, it is not easy to compare the national performances of countries, since these countries may differ in several aspects. to obtain a rather homogenous sample, we rely on the world bank’s income classification and limit our sample only to the hics. yet, as correctly stated by hodgins and saad (2020), countries may differ in terms of their economies, demographics, and geography. future studies may choose to cluster countries depending on these factors to obtain more homogeneous samples. in this way, it will be possible to compare countries with similar characteristics, not only in terms of income, and provide unique managerial implications for each cluster. author contributions: sevgi eda tuzcu: conceptualization, methodology, original draft preparation, reviewing and editing. serap pelin türkoğlu: conceptualization, methodology, data analysis, reviewing and editing. the authors have read and agreed to the published version of the manuscript. funding: this research received no external funding. tuzcu and türkoğlu/decis. mak. appl. manag. eng. 5 (2) (2022) 372-395 392 conflicts of interest: the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. references afshari, a., mojahed, m., & yusuff, r. m. 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(2021). multiple-criteria evaluation model for medical professionals assigned to temporary sars-cov-2 hospitals. decision making: applications in management and engineering, 4(1), 153-173. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 2, issue 2, 2019, pp. 65-80. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame1902119a * corresponding author. e-mail addresses: farhadyusifov@gmail.com (f. yusifov), r.alguliev@gmail.com (r. alguliyev), r.aliguliyev@gmail.com (r. aliguliyev) multi-criteria evaluation + the positional approach to candidate selection in e-voting rasim alguliyev 1, ramiz aliguliyev 1 and farhad yusifov 1* 1 institute of information technology, baku, azerbaijan received: 28 may 2019; accepted: 16 august 2019; available online: 23 august 2019. original scientific paper abstract: e-voting is one of the most important components of e-democracy and includes interesting research topics, such as the mechanisms of participation in elections, technological solutions to e-voting and the efficient application of those in e-voting. currently, there are numerous voting systems adopted in many countries of the world and each of those has specific advantages and problems. the paper explores the e-voting system as one of the main tools of e-democracy and analyzes its advantages and drawbacks. voting results always lead to a broad debate in terms of candidate selection and of whether the candidate elected to a position is suitable for that position. at present, the selection of qualified personnel and their appointment to responsible positions in public administration is one of the topical issues. in the paper, multi-criteria decision-making (mcdm) is proposed for the selection of candidates in e-voting. the criteria for candidate selection are determined and the relationship of each candidate with the other candidates is assessed by using a binary matrix. the candidate rank is calculated according to all the criteria. in a numerical experiment, candidate evaluation is enabled based on the selected criteria and ranked by using a positional ranking approach. the proposed model allows for the selection of a candidate with the competencies based on the criteria set out in the e-voting process and the making of more effective decisions as well. key words: e-government, e-democracy, e-voting, mcdm, candidate selection, election, e-government maturity model, governance. 1. introduction the implementation of information-communication technologies (ict) has an impact on social, economic and political life. especially, the development of ict and emailto:farhadyusifov@gmail.com mailto:r.alguliev@gmail.com mailto:r.aliguliyev@gmail.com alguliyev et al/decis. mak. appl. manag. eng. 2 (2) (2019) 65-80 66 government forming has substantially changed the public governance and political processes. e-democracy is regarded as the engagement of citizens and government bodies in political relations and processes (lee 2010; van der meer et al. 2014). this stage is characterized by the level of the close participation of citizens in socio-political processes and e-citizen problems. the efficiency in governance can be achieved with the close participation of citizens, as well as civil societies, in the process of politicoadministrative decision-making. e-government is forming a new environment in this regard. e-democracy is mentioned as the evolutionary stage of several developmental models of e-government (lee 2010). according to some researchers, a transition must be made from the use of the term ‘e-government’ to the use of the term ‘e-democracy’ (meier 2012; taghavifard et al. 2014) because e-democracy is considered as the evolutionary stage of several developmental models of e-government (lee 2010). the strengthening and development of democratic institutions, the use of ict and the information infrastructure for the expansion of civil participation in public and political processes reflects the essence of e-democracy (anttiroiko 2003; carrizales 2008; strielkowski et al. 2017). currently, the study of the role of e-voting in the countries which have adopted the formation of e-democracy as a priority is deemed as an integral part of investigations in the field of e-democracy (musial-karg 2014). the dynamic development of ict and the enhancement of social media tools have resulted in significant changes in the functioning of modern countries and societies. ict has started to play an important role in practically all fields of human life, including political processes. as one of the important components of e-democracy, e-voting encompasses interesting research topics, such as participation mechanisms in elections, the provision of legitimacy, technological solutions and the efficient application of those in the e-voting process. in this regard, e-voting can be considered as one of the forms of e-democracy (musialkarg 2014). in this study, approaches regarding the development of new e-voting mechanisms are analyzed. nowadays, human resources are considered as the main strategic resource of the government. the selection of qualified personnel at the government level and their appointment to responsible positions are important issues in economic and political processes. candidate selection is understood as a process in which the best candidates are selected for a particular position. different methods and technologies that help decision-makers to predict how successful a candidate will be in the future workplace are applied in the recruitment and selection processes (dursun et al. 2010; kabak et al. 2012; tuan 2017; afshari et al. 2017). in the literature, multi-criteria decisionmaking (mcdm) is widely used in various fields, such as the selection of appropriate personnel in the recruitment process, the choice of equipment in production, the selection of projects, etc. (kabak et al. 2012; kazana et al. 2015; tuan 2017, 2018; mukhametzyanov & pamučar 2018). there are research studies on the comparison and review of mcdm (stanujkic et al. 2013; zavadskas et al. 2014; mardani et al. 2015, khorami et al. 2015). a literature review highlights a few research studies on the application of mcdm for candidate selection in the election process. royes et al. (2001) use fuzzy mcdm in election predictions. the use of a computational system was proposed as a practical means of election forecasting. according to the decision-maker (the system user), the proposed flexible system allows for a choice of fuzzy weights and fuzzy evaluation functions in respect of the selection criteria. kazana (2015) showed that a total of 15 criteria were taken into account when selecting deputy candidates for political parties. the weight of the criteria is evaluated by the party representatives by means of the multi-criteria evaluation + positional ranking approach for candidate selection in e-voting 67 analytical hierarchy process (ahp) by using the fare (factor relationship) method. candidates are assessed based on the criteria selected by applying mcdm. an empirical assessment is carried out in the research work and the candidates to the deputies are ranked through mcdm. the recent research works of the authors alguliyev et al. (2019) have proposed an mcdm model for the selection of candidates in e-voting. the rating of candidates is calculated based on mcdm and candidates are selected based on the importance of the criteria. the proposed approach enables us to select a candidate with more relevant competencies within the framework of selected criteria. in a numerical experiment, the five candidates selected on the three criteria (education, work experience, and professional competencies), are evaluated and the candidates are ranked according to the importance of the criteria. note that the effective functioning of the government is directly dependent on human resources, and the participation of qualified personnel with competencies in governance is an issue of national importance. from this point of view, the selection of a candidate with appropriate competencies for the appointment of elected candidates to administrative positions as a result of e-voting, as well as the criteria and factors that should be considered in the selection process, are referred to as topical issues. the paper considers the application of mcdm in candidate selection in e-voting. 2. the e-voting system as an important component of e-democracy the concept of e-democracy that emerged in the 1990’s has started being perceived as the evidence of changes taking place against the backdrop of democratic principles in government. the support of the application of ict in the political arena has facilitated the emergence of e-democracy, which encompasses new methods of the governance of democratic government. political institutions, parties and politicians utilize ict in the three main processes in the political arena, including the issues of information, communication and voting. e-government maturity models are constituted of a sequence starting at the base stage all the way to the advanced stage, these stages determining the e-government maturity level. the proposal for methods for determining the development level of egovernment and ranking e-government portals is considered as the main advantage of mature models (fath-allah et al. 2014). moreover, mature models may assist organizations in promoting the efficiency of e-government. concha et al. have proposed that mature models of e-government should be categorized into three groups, namely: governmental models, holistic approach models, and e-government maturity models (layne et al. 2001; andersen et al. 2006; concha 2012). according to developed countries’ practices, research in this three categories has shown that e-government maturity models bear large importance from the standpoint of e-democracy development. the analysis of the existing e-government maturity models in the literature shows that there are several models in place, as proposed by layne and lee (2001), wescott (2001), siau et al. (2005), chen et al. (2011), and other researchers and numerous organizations (fath-allah 2014; layne et al. 2001; wescott, 2001; siau et al. 2005; andersen et al. 2006; shahkooh et al. 2008). among those models, the formation of e-democracy has been proposed by several authors, including wescott (2001), siau et al. (2005) and shahkooh et al. (2008), as the last stage of e-government development. while exploring the above-mentioned models, it is evident that e-voting, public forums, open government, the analysis of the public opinion and the development of feedback mechanisms are demonstrated as the alguliyev et al/decis. mak. appl. manag. eng. 2 (2) (2019) 65-80 68 foundation of the formation of e-democracy, which is deemed to be the evolutionary stage of e-government development. in this regard, the development of e-democracy mechanisms and e-voting technologies is necessary in order to boost transparency and efficiency, and constitutes the basis of the open government concept. the evolutionary stage envisions the formation of new requirements and the expansion of the degree of civil participation in processes by altering the relationships between the government and the citizen. the majority of the existing developmental models incorporate democratic processes, such as political participation, eparticipation, wiki democracy, interactive democracy and digital democracy (van der meer et al. 2014). all of these terms pertain to the democratic processes based on the transformation of the relations between citizens and the government. e-democracy has been included in these models as the last stage of a developmental model. logically, the government must complete the preceding information, interaction and transaction/integration stages in order to proceed to the e-democracy stage. as a new concept, the implementation of e-voting is based on reducing errors during election processes and is oriented towards maintaining the integrity of the election process in general. in the scientific literature, e-voting is considered as the use of computers and devices connected to computers in the election process, and more precisely, this term has been adopted so as to characterize elections carried out via the internet (abu-shanab 2010). the e-voting system has offered the election process numerous advantages. for instance, the facilitation of the participation of physically disabled persons, no requests for additional employees to print the election ballot papers, and a costeffective and efficient organization of elections. in general, cost-effectiveness, the expansion of participation and the broadening of voting options, a faster and accurate registration and calculation of votes, as well as accessibility and flexibility against deviations can be considered as the main advantages of e-voting (abu-shanab 2010). research studies on e-voting have gradually become an important issue. the reason for that is a growing number of scientific-research works conducted on the development of new voting methods via the internet and mobile services in european countries and worldwide. as a result, the terms of e-democracy, e-participation and evoting are frequently encountered in the context of e-democracy. in european practice, the studies in the field of e-voting are mainly represented by the empirical studies conducted by estonia, switzerland, poland, norway and other countries (drechsler et al. 2004; braun et al. 2006; trechsel 2002, 2007, 2016; musia-karg 2012, 2014; vassil et al. 2016). despite the growing number of the studies devoted to researching the impact of new technologies on democracy, there is a need for conducting comprehensive research studies in the field of e-voting. in particular, it is essential to analyze the issues such as the implementation of e-participation solutions on the example of european countries and the factors necessitating the rejection of its implementation due to various drawbacks, the application opportunities of e-voting, the existing barriers and effectiveness. hence, the development of mechanisms for and specific technological solutions to e-voting, its effectiveness and a study of undesirable results in comparison with traditional voting are deemed to be the topical research directions. currently, e-voting for elections and referendums at the local, regional and country levels is rapidly developing at the global scale as a more efficient and more feasible alternative to traditional voting and it favorably affects the development of democratic government. alongside, despite the widening international practice regarding the application of the e-voting system, several challenges are still being encountered given the national interests related to legal and social problems and its implementation. multi-criteria evaluation + positional ranking approach for candidate selection in e-voting 69 scientific and public discourse in the field of e-voting has been broadening in the last decade. e-voting systems are categorized as location-bounded and remote voting. in the first case, the voter is required to participate in the election due to the dependence of the voting on the location. remote voting has been applied in various countries, such as estonia, france, the netherlands, switzerland, and so on. e-voting has a great potential for the expansion of the democratic participation of the public by facilitating the participation of non-represented groups in the political life, including youth and physically disabled persons. moreover, e-voting fosters economic effectiveness and facilitates the effective organization of elections in comparison with traditional voting (chondros et al. 2014). in spite of the advantages of the implementation of e-voting, transition to a new technology is accompanied by numerous social, legal and technical problems (wang et al. 2017). among those, equal access to voting points, privacy maintenance, fight against interventions, the verification of information, examination, alteration and other procedures, universal verification, the right to vote, the one-voter-one-vote principle, strictness against errors, etc. can be considered. the necessity of transforming legal obstacles into technical and security solutions can specifically be mentioned amongst these (wang et al. 2017). nowadays, broad discussions are held on holding elections from a legal standpoint; as a result, it is believed that solving legal issues plays a bridging role between the law and technology. 3. mcdm-model-based candidate selection voting is a fundamental tool for decision-making in any consensus-based society and democracy hinges upon the accurate governance of nationwide elections. at present, numerous voting systems are adopted all over the world and each of those has specific advantages and problems. some countries abandoned e-voting due to its risky nature. other countries do not accept the advantages of e-voting in comparison with traditional voting. with the rapid development of the internet, which started in the 1990s, a larger number of politicians, researchers and journalists have started reflecting upon whether e-voting proposes better solutions to elections or a referendum or not. through numerous scientific incentives of non-government organizations at the global scale, the governments of european countries endeavor the use of the voting methods, ict-based solutions, the application of which constitutes the basis of democratic processes (zetter 2008; voting system; trechsel et al. 2016; meserve et al. 2017). nowadays, the majority of countries support e-voting and a growing number of countries consider the e-voting system as useful and practically apply it in their election processes. furthermore, it is to be mentioned that, for the largest part, those efforts are still at the stages of testing and conceptual analysis. the benchmark practice regarding the application of the e-voting system at the global scale can be characterized by the usa’s practice (zetter 2008; voting system; trechsel et al. 2016). at present, new voting technologies are being implemented not only in the usa, but also in several european countries (voting system; trechsel et al. 2016). surely, the efforts to implement the above-mentioned e-voting system result in various outcomes in different countries. for instance, the analysis of e-voting results from the elections to the european parliament, country parliament elections (2011) and municipal elections (2013) shows that the interest in the implementation of a new system has systematically been growing, which is the reason for the conclusion that citizens consider this voting method to be more comfortable and more effective alguliyev et al/decis. mak. appl. manag. eng. 2 (2) (2019) 65-80 70 (zetter 2008; voting system; trechsel et al. 2016; meserve et al. 2017). note that the ratio of the internet voters has grown from 1% in 2005 to 11.4% in 2014 (mona et al. 2013; musia-karg 2014; trechsel et al. 2016; mccormack 2016). the participation of citizens in political processes and the facilitation of voting during the adoption of important decisions, as well as the provision of their direct participation, are considered as the basis of democracy. in spite of the broad implementation of ict in business, various fields of the activity, education, public administration and government entities, the use of ict in the voting process is treated with cautiousness in many countries. in addition, one of the main causes for the postponed implementation of advanced voting technologies is the differences in opinions and skeptical thinking when the internet-based voting in societies is concerned (mona et al. 2013; musial-karg 2014; mccormack 2016). despite the progress made towards a better development of e-voting systems, there is no classification for the purpose of understanding the general characteristics, aims and limitations of these approaches. hence the absence of comparative research or the inaccurate determination of directions for selecting methods appropriate for specific requirements can be shown as the main drawbacks. in this regard, it is topical to develop efficient methods and mechanisms of e-voting by taking democratic processes into consideration. the ability of e-democracy to overcome barriers causing the deterrence or limitation of citizens’ participation in direct decision-making is considered as the main advantage of the development of effective e-voting mechanisms. from this point of view, e-voting is gaining the attention of government entities, political parties and politicians, and is deemed to be a powerful tool for sustaining democratic principles. the conducted research shows that e-voting has become one of the main tools of edemocracy by attaining greater importance (musial-karg 2014). in this regard, the development of e-voting technologies and the study of the implementation opportunities of new technologies are considered as important research topics. the proposed approach to the research is based on the multi-criteria evaluation of the candidates, taking into account the relationship of each candidate with another candidate. assume that, as a result of e-voting, the candidates are elected to be appointed to a relevant position. the intelligence quotient (iq), age, education, work experience, health, conviction, etc. can be attributed to the criteria for the selection of competent candidates. a binary matrix is used for the evaluation of the candidates in the study. the mcdm approach to candidate selection consists of the following stages: let 1 2 ( , ,..., ) n a a a a be the candidates and 1 2 ( , ,..., ) n c c c c be the criteria set. step 1. each candidate constructs an evaluation matrix for the evaluation of the other candidates based on each criterion: 11 12 1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) k k k l l in l k k l ij l k k k n l n l nn l p p p p p p p p   (1) where, j ( ) 1, if according to the opinion of candidate, is superior to according to criterion 0, otherwise k i k ij l l a a a p c       multi-criteria evaluation + positional ranking approach for candidate selection in e-voting 71 the principal diagonal of the k l p matrix consists of zeros, ( ) 0k ii l p  , ( ) ( )k k ij l lij p p , if i j , 0 1, 1 0  . step 2. thereafter, the ( ) l ik l q q outcome matrix is entered and the elements are calculated as below: 1 ( ) ( ) , n k ik l ij l j q p    1, 2,..., ; 1, 2,..., ; 1, 2,...,i n k n l m   , (2) ( ) ik l q reflects the final opinion of the candidate k a on the candidate j a , based on the criterion l c (in comparison with all the candidates): 11 1 1 ( ) ( ) ( ) ( ) l n l l n l nn l q q q q q  (3) step 3. the overall opinion of the candidate k a on all the candidates is based on the criterion l c and is calculated as follows: 1 ( ) , n k l ik l i o q    1, 2,..., ; 1, 2,...,k n l m  (4) step 4. the ranking of the candidate i a based on the criterion l c is determined by applying the following formula: 1 ( ) n l i ik l k r q    1, 2,..., ; 1, 2,...,i n l m  (5) the last relationship expresses the final opinion of all the candidates on the candidate i a based on the criterion l c . step 5. in order to obtain the resulting rank of the alternatives, the resultant rank computed by means of the following formula is used (aliguliyev 2009): s 1 ( 1) resultant rank s s r        (6) where s r denotes the number of the times the method appears in the s -th rank and  is the number of the alternatives. 4. a numerical experiment assume that a total of six candidates are presented based on four criteria (for example, education (c1), work experience (c2), age (c3) and professional competencies (c4)). based on the formula (1), the evaluation of the candidates according to each criterion is given in tables 1-6. based on the formulas (2) and (3), the final opinion of the candidate k a on the candidate j a is calculated according to the criterion l c (in comparison with all the candidates) and is given in table 7. alguliyev et al/decis. mak. appl. manag. eng. 2 (2) (2019) 65-80 72 table 1. the criteria-based evaluation of the candidate 1 a c1 c2 a1 a2 a3 a4 a5 a6 a1 a2 a3 a4 a5 a6 a1 0 0 0 0 1 1 0 1 0 1 0 1 a2 1 0 1 0 0 0 0 0 0 1 1 1 a3 0 0 0 1 0 1 1 1 0 0 0 1 a4 1 0 0 0 0 1 0 0 1 0 0 1 a5 0 0 1 1 0 1 1 0 1 1 0 0 a6 0 0 0 0 0 0 0 0 0 0 1 0 c3 c4 a1 a2 a3 a4 a5 a6 a1 a2 a3 a4 a5 a6 a1 0 0 1 0 0 0 0 0 1 0 0 1 a2 1 0 1 0 1 1 1 0 0 1 0 0 a3 0 0 0 1 1 0 0 1 0 1 0 0 a4 1 1 0 0 1 1 0 0 0 0 0 1 a5 1 0 0 0 0 0 0 0 1 1 0 0 a6 1 0 1 0 1 0 0 1 1 0 1 0 table 2. the criteria-based evaluation of the candidate 2 a c1 c2 a1 a2 a3 a4 a5 a6 a1 a2 a3 a4 a5 a6 a1 0 0 0 1 0 1 0 1 0 1 0 1 a2 1 0 1 0 0 0 0 0 0 1 0 0 a3 1 0 0 0 0 0 1 1 0 0 1 0 a4 0 0 1 0 0 1 0 0 0 0 0 1 a5 0 0 1 0 0 1 1 1 0 1 0 0 a6 0 0 1 0 0 0 0 1 1 0 1 0 c3 c4 a1 a2 a3 a4 a5 a6 a1 a2 a3 a4 a5 a6 a1 0 1 1 0 0 1 0 1 0 0 0 1 a2 0 0 1 0 0 0 0 0 0 1 0 0 a3 0 0 0 1 1 1 0 1 0 0 0 1 a4 1 0 0 0 0 1 0 0 1 0 0 1 a5 1 0 0 1 0 0 0 0 1 1 0 0 a6 0 1 0 0 1 0 0 1 1 0 1 0 multi-criteria evaluation + positional ranking approach for candidate selection in e-voting 73 table 3. the criteria-based evaluation of the candidate 3 a c1 c2 a1 a2 a3 a4 a5 a6 a1 a2 a3 a4 a5 a6 a1 0 1 0 1 0 0 0 1 1 0 1 0 a2 0 0 1 0 0 1 0 0 1 0 0 1 a3 1 0 0 1 1 0 0 0 0 1 1 0 a4 0 1 0 0 0 0 1 1 0 0 0 0 a5 0 0 0 1 0 1 0 1 0 1 0 1 a6 0 0 1 1 0 0 1 0 0 1 0 0 c3 c4 a1 a2 a3 a4 a5 a6 a1 a2 a3 a4 a5 a6 a1 0 1 0 1 0 0 0 0 0 0 0 1 a2 0 0 1 0 1 0 1 0 1 1 0 0 a3 1 0 0 1 0 1 0 0 0 0 1 1 a4 0 0 0 0 1 1 0 0 1 0 0 1 a5 1 0 1 0 0 0 0 0 0 1 0 0 a6 1 1 0 0 1 0 0 1 0 0 1 0 table 4. the criteria-based evaluation of the candidate 4 a c1 c2 a1 a2 a3 a4 a5 a6 a1 a2 a3 a4 a5 a6 a1 0 0 0 0 1 1 0 1 1 0 1 0 a2 1 0 0 0 0 0 0 0 0 0 1 1 a3 1 1 0 0 1 0 0 1 0 1 0 0 a4 1 0 1 0 0 1 1 1 0 0 1 0 a5 0 0 0 1 0 1 0 0 0 0 0 1 a6 0 0 1 0 0 0 0 0 0 1 0 0 c3 c4 a1 a2 a3 a4 a5 a6 a1 a2 a3 a4 a5 a6 a1 0 0 0 0 0 0 0 0 1 1 1 0 a2 1 0 1 0 1 0 0 0 0 0 1 0 a3 1 0 0 0 1 1 0 1 0 1 0 1 a4 1 1 1 0 0 1 0 0 1 0 0 1 a5 1 0 0 1 0 0 0 0 1 1 0 0 a6 1 1 0 0 1 0 1 1 0 0 1 0 alguliyev et al/decis. mak. appl. manag. eng. 2 (2) (2019) 65-80 74 table 5. the criteria-based evaluation of the candidate 5 a c1 c2 a1 a2 a3 a4 a5 a6 a1 a2 a3 a4 a5 a6 a1 0 1 0 0 0 1 0 1 0 0 1 1 a2 0 0 0 0 1 0 0 0 1 0 1 1 a3 1 1 0 1 0 1 1 0 0 1 1 1 a4 1 1 0 0 1 1 1 1 0 0 0 0 a5 1 0 0 0 0 0 0 0 0 1 0 1 a6 0 1 0 0 1 0 0 0 0 1 0 0 c3 c4 a1 a2 a3 a4 a5 a6 a1 a2 a3 a4 a5 a6 a1 0 1 0 0 0 0 0 0 1 0 0 1 a2 0 0 0 0 1 0 1 0 0 0 0 0 a3 1 1 0 0 0 0 0 1 0 1 0 0 a4 0 1 1 0 0 1 1 1 0 0 0 1 a5 0 0 1 1 0 0 0 0 1 1 0 0 a6 0 1 1 0 1 0 0 1 1 0 1 0 table 6. the criteria-based evaluation of the candidate 6a c1 c2 a1 a2 a3 a4 a5 a6 a1 a2 a3 a4 a5 a6 a1 0 1 0 1 0 0 0 0 0 1 0 1 a2 0 0 0 0 1 0 1 0 1 0 0 0 a3 1 1 0 1 0 1 1 0 0 0 0 0 a4 0 1 0 0 0 1 0 0 0 0 0 0 a5 1 0 1 1 0 1 1 0 1 0 0 1 a6 1 0 0 0 0 0 0 1 0 0 0 0 c3 c4 a1 a2 a3 a4 a5 a6 a1 a2 a3 a4 a5 a6 a1 0 0 1 1 0 1 0 1 0 0 1 1 a2 1 0 0 0 1 0 0 0 0 0 1 0 a3 0 1 0 0 0 1 0 1 0 0 0 1 a4 0 1 1 0 0 1 0 1 1 0 0 1 a5 0 0 1 1 0 0 0 0 1 1 0 0 a6 0 1 0 0 1 0 0 1 0 0 1 0 multi-criteria evaluation + positional ranking approach for candidate selection in e-voting 75 table 7. the final opinion of the candidates based on the four criteria c1 c2 a1 a2 a3 a4 a5 a6 a1 a2 a3 a4 a5 a6 a1 2 2 2 2 2 2 3 3 3 3 3 2 a2 2 2 2 1 1 1 3 1 2 2 3 2 a3 2 1 3 3 4 4 3 3 2 2 4 1 a4 2 2 1 3 4 2 2 1 2 3 2 0 a5 3 2 2 2 1 4 3 3 3 1 2 3 a6 0 1 2 1 2 1 1 3 2 1 1 1 c3 c4 a1 a2 a3 a4 a5 a6 a1 a2 a3 a4 a5 a6 a1 1 3 2 0 1 3 2 2 1 3 2 1 a2 4 1 2 3 1 2 2 1 3 1 1 4 a3 2 3 3 3 2 2 2 2 2 3 2 2 a4 4 2 2 4 3 3 1 2 2 2 3 4 a5 1 2 2 2 2 2 2 2 1 2 2 1 a6 3 2 3 3 3 2 3 3 2 3 3 3 based on the formula (4), the overall opinion of the candidate k a on all the candidates is calculated based on the criterion l c , and is given in table 8. table 8. the criteria-based opinion of each candidate (the final opinion) o1 11 10 12 12 14 14 o2 15 14 14 12 15 9 o3 15 13 14 15 12 14 o4 12 12 11 14 13 13 the ranking of the candidate i a is calculated based on the formula (5), respectively based on the criterion l c and presented in table 9. table 9. the ranking of the candidates based on each criterion r1 rank r2 rank r3 rank r4 rank 12 4 17 1 10 6 13 2 9 5 13 4 13 4 9 6 17 1 15 2 15 3 13 2 14 2 10 5 18 1 13 2 14 2 15 2 11 5 11 5 7 6 9 6 16 2 16 1 alguliyev et al/decis. mak. appl. manag. eng. 2 (2) (2019) 65-80 76 using the positional ranking approach, the resultant rank in table 10 was calculated by means of the formula 6. for example, the rank of the alternative 1 a is calculated as follows: 1 6 (6 1) (6 1 1) 1 (6 2 1) 1 (6 3 1) 0 resultant rank ( ) 6 6 6 6s 1 (6 4 1) 1 (6 5 1) 0 (6 6 1) 1 2.500 6 6 6 s rsa                               table 10. the resultant rank of the candidates candidate resultant rank rank no a1 2.500 3 a2 1.500 6 a3 3.333 1 a4 3.000 2 a5 2.333 4 a6 2.167 5 as described in table 10, the candidates are ranked in accordance with the 3 a , 4 a , 1 a , 5 a , 6 a and 2 a sequence. as the result shows in this case, the candidate 3 a has more appropriate competencies needed for the appointment to the position, according to the multi-criteria evaluation of the candidates. the ranking results can be improved by employing the importance of criteria and the fuzzy hybrid approach for the purpose of computing the weights of the criteria (lin 2010; chang et al. 2013; sakthivel et al. 2015). in practice, a different evaluation scale for the multi-criteria selection of candidates in the e-voting process can be used in the proposed model. the tools that enable the selection of a candidate with more relevant competencies within the framework of certain criteria among the candidates can be created by implementing the proposed model. 5. conclusion the paper investigates the approaches, tools, and mechanisms pertaining to the formation of e-democracy as the last stage of the development of e-government. the research results show that e-voting is gradually gaining greater importance and becoming one of the main components of e-democracy. the selection of qualified personnel at the government level and their appointment to responsible positions are important issues in economic and political processes. the candidates who are the best for the vacancy are selected for a particular position. it is worth noting that the effective functioning of the government directly depends on human resources, and the participation of qualified personnel with competencies in governance is an issue of national importance. from this point of view, the selection of candidates with appropriate competencies, the appointment of elected candidates to administrative positions as a result of e-voting, and the criteria and factors to be considered in the selection process are referred to as topical issues. the paper considers the application of the mcdm model in candidate selection in e-voting. multi-criteria evaluation + positional ranking approach for candidate selection in e-voting 77 the approach proposed in the paper is based on candidate evaluation given each candidate’s attitude towards another candidate. the rank of the candidates is calculated based on the mcdm model and the candidates are selected based on the positional ranking approach. the proposed approach enables us to select a candidate with more relevant competencies within the framework of the selected criteria. in the numerical experiment, a total of the six candidates selected based on the four criteria (education, work experience, age, and professional competencies) are evaluated and the candidates are ranked according to the resultant ranking method. the proposed model allows for the selection of the candidate with the competencies based on the criteria set out in the e-voting process and the making of more effective decisions as well. note that, alongside the rapid development of technologies and the enhancement of the implementation of the same in political processes, there is a need for a detailed analysis of the existing practice and for conducting studies oriented towards supporting citizen participation in political processes by applying these technologies. the development of e-voting methods will allow for the creation of a new e-democracy maturity 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(2008). the cost of e-voting, wired magazine, www.wired.com/2008/04/the-cost-of-e-v/ accessed 10 october 2018. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 6, issue 1, 2023, pp. 18-33. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0313052022i * corresponding author. e-mail addresses: i.badi@lam.edu.ly (i. badi), aalossta@eng.misuratau.edu.ly (a. alosta), elmansouri@eng.misuratau.edu.ly (o. elmansouri), aa_shahed@yahoo.com (a. abdulshahed), salem.asreef@eng.misuratau.edu.ly (s. elsharief) an application of a novel grey-codas method to the selection of hub airport in north africa ibrahim badi1*, abdulaziz alosta2, omar elmansouri2, ali abdulshahed2 and salem elsharief 1 1 the libyan academy-misurata, misurata, libya 2 faculty of engineering, misurata university, libya received: 12 march 2022; accepted: 8 may 2022; available online: 14 may 2022. original scientific paper abstract: air transportation and airports are indispensable means of the modern world, where well-being and travel time reliability are pillars of strength. in the past few years, passengers traveling into, within, or out of africa has enormously increased. however, the region lacks the basic facilities of linking african countries to the outside world. the research studies the possibility of assigning the optimal hub airport location in five north african countries, based on five main criteria. the criteria include airport pricing, hard and soft infrastructure, catchment and landside access, as well as other aspects such as markets and airline partners. the study uses a hybrid greycodas approach to decide the final priority of different decision alternatives. the method was implemented in steps to determine the criteria weights. four experts participated in the evaluation to determine the importance of each criterion used for the ranking of suggested airport sites. the suggested sites include cairo airport, tripoli airport, tunisiacarthage airport, algeria houari boumediene airport, and moroccomohammed v international airport. model ranking suggested morocco as the best alternative to locate a hub airport in north africa. key words: air hubs, mcdm, codas, grey theory, location problem. 1. introduction airport hubs are usually used by airlines to concentrate passenger traffic, freight traffic, and flight operations at a single location. passengers on their way to their final destination frequently stop at hubs for a stopover (bakır et al., 2022). if the final destination is not a hub city, airlines operate flights from non-hub cities to hub cities or through hub cities (raghavan & chunyan, 2021). according to the u.s. federal aviation administration (faa), hub airports are grouped into four categories based on the annual commercial enplanements each airport receives. large hub, medium mailto:i.badi@lam.edu.ly mailto:aalossta@eng.misuratau.edu.ly mailto:elmansouri@eng.misuratau.edu.ly mailto:aa_shahed@yahoo.com mailto:salem.asreef@eng.misuratau.edu.ly an application of a novel grey-codas method to the selection of hub airport in north africa 19 hub, small hub and non-hub which respectively receive 1% or more, between 0.25 to 1%, between 0.05 to 0.25%, and less than 0.05% of the annual u.s. passenger boarding's (faa, 2022). the proposition of air transportation industry development subjected to the right selection of a new hub airport enhances the network operational capability, lowers operating costs, and maximizes the network scale efficiency (bellizzi et al., 2020; button & lall, 1999). meanwhile, it is valuable for the proposed expansion to specifically adapt to the air transportation business evolution, more consistent with the enlargement trend of the airline sector as a whole. airline network represents the backbone of airline operations; where the route is connected in a specific way with the aim of optimizing the allocation of airline's resource and improve revenue (kanafani & ghobrial, 1985). aviation industry in developed countries utilizes flexible and useful large coverage areas known as hub structures, which are the main contributor to the transportation industry development (oktal & ozger, 2013). morocco, algeria, tunisia, libya, and egypt are north african countries located in the northern coast of africa, the largest countries in the continent, with an approximate area of 5,589,280 km2. the region is bounded on the north by the mediterranean sea coastline, on the east by the red sea, on the south by the subsaharan africa, and on the west by the atlantic ocean. such location enriches the great strategic importance and, hence, they have a high profile in the area. the major airports of north african countries are: tunis-carthage airport in tunisia, adrar airport in algeria, casablanca mohammed v airport in morocco, cairo airport in egypt, and tripoli airport in libya. most of them publicly owned, for instance, the libyan civil aviation authority (lycaa), a state entity affiliated with the libyan ministry of transportation, owns and operates the airports in libya (elmansouri et al., 2020). hub airport establishment requires a different network design based upon the considered project area. for instance, istanbul ataturk airport serves as a hub airport for turkish air freight airlines. the airport has been designated as the main airport in the region; it handles half of the domestic air freight traffic in the country (oktal & ozger, 2013). however, one advantage of the chosen hub is that freight traffic flows in both directions with all airports in the network (chen et al., 2017). in the same context, maertenz et al. (2014) created a wadp (weighted average distance penalty) measure implemented to tripoli as a prospective hub location. they concentrated on traffic between africa and europe and assessed the feasibility of establishing an air transportation hub in libya. the improvement of aviation will make an important contribution to the economy and social well-being of north african countries. central hubs are essential for accessing many international destinations for business and leisure desires, and are also the most convenient means of traveling to other parts of africa as well as asia and europe. the importance of the location of the north african countries is based on the fact that this region serves as a base station connecting africa and europe. an extension of its geographical interests, climate, as well as other important factors such as workforce etc. as mentioned in the literature (maertens et al., 2014), morocco, algeria, tunisia, libya, and egypt are the least distant from the shipping lanes in the mainland africa. these countries have the potential to become a hub airport as an alternative corridor for other african countries that need airports for trade, and they have not yet taken advantage of their locations. in 2004 the world food program used libya as a corridor to provide assistance to darfur refugees through chad. the aid was transported approximately 2,700 km from benghazi in libya to the city of abéché in chad (ghashat, 2012). badi et al./decis. mak. appl. manag. eng. 6 (1) (2023) 18-33 20 the use of multi-criteria decision-making methods in the aviation industry has increased recently. there are a variety of applications, such as performance evaluation (eshtaiwi et al., 2018), measurement of passenger satisfaction (pandey, 2016), eservice quality (bakır & atalık, 2021), airlines performance (badi & abdulshahed, 2019), and other applications. barros and dieke (2007) used data envelopment analysis methodology (dea) to analyze the financial and operational performance of thirty-one italian airports, while twenty major international airports around the world were investigated by tseng et al. (2008). lu et al. (2019) examined twenty-seven chinese airports efficiency during the period 2014-2018. ulkü (2015) conducted a comprehensive analysis of spanish and turkish airports to evaluate the efficiencies of the operations. abbott (2015) studied the performance efficiency of the three main airports in new zealand. in this perspective, essential decisions have to be made concerning the potential hub airports in north african. finding the optimal location of a hub is a vital decision that represents the success of decision-making process. the difficulty for the prospective hub airport location is considered a strategic planning issue that should be addressed in line with the aviation sector. in this context, a development of such beneficent projects improves the economic situation of the north african region and boosts industries other than the dominating energy sector. the paper aims to examine a new method to find the optimal site selection, a hub airport that provides the best services and connects the north african region and the african continent. the region suffers the absence of major airports to face the evergrowing demand on travel. a case study of five african potential hub airports was undertaken in this study, covering both developing and fluctuating hubs. survey interviews with four experts were conducted and a review of policy and strategic documents. experts like pilot and airport manager having more than thirty years of professional work experiences in the aviation field. methods have been developed to allocate the services taking into account decision-making variables. the authors claim that expansion and investment in hub airports and their correlated models is a nonlinear task. most of the methods used in the past are based on traditional approaches, however, the application of mcdm method is an important technique for helping the air transportation sector. the authors have developed a new technique that has never been used. a hybrid grey-codas (combinative distance-based assessment) model is used to evaluate the criteria weights and rank the alternatives. 2. air transport between europe and africa modern aviation industry has a significant role in promoting economies in longterm bases. this sector employs over 58 million people worldwide and provides over $2.4 billion to the global gross domestic product (gdp). also, around 3.3 billion passengers and $6.4 billion worth in freight are transported each year. since 1977 and in spite of the economic crisis, global aviation industry has grown and is expected to thrive once every 15 years which demonstrate how investments in the aviation industry are crucial to the revival of the economy. global aviation industry is expected to rise at average annual rate of 3.6 percent within duration from 2011 to 2030, in contrast to 3.2 percent from 1990 to 2010. as the aviation industry thriving across the globe, developing countries has been economically benefitting. southeast asian, latin american, and african airlines are gaining a rising percentage of overall air traffic. the primary causes for this increase are the anticipated rise in load factors as well as an application of a novel grey-codas method to the selection of hub airport in north africa 21 increase in aircraft size and capacity. it is predicted that africa would undergo particularly fast expansion in the next years (yao et al., 2014). one of the most important challenges facing the aviation sector is how to accommodate the growing demand for air transportation. as illustrated in figure 1, europe has the highest growing demand on passenger air transportation. while demand for air transportation in europe has increased at an average annual rate of 5.0 percent, the middle east and africa have seen a significant increase in air transportation recently, with an average annual growth rate of 13 percent per year. passenger air traffic in middle east and africa is anticipated to achieve levels equal to those seen in europe (bonnefoy, 2008). figure 1. historical evolution of air transportation activity across eu and africa. source: prepared by the authors based on data from the world bank it is difficult to predict parameters in the field of air transportation and there will usually be some variation from expectations for some parts of the world. the international civil aviation organization (icao) developed two prediction models, one optimistic and the other pessimistic, to assess and predict the future of air transportation. the global average annual growth rate for passenger air transportation is expected to range between 5.1% and 3.6% per year for optimistic and pessimistic scenarios respectively (yao et al., 2014). future expectations for airline registration can be formulated either by region or route group. the majority of world countries assign the local airline companies for their domestic air transportation. cross-border air traffic for a route group will mainly be supplied by regional airlines; however, airlines from different zone may have the rights of the international air transportation for that region. commercial air transportation has had substantial traffic growth over the previous decades, resulting in the formation of several new commercial air transportation companies. subsequently, the need for trained aviation professionals, such as pilots, aircraft maintenance workers, and air traffic controllers, will increase to handle the demand for the following years (yao et al., 2014). africa is a developing continent; this also applies to the aviation market. currently, africa is responsible for only 3% of global air business while 17% of the world’s population lives in this continent. the top 10 largest airports in north africa, for example, three airports from egypt in the cities of cairo, hurghada and sharm el badi et al./decis. mak. appl. manag. eng. 6 (1) (2023) 18-33 22 sheikh, and casablanca’s mohammed v international airport in moroco accounted for more than a third of the total annual departures from this region. during the period 2000-2010, the total number of passengers for the five countries was 310.1 million. figure 2 shows the percentage of the total number of passengers shared by each country. more than one third of the total passengers use egyptian airports to travel, while 26 percent of the total trips lands on morocco. figure 2. percentage of the total number of passengers shared by each country. source: own calculations based on data from eurostat. table 1 shows the top five ranked destinations and number of passengers from five eu airports to africa. cairo airport plays a significant role in connecting europe with africa and operates with different airline partners from all destinations. the number of passengers is calculated for the period 2009-2019. table 1. top five ranked destinations for five main euro airports. source: own calculations based on eurostat. no. origin destination #pax 1 london heathrow airport o.r tambo 2,729,472 2 nairobi/jomo 2,172,302 3 lagos/murtala muhammed 2,166,918 4 cape town 1,993,102 5 cairo 1,974,654 1 roma/fiumicino airport cairo/intl 1,308,059 2 tunis/carthage 1,121,703 3 casablanca/mohammed v 898,795 4 alger/houari boumediene 667,809 5 addis ababa 460,099 1 paris-charles de gaulle airport alger/houari boumediene 2,903,455 2 casablanca/mohammed v 2,251,388 3 tunis/carthage 1,928,780 4 sir seewoosagur ramgoolam 1,916,157 5 cairo/intl 1,583,887 1 frankfurt/main airport cairo/intl airport 1,551,838 2 o.r tambo international airport 1,424,294 3 hurghada / intl airport 1,131,879 an application of a novel grey-codas method to the selection of hub airport in north africa 23 no. origin destination #pax 4 addis ababa/bole com/met/nof airport 892,269 5 tunis/carthage airport 804,138 1 adolfo suarez madrid-barajas airport marrakech/menara airport 1,494,603 2 casablanca/mohammed v airport 1,235,644 3 tanger/ibn batouta airport 982,686 4 cairo/intl airport 651,184 5 dakar/yoff airport 561,756 3. methodology mcdm (multi-criteria decision making) is an approach used by researchers in the making of decisions that comprise prioritizing, ranking, or selecting between preferences (pmucar, 2020). mcdm system integrates preference's conduct across various quantitative, qualitative, or contradicting criteria and effects in a proposition requiring a consensus. skills amassed from various fields, information systems, economics, computer applied science, behavioral decision theory, and others are utilized. various mcdm procedures have been established and prompted efficiently in various areas of necessity (bakır et al., 2021). there are various mcdm methods such as analytical network process, fuzzy decision-making, and data envelopment analysis (durmić et al., 2020). despite various researches implementing the methods, mcdm remains the rapidly developing problem area in diverse departments (bouraima et al., 2021). however, each of the methods has the same ability to make decisions under distrust, and each holds its privilege. grey system theory is a mathematical methodology that was initially introduced by deng in 1982. the theory has been effectively used for modelling problems with limited amount of data and incorporating uncertainty in systems (li et al., 2007). unlike traditional methods which require large number of samples, the grey theory is designed to study and model systems lacking sufficient information. the grey system theory has been successfully used in different research areas such as finance, engineering, social and economics (badi & pamucar, 2020). when all of the information is known, the system is called white, and when all of the information is unknown, the system is called black (abdulshahed et al., 2017). it is called grey system when the information is being incomplete (liu et al., 2012). grey number can be defined as a measure where we only know the range of values rather than the exact value (eshtaiwi et. al., 2017). the unknown parameters of the grey system are expressed by discrete or continuous grey number represented by the symbol⊗. the theory include a variety of features and operations on grey numbers, including the core of the number⊗ ̂, its degree of greyness g°, and the grey number’s whitening degree which indicates how a number prefers to be in the middle of a range of feasible values (badi et al., 2018). the codas method is developed by keshavarz et al. (keshavarz ghorabaee et al., 2016). it is widely used for many multi criteria decision making problems (badi et al., 2018). this research uses a hybrid grey-codas method to examine the assessment of decision makers to determine the proper location for an airport hub. the aim of this research is to implement this hybrid approach for determining the optimal location for an airport hub in northern african countries. five airports were chosen to represent the inspected sites as follows; s1: cairo airport, s2: tripoli airport, s3: badi et al./decis. mak. appl. manag. eng. 6 (1) (2023) 18-33 24 algeria airport, s4: tunis/carthage airport, s5: mohammed v airport, morocco. the priority weights of the alternatives were determined using ms excel macros based on the questionnaire forms that were used to compare major attributes and alternatives. the grey-codas model is conducted on 11 steps as follows: step 1: choosing set of the most crucial attributes and suggest alternatives. step 2: calculate the weight of attributes 𝑊𝑗 using the following equations: ⊗ 𝑊𝑗 = 1 𝐾 [⊗ 𝑊𝑗 1 +⊗ 𝑊𝑗 2 + ⋯ +⊗ 𝑊𝑗 𝐾] (1) ⊗ 𝑊𝑗 𝐾 = [𝑊𝑗 𝐾, 𝑊𝑗 𝐾] (2) step 3: experts assess the alternatives: expert’s feedback will be on either linguistic or verbal factors depending on the criteria. ⊗ 𝐺𝑖𝑗 𝐾, (𝑖 = 1,2, … , 𝑚; 𝑗 = 1,2, … , 𝑛)is the value of the attribute obtained from the kth expert to any of the alternatives which is represented as, ⊗ 𝐺𝑖𝑗 𝐾 = [𝐺𝑖𝑗 𝐾, 𝐺𝑖𝑗 𝐾 ] and calculated using the following formula:⊗ 𝐺𝑗 = 1 𝐾 [⊗ 𝐺𝑗 1 +⊗ 𝐺𝑗 2 + ⋯ +⊗ 𝐺𝑗 𝐾] step 4: forming the grey decision matrix: 𝐺 = [ ⊗ 𝐺11 ⊗ 𝐺12 ⋯ ⋯ ⊗ 𝐺1𝑛 ⊗ 𝐺21 ⊗ 𝐺22 ⋯ ⋯ ⊗ 𝐺2𝑛 ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⊗ 𝐺𝑚1 ⊗ 𝐺𝑚2 ⋯ ⋯ ⊗ 𝐺𝑚𝑛] (3) step 5: normalizing the decision matrix: 𝐷∗ = [ ⊗ 𝐺11 ∗ ⊗ 𝐺12 ∗ ⋯ ⋯ ⊗ 𝐺1𝑛 ∗ ⊗ 𝐺21 ∗ ⊗ 𝐺22 ∗ ⋯ ⋯ ⊗ 𝐺2𝑛 ∗ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⊗ 𝐺𝑚1 ∗ ⊗ 𝐺𝑚2 ∗ ⋯ ⋯ ⊗ 𝐺𝑚𝑛 ∗ ] (4) benefit attribute ⊗ 𝐺𝑖𝑗 ∗ is formed as ⊗ 𝐺𝑖𝑗 ∗ = [ 𝐺𝑖𝑗 𝐺𝑗 𝑚𝑎𝑥 , 𝐺𝑖𝑗 𝐺𝑗 𝑚𝑎𝑥]where 𝐺𝑗 𝑚𝑎𝑥 = 𝑚𝑎𝑥1<𝑖<𝑚{𝐺𝑖𝑗} and a cost attribute ⊗ 𝐺𝑖𝑗 ∗ is formed as ⊗ 𝐺𝑖𝑗 ∗ = [ 𝐺𝑗 𝑚𝑖𝑛 𝐺𝑖𝑗 , 𝐺𝑗 𝑚𝑖𝑛 𝐺𝑖𝑗 ]where 𝐺𝑗 𝑚𝑖𝑛 = 𝑚𝑖𝑛1<𝑖<𝑚{𝐺𝑖𝑗}. step 6: calculating the elements of weighted normalized grey decision matrix using the following formula. ⊗ 𝑉𝑖𝑗 =⊗ 𝐺𝑖𝑗 ∗ 𝑋 ⊗ 𝑊𝑗 forming the weighted normalised grey decision matrix 𝐷𝑊 ∗ . 𝐷𝑊 ∗ = [ ⊗ 𝑉11 ⊗ 𝑉12 ⋯ ⋯ ⊗ 𝑉1𝑛 ⊗ 𝑉21 ⊗ 𝑉22 ⋯ ⋯ ⊗ 𝑉2𝑛 ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⊗ 𝑉𝑚1 ⊗ 𝑉𝑚2 ⋯ ⋯ ⊗ 𝑉𝑚𝑛] (5) an application of a novel grey-codas method to the selection of hub airport in north africa 25 step 7: determine the negative-ideal solution using equation (6). 𝑛𝑠 = [𝑛𝑠𝑗]1×𝑚 𝑛𝑠𝑗 = 𝑚𝑖𝑛𝑖𝑟𝑖𝑗 (6) step 8. applying the negative-ideal solution to compute the euclidean and taxicab distances using equations (7) and (8) respectively. 𝐸𝑖 = √∑ (𝑟𝑖𝑗 − 𝑛𝑠𝑗) 2𝑚 𝑗=1 (7) 𝑇𝑖 = ∑ |𝑟𝑖𝑗 − 𝑛𝑠𝑗| 𝑚 𝑗=1 (8) step 9: forming the relative assessment matrix using equation (9). 𝑅𝑎 = [ℎ𝑖𝑘]𝑛×𝑛 (9) ℎ𝑖𝑘 = (𝐸𝑖 − 𝐸𝑘) + (𝜓(𝐸𝑖 − 𝐸𝑘) × (𝑇𝑖 − 𝑇𝑘)) where k∈ {1, 2, …, n} and 𝜓 indicates a threshold function to judge the equality of the euclidean. 𝜓(𝑥) = { 1 𝑖𝑓 |𝑥| ≥ 𝜏 0 𝑖𝑓 |𝑥| < 𝜏 τ is a threshold parameter that is defined by decision makers, and is set within a range of values of 0.01 and 0.05. the study evaluated alternatives base on taxicab distance, where the variance between euclidean distances of two alternatives does not exceed τ. model calculations were based on a value of τ = 0.02. step 10: calculate the assessment score of each alternative using equation (10). 𝐻𝑖 = ∑ ℎ𝑖𝑘 𝑛 𝑘=1 (10) step 11: put alternatives score (h) on descending order, so that the highest η is the best choice among the alternatives. 4. the case study in this research study, qualitative parameters for the airport hub site selection problem are identified using questionnaire forms. table 2 shows the five different factors that are taken into account. except for the first, which is a cost criterion, all of these criteria are classified as benefit criteria. the model was programmed using macros in microsoft excel to make the procedure easier. table 2. criteria used (maertens et al., 2014) category factors airport expenses airport usage cost (landing, passenger services, ground handling, atc services charges, fuel cost, etc.) deduction, marketing services ‘‘hard’’ airport infrastructure runways and taxiways, terminal and apron facilities/marking possibility of airport expansion passenger transit services, suitable connection flights offering maintenance facilities ‘‘soft’’ airport infrastructure airport slot capacity business hours work regulations catchment and landside access large-scale location (world/continent/country level) minor-scale location (local region, considering local factors such as obstacle clearance) badi et al./decis. mak. appl. manag. eng. 6 (1) (2023) 18-33 26 demand drivers in the catchment area (gdp, population, standard of living, industrial base, socio-economic factors and ethnic factors) competition from nearby airports other airlines facilities at airport (competing carriers etc.) airline partnership availability four experts were invited to evaluate each of the proposed criteria in the examination of potential airport locations. table 3 shows a scale that can be used to express linguistic variables in grey numbers. sites were scored on a grey scale in table 4 for their performance on many attributes. table 3. the importance of grey number for the weights of the criteria. importance abbreviation scale of grey number⊗ 𝑊 very low vl [0.0, 0.1] low l [0.1, 0.3] medium low ml [0.3, 0.4] medium m [0.4, 0.5] medium high mh [0.5, 0.6] high h [0.6, 0.8] very high vh [0.8, 1.0] table 4. linguistic assessment and the associated grey values. performance abbreviation scale of grey number ⊗ 𝑊 very poor vp [0.0, 1.0] poor p [1.0, 2.0] medium poor mp [2.0, 4.0] fair f [4.0, 5.0] medium good mg [5.0, 6.0] good g [6.0, 8.0] very good vg [8.0, 10.] table 5 summarize the expert responses in evaluating the targeted attributes. weights of attributes are calculated using equation 2. table 5. the linguistic assessment of the attributes by experts. ci expert #1 expert #2 expert #3 expert #4 ⊗ 𝑊 whitening degree c1 h ml h h 0.53 0.70 0.6125 c2 vh h vh vh 0.75 0.95 0.8500 c3 mh m h h 0.53 0.68 0.6000 c4 mh mh vh vh 0.65 0.80 0.7250 c5 h mh m mh 0.50 0.63 0.5625 as presented in table 5, the second attribute, which denotes to the airport infrastructures, is ranked as the top priority among all attributes followed by the catchment and landside access attribute. airports have a vital contribution in the economy of a country. the quality of airport infrastructure, which is an important part of the entire transportation network, has a significant role to attract foreign investment. airline companies require affordable, safe and functional airport infrastructures to expand their passenger and cargo services. a successful hub requires a location on or near major traffic flows. detour factors will be too high if a an application of a novel grey-codas method to the selection of hub airport in north africa 27 hub is too far away for major traffic flows. large detours mean higher costs due to longer flight times, greater fuel usage, and more employees and other factors. table 6 shows the experts' linguistic evaluations of each site. as stated in table 3 and equation 3, convert the linguistic variables into grey numbers using the scales of grey numbers. the grey decision matrix d is calculated based on the consequence's assessment. the suggested sites include cairo airport (s1), tripoli airport (s2), tunisiacarthage airport (s3), algeriahouari boumediene airport (s4), and moroccomohammed v international airport (s5). table 6. experts views on suggested sites selection criteria. cj sites expert #1 expert #2 expert #3 expert #4 gij c1 site #1 g g mg mg [5.50 7.00] site #2 vg g vg vg [7.50 9.50] site #3 vp p g g [3.25 4.75] site #4 vg mg vg g [6.75 8.50] site #5 vg g g g [6.50 8.50] c2 site #1 g g vg g [6.50 8.50] site #2 vp g g f [4.00 5.00] site #3 mp p g g [3.75 5.50] site #4 f f g f [4.50 5.75] site #5 vg g vg g [7.00 9.00] c3 site #1 mp g g g [5.00 7.00] site #2 vp g g mg [4.25 5.75] site #3 p p g g [3.50 5.00] site #4 f g vg vg [6.50 8.25] site #5 mg g vg vg [6.75 8.50] c4 site #1 mg f g mg [5.00 6.25] site #2 vg g g vg [7.00 9.00] site #3 mg g vg g [6.25 8.00] site #4 vg g vg vg [7.50 9.50] site #5 vg mg g mg [6.00 7.50] c5 site #1 g g g mg [5.75 7.50] site #2 vp f vg f [4.00 5.25] site #3 g g vg g [6.50 8.50] site #4 f f vg mg [5.25 6.50] site #5 g mg vg g [6.25 8.00] the decision matrix "d" is normalized, so the grey elements range between 0 and 1. 𝐷 ∗ = [ [0.4643 0.5909] [0.7222 0.9444] [0.5882 0.8235] [0.5263 0.6579] [0.6765 0.8824] [0.3421 0.4333] [0.4444 0.6111] [0.5000 0.6765] [0.7368 0.9474] [0.4706 0.6176] [0.6842 1.0000] [0.4167 0.6111] [0.4118 0.5882] [0.6579 0.8421] [0.7647 1.0000] [0.3824 0.4815] [0.5000 0.6389] [0.7647 0.9706] [0.7895 1.0000] [0.6176 0.7647] [0.3824 0.5000] [0.7778 1.0000] [0.7941 1.0000] [0.6316 0.7895] [0.7353 0.9412]] (11) weights of criteria are calculated using equation (6) using grey multiplication. weights allocated to attributes are multiplied by the corresponding elements of the normalized grey decision matrix. badi et al./decis. mak. appl. manag. eng. 6 (1) (2023) 18-33 28 𝐷𝑤 ∗ = [ [0.2438 0.4136] [0.5417 0.8972] [0.3088 0.5559] [0.2763 0.4441] [0.4397 0.7059] [0.1796 0.3033] [0.3333 0.5806] [0.2625 0.4566] [0.3868 0.6395] [0.3059 0.4941] [0.3595 0.7000] [0.3125 0.5806] [0.2162 0.3971] [0.3454 0.5684] [0.4971 0.8000] [0.2007 0.3370] [0.3750 0.6069] [0.4015 0.6551] [0.4145 0.6750] [0.4015 0.6118] [0.2007 0.3500] [0.5833 0.9500] [0.4169 0.6750] [0.3316 0.5329] [0.4779 0.7529]] (12) table 7 contains weights of criteria that have been used to determine the values of normalized performance. then, data utilized to compute the negative-ideal solution, which is subsequently applied to determine the euclidean and taxicab distances of alternatives (badi et al., 2018). table 7 summarizes the findings. table 7. the weighted normalized decision matrix and the negative-ideal solution alternatives airport pricing hard infrastruct ure soft infrastructure landside access others distances euclidean taxica b 0.3287 0.7194 0.4324 0.3602 0.5728 0.3574 0.658 7 0.2415 0.4569 0.3596 0.5132 0.4000 0.1622 0.216 3 0.5296 0.4465 0.3066 0.4569 0.6485 0.3926 0.633 4 0.2689 0.4910 0.5283 0.5447 0.5066 0.3119 0.584 7 0.2754 0.7667 0.5460 0.4322 0.6154 0.4610 0.880 9 negative-ideal solution 0.2415 0.4465 0.3066 0.3602 0.4000 table 7 and equation 7 can be used to compute the relative assessment matrix and the assessment scores (h) of alternatives assuming the value of 𝜏 = 0.02. results are summarized in table 8. table 8. the relative assessment matrix and the assessment scores of alternatives s1 s2 s3 s4 s5 h 0 0.6376 -0.0099 0.11947 -0.326 0.4214 -0.6376 0 -0.6475 -0.51812 -0.963 -2.7665 0.0099 0.6475 0 0.12935 -0.316 0.4708 -0.1195 0.5181 -0.1294 0 -0.445 -0.1759 0.3258 0.9633 0.31588 0.44523 0 2.0502 table 8 shows that site number 5 has the highest value of h. as a result, when it comes to codas approach evaluation, s5 is the best site. also, a sensitivity analysis was performed to assess the results' validity and stability. to examine their effect on site ranking, fourteen values of  ranging from 0.01 to 1.00 were chosen. the values of  and their impact on site ranking are shown in table 9. an application of a novel grey-codas method to the selection of hub airport in north africa 29 table 9. sites ranking with different values of   0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.15 0.30 0.50 1.00 s1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 s2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 s3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 s4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 s5 4 4 4 3 3 3 3 3 3 3 3 3 3 3 figure 3 illustrates that the fifth site (s5) is the best among all sites. also, it can be noted that changing the parameter  has a minimal impact on the ranking of alternatives. figure 3. sites ranking with different values of  5. discussion. most african countries are destitute of a regional hub airport that has the ability to connect them with the outside world. due to the everlasting and continuous growth of demand for air transportation, a new hub airport needs to be constructed in north africa. the area compared to the rest of the continent has the resources and the capability to help the region and the continent to provide such services and to connect the countries of the world with africa. moreover, north africa geostrategic location enhances the benefits of the project, where the region situated at the crossroads of africa, europe, and asia. although the importance of the geographic location when allocating hub airports, the results show that infrastructure is the most important criterion. the number of people and goods transported depends on the basic airport facilities such as airport area, runways, terminals, and the availability of other services, maintenance for instance. therefore, freight and passengers could be transported efficiently with the badi et al./decis. mak. appl. manag. eng. 6 (1) (2023) 18-33 30 least journey time, and maximum flight capacity. nevertheless, the geographic location is ranked second and so forth. on the other hand, maertens et al. (2014) evaluated hub airports in libya, algeria, egypt and other regional countries with respect to their geographic location and passenger traffic. moreover, oktal & ozger (2013) suggested that aircraft range, travel cost, and other aspects are considered as major factors affecting hub locations. the results also indicate that morocco's airport is the best alternative between the five locations. the country in the past few years has invested in infrastructure projects as well as airports. various international airlines provide quality and competitive services to the customers. the political system is very stable, aiming to help tourists visit the country. on the contrary, tripoli is the worst alternative, given the current situation in libya. the airport infrastructure is dilapidated, and the country is in crisis. consequently, international airlines are banned and out of service. unlikely, maertens et al. (2014) ranked cairo and tripoli among the best airports in the region. the five alternatives were studied under different circumstances by conducting a sensitivity analysis. results show that differences between values are insignificant and the alternatives ranking model is valid. libya is located in the center of the north african coast, and the gdp is very high compared to the neighboring countries and to the rest of african countries in general. the country is also an oil exporter, and the fuel prices are very low. hence, new job opportunities will be created based on the substantial investments expected in the future. despite of the ongoing circumstances in libya, the ranking of the five alternatives may alter at any instant. 6. conclusion constructing a new hub airport in north africa is a vital to the region and to the african continent. such crucial projects could assist countries to improve infrastructure, socioeconomic status, and prosperity of the african nations. the study focused on selecting the optimal alternative from five major airport locations located in north africa. the main purpose is to link africa with the rest of the world and to connect long-haul flights between the countries as a transit station taking into account the immense number of passengers travelling from and to african territories. infrastructure, geographic location, and three major criteria have been chosen to evaluate each location, where the infrastructure was ranked as the most significant. the study came to an end that morocco is distinguished as the best choice to allocate the project. the limitation to this research is that it does not guarantee that one obtained location is the optimal, in the case of the criteria has been changed. the topic remains open for further studies using a broader range of samples and applying different modeling hypothesis. given the exceptional situation of libya; deteriorated infrastructure, and instability of the country; the country’s economic situation is still secure. libya also possesses a fleet of aircraft that is the best in the region. author contributions: conceptualization, i. b; methodology, i.b., s.e; validation, o. e.; formal analysis, i.b., s.e; investigation, s.e., o.e; resources, s.e., a.a; writing—original draft preparation, i.b., o.e., a.a; writing—review and editing, a.a.; visualization, a.a.; supervision, i.b.; the authors have read and agreed to the published version of the manuscript. funding: this research received no external funding. an application of a novel grey-codas method to the selection of hub airport in north africa 31 conflicts of interest: the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. references abbott, m. 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(2014). forecasting methods and icao’s vision of 2011-2030 global air traffic. journal of air transport studies, 5(1), 1-22. © 2023 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). decision making: applications in management and engineering vol. 3, issue 1, 2020, pp. 22-29. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame2003132h * corresponding author. e-mail addresses: omernuricam@gmail.com (ö.n. çam), kemal.sezen@altinbas.edu.tr (h.k. sezen). the formulation of a linear programming model for the vehicle routing problem in order to minimize idle time ömer nuri çam 1 and hayrettin kemal sezen 2* 1 uludag university, faculty of economics and administrative sciences, bursa, turkey 2 altınbaş university, school of applied sciences, istanbul, turkey received: 15 august 2019; accepted: 2 december 2019; available online: 23 december 2019. original scientific paper abstract: the paper deals with the question, “what is the vehicle routing problem, which is minimized idle time, and how its linear programming model is written?” in this study, a linear programming (lp) model has been developed for the vehicle routing problem (vrp) in order to minimize the total idle time (mit). this problem was realized while managing the route operations of a company transporting long-distance passengers by bus in turkey. the differences between this problem and other vrps first arise from its objective function. it suggests that vehicles should work more because they make profit if they work. so, its objective function should be defined so as to minimize the sum of the idle time of those vehicles. contrary to the vr problems examined so far, vehicles should work more, sometimes preferring long-distance routes as well. the other two differences pertain to constraints: some locations should be visited more than once in different time periods, and subtours could be allowed in some situations. in order to present the problem, a total of 34 routes of the company which belongs to one of the five subgroups were chosen for the samples. to solve this kind of problems, it is very important that exact methods, such as linear programming or branch and bound, should be used. keywords: vehicle routing problem; minimizing idle time; linear programming; location and time point. 1. introduction the vehicle routing problem (vrp) is an np-hard problem and is encountered in logistics management, which constitutes about 20% of product-service costs (laporte, 2009). researching vr problems grows 6% on average every year (kramer et al., 2015) ⁠. mailto:omernuricam@gmail.com mailto:kemal.sezen@altinbas.edu.tr the formulation of a linear programming model for the vehicle routing problem in... 23 the main components of the vrp are the geographical route network, the customer, the warehouse, vehicles and drivers. different vrps are distinguished according to these basic components by different constraints or types of routes. according to those components, the basic vrps are divided into subcategories, such as capacity limited, distance constrained, time restricted, pick and delivery, open, mixed vehicles, partly carried, periodic, stochastic, fuzzy, green, and so on (cacerescruz et al., 2014). ⁠ day by day, new subcategories emerge, with different constraints and criteria representing real life. according to the literature, it can be said that it is impossible to directly classify some studies into one specific category. for this reason, rich vehicle routing is proposed for solving multiple categories and constraints simultaneously (caceres-cruz et al., 2014) ⁠. green vehicle routing can be seen as a new constraint to a reduction in carbon emissions and environmental pollution (koç & karaoglan, 2016) ⁠. vr problems are generally constraint satisfaction problems (csp). constraint satisfaction problems are combinatorial in their nature (brailsford et al., 2019) ⁠. since it is one of the np-hard type problems, heuristic algorithms are used extensively (cordeau et al., 2007) ⁠. the objective functions of all problems are generally seen to minimize the costs of linking geographical resources to target points in previous research studies (daneshzand, 2011; braekers et al., 2015) ⁠. in the most general sense, vr problems are grouped as said above, but there is also a subproblem related to the definition of the hybrid elements of the basic vr problems (kramer et al., 2015) ⁠. in this study, a new type of the objective function is presented and modeled, which will cast a different light on the aforementioned problems. the problem addressed is that it is different from other problems in terms of targeting the minimization of idle time for geographical points while offering a greater use of vehicles on their route. for this reason, the proposed name for the problem is vrpmit (vehicle routing problem which minimizes idle time). the objective function proposed in this study is similar to the sequencing problems with the minimum idle time in a job shop. the basic approach to the problem already considers each tool as a machine that should be used efficiently. in the literature, it is proposed that the same solution methods may be used because of the similarity of vehicle routing problems to scheduling problems, such as a job shop (beck et al., 2003) ⁠. vehicle scheduling algorithms can be considered as the top category of the vrp. at this stage, vehicles are listed according to dozens of constraints, such as working hours, employee permissions, official requests and customer needs, only to be followed by planning their routes. generally speaking, routing is defined for the shortest route (laporte, 2013) ⁠. at the same time, time window vrp problems were created by adding a time constraint to the problem (vrptw). a vehicle must arrive in time and wait for the departure time at the arrival point (el-sherbeny, 2010) ⁠. the main difference between this study and the vrptw problem is that the whole route will be driven by one vehicle only, and if the departure time is missed, the vehicle will have to wait for the corresponding time next day. in vehicle scheduling problems similar to the scope of this study, it is proposed that public transport vehicles should be scheduled approximately according to customers’ needs. in such a study, the arrival-departure times at the stops can be stretched so that the lowest idle time occurs (wang et al., 2017) ⁠. in this study, all trip times are assumed to be fixed and deterministic. when observing time-targeted or limited problems for the vrp, the literature gives short-term goals, such as the minimum driving salary and the minimum co2 emissions çam and sezen/decis.-mak. appl. manag. eng. 3 (1) (2020) 22-29 24 (braekers et al., 2015) ⁠. it evaluates the issues such as time-related work, traffic conditions and vehicle maintenance, and presents sub-solutions with a certain or indeterminate arrival time. the outputs of these aims may generate similar results, but this study has a different perspective. the objective function addressed in the paper is a timed objective, which is ultimately aimed at achieving the minimum idle time of a vehicle, a device or equipment. from this point of view, vehicles should/may work for longer. as a result, the number of cars operated will be reduced, resulting in less waste in the resource use and increased operating efficiency. when looking at the previous research studies, the proposed solutions to vr problems are exact, heuristic and metaheuristic methods (laporte, 2009) ⁠. there are many solutions using heuristic algorithms due to the easiness of finding a suitable solution (daneshzand, 2011) ⁠. the solutions to be developed for the vrpmit will be effected to direct and indirect costs, such as the number of vehicles, the fleet management, service/maintenance, ticket prices, carbon emissions, waste reduction in both land and airway operations. 2. the definition of the problem the company that provides passenger highway transportation by bus has the operation centers that coordinate and perform passenger transport operations in different settlement locations. these centers independently give their trip decisions related to their operations, can put new trips and cancel or take out the existing trip. a trip is the name of bus passenger transport from a settlement point (a location, called the origin) that will be travelled from at a certain time (the departure time) to another settlement point (the destination) at a certain time. when a bus completes this process, it waits until the most suitable departure time to move on to the target settlement point at this settlement point. this waiting period is called idle time. figure 1 shows the route charts of one of the operation centers of the company on the map during the research period. the idle times between the trips are given in table 1 and are expressed in minutes. table 3 shows how these values are calculated. figure 1. the route charts there can be multiple trips from a given geographical departure point, or more than one trips at different times. each vehicle must end the complete route which the formulation of a linear programming model for the vehicle routing problem in... 25 includes the tours within the subgroups. this tour is called the mevlana tour organized by transport professionals. this problem implies that the company has five subgroups, our interest only being in one of them. thus, we build a tour for one decision center including 34 trips. if we want to join all those subgroups together, we build a tour including about 500 trips. historians called this type of long-distance tours attila tours. hence, costs might be lower and decision-making might be more effective. figure 2 shows the trips managed by one of the company’s operation centers during the research period. each arrow in the figure represents one trip. sp9 arc corresponds to sp9 (trip) in table 2, which represents an expedition from balıkesir at 8 am to the arrival time in alanya at 10 pm, the journey lasting for approximately 14hours. when figure 2 is examined, it is seen that there is only a two-way reciprocal trip to the both sides, but between antalya and izmir, and between fethiye and i̇zmir, there are 6 two-way trips to the both sides reciprocally. figure 1. an operation center’s trips figure 2. as a tsp çam and sezen/decis.-mak. appl. manag. eng. 3 (1) (2020) 22-29 26 in this problem, the aim is to minimize the sum of the idle times (waiting times) between the trips due to the time-dimensional connections of geographical points, taking into account the geographic boundaries, as well as the time dimension. vehicles must make these trips as soon as possible and return to the starting point as they do in the traveling salesman problem. the shortest time of all trips will ensure that the total duration of the tour is the smallest if there is no idle time. in this respect, the problem seems to be like the job shop scheduling problem as well. however, our problem may have sub-routes and some other vrp constraints. table 1. the idle time between the trips; the table is trimmed for the purpose of gaining an easy insight into it trip no sp1-9 sp10 sp11 sp12 sp13 sp14 sp15 sp16 sp17 sp18-34 sp1-9 ... sp10 x sp11 x sp12 x sp13 x sp14 x sp15 x sp16 1090 760 100 1300 1420 280 x sp17 x sp18-34 ... the data structure of this problem can be seen in table 2. to make sure that a vehicle will return to the starting point, every trip needs to be reciprocal. table 2. the data about the trips trip number city departure city destination departure time duration arrival time sp9 balıkesir alanya 8 am 14 hours 10 pm sp23 balıkesir alanya 7 pm 14 hours 9 am sp15 alanya balıkesir 5 pm 14 hours 7 am sp11 alanya balıkesir 11 pm 14 hours 1 pm total time (t) 56 hours for the sake of clarity, the calculation of the vrpmit is shown in table 3. to provide one route, tour i was randomly created from table 2 as: sp23 – sp11 – sp9 – sp15 – sp23 table 3. tour i traveling and idle time tour i journey number arrival time (a) next departure time(d) spare time(d-a) 1 sp23 9 am 11 pm 14 hours 2 sp11 1 pm 8 am 19 hours 3 sp9 10 pm 5 pm 19 hours 4 sp15 7 am 7 pm 12 hours total idle time (b) 64 hours the formulation of a linear programming model for the vehicle routing problem in... 27 efficiency ratio: t / (t+b) 56/120=0.47 the efficiency ratio is about 47%. if the routing sequence is changed to be: sp23– sp15–sp9–sp11–sp23, tour ii is created in table 4. table 4. tour ii traveling and idle time tour ii journey number arrival time (a) next departure time(d) spare time(ad) 1 sp23 9 am 5 pm 8 hours 2 sp15 7 am 8 am 1 hour 3 sp9 10 pm 11 pm 1 hour 4 sp11 1 pm 7 pm 6 hours total idle time (b) 16 hours efficiency ratio: t / (t+b) 56/72=0.78 as shown in tables 3 and 4, the efficiency ratio rose to 78%. in tour i (the first route), the total time was 120 hours, whereas the second route (tour 2) only took 72 hours. a total of 48 hours was obtained, i.e. minimum two buses were saved by means of purchasing. the differences between the performance ratios show that it is very important to find a solution by an exact method, e.g. lp, dp, the branch and bound technique… this sample problem is too small and there were not too many alternatives. real-size problems, however, may be more complex and it might be impossible to find a solution by applying exact methods. 2.1. solution methods the discussed problem is an np-hard, combinatorial type of problem. different methods for solving such problems are proposed in the literature. heuristic, metaheuristic methods, simulation methods generate solutions which are good enough, i.e. sufficiently close to the optimal solution, for which reason they can be considered as satisfactory. moreover, simulation allows us to consider adding more different pillars to the model as well. it has been emphasized above how important using exact methods is for finding out a solution. using mathematical programming methods, such as linear and dynamic programming, might allow us to find out the optimal solution. backtracking and jumptracking, the two versions of the branch and bound technique, might also be very attractive for overcoming problems. when a problem grows in size, however, the effectiveness of the used method and the ability to reach the optimal solution rapidly diminish. 2.2. linear programming model as is known, the vrp differs from the traveling salesman problem (tsp) or hamilton tours, especially so with respect to the fact that it has a larger number of routes. the standard objective function for the tsp is as follows: min ∑ ∑ 𝑐ij 𝑛 𝑗≠i,j=0 𝑛 i=0 𝑥ij (1) where, cij is the cost from i to j; xij is the binary variable providing the information that the trip ended. çam and sezen/decis.-mak. appl. manag. eng. 3 (1) (2020) 22-29 28 the objective function for the proposed vrpmit aimed at finding the minimum idle time reads as follows: min ∑ ∑ 𝑡ij 𝑛 𝑗≠i,j=0 𝑛 i=0 𝑥ij (2) where tij is the idle time. the constraints: every trip should only be taken once: ∑ 𝑋i,j = 1 𝑛 𝑖 (3) ∑ 𝑋i,j = 1 𝑛 𝑗 (4) the limitations for the subtours optionally imposed: ∑ 𝑋i,i = 0 𝑛 𝑖 (5) 𝑋i,j + 𝑥j,i ≤ 1 (6) 𝑋i,j + 𝑥j,1+x1,i ≤ 2 (7) 𝑋i,j + 𝑥j,1 + ...+xz,i ≤ (𝑛 − 1) (8) and 𝑋i,j = 0 or 1 (9) i,j=1,2,...,n (10) sometimes, subtours might be allowed in order to keep the cost lower. in this way, the problem further distances from the tsp. the foregoing model is quite similar to the job shop scheduling (sequencing) model, which considers n jobs and q machines in job shop (sezen et al., 2017). 3. conclusion some solutions to the problem were found out by applying certain heuristic and backtracking methods. some researchers work on solving the problem by applying mathematical programming, different branch and bound method applications, certain heuristic methods and simulation. moreover, the use of these solutions as the core of the operational software related to transport companies, autonomous unmanned vehicles, and automatically controlled vehicles is the subject matter of future work. another paper will deal with solving the lp model of the problem and discussing the result. author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research received no external funding. conflicts of interest: the authors declare no conflicts of interest. the formulation of a linear programming model for the vehicle routing problem in... 29 references beck, j. c., prosser, p., & selensky, e. 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(2016). the green vehicle routing problem: a heuristic based exact solution approach. applied soft computing, 39, 154–164. kramer, r., maculan, n., subramanian, a., & vidal, t. (2015). a speed and departure time optimization algorithm for the pollution-routing problem. european journal of operational research, 247(3), 782–787. laporte, g. (2009). fifty years of vehicle routing. transportation science, 43(4), 408– 416. laporte, g. (2013). scheduling issues in vehicle routing. annals of operations research, 25(2), 1–12. sezen, h.k., (2017). yöneylem araştırması, dora yayınevi, baski, 3(1), 2-24. wang, y., liao, z., tang, t., & ning, b. (2017). train scheduling and circulation planning in urban rail transit lines. control engineering practice, 61, 112–123. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 5, issue 1, 2022, pp. 135-153. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0304042022p # the initial version of the research was published at 2nd international conference on management, engineering and environment (icmnee), obrenovac, serbia. * corresponding authora. e-mail addresses: dragan.pamucar@gmail.com (d. pamučar), dbozanic@yahoo.com (d. božanić), adispuska@yahoo.com (a. puška), dragan.marinkovic@tu-berlin.de (d. marinković) application of neuro-fuzzy system for predicting the success of a company in public procurement dragan pamučar1, darko božanić1*, adis puška2 and dragan marinković3 1 military academy, university of defence in belgrade, belgrade, serbia 2 faculty of agriculture, bijeljena university, bijeljina, bosnia and herzegovina 3 department of structural analysis, technical university of berlin, berlin, germany received: 11 january 2022; accepted: 28 march 2022; available online: 4 april 2022. original scientific paper abstract: the paper presents a neuro-fuzzy system for evaluating and predicting the success of a construction company in public tenders. this model enables companies to operate sustainably by assessing their own position in the market. the model was based on data from a seven-year study, where data from the first six years were used to adjust the model, while data from the last year of the study were used for testing and validation. the neuro-fuzzy model was tuned using the artificial bee colony algorithm key words: fuzzy sets, neuro-fuzzy system, artificial bee colony. 1. introduction the companies operating in the market in any area (marketing, information technology, manufacturing, consulting services, etc) are constantly facing with the problems of existence and continuous maintenance of the level of services, on one hand, as well as with the problem of evaluating competition, on the other hand (pamucar & bozanic, 2018). knowing of the competition is the first step towards winning on the market (rietveld and schilling, 2020). when considering evaluation of competition and predicting results of consulting services in construction, it should be borne in mind that predicting of performance in public tenders is a key mission of a consulting company in order to neutralize negative impact of competition. therefore, in order to quickly adapt to market demands, it is necessary to analyze conditions and changes in the environment, as mailto:dragan.pamucar@gmail.com mailto:dbozanic@yahoo.com mailto:adispuska@yahoo.com mailto:dragan.marinkovic@tu-berlin.de pamučar et al./decis. mak. appl. manag. eng. 5 (1) (2022) 135-153 136 well as to measure changes within the entire investment area. this implies evaluation of the economy stability in the subject area, as well as knowledge of technological innovations. something new needs to be offered, and the technical innovations that companies are developing help (del giudice, et al., 2019). one of the most important questions for sustainable management of consulting companies is how to predict whether the consulting company will win in public bidding with other consulting companies. this implies that it is necessary to evaluate the competition and predict the results of consulting services in the construction industry for a particular company, respectively, the possibility of being awarded a job on a tender realized as public procurement. in order to win a tender in public procurement, company's bid must be more favorable than other bids (hanák, et al., 2021). as a logic consequence, there is the need of analyzing and evaluating the parameters that influence such success over a certain period of time (several years of company's previous business operations), with the tendency to provide a prediction for the following period based on the results obtained by such analysis. in the business system of every company, especially consultants in the field of construction industry and investment, there are uncertainty, subjectivity and imprecision. in certain situations also, decisions are made on the basis of experience, intuition and subjective assessment of some parameters. uncertainty and complexity are caused by the specifics that construction as a business area has, compared to other areas. specificities are a consequence of complex nature of investment activities, external organizational and economic factors and the conditions under which construction production is taking place. complex nature of investment activities initiates a large number of activities and long-term realization. therefore, the selection of investment is very difficult and requires considering many aspects (puška, et al., 2018). therefore, in formulating methodological principles for predicting the success of consulting companies in construction, it is necessary to use mathematical methods which in a satisfactory way treat uncertainty, subjectivity, and imprecision (indeterminacy). thus, the methods based on neuro-fuzzy modeling and generating of rules from data imposed themselves as the most suitable mathematical apparatus. accordingly, the subject of this paper is the selection of parameters that influence winning of consulting jobs in public tenders and formation of adequate strategic decision-support system that enables managers to make quality and sustainable decisions. in this regard, the aim of the research is to create a decision-support model that allows predicting the performance of the observed company in public tenders, respectively, the rank prediction of the company, compared to the competition. consequently, the objective is to find out and conclude which jobs have the best prognosis, which enables undertaking of concrete strategic measures in the longterm policy of a specific consulting company. in order to achieve this goal, research and monitoring of the results of a consulting company during seven-year period (2010 2016) is performed. on the basis of these data, a prediction model is made and its validation is carried out. the prediction model presented in this paper belongs to the group of multicriteria decision-support systems and it is designed using the neuro-fuzzy technique (nf) and the artificial bee colony (abc) algorithm – the nf-abc model. based on seven-year research, three variables are identified that affect the rank prediction of the consulting companies in public tenders. using fuzzy technique, initial fuzzy logic system (fls) is modeled and initial rule base is created. in the following phase of the neuro-fuzzy modeling, the fls is mapped into a five-layer adaptive neural network and the training of the neuro-fuzzy model is carried out using the abc algorithm. application of neuro-fuzzy system for predicting the success of a company in public … 137 before the training of the nf-abc model, the validation of the data collected during the research is performed using the 2 test. the rest of the paper is organized as follows. in the second part of the paper, the analysis of the literature is performed and the overview of the techniques used to rank the bidders in public procurements is provided. in the third and the fourth part of the paper, the architecture of the neuro-fuzzy model and the training procedure using the abc algorithm are presented. in the fifth section of the work, the validation of the model is carried out and the managerial applications of the nf-abc model are shown. in the sixth section of the paper, the concluding observations with directions for future research and model improvement are provided. 2. literature revew the success of business in the construction industry has been the topic dealt by numerous authors, pointing out the importance and the possibility of applying different techniques and methods to solve numerous issues depending on the type of company/business. some papers (maybeck, 1979; singh, 1980; mocler, 1980) point to essential business terms and some specifics in the construction industry. although these papers were published in the second half of the 20th century, findings obtained in them are still relevant in analyzing the impact of changes, distortions in business environment and the importance of choosing business alternatives. the problem of bidder selection in public procurements in various areas conditioned a large number of research procedures that resulted in numerous scientific papers within the previous decades. this problem is present both in procurements implemented in the construction sector, as well as public procurement procedures in the public sector, with almost identical objectives related to the corresponding product quality, timely delivery, the best price, etc (zak, 2015; falagario et al., 2012). the selection of bidders in public procurement procedures is multi-criteria problem, and the application of multi-criteria optimization model is convenient tool for decision making. when implementing this procedure, it is necessary to define first the model that will be applied, and then the criteria in relation to which the selection will be made. the study of the selection of criteria used in procurement procedures began in the 70s of the twentieth century with the papers of a number of authors, among which dickson (1966) stands out. later, similar research procedures were carried out (moore & fearon, 1973; de boer et al., 2001; kahraman et al., 2003; dobi et al., 2020; chai et al., 2013) showing that the criteria such as price, quality, timely delivery, technical capacity of suppliers are most often set as conditions for contract award. chai et al. (2013) carried out the categorization of the literature which deals with the issue of selection of service providers in public tenders considering 123 scientific papers. these papers systematically present a review of the literature regarding the application of multi-criteria models in the period from 2008 to 2014. the research included was divided into seven categories, in which 26 techniques of multi-criteria decision making were applied which were grouped into three sections: (1) multicriteria decision making (mcdm) techniques, (2) mathematical programming (mp) techniques, and (3) artificial intelligence (ai) techniques. the following table (table 1) presents an overview of the models used in the last ten years to define ranks of bidders in public procurement. pamučar et al./decis. mak. appl. manag. eng. 5 (1) (2022) 135-153 138 table 1. overview of literature and models for defining ranks of bidders in public procurement literature dm techniques amid et al., 2011 weighted max–min fuzzy ahp amin and zhang, 2012 multiobjective mixed integer lp buyukozkan and cifci, 2012 anp, topsis, dematel božanić et al., 2021a neuro-fuzzy system bhattacharya et al., 2010 ahp with cost factor measure; qfd technique rodrigues et al., 2021 random forest classifier; bidders recommender algorithm chan and chan, 2010 ahp model for apparel industry chua et al., 2015 ahp crispim and de sousa, 2010 fuzzy topsis; regarding virtual enterprises dotoli et al., 2020 ahp, promethee, dea, maut feng et al., 2011 collaborative utility; tabu search based algorithm ferreira and borenstein, 2012 fuzzy bayesian model; influence diagrams karamaşa et al., 2021 entropy, maut ho et al., 2011 anp; qfd technique hosseini and barker, 2016 bayesian network model ishizaka et al., 2012 ahp-based sorting approach kuo et al., 2015 fuzzy anp and fuzzy topsis labib, 2011 fuzzy ahp linguistic expression; fuzzy logic levary, 2008 ahp liu and zhang, 2011 combine entropy weight and electre-iii mafakheri, et al., 2011 ahpin two-stage dynamic programming razmi et al., 2009 mixed integer nlp rezaei and davoodi, 2012 multiobjective mixed integer nlp rezaei et al., 2016 bwm method sen et al., 2010 max–min method vahdani and zandieh, 2010 extend electre for interval values yu et al., 2012 ahp, mop zhang and liao, 2022 a stochastic cross-efficiency dea zhao and guo, 2014 fuzzy topsis de boer et al. (2001) defined four stages of solving the problem of bidder selection including problem formulation, formulation of criteria, qualification and final selection. they stated that most authors paid the greatest attention to final selection, which was significant for further prediction of a company's activities. from the literature dealing with the issue of bidders’ rank in procurement procedures in the last decade, it can be concluded that the ahp method was particularly used (dobi et al., 2010) made a comparison of the ahp and the anp when making decisions in selecting the most favorable bidder in the procurement procedures and evaluating the criteria. similarly, sameh et al. (2016) are used for pair wise comparison and prioritization of criteria; classical ahp and fuzzy ahp. some authors, such as adil et al. (2014), also use hybrid models that combine multiple multi-criteria models – the topsis (technique for order preference by similarity to an ideal solution) and the copras (complex proportional assessment) when selecting bidders for the needs of maldivian public sector. bana e costa et al. (2007) propose bidder evaluation model application of neuro-fuzzy system for predicting the success of a company in public … 139 using the macbeth (measurement attractiveness by a categorical based evaluation technique) method by which the weights of criteria are not directly evaluated based on relative importance of criteria, but the range of criteria values of variants is considered. according to the author's knowledge, there are rare cases of application of other models for bidder ranking in public bidding, which do not fall under the scope of multi-criteria techniques. ltifi et al. (2016) indicate that data research has great potential in extracting useful knowledge from a large amount of data for dynamic decision making. thus, ltifi et al. (2016) propose using neuro-fuzzy techniques to create dynamic models that have the ability to adapt to changes in the environment. son and kim (2015) demonstrated the possibility of applying data mining techniques for predicting costs and project scheduling based on the level of definition of certain components. they propose a three-step procedure for achieving this objective: previous processing, selection of variables and development of a prediction model. the rapidly growing and large amount of data in the field of construction, together with the necessity for data analysis, created urgent need for powerful tools that could generate knowledge and predictions from large data sets. traditional and hybrid multi-criteria models cannot adequately respond to such requirements (zhun, 2016). zhun et al. (2016) point out the importance of using data mining techniques, both for describing and predicting activities, in various business areas. by analyzing the literature dealing with models for predicting the performance of companies in public procurements, it is noticed that there are no models for predicting the performance of construction companies in public procurements. there is a lack of application of fuzzy logic, adaptive neuro-fuzzy models, linear and dynamic programming, as well as heuristic and meta heuristic models. as previously shown (table 1), most papers discuss the rank of bidders in public procurements using classic multi-criteria decision-making methods. that is why the development of the nf-abc model for the prediction of bidders’ ranks using adaptive neuro-fuzzy techniques and the abc algorithm is a logical step towards overcoming of this gap. 3. neuro-fuzzy model for predicting results of consulting services in construction the nf-abc model for predicting the rank of consulting companies in public tenders uses the benefits of fuzzy sets and fuzzy logic systems, combined with the concept of artificial neural networks. fuzzy logic provides mathematical description of uncertainties that occur in human cognitive processes, such as thinking and reasoning (ghazinoory et al., 2010; božanić et al., 2015; pamučar et al., 2016a; pająk, 2020; jokić et al., 2021). thus, using fuzzy logic and approximate reasoning algorithm it is enabled concluding based on incomplete and insufficiently precise information (pamučar et al., 2016b; pamučar et al., 2016c; gharib, 2020; božanić et al., 2021b; tešić et al., 2022). on the other hand, artificial neural networks possess architecture built on the concept of artificial neurons imitating biological nerve systems in its functioning. one of the most important processes that enable artificial neural networks is example-based learning. combining fuzzy logic concepts, which provide a mechanism for concluding with incomplete and insufficiently precise information, and artificial neural networks which provide learning, adaptation, and generalization possibilities, very powerful hybrid neuro-fuzzy systems are obtained that find its application in solving many real engineering problems. from the above mentioned can be concluded that both concepts have advantages/disadvantages: (1) neural networks can learn from examples pamučar et al./decis. mak. appl. manag. eng. 5 (1) (2022) 135-153 140 automatically, but it is difficult to describe the knowledge acquired in this way and (2) fuzzy logic allows approximate conclusion, but it does not have the property of auto adjusting (adaptability) (božanić et al, 2014). basic idea of neuro-fuzzy adaptive technique is founded on fuzzy modeling and learning methods based on the given data set. due to these advantages, in this paper the authors have chosen to develop models for predicting the success of consulting companies in public tenders in construction based on adaptive neuro-fuzzy techniques. as stated in the previous section, three criteria are identified on the basis of which the prediction of the potential rank of a consultant service provider is carried out: input 1: institution (x1∈[1,4]). this input variable describes the nature of the investor which announces the tender for public procurement of services: state institution/company, public institution/company, municipality, private legal entity and natural person. input 2: work type (x2∈[1,16]). input variable work type refers to a type of consulting services a consultant can perform. this model covers the following consulting services in the construction industry: geodetic recording, testing, assessment act elaboration, elaborate creation, elaboration of project management plan, project elaboration, regulatory plan elaboration, strategy development, study development, measuring, monitoring, designing, spatial planning, auditing and nostrification of technical documentation, expert supervision and technical inspection. input 3: number of participants in public procurement (x3∈[1,7]). input variable number of participants in public procurement refers to the expected number of bidders of consulting services in public procurement. from the neuro-fuzzy model it is obtained the output: rank of the consultant service provider (y ∈[1,7]), respectively, the ranking of the bidders is performed depending on the input parameters. in addition to the mentioned variables (input 1 input 3 and output), the neuro-fuzzy model also had the attribute of financial value of the bid that was not directly considered due to the following restrictions: (1) because of business secrets and 2) based on the available data about the tenders in the period of observation, it is noted that for the investors, the price of the service was the most important when choosing the best bidder. since the prices of services for each individual tender are presented and analyzed through ranks, it can be concluded that this attribute is indirectly included in consideration through the output variable, respectively, the rank of the bid. using descriptive statistics method, non-parametric statistical tests (2 test) and generating rules from data in the analytical monitoring process, the design of the nfabc model for predicting the rank of consulting companies in public tenders is carried out. descriptive statistics is used to evaluate the validity of the collected data. the 2 test is used for: (1) checking statistical significance between observed and theoretical frequencies; (2) checking the membership of independent data sets, expressed as frequencies on the same data population; (3) determining whether there is a significant difference between the groups of data obtained on one sample; (4) determining whether there is a connection between data sets and (5) calculating the degree of connection between the data sets in the form of correlation coefficient. the intervals of the input parameters of the neuro-fuzzy model were determined based on the parameters from the database obtained by the research in seven-year period from 2010 to 2016. during the research, the data were collected and a database was created with the results achieved by particular consulting company during the observed seven-year period. the acquired database was later used to train neuro-fuzzy model to predict the results of consulting services. one part of the data application of neuro-fuzzy system for predicting the success of a company in public … 141 obtained (from 2010 to 2015) was used to train neuro-fuzzy model, while the data from 2016 were used to test and validate the model. in the initial phase of the neuro-fuzzy model design, a set of linguistic rules, types and parameters of membership functions (mf) describing input/output variables of the model are defined. in the neuro-fuzzy model, the gaussian mfs were used to describe the three input variables. gaussian mfs are chosen because: (1) they describe well the input variables, (2) they ensure satisfactory sensitivity of the model, (3) with their adjustment it is provided the smallest output error, and (4) they are easy to manipulate when setting the model. since it is a zero-order sugeno fuzzy logic system (fls) mapped to neural network, the output variable range of the consultant service provider (y) is presented by 15 mfs presented by constants (   y ax by c , 0 a b ). the output variable y is scaled within the interval [1, 7]. by comparing the output parameters of the fls and the desired set of solutions, it was noted that the difference between the expected rank from the database and the output values from the fls was not within the limits of tolerance. changes to the type and parameters of the mf, as well as changes to the rules in the base of the fls did not contribute to the approximation of the output values to the values from the training base. in the next step, the fls was mapped to a five-layer neuro-fuzzy model (figure 1), with the goal of further adjusting and obtaining output data that are closer to the data from the training set. z1=p1,1x1+...+p1,6x3+p1,7z1=p1,1x1+...+p1,6x3+p1,7n1 n2 n7 n14 n15 x1 x2 x3 σ ω1 ω2 ω7 ω14 ω15 ω1*z1 ω2*z2 ω7*z7 ω14*z14 ω15*z15 y layer #1: fuzzy layer layer #2: product layer layer #3: normalized layer layer #4: de-fuzzy layer layer #5: summation layer mf11 mf12 mf13 mf14 mf21 mf22 mf23 mf24 п1 п2 п15 ω1 ω15 mf41 mf42 mf43 mf44 . . . . . . п7 п14 . . . z2=p2,1x1+...+p2,6x3+p2,7z2=p2,1x1+...+p2,6x3+p2,7 z3=p3,1x1+...+p3,6x3+p3,7z3=p3,1x1+...+p3,6x3+p3,7 z15=p15,1x1+...+p15,6x3+p15,7z15=p15,1x1+...+p15,6x3+p15,7 . . . z14=p14,1x1+...+p14,6x3+p14,7z14=p14,1x1+...+p14,6x3+p14,7 figure 1. neuro-fuzzy model for determining the cr value of the network branches the input layer consists of three units: x1, x2 and x3. it simply transfers inputs further via the interconnections to the hidden or first layer. the all units in the input layer (x1, x2, x3) are connected with the four units in the first layer. the strengths of connections between the units in the input layer and the units in the first layer are crisp numbers equal to 1. the first layer consists of 3+4 units representing the number of verbal descriptions quantified by fuzzy sets ("very low", "low", "medium", "high") for each input variable (x1, x2, x3). every unit in the first layer is an adaptive unit with an output being the membership value of the premise part. the number of units in the second layer equals the number of fuzzy rules. every unit in this layer is a fixed unit pamučar et al./decis. mak. appl. manag. eng. 5 (1) (2022) 135-153 142 that calculates the minimum value of incoming two inputs. the outputs from this layer are firing strengths of rules. for example, the output from the first unit in the second layer is:  1 1 2 3min ( ), ( ), ( )    l h mx x x (1) the third layer has 15 adaptive nodes that calculate sections of the fuzzy sets (consequent) with the maximum of the incoming rules' firing strengths. the normalization of the weight coefficient of every neuro-fuzzy model rule,  k ,is calculated using the expression for additive normalization 15 1       k k i i (2) where k ( 1,...,15k  ) presents the number of rules in the neuro-fuzzy model. the single unit in the fourth layer is a fixed unit that computes the overall output of the neuro-fuzzy model:       1 15max ,..., ,  m v vy y y (3) the obtained output is then defuzzified in the single unit in the fifth layer. the selection of the final crisp value can be made in various ways. in this paper calculates the action that is closest to the center of gravity (center-of-gravity method): 15 15 1 15 1 1             k k k k k k k k o overall utput y o y (4) the neuro-fuzzy model parameters were adjusted using the artificial bee colony (abc) algorithm (pamučar et al., 2016d). the abc algorithm showed significantly better results during training when compared to standard algorithms (back propagation and hybrid algorithms) that were implemented in the toolbox of the matlab r2008a. the training of the anfis with back propagation and hybrid algorithm was done under the same conditions that were valid when applying the abc algorithm. the training with back propagation and hybrid algorithms lasted longer than with the abc algorithm, and the error at the end of the training was 2.891 (back propagation algorithm) and 3.542 (hybrid algorithm), compared to the error of 0.13 in the training with the abc algorithm (figure 2). application of neuro-fuzzy system for predicting the success of a company in public … 143 0 205 410 615 820 1025 1230 0 1 2 3 4 5 6 e rr o r traininig iterations error = 2.891 error = 2.542 error = 0.13 abc algorithm backprobagation algorithm hybrid algorithm figure 2. abc, backpropagation i hybrid algoritam traininig error the attempt to increase the number of training epochs (from 60 to 1500) and to change the type of mf in order to additionally reduce the error using back propagation and hybrid algorithm did not lead to any significant improvement in the results. the error remained at the level of the previous testing. this confirmed the author's determination for applying the abc algorithm. in the figure 3 are graphically presented the results of comparing the training data set with the output parameters of the neuro-fuzzy model after the training with the abc algorithm. figure 3. comparison of the training data set with the results of the nfabc model based on the figure 3, it can be concluded that the projected neuro-fuzzy model provides satisfactory prediction of the bidders’ ranks with the negligible error of 0.13. since it is a decision-support system for rank prediction, the error value of 0.13 is negligible and it can be concluded that the neuro-fuzzy model is successfully set. 4. testing and validation of the neuro-fuzzy model the model presented was tested in a total of 239 public procurements in which the consulting company investigated took part in 2016. in the table 2 is presented a part of the data collected during 2016 that were used for testing and validation of the nf-abc model. due to the limited space for presenting the results and the fact that it pamučar et al./decis. mak. appl. manag. eng. 5 (1) (2022) 135-153 144 is a large number of data (a total of 239), in the following table is shown only the part of the data used for model validation. table 2. data collected during 2016 used for neuro-fuzzy model testing number institution work type partici pants no. rank public institution/company measurements 4 3 private legal entity monitoring 3 1 public institution/company project creation 5 2 municipality regulation plan creation 5 2 municipality measurements 4 2 state institution/company project creation 5 2 public institution/company project creation 5 2 municipality elaborate creation 4 2 private legal entity testing 6 3 public institution/company expert supervision 5 1 municipality regulation plan creation 4 2 public institution/company project creation 6 2 public institution/company revision and nostrification 4 2 state institution/company project creation 5 2 state institution/company project creation 5 2 state institution/company project creation 5 2 state institution/company project creation 5 2 municipality management plan creation 4 2 municipality regulation plan creation 3 2 state institution/company project creation 5 2 municipality management plan creation 4 2 public institution/company project creation 6 2 application of neuro-fuzzy system for predicting the success of a company in public … 145 number institution work type partici pants no. rank … … … … … state institution/company expert supervision 4 1 municipality testing 4 3 public institution/company expert supervision 4 2 public institution/company expert supervision 5 1 ... … … … … municipality spatial planning 5 1 state institution/company project creation 6 3 public institution/company revision and nostrification 5 2 public institution/company expert supervision 7 1 state institution/company project creation 5 2 municipality project creation 4 2 municipality expert supervision 6 4 public institution/company monitoring 4 3 state institution/company expert supervision 6 1 public institution/company geodetic recording 5 1 … … … … … public institution/company assessment act creation 4 3 state institution/company project creation 5 3 municipality project creation 5 2 municipality revision and nostrification 4 2 municipality spatial planning 5 1 municipality regulation plan creation 7 1 municipality expert supervision 5 2 by putting input parameters (table 2) through the nf-abc model, the results are obtained which are shown in the table 3. a comparative overview of the results collected for 2016 and the rank prediction given by the nf-abc model are shown in pamučar et al./decis. mak. appl. manag. eng. 5 (1) (2022) 135-153 146 the table 3. in order to have an overview of the quality of the results, in the table 3 the target ranges are also shown next to the outputs of the nf-abc model. table 3. comparative overview of the results of the neuro-fuzzy model and the control data group t a rg e t a n f is t a rg e t a n f is t a rg e t a n f is t a rg e t a n f is t a rg e t a n f is 3 2.85 3 2.77 3 3.09 2 1.99 3 3.00 3 3.00 2 1.82 1 1.00 2 2.00 4 3.98 2 2.00 1 1.04 2 2.00 2 2.00 2 2.00 2 2.01 2 1.98 2 2.27 2 2.00 2 2.00 2 2.00 1 1.06 2 1.98 2 2.00 3 2.95 2 2.00 3 3.00 2 1.98 2 2.00 2 1.86 2 2.00 2 1.90 2 2.00 1 1.00 3 3.00 2 2.00 1 1.32 2 2.00 2 1.89 1 0.81 3 3.00 1 1.00 2 2.00 2 2.00 1 1.50 2 2.00 1 1.00 2 2.00 2 2.00 2 1.98 2 1.86 3 3.00 2 2.00 2 2.00 2 2.12 2 2.00 1 1.04 1 1.00 2 2.02 2 2.00 2 2.00 2 2.00 2 2.00 3 3.02 2 2.02 2 2.00 2 1.89 2 2.00 2 1.99 2 2.00 2 2.00 4 4.01 2 2.00 1 1.00 2 1.90 2 2.00 2 1.98 2 2.00 1 0.93 2 2.00 2 2.00 2 2.00 2 2.00 2 2.00 2 2.00 2 2.00 2 2.00 2 2.00 3 3.00 2 2.02 2 2.00 1 1.00 2 2.00 3 3.00 2 2.00 2 2.00 2 2.00 2 2.00 2 2.02 2 2.00 2 1.98 1 1.04 2 2.00 2 2.06 3 3.00 2 2.00 2 2.00 2 2.00 1 0.99 1 1.89 2 2.00 2 2.00 2 2.00 1 1.00 2 2.00 2 2.00 2 2.02 2 1.98 2 2.00 2 2.00 2 2.00 2 1.98 2 1.99 2 2.00 2 2.00 2 2.00 2 2.00 2 1.89 2 2.00 1 1.00 2 2.00 2 2.00 3 3.04 2 2.00 2 2.00 2 2.00 2 2.00 2 2.00 2 2.00 2 1.64 2 1.90 1 1.06 2 2.00 2 2.01 2 2.00 2 2.00 2 2.02 2 2.00 1 1.00 2 2.00 2 2.00 2 2.02 1 1.01 1 1.00 2 2.00 2 1.89 2 2.21 1 0.99 2 2.00 2 2.00 3 2.94 2 2.00 2 2.00 2 2.00 2 2.00 2 1.99 2 2.00 2 1.89 2 2.26 2 1.90 2 2.02 2 2.00 2 1.89 2 2.00 2 1.90 2 2.02 2 2.00 3 3.00 2 2.00 2 1.90 application of neuro-fuzzy system for predicting the success of a company in public … 147 t a rg e t a n f is t a rg e t a n f is t a rg e t a n f is t a rg e t a n f is t a rg e t a n f is 2 2.02 2 2.00 2 2.00 3 3.00 4 3.98 2 2.02 2 2.00 1 1.00 3 3.00 1 1.00 2 2.00 2 2.00 2 2.00 2 2.00 2 2.12 2 2.02 2 2.00 1 1.00 2 1.99 2 2.00 2 2.00 1 1.06 2 1.89 3 3.00 2 2.00 2 2.00 2 2.01 2 2.00 1 1.04 2 2.00 2 1.86 1 1.00 2 2.02 2 2.00 2 2.00 2 2.00 1 1.04 2 2.00 2 2.00 2 2.00 3 3.00 2 2.00 4 4.00 1 1.00 2 2.00 2 2.00 2 2.00 2 2.26 2 2.00 2 2.00 3 3.00 2 1.89 2 2.00 2 2.00 2 1.98 2 2.01 2 2.00 2 2.00 2 2.00 out of a total of 239 tests, in 89% of the cases, the predicted rank at the output of the neuro-fuzzy model fully coincides with the real rank achieved at the public tender in 2016. in the remaining 11% there are minor deviations, which are eliminated by rounding to integer values. by analyzing the results we can conclude that the nf-abc model successfully predicts rank and the results of consulting services. the model provides the possibility of predicting own performance in the market of consulting services. the results obtained can make a good basis for forecasting the success of the observed company in the forthcoming business period, as well as its position on the market. in addition, the predictions provided by the nf-abc model enable managers to adopt sustainable business strategies in the future. this research has confirmed two research goals of this paper: (1) it is performed quantification of variables that influence the prediction of ranks of consulting companies participating in public tenders; and (2) it is determined the influence of input variables on the rank of the consulting services providers using neuro-fuzzy model and the abc algorithm. specific goal achieved by this paper is the possibility to apply the subject model in other fields of engineering, and not only in construction. this model achieves qualitative planning of further development potentials of companies and understanding and improvement of existing potentials. the results of the research clearly justify the use of presented model in the decision-making process of the consulting companies. on the one hand, the nf-abc model is based on a complex mathematical apparatus and, as such, its application can cause an aversion of the management. however, on the other hand, this model allows obtaining credible results when deciding under uncertain conditions. therefore, the application of this model is significant. firstly, the nf-abc model helps managers deal with their own subjectivity when considering the conditions in the environment and the competition. secondly, with the successful prioritization of the companies, the nf-abc model reduces uncertainty in the decision-making process. thirdly, by considering the variables that affect the prediction of company performance, the nfabc model significantly reduces uncertainty in business of consulting companies. finally, the nf-abc model has the adaptability allowing further modeling based on new real data from the practice. this model provides the management with the most pamučar et al./decis. mak. appl. manag. eng. 5 (1) (2022) 135-153 148 appropriate strategy in accordance with current requirements, while minimizing the risk of decision making. 5. conclusions the paper presents a model of strategic management in which the neuro-fuzzy approach and the abc algorithm are applied. neuro-fuzzy model is trained with the abc algorithm and is used to predict the rank of consulting companies when applying for public procurement. the authors’ opinion is that this new approach in predicting the success of companies in public procurement is qualitative shift forward in the direction of improving strategic management of the potentials of consulting companies in the construction industry. the nf-abc model extends theoretical framework of knowledge in the field of decision-support system. the existing problem is considered with the new methodology, which creates the basis for further theoretical, as well as practical upgrading. this model enables the assessment of future position of consulting companies in the market and the selection of sustainable business strategy in the process of management decision making. this is particularly important in cases where consulting services are provided for the implementation of capital and infrastructure projects with a longer time interval (two years or more). also, the model presented highlights new criteria for predicting the success of companies that have not been considered in the previous models, and are of importance for this issue. by introducing new criteria and their presentation in the model, it is pointed out the need for their consideration in further analyzes of this and similar issues. the main disadvantage of the model, arising from the nature of the criteria, is its application only to the company whose data are used for training the model. this would in particular mean that for other companies it is necessary to re-train the model based on their own input and output data. future research should be directed towards the identification of additional parameters that affect the rank of companies in line with the specifics of the environment in which the companies operate. this allows directing and controlling strategic development of companies and preventing the development of unwanted situations. in this direction of research, the methods of fuzzy linear and dynamic programming joined with heuristic and meta heuristic methods find their application. one of the recommendations is to consider the rank of companies using genetic algorithms, while defining the limits that are considered by fuzzy linear programming. author contributions: research problem, d.p., a.p. and d.m..; methodology, d.p., d.b. and a.p.; formal analysis, d.p., d.b. and a.p.; resources, d.m.; writing original draft preparation, d.p. and d.b.; writing review & editing, d.p., d.b., a.p., and d.m.; project administration, a.p. and d.m.; supervision, d.p. and d.b. all authors have read and agreed to the published version of the manuscript. funding: this research received no external funding. data availability statement: the study did not report any data. acknowledgments: the authors are grateful for the anonymous reviewers for their valuable comments and 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(2014). selecting green supplier of thermal power equipment by using a hybrid mcdm method for sustainability. sustainability, 6(1), 217-235. zhun, j., yu, j. z., haghighat, f., benjamin c. m., & fung, c. m. b (2016). advances and challenges in building engineering and data mining applications for energy-efficient communities. sustainable cities and society, 25, 33–38. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering issn: 2560-6018 eissn: 2620-0104 doi:_https://doi.org/10.31181/dmame0306102022r * corresponding author. e-mail addresses: sadhu.tithli@gmail.com (t. sadhu), sc.19ch1103@phd.nitdgp.ac.in (s. chowdhury), shubhammondal1999@outlook.com (s. mondal), jroy@nitw.ac.in (j. roy), jitamanyu.chakrabarty@ch.nitdgp.ac.in (j. chakrabarty), sandipkumar.lahiri@che.nitdgp.ac.in (s.k.. lahiri) a comparative study of metaheuristics algorithms based on their performance of complex benchmark problems tithli sadhu1,2, somanth chowdhury3, shubham mondal4, jagannath roy5, jitamanyu chakrabarty1 and sandip kumar lahiri3* 1 department of chemistry, national institute of technology durgapur, west bengal, india 2 department of biochemistry, school of agriculture, sr university, hanumakonda, telangana, india 3 department of chemical engineering, national institute of technology durgapur, west bengal, india 4 department of computer science and engineering, institute of engineering and management kolkata, west bengal, india 5 department of mathematics, national institute of technology warangal, telangana, india received: 16 february 2022; accepted: 18 september 2022; available online: 6 october 2022. original scientific paper abstract: metaheuristic approaches with significant improvements are very promising in the solution of intractable optimization problems. the objective of the present study is to test the capability of applications and compare performance of the four selected algorithms from “classical” (simulated annealing (sa), genetic algorithm (ga), particle swarm optimization (pso), and differential evolution (de)) and “new generation” (firefly algorithm (ffa), krill herd (kh), grey wolf optimization (gwo), and symbiotic organism search (sos)) each by solving selected benchmark problems. sos and kh algorithm successfully solved most of all the selected problems by achieving the best solution in minimum execution time. on the other hand, de, and pso also effectively attained the optimal solution which were very close to the best one. therefore, no firm conclusion can be done about the universally best algorithm and their performance may be varied for different benchmark problems. however, “new generation” algorithm exhibited the most promising result and great potential than the “classical” one. this study gives some insights to use sos and kh as best performing algorithm to the novice user who can easily get lost by the plethora of large number of optimization algorithms. mailto:sadhu.tithli@gmail.com mailto:sc.19ch1103@phd.nitdgp.ac.in mailto:shubhammondal1999@outlook.com mailto:jroy@nitw.ac.in mailto:jitamanyu.chakrabarty@ch.nitdgp.ac.in mailto:sandipkumar.lahiri@che.nitdgp.ac.in sadhu et al./decis. mak. appl. manag. eng. (2022) 2 key words: metaheuristic, algorithms, optimization, performance, benchmark problems 1. introduction optimization in every field of science and technology recently gets attention to the researchers as resources, energies are getting limited day by day. optimization refers to a procedure to get a balance between two or more conflicting objectives with respect to design variables under several conditions and restrictions on them (rangaiah, 2010). due to the complexity of modern technologies, the objective function and associated constraints are very complex in real-life applications. gradient-based classical optimization algorithms often trap in local optima of this complex objective function and cannot find the global optima. to overcome the limitation of gradient-based algorithms, metaheuristic, stochastic approaches are introduced. these algorithms have a random component included in their execution which helps them to escape from local minima. there is an iterative computational technique at the heart of these metaheuristic algorithms that selects an optimal solution iteratively and tries to enhance a candidate solution with regard to a particular measure of quality (wang & chen, 2013). in the recent studies, it is referred as nature inspired “metaheuristic” that means the higher level of heuristic that are applied to solve a wide variety of optimization problems. the main advantage of using this metaheuristic algorithm is that it allows the decision makers to obtain near optimal solutions within a relatively shorter period of time even for large size complex problems because of their efficient performance ability (dokeroglu et al., 2019). many complex optimization problems with a large variance, ranging from single to multiobjective, constrained to unconstrained, continuous to discrete, can be solved by a practical and elegant way using metaheuristic approaches (dokeroglu et al., 2019). the majority of state-of-the-art metaheuristic has been created prior to the year 2000 and in this article these can be termed as “classical” metaheuristic algorithms (dokeroglu et al., 2019). some of the major examples of classical algorithms are: simulated annealing (sa) (kirkpatrick et al., 1983), genetic algorithm (ga) (goldberg, 1989), ant colony optimization (dorigo & di caro, 1999), particle swarm optimization (kennedy & eberhart, 1995), differential evolution (de) (storn & price, 1997), chaos optimization method (com) (li & jiang, 1997), variable neighborhood search (vns) (mladenović & hansen, 1997), genetic programming (gp) (banzhaf et al., 1998), tabu search (ts) (glover & laguna, 1998), greedy randomized adaptive search procedure (grasp) (marques-silva & sakallah, 1999), etc. these algorithms are widely used in almost every field of science and technology and proven to be very versatile. the main advantages of the “classical” metaheuristic algorithms as reported in literature are: (1) low execution time (processing time of the program) for large and complex problems, (2) their ability to escape local optima and high probability to getting global optima solutions (beheshti & shamsuddin, 2013). despite the huge success rate of classical metaheuristic algorithms in diverse field, new generation evolutionary algorithms also developed magnificently in last twenty years improving their performance and execution time. recently, researcher find out that nature itself has many efficient optimization processes. various species in nature such as, birds, fish etc. possesses very effective optimization capabilities. from year 2000 to till now, researchers focused on evolutionary or behavioral processes seen in nature and try to mimic them in mathematical algorithms. these give rise the development of new generation, nature-inspired metaheuristic algorithms that can solve all complex reala comparative study of metaheuristics algorithms based on their performance of complex … 3 world problems and acquire the more practical optimal solution in very short execution time compare to “classical” one for some unsolved benchmark problem sets in all perspective, even for very large problem size (dokeroglu et al., 2019). some major examples of “new generation” metaheuristic algorithms are harmony search (hs) (geem et al., 2001), artificial bee colony (abc) (karaboga, 2005), bacterial foraging optimization (bfo) (das et al., 2009), cuckoo search (cs) (yang & deb, 2009), firefly algorithm (ffa) (yang, 2010a), bat algorithm (ba) (yang, 2010b), krill herd (kh) (gandomi & alavi, 2012), grey wolf optimization (gwo) (mirjalili et al., 2014), symbiotic organism search (sos) (cheng & prayogo, 2014), whale optimization (woa) (mirjalili & lewis, 2016) etc. a schematic representation of development of both “classical” and “new generation” metaheuristic optimizations (year-wise) is represented in figure 1(a) and (b), respectively. figure 1. year-wise development of (a) “classical” and (b) “new generation” metaheuristic algorithms due to the availability of large number of metaheuristic algorithms, the novice researchers get confused about the selection of efficient algorithm to solve hard and complex problems. now-a-days, new algorithm is developed every month and is claimed to be the best among the existing one. however, when the algorithms are applied to a new objective function, this claim is not substantiated. this necessitates making comparison of different old and new generation metaheuristic algorithms on benchmark problem sets. this study aims to address the above-mentioned issues by following manners:  first, seven benchmark problems were selected from the published literature and at the time of selection, attention was given that the objective function and associated constraints are very complex with multiple local optima and used before in literature for algorithm testing purpose.  out of fifty major “classical” and “new generation” algorithms, four “classical” and four “new generation” algorithms were chosen from published papers (in last 40 years) on the basis of their potential and applications in diverse fields. sadhu et al./decis. mak. appl. manag. eng. (2022) 4  finally, the main objective of this paper is fulfilled by testing their capability of applications by solving the selected benchmark problems and compare the performance of recent next generation promising algorithms against old generation “classical” algorithms. this paper is organized as follows: section 2 and section 3 describe a brief description of the selected “classical” and “new generation” algorithms, respectively. in result and discussion, section 4.1 covers the detailed information about the selected benchmark problems. the performance of the selected algorithms on benchmark problems and concluding remarks about performance are presented in section 4.2 and 5, respectively. the ultimate concluding remarks about this study are provided in section 6. 2. classical algorithms classical or old generation algorithms refer to all metaheuristic algorithms published before the year 2000. because of their versatility over 40 years, these algorithms are applied in diverse field of science, engineering and medical and proven to be very robust. a schematic representation of major old generation algorithms is given in figure 1(a). due to brevity, in this section comprehensive information describing the main mechanism for optimization, metaheuristic behind execution and pseudocode are provided about the selected “classical” algorithms. 2.1. simulated annealing (sa) sa is a stochastic optimization tool for approximating the global optimum of a given function (rao, 2019). this versatile and successful method of optimization is very beneficial for locating global optima when there are a lot of local optima. sa mimics the certain thermodynamics principals of producing ideal crystal. the term “annealing” refers to a thermodynamic analogue, specifically the cooling and annealing of metals. this algorithm simulates the slow cooling process of molten metal by controlling the parameter such as temperature to get the minimum value of a function in a minimization problem. a selection of cooling system, i.e., the technique of reducing temperature is discussed in detail in rangaiah (2010). let an unconstrained nonlinear minimization problem but with bounds on variables is defined as: min 𝐹(𝑥) (1) subject to 𝑥𝑖 𝑙 ≤ 𝑥𝑖 ≤ 𝑥𝑖 𝑢𝑝 ; 𝑖 = 1, … , 𝑝 (2) basic sa algorithm to solve the above-mentioned problem (eq. 1 and 2) is illustrated in figure 2. eq. 3 and conditions (1) and (2) in the figure are: ∆𝐹𝑘 = 𝐹(𝑥 𝑘 ) − 𝐹(𝑥 𝑘−1) (3) ∆𝐹𝑘 ≤ 0 (condition 1) ∆𝐹𝑘 > 0 and p > 𝑅𝑁𝐷 (condition 2) where, p= probability of accepting a feasible point rnd =random number from uniform distribution between 0 and 1. sa approach is also used for many non-linear problems with continuous variables. sa optimization process is commonly used in chemical and process engineering, especially for combinatorial problems or discrete-valued optimization problems. therefore, sa is applied in discrete, but very large configuration spaces and has a wide range of applications that are still being investigated (rangaiah, 2010). a comparative study of metaheuristics algorithms based on their performance of complex … 5 figure 2. flowchart of basic sa algorithm 2.2. genetic algorithm (ga) ga is the first algorithm in the field of metaheuristics designed by natural inspiration in the search optimization and machine learning processes. ga works by combining the ‘survival of the fittest' principle of natural evolution with genetic propagation of characteristics (lahiri & ghanta, 2010). the benefit of this principle is that it intelligently exploits the random search provided by previous data for exploring the better performance region of the solution space. as a result, this algorithm is commonly used to provide high-quality solutions to optimization and search problems. the details of the procedures of ga are described in lahiri and ghanta (2009, 2010). the pseudo code for ga algorithm is represented in figure 3. sadhu et al./decis. mak. appl. manag. eng. (2022) 6 figure 3. pseudo code of ga algorithm 2.3. particle swarm optimization (pso) particle swam optimization (pso) is another metaheuristic global optimization algorithm which developed from the principle of swarm intelligence and based on the research on bird and fish flock movement behavior. this stochastic optimization technique is very easy to implement and very few particles need to be tuned. pso is rapidly applied to solve complex optimist problems because the most optimist solution is worked out by the cooperation of each individual (bai, 2010). the position of the most optimist particle during its movement (individual experience) and also in its surrounding (near experience) affects the position of each particle in the swarm. by applying its previous experience with flying and nearby particles, each particle adjusts its velocity to locate a better solution. the modification of the process in positions and velocities of the particles is illustrated in detail in lahiri et al. (2012) and in this article shortly depicted in figure 4. easy implementation procedure and simplicity of the algorithm make helpful to wide and successful application of it in many areas such as model classification, function optimization, neural network training, the signal procession, automatic adaptation control, fuzzy system control etc. a comparative study of metaheuristics algorithms based on their performance of complex … 7 figure 4. pseudo code of pso algorithm 2.4. differential evolution (de) de is another most popular old generation metaheuristic algorithms (storn & price, 1997), and has been proved its effectiveness in addressing various real-life optimization problems. the only difference between the ga and de is that selfadaptive de uses mutation as its primary search methods and creates new solution strings using non-uniform crossover and tournament selection operators (enitan & adeyemo, 2011). de only requires few control variables that are usually derived from numerical interval with a well-defined range. de is easy to use because it generates new vectors without relying on an external probability density function with yet to be determined mean and standard deviation. the detail approaches of de for solving the problem are described in storn and price (1997). the pseudo code of de is illustrated in figure 5. because of ease to use nature, promising result in real-world problems de is frequently praised in industrial environments, especially in projects where no optimization specialists are present. sadhu et al./decis. mak. appl. manag. eng. (2022) 8 figure 5. flowchart of de algorithm 3. new generation algorithms despite the successes of the “classical” metaheuristic algorithms, new generation metaheuristic techniques give the best solution for some unsolved hard real world problems by evolutionary or behavioral approaches (dokeroglu et al., 2019). all major new generation algorithms developed in last 20 years are summarized schematically in figure 1(b). in this section, a brief information about the selected algorithms is provided. all of these algorithms are population-based and take inspiration from the characteristics of the natural evolution. 3.1. firefly algorithm (ffa) the firefly algorithm, which mimics the short and idealized flashing behavior of fireflies is proposed by yang (2010a). the rhythmic flashing character can be expressed as a function that can be used to optimize combinatorial algorithms. the following principles idealize the flashing behaviors (yang, 2010a). (1) all fireflies are attracted to other fireflies by flashing irrespective of their sex. (2) the attractiveness is directly related to the brightness of the firefly and both decrease with the increase in their distance. (3) the brightness or light intensity of a firefly is affected by the search space of the optimized objective function. the mathematical formulations are explained in detail by yang (2010a). the pseudocode of the ffa algorithm is represented in figure 6. ffa has established itself as the most promising new generation algorithm and it can strike a delicate balance during the optimization process between the exploration and exploitation of search space. from the literature, it is found that it can solve a wide range of optimization problems in a versatile field. a comparative study of metaheuristics algorithms based on their performance of complex … 9 figure 6. pseudocode of ffa algorithm 3.2. krill herd (kh) gandomi and alavi proposed the kh metaheuristic in 2012. this biologically inspired novel algorithm is based on a simulation of krill herding behavior. the main advantage of the kh is that they can construct large groups. when predators (e.g., penguins, seals, or sea birds) attack a herd, they can eat individual krill and remove them from the herd, resulting in reducing the density of the krill herd and the distance of the krill from the location of the food (gandomi & alavi, 2012). the herding of the krill individuals is a multi-objective global optimization problem that includes two main goals: (1) density-dependent attraction of krill (increasing krill density) and (2) reaching food (areas of high food concentration) (gandomi & alavi, 2012). the following actions decide the time-dependent position of an individual krill in 2d surface: i) motion induced by other krill individuals ii) foraging for food iii) random diffusion a lagrangian model is used to be able to search the whole space with n dimensions: 𝑑𝑋𝑖 𝑑𝑡 = 𝑁𝑖 + 𝐹𝑖 + 𝐷𝑖 (4) where, 𝑁𝑖 =the motion effected by other individuals, 𝐹𝑖 =the act of foraging motion, 𝐷𝑖 =the physical diffusion of the 𝑖-th krill individual (𝑖 = (1,2, … , 𝑛). the position of the individual krill depending on the above-mentioned actions is illustrated in detail in gabdomi and alavi (2012). the pseudocode of kh algorithm is represented in figure 7. kh is widely applied to solve the global numerical optimization problems (dokeroglu et al., 2019; bolaji et al., 2016). figure 7. pseudocode of kh algorithm sadhu et al./decis. mak. appl. manag. eng. (2022) 10 3.3. grey wolf optimization (gwo) mirjalili et al. (2014) developed gwo which is a stochastic and metaheuristic optimization methodology. this bionic optimization algorithm stimulates the rankbased mechanisms and attacking behaviour of the grey wolf pack. the lead wolf helps the other wolves to capture the prey through the surrounding, haunting, and attacking process. this large-scale search methodology centred on 3 best grey wolves, but there is no elimination mechanism. the optimization technique is different from others in terms of modelling. it constitutes a strict hierarchical pyramid. the group size is 5-12 on average. α layer, consisting of a male and a female leader, is the strongest and most capable individual for deciding the team’s predation actions and other activities. β and δ layers are the second and third layers respectively in the hierarchy, responsible for assisting α in the behaviour of group organizations. the bottom ranking of the pyramid is occupied by the majority of the total, named ω. they are mainly responsible for satisfying the entire pack by balancing the internal relationship of the populations, looking after the young, and maintaining the dominance structure (mirjalili et al., 2014). the social hierarchy, encircling, hunting, attacking prey (exploitation), and searching for prey are the main key elements of the gwo model (exploration). the detail mathematical modelling of gwo has been described by mirjalili et al. (2014). this metaheuristic approach is applied in various real world problems because of its efficient and simple performance ability by tuning the fewest operators (emary et al., 2016; kohli & arora, 2018; mirjalili et al., 2016; mittal et al., 2016; qin et al., 2019). recent researches in this regard look forward to the further development of the optimization algorithm (niu et al., 2019). the detail of the gwo algorithm is depicted in figure 8. figure 8. pseudocode of gwo algorithm 3.4. symbiotic organism search (sos) the sos algorithm mimics the interactive behavior among different species of organisms in nature. the greek word “symbiosis” means “living together”. in nature, symbiosis defines the reliance-based interaction between any two distinct species, which may be either obligatory or facultative (cheng & prayogo, 2014). therefore, in nature, symbiosis relationships can be classified as mutualism, commensalism, and parasitism. mutualism means a symbiotic interaction between two different species that benefits them. commensalism is a symbiotic connection that defines one can get a comparative study of metaheuristics algorithms based on their performance of complex … 11 an advantage, and the other is neutral between two species. in parasitism, one benefits and the other is deliberately harmed (dokeroglu et al., 2019). the optimization approach of the sos algorithm in the mutualism, commensalism, and parasitism phase is elaborated by cheng and prayogo (2014). the pseudocode of the sos algorithm is represented in figure 9. the new simple and powerful metaheuristic algorithm, sos is a most potential candidate for solving hard optimization problems despite using fewer control parameters than other competing algorithms. the three phases of the sos algorithm are simple to operate, with only simple mathematical operations to code. it can generate better solutions significantly than other metaheuristic algorithms. figure 9. pseudocode of sos algorithm 4. result discussion 4.1. test problems the selected “classical” and “new generation” metaheuristic algorithms were tested on a number of benchmark problems of mixed integer non-linear programming (minlp) and non-linear programming (nlp) used in literature for algorithm testing purpose (dokeroglu et al., 2019; shopova & vaklieva-bancheva, 2006). while selecting the benchmark problems, attention was given that the objective function and associated constraints are very complex with multiple local optima and tried to solve before in published literature. in this study, seven of them were chosen and illustrated below. the detailed equation of objective function and associated constraint of the problems are represented in table 1. sadhu et al./decis. mak. appl. manag. eng. (2022) 12 table 1. details of test problem for algorithm testing test probl em no. 1 floudas et al. (1989) and summanwar et al. (2002): 𝑀𝐼𝑁 (𝑦1 − 1) 2 + (𝑦2 − 2) 2 + (𝑦3 − 1) 2 − log(𝑦4 + 1) + (𝑥1 − 1) 2 + (𝑥2 − 2)2 + (𝑥3 − 3) 2 subject to the constraints: 𝑦1 + 𝑦2 + 𝑦3 + 𝑥1 + 𝑥2 + 𝑥3 ≤ 5 𝑦3 2 + 𝑥1 2 + 𝑥2 2 + 𝑥3 2 ≤ 5.5 𝑦1 + 𝑥1 ≤ 1.2 𝑦2 + 𝑥2 ≤ 1.8 𝑦3 + 𝑥3 ≤ 2.5 𝑦4 + 𝑥1 ≤ 1.2 𝑦2 2 + 𝑥2 2 ≤ 1.64 𝑦3 2 + 𝑥3 2 ≤ 4.25 𝑦2 2 + 𝑥3 2 ≤ 4.64 𝑥𝑖 ≥ 0, 𝑖 = 1 … . .3 𝑦𝑖 ∈ {0,1}, 𝑖 = 1 … . .4 test probl em no. 2 summanwar et al. (2002): 𝑀𝐼𝑁 5.35785𝑥3 2 + 0.83569𝑥1𝑥5 + 37.2932𝑥1 − 40792.14 subject to the constraints: 85.3344 + 0.0056858𝑥2𝑥5 + 0.0006262𝑥1𝑥4 − 0.0022053𝑥3𝑥5 ≥ 0 85.3344 + 0.0056858𝑥2𝑥5 + 0.0006262𝑥1𝑥4 − 0.0022053𝑥3𝑥5 ≤ 92 80.5125 + 0.0071317𝑥2𝑥5 + 0.0029955𝑥1𝑥2 + 0.0021813𝑥3 2 ≥ 90 80.5125 + 0.0071317𝑥2𝑥5 + 0.0029955𝑥1𝑥2 + 0.0021813𝑥3 2 ≤ 110 9.300961 + 0.0047026𝑥3𝑥5 + 0.0012547𝑥1𝑥3 + 0.0019085𝑥3𝑥4 ≥ 20 9.300961 + 0.0047026𝑥3𝑥5 + 0.0012547𝑥1𝑥3 + 0.0019085𝑥3𝑥4 ≤ 25 78 ≤ 𝑥1 ≤ 102 33 ≤ 𝑥2 ≤ 45 27 ≤ 𝑥𝑖 ≤ 45, 𝑖 = 3, … ,5 test probl em no. 3 summanwar et al. (2002): 𝑀𝐼𝑁𝑥1 2 + 𝑥2 2 + 𝑥1𝑥2 − 14𝑥1 − 16𝑥2 + (𝑥3 − 10) 2 + 4(𝑥4 − 5) 2 + (𝑥5 − 3)2 + 2(𝑥6 − 1) 2 + 5𝑥7 2 + 7(𝑥8 − 11) 2 + 2(𝑥9 − 10) 2 + (𝑥10 − 7) 2 + 45 subject to the constraints: 105 − 4𝑥1 − 5𝑥2 + 3𝑥7 − 9𝑥8 ≥ 0 −10𝑥1 + 8𝑥2 + 17𝑥7 − 2𝑥8 ≥ 0 8𝑥1 − 2𝑥2 − 5𝑥9 + 2𝑥10 + 12 ≥ 0 −3(𝑥1 − 2) 2 − 4(𝑥2 − 3) 2 − 2𝑥3 2 + 7𝑥4 + 120 ≥ 0 −5𝑥1 2 − 8𝑥2 − (𝑥3 − 6) 2 + 2𝑥4 + 40 ≥ 0 −𝑥1 2 − 2(𝑥2 − 2) 2 + 2𝑥1𝑥2 − 14𝑥5 + 6𝑥6 ≥ 0 −0.5(𝑥1 − 8) 2 − 2(𝑥2 − 4) 2 − 3𝑥5 2 + 𝑥6 + 30 ≥ 0 3𝑥1 − 6𝑥2 − 12(𝑥9 − 8) 2 + 7𝑥10 ≥ 0 0 ≤ 𝑥𝑖 ≤ 10, 𝑖 = 1, … ,10 test probl em no. 4 michalewicz (1995) and deb (2000): 𝑀𝐼𝑁5 ∑ 𝑥𝑖 4 𝑖=1 − 5 ∑ 𝑥𝑖 24 𝑖=1 − ∑ 𝑥𝑖 13 𝑖=5 subject to the constraints: 2𝑥1 + 2𝑥2 + 𝑥10 + 𝑥11 ≤ 10 2𝑥1 + 2𝑥3 + 𝑥10 + 𝑥12 ≤ 10 2𝑥2 + 2𝑥3 + 𝑥11 + 𝑥12 ≤ 10 −8𝑥1 + 𝑥10 ≤ 0 −8𝑥2 + 𝑥11 ≤ 0 −8𝑥3 + 𝑥12 ≤ 0 a comparative study of metaheuristics algorithms based on their performance of complex … 13 −2𝑥4 − 𝑥5 + 𝑥10 ≤ 0 −2𝑥6 − 𝑥7 + 𝑥11 ≤ 0 −2𝑥8 − 𝑥9 + 𝑥12 ≤ 0 0 ≤ 𝑥𝑖 ≤ 1, 𝑖 = 1, … ,9 0 ≤ 𝑥𝑖 ≤ 100, 𝑖 = 10,11,12 0 ≤ 𝑥13 ≤ 1 test probl em no. 5 michalewicz (1995) and deb (2000): 𝑀𝐼𝑁𝑥1 + 𝑥2 + 𝑥3 subject to the constraints: 1 − 0.0025(𝑥4 + 𝑥6) ≥ 0 1 − 0.0025(𝑥4 + 𝑥7 − 𝑥4) ≥ 0 1 − 0.01(𝑥8 + 𝑥5) ≥ 0 𝑥1𝑥6 − 833.33252𝑥4 − 100𝑥1 + 83333.333 ≥ 0 𝑥2𝑥7 − 1250𝑥5 − 𝑥2𝑥4 + 1250𝑥4 ≥ 0 𝑥3𝑥8 − 𝑥3𝑥5 + 2500𝑥5 + 1250000 ≥ 0 100 ≤ 𝑥1 ≤ 10000 1000 ≤ 𝑥2, 𝑥3 ≤ 10000 10 ≤ 𝑥𝑖 ≤ 1000, 𝑖 = 4, … ,8 test probl em no. 6 michalewicz (1995) and deb (2000): 𝑀𝐼𝑁(𝑥1 − 10) 2 + 5(𝑥2 − 12) 2 + 𝑥3 4 + 3(𝑥4 − 11.0) 2 + 10𝑥5 6 + 7𝑥6 2 + 𝑥7 4 − 4𝑥6𝑥7 − 10𝑥6 − 8𝑥7 subject to the constraints: 127 − 2𝑥1 2 − 3𝑥2 4 − 𝑥3 − 4𝑥4 2 − 5𝑥5 ≥ 0 282 − 7𝑥1 − 3𝑥2 − 10𝑥3 2 − 𝑥4 + 𝑥5 ≥ 0 196 − 23𝑥1 + 3𝑥1𝑥2 − 2𝑥3 2 − 5𝑥6 + 11𝑥7 ≥ 0 10 ≤ 𝑥𝑖 ≤ 10, 𝑖 = 1, … ,7 test probl em no. 7 floudas et al. (1989) and summanwar et al. (2002): 𝑀𝐴𝑋 − 2𝑥1 − 3𝑥2 − 1.5𝑦1 − 2𝑦2 + 0.5𝑦3 subject to the constraints: 𝑥1 2 + 𝑦1 = 1.25; 𝑥2 1.5 + 1.5𝑦2 = 3 𝑥1 + 𝑦1 ≤ 1.6; 1.333𝑥2 + 𝑦2 ≤ 3 −𝑦1 − 𝑦2 + 𝑦3 ≤ 0 𝑥1, 𝑥2 ≥ 0 𝑦1, 𝑦2, 𝑦3 ∈ {0,1} 4.1.1. test problem 1 it is a minlp minimization problem that includes nine inequality constraints and four binary and three continuous variables (floudas et al., 1989). 4.1.2. test problem 2 it is an nlp minimization problem which comprises six inequality constraints and five continuous variables (summanwar et al., 2002). 4.1.3. test problem 3 it is an nlp minimization problem containing eight inequality constraints and ten continuous variables (summanwar et al., 2002). sadhu et al./decis. mak. appl. manag. eng. (2022) 14 4.1.4. test problem 4 it is a relatively easy problem of nlp minimization with nine inequality constraints and thirteen variables (deb, 2000; michalewicz, 1995). 4.1.5. test problem 5 this nlp minimization problem comprises of six inequality constraints and eight variables (michalewicz, 1995). 4.1.6. test problem 6 it is an nlp minimization problem that consists four non-linear constraints and seven variables (michalewicz, 1995). 4.1.7. test problem 7 it is a minlp maximization problem having two equality and three inequality constraints as well as three binary and two continuous variables (summanwar et al., 2002). 4.2. performance of the selected algorithms on test problems in this study, four old generation (sa, ga, pso, and de) and four new generation (ffa, kh, gwo, and sos) algorithms were chosen on the basis of their potential in diverse field that are published in large number of research papers in last 40 years. table 2 represents the lower limit and upper limit of decision variables of the test problems. codes are developed in matlab for each of eight algorithms and run in matlab r2017a platform. herein, due to stochastic nature of each algorithm, each benchmark problem was run for at least 200 times to obtain the best result and maintain accuracy. the population solutions for 200 runs are summarized in a box plot (figure 10) for test problems 1, 3, 5, and 7. all the simulations were performed on pentium i7 processor. the performance of selected algorithms against the tested problems is presented in table 3 (a), (b), and (c), respectively. their performance indices were chosen to compare their effectiveness: (1) the minimum or maximum objective function value and its proximity with reported global solution, (2) number of constraints violation (ideally all constraints should be obeyed i.e., the value should be zero), (3) required execution time to attain the optimal solution (less time is preferable). there are many meta parameters of the individual evolutionary algorithms which need to set according to the specific problem. judicious selections of these meta parameters improve the solution quality of individual algorithms. however, in this work we have selected the default values of these parameters as suggested by literatures (kirkpatrick et al., 1983; goldberg, 1989; kennedy & eberhart, 1995; storn & price, 1997; yang, 2010a; gandomi & alavi, 2012; mirjalili et al., 2014; cheng & prayogo, 2014). it may possible to improve the final solution reached by the individual algorithms by optimizing these meta parameters. however, this was not tried in the present study as this study focuses on the evaluation of different optimization algorithms at their default parameter settings. a comparative study of metaheuristics algorithms based on their performance of complex … 15 table 2. lower limit and upper limit of decision variables test problem number limit decision variables 𝑋1 𝑋2 𝑋3 𝑋4 𝑋5 𝑋6 𝑋7 𝑋8 𝑋9 𝑋10 𝑋11 𝑋12 𝑋13 1 lower 0 0 0 0 0 0 0 − − − − − − upper 1 1 1 1 2 2 2 − − − − − − 2 lower 78 33 27 27 27 − − − − − − − − upper 102 45 45 45 45 − − − − − − − − 3 lower 0 0 0 0 0 0 0 0 0 0 − − − upper 10 10 10 10 10 10 10 10 10 10 − − − 4 lower 0 0 0 0 0 0 0 0 0 0 0 0 0 upper 10 10 10 10 10 10 10 10 10 10 10 10 10 5 lower 100 1000 1000 10 10 10 10 10 − − − − − upper 10000 10000 10000 1000 1000 1000 1000 1000 − − − − − 6 lower −10 −10 −10 −10 −10 −10 −10 − − − − − − upper 10 10 10 10 10 10 10 − − − − − − 7 lower 0 0 0 0 0 − − − − − − − − upper 1 1 1 10 10 − − − − − − − − table 3(a). performance of algorithms for minlp minimization problem p ro b le m n o . best solution reported result of tested algorithm best fitness value algorithm fitness value no. of constraints violated time required (sec) % of deviation from the best value 1 4.5795 sa 3.5527 0 5.734 1.63 ga 4.0780 0 0.375 16.66 pso 3.6381 1 39.50 4.08 de 3.5161 0 0.219 0.59 ffa 3.4971 0 3.422 0.04 kh 3.6305 0 0.844 3.86 gwo 3.9040 0 0.391 11.68 sos 3.4956 0 0.250 0.00 sadhu et al./decis. mak. appl. manag. eng. (2022) 16 table 3(b). performance of algorithms for nlp minimization problem p ro b le m n o . best solution reported result of tested algorithm best fitness value algorithm fitness value no. of constraints violated time required (sec) % of deviation from the best value 2 -30665.41 sa -30517.33 0 5.875 0.48 ga -30665.50 0 0.203 0.00 pso -30665.55 1 33.93 0.0001 de -30665.50 0 0.203 0.00 ffa -30665.50 0 0.203 0.00 kh -30427.22 0 0.453 0.77 gwo -30536.47 0 0.391 0.42 sos -30665.54 0 0.262 0.0001 3 24.3062 sa 39.3890 0 8.047 58.07 ga 32.1862 0 0.234 29.16 pso 25.3479 0 1.722 1.72 de 26.8428 0 0.250 7.72 ffa 32.1862 0 29.164 29.16 kh 106.9755 0 0.6406 329.30 gwo 51.5601 0 0.4218 106.91 sos 24.9187 0 0.2968 0.00 4 -15 sa -7.6917 0 9.703 61.03 ga -14.9450 0 0.938 24.29 pso -14.6471 2 33.32 25.80 de -14.9959 0 0.234 24.03 ffa 99.2648 2 2.984 402.871 kh -19.7396 0 3.203 0.00 gwo -7.4812 0 0.640 62.10 sos -14.9999 0 0.250 24.01 5 7049.3309 sa 3327.859 0 7.422 156.28 ga 16410.230 0 0.969 1163.75 pso 2100.000 0 28.766 61.72 de 2100.000 0 0.172 61.72 ffa 8038.478 0 2.625 519.04 kh 1298.535 0 0.484 0.00 gwo 2100.000 0 0.391 61.72 sos 2100.000 0 0.250 61.72 6 680.6300 sa 682.014 0 4.719 0.20 ga 760.213 0 0.891 11.69 pso 680.646 0 30.094 0.00 de 681.521 0 0.188 0.13 ffa 680.906 0 2.626 0.04 kh 691.964 0 0.609 1.66 gwo 689.110 0 0.359 1.24 sos 680.697 0 0.297 0.01 a comparative study of metaheuristics algorithms based on their performance of complex … 17 table 3(c). performance of algorithms for minlp maximization problem 4.2.1. performance for test problem 1 the best solution for this minlp minimization problem was 4.97 that reported by deb’s method in the year 2000 and further summanwar et al. (2002) has obtained the best solution of 4.5795 with modified constraints and more complicated algorithm. from the table 3(a), and figure 10(a), it is observed that in this study the obtained optimum solution is 3.4956 by sos with short time among the eight selected algorithms which is far better than the reported solution. ffa also attains very close value to the optimum (3.4971) but with high execution time. among the “classical” algorithms, de shows the most promising result (3.5161) even with shortest execution time. 4.2.2. performance for test problem 2 deb solved this nlp minimization problem with only ga and the best solution was reported -30664.99 and -30665.41 by deb (2000) and summanwar et al. (2002), respectively. herein, we have obtained the best solution of -30665.50 with ga, de and ffa that is commensurable than reported ones (table 3(b)). even all the three algorithms achieve the optimum value with same execution time. “new generation” metaheuristic, sos also achieves optimal solution (-30665.54) which has very close proximity with the best one within short execution time. p ro b le m n o . best solution reported result of tested algorithm best fitness value algorithm fitness value no. of constraints violated time required (sec) % of deviation from the best value 7 -7.66 sa -8.401 0 5.203 9.60 ga -0.798 1 1.359 89.59 pso -0.793 1 36.469 89.65 de -7.675 1 0.203 0.13 ffa -0.783 1 3.453 89.78 kh -7.665 0 0.516 0.00 gwo -0.797 1 0.344 89.60 sos -8.416 0 0.250 9.80 sadhu et al./decis. mak. appl. manag. eng. (2022) 18 figure 10. box plot of solutions for (a) test problem 1; (b) test problem 3; (c) test problem 5; and (d) test problem 7 4.2.3. performance for test problem 3 deb (2000) and summanwar et al. (2002) solved this benchmark problem and obtained the optimum solution of 24.34 and 24.366453, respectively. however, the global optimum reported of the problem is 24.3062. in this study, the best solution of 24.9187 is found by sos algorithm with minimum time and it is also approximately around the reported value (table 3(b) and figure 10(b)). the old algorithm, pso obtains the second-best optimum solution of 25.3479, followed by de among the eight algorithms. 4.2.4. performance for test problem 4 michalewics (1995) stated that all constrained handling methods used to solve this problem have found the optimal solution. in the year 2000, deb found the optimum solution of -15 (deb, 2000). in this study, table 3(b) presents that for test problem 4 the optimum solution of -19.7396 is obtained by kh algorithm in 3.203 sec that is improved sufficiently than the reported best optimum solution. among the eight algorithms, sos obtains the second-best optimum solution (-14.9999) at 0.250 sec. the “classical” one, de also attain the optimum value of -14.9959 (very close proximity with second-best value) even with minimum execution time (0.234 sec). a comparative study of metaheuristics algorithms based on their performance of complex … 19 4.2.5. performance for test problem 5 michalewics reported the optimum solution of 7377.976 by solving the nlp minimization problem and further, in the year 2000, deb attained the best solution of 7060.221 with ga. however, till now, the reported global optimum solution is 7049.3309. however, in the present study, among the eight selected algorithms, kh performs best and reaches the optimum solution of 1298.535 within a very short time (0.484 sec). other algorithms, de and pso from “classical” one, and gwo and sos from “new generation” algorithms, obtain the second-best solution, the value of 2100.00 with different execution time. among those, de takes minimum execution time followed by sos and gwo, whereas, pso requires highest execution time (28.766 sec). the detail of result is presented in table 3(b) and figure 10(c). 4.2.6. performance for test problem 6 the reported best solution for this nlp minimization problem is 680.6300. michalewics found the best result of 680.642 in 1995 with penalty function approach (michalewicz, 1995) and deb reported the best solution as 680.634 by constrained handling method (deb, 2000). herein, for test problem 6, table 3(b) presents that the true optimal solution of 680.646 is obtained by pso algorithm that is very close to the reported best one but the execution time is too high. in contrast, sos and ffa attain the fitness values of 680.697 and 680.906 respectively, which are very close to the best one even with very short time. among the “classical” algorithm, de reaches the optimum value of 681.521 (little bit high than the best value) within very less execution time compare to pso. 4.2.7. performance for test problem 7 the best solutions obtained by deb’s method and summanwar et al. are -7.66718 and -7.667178, respectively for this minlp maximization problem. therefore, the global optimum of -7.66 is reported as best-known solution. eight selected algorithms perform very well against this maximization problem and among all, kh improves the result most and reaches the optimum solution of -7.665 in execution time of 0.516. all other algorithms except sa, and sos, fail to solve this problem as they violate one constraint. on the other hand, sa and sos unable to obtain the optimum solution far better than the reported one. the detail performance of the algorithms and related box plot are presented in table 3(c) and figure 10(d), respectively. 5. overall observation on performance while comparing the performance of various optimization algorithms on test problems, we give priority of objectives as per below: priority 1: algorithms attain the lowest (or highest) objective function value. priority 2: if two or more algorithms attain the optimum value simultaneously, then their lower execution time given priority. priority 3: if any algorithm fails to obey the any constraint that will make it ineligible candidate solution. based on the above criteria, performances of eight algorithms on the seven selected benchmark problems are compared that are presented in table 4 and following concluding remarks are provided: sadhu et al./decis. mak. appl. manag. eng. (2022) 20 1) comparing the best results of optimal values by eight selected algorithms, it can be observed that there is no single optimization algorithm which universally performs best on all the seven selected benchmark problems. 2) from the table 4, it is shown that among the selected eight metaheuristic algorithms, kh has achieved the best solution for three benchmark problems. whereas, sos has performed best for two and ffa, ga, de, and pso each have executed best result for one test problem. 3) while comparing the performance of “classical” and “new generation” metaheuristic algorithms, new generation algorithms performed much better (selected as best performer for six times among seven benchmark problem). 4) among four “classical” algorithms, de, and pso effectively attains the optimal solution which are very close to the best one. in spite of its accurate performance ability, pso takes much more time than any other algorithms. de, pso and sa performed best for four, two and one selected test problems, respectively. 5) among the selected “new generation” algorithm, sos and kh, each performed best for three benchmark problems and ffa showed its best performance for one. therefore, sos and kh are most promising among new generation algorithms. 6) if top two global performer are considered then out of 14 performers, the ranking based on their performance is as follows: sos (5 times)>kh (3 times)>ffa=de=pso (2 times)>ga (1 time) though the kh achieved the best solution for one more benchmark problem compared to sos (already mentioned in observation number 2), but the ranking proved that the consistency of sos performance is better than kh. however, these observations are not universal and the performance of the algorithms may change for other benchmark problems. table 4. overall performance of the selected metaheuristic algorithms test problem number top two performer of “classical” algorithms* top two performer of “new generation” algorithms* top two global performer among “classical” and “new generation” algorithms* 1 2 3 4 5 6 7 de, sa de, ga pso, de de, ga de, pso pso, de sa sos, ffa ffa, sos sos, ffa kh, sos kh, sos sos, ffa kh, sos sos, ffa ffa, de, ga** sos, pso kh, sos kh, de pso, sos kh, sos *arrange them in decreasing order on the basis of performance **all are in same order of sequence from the results of all algorithms on all the test problems, it is evident that there is no single optimization algorithm that universally performs best on all the seven selected benchmark problems. it supports the facts of no free lunch theorem (nflt) (wolpert & macready, 1997). according to the recently discovered no free lunch theorem (nflt) (adam et al., 2019; ho & pepyne, 2001, 2002; wolpert & macready, 1997), no strategy can be predicted to outperform another if we are unable to make any previous assumptions about the optimization problem we are attempting to solve. in other words, there is a comparative study of metaheuristics algorithms based on their performance of complex … 21 no such thing as a general-purpose universal optimization approach. one technique may outperform another only if it is tailored to the problem at hand. what can we assume about likely problem instances, what structural qualities the assumptions suggest, what technique is best matched to that structure, and how sensitive our strategy is to the assumptions are the fundamental questions in optimization practice. therefore, we still need to conduct a great deal of research and develop a better understanding of the implications of “no free lunch theorem” to find a universal metaheuristic optimization algorithm. 6. conclusions in recent years, metaheuristic algorithms were successfully being applied for solving the intractable optimizing problems. the majority of state-of-the-art metaheuristic have been developed before the year 2000 (“classical” algorithms) and then they are become more and more advanced improving their performance and execution time (“new generation” algorithms). the novelty of the present study was to test the capability of applications and compare their performance of the four selected algorithms from “classical” and “new generation” each by solving a number of selected benchmark problems that are used in the literature for algorithm testing purpose and ultimate aim was to find out the universally best algorithm among the selected eight metaheuristic algorithms. however, there was no such universally best algorithm which will perform best in different problem statement. the “new generation” sos and kh algorithm successfully solved most of all the selected test problems and achieved the best solution for most of them. among four “classical” algorithms, de, and pso effectively attained the optimal solution which were very close to the best one. based on the result obtained on the present study, new generation performed much better than old generation. it can be concluded on the basis of the performance of different algorithms that both sos and kh exhibited the most promising result and great potential with respect to execution time also. this study gives some insights to use sos and kh as best performing algorithm to the novice user who can easily get lost by the plethora of large number of optimization algorithms. author contributions: tithli sadhu: formal analysis, methodology, writing-original draft. somnath chowdhury: conceptualization, methodology, writing-review & editing. shubham mondal: methodology, formal analysis. jagannath roy: writingreview & editing. jitamanyu chakrabarty: writing-review & editing. sandip kumar lahiri: conceptualization, writing-review & editing, supervision. all authors have read and agreed to the published version of the manuscript. funding: this research received no external funding. acknowledgement: tithli sadhu and somnath chowdhury would like to thank nit durgapur for their ph.d. fellowship and educational support. conflicts of interest: the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. sadhu et al./decis. mak. appl. manag. eng. 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(2009). cuckoo search via lévy flights, in 2009 world congress on nature & biologically inspired computing (nabic). ieee, 210–214. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame15122022c * corresponding author. e-mail addresses: saikatchatterjee007@gmail.com (s.chatterjee), s_chakraborty00@yahoo.co.in (s.chakraborty) application of the r method in solving material handling equipment selection problems saikat chatterjee1*and shankar chakraborty2 1 department of mechanical engineering, sikkim manipal institute of technology, sikkim manipal university, sikkim, india 2 department of production engineering, jadavpur university, kolkata, india received: 09.03.2022; accepted: 05.12.2022; available online: 15.12.2022. original scientific paper abstract: in manufacturing industries, material handling equipment plays a vital role and is considered as one of the important pillars to increase production efficiency. hence, the selection of appropriate material handling equipment for a specific task is well acknowledged, but the complexity of this selection process drastically increases with the rise in the number of alternative equipment available in the market and a set of conflicting evaluation criteria. to resolve this problem, several multi-criteria decisionmaking (mcdm) techniques have been proposed by past researchers. in this paper, the application potentiality of a newly developed mcdm technique, i.e. r method is explored while solving five material handling equipment selection problems, i.e. conveyor, automated guided vehicle (agv), stacker, wheel loader and excavator. the derived ranking results are contrasted with other popular mcdm techniques to validate its potentiality in shortlisting the candidate alternatives from the best to the worst, which would ultimately help in improving the overall efficiency of the manufacturing processes. key words: material handling equipment; selection; mcdm; r method; ranking 1. introduction the growth of a manufacturing unit largely depends on the resources procured and utilized as monitored by the decision-makers. the material handling systems can increase the profitability of a manufacturing organization at a lower cost of production. selection of suitable equipment for material handling requires knowledge of the complete production process, flow of material etc. since material handling equipment cost shares a substantial amount of the total production cost, its proper selection becomes an important step in facility layout planning and design. moreover, due to the existence of global competitors, quicker delivery of products is most desired mailto:saikatchatterjee007@gmail.com mailto:s_chakraborty00@yahoo.co.in chatterjee and chakraborty/decis. mak. appl. manag. eng. (2022) 2 requiring optimal use of available time and space in the production facilities. it is believed that material handling increases the cost of the product without adding any value, but proper use of time and space can increase the value of the complete process. to develop a robust material handling system based on available facilities, principles, like ergonomics, unit load concept, space utilization and automation etc. need to be addressed. these require a decision on the selection of various handling equipment for the movement of materials available in different forms from one place to another. the competitive nature of the market forces the manufacturers to reduce costs and enhance the quality of their products. benefits of good selection decision include less workforce and associated cost, and a decline in fuel cost, production and delivery times, thus increase in productivity. this complete material handling system mainly consists of activities and deployment of related equipment which constitute a major portion of the factory space, workforce and production time. size, shape, weight and other characteristics of the material considerably affect the decision on handling system for any industrial application. the major categories of material handling equipment, such as transport equipment that are found in industries are positioning equipment, unit load formation equipment, storage equipment, identification and control equipment etc. (chakraborty & banik, 2006). transport equipment help in shifting material to different locations and positioning equipment is utilized to operate at a single location. transport equipment includes conveyors, cranes and industrial trucks. unit load formation equipment confines materials so that they uphold their structure while movement. storage equipment, like automatic storage and retrieval systems, helps to hold excess materials over a period of time. identification and control equipment aid in collecting the information required to maintain the flow of materials. figure 1 provides a list of commonly employed material handling equipment in a typical manufacturing industry. the selection decision of the most apposite material handling equipment to perform a given handing task has now become more complex due to the availability of a wide range of candidate alternatives with varying specifications to serve the same purpose. it compels the deployment of suitable mathematical tools to identify the appropriate material handling equipment in the presence of a set of conflicting criteria, like cost, safety, flexibility, serviceability, speed etc. (saputro et al. 2015). most of small manufacturing organizations usually prefer conventional material handling equipment due to compatibility issues with the existing facilities. the varying flow of materials and design principles of facility layout along with too many choices under various categories of material handling equipment pose a challenging task to the decision-makers. further, the technical and economic feasibility of the application of material handling equipment requires expertise. to resolve a material handling equipment selection problem, the concerned decisionmakers primarily rely on handbooks/catalogues, articles, manuals, experience, opinions and expertise, which is often time-consuming having poor reliability. limited applications of different mathematical models, mainly in the form of multi-criteria decision-making (mcdm) techniques for solving material handling equipment selection problems are available in the literature. although, these methods are quite effective in identifying the most suitable material handling equipment for varying tasks, they have their limitations in solving complex high-dimensional decisionmaking problems. application of the r method in solving material handling equipment selection problems 3 figure 1. commonly used material handling equipment in a typical manufacturing setup karande & chakraborty (2013) explored the application potentiality of the wuta method to identify the best equipment for a given handling task. khandekar & chakraborty (2015) proposed the application of fad principles to identify the most appropriate loading, handling and hauling equipment for surface mines. the derived ranking results were later compared with those of the past researchers to prove the robustness of the adopted approach. hadi-vencheh & mohamadghasemi (2015) first applied the voting approach to determine the corresponding criteria weights which were subsequently converted into a single fuzzy weight based on linguistic variables. f-vikor method was later applied to select the most suitable handling equipment, and the ranking results were finally compared with those derived using f-topsis. bairagi et al. (2015) applied the technique of precise order preference to address rank reversal problems while solving material handling equipment selection problems. nguyen et al. (2016) integrated f-ahp and f-aras methods to select the best conveyor for a specific handling task. saputro & rouyendegh (2016) integrated entropy-based topsis with momilp to solve material handling equipment selection problems. the subjective and objective criteria weights were measured using f-ahp and entropy methods respectively. agarwal & bharti (2018) attempted to solve the agv selection problem using ahp, dematel, topsis, f-ahp, f-dematel and ftopsis methods. rahimdel & bagherpour (2018) applied dematel and topsis methods in the fuzzy environment to select the best haulage system from a set of fixed crushers and trucks, semi-mobile crushers and mobile crushers for an open-pits mine. ulutaş et al. (2020) integrated correlation coefficient and standard deviation values with indifference threshold-based attribute ratio analysis to determine the corresponding criteria weights for a material handling equipment selection problem. marcos method was later employed to rank the candidate alternatives. goswami & behera (2021) investigated the applicability of aras and copras methods to select three material handling equipment for industrial use. horňáková et al. (2021) utilized ahp to evaluate the best material handling technology based on the entry conditions and type of the material. satoglu & turkekul (2021) applied ahp to estimate the corresponding criteria weights and later employed the moora method to rank the pallet truck alternatives. bozanic et al. (2021) presented neuro-fuzzy system to select loader for construction purposes. some recent research work includes the selection of passenger vehicles (biswas et al., 2020); location for emergency medical services (alosta et al., 2021); green supplier (fazlollahtabar & kazemitash, 2021) and appropriate training models (feng, 2021). table 1 presents the expansions of the abbreviations used in this article. this brief literature review indicates the application of mcdm techniques, sometimes integrated with criteria weighting methods based on subjective (depends on decision-maker) or objective (does not depend on the decision-maker; relies on the chatterjee and chakraborty/decis. mak. appl. manag. eng. (2022) 4 established procedure) approaches, along with fuzzy set theory in solving diverse material handling equipment selection problems. the above techniques require quantitative, qualitative, or imprecise performance values to work upon. thus, the objective of the present work is to utilize a simple approach to solve complex decisionmaking problems. although validating the applicability and feasibility of many of the newly developed mcdm techniques in solving material handling equipment selection problems is limited, this paper explores and proposes the application of the r method for the first time in solving five material handling equipment selection problems in a real-time manufacturing environment and calculates the number of computations required to solve the selection problem. being a new approach, the application of the r method in solving mcdm problems is itself very limited and there is a huge opportunity in exploring its application potentiality in dealing with high-dimensional mcdm problems. the rest of this paper is structured as follows: section 2 presents the mathematical steps of the r method along with weight calculations. section 3 demonstrates the application of the r method in solving five real-time material handling equipment selection problems. results and discussions are presented in section 4, sensitivity analysis in section 5 and conclusions are drawn in section 6. table 1. abbreviated terms with elaboration abbreviatio n elaboration aras additive ratio assessment ahp analytic hierarchy process copras complex proportional assessment critic criteria importance through intercriteria correlation dematel decision making trial and evaluation laboratory electre elimination and choice expressing the reality fanma abbreviation derived from name of authors f fuzzy fad fuzzy axiomatic design marcos measurement of alternatives and ranking according to the compromise solution momilp multi-objective mixed integer linear programming moora multi-objective optimization on the basis of ratio analysis topsis technique for order of preference by similarity to ideal solution vikor vlse kriterijumska optimizacija i kompromisno resenje waspas weighted aggregates sum product assessment wuta weighted utility additive 2. r method the r method is a recently developed mcdm technique (rao & lakshmi, 2021), which ranks the alternatives based on their performance scores with respect to each application of the r method in solving material handling equipment selection problems 5 of the evaluation criteria. furthermore, it also ranks the considered criteria based on the opinion of the concerned decision-maker. these assigned ranks are subsequently converted into corresponding weights and composite scores are evaluated using these weights, leading to the final ranking of the alternatives. the procedural steps of the r method are presented below (rao & lakshmi, 2021, 2021a). step 1: construct the decision matrix based on the performance scores of the alternatives against each criterion. step 2: assign ranks (1, 2, 3,…, etc.) to the criteria based on their significance and perception of the decision-maker. assign average rank to equally significant criteria. step 3: assign ranks (1, 2, 3,…, etc.) to the candidate alternatives based on their performance scores against each criterion. allocate average rank to those alternatives having equal performance scores against a specific criterion. step 4: transform the ranks assigned to both the alternatives and criteria into corresponding weights using the information provided in table 2. however, to compute weights from the assigned ranks, eq. (1) can be employed.       n j j k k j k k j r r w 1 1 1 )]/1(1[ )]/1(1[ (1) where wj is the weight of jth alternative or criterion (j = 1,2,...,n), rk is the ranked assigned to kth alternative or criterion (k = 1,2,...,j) and n is the number of alternatives or criteria. step 5: calculate the composite scores of the candidate alternatives by adding up the products of the criteria weights and the corresponding weights of the alternatives. step 6: award ranks to the alternatives based on their composite scores. the alternative with the maximum composite score is the best option. table 2 provides values of the weights calculated from different ranks assigned to the alternatives or criteria based on eq. (1). to illustrate the calculation steps involved in table 2, the computation procedure of weights for four criteria or alternatives (column values under rank 4) is shown as below (rao & lakshmi, 2021, 2021a): 1/reciprocal of rank 1: 1/(1/1) = 1.0000 1/reciprocal of rank up to 2: 1/(1/1 + 1/2) = 0.6666 1/reciprocal of rank up to 3: 1/(1/1 + 1/2 + 1/3) = 0.5454 1/reciprocal of rank up to 4: 1/(1/1 + 1/2 + 1/3 + 1/4) = 0.4800 total = 1.000 + 0.6666 + 0.5454+ 0.4800 = 2.6921 therefore, weight is assigned to rank 1 = 1/2.6921 = 0.3714, similarly for rank 2 = 0.6666/2.6921 = 0.2476, for rank 3 = 0.5454/2.6921 = 0.2026 and for rank 4 = 0.4800/2.6921 = 0.1783. table 2. weight calculations for different assigned ranks rank number of criteria or alternatives 2 3 4 5 6 7 8 9 10 calculated weights 1 0.6 0.452 0.371 0.319 0.283 0.255 0.233 0.215 0.201 2 0.4 0.301 0.248 0.213 0.188 0.17 0.155 0.144 0.134 3 0.247 0.203 0.174 0.154 0.139 0.127 0.117 0.109 4 0.178 0.153 0.136 0.122 0.112 0.103 0.096 5 0.140 0.124 0.112 0.102 0.094 0.088 6 0.115 0.104 0.095 0.088 0.082 7 0.098 0.09 0.083 0.077 chatterjee and chakraborty/decis. mak. appl. manag. eng. (2022) 6 rank number of criteria or alternatives 2 3 4 5 6 7 8 9 10 calculated weights 8 0.086 0.079 0.074 9 0.076 0.071 10 0.068 3. selection of material handling equipment to check the application potentiality of the r method in solving material handling equipment selection problems, the following five demonstrative examples are considered. 3.1 example 1: conveyor selection this example of conveyor selection (kulak, 2005) is a classic and extensively adopted problem in the literature. thus, it is well suited to validate the applicability of r method in accurately ranking the candidate alternatives and comparing the ranking performance with other popular mcdm techniques. this problem consists of four alternative conveyors to be evaluated based on six criteria. the corresponding decision matrix is provided in table 3. among the considered criteria, fixed cost per hour (fic) and variable cost per hour (vac) are non-beneficial attributes requiring their lower values. on the other hand, the higher values for the speed of conveyor (spc), item width (itwi), item weight (itwei) and flexibility (flex) are often preferred. among the four beneficial criteria, flex is expressed subjectively and rao (2007) adopted a linguistic scale to convert the subjective values of this criterion into quantitative measures. now, following the procedural steps of the r method, it is first required to assign the corresponding ranks to the considered alternatives and evaluation criteria. karande & chakraborty (2013) applied ahp method to determine the weights of those six criteria as wfic = 0.1049, wvac = 0.1260, wspc = 0.1260, witwi = 0.2402, witwei = 0.2245 and wflex = 0.1782. based on these weights, rank 1 is assigned to the itwi criterion, followed by itwei, flex, spc, vac and fic, as exhibited in table 4. as vac and spc have the same priority weight, they are assigned their average rank of 4.5. when there is no prior information regarding the criteria weights, the opinion of a decision maker may be sought to assign the corresponding ranks to the set of evaluation criteria. in table 4, based on the type of criterion (beneficial or nonbeneficial), ranks are also assigned to the four alternative conveyors based on their performance against each criterion. for example, in table 4, vac being a nonbeneficial criterion, its lowest value is always desirable. thus, for this criterion, conveyor cb is the best choice and is assigned a rank of 1. as the conveyor’s ca and cc have the same value for vac, they are assigned with an average rank of 2.5 (i.e. an average of 2 and 3). conveyor cd, having the highest vac value, is allotted a rank of 4. similarly, for beneficial criteria, the alternative conveyor with the highest value for the corresponding criterion is assigned with a rank of 1 and so on. now, using eq. (1) and table 2, the corresponding weights for different assigned ranks are calculated for all the alternative conveyors and evaluation criteria, as exhibited in table 5. since, in this decision-making problem, there are four alternatives, the set of weights to be assigned to each alternative is {rank 1 (0.371), rank 2 (0.248), rank 3 (0.203), rank 4 (0.178)}. in the similar direction, for the six evaluation criteria, the set of weights to be assigned is {(rank 1 (0.283), rank 2 (0.188), rank 3 (0.154), rank 4.5 (0.130), rank 6 (0.115)}. to estimate the weight for an average rank, the average of the weights for the application of the r method in solving material handling equipment selection problems 7 corresponding ranks is considered. for example, conveyors ca and cc have the same average rank for the vac criterion. hence, both of them are assigned an average weight of 0.2251 (average of 0.248 and 0.203). the composite score for each of the alternative conveyors is finally calculated by adding the products of criteria weights and corresponding alternative weights, and the candidate conveyors are ranked based on the descending values of this composite score. thus, conveyor cc with the maximum composite score emerges as the best choice for the given handling task, followed by conveyors cb, ca and cd. thus, the complete ranking of the conveyors is derived as cc→cb→ca→cd. conveyor cc also appeared to be the first choice when the same problem was solved using other mcdm techniques, like graph theory and matrix approach, wuta, vikor, promethee, elimination et choice translating reality (electre), evaluation based on distance from average solution (edas), combinative distance-based assessment (codas), weighted aggregated sum product assessment (waspas), moora (karande & chakraborty, 2013; rao, 2007; mathew & sahu, 2018) etc. due to differences in the mathematical treatments in all the considered mcdm techniques, there are slight variations in the intermediate rankings of the conveyors, but, the top-ranked conveyor (cc) exactly matches. it thus proves the potentiality of the r method in identifying the best alternative from a set of feasible options for a given decision-making problem. table 3. decision matrix for conveyor selection problem (kulak, 2005) alternative criteria fic vac spc itwi itwei flex ca 2 0.45 12 15 10 very good (0.745) cb 2.3 0.44 13 20 10 excellent (0.955) cc 2.25 0.45 11 30 20 excellent (0.955) cd 2.4 0.46 10 25 15 very good (0.745) table 4. ranks assigned to the alternatives and criteria for the conveyor selection problem alternative criteria fic vac spc itwi itwei flex ca 1 2.5 2 4 3.5 2.5 cb 3 1 1 3 3.5 1.5 cc 2 2.5 3 1 1 1.5 cd 4 4 4 2 2 2.5 criteria rank 6 4.5 4.5 1 2 3 table 5. assigned weights to the alternatives and criteria for the conveyor selection problem alternative criteria composite score rank fic vac spc itwi itwei flex ca 0.3714 0.2251 0.2476 0.1783 0.1904 0.2251 0.2251 3 cb 0.2026 0.3714 0.3714 0.2026 0.1904 0.3095 0.2607 2 cc 0.2476 0.2251 0.2026 0.3714 0.3714 0.3095 0.3067 1 cd 0.1783 0.1783 0.1783 0.2476 0.2476 0.2251 0.2182 4 chatterjee and chakraborty/decis. mak. appl. manag. eng. (2022) 8 criteria weight 0.115 0.1300 0.1300 0.2830 0.188 0.154 3.2 example 2: agv selection this example deals with the selection of the most suitable agv for a particular industrial application. table 6 depicts the corresponding decision matrix having six criteria and eight alternatives. among the six criteria, except cost (c), all the remaining criteria, i.e. controllability (con), accuracy (acc), range (r), reliability (rel) and flexibility (f) are beneficial. based on a numerical scale provided by rao (2007), all the subjective performance scores of the agvs with respect to the evaluation criteria are first converted into their corresponding numerical values, as shown in table 6. maniya & bhatt (2011) adopted ahp method to calculate the corresponding criteria weights as wcon = 0.346, wacc = 0.168, wc = 0.0584, wr = 0.073, wrel = 0.063 and wf = 0.293, and later applied modified grey relational analysis to rank the candidate agvs. based on the same set of criteria weights, mathew & sahu (2018) also solved this problem using edas, codas, waspas and moora methods. using the procedural steps of the r method, the corresponding ranks are assigned to both the alternative agvs and evaluation criteria, as exhibited in table 7. based on the ahpbased priority weights of the criteria, rank 1 is assigned to con, rank 2 to f and so on. ranks are also assigned to eight alternative agvs based on their performance scores concerning each of the evaluation criteria. applying eq. (1) and the information provided in table 2, the corresponding weights are now allotted to the ranks for both the alternative agvs and criteria, as presented in table 8. this table also provides the calculated values of the composite scores for the agvs and their positions in the final ranking list. the ranking of the candidate agvs is obtained as ag5→ag1→ag4→ag2→ag7→ag6→ag3→ag8. thus, ag5 evolves as the most preferred solution for the specific industrial application, which exactly corroborates the observations of past researchers (mathew & sahu, 2018; maniya & bhatt, 2011). application of the r method in solving material handling equipment selection problems 9 t a b le 6 . d e ci si o n m a tr ix f o r a g v s e le ct io n p ro b le m ( m a n iy a a n d b h a tt , 2 0 1 1 ) a lt e rn a ti v e c ri te ri a c o n a c c c r r e l f a g 1 h ig h a v e ra g e a b o v e a v e ra g e a v e ra g e h ig h b e lo w a v e ra g e a g 2 l o w h ig h h ig h h ig h a v e ra g e a v e ra g e a g 3 l o w l o w h ig h l o w a b o v e a v e ra g e h ig h a g 4 b e lo w a v e ra g e h ig h l o w a v e ra g e a v e ra g e h ig h a g 5 h ig h a v e ra g e l o w a b o v e a v e ra g e b e lo w a v e ra g e a v e ra g e a g 6 a v e ra g e a v e ra g e h ig h l o w a b o v e a v e ra g e a b o v e a v e ra g e a g 7 l o w b e lo w a v e ra g e h ig h l o w h ig h h ig h a g 8 l o w a v e ra g e a b o v e a v e ra g e a v e ra g e a v e ra g e a b o v e a v e ra g e l o w ( 0 .1 1 5 ); b e lo w a v e ra g e ( 0 .2 9 5 ); a v e ra g e ( 0 .4 9 5 ); a b o v e a v e ra g e ( 0 .6 9 5 ); h ig h ( 0 .8 9 5 ) chatterjee and chakraborty/decis. mak. appl. manag. eng. (2022) 10 table 7. ranks assigned to the alternatives and criteria for the agv selection problem alternative criteria con acc c r rel f ag1 1.5 4.5 3.5 4 1.5 8 ag2 6.5 1.5 6.5 1 6 6.5 ag3 6.5 8 6.5 7 3.5 2 ag4 4 1.5 1.5 4 6 2 ag5 1.5 4.5 1.5 2 8 6.5 ag6 3 4.5 6.5 7 3.5 4.5 ag7 6.5 7 6.5 7 1.5 2 ag8 6.5 4.5 3.5 4 6 4.5 criteria rank 1 3 6 4 5 2 table 8. weights allocated to the alternatives and criteria for the agv selection problem alternative criteria composite score rank con acc c r rel f ag1 0.194 0.107 0.1195 0.112 0.194 0.086 0.1406 2 ag2 0.0925 0.194 0.0925 0.233 0.095 0.0925 0.1275 4 ag3 0.0925 0.086 0.0925 0.09 0.1195 0.155 0.1062 7 ag4 0.112 0.194 0.194 0.112 0.095 0.155 0.1400 3 ag5 0.194 0.107 0.194 0.155 0.086 0.0925 0.1428 1 ag6 0.127 0.107 0.0925 0.09 0.1195 0.107 0.1102 6 ag7 0.0925 0.09 0.0925 0.09 0.194 0.155 0.1161 5 ag8 0.0925 0.107 0.1195 0.112 0.095 0.107 0.1035 8 criteria weight 0.283 0.154 0.115 0.136 0.124 0.188 3.3 example 3: stacker selection a manual stacker selection problem (ulutas et al., 2020) for a small warehouse is considered in this demonstrative example, which consists of five evaluation criteria, such as the price of the stacker (p) (in usd), capacity (c) (in kg), lift height (h) (in mm), warranty period (w) (in month) and fork length (l) (in mm), and eight alternatives, as shown in table 9. ulutaş et al. (2020) applied indifference threshold-based attribute ratio analysis approach integrated with correlation coefficient and standard deviation values to determine the criteria weights as wp = 0.1061, wc = 0.3476, wh = 0.3330, ww = 0.1185 and wl = 0.0949, which would be employed here for r method-based ranking of the candidate stackers. to rank those alternatives, ulutaş et al. (2020) proposed the application of the marcos method. in table 10, ranks are first assigned to the five evaluation criteria based on their importance in solving this material handling equipment selection problem. similarly, alternative stackers are also ranked depending on their performance with respect to each of the criteria. in table 11, these ranks assigned to both the criteria and alternative stackers are converted into their corresponding weights. finally, while adding the products of the criteria and alternative weights, the composite scores for all the eight stackers are computed in table 11, which are deployed for their subsequent ranking. table 11 reveals the ranking of the alternative stackers as s8→s1→s3→s4→s5→s7→s2→s6. the application of the r method in solving material handling equipment selection problems 11 emergence of s8 as the most suitable stacker for the considered manual handling task exactly matches the observation of the past researchers (ulutas et al., 2020). table 9. decision matrix for the stacker selection problem (ulutas et al., 2020) alternative criteria p c h w l s1 660 1000 1600 18 1200 s2 800 1000 1600 24 900 s3 980 1000 2500 24 900 s4 920 1500 1600 24 900 s5 1380 1500 1500 24 1150 s6 1230 1000 1600 24 1150 s7 680 1500 1600 18 1100 s8 960 2000 1600 12 1150 table 10. ranks assigned to the alternative and criteria for the stacker selection problem alternative criteria p c h w l s1 1 6.5 4.5 6.5 1 s2 3 6.5 4.5 3 7 s3 6 6.5 1 3 7 s4 4 3 4.5 3 7 s5 8 3 8 3 3 s6 7 6.5 4.5 3 3 s7 2 3 4.5 6.5 5 s8 5 1 4.5 8 3 criteria rank 4 1 2 3 5 table 11. weights assigned to the alternatives and criteria for the stacker selection problem alternative criteria composite score rank p c h w l s1 0.233 0.0925 0.107 0.0925 0.233 0.1367 2 s2 0.127 0.0925 0.107 0.127 0.09 0.1064 7 s3 0.095 0.0925 0.233 0.127 0.09 0.1284 3 s4 0.112 0.127 0.107 0.127 0.09 0.1151 4 s5 0.086 0.127 0.086 0.127 0.127 0.1119 5 s6 0.09 0.0925 0.107 0.127 0.127 0.1059 8 s7 0.086 0.127 0.107 0.0925 0.102 0.1068 6 s8 0.102 0.233 0.107 0.086 0.127 0.1455 1 criteria weight 0.153 0.319 0.213 0.174 0.14 chatterjee and chakraborty/decis. mak. appl. manag. eng. (2022) 12 3.4 example 4: wheel loader selection this problem deals with the selection of an appropriate wheel loader to transport bulk quantities of materials, like debris, gravels and sands at a shorter time on a construction site (prasad et al., 2015). to solve this problem, prasad et al. (2015) developed a software prototype in vbasic based on the quality function deployment (qfd) technique while matching the customers’ requirements with the technical specification of the considered wheel loaders. the performance of seven wheel loaders, i.e. zw140a (wl1), zw150b (wl2), zw180e (wl3), zw250e (wl4), zw40d (wl5), zw50c (wl6), zw80d (wl7) were evaluated based on bucket capacity (bc) (in m3), cost (c) (measured in a relative 1-9 scale), digging depth (dd) (in mm), operating weight (ow) (in ton) and travel speed (ts) (in km/h). table 12 depicts the initial decision matrix for this problem. prasad et al. (2015) estimated the corresponding criteria weights as wbc = 0.1794, wc = 0.1300, wdd = 0.1525, wow = 0.3139 and wts = 0.2242 which would be employed for r method-based solution of this problem. based on the criteria weights and performance scores of the alternative wheel loaders, ranks are assigned to both the criteria and alternatives, as shown in table 13, which are subsequently converted into their related weights in table 14. the composite scores of the candidate wheel loaders are now calculated and they are finally ranked based on descending values of this score in table 14. the complete ranking of the wheel loaders is achieved as wl4→wl3→wl7→wl2→wl1 →wl6→wl5. wheel loader 4 (wl4) evolves out as the best-suited alternative for the given handling task, which exactly matches with the observation of prasad et al. (2015). table 12. decision matrix for the wheel loader selection problem (prasad et al., 2015) alternative criteria bc c dd ow ts wl1 2 5 110 10.29 20 wl2 2.2 5 110 11.8 20 wl3 2.2 6 110 14.71 24 wl4 2.9 8 120 19.89 23 wl5 0.5 3 50 3.375 15 wl6 0.9 3 55 3.66 15.2 wl7 1 4 65 5.27 34 table 13. ranks assigned to the alternative and criteria for the wheel loader selection problem alternative criteria bc c dd ow ts wl1 4 4.5 3 4 4.5 wl2 2.5 4.5 3 3 4.5 wl3 2.5 6 3 2 2 wl4 1 7 1 1 3 wl5 7 1.5 7 7 7 wl6 6 1.5 6 6 6 wl7 5 3 5 5 1 criteria rank 3 5 4 1 2 application of the r method in solving material handling equipment selection problems 13 table 14. weights assigned to the alternatives and criteria for the wheel loader selection problem alternative criteria composite scores rank bc c dd ow ts wl1 0.122 0.117 0.139 0.122 0.117 0.1227 5 wl2 0.1545 0.117 0.139 0.139 0.117 0.1338 4 wl3 0.1545 0.104 0.139 0.17 0.17 0.1531 2 wl4 0.255 0.098 0.255 0.255 0.139 0.2080 1 wl5 0.098 0.2125 0.098 0.098 0.098 0.1139 7 wl6 0.104 0.2125 0.104 0.104 0.104 0.1191 6 wl7 0.112 0.139 0.112 0.112 0.255 0.1461 3 criteria weight 0.174 0.14 0.153 0.319 0.213 3.5 example 5: excavator selection the last demonstrative example considers an excavator selection problem (prasad et al., 2015) for material handling which consists of four evaluation criteria, i.e. battery power (bp) (in ah), cost (c) (in a relative 1-9 scale), operating weight (ow) (in ton) and rated power (rp) (in kw), and six alternative excavators, i.e. zx200lc-3g (ex1), zx225usr-3 (ex2), zx240-3g (ex3), zx350h-3g (ex4), zx350lch-3g (ex5) and zx470lch-3 (ex6). with the help of a software prototype and based on qfd technique, prasad et al. (2015) solved this problem, while identifying zx470lch-3 (ex6) and zx200lc-3g (ex1) as the best and the worst alternatives respectively, and also determined the corresponding criteria weights as wbp = 0.2123, wc = 0.0559, wow = 0.4972 and wrp = 0.2346. table 15 provides the decision matrix for this decisionmaking problem. following the steps of the r method, the considered evaluation criteria and alternative excavators are first ranked depending on their weights and performances with respect to the criteria respectively in table 16. in table 17, these ranks assigned to the criteria and alternatives are transformed into their respective weights. based on the computed composite scores, the alternative excavators are finally ranked from the best to the worst in table 17. like the observations of prasad et al. (2015), zx470lch-3 (ex6) and zx200lc-3g (ex1) emerge as the most and the least preferred alternatives respectively for the considered handling task. the complete ranking of the excavators is derived as ex6→ex5→ ex4→ex3→ex2→ex1. table 15. decision matrix for excavator selection problem (prasad et al., 2015) alternative criteria bp c ow rp ex1 96 3 21.7 110 ex2 88 3 23.3 122 ex3 96 3 23.6 125 ex4 128 4 34.58 184 ex5 128 4 35 184 ex6 170 5 48.1 260 chatterjee and chakraborty/decis. mak. appl. manag. eng. (2022) 14 table 16. ranks assigned to the alternative and criteria for the excavator selection problem alternative criteria bp c ow rp ex1 4.5 2 6 6 ex2 6 2 5 5 ex3 4.5 2 4 4 ex4 2.5 4.5 3 2.5 ex5 2.5 4.5 2 2.5 ex6 1 6 1 1 criteria rank 3 4 1 2 table 17. assigned weights to alternatives and criteria for the excavator selection problem alternative criteria composi te score rank bp c ow rp ex1 0.13 0.188 0.115 0.115 0.1310 6 ex2 0.115 0.188 0.124 0.124 0.1336 5 ex3 0.13 0.188 0.136 0.136 0.1440 4 ex4 0.171 0.13 0.154 0.171 0.1574 3 ex5 0.171 0.13 0.188 0.171 0.1700 2 ex6 0.283 0.115 0.283 0.283 0.2531 1 criteria weight 0.203 0.178 0.371 0.248 4. results and discussion in order to validate the performance of the r method, rankings of the alternative material handling equipment for all the five illustrative examples are contrasted with those derived using other popular mcdm methods, i.e. vikor, waspas, moora, copras and topsis. to maintain uniformity of calculations in all these methods, criteria weights as considered in the r method are employed for solving those examples. it can be interestingly noticed that in all these mcdm techniques, the position of the top-ranked material handling equipment exactly matches. there are marginal deviations in the intermediate rankings of the alternatives which may be attributed to the differences in the mathematical treatments involved in the mcdm techniques. figure 2 plots the spearman’s rank correlation coefficients between r method and other considered mcdm techniques for example 2 (agv selection problem). it can be unveiled from this figure that the r method has high a degree of similarity with the other mcdm technique with respect to the ranking pattern of the alternatives. the correlation coefficient of r with vikor, waspas, moora, copras and topsis is 0.9, 0.81, 0.88, 0.81 and 0.90 respectively. this indicates more similarity in rank with vikor and topsis followed by moora, waspas and copras. similar observations are also noticed for the remaining material handling equipment selection problems (not shown here due to paucity of space). application of the r method in solving material handling equipment selection problems 15 figure 2. rank correlation plot for different mcdm techniques for agv selection problem to prove the simplicity of the r method, numbers of computations involved in different mcdm methods are calculated with respect to computational complexity considering a decision-making problem with m alternatives and n criteria (ghaleb et al., 2020; chatterjee & chakraborty, 2022). table 18 shows the number of computations required for each of the mcdm techniques. on the other hand, table 19 exhibits the actual numbers of computations required by these mcdm techniques while solving the five illustrative material handling equipment selection problems. it becomes clear from table 19 that except for moora, the r method outperforms others with respect to the number of computation steps. table 18. number of computation steps involved in different mcdm methods r vikor waspas step computati on step computation step computati on assigning ranks to the criteria n determinin g the best and the worst values 2n calculation of the normalized matrix m×n assigning ranks to the alternative s based on each criterion m×n calculation of the normalized matrix m×n weighted sum matrix and performan ce score m×n +m assigning weights to the criteria n calculation of the weighted m×n weighted product matrix and m×n +m chatterjee and chakraborty/decis. mak. appl. manag. eng. (2022) 16 normalized matrix performan ce score assigning weights to the alternative s m×n computati on of s, r and q values 3m generalize d weighted aggregatio n m composite score evaluation m computati on of s*, r*, s-, and rvalues 4 total computatio ns required 2mn+2n+ m total computatio ns required 2mn+3m+2n +4 total computatio ns required 3mn+3m table 18. number of computation steps involved in different mcdm methods (continued) moora copras topsis step computati on step computati on step computati on calculation of the normalized matrix m×n calculation of the normalized matrix m×n evaluation of the normalized matrix m×n calculation of the weighted normalized matrix m×n calculation of the weighted normalized matrix m×n evaluation of the weighted normalized matrix m×n calculation of weighted normalized assessment value m computatio n of sums of beneficial criteria and nonbeneficial criteria 2m computatio n of positive distances (m×n)+m determinin g minimum value of nonbeneficial sums 1 evaluation of negative distances (m×n)+m determinin g relative significance and quantitativ e utility 2m determinati on of relative closeness with respect m application of the r method in solving material handling equipment selection problems 17 to the ideal solution total computatio ns required 2mn+m total computatio ns required 2mn+4m+ 1 total computatio ns required 4mn+3m table 19. actual number of computations for different examples method r vikor waspas moora copras topsis example 1 54 66 72 44 57 92 example 2 116 136 168 104 129 216 example 3 98 118 144 88 113 184 example 4 87 112 126 77 99 161 example 5 62 78 90 54 73 108 the r method has also several advantages over the other mcdm techniques. it does not require normalization of the decision matrix having simple and easy-tounderstand calculation steps. the application of the r method, being unaffected by any extraneous tuning parameter, results in quick decision-making with minimum involvement of the decision-maker. it can deal with both qualitative and quantitative information in the decision matrix and has the ability to solve high-dimensional mcdm problems with any number of alternatives or criteria. based on the calculated criteria weights and their importance, it assigns ranks to those criteria. when criteria weights are not available, judgments of the concerned decision-makers may be sought to provide relative importance to the considered criteria. in a similar direction, alternatives are also ranked based on their performance values against each of the criteria. using simple mathematical steps, these ranks allotted to both the criteria and alternatives are subsequently converted into their corresponding weights. after adding the products of the criteria and alternative weights, the candidate alternatives are finally ranked based on their computed composite scores. for its application, the experience and knowledge of the participating decisionmakers are not so much important. 5. sensitivity analysis in-depth sensitivity analysis studies are carried out here to show the ranking stability and robustness of the r technique. figure 3 displays the ranking positions of the options under various scenarios. each scenario considers a new set of weights for the criteria that were determined using the entropy (zou et al., 2006), fanma (srdjevic et al., 2003) and critic (diakoulaki et al., 1995) weighting methods. equal weights, proportionate reduction and rise of the top three and bottom three weighted criteria, and a method of gradual elimination of the least significant criterion are all taken into account in this sensitivity analysis. thus, scenario 1, 2 and 3 shows the rank of the alternatives which considers entropy, critic and fanma weighting methods. the weights are evaluated using these weighting methods and then the r method is applied to rank the criteria based on the computed weights and the rank of the alternatives is determined using these new weights as computed by the r method. in scenario 4, equal weights are considered, and in scenario 5, the top three criteria weights are reduced by 5 % each, and the bottom 3 criteria are increased by 5% each. in scenario 6, eliminating the least important criterion having the minimum weight, i.e. c as previously identified from chatterjee and chakraborty/decis. mak. appl. manag. eng. (2022) 18 the original set of weights, utilized by the previous researcher, and then applying r method weight sets for the remaining criteria using the r method. similarly, in scenario 7, reliability rel is successively eliminated from the evaluation process and a new rank set using the r method is utilized based on the remaining criteria. further, in scenario 8 and 9, r and acc are eliminated and a new rank set is again developed. this procedure continues till only two criteria remain in the evaluation process. for each scenario, the corresponding ranking order of the agv alternatives is derived. it can be observed from figure 3 that these scenarios in the r method-based analysis do not influence the rankings of the top alternative agv; however, minor changes are observed in the intermediate rankings of the other agvs. thus, the position of the best agv (agv5) remains unaffected as they are insensitive to changes in the criteria weights, which proves the consistency and ranking stability of the adopted approach. figure 3. ranking performance of r method at different scenarios 6. conclusions due to the availability of a large set of equally potential alternatives and conflicting evaluation criteria, the selection of the most apposite material handling equipment for a specific handling task is a complicated problem. this paper demonstrates the application of the r method in solving five material handling equipment selection problems taken from the literature. based on the analysis, the following conclusions can be drawn: a) this approach seems to be an effective mcdm strategy for solving problems involving the selection of material handling equipment, resulting in the achievement of the most desirable option. b) the r technique selects conveyor cc as the best equipment for the conveyor selection problem, which is consistent with previous findings. c) this approach produces the automatic guided vehicle agv5, which is consistent with earlier results and the top-ranked alternative. d) this method ranks stacker s8 as the top choice, which is consistent with previous outcomes. e) (e)wheel loader 4 is the top-ranked alternative, according to this methodology, which is consistent with the previous findings. f) this method determines excavator 6 as the top-ranked alternative, which is similar to the results of the past. g) the results from the r approach are compared to those from the vikor, waspas, moora, copras, and topsis methods in this work. there is excellent agreement between the rankings obtained using the r method and other well-liked mcdm strategies for all of the challenges. application of the r method in solving material handling equipment selection problems 19 h) this method offers the most straightforward way to calculate the weights of the criterion. there are minor differences in the intermediate rankings of the alternatives due to differences in the mathematical treatments of the other mcdm methods. the r method can process both quantitative and qualitative criteria, requires few computational steps, is unaffected by tuning parameters, and does not require data normalization. as a result, it can be successfully used to solve both low and highdimensional mcdm problems in a real-time manufacturing environment. in the future, the effectiveness of this method in solving problems associated with parametric optimization can be investigated. decision problems involving too many criteria and alternatives can be problematic. author contributions: saikat.c.: data collection, analysis; draft preparation, review, technical writing; shankar. c.: conceptualization, technical writing, editing. all authors have read and agreed to the published version of the manuscript. funding: this research received no external funding. acknowledgments: the authors would like to thank the editor and the reviewers for their comments which led to considerable improvement in this article. conflicts of interest: the authors declare no conflicts of interest. references alosta, a., elmansuri, o., & badi, i. 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(2003). objektivno vrednovanje kriterijuma performance sistema akumulacija. vodoprivreda, 35, 163-176, (only in serbian). ulutaş, a., karabasevic, d., popovic, g., stanujkic, d., nguyen, p. t., & karaköy, ç. (2020). development of a novel integrated ccsd-itara-marcos decision-making approach for stackers selection in a logistics system. mathematics, 8(10), 1672-1684 zou, z. h., yi, y., & sun, j. n. (2006). entropy method for determination of weight of evaluating indicators in fuzzy synthetic evaluation for water quality assessment. journal of environmental sciences, 18(5), 1020-1023. © 2023 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). decision making: applications in management and engineering vol. 3, issue 1, 2020, pp. 30-42. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame2003065g * corresponding author. e-mail addresses: sganguly@iitg.ac.in (s. ganguly) multi-objective distributed generation penetration planning with load model using particle swarm optimization sanjib ganguly 1 1 department of electronics and electrical engineering, indian institute of technology guwahati, india received: 5 january 2019; accepted: 6 august 2019; available online: 13 august 2019. original scientific paper abstract: the paper presents an approach for simultaneous optimization of distributed generation (dg) penetration level and network performance index to obtain the optimal numbers, sites, and sizes of dg units. two objective functions are formulated. these are: (ii) dg penetration level, (ii) network performance index. the minimization of the first objective reduces the capital investment cost of a distribution network owner (dno) to integrate dg. the minimization of the second objective helps in reduction of network losses and improvement in node voltage profile and line loading. the solution approach provides a set of non-dominated solutions with different values of dg penetration level and network performance index. thus, it offers more flexibility to a dno to choose a final solution from the set of solutions according to its strategic decisions, regulatory directives, and budget restrictions. the solution approach used is multi-objective particle swarm optimization. the approach is validated on a 38-node distribution system. the results are compared with some existing approaches. keywords: distributed generation, multi-objective optimization, paretodominance, particle swarm optimization. nomenclature dg s total dg penetration level; pi network performance index; i y binary decision variable (=1 if there is a dg unit in ith node, otherwise=0); i s size of the dg unit; ilp ( ilq ) real (reactive) power loss index; ilo ( ivd ) line loading (voltage deviation) index; mailto:sganguly@iitg.ac.in multi-objective distributed generation penetration planning with load model using particle... 31 i  weights for ith objective function; ij lo ( ij cl ) line loading (capacity) at line i-j; ( ) l n n total number of lines (nodes); i v voltage at node i; iter iteration number; iter i pv  ( iter i x  ) velocity value (position value) for the  -th dimension of the i-th particle in iteration iter ; 1 2 ( )  learning coefficients/factors of pso; 1 2 ( )r r random number lies within [0,1]; iter i pbest  the best position value for the  -th dimension of i-th particle; iter i guide  the position of the guide particle for  -th dimension of the i-th particle; i d ( d ) distance (mean) between two neighboring solutions of pareto approximation fronts in objective space. 1. introduction the presence of distributed generation (dg) changes the conventional passive power distribution systems to active systems. the dg has a significant impact on the quality of the power supply provided by distribution systems. it can reduce active and reactive power losses, improve node voltage profile, reduce line loading etc (pecas lopes et al., 2007; chiradeja & ramakumar, 2004). also, dg may lessen the impact of future load growth and it can utilize the non-conventional local resources. this becomes a driving force to many power system researchers to investigate its impact on distribution systems (ochoa et al., 2006; singh et al., 2007). simultaneously, a lot of research is going on around the globe to determine a suitable optimization approach for the best allocation of dg units on distribution systems. a suitable optimization approach relies on a realistic problem formulation and an appropriate solution strategy. the existing approaches are basically focused on the determination of optimal site(s) and size(s) of dg unit(s). this is usually done by formulating suitable objective function(s) aiming at the optimization of several features to improve performance of a distribution network, for example real and reactive power losses (singh et al., 2009), voltage profile (singh et al., 2009; mantway & al-muhaini, 2008), line loading (singh et al., 2009; mantway & al-muhaini, 2008), and short circuit capacity (el-zonkoly, 2011) etc. the objective functions formulated in different approaches are minimization of dg installation and operational cost (de-souza & dealbuquerque, 2006; celli et al., 2005), the cost of energy purchase from the grid (celli et al., 2005), and the cost of energy loss (celli et al., 2005; carpinelli et al., 2005) etc. in general, there can be multiple objective functions to be optimized in the optimal dg allocation problem. in some approaches (singh et al., 2009; el-zonkoly, 2011; celli et al., 2005), the multiple objective functions are aggregated with different weights to form a single objective function so as to optimize them. however, if the objective functions conflict with each other there exist a set of trade-off solutions of different objectives, known as non-dominated solutions (deb, 2004). the set of non ganguly /decis. mak. appl. manag. eng. 3 (1) (2020) 30-42 32 dominated solutions is also known as pareto-approximation set. the set of solutions can be determined using different approaches, for example weighted aggregation of objectives with varying weights (mantway & al-muhaini, 2008), ε-constrained method (celli et al., 2005; carpinelli et al., 2005), pareto-dominance method (deb, 2004) etc. the most of the approaches are based on the constant load model, except in (singh et al., 2007; singh et al., 2009), in which it is shown that voltage dependent load model has significant impact on the solutions. practically, the dg penetration level and network performance index conflict with each other up to a certain value of dg penetration level (bollen & hassan, 2011). the network performance in terms of losses, node voltage level etc., improves with increasing dg penetration level. but, it may deteriorate beyond a certain value of dg penetration level. for example, network real and reactive power losses reduce with increasing dg penetration level. however, they may increase after a certain value of dg penetration level. the node voltage and line loading can also be improved with increasing dg penetration level. thus, there is a requirement to have an investigative study so as to determine the optimal value of dg penetration level as well as the optimal network performance index. in this work, the pareto-dominance-based approach is used to simultaneously minimize these two objectives. in general, there are two ways of integrating dg in distribution systems: (i) the distribution network owner (dno) is directly given license to own its generation, and (ii) a distributed generation owner (dgo) is only given license to set up dg units and the dno has to purchase energy from the dgo. in the first case, one of the objectives of the dno would be minimization of the capital investment cost for dg installation and the operational cost of dg as well. in the second case, one of the objectives of the dno would be minimization of the quantity of the energy purchased from the dgo, since the energy provided by dg is comparatively expensive than the energy provided by large central generators (de-souza & de-albuquerque, 2006). in this work, it is assumed that a dno needs to integrate dg into distribution networks. thus, one of the objectives of a dno would be minimization of the dg penetration level into the network. on the other hand, another objective of the dno would definitely be improvement of the performance of its network for the sake of customer satisfaction, which is a key issue in the current competitive power market. thus, the two objective functions formulated in this work are: (i) dg penetration level, (ii) network performance index. the minimization of the first objective reduces the capital investment cost of the dno to integrate dg. the minimization of the second objective helps in minimization of network losses and improvement in node voltage profile and line loading. hence, lower value of the index implies to better performance of a network. the formulation of the second objective function is similar to singh et al. (2009). however, in singh et al. (2009) only this objective is optimized to determine the site and size of single dg unit. on the contrary, in this work, both the objective functions are simultaneously minimized using the paretodominance principle so as to obtain the sites and sizes of multiple dg units. this approach yields a set of non-dominated solutions representing different values of dg penetration level and network performance index. the voltage-dependent load model as reported in (singh et al., 2009) is also used in this work. the solution strategy used in this work is multi-objective particle swarm optimization (mopso). particle swarm optimization (pso) (mantway & al-muhaini, 2008; el-zonkoly, 2011) is a population-based meta-heuristic algorithm, such as genetic algorithm (ga) (singh et al., 2009; celli et al., 2005; carpinelli et al., 2005), evolutionary programming (de-souza & de-albuquerque, 2006) etc. mopso is the multi-objective version of pso. these types of population-based meta-heuristic multi-objective distributed generation penetration planning with load model using particle... 33 algorithms can provide a set of non-dominated solutions in a single run. they do not suffer from the curse-of-dimensionality. in this work, the mopso variant, named as heuristics based selection of guides for mopso (hsg-mopso), proposed by the author in sahoo et al. (2011), is used as the solution strategy and its performance is compared with another two mopso variants (li, 2003; zitzler et al., 2001), i.e., nondominated sorting mopso (ns-mopso) (li, 2003) and strength pareto evolutionary algorithm-ii based mopso (spea2-mopso) (ganguly et al., 2011). these two mopso variants are based on the philosophies of two well known multi-objective optimization algorithms of this kind, i.e., nsga-ii and spea2, respectively (deb, 2004). the proposed approach is validated on the 38-node distribution system reported in singh et al. (2009). the paper is organized as: the multi-objective dg penetration planning formulation and the proposed planning algorithm using hsg-mopso are discussed in sections 2 and 3, respectively. in section 4, the results obtained with the simulation study are presented. section 5 concludes the paper. 2. multi-objective dg penetration planning problem the multi-objective planning problem formulated in this work is aimed at facilitation the decision making of a dno in integrating dg units in distribution networks. thus, the simultaneous optimization of dg penetration level and network performance index is the main focus of the proposed planning approach. the dg penetration level of a network is sum of size of all dg units. network performance index formulation is similar to that of (singh et al., 2009). it is to be noted that these two objectives conflict with each other up to a certain value of dg penetration level. the aim of this planning approach is to determine this value of dg penetration level and to obtain the pareto-approximation set below to this dg penetration level. since each solution in the pareto-approximation set is equally good (deb, 2004), this planning offers more flexibility to the dno to chose a final solution for implementation according to its requirement. the mathematical expressions of these two planning objectives are: 1 n dg i i i s y s    (1) 1 2 3 4 pi ilp ilq ilo ivd       (2) the first objective function is a discrete function with discrete decision variable ( i y ) to be determined to obtain the sites of dg units and continuous variable i s to be determined to obtain the sizes of dg units. the second objective function is a performance index which is a sum of weighted objective functions comprising of network real power loss, reactive power loss, line loading, and voltage deviation. the real power loss index is the ratio of total real power loss with dg ( dg pl ) to the network real power loss without dg (pl). this index shows the improvement in real power loss due to dg penetration. thus, the lower value of this index indicates better performance. its mathematical expression is: dg pl ilp pl  (3) ganguly /decis. mak. appl. manag. eng. 3 (1) (2020) 30-42 34 the reactive power loss index is the ratio of total reactive power loss with dg ( dg ql ) to the network reactive power loss without dg (ql). this index shows the improvement in reactive power loss due to dg integration. the lower value of this index also refers to better performance. it is mathematically expressed as: dg ql ilq ql  (4) the network line loading index is the maximum value of the ratio of line loading to the capacity of each line. this index should be less than one to satisfy thermal limit of each line. the lower value of this index indicates more line capacity available in the network. it can be expressed as: 1 max l ijn ij ij lo ilo cl           (5) the voltage deviation index is the maximum ratio of the voltage deviation of each node to the substation voltage. the node 1 is considered to be the substation node. the numerator represents the voltage deviation of each node with respect to the substation node. the lower value of this index means less voltage deviation, which is desirable. its mathematical expression is: 12 1 max in i v v ivd v           (6) all these indices are added with suitable weights to obtain the network performance index, which is the second objective function in this planning problem, as shown in equation (2). all indices are to be ranked according to the preference of the dno to set their respective weights. the highest weight is to be given to the most preferable index, which the dno wants to optimize. this optimization is subjected to the following constraints: power balance constraint: the demand and supply balance needs to be met in each node. line capacity constraint: the loading should be less than the respective capacity in each line, i.e., ij ij lo cl (7) voltage deviation constraint: the voltage deviation in each node should be less than an allowable limit. 1 limj v v v   (8) the proposed planning approach is done with different voltage dependent load models as described in (singh et al., 2009). the pareto-dominance principle used in simultaneous optimization of these two objective functions is briefly described below. 2.1. pareto-dominance principle in an optimization problem (say, minimization) with m objective functions, a solution x is said to dominate a solution y if the following criteria are satisfied. i ( ) ( ) i i f x f y ,and j uchthat ( ) ( )j jf x f y , [ 1,...,i m ] (9) multi-objective distributed generation penetration planning with load model using particle... 35 the aim is to determine a set of non-dominated solutions, in which no solution is inferior to others. the set of optimal non-dominated solutions is called the paretooptimal set. 3. the multi-objective planning algorithm using hsg-mopso the multi-objective planning approach using hsg-mopso is described in this section. the particle decoding/ encoding scheme, a support subroutine used in this planning algorithm, is also provided. 3.1. multi-objective particle swarm optimization (mopso): an overview in mopso, each particle representing the sites and sizes of dg units in this optimization problem is encoded by a continuous position vector (x) which consists of multi-dimensional information. the position vector is randomly chosen in initial iteration. then, it is iteratively updated with particle’s velocity. the choice of initial velocity for a particle is also random. the velocity vector (pv) for a particle is also iteratively updated with the help of the respective previous best position (pbest) and the position of a guide. the choice of guides depends upon the mopso variants to be used. the particle velocity and position updating equations are given below. the updating equations for the θth-dimension of the ith-particle are taken from sahoo et al. (2011). 1 1 1 2 2 ( ) ( ) iter iter iter iter iter iter i i i i i i pv pv r pbest x r guide x               (10) 1 1iter iter iter i i i x x pv        (11) the guide selection is the most important task for such kind of multi-objective evolutionary algorithms (sahoo et al., 2011). the heuristics-based guide selection technique, hsg-mopso, is followed in this work. in hsg-mopso, a set of potential guides is selected from an iteration and then, the set is iteratively updated using the set of non-dominated solutions and some dominated solutions from some specific regions of the feasible objective space. the dominated solutions are heuristically chosen from such region in the objective space where no non-dominated solutions are obtained. each member of the population either follow the nearest nondominated guide or the nearest dominated guide. the objective is to balance between the exploration and exploitation. 3.2. particle encoding/decoding scheme the position vector of a particle is directly encoded with the information on the decision on dg location at each node and the size of dg units. thus, a particle in the proposed encoding scheme consists of two segments as shown in fig. 1. in the first segment, the binary decision on integrating dg units in each node of a network is encoded. the other segment contains the sizes of dg units. all nodes of a network, except the substation node (i.e., node 1), is considered to be the potential locations for integrating dg units. since a particle is encoded with direct information, its decoding is straight-forward. in the decoding process, if yi at node i is found to be 1 a dg unit with size si is to be integrated in the network. some infeasible solutions are to be heuristically filtered out. for example, if yi is zero and si at node i is found to be non-zero it is forcefully made zero. the size of a dg unit is kept between specified minimum and maximum values. ganguly /decis. mak. appl. manag. eng. 3 (1) (2020) 30-42 36 3.3. constraint handling technique the constraints of this planning problem are handled as given below. the demand and supply balance constraint is met using the forward-backward sweep power flow subroutine which in embedded into the hsh-mopso. if line capacity constraint is violated the solution is to be penalized by adding a suitable penalty factor to the both objective functions. the value of the penalty factor is computed as the product of the maximum ratio of line loading to capacity and a very high integer number. if the voltage limit constraint is seen to be violated in any node, the solution is to be penalized with a suitable penalty factor in a similar way mentioned in sahoo et al. (2015). 3.4. complete planning algorithm the pseudo codes of the complete planning algorithm are given in fig. 2. the nondominated solutions are preserved in an elite archive with a fixed archive size (ηa). the decision on dg location is updated using the concept of binary pso (bpso) (mantway & al-muhaini, 2008). the velocity updating equation in bpso is same as that of continuous version of mopso given in equation (10). the position updating equation follows a sigmoid transformation to restrict the value of position to binary value as shown below. 1 1 1 ( ) 1 exp( ) iter i iter i sig pv pv        (12) 1 1 1, rand(0,1)< ( ) 0, otherwise iter iter i i sig pv x         (13) decisions on the locations of dg units sizes of dg units y2 y3 yn s2 s3 sn figure 1. particle encoding scheme multi-objective distributed generation penetration planning with load model using particle... 37 4. simulation results the proposed multi-objective distribution system planning approach is evaluated via computer simulation studies on a 38-node distribution system. the system data are available in singh et al. (2009). the mopso parameters are optimized sequentially as done in sahoo et al. (2011). these are shown in table 1. the plot of pareto-approximation solutions in objective space is called pareto-approximation front. one sample pareto-approximation front obtained with the hsg-mopso for mixed load model is shown in fig. 3. the comparison of the pareto-approximation fronts obtained with hsg-mopso, spea2-mopso, and ns-mopso is shown in fig. 4. the hsg-mopso is a mopso variant proposed in sahoo et al. (2011). the spea2 is a ga-based multi-objective optimization approach which is originally proposed in zitzler et al. (2001). the idea is borrowed to devise spea2-mopso in ganguly et al. (2011). the ns-mopso is reported in li (2003). the results show that the performances of hsg-mopso and ns-mopso are competitive and better than spea2mopso. a comparison of the pareto-approximation fronts obtained with different load models as shown in fig. 5. this illustrates that there is distinct difference in performance between the practical load models and constant load models. the best solutions in view of the network performance index obtained with the different load models with those given in singh et al. (2009) are compared in table 2. the results show that much better solutions in terms of network performance index are begin // ηpop = population size of hsg-mopso // max_iter = number of maximum iterations randomly generate the set of initial population of position and velocity vectors for hsg-mopso using the particle encoding scheme; decode the particles and calculate the fitness functions; find the set of non-dominated solutions and store them in an elite archive; determine the set of particles to be chosen as guides; itern=1; while itern<= max_iter for i=1,...,ηpop find a guide for the ith particle from the set of guides; update the particle’s velocity and position vectors; decode the particle to obtain the sites and sizes of dg units; perform the forward-backward sweep load flow; calculate the fitness functions using equations (1-2); endfor determine the set of non-dominated solutions update the set of guides; itern=itern+1; endwhile the elite archive consists of the optimal solutions with different number, sizes and sites of dg units; end figure 2. pseudo codes of the complete planning algorithm using hsg-mopso ganguly /decis. mak. appl. manag. eng. 3 (1) (2020) 30-42 38 obtained in the proposed approach. however, the dg penetration level for those solutions is higher than those obtained in singh et al. (2009). in fact, the maximum dg penetration level which is set to 0.63 p.u. in singh et al. (2009), results into suboptimal solutions. thus, all reported optimal solutions in singh et al. (2009) have dg penetration level of 0.62-0.63 p.u. the overall investigation shows that there are certain ranges of dg penetration level and network performance index in which these two objectives conflict with each other. for this 38-node system, these are found to be 0-3.5 p.u. and 0.4-0.8, respectively. these may vary in different systems. the investigation also shows that the planning algorithm should determine these ranges so as to provide many equally good alternative solutions to a dno. table 1. the parameters of different mopso variants studied here parameters hsg-mopso spea2-mopso ns-mopso population size 50 50 50 maximum iteration 200 200 200 learning factors ϕ1=2, ϕ2=1.5 ϕ1=2, ϕ2=1.5 ϕ1=2, ϕ2=1.5 size of elite archive 40 40 40 table 2. comparison among the best solutions in view of the network performance index obtained with the different load models in proposed approach and the ga-based approach reported in singh et al. (2009) algorithms constant load model industri al load model residenti al load model commercia l load model mixed load model ga-based approach 0.6539 0.7629 0.7631 0.7645 0.7647 proposed approach 0.4363 0.4322 0.4662 0.4778 0.4594 figure 3. the pareto-approximation front obtained with hsg-mopso for mixed load 0.4 0.5 0.6 0.7 0.8 0.9 0 0.5 1 1.5 2 2.5 3 network performance index d g p e n e tr a ti o n l e v e l multi-objective distributed generation penetration planning with load model using particle... 39 figure 4. comparison among the pareto-approximation fronts obtained with hsg-mopso, spea2-mopso, and ns-mopso 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0 0.5 1 1.5 2 2.5 3 3.5 network performance index d g p e n e tr a ti o n l e v e l constant load model residential load model 0.4 0.5 0.6 0.7 0.8 0.9 0 0.5 1 1.5 2 2.5 3 3.5 network performance index d g p e n e tr a ti o n i n d e x constant load model commercial load model 0.4 0.5 0.6 0.7 0.8 0.9 0 0.5 1 1.5 2 2.5 3 network performance index d g p e n e tr a ti o n l e v e l constant load model mixed load model 0.4 0.5 0.6 0.7 0.8 0.9 0 0.5 1 1.5 2 2.5 3 3.5 network performance index d g p e n e tr a ti o n l e v e l constant load model industrial load model ( a ) ( b ) ( c ) ( d ) figure 5. comparison between the pareto-approximation fronts obtained with constant load model and: (a) residential load model, (b) commercial load model, (c) industrial load model, and (d) mixed load model obtained with the hsg-mopso 0.4 0.5 0.6 0.7 0.8 0.9 0 0.5 1 1.5 2 2.5 3 3.5 network performance index d g p e n e tr a ti o n l e v e l hsg-mopso spea2-mopso ns-mopso ganguly /decis. mak. appl. manag. eng. 3 (1) (2020) 30-42 40 the performances of the spea2-mopso are assessed with statistical tests and compared with ns-mopso and spea2-mopso. for this purpose, 30 runs are taken separately for those mopso variants. two performance assessment indicators, i.e., hypervolume and diversity indicators (deb, 2004) are used for this comparison. i hypervolume indicator: this is an indicator used to determine the area/volume of objective space being dominated by the pareto-approximation set of solutions. the higher value of hypervolume indicator implies to the larger area (for bi-objective problem) or volume (for a problem with more than 2 objective functions) being dominated by the approximation set of solutions. this indicates comparatively better solutions close to the pareto-optimal set. in this work, the pareto approximation solutions in objective space are normalized with respect to a reference point, i.e., (4,1). the reference point is judiciously chosen in view of the maximum values of the two objective functions obtained in multiple simulation run. the means and variances of the hypervolume indicator for different test systems are given in table 3. the higher hypervolume indicator is preferable because it signifies the solutions close to the pareto-optimal set. ii diversity indicator: the mathematical expression for the diversity indicator () is shown in equation (14). it is used to measure the diversity among the solutions in a set of pareto-approximation solutions.   1 | | / ndfn i ndf j d d n d             (14) the ideal value for diversity indicator is to be zero or close to zero to show that there is good diversity among the pareto solutions. hence, the lower diversity indicator implies to better diversity among the solutions. the results illustrate that the better convergence is obtained with ns-mopso and hsg-mopso. however, the diversity among the solutions obtained with hsg-mopso is reasonably better than ns-mopso and spea2-mopso. the mean execution time of hsg-mopso is found to be reasonably higher as compared to ns-mopso and spea2-mopso. since this is a type of investment planning to decide the dg integration capacity, it needs offline optimization. hence, the execution time may not be a bottleneck to implement the hsg-mopso algorithm. table 3. the results of statistical tests mopso variants hypervolume indicator diversity indicator mean execution time (sec) mean value variance mean value variance hsg-mopso 0.4571 2.32×10-5 0.6215 0.0102 59.2348 spea2-mopso 0.4431 4.56×10-5 0.7656 0.0134 17.5322 ns-mopso 0.4583 5.93×10-5 0.8141 0.0155 16.2845 5. conclusion in the paper, an approach for the simultaneous optimization of dg penetration level and the network performance index has been provided to determine the optimal numbers, sites, and sizes of dg units. this planning yields a set of nondominated solutions with different values of dg penetration level and network performance index. the contributions of this approach are: multi-objective distributed generation penetration planning with load model using particle... 41 the proposed approach offers more flexibility to the dno to choose a final solution from the set of solutions according to its strategic business decisions, regulatory directives regarding the electric service, and budget restrictions. for example, a dno may prefer to reduce the dg penetration level instead of improving network performance or vice-versa. the proposed planning can determine the ranges of dg penetration level and network performance index on which they conflict with each other. the energy provided by dg is relatively expensive than that provided by large central generators. hence, the allocation of the dg units would be worthy for a dno if it leads to a significant reduction on power losses, improvement in voltage levels etc. a comparison between the pareto-approximation sets obtained with different types of load models is carried out so as to bring out the impact of load models on the multi-objective type of problem formulation. the performance comparison between the three mopso variants is also reported in this work. author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research received no external funding. conflicts of interest: the authors declare no conflicts of interest. references bollen, m., & hassan f. (2011). integration of distributed generation in the power system. (1. ed.). new york, usa: wileyinstitute of electrical and electronics engineers press publishing. carpinelli, g., celli, g., mocci, s., pilo, f., & russo, a. (2005). optimisation of embedded generation sizing and siting by using a double trade-off method. iee proceeding -generation, transmission and distribution, 152(4), 503–513. celli, g., ghiani e., mocci, s., & pilo, f. (2005). a multiobjective evolutionary algorithm for the sizing and siting of distributed generation. institute of electrical and electronics engineers transmission power systems, 20(2), 750–757. chiradeja, p., ramakumar, r. (2004). an approach to quantify the technical benefits of distributed generation. institute of electrical and electronics engineers transcations on energy conversion, 19(4), 764–773. deb, k. (2001). multi-objective optimization using evolutionary algorithms. in deb, k. wiley interscience series in systems and optimization (pp. 501-525). new york, usa: john wiley publishing. de-souza, b.a., & de-albuquerque, j. m. c. (2006). optimal placement of distributed generators networks using evolutionary programming. 2006 institute of electrical and electronics engineers /pes transmission and distribution conference and exposition, 1-6. el-zonkoly, a. m. (2011). optimal placement of multi-distributed generation units including different load models using particle swarm optimization. iet generation transmission distribution, 5(7), 760–771. https://ieeexplore.ieee.org/xpl/recentissue.jsp?punumber=2195 https://ieeexplore.ieee.org/xpl/recentissue.jsp?punumber=2195 ganguly /decis. mak. appl. manag. eng. 3 (1) (2020) 30-42 42 ganguly, s., sahoo, n. c., & das, d. 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(2006). evaluating distributed generation impacts with a multiobjective index. institute of electrical and electronics engineers transcations on power delivery, 21(3), 1452–1458. pecas lopes, j. a., hatziargyriou, n., mutale, j., djapic, p., & jenkins, n. (2007). integrating distributed generation into electric power systems: a review of drivers, challenges and opportunities. electric power systems research, 77, 1189–1203. sahoo, n. c., ganguly, s., & das, d. (2011). simple heuristics-based selection of guides for multi-objective pso with an application to electrical distribution system planning. engineering applications of artificial intelligence, 24(4), 567–585. singh d., singh d., & verma k. s. (2009). multiobjective optimization for dg planning with load models. institute of electrical and electronics engineers transcations on power systems, 24(1), 427–436. singh, d., misra, r. k., & singh, d. (2007). effect of load models in distributed generation planning. institute of electrical and electronics engineers transcations on power systems, 22(4), 2204–2212. zitzler, e., laumanns, m., & thiele, l. (2001). spea2: improving the strength pareto evolutionary algorithm. technical report 103, computer engineering and communication networks lab (tik), swiss federal institute of technology (eth) zurich, 4-18. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). https://www.researchgate.net/publication/321555987_genetic_and_evolutionary_computation_-_gecco_2003_genetic_and_evolutionary_computation_conference_chicago_il_usa_july_12-16_2003_proceedings_part_i https://www.researchgate.net/publication/321555987_genetic_and_evolutionary_computation_-_gecco_2003_genetic_and_evolutionary_computation_conference_chicago_il_usa_july_12-16_2003_proceedings_part_i https://www.researchgate.net/publication/321555987_genetic_and_evolutionary_computation_-_gecco_2003_genetic_and_evolutionary_computation_conference_chicago_il_usa_july_12-16_2003_proceedings_part_i plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0321052022v * corresponding author. e-mail addresses: mechieashutosh@gmail.com (a. verma), *sushant.tripathy@gmail.com (s. tripathy), dsinghal.iiit@gmail.com (d. singhal) the significance of warehouse management in supply chain: an ism approach ashutosh verma1, sushanta tripathy1 and deepak singhal1 1 school of mechanical engineering, kiit university, india received: 2 april 2022; accepted: 21 may 2022; available online: 21 may 2022. original scientific paper abstract: warehouse management is the key aspect for an uninterrupted flow of products within a supply chain. this paper deals with the critical factors that are responsible for creating an impactful influence on the working of warehouse management. the analysis involves the selection of critical factors then applying interpretive structural modelling (ism) methodology to them in order to get the level partition and final ism model. this research also involves the micmac analysis on the factors which classifies all the selected factors into four groups namely, autonomous variables, dependent variables, linkage variables and driver variables. this research will help the supply chain architects to establish a better and reliable warehouse system. as this research involves analysis of multiple domains that is why a variety of users can refer to this work for their businesses, also the ism approach gives a good accuracy of the hierarchy of the factors which helps in deciding the most effective chronology of the implementation of various warehousing operations. researchers can also refer to this work to get insights of the significance of warehouse management in the supply chain and also the complete working of the ism methodology. key words: warehouse, supply chain management, critical factors, ism. 1. introduction one of the most critical parts of any supply chain is the warehouse. the success or failure of a business significantly depends on its supply chain structure. the supply chain consists of multiple stages that contain multiple nodes, each having different functionalities and objectives (handfield & nicholas jr., 1999). similarly, warehouses occupy their places in the supply chain at multiple levels. there is a need for warehouses at the supplier side, at the manufacture as well as the retailer end. the main function of a warehouse is to provide buffer storage to maintain the uninterrupted flow of the supply chain. warehouses stores goods on a temporary basis in order to deal with the variability in the market, interruption in the flow of products, verma et al./decis. mak. appl. manag. eng. (2022) 2 the delivery criterion of the customers, value addition processes, customization of products, etc.. markets are experiencing rapid variations in customer demands. products get updated frequently, new products are also being introduced very rapidly and due to large competition, there is also a need for quick delivery of the demanded products. to cope up with the changing market structure, businesses are also adapting to the new production methods like the just in time (jit) method or lean production methods (gu et al., 2007). to achieve these production levels, a well-equipped supply chain is needed and for a good supply chain, there is a necessary requirement of highly efficient and effective warehouses. these warehouses must include a better storage facility, faster input and output processes, technologically advanced infrastructure, an efficient warehouse management system, skilled human resources and all other required elements. along with this, automation is also playing an important role in the success of warehouses, it ensures better resource utilization with improved reliability. the main constraint to this increasing influence of it in supply chain management is the lack of adaptability of the old conventional system into a new dynamic system. in this way, there are multiple factors that are responsible for the success of the warehouse system (baker & canessa, 2009). this paper deals with the identification of the critical factors that influences the establishment and working of a warehouse. a number of literature reviews have been explored for a better understanding of the topic. in this work, one multiple criterion decision making (mcdm) approach is used to define a model that can be referred to while building a warehouse system. the mcdm technique which has been used here is interpretive structural modelling (ism) methodology. ism focuses on the complex interrelationships between the selected factors and gives out the best and analytically correct model for implementation. 2. literature review the most critical phase of any mcdm technique is the determination of factors that influence the functioning as well as the growth of the selected domain. in this research, we have considered a domain comprising of advanced warehouse activities. after a thorough review of the earlier literature, we have enlisted all the important factors that are most influential during the functioning as well as determining the growth of a modern warehouse, in table 1. table 1. critical factors affecting the modern warehouse identified by previous studies. important factors research articles optimal warehouse storage leng, j., yan, d., liu, q., zhang, h., zhao, g., wei, l., ... & chen, x. (2021). digital twin-driven joint optimisation of packing and storage assignment in large-scale automated high-rise warehouse product-service system. international journal of computer integrated manufacturing, 34(7-8), 783-800. inventory management au, y. h. n. (2009). warehouse management system and business performance: case study of a regional distribution centre. the significance of warehouse management in supply chain: an ism approach 3 important factors research articles yerpude, s., & singhal, t. k. (2018). smart warehouse with internet of things supported inventory management system. international journal of pure and applied mathematics, 118(24), 1-15. order picking system custodio, l., & machado, r. (2020). flexible automated warehouse: a literature review and an innovative framework. the international journal of advanced manufacturing technology, 106(1), 533-558. lee, c. k., lv, y., ng, k. k. h., ho, w., & choy, k. l. (2018). design and application of internet of things-based warehouse management system for smart logistics. international journal of production research, 56(8), 2753-2768. warehouse management system ramaa, a., subramanya, k. n., & rangaswamy, t. m. (2012). impact of warehouse management system in a supply chain. international journal of computer applications, 54(1). transportation kondratjev, j. (2015). logistics. transportation and warehouse in supply chain. manpower tonape, s., patil, k., & karandikar, v. (2016). manpower optimization and method improvement for a warehouse. safety measures glickman, t. s., & white, s. c. (2007). safety at the source: green chemistry's impact on supply chain management and risk. international journal of procurement management, 1(1-2), 227-237. rajaprasad, s. v. s., & chalapathi, p. v. (2015). factors influencing implementation of ohsas 18001 in indian construction organizations: interpretive structural modeling approach. safety and health at work, 6(3), 200205. overall running cost varila, m., seppänen, m., & suomala, p. (2007). detailed cost modelling: a case study in warehouse logistics. international journal of physical distribution & logistics management. speh, t. w. (2009, june). understanding warehouse costs and risks. in ackerman warehousing forum (vol. 24, no. 7, pp. 1-6). warehouse productivity staudt, f. h., alpan, g., di mascolo, m., & rodriguez, c. m. t. (2015). warehouse performance measurement: a literature review. international journal of production research, 53(18), 5524-5544. location jha, m. k., raut, r. d., gardas, b. b., & raut, v. (2018). a sustainable warehouse selection: an interpretive structural modelling approach. international journal of procurement management, 11(2), 201-232. green initiatives bartolini, m., bottani, e., & grosse, e. h. (2019). green warehousing: systematic literature review and bibliometric analysis. journal of cleaner production, 226, 242-258. verma et al./decis. mak. appl. manag. eng. (2022) 4 important factors research articles use of right storage solution lee*, m. k., & elsayed, e. a. (2005). optimization of warehouse storage capacity under a dedicated storage policy. international journal of production research, 43(9), 1785-1805. order fulfilment reaidy, p. j., gunasekaran, a., & spalanzani, a. (2015). bottom-up approach based on internet of things for order fulfillment in a collaborative warehousing environment. international journal of production economics, 159, 29-40. automation atieh, a. m., kaylani, h., al-abdallat, y., qaderi, a., ghoul, l., jaradat, l., & hdairis, i. (2016). performance improvement of inventory management system processes by an automated warehouse management system. procedia cirp, 41, 568-572. van geest, m., tekinerdogan, b., & catal, c. (2021). design of a reference architecture for developing smart warehouses in industry 4.0. computers in industry, 124, 103343. forecasting mohsen, & hassan, m. d. (2002). a framework for the design of warehouse layout. facilities, 20(13/14), 432440. suesut, t., gulphanich, s., nilas, p., roengruen, p., & tirasesth, k. (2004, november). demand forecasting approach inventory control for warehouse automation. in 2004 ieee region 10 conference tencon 2004. (pp. 438-441). ieee. 3. identified critical factors there were multiple factors that are affecting the warehousing operations. as mentioned in the preceding section, 15 of the most important factors have been chosen for further investigation. the significance of all these factors is discussed in detail below. 3.1. optimal warehouse storage the storage capacity of a warehouse is critical to the overall operation of warehousing. someone can think that the whole available storage must be used for storage purposes, but in reality, this is one of the blunder mistakes that leads to the failure of many supply chains. the proper approach is we should focus on utilizing the available space in such a way that smooth conduct of other warehousing activities apart from storage, like the ease of movement of workforce and machinery, proper accounting units, can be assured (leng et al., 2021). some key features of optimal warehouse storage are 1. optimal storage utilization for maximum efficiency. 2. storage occupancy within 22% to 27% of the total usable space is maintained. 3. it can handle seasonal growth in demands. 4. it ensures optimal clearance height. the significance of warehouse management in supply chain: an ism approach 5 5. ease in the movement of employees and robots. 6. can provide high flexibility and agility to the warehouse depending upon the change in product. 3.2. inventory management every organization requires inventories to ensure the efficient and de-stressed flow of supply chains. inventory accounts for safeguarding the organizations from demand fluctuations, the unreliability of supply, price variations, lead time variations, etc. (au, 2009). the supervision and controlling of this inventory with proper management techniques are termed inventory management. inventory management is a vital aspect that heavily contributes to the success of the industry (yerpude & singhal, 2018). some key features of inventory management are 1. control over inventory for achieving optimal storage and capital utilization. 2. fulfilment of volatile customer demands. 3. tackles the problem of erratic supplies. 4. safeguarding organizations from price fluctuations. 5. ensures real-time storage usage. 3.3. order picking system order picking is the process of retrieving products from storage in response to specific customer demand (custodio & machado, 2020). it is one of the most labour intensive and capital consuming activities in the whole warehousing operations. it has been found from the earlier studies that order picking activities almost constitutes 55% of the total warehousing costs. any shortcoming in the order picking process results in unsatisfactory customer reviews and an increase in the cost of maintaining operations activities (lee et al., 2018). during the design of the order picking system, special attention should be paid to the fact that it must be both sturdy and efficient in operation. some key features of the order picking system are 1. reduces the lead time. 2. supports the just in time approach of the supply chain. 3. helps in handling the changing market conditions. 4. ease in accumulation of items in the storage as well as their proper labelling can be assured (de koster et al., 2007). 5. ease in the accessibility of all the items with optimal storage utilization. 3.4. warehouse management system warehouse management system is used to manage processes, resources, people and equipment on the operational level within a warehouse. there are basically three types of warehouse management systems (mao et al., 2018). first is the basic wms, it supports the stock and location controls of the product. its main functions are registering, storing and picking information about the products. the second is the advanced wms, which has some added advantages over the basic wms like it can do resource planning activities to synchronize the flow of items. basically, it focuses on the throughput, stocks and capacity analysis within a warehouse. the third is the complex wms, it can organize and optimize a group of warehouses simultaneously (ramaa et al., 2012). its advanced additional verma et al./decis. mak. appl. manag. eng. (2022) 6 functionalities include transportation planning, value-added logistics planning and most importantly helps in smooth communication between the different units of the supply chain. some key features of the warehouse management system are 1. tracking of the flow of items within the supply chain. 2. sharing of the information at different workstations. 3. synchronization of all the logistics processes. 4. improving the efficiency of resource planning. 5. ensuring proper connectivity at all the nodes within the supply chain. 3.5. transportation transportation is one of the basic parts of any economic activity, which is associated with an improvement in the satisfaction level of people and businesses by altering the location of goods and services (kondratjev, 2015). moving products for fulfilling customer requirements is the key aspect of transportation in logistics. there are various means of transportation such as roadways, railways, waterways, airways, etc. are involved in supply chain management (žunić et al., 2018). some key features of transportation are 1. forwarding, cargo handling and transfer of products. 2. sequencing of the utilization of different modes of transport. 3. ensuring risk-free maneuver of products. 4. proper scheduling of delivery for reducing time consumption within the supply chain. 3.6. manpower for all firms, trained staff is a requirement of the day. the efficiency and effectiveness of the human resource involved in a warehouse are critical to its growth and development. manpower is the key requirement for every workstation, especially at the loading, unloading and packaging units (tonape et al., 2016). a good working environment gives better output in comparison to the overcrowded and overloaded facilities. and finally, all the critical decision-making lies with the skilled and experienced personnel working in the supply chain. some features of manpower activities are 1. responsible for unpacking and arranging goods when they arrive at the storage facility. 2. labelling and packing are also carried out by the employees with the help of some tools and technologies. 3. customization of special orders is mostly handled by the manpower working there. 4. a good set of technically skilled employees can give a lot of flexibility to the warehouse. 5. all the final stage inspections are carried out by the employees themselves. 3.7. safety measures the goal of putting in place safety measures in a company is to protect the employees' health, safety, and well-being, as well as the valuable commodities that are kept there. in the event of an emergency, safety precautions are critical (glickman & white, 2007). mis happenings are a part of the working environment but by applying safety measures we can reduce it drastically. the significance of warehouse management in supply chain: an ism approach 7 some of the key safety measures are 1. fireproofing of the whole warehousing facility. 2. strictly following checkboxes that are related to the safety measures (rajaprasad & chalapathi, 2015). 3. installation of advanced and reliable alarming system. 4. proper arrangement of security personnel for the safeguarding of the warehouse. 5. availability of first aid kits and fire extinguishers at every possible location. 3.8. overall running cost running cost calculation is the activity that assists the decision making, planning and control strategies. this task needs to be done very accurately because almost all the major decisions are taken based on the overall cost (varila et al., 2007). the location and size of the project play a vital role in the determination of overall running cost. it incorporates almost all the expenditures such as power bills, the salary of employees, transportation costs, maintenance costs, etc. (speh, 2009). some key characteristics of the overall running cost are 1. it helps in finding the cost per order. 2. important operational expenditures like the cost of equipment and maintenance costs can be determined. 3. maximum attempts are made in the direction of reducing the running costs. 4. lastly, it helps the organization to compare the total cost incurred during the running of the warehouse in comparison with the total revenue generated by the warehouse. 3.9. warehouse productivity metric analysis of the warehouses is required for determining the overall throughput and productivity. due to the increasing complexities in the warehousing operations, it becomes very difficult to quantify all the aspects for easy understanding (staudt et al., 2015). the performance analysis helps the managers to get a clear view of the productivity of a warehouse. warehouse productivity includes 1. analysis of orders dispatched per hour. 2. analysis of lines cleared per hour. 3. analysis of items handled per hour. 4. calculates the total throughput of the warehouse. 5. helps in determining the working layout (for e.g., shifting from multichannel distribution to omnichannel distribution). 3.10. location selection of the location for a warehouse is a very critical decision. as it is a onetime decision to be made so a lot of other factors are considered while selecting a strategic location. some of these factors are good accessibility to the required modes of transport (roadways, railways, waterways, airways, etc.), land acquired must be free from any legal conflicts, the location should be isolated from the people-centric city areas, there should be a continuous supply of water and electricity, favorable legal policies for the uninterrupted working of the warehouse, it must be a properly planned node point on the supply chain, must be at a product and employee-friendly climatic conditions, etc. (jha et al., 2018). verma et al./decis. mak. appl. manag. eng. (2022) 8 some additional features of the strategic location are 1. it accounts for the quick responsiveness towards the customer demands. 2. it must avoid choking up the supply chain. 3. it should ensure an optimal supply chain path. 4. it must satisfy the basic needs of transportation, power and water supply. 5. it should work in accordance with government policies and regulations. 3.11. green initiatives unfortunately, warehouses are the major contributors to the emission of greenhouse gases. also, a large portion of the human resource working in the supply chain is involved in various warehousing activities. that is why it becomes very essential to provide a sustainable working environment within the warehouses (bartolini et al., 2019). accordingly, implementation of green initiatives becomes an integral part of the warehousing operations. green initiatives basically are the activities that an organization undertakes in order to reduce its carbon footprint and to improve the work environment for the employees. some of the key green initiatives are 1. establish a green environment around the workplace. 2. reduce fossil fuel consumption from the various warehousing operations. 3. promote utilization of recyclable and biodegradable resources. 4. take account of the moral and social responsibilities. 5. develop a feel-good factor within the workplace. 3.12. use of right storage solution warehouses are large volumes of spaces that are used for storage purposes for a supply chain. if the internal space within the warehouse is not managed properly then it would lead to major setbacks for the whole supply chain. to ensure the optimal utilization of the available space, the use of the right storage solution is required (lee* & elsayed, 2005). some of the right storage solution practices are 1. use of pallet racks for storage. 2. flexible and situation-specific implementation of lifo and fifo arrangements. 3. use of mobile racks, conveyors and forklifts for the movement of products within the warehouse 4. ensuring proper management of all the stock-keeping units (skus). 3.13. order fulfilment order fulfilment is one of the parameters that give a direct insight into customer satisfaction for an organization. competition in the market is increasing day by day, firms that are able to fulfil the needs of their clients at a satisfactory rate can only sustain in the business for a longer period of time (reaidy et al., 2015). order fulfillment at a rapid rate gives the organizations a push towards expanding the number of customers for their products. some of the key features of order fulfilment are 1. orders were delivered on time according to the client’s requested date. 2. optimum utilization of the shipment facility. 3. proper handling of orders while picking, packaging and shipping operations. 4. lead time monitoring of all the orders. the significance of warehouse management in supply chain: an ism approach 9 3.14. automation when compared to a manual system, a well-automated warehousing facility requires fewer efforts, is more efficient and produces more consistent results. there are a lot of repetitive tasks that are involved in warehousing activities. now, with the help of advanced technologies, one can easily automate these tasks in order to get very high efficiency with very little effort (atieh et al., 2016). another reason to automate the warehouse is that manual activities might result in inadvertent human errors, lowering warehouse productivity. the basic steps towards automating a warehouse are the identification and reengineering of the processes which are best suited to get automated. followed by detailed modelling of the activity, then building the framework, integration of the software to it and finally implementing it in the warehouse. some of the key features of automation are 1. data collection of the products as soon as they arrive at the warehouse. 2. implementing robots in the warehouse to ease up the tasks for the employees. 3. implementation of automated guided vehicles (agvs) for the automated flow of products. 4. implementation of iot, ar, and rfid technologies for various activities (van geest et al., 2021). 3.15. forecasting forecasting is used for preparing the capacity of the warehouse and collecting the necessary information which would be utilized for determining the inventory capacity, equipment requirements and proper allotment of storage locations (hassan, 2002). without forecasting the organizations are not able to keep a track of the future demand trends which can lead them towards catastrophic failures. some of the key aspects of forecasting are 1. tracking the trends of future demands. 2. determining the order limit according to the warehouse capacity. 3. segregation of business demands according to the local and global market (suesut et al., 2004). 4. identification of the seasonal market along with their demand pattern 5. helps in stock planning activities. 4. the proposed ism methodology determination of the appropriate relationships between the selected factors and their impact on the whole project leads to better decision making and project planning. interpretive structural modelling (ism) is a tool (an algorithm) that creates solutions for complicated issues by establishing the contextual relationships between the elements, and when combined with the micmac analysis it also gives the classification and ranking of the components in terms of their driving force. the requirements for ism analysis are the identified elements according to the problem statement and defining a proper contextual relationship between them. these elements can either be qualitative or quantitative. the scope of the analysis doesn’t get bounded by the measurable entities only and because of this ism possesses a lot of flexibility than other methods which are only based on quantitative analysis (singh & kant, 2008). after selection of the factors, the maximum efforts are required verma et al./decis. mak. appl. manag. eng. (2022) 10 in determining the complex relationships between these factors and this task is carried out by a set of professionals who have the expertise in the designated sectors. in this study, 15 factors have been identified after thorough literature reviews. now the task is to implement ism methodology on these selected factors and further discuss the results obtained. the steps involved in ism methodology are, a. creation of structural self-interaction matrix (ssim). b. development of the reachability matrix. c. execution of the level partitions. d. classification of the selected factors. e. development of the ism model. 4.1. creation of structural self-interaction matrix (ssim) the most complex and demanding phase of ism is the structural self-interaction matrix. a variety of specialists from various sectors and academies were consulted during this step to construct the intricate contextual relationship between the specified elements (pfohl et al., 2011). the following four symbols are used to determine the direction of a relationship between two elements (i and j). v: factor i will help to achieve factor j a: factor j will help to achieve factor i x: factor i and j will help each other, and o: both factors i and j are unrelated for the sake of clarity, each factor has been allocated a notation. table 2 contains a list of all the notations. table 2. factors with their notations. sl. no. selected factors notations 1. optimal warehouse storage ows 2. inventory management im 3. order picking system ops 4. warehouse management system wms 5. transportation tr 6. manpower man 7. safety measures sm 8. overall running cost orc 9. warehouse productivity wpr 10. location loc 11. green initiatives gi 12. use of right storage solution urs 13. order fulfillment of 14. automation aut 15. forecasting fr the structural self-interaction matrix (ssim) is generated in table 3 based on the determined associations. the significance of warehouse management in supply chain: an ism approach 11 table 3. structural self-interaction matrix (ssim) facto rs o ws i m o ps w m s t r m a n s m o r c w p r l o c gi u r s o f a u t f r ows a v a a a a v v a a a v a a im v a a a a v v a o x v a a ops a a a a o v a a a v a a wms a a v v v a v v v x a tr o v v v o v v v v o man v v v o v v v v o sm v v a v v v o a orc v a a a v a a wpr a a a a a a loc v v v v o gi x v a a urs v a a of a a aut a fr 4.2. development of the reachability matrix the ssim has been turned into a binary matrix, known as the initial reachability matrix, at this phase (aich & tripathy, 2014). the reachability matrix is created by replacing v, a, x, and o with 1 and 0. the substitution of 1s and 0s is accomplished using the rules below: 1. if the ssim entry i j) is v, the reachability matrix i j) entry becomes 1 and the (j, i entry becomes 0. 2. if the ssim entry i j) is a, the reachability matrix i j) entry becomes 0 and the (j, i entry becomes 1. 3. if the ssim's i j) entry is x, then the reachability matrix's i j) and (j, i entries are both 1. 4. if the ssim's i j) entry is o, then the reachability matrix's i j) and (j, i entries become 0 as well. by applying the above-mentioned rules, the reachability matrix has been developed and shown in table 4. verma et al./decis. mak. appl. manag. eng. (2022) 12 t a b le 4 . r e a ch a b il it y m a tr ix ( m a tr ix m ) f a ct o rs o w s im o p s w m s t r m a n s m o r c w p r l o c g i u r s o f a u t f r d ri v in g p o w e r o w s 1 0 1 0 0 0 0 1 1 0 0 0 1 0 0 5 im 1 1 1 0 0 0 0 1 1 0 0 1 1 0 0 7 o p s 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 3 w m s 1 1 1 1 0 0 1 1 1 0 1 1 1 1 0 1 1 t r 1 1 1 1 1 0 1 1 1 0 1 1 1 1 0 1 2 m a n 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 2 s m 1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 9 o r c 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 3 w p r 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 l o c 1 1 1 1 0 0 1 1 1 1 1 1 1 1 0 1 2 g i 1 0 1 0 0 0 0 1 1 0 1 1 1 0 0 7 u r s 1 1 1 0 0 0 0 1 1 0 1 1 1 0 0 8 o f 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 2 a u t 1 1 1 1 0 0 0 1 1 0 1 1 1 1 0 1 0 f r 1 1 1 1 0 0 1 1 1 0 1 1 1 1 1 1 2 d e p e n d e n c e p o w e r 1 1 9 1 2 6 1 1 6 1 2 1 5 1 9 1 0 1 4 6 1 the significance of warehouse management in supply chain: an ism approach 13 table 4 also displays the driving and dependence powers of each element. the driving power of a selected factor refers to the number of variables it assists in achieving or making them fulfil the aim, whereas the dependence power of a selected factor refers to the number of elements it relies on to accomplish its goal. with the use of micmac analysis, these driving and dependence powers will be used to classify all of the components. 4.3. execution of the level partitions the level division is carried out after the production of the final reachability matrix (m). to begin, each factor's reachability set, antecedent set, and intersection set are determined. furthermore, the component with the same reachability set and intersection set gains the topmost level in the level partition (singhal et al., 2018). after the top-level factor is established, it is removed from the list of other factors, and the next level of factors is determined using the same procedures. after all of the components' levels have been determined, they are grouped in a digraph according to their levels. this digraph showcases the structure of the final ism model that is desired to be achieved. table 5 shows the final level partition of all the selected critical factors. table 5. final level partition factor notation reachability set antecedent set intersection set level 1. ows 1, 3, 8, 9, 13 1, 2, 4, 5, 6, 7, 10, 11, 12, 14, 15 1 4 2. im 1, 2, 3, 8, 9, 12, 13 2, 4, 5, 6, 7, 10, 12, 14, 15 2, 12 5 3. ops 3, 9, 13 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 14, 15 3 3 4. wms 1, 2, 3, 4, 7, 8, 9, 11, 12, 13, 14 4, 5, 6, 10, 14, 15 4, 14 6 5. tr 1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 14 5 5 7 6. man 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14 6 6 7 7. sm 1, 2, 3, 7, 8, 9, 11, 12, 13 4, 5, 6, 7, 10, 15, 7 6 8. orc 8, 9, 13 1, 2, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15 8 3 9. wpr 9 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 9 1 10. loc 1, 2, 3, 4, 7, 8, 9, 10, 11, 12, 13, 14 10 10 7 11. gi 1, 3, 8, 9, 11, 12, 13 4, 5, 6, 7, 10, 11, 12, 14, 15 11, 12 5 12. urs 1, 2, 3, 8, 9, 11, 12, 13 2, 4, 5, 6, 7, 10, 11, 12, 14, 15 2, 11, 12 5 verma et al./decis. mak. appl. manag. eng. (2022) 14 factor notation reachability set antecedent set intersection set level 13. of 9, 13 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15 13 2 14. aut 1, 2, 3, 4, 8, 9, 11, 12, 13, 14 4, 5, 6, 10, 14, 15 4, 14 6 15. fr 1, 2, 3, 4, 7, 8, 9, 11, 12, 13, 14, 15 15 15 7 4.4. classification of the selected factors in this phase, all the 15 selected factors will get classified into four groups namely, driver variable, dependent variable, autonomous variable and linkage variable. the classification is shown in figure 1. figure 1. classification of factors (driver-dependence graph) the first quadrant, ‘autonomous variable’ refers to the group of elements that nave the least driving as well as dependence power. autonomous variables usually don’t have a significant impact on the working domain. the second quadrant, ‘dependent variable’ refers to the group of elements that have a high dependence on other factors. the third quadrant, referred to as 'linkage variable,' refers to elements with a strong driving and dependence power. these are the most insecure and problematic characteristics in any working environment. the fourth quadrant, ‘driver variable’ refers to the elements that have a very high driving influence on the other factors (aich & tripathy, 2014). figure 1 shows that there are no factors that come under the autonomous variable category, implying that no insignificant factors were chosen for our investigation. in the dependent variable region, there are 7 factors among which ‘warehouse productivity’ has the highest dependence. in the linkage region, the only factor is there, that is ‘use of right storage solution’. this factor needs to be optimized in order to make the supply chain stable. in the driving variable region, there are 7 factors among which ‘transportation’, ‘manpower’, ‘location’ and ‘forecasting’ have the highest driving power. the significance of warehouse management in supply chain: an ism approach 15 4.5. development of the ism model the construction of the ism model is the final stage of the ism study. the final reachability matrix (table 4) and the level partition of the entire model can be used to create this model. if there is a link between factors i and j, arrows and points will be used to depict that relationship (singhal et al., 2018). the initial digraph is also known as the initially directed graph. the final ism model is created after deleting all transitivities among the variables, as shown in figure 2. 5. results and discussions the warehouse is well-known for playing a critical role in the entire supply chain. establishing an efficient warehousing facility for the supply chain's profitable output is one of the most important jobs. from the earlier studies of the analysis, 15 factors have been found out which influences the establishment and working of a warehouse. after the identification of the critical factors, ism analysis was carried out and the corresponding results were obtained. from the gathered outcomes, it can be deduced that, 1. there is no autonomous variables or factors. autonomous variables are usually those elements that have a very weak influence over the selected domain. in this case, there are no factors that have an insignificant effect on the warehousing activities or supply chain. also because of the less driving as well as less dependence power, the autonomous factors doesn't participate in the decision-making activities. 2. seven factors fall under the dependent variable region. these factors are namely inventory management, green initiatives, optimal warehouse storage, order picking system, overall running cost, order fulfilment and warehouse productivity. those elements with a high dependence power but low driving power are known as dependent variables. usually, these elements occupy the topmost levels of the ism model and are hence also known as the long-term objectives. these are basically the desired outcomes of the warehousing activities. warehouse productivity has the largest dependence power among all the other factors. 3. one factor falls under the linkage variable region. linkage variables are the most unstable entities within the selected domain. they need to be stabilized in order to establish an efficient and effective warehouse. from the study, use of right storage solution is the only factor that is unstable. if any change occurs in this factor, then it will cause an effect on the whole supply chain, as it has a high driving power as well as high dependence power. 4. seven factors fall under the driver variable region. these factors are namely transportation, manpower, location, forecasting, warehouse management system, automation and safety measures (wang et al., 2017). the elements with a high driving power and a low dependence power are known as driver variables. as the warehouse's key driving forces, these factors are accountable for accomplishing the necessary long-term objectives. these factors occupy the bottom-most levels of the ism model because of their high driving influence and hence also known as the short-term objectives. verma et al./decis. mak. appl. manag. eng. (2022) 16 transportation, manpower, location and forecasting are the largest driving factors among all the other factors. figure 2. classification of factors (driver-dependence graph) according to the ism analysis, optimal warehouse storage is a major bottleneck that has a direct influence on long-term objectives. according to studies, a lack of planning when determining actual utilizable space causes major setbacks in the warehouse's economic stability (rebelo et al., 2021). along with this, a stable workforce schedule increases the overall effectiveness of all warehouse activities (popović et al., 2021). the 5m (man, method, material, machine, and measurement) are the most common sources of errors, and they require continuous maintenance (hajej et al., 2021). following the implementation of all remedies, performance metrics such as throughput and turnover should be monitored (karim et al., 2020). it must have a long-term favourable trend, if not, then there must be some variables causing problems with the warehouse's operation. 6. conclusions the architects of the supply chain are well aware of the challenges that are encountered during the establishment and working of a warehouse within the system. to give a clear and more specific insight into the factors that influence the warehousing activities, this mcdm technique, i.e., ism analysis was carried out. 15 critical factors have been identified that have a significant impact on the supply chain. the most crucial task is to figure out how these factors interact in complex ways, and that is why this task was carried out by a set of experts after which the rest of the ism analysis was carried out. the outcomes show that there is a need to stabilize the instability of the use of right storage solution. factors like warehouse productivity the significance of warehouse management in supply chain: an ism approach 17 and order fulfilment are the long-term objectives of the warehouse whereas factors like transportation, manpower, location and forecasting are the primary drivers for the working of warehouses. this work will help the supply chain architects to properly plan an effective and efficient warehousing facility by following the ism model. in order to attain long-term goals, more emphasis should be placed on the bottom-level driver variables. references handfield, r. b., & nichols jr, e. l. 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(2020). revising the warehouse productivity measurement indicators: ratio-based benchmark. maritime business review. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 5, issue 2, 2022, pp. 362-371 issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0316102022p * corresponding author. e-mail addresses: dpamucar@gmail.com (d. pamucar), zizovic@gmail.com (m. žižović), dragandjurcic@gmail.com (d. đuričić) modification of the critic method using fuzzy rough numbers dragan pamucar1*, mališa žižović2 and dragan đuričić2 1 faculty of organizational sciences, university of belgrade, belgrade, serbia 2 university of kragujevac, kragujevac, serbia received: 12 june 2022; accepted: 18 september 2022; available online: 16 october 2022. original scientific paper abstract: this paper presents a new approach in the modification of the criteria importance through intercriteria correlation (critic) method using fuzzy rough numbers. in the modified critic method (critic-m), the normalization procedure of the home matrix elements was improved and the aggregation function for information processing in the normalized home matrix was improved. by introducing a new way of normalization, smaller deviations between normalized elements are obtained, which affects smaller values of standard deviation. thus, the relationships between the data in the initial decision matrix are presented in a more objective way. the introduction of a new way of aggregating the values of weights in the criticm method enables a more comprehensive view of information in the initial decision matrix, which leads to obtaining more objective values of weights. a new concept of fuzzy rough numbers was used to address uncertainties in the critic-m methodology. key words: mcdm, fuzzy sets, rough sets, fuzzy rough numbers, critic-m. 1. introduction determining criterion weights is one of the key problems that arises in multicriteria optimization models. in order to develop effective methods for determining the weight of the criteria, researchers around the world in recent years in the literature pay considerable attention to this problem. most authors suggest dividing the model for determining the weights of criteria into subjective and objective (zhu et al., 2015). subjective approaches reflect the subjective opinion and intuition of the decision maker. in this approach, the weight of the criteria are determined based on the preferences of the decision maker. traditional methods of determining weights of criteria include tradeoff method (keeney & raiffa, 1976), proportional (ratio) method, swing method (weber et al., 1988) and conjoint method (green & srinivasan, mailto:dpamucar@gmail.com mailto:dragandjurcic@gmail.com modification of the critic method using fuzzy rough numbers 363 1990), analytic hierarchy process (ahp) model (saaty, 1980), smart (the simple multi attribute rating technique) method (keeney & raiffa, 1976), macbeth (measuring attractiveness by categorical based evaluation technique) method (bana e costa & vansnick, 1994), direct point allocation method (poyhonen & hamalainen, 2001), ratio or direct significance weighting method (weber & borcherding, 1993), resistance to change method (rogers & bruen, 1998), wls (weighted lest square) method (graham, 1987) and fpp (the fuzzy preference programming method) method (mikhailov, 2000). recent subjective methods include multipurpose linear programming (costa & climaco, 1999), linear programming (mousseau et al., 2000), swara (step‐wise weight assessment ratio analysis) method (valipour et al., 2017), bwm (best worst method) (rezaei, 2015) and fucom (full consistency method) (pamučar et al., 20018). among the most known objective methods are the following: entropy method (shannon & weaver, 1947), critic method (criteria significance through intercriteria correlation) (srđević et al., 2003) and fanma method whose name was derived from the names of the authors of the method (žižović et al., 2020). the critic method is one of the most well-known and most frequently used objective methods. the critic method belongs to the group of correlation methods, which uses standard deviations of the standardized criterion values of variants to determine the contrast of criteria, as well as the correlation coefficients of all pairs of columns. in this study, certain limitations were identified when applying the classical critic method and a modification of the critic method (critic-m) in a fuzzy rough environment was proposed. the rest of the work is organized as follows. the following section shows the preliminary settings for fuzzy rough numbers. section 3 presents the mathematical foundations of the classical critic method. while section 4 shows a modification of the ctiric method in a fuzzy rough environment. the fifth section of the paper presents the application of the fuzzy rough critic-m method through an example from the literature. concluding remarks and directions for future research are given in section 6. 2. preliminaries on fuzzy rough numbers in the fuzzy rough concept, fuzzy theory was used to represent uncertainty in information, while rough theory was used to create flexible boundary intervals of fuzzy numbers. the use of hybrid fuzzy rough numbers eliminates the limitation of classic fuzzy type 2 numbers that have a predefined imprint of uncertainty. we assume that u universe contains all of the objects and let y be an arbitrary object from u. we assume there is a set of k classes which represent the preferences of the dm, * 1 2 ( , ,..., ) k g a a a , with the condition that they belong to a series which satisfies the condition 1 2 ,..., k a a a   . all objects are defined in the universe and connected with the preferences of the dm. each element ai ( 1 i k  ) represents a fuzzy number that is defined as 1 2 3 ( , , ) q q q q a a a a . since element ai from the class of objects *g is represented as fuzzy number 1 2 3( , , )q q q qa a a a , for each value 1qa , 2 qa and 3 q a we obtain one class of objects that is represented in the interval  1 1 1( ) ( ) , ( )q lq uqi a i a i a ,  2 2 2( ) ( ) , ( )q lq uqi a i a i a and  3 3 3( ) ( ) , ( )q lq uqi a i a i a where the condition is fulfilled that ( ) ( ) j lq j uq i a i a ( 1, 2,3; 1j q k   ), as well as the condition d. pamucar et al./decis. mak. appl. manag. eng. 5 (2) (2022) 362-371 364 * 1 2 3 ( ) , ( ) , ( ) q q q i a i a i a g . then ( )j lqi a and ( )j uqi a ( 1, 2,3; 1j q k   ) respectively represent the lower and upper border of the intervals of the q-th class of objects. if both limits of the classes of objects (upper and lower limits) respectively are compared so that * * * * * * 1 2 1 2 ( ) ( ) ,..., ( ) ; ( ) ( ) ,..., ( ) j l j l j ls j u j u j um i a i a i a i a i a i a      ( 1, 2,3j  ; 1 ,s m k  ), then for any of the classes of objects * *( ) j lq i a g and * *( ) j uq i a g ( 1, 2,3j  ; 1 q k  ) we can define the lower approximation * ( ) j lq i a using the following equations      * * *1 1( ) / ( ) ( ) ; 1lq lqapr i a y u g y i a q k     (1)      * * *2 2( ) / ( ) ( ) ; 1lq lqapr i a y u g y i a q k     (2)      * * *3 3( ) / ( ) ( ) ; 1lq lqapr i a y u g y i a q k     (3) and the upper approximation of * ( ) j uq i a using the following equations      * * *1 1( ) / ( ) ( ) ; 1uq uqapr i a y u g y i a q k     (4)      * * *2 2( ) / ( ) ( ) ; 1uq uqapr i a y u g y i a q k     (5)      * * *3 3( ) / ( ) ( ) ; 1uq uqapr i a y u g y i a q k     (6) both classes of objects (object classes * ( ) j lq i a and * ( ) j uq i a ) are defined by their lower limits  * ( )j lqlim i a ; 1, 2, 3j  , and upper limits   * ( ) j uq lim i a ; 1, 2, 3j  . the lower limits are defined by the following equations       1 * * * 1 1 ( ) 1 ( ) ( ) ( ) ; 1 lq lq l a lim i a g y y apr i a q k m     (7)       2 * * * 2 2 ( ) 1 ( ) ( ) ( ) ; 1 lq lq l a lim i a g y y apr i a q k m     (8)       3 * * * 3 3 ( ) 1 ( ) ( ) ( ) ; 1 lq lq l a lim i a g y y apr i a q k m     (9) where 1( )l a m , 2( )l a m and 3( )l a m respectively represent the number of objects included in the lower approximation of the classes of objects * 1 ( ) lq i a , * 2 ( ) lq i a and * 3 ( ) lq i a . the upper limits  * ( )j uqlim i a ; 1, 2, 3j  are defined by equations (10)-(12)       1 * * * 1 1 ( ) 1 ( ) ( ) ( ) ; 1 uq uq u a lim i a g y y apr i a q k m     (10)       2 * * * 2 2 ( ) 1 ( ) ( ) ( ) ; 1 uq uq u a lim i a g y y apr i a q k m     (11)       3 * * * 3 3 ( ) 1 ( ) ( ) ( ) ; 1 uq uq u a lim i a g y y apr i a q k m     (12) where 1( )u a m , 2( )u a m and 3( )u a m respectively represent the number of objects that are contained in the upper approximation of the classes of objects * 1 ( ) uq i a , * 2 ( ) uq i a and * 3 ( ) uq i a . as we see, each class of objects 1 ( ) q i a , 2 ( ) q i a and 3 ( ) q i a is defined by means of its own lower and upper limits, which make up the interval fuzzy-rough number a figure 1, defined as modification of the critic method using fuzzy rough numbers 365                   * * * * 1 2 2 3 1 * * * * 1 2 2 3 2 ( ) , ( ) , ( ) , ( ) ; ( ) , ( ) , ( ) , ( ) , ( ) ; ( ) l uq lq uq lq q l u q q u lq lq uq uq q lim i a lim i a lim i a lim i a w a a a a lim i a lim i a lim i a lim i a w a                (13) where l q a and u q a respectively represent the upper and lower trapezoidal fuzzyrough number which meets the condition that l u q q a a , while 1 ( ) l q w a and 2 ( ) u q w a respectively represent the maximum values of interval fuzzy-rough number a . 1 1 2 ( ) ( ) 1 l u q q w a w a   * 1( )lqlim i a   * 1 ( ) uq lim i a  * 2( )lqlim i a   * 2 ( ) uq lim i a  * 3( )lqlim i a   * 3 ( ) uq lim i a u q a l q a 0 figure 1. interval fuzzy-rough number a from figure 1 we observe that for interval-valued fuzzy-rough number a it is valid that 1 2 ( ) ( ) 1 l u q q w a w a  . on this basis we can write equation (13) in the following form:      1 1 2 2 3 3, , , , , , l u l u l u l u q q q q q q q q a a a a a a a a a        (14) where  * ( )ljq j lqa lim i a and     * ( ) ; 1, 2, 3; 1 u jq j uq a lim i a j q k    . if there is consensus among the decision makers on the assignment of specific values from the linguistic fuzzy scale then 1 1 l u q q a a , 2 2 l u q q a a and 3 3 l u q q a a . then interval-valued fuzzy-rough number a becomes fuzzy number a type-1. 3. critic method the critic method (criteria importance through intercriteria correlation) (žižović et al., 2020) is a correlation method. standard deviations of ranked criteria values of options in columns, as well as correlation coefficients of all paired columns are used to determine criteria contrasts. step 1: starting from an initial decision matrix, ij m n x      , we normalize the element of the initial decision matrix and form the normalized matrix ij m n x       . d. pamucar et al./decis. mak. appl. manag. eng. 5 (2) (2022) 362-371 366 1 2 11 12 11 2 21 22 2 1 2 n n n m m m mn m n c c c a a x a                          (15) the normalization of matrix elements ij m n x       is done by applying (16) and (17): a) for maximizing criteria: min max min , 1, 2,..., ; 1, 2,..., ; ij j ij j j i n j m           (16) b) for minimizing criteria: max max min , 1, 2,..., ; 1, 2,..., ; j ij ij j j i n j m           (17) where    max min1 2 1 2max , ,..., ; min , ,...,j j j mj j j j mj jj          . upon normalizing criteria of the initial decision matrix, all elements ij  are reduced to interval values [0, 1], so it can be said that all criteria have the same metrics. step 2: for criterion j c  1, 2,...,j n we define the standard deviation j , that represents the measure of deviation of values of alternatives for the given criterion of average value. standard deviation of a given criterion is the measure considered in the further process of defining criteria weight coefficients. step 3: from the normalized matrix ij m n x       we separate the vector  1 2, ,. .., j j j mj    that contains the values of alternatives  1, 2,..,ia i m for the given criterion j c  1, 2,...,j n . after forming the vector  1 2, ,. .., j j j mj    , we construct the matrix jk n n l l      , that contains coefficients of linear correlation of vectors j  and k  . the quantity of data j w contained within criterion j is determined by combining previously listed measures j  and jk l as follows: 1 (1 ) n j j j j kj k w l        (18) based on the previous analysis we can conclude da a higher value j w means a larger quantity of data received from a given criterion, which in turn increases the relative significance of the given criterion for the given decision process. step 4: objective weights of criteria are reached by normalizing measures j w : 1 j j m k k w w w    (19) modification of the critic method using fuzzy rough numbers 367 4. fuzzy rough critic method the modification of the critic method presented in this section is based on three starting points: 1) modification of the initial decision matrix data normalization method, 2) modification of the expression for determining the final values of criterion weights, and 3) extension of the modified critic method using fuzzy rough numbers. in the following part, the modified fuzzy rough critic method algorithm is presented and testing is performed on an example from the literature. step 1. construct the basic fuzzy rough decision matrix (  ). we will assume that the evaluation of alternatives was performed by e experts using the fuzzy scale. also, we will assume that expert preferences are presented in the home matrix b b ij m n         where 1 b e  ; i=1,...,m; j=1,...,n; and  ( ) ( ) ( ), , b b l b m b u ij ij ij ij     represent linguistic variables from the fuzzy scale used by expert e. for each element ( )e l ij  , ( )e m ij  and ( )e u ij  from b b ij m n         we form matrices of the aggregated sequences of experts ( ) ( ) b l b l ij m n         , ( ) ( ) b m b l ij m n         and ( ) ( ) b u b l ij m n         . using expressions (1)-(12) sequence ( )e l ij  , ( )e m ij  and ( )e u ij  are transformed into fuzzy rough number ( ) ( ) ( ) ( ) ( ) ( ) , , , , , b b l b l b m b m b u b u ij ij ij ij ij ij ij                                 ; 1 b e  . for fusion fuzzy rough values b ij ( 1 b e  )the fuzzy rough weighted geometric bonferroni function was used. this is how the aggregated fuzzy rough matrix ij m n        is defined. step 2. the elements of the matrix ij m n        are normalized as follows: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , , , , ; , , , , , , l l m m u u ij ij ij ij ij ij u u u u u u j j j j j j ij u u u u u u j j j j j j u u m m l ij ij ij ij ij if j b                                                                                              ( ) ; l ij if j c                      (20) where ( ) 1 max( ) u u i ij i m        and ( ) 1 min( ) l l i ij i m        . step 3: construct a matrix of linear correlations. for each criterion j c from the normalized matrix n ij m n        , the vector  1 2, ,. .., j j j mj    is defined, and linear correlations of the vectors j  and k  are calculated. by summing the linear correlations by criteria, we obtain the measure of the conflict of criteria 1 (1 ) n j jk k l    . the amount of information jw contained in criterion j is determined by applying expression (21): 1 (1 ) n j j kj k w l    (21) korak 4: determination of weight coefficients of criteria. objective weights of criteria are obtained by applying expression (22): d. pamucar et al./decis. mak. appl. manag. eng. 5 (2) (2022) 362-371 368 1 1 1 j j j j n j j j j w w w                 (22) example: we will assume that the multi-criteria model considers the evaluation of three alternatives under five criteria. we will also assume five experts evaluated the alternatives using the fuzzy scale presented in table 1. table 1. fuzzy scale linguistic terms membership function absolutely low (al) (1, 1.5, 2.5) very low (vl) (1.5, 2.5, 3.5) low (l) (2.5, 3.5, 4.5) medium low (ml) (3.5, 4.5, 5.5) equal (e) (4.5, 5.5, 6.5) medium high (mh) (5.5, 6.5, 7.5) high (h) (6.5, 7.5, 8.5) extremly high (eh) (7.5, 8.5, 9.5) absolutely high (ah) (8.5, 9, 10) experts' assessments of alternatives are presented in table 2. table 2. expert evaluation of alternatives a1 a2 a3 c1 eh,eh,eh,ah,ah h,eh,h,mh,h vl,l,l,l,l c2 ah,ah,ah,ah,eh e,ml,ml,ml,e ah,ah,ah,h,ah c3 eh,ah,ah,eh,ah h,eh,h,eh,h eh,h,h,eh,ah c4 eh,ah,ah,h,ah h,h,h,h,eh mh,mh,e,mh,e c5 vl,vl,al,vl,vl e,e,ml,ml,ml al,al,vl,vl,al by applying expressions (1) (12) the expert estimates were transformed into fuzzy rough values, table 3. modification of the critic method using fuzzy rough numbers 369 table 3. fuzzy rough home matrix crit. a1 a2 c1 ([7.56,8.28],[8.53,8.89],[9.53,9.89]) ([5.96,6.75],[6.97,7.76],[7.97,8.76]) c2 ([7.97,8.50],[8.74,9.00],[9.74,10.0]) ([3.56,4.13],[4.56,5.13],[5.56,6.14]) c3 ([7.70,8.43],[8.60,8.97],[9.60,9.97]) ([6.56,7.14],[7.56,8.14],[8.56,9.14]) c4 ([7.16,8.43],[8.07,8.97],[9.07,9.97]) ([6.50,6.81],[7.5,7.81],[8.50,8.810]) c5 ([1.22,1.50],[1.93,2.50],[2.95,3.50]) ([3.56,4.13],[4.56,5.13],[5.56,6.14]) crit. a3 c1 ([1.93,2.44],[2.95,3.44],[3.96,4.45]) c2 ([7.41,8.39],[8.19,8.92],[9.20,9.92]) c3 ([6.70,7.64],[7.70,8.49],[8.70,9.49]) c4 ([4.70,5.33],[5.70,6.33],[6.70,7.33]) c5 ([1.03,1.31],[1.55,2.11],[2.56,3.12]) using the expression (20), the elements from table 3 were normalized. then, using the expressions (21) and (22), the matrices of linear correlations of fuzzy rough elements were defined and the final values of the weighting coefficients were determined as follows: 1 2 3 4 5 0.153; 0.380; 0.189; 0.118; 0.160. w w w w w      5. conclusion this research presents a modification of the critic method using fuzzy rough numbers. fuzzy rough numbers are applied because part of the uncertainty and subjectivity are neglected in the classic fuzzy and rough models. given the wellknown performance of fuzzy sets in representing uncertainties and confirmed advantages of rough numbers in subjectivity manipulation, a modification of the critic method based on information processing using hybrid fuzzy rough numbers is proposed. also, the application of the fuzzy rough critic method is shown in an example that considers the evaluation of three alternatives under five criteria. author contributions: conceptualization, d.p. and m.ž.; methodology, d.p., m.ž. and d.đ.; software, d.p., m.ž. and d.đ.; validation, d.p., m.ž. and d.đ.; formal analysis, d.p.; investigation, d.p., m.ž. and d.đ.; writing—original draft preparation, d.p. and m.ž.; writing—review and editing, d.p., m.ž. and d.đ.; visualization, d.p.; supervision, d.p. all authors have read and agreed to the published version of the manuscript. funding: this research received no external funding. conflicts of interest: the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. d. pamucar et 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(2020). objective methods for determining criteria weight coefficients: a modification of the critic method. decision making: applications in management and engineering, 3(2), 149-161. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0304052020b * corresponding author. e-mail address: bipradasbairagi79@gmail.com (b. bairagi) a fuzzy interval based multi-criteria homogeneous group decision making technique: an application to airports ranking problem bipradas bairagi*1 1 department of mechanical engineering, haldia institute of technology, india received: 26 march 2022; accepted: 5 may 2022; available online: 13 may 2022. original scientific paper abstract: this paper aims to develop and introduce a fuzzy interval based multi-criteria homogeneous group decision making technique (imgd) to make appropriate decision under fuzzy environment. in fuzzy multi-criteria group decision making process, a group of decision makers often considers several subjective criteria for ranking a set of alternatives. due to vague and imprecise information, decision makers generally utilize linguistic variables which are mandatorily converted into triangular or trapezoidal fuzzy numbers. the total process then becomes very complex and time consuming. the current investigation advocates fuzzy intervals instead of triangular or trapezoidal fuzzy numbers for simplification of the complex situation and ease of calculation. in this method, fuzzy intervals of performance ratings and weights assessed by homogeneous group decision makers under subjective criteria are converted into first mean fuzzy intervals then into normalized crisp numbers. the normalized crisp performance ratings and normalized crisp weights are combined together to determine initially individual contribution and then into aggregate contribution to each alternative for final ranking and selection of the alternative. the new model is demonstrated with an application to airports ranking and selection problem for better clarification and verification. the outcome of the proposed is validated with the results obtained by well-known existing mcdm techniques. the analysis shows that the proposed method is applicable, useful and effective for appropriate decision making under fuzzy mcdm environment. key words: airport selection, decision making under uncertainty, homogeneous group decision making, fmcdm. 1. introduction the ranking and selection procedure of airports in general involves multiple alternatives, multiple conflicting subjective criteria and a group of experts. therefore multi criteria decision making techniques are required to employ for finding the bairagi/decis. mak. appl. manag. eng. (2021) 2 ranking order of airports. the decision making procedures becomes hard while decision is to make under fuzzy environments considering many subjecting criteria. it becomes more complex while multiple decision makers’ perception are required to incorporate in the decision. an airport ranking problem may be different from the others by type and nature of criteria, involvement of decision makers and number of alternative. past researchers have suggested several techniques for ranking and selection of airports. a detailed literature survey on airport ranking and selection proposed and suggested by past researchers is carried out and presented in this section. zhao et al. (2019) chooses civil airport site considering bird economical preservation by expert base selection in dalian, china. hammad et al. (2017) proposed a model for multi-objective optimization. mixed integer linear programming model has been applied for solving a bi-level program. fu et al. (2016) investigated the relocation case of a china airport considering the perspective towards risk of bird strike. markov chain is applied to analyze the birds’ flying procedure and to frame new algorithm in estimating bird strikes with aircraft. merkisz-guranowska (2016) developed a method having multiple layers for solving the location selection problem on airport that involves three methods to allow significant progress than the already available approaches. the initial method extends the problem adding up criteria and constructing genetic algorithm. the second method applies theory of fuzzy set whereas the third method advocates the proposed min approach. yang et al. (2016) extended ease of access for indicator to airports by transportation on surface and airside. the relation between airport size and scale of aircraft network is determined by structural equation model. bo et al. (2014)obtained the advantage of the multiple phase layer fuzzy logic approach which has the ability to remove evaluation disenchantment due to nonsensicalness in the brainpower of human being while multidirectional changes occurs. liao and bao (2014) solved the dilemma of airport location selection as a madm problem and the perception of expert group were expressed with crisp decision matrix affecting the tfn to elucidate preference of the decision makers and built a numerical approach to evaluate, rank and select alternatives. yang et al. (2014) applied a pair of ranking techniques to assess and ranking airport locations. the primary technique (wlsm) was accepted by the decision makers to calculate the weight and change the linguistic terms into crisp numbers. the subsequent technique was topsis. it was used to compute closeness coefficients to find the ranking order of the alternatives. fuzzy electre-i and fuzzy topsis were used for ranking and selection of the airports (belbag et al., 2013). the authors considered multiple important criteria such as costs, climatic condition, environmental condition, geographical condition, potential demand, infrastructure, social effects, the extension possibility, and legal regulations and restrictions. carmona-benítez et al. (2013) solved the issues of airport location to maximize the total anticipated aircraft passenger requirement as the indispensable aspects by utilizing the wealth index for computing passengers’ requirements. huang b. et al. (2013) employed the gis method to rank and choose airport location by collecting the geographic information of the intricate airspace area and using programming of super-map to make the geographic catalog. the airspace construction of multifaceted airspace region was compared. subsequently, the locations of airport were ranked and selected. zhao and sun (2013) compared newfangled airport location selections by two different index methods. the authors measured the evaluating index relative weight and calculated total numerical differences of the schemes. a fuzzy interval based multi-criteria homogeneous group decision making technique… 3 postorino and praticò (2012) applied the multi-criteria decision making model in determination of the position of airports contained by a multi-airport organization. sur and majumder (2012) applied the entropy weighting method for evaluation and selection of airport location. construction related cost per individual was considered a criterion in the model. the gap analysis of the above literature survey clearly explores that though previous researchers have applied some existing mcdm approaches, still there exists absolute necessity of introducing new mcdm model for solving and making appropriate decision regarding airport ranking and selection under new criteria and specific environment. the objective of the current paper is to develop and introduce a novel mcdm model under fuzzy environment for making appropriate decision in industrial application and demonstrated by illustration suitable example on airport ranking and selection. the paper is presented by dividing it into some sections for better illustration. section 1 presents the short introduction and literature survey. section 2 presents the proposed mathematical algorithm which is the heart of the paper. section 3 covers the numerical example along with solution and discussions. section 4 furnishes some essential concluding remarks and scope for further research. 2. proposed algorithm let, there is a decision making problem involving multiple alternatives, multiple subjective criteria with vague information and a group of homogeneous decision making experts. for solving such a decision making problem under fmcdm the following algorithm is constructed and proposed. step 1: formation of decision making committee comprising of experts from different important sections of the organizations. the member of the decision making committee can be expressed as follows. 1 ... ...k pd d d d    (1) here, id denotes the i th decision maker or expert. whereas, ‘p’ is the number of decision makers. step 2: make a list of the available feasible alternatives. the set of listed alternatives are under consideration for performance assessment. the alternatives can be represented in the form of a row matrix as shown in the eq. (2). 1 ... ...i ma a a a   (2) here, ia denotes the i th alternative. m is the number of alternatives. step 3: identify the significant criteria for decision making regarding evaluation and selection of the alternatives. the set of decision criteria can be represented in the form of the transpose of a row matrix as shown in the eq. (3). 1 ... ... t j nc c c c    (3) here, jc denotes the j th criterion. step 4: decision matrix: formation of decision matrix involves alternatives, criteria, decision makers and performance ratings. each alternative is assessed with respect to each criterion as per the preference of the decision makers in terms of linguistic variables. if all criteria are subjective, then only linguistic variables are used by the decision makers for estimating the performance rating of the alternatives with bairagi/decis. mak. appl. manag. eng. (2021) 4 respect to criteria. the decision matrix is formed by the decision maker applying their knowledge, preference and perceptions. 1 1 1 1 11(1) 11( ) 1 (1) 1 ( ) 1 (1) 1 ( )1 1(1) 1( ) (1) ( ) (1) ( ) ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... i m p p p p i i p m m pl n n n p ni ni p nm nm p a a a d d d d d d x x x x x xm c c x x x x x x                  (4) here, ( )ji kx denotes the linguistic performance rating of i th alternative with respect jth criterion, assessed by kth decision maker. step 5: decision matrix in interval: linguistic terms of decision matrix are transformed into intervals. an interval is expressed by two values viz. lower and upper. it is required for quantification of the assessment of the alternatives with respect to criteria. the decision matrix in interval can be represented in following matrix form. 1 1 1 1 11(1) 11( ) 1 (1) 1 ( ) 1 (1) 1 ( )1 1(1) 1( ) (1) ( ) (1) ( ) ... ... ... ... ... ... ... ... ... ... ... ...m ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... i m p p p p i i p m m pin n n n p ni ni p nm nm p a a a d d d d d d r r r r r rc c r r r r r r                  (5) ( )ji kr denotes the fuzzy interval expressing the performance rating of i th alternative under jth criterion by kth decision maker. step 6 determine the geometric mean of performance rating using the following eq. (6). 1 1 ( ) ( ) ( )( ) ( )( ) 1 1 , , p pp p ij l ij l ji k l ji k u k k r r r r                                 (6) step 7: determine the mean crisp performance rating using the eq. (7a) ( ) ( )ji ij l ji lt r r  (7a) the mean crisp performance rating s of the alternative with respect to criteria are accommodated in a matrix as provided below. 1 1 11 1 1 1 1 ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... i m i m j j ji jmcd n t ni nm a a a c t t t c t t tm c t y t                 (7b) here jit is mean crisp performance rating of the i th alternative with respect to jth criterion. a fuzzy interval based multi-criteria homogeneous group decision making technique… 5 step 8: construct the weight matrix in linguistic variables. importance weights of the different criteria may vary from criteria to criteria, decision maker to decision maker and problem to problem. in the current problem each decision maker estimates impotence weight for each criterion based on own experience, knowledge and perception. varying degrees of linguistic variables are used for the purpose of measuring the importance weights of the criteria which are accommodated in the following matrix. 1 1 11 1 1 1 1 ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... k m k m j j jk jmlv n n nk nm d d d c y y y c y y yw c y y y                 (8) here, yjk is the linguistic weight of jth criterion provided by the kth decision maker. here, m is the number of decision makers and n is the number of criteria. step 9: conversion of linguistic weights into corresponding intervals. this conversion is absolutely necessary for quantification of assessment. the importance weights of the criteria in terms of interval can be represented in the following matrix. 1 11 1 11 1 1 ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... k p k p j j jk jpi n n n nk np d d d z z zc c z z zw c z z z                  (9)   ( ) ( ) , jk l jk ujk z z z denotes the importance weight of the jth criterion assigned by kth decision maker. step 10: calculate the average criteria weight in interval by calculating the arithmetic mean of the lower and upper values separately by using eq. (10).   ( ) ( )( ) ( ) 1 1 1 1 , , jk l jk u p p j l j u i i u u z z p p             (10) step 11: compute the crisp weight for each criterion using the eq. (11). ( ) ( )j j l j uv u u  (11) ( )j lu denotes the lower value of the interval and ( )j uu denotes the upper value of the interval. jv is the geometric mean of ( )j lu and ( )j uu . step 12: measure the normalized crisp weight using the following normalization eq.(12). 1 j j n j i v w v    (12) here, jw denotes the normalized crisp weight of the jth criterion. bairagi/decis. mak. appl. manag. eng. (2021) 6 step 13: determine individual contribution. this investigation suggests implementation of trigonometric functions for measuring the contribution of individual criterion towards the performance evaluation of the alternatives under consideration. individual contribution of each criterion to each alternative is computed by applying the eq. (13).  1 cos sinij j jis w t    (13)  is a modifier. if the modifier is less than unity it can be termed as reducer. if the modifier is greater than unity it can be termed as amplifier. the exact value depends upon the data of the associated problem and the decision of the decision makers. step 14: determine total contribution. it is the aggregate of the total individual contribution of all the criteria under consideration applying the eq. (14).   1 1 cos sin n i j ji j s w t      (14) arrange the alternative according to decreasing order of the total contribution of. select the best alternative with the highest total contribution. 3. illustrative example the proposed algorithm has been illustrated by a suitable decision making problem on airport selection. the problem is discussed by subdivided it into two subsections viz. problem definition, calculation and discussion as described below. 3.1 problem definition the proposed algorithm is demonstrated by illustrating a suitable example on airport selection considering subjective criteria though homogeneous group decision making. this example is partially cited from wang et al. (2007). in this example, a decision making committee is formed with four rational decision makers having necessary knowledge in the domain. the decision makers are denoted by d1, d2, d3 and d4. the members of the decision making committee unanimously decided to consider a set of 15 subjective criteria vz. c1: return to capital (operation profit), c2: cleanness and comfort at terminal, c3: trolley move toward travelers, c4: direction and signal, c5: aerodrome controlling system, c6: security, c7: check-in and check-out system and time, c8: take-off and loading time, c9: traffic connecting city, c10: crew courtesy, c11: airport scale, c12: parking lots, c13: noise pollution system, c14: navigation controlling system, and c15: aircraft safety control. three alternative airports are initially chosen for further evaluation. the airports are designated by a1, a2 and a3. the proposed multi-criteria decision making algorithm is applied for evaluation, ranking and selection of the airport under consideration. the solution procedure of the airport selection problem is illustrated through the demonstration of the developed and proposed paradigm in the following subsection. 3.2 calculation and discussions in the current decision making problem, there are three alternative airports, fifteen criteria and four decision makers. all criteria are subjective with imprecision, vagueness and ambiguity. hence linguistic variables are used by the decision makers a fuzzy interval based multi-criteria homogeneous group decision making technique… 7 for estimating the associated performance rating of the alternative airports. seven degrees of linguistic variables viz. very poor, poor, medium poor, fair, medium good, good and very good are used for estimation of performance ratings. for quantification of each linguistic variable specific fuzzy interval is used. the linguistic variables, abbreviations and corresponding fuzzy intervals for measuring performance rating are represented in table 1. in ranking and selection of alternative airports, various decision criteria are given varying importance weight by the experts based on their significance as per the decision makers’ experience, knowledge and perceptions. for the purpose of extracting this importance weights decision makers generally prefer linguistic terms. the present investigation advocates five degrees of linguistic terms viz. very low, low, medium, high and very high. the linguistic terms, abbreviation and associate fuzzy intervals are accommodated in table 2. table 2: linguistic variables, abbreviations and intervals for criteria weight linguistic variables abbreviations intervals very low vl (0, 0.3) low l (0, 0.5) medium m (0.3, 0.7) high h (0.5, 1) very high vh (0.7, 1) three alternative airports are assessed by four decision makers using the prescribed seven degrees of linguistic variables which are regarded as the performance ratings of the alternatives. it is seen that decision makers d1, d2, d3 and d4 estimate alternative a1 with mg, g, g, vg with respect to criterion c1. alternative a2 is assessed with vg, g, mg, mg. here, vg implies very good, g means good, mg implies medium good. all the other abbreviation bears the similar meanings as described earlier. the decision matrix containing performance rating in terms of linguistic variable is presented in table 3. table1. linguistic variable, abbreviation and interval for performance rating linguistic variables abbreviations intervals very poor vp [0,0.2] poor p [0.1, 0.3] medium poor mp [0.2,0.4] fair f [0.4, 0.6] medium good mg [0.6,0.8] good g [0.7, 0,9] very good vg [0.8, 1] bairagi/decis. mak. appl. manag. eng. (2021) 8 table 3. decision matrix in linguistic variables expressing performance ratings a1 a2 a3 ci d1 d2 d3 d4 d1 d2 d3 d4 d1 d2 d3 d4 c1 mg g g vg vg g mg mg mg f mg f c2 mg vg g mg g g vg g g vp g g c3 vg f f mg vg g mg g vg vp g g c4 vg g vg vg f mg mg mg mg mg g mg c5 g mg f g mg f f g f vg g mg c6 vg g vg vg mg vg g g g f mg g c7 f g mg g g mg vg mg vg mg vg g c8 mg vg mg g vg f vg g g g vg mg c9 vg g g vg mg g g vg vg g vg vg c10 g g g f g mg g g g vg g mg c11 g vg mg mg vg mg g mg vg mg g g c12 g vg g mg vg g vg g g g vg mg c13 f mg mg g f mg f mg g g vg vg c14 vg mg mg vg mg mg g vg f mg g mg c15 g vg f g mg f vg g f f f f the linguistic terms expressing performance ratings are converted into fuzzy interval as per prescribed conversion scale. each fuzzy interval has two value viz. lower value and upper value. application of fuzzy interval value is recommended for simplicity in calculation and having capability of conveying information. geometric mean of the performance rating is determined using the eq.(6). the fuzzy intervals of the alternative a1 with respect to criterion c1 assessed by the four decision makers d1, d2, d3 and d4 are [0.6, 0.8], [0.7, 0.9], [0.7, 0.9] and [0.8, 1] respectively. therefore, the geometric mean of the performance rating in fuzzy interval is calculated as 1/ 4(0.6 0.7 0.70.8) (0.4141, 0.5045)   . the other mean performance ratings in fuzzy intervals are similarly calculated and accommodated in table 4. table 4. mean performance ratings in fuzzy intervals alternative airports criteria a1 a2 a3 a4 c1 (0.4141, 0.5045) (0.6447, 0.8459) (0.6447, 0.8459) (0.4899, 0.6928) c2 (0.6701, 0.8239) (0.7238, 0.9240) (0.7238, 0.9240) (0.7238, 0.9240) c3 (0.4757, 0.6447) (0.6964, 0.8972) (0.6964, 0.8972) (0.7483, 0.9487) c4 (0.8181, 0.9740) (0.5422, 0.7445) (0.5422, 0.7445) (0.6236, 0.8239) c5 (0.6236, 0.7896) (0.5091, 0.6447) (0.5091, 0.6447) (0.6055, 0.8107) c6 (0.8181, 0.9740) (0.6964, 0.8426) (0.6964, 0.8426) (0.5856, 0.7896) c7 (0.6236, 0.7896) (0.6701, 0.8239) (0.6701, 0.8239) (0.7200, 0.5180) c8 (0.7135, 0.8712) (0.6506, 0.8107) (0.6506, 0.8107) (0.6964, 0.8972) c9 (0.7913, 0.9487) (0.6964, 0.8426) (0.6964, 0.8426) (0.7737, 0.974) c10 (0.6735, 0.8132) (0.6735, 0.8207) (0.6735, 0.8207) (0.6964, 0.8972) c11 (0.7200, 0.8712) (0.6701, 0.8181) (0.6701, 0.8181) (0.6964, 0.8972) c12 (0.7483, 0.8972) (0.7483, 0.8972) (0.7483, 0.8972) (0.6964, 0.8972) a fuzzy interval based multi-criteria homogeneous group decision making technique… 9 table 4. mean performance ratings in fuzzy intervals alternative airports criteria a1 a2 a3 a4 c13 (0.6000, 0.7667) (0.4899, 0.6260) (0.4899, 0.6260) (0.7483, 0.9487) c14 (0.6928, 0.8459) (0.6701, 0.8181) (0.6701, 0.8181) (0.5635, 0.7667) c15 (0.6701, 0.8349) (0.6055, 0.7667) (0.6055, 0.7667) (0.4000, 0.6000) mean performance rating in crisp numbers is calculated by using eq. (7a) in the manner 0.4141 0.5045 0.4571  for the alternative a1 under criterion c1. the remaining values are similarly calculated and have been put in table 5. table 5. mean performance rating in crisp numbers alternative airports ci a1 a2 a3 a4 c1 0.4571 0.7385 0.7385 0.5826 c2 0.7430 0.8178 0.8178 0.8178 c3 0.5538 0.7905 0.7905 0.8426 c4 0.8927 0.6353 0.6353 0.7168 c5 0.7017 0.5729 0.5729 0.7006 c6 0.8927 0.7660 0.7660 0.6800 c7 0.7017 0.7430 0.7430 0.6107 c8 0.7884 0.7263 0.7263 0.7905 c9 0.8664 0.7660 0.7660 0.8681 c10 0.7401 0.7435 0.7435 0.7905 c11 0.7920 0.7404 0.7404 0.7905 c12 0.8194 0.8194 0.8194 0.7905 c13 0.6783 0.5538 0.5538 0.8426 c14 0.7655 0.7404 0.7404 0.6573 c15 0.7480 0.6814 0.6814 0.4899 linguistic weights of criteria assessed by the decision makers using their own experience as well as knowledge ate presented in table 6. table 6. linguistic weights of criteria assessed by the decision makers ci d1 d2 d3 d4 c1 medium very high medium high c2 high high medium very high c3 medium medium high medium c4 low medium very high medium c5 very high very high very high very high c6 very high high very high very high c7 high very high medium high c8 medium high very high medium c9 medium medium high medium c10 low medium high very high bairagi/decis. mak. appl. manag. eng. (2021) 10 table 6. linguistic weights of criteria assessed by the decision makers ci d1 d2 d3 d4 c11 very high high very high medium c12 high high medium low c13 high medium high high c14 medium high medium high c15 high very high high very high the linguistic variables expressing weights of criteria are required to transform into corresponding fuzzy interval to initiate the process towards quantification of weights of criteria. the weights in fuzzy intervals are inserted in table 7. table 7. weights of criteria in fuzzy intervals ci d1 d2 d3 d4 c1 (0.3, 0.7) (0.7, 1) (0.3, 0.7) (0.3, 0.7) c2 (0.5, 1.0) (0.3, 0.7) (0.3, 0.7) (0.7, 1) c3 (0.3, 0.7) (0.3, 0.7) (0.3, 0.7) (0.3, 0.7) c4 (0.0 0.5) (0.3, 0.7) (0.7, 1) (0.3, 0.7) c5 (0.7, 1.0) (0.7, 1) (0.7, 1) (0.7, 1) c6 (0.7, 1.0) (0.3, 0.7) (0.7, 1) (0.7, 1) c7 (0.3, 0.7) (0.7, 1) (0.3, 0.7) (0.3, 0.7) c8 (0.3, 0.7) (0.3, 0.7) (0.7, 1) (0.3, 0.7) c9 (0.3, 0.7) (0.3, 0.7) (0.3, 0.7) (0.3, 0.7) c10 (0.0, 0.5) (0.3, 0.7) (0.3, 0.7) (0.7, 1) c11 (0.7, 1.0) (0.3, 0.7) (0.7, 1) (0.3, 0.7) c12 (0.3, 0.7) (0.3, 0.7) (0.3, 0.7) (0, 0.5) c13 (0.3, 0.7) (0.3, 0.7) (0.3, 0.7) (0.3, 0.7) c14 (0.3, 0.7) (0.3, 0.7) (0.3, 0.7) (0.3, 0.7) c15 (0.3, 0.7) (0.7, 1) (0.3, 0.7) (0.7, 1) mean (arithmetic) weights of criteria in fuzzy interval is calculated by using eq. (10) and the conversion of crisp weights is accomplished by using eq. (11). mean fuzzy weights and normalized crisp weight are presented in table 8. table 8. mean weights in interval and in crisp numbers. criteria weight in interval gm weights in crisp c1 (0.4, 0.775) 0.5568 0.0679 c2 (0.45, 0.85) 0.6185 0.0754 c3 (0.3, 0.7) 0.4583 0.0558 c4 (0.325, 0.725) 0.4854 0.0592 c5 (0.6, 0.775) 0.6819 0.0831 c6 (0.6, 0.925) 0.7450 0.0908 c7 (0.4, 0.775) 0.5568 0.0679 c8 (0.4, 0.775) 0.5568 0.0679 c9 (0.3, 0.7) 0.4583 0.0558 c10 (0.325, 0.725) 0.4854 0.0592 a fuzzy interval based multi-criteria homogeneous group decision making technique… 11 table 8. mean weights in interval and in crisp numbers. criteria weight in interval gm weights in crisp c11 (0.5, 0.85) 0.6519 0.0794 c12 (0.225, 0.65) 0.3824 0.0466 c13 (0.3, 0.7) 0.4583 0.0558 c14 (0.3, 0.7) 0.4583 0.0558 c15 (0.5, 0.85) 0.6519 0.0794 table 9. weighted individual contribution ci a1 a2 a3 a4 c1 0.1016 0.1549 0.1266 0.1266 c2 0.1921 0.2071 0.0000 0.0000 c3 0.0936 0.1108 0.0000 0.0000 c4 0.1362 0.1038 0.1149 0.1149 c5 0.2228 0.1871 0.2225 0.2225 c6 0.3207 0.2855 0.2590 0.2590 c7 0.1485 0.1557 0.1320 0.1320 c8 0.1632 0.1528 0.1635 0.1635 c9 0.1188 0.1081 0.1190 0.1190 c10 0.1180 0.1184 0.1243 0.1243 c11 0.2245 0.2128 0.2242 0.2242 c12 0.0793 0.0793 0.0772 0.0772 c13 0.0978 0.0820 0.1164 0.1164 c14 0.1080 0.1052 0.0952 0.0952 c15 0.2145 0.1987 0.1484 0.1484 normalized crisp weights and normalized fuzzy performance ratings are integrated together to compute contribution by individual criterion using eq. (13) and the calculated weighted individual contribution are depicted in table 9. aggregate performance score (aps) of the airports are determined from the algebraic summing up of the individual contributions for each alternative airport. aps for each alternative is presented in table 10. table 10. aggregate performance score (aps) of the airports a1 a2 a3 aps 2.3396 2.2621 1.9230 rank 1 2 3 the aps for the alternatives a1, a2 and a3 in decreasing order are 2.3396, 2.2621 and 1.9230 respectively. therefore the ranking orders of the airports a1, a2, a3 are 1, 2 and 3 respectively. a1 is selected as the best airport and a3 as the worst airport. the result is compared with the results obtained by two well-known and well established existing techniques viz. topsis and saw. pis, nis, closeness coefficients and ranks of the airports by topsis method are represented in table 11. bairagi/decis. mak. appl. manag. eng. (2021) 12 table 11. pis, nis, closeness coefficient and rank of the airports by topsis ci a1 a2 a3 pis nis c1 0.0310 0.0501 0.0395 0.0501 0.0310 c2 0.0560 0.0616 0.0000 0.0616 0.0000 c3 0.0360 0.0441 0.0000 0.0441 0.0000 c4 0.0528 0.0376 0.0424 0.0528 0.0376 c5 0.0583 0.0476 0.0582 0.0583 0.0476 c6 0.0810 0.0695 0.0617 0.0810 0.0617 c7 0.0476 0.0504 0.0414 0.0504 0.0414 c8 0.0535 0.0493 0.0536 0.0536 0.0493 c9 0.0484 0.0428 0.0485 0.0485 0.0428 c10 0.0438 0.0440 0.0468 0.0468 0.0438 c11 0.0629 0.0588 0.0628 0.0629 0.0588 c12 0.0382 0.0382 0.0368 0.0382 0.0368 c13 0.0379 0.0309 0.0471 0.0471 0.0309 c14 0.0428 0.0413 0.0367 0.0428 0.0367 c15 0.0594 0.0541 0.0389 0.0594 0.0389 s+ 0.0238 0.0411 0.1585 ---- s0.155 0.131813 0.021726 ---- cc 0.867 0.762292 0.12056 ---- rank 1 2 3 ---- the same problem is solved by saw method. calculated composite scores and ranking orders of the alternative airports obtained by applied saw method are decorated in table 12. it is seen that the composite scores of the airports a1, a2, a3 are 0.750, 0.720 and 0.615 reactively. since, the higher composite score is better, airport a1 is ranked 1, a2 is ranked 2 and a3 is ranked is 3. table 12. composite score and ranking by saw method airports a1 a2 a3 composite score 0.750 0.720 0.615 rank 1 2 3 table 13. comparison of ranking order methods a1 a2 a3 rank by proposed method 1 2 3 rank by topsis method 1 2 3 ranking by saw method 1 2 3 comparison of ranking orders obtained by the proposed method with topsis and saw are shown in table 13. aggregate performance score of the airports are graphically represented in figure 1. for better visibility and demonstration. a fuzzy interval based multi-criteria homogeneous group decision making technique… 13 figure 1. aggregate performance score of the airports the ranking orders of the airports is depicted in figure 2. it is observed that airport a1is ranked 1, airport a2 is ranked 3 and airport a3 is ranked 3. therefore the preferene order is a1>a2>a3. figure 2. ranking orders of the airports 4. conclusion this research work aims to develop and implement a new framework for evaluating, ranking and selecting the best airports considering multiple conflicting criteria incorporating group homogeneous decision makers’ experience, opinion, knowledge and perception. the proposed method has been demonstrated through the illustration of a airport selection problem containing three feasible airports, fifteen subjective criteria and four rational decision makers. the result clearly indicates the best airport ensuring the better applicability of the method. the same problem on airport selection is also solved and the result is compared with that of the proposed approach. it is found that the result obtained by the proposed method completely matches with those of the existing approaches. the proposed interval based multi-criteria homogeneous group decision making technique (imgd) can also bairagi/decis. mak. appl. manag. eng. (2021) 14 be applied for solving similar decision making problem under fmcdm. the approach may be useful fmcdm tool for individual as well as managerial decision makers. heterogeneous group decision making by considering interdependent multiple conflicting criteria may be the direction of future research. author contributions: conceptualization, b.b.; methodology, b.b.; validation, b.b.; formal analysis, b.b.; investigation, b.b.; resources, b.b.; writing—original draft preparation, b.b.; writing—review and editing, b.b.; visualization, b.b.; supervision, b.b.; the author has read and agreed to the published version of the manuscript. funding: this research received no external funding. conflicts of interest: the author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. references bao, f. 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(2012). an application of the multi-criteria decision ‐ making analysis to a regional multi-airport system. research in transportation business & management (rtbm), 4, 44–52. a fuzzy interval based multi-criteria homogeneous group decision making technique… 15 sur, k.k., & majumder, s.k. (2012). construction of a new airport in a developing country, using entropy optimization method to the model. icsrs publication, 8(1), 29–34. yang, c., wu, t., & liao, y. (2014). evaluation for the location selection of airport based on wlsmtopsis method. applied mechanics and materials, 1823–1827. yang, z., yu, s., & notteboom, t. (2016). airport location in multiple airport regions (mars): the role of land and airside accessibility. journal of transport geography jtrg, 52, 98–110. wang, y.j., & lee, h.s. (2007). generalizing topsis for fuzzy multiple-criteria group decision-making. computers and mathematics with applications, 53, 1762–1772. zhao, s., & sun, p. (2013). scheme comparison of new airport site selection based on lattice order decision making method in the integrated transportation system. international journal of online engineering, 9, 95–99. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 5, issue 1, 2022, pp. 1-26. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0326022022a * corresponding author. e-mail addresses: arsyad.andi-arniaty825@mail.kyutech.jp (a. a. arsyad), widayat.irawanwidi389@mail.kyutech.jp (i. w. widayat), mkoeppen@ieee.org (m. köppen). supporting farming smart documentation system by modular blockchain solutions andi arniaty arsyad1, irawan widi widayat1 and mario köppen1* 1 departement of computer science and systems engineering, graduate school of creative informatics, kyushu institute of technology, japan received: 6 january 2022; accepted: 25 february 2022; available online: 26 february 2022. original scientific paper abstract: for more than a decade, various farm-specific models have been developed by collaborating and integrating sensing technologies as a step toward successful data-farm documentation and effective decision-making. however, the stored and gathered data continues to rely on cloud infrastructure or centralized platform control, which is particularly vulnerable to threats such as data tampering, data distortion, confidentiality, and manipulation, which caused the farm product data difficult to trace to its provenance. the objective study in this paper proposes a farm transaction model by demonstrating a flow of farm transaction simulation implicated by modular block chain(mbc) sensing instrument with an array of sensors, controllers, networking hardware, computing equipment, and internal memory functions to enhance data integrity and security farm object. based on the proposed model, a proof-of-concept experimental system called encapsulating block mesh (ebm) integrates blockchain technology with the specific application case of cocoa production has been implemented. results have shown that farm objects represented by mbc take turn recording information on the process of generating, transacting, and consuming a farm product and encrypting it into a block was validated and linked in the ebm with the hash of transaction data that connected to each cocoa farm object in a simulation environment. the findings from this study are twofold: the approach has been shown to be feasible and effective, but also capable to be expanded to other stages of the food supply chain, such as manufacturing, supermarkets, food consumption (consumers), and recycling in real-world environments. key words: smart farming, blockchain, iot, smart documentation, virtual world. mailto:arsyad.andi-arniaty825@mail.kyutech.jp mailto:widayat.irawan-widi389@mail.kyutech.jp mailto:widayat.irawan-widi389@mail.kyutech.jp mailto:mkoeppen@ieee.org arsyad et al./decis. mak. appl. manag. eng. 5 (2) (2022) 1-26 2 1. introduction these days, a lot of development of tools and applications helping the farmer to handle the various aspect of farming implicated by sensing technology such as data processing, water management, field monitoring, soil condition monitoring, crop yield analysis, and disease management (sreekantha, d.k. and kavya, a.m., 2017). with these smart technologies, farmers can become strategic and efficient in their daily farm-related tasks and responsibilities. centralizes, manages and optimizes farm production activities and operations, automates the recording and storage of farm data, monitors and analyzes farm activities and consumption, and tracks business expenses and farm budgets (kutter, t et.al., 2011). however, complex food supply chains characterized by production, distribution, transportation, processing, retail, and food consumption are increasingly exposed to a wide range of risks (chammem, n et.al., 2018), including contamination, domino effects, resource depletion, difficulty in the following origination, and quality disputes, create this technology ineffective in terms to the security data stored which is automatically vulnerable to distortion, manipulation, and so on. simultaneously, farming as the vital component of the overall goal agri-food chain causes such complexities to be triggered, somehow guiding back to the production level. a few reported food safety risks headlines, such as horse dna (the telegraph, 2013) were detected in various food products worldwide, which took some time to trace back to where the horse meat came into contact with the other meat products. other reports address the effect of china's covid-19 crisis and coronavirus shutdowns affecting american farmers (legal insurrection, 2020). farmers across the country have a surplus of produce, milk, eggs and so on that they are dumping, letting rot, or plowing under due to drastically reduced demand, which has also caused global problems in other parts of michigan, forcing the shutdown of their restaurants, resulting in a domino effect on their profit (g.m.america, 2020). the supply chain is already jeopardized as a result of these challenges. this indicates that even if they are not a link in the pandemic chain, interaction has already occurred in this supply chain. in other words, it isn't easy to trace back any source of the origins by natural influences or different environments to any of the challenges described. another point of view is that, while modern technology provides enormous agricultural benefits, its adoption and utilization in rural areas are limited. this is due to a lack of education among farmers and the high cost of maintaining this technology. therefore, it is essential to establish a farm transaction model that incorporates a sensing instrument to face the challenge of tracking the origin of farm products by integrating unique design security systems with a proper encryption mechanism to ensure the tracking back of product provenance while maintaining data confidentiality within these tools. so that captured event data is traceable and distributed in a secure, comprehensive manner, making it nearly impossible for attackers to infiltrate all supply chain nodes. in this study, we propose a farm transaction model by demonstrating a flow of farm transaction simulation representing a 'versatile smart instrument' with an array of sensors, controllers, networking hardware, computing equipment, and internal memory functions to enhance data integrity and security farm object. each farm object is linked in a secure block system, ensuring that farming data is monitored, stored safely, integrated, and is difficult to manipulate due to the encryption provided by a hash value in each object. for this purpose, we specially design a unique blockchain concept in the form of blockchain modularity in which farm objects take turns recording information on the process of generating, transacting, and consuming a farm product into a block. the block contains a record of every transaction ever made and supporting farming smart documentation by modular blockchain solutions 3 provides a hash value of its contents, including the previous block's hash. each block is then encapsulated and linked to the previous farm transaction block and keeps the hash of that block. any modification to any block in that chain will break the chain later on by having an invalid hash value. based on the proposed model, a proof-of-concept experimental system called encapsulating block mesh (ebm) integrates blockchain technology with the specific application case of cocoa production has been implemented and validated on the simulation environment. subsequently, the block contents data are later transmitted to the central terminal as the corresponding entry of the events log, which monitors the overall activity of transaction events. the novelty of this study is developing and implementing sensing instruments in special farming transaction models with the integrity of the special security design using the blockchain concept. offer farm management for farmers to monitor their farms with a detailed level of security, high accuracy, and can be traced with legality valid information from each farm object in real-time. the block security system is interconnected so that it can be ensured that the farming data collected can be monitored, stored securely, integrated, and difficult to manipulate because it is encapsulated in encrypted blocks in each farm object. the remaining part of this paper is structured as follows. section 2 presents reviews of published literature. section 3, the system architecture, including a brief the blockchain transaction validation model, followed the proposed farm transaction model, the actual simulation operation of the proposed ebm in detail, as well as how the concept of blockchain is suitable in the context of our research. section 4 contains the result and discussion. finally, in section 5, the conclusions are presented. 2. related work and motivation for over a decade now, sensor and sensing technology has been integrated into the supply chain for the smart farming practices. citrus fruit production (lee & ehsani, 2015), uavs for vineyards (candiago et.al., 2015), and using multi-purpose satellite systems to enhance cotton cultivation (huang & thomson, 2015) are just a few examples of sensor deployments for specialty crops. in the instance of cow health (helwatkar et.al., 2014) discovered a number of prevalent disorders in dairy cattle that may be detected using non-invasive, low-cost sensor technology. there are more advanced sensor platforms available, such as camera systems that detect back position (viazzi et.al., 2014) and ingestible tablets for heart rate assessment (warren et.al., 2008). caja et al. (2016) and rutten et al. (2013) examined the literature in terms of the documented usage of sensors for managing the health of dairy herds in agri-food sectors such as dairy farms. sensors that monitor arbitrary features of a cow or aggregate sensor data to offer information such as estrus predominant. on sensor networks, specifically wireless sensor networks (wsns), have seen widespread use in agriculture (ojha et.al., 2015)(abbasi et.al., 2014) and the food business (wang et.al., 2015)(garcia et.al., 2009). application domains include crop management (juul et.al., 2015), phenotypic assessment (greenwood et.al., 2014), rustle prevention (nkwari et.al., 2014), and greenhouse management (srbinovska et.al., 2015) are only a few examples of application domains. in applications such as irrigation control, wireless sensor and actuator networks (wsans) are gaining traction (nikolidakis et.al., 2015) (chikankar et.al., 2015). moosense (sarangi et.al., 2014) is a wsn that uses both ground-based and animal-mounted sensors to control a variety of animal characteristics such as ambient environment factors and nutrition intake (customized food auger and fluid kiosk). gonzalez et.al. (2014) illustrated the arsyad et al./decis. mak. appl. manag. eng. 5 (2) (2022) 1-26 4 potential of a heterogeneous wsn in providing data in real-time to help in the analysis of animal behavior and allow effective herd management. each of these evaluations offers farmers a variety of farming models with integrated technology for targeted, effective decision making and the option to monitor their farms in real-time with an unparalleled degree of detail. however, none expressly address whether the obtained information is securely stored, distributed, and traceable, including models developed to simulate such circumstances. all indicate the effectiveness of sensor and sensing technology usability to detect and monitor the physical, chemical, or biological property quantities and characteristics of various farming products while disregarding the requirement for data privacy, confidentiality, and integrity. those few studies with model approach and technique used are outlined in table 1. motivated by the above, our basic idea here is to implement a farm-specific model, i.e., a farm transaction model based on "the bucket principle." since the applied concept gives a visual representation of a bucket-based transaction that occurs from interactions with other farm objects and following processes in the farm food life-cycle from field to consumer implicated by sensing instrument. we believe in an integrative approach that can mediate the actual constraints of farm operation information recorded such as security, durability, integrity, and traceability by improvising the utilization of a one-of-a-kind blockchain plan. 3. system architecture this section first introduced the proposed farm transaction model, which encompasses the key aspect implicated in the cycle farm transaction, such as physical assets and objects that we named 'the bucket principle,' complemented by a brief overview of blockchain transaction. a bucket-based transaction was first presented with the “satyr farm” farm simulation game operating in the opensimulator-driven simulation environment. the system model is then presented with the specific application case of cocoa production, followed by described encapsulation block mesh (ebm) concept and subsequently the farm simulation operation of the proposed ebm described in detail. 3.1 the bucket principle in other blockchain applications, the validity of a transaction is usually a rather straightforward task. consider bitcoin transactions as shown in figure 1. alice wants to transfer an amount of n bitcoins to bob, where she has a wallet of m bitcoins. so, the only condition here is that while other circumstances of the transaction do not matter: like the true identity of either alice or bob, their location, day of time the transaction took place or the weather. the story is different for such transactions in the farming and general food supply chain environment. we may point out a number of aspects that make such transactions appear different. assume a simplified model where a farmer brings the apple harvest from some trees to a storage site. table 1. summary of related works with the main objective of the model. authors main objective of the model technique used lee & ehsani (2015) defines sensing systems, including a yield mapping system that uses fruit recognition disease detection sensors that are carried by groundand aerial-based platforms in the citrus fruit production. nir and raman spectroscopy supporting farming smart documentation by modular blockchain solutions 5 authors main objective of the model technique used candiago et.al (2015) huang & thomson (2015) helwatkar et.al. (2014) viazzi et.al. (2014) warren et.al. (2008) caja et al. (2016) rutten et al. (2013) ojha et.al. (2015) abbasi et.al. (2014) wang et.al. (2015) garcia et.al. (2009) juul et.al. (2015) greenwood et.al. (2014) demonstrates high-resolution unmanned aerial vehicles (uav)-based remote sensing and photogrammetric techniques applied in the agriculture framework to collect multispectral images. discusses using multi-purpose satellite systems to enhance cotton cultivation, including growth monitoring, insect control, yield prediction. identifies specific diseases common in dairy animals and the development of the next generation of health monitoring systems which can be identified through non-invasive, low-cost sensor technology. evaluate a two-dimensional and three-dimensional camera system to measure dairy cows' back posture automatically. design a pill that can remain in an animal’s reticulum and electrocardiographic techniques to ascertain and automate heart rate determination. reviews the literature in terms of the documented usage of sensors for managing the health of dairy herds in agri-food sectors such as dairy farms that are expected to produce dramatic changes in traditional dairy farming systems. provides an overview of the published sensor systems for dairy health management, including techniques that measure something about the cow activity, interpretations that summarize changes in the sensor data (e.g., increase in movement), to produce information about the cow’s status (e.g., estrus), integration of information where sensor information is supplemented with other information (e.g., economic information) to produce advice (e.g., whether to inseminate a cow or not), and the farmer makes a decision, or the sensor system makes the decision autonomously. evaluate the network and node architectures of wsns, the associated factors, and classification according to different applications, including the various available wireless sensor nodes and the different communication techniques followed by these nodes. evaluate the need for wireless sensors in agriculture, wsn technology, and their applications in different aspects of agriculture and existing system frameworks in the agriculture domain. designs and implements reconfigurable, low data rate, cost-efficient, and low-power wsn nodes and developed a real-time monitoring system for perishable food supply chain management, including the environmental parameters the state of motion of perishable food. uav data and photogrammetric techniques remote sensing non-invasive sensor three-dimensional camera electrocardiographic pill sensor devices sensor systems wireless sensor networks (wsns) wireless sensor networks (wsns) wireless sensor networks (wsns) wsn and rfid wireless sensor networks (wsns) wireless sensor networks (wsns) arsyad et al./decis. mak. appl. manag. eng. 5 (2) (2022) 1-26 6 authors main objective of the model technique used nkwari et.al. (2014) srbinovska et.al. (2015) nikolidakis et.al. (2015) chikankar et.al. (2015) sarangi et.al. (2014) gonzalez et.al. (2014) evaluate the standards and the numerous applications that utilize wireless sensor technologies (wst) in the agriculture and food industry and classify them into appropriate categories. introduces a system comprised of a wsn and a user interface that presents the measurements to the user in an accessible way. the system helps farmers avoid losses and achieve a more consistent quality of crops by monitoring environmental variables such as temperature and humidity during long storage periods. discusses issues underlying the need for new and novel phenotyping methods and the establishment of the wsn and pasture intake research platforms to predict feed intake and feed efficiency of individual grazing animals. determines the effectiveness of using the continuous-time markov process to determine if a cow is being stolen or not and determine anomalies in behavior that could indicate the presence of the thieves. a wireless sensor node with gps was designed to sense the position and speed of a cow. presents the development of a wsn application for precision agriculture, which is deployed in a pepper vegetable greenhouse to achieve scientific cultivation and lower management costs from environmental monitoring, including the temperature, humidity, and illumination. presents an integrated architecture based on wsns, for automated irrigation management to achieve effective and prompt irrigation of parcels with excellent energy efficiency due to the utilization of a novel routing protocol echerp. describes an effective irrigation management system for the container-grown crops that effectively utilize water resources for agriculture and crop growth monitoring using gsm and zigbee technology. presents moosense, a wsn that uses ground-based and animal-mounted sensors to control various animal characteristics such as ambient environment factors and nutrition intake (customized food auger and fluid kiosk). presents potential of a heterogeneous wsn in providing data in real-time to help in the analysis of animal behavior and allow effective herd management with sufficient frequency to increase understanding of animal biology and improve productivity. wireless sensor node with a gps wireless sensor networks (wsns) wireless sensor and actuator networks (wsans) wireless sensor networks (wsns) wireless sensor networks (wsns) wireless sensor networks (wsns)  the transaction involves a physical object, subject to physical and subsequent features, weight, volume, freshness etc. supporting farming smart documentation by modular blockchain solutions 7  identity of the agent’s matter. for example, the agents need to be spatially close to the “wallets” (that appear to be physical assets as well, trees, plants, soil, container etc.).  the transactions are inevitably linked to other transactions, foregoing that transaction, accompanying it, or, with some delay “unlocking” it.  the transaction can be sensed and recorded, e.g., by a video camera or subject witness.  the transaction has quality means on their own. in our example, the apples have been carried by hand, or within a large container lowering apple quality by inflicting damages. in a similar way, the transaction includes aging.  generally, such transactions need energy to be maintained, due to the physical nature of its constituents. at the same time, it needs human cognition and intervention to become subject of documentation and recording.  a classical transaction e.g., of a bitcoin transfer can only be valid or invalid. farming transactions appear valid to some degree as the outcome of the transaction at the goal site can vary even having same starting point, while there can be unknown, lost, or manipulated circumstances at the origin.  the transactions expire in some sense. at one point in time, all products made from an apple have been eaten by a living being. there is no primary need to log transactions forever if there is nothing to do or to conclude from the information in a ledger anymore.  farming doesn't happen on a terminal, so the interfacing among physical "not connected" objects need to be organized. figure 1. classical alice-bob bitcoin transaction vs farm transaction. already by number, those aspects may put into question if blockchain is a suitable concept here at all, compared to the surveying of a number of virtual wallets. of course, it can be argued that this just increases the number of data that has to be stored within a block and nothing else. this can indeed account for a number of aspects, but not all. most at all, the mutual influence of transactions, and the locality aspect. arsyad et al./decis. mak. appl. manag. eng. 5 (2) (2022) 1-26 8 in that regard, and to overcome those problems, we propose a transaction model called “bucket principle” that reflects the transactions in a farming (and subsequent food supply chain) in a way that makes it accessible to the integration in a specially designed blockchain. the principle can be stated as: bucket principle: all transactions in farming and the subsequent food supply chain can be modeled by the means of transport of an item s from site a to site b within abstract bucket b. the bucket alone can maintain all documentation tasks while logging related events: creation or filling of a bucket, picking up or dropping of a bucket, replenishing, emptying, or consumption of a bucket's content. to reconsider above example the apple harvest will be filled into a basket, for example. now, this can be the proverbial wicker basket, and it is brought by hand to a nearby storage site, keeping it cool until the storage site is filled up to load a truck. many other such farm activities would refer to real-world buckets. but it is also an abstraction: pruning a tree, the farmer will have to bring tools for pruning to the tree. for watering, one might to have to bring a hose nearby, or dig a ditch to guide water flow it where the bucket then is the channel leading to the tree. the bucket principle has several implications. first of all, it can be seen as a “smart tool” operating in a communication environment like edge computing or an iot infrastructure, which can have sensors, controllers, networking hardware, computing facilities, and internal memory. we agree that the focus in “smart farming" so far was on the sensor equipment, and not much engineering efforts have been put into “tools iot” so far. but the argumentation so far gives a good hint that such developments are overdue, and obviously feasible. on the bottom line, even carrying items by hand can become an abstract bucket transaction by using a specially designed data glove. a bucket has content, that is at the same time a symbolic name and a physical entity. while the latter is obvious, the former can cause documentation problems. receiving site b might call the content of the bucket differently than the originating site a according to documentation and registration needs. it implies the need for a bucket (in conjunction with its filling) to have a unique id, thus it needs a digital identity. it also has to be added that the action commonly referred to as transport, e.g., by a cargo ship, isn't such a bucket here. that transport has to be modeled as a remote storage: there is a bucket that brought the content of a storage that became subject of being moved to another location. arriving there, buckets will be filled with content of the same storage, just at a different location. however, while not applying to transport in classical meaning, it extends to many other subsequent transactions of farm items: cooking, placing items in a supermarket shelf or market booth, consuming like eating, after all, yes. while here is not the place to consider the universality of the bucket principle, as we are primarily interested in its implications for the design of a monitoring blockchain. the bucket principle, up to our best knowledge, was initially introduced along the “satyr farm” farm simulation game (satyr farm, 2020) running in the opensimulator driven simulation environments (opensimulator, 2020) and that was introduced in the opensimulator driven hypergrid around 2018. the simulation provides a visual cue of a bucket-based transaction that is the result of interactions with farm objects. for example, as shown in figure 2, to water a tree the avatar mediating user control in the simulation interacts with a well that renders a 3d model of a water bucket and that starts following the avatar as it moves to the tree to water. there, the water is replenished to the tree, and the bucket itself dismissed. the whole satyr farm follows this method in all transactions. in subsequent developments, we could demonstrate supporting farming smart documentation by modular blockchain solutions 9 that the same principle allows extension of the farm to the various stages of the food supply chain, including factory, supermarket, and food consumption and recycling. (a) (b) (c) (d) (e) (f) figure 2. a bucket-based transaction in the used simulation environment. 3.2 encapsulating block mesh (ebm) the bucket principle explanation aforementioned gives the base for a smart farming documentation system that integrates blockchain technology. we will point out three major practical requirements:  concurrency: a farming transaction does not happen in isolation but under the concurrent ongoing of other transactions that are either a condition for the proper fulfilment of current transaction, or an influencing factor, up to issues of resource sharing and replenishment.  locality: there is no reasonable method to link farm items over long distances, especially among different, maybe even competing farms. the documentation should be done close to the related sites and only based on near field communication in a technical way.  sparsity: we can’t assume to have blockchain recording all over a farm. this is accompanied by the circumstance that several factors affecting trust and quality of farm products are not single-point events but spread over some area and time. hardware for recording and documenting can be damaged, stolen, or being tampered with. so, the means of validating a blockchain has to account for gaps. the degree of trust into the validation has to be related by the depth to which the events proceeding the current transaction can be safely traced back. taking those three requirements into account, we propose the following concept of “encapsulated block mesh” extending the cocoa production blockchain concept. 3.2.1. cocoa production transaction model we selected the cocoa production model, specifically locally cocoa processing, as cocoa processing has a number and unique stages and aspects, quite a straightforward chain, and a well-defined structure. the steps involved in primary cocoa processing are also the corresponding variables in the chain, including planting, harvesting, fermentation, drying, and bagging/storage (guda & gadhe, 2017) (saltini, et.al., 2013). arsyad et al./decis. mak. appl. manag. eng. 5 (2) (2022) 1-26 10 in the system model as shown in figure 3, we consider that each cocoa farm object, including well, tree, fermenter, drier, and bagging in the storage, has a built-in sensing device called mbc, where the buckets, as the result of finishing a processing stage, processes events such as event logging and documentation hence enabling any farm item to receive messages and links to other farm items. all data from the buckets are later transmitted to the central terminal/cloud storage as the corresponding entry of the events log, which monitors the overall activity of the buckets. figure 3. cocoa production transaction. 3.2.2. modular blockchain (mbc) as previously stated, a bucket-based transaction has various implications, such as a “smart tool” operating in a communication environment such as iot infrastructure and including sensors, controllers, networking hardware, and internal memory. it is referred to as mbc. in this scenario, mbc serves as a smart sensor integrated on a farm object to record, collect, and store essential farm product information which encapsulating it in a strong cryptographic proof for data authenticity and integrity with a block-generated contract within. figure 4. a block-generated contract within mbc. figure 4 depicts a contract produced by a block within mbc. initially, a stand-alone mbc was used, which was later integrated into the farm object (well). furthermore, the mbc-connected well will generate a bucket of water, which will carry a record log supporting farming smart documentation by modular blockchain solutions 11 of events, which will subsequently be transferred and consumed to the next farm object, i.e., a tree. there are two sorts of blocks created based on the events that occur.  a bucket of water generated by a well has information stored in blocks. as described in algorithm 1, the previous block, hash 1, is the information generated in each transaction. an entry "nope" indicates that no specific information is stored here for the case of produce events. the block event was unrelated to consuming a bucket, referred by hash 2, and hash 3 is the information generated each time the farm object produces a bucket. next, create a new block, and it is classified as a produced event.  in the delivery of the bucket of water, the bucket will carry the amount of water and confirmation of the content transfer. as described in algorithm 2, the hash of the very first block is generated in each transaction as hash 1. hash 2 is content transfer originating from the bucket of produce event, and hash 3 is information generated each time the farm object consumes a bucket. the farm object, particularly the tree, will absorb the water, triggering mbc to form a new block. it is categorized as a consumed event. next, the pseudo-codes of blocks that are generated based on the events that transpired. algorithm 1 mbc produce event 1: function mbcproduces(block): 2: get last previous block, blast = block(last) 3: hash1 = hash(blast) 4: hash2 = “nope” 5: hash3 = hash(eventinformation) 6: bnew = [hash1, hash2, hash3] 7: return bnew algorithm 2 mbc consume event 1: function mbcconsume(block): 2: get last previous block, blast = block(last) 3: hash1 = hash(blast) 4: hash2 = hash(mbcproduce()) 5: hash3 = hash(eventinformation) 6: bnew = [hash1, hash2, hash3] 7: return bnew since our model is primarily concerned with documenting all farm operations in nearby areas, we consider mbc compatible with the nfc tag and solely interact with the nfc. nfc serves as a gateway, allowing access to data stored in the cloud. the mbc mesh's purpose is to secure the event log recordings and does not require internet connectivity. hence, there is a requirement to employ nfc as a peer to transfer information and store data on tags to prevent mbc from being hacked. so, if the enduser wants to access sensor data from outside, it has to ask nfc to get the required data. arsyad et al./decis. mak. appl. manag. eng. 5 (2) (2022) 1-26 12 3.3 implementation principles to put the ebm into action, we simulate local cocoa processing activities, including the primary cocoa bean processing chain, especially harvesting, fermentation, drying, and storage. this activity process represents mbc which collects, stores, manages key product information of each farm product. the following illustrates the mbc are linked in the ebm in figure 5.  harvesting. this process starts with flowers and ends with cocoa beans growing in pods. the cocoa tree consumes water during its growing period and eventually generates cocoa pods as the produce event.  fermentation. the fermentation process should begin after pod shattering. the box fermentation collects cocoa beans per pod and finishes with the beans being equally fermented. this process is categorized as consuming and producing events.  drying. after fermentation, the fermented cocoa seeds must be dried, and the fermented seeds must be spread on trays exposed to sunlight. the drying plate receives a bucket of fermented cocoa, and the process ends with a bucket of dried cocoa. this process is classified as consuming and producing events.  storage. the dried beans are packed into sacks for storage in a warehouse. the storage is obtained in a bucket of dried cocoa and then transferred to various locations. the final stage of cocoa processing is categorized as a consume event. figure 5. block mesh. 3.4 farm simulation in this section, we have implemented the ebm for cocoa production on simulation platform as depicted in figure 6 followed by a detailed explanation. an open simulator environment was employed to illustrate and demonstrate the farm transaction by encapsulating block mesh (ebm). a cloud-hosted instance of the opensimulator server (version 0.9.1). this opensimulator provides an appropriate research environment by supporting several frameworks such as server-client architecture, grid architecture, avatar-based control, concurrency, and scripting support. it became feasible to develop an experimental framework for conducting simulations that can be evaluated, analyzed, and enhanced through multi-institutional collaboration inside the so-called hypergrid connecting the various server simulations globally (delp et.al., 2007). experimental environment cocoa processing involves several vital aspects before it is transformed into a cocoa product. in this experiment, we confine the modeling of the cocoa production process to watering, harvesting, fermenting, drying, and bagging. the following scenarios are considered. supporting farming smart documentation by modular blockchain solutions 13  watering the cocoa plant. at this point, the well serves as a water supplier as one of the vital components to ensure healthy growth and cocoa yield. this stage, which represents the first activity of the cocoa plant growth process, is classified as a produce event in which the well generates water consumed by the cocoa tree.  harvesting. which begins with flowering and finishes with cocoa beans growing in pods. this stage is divided into two parts: consume events and produce events, in which the cocoa tree consumes water during its growth phase and ends up by generating cocoa pods.  fermentation. after pod breakage, fermentation should take place. this step is also divided into consume and produce events, in which the box fermentation receives cocoa beans per pod and finishes with the beans being evenly fermented.  drying. the fermented cocoa seeds must be dried after fermentation, and the fermented seeds must be spread on trays under sun exposure before they are shipped to the storage in the warehouse. this step is divided between consume and produce events. the drying plate receives a bucket of fermented cocoa, and the process culminates in producing a bucket of dried cocoa.  bagging/storage. the dried beans are now packed into barrels/sacks for storage in a warehouse. this step is classified as a consume event in which the storage is obtained in a bucket of dried cocoa and then transferred to various locations. ebm-specific parameters and settings will be described in table 2 as follow. table 2. application-specific parameters and settings of simulation. process of event description 1. produce event • code 96 • uuid’s bucket • hash of former block • hash of “nope” • hash of create event farm object generates a bucket. the code used by the farm object to send to mbc as a create event. initial information in the form of the identity of the farm bucket received by mbc. the hash of the very first block is generated in each transaction. nope signifies the block event was not related to consuming a bucket. the hash that is generated each time the farm object produces a bucket. 2. consume event • bucket of content transfer • code 97 • hash of former block • hash of origin block • hash of consume event farm object consumes a bucket. content transfer originating from the bucket of produce event. the code used by the farm object to send to mbc as a consume event. the hash of the very first block is generated in each transaction. the origin identifier of previous blocks. the hash that is generated each time the farm object consumes a bucket the following ebm for cocoa processing will be described in detail as shown in figure 6.  after connecting to mbc, each farm item, including well, cocoa tree, fermentation, drying, and storage, is assigned a unique channel number and block count. the arsyad et al./decis. mak. appl. manag. eng. 5 (2) (2022) 1-26 14 mbc of each farm object contains a block component structure that includes a former block hash, hash block origin, and event hash.  the well will generate blocks, which will subsequently be carried and absorbed by the cocoa tree. the previous hash block, produced from a bucket of water, is then used as the identifier or original hash block for the following event object.  the cocoa tree, which is the next consumption event, will receive several buckets of water and be consumed. each tree has transaction records for tree watering, one of which results from the cocoa harvesting stage, specifically cocoa pods. the block contains hash data from water activities, while the other block contains hash data from cocoa pods. all blocks have an identical block component format, consisting of a hash of the previous block of transactions, the origin of the block hash as an identifier of the origin of the previous block from well's mbc, and a new hash event for consuming events already consumed by the cocoa tree. in event fermentation, the hash of the previous block of the resulting cocoa pod becomes the identifier or block hash of origin for the next event object.  at the fermentation stage, as in the following bucket, the cocoa pod provides information on the transfer of event data content from the tree, transmitted to the fermentation stage. each fermentation box comprises a transaction records block in which the cacao pod is poured into the fermentation box while another block of fermented cocoa is generated. the former block hash of each generated fermented cocoa becomes a further identity for the object in the drying event.  similar to the process in the fermentation stage, in the drying stage, each drier plate comprises a record of transactions where a fermented cocoa bucket as a result of a fermentation process is poured into the drier while other blocks are generated dried cocoa. the former block hash of each generated dried cocoa becomes the identifier for the object in the final storage of the cocoa processing journey.  storage as the cocoa processing series' final generates has the same block component structure as the previous stages of cocoa processing. the transaction records in each block contain event log data from a bucket of dried cocoa that will be utilized as a secondary identification for the event's next step. supporting farming smart documentation by modular blockchain solutions 15 figure 6. ebm for cocoa production. 3.5 mbc event log in this section, we constructed gateway communication between mbc to the cloud storage, specifically transferring the farming transaction data recorded on mbc until it is stored in cloud storage. in this case, we are using google cloud. each mbc will record and keep every transaction that occurs. therefore, we construct a storage method that takes advantage of cloud storage, in this case, google sheets as storage media. every transaction performed by mbc will be broadcast to a certain communication channel specified in the script. to listen to every transaction that occurs on each mbc, we improve the communication gateway in the form of a script placed in a farming simulation. this script's function is to record and report every transaction from each mbc's farming simulation to the google cloud. some parameters such as key_id and http_requests function are needed to construct the communication between farming simulation and google spreadsheet. however, like other cloud storage media, this is a plain external log that is mostly unprotected. hence the event log protection derives from its connection to the reliably ebm. 4. result and discussion 4.1 result we now have a few records log of transactions from mbc’s. since our transaction model runs in parallel from one transaction to the next, thus necessitating authentication and validation of the farm product data as the block updates each time arsyad et al./decis. mak. appl. manag. eng. 5 (2) (2022) 1-26 16 and proceeds to the next block. we then investigate the certain mbc of the block on the farm object. 4.1.1. validation figure 7 illustrates the detailed block of mbc, with each block identified and tracked through the operational procedure of the proposed ebm. the following is an overview of the transaction validation procedure for cocoa processing.  a bucket of water is produced by the well that initiates the cocoa production process. a bucket of water with the bucket's information content distributes it to mbc. the contents are the bucket's uuid, the token, and the event. from here, the transaction data is hashed into blocks, and mbc generates a block and then proceeds to send back the new block to the water bucket, and it becomes the first block of the bucket. the validation process occurs if the transaction data is traced by tracing the identity of the hash. here, between the existing hash block origin and the previous block hash of mbc. if the hash values fully comply, the transaction is considered valid.  assume that the cocoa tree requires transaction data from a bucket of water at an early stage. the bucket of water then transports a certain amount of water and other content transfer information to the cocoa tree for consumption. the cocoa tree then provides information comprising hash origin and consumption events to mbc, following which transaction data is hashed into blocks until mbc creates new blocks. if both hash values match, the transaction is considered valid. in such situation, the validation procedure is carried out by tracing the identity of the hash, which matches the previous hash block of the bucket of water and the origin of the hash block in the cocoa tree.  regarding transaction validation during the fermentation stage, drying to storage, as aforementioned, it appears to follow the same workflow in terms of producing blocks in each transaction. produce and consume events are the two sorts of blocks that are created. validation is accomplished during transaction block tracing by finding and matching the identity of the hash between the previous hash block of the cocoa tree and the hash block origin at the fermentation level. if the hash values of the two transactions are matched, the transaction is considered valid. something similar will occur at the drying and storage levels. 4.1.2. investigate the mbc’s the investigate step is taken to monitor the block on the harvested cocoa, transporting it to the fermentation box to obtain fermented cocoa, which is subsequently transported to the drier plate to get dried cocoa. the distribution process finishes with warehouse storage.  investigation 1: inquire about the origin of the dried cocoa in the warehouse storage, in this case, randomly selected storage rack at block 2. (table a.16) mbc block 2: former block hash: be12382f98fd9240ed80fe1a322175a9 hash block at origin: d9e1e110f5cf22ed97f4a05ccd15a932 event hash: fe7ccaa75cebbf540c5c9517731d4f7e supporting farming smart documentation by modular blockchain solutions 17 as can be seen, the hash of the origin block is: d9e1e110f5cf22ed97f4a05ccd15a932. now double-check with the drier: after calculating the hash, the result of the retrieve hash block origin is d9e1e110f5cf22ed97f4a05ccd15a932 which corresponds to the second entry in block 2 of drier plate two. it signifies that the dried cocoa in the warehouse storage originated the dried plate two and allowed the uuid of the mbc of the drier plate to be verified.  investigation 2: after double-checking the origins of the dried cocoa, the next step is to inquire about the origins of the fermented cocoa in drier plate two block 1. (table a.13) mbc block 1: former block hash: fff7a973d18eac54302e41ce70530816 hash block at origin: 3616c124ebd63803f088017de4b85c55 event hash: f5551d1423cc272844822b938b020d74 as can be seen, the hash of the origin block is: 3616c124ebd63803f088017de4b85c55. double-check with the fermentation box: after calculating the hash, the result of the retrieve hash block origin is 3616c124ebd63803f088017de4b85c55 which corresponds to the first entry in the block 1 of drier plate two. it indicates that the fermented cocoa in the drier plate originated in fermentation box four block 2 and allowed the uuid of the mbc of the fermentation box to be verified.  investigation 3 inquire where the cocoa pod originated from in fermentation box four block 1. (table a.11) mbc block 1: former block hash: fff7a973d18eac54302e41ce70530816 hash block at origin: 017a6b68a8299277067841085d79b803 event hash: 31a73da207677efd5f1603b9ef1e7d39 as can be seen, the hash of the origin block is: 017a6b68a8299277067841085d79b803. double-check with the cocoa tree: after calculating the hash, the result of the retrieve hash block origin is 017a6b68a8299277067841085d79b803 same as the first entry in the block 1 of fermentation box four. it indicates that the cocoa pod in the fermentation box four arsyad et al./decis. mak. appl. manag. eng. 5 (2) (2022) 1-26 18 originated in harvest cocoa of tree 1 and allowed the uuid of the mbc of the cocoa tree to be verified.  investigation 4: since the well has 30 blocks while we got the last block traced from the tree 1, the tree has been watered in between. thus, we can read the "former block hash" directly. now we investigate the former block hash of the bucket of water that was watered the cocoa tree 1. (table a.2) mbc block 1: former block hash: fff7a973d18eac54302e41ce70530816 hash block at origin: f77bc30541130cb9d10e6afc4ebd9ccf event hash: 327e5bcd1e522ff90d3da725c9d6454c as can be seen, the hash of the origin block is: f77bc30541130cb9d10e6afc4ebd9ccf. double-check with the well: as a result of the calculating hash, hash block origin is f77bc30541130cb9d10e6afc4ebd9ccf, which is the same as the former block hash of the well. the uuid of the mbc of the well can be verified. to put it another way, we have securely identified one source of the water that watered the tree, finally leading to the examined cocoa beans in warehouse storage. 4.1.3. corresponding entry of event log the cocoa processing encapsulating block mesh is used as the foundation for capturing transaction data of cocoa processing in order to ensure data integration, safety, and traceability of the overall activity of transaction events. the cocoa processing encapsulating block mesh method's final output can subsequently be saved on the google cloud storage. figure 8 shows a screenshot of the output page of the corresponding event of the data log. supporting farming smart documentation by modular blockchain solutions 19 figure 7. detailed traced block of the mbc arsyad et al./decis. mak. appl. manag. eng. 5 (2) (2022) 1-26 20 figure 8. corresponding entry of event log 4.2 discussion our proposed ebm has been evaluated in terms of its secure data documentation. in particular, the protection of records data in the event log and the degree of trust of transaction validation have been analyzed by encapsulating the blocks for all farm entities involved in the process. our farm model works in parallel from one transaction to the next as a condition for effectively fulfilling ongoing circumstances and transactions. it demonstrates that it is valid to utilize the value of one entity to integrate another entity at a different goal site even if it has the same starting point. hence the sustainability, resilience of relatedness, and reliability can be proven of the whole product tracing while avoiding manipulation of data-related decision-making. considering the possibility that mbc may be damaged, stolen, or tampered with, it may be assumed that there is no guarantee that the blockchain can record throughout a farm. however, authenticating blockchain related to the depth to which the events preceding the current transaction can be reliably tracked back is of great concern here for the trustworthiness of this method. in other words, the hash value match and relatedness blocks in each event where an event that continues the current transaction occurs and wherever its area is a benchmark for the legitimacy of a transaction in the block. after simulating ebm, the interfacing among physical objects unconnected has been constructed and organized. it is evident that the documentation of farming transaction records is more structured, complete, secured, and protected, and each farm product is identified, integrated, and verified. it implies that the proposed blockchain approach ebm corresponds to the conditions of a farming transaction. 5. conclusion we have proposed the encapsulating block mesh (ebm) for cocoa production by integrating a unique designed blockchain and applying the principle of a bucket-based transaction implicated by mbc's sensing instruments. supporting farming smart documentation by modular blockchain solutions 21 the transaction documentation model is recorded at each level of cocoa processing and will be connected to events that occur on other farm objects chained together in blocks using a strong cryptographic method. first and foremost, the overall practical system requirements have been defined. then, we introduced a protocol that allows each cocoa farm stage to validate and authenticate product farm transaction data through chained hash values in the blocks. a remarkable feature of the proposed ebm is protecting the recorded data in the event log to eliminate data tampering and distortion, allowing data to be monitored safely and transparently. the claim procedure validation and mbc analysis demonstrated the validity of our ebm. from the finding of this study, we may refer to a realistic portrayal of a flow of farm transaction model in which farm objects take turns recording information on the process of producing, transacting, and consuming a farm product into a block. each block is then encapsulated, linked, and verified with the previous farm transaction block. this remarkable simulation model is appropriate for implementing agricultural system documentation in the real-world environment. the simulation approach is also expected to address farming documentation issues at several levels of the food supply chain, including factory, supermarket, and food consumption. farm management may use this simulation to manage or develop a documentation system employing a specifically built security of sensing instrument. there are some open problems which are attractive to be explored further. the first relates to the method used, which is still constrained in the simulation environment, and the data distribution system, which is still at the farm level. so, for future works related to ebm, it will be highly fascinating to examine how the same principle allows agriculture to be expanded to other stages of the food supply chain, such as manufacturing, supermarkets, food consumption (consumers), and recycling in real-world environments. author contributions: author contributions: conceptualization, a.a.a, i.w.w, and m.k.; methodology, m.k.; software, m.k. and i.w.w.; validation, a.a.a and m.k.; formal analysis, a.a.a and m.k.; investigation, a.a.a and m.k.; resources, a.a.a. and m.k.; data curation, a.a.a. and m.k.; writing—original draft preparation, a.a.a; writing-review and editing, m.k.; visualization, a.a.a.; supervision, m.k.; all authors have read and agreed to the published version of the manuscript. acknowledgments: the authors of this article would like to thank kyushu institute of technology for their financial and educational support. conflicts of interest: the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. arsyad et al./decis. mak. appl. manag. eng. 5 (2) (2022) 1-26 22 appendix table a.1: well block former block hash hash block origin event hash 1 fff7a973d18eac54302e41ce70530816 nope e57dd29e50f7b4d4676fcbd360a2a3f8 2 2b686a7c8601b901ab8dfd8a9d255ac8 nope 657d125c2693b2718350bd556b493ffb 3 f77bc30541130cb9d10e6afc4ebd9ccf nope 6823b54bfff5969f432f9787de991fab 4 e5e130b88e093499fb8f6aaa300400b6 nope 46be480f59381b4b16c571f4e0e8a64b 5 4c760f7a1b0ca3542fe5d2aba8ad017e nope 872604e1f8d8ffa837468202c27fbd53 6 127621e1abf18c792c6c1f566fba243f nope 529813506deec4e77493f8651d04ed13 7 b45be60147a7cc2daa34cef71942bd6d nope 9c9a63fb5699170175e9f19015e81a04 8 ab6b314704fca1ec8ec2eeba7946d225 nope 28886af72c1ffafb52e36dcf83da9aa3 9 f9f14141fd090b38ce86cee8cd10208e nope 23134c7962420edd64b9000c6237b20b 10 6336806085e3b1b398b9d795a78ac28a nope e025b3ea1c595c599e3e9542755fbf30 11 c304a81c184818778e12e2a465743572 nope 3c6316bb026e20f4b497ebcd186c6f1c 12 2a418e361721c0c35722dc3f44d97fd5 nope 18dac09d1ce818f3e80bb939c312c94e 13 21ca390e3a5260596028af93927dbf6f nope ceb55a4f9cf6f36a4d6533ab3af2f344 14 758e14ecd188acaa352ee73fb565dd33 nope 3caa7e7eedebbe619f7516c88232e08d 15 be1c438246d6471521a9fdf4b799c3ea nope 5463688b1f6f766cd0335b49c950b27f 16 7538340e74dcc686bcbe33c5b0a1e6ec nope 2b19ec927e2076d03b51806987453316 17 f49444fe520d3aa73a2a3849caf36232 nope 33a0c93dcd5ec8585384bec6c1b7ed01 18 c60b5ea0d579c1aeba7312393d61b8f9 nope 6f6c9b7fec611acf13ae1bed3f0e10a7 19 64b09997c23c5651a761a90d5e6a3273 nope ca848ccd11bc9ddb90a3a6b82979e595 20 88e875c13fc12794f273e657729238c4 nope 3f4ce43619d24e580c251b7389f9d8dc 21 0c632c4845b349fa993f5878a3575737 nope 2f0f870ff0f2826c6e8fe572db69f490 22 22cf3686af45b3e5a6093f040e178b01 nope 847968257d0adb1184188e5d286ebda3 23 998a8b241021d888f03f09383878688c nope 62a8e5a40acd2862a4340ba5596d0701 24 54840d45afffa4332dd35a080b5746d5 nope 5efbbcd965a092fc43d05644849c427e 25 8559095dbb33dd22fb15675d000023a1 nope 24b2d8beaced5aecebadbad1f393077a 26 98b010bd5c2641257a8f06c443e7d1cd nope e9937b640838f082970b67b05dc094e3 27 63d8ab9a53527ff864f7951d4c2ffc5e nope 66d91d3bba59deb5a021c18b80ca7923 28 330a7177f55f2cbe10e58bdc4720a855 nope 2005017c54b4092c4335cff78abcfe27 29 408519e6139b07849f2585b2c29b3421 nope 279afb0ecb2bf9223d94e23b60172d5b 30 2db1e2cc196f3b78d6d0bc6a8ba02678 nope 38000344f4d52878b1d4396cc92f7807 table a.2: tree 1 block former block hash hash block origin event hash 1 fff7a973d18eac54302e41ce70530816 f77bc30541130cb9d10e6afc4ebd9ccf e57dd29e50f7b4d4676fcbd360a2a3f8 2 dd2a481807268e16e510fd1dc587ce20 c304a81c184818778e12e2a465743572 657d125c2693b2718350bd556b493ffb 3 3e55384c6d35f3d864ba3713c8100a15 7538340e74dcc686bcbe33c5b0a1e6ec 6823b54bfff5969f432f9787de991fab 4 f2aca1105308dfa11a898473997699ee 998a8b241021d888f03f09383878688c 46be480f59381b4b16c571f4e0e8a64b 5 15c54c6970ede71b5959429a4b11c648 330a7177f55f2cbe10e58bdc4720a855 872604e1f8d8ffa837468202c27fbd53 6 ff682f6161312e1117ff16422dc46dca nope 529813506deec4e77493f8651d04ed13 table a.3: tree 2 block former block hash hash block origin event hash 1 fff7a973d18eac54302e41ce70530816 2b686a7c8601b901ab8dfd8a9d255ac8 deb41d87667b21468f04b5acb79fa085 2 31636d37495e8277c13c95148de7dc93 6336806085e3b1b398b9d795a78ac28a 3e78a147bb315dc706af478f6025fe1e 3 6af327fca1139f5f2073b8ad599e33fa 758e14ecd188acaa352ee73fb565dd33 90a171f92058f05a92d0c93ae41c5477 4 254828ec0945efa3fa6959554ad39d8c 54840d45afffa4332dd35a080b5746d5 9c62cfa31101717ddb20edf09e7e38e3 5 899e5b9f4034bac0502e53e13226c32f 63d8ab9a53527ff864f7951d4c2ffc5e fe25d67eb432f707874cf689bd7d4151 6 38a8e1d31f7df1715b54176d8af01189 nope ad6b11736be1b88503a7c1d5f230ba24 table a.4: tree 3 block former block hash hash block origin event hash 1 fff7a973d18eac54302e41ce70530816 e5e130b88e093499fb8f6aaa300400b6 91507f9d2feec89e4347f8ef4e10c439 2 ab638335c76fde17260bf6775b03f3fd 2a418e361721c0c35722dc3f44d97fd5 2054e6caeb0f20aada1fb8e8e2611e56 3 6da1e0f9b74cd56c7e21b00e803f2ce3 be1c438246d6471521a9fdf4b799c3ea 1b61ed771e63a3c04f9d9477403e74a4 4 b17a82d1231ff9a3be8330e82358e94a 8559095dbb33dd22fb15675d000023a1 c0e31da9fee418c18a739c57753d3f00 5 def5decbbdc59498e886390c6e1520be 98b010bd5c2641257a8f06c443e7d1cd f444853e0ac54ef077e5034b18b7069d 6 5a6b0916feb3b5edcd89da71516c047d nope 4f2367931d670fe0cfeaf11403882e25 table a.5: tree 4 block former block hash hash block origin event hash 1 fff7a973d18eac54302e41ce70530816 4c760f7a1b0ca3542fe5d2aba8ad017e a1f915bd02b4055a1ed6e9cccbad809f 2 7c33a136fee6f09a4bf4978bd9b5e2f2 21ca390e3a5260596028af93927dbf6f 166c09a6f0bfa0d8f897090a54d42dac 3 2f5522a8e7313839f87d12fc0d24df94 c60b5ea0d579c1aeba7312393d61b8f9 cc790b20420e7634da462be40bd3dc10 4 192fc2038e5db537bdba7aa44a3263cd 88e875c13fc12794f273e657729238c4 2430e43fb02616071bb7a4114a2c5f1b 5 9f5e77abc5d3a3a04385ee03cd429da5 8f25e89ca0b897e1e099af7c25270214 582cf172fa340fa4af335caf4ace54bc supporting farming smart documentation by modular blockchain solutions 23 6 33ecc21f27310dbc28013d8699013a38 nope 456b456115b5e7ef06f61e8d8f78f7f6 table a.6: tree 5 block former block hash hash block origin event hash 1 fff7a973d18eac54302e41ce70530816 127621e1abf18c792c6c1f566fba243f 285dda67e2a5a44c5a25a029f174ffc5 2 b1845125e31771b19cd3ffa99681d2c3 f9f14141fd090b38ce86cee8cd10208e cc8a3008f4b185c9ce004f723a12f360 3 efab4637d548cdfd70b8bd13bf2f9afc f49444fe520d3aa73a2a3849caf36232 e475c880dcc561805d764c9ed9ce5416 4 f2455f916dd1ff690259ce913b559e6c 0c632c4845b349fa993f5878a3575737 51e0e1133246d68e266668032c5cd724 5 f55a38001f7a682903c91eededcfa5c1 408519e6139b07849f2585b2c29b3421 f6967fb451bc928d2f3d7f99003410d5 6 4ccdb0cf56cd364cf7d36f078df75e64 nope 9b55bbaffbd3bdc1d69c17ccf3b8bfb4 table a.7: tree 6 block former block hash hash block origin event hash 1 fff7a973d18eac54302e41ce70530816 b45be60147a7cc2daa34cef71942bd6d 764a40463057bd5278365150e12d348f 2 1cc89f7df6b8ec667320359a78ab5d28 ab6b314704fca1ec8ec2eeba7946d225 9d2016c662ff5d31b8e428080835fd16 3 ede95a11abb3fe5d0971772f326b458a 64b09997c23c5651a761a90d5e6a3273 b9967b72fdd5ac39f84049831cdb57fb 4 74bbe8e39abd392d2e16c3b19cf3bf89 22cf3686af45b3e5a6093f040e178b01 3001b9c6ecf6e0def4a871591ba055d9 5 1c5b07bbd20c3fd8501cdbbee048a2ea 2db1e2cc196f3b78d6d0bc6a8ba02678 f0eb3fb96d8786882a7fa5319a351672 6 de4445c42470314ff6aaf4ff6eca5905 nope 9b28d36a31d6e0204912267338f6a986 table a.8: fermentation box 1 block former block hash hash block origin event hash 1 fff7a973d18eac54302e41ce70530816 8cd261c40ebb9cefaae17fa3795a1435 3c047473a8ec1b43e25589da561c4fa4 2 341f7c950afc7fb316f3372da690d597 nope b7907fbf5bbc22f0e4f0a82a8731bc95 3 85807726a37a81e239354c085dc58911 210f9fdcf44974c6b8118b856265b72c 3b9999e23c9a189c52c4b40f36b247d6 4 10691e511c294ab745d4d2d898b0bb9a nope 26d60839f9d28bb3889109bf60b4ac3b table a.9: fermentation box 2 block former block hash hash block origin event hash 1 fff7a973d18eac54302e41ce70530816 bae871823b83c1f96f8946db58f79e26 58a572742e9594aa74a87b595a3bda7f 2 a859101fbe8c23fcdc9e4e5e65b282f4 nope 857e15fd48c1a236737b24bd5ecfddca 3 2629b590e584815f46c1515c25aaea4a 3c6652bdb2402277f2624f0b6146e377 b1f500fc24300657d057a0038548cb08 4 f9c04d83be42e64782df33f196a37159 nope 2822d2b574bfbcc6a4eeddf7e994abff table a.10: fermentation box 3 block former block hash hash block origin event hash 1 fff7a973d18eac54302e41ce70530816 028ba03293f3a90bc88e25e356e766c8 604f99b6cb01c2149bfdc0331dd3c9ed 2 1d5a18141d76c51bc63a2d3433ed16f3 nope 7bf950c7903e0b5bbc3f4b455a609a07 table a.11: fermentation box 4 block former block hash hash block origin event hash 1 fff7a973d18eac54302e41ce70530816 017a6b68a8299277067841085d79b803 31a73da207677efd5f1603b9ef1e7d39 2 b466b91076cfc90f39f9e88905141f84 nope 3ab5ce6471ea7bcd39154d434054a138 table a.12: drier plate 1 block former block hash hash block origin event hash 1 fff7a973d18eac54302e41ce70530816 2629b590e584815f46c1515c25aaea4a cd829eeb4a05265ed7c4e5bae10bb79f 2 432fed641dab4733ba24bf81e9665023 nope e7d0dd071eca2c95df9235167ae9b3f2 3 8e37264acd37c4ad2f19a28c7ce32ef7 9ff389b611e36a118a135c876a1ec5b6 8fe6762da16a69f2bc4241e18ddec6ce 4 a8a4be40a5339d596c32172f0d9091c8 nope ea9a2ff0bb42cd6d3f6aad3e414902ba table a.13: drier plate 2 block former block hash hash block origin event hash 1 fff7a973d18eac54302e41ce70530816 3616c124ebd63803f088017de4b85c55 f5551d1423cc272844822b938b020d74 2 f20d259f742a16387f98a587344c94f6 nope 9efdbf2ce2df9d88ab457d9267d46371 table a.14: drier plate 3 block former block hash hash block origin event hash 1 fff7a973d18eac54302e41ce70530816 85807726a37a81e239354c085dc58911 5b824e46784dbaf44e5f16b9e130dbcd 2 08d7037538bf163c48e6a5b1b0234e1a nope 2cd56eeae9afeba963a9690b0f78d601 3 54e7154d726af7b521ce456085f93477 bd3c7c82e1d2324ffc4d120a1594d9d5 0a1ca99f12a25f270c747d22fe40f1c2 4 d65b6b98ee4b8297ed48fd018c1f50bc nope 3bbd9af9881033ac963452790c56472e table a.15: drier plate 4 block former block hash hash block origin event hash 1 fff7a973d18eac54302e41ce70530816 1c95778108c10289c60a5867ca48e37d 997c0c644449bfc37fa32e9afa9f2d5d 2 50e1ffae45052a9206023e6b0dacd2c3 nope 57e9a39524d427a2c56f4931ef789063 arsyad et al./decis. mak. appl. manag. eng. 5 (2) (2022) 1-26 24 table a.16: storage block former block hash hash block origin event hash 1 fff7a973d18eac54302e41ce70530816 8e37264acd37c4ad2f19a28c7ce32ef7 1b29eb80901752ba79e10a98501491c8 2 be12382f98fd9240ed80fe1a322175a9 d9e1e110f5cf22ed97f4a05ccd15a932 fe7ccaa75cebbf540c5c9517731d4f7e 3 ec2fb067490be6e3bb004d0528c4b1bf 54e7154d726af7b521ce456085f93477 9aeeb41a3d8d9e26f65271c3bfba6894 4 95301626934335f5e76aa2d10f715744 f935e4a5a03e75007688326cc2241264 5c380d674e523499c94d31610cd7831e 5 d73fabb43416561382cbe55a5db95d8f 603a189b5a8dd9c8400de137c19e3968 26ca7014e41a97bc894376c9ef63e75a 6 951957d29fbf48930f3a481e0bef1365 7035e45e5e31c51d1183049f6dadfd38 23609fd5eb455a4a2b67480b76787bc8 references abbasi, a. z., islam, n., shaikh, z.a. 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(2008). electrocardiographic pill for cattle heart rate determination, in 2008 30th annual international conference of the ieee engineering in medicine and biology society. ieee, 4852–4855. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). https://www.telegraph.co.uk/foodanddrink/9857136/horse-meat-scandal-timeline.html https://www.telegraph.co.uk/foodanddrink/9857136/horse-meat-scandal-timeline.html plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 5, issue 2, 2022, pp. 396-412 issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0331102022n * corresponding author. e-mail addresses: nassar.samia.rasmy@hallgato.uni-szie.hu (s. nassar), shohan_bd13@yahoo.com (m. b. hossain) naarne.toth.zsuzsanna.eva@uni-mate.hu (e. z. t. naárné), laszlo.vasa@ifat.hu (l. vasa) the mediating effect of organizational and coworkers support on employee retention in international non-governmental organizations in gaza strip samia nassar1,*, md billal hossain1, éva zsuzsanna tóth naárné1 and lászló vasa2 1 hungarian university of agriculture and life sciences (mate), godollo, hungary 2 széchenyi istván university, hungary received: 30 august 2022; accepted: 31 october 2022; available online: 31 october 2022. original scientific paper abstract: because of the enormous beneficial influence that employee retention has on work-related outcomes, it has been the focus of much research for a long time and especially in the field of international nongovernmental organizations (ingos). the importance of highlighting elements that may enhance the beneficial effect of workplace fun (wf) and work-life balance (wlb) in increasing employee retention (er). so this research examines the influence of worklife balance and workplace fun on employee retention, as well as the mediating effect of perceived organizational support (pos) and perceived coworker support (pcs) on employee retention. the obtained data were analyzed using a conceptual model. an online survey was used to collect the information. more than 358 surveys and analyses have been conducted utilizing the amos. in terms of employee retention, the study found a favorable correlation between workplace fun and work-life balance. the association between workplace fun, work-life balance, and employee retention was mediated by supervisor and coworker support. to assist firms to recognize the value of supervisor support in minimizing bad work outcomes for employees, these findings will be useful. key words: organizational support, coworker support, workplace fun, work-life balance and employee retention. mailto:nassar.samia.rasmy@hallgato.uni-szie.hu mailto:shohan_bd13@yahoo.com mailto:naarne.toth.zsuzsanna.eva@uni-mate.hu mailto:laszlo.vasa@ifat.hu the mediating effect of organizational and co-workers support on employee retention in… 397 1. introduction work-life balance is examined as a means of promoting engagement in employee development programs. in light of the research, it was predicted that work-life balance and involvement in employee development activities would have a positive correlation and that this link would be somewhat mediated by organizational and coworker support. work-life balance and work engagement were shown to be positively associated with the degree of interaction between leaders and employees, although this relationship would be more pronounced with greater levels of interchange according to zeffane and melhem (2017). it's common for a workplace fun to have a laid-back, supportive atmosphere. in order to boost employee morale, these organizations often offer a range of official and unofficial events. when employees know that they are appreciated by their bosses, colleagues, and the organization, they feel more positive about their job. employee turnover is one of the most difficult issues facing managers today, especially in international non-governmental organizations. having a high employee turnover costs money and causes problems for international non-governmental organizations (ingos). management faces a never-ending recruiting, selection, and training cycle that puts pressure on ingos finances and efficiency. high employee turnover and the desire to fill positions quickly lead many managers to settle for recruiting "warm bodies," which results in poor employees’ experience. it's no surprise that controlling international non-governmental organizations staff turnover is a major source of stress for managers. determining the best ways to keep entry-level staff is thus necessary according to utami et al. (2021). for the purpose of this research, it will focus on employees, who tend to work with lot of social interaction. the study's other goal is to prove that workplace fun and work-life balance contributes to employee turnover among ingos workers. this study will look at the impact of workplace fun and work-life balance on employee retention taking perceived organizational support and coworker support as mediators. the research used a relational perspective on turnover to argue that having fun at work may enhance the social experience and make it easier to build good working connections that help people feel a part of the company. accordingly, the research will also assess perceived organizational support and perceived coworker support as a mediator between workplace fun and work-life balance and their impact on employee retention in the workplace. common characteristics of an enjoyable workplace include a relaxed and encouraging vibe. organizations like this often host both formal and informal activities for their staff in an effort to increase morale. when workers feel valued by their superiors, peers, and the company as a whole, they are more likely to put up their best effort. the main aim behind this research is to focus on the most challenging problems confronting managers today in international non-governmental organizations since having a high turnover rate forced managers to conduct a continuous cycle of hiring, screening, and training places a strain on the resources and productivity of nongovernmental organizations (ingos) and their management. 2. impact of workplace fun on employee retention for millennials, this research focuses on the impact of workplace fun and other components of the working experience on people's job embeddedness, which has emerged as a crucial construct for promoting employee retention and lowering nassar et al./decis. mak. appl. manag. eng. 5 (2) (2022) 396-412 398 turnover. employees who are deeply entrenched in, integrated with, and bound to their workplaces have a high degree of embeddedness. meta-analysis by thanh and toan (2018) found that embeddedness was associated with both turnover intentions and actual turnover, even after adjusting for emotional commitment, work satisfaction, and employment alternatives. the lack of organizational commitment among millennials according to dahkoul (2018) calls for a look at how embeddedness might be improved. chouhan et al. (2016) asserts that there are three key ways to improve embeddedness: fit, linkages, and sacrifice. fit refers to a person's ability to function well in their current position and work environment. the best fit is achieved when a person's strengths, career ambitions, and values are in relation with the demands of a position and the culture of the organization. a link is a relationship that someone has with another individual. the more official and informal relationships one has, the more entangled one gets. finally, sacrifice is a reflection of the anticipated monetary and psychological rewards that turnover may result in. people get ingrained when quitting would result in considerable losses. the study's focus factors, such as fun, may make it easier for participants to fit in, form ties, and make sacrifices, which might have a significant impact on embeddedness among millennials according to dahkoul (2018). because social interaction is such an important aspect of workplace fun, it has the ability to strengthen ties with employees and create a sense of belonging. employees may build stronger bonds with one another and with their superiors if they participate in entertaining activities, network with their colleagues, and get management backing for such activities. these channels allow individuals to meet and create relationships that go beyond the limits of their jobs. in the workplace, it's not only about getting things done; it's also about forming relationships with coworkers and colleagues. an important point to take away from this research is the likelihood that millennials who work for international non-governmental organizations prioritize having fun. those that work in international nongovernmental organizations are more likely to be outgoing, thus they may put a bigger priority on having a good time, which is in line with previous studies demonstrating that extroverts place a higher value on good times (danaeifar et al, 2017). based on the above literature, the following hypothesis can be formulated: h1. workplace fun tends to impact employee retention 3. impact of work-life balance on employee retention studies show that workers prefer work-life balance, having the opportunity to have a personal life outside of work, and the chance to grow quickly in their careers (hair et al, 2017). employers' views on work-life balance as a predictor of embeddedness and pleasure in the workplace are examined in the current research, which draws on these ideas. these features of working experience have been linked to better workplace attitudes and conduct in many studies. the results of this research will help to solidify their connections to embeddedness. more crucially, this study will look at how embeddedness affects enjoyment, a concept that has received far less attention (li et al, 2021). in order for millennials to achieve both professional and personal objectives, it is believed that work-life balance is critical (karácsony et al, 2021). it has been customary to explore concerns of work-life balance in relation to lessening job expectations in order for people to better respond to their family responsibilities (mahmoud & grigoriou, 2017). millennials, on the other hand, may place a higher value on work-life balance since they want to pursue their personal the mediating effect of organizational and co-workers support on employee retention in… 399 and leisure interests in addition to their professional ones. millennials place a high priority on finding a work-life balance while still wanting to have a good time outside of work. work-life balance is said to encourage employee connection because it enables employees to strike a balance between work and leisure outside the office, something they would not be able to do elsewhere. employees who have a better work-life balance are more likely to remain entrenched, but those who quit their jobs will have to renounce and forsake such balance. employees who are overcommitted at work will have fewer opportunities to engage fully in their personal life (bite et al, 2020). besides looking at time-based conflict in general, this study will look at weekend work and the lack thereof in terms of work-life balance. weekends are when most people engage in personal and leisure activities, therefore those who work longer hours than the average sunday-through thursday schedule have less time to socialize. weekend work, according to lin and kellough (2019) qualitative study, adds to partner unhappiness and deteriorates family ties. evening and weekend employment might have a negative impact on personal life activities such as socializing and relaxing. based on the above literature, the following hypothesis can be formulated: h2: work-life balance tends to have a direct impact on employee retention 4. perceived organizational support mediates the relationship between workplace fun and employee retention the amount to which supervisors enable and encourage their workers to have fun at work is defined as management support for fun (kurdi et al, 2020). a manager's support for pleasure is based on an idea similar to kampkötter (2017) concept of personal liberties, but it focuses on the support offered by managers explicitly. when employees socialize, they are described as cordial, outgoing, and looking for one another's company. work tasks that are personally pleasurable, meaningful, and in line with a person's own interests are referred to as pleasant work duties similar to abela and debono (2019). while these dimensions are not all-inclusive, we feel they cover the most common aspects of having a good time at work. due to the fact that it is in line with millennials' ideals, having fun may help with embeddedness. fun may be more important to millennials since they are younger and appreciate youth more than older people do. millennials, who have grown up in a relatively carefree environment, may place a higher priority on having a good time than previous generations (asikhia et al, 2020). in the promotion of embeddedness, enjoyable activities, although vital, are said to be less significant than management support for fun, colleague socialization, and pleasant job duties. because they are less frequent and do not affect workers' day-today lives, fun activities may be considered less significant. workers' work environments are characterized by a variety of elements such as supervisors, colleagues, and job duties. these components of enjoyment have a stronger influence on the quality of one's work experiences since they are more frequent occurrences than enjoyable activities. workplace socializing may be even more vital than having fun since it is less fabricated and formal, and it may be more genuinely fulfilling than engaging in entertaining activities. abela and debono (2019) asserts the significance of unstructured, organic enjoyment above more structured, manufactured enjoyment. as previously mentioned, asikhia (2020) found that manager support for fun, coworker socializing, and fun job responsibilities were more important than fun activities when looking at nassar et al./decis. mak. appl. manag. eng. 5 (2) (2022) 396-412 400 applicant attraction and employee turnover. this supports the argument made by bakker and demerouti (2017) as well. based on the above literature, the following hypothesis can be formulated: h3: perceived organizational support mediates the relationship between workplace fun and employee retention 5. co-workers’ support mediates the relationship between workplace fun and employee retention high-quality interpersonal ties "enmesh people into a relational web, making them less sensitive to factors that may remove them from their organization," as claimed by hair et al. (2017). positive interpersonal interactions act as a buffer when presented with unpleasant aspects of the workplace, helping to keep workers loyal to the company. according to huynh and nguyen (2019), when people build deeper, more meaningful connections, they become more attached to and integrated into the organization. the proof that colleague assistance reduces turnover has yet to be found, which is unexpected. there has been mixed empirical evidence for the relational concept that colleague assistance helps reduce turnover. a sample of health care personnel was used in utami et al. (2021) recent study to assess the influence of four relationship factors on turnover. over a five-year period, their findings showed a link between turnover and network centrality and interpersonal civic behavior. coworker support's effect on employee retention has been the subject of conflicting studies in the past. millennials were up at a time of economic abundance, with parents who cared deeply about their children's well-being and who encouraged participation in extracurricular activities. however, millennials have been described as craving social interaction and a balance between work and pleasure, despite their willingness to put in the effort (kurdi et al, 2020). more broad and better quality connections may be fostered by fun's non-task characteristics, resulting in more interpersonal linkages. one of the primary factors that embedded workers, according to thanh and toan (2018), is constituent attachment a worker's connection to other members of the company. when it comes to encouraging embeddedness among millennials, building relationships with colleagues may be even more important, since younger people place a premium on developing connections in the workplace as they forge their adult identities. coworker support, we contend, may be important, but employee characteristics, the employment situation, the time period in which turnover is evaluated, and the dimensions of coworker support should all be taken into account. support from coworkers may be more critical in certain situations than in others. researchers on turnover suggest that context is important, although it has been largely ignored in past studies (huynh & nguyen, 2019). when it comes to coworker support, it is possible that some workers value it more than others. additionally, it's possible that in certain workplaces, colleague assistance is more critical than in others. because of the features of job experience and the workers that predominate in the international non-governmental organizations, we do not know whether or not coworker assistance has been explored with entry-level employees in that sector. another option is to see how employee assistance initiatives affect turnover over a short time period. coworker support and accompanying perceptions of support have a better chance of changing over a longer time period. based on the above literature, the following hypothesis can be formulated: the mediating effect of organizational and co-workers support on employee retention in… 401 h4: coworkers support mediates the relationship between workplace fun and employee retention 6. methodology sample the poll was conducted online throughout the months of september and october 2021 using google forms. in gaza strip, the study sought to find people who worked in international non-governmental organizations. the survey was done in arabic, which is widely spoken in gaza strip since it is the main language spoken in the country and then translated into the english language to code it using spss statistical tool and validate the research hypothesis. furthermore, this study included 358 participants (i.e., n=358). the following sections go into the different techniques used to collect data on the study's findings. the sample being used is convenience sampling where the sample is selected from the population because it is conveniently available to the researcher 7. instruments allows respondents to choose whether or not to answer the questionnaire and assures that their identities remain anonymous in order to uphold research ethics in order to protect confidentiality. questionnaires were used to determine if perceived organizational support, coworker support, workplace fun and work-life balance had an influence on employees' retention. the survey was broken down into eight sections. the first section was expanded after the approval form to include 5 demographic questions. this questionnaire asked about gender, age, marital status, and years of work experience, as well as job title and position. in the second part, perceived organizational support was evaluated using the scale of huynh and nguyen (2019). answers ranging from 1 (never) to 5 (always) their involvement activity were evaluated at a 5-point likert scale. in addition, the third part of the questionnaire related to perceived coworker support using the scale of hair et al. (2017). three questions with 5-point scale ranging from 1 (never) to 5 were addressed to participants (always). sample questions included: “i find my coworkers very helpful in performing my duties”, and “when performing my duties, i rely heavily on my coworkers”. the fourth section of the questionnaire related to work-life balance using the scale of danaeifar et al. (2017). eleven questions with 6-point scale ranging from 0 (never) to 6 (always) were addressed to participants. sample items included “when i get up in the morning, i feel like going to work.” and “i feel happy when i am working intensely”. the fifth section was constructed based on dahkoul (2018) likert scale in which the items varied from 1 (never) to 5 (always). the sample items included: “how often have you considered leaving your job?" and "to what extent is your current job satisfying your personal needs?" the sixth section measured the questions of workplace fun based on the scale of chouhan et al. (2016). the items were evaluated using a likert scale of 6 ranging from 0 (never) to 6(always). example items are "at my work, i feel bursting with energy" and" i find the work that i do full of meaning and purpose." nassar et al./decis. mak. appl. manag. eng. 5 (2) (2022) 396-412 402 8. reliability and validity analysis cronbach's alpha was used to gauge the resiliency of the sample variables and was also taken into account for validity. to see how correct the constructs are, this tool is often used in conjunction with likert scales. 9. descriptive statistics it can be noted that 202 respondents are females and 156 respondents are males as shown in table 1. table 1. gender frequency percent valid percent cumulative percent valid female 202 56.4 56.4 56.4 male 156 43.6 43.6 100.0 total 358 100.0 100.0 it can be noted that 2 respondents are divorced, 5 respondents are widowed,10 respondents are separated, 135 respondents are married and 206 respondents are single as shown in table 2. table 2. marital status frequency percent valid percent cumulative percent valid divorced 2 .6 .6 .6 widow 5 3.9 3.9 4.5 separated 10 5 20 24.5 married 135 38 38 62.5 single 206 53 53 100.0 total 358 100.0 100.0 it can be noted that 166 respondents have less than 5 years of experience and 132 respondents have between 5 and 10 years of experience and 52 respondents have between 11and 15-years’ experience, and 6 respondents have between 16 and 20 years of experience and 2 respondents have more than 20 years of experience as shown in table 3. table 3. years of experience frequency percent valid percent cumulative percent valid less than 5 years 166 46.4 46.4 99.4 5-10 years 132 36.9 36.9 53.1 11-15 years 52 14.5 14.5 14.5 16-20 years 6 1.7 1.7 16.2 more than 20 years 2 .6 .6 100.0 total 358 100.0 100.0 the mediating effect of organizational and co-workers support on employee retention in… 403 10. regression one: relationship between work-life balance, workplace fun and employee retention the regression analysis aims to test the relationship between the dependent and independent variables based on a margin error of 5%. if the significance level showed a margin error lower than 5% then h0 will be rejected and h1 will be accepted. in the table 4, workplace fun showed a beta = 0.211, and t = 4.405 which is the result of dividing the b over the standard error, and showed f-significant of 0.00 which is lower than 0.05. as for the variable work life balance it showed a beta = 0.524, t = 10.951 and f significant = 0.00 which is also lower than 0.05. this can lead us to validate the following hypothesis: h1: there is a direct relationship between workplace fun and employee retention h2: there is a direct relationship between work life balance and employee retention it can be noted that this model scored r = 66.5% which means that the addressed variables represent 66.5% of the variables which affect the dependent variables and the remaining 33.5% are the variables which are not mentioned in the model. as for the r2 it scored 44.2% which means that workplace fun and work-life balance tends to impact employee retention by 44.2% which is considered a moderate strength since the r2 falls between 25% and 50% from the above regression it can be noted that: for every 1% increase in workplace fun, the employee retention tends to increase by 21.1% for every 1% increase in work-life balance, the employee retention tends to increase by 52.4%. table 4. relationship between workplace fun, work-life balance and employee retention model r r square adjusted r square std. error of the estimate 1 .665a .442 .439 1.231 a. predictors: (constant), work life balance, workplace fun and employee retention’ model unstandardized coefficients standardized coefficients t sig. b std. error beta (constant) .793 .197 4.018 .000 1 workplace fun .243 .055 .211 4.405 .000 work-life balance .575 .052 .524 10.951 .000 a. dependent variable: employee retention 11. regression two: perceived organizational support mediates the relationship between workplace fun, work-life balance and employee retention the regression had been conducted to study the mediation of perceived organizational support on the relationship between work-place fun, work life balance nassar et al./decis. mak. appl. manag. eng. 5 (2) (2022) 396-412 404 and employee retention. referring to the above model, it can be noted that workplace fun scored a b = 0.151, and t = 3.128 and f (sig) = 0.002, and the work-life balance scored b = 0.403, t = 7.553 and f(sig) = 0.00, and at last the pos variable scored b = 0.242, t = 4.643 and f(sig) = 0.00 as shown in table 5. it can also be noted that the model scored r of 68.9% which is higher than the r scored in regression 1 (66.5%), and the r2 in model 2 is 47.4% which is also higher than the r2 in model 1. this can lead us to validate the following hypothesis: h3: perceived organizational support mediates the relationship between workplace fun, work life balance and employee retention. since the r2 had increased from 44.2% in model 1 to 47.4% in model 2, this means that pos tends to mediate the relationship between the independent variable by 3.2%. table 5. mediation effect of perceived organizational support on the relationship between work-place fun, work-life balance and employee retention model r r square adjusted r square std. error of the estimate 1 .689a .474 .470 1.197 a. predictors: (constant), workplace fun, work-life balance, perceived organizational support and employee retention model unstandardized coefficients standardized coefficients beta t sig. b std. error 1 (constant) .628 .195 3.213 .001 work-place fun .174 .056 .151 3.128 .002 work-life balance .442 .059 .403 7.553 .000 perceived organizational support .254 .055 .242 4.643 .000 a. dependent variable: employee retention 12. regression three: perceived co-worker support mediates the relationship between workplace fun, work-life balance and employee retention as for model 3 it scored r of 67.9% which is higher than model 1 (66.5%) and lower than model 2 (68.9%) and r2 of (46.1%) which is higher than model 1(44.2%) and lower than model 2 (47.4%) as shown in table 6. this can lead us to validate the following hypothesis: h4: perceived coworker support mediates the relationship between workplace fun, work life balance and employee retention. perceived coworker support mediates the relationship between workplace fun, work-life balance and employee retention by 1.9% which means that perceived organizational support tends to have a higher mediation effect on the relationship of workplace fun, work-life balance and employee retention. the mediating effect of organizational and co-workers support on employee retention in… 405 table 6. mediation effect of perceived coworker support on the relationship between workplace fun, work life balance and employee retention model r r square adjusted r square std. error of the estimate 1 .679a .461 .457 1.212 a. predictors: (constant), workplace fun, work-life balance, perceived coworker support and employee retention model unstandardized coefficients standardized coefficients t sig. b std. error beta 1 (constant) .659 .198 3.324 .001 workplace fun .166 .059 .144 2.830 .005 work-life balance .443 .064 .404 6.922 .000 perceived co-worker support .228 .065 .216 3.499 .001 a. dependent variable: employee retention the regression 3 shows that workplace fun scored b = 0.144, t = 2.830 and f(sig) = 0.005, as for work life balance scored b = 0.404, t = 6.922 and f(sig) = 0.000, and perceived coworker support scored b = 0.216, t = 3.499 and f(sig) = 0.001. 13. structure equation model analysis the structure equation model in figure 1 outlined the following variables and hypothesis will be listed: dependent variable: employee retention independent variables: workplace fun and work-life balance mediators: perceived organizational support and perceived coworker support as for the hypothesis: h1: there is a direct relationship between workplace fun and employee retention h2: there is a direct relationship between work-life balance and employee retention h3: perceived organizational support mediates the relationship between work-life balance, workplace fun and employee retention h4: perceived coworker support mediates the relationship between work-life balance, workplace fun and employee retention it can be noted that workplace fun is measured by three variables (wf_1, wf_2 and wf_3) as for the work-life balance is measured by three variables (wlb_1, wlb_2 and wlb_3). as for the mediators which are perceived organizational support and coworker support they are measured by (pos_, pos_2 and pos_3) and (pcs_1, pcs_2 and pcs_3) consecutively. at the end, the dependent variable employee retention is measured by er_1, er_2 and er_3. as shown in the model, workplace fun tends to impact pos by 0.66 and impact pcs by 0.33, and in turn pos affect employee retention by 0.49. however, workplace fun as for the work life balance tends to impact pos by 0.57 and pcs by 0.77 and in turn pcs impact employee retention by 0.46. thus, as it can be noted, both perceived organizational support, and perceived coworker support mediates the relationship between workplace fun, work-life balance and employee retention, but perceived organizational support tends to have higher effect on the mediating relationship among the variables. nassar et al./decis. mak. appl. manag. eng. 5 (2) (2022) 396-412 406 figure 1. structure equation model 14. findings 14.1 perceived organizational support mediates the relationship between workplace fun, work-life balance and employee retention the support of the organization is critical in assisting workers in both their professional and personal lives. it aids in the reduction of conflict among workers and the achievement of the greatest possible balance between work and family responsibilities, aiding in the enrichment of family life via work. perceived organizational support is regarded as a good trait that ensures workers that the company would help them in tough tasks when they need it the most. as far as workers are concerned, perceived organizational support is a measure of how much they believe that their employers value their contributions as well as how much they care about the people inside those organizations. the findings of this study are aligned with the findings of mahmoud and grigoriou (2017) who stated that perceived organizational support mediates the relationship between workplace fun, work life balance and employee retention. it demonstrates how much companies value their workers and put their well-being first. employee well-being refers to the mental, bodily, and emotional health of workers, and it is taken to mean that employees have a favorable opinion of their work experience as it relates to wellbeing. according to the findings, employees have a proclivity to express thoughts about how much their company values their contributions and cares about their wellbeing. according to the research, perceived organizational support is critical to both corporate and individual well-being. organizations may improve perceived the mediating effect of organizational and co-workers support on employee retention in… 407 organizational support and, as a result, their employees' physical health in a variety of ways. the employment of fairways by employers in interacting with employees, for example, has been demonstrated to be a precursor of perceived organizational support. according to the findings, organizations should provide assistance to employees who plan to take advantage of paid sick time without fear of being dismissed. if people lose their employment because of legal measures, they won't be able to enjoy the full assistance of their employers during these downtimes. individuals' rights may be safeguarded by organizations via the use of policies. perceived organizational support is also linked to employee job engagement on a weekly basis, which in turn enhances workers' weekly well-being, according to the data. 14.2 perceived coworker support mediates the relationship between workplace fun, work life balance and employee retention in various respects, the findings of this study added to the body of knowledge on workplace fun. using the research findings proved that coworker support mediates the relationship between workplace fun, work-life balance and employee retention. to show other ways international non-governmental organizations may assist embed their members, we included the fun into these relational approaches. it studied the role of perceived coworker support in the link between fun and retention. our belief is that fun has a significant effect on employee retention via the formation of high-quality connections, and constituent attachment may be a crucial process variable through which fun impacts workers' choices to stay or leave the international non-governmental organizations in gaza strip. there is a strong correlation between enjoyable activities, management support for fun, and colleague networking, but these concepts are separate. employees should be able to have a good time at work while also strengthening the bonds that bind them to the company. however, the amount of formality of these aspects might be compared. the most formal and "produced" fun events are those that the organization plans and sponsors. since socializing is more informal and voluntary, colleague socialization tends to be more spontaneous. somewhere in the center is manager support for having a good time. coworkers' support for pleasure might be seen as formal since they oversee their subordinates. coworkers' support for fun, on the other hand, is a way of allowing workers the flexibility to have fun that may encourage informal and unplanned enjoyment. whether or whether these aspects of enjoyment occur at the same time is up to you. coworkers may socialize while participating in a pleasant activity supervised by management, but they may also socialize on their own. managers might also be passionate about entertaining events, yet other times they would not show any support. manager support for fun and colleague interaction may be connected, but both aspects of fun are not a one-size-fits-all proposition. since these dimensions are separate structures, there is a benefit to be gained by concentrating on them. the findings of this study are aligned with the findings of huynh and nguyen (2019) who stated that perceived coworker support mediates the relationship between workplace fun, work life balance and employee retention. 15. comparison between research findings and previous literature the summary of the hypotheses and the result shown in table 7. nassar et al./decis. mak. appl. manag. eng. 5 (2) (2022) 396-412 408 table 7. summary hypothesis p value r validation complies with author h1: there is a direct relationship between workplace fun and employee retention 0.00 0.665 accepted thank and toan (2018) h2: there is a direct relationship between work-life balance and employee retention 0.00 0.665 accepted utami et al (2021) h3: perceived organizational support mediates the relationship between work-life balance, workplace fun and employee retention 0.00 0.689 accepted with strong mediation mahmoud and grigoriou (2017) h4: perceived coworker support mediates the relationship between work-life balance, workplace fun and employee retention 0.001 0.679 accepted with moderate mediation huynh and nguyen (2019) 16. conclusions, implications and limitations this research aimed to discover the experience of workplace fun, and work-life balance in international non-governmental organizations in gaza strip by addressing employees. the present study focused on quantitative methods shows that perceived organizational support and coworker support tends to mediate the relationship between workplace fun, work-life balance and employee retention in the international non-governmental organizations in gaza strip. until now, this has been the only research of its sort in a developing nation that concentrates on workplace fun, and it is also the first study to target the international non-governmental organizations in a major developing country. models for measuring workplace fun, work-life balance, perceived organizational support, coworkers’ support and employee retention were developed by the author, who included workplace reality, management practices, workplace behavior, and meaning-related hurdles into his work. keeping an eye on these four areas offers a solid basis for firms to build and maintain a fun and witty workplace culture. 16.1. theoretical contribution gaza strip's first theoretical contribution is broadening the conversation about workplace fun and employee retention. today's workers are demanding and have high expectations of their employer, and the author not only emphasizes the importance of creating, developing, and maintaining managed and task-driven fun in international non-governmental organizations gaza strip. in addition, the findings the mediating effect of organizational and co-workers support on employee retention in… 409 discussed how working conditions have changed in international non-governmental organizations. there is a chance that this will motivate the targeted organizations to reexamine their internal behavior and include workplace fun as a part of their hr strategy and policies that will follow. budhwar et al. (2009) emancipation theory was extended by demonstrating that structural obstacles may be removed if workplace fun is properly implemented and maintained as the second theoretical contribution. once this happens, social bonds will start to form between the workers themselves and their bosses. workers (in this example, international non-governmental organization employees) are regularly invited to chat, express and communicate via the use of social events, after-work parties, barbecue meetings and birthday celebrations. it is important to note that according to budhwar et al. (2009) theory of emancipation, human empowerment and the elimination of structural obstacles may be achieved through ensuring organizational inclusiveness and reducing negative prejudice among workers. a further perspective on this hypothesis is provided by workplace entertainment activities. 16.2. practical implications increasing job problems in international non-governmental organizations in gaza strip led to stress and burnout for personnel. humor among workers and leaders has been shown to be associated with better health outcomes for employees, such as reduced stress and job happiness, as well as leader effectiveness, such as approbation from subordinates. as a result, the author advises the company leaders in this research to create a unit dedicated to making the workplace more entertaining and to include it in the hr department. one way to do this is to choose or pick one or two hr staff members who will be in charge of managing fun in the workplace, similar to those who are in charge of recruitment or managing organizational learning, and all of them are hr department workers. the author suggests that companies secure intensive coaching, seminars and training sessions on organizational inclusion to avoid or reduce the possibility of pranks or jokes being used to harm or hurt the feelings of minority affiliated members like women and christians in order to alleviate any negative consequences when adopting workplace fun. results from this study reveal that having a strong degree of organizational support is linked to having a greater work-life balance. as a result, the company should provide services to employees to help them achieve a better work-life balance. organizations must create a positive working atmosphere in order to prevent negative attitudes among their staff. as a result, human resource and organizational management professionals should implement policies to help workers. they may, for example, adopt rules aimed at helping workers deal with issues at work. this may assist them in lessening workplace friction, which will have a positive impact on their personal life as well. managers may improve work-life balance by implementing policies that include a variety of flexible work alternatives. these arrangements may make it easier for workers to balance home and work life. in order to improve work-life balance, organizations should implement family-friendly policies. coworker support programs may be developed by companies to improve employee well-being. assistance programs for workers are seen as one of the most important tools for enhancing employee well-being. another factor is that workers who feel well supported by their company may anticipate that the corporation will provide them nassar et al./decis. mak. appl. manag. eng. 5 (2) (2022) 396-412 410 with resources to aid in resolving the problem. with these approaches, employee well-being will improve. the research aimed at investigating the relationship between perceived organizational support and perceived coworker support by modeling both work-life balance as well as workplace fun concurrently. workplace fun and perceived organizational support are linked together based on the "social support theory." the research assumed that when employees have fun in the workplace the higher their productivity will be. this, in turn, increases employee retention and minimizes the employee turnover rate. perceived organizational support had a greater impact on employee retention than coworker support according to the research findings. in addition to reducing employee turnover between their job and home responsibilities, organizational support also encourages workers to have a healthy work-life balance. work-life balance and workplace fun are both linked to higher employee retention, according to the findings of this research. it is important for human resources managers and practitioners to understand how workers feel about work and family to treat them fairly while still satisfying their company's demanding criteria. the introduction of rules aimed at improving employee retention is thus critical. to summarize, organizational support may play a critical role in improving employee retention by helping to enhance work-life balance and reducing conflict between work and family obligations. finally, the author believes that the most appropriate and relevant choice for organizations were controlled workplace fun, arranged and launched by managers and executives since the employees suffer from the lack of engagement, long working hours, and inadequate monetary incentives as a result, management involvement would be preferable to employee initiative in this situation. 16.3. limitations the study's biggest shortcoming was that it focused just on international nongovernmental organizations in gaza strip, leaving out other organizations. organizational culture, working conditions, growth possibilities, and economic incentives vary widely from place to place in gaza strip and the middle east as a whole. this constraint makes it difficult for the author to extrapolate the findings of the study. the current paper's conclusions may be difficult to generalize if it simply considers workers without taking into account their supervisors. 16.4. future research for a deeper knowledge of workplace fun, the author of this report invites other organizational and human resources management experts to do similar studies in underdeveloped countries. other institutions in developing countries, such as colleges, small and medium-sized organizations, and non-profits, might benefit from asking the same research questions. lastly, the author recommends that researchers in human resources management collaborate with colleagues from the fields of interdisciplinary studies on how to initiate workplace fun in different public and private organizations, such as applied psychology, humanities, public policy, organizational psychology, and sociology. organizational culture, reward, promotion choices, workplace and corporate justice all play a vital role in motivating employees to better retain staff. when money is used as an external incentive, intrinsic motivation frequently decreases, but intrinsic motivation improves when verbal praises and constructive feedback are used. employee retention has been proven to be enhanced by factors such as job the mediating effect of organizational and co-workers support on employee retention in… 411 significance, safety, positive feedback, a diverse work environment, and a high degree of freedom and power. workers' quality of life may increase if they have a reasonable amount of fun, but excessive enjoyment might be detrimental. since one of the primary motivations for seeking employment in the international non-governmental organizations is to have a good time, having fun may be key to reducing employee turnover. seeing this offers employees faith that their increased performance will be recognized and that their well-being will be maintained. it's less probable that motivated employees would quit a firm thanks to the pos system. when the pos is high, absenteeism is minimized and employee retention is enhanced. dispelling bad feelings and enhancing productivity and long-term connections are some of the benefits of using perceived organizational support. according to the study, it is possible to develop friendships in the workplace by encouraging employees to participate in fun activities and providing support for fun managers. fun activities and the support of managers for fun have a significant impact on turnover because of these reasons. author contributions: conceptualization, s.n. and m.b.h.; methodology, s.n. and m.b.h.; software, s.n.; validation, s.n.; formal analysis, s.n. and m.b.h.; investigation, s.n.; resources, s.n. and m.b.h.; data curation, s.n.; writing—original draft preparation, s.n.; writing—review and editing, s.n.; visualization, s.n. and m.b.h.; supervision, e.z.t.n. and l.v.; project administration, e.z.t.n. and l.v. all authors have read and agreed to the published version of the manuscript. funding: this research was funded by hungarian university of agriculture and life sciences (mate) through stipendium hungaricum scholarship acknowledgments: the authors of this article would like to thank hungarian university of agriculture and life sciences (mate) for their financial and educational support. conflicts of interest: the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. references abela, f., & debono, m. 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(2017). trust, job satisfaction, perceived organizational performance and turnover intention. employee relations, 39(7), 1148-1116. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.13106/jafeb.2018.vol5.no3.179 https://doi.org/10.13106/jafeb.2018.vol5.no3.179 plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 5, issue 2, 2022, pp. 176-200. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0316102022n * corresponding author e-mail addresses: padmanabhanpadhu2024@gmail.com(padmanabhan.s), nitsk15@gmail.com(nitish.s), raghuram.mechanical@gmail.com(raghuram.p), m_thenarasu@cb.amrita.edu (thenarasu. m) job shop scheduling using heuristics through python programming and excel interface padmanabhan sowmia narayanan1, nitish shankar kumar1, raghuram potluru2 and thenarasu mohanavelu1* 1 department of mechanical engineering, amrita school of engineering, india 2 maryland health benefit exchange, baltimore, usa received: 4 july 2022; accepted: 29 september 2022; available online: 6 october 2022. original scientific paper abstract: job shop scheduling problem (jssp) has remained a challenge both for the practitioners and the researchers. a jssp consists of multiple number of machines (m) and jobs (n). as the number of jobs increases, the complexity of the problem increases exponentially and it becomes difficult to schedule manually. many papers in the literature discuss heuristic and metaheuristic solutions to solve job shop scheduling problems. but there is no ease of use for practitioners who rely on their experience to schedule jobs in ad hoc sessions resulting in inefficient allocation of jobs and machines. in this paper, a job shop scheduling problem under static and dynamic conditions is solved using heuristic approaches using python programming with an ms excel user interface. for a supplier of automotive parts with a set of jobs and machines, priority dispatching rules, viz., shortest processing time (spt), earliest due date (edd), first-in first-out (fifo), critical ratio (cr) and slack per remaining operation (s/ro) are evaluated. the obtained performance metrics such as makespan, and tardiness are compared between the heuristics to select an optimal schedule by the job shop. the user inputs the jobs, machines, start and due dates through the ms excel interface and obtains faster, practically usable results. this reduces the time taken for job scheduling and helps in making faster productivity-based decisions to maximize resource utilization and the total time to produce the product. integrating python at the backend and excel at the front end will encourage many msmes to perform optimized scheduling using heuristics thereby reducing the throughput time. key words: job shop scheduling problem, priority dispatching rules, python programming, micro small and medium enterprises. mailto:nitsk15@gmail.com(nitish.s),%20raghuram.mechanical@gmail.com(raghuram.p) padmanabhan et al./decis. mak. appl. manag. eng. 5 (2) (2022) 201-218 202 1. introduction a job shop scheduling problem is a complex combinatorial optimization problem that needs a practically usable solution for msmes (amaro, 2022). a lean and flexible operations can be underutilized with an ineffective scheduling process. a typical job shop consists of multiple jobs (n) and machines (m) with varying routes and different setup, and processing times. each machine can work on only one job at a time and each job is processed in a particular order (bakuli, 2006). the objective of the job shop scheduling is to find the best order of the jobs and operations, keeping in mind the varied routing requirements. exact algorithms used to solve jssps make it necessary to find a precise algorithm to optimize the problem having finite instances. job shop scheduling belongs to a class of np-hard problems that are non-deterministic polynomial (lenstra et al., 1977). this class of problems do not have an exact algorithm to find the solution in definite polynomial time. therefore, large-scale application of these methods found it impractical in many cases because of its high computational time (vinod and sridharan, 2011). because of the deficiency in computational power for solving large-scale operations, it was recognized that precise or exact methods are ineffective, therefore research for obtaining approximate solutions based on heuristic methods was developed. in today’s production environment, delivery time is critical to dealing with market competitive pressures which means industries have to deliver a wide range of products within expected delivery dates, as failure to meet deadlines can lead to loss of customers and markets. while dealing with planning problems in actual production shop floors, uncertainties hinder the use of rules based on ideal assumptions (romero-silva, 2022). considering the prevailing production problems and supply uncertainties, proper scheduling has to be done, especially for small and medium enterprises (smes). for example, uncertainties like the lack of resources and equipment availability leads to an increase in tardiness of jobs. with disruptions in a production environment such as processing overtime or ahead, an emergency order to join, and inaccurate processing time estimates, etc., the planned schedule becomes obsolete in an actual production scenario (raghuram and harishankar, 2021; cowling and johansson, 2002). many of the researchers consider only static jssps, which are impractical as job flow is dynamic in practice (wang et al. 2019). as a result, we must constantly adjust the scheduling plan in response to changes in real conditions, which is referred to as dynamic scheduling. fulfilling customer demands responsively while scheduling is crucial in retaining customers (raghuram and saleeshya, 2021). dynamic scheduling, on the other hand, is clearly more complex and difficult to solve. the dynamic events are classified into four categories, job related, machine related, process related events and other occurrences (suresh and chaudhuri, 1993). as it is important to consider practical dynamic conditions, this research paper uses heuristics with a practical interface to solve both static and dynamic jssps. 2. literature review production scheduling is the allocation of limited production facilities such as labor, machinery, and tools to complete a variety of activities (jiang and zhang, 2018). job shop scheduling problem (jssp) is one of the most essential manufacturing problems (asadzadeh, 2015) because of its impact on overall firm and supply chain productivity. the jssp entails sequencing a series of tasks, each with its own chain of processes, to be performed in given machines for a certain amount of time. according to varied production settings, there are various types of shop scheduling simulation analysis and development of priority dispatching rules for a partial flexible job… 203 patterns, which may be further classified as single-objective or multi-objective (xiong et al. 2022). due to the great complexity of job-shop setups, finding a perfect solution to these problems in a reasonable length of time is difficult. jssps are classified as nphard, due to the combinatorial growth of effective options (ghedira and ennigrou, 2000; garey and johnson, 1979). as a result, there are several techniques and strategies for dealing with jssp, and each has a distinct and direct impact on the quantity, regularity, and severity of information exchange in the shop, as well as the scheduling quality. the scheduling parameters are often evaluated using criteria such as due date sensitivity, operating costs, and setup durations (kim and bobrowski, 1994). under these wide range of problems, precise or optimum approaches that yield optimum answers may take much longer to estimate, but approximation methods yield near-ideal solutions in less time (liaqait et al. 2021). according to delgoshaei et al. (2021), job shop scheduling is the arrangement of resources available to optimize given performance measures. the scheduling framework includes a succession of jobs and units while mediating an optimum solution job sequence on each machine under specified limitations. the shop scheduling becomes increasingly complex to solve when several performance measurements are taken into account (admi syarif et al. 2021). there are several approaches for solving jssp as described in the following section. 2.1 approaches to solving job shop scheduling to solve scheduling problems, various methods can be used. these methods are divided into two groupings: exact and approximate methods. in jssp, the exact methods cannot produce solutions for large-scale problems (fox and smith, 1984). hence heuristic methods that return approximate solutions are used. the time required to produce approximate solutions is always less than the time required to reach exact solutions (bulbul and kaminsky, 2013). in the real world, obtaining an optimal solution is practically impossible. therefore, obtaining a high-quality approximate solution will be satisfactory (kapanoglu and alikalfa, 2011). hence, researchers focus on developing heuristics algorithms that can produce solutions close to optimal solutions in the least possible time. by converting production scheduling problems to equality or inequality constraints, approximate methods create one or more optimization models of the target function to arrive at an optimal solution. most of the methods proposed in extant literature are hybrid methods, that is a combination of two or more best performing rules that were previously developed. the aim of scheduling using heuristics is to optimize the value of performance measures. approximate methods can also be used in scheduling dynamic jssp (gupta and sivakumar, 2006). the complexity of the problem increases as the number of machines and flow sequences in the scheduling problem grow. the feasible solution grows exponentially, and approximate methods can find an optimal solution in a reasonable amount of time. as a result, approximate methods can be used to solve practical problems. usually in small-scale industries the jobs arrive dynamically and each set of jobs have different due dates. the jobs have to be prioritized and completed before the corresponding due dates with the existing set of jobs. in the following sections we review the priority dispatching rules and describe a practical jssp scenario from a small-scale industry. padmanabhan et al./decis. mak. appl. manag. eng. 5 (2) (2022) 201-218 204 2.1.1 metaheuristics in solving jssp there are numerous papers that offer different methodologies and solutions to job shop scheduling problems using metaheuristics. optimization conceptual researchers are researching optimization methods based on nature that could be used as optimization techniques for engineering problems. uniyal et al. (2022) presented an overview of the most intriguing class, the nature-inspired optimization algorithms that evolved over time and with inspiration from nature. optimization-based procedures and approaches could help to expand, develop, and generate appropriate designs and operations (kumar et al. 2021a). kumar et al. (2022) have provided an indepth examination of the most widely used and explored meta-heuristic optimization methods and nature-inspired algorithms. these have wider practical applications and remain a popular research topic and an efficient tool for solving complex optimization problems. meloni et al. (2004) proposed an alternative graph solution algorithm for a general formulation of the jssp in presence of blocking and/or no-wait constraints. artificial ants are defined such that they are easily modifiable to include new constraints and can be reconfigured for multi-objective cases. kahraman, (2006) proposed a modified ant colony algorithm to solve jssp in an acceptable amount of time. to assess the system's reliability when the available information is uncertain, use of fuzzy reliability function will be useful (chaube et al. 2018). pongchairerks (2019) proposed a novel two-level viz. upper and lower levels, metaheuristic algorithm to solve jssp. the former is a population-based algorithm that acts as a parameter controller for the latter, whereas the latter seeks optimality. kumar et al. (2021b) worked to minimize the cost while satisfying the system's availability constraints. they focused on increasing the operational time of the individual components of a system to maintain higher system reliability and improve productivity and profit by the application of nature-inspired optimization techniques such as grey wolf optimization (gwo) and the cuckoo search algorithm (csa). uniyal et al. (2020) reviewed nature-inspired optimization along with a background of fundamentals, classification, and their reliability applications. they also demonstrate the difference between multi-objective optimization and singleobjective optimization. the article provides a foundation for a few nature-inspired optimization techniques and their reliability applications. negi et al. (2021a) provide an up-to-date review of the gwo algorithm and its usefulness in more complex realworld problem-solving. jit-jss, a variant of the job-shop scheduling problem in which each operation has a distinct due date was studied by ahmadian et al. (2021). in this method, any deviation of the operation completion time from its due date incurs an earliness or tardiness penalty. the authors solved this using a variable neighbourhood search (vns) algorithm. a jssp with blocking (bjss) constraints was provided by pranzo and pacciarelli, (2016). blocking constraints simulated the absence of buffers (zero buffer), whereas buffers had infinite capacity in the traditional job shop scheduling model. particle swarm optimization (pso) has gained popularity as one of the most popular algorithms for solving jssp. researchers have attempted to improve this algorithm by introducing hybrid methods. pant et al. (2017) presented a new and improved particle swarm optimization algorithm, abbreviated mpso, for both constrained and unconstrained nonlinear optimization problems. negi et al. (2021b) proposed a framework for implementing a hybrid pso-gwo algorithm (hpsogwo) for solving reliability allocation and optimization problems in a space capsule's simulation analysis and development of priority dispatching rules for a partial flexible job… 205 complex bridge system and life support system. in the current research work, heuristic methods are considered and will be discussed in the next section. 2.1.2 priority dispatching rules the priority dispatching rules (pdrs) are used to prioritize work in a job shop in order to improve performance measures (thenarasu et al. 2022). sels et al. (2012) have discussed various priority rules for jssp which are compared and validated using different objective functions. the ranking of priority rules is checked applying it to larger problems, on the extension of multiple machines per job as well as on the introduction of sequence-dependent setup times. dynamic arrival of jobs was also tested for the ranking of the priority rules. there are numerous papers that give an insight on what and how, jssp with various constraints, have been solved with the help of heuristics and improved techniques, over the course of time. kalita et al. (2016) worked on providing a heuristic approach for determining the best machine loading sequence while minimizing makespan and other performance measures. abbas et al. (2016) employed heuristics such as shortest processing time (spt) and longest processing time (lpt) with no delays and incorporating fatigue was employed and studied. they used a case study with a variety of jobs with different production sequences using time and motion studies and found that spt rule provides lower makespan values when no scheduled breaks occur, while lpt performs better for scheduled breaks. snyman and bekker (2019) applied various dispatching rules for a dynamic jssp. a simulation model of an auto ancillary unit to prioritize jobs, using an analytic hierarchy process (ahp) based priority rules in a press shop was developed and designed (mohanavelu et al. 2017). thenarasu et al. (2019), also proposed using an arena simulation model to evaluate performance measures, integrating pdrs and multi-criteria decision-making (mcdm) with technique for order of preference by similarity to ideal solution (topsis) approach (thenarasu et al. 2020). further, ashwin et al. (2022) utilized lekin software to analyze, compare and evaluate various pdrs to improve performance measures. 2.2 identified research gaps two major research gaps were identified in the extant literature and industry. firstly, solving static jssp problems do not offer practically viable solutions to the industry. over the years, researchers have been attempting to find different approaches to solve jssps. the intricacies of jssp make it difficult to develop an appropriate and effective technique. an effective strategy that produces an improved performance for a given job shop configuration cannot generate the same outcome with another configuration. secondly, many of the researchers use simulation and optimization algorithms solved using software, that are not affordable by msmes. currently, most of the industries do not own any scheduling software because of the high licensing costs involved, and the difficulty in learning and using it on a daily basis. so, it becomes difficult for the operations team to develop effective and optimal schedules. 3. description of case study and problem statement small and medium scale industries use manual scheduling which is not effective way of scheduling jobs. commercial scheduling software available in the market are padmanabhan et al./decis. mak. appl. manag. eng. 5 (2) (2022) 201-218 206 costly and not affordable by smes. hence most of these companies perform scheduling operations based on experience. table 1. jobs with machine sequence and processing times jobs operations machines (m/c) processing time (s) terminal turning lathe 360 facing dro 180 drilling drill 60 taping taper 60 shutter body turning lathe 720 counter boring dro 300 drilling drill 180 taping taper 90 top cover facing dro 180 drilling drill 240 taping taper 300 cover turning lathe 240 drilling drill 360 pole shoe cropping mechanical press 20 shot blasting shot blast 900 sizing hydraulic press 10 extrusion hydraulic press 15 annealing furnace 7200 clipping mechanical press 10 deburring vibratory deburrer 300 bending hydraulic press 15 drilling drill 25 chamfering chamfer 10 threading thread 10 flange turning (od) lathe 70 turning (id) lathe 30 yoke cutting mechanical press 300 deburring vibratory deburrer 120 turning lathe 40 chamfering chamfer 40 slotting slotting 28 polishing polishing 20 gimping gimping 66 crank shaft finishing dro 22 drilling drill 22 threading thread 60 turning lathe 60 crank case turning lathe 15 boring bore 15 grooving dro 15 milling milling 75 drilling drill 75 chamfering chamfer 75 tapering taper 75 simulation analysis and development of priority dispatching rules for a partial flexible job… 207 production and supply chain risks such as machine breakdowns, lack of labor availability, lack of raw materials and arrival of new jobs also result in repeated planning and scheduling activities. the time and effort spent in scheduling makes it harder for the managers to focus on production related activities. also, manual scheduling results in wastage of time and resources, as they cannot develop an optimized schedule. hence, there is a need to produce an optimal schedule faster in a real-time production environment with an easy-to-use and affordable interface. a number of jobs can arrive at a shop floor within a given time horizon. table 1 shows the different jobs that are available for scheduling on a particular day. the sequence of machines through which these jobs travel, in different paths through the machines in the shop floor are also depicted in order. the processing times of the jobs in each machine are also given. 4. methodology the methodology followed to solve both static and dynamic job shop scheduling problems using various heuristic methods is as follows. data was collected from a small-scale job shop, in automotive industry, located in chennai. a static instance of the problem was solved using five priority rules, viz., spt, edd, fifo, s/ro and cr. the performance measures, viz., makespan, maximum tardiness, number of tardy jobs and total tardiness are obtained and compared to find the best solution. as jobs may be introduced after the production process has started, the problem becomes dynamic in nature. the resulting djssp was also solved using priority rules. python programming was used to code the algorithms of heuristics optimization. the performance measures obtained were compared using python and the output given through ms excel. a user interface for data collection and display of resulting schedule and performance measures was developed in ms excel with python programming in the backend. 4.1. solving dynamic jssp dispatching rules are used to handle job sequencing on machines. the jobs to be performed are organized for each of machines by using a job priority rule. jobs are queued and whenever a machine is available, it must be chosen after checking which of the queued tasks will be executed on the machine. priority rules are assigned depending on the pending jobs in order to choose the job to be processed next. five of the pdrs are employed in this paper: spt, edd, fifo, cr & s/ro. 4.1.1 pseudo code of heuristic approach pseudocode of heuristic approach used for prioritizing of jobs is given in this section, with comments in brackets. step 1: read the input excel file step 2: assign job, m [job and no of machines] step 3: create a 3d array [processing time of jobs, allocated machines, and a flag value are initialized to zero] step 4: opr = [] step 5: mac = [] step 6: sum (opr) step 7: ddj = sum (ptj)* k padmanabhan et al./decis. mak. appl. manag. eng. 5 (2) (2022) 201-218 208 step 8: flag1, flag2 = 0 step 9: while flag1! = totalopr or flag2! = m: step 9(a): time.sleep(0.2) if keyboard.is_pressed('s’): [adding new jobs] file_name = str(input("enter the filename with ext: ")) f = open(file_name,'r') f_read = f.readlines() f.close() [increment the no of jobs, operation array, update the due date, processing time, job completion time array] step 9(b): for k in range (m): if flag value == 0 and arr[i][j][0] < dispatching rule cond: [update the mac array by checking which jobs operation (in the queue) has the smallest processing time to be performed in that particular machine and accordingly assign that jobs operation to the machine] step 9(c): var = min (i for i in mac if i > 0) for k in range (m): if mac[k] > 0: mac[k] = mac[k] – var macult[k] = macult[k] + var cmax = cmax + var [reduce the processing times assigned in the mac array by the min value in the mac array thus indicating that the job is being processed] step 9(d): cmax, cseq, cj, um [calculate the make span, job completion sequence, job completion time, mac utilization time accordingly] step 10: end while step 11: nj, sum (cj), max (tj) [calculate the no of tardy jobs, total job completion time, max tardiness and other performance measures accordingly] step 12: print the performance measures thus obtained figure 1. constitutes diagrammatic representation of the algorithm developed. the flowchart deals with application of spt dispatching rule for solving the jssp problem. this flow chart can be adapted for each of the dispatching rules. the performance measures obtained are compared and the best performing rule is chosen for the given products. 4.2. ms excel user interface one of the main purpose of this paper is to develop an easy to understand-anduse interface to input the data as well as to receive the required output. for this purpose, a user interface developed using ms excel, with python programming executing the heuristics in the backend is created. it is therefore not necessary for a production scheduler to have an in-depth knowledge of either python programming or ms excel. the excel interface consists of two input sheets and an output sheet. in the first sheet named ‘instructions’, the general information, and scheduling simulation analysis and development of priority dispatching rules for a partial flexible job… 209 information are collected, and instructions for data input are provided. in the second sheet, ‘scheduling’, the data table is provided with various fields to input jobs, machines, start dates, due dates. once the data is entered, the scheduler will process the schedule using the ‘run’ button to obtain the results. figure 2. shows the ‘instructions’ sheet and figure 3. shows the ‘scheduling’ sheet. the number of machines for any product can be added as needed. the program automatically recognizes the number of machines for each product and adds it to the sequence. figure 1. flow chart representation of the proposed heuristic algorithm padmanabhan et al./decis. mak. appl. manag. eng. 5 (2) (2022) 201-218 210 figure 2. ms-excel user interface information sheet figure 3. ms-excel user interface scheduling sheet simulation analysis and development of priority dispatching rules for a partial flexible job… 211 5. results and discussion the results obtained are based on the input data provided in two benchmark instances, 15x15 and 30x20 (tasgetiren et al., 1993) and from the case company (9x16) for static instance. the total program run time taken for the heuristic methods is less than a minute for all instance. the job completion sequence, for 15x15 and 30x20 instances, using the aforementioned dispatching rules are displayed in table 2 and table 3 respectively. the performance measures using different heuristic methods are compared and presented in figure. 4 figure. 5 for 15x15 and 30x20 instances respectively. table 2. job completion sequence (15x15) – jobs 1-15 arranged in order of operations spt 8 3 12 10 7 9 2 14 1 15 5 13 6 11 4 fifo 10 1 11 3 9 14 15 8 7 2 6 13 5 4 12 edd 8 10 14 2 3 15 6 5 7 12 9 4 11 1 13 cr 12 10 7 2 5 14 15 8 1 9 4 6 3 13 11 s/ro 7 10 8 1 5 11 9 15 4 14 3 2 13 12 6 figure 4. graph of performance measures vs. dispatching rules (15x15) the simulation was carried out for the benchmark instances, 15x15 and 30x20, and the performance measures were obtained. the obtained results were cross verified and thus the code proved effective. padmanabhan et al./decis. mak. appl. manag. eng. 5 (2) (2022) 201-218 212 table 3. job completion sequence (30x20) – jobs 1-30 arranged in order of operations spt 23 18 10 7 21 13 30 6 29 5 12 3 27 14 26 2 12 17 28 25 19 11 8 24 22 1 9 16 20 4 fifo 10 6 22 18 23 7 13 4 2 1 3 19 17 12 11 27 5 30 16 9 14 29 26 8 24 21 28 15 20 25 edd 6 10 7 23 27 2 14 18 9 21 29 11 22 12 4 16 13 1 17 3 19 5 24 28 8 26 25 15 20 30 cr 11 17 26 16 4 20 3 5 13 18 9 27 19 8 1 6 14 25 7 21 12 15 29 24 28 22 2 23 10 30 s/ro 5 23 7 18 1 19 1 13 11 25 9 15 20 16 26 27 8 2 3 17 4 6 30 24 28 22 29 12 21 10 figure 5. graph of performance measures vs. dispatching rules (30x20) 5.1 case study data the simulation was run for the data obtained from case company. there were nine different jobs, each with its comprising of 44 operations in total, to be completed within the specified due date. the job completion sequence, for 9x16 instance, using the different heuristic methods is displayed in table 4 and the results are compared and presented in figure. 6 for the same instance. to identify the best approach, four performance measures were taken and reviewed. from the graphical representation, it is observed that while considering makespan, spt, fifo and cr outperform the other dispatching rules. for maximum tardiness, spt has least maximum tardiness thereby outperforming the rest. for number of tardy jobs and total tardiness, spt simulation analysis and development of priority dispatching rules for a partial flexible job… 213 gives the least value making it better than the rest. hence, for this particular problem instance, shortest processing time dispatching rule is recommended. table 4. job completion sequence (9x16) – jobs 1-9 arranged in order of operations spt 6 8 3 9 4 7 1 2 5 fifo 3 1 8 6 7 4 2 9 5 edd 6 3 8 1 7 4 9 2 5 cr 3 1 7 8 6 4 2 9 5 s/ro 3 9 7 2 1 8 6 4 5 figure 6. graph of performance measures vs. dispatching rules (9x16) to validate the same, few benchmark instances were also reviewed to obtain the performance measures. to comprehend better, table 5 to table 7 provide the comparison of performance measures for various heuristic approaches utilized to deal with the instances 15x15, 30x20 and 9x16 respectively. table 5. comparison of dispatching rules with performance measures (15x15) makespan max tardiness no. of tardy jobs total tardiness spt 1462 14 1 14 fifo 1612 94 4 149 edd 1501 0 0 0 cr 1524 125 1 125 s/ro 1635 160 3 246 padmanabhan et al./decis. mak. appl. manag. eng. 5 (2) (2022) 201-218 214 table 6. comparison of dispatching rules with performance measures (30x20 benchmark instance) makespan max tardiness no. of tardy jobs total tardiness spt 2499 455 11 2015 fifo 2529 473 18 3166 edd 2957 669 12 2505 cr 2909 1067 21 7279 s/ro 2801 809 23 7442 for the 15x15 benchmark instance, solved using heuristics, it can be seen that spt rule is producing better results for the performance measure, makespan. whereas, edd produces better results with the remaining performance measures compared to all the other heuristics. for the 30x20 benchmark instance, solved using heuristics, it can be seen that the spt rule is producing better results with all performance measures compared to all the other heuristics. table 7. comparison of dispatching rules with performance measures (9x16) makespan max tardiness no. of tardy jobs total tardiness spt 141.9 0.0 0 0.0 fifo 141.9 15.6 5 50.1 edd 146.9 1.40 1 1.4 cr 141.9 131.9 6 176.7 s/ro 146.9 116.9 6 139.5 with the job shop configuration, 9x16 sjssp considered for the case company, spt and edd outperform the rest of the dispatching rules with least maximum tardiness. considering makespan, spt, fifo and cr outperform the rest. spt gives the least number of tardy jobs and total tardiness. hence, for this case, shortest processing time dispatching rule is recommended in solving the static job shop problem, using heuristics. it is observed that different dispatching rules affect each performance measures based on the priority given. in our case problem (9x16), spt rule is found to be performing better for all measures because it can determine the status of specific job, establish relative priority among jobs on a common basis, relate both stock and make-to-order jobs on a common basis and also dynamically track job progress and location. 6. conclusions a precise schedule for static and dynamic job shop scheduling problem using a comparison of heuristic (priority rules) methods were developed using python programming. both static and dynamic jssp were solved for 9x16, 15x15 and 30x20 problems using priority dispatching rules, viz., spt, edd, fifo, s/ro and cr. the program was run, with a case company data and benchmark problems and the following conclusions are drawn. a simple and easy to use job shop scheduler has been developed for msmes. it offers msmes, an affordable and easy-to-use solution for a time consuming, manual planning task. the above solution needs to be properly packaged for distribution to the msme industry. as a future research work, the simulation analysis and development of priority dispatching rules for a partial flexible job… 215 authors are developing an excel and python-based scheduler with metaheuristics for multiple jobs-multiple machines scenario, which can provide exact optimal solutions. it can be an effective tool in increasing the profits of the industry, as it lessens the planning time as well as improves the productivity through an optimized schedule. 6.1. practical implications as there is no learning curve involved in using the program and no prior knowledge of python programming is necessary, only a basic working knowledge of ms excel is necessary to use the interface. dynamic jobs can be added, as in real-time, considering the current status of work-in-process and this is a major outcome of this research work. author contributions: methodology, 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(2022). a survey of job shop scheduling problem: the types and models. computers & operations research, 142, 105731. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 2, issue 2, 2019, pp. 126-137. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame1902102z * corresponding author. e-mail addresses: zizovic@gmail.com (m. zizovic), dpamucar@gmail.com (d. pamucar) new model for determining criteria weights: level based weight assessment (lbwa) model mališa žižović 1 and dragan pamučar 2* 1 faculty of technical sciences in čačak, university of kragujevac, serbia 2 department of logistics, military academy, university of defence, serbia received: 10 april 2019; accepted: 09 august 2019; available online: 09 august 2019. original scientific paper abstract: this paper presents new subjective model for determining weight coefficients in multi-criteria decision-making models. the new level based weight assessment (lbwa) model enables the involvement of experts from different fields with the purpose of defining the relations between criteria and providing rational decision making. the method can be applied in practical cases in specialized decision-making support systems, as well as in alternative dispute resolutions in virtual environment. the lbwa model has several key advantages over other subjective models based on mutual comparison of criteria, which include the following: (1) the lbwa model allows the calculation of weight coefficients with small number of criteria comparisons, only n-1 comparison; (2) the algorithm of the lbwa model does not become more complex with the increase of the number of criteria, which makes it suitable for use in complex multi-criteria (mcdm) models with a large number of criteria; (3) by applying the lbwa model, optimal values of weight coefficients are obtained with simple mathematical apparatus that eliminates inconsistencies in expert preferences, which are tolerated in certain subjective models (best worst method bwm and analytic hierarchy process ahp); (4) the elasticity coefficient of the lbwa model enables, after comparing the criteria, additional corrections of the values of the weight coefficients depending on the preferences of the decision makers. this feature of the lbwa model enables sensitivity analysis of the mcdm model by analyzing the effects of variations in the values of the weights of criteria on final decision. key words: multi-criteria decision making, criteria weights; lbwa model. 1. introduction determining weights of criteria is one of the key problems arising in the models of multi-criteria analysis to which the problem being solved in this paper belongs to. the new model for determining criteria weights: level based weight assessment (lbwa) model 127 absence of unique precise definition of the notion of the weight of criteria and the problem of selecting appropriate method for determining weights of criteria in specific decision-making situation are among the most important factors that make the problem of determining weights of criteria significantly more complex. taking into account the fact that the weights of criteria can significantly influence the outcome of the decision-making process, it is clear that special attention must be paid to the models for determining weights of criteria. most authors suggest the division of models for determining weights of criteria on subjective and objective models (zhu et al, 2015). subjective approaches reflect subjective thinking and intuition of a decision maker. in such approach the weights of criteria are determined on the basis of information obtained from the decision makers or from the experts involved in the decisionmaking process. traditional methods of determining weights of criteria include tradeoff method (keeney and raiffa, 1976), proportional (ratio) method, swing method (weber, 1988) and conjoint method (green and srinivasan, 1990), analytic hierarchy process model (ahp) (saaty, 1980), smart method (the simple multi attribute rating technique) (edwards & barron, 1994), macbeth method (measuring attractiveness by categorical based evaluation technique) (bana e costa & vansnick, 1994), direct point allocation method (poyhonen & hamalainen, 2001), ratio or direct significance weighting method (weber & borcherding, 1993), resistance to change method (rogers & bruen, 1998), ahp method (saaty, 1980), wls method (weighted lest square) (graham, 1987) and fpp method (the fuzzy preference programming method) (mikhailov, 2000). recent subjective methods include multipurpose linear programming (costa and climaco, 1999), linear programming (mousseau et al, 2000), dematel (decision making trial and evaluation laboratory) method (gabus i fontela, 1972), swara (step‐wise weight assessment ratio analysis) method (valipour et al. 2017), bwm (best worst method) (rezai, 2015) and fucom (full consistency method) (pamucar et al., 2018). on the other hand, objective approaches ignore decision makers' opinion and are established on determining weights of criteria based on the information contained in decision-making matrix using certain mathematical models. among the most known objective methods are the following: entropy method (shannon and weaver, 1947), critic method (criteria significance through intercriteria correlation), (diakoulaki, et al, 1995) and fanma method whose name was derived from the names of the authors of the method (srđević et al, 2003). according to zhu et al (2015) the most commonly used models for determining weight coefficients of criteria are subjective models with pair comparisons of criteria. in the models with pair comparisons, decision makers compare each criterion with other criteria and determine the level of preferences for each pair of criteria. as a support in determining the size of the preference of a criterion over another one it is used the ordinal scale. the most commonly used methods based on pair comparisons include (zavadskas et al, 2016) ahp method, bwm and dematel method. zavadskas et al (2016) have shown in their research that the ahp method is the most commonly used method for determining weights of criteria in the literature. however, in the ahp method needs to be performed comparison in pairs of criteria. a large number of comparisons makes the application of the model more complex, especially in cases of a large number of criteria. according to zhu et al (2015) in the ahp method it is almost impossible to perform fully consistent comparisons in pairs with over nine criteria. this problem is often overcome by dividing the criteria into subcriteria, which further makes the model more complex.  1 / 2n n  zizovic and pamucar/decis. mak. appl. manag. eng. 2 (2) (2019) 126-137 128 the dematel method is also used in numerous studies, but its main disadvantage is a large number of comparisons in pairs which is  1n n  . therefore, the dematel method is mostly used to determine the interaction between the criteria and the relationship diagram (parezanovic et al., 2019). the method that has become widely used in a short time is the bwm method. its biggest advantage compared to the ahp model is smaller number of pair comparisons ( 2 3n  ). however, a large number of comparisons in pairs of criteria, defining the limitations for solving nonlinear model and solving non-linear model make the application of the bwm significantly more complex. therefore, this model is still unacceptable to a large number of researchers. taking into consideration the stated deficiencies of the presented models, the need arises to provide for a method whose algorithm requires small number of comparisons in pairs of criteria and which has rational and logical mathematical algorithm. starting from this point, a level based weight assessment model (lbwa) has been developed. the first goal of the paper is to present the new model for determining weights of criteria which requires small number of criteria comparisons, just 1n  comparison. the second goal of the paper is to present practical model for solving complex mcdm models, regardless of the number of evaluation criteria. one of significant characteristics of the lbwa model is to maintain simple algorithm regardless of the complexity of the model. the third goal is to define a model which allows the calculation of reliable values of weight coefficients of criteria that contribute to rational judgment. the fourth goal of the paper is the development of a model that can be easily presented/explained to decision-makers, and therefore easily implemented in solving practical problems. the remaining part of the paper is organized in the following way. in the second section of the paper, the lbwa model algorithm is presented. in the third section of the paper, the lbwa model is tested with two examples from the literature. the fourth chapter provides concluding observations and directions for future research. 2. level based weight assessment (lbwa) model let us consider a multi-criteria model with n criteria  1 2, , , ns c c c . suppose that weight coefficients associated to these criteria are to be determined, i.e., they are not given in advance. in the following part it is presented the process of obtaining the weight coefficients of criteria by applying the lbwa model: step 1. determining the most important criterion from the set of criteria  1 2, , , ns c c c . let the decision maker determine the most important criterion, i.e., let the criterion 1 c be the criterion in the set of criteria  1 2, , , ns c c c that is the most significant for the decision-making process. step 2. grouping criteria by levels of significance. let the decision maker establish subsets of criteria in the following way: level 1 s : at the level 1 s group the criteria from the set s whose significance is equal to the significance of the criterion 1 c or up to twice as less as the significance of the criterion 1 c ; level 2 s : at the level 2 s group the criteria from the set whose significance is exactly twice as less as the significance of the criterion 1 c or up to three times as less as the significance of the criterion 1 c ; s new model for determining criteria weights: level based weight assessment (lbwa) model 129 level 3 s : at the level 3 s group the criteria from the set s whose significance is exactly three times as less as the significance of the criterion 1 c or up to four times as less as the significance of the criterion 1 c ; … level k s : at the level k s group the criteria from the set s whose significance is exactly times as less as the significance of the criterion 1 c or up to 1k  as less as the significance of the criterion 1 c . by applying the rules presented above, the decision maker establishes rough classification of the observed criteria, i.e., groups the criteria according to the levels of significance. if the significance of the criterion j c is denoted by ( ) j s c , where  1, 2, ,j n , then we have 1 2 ks s s s    , where for every level  1, 2, ,i k , the following applies     1 2 , , , : ( ) 1 si i i i j j s c c c c s i s c i      (1) also, for each  , 1, 2, ,p q k such that p q holds p qs s   . thus, in this way is well defined partition of the set of criteria s. step 3. within the formed subsets (levels) of the influence of the criteria it is performed the comparison of criteria by their significance. each criterion pi i c s in the subset   1 2 , , , si i i i s c c c is assigned with an integer  0,1, , pi i r such that the most important criterion 1 c is assigned with 1 0i  , and if pi c is more significant than qi c then p q i i , and if pi c is equivalent to qi c then p q i i . where the maximum value on the scale for the comparison of criteria is defined by applying the expression (2)  1 2max , , , kr s s s (2) step 4. based on the defined maximum value of the scale for the comparison of criteria (r), the equation (2), it is defined the elasticity coefficient 0 r n (where n presents the set of real numbers) which should meet the requirement where 0 r r ,  1 2max , , , kr s s s . step 5. calculation of the influence function of the criteria. the influence function :f s r is defined in the following way. for every criterion pi i c s can be defined the influence function of the criterion 0 0 ( ) p p i i r f c i r i    (3) where i presents the number of the level/subset in which is classified the criterion, 0 r presents the elasticity coefficient, while  0,1, , pi i r presents the value assigned to the criterion pi c within the observed level. step 6. calculation of the optimum values of the weight coefficients of criteria. by applying the equation (4) it is calculated the weight coefficient of the most significant criterion: 1 2 1 1 ( ) ( ) n w f c f c     (4) k zizovic and pamucar/decis. mak. appl. manag. eng. 2 (2) (2019) 126-137 130 the values of the weight coefficients of the remaining criteria are obtained by applying the expression (5) 1 ( ) j j w f c w  (5) where 2,3, ,j n , and n present total number of criteria. 3. application of the lbwa model in the following section it is presented the application of the lbwa model in determining weight coefficients of criteria in the multi-criteria problems discussed in the literature. in the first section, the multicriteria problem of prioritizing railway level crossings for safety improvements is presented (pamucar et al., 2013), while in the second section the problem of determining weight coefficients in evaluating the performance of suppliers is considered (chatterjee et a., 2018). example 1. determination of the weight coefficients of criteria for the evaluation of level crossings in the research conducted by pamucar et al (2013), eight criteria were identified that influence the selection of the level crossings for the installation of necessary equipment for increasing traffic safety at the observed crossing: c1 rail traffic frequency at the observed crossing, c2 road traffic frequency at the observed crossing, c3 number of tracks at the observed crossing, c4 maximum allowed train speeds at the crossing chainage, c5 rail and road crossing angle, c6 number of extraordinary events at the observed crossing in the past year, c7 sight distance of the observed crossing from the aspect of road traffic and c8 investment value of the activities in terms of the width of the crossing. the following section presents the application of the lbwa model in calculating the weight coefficients of criteria for the evaluation of level crossings: step 1. determining the most important criterion from the set of criteria  1 2 8, , ,s c c c . in the defined problem, the criterion 2c is selected as the most important/influential criterion. step 2. grouping criteria by levels of significance. in accordance with the preferences of the decision makers, the criteria are grouped in the following subsets/levels: level : the criteria 1 3 5 6 , , ,c c c c and 7 c are up to twice as less significant as the criterion 2 c and level : (2) the criteria 4 c and 8 c are between twice and three times less significant than the criterion 2 c . than, based on the preferences mentioned the criteria can be grouped in the following subsets/levels:     1 2 1 3 5 6 7 2 4 8 , , , , , , , . s c c c c c c s c c   step 3. within the formed subsets/levels of criteria influence, a comparison of the criteria with respect to their significance is made. based on the equation (2), it is defined the maximum value of the scale for comparing the criteria       1 2 1 3 5 6 7 1 2 2 4 8 , , , , , max , 6 , s c c c c c c r s s s c c        1 s 2 s new model for determining criteria weights: level based weight assessment (lbwa) model 131 on the basis of the obtained value can be concluded that the scale for comparing the criteria ranges in the interval  0,1, ,6 pi i  . applying previously defined relations can be performed the comparison of criteria within each individual set of criteria: level : based on the preferences of the decision makers, the following relations can be defined: 2 0i  , 5 2i  , 7 3i  , 6 4i  , 1 4i  , 3 5i  . considering that the criterion 2 c has the largest influence, its value assigned is 1 0i  . to the remaining criteria are assigned the values from the predefined scale  0,1, ,6 pi i  , under the condition where if the criterion pi c has higher weight coefficient than the criterion qi c , then the condition p q i i is met. level : based on the preferences of the decision makers, the following relations can be defined: 8 1i  i 4 2i  . step 4. based on the defined maximum value of the scale for comparing the criteria 6r  , it is defined the elasticity coefficient where 0 r r , respectively, 0 7r  . step 5. defining the influence function of the criteria. if it is known that 0 6r  , it is arbitrarily determined the value 0 7r  . by applying the equation (3) the influence functions of the criteria are calculated. 2 5 7 6 1 3 8 4 7 7 7 7 7 7 7 7 ( ) 1; ( ) ; ( ) ; ( ) ; 1 7 0 7 1 7 2 9 1 7 3 10 1 7 4 11 7 7 7 7 7 7 7 7 ( ) ; ( ) ; ( ) ; ( ) . 1 7 4 11 1 7 5 12 2 7 1 15 2 7 2 16 f c f c f c f c f c f c f c f c                                  step 6. calculation of the optimum values of the weight coefficients of criteria. by applying the equation (4) it is calculated the weight coefficient of the most influential criterion 2 0.778 0.700 1 ... 0 8 1 0.19 1 .43 w       the values of the weight coefficients of the remaining criteria are obtained by applying the equation (5). therefore, for the criterion 1 c it is obtained the weight coefficient 1 1 2 ( ) 0.636 0.191 0.121w f c w     . in the similar way are obtained the values of the weight coefficients of the remaining criteria which meet the condition where 1 1 n jj w   . 3 3 2 4 4 2 5 5 2 7 7 2 6 6 2 8 8 2 ( ) 0.583 0.191 0.111; ( ) 0.438 0.191 0.084; ( ) 0.778 0.191 0.148; ( ) 0.700 0.191 0.134; ( ) 0.636 0.191 0.121; ( ) 0.467 0.191 0.089. w f c w w f c w w f c w w f c w w f c w w f c w                               finally, it is obtained the vector of the weight coefficients  0.121,0.191,0.111,0.084,0.148,0.121,0.134,0.089 t j w  . by comparing the values of the weight coefficients obtained using the lbwa model with the weight coefficient values from the study made by pamucar et al (2013), it can be noted that almost identical weight values are obtained, which confirms successful validation of the lbwa model. 1 s 2 s zizovic and pamucar/decis. mak. appl. manag. eng. 2 (2) (2019) 126-137 132 considering that the value of the elasticity coefficient 0 r in this example is defined arbitrarily as 0 7r  , in the following part (figure 1) is presented the influence of the value 0 r to the change of the values of the weight coefficients of criteria. 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 r0=7 r0=8 r0=9 r0=10 r0=11 r0=12 r0=13 w1 w2 w3 w4 w5 w6 w7 w8 figure 1. influence of the value of 0 r to the change of the weight coefficients values from the figure 1 it can be noted that the parameter 0 r in certain measure can cause smaller changes of the weight coefficients values. the parameter 0 r allows decision makers to make fine adjustments of the weight coefficients values in accordance with their own preferences. the authors recommend the initial values of the weight coefficients to be defined on the basis of the value of the parameter 0 1r r  . after the definition of initial values, decision makers can make additional adjustment of weight coefficients by changing the parameter 0 r . example 2. determination of the weight coefficients of criteria in the evaluation of the work of advisors in the transport of dangerous goods in the research carried out by pamucar et al (2019), nine criteria were identified for the evaluation of the work of advisors in the transport of dangerous goods: c1 knowledge of regulations and professional development, c2 analytic processing of established requirements, c3 quality of proposed measures, c4 level of realization of the proposed measures, c5 quality of professional training of employees, c6 response method in emergency situations, c7 document preparation, c8 method of solving professional questions and c9 activity in professional institutions. the weight coefficients of the criteria for evaluating the work of advisors in the transport of dangerous goods are defined using the lbwa model: step 1. determining the most important criterion from the set of criteria  1 2 9, , ,s c c c . as the most significant/influential criterion, it is selected the criterion 5 c within the defined problem. new model for determining criteria weights: level based weight assessment (lbwa) model 133 step 2. grouping criteria by levels of significance. in accordance with the preferences of the decision makers, the criteria are grouped in the following subsets/levels: level : the criteria 1 8 ,c c and 9 c are up to twice as less important as the criterion 5 c , level : the criteria 3 c and 4 c are between twice or three times as less important as the criterion 5 c level : the criterion 6 c is between four or five times as less important as the criterion 5 c , level : the criteria 2 c and 7 c are between seven or eight times as less important as the criterion 5 c then, based on the mentioned preferences of the decision makers the criteria can be grouped in the following subsets/levels:         1 1 5 8 9 2 3 4 3 4 6 5 6 7 2 7 , , , , , , , , , , . s c c c c s c c s s c s s s c c          step 3. based on the equation (2), it is defined the maximum value of the scale for the comparison of criteria           1 1 5 8 9 2 3 4 3 1 2 4 7 4 6 5 6 7 2 7 , , , , , , , max , , , 4 , , , . s c c c c s c c s r s s s s s c s s s c c                    based on the maximum value of the scale for comparison, it can be concluded that the scale for comparing the criteria ranges in the interval  0,1, , 4 pi i  . based on the scale and the pre-defined set of criteria, it can be performed the comparison of criteria within each individual set: level 1 s : based on the preferences of the decision makers, the following relation are defined: 5 0i  , 8 1i  , 9 2i  and 1 4i  . level 2 s : within the set 2 s the following relations are defined: 3 1i  and 4 2i  . level 4 s : within the set 4 s the following relation is defined 6 2i  . level 7 s : within the set 7 s the following relations are defined: 2 1i  and 7 3i  . step 4. based on the defined maximum value of the scale for the comparison of criteria 4r  , it is defined the elasticity coefficient such that 0 r r , respectively, 0 4.r  step 5. defining the influence function of the criteria. if it is known that 0 4r  , arbitrarily is determined the value 0 5r  . by applying the equation (3) the influence functions of the criteria are calculated. 1 s 2 s 4 s 7 s zizovic and pamucar/decis. mak. appl. manag. eng. 2 (2) (2019) 126-137 134 5 8 9 1 3 4 6 2 7 5 5 5 5 5 ( ) 1; ( ) ; ( ) ; 1 5 0 1 5 1 6 1 5 2 7 5 5 5 5 5 5 ( ) ; ( ) ; ( ) ; 1 5 4 9 2 5 1 11 2 5 2 12 5 5 5 5 5 5 ( ) ; ( ) ; ( ) . 4 5 2 22 7 5 1 36 7 5 3 38 f c f c f c f c f c f c f c f c f c                                     step 6. calculation of the optimum values of the weight coefficients of criteria. by applying the equation (4) it is calculated the value of the weight coefficient of the most influential criterion 5 0.833 0.714 4 ... 0 2 1 0.22 1 .13 w       by applying the equation (5) are obtained the values of the weight coefficients of the remaining criteria: 8 8 5 4 4 5 9 9 5 6 6 5 1 1 5 2 2 5 3 3 5 ( ) 0.883 0.224 0.186; ( ) 0.417 0.224 0.093; ( ) 0.714 0.224 0.160; ( ) 0.227 0.224 0.051; ( ) 0.556 0.224 0.124; ( ) 0.139 0. ( ) 0.455 0.224 0.102; w f c w w f c w w f c w w f c w w f c w w f c w w f c w                                   7 7 5 224 0.031; ( ) 0.132 0.224 0.029.w f c w       finally, the vector of the weight coefficients is obtained  0.124;0.03;0.102;0.093;0.224;0.051;0.029;0.186;0.160 t j w  . in this example, the value of the elasticity coefficient 0 r is arbitrarily defined as 0 5r  , and in the following part (figure 2) is presented the influence of the value 0 r to the change of the weight coefficients of criteria. w1 w2 w3 w4 w5 w6 w7 w8 w9 0 0.05 0.1 0.15 0.2 0.25 r0=5 r0=5 r0=7 r0=8 r0=9 r0=10 r0=11 r0=12 r0=13 figure 2. influence of the value to the change of the weight coefficients from the figure 2 can be observed that the changes in the elasticity coefficient lead to minor changes in the weight coefficients of criteria. this feature of the lbwa model allows additional adjustment of the weight coefficients in accordance with the decision makers preferences. 0 r new model for determining criteria weights: level based weight assessment (lbwa) model 135 4. discussion of results and conclusion literature review and the analysis of the models for determining weight coefficients of criteria present in the literature so far clearly indicate the need for the development of a new credible model for determining weight coefficients of criteria. therefore, in this paper is presented a new model, the lbwa model, which is characterized by simple and rational mathematical algorithm. the results of this study have shown that the lbwa model allows obtaining credible and reliable weight coefficients that contribute to rational judgment, and thus to obtaining credible results in decision-making process. based on the results presented can be outlined the following advantages of the lbwa model: (1) the lbwa model allows the calculation of weight coefficients with small number of criteria comparisons, only comparison; (2) the lbwa model algorithm does not become more complex with the increase of the number of criteria, which makes it suitable for use in complex mcdm models with a larger number of evaluation criteria; (3) the lbwa model allows decision makers to present their preferences through logical algorithm when prioritizing criteria. using the lbwa model, optimal values of weight coefficients are obtained with simple mathematical apparatus that eliminates inconsistencies in expert preferences, which are tolerated in certain subjective models (bwm and ahp); (4) flexibility of the model in terms of using all the values from the predefined scale, i.e., it is not limited to integer values from the defined interval. in addition to the mentioned advantages, it is necessary to emphasize the flexibility of the lbwa model in terms of additional corrections of weight coefficients values by the elasticity coefficient ( ). the elasticity coefficient allows decision makers to further adjust weight coefficients values in accordance with their own preferences. in addition, the elasticity coefficient allows the analysis of the robustness of the mcdm model by defining the effect of the change of the criteria weight coefficients on the final decision. in order to approach users and exploit all the advantages of the lbwa model, the need for software development and implementation in real-world applications is imposed. one of the directions of future research should cover the extension of the algorithm for the application in group decision making. also, one of the directions of future research should be the extension of the lbwa model using different uncertainty theories (neutrosophic sets, fuzzy sets, rough numbers, gray theory, etc.). the implementation of the lbwa model in uncertain environment will enable the processing of expert preferences, even in cases where the information about the considered problem are partially accessible or even very little known. this would enable more objective expression of the decision makers' preferences by respecting subjectivity and lack of information on certain phenomena. literature bana e costa, c., & vansnick, j.c. 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(2015). an integrated ahp and vikor for design concept evaluation based on rough number, advanced engineering informatics, 29, 408–418. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering issn: 2560-6018 eissn:2620-0104 doi: https://doi.org/10.31181/dmame0306102022c * corresponding author. e-mail addresses: divya.jain@juetguna.in (d. jain), hrishi4676@gmail.com (r. chaurasiya). hybrid mcdm method on pythagorean fuzzy set and its application rishikesh chaurasiya1 and divya jain1 * 1 department of mathematics, jaypee university of engineering and technology, india received: 8 may 2022; accepted: 14 july 2022; available online: 6 october 2022. original scientific paper abstract: here in this article a hybrid mcdm method on the pythagorean fuzzy-environment is presented. this method is based on the pythagorean fuzzy method based on removal effects of criterion (pf-merec) and stepwise weight assessment ratio analysis (swara) approaches. the objective and subjective weights are assessed by pf-merec, swara model and the preference order ranking of the various alternatives is done through complex proportional assessment (copras) framework on the pfs. the proposed method is the hybrid model of merec, swara and copras methods. further, the proposed model is used to identify the best banking management software (bms) so that the bank can choose the robust bank management software tool to enhance its efficiency and excellence. thereafter, sensitivity analysis and comparative discussion of the proposed model is done with the existing techniques to judge the reasonability and efficiency of the proposed model. key words: pythagorean fuzzy set, decision-making, merec, swara, copras, banking management software. 1. introduction there are many uncertain, fuzzy and incomplete problems in the real world. hence, the fuzzy set theory, originated by zadeh (1965) is a successful and vigorous tool for determining many same issues. to overcome its primary extension and shortcomings, intuitionistic fuzzy sets (ifs) has been established by atanassov (1986), it satisfies the requirement of sum total of membership function (mf) and non-mf (n-mf) is less than or equal to one. nevertheless, there may be difficulties in the policymaking procedure when both the fs and the ifs theories are not capable of addressing the uncertain and incompatible data. viz., if a decision expert assigns 0.8 and 0.4 as his preference of belonging and non-belongingness of any object, then plainly, it can be easily seen that 0.8 + 0.4 > 1. hence, this situation is not handled by ifss. to beat these shortcomings of ifss, initially, yager (2013) presented the fundamental of the mailto:divya.jain@juetguna.in mailto:hrishi4676@gmail.com chaurasiya et al./decis. mak. appl. manag. eng. (2022) 2 pfss. in a pythagorean fuzzy set mf and n-mf satisfies the condition (0.8)2 + (0.4)2 ≤ 1. in the pfs is a good device for expressing uncertain information ascending in practical, complicated mcdm problems. it has the same provision as ifss, however has a lot of flexibility and more space to express fuzzy information than ifs. in this regard, pfs has attracted the eye of many scholars and has been studied extensively in management. some pf-aggregation operators are also presented such as pf-weighted averaging operators (garg, 2019; pamucar & jankovic, 2020; rong et al., 2020; akram et al., 2021; farid & riaz, 2022) to help tackle mcdm problems. “einstein geometric aggregation operators employing a new complex-ivpfs” (ali et al., 2021). some researchers presented the score functions on pfs (zhang & xu, 2014; peng & yuan, 2017) can accurately rank general choices and also has a strong sense of partiality by taking hesitant information into account. moreover, some researchers focused on pythagorean fuzzy objective weighting methods (biswas & sarkar, 2019; ozdemir & gul, 2019) and subjective weight (wei, 2019; chen, 2019; wang et al., 2019; zavadskas et al., 2020). the subjective weights are submitted by dms supported in their own knowledge, whereas they neglect the primary weight info explained by the valuation data. some novel approaches to obtaining objective weight from assessment data don’t take into account the des’ preferences. so, a combined weighting approach is submitted, which may amalgamate each subjective and objective weight (ow). many multi-criteria decision-making (mcdm) approaches are dealing with a massive quantity of problems and estimating alternative and help the user in mapping the problem. criterion weights has an important role in the decision-making (dm) procedure, as the suitable selection of criterion weights is best for ranking of alternatives. thereafter, it’s vital to discover a method to define the weights. some approaches have been available in the literature. as a result, many scholars have studied the ow by criteria importance through intercriteria correlation (critic) and entropy measure of pfss. xu et al. (2020) proposed an entropy measure on pfs to solve mcdm problems. chaurasiya and jain (2021) proposed marcos method on ifss. the authors have applied the predictable mcdm method in various fields (rani & jain, 2019; petrovic et al., 2019; eiegwa, 2020; mishra et al., 2021; li et al., 2022; yildirim & yildirim, 2022). in addition, criterion weight is very significant in solving mcdm difficulties. therefore, the authors have moved their attention to methods related to criterion weight. keshavarz et al. (2021) developed merec technique is one of the powerful approaches for defining the objective criterion weights (ocws). whereas, among the innovative technique to determining criteria weight (zizovic & pamucar, 2019). hadi and abdullah (2022) presented integrate merec-topsis (technique for order of preference by similarity to ideal solution) method for iot-based hospital place selection. hezam et al. (2022) proposed an if-merec-ranking sum-double normalization-based multi-aggregation method for evaluating alternative fuel vehicles concerning sustainability. marinkovic et al. (2022) employed the mereccombined compromise solution multi-criteria method to evaluate the application of waste and recycled materials to production. integrated merec method on fermatean fuzzy environment proposed rani et al. (2022), merec-multimoora (mishra et al., 2022), merec-marcos (nguyen et al., 2022), level based weight assessment-zmairca method (bozanic et al., 2020). kersuliene et al. (2010) has established the swara approach to be an effective device for calculating the scws. alipour et al. (2021) employed a combined swara and copras technique to assess the supplier selection of fuel cell and hydrogen constituents in the pfs domain. saraji et al. (2021) proposed the hesitant fuzzyhybrid mcdm method on pythagorean fuzzy set and its application 3 swara-multimura method for online education. some researchers have drawn attention to integrated methods to solve the mcdm problems, such as (rani et al., 2020) developed a new integrate swara-aras method on pfs for healthcare waste treatment problem. since at present, many scholars have developed the following ranking methods to solve mcdm problems. for examples, badi and pamucar (2020) proposed integrate grey-marcos methods for supplier selection. durmic et al. (2020) proposed an integrated full consistency method-rough-simple additive weighting (fucom-rsaw) method has been employed to choose sustainable suppliers. tesic et al. (2022) presented dibr-fuzzy-marcos framework. puska et al. (2020) suggested a way for measurement of alternatives and ranking according to compromise solution (marcos) method for project management software. some researchers applied mcdm methods such as (kaya, 2020; pamucar, 2020; keshavarz-ghorabaee, 2021; ashraf et al., 2022). the copras method, established by zavadskas et al. (1994) is one of the practical well-orderly approaches to solve intricate mcdm difficulties. the main objective of the copras approach, includes: (i) it is an appropriate and assess method to obtain the solution to the dm issue. (ii) it considers the ratio of the worst and the best outcome; (iii) it provides results in a short-time as compared to other mcdm methods. several researchers have used the copras technique for various applications (mishra et al., 2020). these days, various academicians have expanded the traditional copras technique under a range of vague environments. zheng et al. (2018) studied a hesitant fuzzy (hf) copras approach to solving the health decision-making problem. thereafter, mishra et al. (2019) proposed the integrated hf-copras method to solve service quality problems. the pf-copras approach has been used by (rani et al., 2020a) to appraise pharmaceutical therapy for type-2 diabetic disease. song and chen (2021) proposed the copras method on the probabilistic hfs, which is based on the new distance measures of probabilistic hf-elements. for waste-to-energy technology selection, (mishra et al., 2022) suggested the copras approach based on the ivpfsimilarity measure. currently, chaurasiya and jain (2022) have submitted the copras technique on pfs in the mcdm problems which, competently launch the interrelation among criteria & permits decision experts (de’s) to catch the uncertainty elaborate in judgments of numerous incompatible criterions. the main motivation for this study is, a new hybrid pf-merec-swara-copras method is established that can efficiently deal with the implicit vagueness and uncertainty concerned with de’s judgment. therefore, the summary of the article is as follows: 1) to develop a novel hybrid pf-merec-swara-copras method under the pfdomain. 2) we calculate the decision experts’ weights in pfs based on (boran et al., 2009) formula. 3) to calculate objective criterion weights, by new merec and subjective criterion weight by swara method. thereafter, we calculate combined criterion weights. 4) the proposed technique is employed to solve the problem of selecting banking management software. subsequently, its method is compared with other existing methods and sensitively analysis by taking a set of criterion weights. the paper is planned as follows: in section 2, we describe fundamental on pfss. section 3, presents a novel hybrid mcdm method on pythagorean fuzzy set. section 4, a case study of banking management software selection, which illustrates the chaurasiya et al./decis. mak. appl. manag. eng. (2022) 4 efficiency and applicability of the advanced method. along with it, sensitivity analysis and the results are compared with already existing methods to validate. finally, in 5th section, conclusion and future outlook is considered. 2. basic concept of pfs this section delivers a brief-overview of the pfs. definition 2.1. (yager, 2013) a pfs 𝐴 ⊂ 𝑈 in a fixed set defined as: 𝐴 = {〈𝑢𝑖,𝜇𝐴(𝑢𝑖),𝜈𝐴(𝑢𝑖)〉| 𝑢𝑖 ∈ 𝑈} (1) where 𝜇𝐴(𝑢𝑖):𝑈 → [0,1] indicate the mf and 𝜈𝐴(𝑢𝑖):𝑈 → [0,1] indicate the n-mf that mollify the state 0 ≤ 𝜇𝐴 2(𝑢𝑖) + 𝜈𝐴 2(𝑢𝑖) ≤ 1. the hesitancy function 𝜋𝐴(𝑢𝑖) is denoted by 𝜋𝐴(𝑢𝑖) = √1 − 𝜇𝐴 2(𝑢𝑖) − 𝜈𝐴 2(𝑢𝑖), then it is pythagorean fuzzy-index. definition 2.2. (peng & li, 2019) let 𝛽 = (μ𝛽,ν𝛽) be a pfn. the modified normalized score and accuracy function of 𝛽 is given as: 𝒮∗(𝛽) = 2(μ𝛽) 2 +(1−(ν𝛽) 2 )+((μ𝛽) 2 ) 2 4 and ℏ°(𝛽) = 1 − ℏ(𝛽), (2) where 𝒮∗(𝛽),ℏ°(𝛽) ∈ [0,1]. definition 2.3. (yager, 2013a, b) assume 𝛽 = (𝜇𝛽,ν𝛽),𝛽1 = (𝜇𝛽1,ν𝛽1) and 𝛽2 = (𝜇𝛽2,ν𝛽2) be pfns. where the operations on the pfns are depicted below as: (i) 𝛽𝑐 = (μ𝛽,ν𝛽); (ii) 𝛽1 ⊕ 𝛽2 = (√μ𝛽1 2 + μ𝛽2 2 − μ𝛽1 2 μ𝛽2 2 , ν𝛽1ν𝛽2 ); (iii) 𝛽1⨂ 𝛽2 = (μ𝛽1μ𝛽2 , √ν𝛽1 2 + ν𝛽2 2 − ν𝛽1 2 ν𝛽2 2 ); (iv) 𝜆𝛽 = (√1 − (1 − μ𝛽 2) 𝜆 , (ν𝛽) 𝜆 ),𝜆 > 0; (v) 𝛽𝜆 = ((μ𝛽) 𝜆 , √1 − (1 − ν𝛽 2) 𝜆 ) ,𝜆 > 0. 3. pythagorean fuzzy merec-swara-copras method in this section, we have developed a new decision-making scheme, as hybrid pfmerec-swara-copras method, to deal with the mcdm problems on pfs domain. the present method uses the merec method to evaluate the ocws. this method uses the removal effect of each criterion on the performance of the alternatives to calculate the objective criterion weights. the swara method is an effective tool for evaluating scws. thereafter, we have calculated criteria weights by combined formula. whereas, the copras technique uses the notion of relative degree to assess the importance of the ranking of the alternatives. during this method, the relative degree that describes the complex relative proficiency best selection is directly proportional to the comparative outcome and criterion weights pondered in the decision-making issue. so, we combine these three methods on pfss to get additional precise and suitable judgments in an ambiguous reference. it is based on merec-swara and copras hybrid mcdm method on pythagorean fuzzy set and its application 5 method under the pfs. the working procedure of the hybrid framework is as given below (see figure 1): step 1. for a mcdm problem under pf-domain, assume alternatives 𝑇 = {𝑇1,𝑇2,… ,𝑇𝑚} and the features/criteria 𝐹 = {𝐹1,𝐹2,… ,𝐹𝑛}. a group of decision expert’s (de’s) 𝐸 = {𝐷𝐸1,𝐷𝐸2,… ,𝐷𝐸𝑙 } represents their ideas on each alternative 𝑇𝑖 with respect to each criterion 𝐹𝑗 in terms of linguistic values (lvs). let 𝑋 = (𝑥𝑖𝑗 (𝑘) ) be a linguistic decision matrix recommended by the de’s, where 𝑥𝑖𝑗 (𝑘) present to the assessment of an alternative 𝑇𝑖 regarding a criterion 𝐹𝑗 in forms of lvs for 𝑘 𝑡ℎ de. step 2. calculate a primary de’s weights (𝜆𝑘). for the judgements of the 𝑘 𝑡ℎ de’s weight, let 𝐸𝑘 = (𝜇𝑘,𝜈𝑘,𝜋𝑘) be a pfns, then 𝜆𝑘 = (𝜇𝑘 2 + 𝜋𝑘 2 × ( 𝜇𝑘 2 𝜇𝑘 2 + 𝜈𝑘 2)) ∑ (𝜇𝑘 2 + 𝜋𝑘 2 × ( 𝜇 𝑘 2 𝜇𝑘 2 + 𝜈𝑘 2)) ℓ 𝑘=1 (3) here 𝜆𝑘 ≥ 0, ∑ 𝜆𝑘 = 1 ℓ 𝑘=1 . step 3. define the aggregated pythagorean fuzzy decision matrix (apf-dm), corresponding to expert’s weight. let ℕ = ( 𝑖𝑗)𝑚×𝑛 be the apf-dm, where 𝑖𝑗 = (√1 − ∏ (1 − 𝜇𝑘 2)𝜆𝑘 ℓ 𝑘=1 , ∏ (𝜈𝑘) 𝜆𝑘 ℓ 𝑘=1 ) (4) step 4. determination of criteria weights (cws) step 4.1. estimate objective criteria weights (ocws) by merec technique using following steps: step 4.1a. evaluate the score matrix 𝒮∗( 𝑘𝑗) = ( 𝑖𝑗)𝑚×𝑛 using equation (2) of each pfn 𝑖𝑗. step 4.1b. normalize the apf-dm (ℕ1) = 𝑛𝑖𝑗 𝑥 . the decision-matrix components are scaled using a linear normalization. the elements of the normalized-dm are denoted by 𝑛𝑖𝑗 𝑥 . here 𝐹𝑏 represents beneficial criteria and 𝐹𝑐 represents cost criteria. 𝑛𝑖𝑗 𝑥 = { 𝑚𝑖𝑛 𝑘 𝑥𝑘𝑗 𝑥𝑖𝑗 , 𝑖𝑓 𝑗 ∈ 𝐹𝑏 𝑥𝑖𝑗 𝑚𝑎𝑥 𝑘 𝑥𝑘𝑗 , 𝑖𝑓 𝑗 ∈ 𝐹𝑐 (5) step 4.1c. compute the entire performance of the alternatives (ω𝑖). a logarithmic function with identical cws is employed to get alternative entire performance. ω𝑖 = ln(1 + ( 1 𝑛 ∑ |ln(𝑛𝑖𝑗 𝑥 )| 𝑛 𝑗=1 )) (6) step 4.1d. estimate the behavior of the alternatives by eliminating each criterion. the same logarithmic function as in step 4.1c is employed, the only difference is, the alternative appraisals are calculated on the basis of eliminating each criterion individually in this step. hence, we have n sets of appraisals corresponding to 𝑛 criteria. assume ω𝑖𝑗 ′ represent the entire evaluation of 𝑖𝑡ℎ alternative for eliminating the jth criterion. the following process of appraisal using eq. (7): chaurasiya et al./decis. mak. appl. manag. eng. (2022) 6 ω𝑖𝑗 ′ = ln(1 + ( 1 𝑛 ∑ |ln(𝑛𝑖𝑘 𝑥 )| 𝑘,𝑘≠𝑗 )) (7) step 4.1e. calculate the summation of absolute deviations (𝐷𝑗). we use the eqs. (6), (7) 𝐷𝑗 = ∑ |𝛺𝑖𝑗 ′ − ω𝑖| 𝑚 𝑖=1 (8) step 4.1f. evaluate final ocws. the 𝐷𝑗 is employed to compute the objective weight of each criterion in this step. the process is applied to calculate ϖj. ϖj = 𝐷𝑗 ∑ 𝐷𝑗 𝑛 𝑗=1 (9) step 4.2. determine the subjective criteria weights (scws) by swara technique. the procedures for assessment of the scws using the swara technique is given follow as: step 4.2a. analyze the conventional values. primary, score values 𝒮∗( 𝑘𝑗) of pfns by (2) are calculated using apf-dm. step 4.2b. compute the rank of criteria by the expert’s insight from the greatest significant to the smallest significant criteria. step 4.2c. find the relative significance (𝑠𝑗) of the mean value. relative position is evaluated from the criteria that are placed at second location. the subsequent relative importance is obtained by comparing the criteria located at 𝐹𝑗 to 𝐹𝑗−1. step 4.2d. evaluate the relative coefficient (𝑐𝑗) by eq. (10) 𝑐𝑗 = { 1 , 𝑗 = 1 𝑠𝑗 + 1, 𝑗 > 1 (10) where, 𝑠𝑗 is relative significance. step 4.2e. calculate the weights (𝑝𝑗), as given by eq. (11). 𝑝𝑗 = { 1 , 𝑗 = 1 𝑐𝑗−1 𝑐𝑗 , 𝑗 > 1 (11) step 4.2f. compute scaled weight. in common, the criterion weights are discussed by the expression. 𝜔𝑗 = 𝑝𝑗 ∑ 𝑝𝑗 𝑛 𝑗=1 (12) step 4.3. evaluate the combining cws. in the mcdm technique, all criteria have varying degrees of significance. let 𝑤 = (𝑤1,𝑤2,…,𝑤𝑛) 𝑇 be a set of cws with ∑ 𝑤𝑗 = 1 𝑛 𝑗=1 and 𝑤𝑗 ∈ [0,1], given as: 𝑤𝑗 = ϖj∗𝜔𝑗 ∑ ϖj∗𝜔𝑗 𝑛 𝑗=1 (13) step 5. ranking of the alternative by copras method. the values of benefit-(𝜎𝑖) and cost-(𝜑𝑖) type criteria, 𝑖 = 1(1)𝑚 is given as: 𝜎𝑖 =⊕ 𝑗=1 𝑛 𝑤𝑗 𝑖𝑗, (for benefit-type) (14) 𝜑𝑖 = ⊕ 𝑗=𝑙+1 𝑛 𝑤𝑗 𝑖𝑗, (for cost-type) (15) hybrid mcdm method on pythagorean fuzzy set and its application 7 step 6. evaluate the relative degree (𝛿𝑖) of each alternative as follows: 𝛿𝑖 = 𝒮 ∗(𝜎𝑖) + ∑ 𝒮∗(𝜑𝑖) 𝑚 𝑖=1 𝒮∗(𝜑𝑖) ∑ 𝒮 ∗(𝜑𝑖) 𝑚 𝑖=1 (16) where 𝒮∗(𝜎𝑖) and 𝒮 ∗(𝜑𝑖) represents the score values of 𝜎𝑖 and 𝜑𝑖. step 7. compute the utility degree (𝛾𝑖). using eq. (17) 𝛾𝑖 = 𝛿𝑖 max (𝛿𝑖) × 100% (17) step 8. find the best ranking of alternatives. figure 1. representation of the pf-hybrid method for bms selection 4. application in banking management software all over the world, the banks are being digitized with the assistance of information technology tools. it provides extraordinary speed to banking operations. thus, to be successful in banking services, one has to offer the best banking software choosing the best banking software requirements. the opinion of banking experts needs to become technologically more innovative to meet all the requirements and expectations of the clients. banking software is a means of communication between the bank and the user. chaurasiya et al./decis. mak. appl. manag. eng. (2022) 8 it serves to improve the workflow within the company and its branches, for easier investment policies, and to provide services that address the necessity of the users. which is offers greater functionality, convenience, flexibility, reliability, security, instant transfers, mobile apps, the ability to remain adaptable and modern to meet the changing nature of market needs and competitiveness. innovations in information communication technology (ict) and globalization are constantly changing business processes. these alterations range from easy structural changes to paradigm shifts laudon and laudon (2015). the bank’s goal is to alleviate costs, increase efficiency and guarantee client holding with the use of technology. in the banking sector, the relationship among organizations and its clients is vital. technological advancement enables closer and longer-terms affinities with clints. the cbs developed in the 1970s and has undergone important changes over time. the upgraded core banking system has capability of real-time processing and multichannel unification (kreca & barac, 2015). due to the growing issues of electronic payments, some researchers and managers have turned their attention to banking software. for this mcdm methods are best suited that can based on numerous criteria. recently, due to digitization in the banking sector, it became very important to select the best banking management software. it provides extraordinary speed to banking operations. thus, to be successful in banking services, one has to offer the best banking software choosing the best banking software requirements. here, a case study of bms for a banking area in india is measured to demonstrate the applicability and practicality of the evolved pf-merec-swara-copras method. in the procedure of existing method, the bank shaped a team entailing of four decision experts who are responsible for bms. let the various banking tools available with us are (figure 2): mambu (𝑇1), temenos (𝑇2), oracle flexcube (𝑇3), finastra (𝑇4) and finacle (𝑇5). we have to identify the best software tool for any banking management based on the following important features (criteria’s): customizable interfaces (𝐹1), data management and history tracking (𝐹2), documentation (𝐹3), live customer support (𝐹4), online payments and bills (𝐹5), mobile version (𝐹6), self-service options for clients (𝐹7), transaction processing (𝐹8). figure 2. the selection of bms https://www.softwaresuggest.com/caresoft-his https://www.softwaresuggest.com/hospital-mgt-sy-by-genipulse hybrid mcdm method on pythagorean fuzzy set and its application 9 table 1 presents the lvs given in pfns for the relative behavioral rating of weights. table 1. linguistic values (lvs) in terms of pfns lvs pfns thrillingly significant (ths) (0.90, 0.10) typically significant (ts) (0.80, 0.20) noteworthy (n) (0.60, 0.40) reasonable (r) (0.50, 0.50) inconsequential (ic) (0.45, 0.55) trivial (tr) (0.30, 0.75) pitty (p) (0.10, 0.90) table 2 shows the weight of each de’s as calculated using eq. (3). table 2. decision expert weights decision experts lvs pfns weights (𝝀𝒌) de1 n (0.60,0.40) 0.2749 de2 r (0.50,0.50) 0.2257 de3 ts (0.80,0.20) 0.3058 de4 ic (0.45,0.55) 0.1936 for assessing the alternatives linguistic values are transformed in terms of pfns. table 3. lv for assessing the alternatives lvs pfns extremely small (es) (0,1) very small (vs) (0.10, 0.90) small (s) (0.20, 0.80) slightly small (ss) (0.30, 0.70) below intermediate (bi) (0.40, 0.60) intermediate (i) (0.50, 0.50) above intermediate (ai) (0.60, 0.40) slightly big (sb) (0.70, 0.30) big (b) (0.80, 0.20) very big (vb) (0.90, 0.10) extremely big (eb) (1, 0) here, table 4 represents the ideas of de’s on each of the alternative 𝑇𝑖 respect to each criterion 𝐹𝑗 in terms of lvs defined in table 3. table 4. the lv’s calculation of alternatives given by de’s alternative des criteria 𝐹1 𝐹2 𝐹3 𝐹4 𝐹5 𝐹6 𝐹7 𝐹8 𝑻𝟏 de1 vb vb b b vb b vb bi de2 b sb sb i b vb b ss de3 b vb vb sb vb b vb i chaurasiya et al./decis. mak. appl. manag. eng. (2022) 10 de4 vb vb b b ss b b ai 𝑻𝟐 de1 vb b sb sb vb vb vb bi de2 b b i sb b sb b ai de3 vb vb b ai b i b i de4 b b vb vb ai vb sb bi 𝑻𝟑 de1 vb b sb sb b b vb i de2 vb b sb ai ai sb b ss de3 b vb vb i vb sb b bi de4 vb ai bi vb i b bi ai 𝑻𝟒 de1 b b a i sb b b vb i de2 b b sb ai ss sb i ai de3 b b sb sb ai ai sb bi de4 b vb vb ss sb s bi i 𝑻𝟓 de1 vb b sb sb b b b i de2 b b b bi sb bi ai s de3 b b ai sb b b sb i de4 vb sb vb i ai ai ai sb in table 5, the lv’s of alternatives given by de’s in table 4 is converted to apfdm using eq. (4). table 5. computed apf-dm 𝑭𝟏 𝑭𝟐 𝑭𝟑 𝑭𝟒 𝑭𝟓 𝑭𝟔 𝑭𝟕 𝑭𝟖 𝑻𝟏 (0.8562 , 0.1445) (0.8733, 0.1281) (0.8244, 0.1773) (0.7262, 0.2784) (0.8383, 0.1704) (0.8297, 0.1710) (0.8669, 0.1337) (0.4649, 0.5432) 𝑻𝟐 (0.8669 , 0.1337) (0.8389, 0.1618) (0.7660, 0.2404) (0.7406, 0.2648) (0.8139, 0.1890) (0.7992, 0.2096) (0.8228, 0.1788) (0.4867, 0.5178) 𝑻𝟑 (0.8769 , 0.1236) (0.8179, 0.1850) (0.7646, 0.2452) (0.7078, 0.3025) (0.7819, 0.2259) (0.7529, 0.2481) (0.8026, 0.2045) (0.4619, 0.5463) 𝑻𝟒 (0.8266 , 0.1710) (0.8258, 0.1749) (0.7427, 0.2625) (0.6321, 0.3772) (0.6578, 0.3547) (0.6622, 0.3543) (0.7299, 0.2847) (0.5009, 0.5027) 𝑻𝟓 (0.8562 , 0.1445) (0.7841, 0.2163) (0.7633, 0.2417) (0.6204, 0.3873) (0.7514, 0.2506) (0.7161, 0.2931) (0.7002, 0.3028) (0.5139, 0.5036) 4.1. merec technique this measure reflects the difference between the performance of the composite option and its performance in removing the criterion. the following steps are used to calculate the ocws by merec method: we compute the score matrix using eq. (2). as {𝑭𝟓,𝑭𝟖} a set of cost/non-benefit and others are benefit type of criteria, so, we normalized-apf-dm using eq. (5) and shown in table 6. hybrid mcdm method on pythagorean fuzzy set and its application 11 table 6. normalized apf-dm 𝑭𝟏 𝑭𝟐 𝑭𝟑 𝑭𝟒 𝑭𝟓 𝑭𝟔 𝑭𝟕 𝑭𝟖 𝑻𝟏 0.9405 0.8286 0.8383 0.9689 0.6713 0.6888 0.6981 0.8804 𝑻𝟐 0.9197 0.8903 0.9506 1.0000 0.7064 0.7358 0.7658 0.9384 𝑻𝟑 0.9008 0.9308 0.9541 0.9286 0.7569 0.8119 0.7999 0.8729 𝑻𝟒 1.0000 0.9150 1.0000 0.7804 1.0000 1.0000 0.9386 0.9759 𝑻𝟓 0.9405 1.0000 0.9559 0.7594 0.8078 0.8823 1.0000 1.0000 to obtain the ocws by merec method, we compute the overall performance of the table 7. calculate the performance of the alternatives by removing each criterion. 𝑭𝟏 𝑭𝟐 𝑭𝟑 𝑭𝟒 𝑭𝟓 𝑭𝟔 𝑭𝟕 𝑭𝟖 𝑻𝟏 0.1879 0.1747 0.1760 0.1910 0.1524 0.1551 0.1566 0.1811 𝑻𝟐 0.1345 0.1309 0.1381 0.1436 0.1052 0.1098 0.1143 0.1367 𝑻𝟑 0.1221 0.1257 0.1284 0.1255 0.1027 0.1105 0.1089 0.1186 𝑻𝟒 0.0517 0.0411 0.0517 0.0218 0.0517 0.0517 0.0442 0.0488 𝑻𝟓 0.0792 0.0862 0.0810 0.0541 0.0614 0.0718 0.0862 0.0862 alternative values from eq. (6), given as (ω1 = 0.1943, ω2 = 0.1436, ω3 = 0.1336, ω4 = 0.0517, ω5 = 0.0862). apply the eq. (7), we appraise alternatives overall performances (𝛺𝑖𝑗 ′ ) in removing criterion and are given in table 7. afterward, we compute the absolute deviation (𝐷𝑗) values from eq. (8). finally, we compute ocws (𝜛𝑗) using eq. (9) by merec method. absolute deviation (𝐷𝑗) = (0.0340, 0.0508, 0.0342, 0.0734, 0.1360, 0.1105, 0.0992, 0.0380). objective weight (ϖj) = (0.0590, 0.0882, 0.0594, 0.1274, 0.2361, 0.1918, 0.1722, 0.0659). 4.2. subjective weights by swara technique the following steps are used to compute the scws by swara method. table 8. evaluate of criteria weights by de’s criteria de1 de2 de3 de4 aggregated pfns crisp values 𝓢∗(𝜺𝒌𝒋) 𝑭𝟏 vb b sb vb (0.8385, 0.1636) 0.7184 𝑭𝟐 vb sb i b (0.7688, 0.2397) 0.6185 𝑭𝟑 b ai i sb (0.6717, 0.3348) 0.4985 𝑭𝟒 sb b bi ai (0.6528, 0.3578) 0.4765 𝑭𝟓 vb i sb ss (0.7250, 0.2933) 0.5604 𝑭𝟔 b sb b ai (0.7514, 0.2506) 0.5963 𝑭𝟕 vb ai bi i (0.6974, 0.3230) 0.5262 𝑭𝟖 bi b i sb (0.6257, 0.3872) 0.4466 chaurasiya et al./decis. mak. appl. manag. eng. (2022) 12 table 9. criteria weights evaluated by swara method criteria crisp values relative significance (𝐬𝒋) relative coefficient (𝒄𝒋) recalculate d weight (𝒑𝒋) criteria weight (𝝎𝒋) 𝑭𝟏 0.7184 1.0000 1.0000 0.1459 𝑭𝟐 0.6185 0.0999 1.0999 0.9092 0.1327 𝑭𝟔 0.5963 0.0222 1.0222 0.8895 0.1298 𝑭𝟓 0.5604 0.0359 1.0359 0.8587 0.1253 𝑭𝟕 0.5262 0.0342 1.0342 0.8303 0.1211 𝑭𝟑 0.4985 0.0277 1.0277 0.8079 0.1179 𝑭𝟒 0.4765 0.0220 1.0220 0.7905 0.1153 𝑭𝟖 0.4466 0.0299 1.0299 0.7676 0.1120 (𝜔𝑗) = (0.1459,0.1327,0.1179,0.1153,0.1253,0.1298,0.1211,0.1120). there after we calculated the weights of the criteria by eqs. (13). combined weight (𝑤𝑗 ) = (0.0690, 0.0938, 0.0562, 0.1178, 0.2372, 0.1996, 0.1672, 0.0592) t. table 10. calculate the values from 𝜎𝑖 and 𝜑𝑖 𝝈𝒊 𝝋𝒊 𝑺 ∗(𝝈𝒊) 𝑺 ∗(𝝋𝒊) 𝜹𝒊 𝜸𝒊 𝑻𝟏 (0.8048, 0.2161) (0.3124, 0.8296) 0.6671 0.1291 0.7758 100.0 𝑻𝟐 (0.7729, 0.2504) (0.3223, 0.8224) 0.6222 0.1356 0.7257 93.54 𝑻𝟑 (0.7475, 0.2786) (0.3027, 0.8381) 0.5880 0.1223 0.7027 90.58 𝑻𝟒 (0.6753, 0.3593) (0.2726, 0.8559) 0.4977 0.1054 0.6307 81.30 𝑻𝟓 (0.7036, 0.3231) (0.2695, 0.8587) 0.5327 0.1033 0.6685 86.17 from equations (14)-(17), the values of 𝜎𝑖,𝜑𝑖,𝑆 ∗(𝜎𝑖),𝑆 ∗(𝜎𝑖),𝛿𝑖 𝑎𝑛𝑑 𝛾𝑖 of 𝑇𝑖 are assessed with respect to criteria 𝐹𝑗, shown in table 10. displayed in table 10, the rank descending sequence of the banking management software choice is 𝑇1 ≻ 𝑇2 ≻ 𝑇3 ≻ 𝑇5 ≻ 𝑇4. thus, alternative 𝑇1 is the best selection. 4.3. sensitivity analysis here sensitivity analysis is undertaken to calibrate the presented methods behavior. eight different cw sets are taken and displayed in table 11. the table shows for each set, one of the criteria has the highest weight, whereas the others have lesser weights. using this procedure, a sufficient range of criterion weights has been built to examine the sensitivity of the evolved method to variants of cws. the ranking outcomes of bms amenity alternative and the relative degree 𝛿𝑖 of various criteria weight, according to the sensitivity analysis outcomes are displayed in table 12 and figure 3. when the de’s provide weighting table 11. diverse criteria weight sets for bms alternative set-i set-ii set-iii set-iv set-v set-vi set-vii set-viii 𝑭𝟏 0.0690 0.0938 0.0562 0.1178 0.2372 0.1996 0.1672 0.0592 𝑭𝟐 0.0938 0.0562 0.1178 0.2372 0.1996 0.1672 0.0592 0.0690 𝑭𝟑 0.0562 0.1178 0.2372 0.1996 0.1672 0.0592 0.0690 0.0938 𝑭𝟒 0.1178 0.2372 0.1996 0.1672 0.0592 0.0690 0.0938 0.0562 hybrid mcdm method on pythagorean fuzzy set and its application 13 set-i set-ii set-iii set-iv set-v set-vi set-vii set-viii 𝑭𝟓 0.2372 0.1996 0.1672 0.0592 0.0690 0.0938 0.0562 0.1178 𝑭𝟔 0.1996 0.1672 0.0592 0.0690 0.0938 0.0562 0.1178 0.2372 𝑭𝟕 0.1672 0.0592 0.0690 0.0938 0.0562 0.1178 0.2372 0.1996 𝑭𝟖 0.0592 0.0690 0.0938 0.0562 0.1178 0.2372 0.1996 0.1672 table 12. relative degree for bms alternatives for different criteria weight sets set-i set-ii set-iii set-iv set-v set-vi set-vii set-viii t1 0.7758 0.7838 0.7804 0.8156 0.7678 0.7481 0.7719 0.7592 t2 0.7257 0.7311 0.7106 0.6862 0.7152 0.7143 0.7198 0.6987 t3 0.7027 0.7261 0.7160 0.6996 0.7232 0.7263 0.7188 0.6812 t4 0.6307 0.6789 0.6847 0.6717 0.6724 0.6639 0.6473 0.5987 t5 0.6685 0.7277 0.7139 0.6966 0.6891 0.6767 0.6638 0.6239 sets i, vii, and viii, the bms ranks them in the same order, while for other sets its different. according to the description above, the bms selection is dependent on, and sentient to, these cw sets, as the proposed method is stable with a variety of weight sets. figure 3. outcome of 𝛿𝑖 for each alternative with various weight sets of criteria chaurasiya et al./decis. mak. appl. manag. eng. (2022) 14 table 13 the comparative study with existing techniques methods standar d expert’s weight criteria weights ranking bms alterna tive zhang and xu (2014) pftopsis method not evaluate assumed 𝑇1 ≻ 𝑇2 ≻ 𝑇3 ≻ 𝑇4 ≻ 𝑇5 𝑇1 kumari and mishra (2020) ifcopras method evaluate completely unknown 𝑇1 ≻ 𝑇3 ≻ 𝑇2 ≻ 𝑇5 ≻ 𝑇4 𝑇1 peng et al. (2020) pfcocoso method not evaluate assumed 𝑇1 ≻ 𝑇2 ≻ 𝑇3 ≻ 𝑇5 ≻ 𝑇4 𝑇1 proposed method pf copras evaluate merecswara combined method 𝑇1 ≻ 𝑇2 ≻ 𝑇3 ≻ 𝑇5 ≻ 𝑇4 𝑇1 4.5. comparison and discussion in this section, now, we see that the framework submitted here has a lot of similarities with the existing methods. the pf-merec-swara-copras method is found to be proficient for handling qualitative and quantitative mcdm issues, especially in cases where there are many conflicting criteria. the advantages or features of the presented framework can be discussed as follows:  the method pf-topsis (zhang & xu, 2014) and pf-cocoso (peng et al., 2019) and the proposed pf-hybrid method are submitted in the context of pfs, whereas (kumari & mishra, 2020) have described if-copras method is used.  in the developed pf-hybrid method, we have evaluated expert weights on the basis of expert opinion, leaving no space to treat vagueness, whereas pftoopsis (zhang & xu, 2014) and pf-cocoso (peng et al., 2019) the procedure does not involve expert opinion.  pf-copras outperformed pf-topsis and if-copras in terms of effectiveness and proficiency. in addition, the hybrid copras method is more powerful and stable in terms of criterion weight disparity than pf-cocoso (peng et al., 2019).  the practical outcomes of the presented method provide some significant perceptions related to the evaluation criteria and the alternative for bms in india. as may be displayed in table 10, the most significant is the effectiveness of the bms. we find the best alternative among the existing ones. the problem of banking management can be solved to a great extent by seeing the outcomes of this paper. we also analyzed the performance of bms alternatives and compared the results for each criterion evaluated. according to the results, mambu (𝑇1) first rank among all alternatives and (𝑇4) is the last in the ranking. therefore, (𝑇1) can be chosen as the best alternative meeting all the valuation. 5. conclusions currently, with the speedy growth of it, it is a composite problem to select the best software for the diverse work of bank. mcdm is the best tool to deal with it. the key hybrid mcdm method on pythagorean fuzzy set and its application 15 purpose of the present paper is to develop an mcdm method in a pythagorean fuzzy environment. to do this, we first submitted a new merec method and score function on pfs. the pfss provide a precise and practical solution of the ambiguous real-life dm difficulties; consequently, a new hybrid pf-merec-swara-copras method has been developed under pfs. finally, the pf-copras methodology is proposed for ranking the alternatives. in addition, the discussion of comparative study of the presented method with the existing methods. based on a comparison with existing method, it is worth saying that the pf-copras method provide an effortless calculation with accurate and effective outcomes for the development of mcdm difficulties. the application of the proposed hybrid method on selecting the optimal banking software tool helps in finding the best bms.  a new normalization score function for pfn is submitted, which minimizes intimation loss by taking uncertainty information into account. compared to existing score functions, it has a more robust ability to differentiate when comparison two pfns.  the combined weight framework has been submitted on the basis of merec and swara weighted extensive methods, which consider both objective and subjective weight.  merec presented a novel pf-decision-making technique basis on the copras method, which can get the best alternative without any adverse events, get the outcome of the decision without segmentation, and has a robust ability and stability. some short comings of the projected structure are significant. a practical problem is that dm necessity skilled in the flexibility and ability to properly use the preferred style of pfs. the projected structure will help as a useful device for selecting the best bms under multiple-criteria situations and ambiguous environments. in the future, the evolved mcdm method may be further proceed to fermatean-fss, interval-valued pfs, and hesitant pfs. in addition, the researchers can extend our research via various mcdm platforms (for example, mixed aggregation by comprehensive normalization technique (macont), gained and lost dominance score (glds), mairca, and cocoso) to choose the most suitable bms selection. the limitation of the current study is that only a small number of de’s were included, and it does not take into account the interrelationships among the criteria, which somehow limits the scope of the application of the proposed framework. consequently, further research is still needed, which considers huge number of decision experts. author contributions: research problem, r.k.c. and d.j.; methodology, r.k.c.; formal analysis, r.k.c.; writing-original draft preparation, r.k.c; modification, d.j.; writing-review & editing, d.j ackno wledgme nt: the authors would like to express their gratitude to the editors and anonymous referees for their informative, helpful remarks and suggestions to improve this paper as well as the important guiding significance to our researches. conflicts of interest: there is no conflict of interest. funding: this research received no external funding. data availability statement: not applicable. chaurasiya et al./decis. mak. appl. manag. eng. 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(2019). new model for determining criteria weights: level based weight assessment (lbwa) model. decision making: applications in management and engineering, 2(2), 126-137. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 1, number 1, 2018, pp. 38-50 issn: 2560-6018 doi: https://doi.org/10.31181/dmame180138r * corresponding author. e-mail addresses: a.rikalovic@uns.ac.rs (a. rikalović), gerson@inf.ufsm.br (g. a. soares), ignjaticjelena@gmail.com (j. ignjatić) spatial analysis of logistics center location: a comprehensive approach aleksandar rikalović1*, gerson antunes soares2, jelena ignjatić3 1 university of novi sad, faculty of technical sciences, department of industrial engineering and management, novi sad, serbia 2 universidade luterana do brasil, department of computer science, rio grande do sul, brazil 3 urban institute of vojvodina, novi sad, serbia received: 13 november 2018; accepted: 26 january 2018; published: 15 march 2018. original scientific paper abstract. to select a suitable logistic center (lc) location it is necessary to do a comprehensive spatial analysis. geographic information systems (gis) are ideal for this type of spatial analysis which provides functionality to capture, store, and query, as well as analyze geographic information. the paper presents novel methodology for a lc location analysis based on gis and swot analyses. the proposed method uses a gis for data collection, spatial analysis, generating alternatives, and producing maps for further analysis. we used the gis to support decision-making and for an analysis of the location attributes, including a number of relevant factors, namely spatial position, intermodal connections (road, water, air and rail transport), the size of the available location, topography, local traffic connections, environment, ecological aspect of the location, ownership structure, equipment of communal infrastructure, constraints in the area, etc. we used the swot analysis to determine strengths, weaknesses, opportunities, and threats of the local area for attracting capital, knowledge and innovation. experimental results on real-world problems, i.e., application of the proposed comprehensive methodology in a case study of a location analysis for the logistic center in the municipality of apatin, vojvodina, serbia, show that the proposed methodology provides upstanding results in the spatial analysis. key words: geographic information systems, location analysis, logistic center, spatial analysis, spatial data mining, swot analysis key. 1. introduction the recent development of logistics management in companies has caused the need for streamlining business operations in the most efficient way, which implies cost mailto:a.rikalovic@uns.ac.rs mailto:gerson@inf.ufsm.br mailto:ignjaticjelena@gmail.com spatial analysis of logistics center location: a comprehensive approach 39 reduction and service quality improvement in order to satisfy the growing customers' demands. the council of supply chain management professionals defines the term logistics management as follows: "logistics management is that part of supply chain management that plans, implements, and controls the efficient, effective forward and reverses flow and storage of goods, services and related information between the point of origin and the point of consumption in order to meet customers' requirements" (cscmp, 2012). thus, the importance of logistics centers (lc) is reflected in linking economic activities both locally and globally, where lcs become places of circulation of production, storage and transportation of goods. the lc is a separate entity operating in a secured area, within which all logistics activities (transportation and forwarding, material handling, warehousing, inventory management, cross-docking, intermodal trans-shipment, physical distribution of goods) are carried out on a commercial basis (żak & weglinski, 2014). according to the european transport ministers' conference (ecmt) the following criteria relating to site selection are common to all lcs (ecmt, 1996):  the specific location of a lc reflects the need to optimize the quantity and quality of local and long-distance traffic, so that flows can be efficiently grouped together in terms of traffic volume. of utmost importance is that the location is able to optimize the inevitably broken local and long-distance flow by achieving a cluster effect. topographical factors are important in this context, as is the possibility of using existing infrastructure developed by different transport modes.  the size of logistics centers must be given considerable thought. firms looking for a location to generate or organize transport operations regard networked traffic nodes as highly attractive. in all events, a sufficient area of land must be set aside for the centre so that any future growth in traffic can be absorbed.  existing infrastructure has already been identified as an important factor when choosing the site of a logistics centre. in the case of the european inland transport, centers must at least offer the possibility of linking up with the combined transport network of road/rail services. links with other transport modes, such as the inland waterways, air transport or, in coastal areas, maritime transport, can of course also be incorporated. the attractiveness of these logistics nodes will depend mainly on the quality of the existing infrastructure, in other words, available capacity and, in particular, the possibility of extending capacity where necessary.  another important factor for developing a network of logistics centers is compatibility, particularly at the level of transshipment procedures, containers, etc. a certain amount of standardization of technical equipment is therefore required. in addition, telematics networks of information and communications systems are needed so that firms can have on-line access to any important data required for rapid decision-making (e.g. data on consignments, stocks, status, etc.). a lc location is a strategic decision and a critical point of the entire supply network, from the manufacturer to the consumer, whereby one of the main factors determining the location is the transportation cost. according to rikalovic et al., the transportation costs, taxes, salaries, raw materials, renting costs, etc. are directly under the influence of the location and can make up 25% of the purchase product price. when all costs are summarized the location may change the overall operating costs by 50% (rikalović et al., 2014). as a result, the optimal location of a lc may lead to reduced transportation costs, promote synchronization between production and consumption, ensure a rikalović et al./decis. mak. appl. manag. eng. 1 (1) (2018) 38-50 40 balanced development of transportation systems, and achieve better overall benefits (gao & dong, 2012; lium et al., 2009). many mathematical approaches to solving a lc location problem have been reported, including: mathematical programming (klose & drexl, 2005), evaluation and ranking of considered sites (özcan, et al., 2011; awasthi et al., 2011; żak & weglinski, 2014). the location problem plays a crucial role in logistics, where it refers to finding the most desirable (proper) location of lc (thai & grewal, 2005; kayikci, 2010). recent developments in the field of decision-making have led to dramatic improvements in the capabilities of gis for a lc location analysis. geographic information systems (gis) are powerful tools designed for spatial analysis which provides functionality to capture, store, query, analyze, display and output geographic information (bolstad, 2002). the power of geographic information systems is reflected in their simultaneous analysis of the geographic space and the information (attributes) linked to the space (rikalović et al., 2014). to address all of the above aspects we propose a novel comprehensive methodology for a location analysis based on gis, which provides for an innovative approach to finding a desirable location efficiently. this paper is organized as follows. in section 2 we summarize the literature overview in the field of logistics centers, pointing out the role of a multi-criteria decision analysis and the geographic information systems in the location analysis. in section 3 we present innovative comprehensive methodology for a spatial analysis of the logistics center location, with a study case to show implementation choices. in section 4 we present a swot analysis of the competitiveness of the local area for attracting capital, knowledge and innovation. section 5 presents a discussion and analysis of the application of the proposed approach. section 6 derives some conclusions and directions for future research. 2. background the location problem consists in determining proper placement of an infrastructural component (ground, site, facility) in a considered area, taking into account the decision-maker’s preferences and existing constraints (owen & daskin, 1998; farahani & hekmatfar, 2009). one of the typical logistics location problems refers to the proper placement of a lc (thai & grewal, 2005; kayikci, 2010). logistics centers should be located at the intersection of large streams of flow, or very close to major transport links, especially in order to take advantage of multimodality. the advantage of multimodal transport in addition to optimizing transport times and reducing transport costs represents greater justification regarding environmental protection (gao & dong, 2012). decisions related to a lc location have an impact on operating costs and revenues. for instance, a poor choice of location might result in excessive transportation costs, a shortage of qualified workers, loss of competitive advantage, inadequate supplies of raw materials, or some similar condition that would be detrimental to operations (stevenson, 1993). while the optimal location selection in the logistics sector decreases transportation costs, it increases the job performance, competitive capacity and profitability of companies (thai & grewal, 2005). lee et al. (2009) point out the need to consider geographic, topographic, and hydrological aspects of the available land for the logistics center installation. in the literature the site selection processes generally disaggregate the problem into several levels. zelenović (2003) distinguishes macro and micro locations. the spatial analysis of logistics center location: a comprehensive approach 41 macro location is the geographical area, which can meet the basic requirements for construction and development of the lc with minimal operating costs. the micro location is the specific place in the macro location that meets technical, infrastructural and working process requirements (zelenović, 2003).the basic criteria for analyzing the lc micro-location include an analysis of all the parameters from the micro and macro aspect with the coordination of all the entities in the multimodal transportation process. the factors that influence the lc location selection, mentioned in the literature, can be divided into two categories: factors that influence the selection of a macro location of a facility and factors that influence the selection of the microlocation. in the selection of the macro location of facilities, the factors are considered that are related to the role of the facilities in the logistic system – market characteristics, type and characteristics of goods in the distribution chain, transport possibilities and the availability of the expert personnel. in determining the microlocation of the facility, the following characteristics are considered (fig. 1.): the area of the terrain, fitting of the location into urban plans, condition of the traffic infrastructure in the observed area, price of the land, attitudes and exemptions for the local community, etc. (pašagić-škrinjar et. al., 2012). (a) (b) figure 1. micro location (from the urban plan): (a) land use and (b) infrastructure in the modern concepts of 3pl and 4pl the lc plays an important role of a logistics service integrator that provides customers with a comprehensive and high quality service besides connecting all possible transportation modes (road, rail, water and air). the concept of a location planning is based on the technological-functional requirements of the work organization such as a multimodal transport node (rushton et al., 2006). also, the transport and technological requirements of goods flows and the global and domestic experience in determining the technology, structure and size of individual subsystems of the lc have a significant influence on the planned solution. the development of multimodal transport corridors and logistics services are promoted as part of the overall development transport strategy (world bank, 2008). with the development of information technology, spatial databases have become an irreplaceable tool in management systems. it has been estimated that 80% of data used by managers and decision-makers are geographical (spatial) in nature (worral, 1991). a large number of decisions have geographic (spatial) character, and finding a winning formula for the optimal location selection is the challenge for engineers and scientists. besides a standard spatial analysis, gis is increasingly used in the decisionmaking process. experts in the field of industry and trade consider a spatial analysis using gis technology as a reliable approach in the location analysis as well as in overall rikalović et al./decis. mak. appl. manag. eng. 1 (1) (2018) 38-50 42 marketing strategy of the company (beaumont, 2000). several studies have used this type of gis function for a location analysis. one of the papers in which the gis methodology is applied is the role of gis in the industrial site of analysis (rikalović et al., 2014). previous research on the location analysis has determined that gis provides a methodological framework for a quality, reliable and efficient decision-making and development of the overall strategy for the development of a certain space or within a particular company. 3. spatial analysis of logistics center location: a comprehensive approach in this section we introduce an innovative approach to the logistics center analysis: a gis based approach. the proposed comprehensive method uses a gis for generating location alternatives and spatial data mining. we used gis to support decision-making and for the analysis of the location attributes, including a number of relevant factors: spatial position, intermodal connections (road, water, air and rail transport), the size of the available location, topography, local traffic connections, environment, ecological aspect of the location, ownership structure, equipment of communal infrastructure, constraints in the area, etc. figure 2. process of creating a database we have divided the proposed methodology into eleven phases (fig. 2): 1. creating (preparing) graphics in autocad in relevant layers 2. graphic transformation in gauss-krieger's projection (hermannskogel's date) 3. controlling the graphics before entering arcgis import data into arcgis 4. transformation of graphics from hermannskogel to wgs84 (world geodetic system worldwide reference date-applied by google) using geocentric translation 5. inserting a satellite reference folder (which is in wgs84) spatial analysis of logistics center location: a comprehensive approach 43 6. controlling the position of the brownfield location relative to the reference map 7. graphics processing in arcgis / geo-visualization 8. relational connection of spatial data with alpha-numeric data 9. entering attribute (alpha-numeric) data into the database 10. controlling of compliance of attribute and spatial data 11. importing already prepared layers of infrastructure (traffic, electricity, thermal power, water management, etc.) on the territory of the province of vojvodina, serbia, in the form of information (fig. 3):  review and analysis of infrastructure data in the territory of the province of vojvodina, serbia, created in the process of drafting the regional spatial plan of the province of vojvodina,  graphic transformation from hermannskogoel to wgs84 (world geodetic system world reference date-applied by google) using geocentric translation,  control of compliance of spatial and attribute data, and,  final control. figure 3. traffic, electricity, thermo power and water management layer on the territory of the province of vojvodina macro view gis techniques & procedures have an important role in analyzing decision problems in the form of a spatial decision support system for location analysis especially in the screening phase. in the screening phase the role of gis is to analyze feasible alternatives that will be later considered in the evaluation phase. the first step is the collection, input and processing of data such as: geographical characteristics, urban parameters, ecological aspect of the site, ownership structure, infrastructure facilities, architectural heritage and other relevant data. rikalović et al./decis. mak. appl. manag. eng. 1 (1) (2018) 38-50 44 the analysis and assessment based on the collected data are extremely important because they represent the basis which enables us to determine the extent of the potential and risk posed by the location of logistic centre. based on the information provided by the municipality as well as spatial plans, we are able to look at and adequately address the spatial-technical-technological aspects of the site and set up the database. for generating location alternatives and spatial analyses we develop a gis, using the arcgis infrastructure. to provide a useful set of location alternatives, we have developed a gis to mine data and generate location alternatives. our gis is based on the spatial database that contains data for all the selected features in the region of interest. the spatial data mining process focuses on the process of discovering interesting and potentially useful patterns for generating industrial locations alternatives from a spatial dataset by using a multilayer framework, i.e., an infrastructure where the spatial database is organized in multiple layers (thematic maps) that represent the selected features layer by layer (map by map) (fig 4.). using our gis, we can separately analyze each layer of the spatial data in the region of interest and of the overlapping specific layers in order to obtain useful information as well. in fig. 4 we are presenting application of the proposed methodology, i.e., spatial data mining process in a case study of a location analysis for the logistic center in the municipality of apatin, vojvodina, serbia in google earth environment. (a) (b) (c) (d) figure 4. spatial data mining process: (a) zone of logistic centre, (b) land use, (c) property and (d) infrastructure spatial analysis of logistics center location: a comprehensive approach 45 4. swot analysis the analysis includes a quantitative and qualitative approach to data processing, but the stakeholder involvement in identifying, analyzing and ranking the problem is essential. the analysis of the situation encompasses the following sources: reports, information, studies, plans, earlier analysis, strategic documents, analysis questionnaires, interviews with key experts. after collecting data and analyzing the situation, the swot analysis (strengths, weaknesses, opportunities, threats) is established. the swot analysis is a method that gives the opportunity to establish a balance between internal capabilities and external possibilities. it is a set of analytical methods based on which the comparison of their strengths and weaknesses with opportunities and threats in the environment is conducted. a swot analysis (table 1) of the competitiveness of the local area for attracting capital, knowledge and innovation as key determinants for entering the upward path of sustainable development provides a synthesis of the established state of endogenous factors and processes in their interaction with external events. table 1. swot analysis of the lc location in apatin1 internal factors strengths weaknesses a favorable traffic and strategic position of the lc (easy access to pan-european corridors x and vc, the position on paneuropean corridor vii danube river) and higher intermodal transport potentials -by water, rail, road, air multiple natural potentials (land, water, climate) as motors for economic development good infrastructure network growing public-private sector partnership relatively trained, affordable and low-cost labor (tradition and quality) the realization of the project of constructing apatin port complex with dock is in progress completion of work on the corridor v (budapest ploce) section good electrical connection good telecommunication network, mobile, cable distribution systems, internet, wireless tradition in the manufacturing industry, wholesale and retail trade a trend of increasing exports relatively favorable business expenses negative demographic trends lack of new innovative and information technologies, know-how and their slow integration into business flows information and innovative technologies are not sufficiently integrated into business flows and productive sectors obsolete technology and equipment poor cross-sectoral links between enterprises and scientific and research facilities and other development institutions at the regional level insufficient marketing activities poor condition of the railway infrastructure lack of regional/state roads of the highest rank insufficient utilization of the danube river lack of a resource database preparation for land and plant purposes the lack of institutions dealing with scientific-research activities in the field of economy, as well as low level of cooperation with scientific-research facilities and development institutions 1 in the swot analysis were used the data from the strategy of sustainable development of apatin municipality 2009-2019, (2009). rikalović et al./decis. mak. appl. manag. eng. 1 (1) (2018) 38-50 46 external factors opportunities threats the danube river (corridor vii) as the most important european river roadway, as well as a better use of the potentials of corridor iv, vc and x access to trade and other integration models strengthening of cooperation with other regions in order to promote socio-economic development and create a regional brand institutional and investment support in the development and modernization of existing capacities openness of the municipal administration towards potential investors, through making of higher-order planning documents construction of the highway vc (budapest ploce) development of freight transport facilities and associated content related to corridor vii; reconstruction of the passenger harbor, construction of a marina for the reception of smaller vessels revitalization of the railroads and establishment of regular traffic on the so-apatin factory -sonta route this would enable the inclusion of apatin on the railway corridor x; further construction of industrial tracks and free zone the existence of the logistic concept for ap vojvodina, west bačka county and municipality of apatin, as well as spatial planning documents construction of a road bypass around the city existing plans for improvement of the existing production which have the goal to create better and more competitive products eu market proximity participation in the cross-border cooperation programs openness of the municipal administration to potential investors establishing of a cluster network of businesses improvement of cooperation with scientificresearch institutions lower labor costs in relation to prices in the eu тax and financial incentives for foreign investors potentials for the development of a specialized business incubator transport logistics unstable economic and political situation strengthening the competitive advantages of the eu market absence of interregional cooperation in the field of goods and services flow planning with the goal to exploit potential economic capacities migration movements of highly educated people from the area postponement of the beginning of the construction of a road bypass around apatin undefined role of the airport in the traffic vision of the region complicated administrative procedures in the near future, the municipality of apatin recognizes itself as a micro-regional, multimodal transport hub, with well-developed water management, electricity, gas and telecommunications infrastructure. in this sense, the first goal is to ensure all the necessary conditions for the development and continuous improvement of the infrastructure capacities, contents and supporting elements. the realization of this goal, the accompanying measures and spatial analysis of logistics center location: a comprehensive approach 47 projects are in the function of more complete and better utilization of local economic potentials. in addition to the difficult transport of goods and transport of people, the marginal role of the railway can be a potential bottleneck in the functioning of the multimodal traffic system in the territory of the municipality of apatin. the expressed interest of the local community for the revitalization of the sombor-apatin factory-sonta route, for the development of commodity terminals and for the construction of industrial tracks in the logistics center apatin, gives this measure a priority status. due to its natural characteristics, the danube is invaluable, not only for the observed location, but also for a wider environment. the construction, completion and smooth functioning of the lc in its vicinity can open the doors to all the major european metropolises located on the danube coast. by implementing this measure, the municipality of apatin would, in a short period of time, experience direct and positive impacts of better and more extensive commodity traffic. 5. discussion and analysis the significance of the lc is reflected in the geographical position, infrastructure and traffic connections in the distribution chain. logistics today is one of the most important factors that arise in the market economy. companies use lc services to speed up the process of starting up goods while reducing transportation costs. the possibility of connecting with the environment, as well as that of communal equipping, are very important parameters for the lc functioning. we used gis for data mining, generating alternatives, spatial analysis and presentation of the layers obtained by analysis in the form of maps. after analyzing the geo-traffic position and infrastructure, as the most important factors for the development of a lc, in the case of the municipality of apatin we can state:  the direct support of the lc to the danube aquarium, which represents the most important river road in europe (by commissioning the rhine-main-danube channel, all the danube countries, with good interconnections, the exit to the black sea and the north sea) provides an excellent geo-traffic position of the lc. apatin is located in the central part of this waterway, and it is also the first convenient location for the formation of a freight transport center, downstream of the hungarian border, through which the entire central and north-western area of bačka can reach an exit to this important international river road, known as the pan-european corridor vii;  railway line apatin factory sonta in sonta comes to the main road szeged (hungary) -subotica, sombor (serbia) -vinkovci, osijek (croatia), so it has a direct connection with european corridors vii, x, iv and v;  apatin is not directly connected with neighboring croatia because the danube river and kopački rit represent a natural obstacle for the construction of road and railroads, and the traffic connection with croatia is realized in the hinterland over the state road (sombor-apatin bridge over the danube near bogojevo);  the apatin settlement has an eccentric position in relation to the area of the municipality of apatin, and also to the wider hinterland where the main landing traffic routes are located, which in some way makes it separate; yet this is usually a common position for coastal settlements and can be completely compensating for the activation of river transport potentials and the formation rikalović et al./decis. mak. appl. manag. eng. 1 (1) (2018) 38-50 48 of an appropriate network of road and railroads, through which they will be linked with their immediate, but also wider environments. a direct access to the international waterway the danube river, as a paneuropean corridor vii, an indirect rail link with the main railway line through the local railway line, an exit by road connection to the state road iia of order no. 107, are elements that give this space extraordinary spatial-traffic predispositions for smooth development, especially in the domain of integral transport. in addition to the analyzed infrastructure from the aspect of traffic, the subject location is also analyzed from the aspect of the utility equipment, which is assessed as suitable for placing space in function since the infrastructure of electricity, water supply, gas pipelines and electronic communication infrastructure has been built. 6. conclusion this paper proposes an innovative approach to a logistics center analysis that adopts a gis based approach. the proposed methodology uses a developed gis for generating location alternatives and spatial data mining. we used a gis for data collection, spatial analysis, generating alternatives and producing maps for further analysis. the results of the location analysis in the study case of the municipality of apatin, vojvodina, serbia show that a suitable geostrategic position is the biggest advantage for apatin's future development priority in the field of economy, infrastructure and logistics development. the position of the lc, on the very bank of the danube, provides an unhindered access from the corridor of the waterway and service of all sub-systems and sub-stations of the lc. the international port gives a quality basis to enable dispatch and delivery of all types of products across europe through the danube (international waterway) with the maximum application of an integral transport system. the apatin intermodal hub, as a sublimate of all available, natural and infrastructural benefits, with water transport as the carrier of the entire system, will contribute to increasing system efficiency, reducing transport costs, increasing the level of transport services, applying modern transport technologies with ecologically minimal negative impacts and integrating all traffic types. the lc in apatin will enable the rationalization of the distribution of all types of product on the micro and macro plan, including all modes of transport in the observed area, which will contribute to reducing the cost of shipping delivery and overall reduction in transport costs. in further research the comprehensive method for industrial site selection will be studied at the micro-location level. the results show that our comprehensive methodology constitutes an efficient and highly accurate tool for spatial decision support. in future research, our methodology will be tested on different cases with larger datasets. moreover, we plan to study strategies such as the use of intelligent systems to improve the decision support effectiveness. acknowledgments the authors would like to thank the urban institute of vojvodina, novi sad, serbia, which provided highly valuable data of lc locations in vojvodina. the authors would also like to thank j. lovric and d. moraca for their expert advice on performing lc locations analysis. spatial analysis of logistics center location: a comprehensive approach 49 references ***(2009). strategy of sustainable development of apatin municipality 2009-2019 (only in serbian: strategija održivog razvoja opštine apatin 2009-2019). apatin municipality, serbia. awasthi, a.,chauhan, s., & goyal, s. 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(2003). the design of production systems, 2nd ed., novi sad (serbia): faculty of technical sciences. plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 5, issue 2, 2022, pp. 30-45. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0305102022d * corresponding author. e-mail addresses: djeniseles11@gmail.com (jeni seles martina. d), deepa.g@vit.ac.in (deepa. g*) the energy of rough neutrosophic matrix and its application to mcdm problem for selecting the best building construction site jeni seles martina donbosco1 and deepa ganesan1* 1 department of mathematics, vit university, vellore, india received: 10 august 2022; accepted: 26 september 2022; available online: 5 october 2022. original scientific paper abstract: an approach to data processing for relational databases is called a rough set theory. it is an interesting area of uncertainty mathematics that is mainly related to fuzzy theory. rough set theory and neutrosophic set theory can be joined to create a powerful tool for dealing with indeterminacy. neutrosophic matrices help decision-makers deal with multi-criteria decisionmaking by providing them with more useful and practical when we apply the concept of matrix energy. in this paper, we defined a rough neutrosophic matrix and its energy. some propositions, lower and upper limits of the rough neutrosophic matrix's energy were derived. the proposed energy of the rough neutrosophic matrix was applied in multi-criteria decision-making problems. the problem is to select the best place for constructing the school building. applying the energy method to the mcdm problem became more relatable and produced good results. keywords: rough set, rough neutrosophic set, rough neutrosophic matrix, energy of rough neutrosophic matrix, multi-criteria decision making. 1. introduction fuzzy sets, fuzzy membership functions, and fuzzy logic were first introduced (zadeh 1965). fuzzy matrix theory, which focused on the convergence of fuzzy matrices’ powers, was first presented by michael g. thomason in 1977. it can be applied in several circumstances. it is commonly known that the matrix representation offers an additional advantage in resolving the issue. intuitionistic fuzzy matrices were first introduced (khan et al., 2002). it is difficult to determine the value of membership or non-membership as a point, though. the neutrosophic set was first introduced (smarandache 1998). he put out the concepts of neutrosophic set, probability, and logic to specifically address the issue of indeterminacy. he also created hesitant and dual hesitant neutrosophic sets, single and interval valued mailto:djeniseles11@gmail.com mailto:deepa.g@vit.ac.in the energy of rough neutrosophic matrix and its application to mcdm problem for selecting … 31 neutrosophic sets, and multi-valued neutrosophic sets. after that the introduction of fuzzy relational maps and neutrosophic relational maps was presented (smarandache et al., 2004). in this, they included square neutrosophic matrices. the neutrosophic matrix and associated algebraic operations were created (smarandache et al., 2014). christi distefano and colleagues introduced the idea of matrix energy in 2009. they devised the equation for the matrix’s energy. a generalization of the energy of a graph is the energy of a matrix. a paper titled energy of matrixes was proposed (bravo et al., 2017). they produced a number of theorems on matrix energy as well as upper and lower bounds. the notion of matrix energy does not hold true in a neutrosophic setting or for mcdm problems, however, the notion of graph energy will gain in popularity. we, therefore, examine energy in neutrosophic matrices and how it might be used in mcdm in this paper. the idea of a rough set was first introduced by pawlak (1982). its foundation is the approximation of sets by a pair of sets known as the lower and upper approximations of a set. in this instance, the equivalence relation is the basis for those approximations. then he compares the fuzzy set with the rough set concept (pawlak 1985). then the rough set techniques for incomplete information systems were presented, along with fuzzy rough sets (kryszkiewicz 1998). the rough intuitionistic fuzzy set was proposed (rizvi et al., 2002) they defined the rough intuitionistic fuzzy set and its properties. the generalized fuzzy rough sets were introduced (wei-zhi wu et al., 2003). this paper studies fuzzy rough sets using both constructive and axiomatic approaches. then the rough set and fuzzy rough set on interval-valued fuzzy was proposed (gong et al., 2008). both axiomatic and constructive methods to develop a complete framework for the study of interval type-2 rough fuzzy sets are used (zhang 2012). a concept of the rough fuzzy set model for a set-valued ordered fuzzy decision system was presented (bao et al., 2014). in order to create decision rules for long-term forecasting of air passengers suggested a novel hybrid method based on rough set theory (sharma et al., 2018). then they used a combination technique to evaluate india’s sugarcane production based on the rough set approach (sharma et al., 2021). they also provide a basic decision-making process based on a set theory for assessing the performance of delhi hotels (sharma et al., 2022). they develop a rough set theory to offer a set of decision rules and significant feature sets. rough set theory and neutrosophic set theory will both be useful methods for handling incomplete, ambiguous, uncertain, and incorrect data. (said broumi et al., 2014) introduced the idea of the rough neutrosophic set. they outlined the rough neutrosophic sets and their operations in this study. then they proposed the interval valued neutrosophic rough set (said broumi et al., 2015). a rough grey relational analysis-based strategy for neutrosophic multi-attribute decision-making is illustrated (kalyan mondal et al., 2015). in this work, the rough neutrosophic decision matrix is defined, and an mcdm issue is solved using this matrix. a number of authors offered various ideas for a rough neutrosophic field (alias et al., 2017, yang et al., 2017, pramanik et al., 2017). they studied mcdm in rough neutrosophic sets with coefficient correlation, rough single-valued neutrosophic sets, and rough neutrosophic multisets. on novel multi-granulation neutrosophic rough set on a single value and its uses was discussed (bo et al., 2018). rough neutrosophic set is used in medical diagnosis (samuel et al., 2018). this paper discusses the use of medical diagnostics to identify the patient’s health. an article with the title medical diagnosis focused on singlevalued neutrosophic uncertain rough multisets over two universes was published in the same year (zhang et al., 2018). rough neutrosophic sets pi-distance for medical diagnosis was presented (samuel et al., 2019). the objective of the study is to establish a causal relationship between the illness and the patient’s symptoms and to examine jeni et al./decis. mak. appl. manag. eng. 5 (2) (2022) 30-45 32 the patient’s state using a rough neutrosophic set. the notions of the neutrosophic soft set with rough set theory (das et al., 2021), neutrosophic single-valued rough sets, and including topology (jin et al., 2021; almathkour et al.,, 2022) will be further developed. the rough set is important in every field of the neutrosophic environment as result. multi-criteria decision making (mcdm) is a key and quickly developing subject in operations research. indeterminacy should be handled in the modeling approach of challenges since mcdm problems are well addressed in fuzzy. the development of the mcdm field in a fuzzy environment led to the proposal of the neutrosophic fuzzy mcdm, which was used in numerous methods (otay 2022, wang et al., 2022). there are some more different methods in mcdm to select the best alternatives. recently, several researchers work on many types of methods (gorcun et al., 2021, arora et al., 2022). in this study, we added a new step to the method for resolving mcdm issues with a rough neutrosophic matrix by determining its energy. in section 2, the fundamental definitions are provided. in section 3, we introduced the energy of the rough neutrosophic matrix together with its hypotheses, and upper and lower bounds. a new strategy named the rough neutrosophic energy method was introduced in section 4 and described in detail. the numerical example of the suggested method was resolved in section 5. then, a conclusion was given. 2. preliminaries definition 2.1. rough set (pawlak, 1982) let u be the universal set and r be an equivalence relation on u (this is called an indiscernibility relation). the collection of all equivalence classes of u under 𝑅 is defined as 𝐴 = 𝑈/𝑅, which is called an approximation space. let 𝑋 ⊆ 𝑈 be a subset of u. we define lower and upper approximation of x in a, denoted 𝐴(𝑋) and 𝐴(𝑋) respectively, as follows 𝐴(𝑋) = {𝑎 ∈ 𝑈 ∶ [𝑎]𝑅 ⊆ 𝑋} 𝐴(𝑋) = {𝑎 ∈ 𝑈 ∶ [𝑎]𝑅 ∩ 𝑋 ≠ ∅} where [𝑎]𝑅 denotes the equivalence class of r containing an element a. the pair 𝐴(𝑋) = (𝐴(𝑋), 𝐴(𝑋)) is called the rough set of x in a. definition 2.2. neutrosophic set (smarandache, 1998) let u be the universal set and every element 𝑎 ⊆ 𝑈 has a degree of true, indeterminacy, false membership in neutrosophic set. it is denoted by s. then it can be written as 𝑆 = { ⟨ 𝑎,𝑇𝑆(𝑎), 𝐼𝑆(𝑎),𝐹𝑆(𝑎)⟩ ∶ 𝑎 ∈ 𝑈} where, 0 ≤ 𝑇𝑆(𝑎) + 𝐼𝑆(𝑎) + 𝐹𝑆(𝑎) ≤ 3 and truth membership function 𝑇𝑆: 𝑈 → [0,1], indeterminacy membership function 𝐼𝑆: 𝑈 → [0,1], false membership function 𝐹𝑆: 𝑈 → [0,1]. definition 2.3. rough neutrosophic set (said broumi et al., 2014) let u be the universal set and every element 𝑎 ∈ 𝑈. let r be an equivalence relation on u and s be the neutrosophic set in u with truth membership function 𝑇𝑆, indeterminacy function 𝐼𝑆 and false membership function 𝐹𝑆. the lower and upper the energy of rough neutrosophic matrix and its application to mcdm problem for selecting … 33 approximations of 𝑆 in 𝑈/𝑅 is denoted by 𝑁(𝑋) and 𝑁(𝑋) and they are defined as follows, 𝑁(𝑆) = {⟨ 𝑎,𝑇𝑁(𝑆)(𝑎),𝐼𝑁(𝑆)(𝑎),𝐹𝑁(𝑆)(𝑎)⟩: 𝑏 ∈ [𝑎]𝑅,𝑎 ∈ 𝑈} 𝑁(𝑆) = {⟨ 𝑎,𝑇𝑁(𝑆) (𝑎),𝐼𝑁(𝑆) (𝑎),𝐹𝑁(𝑆) (𝑎)⟩ ∶ 𝑏 ∈ [𝑎]𝑅,𝑎 ∈ 𝑈} where, 𝑇𝑁(𝑆)(𝑎) = ⋀ 𝑇𝑆(𝑏) 𝑏∈ [𝑎]𝑅 𝑇𝑁(𝑆)(𝑎) = ⋁ 𝑇𝑆(𝑏) 𝑏∈ [𝑎]𝑅 𝐼𝑁(𝑆)(𝑎) = ⋁ 𝐼𝑆(𝑏) 𝑏∈ [𝑎]𝑅 𝐼𝑁(𝑆)(𝑎) = ⋀ 𝐼𝑆(𝑏) 𝑏∈ [𝑎]𝑅 𝐹𝑁(𝑆)(𝑎) = ⋁ 𝐹𝑆(𝑏) 𝑏∈ [𝑎]𝑅 𝐹𝑁(𝑆)(𝑎) = ⋀ 𝐹𝑆(𝑏) 𝑏∈ [𝑎]𝑅 where, 0 ≤ 𝑇𝑁(𝑆)(𝑎) + 𝐼𝑁(𝑆)(𝑎) + 𝐹𝑁(𝑆)(𝑎) ≤ 3 and 0 ≤ 𝑇𝑁(𝑆)(𝑎) + 𝐼𝑁(𝑆)(𝑎) + 𝐹𝑁(𝑆)(𝑎) ≤ 3. where, ⋁ means ‘max’ and ⋀ means ‘min’ and 𝑇𝑆(𝑎),𝐼𝑆(𝑎),𝐹𝑆(𝑎) are truth, indeterminacy, false membership function of a on s. therefore 𝑁(𝑆) and 𝑁(𝑆) are two neutrosophic sets in u. the pair (𝑁(𝑆),𝑁(𝑆)) is called the rough neutrosophic set in u/r. if 𝑁(𝑆) = 𝑁(𝑆) for any 𝑎 ∈ 𝑈, then s is called definable neutrosophic set. definition 2.4. energy of matrix (bravo et al., 2017) let 𝑀𝑛(ℂ) denote the space of 𝑛 × 𝑛 matrices with entries in ℂ and p be a matrix in 𝑀𝑛(ℂ). we define the energy of a as 𝐸(𝑃) = ∑|𝜆𝑖 − 𝜇| 𝑛 𝑖=1 where, 𝜆1,𝜆2,… 𝜆𝑛 are the eigenvalues of p and 𝜇 is the mean of eigenvalues. if 𝜇 = 0 or p is the adjacency matrix of a graph g then e(p) is precisely the energy of the graph g. definition 2.5. energy of neutrosophic matrix let p(n) be the neutrosophic matrix with the order of 𝑛 × 𝑛 (square matrix). it can be expressed as three matrices, the first matrix contains the entries 𝑎𝑖𝑗 as truth membership values, the second contains the entries 𝑏𝑖𝑗 as indeterminacy membership values and the third matrix contains the entries 𝑐𝑖𝑗 as false membership values. it is denoted as 𝑃(𝑁) = ⟨ 𝑃(𝑇𝑖𝑗),𝑃(𝐼𝑖𝑗),𝑃(𝐹𝑖𝑗)⟩𝑛× 𝑛 and 𝑎𝑖𝑗 ∈ 𝑃(𝑇𝑖𝑗)𝑛× 𝑛 ,𝑏𝑖𝑗 ∈ 𝑃(𝐼𝑖𝑗)𝑛× 𝑛 𝑎𝑛𝑑 𝑐𝑖𝑗 ∈ 𝑃(𝐹𝑖𝑗)𝑛× 𝑛 the energy of a neutrosophic matrix is defined as 𝐸[𝑃(𝑁)] = ⟨ 𝐸[𝑃(𝑇𝑖𝑗)],𝐸[𝑃(𝐼𝑖𝑗)],𝐸[𝑃(𝐹𝑖𝑗)]⟩ = 〈∑|𝜆𝑖 − 𝜇| 𝑛 𝑖=1 ,∑|𝜁𝑖 − 𝜇| 𝑛 𝑖=1 ,∑|𝜂𝑖 − 𝜇| 𝑛 𝑖=1 〉 jeni et al./decis. mak. appl. manag. eng. 5 (2) (2022) 30-45 34 where, 𝜆𝑖,𝜁𝑖 and 𝜂𝑖, (𝑖 = 1,2,… 𝑛) are the eigenvalues of truth, indeterminacy, and false membership values respectively and 𝜇𝜆, 𝜇𝜁, and 𝜇𝜂 are the mean values of 𝜆𝑖,𝜁𝑖 and 𝜂𝑖 respectively. 3. energy of rough neutrosophic matrix definition 3.1. energy of rough neutrosophic matrix let 𝐷(𝑁) = ⟨𝐷(𝑁𝑖𝑗(𝑆)),𝐷(𝑁𝑖𝑗(𝑆))⟩ be the rough neutrosophic matrix with the order 𝑛 × 𝑛. where, 𝐷(𝑁𝑖𝑗(𝑆)) and 𝐷(𝑁𝑖𝑗(𝑆)) are a lower and upper approximation of the neutrosophic set s. the rough neutrosophic matrix can be expressed as 6 matrices, first 3 matrices are under lower approximation which contains the elements 𝑎𝑖𝑗, 𝑏𝑖𝑗, 𝑐𝑖𝑗, another 3 matrices are under upper approximation which contains the elements 𝑎𝑖𝑗,𝑏𝑖𝑗, 𝑐𝑖𝑗. where, 𝑎𝑖𝑗,𝑎𝑖𝑗 are truth membership values, 𝑏𝑖𝑗,𝑏𝑖𝑗 are indeterminacy membership values and 𝑐𝑖𝑗, 𝑐𝑖𝑗 are false membership values. which is denoted as, 𝐷(𝑁) = ⟨ 𝐷(𝑁𝑖𝑗(𝑆)),𝐷(𝑁𝑖𝑗(𝑆))) ⟩ = ⟨ (𝐷(𝑇𝑖𝑗(𝑆)),𝐷(𝐼𝑖𝑗(𝑆)) ,𝐷(𝐹𝑖𝑗(𝑆))),(𝐷(𝑇𝑖𝑗(𝑆)) ,𝐷(𝐼𝑖𝑗(𝑆)) ,𝐷(𝐹𝑖𝑗(𝑆)))⟩ where the elements, 𝑎𝑖𝑗 ∈ 𝐷(𝑇𝑖𝑗(𝑆)) ,𝑏𝑖𝑗 ∈ 𝐷(𝐼𝑖𝑗(𝑆)),𝑐𝑖𝑗 ∈ 𝐷(𝐹𝑖𝑗(𝑆)), 𝑎𝑖𝑗 ∈ 𝐷(𝑇𝑖𝑗(𝑆)), 𝑏𝑖𝑗 ∈ 𝐷(𝐼𝑖𝑗(𝑆)),𝑐𝑖𝑗 ∈ 𝐷(𝐹𝑖𝑗(𝑆)). then the energy of rough neutrosophic matrix defined as 𝐸[𝐷(𝑁)] = ⟨ (𝐸[𝐷(𝑇𝑖𝑗(𝑆))],𝐸[ 𝐷(𝐼𝑖𝑗(𝑆))],𝐸[𝐷(𝐹𝑖𝑗(𝑆))]), (𝐸[𝐷(𝑇𝑖𝑗(𝑆))],𝐸[𝐷(𝐼𝑖𝑗(𝑆))],𝐸[𝐷(𝐹𝑖𝑗(𝑆))])⟩ 𝐸[𝐷(𝑁)] = 〈 (∑|𝜆𝑖 − 𝜇𝜆| 𝑛 𝑖=1 ,∑|𝜁𝑖 − 𝜇𝜁| 𝑛 𝑖=1 ,∑|𝜂𝑖 − 𝜇𝜂| 𝑛 𝑖=1 ) , (∑|𝜆𝑖 − 𝜇𝜆| 𝑛 𝑖=1 ,∑|𝜁 𝑖 − 𝜇 𝜁 | 𝑛 𝑖=1 ,∑|𝜂 𝑖 − 𝜇𝜂| 𝑛 𝑖=1 ) 〉 where, 𝜆𝑖, 𝜁𝑖, 𝜂𝑖 are the eigenvalues of truth, indeterminacy, and false values of lower approximation matrices and 𝜆𝑖, 𝜁𝑖, 𝜂𝑖 are the eigenvalues of truth, indeterminacy, and false values of upper approximation matrices. 𝜇𝜆, 𝜇𝜁, 𝜇𝜂, 𝜇𝜆, 𝜇𝜁 and 𝜇𝜂 are mean values of the eigen values 𝜆𝑖, 𝜁𝑖, 𝜂𝑖 , 𝜆𝑖, 𝜁𝑖 and 𝜂𝑖 respectively. example: let d be the rough neutrosophic matrix with the order of 3× 3. 𝐷 = [ ⟨(.8, .6, .7),(.9, .3, .4)⟩ ⟨(.4, .6, .5),(.7, .3, .2)⟩ ⟨(.3, .5, .6),(.7, .2, .1)⟩ ⟨(.5, .7, .7),(.6, .4, .5)⟩ ⟨(.6, .6, .7),(.8, .5, .3)⟩ ⟨(.2, .7, .8),(.9, .2, .3)⟩ ⟨(.1, .4, .5),(.5, .2, .3)⟩ ⟨(.5, .7, .8),(.6, .1, .2)⟩ ⟨(.4, .6, .9),(.7, .4, .3)⟩ ] 𝑛×𝑛 d can be expressed as 6 matrices. the energy of rough neutrosophic matrix and its application to mcdm problem for selecting … 35 𝐷(𝑇𝑖𝑗) = ( 0.8 0.5 0.1 0.4 0.6 0.5 0.3 0.2 0.4 ) 𝐷(𝐼𝑖𝑗) = ( 0.6 0.7 0.4 0.6 0.6 0.7 0.5 0.7 0.6 ) 𝐷(𝐹𝑖𝑗) = ( 0.7 0.7 0.5 0.5 0.7 0.8 0.6 0.8 0.9 ) 𝐷(𝑇𝑖𝑗) = ( 0.9 0.6 0.5 0.7 0.8 0.6 0.7 0.9 0.7 ) 𝐼𝑖𝑗(𝑆) = ( 0.3 0.4 0.2 0.3 0.5 0.1 0.2 0.2 0.4 ) 𝐹𝑖𝑗(𝑆) = ( 0.4 0.5 0.3 0.2 0.3 0.2 0.1 0.3 0.3 ) energy of d matrix, 𝐸(𝐷) = ⟨(1.4749,2.4105,2.6262),(2.6387,0.9544,1.0003)⟩ theorem 3.3. let d(n) be the rough neutrosophic matrix. if 𝜆𝑖, 𝜁𝑖, 𝜂𝑖 , 𝜆𝑖, 𝜁𝑖 and 𝜂𝑖, (𝑖 = 1,2,… 𝑛) are the eigenvalues of lower approximation of truth 𝐷(𝑇𝑖𝑗), indeterminacy 𝐷(𝐼𝑖𝑗), and false 𝐷(𝐹𝑖𝑗) and upper approximation of truth 𝐷(𝑇𝑖𝑗), indeterminacy 𝐷(𝐼𝑖𝑗), and false 𝐷(𝐹𝑖𝑗) membership values respectively. 1)∑|𝜆𝑖 − 𝜇𝜆| 𝑛 𝑖=1 = ∑|𝑎𝑖𝑖 − 𝜇𝜆| 𝑛 𝑖=1 = ∑|𝜆𝑖 − 𝜇𝜆| 𝑛 𝑖=1 = ∑|𝑎𝑖𝑖 − 𝜇𝜆| 𝑛 𝑖=1 = 0 ∑|𝜁𝑖 − 𝜇𝜁| 𝑛 𝑖=1 = ∑|𝑏𝑖𝑖 −𝜇𝜆| 𝑛 𝑖=1 = ∑|𝜁 𝑖 − 𝜇 𝜁 | 𝑛 𝑖=1 = ∑|𝑏𝑖𝑖 − 𝜇𝜁| 𝑛 𝑖=1 = 0 ∑|𝜂𝑖 − 𝜇𝜂| 𝑛 𝑖=1 = ∑|𝑐𝑖𝑖 − 𝜇𝜂| 𝑛 𝑖=1 = ∑|𝜂 𝑖 − 𝜇𝜂| 𝑛 𝑖=1 = ∑|𝑐𝑖𝑖 − 𝜇𝜂| 𝑛 𝑖=1 = 0 2) ∑(𝜆𝑖 − 𝜇𝜆) 2 = 𝑛 𝑖=1 ∑𝑎𝑖𝑖 2 + 2 ∑ 𝑎𝑖𝑗𝑎𝑗𝑖 − 𝑛 1≤𝑖<𝑗≤𝑛 𝜇𝜆 2 𝑛 𝑖=1 , ∑(𝜆𝑖 −𝜇𝜆) 2 = 𝑛 𝑖=1 ∑𝑎𝑖𝑖 2 + 2 ∑ 𝑎𝑖𝑗𝑎𝑗𝑖 − 𝑛 1≤𝑖<𝑗≤𝑛 𝜇 𝜆 2 𝑛 𝑖=1 ∑(𝜁𝑖 −𝜇𝜁) 2 = 𝑛 𝑖=1 ∑𝑎𝑖𝑖 2 + 2 ∑ 𝑎𝑖𝑗𝑎𝑗𝑖 − 𝑛 1≤𝑖<𝑗≤𝑛 𝜇𝜁 2 𝑛 𝑖=1 , ∑(𝜁 𝑖 − 𝜇 𝜁 ) 2 = 𝑛 𝑖=1 ∑𝑎𝑖𝑖 2 + 2 ∑ 𝑎𝑖𝑗𝑎𝑗𝑖 −𝑛 1≤𝑖<𝑗≤𝑛 𝜇 𝜁 2, 𝑛 𝑖=1 ∑(𝜂𝑖 − 𝜇𝜂) 2 = 𝑛 𝑖=1 ∑𝑎𝑖𝑖 2 + 2 ∑ 𝑎𝑖𝑗𝑎𝑗𝑖 − 𝑛 1≤𝑖<𝑗≤𝑛 𝜇𝜂 2 𝑛 𝑖=1 , ∑(𝜂 𝑖 − 𝜇𝜂) 2 = 𝑛 𝑖=1 ∑𝑎𝑖𝑖 2 + 2 ∑ 𝑎𝑖𝑗𝑎𝑗𝑖 − 𝑛 1≤𝑖<𝑗≤𝑛 𝜇𝜂 2 𝑛 𝑖=1 theorem 3.4. let 𝐷(𝑁) = ⟨(𝐷(𝑇𝑖𝑗),𝐷(𝐼𝑖𝑗),𝐷(𝐹𝑖𝑗)) ,(𝐷(𝑇𝑖𝑗),𝐷(𝐼𝑖𝑗),𝐷(𝐹𝑖𝑗))⟩ be the rough neutrosophic matrix. then the lower and upper bound of each energy is as follows jeni et al./decis. mak. appl. manag. eng. 5 (2) (2022) 30-45 36 𝑖)√(∑|𝜆𝑖 − 𝜇𝜆| 𝑛 𝑖=1 ) 2 − 2 ∑ |𝜆𝑖 − 𝜇𝜆||𝜆𝑗 − 𝜇𝜆| 1≤𝑖<𝑗≤𝑛 + 𝑛(𝑛 − 1)[|𝐷 − 𝜇𝜆|] 2 𝑛 ≤ 𝐸(𝐷(𝑇𝑖𝑗)) ≤ √2[(∑|𝜆𝑖 − 𝜇𝜆| 𝑛 𝑖=1 ) 2 − 2 ∑ |𝜆𝑖 − 𝜇𝜆||𝜆𝑗 − 𝜇𝜆| 1≤𝑖<𝑗≤𝑛 ] 𝑖𝑖)√(∑|𝜁𝑖 −𝜇𝜁| 𝑛 𝑖=1 ) 2 − 2 ∑ |𝜁𝑖 − 𝜇𝜁| |𝜁𝑗 − 𝜇𝜁| 1≤𝑖<𝑗≤𝑛 +𝑛(𝑛 − 1)[|𝐷 − 𝜇𝜁|] 2 𝑛 ≤ 𝐸(𝐷(𝐼𝑖𝑗)) ≤ √2[(∑|𝜁𝑖 − 𝜇𝜁| 𝑛 𝑖=1 ) 2 − 2 ∑ |𝜁𝑖 − 𝜇𝜁| |𝜁𝑗 − 𝜇𝜁| 1≤𝑖<𝑗≤𝑛 ] 𝑖𝑖𝑖)√(∑|𝜂𝑖 − 𝜇𝜂| 𝑛 𝑖=1 ) 2 − 2 ∑ |𝜂𝑖 −𝜇𝜂| |𝜂𝑗 − 𝜇𝜂| 1≤𝑖<𝑗≤𝑛 + 𝑛(𝑛 − 1)[|𝐷 −𝜇𝜂|] 2 𝑛 ≤ 𝐸(𝐷(𝐹𝑖𝑗)) ≤ √2[(∑|𝜂𝑖 −𝜇𝜂| 𝑛 𝑖=1 ) 2 − 2 ∑ |𝜂𝑖 −𝜇𝜂| |𝜂𝑗 − 𝜇𝜂| 1≤𝑖<𝑗≤𝑛 ] 𝑖𝑣)√(∑|𝜆𝑖 − 𝜇𝜆| 𝑛 𝑖=1 ) 2 − 2 ∑ |𝜆𝑖 − 𝜇𝜆||𝜆𝑗 − 𝜇𝜆| 1≤𝑖<𝑗≤𝑛 +𝑛(𝑛 − 1)[|𝐷 − 𝜇𝜆|] 2 𝑛 ≤ 𝐸(𝐷(𝑇𝑖𝑗)) ≤ √2[(∑|𝜆𝑖 −𝜇𝜆| 𝑛 𝑖=1 ) 2 − 2 ∑ |𝜆𝑖 − 𝜇𝜆||𝜆𝑗 − 𝜇𝜆| 1≤𝑖<𝑗≤𝑛 ] 𝑣)√(∑|𝜁 𝑖 −𝜇 𝜁 | 𝑛 𝑖=1 ) 2 − 2 ∑ |𝜁 𝑖 − 𝜇 𝜁 | |𝜁 𝑗 − 𝜇 𝜁 | 1≤𝑖<𝑗≤𝑛 + 𝑛(𝑛 − 1)[|𝐷 −𝜇𝜁|] 2 𝑛 ≤ 𝐸(𝐷(𝐼𝑖𝑗)) ≤ √2[(∑|𝜁𝑖 − 𝜇𝜁| 𝑛 𝑖=1 ) 2 − 2 ∑ |𝜁 𝑖 −𝜇 𝜁 | |𝜁 𝑗 − 𝜇 𝜁 | 1≤𝑖<𝑗≤𝑛 ] the energy of rough neutrosophic matrix and its application to mcdm problem for selecting … 37 𝑣𝑖)√(∑|𝜂 𝑖 −𝜇𝜂| 𝑛 𝑖=1 ) 2 − 2 ∑ |𝜂 𝑖 − 𝜇𝜂| |𝜂𝑗 − 𝜇𝜂| 1≤𝑖<𝑗≤𝑛 + 𝑛(𝑛 − 1)[|𝐷 − 𝜇𝜂|] 2 𝑛 ≤ 𝐸(𝐷(𝐹𝑖𝑗)) ≤ √2[(∑|𝜂𝑖 − 𝜇𝜂| 𝑛 𝑖=1 ) 2 − 2 ∑ |𝜂 𝑖 − 𝜇𝜂| |𝜂𝑗 − 𝜇𝜂| 1≤𝑖<𝑗≤𝑛 ] 4. the rough neutrosophic energy method in this section, we present a new approach to multi-criteria decision-making for selecting the best alternative using rough neutrosophic matrix energy. determine the set of k alternatives over m criteria. the alternatives are evaluated by n decision makers. so, we set 𝐴 = {𝐴1,𝐴2,… 𝐴𝑘}, 𝐶 = {𝐶1,𝐶2,… 𝐶𝑚} and 𝐷𝑀 = {𝐷𝑀1,𝐷𝑀2,… 𝐷𝑀𝑛} step 1: the rating values of each alternative on every criterion and the weighted values of m criteria were given by each decision-maker. we take each alternative rating and weight value as a matrix. consider the ratings of m criteria given by n decision-makers as a 𝑚 × 𝑛 matrix for weight w. ( 𝐷𝑀1 𝐷𝑀2 … 𝐷𝑀𝑛 𝐶1 ⟨ 𝛼{11} ,𝛽{11},𝛾{11}⟩ ⟨ 𝛼{12} ,𝛽{12},𝛾{12}⟩ … ⟨ 𝛼{1𝑛} ,𝛽{1𝑛},𝛾{1𝑛}⟩ 𝐶2 ⟨ 𝛼{21} ,𝛽{21},𝛾{21}⟩ ⟨ 𝛼{22} ,𝛽{22},𝛾{22}⟩ … ⟨ 𝛼{2𝑛} ,𝛽{2𝑛},𝛾{2𝑛}⟩ ⋮ ⋮ 𝐶𝑚 ⟨ 𝛼{𝑚1} ,𝛽{𝑚1},𝛾{𝑚1}⟩ ⟨ 𝛼{𝑚2} ,𝛽{𝑚2},𝛾{𝑚2}⟩ …⟨ 𝛼{𝑚𝑛} ,𝛽{𝑚𝑛},𝛾{𝑚𝑛}⟩ ) (1) consider the ratings of n decision-makers over n criteria as a 𝑛 × 𝑚 matrix in alternative 𝐴1. ( 𝐶1 𝐶2 … 𝐶𝑚 𝐷𝑀1 ⟨ 𝑎{11} ,𝑏{11}, 𝑐{11}⟩ ⟨ 𝑎{12} ,𝑏{12}, 𝑐{12}⟩ …⟨ 𝑎{1𝑚} ,𝑏{1𝑚}, 𝑐{1𝑚}⟩ 𝐷𝑀2 ⟨ 𝛼{21} ,𝛽{21},𝛾{21}⟩ ⟨ 𝛼{22} ,𝛽{22},𝛾{22}⟩ …⟨ 𝛼{2𝑚} ,𝛽{2𝑚},𝛾{2𝑚}⟩ ⋮ ⋮ 𝐷𝑀𝑛 ⟨ 𝛼{𝑛1} ,𝛽{𝑛1},𝛾{𝑛1}⟩ ⟨ 𝛼{𝑛2} ,𝛽{𝑛2},𝛾{𝑛2}⟩ … ⟨ 𝛼{𝑛𝑚} ,𝛽{𝑛𝑚},𝛾{𝑛𝑚}⟩ ) (2) step 2: determine the weights of decision makers. let 𝐷𝑀1,𝐷𝑀2,… 𝐷𝑀𝑛 be the decision makers, they have individual’s weights. consider 𝐷𝑀1 = ⟨ 𝑥1,𝑦1,𝑧1⟩ , 𝐷𝑀2 = ⟨ 𝑥2,𝑦2,𝑧2⟩, …, 𝐷𝑀𝑛 = ⟨ 𝑥𝑛,𝑦𝑛,𝑧𝑛⟩ step 3: determine rough neutrosophic matrix for criteria and alternatives. the relation between the weight of decision makers and criteria is formed as a rough neutrosophic matrix for criteria. 𝑊(𝐶1𝐷𝑀1) = ⟨ (𝑚𝑖𝑛(𝑥1,𝛼11),𝑚𝑎𝑥(𝑦1,𝛽11),𝑚𝑎𝑥(𝑧1,𝛾11)),(𝑚𝑎𝑥(𝑥1,𝛼11),𝑚𝑖𝑛(𝑦1,𝛽11),𝑚𝑖𝑛(𝑧1,𝛾11))⟩ 𝑊(𝐶1𝐷𝑀1) = ⟨ (𝛼11,𝛽11,𝛾11),(𝛼11,𝛽11,𝛾11)⟩ (3) jeni et al./decis. mak. appl. manag. eng. 5 (2) (2022) 30-45 38 𝑊 = ( 𝐷𝑀1 … 𝐷𝑀𝑛 𝐶1 ⟨ (𝛼11,𝛽11,𝛾11),(𝛼11,𝛽11,𝛾11)⟩ … ⟨ (𝛼1𝑛,𝛽1𝑛,𝛾1𝑛),(𝛼1𝑛,𝛽1𝑛,𝛾1𝑛)⟩ 𝐶2 ⟨ (𝛼21,𝛽21,𝛾21),(𝛼21,𝛽21,𝛾21)⟩ … ⟨ (𝛼2𝑛,𝛽2𝑛,𝛾2𝑛) ,(𝛼2𝑛,𝛽2𝑛,𝛾2𝑛)⟩ ⋮ ⋮ 𝐶𝑚 ⟨ (𝛼𝑚1,𝛽𝑚1,𝛾𝑚1),(𝛼𝑚1,𝛽𝑚1,𝛾𝑚1)⟩ … ⟨ (𝛼𝑚𝑛,𝛽𝑚𝑛,𝛾𝑚𝑛),(𝛼𝑚𝑛,𝛽𝑚𝑛,𝛾𝑚𝑛)⟩ ) the relation between the weight of criteria and alternatives is formed as a rough neutrosophic matrix for alternative. 𝐴1(𝐷𝑀1𝐶1) = ⟨ (𝑚𝑖𝑛(𝛼11,𝑎11),𝑚𝑎𝑥(𝛽11,𝑏11),𝑚𝑎𝑥(𝛾11 , 𝑐11)), (𝑚𝑎𝑥(𝛼11,𝑎11),𝑚𝑖𝑛(𝛽11,𝑏11),𝑚𝑖𝑛(𝛾11, 𝑐11))⟩ 𝐴1(𝐷𝑀1 𝐶1) = ⟨ (𝑎11,𝑏11, 𝑐11),(𝑎11,𝑏11, 𝑐11)⟩ (4) 𝐴1 = ( 𝐶1 … 𝐶𝑚 𝐷𝑀1 ⟨ (𝑎11,𝑏11, 𝑐11),(𝑎11,𝑏11, 𝑐11)⟩ … ⟨ (𝑎1𝑚,𝑏1𝑚, 𝑐1𝑚),(𝑎1𝑚,𝑏1𝑚, 𝑐1𝑚)⟩ 𝐷𝑀2 ⟨ (𝑎21,𝑏21, 𝑐21),(𝑎21,𝑏21, 𝑐21)⟩ … ⟨ (𝑎2𝑚,𝑏2𝑚, 𝑐2𝑚),(𝑎2𝑚,𝑏2𝑚, 𝑐2𝑚)⟩ ⋮ ⋮ 𝐷𝑀𝑛 ⟨ (𝑎𝑛1,𝑏𝑛1, 𝑐𝑛1),(𝑎𝑛1,𝑏𝑛1, 𝑐𝑛1)⟩ … ⟨ (𝑎𝑛𝑚,𝑏𝑛𝑚,𝑐𝑛𝑚),(𝑎𝑛𝑚,𝑏𝑛𝑚, 𝑐𝑛𝑚)⟩ ) step 4: in this step, we convert the non-square matrix into a square matrix. from the above w, the matrix is expressed as 6 matrices which are truth, indeterminacy, false matrix of lower approximation and truth, indeterminacy, false matrix of upper approximation which are denoted by 𝑊(𝑇 𝑖𝑗 ), 𝑊(𝐼 𝑖𝑗 ), 𝑊(𝐹 𝑖𝑗 ) and 𝑊(𝑇𝑖𝑗), 𝑊(𝐼𝑖𝑗), 𝑊(𝐹𝑖𝑗). similarly, 𝐴1 matrix expressed as 𝐴1(𝑇𝑖𝑗), 𝐴1(𝐼𝑖𝑗), 𝐴1(𝐹𝑖𝑗) and 𝐴1(𝑇𝑖𝑗), 𝐴1(𝐼𝑖𝑗), 𝐴1(𝐹𝑖𝑗). 𝐴1 (𝑇𝑖𝑗)𝑛×𝑚 ∗ 𝑊(𝑇 𝑖𝑗 ) 𝑚×𝑛 = ( 𝛼𝑎 11 𝛼𝑎 21 ⋯ 𝛼𝑎 𝑛1 ⋮ ⋱ ⋮ 𝛼𝑎 1𝑛 𝛼𝑎 2𝑛 ⋯ 𝛼𝑎 𝑛𝑛 ) 𝑛×𝑛 (5) step 5: using the definition of rough neutrosophic matrix energy, calculate the energy of the matrix. we got six energies for truth, indeterminacy, and false matrices of lower and upper approximation for one alternative. 𝐸(𝐴1) = ⟨ (𝐸(𝐴1(𝑇)),𝐸(𝐴1(𝐼))𝐸(𝐴1(𝐹))),(𝐸(𝐴1(𝑇)) ,𝐸(𝐴1(𝐼))𝐸(𝐴1(𝐹)))⟩ (6) step 6: continue this process for k alternatives. for each alternative, we got rough neutrosophic matrix energies of 𝐸(𝐴1),𝐸(𝐴2)… 𝐸(𝐴𝑘). step 7: for ranking the energy values we determine the average values of lower and upper approximation values. then we get, the energy of rough neutrosophic matrix and its application to mcdm problem for selecting … 39 𝐸(𝐴1) = ⟨ 𝐸(𝐴1(𝑇)),𝐸(𝐴1(𝐼)),𝐸(𝐴1(𝐹)) ⟩ 𝐸(𝐴2 ) = ⟨ 𝐸(𝐴2 (𝑇)),𝐸(𝐴2(𝐼)),𝐸(𝐴2(𝐹)) ⟩ ⋮ 𝐸(𝐴𝑘) = ⟨ 𝐸(𝐴𝑘(𝑇)),𝐸(𝐴𝑘(𝐼)),𝐸(𝐴𝑘(𝐹))⟩ finally, we rank the alternatives according to their truth values. the alternative that has the highest truth energy value will be the best. 5. numerical example we solve the problem by using our proposed method to choose the best place to construct the school building in a particular town. in this problem, the decision makers are project manager (𝐷𝑀1), approval officer (𝐷𝑀2), engineer (𝐷𝑀3), and public representative (𝐷𝑀4). the following are the criteria for deciding where to build: 𝐶1 land clearance, 𝐶2land title, 𝐶3zonal clearance, 𝐶4cost, 𝐶5transport facility, and 𝐶6building plan. the decision-makers choose the best place from the following alternatives based on the above criterion, place a, place b, place c, place d, place e, and place d. the decision makers give their ratings in terms of linguistic variables. it is shown in table 1. table 1. linguistic variable for svnn s.no linguistic variable neutrosophic numbers 1 very poor (vp)/ very low (vl) ⟨ 0.1,0.8,0.9 ⟩ 2 poor (p)/ low (l) ⟨ 0.35,0.6,0.7⟩ 3 medium (m)/ fair (f) ⟨ 0.5,0.4,0.45 ⟩ 4 good(g)/ high (h) ⟨ 0.8,0.2,0.15⟩ 5 very good (vg)/ very high (vh) ⟨0.9,0.1,0.1 ⟩ step: 1 the decision makers evaluate the criteria and each alternative by the linguistic variable. it is shown in table 2 and 3 respectively. table 2. weights of criteria criteria 𝐷𝑀1 𝐷𝑀2 𝐷𝑀3 𝐷𝑀4 𝐶1 𝑉𝐺⟨ .9, .1, .1⟩ 𝐺⟨ .8, .2, .15 ⟩ 𝑉𝐺⟨ .9, .1, .1⟩ 𝑀⟨ .5, .4, .45 ⟩ 𝐶2 𝑀⟨ .5, .4, .45 ⟩ 𝐺⟨ .8, .2, .15 ⟩ 𝑀⟨ .5, .4, .45 ⟩ 𝐺⟨ .8, .2, .15 ⟩ 𝐶3 𝐺⟨ .8, .2, .15 ⟩ 𝑉𝐺⟨ .9, .1, .1⟩ 𝐺⟨ .8, .2, .15 ⟩ 𝐺⟨ .8, .2, .15 ⟩ 𝐶4 𝐻⟨ .8, .2, .15 ⟩ 𝐻⟨ .8, .2, .15 ⟩ 𝐹⟨ .5, .4, .45 ⟩ 𝑉𝐻⟨ .9, .1, .1 ⟩ 𝐶5 𝑀⟨ .5, .4, .45⟩ 𝐺⟨ .8, .2, .15 ⟩ 𝐺⟨ .8, .2, .15 ⟩ 𝑀⟨ .5, .4, .45 ⟩ 𝐶6 𝑉𝐺⟨ .9, .1, .1⟩ 𝑉𝐺⟨ .9, .1, .1⟩ 𝐺⟨ .8, .2, .15 ⟩ 𝐺⟨ .8, .2, .15 ⟩ table 3. ratings in terms of linguistic variables for each alternative alt 𝐷𝑀 𝐶1 𝐶2 𝐶3 𝐶4 𝐶5 𝐶6 a 𝐷𝑀1 𝐺⟨.8, .2, .15⟩ 𝑉𝐺⟨.9, .1, .1⟩ 𝐺⟨.8, .2, .15⟩ 𝐹⟨.5, .4, .45⟩ 𝑀⟨.5, .4, .45⟩ 𝐺⟨.8, .2, .15⟩ 𝐷𝑀2 𝑀⟨.5, .4, .45⟩ 𝐺⟨.8, .2, .15⟩ 𝑀⟨.5, .4, .45⟩ 𝑉𝐻⟨.9, .1, .1⟩ 𝑉𝐺⟨.9, .1, .1⟩ 𝑃⟨.35, .6, .7⟩ jeni et al./decis. mak. appl. manag. eng. 5 (2) (2022) 30-45 40 𝐷𝑀3 𝑃⟨.35, .6, .7⟩ 𝑀⟨.5, .4, .45⟩ 𝐺⟨.8, .2, .15⟩ 𝐻⟨.8, .2, .15⟩ 𝑀⟨.5, .4, .45⟩ 𝑉𝐺⟨.9, .1, .1⟩ 𝐷𝑀4 𝑀⟨.5, .4, .45⟩ 𝑃⟨.35, .6, .7⟩ 𝑀⟨.5, .4, .45⟩ 𝐻⟨.8, .2, .15⟩ 𝑉𝐺⟨.9, .1, .1⟩ 𝐺⟨.8, .2, .15⟩ b 𝐷𝑀1 𝑀⟨.5, .4, .45⟩ 𝐺⟨.8, .2, .15⟩ 𝑃⟨.35, .6, .7⟩ 𝐿⟨.35, .6, .7⟩ 𝐺⟨.8, .2, .15⟩ 𝑀⟨.5, .4, .45⟩ 𝐷𝑀2 𝑉𝑃⟨.1, .8, .9⟩ 𝑀⟨.5, .4, .45⟩ 𝐺⟨.8, .2, .15⟩ 𝐿⟨.35, .6, .7⟩ 𝑀⟨.5, .4, .45⟩ 𝐺⟨.8, .2, .15⟩ 𝐷𝑀3 𝑃⟨.35, .6, .7⟩ 𝑃⟨.35, .6, .7⟩ 𝐺⟨.8, .2, .15⟩ 𝐹⟨.5, .4, .45⟩ 𝑃⟨.35, .6, .7⟩ 𝑀⟨.5, .4, .45⟩ 𝐷𝑀4 𝐺⟨.8, .2, .15⟩ 𝑉𝑃⟨.1, .8, .9⟩ 𝐺⟨.8, .2, .15⟩ 𝐹⟨.5, .4, .45⟩ 𝑉𝑃⟨.1, .8, .9⟩ 𝑃⟨.35, .6, .7⟩ c 𝐷𝑀1 𝑉𝐺⟨.9, .1, .1⟩ 𝐺⟨.8, .2, .15⟩ 𝑉𝐺⟨.9, .1, .1⟩ 𝐹⟨.5, .4, .45⟩ 𝐺⟨.8, .2, .15⟩ 𝐺⟨.8, .2, .15⟩ 𝐷𝑀2 𝑀⟨.5, .4, .45⟩ 𝑉𝐺⟨.9, .1, .1⟩ 𝐺⟨.8, .2, .15⟩ 𝐻⟨.8, .2, .15⟩ 𝑉𝐺⟨.9, .1, .1⟩ 𝑀⟨.5, .4, .45⟩ 𝐷𝑀3 𝐺⟨.8, .2, .15⟩ 𝑀⟨.5, .4, .45⟩ 𝐺⟨.8, .2, .15⟩ 𝐹⟨.5, .4, .45⟩ 𝑀⟨.5, .4, .45⟩ 𝐺⟨.8, .2, .15⟩ 𝐷𝑀4 𝑉𝐺⟨.9, .1, .1⟩ 𝑉𝐺⟨.9, .1, .1⟩ 𝑀⟨.5, .4, .45⟩ 𝐻⟨.8, .2, .15⟩ 𝐺⟨.8, .2, .15⟩ 𝑀⟨.5, .4, .45⟩ d 𝐷𝑀1 𝐺⟨.8, .2, .15⟩ 𝑀⟨.5, .4, .45⟩ 𝐺⟨.8, .2, .15⟩ 𝐿⟨.35, .6, .7⟩ 𝑃⟨.35, .6, .7⟩ 𝑉𝑃⟨.1, .8, .9⟩ 𝐷𝑀2 𝑃⟨.35, .6, .7⟩ 𝑉𝑃⟨.1, .8, .9⟩ 𝐺⟨.8, .2, .15⟩ 𝐹⟨.5, .4, .45⟩ 𝑉𝐺⟨.9, .1, .1⟩ 𝑃⟨.35, .6, .7⟩ 𝐷𝑀3 𝑀⟨.5, .4, .45⟩ 𝑃⟨.35, .6, .7⟩ 𝑃⟨.35, .6, .7⟩ 𝐻⟨.8, .2, .15⟩ 𝑃⟨.8, .2, .15⟩ 𝑉𝑃⟨.1, .8, .9⟩ 𝐷𝑀4 𝑃⟨.35, .6, .7⟩ 𝑀⟨.5, .4, .45⟩ 𝑉𝑃⟨.1, .8, .9⟩ 𝐻⟨.8, .2, .15⟩ 𝑀⟨.5, .4, .45⟩ 𝑀⟨.5, .4, .45⟩ e 𝐷𝑀1 𝐺⟨.8, .2, .15⟩ 𝑃⟨.35, .6, .7⟩ 𝑃⟨.35, .6, .7⟩ 𝐹⟨.5, .4, .45⟩ 𝑀⟨.5, .4, .45⟩ 𝐺⟨.8, .2, .15⟩ 𝐷𝑀2 𝑃⟨.35, .6, .7⟩ 𝐺⟨.8, .2, .15⟩ 𝑉𝐺⟨.9, .1, .1⟩ 𝐻⟨.8, .2, .15⟩ 𝑀⟨.5, .4, .45⟩ 𝐺⟨.8, .2, .15⟩ 𝐷𝑀3 𝑀⟨.5, .4, .45⟩ 𝑃⟨.35, .6, .7⟩ 𝑉𝑃⟨.1, .8, .9⟩ 𝐿⟨.35, .6, .7⟩ 𝐺⟨.8, .2, .15⟩ 𝑃⟨.35, .6, .7⟩ 𝐷𝑀4 𝑀⟨.5, .4, .45⟩ 𝑃⟨.35, .6, .7⟩ 𝐺⟨.8, .2, .15⟩ 𝐿⟨.35, .6, .7⟩ 𝑀⟨.5, .4, .45⟩ 𝑃⟨.35, .6, .7⟩ step: 2 weights of decision makers. 𝐷𝑀1 = 𝐺 ⟨ 0.8,0.2,0.15 ⟩ , 𝐷𝑀2 = 𝑉𝐺⟨ 0.9,0.1,0.1⟩, 𝐷𝑀3 = 𝑀 ⟨ 0.5,0.4,0.45 ⟩ and 𝐷𝑀4 = 𝐺 ⟨ 0.8,0.2,0.15 ⟩ step: 3 determine rough neutrosophic matrix for criteria. table 4 shows that the relation between the weight of decision makers and criteria are formed as rough neutrosophic matrix for criteria 𝑊(𝐶1𝐷𝑀1) = ⟨ (min (0.8,0.9),max (0.2,0.1),max (0.15,0.1)), (𝑚𝑎𝑥(0.8,0.9),𝑚𝑖𝑛(0.2,0.1),𝑚𝑖𝑛(0.15,0.1))⟩ 𝑊(𝐶1𝐷𝑀1) = ⟨ (0.8,0.2,0.15),(0.9,0.1,0.1)⟩ table 4. rough neutrosophic matrix of criteria c 𝐷𝑀1 𝐷𝑀2 𝐷𝑀3 𝐷𝑀4 𝐶1 ⟨(.8, .2, .15),(.9, .1, .1) ⟩ ⟨(.8, .2, .15),(.9, .1, .1) ⟩ ⟨(.5, .4, .45),(.9, .1, .1)⟩ ⟨(.5, .4, .45),(.8, .2, .15)⟩ 𝐶2 ⟨(.5, .4, .45),(.8, .2, .15)⟩ ⟨(.8, .2, .15),(.9, .1, .1) ⟩ ⟨(.5, .4, .45),(.5, .4, .45)⟩ ⟨(.8, .2, .15),(.8, .2, .15)⟩ 𝐶3 ⟨(.8, .2, .15),(.8, .2, .15)⟩ ⟨(.9, .1, .1),(.9, .1, .1)⟩ ⟨(.5, .4, .45),(.8, .2, .15)⟩ ⟨(.8, .2, .15),(.8, .2, .15)⟩ 𝐶4 ⟨(.8, .2, .15),(.8, .2, .15)⟩ ⟨(.8, .2, .15),(.9, .1, .1) ⟩ ⟨(.5, .4, .45),(.5, .4, .45)⟩ ⟨(.8, .2, .15),(.9, .1, .1)⟩ 𝐶5 ⟨(.5, .4, .45),(.8, .2, .15)⟩ ⟨(.8, .2, .15),(.9, .1, .1) ⟩ ⟨(.5, .4, .45),(.8, .2, .15)⟩ ⟨(.5, .4, .45),(.8, .2, .15)⟩ 𝐶6 ⟨(.8, .2, .15),(.9, .1, .1)⟩ ⟨(.9, .1, .1),(.9, .1, .1)⟩ ⟨(.5, .4, .45),(.8, .2, .15)⟩ ⟨(.8, .2, .15),(.8, .2, .15)⟩ table 5 shows that the relation between the weight of criteria and alternatives are formed as rough neutrosophic matrix for alternative 𝐴1(𝐷𝑀1𝐶2) = ⟨ (𝑚𝑖𝑛(0.5,0.9),𝑚𝑎𝑥(0.4,0.1),𝑚𝑎𝑥(0.45,0.1)), (𝑚𝑎𝑥(0.5,0.9),𝑚𝑖𝑛(0.4,0.1),𝑚𝑖𝑛(0.45,0.1))⟩ 𝐴1(𝐷𝑀1𝐶2) = ⟨ (0.5,0.4,0.45),(0.9,0.1,0.1) ⟩ the energy of rough neutrosophic matrix and its application to mcdm problem for selecting … 41 table 5. rough neutrosophic matrix of alternative 1 dm rough neutrosophic values of each criterion for 𝐴1 𝐷𝑀1 𝐶1⟨(.8, .2, .15),(.9, .1, .1)⟩ 𝐶2⟨(.5, .4, .45),(.9, .1, .1)⟩ 𝐶3⟨(.8, .2, .15),(.8, .2, .15)⟩ 𝐶4⟨(.5, .4, .45),(.8, .2, .15)⟩ 𝐶5⟨(.5, .4, .45),(.5, .4, .45) ⟩ 𝐶6⟨(.8, .2, .15),(.9, .1, .1)⟩ 𝐷𝑀2 𝐶1⟨(.5, .4, .45),(.8, .2, .15)⟩ 𝐶2⟨(.8, .2, .15),(.8, .2, .15)⟩ 𝐶3⟨(.5, .4, .45),(.9, .1, .1)⟩ 𝐶4⟨(.8, .2, .15),(.9, .1, .1)⟩ 𝐶5⟨(.8, .2, .15),(.9, .1, .1)⟩ 𝐶6⟨(.35, .6, .7),(.9, .1, .1)⟩ 𝐷𝑀3 𝐶1⟨(.35, .6, .7),(.9, .1, .1)⟩ 𝐶2⟨(.5, .4, .45),(.5, .4, .45)⟩ 𝐶3⟨(.8, .2, .15),(.8, .2, .15)⟩ 𝐶4⟨(.5, .4, .45),(.8, .2, .15)⟩ 𝐶5⟨(.5, .4, .45),(.8, .2, .15)⟩ 𝐶6⟨(.8, .2, .15),(.9, .1, .1)⟩ 𝐷𝑀4 𝐶1⟨(.5, .4, .45),(.5, .4, .45)⟩ 𝐶2⟨(.1, .8, .9),(.8, .2, .15)⟩ 𝐶3⟨(.5, .4, .45),(.8, .2, .15)⟩ 𝐶4⟨(.8, .2, .15),(.9, .1, .1)⟩ 𝐶5⟨(.5, .4, .45),(.9, .1, .1)⟩ 𝐶6⟨(.8, .2, .15),(.8, .2, .15)⟩ step 4: we convert the non-square matrix into a square matrix. from table 4 and 5, we expressed the both matrices into 6 matrices. now we consider the truth lower approximation matrix of both tables 𝐴1 (𝑇𝑖𝑗)𝑛×𝑚 = [ 0.8 0.5 0.8 0.5 0.5 0.8 0.5 0.8 0.5 0.8 0.8 0.35 0.35 0.5 0.8 0.5 0.5 0.8 0.5 0.1 0.5 0.8 0.5 0.8 ] 𝑊(𝑇 𝑖𝑗 ) 𝑚×𝑛 = [ 0.8 0.5 0.8 0.8 0.5 0.8 0.8 0.8 0.9 0.8 0.8 0.9 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.8 0.8 0.8 0.5 0.8 ] 𝐴1 (𝑇𝑖𝑗)𝑛×𝑚 ∗ 𝑊(𝑇 𝑖𝑗 ) 𝑚×𝑛 = [ 2.82 3.28 1.95 2.73 2.52 3.085 1.875 2.61 2.46 2.92 1.725 2.505 2.38 2.69 1.6 2.26 ] step 5: calculating the energy of rough neutrosophic matrix eigen values of the above matrix 9.8836, 0.0176 + 0.0579i, 0.0176 0.0579i, 0.0287 and mean of the eigen values is 2.4725. 𝐸(𝑇 𝑖𝑗 ) = |9.8836 − 2.4725| + |0.0176 + 0.0579𝑖 − 2.4725| + |0.0176 − 0.0579𝑖 − 2.4725| + |−0.0287 − 2.4725| = 14.8235 step 6: similarly, we can find the energy for all the matrices of lower and upper approximation of truth, indeterminacy, and false matrices. energy of rough neutrosophic matrices place a = [(14.8236, 3.6111, 3.8090), (23.7794, 1.1667, 0.9430)] place b = [(11.4882, 4.8871, 5.5057), (23.2477, 1.3475, 1.1150)] place c = [(16.1570, 2.9861, 3.0163), (24.2484, 1.1979, 1.0370)] place d = [(11.0676, 4.9059, 5.4638), (22.5909, 1.3974, 1.1322)] place e = [(13.4987, 4.3102, 4.6900), (22.1196, 1.4901, 1.2752)] jeni et al./decis. mak. appl. manag. eng. 5 (2) (2022) 30-45 42 step 7: calculate the average value of the lower and upper approximation of energy of rough neutrosophic sets, then the ranking of alternatives will be decided by the truth values. average energy of each alternative is given below place a = [(19.3015, 2.3889, 2.376)] place b = [(17.3679, 3.1173, 3.3103)] place c = [(20.2027, 2.092, 2.0266)] place d = [(16.8292, 3.1516, 3.298)] place e = [(17.8091, 2.9001, 2.9826)] the average energy of truth and ranking order of alternatives presented in table 6. table 6. ranking order alternatives truth energy ranking order place a 19.3015 ii place b 17.3679 iv place c 20.2027 i place d 16.8292 v place e 17.8091 iii the ranking order of the alternatives is c > a > e > b > d. place c is the best location to build the school construction in the town. 6. conclusion the energy of the matrix helps to determine the matrix's weight. we apply this idea of energy to the rough neutrosophic matrix. the energy of rough neutrosophic matrix contains truth indeterminacy and false energy for the lower and upper approximations of each matrix. the final energy was determined by averaging the lowest and upper approximations of each energy. in that, the ranking of alternatives is evaluated using truth value. in our taken problem, the decision-maker chooses the perfect spot for the construction of the school. the building should be constructed at place c. it satisfies all requirements. as a result, the energy of rough neutrosophic matrix will be used in every situation and our proposed energy method helps to solve the multi-criteria decision-making problems. compared to other mcdm methods our presented method simplifies the work and also give more effective result. further, we will extend the rough neutrosophic energy concept to other types of rough neutrosophic matrices such as interval-valued, multi-valued and so on. author contributions: conceptualization, j.s.m. and d.g.; methodology, j.s.m. and d.g.; software, j.s.m. and d.g.; validation, j.s.m. and d.g.; formal analysis, j.s.m. and d.g.; investigation, d.g.; writing—original draft preparation, j.s.m; writing—review and editing, d.g.; visualization, j.s.m. and d.g.; supervision, d.g. all authors have read and agreed to the published version of the manuscript. the energy of rough neutrosophic matrix and its application to mcdm problem for selecting … 43 funding: this research received no external funding. acknowledgments: we are grateful to vellore institute of technology, vellore for giving us this opportunity. data availability statement: not applicable. conflicts of interest: the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. references alias, s., mohamad, d., & shuib, a. 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(2018). medical diagnosis based on singlevalued neutrosophic probabilistic rough multisets over two universes. symmetry, 10(6), 213. zhang, z. (2012). on interval type-2 rough fuzzy sets. knowledge-based systems, 35, 1-13. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 5, issue 2, 2022, pp. 120-139. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0305102022v * corresponding author. e-mail addresses: lukacs.eszter@sze.hu (e. lukács), volgyi.katalin@sze.hu (k. völgyi), kovacsn@sze.hu (n. kovács), totha@sze.hu (á. tóth) american versus domestic digital companies in the chinese market eszter lukács1, katalin völgyi1,2*, norbert kovács1 and árpád tóth1 1 széchenyi istván university, hungary 2 elkh centre for economic and regional studies, institute of world economics, hungary received: 29 july 2022; accepted: 17 september 2022; available online: 5 october 2022. original scientific paper abstract: the digital economy has become an increasingly important part of the world economy. it is vastly concentrated in two economies, namely, the united states and china. the main aim of our study is to investigate chinese digital companies and government policy enabling the rapid development of the country’s digital economy and the largest american digital companies’ performance in the chinese market. our findings show that the largest american digital companies, which are globally active players, usually have a very limited market share in different segments of the chinese digital economy or have been forced to leave the chinese market after a short period of operation. in the future, chinese government policy will continue to ensure the priority role of domestic digital companies in the upgrading and structural transformation of the chinese economy driven by services, high-tech sectors, and consumption while limiting the role of american digital companies. keywords: digital companies; china; usa; government policy. 1. introduction since the global financial and economic crisis of 2008-2009, china’s economic growth has significantly decelerated. the crisis has resulted – among others – in the ‘launching of china’s industrial upgrading and economic transformation from an export, manufacturing, and investment-driven growth model to a consumption, innovation, and service sector-driven one’ (völgyi & lukács, 2021, p. 172). ‘the services sector, in particular producer services (transportation, logistics, warehousing, posts and telecommunications, finance, computer services and software, and leasing and business services, etc), and the high-tech sector will be the core drivers of growth’ (kpmg, 2018, p. 23) in the new development stage of the chinese economy. in 2015, the services sector became the ‘foremost driver of growth mailto:lukacs.eszter@sze.hu mailto:volgyi.katalin@sze.hu mailto:kovacsn@sze.hu mailto:totha@sze.hu american versus domestic digital companies in the chinese market 121 in the chinese economy. its contribution to gdp growth surpassed the 50 percent mark for the first time since the beginning of the “reform and opening up”’ (kpmg, 2018, p. 23). in 2020, the services sector’s contribution to china’s economic growth reached 60 percent (wong, 2020). ‘the high-tech sector serves as an important driver of industrial growth. since 2012, growth in value added from high-tech has consistently outpaced the manufacturing sector. technology now also clearly outperforms the traditional manufacturing sector in terms of its contribution to total economic growth’ (kpmg, 2018, p. 23). from the expenditure approach of gdp, we can say that in the ‘years since 2011, and with the exception of 2013 and 2020, consumption has exceeded investment in terms of its annual contribution to economic growth’ (kpmg, 2018, p. 13). in 2021, consumption had again the largest contribution (5.3 percentage points) to gdp growth rate (8.1 percent) (national bureau of statistics of china, 2021, 2022). in other words, domestic consumption has become the main driver of growth in the chinese economy. in our study, we investigate the largest chinese digital companies that have been playing a significant role in the country’s new growth model and development path. the digital economy reached 30 percent of the chinese gdp by 2017 (unctad, 2019). the market capitalization of chinese digital companies is comparable to their american counterparts (e.g., alphabet (google), amazon, apple, facebook, ebay, booking), which together have been dominating the list of top 20 internet companies in the world. the further rapid development of the chinese digital economy is supported by the 14th five-year economic plan and several recently launched elements of the state industrial policy, such as ‘internet plus’, ‘made in china 2025’ and ‘national informatization development strategy’. chinese government policy gives an advantage to domestic companies in the chinese digital market and strives to preserve the benefits from the upgrading and structural transformation of the chinese economy driven by services, high-tech sectors, and consumption exclusively for them by limiting the role of american digital companies or even excluding them from the domestic market. 2. the evolution of the digital economy ‘the digital economy is becoming an ever more important part of the global economy’ (unctad, 2017, p. iv). the development of the digital economy can be divided into two phases. the first phase started in the 1980s, whereas the second began in the 2000s. table 1 gives a clear overview of the main features of the digital economy 1.0, and the new ones brought in by the second phase of the digital economy. table 1. the evolution of the digital economy (li, 2017) digital economy 1.0 digital economy 2.0 supporting technology computer, software, communication, internet mobile internet, big data, cloud computing, internet of things (iot), artificial intelligence (ai), robots, 3d printing, virtual/augmented/mixed reality (vr/ar/mr) characteristi cs consumption internet, commercial internet industrial internet field of use news, search engine, email, shopping, supply chain management, manufacturing, customized services, smart city, smart transportation, lukács et al./decis. mak. appl. manag. eng. 5 (2) (2022) 120-139 122 communications, games smart healthcare, smart home devices, fintech, wearable devices business model web portal, ecommerce, instant messaging, online music, video, games, etc. mobile e-commerce, mobile social network, sharing economy, online-tooffline (o2o) commerce, crowdsourcing, self-media, locationbased services (lbs), etc. the global advance of the digital economy can be measured by several indicators. the ict sector accounted for 4.5 percent of the global gdp in 2017 (unctad, 2019). between 2017 and 2019, the estimated value of global e-commerce sales grew from 23.8 trillion us dollars to 26.7 trillion us dollars (unctad, 2020, 2021a). between 2015 and 2020, exports of ict services increased by 49 percent, to 676 billion us dollars. trade in ict goods had exceeded 2.354 trillion usa dollars by 2020. in 2020, due to the global pandemic, there was a drop in total trade in goods and services, but the trade of ict goods and services increased by growing demand for ‘accelerating digitalization and deepening reliance on digital technologies’ (unctad, 2021b, p. 2). ‘global internet protocol (ip) traffic grew from about 100 gigabytes per second in 2002 to some 88,000 gigabytes per second in 2020’ (un, 2021, p. 13). in 2020, ‘the global pandemic had a dramatic impact on internet traffic, as most activities increasingly took place online. the global internet traffic in 2022 is expected to exceed all the internet traffic up to 2016’ (unctad, 2021, p. 17). there are significant differences in the development of the digital economy between developed and developing countries (digital divide). for example, ‘in 2021, an estimated 37 percent of the global population did not have internet access. of the 2.9 billion people still offline, an estimated 96 percent lived in developing countries’ (itu, 2021a). 90.3 percent of the population in developed countries and 57.1 percent in developing countries used the internet (itu, 2021b). nevertheless, between 2005 and 2021, the number of global internet users expanded by 3.9 billion people, of which 3.3 billion people are from developing countries (itu, 2021b). in the case of the most populous country, namely china, the contribution to this expansion was significant where the share of internet users in the total population increased from 8.5 percent in 2005 to 71.6 percent in 2021 (itu, 2021c; cnnic, 2021). in addition, we have to emphasize the effect of the pandemic on global internet usage. a ‘covid connectivity boost’ has brought an estimated 782 million additional people online since 2019 (itu, 2021a). the rapid increase of global e-commerce (b2b, b2c) is also evidence for the headway of the digital economy. 59 percent of the population aged 15 years or older in high-income countries bought something on the internet in 2017, while less than 2 percent of the population aged 15 years or older in low-income countries did the same (unctad, 2019). in 2019, the united states had the largest e-commerce market, followed by japan and china in the global ranking. however, if only b2c commerce is taken into consideration, then china has the largest market, followed by the usa in second place. the global ranking of cross-border b2c e-commerce sales is led by china, the united states, and the united kingdom. cross-border b2c e-commerce sales (440 billion us dollars) accounted for 9 percent of the global b2c e-commerce sales in 2019 (unctad, 2021a). the advancement of the digital economy is also underpinned by the rapid spread of robots and 3d printers. according to data of the international federation of robotics, sales of industrial robots grew worldwide by an average 9 percent per year between 2015 and 2020. the number of newly installed robots grew from 254,000 units in 2015 to 422,000 units in 2018. this number decreased to 382,000 units in 2019, which ‘reflected the difficult times the two main consumer industries, american versus domestic digital companies in the chinese market 123 automotive and electrical/electronics, experienced and the trade conflict between china and the united states’ (ifr, 2021, p. 12). however, despite the global pandemic situation, it slightly increased to 384,000 units in 2020. a further increase is expected to 518,000 units in 2024. the stock of industrial robots operating worldwide grew from 1.6 million units in 2015 to 3 million units in 2020. the number of robots per 10,000 persons employed in manufacturing has the highest level in south korea, singapore, japan, and germany. nevertheless, according to sales data, since 2013 the largest robot market has been china, which accounted for 44 percent of total installations in 2020 (ifr, 2021). besides the aforementioned indicators measuring the development of the digital economy, it is worth mentioning the so-called digital intelligence index that ‘encompasses several scorecards measuring various aspects of the global digital economy’ (chakravorti et al., 2020, p. 16). digital evolution scorecard (2020) is one of them, which ‘tracks the state and historical momentum of 90 economies – comprising 95 percent of the world’s online population – over twelve years (2008-2019)’ (chakravorti et al., 2020, p. 16). it primarily captures the four driving forces of the digital economy: supply conditions (how developed is the infrastructure that facilitates digital interactions and transactions); demand conditions (how inclined and able are consumers to participate in the digital economy); institutional environment (how do government policies and regulations promote digital development); and innovation and change (what kind of innovations took place in a country’s digital economy). according to the digital evolution scorecard results of 2020, singapore, the united states, hong kong, finland, denmark, and switzerland have the most advanced digital economy. if we take the development performance of the digital economy of different countries between 2008 and 2019 into consideration, we can find china, azerbaijan, indonesia, india, vietnam, and iran in the top six places (chakravorti et al., 2020). ‘as in the two previous editions of digital evolution (2014, 2017), china remained the fastest-moving economy in terms of the pace of change in its digital evolution – i.e., digital momentum’ (chakravorti et al., 2020, p. 16) in 2020. a short analysis of the development of the global digital economy cannot omit the introduction of digital and ict companies spurring the process itself. according to the definition of unctad, digital companies are purely digital players, which operate in the digital dimension, or mixed players, which operate in both the digital and physical dimensions. examples of the former are internet platforms (e.g., search engines, social networks, sharing platforms, etc.) and digital solution providers (e.g., electronic and digital payment operators, cloud players, etc.). examples of the latter are e-commerce companies (e.g., internet retailers, online travel agencies, etc.), and producers and distributors of digital content (e.g., videos, music, e-books, games, data, etc.). ict companies provide infrastructure and access to the internet to consumers and companies. they include telecom companies and it companies producing software and hardware (unctad, 2017a). table 2 exemplifies each activity with specific companies. each category includes three companies listed on the stock exchange with the world’s largest operating revenues (in 2015). (classification was made according to the most important activity of each company. nevertheless, most companies are usually present in other digital sectors as well.) lukács et al./decis. mak. appl. manag. eng. 5 (2) (2022) 120-139 124 t a b le 2 . t h e m o st s ig n if ic a n t d ig it a l a n d i c t c o m p a n ie s (u n c t a d , 2 0 1 7 a ) d ig it a l co m p a n ie s in te rn e t p la tf o rm s e -c o m m e rc e se a rc h e n g in e s a lp h a b e t (g o o g le ), b a id u , y a h o o ! ja p a n in te rn e t re ta il e rs a m a z o n , j d .c o m , a li b a b a so ci a l n e tw o rk s f a ce b o o k , n e te a se , i a c / in te ra ct iv e o th e r e -c o m m e rc e p ri ce li n e g ro u p ( n o w b o o k in g h o ld in g ), e x p e d ia , a m a d e u s it o th e r p la tf o rm s e b a y , r e d h a t, g ro u p o n d ig it a l so lu ti o n d ig it a l co n te n t e le ct ro n ic p a y m e n ts f ir st d a ta , p a y p a l, w o rl d p a y d ig it a l m e d ia c o m ca st , t im e w a rn e r, 2 1 st c e n tu ry f o x o th e r d ig it a l so lu ti o n a d p , s a le sf o rc e , v m w a re g a m e s t e n ce n t, a ct iv is io n b li z z a rd , e le ct ro n ic a rt s in fo rm a ti o n a n d d a ta t h o m so n r e u te rs , a ll ia n ce d a ta s y st e m s, n ie ls e n ic t c o m p a n ie s it so ft w a re a n d s e rv ic e s m ic ro so ft , h p e n te rp ri se , o ra cl e d e v ic e s a n d c o m p o n e n ts a p p le , s a m su n g , h o n h a i te le co m a t & t , v e ri z o n , c h in a m o b il e american versus domestic digital companies in the chinese market 125 in pricewaterhousecooper’s list of global top 100 companies ranked by market capitalization, we can find several digital and ict companies. in the forefront, in the first ten places, us companies such as apple, microsoft, amazon, alphabet, and facebook and chinese companies such as alibaba and tencent are listed (pwc, 2021). according to the survey of van alstyne (2016), the share of north america, asia (mainly china), europe, africa, and latin america in the total market capitalization of digital and ict companies with more than 1 billion us dollar market capitalization was 75.8, 18.1, 4.4, 1.6 and 1.6 percent, respectively. in other words, it means that the headquarters of most of these companies are located in the united states or china. figure 1, which contains the ranking of the most significant internet companies by market capitalization, also shows the dominance of the aforementioned us and chinese companies on which this study focuses. (apple is not listed because it is primarily considered as an ict software and hardware company. however, we examine this company because it provides several digital services). figure 1. the largest internet companies by market capitalization, february 2021 (billion us dollars) (markinblog, 2021) 3. the digital economy of china according to calculations of unctad, the global digital economy accounted for 4.5 percent of the global gdp, if we take the narrow scope of the digital economy into consideration that covers telecommunications, information services, software and it consulting, hardware manufacturing, as well as digital and platform-based services. the broader scope of the digital economy, which additionally covers the use of various digital technologies for performing different economic activities (e.g., sharing economy, gig economy, e-business, e-commerce, industry 4.0, etc.), accounted for 15.5 percent of global gdp in 2017 (unctad, 2019). the global digital economy is vastly concentrated in two economies, namely, the united states and china. according to the broader definition, china’s digital economy reached 30 percent of the country’s total gdp in 2017. in the case of the united states, the same indicator was 21.6 percent. nevertheless, estimations based on the narrow definition showed a slight advantage 1662 1392 759 571 461 295 239 231 162 85 78 53 52 51 51 46 44 44 43 0 200 400 600 800 1000 1200 1400 1600 1800 lukács et al./decis. mak. appl. manag. eng. 5 (2) (2022) 120-139 126 to the us with a 6.9 percent share compared to a 6 percent share for china in 2017 (unctad, 2019). in 2021, 4.9 billion people used the internet worldwide of which 1.01 billion people lived in china, in other words china had the largest number of internet users in the world (itu, 2021a; cnnic, 2021). in june 2021, the internet penetration rate stood at 71.6 percent in china. the proportion of mobile internet users to the total number of internet users was 99.6 percent (cnnic, 2021). figure 2. the number of chinese internet users (10,000 persons) and the internet penetration rate (cnnic, 2021) figure 3. the number of mobile internet users in china (10,000 persons) and its proportion to total internet users (cnnic, 2021) in china, we can see a rapid increase in the number of mobile internet users in the last 15 years and a growing reliance on mobile internet among internet users as the high penetration rate reflects. ‘the number of mobile payment users in china grew from 125 million people in 2013 to 852 million people in 2020’ (cnnic, 3rd february, american versus domestic digital companies in the chinese market 127 2021). ‘the adoption rate of mobile payment among mobile internet users reached 86 percent’ (sina.com.cn, 3rd february 2021). ‘in 2018, the market share of mobile payments in china accounted for 83 percent of all payments, indicating an explosive growth from 3.5 percent in 2011’ (daxue consulting, 10th may 2019). china is the world’s largest retail e-commerce market, as we have already mentioned above. according to estimates, in 2021, the share of china in global online retail sales reached 52.1 percent, exceeding that of the united states (19 percent), the united kingdom (4.8 percent), japan (3 percent), south korea (2.5 percent) and germany (2.1 percent) combined (emarketer, 14th july 2021). ‘86.1 percent of all transactions in china’s online shopping market (b2c, c2c)’ (iresearch, 30th june 2020) are carried out via mobile phones. china is a key player in the global digital economy not only from the side of consumption but also from that of business (he et al., 2021). in china, baidu, alibaba, and tencent can be considered as the most significant digital companies, which are often simply referred to as bat. the initial activity of each company focused only on one particular segment of the digital economy, but since then they have considerably diversified their activities. the core activity of alibaba is e-commerce. ‘56 percent of all chinese e-commerce went to alibaba in 2020’ (buchholz, 2020). alibaba.com was established as a b2b website in 1999, followed by the creation of an online consumer website (c2c platform), taobao, in 2003. one year later, the company introduced digital payment services through alipay. in 2008, it created its b2c platform, taobao mall (now tmall.com). in 2009, it launched alibaba cloud, which provides cloud computing services. in 2014, alihealth was introduced which is now ‘a leading online retailer of medical prescriptions and over-the-counter medicines in china’ (m&g investments, 2021). alibaba is also present ‘in digital wealth management through yu’e bao and in entertainment through the acquisition of youku tudou in 2016 which is a major video streaming and internet tv player’ (woetzel at al., 2017b, p. 9). baidu started its activity as a search engine, which nowadays controls 76 percent of search volume in china (buchholz, 2020). later, it gradually expanded its activities towards mobile services. in the last few years, it has invested significant amounts in o2o (online-to-offline) services such as food delivery, financial products, and group buying. moreover, the company puts a growing emphasis on development activities related to artificial intelligence and their application in the business sector (e.g., automotive industry [autonomous vehicles]). the main profile of tencent is social network. one of its most important services is the wechat messaging application, which was introduced in 2011 and, nowadays, ‘78 percent of all chinese internet users have accounts on wechat’ (buchholz, 2020). after social media, tencent has started to expand towards segments such as digital payment (tenpay), online banking services (webank), and dining services (meituan-dianping) (woetzel at al., 2017b). wechat and alipay can be called super applications in which they multiplied their original functions in just a few years. wechat and alipay cover several functions in different fields such as public services, social media, finance, education, communications, purchasing, entertainment, travel, etc. both belong to the applications most frequently used by chinese internet users. besides these, qq, iqiyi, and taobao also can be found in the top 5 list (figure 4). qq is a messaging service of tencent as well. beyond the messaging service, qq provides online social games, music, shopping, and movie related services. iqiyi is an online video platform owned by baidu. lukács et al./decis. mak. appl. manag. eng. 5 (2) (2022) 120-139 128 figure 4. monthly active users of the leading apps in china in january 2022 (in millions) (analysys, 2022) besides the three digital giants, there are several other digital companies operating in china. nevertheless, these three companies have been playing an important role in the development of further digital companies (woetzel et al., 2017a). for example, in figure 1 we can find apart from bat other chinese companies such as bytedance, jd.com, meituan-dianping, and pinduoduo. tencent has been the largest shareholder of jd.com until recently, which is ‘china’s second largest e-commerce company’ (he, 2021) after alibaba. tencent is also a major shareholder in pinduoduo, which was founded in 2015 and ‘attained a gross merchandise value of 15 billion us dollars only two years from launch, a milestone that took incumbents alibaba and jd.com 5 and 10 years to achieve’ (natason, 2019). pinduoduo managed to overtake alibaba in terms of active users in 2020 (chen 2021). tencent also invested heavily into the merger of two startups meituan and dianping in 2015 (osawa & carew, 2019) and in 2021; it increased its stake in meituan-dianping to 17.2 percent (fu, 2021). bytedance founded in 2012 ‘is the creator of the short video app tiktok and news aggregator service toutiao and one of the most valuable private technology firms in the world’ (kharpal, 2019a). its aim is to replicate the growth model of tech giants such as alibaba and tencent, ‘using buyouts and investments to push into new business areas’ (kharpal & cheng, 2022). china is a very active player in the global digital economy in terms of investments and the launching of new companies. a growing part of venture capital investments is directed towards digital technologies. china belongs to the world’s three most significant venture capital investors in the fields of ‘virtual reality, autonomous vehicles, 3d printing, robotics, drones, and artificial intelligence’ (woetzel et al., 2017b, p. 3). 4. american digital companies in the chinese market the large domestic market, china, has facilitated the establishment of domestic giant digital companies. we can easily identify the chinese counterparts of every significant american digital company, which are dominating different segments of the digital market, leaving only few business opportunities for american companies. for example, alphabet (google), facebook, youtube, and netflix have not been available at all in china due to government censorship. baidu search engine could be considered american versus domestic digital companies in the chinese market 129 as the chinese counterpart of google, which has a 76 percent share in the chinese market. like google, as mentioned before, baidu has been investing significant amounts in the development of autonomous cars. tencent is china’s facebook, which is well-known for its messaging and social media platforms. of these, wechat stands out with nearly 1 billion monthly users. it is china’s largest social services provider (dunn, 2017). chinese counterparts of amazon are alibaba and jd.com. alibaba’s activities expand well beyond e-commerce, but the company still relies mainly on its online shopping sites such as tmall.com and taobao. the former is china’s largest retail e-commerce site, whereas the latter could be considered the chinese ebay. behind tmall.com, jd.com is the second largest b2c e-commerce platform in china. similar to amazon, it has been investing large amounts in the development of drone delivery. we have already mentioned alibaba-owned youku tudou, which is the chinese counterpart of the american youtube. last, but not least, we can highlight iqiyi owned by baidu as well, which is china’s netflix (dunn, 2017). most of the american giant digital companies (e.g., amazon, alphabet [google], ebay, facebook) tried to conquer china’s large market with great growth potential, but they have not proved to be successful. amazon is almost synonymous with online shopping in the united states, however in the chinese online retail market in 2016, it had only a 1.3 percent share (keyes, 2017). this share further eroded to less than 1 percent in 2019 (kharpal, 2019b). amazon entered the chinese market in 2004 when it acquired online bookseller joyo. the company was renamed amazon china in 2011. in october 2016, amazon china launched its premium membership program. its premium subscription offerings did not make the company outstanding among competitors. offerings of chinese local companies were at least as favorable as, or even better than, that of amazon. additionally, its premium video service was banned by chinese government censorship, which further decreased the merit of the premium subscription. so it is easy to understand amazon’s unsuccessful performance in the chinese market. furthermore, the other reason for the failure of amazon was that its mobile application design fell behind the chinese mobile applications (keyes, 2017); it is widely known that more than 80 percent of retail e-commerce transactions are made via mobile phone in china. in addition, amazon china was less visible in terms of marketing. finally, in 2019, amazon decided to ‘shut down its domestic e-commerce marketplace business in china’ (kharpal, 2019b). stiff competition from the largest market players, alibaba and jd.com, was a significant factor that sealed the fate of amazon china. besides e-commerce, amazon has also entered the market of cloud services in china in 2013, which is mostly dominated by chinese companies. according to chinese regulations, amazon web services can provide cloud services only with a local partner. chinese cyber security regulations that came into effect in the summer of 2017, which were intended to control the cross-border flow of data more rigorously, required local storage of data, which forced amazon to sell a part of its physical infrastructure of cloud services to its local partner, sinnet. these regulations have been limiting not only amazon’s chinese activity in the field of cloud services, but also that of other american companies such as apple, microsoft, and oracle (cadell, 2017). amazon web services started to operate with its second local partner, nwdc since december 2017, in full compliance with the aforementioned chinese regulations. moreover, chinese regulators are rigorously stepping up against the use of vpns and other services (and their providers) that give access to foreign sites, such as facebook, twitter, and google, which are blocked by the chinese ‘great firewall’. local partners of amazon web services limit access to this aforementioned software, which secures further operation for its web services in the chinese market. lukács et al./decis. mak. appl. manag. eng. 5 (2) (2022) 120-139 130 google established its chinese subsidiary google china in 2005. it was the third largest search engine behind baidu and soso.com in the chinese market. in 2009, the chinese government blocked the use of google because of content directed against the communist leadership. in 2010, google pulled out of the search engine market of china, citing government censorship and continuous hacker attacks as the reason. however, according to market analysts, the intervention of the chinese government was not the only reason for google’s exit. it is also true that google could not manage to demolish the view of chinese consumers according to which baidu is created for the chinese community, but google is rather established for foreigners (pierson, 2016), namely, google was not well-positioned in the chinese market. facebook entered china in 2005. the chinese government blocked facebook in 2009 when it turned out that activists participating in the riots of ürümqi used facebook for communication. since then, the website has not been available. in 2003, ebay appeared on the chinese market with the acquisition of eachnet, which was the largest online auction site at that time, and in response to this, alibaba established the previously mentioned taobao. taobao overtook ebay in the chinese market in a very short period. between 2003 and 2005, the market share of taobao increased from 8 percent to 59 percent, while the market share of ebay decreased from 79 percent to 36 percent. in 2006, ebay closed down its online auction activities in china. the failure of ebay can be mainly explained by its misunderstanding of how the chinese market operates (wang, 2010). booking (priceline before 2018) is the eighth largest american digital company based on market capitalization (see figure 1). ‘it is the world’s leading provider of online travel and related services’ (accommodation, car hire, air tickets, etc.) (booking holdings, 2022). before the pandemic, the value of gross travel bookings through booking grew from 7.4 billion us dollars in 2008 to 96.4 billion us dollars in 2019 (booking holdings, 2022). the growing middle class of china has become an increasingly important market for booking. the american company does not want to compete directly with chinese travel websites; therefore, it acquired a stake in the chinese ctrip.com in 2012, through which the offerings of booking.com reach chinese consumers. ctrip uses booking.com for outbound international travel. in 2017, the american company bought shares in meituan-dainping as well, the travel division of which cooperates with agoda.com (o’neill, 2017). in addition, it invested 500 million us dollars in the chinese car-hailing giant didi chuxing in 2018. ‘didi supplies rides to customers of booking’s apps around the world. its american partner, in turn, allows didi users to make hotel reservations on booking.com and agoda.com’ (chan, 2018). executives of booking consider the chinese and – in a broader sense – asian market as the biggest business opportunity, but at the same time this is where they meet the most challenges. therefore, booking has chosen to acquire shares in domestic companies to enter china instead of launching its subsidiaries and building up its own business activities. apple, which is the world’s largest ict company based on market capitalization (pwc, 2021), started to sell iphone devices in china in 2009. in 2015, china became the world’s largest market of activated iphones (horowitz, 2016). in china, the sales of iphones reached a peak of 71.2 million us dollars in 2015, but since then they have fallen to 31.4 million us dollars in 2019. in 2020, they showed only a slight increase to 34.9 million us dollars (curry, 2022). in 2020, china was the world’s third largest market for iphone sales behind the usa and europe, but still, only 18.1 percent of smartphone owners use ios in china (kantar worldpanel, 2022). in the peak year of iphone sales, 2015, apple launched itunes movies, ibooks, and apple music to china and some months later, apple pay. however, six months after the rollout, on orders of american versus domestic digital companies in the chinese market 131 the chinese authorities, apple was forced to close down itunes movies and ibooks services. services such as apple music and apple pay could still operate further (hsu, 2017). in 2016, apple secretly signed an agreement (about 275 billion us dollars) ‘with the chinese government that allowed apple to grow most of its operations in the country. in turn, apple promised it would do its part to develop china’s economy and technological prowess through investments, business deals and working training’ (scherr, 2021). to woo the chinese government, apple opened two data centers and invested 1 billion us dollars into ride hailer didi chuxing (van boom, 2018). in order to boost the performance of apple music and apple pay in the chinese market, apple has started to cooperate with chinese companies, such as meituan-dianping, ant financial (alipay), and tencent music entertainment (all tech asia, 2017; van boom, 2018; grogan, 2021, ouyang et al., 2022). in the last five years, apple was requested several times by the chinese government to remove apps from the app store (gurman, 2019; gallagher, 2020), which means apple has been operating under strong censorship in china. despite all difficulties, china is an important market for apple; it was the third largest in terms of revenues in 2020 (curry, 2022). as mentioned above, the largest american digital companies have faced significant challenges in the chinese market, which can be traced back to several reasons. firstly, most of them did not understand the needs of chinese customers, which led to market failure. secondly, almost every american digital company has a chinese counterpart (strong competition in the market). these chinese digital companies that rely on the large domestic market could and can grow rapidly; furthermore, they are also supported by government policy giving advantages and subsidies to domestic companies and limiting foreign competition. the chinese government often prevented leading american digital companies from acquiring chinese companies in the initial stage of their development (heilmann, 2017). thirdly, the chinese government often uses the tool of censorship against foreign companies if it is in their strategic interests. 5. chinese government policy for promoting domestic digital companies since the global economic and financial crisis, there has been a change in the economic strategy of the chinese government. instead of exports and foreign direct investments, the government has started to promote domestic innovations and companies (pierson, 2016). in the future, the economic growth of china will rely increasingly on services, the high-tech sector, and consumption. the chinese government strives to preserve the most lucrative market opportunities for domestic companies. in the 13th five-year economic plan (2016-2020), the chinese government assigned the development of the ict sector as the upmost priority field (hong, 2017), which has been playing a significant role in the transition of china’s aforementioned economic growth model. according to wang (as quoted in drinhausen, 2018, p. 3), ‘the chinese government understood the importance of the digital economy and included specific goals and measures for its promotion in the 13th plan.’ during the period of the 13th five-year plan, several ongoing elements of the chinese state industrial policy targeting the development of digital economy were introduced such as ‘internet plus’, ‘made in china 2025’, and ‘national informatization development strategy’. the main aim of ‘internet plus’, initiated by the chinese government in 2015, is to modernize and transform the operation of the traditional manufacturing sector and society through the help of the internet. the ‘internet plus’ action plan ‘identifies mobile devices, cloud computing, big data, and the internet of things (iot) as key tools lukács et al./decis. mak. appl. manag. eng. 5 (2) (2022) 120-139 132 for this transformation’ (borst, 2018, p. 9). it ‘maps development targets and supportive measures for key fields where the government hopes that it can establish new industrial modes by integration with internet technologies, including mass entrepreneurship and innovation, manufacturing, agriculture, energy, finance, public services, logistics, e-commerce, transportation, environment protection, and artificial intelligence. according to the plan of the chinese government, ‘internet plus’ will become a new economic model and an important driving force for economic and social innovation and development by 2025’ (china daily, 2015). moreover, it is worth highlighting that the action plan is definitely aimed at decreasing china’s dependence on foreign technology innovations and providing more state support for domestic business development (chang, 2016). for example, the government supports the establishment of new digital companies with tax allowance and initial capital. in 2016, approximately 2500 incubator facilities were ready across china to help with the start of new businesses. the chinese government established a 30-billion-dollar venture capital fund in shenzhen in 2016, which also supports start-ups. in 2017, ‘beijing zhongguancun inno way, a high-tech community known as china’s silicon valley, launched its first 75.3 million us dollar venture capital targeting ai-related startups’ (woetzel et al., 2017b, p. 32). in the same year, ‘the cyberspace administration of china and the ministry of finance launched a 100-billion-renminbi fund to support digital companies and the ‘internet plus’ action plan through equity investments. several chinese banks also pledged to back up the initiative through providing credit’ (borst, 2018, p. 9). the concept of ‘internet plus’ originates from tencent ceo ma huateng. the other significant (private) digital companies (e.g., alibaba, baidu) also participated in working out the guidelines of the concept (stepan & shih, 2016). unlike ‘internet plus’, the plan of ‘made in china 2025’ is characterized by a top to bottom approach and was adopted by the chinese government in 2015 to develop smart manufacturing (which means the use of automatization and digitalization technologies in industrial production and organizations). the industrial production of china still shows backwardness in comparison to that of developed countries. low level of automatization and lack of digitalization are characteristics of most chinese factories. however, a rapid increase in demand for automatization and digitalization technologies is generated by chinese industry. the plan of ‘made in china 2025’ is different from former industrial strategies because it disposes of larger financial sources and includes several efforts, which were not previously synchronized. with its plan, the chinese government would like to support the technological upgrading of small and large as well as private and state-owned companies. moreover, the other main aim of the government is to replace and substitute foreign technologies with domestic technologies. in ten high-tech sectors (new generation information technology, high-end computerised machines and robots, space and aviation technology, maritime equipment and high-tech ships, advanced railway transportation equipment, new energy and energy-saving vehicles, energy equipment, agricultural machines, new materials, biopharma, and high-tech medical devices), the chinese government promotes chinese companies (national champions) ‘to create innovative technology solutions and replace their foreign competitors in the domestic market’ (wübbeke et al., 2016, p. 20) and to expand abroad as well. the ‘national informatization development strategy’, which partially covers the two aforementioned plans, was adopted by the chinese government in 2016. it is an adjustment and development of the ‘national informatization development strategy 2006-2020’ (china copyright and media, 2016). this strategy projects the broad it goals of the chinese government and state regulation and control of cyberspace for the next ten years. the most important goal of the strategy is to boost the domestic it american versus domestic digital companies in the chinese market 133 sector with the ‘upbringing’ of internationally competitive digital and ict companies, as well as to overtake leading countries such as the usa and germany. according to the strategy, between 2016 and 2020, china’s aim was to strengthen the domestic industry (e.g., integrated circuit, software services) in the field of some core technologies and expand 3g and 4g services in the whole country, and promote 5g technology related r&d activities. by 2025, china would like to build up mobile networks similar to that of developed countries. foreign expansion of leading chinese companies and innovations are also included in the goals. the value of it products and services and e-commerce are planned to be quadrupled by 2025. the strategy is also aimed at relying more on domestic technology innovations and decreasing dependence on foreign ones. the government supports research cooperation among universities, research institutions, and companies. highlighted fields are mobile internet technology, cloud computing, big data, the internet of things (iot), etc. (zhao & heatley, 2016). the strategy also mentions further severity of domestic regulation of cyber security, which has a disadvantageous impact on the activities of foreign companies in the chinese market. digitalization has remained a key priority in the latest five-year economic plan (2021-2025) ‘which is now pushing the general application of digital and smart solution in the economy, governance and the social sector’ (grünberg & brussee, 2021). moreover, the 14th five-year economic plan has reinforced china’s efforts to attain self-reliance in science and technology. ‘despite calls for continued openness and international cooperation, china is looking internally to ensure its future development’ (dudley, 2021) or in other words, the 14th plan prioritizes the ‘internal cycle’ of china’s so-called dual circulation strategy. 6. conclusion the digital economy is a rapidly growing segment of the global economy. this process is highly supported and enabled by the so-called digital and ict companies. the global list of the top 20 internet companies ranked by the size of market capitalization is mainly dominated by american and chinese companies. according to the broader definition of unctad, the share of the digital economy in gdp is higher in china than in the usa. our study focuses on the latest developments of the chinese digital economy, especially the activities of the most important market players, namely, alibaba, tencent, and baidu. our major findings are that the largest american digital companies (such as alphabet (google), amazon, apple, facebook, ebay, booking), which are globally active players, usually have a very limited market share in the different segments of the chinese digital economy, or have been forced to leave the chinese market after a short period of operation. in its catching-up phase, china has brought up its national champions now dominating the domestic digital market. we can easily identify the chinese counterparts of every american digital giant: baidu – google; tencent – facebook; tmall.com (alibaba) and jd.com – amazon; taobao (alibaba) – ebay; youku tudou (alibaba) – youtube; iqiyi (baidu) – netflix, etc. in the future, the 14th five-year economic plan and the recent elements of the state industrial policy, such as ‘internet plus’, ‘made in china 2025’ and ‘national informatization development strategy’, will continue to ensure the priority role of domestic digital companies in the upgrading and structural transformation of the chinese economy driven by services, high-tech sectors, and consumption. lukács et al./decis. mak. appl. manag. eng. 5 (2) (2022) 120-139 134 author contributions: conceptualization, e. lukács; writing—original draft preparation, k. völgyi; writing—review and editing, e. lukács, k. völgyi, n. kovács, á. tóth; supervision, n. kovács; project administration, á. tóth. all authors have read and agreed to the published version of the manuscript. funding: this research received no external funding. conflicts of interest: the authors declare that they have no known competing financial interests or personal 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(2016). china’s master plan for it dominance. the diplomat, august 11. https://thediplomat.com/2016/08/chinas-master-plan-for-itdominance/ © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). https://unctad.org/system/files/official-document/tn_unctad_ict4d19_en.pdf https://unctad.org/system/files/official-document/tn_unctad_ict4d19_en.pdf https://unctad.org/system/files/official-document/der2021_en.pdf https://unctad.org/system/files/official-document/der2021_en.pdf https://www.youtube.com/watch?v=8ofrd66pi0y https://www.cnet.com/tech/mobile/apple-teams-alipay-win-over-china/ https://www.forbes.com/sites/china/2010/09/12/how-ebay-failed-in-china/2/#6cbd4d8242f8 https://www.forbes.com/sites/china/2010/09/12/how-ebay-failed-in-china/2/#6cbd4d8242f8 https://www.mckinsey.com/featured-insights/china/digital-china-powering-the-economy-to-global-competitiveness https://www.mckinsey.com/featured-insights/china/digital-china-powering-the-economy-to-global-competitiveness https://www.mckinsey.com/featured-insights/china/chinas-digital-economy-a-leading-global-force https://www.mckinsey.com/featured-insights/china/chinas-digital-economy-a-leading-global-force https://www.china-briefing.com/news/chinas-services-sector-expansion-beijing-9-industry-reforms/ https://www.china-briefing.com/news/chinas-services-sector-expansion-beijing-9-industry-reforms/ https://www.merics.org/sites/default/files/2017-09/mpoc_no.2_madeinchina2025.pdf https://www.merics.org/sites/default/files/2017-09/mpoc_no.2_madeinchina2025.pdf https://thediplomat.com/2016/08/chinas-master-plan-for-it-dominance/ https://thediplomat.com/2016/08/chinas-master-plan-for-it-dominance/ plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering issn: 2560-6018 eissn: 2620-0104 doi:_https://doi.org/10.31181/dmame0310112022j * corresponding author. e-mail addresses: sudhanshu.joshi@uts.edu.au (s. joshi), manu.sharma@geu.ac.in (m. sharma), dr.prasanjitchatterjee6@gmail.com (p. chatterjee) omnichannel retailing for enhancing customer engagement amidst supply chain disruption: an emerging market perspective sudhanshu joshi1,2*, manu sharma3,4 and prasanjit chatterjee,5 1 operations and supply chain management research laboratory, school of management, doon university, dehradun, india , sudhanshujoshi@doonuniversity.ac.in 2australian artificial intelligence institute (aaii), school of computer science, faculty of engineering & information technology, university of technology sydney, sydney, australia, sudhanshu.joshi@uts.edu.au 3 department of management studies, graphic era deemed to be university, dehradun, india, manu.sharma@geu.ac.in 4guidhall school of business and law, london metropolitan university, london 5mckv institute of engineering, west bangal, india received: 3 august 2022; accepted: 18 october 2022; available online: 10 november 2022. original scientific paper abstract: the research aims to explore the strength of enablers and adoption barriers in omnichannel retailing (ocr) and discuss how organizations may focus on redesigning their business models in emerging markets to manage the disruptive environment. the major enablers may enhance the omnichannel' performance to deliver a unified experience across all channels during the pandemic. the paper has used hybrid multi-criteria decisionmaking (mcdm) methods. organizations widely use these methods to explore the interrelationship among barriers and enablers affecting their performance. in the current study, 18 experts from different domains have examined and evaluated the 10 barriers and 7 enablers. the study reveals that integration, visibility, internet accessibility, and advanced distribution centers are the primary enablers and driving the customer analytics enabler to strengthen their customer engagement and providing a unified experience to the. during the pandemic time the usage of the online channels have increased and thus retail channels may consider these enablers to enhance the unified experience level of the customers. the study also shows that inconsistency in price is the main adoption barrier followed by inconsistency in product discounts that should be minimized to engage customers effectively. the retail organizations need to understand the roadblocks in adopting ocr and should take relevant actions to minimize them. the retail organization or marketers joshi et al./decis. mak. appl. manag. eng. (2022) 2 may redesign their existing strategies based on price consistency, integration, visibility, information systems, and coordination to develop a unified experience across channels during the pandemic situation. key words: omni-channel retailing (ocr); supply chain disruption; emerging markets; interpretative structural modeling (ism); fuzzy micmac. 1. introduction the global disruption, digital technologies progression, pandemic and enhanced usage of smart devices have reformed the countenance of retailing all across the world, subsequently engaging customers using multiple touch points and adopting ocr strategies to augment their experience (li et al., 2020; sharma et al., 2021a). the explosive growth rate of online customers in south asia, europe, uk and usa is carrying an irregular change to these marketplaces and giving a chance to companies to necessarily reimagine their business models. developing economies like india and china are showing high encouragement through more engagement of online customers, and online purchases upsurge to $3.9 trillion value (wef, 2020). approximately 3 billion consumers from the emerging market will be online by 2022, exposing the opportunity for retail organizations to plan and target customer engagement appropriately (nguyen et al., 2019). it is expected to achieve 1000 million target by 2030 (wef, 2020). moreover, the impact of digital influence can be understood by an example of africa where e-commerce is limited to 1 % only but the digital influence is skyrocketing (bcg, 2018). in terms of growth perspective, online consumers in emerging markets represent an enormous opportunity. technically, ocr raises to combine multiple points for online customers. the consumer decides where and when to shop and through which device (ieva & ziliani, 2018). if consumers do not purchase directly on the internet, they still search for information on their mobile phones, which often influences their purchases. the value of digitally influenced spending in emerging markets is expected to reach $4 trillion (bcg, 2018). in the past years, emerging economies have shown exceptional growth. during 2000-2018, the share of these countries is from 11 to 28% in the world's gross domestic product and 11 to 24 % in global household consumption expenditures. the price fall of smartphones by 40% in emerging markets has impelled these devices into the hands of millions of people who were previously unable to afford them. the arrival of high-speed data networks has enabled these markets to achieve spectacular expansions in the connectivity and due to this half of the population is now connected to the internet in emerging markets, mainly in the parts of southeast asia, russia, turkey and brazil. the generations are spending time online, and thus marketers should explore new ways to reach digitally millennial consumers on modern platforms. both online and offline retailers in emerging markets are highly motivated and aim to serve customers better through seamless experience and creating the appropriate content for each segment, communicating each segment through proper channels. retailers in the emerging market need to adopt ocr as their businesses cannot be long-lasting if consumers are connected in a unidirectional way. with the advancement in information technology, consumers quickly disseminate and access information through multiple channels (cai & lo, 2020; joshi et al., 2021; joshi & sharma, 2021). but, purchase actions happen as per consumer convenience and choices (park & lee, 2017; chatterjee & kumar, 2017). thus, retail organizations need to understand what factors drive and restrict consumers' convenience during switching channels and completing their purchase actions. previous research also omni-channel retailing for enhancing customer engagement amidst supply chain disruption 3 highlights that discounts are the significant triggers for online purchases, but in emerging markets, discounts are not the only thing that matters (chopra, 2016; chopra et al., 2019; arslan et al., 2021; joshi et al., 2021). the other form of retail, i.e., offline retail organization, has a limitation of consumer's time constraint that restricts him from visiting the store. also, a limited range of product availability is overcome by online retailers. thus, amidst the pandemic, retail organizations need to adopt a hybrid business model where online and offline formats will merge and engage consumers by giving them a choice to decide when, where, and how to shop. online retail organizations have some limitations, such as tangibility, waiting for product delivery, delivery delay, dynamic pricing, etc. (yang et al., 2019; sharma et al., 2022a). there is still a population who believes in the visualization of products before purchase, and thus e-commerce companies need to open brick-and-mortar stores to capture this segment. lens kart is one of the recent examples of ocr in the indian market. emerging market economies are expanding, boosted by educated younger, healthier populations with rising incomes, fueling a substantial increase in goods and services consumption. the spending by consumers in these economies is projected to be more than the developed nations. thus, ocr is essential for retailers to reach targeted consumers efficiently by adopting digital technologies with minimum cost. offline and online channels will complement each other to develop an efficient ocr system, breaking the wall between all the channels to provide a unified brand experience (sharma et al., 2020a; sharma & joshi, 2020). the development of ocr will be dependent on the infrastructural development, technologies and digital transformations, and retailers' decisions to manage the current issues such as price inconsistency, order management, customer expectations, and others to provide a flawless experience across all the channels (picot-coupey et al., 2016; ewerhard et al., 2019). there is a need to evaluate the existing scenario of the retailing industry in emerging markets as the future lies in the young and educated populations. there are research contributions in the area of ocr. still, little attention has been given to the challenges or bottlenecks handled by the retail firms in the acceptance of ocr and also the influence of enablers enhancing its acceptability. this study is significant for mainly three reasons. first, prior research in the context of ocr in emerging markets is limited and focused only on the basic understanding and comparison with multichannel retailing. second, there are insufficient information regarding ocr's challenges and adoption barriers and enablers (salvietti et al., 2022; sharma et al., 2020a; solem et al., 2022). the past research has not examined the challenges and adoption barriers existing in ocr influencing customer purchases. lastly, the merit of inter-relationships among the identified adoption barriers and enablers of ocr is still unknown. thus, to bridge the above research gaps, the present study intends to determine the enablers and adoption roadblocks in the ocr ecosystem influencing customers' journeys, choices, and unified experience. all the enablers and barriers are to be analyzed to explore the strength and weaknesses of the current channels of retail organizations that restrict or drive the customer to adopt ocr. the customers' switching of channels to complete their purchase action has questioned the organizations to investigate the reasons behind their behavior. this study provides the basis of acceptance and rejection of the omnichannel based on their attributes. this study also explores the intensity of enablers and barriers using hybrid mcdm methods for understanding the interrelationship among them. thus, it helps the policymakers develop their ocr strategies based on the critical obstacles and enablers prioritized by the experts. thus, the study framed key objectives to demarcate pressing strategic challenges that could set the pathway for joshi et al./decis. mak. appl. manag. eng. (2022) 4 retail management and can further contribute to existing theories. the theoretical background of the research is carried out from dynamic capabilities theory, the combination of the technology acceptance model (tam) technologyorganizationenvironment model(toem), and resource-based view theory. retail chains' dynamic capabilities demonstrate their ability to develop and adopt the omni channel framework for creating agile and responsive supply chains and improving operational excellence. also, there is an urgent need to develop a strategic roadmap to bridge the implementation and research gaps. the research is an attempt to overall these issues. based on these arguments, the study proposes the following objectives. ro1: investigating the enablers and adoption barriers of ocr in emerging markets. ro2: to develop the hierarchical structure of enablers and barriers using the ism approach. ro3: to investigate interrelationships among enablers and barriers using fuzzy micmac and dematel methods. the decision to design ocr is a complex problem. it has multiple levels, and thus hybrid mcdm approach has been employed to achieve the above objectives. the rest of the research work is structured as follows. section 2 elaborates on existing works on ocr, enablers, and barriers. section 3 explained the research methodology and steps for ism, fuzzy micmac and dematel methods. section 4 presents the method applications, followed by the findings and discussions in section 5. section 6 elaborates the inferences and future work directions. section 7 summarizes the study. 2. literature review based on the scopus database, a comprehensive review of literature has been carried out on the relevant research works on omnichannel retailing and emerging markets in this domain. as depicted in table 1, a search protocol was used using multiple keywords: "omnichannel retailing" and "emerging markets," and "pandemic” or “covid-19". a systematic literature review process was followed to evaluate the prominent publications on omni channel strategy implementation challenges and the digitalization of retailing in emerging economies amidst the pandemic. for the slr, the selected timeline was 2019–2022; the research results are articles. after the standard systematic literature review process, 37 papers were selected for final review. table 1. search protocol for systematic literature review dimensions detailed explanation keywords/ terms used “omnichannel retailing” and “emerging markets” and “pandemic” or “covid-19" timeline 2019-2022 field covered title, keywords, detailed abstract inclusion criteria scopus database exclusion criteria non-english articles 2.1 omni-channel retailing: opportunities and challenges ocr can be explained as a supply chain system where information, material, and fund flow happen by using several channels to coordinate, interact and fulfill customer omni-channel retailing for enhancing customer engagement amidst supply chain disruption 5 demand (rai et al., 2019; chopra, 2016; chopra, 2019; galipoglu et al., 2018). it has caused interventions from many fields, such as decision sciences, virtual reality (vr), visual displays and merchandising decisions, engagement patterns, big data analytics, and profitability (payne et al., 2017; farah et al., 2019). retail has evolved throug h multi-channel and has enhanced supply chain networks (prabhuram et al., 2020). the traditional retailers of various products are extending their channels through virtual stores to derive the benefits of online media. also, online retailers extend their reach through physical stores (pan et al., 2017). this indicates the need for channel integration (bayram & cesaret, 2021). the integration in ocr has been categorized from three perspectives i) ocr stages ii) ocr types iii) ocr agents (saghiri et al., 2017). the core enablers of integration and consistency in ocr strategy (melero et al., 2016; shen et al., 2018; mirzabeiki & saghiri, 2020).the omnichannel need enablers, including broadband internet accessibility (ye et al. 2018); well located and well designed distribution centers (melacini et al., 2018; mkansi et al., 2018); efficient & extensive logistics (kembro et al., 2018; murfield et al., 2017; saghiri et al., 2017; daugherty et al., 2018; hazen & ellinger, 2019); customer analytics (lekhwar et al., 2019;vakhutinsky et al., 2019; zaki & neely, 2019); visibility to customers (ewerhard et al., 2019; gawor & hoberg, 2018); information system (kembro et al., 2018; kembro & norrman, 2019) and product digitization (cortiñas, et al., 2010; ainsworth & ballantine, 2017). few researchers have underlined integration and visibility as essential enablers for ocr (ewerhard et al., 2019; verhoef et al., 2015). the challenges in ocr are discussed by picot-coupey (2016) and divided into strategy-related and development-related levels. the organizational, managerial, and cultural were included in strategy-related and product mix, and information systems in development-related (rai et al., 2019; niranjan et al., 2019). achieving demand, inventory, and in a single view is one of the most critical challenges for omnichannel. the objective of ocr is to transform the current business models, consumer behavior, and advancements in technology (marchet et al., 2018). 2.2 omnichannel retailing in emerging markets ocr is changing the retailing landscape in emerging economies. the incremental growth of online retailing in association with small offline retailers is bringing profits for these economies. the integration of physical and online channels will create a winwin situation, such as a reduction in distribution costs and a wide variety of product availability. however, the success of this hybrid model depends on the development of complementary strengths of both channels to create a cost-efficient omnichannel and more responsive to consumer needs (chopra, 2019). retailers have put significant efforts into providing information access to customers using a number of channels and devices in the developed markets (mrutzek-hartmann, 2022). an e-retailer can send only sensory and digital information, which is a significant factor in the existence of physical retail organizations in emerging markets as still, many consumers use the offline channel for shopping (asmare & zewdie, 2022; chopra, 2016; lin et al., 2022; yin et al., 2022). the omnichannel structure is not creating all the capabilities in each channel but rather assigns products and tasks to channels on the basis of effective handling (chen et al., 2014;chen et al., 2022). this structure is more appropriate in emerging markets where interim retailing models such as borders and circuit way have not yet been developed, and governments are still struggling to cope with the impact of online retailing on offline retailers (ishfaq et al., 2022; teixeira et al., 2022). the ocr will offer an exclusive opportunity that merges the online and offline model’s advantages to bring mutual benefits. joshi et al./decis. mak. appl. manag. eng. (2022) 6 2.3 research gaps the shift to ocr has been well familiar in the research literature (park & lee, 2017; park & kim, 2018). in the last decade, multi-channel retailing has grown into a standard approach (schramm-klein et al., 2011). recent studies advocate the transition stage (park & kim, 2018; zhang et al., 2019). due to the prominent role of the physical retail format in the buying process, hybrid strategies for ocr have also been proposed (huang & jin, 2020).the previous research suggests customer wants a seamless experience during online purchases. many studies have been conducted to understand the scenario of e-commerce in emerging markets in the context of dynamic pricing (cavallo, 2017; dan et al., 2012). the fulfillment and returns are also discussed by many researchers representing the consumers' view towards the e-commerce process in the context of the omnichannel environment (bayram & cesaret, 2021; ewerhard et al., 2019). moreover, the increasing synergy between both channels has been analyzed and highlights that channel integration is one of the key issues discussed (zhang et al., 2019). past research is limited to the understanding of omnichannel. but what are the barriers and enablers that may affect the ocr framework that is still missing? the presence of enablers like coordination, infrastructure, analytics, etc., can enhance the ocr results, whereas the barriers like prince inconsistency and others may bring failure for omnichannel strategies. the literature also reveals that emerging markets like india and omnichannel are providing a wide variety of products to customers at a lesser cost, and thus, the strengths of both channels can be combined to develop a strong omnichannel structure (chopra, 2016). this study establishes an ocr framework considering the barriers and enablers present in the retail environment that need to be considered by the retailers to build a strong and robust ocr framework where customers can be engaged and influenced to purchase products without any discrepancy among the channels. the conceptual framework is developed and illustrated in figure 1. the literature review has identified seven main enablers and ten critical barriers exhibited in table 2. the enablers are supporting ocr to enhance the customer's experience during purchase, including internet accessibility (wang, 2013; yu et al., 2016), internet-enabled distribution centers (chatterjee et al., 2002; chen et al., 2014), efficient and extensive logistics (chen et al., 2022;yan and pei 2011; blázquez, 2014), customer analytics (chatterjee et al., 2002), visibility to customers (agatz et al., 2008; bahn & fischer, 2003; berman & thelen,2018; cassab & maclachlan, 2009), product digitization (bernon et al., 2016; verhoef et al., 2015), and integration (channel types, channel agents, and channel stages) (saghiri et al., 2017). whereas the barriers are restricting the customer to use omnichannel including low coordination among channel partners (fulgoni, 2014; hübner et al., 2016; picot-coupey et al., 2016), variation in pricing (shankar et al., 2003; neslin et al., 2006; verhoef et al., 2015), product unavailability (bernon et al., 2016; chopra, 2016; hübner et al., 2016; ishfaq et al., 2022; huang & jin, 2020), inconsistent contents (verhoef et al., 2015), central product (balasubramanian et al, 2005; verhoef et al., 2015), data security issues (piotrowicz & cuthbertson 2014) and non-understanding young customer habits (verhoef et al., 2015), order fulfillment (chopra, 2016), inconsistent product discount (sousa & voss, 2006), and time constraint (picot-coupey et al., 2016; neslin et al., 2006). omni-channel retailing for enhancing customer engagement amidst supply chain disruption 7 figure 1. conceptual framework of omnichannel retailing the paper intends to discuss the influence of strong enablers on consumer’s experience towards ocr. the enablers and barriers need to be analyzed so that the decisionmakers can enhance the organizational performance through omnichannel. table 2. enablers & barriers identified from literature enablers references 1.broadband internet accessibility (wang et al., 2013; ye et al., 2018; yu et al., 2016) 2. internet-enabled distribution centers (chatterjee et al., 2002; chopra, 2016; sharma et al., 2020c) 3. efficient and extensive logistics (zhang et al., 2019; yan and pei 2011; blázquez, 2014) 4. customer analytics (chatterjee & kumar, 2017; berman & thelen, 2013) 5. visibility to customers (agatz et al., 2008; bahn & fischer, 2003; (berman & thelen, 2013; cassab & maclachlan, 2009) 6. product digitization (berman & thelen, 2004; verhoef et al., 2015) 7. integration (channel types, channel agents, and channel stages) (saghiri et al., 2018;jocevski et al., 2019; shanker et al., 2022;sharma et al., 2020b; sharma et al., 2022c). omni-channels retailing enab2 enab3 enab7 enab4 enab5 enab6 enab1 b3 b4 b5 b6 b7 b8 b9 b2 b10 b11 b1 b12 joshi et al./decis. mak. appl. manag. eng. (2022) 8 enablers references barriers references 1. low coordination among channel partners (fulgon 2014; hübner et al.,2016; picot-coupey et al., 2016) 2. variation in pricing (shankar et al., 2003; neslin et al., 2006; verhoef et al., 2015). 3. product unavailability (bernon et al., 2016; chopra, 2016; hübner et al., 2016; huang & jin, 2020) 4. inconsistent contents (clinton and whisnant, 2019; sousa & voss, 2006; verhoef et al., 2015) 5. central product (balasubramanian et al, 2005; verhoef et al., 2015) 6. data security issues (piotrowicz and cuthbertson 2014; verhoef, & agrawal, 2004) 7. non-understanding young customer habits (verhoef et al., 2015; picot-coupey et al., 2016) 8. order fulfillment (chopra, 2016) 9 . inconsistent product discount (chopra & whisnant 2019) 10. time constraints (neslin et al., 2006) 3. research methodology this study proposes a framework of enablers as well as barriers of ocr based on literature and experts’ responses. data is collected through experts’ interviews, reviews, databases, and reports of the retailing industry. various databases like wos, scopus, emerald insight, and google scholar are extracted for identifying the enablers and barriers. the experts validated the enablers and barriers and evaluated them for developing hierarchical levels using ism methodology. it is elaborated in two phases. phase i includes the identification of enablers and barriers of ocr and employing interpretative structuring modeling (ism) to develop a multi-level structure. the relationships among the variables vary, sometimes strong, weak equal, or not equal; thus, fuzzy micmac and dematel methods are used to compute the strengths of the enablers and barriers. 3.1 phase i 3.1.1. data collection the retail experts working in different capacities are selected for collecting data. the pool of experts includes consultants, cios, digital marketing, and supply chain management professionals. three experts from the supply chain function of the retail organizations with a working experience of more than ten years, two academicians associated with the retail management program, two experts in the marketing domain from an online retail store, and three experts from operation management with five years of experience are selected for the panel. a questionnaire was circulated among the experts to collect data for this study. 3.1.2 interpretative structural modeling (ism) method this method is used to describe the relationship between the variables through hierarchical levels (sharma et al., 2019). the steps of the ism model, postidentification of barriers and enablers, are described as follows. omni-channel retailing for enhancing customer engagement amidst supply chain disruption 9 i. seven enablers and ten barriers are identified and validated by experts’ judgment. ii. established a relationship among all the identified enablers and barriers. iii. a structural self interaction matrix (ssim) is formed, and the relationship is represented in the form of four symbols. v: enabler i will ameliorate enabler j; a: enabler i will be ameliorated by enabler j; x: enablers i and j will ameliorate each other; and iv. an initial reachability matrix (irm) is formed, and transitivity is checked v. the final reachability matrix is developed after checking for transitivity. vi. a digraph is made based on contextual relationships. vii. nodal elements are then replaced by the statement. viii. the established model has assessed any conceptual inconsistencies. 3.2 phase ii 3.2.1 fuzzy micmac and dematel methods this phase includes fuzzy micmac and dematel to explore the strength of enablers and barriers. the fuzzy micmac method derives the driving and dependence value of the variables that help to understand the interrelationship among the variables. the relationships among the enablers or barriers vary, weak, equal, or sometimes stronger. thus, this method helps to categorize the enablers and barriers on the basis of their driving and dependence powers. the following steps are used to obtain results (sharma & joshi, 2020). i. establishing binary direct relationship matrix ii. developing fuzzy binary direct relationship matrix (fbdrm) iii. developing fuzzy-micmac stabilized matrix in the recent literature, multi-criteria decision methods are used for a variety of research in the area of marketing operations, viz online shopping for analyzing the change in purchasing behavior (sharma et al., 2020a; sharma et al., 2020b; sharma et al., 2022a); to develop marketing strategies for alliance development (tang et al., 2022), technological interventions in marketing and retailing (kamble et al., 2019; singh et al., 2020), waste management (sharma et al., 2019; sharma et al., 2020a; sharma et al., 2020b) and product development and its supply chain management (panchal and kumar, 2017; panchal et al., 2022; sharma et al., 2020c; tyagi et al., 2019). specifically, the dematel method has been employed in various domains such as marketing, supply chains, waste management, technology management, and reverse logistics (chauhan et al., 2020;mousavizade & shakibazad, 2019; sharma et al., 2020c). the method is described as follows: step 1: average matrix computation the experts are asked to rate the variables on the scale of 0 – 4, where 0 indicates ‘no influence’, 4 indicates ‘very high influence’. a n x n matrix is developed as xk= [𝑥𝑖𝑗 𝑘 ] on the basis of the expert responses. the responses are incorporated from h respondents, direct relation matric ‘aij’ is formed through equation 1. 𝑎𝑖𝑗 = 1 𝐻 ∑ 𝑥𝑖𝑗 𝑘 𝐻𝐾=1 (1) where, k= number of respondent with 1≤ ik ≤ h n= number of criteria joshi et al./decis. mak. appl. manag. eng. (2022) 10 step 2: calculating the normalized initial directrelation matrix d= m x b b = min [ 1 𝑀𝑎𝑥 ∑ 𝑎𝑖𝑗 𝑛 𝑗=1 , 1 𝑀𝑎𝑥 ∑ 𝑎𝑖𝑗 𝑛 𝑖=1 ] (2) step 3: calculating the total relation matrix by the following equation t is calculated as 𝑇 = 𝑁(𝐼 − 𝑁)−1 (3) i denote the identity matrix. step 4: drawing the diagraph the sum of rows [ri]n x and columns [cj]1 x n denotes the vectors. values of (ri + cj) and (ri cj) are calculated. (if the value of (ri -ccj) is positive, then the enabler is categorized as causal group variables, and if the value of (ricj) is negative, then the enablers are categorized as effect group variables. 4. models application the integrated ism-fuzzy micmac-dematel elaborated in section 3 is followed in this section for obtaining dependence and driving powers. the ten barriers are classified into six hierarchical levels and seven enablers after iterations shown in tables 3 and 4. the hierarchical levels of enablers and barriers are exhibited in figures 2 and 3. these barriers and enablers are taken into phase two for further analysis to explore the inter-relationships. table 3. irm -enablers enablers ocre7 ocre6 ocre5 ocre4 ocre3 ocre2 ocre1 ocre1 v x v v v v ocre2 x v v a v ocre3 o v o x ocre4 v v v ocre5 a a ocre6 v ocre7 table 4. irmbarriers ado b10 ado b9 ado b8 ado b7 ado b6 ado b5 ado b4 ado b3 ado b2 ado b1 adob1 v v v v o v v v v adob2 o x o o o o x o adob3 a o v o o a v adob4 o a v a o v adob5 v v v v o adob6 o o o v adob7 o o v adob8 a a adob9 v adob10 omni-channel retailing for enhancing customer engagement amidst supply chain disruption 11 figure 2. driving and dependence power diagram in phase ii, a binary direct reachability matrix (bdrm) is obtained and the diagonal entries are converted to zero. fuzzy set theory (eq. 4) is used to enhance the responsiveness of micmac. 𝐶 = 𝐴, 𝐵 = max 𝑘[(min(𝑎𝑖𝑘 , 𝑏𝑘𝑗 ))] where𝐴 = [𝑎𝑖𝑘 ] and 𝐵 = [𝑏𝑘𝑗 ] (4) the final matrix for enablers and barriers are obtained and exhibited in table 5 and 6. table 5. irmenablers ocre1 ocre2 ocre3 ocre4 ocre5 ocre6 ocre7 ocre1 1 1 1 1 1 1 1 ocre2 0 1 1 0 1 1 1 ocre3 0 0 1 1 0 1 0 ocre4 0 1 1 1 1 1 1 ocre5 0 0 0 0 1 0 0 ocre6 1 0 0 0 1 1 1 ocre7 0 1 0 0 1 0 1 enablers: broadband internet accessibility orce1;well-located and well-designed distribution centers (ocre2);efficient and extensive logistics network(orce3);cross-channel integration (orce4);customer analytics(orce5);omni-channel visibility to customers (orce6);product digitization (orce7). table 6. irmbarriers ado b1 ado b2 ado b3 ado b4 ado b5 ado b6 ado b7 ado b8 ado b9 ado b10 adob1 1 1 1 1 1 0 1 1 1 1 adob2 0 1 0 1 0 0 0 0 1 0 adob3 0 0 1 1 0 0 0 1 0 0 adob4 0 1 0 1 1 0 0 1 0 0 joshi et al./decis. mak. appl. manag. eng. (2022) 12 ado b1 ado b2 ado b3 ado b4 ado b5 ado b6 ado b7 ado b8 ado b9 ado b10 adob5 0 0 1 0 1 0 1 1 1 1 adob6 0 0 0 0 0 1 1 0 0 0 adob7 0 0 0 1 0 0 1 1 0 0 adob8 0 0 0 0 0 0 0 1 0 0 adob9 0 1 0 1 0 0 0 1 1 1 adob10 0 0 1 0 0 0 0 1 0 1 adob1: lack of coordination and information among channels; adob2: price inconsistency; adob3: product unavailability; adob4: content inconsistency; adob5:lack of centralized product assortment (cpa); adob6:data privacy; adob7:non-understanding young customer habits; adob8: order fulfillment; adob9:inconsistent product discount; adob10:time constraint. the relationship among the seven enablers, as well as the ten adoption barriers, have been developed using dematel. by steps 1,2,3,4, and 5 of the dematel process demonstrated in section 3, the direct influence matrix, normalized direct influence matrix, total relation matrix, and degree of influences are developed. tables 7 and 8 represent the direct influences of enablers and barriers. table 9 (a) and table 9 (b) demonstrated direct influences – enablers and direct influences – barriers using dematel. table 7. transitivity matrix -enablers ocre1 ocre2 ocre3 ocre4 ocre5 ocre6 ocre7 ocre1 1 1 1 1 1 1 1 ocre2 *1 1 1 *1 1 1 1 ocre3 *1 *1 1 1 1 1 *1 ocre4 *1 1 1 1 1 1 1 ocre5 0 0 0 0 1 0 0 ocre6 1 1 *1 1 1 1 1 ocre7 0 1 *1 0 1 *1 1 enablers: broadband internet accessibility orce1;well-located and well-designed distribution centers (ocre2);efficient and extensive logistics network(orce3);cross-channel integration (orce4);customer analytics(orce5);omni-channel visibility to customers (orce6);product digitization (orce7). table 8. transitivity matrix-barriers ado b1 ado b2 adob 3 adob 4 ado b5 ado b6 ado b7 ado b8 ado b9 adob 10 adob1 1 1 1 1 1 0 1 1 1 1 adob2 0 1 0 1 *1 0 0 *1 1 *1 adob3 0 *1 1 1 *1 0 0 1 0 0 adob4 0 1 0 1 1 0 *1 1 *1 *1 adob5 0 *1 1 *1 1 0 1 1 1 1 adob6 0 0 0 *1 0 1 1 *1 0 0 adob7 0 *1 0 1 *1 0 1 1 0 0 adob8 0 0 0 0 0 0 0 1 0 0 adob9 0 1 *1 1 *1 0 0 1 1 1 adob10 0 0 1 *1 0 0 0 1 0 1 adob1: lack of coordination and information among channels; adob2: price inconsistency; adob3: product unavailability; adob4: content inconsistency; adob5: lack of centralized product assortment (cpa); adob6: data privacy; adob7: non-understanding young customer habits; adob8: order fulfillment; adob9: inconsistent product discount; adob10: time constraint. omni-channel retailing for enhancing customer engagement amidst supply chain disruption 13 table 9 (a). direct influences enablers row total (d) column total (r) d+r values d-r values ocre1 1.569 0.280 1.849 1.289 ocre2 0.815 0.638 1.453 0.177 ocre3 0.523 0.615 1.138 -0.093 ocre4 0.974 0.390 1.364 0.585 ocre5 0.000 1.384 1.384 -1.384 ocre6 0.782 0.962 1.743 -0.180 ocre7 0.402 0.796 1.199 -0.394 table 9 (b). direct influences – barriers row total (d) column total (r) d+r values d-r values adob1 0.294 0.000 0.294 0.294 adob2 0.032 0.204 0.236 -0.172 adob3 0.013 0.244 0.258 -0.231 adob4 0.128 0.239 0.367 -0.112 adob5 0.179 0.183 0.363 -0.004 adob6 0.001 0.000 0.001 0.001 adob7 0.013 0.125 0.138 -0.111 adob8 0.000 0.454 0.454 -0.454 adob9 0.175 0.199 0.373 -0.024 adob10 0.002 0.092 0.094 -0.090 5. results and discussion the ism and fuzzy micmac results demonstrate the hierarchical structure and categorization of the enablers and barriers. the enablers have shown a two-level structure from the ism method application, whereas the barriers show a six-level structure in figures 2 and 3. customer analytics (ocre5) is the top-level enabler at the hierarchical level. all other enablers are on the second level, which exhibits the two hierarchical levels structure for enablers, whereas there are six hierarchical levels in the adoption barriers. order management (ocrb8) is on the top level, followed by three barriers, namelyinconsistency in content (ocrb4), inconsistency in product information (ocrb5), and lack of information about consumers (ocrb7). the third level of barriers has product unavailability (ocrb3) and time constraint (ocrb10). data privacy and security (ocrb6) is present at the fourth level. the inconsistency in price (ocrb2) is the most critical adoption barrier in the ocr, followed by a lack of coordination & information sharing (ocrb1) and inconsistency in price discounts (ocrb9). the levels are exhibited in figures 2 and 3, illustrating the multi-levels of the enablers and barriers. these levels are further validated by the fuzzy micmac and dematel and reveal that inconsistency in price, discounts, and information sharing are the most critical barriers in the ocr framework. the fuzzy micmac results are exhibited in figure 2, showing the four clusters consisting of enablers and barriers as per their dependence and driving powers. the joshi et al./decis. mak. appl. manag. eng. (2022) 14 seven enablers are categorized into two clusters only, whereas adoption barriers are classified into 3 clusters. cluster i reflects weak driving and dependence power. the absence of any enablers or barriers in this cluster suggests that all the enablers and barriers undertaken in the study are significant. the enabler -customer analytics (ocre5), and three barriers order management (ocrb8), inconsistency in content (ocrb4) and inconsistency in product information (ocrb5), and lack of information about customers (ocrb7) are included in cluster ii (dependent barriers). this cluster has strong dependence and weak driving power. the strength of these variables (enablers & barriers) shows that the other variables need support to minimize their effect. these enablers and barriers are critical and need to be addressed by the retail organizations or decision-makers as a priority. the ocr should be more effective and efficient if the adoption barriers are minimized. no enabler is present in cluster iii (linkage barriers), indicating that the enablers are either dependent or driving. this cluster has three barriers having high driving and dependence power, making it sensitive. these barriers are highly volatile and impede the execution of adoption among omnichannel. data privacy and security (ocrb6), product unavailability (ocrb3) and time constraint (ocrb10) are linkage barriers. cluster iv (driving barriers) has high driving barriers and low dependence on power. this cluster includes enablersinternet accessibility (ocre1), well-located distribution centers (ocre2), integration (ocre3), integration across channels (ocre4), visibility (ocre6) and digitization (ocre7). omni-channel retailers should focus on integration among channels, visibility to customers, and accessibility to understand and develop strategies for enhancing the customer's experience. the intention of retail firms is to provide a single view of products, services, and inventory to the customers, possible only when all the operations of channels are integrated and synchronized. thus, retailers need to integrate their entire value chain, including supply chains, operations, e-commerce, and order fulfillment, which will lead to enhanced visibility and transparency among the channels regarding customers' orders, information and purchase (liu et al., 2020). the barriers present in this cluster are – inconsistency in price (ocrb2), lack of coordination & information sharing among channels (ocrb1), and inconsistency in product discounts (ocrb9). the adoption barriers of ocr, main inconsistency in price, discounts and information sharing, can deteriorate the unified experience of the customer (sharma et al., 2019; sharma & joshi, 2020). the dematel findings from table 7 validate the results obtained from ism and fuzzy micmac methods application. the value for r-c shows that the single enabler customer analytics (-0.647) is the only dependent enabler and should be treated as the effect factor group, whereas the four barriersorder management (ocrb8); inconsistency in content (ocrb4), inconsistency in product information (ocrb5); lack of information about customers (ocrb7). dematel results also signify that inconsistency in price (ocrb2), lack of coordination among channels & information sharing (ocrb1) and inconsistency in price discounts (ocrb9) are the cause group variables. the results of dematel signify that the ism results are valid and the levels of enablers and barriers should be considered by the retail organizations while designing their strategies. the firms should take action to remove inconsistencies among the price, discounts and information sharing, as on these driving barriers, the other barriers are dependent (sharma et al., 2020a; sharma et al., 2022b). for example, if the inconsistency in price and discount exists, it will affect the order management of the product. three barriers, namely inconsistency in price (ocrb2), lack of coordination & information sharing among channels (ocrb1), and inconsistency in product discounts (ocrb9), are the driving barriers affecting all the other barriers. inconsistency in price has the highest omni-channel retailing for enhancing customer engagement amidst supply chain disruption 15 r-c value (1.162), driving all the other barriers, and thus organizations need to remove inconsistency among prices across their channels. this study tries to develop a theoretical understanding of the ocr in relation to the adoption barrier faced by retail firms as well as the enablers to enhance ocr adoption. the driving barriers (ocrb1, ocrb2, ocrb9) are validated integrated ism-fuzzy micmac-dematel methods and proven that these are the most significant barriers that need to be minimized in the ocr framework. also, the retail organization can perform better if they focus on the prominent enablers like integration, efficient logistics system and digitization (cao and li, 2015; li et al., 2020). 6. implications the emerging markets currently have the highest market potential, which may be targeted at young, millennial, and educated populations who are making their buying choices at their convenience through multiple channels like smartphones, the internet, and mom-and-pop stores. consumers are becoming agnostic today and want a unified experience across all the multiple points, but retailers are not upgraded yet and need much more integration in their back-end systems. traditional retail firms are well aware that their systems, like merchandise planning, inventory planning, order management and others, are not compatible with omnichannel. thus, enablers like integration, visibility, robust information systems, information sharing, and the wide accessibility of the internet need to be strengthened by the retail firms to engage their customers effectively and enhance their customer experience. previously the studies have focused on the relationship between the physical shop's retail metrics and firm performance (ailawadi and farris, 2017; sharma et al., 2019; sharma et al., 2021). but, today, the turbulence in the current retail industry enhances the need to select the right metrics at the right time to predict product purchases (caro and sadr 2019; caboni and hagberg, 2019; sharma et al., 2021; sharma et al., 2022a; hagberg et al., 2017). thus, a more focused approach is needed to unravel the challenges in a more changing environment like ocr, which saves time and cost for the retailer and the customers (larke et al., 2018; galipoglu et al., 2018; jocevski et al., 2019). the study also reveals customer analytics is the only dependent enabler. analytics is not limited to online interactions only. rather, physical stores can use advanced analytics through their robust information system that will help to learn how customers navigate their purchasing journey (sharma et al., 2021). moreover, analytics provides real-time information regarding the assortment and merchandise of the organizations and thus helps the retailer to improve the experience through optimized assortment decisions. the other benefit is personalization, one of the imperatives of the ocr and key to engaging customers effectively. it customizes the customer's experience by presenting only the most relevant choices, content, and offers. data analytics is the emerging area of marketing to employ information management tools, which help retailers to engage consumers appropriately with more personalization. cross-channel analytics seeks to correlate and analyze customer interaction across channels. it helps to track the performance of the multiple channels, i.e., how effective and attractive channels are performing at certain risks or generating positive results. retailers need to develop their logistics systems through real-time monitoring, sensors etc., for better control of the supply chains. also, a welldistributed warehouse helps retail organizations to control their order management and delivery. integration and visibility are the main enablers nowadays, as customers are more aware and access multiple devices. the retailers know it very well that joshi et al./decis. mak. appl. manag. eng. (2022) 16 without omnichannel transformation, the customers cannot be retained in the long run. the decision-makers and retailers can track customers' behavior at various touch points, indicating how organizations may improve the experience throughout the customer's journey through all these enablers. the analysis of clickstreams and product searches can also provide knowledge of the purchase journey, gauge demand, and upsell opportunities to the retailers for their future strategies. nowadays, software is also employed, such as 'shopping assistant,' to ease and direct the customer journey. this study has focused on the critical role of cross-channel digital technologies, internet accessibility, distribution centers and inconsistency among price, discounts and content in the ocr structure. this study has indicated various key factors that need to be addressed by the managers and practitioners to transform the retail form into ocr. 6.1 research limitations and future directions the results from ism show the multi-level structure that can be further extended by tism for exploring the strength of the enablers and the barriers. firstly, ocr is an evolving area, and thus new approaches would be welcomed to analyze the effect on consumer engagement in emerging markets. secondly, prominent adoption barriers and enablers identified in this study can be further built for industry-based studies such as consumer goods, automobiles, fmcg, and online retail. this study has undertaken the ocr as a whole, which can be further broken into distinct segments of products and services. future studies may work on products and services that may help to redesign strategies specifically for them appropriately. thirdly, the framework developed can be further empirically validated in future research works. 7. conclusion the evolving ocr unifies all the customer touch points. there is a lack of theoretical embeddedness of research in ocr, and therefore, this study highlights the comprehensive structure of ocr considering enablers and adoption barriers influencing the performance of retail organizations in emerging markets like india retail organizations can perform better if they focus on prominent enablers like integration, efficient logistics system and digitization. advanced technologies like artificial intelligence, predictive modeling, machine learning, and real-time monitoring etc. may help retailers to develop their competitive advantage in engaging customers effectively. digital technologies have challenged traditional retail organizations to transform their business models. retail organizations need to adopt advanced analytical models to cope with the pandemic situation, as their traditional models are weak in handling customer's choices and expectations and managing customer's journeys efficiently. this study highlights the importance of integration across the channels with consistency, which may enhance the customer's intentions for future purchases. due to the upsurge in mobile technology usage, new systems need to be developed with better integration and interchangeability. from the analysis of this study, inconsistent price (ocrb2) is the most crucial barrier in the ocr adoption process. organizations should take action to eradicate inconsistency among prices across all channels. ocr does not aim to develop all the capabilities in each channel but rather assign products and tasks to channels on the basis of effective handling. the study also reveals that customer analytics is dependent on all the other enablers, which implies that the retailers need to upgrade their current sub-systems if they need omni-channel retailing for enhancing customer engagement amidst supply chain disruption 17 to best fit the customer choices with multiple channels. the finding of the research work shall help in decision-making to the practitioners also as they can involve these enablers and barriers while adopting the ocr. more specifically, the practitioners should concentrate on the enablers and barriers to successfully adopting ocr. both retail channels in emerging markets are highly motivated and aim to serve their customers better by facilitating them with a seamless purchasing experience. the digitization and hybrid business models are creating a competitive environment for firms where ocr combat the prevailing barriers and design strategies to strengthen their enablers. the implication of ocr suggests that retailers become ubiquitous. author contributions: conceptualization, s.j and m.s.; methodology, m.s.; software, m.s. and p.c.; validation, s.j and m.s.; formal analysis, s.j.; investigation, s.j.; resources, s.j.; data curation, p.c.; writing—original draft preparation, m.s.; writing—review and editing, s.j and p.c..; visualization, m.s..; supervision, p.c..; project administration, m.s.. all authors have read and agreed to the published version of the manuscript. funding: this research received no external funding. acknowledgments: the authors are thankful to the reviewers and editorial members of the dmame. conflicts of interest: the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. references agatz, n. a., fleischmann, m., & van nunen, j. a. 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(2019). “preorder-online, pickup-in-store” strategy for a dual-channel retailer. transportation research part e: logistics and transportation review, 122, 27-47. https://doi.org/10.1016/j.tre.2018.11.001 © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1016/j.jretconser.2022.103070 https://doi.org/10.1016/j.procir.2016.08.002 https://doi.org/10.1016/j.tre.2018.11.001 decision making: applications in management and engineering vol. 3, issue 1, 2020, 108-125. issn: 2560-6018 eissn:2620-0104 doi: https://doi.org/10.31181/dmame2003015g * corresponding author. e-mail addresses: bcgiri.jumath@gmail.com (b.c. giri), sushildeyju@gmail.com (s.k. dey) game theoretic models for a closed-loop supply chain with stochastic demand and backup supplier under dual channel recycling bibhas chandra giri 1* and sushil kumar dey 1 1 department of mathematics, jadavpur university, kolkata, west bengal, india received: 12 january 2019; accepted: 11 june 2019; available online: 27 june 2019. original scientific paper abstract: in this paper, a dual-channel closed-loop supply chain is consi-dered for waste recycling. the manufacturer produces the finished product using recycled and recyclable waste materials as well as fresh raw materials. the recyclable wastes collected by the collector are supplied to the manufacturer directly or indirectly via a third party (recycler). two different game models are considered for two different cases of recycling: recycling by the manufacturer and recycling by the recycler. if the collector fails to supply the required amount of waste materials, the backup supplier meets up the shortfall by supplying fresh raw materials. the customer demand is assumed to be stochastic. optimal results for the two game models are obtained through numerical examples. it is seen that ex-ante pricing commitment i.e, fixed markup strategy is beneficial for the whole supply chain as well as the supply chain entities, compared to the decentralized policy. from the numerical study, it is also observed that when the recyclability degree of wastes increases, the expected total profit increases for the whole supply chain. a higher price sensitivity of customer demand leads to lower profit for the chain members. keywords: supply chain management, closed-loop supply chain, recycling, markup-strategy. 1. introduction one of the biggest concerns of our society today is degradation of environment. increasing consumption, richer lifestyle, higher level of logistics and transportation have led to higher carbon emissions and as a consequence, all these are raising important questions about environmental sustainability. most of the supply chains in today’s business scenario are attentive to sustainability, not only for the present age but also for the future generations. various ways like remanufacturing, reusing, green mailto:bcgiri.jumath@gmail.com mailto:sushildeyju@gmail.com game theoretic models for a closed-loop supply chain with stochastic demand and backup... 109 purchasing, recycling etc. are being used to achieve environmental sustainability. recycling is one of the most suitable way to adopt in manufacturing industries because recovery of materials and recycling from used product is one of the major avenues to reduce the usage of fresh materials. for example, plastic is the third highest manufacturing sector in the united states where over million of workers are working but for the conscience of environmental sustainability they have installed about 30,000 recycling drop-off points nationwide and plastic film recycling is continuing to grow (www.nytimes.com). kreiger et al. (2014) studied about the recycling of hybrid polyethelene for 3-d painting. bing et al. (2014) also studied in the same line concerning the household plastics of different types. the uk has a recycling rate of approximately 60% for iron and steel. most of this waste comes from scrap vehicles, cooker, fridges and other kitchen appliances and, in germany, the recycling rate for plastic is 70% (giri & dey, 2019). measuring of reduction limit of repeated recycling for paper flow was analyzed by chen et al. (2015). sheu et al. (2012) analyzed the effect of governmental financial intervention on a green supply chain management using a three stage game-theoretic model. strategically low wholesale price is suggested to recycled-component suppliers to stimulate the manufacturer’s intention of green production under green taxation. in case of waste recycling, some european countries like sweden and germany achieved great results of success even though recycling rate is lower than most of the other countries (zhang et al., 2014). jafari et al. (2016) studied dual channel recycling in a three-echelon supply chain with game theoretic approach. ragaert et al. (2017) studied about mechanical and chemical recycling of solid plastic waste. their discussion was about the main challenges and some potential remedies to the recycling strategies. wan et al. (2017) reviewed solid state recycling of aluminium chips. sultan et al. (2017) studied an integrated model for product recycling desirability. texas instruments makes significant investments to efficiently use, reuse, or recycle materials across its operations, and reduces its potential environmental impact by sourcing materials responsibly, as well as appropriately managing waste handling and disposal (giri & dey, 2019) in this paper, for a multi-echelon closed-loop supply chain with price dependent stochastic customer demand, we investigate the optimal decisions for pricing and corresponding profit for each player using game theoretic approach. dual channel recycling (recycling by the manufacturer and recycling by the third party i.e, recycler) is adopted in this paper in two different models. the paper is organised as follows: review of relevant literature is given in the next section. notations and problem description are provided in section 3. two different game models and their analytical results are discussed in section 4. in section 5, numerical demonstration along with sensitivity analysis is given. finally, the paper is concluded with future research directions in section 6. 2. literature review in this section, a brief review of three different streams of research such as dual channel supply chain, sustainable development in supply chain and markup policies are given. 2.1. dual channel supply chain in traditional dual-channel supply chain, researchers customarily use online channel and offline channel. yao and liu (2005), mukhopadhyay et al. (2008), liu et al. (2010), zhang et al. (2012), cao et al. (2013) examined the optimal pricing giri and dey/decis. mak. appl. manag. eng. 3 (1) (2020) 108-125 110 decisions for asymmetric information scenario through a dual channel structure. a profit maximization strategy in a dual-channel was derived by batarfi et al. (2016). zhang et al. (2017) studied about the retailer’s channel structure choice; whether he would chose a online channel, offline channel or dual-channel. some recent works on pricing, service and quality decisions in dual channel have been done by wang et al. (2017) and li et al. (2017). chen et al. (2017) studied the impact of adding a new channel on price, quality and profit’s change. zhao et al. (2017) analyzed pricing policies for complementary products in a dual-channel supply chain where one among the two manufacturers uses dual channel. wei et al. (2018) analyzed a dual collecting channel with dynamic nature of life-cycle of wastes. the effects of profit discount and collection competition on firms pricing decisions, collection rates and profits were studied and the remanufacturer’s optimal strategy of maximizing its profit or maximizing the collection rate was revealed. recently, zhang et al. (2019) developed a new strategy for a dual-channel retailer to identify whether the strategy is always beneficial for improving the dual-channel retailer’s profit or market share. 2.2. sustainable development in supply chain in recent years, mainly in the last decade, sustainability has become one of the biggest global business issues. business environment has become more complex in nature. environmental complexity due to unsustainable resources and activities is raising high. hence, in this situation, large industries as well as small business firms are concerned for sustainability due to governmental pressure and instructions, as well as for their own concern about a greener world. navinchandra (1990) first proposed the idea of green product design. this means the improvement of the product’s compatibility with the environment, without harming its quality or its function. an empirical study for sustainable supply chain management was proposed by ageron et al. (2012). ahi and searcy (2013) analyzed a competitive literature of definitions for green and sustainable supply chain management. impacts of lean, resilient and green practices on social, economic and environmental sustainability of supply chains were proposed by govindan et al. (2014). an optimization oriented brief review of social and environmental sustainability was presented by eskandarpour et al. (2015). li et al. (2015) determined the pricing policy in a competitive dual-channel green supply chain. yu and solvang (2016) discussed the recycling with environmental considerations. jafari et al. (2017) studied dual-channel waste recycling under deterministic scenario. 2.3. markup pricing strategy a simple but often used pricing policy is to include a fixed markup over the wholesale price of each item. according to liu et al. (2006) retail fixed markup (rfm) simply exists as an “agreement" more than a formal written code. markup can also be defined as the difference between the wholesale price and the retail price. two types of markup are commonly used by the retailers: (i) fixed-price markup and (ii) fixed percentage markup. wang et al. (2015) analyzed the performance of this two types of markup startegy under a chain to chain competition with dominant retailer. for instance, under a keystone markup, the retailer simply doubles the production cost to settle the retail price. so markup actually specifies pricing policies among different entities involved in a supply chain. in general, a contract is for long term and it varies over time and product but retail price markup remains fixed for the duration of the specified supply chain. gasoline dealers or some grocers also use traditional fixed markup policy. liu et al. (2009) studied vertically restrictive pricing using markup game theoretic models for a closed-loop supply chain with stochastic demand and backup... 111 strategy. for an integrated supply chain with price dependent demand, they could not find a closed form solution under any general distribution of the stochastic customer demand. they also showed that pareto-improving rfm solution exists in a deterministic scenario but it is not always possible to find when demand is stochastic. a two-way price commitment for the retailer and the manufacturer was studied by liu et al. (2013). they assumed fixed markup contract for the retailer and price protection contract for the manufacturer. maiti and giri (2016) proposed a model with both variable and fixed markup. giri et al. (2017) analyzed pricing policies for a threeechelon supply chain with sub-supply chain and rfm strategy. 3. notations and problem description we use the following notations throughout the paper: 𝐶𝑐 unit collection cost of recyclable wastes to the collector. 𝐶𝑟 unit recycling cost of recyclable wastes to the recycler. 𝐶𝑠 unit procurement cost of recycled waste to the back-up supplier. 𝐶𝑚 unit recycling cost of recyclable wastes to the manufacturer. 𝐶𝑝 unit production cost of the finished product to the manufacturer. u per unit shortage penalty cost of the manufacturer. v per unit salvage value of the manufacturer. 𝜖 random part of the demand. 𝜃 recyclability degree of waste denoting the portion of waste that can be recovered and turned into new products. (0 < 𝜃 < 1) 𝛾 quantity of recycled materials required to produce one unit of the finished product. (𝛾 > 1) q quantity of finished items produced by the manufacturer. 𝛾 𝜃 quantity of recyclable waste required to produce one unit of the finished product. a maximum possible demand faced by the manufacturer for the finished product. 𝑏 price sensitivity of the customer’s demand. (𝑏 > 0) 𝜆 fractional part of the manufacturer’s requirement of recycled materials supplied by the collector. (0 < 𝜆 < 1) z stocking factor for the stochastic demand. 𝑃𝑑 wholesale price charged by the collector to the manufacturer for one unit of recyclable waste. 𝑃𝑐 wholesale price charged by the collector to the recycler for one unit of recyclable waste. 𝑃𝑟 wholesale price charged by the recycler to the manufacturer for one unit of recycled material. 𝑃𝑠 wholesale price charged by the supplier to the manufacturer for one unit of fresh raw material. p retail price charged by the manufacturer to the customers for one unit of finished product. d customer demand of the finished product at the manufacturer. 𝐷𝑐 quantity of raw materials supplied by the collector to the recycler. 𝐷𝑟 quantity of recycled waste supplied by the recycler to the manufacturer. 𝐷𝑠 quantity of fresh raw materials supplied by the backup supplier to the manufacturer. π𝑐 collector’s profit. giri and dey/decis. mak. appl. manag. eng. 3 (1) (2020) 108-125 112 π𝑟 recycler’s profit. π𝑠 supplier’s profit. π𝑚 manufacturer’s profit. π𝑐𝑟 profit obtained from the co-ordination established between the collector and the recycler the proposed closed-loop supply chain consists of one manufacturer, one collector, one recycler and one backup supplier. the manufacturer may get the recycled materials from the recycler as well as recyclable wastes from the collector. a dual channel is considered to receive recyclable wastes and recycled materials from the collector and the recycler, respectively (see figure 1). when the collector or the recycler fails to satisfy the manufacturer’s need (𝑞𝛾 units), the manufacturer needs help of a backup supplier. the manufacturer then buys fresh raw materials from the backup supplier at a high price to make up the shortfall. figure 1. material flow diagram we assume that the customer’s demand 𝐷 is linear, price-dependent and random in nature. we take 𝐷 = 𝑎 − 𝑏𝑃 + 𝜖 where 𝑏 > 0, 𝑃 < 𝑎 𝑏 and 𝜖 is the random part of the customer demand. here 𝑎 denotes market’s total potential demand but actual demand is d and b represents the price sensitivity for customer demand. in supply chain literature, this type of demand function is common where the customer demand depends on retail price and demand decreases with the increment of retail price (jafari et al., 2017; petruzzi & dada , 1999). unit shortage penalty cost and salvage value are also incurred in the model setting as well. 4. model development under the problem scenario mentioned above, we develop two models (see fig. 1) depending upon two different situations : model i: in this model, the collector collects the recyclable wastes from the end customers and then supplies to the manufacturer. the manufacturer first recycles the waste materials and then produces finished goods for the end customers. any shortfall of wastes is meet up by a backup supplier by supplying fresh raw materials. game theoretic models for a closed-loop supply chain with stochastic demand and backup... 113 model ii: this model includes a recycler. the collector collects the wastes but the recycling is done by the recycler. the recycler recycles the wastes and then sends to the manufacturer for production of finished goods. any shortfall of recycled material is meet up by a backup supplier by supplying fresh raw materials. 4.1. model i: the manufacturer gets recyclable wastes from the collector here, we assume that the collector may or may not satisfy the manufacturer’s demand of recyclable wastes. the natural disasters, communication problems or unavailability of resources may be the reasons behind this. we assume that the manufacturer estimates an amount of 𝑞𝛾 units of raw materials to produce 𝑞 units of finished product. let us suppose that the collector can supply 𝑞𝛾𝜆 𝜃 units of wastes where 0 < 𝜆 ≤ 1 and 𝜃 (0 < 𝜃 < 1) is the recyclability degree of waste. so, we have in this case, 𝐷𝑐 = 𝑞𝛾𝜆 𝜃 , 0 < 𝜆 ≤ 1 and 𝐷𝑠 = 𝑞𝛾(1 − 𝜆). when 𝜆 = 1, the manufacturer’s demand for recyclable wastes is completely meet up by the collector and h ence there is no need of any action from the backup supplier. in this model, we develop three game theoretic approaches, viz. centralized game, decentralized game and fixedmarkup game. 4.1.1. centralized game since the market demand is stochastic, so sometimes the estimated inventory of the manufacturer may be less than the market demand or sometimes there may be some left over inventory in hand. we assume that the manufacturer sells the left over inventory in a secondary market with a salvage value 𝑣 per unit. unit shortage penalty cost is 𝑢. then the expected profits of the manufacturer, the collector and the supplier are given by π𝑚 = 𝐸[𝑃 𝑚𝑖𝑛(𝑞, 𝐷) − 𝑢(𝐷 − 𝑞) + + 𝑣(𝑞 − 𝐷)+ − 𝑃𝑠 𝐷𝑠 − (𝑃𝑑 + 𝐶𝑚 )𝐷𝑐 − 𝑞𝐶𝑝], (1) π𝑐 = 𝐸[(𝑃𝑑 − 𝐶𝑐 )𝐷𝑐 ], 𝑎𝑛𝑑 (2) π𝑠 = 𝐸[(𝑃𝑠 − 𝐶𝑠)𝐷𝑠 ], (3) respectively, where 𝑋+ = 𝑚𝑎𝑥(𝑋, 0) and the subscripts 𝑐, 𝑠 and 𝑚 stand for the collector, the supplier and the manufacturer, respectively. we replace 𝑧 = 𝑞 − 𝑦(𝑃), where 𝑧 is the stocking factor on which the shortage or overage depends. stocking factor is also sometimes called safety stock factor. our objective is to find the optimal selling price, stocking factor rather than selling price and the stocking quantity. the expected total profit in the centralized game is given by π = π𝑚 + π𝑐 + π𝑠 = 𝐸[𝑃 min (𝑞, 𝐷) − 𝑢(𝐷 − 𝑞)+ + 𝑣(𝑞 − 𝐷)+ − 𝑃𝑠 𝐷𝑠 (𝑃𝑑 + 𝐶𝑚 )𝐷𝑐 +(𝑃𝑑 − 𝐶𝑐 )𝐷𝑐 + (𝑃𝑠 − 𝐶𝑠 )𝐷𝑠 ] = (𝑃 − 𝐶𝑝)[𝑦(𝑝) + 𝜇] − (𝐶𝑝 − 𝑣)𝜙(𝑧) − (𝑃 + 𝑢 − 𝐶𝑝)𝜓(𝑧) −(𝐶𝑚 + 𝐶𝑐 ) [𝑧 + 𝑦(𝑃)] 𝛾𝜆 𝜃 , where 𝜙(𝑧) = ∫ 𝑧 0 (𝑧 − 𝑡)𝑓(𝑡)𝑑𝑡 and 𝜓(𝑧) = ∫ ∞ 𝑧 (𝑡 − 𝑧)𝑓(𝑡)𝑑𝑡. giri and dey/decis. mak. appl. manag. eng. 3 (1) (2020) 108-125 114 now, our problem becomes 𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑧;𝑃 π. we consider the first and second order partial derivatives of π with respect to 𝑧 and 𝑃. when 𝑧 is fixed, we get the optimal value of 𝑃 as 𝑃∗(𝑧) = 𝑎+𝑏𝐶𝑝+𝜇+(𝐶𝑚 +𝐶𝑐) 𝑏𝛾𝜆 𝜃 − 𝜓(𝑧) 2𝑏 and optimal value of 𝑧 for a fixed 𝑃 is 𝑧∗(𝑃) = 𝐹−1 {1 − (𝐶𝑝−𝑣)+(𝐶𝑚 +𝐶𝑐) 𝛾𝜆 𝜃 𝑃+𝑢−𝑣 }. corollary 1. the profit function π is concave in z for a given value of p and concave in p for a given value of z. proof: see appendix a. proposition 1. (i) the optimal retail price increases with the stocking factor and (ii) the optimal stocking factor of the manufacturer is also an increasing function of the retail price. proof: (i) we have the optimal retail price 𝑃∗(𝑧) = 𝑎+𝑏𝐶𝑝 +𝜇+(𝐶𝑚+𝐶𝑐) 𝑏𝛾𝜆 𝜃 − 𝜓(𝑧) 2𝑏 . then clearly, 𝑑𝑃∗(z) 𝑑𝑧 = − ( 1 2𝑏 ) 𝑑 𝑑𝑧 𝜓(𝑧) = 1 2𝑏 ∫ ∞ 𝑧 𝑓(𝑡)𝑑𝑡 > 0, since 𝑓(𝑡) ≥ 0 for all 𝑡. (ii) for the optimal stocking factor 𝑧∗, we have 𝐹(𝑧∗) = 1 − (𝐶𝑝 −𝑣)+(𝐶𝑚 +𝐶𝑐) 𝛾𝜆 𝜃 𝑃+𝑢−𝑣 . differentiating partially with respect to 𝑃, we get 𝑓(𝑧∗) 𝑑𝑧 𝑑𝑃 = (𝐶𝑝−𝑣)+(𝐶𝑚 +𝐶𝑐) 𝛾𝜆 𝜃 (𝑃+𝑢−𝑣)2 which implies 𝑑𝑧 𝑑𝑃 = 1 𝑓(𝑧 ∗) (𝐶𝑝 − 𝑣) + (𝐶𝑚 + 𝐶𝑐 ) 𝛾𝜆 𝜃 (𝑃 + 𝑢 − 𝑣)2 > 0, 𝑠𝑖𝑛𝑐𝑒 𝑓(𝑧) ⩾ 0. proposition 2. under linear additive demand function, the supply chain’s centralized solution is to set quantities z∗, p∗ and to order a − bp∗ + z∗ such that (i) if f(⋅) is an arbitary distribution, then the entire support must be searched to find z∗, (ii) if f(⋅) satisfies 2r(z)2 + dr(z) dz > 0 where r(z) = f(z) 1−f(z) is the hazard rate, then z∗ is the largest z satisfying the first order condition. proof: for details of the proof see appendix a. 4.1.2. decentralized game here our objective is to maximize separately the expected profits of the manufacturer, the supplier and the collector, which are as follows: π𝑚 = 𝐸[𝑃 𝑚𝑖𝑛(𝑞, 𝐷) − 𝑢(𝐷 − 𝑞) + + 𝑣(𝑞 − 𝐷)+ − 𝑃𝑠 𝐷𝑠 − (𝑃𝑑 + 𝐶𝑚 )𝐷𝑐 − 𝑞𝐶𝑝] π𝑐 = 𝐸[(𝑃𝑑 − 𝐶𝑐 )𝐷𝑐 ] π𝑠 = 𝐸[(𝑃𝑠 − 𝐶𝑠 )𝐷𝑠 ] now, we suppose that the profit margins for the players in this game are same i.e. 𝑃𝑑 = 𝑃+𝐶𝑐 2 and 𝑃𝑠 = 𝑃+𝐶𝑠 2 . using these relations, we derive the optimal values of 𝑃 and 𝑧 as, 𝑃∗(𝑧) = 𝑎+𝜇+𝑏𝐶𝑝−𝜓(𝑧)+𝑏[(𝑃𝑑+𝐶𝑚 ) 𝛾𝜆 𝜃 +𝑃𝑠𝛾(1−𝜆)] 2𝑏 game theoretic models for a closed-loop supply chain with stochastic demand and backup... 115 and, 𝑧∗(𝑃) = 𝐹−1 {1 − (𝐶𝑝 − 𝑣) + (𝑃𝑑 + 𝐶𝑚 ) 𝛾𝜆 𝜃 + 𝑃𝑠 𝛾(1 − 𝜆) 𝑃 + 𝑢 − 𝑣 } 4.1.3. fixed markup strategic game in the fixed markup strategic game, we assume that the supplier’s wholesale price 𝑃𝑠 = (1 − 𝛼1)𝑃 where 0 < 𝛼1 < 1 and the collector’s wholesale price is 𝑃𝑑 = (1 − 𝛼2)𝑃 where 0 < 𝛼2 < 1 and that 0 < 𝛼1 ⩽ 𝛼2 < 1. using these relations in the profit functions, we get π𝑚 = 𝐸[𝑃 𝑚𝑖𝑛(𝑞, 𝐷) − 𝑢(𝐷 − 𝑞) + + 𝑣(𝑞 − 𝐷)+ − 𝑃𝑠 𝐷𝑠 − (𝑃𝑑 + 𝐶𝑚 )𝐷𝑐 − 𝑞𝐶𝑝], π𝑐 = 𝐸[(𝑃𝑑 − 𝐶𝑐 )𝐷𝑐 ], and π𝑠 = 𝐸[(𝑃𝑠 − 𝐶𝑠 )𝐷𝑠 ]. we get the optimal price of the manufacturer as 𝑃∗(𝑧) = 𝑎+𝜇+𝑏𝐶𝑝−𝜓(𝑧)+𝑏[𝐶𝑚 𝛾𝜆 𝜃 −𝐶𝑠𝛾(1−𝜆)]−(𝑧−𝑎)[(1−𝛼2) 𝛾𝜆 𝜃 +(1−𝛼1)𝛾(1−𝜆)] 2𝑏−2𝑏(1−𝛼2) 𝛾𝜆 𝜃 −2𝑏(1−𝛼1)𝛾(1−𝜆) and optimal stocking factor as 𝑧∗(𝑃) = 𝐹−1 {1 − (𝐶𝑝 − 𝑣) + ((1 − 𝛼2)𝑃 + 𝐶𝑚 ) 𝛾𝜆 𝜃 + ((1 − 𝛼1)𝑃 − 𝐶𝑠 )𝛾(1 − 𝜆) 𝑃 + 𝑢 − 𝑣 } the optimal wholesale prices of the supplier and the collector are given by the relations 𝑃𝑠 ∗ = (1 − 𝛼1)𝑃 ∗ and 𝑃𝑑 ∗ = (1 − 𝛼2)𝑃 ∗. 4.2. model ii: the manufacturer gets the recycled materials from the recycler here, we consider the situation where the collector supplies recyclable wastes to the recycler for recycling. however, the recycler may or may not satisfy the manufacturer’s demand of recycled materials. in case of any shortfall of recycled materials, the manufacturer purchases the required amount of fresh raw materials from the backup supplier. therefore, in this case we have 𝐷𝑐 = 𝑞𝛾 𝜃 𝜆, 0 < 𝜆 ≤ 1 𝐷𝑟 = 𝑞𝛾𝜆 𝐷𝑠 = 𝑞𝛾(1 − 𝜆) 4.2.1. centralized game like the previous model, here we assume that the manufacturer needs total 𝑞 units of finished product to satisfy customer demand. if the market demand exceeds the order quantity, shortage occurs and the shortage penalty cost of the manufacturer is then 𝑢(𝐷 − 𝑞). on the other hand, if the market demand is less than the total quantity 𝑞, the leftover inventory is sold in a secondary market at a lower cost 𝑣. then the total revenue from the leftover inventory is 𝑣(𝑞 − 𝐷). thus the expected profits of the manufacturer, the collector, the recycler and the supplier are given by giri and dey/decis. mak. appl. manag. eng. 3 (1) (2020) 108-125 116 π𝑚 = 𝐸[𝑃 𝑚𝑖𝑛(𝑞, 𝐷) − 𝑢(𝐷 − 𝑞) + + 𝑣(𝑞 − 𝐷)+ − 𝑃𝑠 𝐷𝑠 − (𝑃𝑑 + 𝐶𝑚 )𝐷𝑐 − 𝑞𝐶𝑝], (4) π𝑐 = 𝐸[(𝑃𝑐 − 𝐶𝑐 )𝐷𝑐 ], (5) π𝑟 = 𝐸[𝑃𝑟 𝐷𝑟 − (𝑃𝑐 + 𝐶𝑟 )𝐷𝑐 ], (6) π𝑠 = 𝐸[(𝑃𝑠 − 𝐶𝑠 )𝐷𝑠 ], (7) where 𝑋+ = 𝑚𝑎𝑥(𝑋, 0) and the subscripts 𝑐, 𝑠, 𝑟 and 𝑚 stand for the collector, the supplier, the recycler and the manufacturer, respectively. the expected total profit in the centralized game is π = π𝑚 + π𝑐 + π𝑟 + π𝑠 = 𝐸[𝑃 𝑚𝑖𝑛(𝑞, 𝐷) − 𝑢(𝐷 − 𝑞)+ +𝑣(𝑞 − 𝐷)+ − 𝑃𝑠 𝐷𝑠 − (𝑃𝑑 + 𝐶𝑚)𝐷𝑐 + (𝑃𝑑 − 𝐶𝑐 )𝐷𝑐 + (𝑃𝑠 − 𝐶𝑠)𝐷𝑠 ] = (𝑃 − 𝐶𝑝)[𝑦(𝑃) + 𝜇] − (𝐶𝑝 − 𝑣)𝜙(𝑧) − (𝑝 + 𝑢 − 𝐶𝑝)𝜓(𝑧) − (𝐶𝑟 + 𝐶𝑐 )[𝑧 + 𝑦(𝑃)] 𝛾𝜆 𝜃 , where 𝜙(𝑧) = ∫ 𝑧 0 (𝑧 − 𝑡)𝑓(𝑡)𝑑𝑡 and 𝜓(𝑧) = ∫ ∞ 𝑧 (𝑡 − 𝑧)𝑓(𝑡)𝑑𝑡. when 𝑧 is fixed, we derive the optimal value of 𝑃 as 𝑃∗(𝑧) = 𝑎+𝑏𝐶𝑝 +𝜇+(𝐶𝑟+𝐶𝑐) 𝑏𝛾𝜆 𝜃 − 𝜓(𝑧) 2𝑏 and for a fixed 𝑃, the optimal value of 𝑧 as 𝑧∗(𝑃) = 𝐹−1 {1 − (𝐶𝑝−𝑣)+(𝐶𝑟+𝐶𝑐) 𝛾𝜆 𝜃 − 𝜓(𝑧) 𝑃+𝑢−𝑣 }. corollary 2. the profit function 𝛱 is concave in 𝑧 for a given value of 𝑃 and concave in 𝑃 for a given value of 𝑧. proof: see appendix b. 4.2.2. decentralized game the decentralized game is considered when all the members in the supply chain have similar decision powers and they are not interested for a collaborative business together. there may be some mutual agreements between a pair of members but they will never collaborate all together like a centralized model. so, our problem is now to maximize separately the expected profits of the manufacturer, the collector and the supplier, which are π𝑚 = 𝐸[𝑃 𝑚𝑖𝑛(𝑞, 𝐷) − 𝑢(𝐷 − 𝑞) + + 𝑣(𝑞 − 𝐷)+ − 𝑃𝑠 𝐷𝑠 − (𝑃𝑑 + 𝐶𝑚 )𝐷𝑐 − 𝑞𝐶𝑝], π𝑐 = 𝐸[(𝑃𝑑 − 𝐶𝑐 )𝐷𝑐 ], π𝑠 = 𝐸[(𝑃𝑠 − 𝐶𝑠 )𝐷𝑠 ]. similar to the previous model, we now suppose that the profit margins for the players in this game are same i.e., 𝑃𝑑 = 𝑃+𝐶𝑐 2 and 𝑃𝑠 = 𝑃+𝐶𝑠 2 . then the optimal values of 𝑃 and 𝑧 are given by 𝑃∗(𝑧) = 𝑎+𝜇+𝑏𝐶𝑝−𝜓(𝑧)+𝑏[(𝑃𝑑+𝐶𝑚) 𝛾𝜆 𝜃 +𝑃𝑠𝛾(1−𝜆)] 2𝑏 𝑎𝑛𝑑 𝑧∗(𝑃) = 𝐹−1 {1 − (𝐶𝑝 − 𝑣) + (𝑃𝑑 + 𝐶𝑚 ) 𝛾𝜆 𝜃 + 𝑃𝑠 𝛾(1 − 𝜆) 𝑃 + 𝑢 − 𝑣 }. proposition 3. the joint profit for all the members in the supply chain in model ii is greater than that of model i if 𝐶𝑚 > 𝐶𝑟 . proof: in model i, the expected total profit of the supply chain is game theoretic models for a closed-loop supply chain with stochastic demand and backup... 117 π𝐷 𝐼 = π𝑚 + π𝑠 + π𝑐 = (𝑃 − 𝐶𝑝)[𝑦(𝑃) + 𝜇] − (𝐶𝑝 − 𝑣)𝜙(𝑧) − (𝑃 + 𝑢 − 𝐶𝑝)𝜓(𝑧) − [(𝐶𝑚 + 𝐶𝑐 ) 𝛾𝜆 𝜃 +𝐶𝑠 𝛾(1 − 𝜆)][𝑧 + 𝑦(𝑃)], and the expected total profit of the supply chain in model ii is π𝐷 𝐼𝐼 = π𝑚 + π𝑠 + π𝑐 + π𝑟 = (𝑃 − 𝐶𝑝)[𝑦(𝑃) + 𝜇] − (𝐶𝑝 − 𝑣)𝜙(𝑧) − (𝑃 + 𝑢 − 𝐶𝑝)𝜓(𝑧) − [(𝐶𝑟 + 𝐶𝑐 ) 𝛾𝜆 𝜃 + 𝐶𝑠 𝛾(1 − 𝜆)][𝑧 + 𝑦(𝑝)]. clearly, π𝐷 𝐼𝐼 > π𝐷 𝐼 whenever 𝐶𝑚 > 𝐶𝑟 . 4.2.3. fixed markup strategic game in the fixed markup strategic game also, each of the players wants to maximize its own profit individually. each downstream player wants to fix his wholesale price grater than the preceding upstream member. hence we assume that the collector’s wholesale price 𝑃𝑐 = (1 − 𝛼3)𝑃𝑟 , the recycler’s wholesale price 𝑃𝑟 = (1 − 𝛼4)𝑃, and the supplier’s wholesale price 𝑃𝑠 = (1 − 𝛼5)𝑃, and that 0 < 𝛼5 ⩽ 𝛼4 ⩽ 𝛼3 < 1. using the above relations in the profit functions π𝑚 = 𝐸[𝑃 𝑚𝑖𝑛(𝑞, 𝐷) − 𝑢(𝐷 − 𝑞) + + 𝑣(𝑞 − 𝐷)+ − 𝑃𝑠 𝐷𝑠 − 𝑃𝑟 𝐷𝑟 − 𝑞𝐶𝑝] π𝑐 = 𝐸[(𝑃𝑐 − 𝐶𝑐 )𝐷𝑐 ] π𝑟 = 𝐸[𝑃𝑟 𝐷𝑟 − (𝑃𝑐 + 𝐶𝑟 )𝐷𝑐 ] π𝑠 = 𝐸[(𝑃𝑠 − 𝐶𝑠 )𝐷𝑠 ], we get the optimal price of the manufacturer as 𝑃∗(z) = 𝑎+𝜇+𝑏𝐶𝑝−𝜓(𝑧)−(𝑎+𝑧)[(1−𝛼4)𝛾𝜆+(1−𝛼5)𝛾(1−𝜆)] 2𝑏[1−(1−𝛼4)𝛾𝜆−(1−𝛼5)𝛾(1−𝜆)] and the optimal value of the stocking factor 𝑧 as 𝑧 ∗(𝑃) = 𝐹−1 {1 − (𝐶𝑝 − 𝑣) + 𝑃[(1 − 𝛼4)𝛾𝜆 + (1 − 𝛼5)𝛾(1 − 𝜆)] 𝑃 + 𝑢 − 𝐶𝑝 }. 5. numerical examples 5.1. example 1 for model i in this example, we set the parameter-values for model i. we assume that the random demand follows (𝑖) exponential distribution i.e., 𝑓(𝛼, 𝑥) = 𝛼𝑒 −𝛼𝑥 , 𝑥 > 0 with 𝛼 = 0.02, mean 𝜇 = 50; and (𝑖𝑖) uniform distribution i.e., 𝑓(𝑧) = 1 100 , 0 ⩽ 𝑧 ⩽ 100, with mean 𝜇 = 50. we consider the other parameter-values as follow: 𝐶𝑝 =5, 𝜃 = 0.7, 𝛾 = 1.3, 𝑎 = 1000, 𝑏 = 1.3, 𝐶𝑐 = 15, 𝐶𝑚 = 65, 𝐶𝑠 = 100, 𝜇 = 10, α1 = 0.65, α2 = 0.65, 𝜆 = 0.6, 𝑢 = 3, 𝑣 = 4 in appropriate units. for this set of data, we obtain the optimal price, optimal stocking factor and profit for each player in different games. the optimal results for exponential and uniform demand distributions are shown in tables 1 and 2, respectively. giri and dey/decis. mak. appl. manag. eng. 3 (1) (2020) 108-125 118 table 1. optimal results in model i for exponential distribution optimal results centralized game decentralized game rfm strategy 𝑃∗ 471.103 498.87 479.14 𝑧∗ 59.81 15.10 15.96 π𝑚 ∗ 45336.70 48973.6 π𝑐 ∗ 63836.90 66882.3 π𝑠 ∗ 18989.40 13837.7 expected total profit 1,33,691.0 1,28,163.0 1,29,693.6 table 2. optimal results in model i for uniform distribution optimal results centralized game decentralized game rfm strategy 𝑃∗ 475.19 501.98 482.24 𝑧∗ 70.024 26.18 27.42 π𝑚 ∗ 46103.2 49776.7 π𝑐 ∗ 65495.7 68630.8 π𝑠 ∗ 19555.7 14325.2 expected total profit 1,37,255.0 1,31,154.6 1,32,732.7 tables 1 and 2 show the optimal results of each of the players under exponential and uniform distributions. expected total profits for all the gaming approaches are higher in case of uniform demand distribution compared to the respective models in exponential demand distribution. the optimal retail price of the product is lower in case of fixed markup strategy which results in higher customer demand and higher profit. the optimal profits of the manufacturer and the collector are higher in case of the fixed markup strategy than those in decentralized policy. 5.2. example 2 for model ii here also we consider two types of demand distribution as given below : (𝑖) exponential distribution i.e., 𝑓(𝛼, 𝑥) = 𝛼𝑒 −𝛼𝑥 , 𝑥 > 0 with 𝛼 = 0.02, mean 𝜇 = 50; (𝑖𝑖) uniform distribution i.e., 𝑓(𝑧) = 1 100 , 0 ⩽ 𝑧 ⩽ 100, with same mean 𝜇 = 50. we consider the parameter-values as follow: 𝐶𝑝 =5, 𝜃 = 0.7, 𝛾 = 1.3, 𝑎 =1000, 𝑏 = 1.3, 𝐶𝑐 = 15, 𝐶𝑚 = 65, 𝐶𝑟 =10, 𝐶𝑠 = 100, 𝜇 = 10, α3 = 0.45, α4 = 0.40, α5 = 0.35, 𝜆 = 0.6, 𝑢 = 3, 𝑣 = 4 in appropriate units. for this set of data, we obtain the optimal price, optimal stocking factor and expected profit of each player in different gaming approaches, as shown in tables 3 and 4. table 3. optimal results in model ii for exponential distribution optimal results centralized game decentralized game rfm strategy 𝑃∗ 442.75 465.98 402.32 𝑧∗ 84.90 8.24 8.15 π𝑚 ∗ 26788.0 27448.9 π𝑐 ∗ 67415.1 70915.1 π𝑠 ∗ π𝑟 ∗ 38296.1 20463.4 40743.6 14526.3 game theoretic models for a closed-loop supply chain with stochastic demand and backup... 119 expected total profit 1,62,928.0 1,52,964.6 1,53,633.9 table 4. optimal results in model ii for uniform distribution optimal results centralized game decentralized game rfm strategy 𝑃∗ 445.639 465.98 403.72 𝑧∗ 81.80 15.21 15.04 π𝑚 ∗ 27042.9 27663.9 π𝑐 ∗ 68464.8 65064.7 π𝑠 ∗ π𝑟 ∗ 38928.0 20808.0 41584.4 14469.9 expected total profit 1,66,501.0 1,55,244.0 1,56,534.0 tables 3 and 4 depict the optimal results for each player as well as for the whole supply chain in model ii. optimal retail prices are lower in this model compared to those in model i which corresponds to higher demand. here also optimal values of the profits are greater for the uniform distribution and, for both the distributions, the expected total profit of the supply chain is improved in the markup policy, compared to the decentralized game. the expected total profits of the supply chain for the two decentralized cases in this model are higher than those of the respective cases in model i due to the lower recycling cost of the recycler (proposition 3). 5.3. sensitivity analysis now, in particular for the exponential distribution, we examine the sensitivity of the key parameters 𝜃, 𝑏 and 𝛾 on the optimal prices as well as the expected profit of the supply chain in different strategies of both the game models. 5.3.1. sensitivity with respect to 𝜃 as the value of 𝜃 increases, in model i, the supply chain’s expected total profit increases for the centralized, decentralized and markup strategic game models. this happen because higher recyclability degree results in higher quality value of the used wastes and this reduces the recycling cost and also the usage of total wastes. we see that the profit of the manufacturer increases as 𝜃 increases but the collector and the backup supplier’s optimal profits are obtained for their respective specific values of 𝜃. in model ii also, the expected total profit increases with 𝜃. the expected total profit is maximum in the centralized model, which is the benchmark case. for the markup strategy, the expected total profit is higher compared to that of the decentralized gaming strategy (see figure 2). giri and dey/decis. mak. appl. manag. eng. 3 (1) (2020) 108-125 120 (a) 𝜃 vs. total profit (model i) (b) 𝜃 vs. total profit (model ii) figure 2. sensitivity w. r. to 𝜃 5.3.2. sensitivity with respect to 𝛾 as the value of 𝛾 increases, the manufacturer requires more recyclable wastes to produce 𝑞 units of finished product. hence the production cost will increase for the manufacturer and that leads to lower profit. however, the collector and the backup supplier attain higher profits for increasing 𝛾, as they will have to supply more raw materials. if the value of 𝛾 increases, the amount of recycled materials to be supplied by the recycler to the manufacturer increases. so, in that case, the expected profit decreases in all the three types of gaming approaches. because of ex-ante price markup commitment, the expected total profit in case of markup policy is higher compared to that of the decentralized policy (see figure 3) (a) 𝛾 vs. total profit (model i) (b) 𝛾 vs. total profit (model ii) figure 3. sensitivity w. r. to 𝛾 5.3.3. sensitivity with respect to 𝑏 for higher values of 𝑏 (price sensitivity of customer demand), the customer demand is lower. as a result, the profits of all individual entities decrease for higher values of 𝑏. in the markup policy, the supply chain’s optimal expected profit becomes higher than that in the decentralized game (see figure 4). game theoretic models for a closed-loop supply chain with stochastic demand and backup... 121 similar observation is made for the game model ii. the expected total profit of the supply chain as well as individual profits gradually decrease with higher values of 𝑏. (a) 𝑏 vs. total profit (model i) (b) 𝑏 vs. total profit (model ii) figure 4. sensitivity w. r. to 𝑏 6. conclusion in this paper, we have studied a closed-loop supply chain scenario where recycling is the main concern for environmental sustainability. a manufacturer performs recycling using two different channel of recycling, directly by his own and also by the help of another recycler. for two different game models, depending on different ways of recycling, we have analyzed the optimal pricing strategy of all the supply chain members. for stochastic demand, it is not always easy to get closed form solution of the model. so, numerically we obtain the optimal solutions for two types of demand distribution uniform and exponential. from the sensitivity analysis, we have the following observations: i. ex-ante markup strategy is beneficial (compared to decentralized model) for the supply chain entities, specially the manufacturer. however, profit is not always pareto-improving in case of stochastic demand scenario (here specially for the backup supplier in model i and recycler in model ii), which supports the result of liu et al. (2009). ii. for higher value of 𝜃, the supply chain will gain higher profit. the individual entities will also gain higher profit for a particular range of 𝜃. iii. when the recycler recycles the wastes at a cost lower than the manufacturer, the expected total profit of the supply chain is higher. iv. a higher price sensitivity of customer’s demand decreases customer demand, and hence it leads to lower profit for the manufacturer. several future studies can be done using different contract policies among the members. one can also assume multiplicative form of stochastic demand. instead of fixed markup, the entities can go for variable markup policy also. acknowledgement: the authors are sincerely thankful to the esteemed reviewers for their comments and suggestions based on which the manuscript has been improved. the first author gratefully acknowledges the research support from csir, govt. of india (grant ref. no. 25(0282)/18/emr-ii). appendix a giri and dey/decis. mak. appl. manag. eng. 3 (1) (2020) 108-125 122 proof of corollary 1: the expected total profit in the centralized model i is, π = (𝑃 − 𝐶𝑝)[𝑦(𝑝) + 𝜇] − (𝐶𝑝 − 𝑣)𝜙(𝑧) − (𝑃 + 𝑢 − 𝐶𝑝)𝜓(𝑧) −(𝑧 + 𝑦(𝑃))(𝐶𝑚 + 𝐶𝑐 ) 𝛾𝜆 𝜃 taking first and second order partial derivatives of π with respect to 𝑧 and 𝑃 we get, 𝜕π 𝜕𝑧 = −(𝐶𝑝 − 𝑣) + (𝑃 + 𝑢 − 𝑣)[1 − 𝐹(𝑧)] − (𝐶𝑚 + 𝐶𝑐 ) 𝛾𝜆 𝜃 𝜕2π 𝜕𝑧2 = −(𝑃 + 𝑢 − 𝑣)𝑓(𝑧) < 0, since 𝑣 < 𝑃. 𝜕π 𝜕𝑝 = (𝑃 − 𝐶𝑝)(−𝑏) + (𝑎 − 𝑏𝑃 + 𝜇) − (𝐶𝑚 + 𝐶𝑐 ) 𝛾𝜆 𝜃 (−𝑏) − 𝜓(𝑧) 𝜕2π 𝜕𝑝2 = −2𝑏 < 0, since 𝑏 > 0. proof of proposition 2: we have 𝑑π 𝑑𝑧 = −(𝐶𝑝 − 𝑣) + (𝑃 + 𝑢 − 𝑣)[1 − 𝐹(𝑧)] − (𝐶𝑚 + 𝐶𝑐 ) 𝛾𝜆 𝜃 let 𝑅(𝑧) = 𝑑π 𝑑𝑧 now, 𝑑𝑅(𝑧) 𝑑𝑧 = 𝑑 𝑑𝑧 [ 𝑑π 𝑑𝑧 ] = 𝑑 𝑑𝑧 [−(𝐶𝑝 − 𝑣) − (𝐶𝑚 + 𝐶𝑐 ) 𝛾𝜆 𝜃 ] + 𝑑 𝑑𝑧 [(𝑃 + 𝑢 − 𝑣)(1 − 𝐹(𝑧))], where 𝑃(𝑧) = 𝑎+𝑏𝐶𝑝+𝜇+ 𝑏𝛾𝜆 𝜃 − 𝜓(𝑧) 2𝑏 = 𝑃0 − 𝜓(𝑧) 2𝑏 , where 𝑃0 = 𝑎+𝑏𝐶𝑝+𝜇 2𝑏 + (𝐶𝑚 + 𝐶𝑐 ) 𝛾𝜆 2𝜃 so, 𝑑𝑅(𝑧) 𝑑𝑧 = 𝑑 𝑑𝑧 [(𝑃0 − 𝜓(𝑧) 2𝑏 + 𝑢 − 𝑣)(1 − 𝐹(𝑧))] = 1 2𝑏 [1 − 𝐹(𝑧)]2 − (𝑃0 + 𝑢 − 𝑣 − 𝜓(𝑧) 2𝑏 )𝑓(𝑧) = 𝑓(𝑧) 2𝑏 {2𝑏(𝑃0 = 𝑢 − 𝑣) − 𝜓(𝑧) − 1−𝐹(𝑧) 𝑟(𝑧) }, where 𝑟(𝑧) = 𝑓(𝑧) 1−𝐹(𝑧) , the hazard rate. again, 𝑑2𝑅(𝑧) 𝑑𝑧2 = 𝑑 𝑑𝑧 { 𝑑𝑅(𝑧) 𝑑𝑧 } = 𝑑𝑅(𝑧)/𝑑𝑧 𝑓(𝑧) . 𝑑𝑓(𝑧) 𝑑𝑧 − 𝑓(𝑧) 2𝑏 {(1 − 𝐹(𝑧)) + 𝑓(𝑧) 𝑟(𝑧) + (1−𝐹(𝑧))[ 𝑑𝑅(𝑧) 𝑑𝑧 ] [𝑟(𝑧)]2 } hence, 𝑑2𝑅(𝑧) 𝑑𝑧2 |𝑑𝑅(𝑧) 𝑑𝑧 =0 = −𝑓(𝑧)(1 − 𝐹(𝑧)) 2𝑏[𝑟(𝑧)]2 [2[𝑟(𝑧)]2 + 𝑑𝑟(𝑧) 𝑑𝑧 ] now if 𝐹(⋅) be a probability distribution function which satisfies, 2𝑟(𝑧)2 + 𝑑𝑟(𝑧) 𝑑𝑧 > 0 then it follows that 𝑅(𝑧) is either monotone or unimodal implying that 𝑅(𝑧) = 𝑑π[𝑧,𝑃(𝑧)] 𝑑𝑧 has at most two roots. again, 𝑅(𝑧) lim𝑧→∞ = −(𝐶𝑝 − 𝑣) − (𝐶𝑚 + 𝐶𝑐 ) 𝛾𝜆 𝜃 < 0. so, if 𝑅(𝑧) has only one root then it gives the maximum value of π(𝑧, 𝑃) and if it has two roots then the larger of them corresponds to the maximum value of π(𝑧, 𝑃). game theoretic models for a closed-loop supply chain with stochastic demand and backup... 123 appendix b proof of corollary 2 : the expected total profit in the centralized model ii is π = (𝑃 − 𝐶𝑝)[𝑦(𝑃) + 𝜇] − (𝐶𝑝 − 𝑣)𝜙(𝑧) − (𝑃 + 𝑢 − 𝐶𝑝)𝜓(𝑧) −(𝐶𝑟 + 𝐶𝑐 ) (z + 𝑦(𝑃)] 𝛾𝜆 𝜃 so, 𝜕π 𝜕𝑧 = −(𝐶𝑝 − 𝑣) + (𝑃 + 𝑢 − 𝑣)[1 − 𝐹(𝑧)] − (𝐶𝑟 + 𝐶𝑐 ) 𝛾𝜆 𝜃 𝜕2π 𝜕𝑧2 = −(𝑃 + 𝑢 − 𝑣)𝑓(𝑧) < 0, 𝑠𝑖𝑛𝑐𝑒 𝑣 < 𝑃 and 𝑓(𝑧) ≥ 0 𝜕π 𝜕𝑝 = (𝑃 − 𝐶𝑝)(−𝑏) + (𝑎 − 𝑏𝑃 + 𝜇) − (𝐶𝑟 + 𝐶𝑐 ) 𝛾𝜆 𝜃 (−𝑏) − 𝜓(𝑧) 𝜕2π 𝜕𝑝2 = −2𝑏 < 0, 𝑠𝑖𝑛𝑐𝑒 𝑏 > 0. author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research received no external funding. conflicts of interest: the authors declare no conflicts of interest. references ageron, b., gunasekaran, a., & spalnzani, a. 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(2017). pricing policies for complementary products in a dual-channel supply chain. applied mathematical modelling, 49, 437-451. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1016/j.apm.2017.04.023 https://www.sciencedirect.com/science/journal/13665545 https://www.sciencedirect.com/science/journal/13665545 plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 5, issue 2, 2022, pp. 46-77. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame03051020224k * corresponding author. e-mail addresses: s.kalantari1989@gmail.com (s. kalantari), h.kazemipoor@iauctb.ac.ir (h. kazemipoor)*, f-movahedi@srbiau.ac.ir (f. movahedi sobhani), molana@srbiau.ac.ir (sm. hadji molana) a neutrosophical model for optimal sustainable closed-loop supply chain network with considering inflation and carbon emission policies saeid kalantari 1, hamed kazemipoor 1*, farzad movahedi sobhani 2 and seyed mohammad hadji molana 2 1 department of industrial engineering, central tehran branch, islamic azad university, iran 2 department of industrial engineering, science and research branch, islamic azad university, iran received: 27 july 2022; accepted: 27 september 2022; available online: 5 october 2022. original scientific paper abstract: in this paper, a stable clsc problem is modeled in conditions of uncertainty and indeterminacy. the scn is designed to maximize npv and minimize carbon releases by maintaining environment friendly policies and accounting for increase. to achieve a suitable model for designing a stable clscn and making important decisions such as selecting the right suppliers, selecting the type of transport, initial the facility, the optimal flow between facilities and accomplishing an efficient solution to the problem decision making, the neutrosophic optimization method is used. the results of experiments that discuss and evaluate different scenarios confirm the efficiency and validity of the proposed model. the findings also show that the effective improvement of the obtained solutions by reducing the solution time up to twenty percent can be responsible for large-scale problems in different scenarios. this paper uses a neutrosophic optimization method to solve the problem of designing a stable clscn under uncertainty and indeterminacy. key words: sustainability, closed-loop supply chain network (clscn), supply chain management (scm), net present value (npv), neutrosophic optimization, neutrosophic logic. a neutrosophical model for optimal sustainable closed-loop supply chain network … 47 1. introduction a sustainable sc is envisaged based on the tbl, including economic, social and environmental. many academics and artisans have considered sscm in recent decades. sscm helps companies reduce their environmental pollution and risks, improve environmental performance, create stronger market benefits, increase brand equity and reputation, lower overall costs, consider wet, and bring better relationships with consumers (saberi et al., 2019; ansari et al., 2017). while the triple bottom line approach has been adopted in research on performance sustainability in sscm, research on environmental dimensions has been more prominent (gimenez et al., 2012; acquaye et al., 2018). inquiry on multiple dimensions of stability is very important, because these dimensions simultaneously affect each other and the stability performance (homayouni et al., 2021). waste (ayvaz et al., 2015), development of new products (jahani et al., 2017), and humanitarian procurement is considered. on the other hand, with more consumer awareness of environmentally friendly products, adopting ways to reduce adverse environmental impacts on production activities has become essential. this is the problem also in other sectors such as the electronics industry, transportation of hazardous materials (golpîra et al., 2021; mohabbati-kalejahi and vinel, 2021), medicine distribution (low et al., 2016), oil (paydar et al., 2017), and disposable goods (gholizadeh et al., 2020) are discussed. on the other hand, due to the short product life cycle and lower profit margins, companies’ closed-loop sc is now a necessity. at the closed-loop sc, forward and reverse logistics systems are integrated simultaneously. inverted sc logistics are responsible for managing product returns for recycling, reuse or disposal. reverse logistics also helps create a competitive advantage and increases the company's profit margin by re-targeting the products used (govindan et al., 2017). in the same way, researchers argue that reverse logistics is essential to the supply chain's environmental and economic issues in today's unstable market (polo et al., 2019). more interaction between pd and pr is needed with more emphasis on sustainable production. pr helps improve the organization's environmental compatibility and reduce production costs. research on clscn design is important for sscm (govindan et al., 2017). an efficient product recovery process and related closed-loop supply chain configuration encourages customers to return their products at the end of their useful life and, through integrated planning, reduce the environmental impact of landfills and product recovery. closed-loop sc meets demand through chain operations and value-added processes by collecting return products intended for reuse, recycling or disposal (darbari et al., 2019). in closed-loop sc design, the management of reverse logistics operations becomes more difficult when many decision variables and their uncertainties affect the environment. (sel et al., 2015; morganti et al., 2015). in this regard, the design of clscn in areas such as the dairy industry (yavari et al., 2019), plastic recycling industry (de vargas mores et al., 2018), energy industry (mohtashami et al., 2020), gasoline industry (saedinia et al., 2019), perishable products (yavari et al., 2020), and engine oil industry (paydar et al., 2017) were used. 0n the contrary, designing a sustainable clscn is a strategic decision. it is difficult to accurately estimate some parameters, such as demand, due to changes in the business environment (yun et al., 2020). therefore, some important parameters such as customer demands are completely unclear and appropriate planning to deal with this uncertainty seems necessary. in this connection, studies on the design of a stable clsc under uncertainty have attracted the attention of researchers, which can be referred to (de et al., 2020; alegoz et al., 2020; pham, 2021). in a stable clsc, financial s. kalantari et al./decis. mak. appl. manag. eng. 5 (2) (2022) 46-77 48 factors are important because they have a greater impact on supply, production, distribution, and recycling. all sc decisions affect how allocation and financing are funded (yang et al., 2021). the inflation expectations, how the budget deficit is met, and the monetary and liquidity base change are important for the industry, as the inflation from exchange rate fluctuations to supply raw materials and production or recycling technology strongly influences sclsc policies (wan et al., 2019). based on what has been said, the contribution of this paper in the field of optimization-based decision models, on the one hand, is to consider sustainability factors (economic and environmental) that both sustainability factors with the assumption that facilities can be affected by these goals. in addition, the development of a robust fuzzy optimization model, using appropriate risk measurements, is used to formulate uncertainty changes in the face of market fluctuations, set uncertain parameters whose distribution functions are unknown, and then use a neutrosophical approach for the total problem. on the other hand, this model selects the most appropriate options according to the existing and potential activities such as supplier selection and selection of the transportation modes and problems related to carbon emission policies. total relationships, changes, and purchases should be related to process structure and cost. to calculate economic criteria for companies' options, fixed costs include opening or using facilities or technology costs, and variable costs include emissions, transportation, production, distribution, waste disposal, and recycling. on the other hand, the net present value is an optimization criterion in the utility function. because the npv compares investment options, where period, cost of capital and inflation are outstanding criteria for evaluating recycled products. because new recycled products have little or no market access at first, it is important to evaluate different periods in this study. therefore, considering the demand based on multi-period conditions, it is possible to use the annual compound growth rate attached to certain products, affecting the opening up of technology. this study provides an operational and strategic framework for designing a stable clsc under uncertainty, minimizing environmental impacts while maximizing the proposed network npv. this article considers a correct momp model with uncertainty with a case study on electronic component chains to achieve these goals. on the other hand, due to the complexity of the mop nature of the proposed model, a new hm based on a ha with a neutrosophical approach to solve these problems is proposed. in the continuation of this paper, in section two, the literature review is reviewed, and the research gap is identified. in section three, the structure of the problem is examined, and the proposed model is shown according to the hypotheses. in section four, the uncertain optimization model is discussed. section five proposes the neutrosophical model and defines the hybrid method for the solution. section six also discusses the analysis of the proposed model and the proposed solution method. at the end, in section seven, conclusions and future suggestions are stated. 2. literature review 2.1. neutrosophic logic crisp set theory can only grip the data having no ambiguity or uncertainty. the crisp set theory comes into play only when the boundary of a piece of information is clear-cut or has sharp boundaries. there is no uncertainty about the location of the set boundaries. the concept of fuzzy set theory is an extension of characteristic functions a neutrosophical model for optimal sustainable closed-loop supply chain network … 49 of crisp sets by enlarging the truth value set of ' grade of membership' from the twovalue set {0, 1} to the unit interval [0, 1] of infinite real numbers. many applications need fuzzy sets (mendel, 1995; ross, 2005; zimmermann, 2011; merigó, 2015; feng and chen, 2018; voskoglou, 2020; das et al., 2021; sorourkhah and edalatpanah, 2022). although, it has a shortcoming, i.e., it only addresses membership value and is unable to address the non-membership value. it is a fact that not all logical and actual models depend only on evident evaluations of participation and the true value of membership. there may emerge a state of affairs where the level of non-enrollment and grades of non-membership is additionally needed with membership degree. atanassov (1986) introduced the new idea of a new theory known as the intuitionistic fuzzy set theory that combined this strength of rejection with acceptance strength in new sets. molodtsov (1999) installed the conception of the theory of soft set that plays an important role in every field of mathematics. yager (2013) introduces the pythagorean fuzzy set. these ideas opened a new era towards generalizations of fuzzy sets and were applied in numerous applications (torra, 2010; wei, 2016; vellapandi and gunasekaran, 2020; eyoh, 2020; akram et al., 2021; rayappan and mohana, 2021). although these theories can deal with incomplete data in various real-world situations, they cannot deal with all sorts of uncertainty, such as indeterminate and inconsistent data. florentin smarandache's neutrosophic theory (smarandache, 1999) is a further expansion of the previously discussed fuzzy extension sets. the neutrosophic set (ns) can deal with uncertain, indeterminate, and discordant information, where the indeterminacy is explicitly quantified, and truth membership, indeterminacy membership, and falsity membership are all fully independent. after that, some types of nss have been proposed (lupiáñez, 2009; wang et al., 2010; ye, 2014; ulucay et al., 2018; liu and cheng, 2020; edalatpanah, 2020; luo al., 2022). furthermore, in neutrosophical literature, several applications such as facility location and routing (deveci et al., 2021), ssp (pamucar et al., 2020), social failure detection (torkayesh et al., 2022), linear programming (edalatpanah, 2019; das et al., 2021) and (kumar et al., 2021), time series models (pattanayak et al., 2022), dea ( mao et al., 2020), spp (kumar et al., 2019), mcdm (deli et al., 2021), etc. have been addressed. 2.2. clscn the design of a clscn for decisions such as the flow of materials and products, location of facilities, production, distribution and product recycling has attracted the attention of many researchers in the last decade. however, some studies have focused on maximizing profits in the clsc. others have focused on minimizing a clscn (nayeri et al., 2020). in addition, several researchers have focused on minimizing environmental impacts (talaei et al., 2016), minimizing the loss of working days (soleimani et al., 2017), maximizing social accountability (heydari, j., & rafiei, p., 2020), maximizing demand coverage (wang et al., 2015), maximizing economic value added (polo et al., 2019), maximizing net present value (amin et al., 2017; polo et al., 2019), maximizing recycling production and production quality (liu et al., 2019). most studies have examined network costs, namely location/relocation costs/facility allocation, operating cost, transportation cost, and other air pollution costs (fathollahi-fard et al., 2018). some studies have considered variables such as the number of raw materials required and unmet demand (farrokh et al., 2018). (yun et al., 2020) recently designed a stable clscn for mobile phones. they proposed a mom to maximize grid profit, minimize total carbon emissions, and maximize social impact to increase the stability of the proposed grid. they considered three types of distribution channels and proposed a hybrid ga to solve the model. their review proved the validity of the proposed model and network and showed the s. kalantari et al./decis. mak. appl. manag. eng. 5 (2) (2022) 46-77 50 effect of distribution channels on the stated goals. (ghahremani-nahr et al., 2021) developed a hybrid approach based on fdm and mp to assimilate supplier selection and sclsc design problems. the authors used the fuzzy method to solve the proposed model. their study showed the impact of suppliers on sustainable network design. also, they showed that by choosing a supplier, sustainable growth could improve net profits. in a recent study, (zahedi et al. 2021) designed a clscn related to the walnut industry and focused on the agricultural chain. they proposed a complex ilpm to reduce the total cost of the walnut industry and used metaheuristic and ha to solve the proposed model. their research showed that the proposed reverse flow network in the walnut industry, in addition to meeting market demands, prepares the returned product with usability and efficiency for reuse. (gholizadeh et al., 2021), in a study focused on designing a sclscn for the dairy industry to maximize profits and minimize environmental impacts in different scenarios, following the possible impact of optimistic scenarios and pessimistically focused on recycling dairy products with a case study in iran. they solved their complex model by presenting a two-objective milpm with an improved heuristic and ε-constraint combined method. then they showed their results with the efficiency of the proposed solution method and the effects of scenario probabilities. to demonstrate the importance of sustainable development (nayeri et al., 2020), they proposed a sclscn for the water reservoir and, at the same time, examined the optimization of financial, environmental and social impacts on the sustainable reservoir network. in addition, their proposed network was subject to uncertainties in the cost and demand parameters of transportation. they used strong fuzzy optimization and three different purposes of the multi-option ideal planning method with the utility function to solve this problem. this study showed the impact of sustainable development on scn design. since achieving competitive advantages over competitors in the market requires balancing the supply chain's social, environmental, and economic aspects, (pourmehdi et al., 2020) considered the design of a stable closed-loop sc in the steel industry under uncertain conditions. their research showed that reducing the profit of the proposed chain up to 1% reduces carbon emissions up to 5%. 2.3. the research gap after reviewing the recent literature on sclsc design, the main contribution of this article is summarized as follow:  to our knowledge, no studies have simultaneously considered the concepts of supplier selection, shipping modes, and carbon emission policies under uncertainty and indeterminacy.  one of the most important features of this study is the consideration of net present value and inflation in the design of a sscn under uncertainty, which makes this study special compared to the recent literature. 3. problem definition here's a clsc for computer mouse electronics. the proposed operational network consists of four levels at the forward operational levels (raw material suppliers, manufacturers, distributors, and customers). the operational process with the supply of materials, production and distribution and customer demand. reverse network operating levels include four levels (collection centers, separation centers, recycling centers, disposal centers) in which, in reverse flow, mainly electronic components a neutrosophical model for optimal sustainable closed-loop supply chain network … 51 related to computer mice, in separation centers to reusable components and equipment and recycling and is not usable, which is used in other centers. however, the price of these parts or materials is affected by inflation and supply and demand. therefore, the uncertainty of the parameters is inevitable. because the decision to separate, reuse, or recycle with a particular technology is useful (each technology has its opening costs, operating costs, and carbon emissions), present value analysis to control the cost and cost margin. useful for making decisions about separation, reuse or recycling decisions. in addition, the transfer of components of this product through reverse flow requires a comprehensive model of material transportation. a percentage of these components are transferred to recycling centers, a percentage to production centers for reuse and a percentage to disposal centers. the recycled materials are then sent to supply centers and the rest to disposal centers. on the other hand, capacity and trade policies have been used to limit the organization to carbon emissions. it restricts this policy to production, recycling, transportation and even landfill activities that lead to carbon emissions. this allows the organization to sell the number of unused carbon emissions in proportion to its designated carbon capacity. also, when the carbon emitted by the organization exceeds the designated carbon capacity, a carbon emission credit is purchased for supply chain activities. economic stability and reducing inflation can provide the basis for improving productive performance and significantly reducing environmental corruption. in the current situation, green gdp is calculated in some developed countries. of course, some countries do not announce this figure, but they inform the official. in this regard, in addition to direct and indirect tax policies have also been used; like the tax on carbon and the tax on energy rates, this tax is charged on the inputs of production or consumer goods, the use of which is detrimental to the environment and can be seen as a signal for achieving an efficient level of social emissions. environmental change is also important for most of these factories, and they demand the least conflict of interest with environmental officials. hence, we are prepared for economic tensions such as exchange rate instability, sudden changes in energy rates, etc. in this study, we used the limitation of point-to-point inflation calculation, which in addition to considering the benefits of the plant, can optimize the frequency of purchases, reduce transportation and, consequently, reduce carbon emissions. to model the proposed network, the following assumptions are considered:  all facilities face capacity constraints.  uncertainty in tactical parameters of demand, costs and carbon emission capacity.  the cost of purchasing technology and raw materials and the price of products are affected by inflation.  the location of the facility in the proposed network is predetermined.  issues of carbon emissions are addressed under carbon capacity policies and trade policy. 3.1. mathematical model in this section, indicators, sets, variables, and parameters are defined, then according to the problem ahead, and with the defined assumptions, the necessary model and constraint are written. s. kalantari et al./decis. mak. appl. manag. eng. 5 (2) (2022) 46-77 52 indicators 𝑆 a set of suppliers marked by 𝑠 index 𝑀 a set of manufacturers marked by 𝑚 index 𝐷 a set of distribution centers with index 𝑑 𝐾 a set of customers with index k 𝐶 collection centers with index c 𝐵 a set of separation centers with index b 𝑅 a set of recycling centers marked by index r 𝐼 set of raw materials marked by index i 𝐽 a set of disposal centers marked by index j 𝑇 a set of recycling technologies marked by index t 𝑄 a set of transport modes with index q 𝑃 a set of products with index p 𝐿 time set with index l 𝐺 a set of product components with index g 𝑒,𝑒′ a set of all levels of the chain 𝑒,𝑒′ ∈ {𝑠,𝑚,𝑑,𝑘,𝑐,𝑟,𝑏, 𝑗,𝑓} 𝑓,𝑓′ set of facilities at chain levels 𝑓,𝑓′ ∈ {1,…,𝐹𝑒} parameters 𝑓𝑐𝑠𝑠 fixed cost related to supplier s 𝑝𝑐𝑡𝑡𝑙 cost of purchasing technology t in the period l 𝑓𝑐𝑟𝑟𝑡 fixed cost of opening recycling centers r marked by technology t 𝑓𝑐𝑜𝑓𝑒 fixed cost of opening the facility (𝑓𝑒|𝑒 ∈ {𝑑,𝑐,𝑏}) 𝐶𝑎𝑝𝑓𝑒𝑙 facility capacity 𝑓𝑒|𝑒 ∈ {𝑠,𝑚,𝑑,𝑐,𝑏,𝑟, 𝑗} in time period l 𝐷𝑒𝑘𝑝𝑙 customer demand k for product p in time period l 𝑠𝑝𝑝𝑙 the sales price of product p in time period l 𝑝𝑟𝑖𝑠𝑙 price of raw materials i purchased from suppliers s in period l 𝑠𝑝𝑟𝑠𝑙 the sale price of recycled materials to suppliers s in period l 𝐶𝑜𝑓𝑒𝑙 operating costs in facilities 𝑓𝑒|𝑒 ∈ {𝑚,𝑑,𝑐,𝑏,𝑗} in time period l 𝐶𝑟𝑟𝑡𝑙 recycling cost of recycling center r marked by technology t in time period l 𝑇𝐶𝑒𝑒′𝑞𝑙 shipping cost from facility 𝑓𝑒 to facility 𝑓𝑒′ with shipping mode q، 𝑒,𝑒 ′ ∈ {𝑠,𝑚,𝑘,𝑏,𝑐,𝑟, 𝑗,𝑑} in time period 𝑙 𝑑𝑖𝑠𝑒𝑒′ distance between facilities 𝑓𝑒 and facilities 𝑓𝑒′( 𝑒,𝑒 ′ ∈ {𝑠,𝑚,𝑘,𝑏,𝑐,𝑟, 𝑗,𝑑}) 𝜎𝑖𝑝𝑙 consumption of raw materials i per unit of product p in time period l 𝛼𝑝𝑘𝑙 product return rate p from customer k in time period l 𝛽𝑔𝑏𝑙 number of reusable components g in the separation center b in time period l 𝜏𝑔𝑏𝑙 no, of recyclable components g separation center b in tp l 𝜇𝑔𝑏𝑙 no, of disposable components g in the separation center b in tp l 𝛿𝑔𝑟𝑡𝑙 the recycling rate of g components in recycling center r with t technology in tp l 𝜃 𝑔𝑟𝑡𝑙 the rate of waste of components g in the recycling center r with technology t in time period l 𝑇𝐶𝑎𝑝𝑒𝑒′𝑞 transport mode capacity 𝑞 between facility 𝑓𝑒and facility 𝑓𝑒′( 𝑒,𝑒 ′ ∈ {𝑠,𝑚,𝑘,𝑏,𝑐,𝑟, 𝑗,𝑑}) 𝜆𝑚 carbon emission rate from the manufacturer 𝑚 𝜆𝑟𝑡 carbon emission rate from recycling center r with t technology. 𝜆 𝑒𝑒′ 𝑞 carbon emission rate from transport from facility 𝑓𝑒to facility 𝑓𝑒′ with transport mode 𝑒,𝑒′ ∈ {𝑠,𝑚,𝑘,𝑏,𝑐,𝑟,𝑗,𝑑} 𝛿+ the purchase price per carbon credit unit 𝛿− the selling price of each carbon credit unit 𝐶𝑎𝑝𝐶𝐸 carbon capacity on carbon emissions on the planning horizon 𝜋 interest rate 𝑀𝑏𝑖𝑔 a large number above the demand limit 𝑆𝐼 the inflation rate variables 𝑄 𝑒𝑒′𝑝 𝑞𝑙 the value of products p is transferred from the facility 𝑓𝑒 to facility 𝑓𝑒′transport mode 𝑒,𝑒′ ∈ {𝑚,𝑑,𝑘,𝑐,𝑏} in time l 𝑄 𝑒𝑒′𝑔 𝑞𝑙 the value of the components g is transferred from the facility 𝑓𝑒 to the facility 𝑓𝑒′ with the transport mode 𝑒,𝑒′ ∈ {𝑏,𝑟,𝑚,𝑗} in time interval l 𝑄𝑖𝑠𝑚 𝑞𝑙 the amount of raw material i that is transferred from the supplier s to the manufacturer m with the transport mode q in period l 𝑄𝑖𝑟𝑡𝑗 𝑞𝑙 the amount of raw material i that is transferred from the recycling center r by technology t with the transport mode q to the disposal center j in time period l a neutrosophical model for optimal sustainable closed-loop supply chain network … 53 𝑄𝑖𝑟𝑡𝑠 𝑞𝑙 the amount of raw material i that is transferred from the recycling center r by technology t with transport mode q to the supplier s in period l 𝑒+ validity of carbon purchased 𝑒− carbon sales credit 𝑆𝑢𝑠 a binary variable, if supplier s is selected 1 otherwise, 0 𝐹𝑌𝑓𝑒 a binary variable, if the facility fe|e{ 𝑑,𝑐,𝑏 } is opened 1, otherwise 0 𝐹𝑌𝑟𝑡 a binary variable, if the recycling center r is opened with technology t 1, otherwise 0 𝑇𝑋𝑒𝑒′𝑞 a binary variable, if transport mode 𝑞 is used between facility 𝑓𝑒 to facility 𝑓𝑒′،, 1 otherwise, 0 3.1.1. economic objective this section examines and defines the first obj function of the problem ahead, namely the maximized npv of the proposed network shown in equation (1). 𝑀𝑎𝑥𝑍1 = (𝑇𝑜𝑡𝑎𝑙 𝑟𝑒𝑣𝑒𝑛𝑢𝑒𝑙 − 𝑇𝑜𝑡𝑎𝑙 𝑐𝑜𝑠𝑡𝑙) (1 + 𝜋)𝑙 (1) according to equation (2), total revenue is relative to the positive cash flow from the sale of manufactured products, carbon to customers and recycled materials to suppliers. the amount of product or carbon transferred to customers is multiplied by its price and inflation. total revenue = ∑∑∑∑∑(𝑠𝑝𝑝𝑙 + 𝑆𝐼.𝑠𝑝𝑝𝑙). 𝑙 𝑄𝑑𝑘𝑝 𝑞𝑙 𝑞𝑝𝑘𝑑 + ∑∑∑∑∑∑(𝑠𝑝𝑟𝑠𝑙 + 𝑆𝐼.𝑠𝑝𝑟𝑠𝑙). 𝑡𝑙 𝑄𝑖𝑟𝑡𝑠 𝑞𝑙 𝑞𝑠𝑟𝑖 + 𝛿−.𝑒−.𝑆𝐼 (2) according to equation three, the total cost has four sub-sections. the first part includes fixed costs related to reopening facilities (separation centers, collection centers, distribution centers and recycling centers with different technologies) and fixed costs of cooperation with suppliers. is. the second section shows the operating costs incurred in each facility. in the third section, transportation costs between facilities are discussed. the fourth section considers the cost of purchasing technology, raw materials and carbon credibility. s. kalantari et al./decis. mak. appl. manag. eng. 5 (2) (2022) 46-77 54 total cost = ∑ 𝑓𝑐𝑜𝑓𝑒.𝐹𝑌𝑓𝑒 𝑓𝑒∈{𝑑,𝑐,𝑏} + ∑∑𝑓𝑐𝑟𝑟𝑡.𝐹𝑌𝑟𝑡 𝑟𝑡 + ∑𝑓𝑐𝑠𝑠 𝑠 .𝑆𝑢𝑠 + ∑∑ ∑ ∑𝐶𝑜𝑓𝑒𝑙.𝑄𝑒𝑒′𝑝 𝑞𝑙 𝑙𝑓𝑒∈{𝑚,𝑑,𝑐}𝑝𝑞 + ∑∑ ∑ ∑𝐶𝑜𝑓𝑒𝑙.𝑄𝑒𝑒′𝑔 𝑞𝑙 𝑙𝑓𝑒∈{𝑏,𝑗}𝑞𝑔 + ∑∑∑∑∑∑𝐶𝑟𝑟𝑡𝑙.𝑄𝑖𝑟𝑡𝑗 𝑞𝑙 𝑙𝑖𝑞𝑡𝑗𝑟 + ∑∑∑∑∑∑𝐶𝑟𝑟𝑡𝑙.𝑄𝑖𝑟𝑡𝑠 𝑞𝑙 𝑙𝑖𝑞𝑡𝑠𝑟 + ∑∑ ∑ ∑𝑇𝐶𝑒𝑒′𝑞𝑙.𝑄𝑒𝑒′𝑝 𝑞𝑙 .𝑑𝑖𝑠𝑒𝑒′ 𝑙𝑓𝑒∈{𝑚,𝑑,𝑘,𝑐,𝑏}𝑝𝑞 + ∑∑ ∑ ∑𝑇𝐶𝑒𝑒′𝑞𝑙.𝑄𝑒𝑒′𝑔 𝑞𝑙 .𝑑𝑖𝑠𝑒𝑒′ 𝑙𝑓𝑒∈{𝑏,𝑟,𝑚,𝑗}𝑔𝑞 + ∑∑∑∑∑∑𝑇𝐶𝑟𝑗𝑞𝑙.𝑄𝑖𝑟𝑡𝑗 𝑞𝑙 .𝑑𝑖𝑠𝑟𝑗 𝑙𝑖𝑞𝑡𝑗𝑟 + ∑∑∑∑∑∑𝑇𝐶𝑟𝑠𝑞𝑙.𝑄𝑖𝑟𝑡𝑠 𝑞𝑙 .𝑑𝑖𝑠𝑟𝑠 𝑙𝑖𝑞𝑡𝑠𝑟 + ∑∑∑∑∑𝑇𝐶𝑠𝑚𝑞𝑙.𝑄𝑖𝑠𝑚 𝑞𝑙 .𝑑𝑖𝑠𝑠𝑚 𝑙𝑞𝑚𝑠𝑖 + ∑∑∑∑∑(𝑝𝑟𝑖𝑠𝑙.𝑆𝐼 + 𝑝𝑟𝑖𝑠𝑙).𝑄𝑖𝑠𝑚 𝑞𝑙 𝑙𝑞𝑚𝑠𝑖 + ∑∑∑∑∑∑(𝑝𝑐𝑡𝑡𝑙.𝑆𝐼 + 𝑝𝑐𝑡𝑡𝑙).𝑄𝑖𝑟𝑡𝑠 𝑞𝑙 𝑙𝑖𝑞𝑡𝑠𝑟 + 𝛿+.𝑒+.𝑆𝐼 (3) 3.1.2. environmental purpose this section, the second obj function of the problem is the equation (4), i.e., minimizing carbon emissions in the proposed clscn, which includes carbon emissions through transport between facilities, production operations and recycling. a neutrosophical model for optimal sustainable closed-loop supply chain network … 55 𝑚𝑖𝑛 𝑍2 = ∑∑∑∑∑𝜆𝑚.𝑄𝑚𝑑𝑝 𝑞𝑙 𝑙𝑞𝑝𝑑𝑚 + ∑∑∑∑∑∑𝜆𝑟𝑡.𝑄𝑖𝑟𝑡𝑠 𝑞𝑙 𝑙𝑖𝑞𝑡𝑠𝑟 + ∑∑∑∑∑∑𝜆𝑟𝑡.𝑄𝑖𝑟𝑡𝑗 𝑞𝑙 𝑙𝑖𝑞𝑡𝑗𝑟 + ∑∑∑ ∑ 𝜆 𝑒𝑒′ 𝑞 𝑓𝑒∈{𝑚,𝑑,𝑘,𝑐,𝑏} . 𝑙𝑞 𝑄 𝑒𝑒′𝑝 𝑞𝑙 .𝑑𝑖𝑠𝑒𝑒′ 𝑝 + ∑∑∑ ∑ 𝜆 𝑒𝑒′ 𝑞 𝑓𝑒∈{𝑏,𝑟,𝑚,𝑗} . 𝑙𝑔 𝑄 𝑒𝑒′𝑔 𝑞𝑙 .𝑑𝑖𝑠𝑒𝑒′ 𝑞 + ∑∑∑∑∑𝜆𝑠𝑚 𝑞 .𝑄𝑖𝑠𝑚 𝑞𝑙 .𝑑𝑖𝑠𝑠𝑚 𝑙𝑞𝑚𝑠𝑖 + ∑∑∑∑∑∑𝜆𝑟𝑗 𝑞 .𝑄𝑖𝑟𝑡𝑗 𝑞𝑙 .𝑑𝑖𝑠𝑟𝑗 𝑙𝑖𝑞𝑡𝑗𝑟 + ∑∑∑∑∑∑𝜆𝑟𝑠 𝑞 .𝑄𝑖𝑟𝑡𝑠 𝑞𝑙 .𝑑𝑖𝑠𝑟𝑠 𝑙𝑖𝑞𝑡𝑠𝑟 (4) 3.1.3. model constraints in this section, we discuss model constraints, including facility capacity constraints, flow balance constraints, facility location constraints, transport mode capacity constraints, carbon policy constraints, and inflation constraints. ∑∑𝑄𝑖𝑠𝑚 𝑞𝑙 𝑞𝑚 ≤ 𝐶𝑎𝑝𝑠𝑖𝑙.𝑆𝑢𝑠 ∀𝑖,𝑠, 𝑙 (5) ∑∑𝑄𝑚𝑑𝑝 𝑞𝑙 𝑞𝑑 ≤ 𝐶𝑎𝑝𝑚𝑝𝑙 ∀𝑚,𝑝,𝑙 (6) ∑∑𝑄𝑑𝑘𝑝 𝑞𝑙 𝑞𝑘 ≤ 𝐶𝑎𝑝𝑑𝑝𝑙.𝐹𝑌𝑑 ∀𝑑,𝑝, 𝑙 (7) ∑∑𝑄𝑘𝑐𝑝 𝑞𝑙 𝑞𝑘 + ∑∑𝑄𝑐𝑏𝑝 𝑞𝑙 𝑞𝑏 ≤ 𝐶𝑎𝑝𝑐𝑝𝑙.𝐹𝑌𝑐 ∀𝑐,𝑝, 𝑙 (8) ∑∑𝑄𝑏𝑚𝑔 𝑞𝑙 𝑞𝑚 + ∑∑𝑄𝑏𝑟𝑔 𝑞𝑙 𝑞𝑟 + ∑∑𝑄𝑏𝑗𝑔 𝑞𝑙 𝑞𝑗 ≤ 𝐶𝑎𝑝𝑏𝑔𝑙.𝐹𝑌𝑏 ∀𝑏,𝑔, 𝑙 (9) ∑∑𝑄𝑟𝑠𝑖𝑡 𝑞𝑙 𝑞𝑠 + ∑∑𝑄𝑟𝑗𝑖𝑡 𝑞𝑙 𝑗𝑞 ≤ 𝐹𝑌𝑟𝑡.𝐶𝑎𝑝𝑖𝑟𝑡𝑙 ∀𝑟,𝑖, 𝑡, 𝑙 (10) ∑𝐹𝑌𝑟𝑡 𝑡 ≤ 1 ∀𝑟 (11) as we can see, the constraints (5, 6) refer to the supplier and producer capacity. constraint (7) states that the amount of product sent to the distributor cannot exceed the capacity of the distribution center. constraint (8) refers to the capacity of collection centers. constraint (9) indicates the capacity of disassembly centers. constraint (10) deals with the capacity of recycling centers according to the type of technology. constraint (11) ensures that a recycling center opens with only one technology. ∑∑∑𝑄𝑠𝑚𝑖 𝑞𝑙 𝑞𝑚𝑠 = ∑∑∑𝜎𝑖𝑝𝑙.𝑄𝑚𝑑𝑝 𝑞𝑙 𝑞𝑑𝑚 ∀𝑖,𝑝, 𝑙 (12) s. kalantari et al./decis. mak. appl. manag. eng. 5 (2) (2022) 46-77 56 ∑∑𝑄𝑑𝑘𝑝 𝑞𝑙 . 𝑞 𝛼𝑝𝑘𝑙 𝑑 = ∑∑𝑄𝑘𝑐𝑝 𝑞𝑙 𝑞𝑐 ∀𝑝,𝑘, 𝑙 (13) ∑∑∑𝑄𝑚𝑑𝑝 𝑞𝑙 𝑞𝑑𝑚 = ∑∑∑𝑄𝑑𝑘𝑝 𝑞𝑙 𝑞𝑘𝑑 ∀𝑝,𝑙 (14) ∑∑∑𝑄𝑐𝑏𝑝 𝑞𝑙 𝑞𝑏𝑐 = ∑∑∑𝑄𝑏𝑚𝑔 𝑞𝑙 𝑞𝑚𝑏 + ∑∑∑𝑄𝑏𝑟𝑔 𝑞𝑙 𝑞𝑟𝑏 + ∑∑∑𝑄𝑏𝑗𝑔 𝑞𝑙 𝑞𝑗𝑏 ∀𝑝,𝑔, 𝑙 (15) ∑∑∑𝑄𝑏𝑚𝑔 𝑞𝑙 𝑞𝑚𝑏 = ∑∑∑𝑄𝑐𝑏𝑝 𝑞𝑙 𝑞𝑏𝑐 .𝛽𝑔𝑏𝑙 ∀𝑝,𝑔, 𝑙 (16) ∑∑∑𝑄𝑏𝑟𝑔 𝑞𝑙 𝑞𝑟𝑏 = ∑∑∑𝑄𝑐𝑏𝑝 𝑞𝑙 𝑞𝑏𝑐 . 𝜏𝑔𝑏𝑙 ∀𝑝,𝑔, 𝑙 (17) ∑∑∑𝑄𝑏𝑗𝑔 𝑞𝑙 𝑞𝑗𝑏 = ∑∑∑𝑄𝑐𝑏𝑝 𝑞𝑙 𝑞𝑏𝑐 .𝜇𝑔𝑏𝑙 ∀𝑝,𝑔, 𝑙 (18) ∑∑∑𝑄𝑟𝑠𝑖𝑡 𝑞𝑙 𝑞𝑠𝑟 = ∑∑∑𝑄𝑏𝑟𝑔 𝑞𝑙 𝑞𝑟𝑏 .𝛿𝑔𝑟𝑡𝑙 ∀𝑖,𝑔, 𝑙, 𝑡 (19) ∑∑∑𝑄𝑟𝑗𝑖𝑡 𝑞𝑙 𝑞𝑗𝑟 = ∑∑∑𝑄𝑏𝑟𝑔 𝑞𝑙 𝑞𝑟𝑏 .𝜃 𝑔𝑟𝑡𝑙 ∀𝑖,𝑔, 𝑙, 𝑡 (20) ∑∑𝑄𝑑𝑘𝑝 𝑞𝑙 𝑞𝑑 ≤ 𝐷𝑒𝑘𝑝𝑙 ∀𝑘,𝑝, 𝑙 (21) the flow constraints for the proposed model equations (12 to 21) are presented to achieve the goals. the constraint (12) shows the number of raw materials sent to the manufacturer, proportional to the number consumed in each product. constraint (13) refers to the amount of product returned from customers. constraints (14 to 20) show the flow balance between facilities, considering the set rate. constraint (21) refers to customer demand, i.e., the relationship between the amount of product delivery from the distributor to the customer relative to customer demand. ∑∑𝑄 𝑒𝑒′𝑝 𝑞𝑙 𝑙𝑝 ≤ 𝑇𝑐𝑎𝑝𝑒𝑒′𝑞.𝑇𝑋𝑒𝑒′𝑞 ∀𝑒,𝑒′ ∈ {𝑚,𝑑,𝑘,𝑐,𝑏} (22) ∑∑𝑄 𝑒𝑒′𝑔 𝑞𝑙 𝑙𝑔 ≤ 𝑇𝑐𝑎𝑝𝑒𝑒′𝑞.𝑇𝑋𝑒𝑒′𝑞 ∀𝑒,𝑒′ ∈ {𝑏,𝑟,𝑚,𝑗} (23) ∑∑𝑄𝑠𝑚𝑖 𝑞𝑙 𝑙 ≤ 𝑇𝑐𝑎𝑝𝑠𝑚𝑞.𝑇𝑋𝑠𝑚𝑞 𝑖 ∀𝑞,𝑠,𝑚 (24) ∑∑∑𝑄𝑟𝑗𝑖𝑡 𝑞𝑙 𝑙𝑡 ≤ 𝑇𝑐𝑎𝑝𝑟𝑗𝑞.𝑇𝑋𝑟𝑗𝑞 𝑖 ∀𝑟,𝑗,𝑞 (25) ∑∑∑𝑄𝑟𝑠𝑖𝑡 𝑞𝑙 𝑙𝑡 ≤ 𝑇𝑐𝑎𝑝𝑟𝑠𝑞.𝑇𝑋𝑟𝑠𝑞 𝑖 ∀𝑞,𝑟,𝑠 (26) as you can see, the constraints (22 to 26) indicate the capacity of material and product transport modes between facilities. constraint (22) refers to the forward current of the network, and constraints (23 to 26) refer to the reverse forward current (reverse current) of the proposed network. a neutrosophical model for optimal sustainable closed-loop supply chain network … 57 ∑∑∑∑∑𝜆𝑚.𝑄𝑚𝑑𝑝 𝑞𝑙 𝑙𝑞𝑝𝑑𝑚 + ∑∑∑∑∑∑𝜆𝑟𝑡.𝑄𝑖𝑟𝑡𝑠 𝑞𝑙 𝑙𝑖𝑞𝑡𝑠𝑟 + ∑∑∑∑∑∑𝜆𝑟𝑡.𝑄𝑖𝑟𝑡𝑗 𝑞𝑙 𝑙𝑖𝑞𝑡𝑗𝑟 + ∑∑∑ ∑ 𝜆 𝑒𝑒′ 𝑞 𝑓𝑒∈{𝑚,𝑑,𝑘,𝑐,𝑏} . 𝑙𝑞 𝑄 𝑒𝑒′𝑝 𝑞𝑙 .𝑑𝑖𝑠𝑒𝑒′ 𝑝 + ∑∑∑ ∑ 𝜆 𝑒𝑒′ 𝑞 𝑓𝑒∈{𝑏,𝑟,𝑚,𝑗} . 𝑙𝑔 𝑄 𝑒𝑒′𝑔 𝑞𝑙 .𝑑𝑖𝑠𝑒𝑒′ 𝑞 + ∑∑∑∑∑𝜆𝑠𝑚 𝑞 .𝑄𝑖𝑠𝑚 𝑞𝑙 .𝑑𝑖𝑠𝑠𝑚 𝑙𝑞𝑚𝑠𝑖 + ∑∑∑∑∑∑𝜆𝑟𝑗 𝑞 .𝑄𝑖𝑟𝑡𝑗 𝑞𝑙 .𝑑𝑖𝑠𝑟𝑗 𝑙𝑖𝑞𝑡𝑗𝑟 + ∑∑∑∑∑∑𝜆𝑟𝑠 𝑞 .𝑄𝑖𝑟𝑡𝑠 𝑞𝑙 .𝑑𝑖𝑠𝑟𝑠 𝑙𝑖𝑞𝑡𝑠𝑟 + 𝑒− ≤ 𝐶𝑎𝑝𝐶𝐸 + 𝑒+ (27) constraint (27) reflects the carbon policies adopted in this study, based on carbon capacity and trade policies. ∑∑(𝑠𝑝𝑝𝑙 + 𝑆𝐼.𝑠𝑝𝑝𝑙). 𝑞 𝑄𝑑𝑘𝑝 𝑞𝑙 𝑑 ≤ 𝐷𝑒𝑘𝑝𝑙 ∀𝑘,𝑝, 𝑙 (28) ∑∑∑∑(𝑝𝑟𝑖𝑠𝑙.𝑆𝐼 + 𝑝𝑟𝑖𝑠𝑙).𝑄𝑖𝑠𝑚 𝑞𝑙 𝑙𝑞𝑚𝑖 ≤ 𝑀𝑏𝑖𝑔.𝑆𝑢𝑠 ∀𝑠 (29) following the effect of inflation on the selling price of the product and the purchase of raw materials, we examine the effect of inflation on the constraints (28 and 29). 𝑇𝑋𝑒𝑒′𝑞 ≤ 𝐹𝑌𝑒" ∀𝑒,𝑒 ′ ∈ {𝑚,𝑑,𝑘,𝑐,𝑏} ∀𝑒" ∈ {𝑑,𝑐,𝑏} (30) 𝑇𝑋𝑟𝑠𝑞 ≤ ∑𝐹𝑌𝑟𝑡 𝑡 ∀𝑟,𝑠,𝑞 (31) 𝑇𝑋𝑟𝑗𝑞 ≤ ∑𝐹𝑌𝑟𝑡 𝑡 ∀𝑗,𝑟,𝑞 (32) 𝑄 𝑒𝑒′𝑝 𝑞𝑙 , 𝑄 𝑒𝑒′𝑔 𝑞𝑙 , 𝑄𝑖𝑠𝑚 𝑞𝑙 , 𝑄𝑖𝑟𝑠𝑡 𝑞𝑙 , 𝑄𝑖𝑟𝑗𝑡 𝑞𝑙 , 𝑒+, 𝑒− ≥ 0 (33) 𝑆𝑢𝑠, 𝐹𝑌𝑓𝑒, 𝑇𝑋𝑒𝑒′𝑞, 𝐹𝑌𝑟𝑡 ∈ {0,1} (34) constraints (30-32) show the relationship between transport of facilities and the reopening of facilities, i.e., the flow of transport occurs when the facility is opened. finally, the range of decision variables of the proposed model is shown in constraints (33 and 34). 4. uncertainty of the model given the fluctuations in the business environment, such as demand and operational and tactical costs, the nature of uncertainty in the design of a stable clscn is undeniable. on the other hand, different types of uncertainties are divided into epistemological, random and deep uncertainties based on access to data. in this study, epistemological and random uncertainties are used. solid optimization is used to deal with this type of uncertainty. using studies (talaei et al., 2016), this study offers an efficient approach based on a robust fuzzy planning approach that allows decision s. kalantari et al./decis. mak. appl. manag. eng. 5 (2) (2022) 46-77 58 makers to control conservatism to satisfy constraints. the approach used in this study is a broad form of the chance-limited fuzzy planning model, which is implemented on the proposed model as follows, which is taken from the study (talaei et al., 2016). total cost = ∑ 𝑓𝑐𝑜𝑓𝑒.𝐹𝑌𝑓𝑒 𝑓𝑒∈{𝑑,𝑐,𝑏} + ∑∑𝑓𝑐𝑟𝑟𝑡.𝐹𝑌𝑟𝑡 𝑟𝑡 + ∑𝑓𝑐𝑠𝑠 𝑠 .𝑆𝑢𝑠 + ∑∑ ∑ ∑[ 𝐶𝑜𝑓𝑒𝑙(1) + 𝐶𝑜𝑓𝑒𝑙(2) + 𝐶𝑜𝑓𝑒𝑙(3) + 𝐶𝑜𝑓𝑒𝑙(4) 4 ].𝑄 𝑒𝑒′𝑝 𝑞𝑙 𝑙𝑓𝑒∈{𝑚,𝑑,𝑐}𝑝𝑞 + ∑∑ ∑ ∑[ 𝐶𝑜𝑓𝑒𝑙(1) + 𝐶𝑜𝑓𝑒𝑙(2) + 𝐶𝑜𝑓𝑒𝑙(3) + 𝐶𝑜𝑓𝑒𝑙(4) 4 ].𝑄 𝑒𝑒′𝑔 𝑞𝑙 𝑙𝑓𝑒∈{𝑏,𝑗}𝑞𝑔 + ∑∑∑∑∑∑[ 𝐶𝑟𝑟𝑡𝑙(1) + 𝐶𝑟𝑟𝑡𝑙(2) + 𝐶𝑟𝑟𝑡𝑙(3) + 𝐶𝑟𝑟𝑡𝑙(4) 4 ].𝑄𝑖𝑟𝑡𝑗 𝑞𝑙 𝑙𝑖𝑞𝑡𝑗𝑟 + ∑∑∑∑∑∑[ 𝐶𝑟𝑟𝑡𝑙(1) + 𝐶𝑟𝑟𝑡𝑙(2) + 𝐶𝑟𝑟𝑡𝑙(3) + 𝐶𝑟𝑟𝑡𝑙(4) 4 ].𝑄𝑖𝑟𝑡𝑠 𝑞𝑙 𝑙𝑖𝑞𝑡𝑠𝑟 + ∑∑ ∑ ∑[ 𝑇𝐶𝑒𝑒′𝑞𝑙(1) + 𝑇𝐶𝑒𝑒′𝑞𝑙(2) + 𝑇𝐶𝑒𝑒′𝑞𝑙(3) + 𝑇𝐶𝑒𝑒′𝑞𝑙(4) 4 ].𝑄 𝑒𝑒′𝑝 𝑞𝑙 .𝑑𝑖𝑠𝑒𝑒′ 𝑙𝑓𝑒∈{𝑚,𝑑,𝑘,𝑐,𝑏}𝑝𝑞 + ∑∑ ∑ ∑[ 𝑇𝐶𝑒𝑒′𝑞𝑙(1) + 𝑇𝐶𝑒𝑒′𝑞𝑙(2) + 𝑇𝐶𝑒𝑒′𝑞𝑙(3) + 𝑇𝐶𝑒𝑒′𝑞𝑙(4) 4 ].𝑄 𝑒𝑒′𝑔 𝑞𝑙 .𝑑𝑖𝑠𝑒𝑒′ 𝑙𝑓𝑒∈{𝑏,𝑟,𝑚,𝑗}𝑔𝑞 + ∑∑∑∑∑∑[ 𝑇𝐶𝑟𝑗𝑞𝑙(1) + 𝑇𝐶𝑟𝑗𝑞𝑙(2) + 𝑇𝐶𝑟𝑗𝑞𝑙(3) + 𝑇𝐶𝑟𝑗𝑞𝑙(4) 4 ].𝑄𝑖𝑟𝑡𝑗 𝑞𝑙 .𝑑𝑖𝑠𝑟𝑗 𝑙𝑖𝑞𝑡𝑗𝑟 + ∑∑∑∑∑∑[ 𝑇𝐶𝑟𝑠𝑞𝑙(1) + 𝑇𝐶𝑟𝑠𝑞𝑙(2) + 𝑇𝐶𝑟𝑠𝑞𝑙(3) + 𝑇𝐶𝑟𝑠𝑞𝑙(4) 4 ].𝑄𝑖𝑟𝑡𝑠 𝑞𝑙 .𝑑𝑖𝑠𝑟𝑠 𝑙𝑖𝑞𝑡𝑠𝑟 + ∑∑∑∑∑[ 𝑇𝐶𝑠𝑚𝑞𝑙(1) + 𝑇𝐶𝑠𝑚𝑞𝑙(2) + 𝑇𝐶𝑠𝑚𝑞𝑙(3) + 𝑇𝐶𝑠𝑚𝑞𝑙(4) 4 ].𝑄𝑖𝑠𝑚 𝑞𝑙 .𝑑𝑖𝑠𝑠𝑚 𝑙𝑞𝑚𝑠𝑖 + ∑∑∑∑∑(𝑝𝑟𝑖𝑠𝑙.𝑆𝐼 + 𝑝𝑟𝑖𝑠𝑙).𝑄𝑖𝑠𝑚 𝑞𝑙 𝑙𝑞𝑚𝑠𝑖 + ∑∑∑∑∑∑(𝑝𝑐𝑡𝑡𝑙.𝑆𝐼 + 𝑝𝑐𝑡𝑡𝑙).𝑄𝑖𝑟𝑡𝑠 𝑞𝑙 𝑙𝑖𝑞𝑡𝑠𝑟 + 𝛿+.𝑒+.𝑆𝐼 (35) s.t: ∑∑𝑄𝑑𝑘𝑝 𝑞𝑙 𝑞𝑑 ≤ (1 − 𝛼1).𝐷𝑒𝑘𝑝𝑙(2) + 𝛼1.𝐷𝑒𝑘𝑝𝑙(1) ∀𝑘,𝑝, 𝑙 (36) a neutrosophical model for optimal sustainable closed-loop supply chain network … 59 ∑∑∑∑∑𝜆𝑚.𝑄𝑚𝑑𝑝 𝑞𝑙 𝑙𝑞𝑝𝑑𝑚 + ∑∑∑∑∑∑𝜆𝑟𝑡.𝑄𝑖𝑟𝑡𝑠 𝑞𝑙 𝑙𝑖𝑞𝑡𝑠𝑟 + ∑∑∑∑∑∑𝜆𝑟𝑡.𝑄𝑖𝑟𝑡𝑗 𝑞𝑙 𝑙𝑖𝑞𝑡𝑗𝑟 + ∑∑∑ ∑ 𝜆 𝑒𝑒′ 𝑞 𝑓𝑒∈{𝑚,𝑑,𝑘,𝑐,𝑏} . 𝑙𝑞 𝑄 𝑒𝑒′𝑝 𝑞𝑙 .𝑑𝑖𝑠𝑒𝑒′ 𝑝 + ∑∑∑ ∑ 𝜆 𝑒𝑒′ 𝑞 𝑓𝑒∈{𝑏,𝑟,𝑚,𝑗} . 𝑙𝑔 𝑄 𝑒𝑒′𝑔 𝑞𝑙 .𝑑𝑖𝑠𝑒𝑒′ 𝑞 + ∑∑∑∑∑𝜆𝑠𝑚 𝑞 .𝑄𝑖𝑠𝑚 𝑞𝑙 .𝑑𝑖𝑠𝑠𝑚 𝑙𝑞𝑚𝑠𝑖 + ∑∑∑∑∑∑𝜆𝑟𝑗 𝑞 .𝑄𝑖𝑟𝑡𝑗 𝑞𝑙 .𝑑𝑖𝑠𝑟𝑗 𝑙𝑖𝑞𝑡𝑗𝑟 + ∑∑∑∑∑∑𝜆𝑟𝑠 𝑞 .𝑄𝑖𝑟𝑡𝑠 𝑞𝑙 .𝑑𝑖𝑠𝑟𝑠 𝑙𝑖𝑞𝑡𝑠𝑟 + 𝑒−(1 − 𝛼 ).𝐶𝑎𝑝𝐶𝐸(2) + 𝛼2.𝐶𝑎𝑝𝐶𝐸(1) + 𝑒 + (37) ∑∑(𝑠𝑝𝑝𝑙 + 𝑆𝐼.𝑠𝑝𝑝𝑙). 𝑞 𝑄𝑑𝑘𝑝 𝑞𝑙 𝑑 ≤ (1 − 𝛼3).𝐷𝑒𝑘𝑝𝑙(2) + 𝛼3.𝐷𝑒𝑘𝑝𝑙(1) ∀𝑘,𝑝, 𝑙 (38) 0.5 ≤ 𝛼i ≤ 1, i = 1,2,3 (39) set of of constraints (5 -21)-(22-26), (29)-(30 34) according to the study by talaei et al. (2016), the uncertainty model is formulated as follows: 𝑚𝑎𝑥 𝐸[𝑍1] + 𝜂(𝑍𝑚𝑎𝑥 − 𝐸[𝑍1]) + 𝜉1.(∑∑∑(1 − 𝛼1).𝐷𝑒𝑘𝑝𝑙(2) + 𝛼1.𝐷𝑒𝑘𝑝𝑙(1) 𝑙𝑝𝑘 − 𝐷𝑒𝑘𝑝𝑙(1)) + 𝜉2.((1 − 𝛼2).𝐶𝑎𝑝𝐶𝐸(2) + 𝛼2.𝐶𝑎𝑝𝐶𝐸(1) − 𝐶𝑎𝑝𝐶𝐸(1)) + 𝜉3.(∑∑∑(1 − 𝛼3).𝐷𝑒𝑘𝑝𝑙(2) + 𝛼3.𝐷𝑒𝑘𝑝𝑙(1) 𝑙𝑝𝑘 − 𝐷𝑒𝑘𝑝𝑙(1)) (40) set of constraints (36)-(40) where 𝐸[𝑍1] is the expected value of the first obj function, η and ξ represent coefficients that regulate the optimal strength and feasibility of the solution vector, respectively 𝑍𝑚𝑎𝑥 also defines: s. kalantari et al./decis. mak. appl. manag. eng. 5 (2) (2022) 46-77 60 𝑍𝑚𝑎𝑥 = ∑ 𝑓𝑐𝑜𝑓𝑒.𝐹𝑌𝑓𝑒 𝑓𝑒∈{𝑑,𝑐,𝑏} + ∑∑𝑓𝑐𝑟𝑟𝑡.𝐹𝑌𝑟𝑡 𝑟𝑡 + ∑𝑓𝑐𝑠𝑠 𝑠 .𝑆𝑢𝑠 + ∑∑ ∑ ∑𝐶𝑜𝑓𝑒𝑙(4).𝑄𝑒𝑒′𝑝 𝑞𝑙 𝑙𝑓𝑒∈{𝑚,𝑑,𝑐}𝑝𝑞 + ∑∑ ∑ ∑𝐶𝑜𝑓𝑒𝑙(4).𝑄𝑒𝑒′𝑔 𝑞𝑙 𝑙𝑓𝑒∈{𝑏,𝑗}𝑞𝑔 + ∑∑∑∑∑∑𝐶𝑟𝑟𝑡𝑙(4).𝑄𝑖𝑟𝑡𝑗 𝑞𝑙 𝑙𝑖𝑞𝑡𝑗𝑟 + ∑∑∑∑∑∑𝐶𝑟𝑟𝑡𝑙(4).𝑄𝑖𝑟𝑡𝑠 𝑞𝑙 𝑙𝑖𝑞𝑡𝑠𝑟 + ∑∑ ∑ ∑𝑇𝐶𝑒𝑒′𝑞𝑙(4).𝑄𝑒𝑒′𝑝 𝑞𝑙 .𝑑𝑖𝑠𝑒𝑒′ 𝑙𝑓𝑒∈{𝑚,𝑑,𝑘,𝑐,𝑏}𝑝𝑞 + ∑∑ ∑ ∑𝑇𝐶𝑒𝑒′𝑞𝑙(4).𝑄𝑒𝑒′𝑔 𝑞𝑙 .𝑑𝑖𝑠𝑒𝑒′ 𝑙𝑓𝑒∈{𝑏,𝑟,𝑚,𝑗}𝑔𝑞 + ∑∑∑∑∑∑𝑇𝐶𝑟𝑗𝑞𝑙(4).𝑄𝑖𝑟𝑡𝑗 𝑞𝑙 .𝑑𝑖𝑠𝑟𝑗 𝑙𝑖𝑞𝑡𝑗𝑟 + ∑∑∑∑∑∑𝑇𝐶𝑟𝑠𝑞𝑙(4).𝑄𝑖𝑟𝑡𝑠 𝑞𝑙 .𝑑𝑖𝑠𝑟𝑠 𝑙𝑖𝑞𝑡𝑠𝑟 + ∑∑∑∑∑𝑇𝐶𝑠𝑚𝑞𝑙(4).𝑄𝑖𝑠𝑚 𝑞𝑙 .𝑑𝑖𝑠𝑠𝑚 𝑙𝑞𝑚𝑠𝑖 + ∑∑∑∑∑(𝑝𝑟𝑖𝑠𝑙.𝑆𝐼 + 𝑝𝑟𝑖𝑠𝑙).𝑄𝑖𝑠𝑚 𝑞𝑙 𝑙𝑞𝑚𝑠𝑖 + ∑∑∑∑∑∑(𝑝𝑐𝑡𝑡𝑙.𝑆𝐼 + 𝑝𝑐𝑡𝑡𝑙).𝑄𝑖𝑟𝑡𝑠 𝑞𝑙 𝑙𝑖𝑞𝑡𝑠𝑟 + 𝛿+.𝑒+.𝑆𝐼 (41) 5. proposed approach: neutrosophic model since our model is a multi-objective problem, we establish a neutrosophical strategy to solve it. the most prevalent mathematical model with competing goals is the multi-objective model (mom). the goal in such instances is to obtain the optimal value of all conflicting objective functions concurrently. in such situations, the decision-maker conveys the importance of their preferences by giving each objective function an ideal weight between zero and one. as a result, decision-maker preferences in an objective function with a high weight value are higher in that objective function. recently, living circumstances have been noticed to have neutral thoughts regarding an element in the information. thoughts concerning the elements that are neutral or indeterminate fall somewhere between falsehood and truth. as a result, smarandache (1999) developed the neutrosophic logic, which consists of three membership sets: truth (membership degree), indeterminacy, and falsehood (nonmembership degree). the model of a sclscn with inflation and carbon emission policies with some conflicting objective functions is addressed in this section using the concept of neutrosophic programming. as a result, each obj function performs three tasks: falsehood function (f), truth membership function (t), and indeterminacy function (i). as a result, the neutrosophic programming method plays a significant and relible role in mom by considering neutral thoughts. a neutrosophical model for optimal sustainable closed-loop supply chain network … 61 consider a mom in which (nd) represents a set of neutrosophic decisions, (nof) a set of neutrosophic obj functions, and neutrosophic constraints (nc). therefore, the set of neutrosophic decisions is represented as: (42) 𝑁𝐷 = (⋂𝑁𝑂𝐹𝑖 𝑚 𝑖=1 )(⋂𝑁𝐶𝑗 𝑛 𝑗=1 ) = (𝑥,𝑇𝑁𝐷(𝑥),𝐼𝑁𝐷(𝑥),𝐹𝑁𝐷(𝑥)) 𝑠.𝑡.:   1 2 1 2 ( ) min , , , ; , , , , m nnd nof nof nof nc nc nc t x t t t t t t   1 2 1 2 ( ) max , , , ; , , , , m nnd nof nof nof nc nc nc i x i i i i i i   1 2 1 2 ( ) max , , , ; , , , . m nnd nof nof nof nc nc nc f x f f f f f f where, m and n represent the number of objective functions and constraints, respectively. also, 𝑇𝑁𝐷(𝑥),𝐼𝑁𝐷(𝑥), and 𝐹𝑁𝐷(𝑥) are the truth membership, indeterminacy, and falsehood membership functions, respectively. the upper and lower ranges for each obj function are generated by the solution of each single goal under the provided set of constraints and are denoted as 𝑈𝑖 and 𝐿𝑖 with a set of decision variables 𝑋, respectively. (43) 𝑈𝑖 = 𝑚𝑎𝑥(𝑍𝑖(𝑋)), ∀𝑖 = 1,2,…,𝑚 𝐿𝑖 = 𝑚𝑖𝑛(𝑍𝑖(𝑋)), ∀𝑖 = 1,2,…,𝑚 under the neutrosophic situation, for truth, non-determination, and falsehood membership functions, the upper and lower bounds for m objective function can be obtained as follows: (44) 𝑈𝑖 𝑇 = 𝑈𝑖, 𝐿𝑖 𝑇 = 𝐿𝑖 𝑈𝑖 𝐼 = 𝐿𝑖 𝑇 + 𝑝𝑖, 𝐿𝑖 𝐼 = 𝐿𝑖 𝑈𝑖 𝐹 = 𝑈𝑖 𝑇, 𝐿𝑖 𝐹 = 𝐿𝑖 𝑇 + 𝑞𝑖 wherein the relationship mentioned above, 𝑝𝑖 and 𝑞𝑖 are predetermined values between 0 and 1. so, based on the preceding components, the linear membership function for a neutrosophic context is as follows: (45) 𝑁𝐷 = (⋂𝑁𝑂𝐹𝑖 𝑚 𝑖=1 )(⋂𝑁𝐶𝑗 𝑛 𝑗=1 ) = (𝑥,𝑇𝑁𝐷(𝑥),𝐼𝑁𝐷(𝑥),𝐹𝑁𝐷(𝑥)) 𝑠.𝑡.: 𝑇𝑖(𝑍𝑖(𝑋)) = { 1 𝑖𝑓 𝑍𝑖 (𝑋) < 𝐿𝑖 𝑇 𝑈𝑖 𝑇 − 𝑍𝑖(𝑋) 𝑈𝑖 𝑇 − 𝐿𝑖 𝑇 𝑖𝑓 𝐿𝑖 𝑇 ≤ 𝑍𝑖(𝑋) ≤ 𝑈𝑖 𝑇 0 𝑖𝑓 𝑍𝑖(𝑋) > 𝑈𝑖 𝑇 } s. kalantari et al./decis. mak. appl. manag. eng. 5 (2) (2022) 46-77 62 𝐼𝑖(𝑍𝑖(𝑋)) = { 1 𝑖𝑓 𝑍𝑜 (𝑋) < 𝐿𝑖 𝐼 𝑈𝑖 𝐼 − 𝑍𝑖(𝑋) 𝑈𝑖 𝐼 − 𝐿𝑖 𝐼 𝑖𝑓 𝐿𝑖 𝐼 ≤ 𝑍𝑖(𝑋) ≤ 𝑈𝑖 𝐼 0 𝑖𝑓 𝑍𝑖(𝑋) > 𝑈𝑖 𝐼 } 𝐹𝑖(𝑍𝑖(𝑋)) = { 1 𝑖𝑓 𝑍𝑖 (𝑋) > 𝑈𝑖 𝐹 𝑍𝑖(𝑋) − 𝐿𝑖 𝐹 𝑈𝑖 𝐹 − 𝐿𝑖 𝐹 𝑖𝑓 𝐿𝑖 𝐹 ≤ 𝑍𝑖(𝑋) ≤ 𝑈𝑖 𝐹 0 𝑖𝑓 𝑍𝑖(𝑋) < 𝐿𝑖 𝐹 } fundamentally, the goal of establishing multiple accomplishment functions is to reach the highest degree or level of pleasure based on the decision maker's preferences. as a result, we have conveniently specified the individual completion factors for each membership function, such as maximization of truth membership, maximization of indeterminacy degree, and minimization of a falsity degree. therefore, the controlled neutrosophic mathematical programming paradigm using linear truth, indeterminacy, and falsity membership functions in neutrosophic surroundings can be expressed as follows: (46) 𝑚𝑎𝑥 ∑(𝜏𝑖 + 𝜄𝑖 − 𝜉𝑖) 𝑚 𝑖=1 𝑠.𝑡. ∶ 𝑇𝑖(𝑍𝑖(𝑋)) ≥ 𝜏𝑖, ∀𝑖 𝐼𝑖(𝑍𝑖(𝑋)) ≥ 𝜄𝑖, ∀𝑖 𝐹𝑖(𝑍𝑖(𝑋)) ≤ 𝜉𝑖, ∀𝑖 𝜏𝑖 ≥ 𝜄𝑖, ∀𝑖 𝜏𝑖 ≥ 𝜉𝑖, ∀𝑖 0 ≤ 𝜏𝑖 + 𝜄𝑖 + 𝜉𝑖 ≤ 3, ∀𝑖 𝜏𝑖, 𝜄𝑖,𝜉𝑖 ∈ (0,1) 𝐸𝑞𝑠(43 − 44) where𝜏𝑖,𝜄𝑖, and 𝜉𝑖 are the truth, indeterminacy, and falsity membership functions' auxiliary accomplishment variables, respectively. therefore, this neutrosophical approach is a well-suited contemporary optimization technique preferred solely by others due to its degree of independent indeterminacy. moreover, for solving the model efficiently, we presented an innovative strategy to the proposed model called the hybrid method. since the computation time for each model of mil /nlp increases with increasing variables and the presence of data, no acceptable solution is obtained even in some cases. hence, an exploratory method based on the relaxation of a binary variable is proposed. at the first, we consider the binary variable greater than zero and solve the optimization model with only non-zero binary variables. then, as a new constraint, we add a mi to the model and solve the optimization model again. the main advantages of this method are that it leads to a drastic reduction of problem-solving time and can also achieve higher quality solutions. the steps of the hybrid approach: step (1): release the zero and one constraint by converting the proposed binary variables to a continuous positive variable. solve the released model. step (2): then hold the binary variable as a continuous variable and use it in the new model and solve it. a neutrosophical model for optimal sustainable closed-loop supply chain network … 63 step (3): report or record all non-zero values for the released variable step (4): set any non-zero values of the released variables to 1 and place them in the original mixed linear programming model. and solve the model again. step (5): finally, report the decision variables. 6. numerical experiments one of the electronic devices we are constantly dealing with is the computer and laptop that we use to improve our work. a set of computer and laptop components such as keyboards, mice, and cables break down after a while, called computer junk. in general, there is a plastic material in computer accessories and even the computer monitor itself that can be reused by removing all computer parts and recycling computer waste and plastic, silicon, iron, lead, and even. it is interesting to know that precious metals such as gold, silver, and palladium in computer parts can be recycled to extract gold, silver, and other metals such as copper. this article considers one of the international companies in iran as its case study, located in south tehran province, the center of iran. the study company is a private iranian company that designs, manufactures, and markets computers, electronics, and information technology equipment. the company's products are frames, mini bags, tablets, speakers, mice and keyboards, headphones and headsets, power banks, and even products such as digital receivers. this study focuses specifically on one of the company's products, the computer mouse, and designs and examines a stable clsc for this product. the data needed to solve the proposed model, such as problem size, number of transport modes with capacity, and carbon emissions, are described below. it should be noted that it was not possible to share information about it due to company policies. therefore, according to the performance of company’s data, we use random data based on the uniform distribution function to implement the proposed model. it should be noted that this distribution is based on real data of the company. table 1 shows the production of real data based on the data behavior of the company. furthermore, the proposed chain for the company has (potential suppliers (s = 5), manufacturer (m = 1), potential distribution centers (d = 5), customers (k = 15), potential collection centers (c = 5)) the probable centers of separation are (b = 4), recycling centers (r = 2), disposal centers (j = 1)). the company uses four modes of transport for its chain, including (nissan with capacity (kg) = 2000 and with carbon emission coefficient (kg / km) = 0.031, light truck with capacity (kg) = 30.000 and with carbon emission coefficient (kg / km) = 0.048, medium truck with capacity (kg) = 60,000 and with carbon emission coefficient (kg / km) = 0.0252 and heavy trucks with capacity (kg) = 100000 and with carbon emission coefficient (kg) / km) = 0.297. s. kalantari et al./decis. mak. appl. manag. eng. 5 (2) (2022) 46-77 64 table 1. production of problem data based on the actual behavior of company data period parameters fuzzy parameter (𝜃 = 𝜃(1),𝜃(2),𝜃(3),𝜃(4)) 𝜃(4) 𝜃(3) 𝜃(2) 𝜃(1) du[650 800] du[450 600] du[250 400] du[100 250] 𝐷𝑒𝑘𝑝𝑙 du[0.265 0.270] du[0.260 0.265] du[0.255 0.260] du[0.250 0.255] 𝐶𝑜𝑓𝑒𝑙($*10 3) du[0.240 0.245] du[0.235 0.240] du[0.230 0.235] du[0.225 0.230] 𝐶𝑟𝑟𝑡𝑙($*10 3) du [0.95 0.100] du [0.85 0.90] du [0.80 0.85] du [0.75 0.80] nissan 𝑇𝐶𝑒𝑒′𝑞𝑙($*10 3) du [0.135 0.140] du [0.125 0.130] du [0.120 0.125] du [0.115 0.120] light truck du [0.160 0.165] du [0.150 0.155] du [0.135 0.145] du [0.125 0.135] medium truck du [0.165 0.170] du [0.155 0.160] du [0.145 0.150] du [0.140 0.145] heavy truck du [60 65] du [50 60] du [40 50] du [30 40] 𝐶𝑎𝑝𝐶𝐸 (ton) du[0.00714 0.0142] 𝑓𝑐𝑠𝑠($*10 3) du[9.55 11.32] 𝑝𝑐𝑡𝑡𝑙($*10 3) du[19.2 22.1] 𝑓𝑐𝑟𝑟𝑡($*10 3) du[15.5 18.6] 𝑓𝑐𝑜𝑓𝑒($*10 3) du[3 15] 𝑠𝑝𝑝𝑙($*10 3) du[3 15] 𝑝𝑟𝑖𝑠𝑙($*10 3) du[3 15] 𝑠𝑝𝑟𝑠𝑙($*10 3) du[3 15] 𝑑𝑖𝑠𝑒𝑒′(km) 0.65 𝜎𝑖𝑝𝑙 0.19 𝛼𝑝𝑘𝑙 0.4 𝛽𝑔𝑏𝑙 0.4 𝜏𝑔𝑏𝑙 0.2 𝜇𝑔𝑏𝑙 0.6 𝛿𝑔𝑟𝑡𝑙 0.4 𝜃𝑔𝑟𝑡𝑙 du [0.185 0.225] 𝜆𝑚(kg / km) du [0.234 0.258] 𝜆𝑟𝑡(kg / km) du [0.0120 0.0125] 𝛿+($*103) du [0.0125 0.0130] 𝛿−($*103) this study is based on a hypothesis for two types of technology used in recycling centers. the first type of technology has low cost but high carbon emissions, and the second type of technology has a high cost but low carbon emissions. this assumption is based on the opinions of company experts. a neutrosophical model for optimal sustainable closed-loop supply chain network … 65 6.1. validation of the model in this section, the validity of the proposed model is first examined. for this purpose, 20 experimental problems for the proposed model are solved by the neutronsophical model (46) and the proposed hybrid method, and the outcomes are shown in table 2. to display the efficiency of the proposed processes, two essential factors of model solution time and the optimal gap based on equation (47) are obtained. 𝐻𝑦𝑏𝑟𝑖𝑑𝑠𝑜𝑙 − 𝑀𝐶𝐺𝑃 − uf𝑠𝑜𝑙 𝑀𝐶𝐺𝑃 − uf𝑠𝑜𝑙 × 100 (47) according to table 2, it can be seen that the results of 20 test samples show the validity of the proposed models. on the other hand, to show the optimal gap between the solutions obtained from the two methods according to the results of table 3 and figure 1, we have shown that the average percentage of the optimal gap in the acceptable range is less than 5% and this proves the validity of the proposed methods. in figure 2, a comparison of the model solution time between two models shows that the hybrid method has reduced the solution time by 25%. table 2. tests performed on the case study (𝐺) (𝑆) (𝑀) (𝐷) (𝐾) (𝐶) (𝐵) (𝑅) (𝐼) (𝐽) (𝑄) (𝑃) (𝐿) test 12 2 1 4 1 3 1 2 2 3 1 1 2 1 12 2 1 4 1 3 1 2 2 3 1 1 2 2 12 3 1 4 1 3 2 2 2 5 1 1 3 3 12 3 1 4 2 5 2 2 3 5 2 2 3 4 12 4 1 4 2 5 2 3 3 8 2 2 3 5 12 4 2 4 2 5 3 3 4 8 2 2 4 6 12 5 2 4 3 7 3 3 4 12 3 3 4 7 12 5 2 4 3 7 3 4 4 12 3 3 4 8 12 6 2 4 3 7 2 4 5 15 3 3 4 9 12 6 2 4 4 7 2 4 5 15 4 3 5 10 12 7 2 4 4 7 4 10 4 17 4 4 5 11 12 7 3 4 4 7 4 10 4 17 5 4 5 12 12 8 3 4 5 8 6 10 5 17 5 4 5 13 12 8 3 4 5 8 6 14 5 20 5 5 5 14 12 9 3 4 5 9 8 14 7 20 6 5 6 15 12 9 3 4 6 9 8 16 7 20 6 5 6 16 12 10 4 4 6 10 8 16 9 20 8 5 6 17 12 10 4 4 6 10 10 20 9 23 8 6 7 18 12 12 4 4 7 12 10 20 10 23 10 6 7 19 12 12 4 4 7 12 10 20 10 23 10 6 7 20 s. kalantari et al./decis. mak. appl. manag. eng. 5 (2) (2022) 46-77 66 table 3. results from solving the proposed model in different dimensions model(46) hm sqa test 1st. obj 2st. obj solving time 1st. obj 2st. obj solving time gap1% gap2% 1 1058204.24 182.14 20.3 1058204.24 182.14 18.3 0.0 0.0 2 1072158.18 194.23 21.1 1072158.18 194.23 18.8 0.0 0.0 3 1113640.21 205.47 22.6 1123645.05 213.12 19.5 0.9 3.7 4 1154819.36 227.07 36.14 1158590.17 235.17 19.2 0.3 3.6 5 1277482.08 260.21 76.5 1328381.21 269.05 21.4 4.0 3.4 6 1555510.19 289.39 103.2 1616710.23 295.33 22.3 3.9 2.1 7 1766083.45 305.12 195.7 1814110.11 315.25 22.8 2.9 3.3 8 1837205.37 320.05 348.5 1857256.50 325.15 21.9 1.1 1.6 9 1859876.68 331.19 736.1 1878800.51 339.36 24.6 1.0 2.4 10 20147688.55 365.20 1053.8 20979820.7 371.14 26.7 4.1 1.6 11 22047205.15 378.68 1766.5 22897805.1 385.61 28.5 3.9 1.8 12 25357456.28 389.51 2623.4 25887616.3 397.47 30.1 2.1 2.1 13 28147084.15 410.25 3415.9 29158990.3 421.13 31.6 3.6 2.7 14 34308460.41 435.10 4896.1 35459806.5 441.18 33.4 3.4 1.4 15 39547371.36 462.19 5623.5 41446691.2 475.29 35 4.8 2.8 16 45149578.12 482.15 6854.2 47098990.6 493.34 38.6 4.3 2.3 17 50648476.45 508.07 8069.6 53038970.6 528.17 42.1 4.7 3.9 18 9345.3 59113215.1 682.14 44.5 19 10560.5 63154890.4 754.23 48.3 20 12038 69278482.2 805.47 52.7 figure 1. comparison of the optimal gap between the two methods 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 g a p problem gap1 gap2 a neutrosophical model for optimal sustainable closed-loop supply chain network … 67 g p gp hm h m no. of tests figure 2. comparison of solution time between two different methods 6.2. sensitivity analysis this segment discusses the effect of some critical parameters of the proposed model on the designed problem. the three most important parameters studied are demand, inflation, and capacity. we first look at the effect of changes in the inflation parameter. then the changes in demand and the changes in the capacity parameter are considered. on the other hand, we analyze the robustness of the uncertainty model and its impact on network design. finally, we discuss sensitivity analysis results and propose appropriate management decisions. 6.2.1. inflation parameter this section examines the effect of inflation on different price strategies (mode 1 = concave price; mode 2 = reference price; mode 3 = convex price). in this way, we solve the problem for different amounts of inflation (from -5% to + 10%). the results of the sensitivity analysis are shown in fig 3. according to the results in fig 3, the pricing strategy for suppliers and producers in modes 1 and 3, when inflation values are smaller than 0.195, 0.210, 0.087, 0.090, 0.105, 0.225, respectively, the objectives of the problem improve. as shown in figure 3, if the other parameters of the model are constant and inflation changes, for markets that are more sensitive to higher prices, the increasing trend of environmental impacts and the declining trend of npv in mode 3 under uncertainty are shown. on the other hand, by reducing the value of inflation, mode 3 improves the target functions. as a result, by opting for mode 3 pricing policy for markets with inflation changes of -5%, the npv performance improved by 2.33%, and the environmental performance improved by 1.46% compared to mode 1. the choice of pricing policy mode 1 for markets with inflation + 5% has an approximately 1.26% improvement in npv performance and 0.84% improvement in environmental performance compared to mode 3. s. kalantari et al./decis. mak. appl. manag. eng. 5 (2) (2022) 46-77 68 origin 1st. obj function 2st. obj function m o d e 1 m o d e 2 m o d e 3 m o d e 1 m o d e 2 m o d e 3 m o d e 1 m o d e 2 m o d e 3 m o d e 1 m o d e 2 m o d e 3 m o d e 1 m o d e 2 m o d e 3 figure 3. the effect of inflation on objectives 6.2.2. demand parameter figure 4 examines the results of the demand parameter analysis relative to the targets. according to this fig, a 5% increase in demand has increased the npv, but onwards 5%, the target values have been constant, indicating an increase in network costs, and its impact on npv is targeted. but then, increasing demand is directly related to increasing environmental impact. this is the increase in the amount of transportation with transportation methods, which has led to more environmental impacts. 1st. obj function 2st. obj function origin 1 s t . o b j fu n ct io n 2 s t . o b j fu n ct io n figure 4. demand parameter sensitivity analysis 6.2.3. capacity parameter as shown in figure 5, the objective functions are directly related to capacity changes. with a 20% increase in capacity, the first objective function increases by 61.59% and the second objective function by 50%. this can be increased production and transportation, but the rate of profitability is affected much more than the environmental goal. a neutrosophical model for optimal sustainable closed-loop supply chain network … 69 1st. obj function 2st. obj function origin of capacity 1 s t . o b j fu n ct io n 2 s t . o b j fu n ct io n figure 5. sensitivity analysis of facility capacity on profitability and environmental impact 6.2.4. robustness analysis of uncertainty model in this section is dedicated to the sensitivity analysis of the parameters of the proposed optimization model. the penalty coefficients used in the rfom for the first objective function were investigated. as shown in table 4, as the value of η increases, the value of npv decreases. also, no change in npv is made by changing ξ. table 4. sensitivity analysis of robust fuzzy optimization model parameters 𝜉𝑖 = 1000 0.9  0.6  0.3  15110325.15 20220650.3 25275812.88 |𝑍1| 0.6  𝜉𝑖 = 2000 𝜉𝑖 = 1000 𝜉𝑖 = 100 20220650.3 20220650.3 20220650.3 |𝑍1| according to the analysis, technological foresight informs managers of the challenges and opportunities for expected strategic decisions. one of the essential opportunities created for recycling e-waste in the technological dimension is the creation of suitable opportunities such as rate of return on investment, export potential of local recycling methods, and added value of the investment. investing in improving production and recycling processes, in the long run, increases the profitability and value of npvs, but may also increase costs in the short term. of course, the risk of technology financing for developing countries should not be forgotten because it is one of the government’s problems. on the other hand, government incentive policies will be more effective in helping industries manage their financing risk at an annual growth rate. therefore, the efficiency of sclsc recycled products is typical, in particular on electronics industry, and can be managed better for npv value. in addition, an appropriate pricing strategy given the inflation rate in iran's volatile economy for more sensitive markets to higher prices, inflation can be a factor in determining pricing strategies. in other words, determining the rate of inflation and pricing leads to the choice of profitability range, which considers the environmental and even social effects. s. kalantari et al./decis. mak. appl. manag. eng. 5 (2) (2022) 46-77 70 regarding the policies used to encourage manufacturers to design a stable clscn for electronic components, essential elements are proposed to establish an efficient recovery system by formulating the implementation of market-based policy plans. this policy is not macro-level to recover damaged and second-hand equipment by manufacturers or retailers so that at the time of sale, an amount is received as a return from customers. this strengthens and develops the design of a stable clscn to recycle electronic components. but on the other hand, this puts more financial burden on customers. therefore, the role of the government in determining financial incentives for customers to create a green culture in electronics and encourage them to give old and second-hand equipment to manufacturers or retailers is very important. at the macro level, such a policy creates excellent economic opportunities in terms of income, foreign exchange savings, and the country's economic growth. on the other hand, tax credit policy can also create a direct economic incentive for the tax credit to encourage manufacturers to design a sclscn. of course, given the current economic conditions of the society, the producers may resist the implementation of such a tax. therefore, the government has used an incentive system in macro-tax policies that forgives producers' vat for such problems. establishing a deposit system is an effective policy in developed countries to recycle electronic products, but manufacturers are reluctant to implement such a system due to the inflation in the country. because they are often afraid of declining sales due to the amount of deposits, this increases the cost of collection and transportation. 7. discussion and conclusion this article presents an integrated momilp model concerning multi-product and multi-period characteristics under uncertainty for configuring a stable clsc of a computer mouse. the chain sought major back-and-forth stakeholders to recycle components of computer mouse products in response to demand. in addition, the proposed model showed the effect of transportation modes, inflation, carbon emission policies, and supplier selection on the study network. stable fuzzy optimization was used to deal with the uncertainty of the model parameters. and a new hybrid method was used to solve the complexity and mo nature of the proposed model. the resulting cilp model was systematically solved for a participation in iran. several test samples in different dimensions were examined to validate the proposed method. the results were compared with the two factors of optimal gap and solution time, which showed the proper performance of the proposed method. then, the tactical results and model strategy were presented for a case study. the optimal flow between facilities, selection of suitable suppliers, transportation type, and facilities opening were shown. finally, sensitivity analysis on important problem parameters was discussed. the results showed that there is an upward trend in environmental impacts for markets that are more sensitive to higher prices and a downward trend in npv due to the convex price relative to the reference price. this means that reducing inflation for the convex price improves target performance. in addition, when the concave selling price is equal to the reference price, it enhances the performance of the target functions by increasing inflation. as a result, the impact of green levels on pricing strategy concerning inflation provides the flexibility of current prices in previous periods. considering the social effects of designing a stable clsc for electronic components can be a new challenge for future research. a neutrosophical model for optimal sustainable closed-loop supply chain network … 71 abbreviations sscm sustainable supply chain management pd product development pr product recovery momp multi-objective mixed programming mop multi-objective problem hm hybrid method ha heuristic algorithm ssp supplier selection problem dea data envelopment analysis spp shortest path problems mcdm multicriteria decision making mom multi-objective model ga genetic algorithm fdm fuzzy decision making mp mathematical programming ilpm integer linear programming model milpm mixed-integer linear programming model obj objective mil mixed-integer linear nil nonlinear programming sqa solution quality assessment mi mixed-integer rfopm robust fuzzy optimization model momilp multi-objective mixed-integer linear programming mo multi-objective cilp complex integer linear programming author contributions: research problem, s.k.; methodology, s.k. and h.k.; formal analysis, s.k. and h.k.; resources, s.k., h.k., f.m.s. and s.m.h.m.; writing – original draft preparation, s.k.; writing – review & editing, s.k., h.k., f.m.s and s.m.h.m. funding: this research received no external funding. acknowledgments: we would like to thank the editor-in-chief, editor and anonymous reviewers for their constructive and helpful comments on the earlier version of this manuscript. conflicts of interest: the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. references acquaye, a., ibn-mohammed, t., genovese, a., afrifa, g. a., yamoah, f. a., & oppon, e. 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(2011). fuzzy set theory—and its applications. springer science & business media. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 3, issue 2, 2020, pp. 37-47 issn: 2560-6018 eissn: 2620-0104 doi:_ https://doi.org/10.31181/dmame2003037b * corresponding author. e-mail addresses: i.badi@lam.edu.ly (i. badi), dpamucar@gmail.com (d. pamucar) supplier selection for steelmaking company by using combined grey-marcos methods ibrahim badi 1* and dragan pamucar2 1 libyan academy, department of mechanical engineering, misurata, libya 2 university of defence in belgrade, department of logistics, belgrade, serbia received: 5 april 2020; accepted: 5 june 2020; available online: 10 june 2020. original scientific paper abstract: selecting the right suppliers should saw as more than simply scanning a set of available price lists and comparing them, but rather as including a wide range of different criteria, whether qualitative or quantitative. in contemporary supply chain management, potential supplier performance is based on multiple criteria rather than considering cost as the main criterion in decision-making. this makes the process of selecting the best supplier from a group of suppliers a complex and laborious process, due to the multiplicity of criteria that must be taken into account in the evaluation process. this study aims to implement a hybrid grey theory-marcos method for decision-making regarding the selection of suppliers in the libyan iron and steel company (lisco) to help it compete. this hybrid model is divided into two phases: the first consists of determining the weights of the criteria that contribute to decision-making, which has done using the grey theory, and the second phase consists of selecting the best supplier from among the six suppliers, which has completed using the marcos model. the effectiveness of the model has compared to three other methods, codas, topsis, and vikor. the results showed that the proposed method effectively selected the best supplier among the six alternative suppliers. key words: mcdm, marcos, grey, supplier selection, lisco 1. introduction before the advent of multi-criteria methods, decision-making problems most often depended on a single criterion or objective function, maximizing profits or reducing costs. however, in reality, economic problems do not depend on a single objective but go beyond it. so it has been more appropriate to resort to methods with several criteria or restrictions, which are multi-criteria methods. these methods may include both quantitative and qualitative criteria, but their effect on decision-making varies from one criterion to another (moslem and duleba, 2018; kiracı and bakır, 2018). when making mailto:i.badi@lam.edu.ly mailto:dpamucar@gmail.com badi and pamucar/decis. mak. appl. manag. eng. 3 (2) (2020) 37-47 38 decisions, the selection of ineffective or inappropriate criteria will have negative economic impacts on the enterprise (chatterjee and stević, 2019). the purchasing function has garnered much attention in supply chains due to several factors such as globalization and accelerating technological change (erceg and mularifović, 2019). perhaps one of the essential activities of the purchasing function is to choose the right supplier, as it allows the company to achieve significant savings by maintaining a long-term partnership with suppliers, and thus dealing with a smaller number of these trusted suppliers. many industrial projects suffer from numerous problems and obstacles that cause them to deviate from their specific objectives. cost, time, and quality are the main objectives of any engineering project, and its achievement is the primary indicator for evaluating performance and ensuring the success of the project. these deviations are represented either in an overrun of the specified cost, an increase in time, or a low level of quality, where the implementing supplier plays a significant role in these deviations. the method used by many industrial companies to select the suppliers often depends on ineffective methods, where the tender is awarded to those offering the lowest price in the invitation to tender, even though the lowest price is not a sufficient indication to select the supplier that is capable of executing the contract and achieving its objectives. methods based on the principle of multi-criteria analysis and which take into account all the criteria necessary to select the best supplier are the most appropriate for the evaluation and selection of suppliers (durmić, 2019). its strength lies in the fact that it applies to decision situations involving multiple criteria that it uses both qualitative and quantitative data, and that provides metrics and indicators for preference selection. from this standpoint, and because the cost of raw materials is the essential component of industrial costs in many plants, the subject of choosing between suppliers has received a great deal of attention from researchers in recent years. competitors in the manufacturing sector today have the challenge of providing highquality products in addition to competitive prices. the total cost of raw materials usually constitutes the central portion of the product's final cost, which causes companies to pay more attention to supplier management (vasiljević et al., 2018). this places the financial department a significant role in reducing products' final cost by selecting the right suppliers. thus, selecting the best suppliers involves more than simply scanning a series from the price list, but goes beyond it to choose the right criteria to compare these suppliers. recently, supplier evaluation and selection have received significant attention from various researchers in the literature (badi and ballem, 2018; chatterjee and stević, 2019; mishra et al., 2019). generally, the criteria for supplier selection are highly dependent on individual industries and companies. therefore, different companies have different management strategies, enterprise culture, and competitiveness. furthermore, company background causes a considerable difference and impacts supplier selection. thus, the identification of supplier selection criteria mostly requires a domain expert's assessment and judgment. to select the best supplier, it is necessary to make a trade-off between these qualitative and quantitative factors (weights), some of which may conflict (ghodsypour and o'brien, 1998). traditional supplier selection methods are often based on the quoted price, which ignores the significant direct and indirect costs associated with quality, delivery, and service cost of purchased materials. uncertainty occurs because the future can never be predicted. one of the critical problems in supplier selection is to find the best supplier among several alternatives according to various criteria, such as service, cost, risk, and others. after identifying the criteria, a systematic methodology requires to integrate experts' assessments in order supplier selection for steelmaking company by using combined grey-marcos methods 39 to find the best supplier. various methods have been used for supplier selection (porrasalvarado et al., 2017). the combined grey theory and the measurement of alternatives and ranking according to the compromise solution (marcos) method will be implemented to evaluate the suppliers of raw materials to the libyan iron and steel company (lisco). lisco is a large scale, a government-owned company. the company's production capacity is about 1,324,000 tons of liquid steel (badi et al., 2017). in the last two decades, the company had almost met the demand for its products in the local market and managed to compete globally. it has started to export its products to egypt, tunisia, qatar, and others. lisco is working against the odds to rebuild the country's economy after the 2011 revolution and is doing so with a carefully considered strategy to expand its 60% iron and steel its market share in libya. the importation of raw material is an important step to maintain and improve its market share in a competitive environment (badi et al., 2017). the quality and cost of the final products are intimately connected to the proper selection of a sponge-iron supplier to the direct reduction, mega-scale factories. lisco usually imports sponge iron from india, brazil, canada, and sweden. suppliers from other countries also consider lisco as a potential customer. since suppliers have variable strengths and weaknesses, careful assessment and evaluation by the client are crucial before orders could be placed. 2. research methodology the methodology used in this research is illustrated in figure 1, where the weights of the criteria were determined using grey theory. after that, suppliers were evaluated using the marcos technique. finally, the results are compared with the other three multi-criteria methods. figure 1. methodology of the research 2.1 grey theory criteria weights are often difficult to determine precisely because of uncertainty, which can be addressed by linguistic terms such as “good,” “weak,” “important” or “very important” and other similar terms. realistic, multi-criteria decision-making applications require inaccurate, uncertain, qualitative, or ambiguous data processing. one effective method for modeling uncertainty and inaccuracy is using the grey theory, identifying research necessity defining of the aims of the research formatting criteria and alternatives conducting the assessment determining the criteria weights using grey theory ranking the suppliers using mrcos method results comparison badi and pamucar/decis. mak. appl. manag. eng. 3 (2) (2020) 37-47 40 which developed by deng (deng, 1982). it provides the flexibility to represent and deal with uncertainty and inaccuracy resulting from a lack of knowledge or inaccurate information. it uses a black-grey-white color to describe complex systems (liu et al., 2011). the concepts of a grey system can be illustrated as in figure 2. sy rebmun yerg a kind of figure that we only know the range of values, and do not know an exact value. this number can be an interval or a general number set to represent the degree of uncertainty of information. this section describes the basics of grey systems theory and grey numbers in order to understand the model. figure 2. the concept of grey system (badi et al., 2018b; abdulshahed et al., 2017b) let x is the universal set. then a grey set g of x is defined by its two mappings µ 𝐺 (x) and µg(x): µ𝐺 (x): 𝑋 ⟶ [0,1] and µg(x): 𝑋 ⟶ [0,1] such that µ𝐺 (x) ≥ µg(x), ×∈ 𝑋. since the lower limit ⊗ 𝐺 = [𝐺,∞) and upper limit ⊗ 𝐺 = (−∞,𝐺] can possibly be estimated, g is defined as an interval grey number ⊗ 𝐺 = [𝐺,𝐺] where 𝐺 > 𝐺. let t be the information, 𝐺 the upper, 𝐺 the lower limit then 𝐺 ≤ 𝑡 ≤ 𝐺 if 𝐺 = 𝐺 then ⊗ 𝐺 is a white number with a crisp value which shows the existence of full knowledge. on the contrary, a black number is a grey number one known nothing about it (liu et al., 2012). the arithmetic of grey numbers is similar to interval value (liu et al., 2012, li et al., 2007) and the operation rules of general grey numbers can defines as operation rules of real numbers (liu et al., 2012; badi et al., 2019). addition: ⊗ 𝐺1 +⊗ 𝐺2 = [𝐺1 + 𝐺2, 𝐺1 + 𝐺2] subtraction: ⊗ 𝐺1 − ⊗ 𝐺2 = [𝐺1 − 𝐺2, 𝐺1 − 𝐺2] multiplication:⊗ 𝐺1 ×⊗ 𝐺2 = [𝑚𝑖𝑛(𝐺1𝐺2, 𝐺1𝐺2, 𝐺1𝐺2, 𝐺1 𝐺2), 𝑚𝑎𝑥(𝐺1𝐺2, 𝐺1𝐺2, 𝐺1𝐺2, 𝐺1 𝐺2)] division: ⊗ 𝐺1 ÷⊗ 𝐺2 = [𝐺1, 𝐺1] × [ 1 𝐺2 , 1 𝐺2 ] length of grey number: 𝐿(⊗ 𝐺) = ⌊𝐺 − 𝐺⌋ comparison of grey numbers: the possibility degree of two grey number expressing as: 𝑃{⊗ 𝐺1 ≤⊗ 𝐺2} = 𝑚𝑎𝑥 (0, 𝐿∗ − 𝑚𝑎𝑥(0, 𝐺1 − 𝐺2)) 𝐿∗ where 𝐿∗ = 𝐿(⊗ 𝐺1) + 𝐿(⊗ 𝐺2) according to this comparison of two grey numbers, there may be four distinct outcomes: if ⊗ 𝐺1 = ⊗ 𝐺2 then 𝑃{⊗ 𝐺1 ≤⊗ 𝐺2} = 0.5 if 𝑃{⊗ 𝐺1 >⊗ 𝐺2} then 𝑃{⊗ 𝐺1 ≤⊗ 𝐺2} = 1 supplier selection for steelmaking company by using combined grey-marcos methods 41 if ⊗ 𝐺1 < ⊗ 𝐺2 then {⊗ 𝐺1 ≤⊗ 𝐺2} = 0 if 𝑃{⊗ 𝐺1 ≤⊗ 𝐺2} > 0.5 then ⊗ 𝐺2 > ⊗ 𝐺1 otherwise if 𝑃{⊗ 𝐺1 ≤⊗ 𝐺2} < 0.5 then ⊗ 𝐺2 < ⊗ 𝐺1 attribute weight 𝑊𝑗 can be calculated as follows (li et al., 2007): ⊗ 𝑊𝑗 = 1 𝐾 [⊗ 𝑊𝑗 1 +⊗ 𝑊𝑗 2 + ⋯ +⊗ 𝑊𝑗 𝐾 ] (1) ⊗ 𝑊𝑗 𝐾 = [𝑊𝑗 𝐾 , 𝑊𝑗 𝐾 ] (2) 3.2 the measurement of alternatives and ranking according to compromise solution (marcos) method the marcos method is based on defining the relationship between alternatives and reference values (ideal and anti-ideal alternatives) (stević et al., 2020). decision-making preferences are defined based on utility functions. a utility function is the position of an alternative concerning the ideal and anti-ideal solutions (stanković et al., 2020). the best alternative is that closest to the ideal point and farthest from the anti-ideal point. the marcos method is implemented through the following steps (puška et al., 2020): step 1. the formation of the initial decision matrix. step 2. the formation of an extended initial matrix. this step defines the ideal and anti-ideal solutions. the ideal solution is an alternative with the best alternative for specific criteria, whereas the anti-ideal solution is the worst alternative. this is based on the following equations: 𝐴𝐴𝐼 = min 𝑗 𝑥𝑖𝑗 𝑖𝑓 𝑗 ∈ 𝐵 𝑎𝑛𝑑 𝐴𝐴𝐼 = max 𝑗 𝑥𝑖𝑗 𝑖𝑓 𝑗 ∈ 𝐶 (3) 𝐴𝐼 = max 𝑗 𝑥𝑖𝑗 𝑖𝑓 𝑗 ∈ 𝐵 𝑎𝑛𝑑 𝐴𝐴𝐼 = min 𝑗 𝑥𝑖𝑗 𝑖𝑓 𝑗 ∈ 𝐶 (4) where b stands for the criteria to be maximized, and c stands for the criteria to be minimized. step 3. the normalization of the extended initial matrix. normalization is performed by using the following equations: 𝑛𝑖𝑗 = 𝑥𝑎𝑖 𝑥𝑖𝑗 𝑖𝑓 𝑗 ∈ 𝐶 (5) 𝑛𝑖𝑗 = 𝑥𝑖𝑗 𝑥𝑎𝑖 𝑖𝑓 𝑗 ∈ 𝐵 (6) where the elements 𝑥𝑖𝑗 and 𝑥𝑎𝑖 represent the elements of the initial decision matrix. step 4. the determination of a weighted matrix. aggravation is performed by multiplying normalized matrix values by corresponding weights. step 5. the calculation of the utility degree of the alternatives ki. the utility degree is determined by applying the following equations: 𝐾𝑖 − = 𝑆𝑖 𝑆𝑎𝑎𝑖 (7) 𝐾𝑖 + = 𝑆𝑖 𝑆𝑎𝑖 (8) where si (i=1,2,..,m) represents the sum of the elements of a weighted matrix 𝑆𝑖 = ∑ 𝑣𝑖𝑗 𝑛 𝑖=1 (9) step 6. the formation of the utility function of the alternatives f(ki). the utility function is calculated by using the following equation: 𝑓(𝐾𝑖 ) = 𝐾𝑖 + +𝐾𝑖 − 1+ 1−𝑓(𝐾 𝑖 +) 𝑓(𝐾 𝑖 +) + 1−𝑓(𝐾 𝑖 −) 𝑓(𝐾 𝑖 −) (10) where f(𝐾𝑖 −) is the utility function versus the anti-ideal solution, while f(𝐾𝑖 +) is the utility function versus the ideal solution. the utility functions are calculated using the following equations: 𝑓(𝐾𝑖 −) = 𝐾𝑖 + 𝐾𝑖 ++𝐾𝑖 − (11) badi and pamucar/decis. mak. appl. manag. eng. 3 (2) (2020) 37-47 42 𝑓(𝐾𝑖 +) = 𝐾𝑖 − 𝐾𝑖 ++𝐾𝑖 − (12) step 7. ranking the alternatives. a rank is formed based on the final value of the utility function. the alternative should have the most significant value of the utility function. 3. case study the proposed model has been applied to evaluate the lisco’s suppliers. lisco is one of the largest national companies in libya. in order to maintain its competitive advantage, the import of raw materials is an important step that should be managed carefully. the quality and cost of the finished products are intimately related to the appropriate selection of sponge iron suppliers. lisco imports sponge iron from several countries, the most potential of which is brazil. the data used in this paper was based on the two models used in (badi et al., 2018a) and (abdulshahed et al., 2017a), which aimed to choose the best suppliers for lisco. four different criteria which are considered: quality (in points) direct cost (in $), lead time (in days), logistics services (in points). quality and logistics services criteria are defined as benefit criteria, while the cost and lead time are cost criteria. table 1 shows the details of these criteria. there are six suppliers. table 1. qualitative criteria for supplier evaluation. evaluation criteria description measuring principle criteria status quality (c1) poor quality materials are found during incoming inspection total number of rejected items in each batch benefitcriteria direct cost (c2) direct cost of the material reasonable direct cost cost-criteria lead time (c3) the supplier capability to timely meet the demand this can be measured by percentage of demand meet in each period cost-criteria logistics service (c4) logistics service used by suppler and transportation time this can be analysed by percentage of demand meet in each period benefitcriteria the first stage is to determine the criteria weights. four experts have been invited to participate in the determination of the importance of each criterion for the evaluation of suppliers. the linguistic variables can be expressed in grey numbers by a scale shown in table 2 (abdulshahed et al., 2017a). the suppliers were rated for their performances of attributes on grey scales shown in table 3. supplier selection for steelmaking company by using combined grey-marcos methods 43 table 2. the importance of grey number for the weights of the criteria. importance abbreviation scale of grey number ⊗ 𝑊 very low vl [0.0, 0.1] low l [0.1, 0.3] medium low ml [0.3, 0.4] medium m [0.4, 0.5] medium high mh [0.5, 0.6] high h [0.6, 0.9] very high vh [0.9, 1.0] table 3. linguistic assessment and the associated grey values. performance abbreviation scale of grey number ⊗ 𝑊 very poor vp [0.0, 0.1] poor p [0.1, 0.3] medium poor mp [0.3, 0.4] fair f [0.4, 0.5] medium good mg [0.5, 0.6] good g [0.6, 0.9] very good vg [0.9, 1.0] collect the evaluation of the experts' attributes by using linguistic variables, as shown in table 4. next, the attributes can be weighted using equation 1. table 4. the linguistic assessment of the attributes by experts. ci expert #1 expert #2 expert #3 expert #4 ⊗ 𝑊 whitening degree c1 vh h h h 0.67 0.92 0.80 c2 h vh vh h 0.75 0.95 0.85 c3 mh h h mh 0.55 0.75 0.65 c4 m m mh mh 0.45 0.55 0.50 after the criteria weights were calculated, the suppliers are ranked using the marcos method. based on the data collected (badi et al., 2018a), an initial decision matrix was prepared (table 5). table 5. the initial decision matrix weights of criteria 0.80 0.85 0.65 0.50 alternatives suppliers quality direct costs ($) lead time (days) logistics service s1 45 3,600 45 0.9 s2 25 3,800 60 0.8 s3 23 3,100 35 0.9 s4 14 3,400 50 0.7 badi and pamucar/decis. mak. appl. manag. eng. 3 (2) (2020) 37-47 44 s5 15 3,300 40 0.8 s6 28 3,000 30 0.6 max 45 3,800 60 0.9 the next step is to normalize the data to be uninformed. for this purpose, a simple linear normalization (equation 5) was applied to the marcos method. the maximum value of the criteria is determined, as required for all criteria to be maximized. the normalization of the initial decision matrix is step 3 of the marcos method (table 6). table 6. the normalized decision matrix alternatives quality direct costs ($) lead time (days) logistics service s1 1.000 1.056 1.333 1.000 s2 0.556 1.000 1.000 0.889 s3 0.511 1.226 1.714 1.000 s4 0.311 1.118 1.200 0.778 s5 0.333 1.152 1.500 0.889 s6 0.622 1.267 2.000 0.667 the fourth step after the normalization of the initial matrix is the calculation of the aggregated values using the weighting coefficients. the fifth step is to calculate the utility degree. in order to perform this step, it was first necessary to determine the ideal and anti-ideal solutions. the ideal solution represents the maximum value of a specific criterion, whereas anti-ideal values represent the minimum value of a specific criterion. then, the values for the individual alternatives and the ideal and anti-ideal solutions were summed up, and the utility degrees were calculated (equations 7 and 8). table 7. the weighted normalized decision matrix and the negative-ideal solution alternatives quality direct costs ($) lead time (days) logistics service sum s1 0.800 0.897 0.867 0.500 3.064 s2 0.444 0.850 0.650 0.444 2.389 s3 0.409 1.042 1.114 0.500 3.065 s4 0.249 0.950 0.780 0.389 2.368 s5 0.267 0.979 0.975 0.444 2.665 s6 0.498 1.077 1.300 0.333 3.208 ideal 0.800 1.077 1.300 0.500 3.677 anti-ideal 0.249 0.850 0.650 0.333 2.082 the sixth step of the marcos method was to form the utility function of the alternatives. the utility function was calculated by using equation 10. to calculate the utility function of the alternatives, it was necessary to calculate the utility function concerning the ideal and anti-ideal solutions. the inclusion of these values generated the final value for the alternatives (table 8) and determined the ranking of the suppliers. supplier selection for steelmaking company by using combined grey-marcos methods 45 table 8. the relative assessment matrix and the assessment scores of alternatives supplier 𝐾𝑖 − 𝐾𝑖 + f(ki) rank s1 1.471 0.833 0.692 3 s2 1.147 0.650 0.539 5 s3 1.472 0.834 0.692 2 s4 1.137 0.644 0.535 6 s5 1.280 0.725 0.602 4 s6 1.541 0.872 0.724 1 as can be seen from table (8), s6 is the best supplier concerning the assessment of the mrcos method. besides, a comparative analysis has been conducted to demonstrate the validity and stability of the mrcos method. three different multi-criteria methods are used, which are the combinative distance-based assessment (codas) model, technique for order performance by similarity to ideal solution (topsis), and višekriterijumsko kompromisno rangiranje (vikor) method. figure 2 shows the results obtained by these methods. figure 2. results comparison 4. conclusion it is well known that mcdm techniques are gaining popularity in solving supplier evaluation and selection problems. this paper provides a supporting tool for multicriteria decision-making to evaluate suppliers using hybrid grey-marcos methods. grey theory has been used to weigh criteria, which is an appropriate method for dealing with uncertainty. suppliers have been ranked using the marcos method. therefore, in the future, this method can be used to deal with uncertainty in multi-criteria decisionmaking problems such as project selection, manufacturing systems, staff selection, and badi and pamucar/decis. mak. appl. manag. eng. 3 (2) (2020) 37-47 46 many other areas related to management decisions. furthermore, the marcos method can be used in the future for other applications of mcdm. author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research received no external funding. conflicts of interest: the authors declare no conflicts of interest. references abdulshahed, a. m., badi, i. a. & blaow, m. m. 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(2020). sustainable supplier selection in healthcare industries using a new mcdm method: measurement of alternatives and ranking according to compromise solution (marcos). computers & industrial engineering, 140, 106231. vasiljević, m., fazlollahtabar, h., stević, ž. & vesković, s. (2018). a rough multicriteria approach for evaluation of the supplier criteria in automotive industry. decision making: applications in management and engineering, 1 (1), 82-96. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 1, number 1, 2018, pp. 1-12 issn: 2560-6018 doi: https://doi.org/10.31181/dmame180101b * corresponding author. e-mail addresses: i.badi@eng.misuratau.edu.ly (i. badi), mohamed.ballem@eng.misuratau.edu.ly (a.m. abdulshahed), a.shetwan@eng.misuratau.edu.ly (a.g. shetwan) a case study of supplier selection for a steelmaking company in libya by using the combinative distance-based assessment (codas) model ibrahim ahmed badi1*, ali m. abdulshahed2, ali g. shetwan3 1 misurata university, faculty of engineering, mechanical engineering department, libya 2 misurata university, faculty of engineering, electrical engineering department, libya 3 college of industrial technology, industrial engineering department, libya received: 6 november 2017; accepted: 30 january 2018; published: 15 march 2018. original scientific paper abstract: multi-criteria decision making (mcdm) problems have received considerable attention from various researchers over the past decades. a great variety of methods and approaches has been developed in this field. the aim of this paper is to use a new combinative distance-based assessment (codas) method to handle mcdm problems for a steelmaking company in libya. so far no literature dealing with supplier selection using the (codas) method in the steelmaking company in libya has been found. the concept of this method is based on computing the euclidean distance and the taxicab distance in order to determine the desirability of an alternative. the euclidean distance is used as a primary measure, while the taxicab distance as a secondary one. the developed method was applied to a real-world case study for ranking the suppliers in the libyan iron and steel company (lisco). an attempt in this regard could enhance a decision-making technique for selecting the best suppliers for the selected case company. the results showed that the proposed method was effectively able to select the best supplier among six alternative ones. key words: criteria, codas, combinative, supplier, selection, assessment. 1. introduction today’s competitive manufacturing sector presents a challenge to provide high quality products while offering competitive prices to the final customers. goffin et al., (goffin et al., 1997) have indicated that the supplier management is one of the key issues of the supply chain management because the total cost of raw materials constitutes the final cost of a product, and most of the companies have to spend a mailto:i.badi@eng.misuratau.edu.ly mailto:mohamed.ballem@eng.misuratau.edu.ly https://www.researchgate.net/profile/ali_abdulshahed mailto:a.shetwan@eng.misuratau.edu.ly https://www.researchgate.net/profile/ali_abdulshahed badi et al./decis. mak. appl. manag. eng. 1 (1) (2018) 1-12 2 considerable amount of their budget on them. in most industries, the cost of raw materials constitutes the main cost of a product so that in some cases it can account for up to 70% (kilincci & onal, 2011). thus, the financial department can play a key role in the firm’s efficiency and effectiveness since this department has a direct effect on the final cost reduction and profitability of a company by selecting suitable suppliers. hence, selecting the best suppliers involves much more than scanning a series of price lists, and choices will depend on a wide range of criteria which involve both qualitative and quantitative ones. recently, supplier evaluation and selection have received significant attention from various researchers in the literature (de boer et al., 2001; govindan et al., 2015; chai et al., 2013; prakash et al., 2015; ghorabaee et al., 2015). supplier selection is a multi-criteria problem which includes both qualitative and quantitative factors (liang et al., 2013). generally, the criterion for supplier selection is highly dependent on individual industries and companies. therefore, different companies have different management strategies, enterprise culture and competitiveness. furthermore, company background causes a huge difference and impacts supplier selection. thus, the identification of the supplier selection criteria largely requires the domain expert’s assessment and judgment. to select the best supplier, it is necessary to make a trade-off between these qualitative and quantitative factors (weights) some of which may conflict (ghodsypour & o'brien, 1998). the traditional supplier selection methods are often based on the quoted price, which ignores significant direct and indirect costs associated with quality, delivery, and service cost of purchased materials. uncertainty occurs because the future can never be predicted. the selection of the best supplier is based on quoted price and considering all the possibilities of the analysis. however, there is always uncertainty about indirect costs associated with quality, delivery time, and others. one of the key problems in supplier selection is to find the best supplier among several alternatives according to various criteria, such as service, cost, risk, and others. after identifying the criteria, a systematic methodology is required to integrate experts’ assessments in order to find the best supplier. at present, various methods have been used for supplier selection such as analytic network process (anp) (porras-alvarado et al., 2017) and analytical hierarchy process (ahp) (porrasalvarado et al., 2017).the following paragraphs will introduce a discussion of important and widely used mcdm techniques most of which aim at selecting the best supplier. ahp is a common multi-criteria decisionmaking (mcdm) method. it is developed by saaty (saaty, 1990; saaty, 1979) to provide for a flexible and easily understood way of analyzing complex problems. it breaks a complex problem into hierarchy or levels, and then makes comparisons between all possible pairs in a matrix to give a weight for each factor and a consistency ratio. according to (chai et al., 2013) the ahp method is found to have been used more than any other mcdm method. however, the ahp methodology is focused on weighting relative importance of the criteria, while dependencies among them are neglected. chan and chan (2004) have used the ahp to evaluate and select suppliers. the ahp hierarchy consists of six evaluating criteria and 20 sub-factors, of which the relative importance ratings are calculated based on the customer needs. chan et al. (2007) also researched this area by developing an ahp approach to solve the supplier selection problem. possible suppliers were evaluated based on fourteen criteria. a sensitivity analysis using the “expert choice” software was performed to examine the response of alternatives when the relative importance rating of each criterion was changed. jiang et al. (jiang et al., 2007) developed an ahp-based decision support system for the supplier selection problem in a mass customization environment. factors from external and a case study of supplier selection for a steelmaking company in libya by using the ... 3 internal influences were taken into consideration in order to meet the needs of the markets in the global changing environment. each cluster at the same level in a pairwise manner was compared by experts based on their own knowledge (ho et al., 2010). mathematically and philosophically, the ahp is capable of providing for an easily understandable method to practitioners; however, it is insufficient to explain uncertain conditions in an especially pair-wise comparison stage. most of human’s judgments are not represented as exact numbers. since some of the evaluation criteria are subjective and qualitative in nature, it is very difficult for the decisionmaker to express his preferences in exact numerical values and to provide exact pairwise comparison judgments. as a result, to tackle these problems, the ahp has been integrated with other methods, including the ann (kuo et al., 2002), fuzzy set theory (jain et al., 2016; gold & awasthi, 2015; pamučar et al., 2016; božanić et al. 2016), grey relational analysis (liang et al., 2015; yang & chen; 2006, bali et al., 2013), and a combination of different methods (zakeri & keramati, 2015). it seems, however, that the growth of ahp applications may derive more from a simplification perspective rather than from a robust theoretical mathematical perspective. the grey systems theory, introduced by deng in the early 1980s (deng, 1982), is another methodology that focuses on solving problems involving incomplete information. the technique works on uncertain systems with partially known information by generating, mining, and extracting useful information from available data. the grey theory considers that although the objective system appears complex, with a small amount of data, it always has some internal laws governing the existence of the system and its operation (liu et al., 2010). a grey number is a kind of figure whose range of values we know only – without knowing an exact value (liu et al., 2012). this number can be an interval or a general number set to represent the degree of uncertainty of information. grey systems theory in a decision-making process is very useful to tackle the disadvantage of ahp. abdulshahed et al. (abdulshahed et al., 2017)used an integrated model by combining the grey model and grey numbers and examined the feasibility of their approach to select the best suppliers. they applied a grey model to calculate relative importance weightings of qualitative criteria. a supplier with the highest value was regarded as the best supplier in an outsourcing manufacturing organization. all the above-mentioned mcdm methods have their own privilege, strength, and weakness for certain applications; however, their estimate is not the aim of this paper. in general, the best supplier selection is still an ill-defined problem. it generally relies on uncertain information, which is not easy to model and is based on the experiences of specialists. this work uses a new developed method to handle multi-criteria decision making problems by ghorabaee et al. (2016); this method is named codas, and has a number of features that have not been considered in the other mcdm methods. for instance, in (keshavarz ghorabaee et al., 2016), the codas method has been compared with some of the existing mcdm methods. according to their analysis, the codas method was efficient to deal with mcdm problems. ghorabaee et al. (2017) also used an integrated model by combining the fuzzy logic theory and the codas method to select the best suppliers. in their work, a fuzzy extension of the codas method was developed to deal with multi-criteria decision-making problems in an uncertain environment. they used linguistic variables and trapezoidal fuzzy numbers to extend the codas method and propose a multi-criteria group decision-making approach. a numerical example of a shoe company was utilized to show the applicability of their method in multi-criteria market segment evaluation and selection. the results indicate that the fuzzy codas badi et al./decis. mak. appl. manag. eng. 1 (1) (2018) 1-12 4 method was consistent with the results of the other method in the literature. panchal et al. (2017) applied an integrated mcdm framework based on the fuzzy ahp and a fuzzy codas approach for solving the maintenance decision problem in a process industry. in order to overcome vagueness in human judgment, they have incorporated a fuzzy set theory within the proposed framework. the sensitivity results confirmed the stability of their framework. in the codas method, the overall performance of an alternative is measured by the euclidean and taxicab distances from the negative-ideal point. the codas use the euclidean distance as the primary measure of assessment. if the euclidean distances of two alternatives are very close to each other, then the taxicab distance is used to compare them. the degree of closeness of euclidean distances is set by a threshold parameter. the euclidean and taxicab distances are measures for norm and norm indifference spaces, respectively. therefore, in the codas method, first the alternative in a norm indifference space is assessed. if the alternatives are not comparable in this space, then an -norm indifference space is chosen. to perform this process, each pair of alternatives should be compared. in this study, the codas method is presented in detail, and a numerical example will be illustrated. moreover, a comparative sensitivity analysis is performed to measure the validity and stability of this method. the proposed codas method will be implemented to evaluate the suppliers of raw materials to the libyan iron and steel company (lisco). lisco is a large scale, government owned company. the production capacity of the company is about 1,324,000 tons of liquid steel (taib, 2011). in the last two decades, the company had almost met the demand for its products in the local market, and managed to compete globally. it has started to export its products to egypt, tunisia, qatar and others. lisco is working against the odds to help rebuild the country’s economy after the 2011 revolution and is doing so with a carefully considered strategy to expand its 60% iron and steel market share in libya. the importation of raw material is an important step towards maintaining and improving its market share in a competitive environment. quality and cost of its final products are intimately connected to the proper selection of a supplier of sponge iron to the direct reduction, mega-scale factories. lisco usually imports sponge iron from india, brazil, canada, sweden. suppliers from other countries also consider lisco as a potential customer. since suppliers have variable strengths and weaknesses, careful assessment and evaluation by the client is crucial before orders could be placed. 2. research methodology in this section, a new method (codas) is introduced to deal with multi-criteria decision-making problems. in this method, the desirability of alternatives is determined by using two measures. the main and primary measure is related to the euclidean distance of alternatives from the negative-ideal. using this type of distance requires an -norm indifference space for criteria. the secondary measure is the taxicab distance, which is related to the -norm indifference space. clearly, the alternative which has greater distances from the negative-ideal solution is more desirable. in this method, if two alternatives are incomparable according to the euclidean distance, then the taxicab distance is used as a secondary measure. although the -norm indifference space is preferred in the codas, two types of indifference space could be considered in its process. based on the assumption that alternatives and criteria are available, the steps of the proposed method can then be presented as follows: a case study of supplier selection for a steelmaking company in libya by using the ... 5 step 1. construct the decision-making matrix as follows: 𝑋 = [𝑥𝑖𝑗 ]𝑛×𝑚 = [ 𝑥11𝑥12 … 𝑥1𝑚 𝑥21𝑥22 … 𝑥2𝑚 . . . . 𝑥𝑛1𝑥𝑛2 … 𝑥𝑛𝑚 ] where xij (xij≥ 0) denotes the performance value of i th alternative on j th criterion (i∈ {1,2,… ,n} and j∈ {1,2,… ,m}). step 2. calculate the normalized decision matrix. linear normalization of performance values is used as given by equation (1). 𝑛𝑖𝑗 = { 𝑥𝑖𝑗 𝑚𝑎𝑥𝑖𝑥𝑖𝑗 𝑖𝑓 𝑗𝜖𝑁𝑏 𝑚𝑖𝑛𝑖𝑥𝑖𝑗 𝑥𝑖𝑗 𝑖𝑓 𝑗𝜖𝑁𝑐 (1) where nb and nc represents the sets of benefit and cost criteria, respectively. step 3. calculate the weighted normalized decision matrix. the weighted normalized performance values are calculated as given by equation (2). 𝑟𝑖𝑗 = 𝑤𝑗 𝑛𝑖𝑗 (2) where wj (0< wj< 1) denotes the weight of j th criterion, and 1 1 m jj w   . step 4. determine the negative-ideal solution (point) as given in equation (3). 𝑛𝑠 = [𝑛𝑠𝑗 ]1×𝑚 𝑛𝑠𝑗 = 𝑚𝑖𝑛𝑖 𝑟𝑖𝑗 (3) step 5. calculate the euclidean and taxicab distances of alternatives from the negative-ideal solution as given in equations (4) and (5), respectively. 𝐸𝑖 = √∑ (𝑟𝑖𝑗 − 𝑛𝑠𝑗 ) 2𝑚 𝑗=1 (4) 𝑇𝑖 = ∑ |𝑟𝑖𝑗 − 𝑛𝑠𝑗 | 𝑚 𝑗=1 (5) step 6. construct the relative assessment matrix as given in equation (6). 𝑅𝑎 = [ℎ𝑖𝑘 ]𝑛×𝑛 (6) ℎ𝑖𝑘 = (𝐸𝑖 − 𝐸𝑘 ) + (𝜓(𝐸𝑖 − 𝐸𝑘 ) × (𝑇𝑖 − 𝑇𝑘 )) where k∈ {1, 2,…, n} and 𝜓 denotes a threshold function to recognize the equality of the euclidean. 𝜓(𝑥) = { 1 𝑖𝑓 |𝑥| ≥ 𝜏 0 𝑖𝑓 |𝑥| < 𝜏 in this function, 𝜏 is the threshold parameter that can be set by the decisionmaker. it is suggested to set this parameter at a value between 0.01 and 0.05. if the difference between euclidean distances of two alternatives is less than 𝜏, these two alternatives are also compared by the taxicab distance. in this study, it is assumed that 𝜏= 0.02 for the calculations. step 7. calculate the assessment score of each alternative as given by equation (7). 𝐻𝑖 = ∑ ℎ𝑖𝑘 𝑛 𝑘=1 (7) step 8. rank the alternatives according to the decreasing values of assessment score (η). the alternative with the highest η is the best choice among the alternatives. to describe the proposed method, a simple situation with seven alternatives and two criteria is used. suppose that weighted normalized performance values (rij) have badi et al./decis. mak. appl. manag. eng. 1 (1) (2018) 1-12 6 been calculated. these values are dimensionless and between 0 and 1. fig. 1 shows the position of all alternatives according to these values. figure 1. a simple graphical example with two criteria(keshavarz ghorabaee et al., 2016) it can be seen in figure 1, that a2 has greater taxicab distance from the negativeideal point. this fact is clear according to the indifference curves, which is presented in figure 1. therefore, we can say that a2 is more desirable than a4, and the final ranking is 𝐴3 < 𝐴1 < 𝐴5 < 𝐴4 < 𝐴2 < 𝐴6 < 𝐴7. 3. results establishing the criteria is the first step in the process of supplier selection. in this paper, qualitative criteria are identified based on questionnaire forms. in order to facilitate the solution process for the supplier selection problem, macros in ms excel were used to compute the model based on the questionnaire forms that have been filled in by the experts and managers who work in lisco. four different criteria which are considered in this supplier selection problem are: quality (in points) direct cost (in $), lead time (in days), logistics services (in points). all these criteria are defined as benefit criteria, except that the cost is defined as a cost criterion. this problem consists of six suppliers, and the corresponding data are given in table 1. every criterion has been given weight by experts, and the total weight of all criteria is 1.0. experts also give weights for the suppliers for each criterion. based on table 1, the decision matrix can be constructed. then the normalized decision matrix is calculated as shown in table 2. a case study of supplier selection for a steelmaking company in libya by using the ... 7 table 1. data of the case study weights of criteria 0.2857 0.3036 0.2321 0.1786 alternatives suppliers quality direct costs ($) lead time (days) logistics service s1 45 3,600 45 0.9 s2 25 3,800 60 0.8 s3 23 3,100 35 0.9 s4 14 3,400 50 0.7 s5 15 3,300 40 0.8 s6 28 3,000 30 0.6 for each criterion, this can be done by dividing each weight of the suppliers by the maximum weight of this criterion. table 2. the normalized decision matrix alternatives quality direct costs ($) lead time (days) logistics service 1.000 0.833 0.750 1.000 0.556 0.789 1.000 0.889 0.511 0.968 0.583 1.000 0.311 0.882 0.833 0.778 0.333 0.909 0.667 0.889 0.622 1.000 0.500 0.667 using weights of criteria that are given in table 1, the weighted normalized performance values can be calculated, and then the negative-ideal solution is determined. according to the obtained values, the euclidean and taxicab distances of alternatives from the negative-ideal solution are also computed. the results are presented in table 3. table 3. the weighted normalized decision matrix and the negative-ideal solution alternatives quality direct costs ($) lead time (days) logistics service distances euclidean taxicab 0.2857 0.2530 0.1741 0.1786 0.2141 0.3277 0.1587 0.2397 0.2321 0.1588 0.1411 0.2256 0.1460 0.2938 0.1354 0.1786 0.1006 0.1901 0.0889 0.2679 0.1934 0.1389 0.0847 0.1254 0.0952 0.2760 0.1547 0.1588 0.0666 0.1210 0.1778 0.3036 0.1161 0.1191 0.1095 0.1528 negativeideal solution 0.0889 0.2397 0.0719 0.1191 badi et al./decis. mak. appl. manag. eng. 1 (1) (2018) 1-12 8 the relative assessment matrix and the assessment scores (η) of alternatives can be calculated by using table 3 and eq. (6). table 4 represents the results. it should be noted that the calculations are performed with 𝜏= 0.02. table 4. the relative assessment matrix and the assessment scores of alternatives s1 s2 s3 s4 s5 s6 h 0.00 0.175 0.2511 0.3317 0.3540 0.2790 1.3914 -0.175 0.00 0.0760 0.1566 0.1790 0.1040 0.3411 -0.2511 -0.0760 0.00 0.0159 0.1030 -0.0090 -0.2170 -0.3316 -0.1570 -0.0159 0.00 0.0180 -0.0520 -0.5381 -0.3542 -0.1790 -0.1031 -0.0181 0.00 -0.075 -0.7292 -0.2795 -0.104 0.0089 0.0521 0.0750 0.00 -0.2481 as can be seen from table (4), the highest η is supplier1. therefore, s1 is the best supplier with respect to the assessment of the codas method. in addition, a sensitivity analysis has been conducted to demonstrate the validity and stability of the codas method. according to the results of the sensitivity analysis, it was found that the codas method is stable and efficient to deal with multi-criteria decisionmaking problems. fourteen values of  ranged between 0.01 and 1.00 are used to evaluate their effect on suppliers ranking. table 5 shows the values of  and their effect on suppliers ranking. table 5. suppliers ranking with different values of  s ce n .  0.01 0.02 0.03 0.04 0.05 ... 0.10 0.15 0.30 0.50 1.00 s1 1 1 1 1 1 ... 1 1 1 1 1 s2 2 2 2 2 2 2 2 2 2 2 s3 3 3 3 4 4 4 4 4 4 4 s4 5 5 5 5 5 5 5 5 5 5 s5 6 6 6 6 6 6 6 6 6 6 s6 4 4 4 3 3 3 3 3 3 3 figure 2 shows the effect graphically, which is clear that the first supplier (s1) is the best regardless of  value. changing parameter  has a minor effect on the ranking of alternatives that can undermine the validity of the results. a case study of supplier selection for a steelmaking company in libya by using the ... 9 figure 2. suppliers ranking with different values of  4. conclusion the aim of this paper is to select the best supplier in the lisco in libya using the codas method. it is well known that mcdm techniques are gaining popularity in solving supplier evaluation and selection problems. this work includes both quantitative and qualitative criteria though some of them may include uncertainty and sometimes they may be conflicting. the codas method has some features that have not been considered in the other mcdm methods. in this paper, the codas method is applied to a real-world case study for ranking the suppliers in the lisco. the results have revealed that supplier s1 was the most suitable choice with respect to all recognized criteria, as seen in the relevant sensitivity analysis. the performance of the suppliers based on the criteria mentioned earlier is a robust one similar to the synthesis results. consequently, the codas method is capable of enhancing quality decision by making its process more rational, explicit and efficient. furthermore, the codas method can be used in the future for other applications of mcdm. references abdulshahed, a. m., badi, i. a., & blaow, m. m. 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(2015). systematic combination of fuzzy and grey numbers for supplier selection problem. grey systems: theory and application, 5, 313-343. decision making: applications in management and engineering vol. 3, issue 2, 2020, pp. 1-18. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame2003102g * corresponding author. e-mail addresses: prasantakumarghosh43@gmail.com (p.k. ghosh), jkdey1971@gmail.com (j.k. dey) imperfect production inventory model with uncertain elapsed time prasanta kumar ghosh 1 and jayanta kumar dey 2* 1 yogoda satsanga palpara mahavidyalaya, purba medinipur, west bengal, india 1 mahishadal raj college,mahishadal, purbamedinipur, west bengal, india received: 25 january 2019; accepted: 15 july 2019; available online: 23 august 2019. original scientific paper abstract: most of the classical inventory control model assumes that all items received conform to quality characteristics. however, in practice, items may be damaged due to production conditions, transportation and environmental conditions. modelling such real world problems involve various indeterminate phenomena which can be estimated through human beliefs. the uncertainty theory proposed by liu (2015) is extensively regarded as an appropriate tool to deal with such uncertainty. this paper investigates the optimum production run time and optimum cost in an imperfect production process, where the rate of imperfect items are different in different states of the process. the process may be shifting from ‘in-control’ state to the ‘out-of-control’ state is an uncertain variable with certain uncertainty distribution. some propositions are derived for the optimal production run time and optimized the expected total cost function per unit time. finally, numerical examples have been illustrated to study the practical feasibility of the model. keywords: inventory, imperfect production, uncertain variables, uncertain distribution, expected value model. 1. introduction in some real uncertain situation, we have to depend on domain experts to represent the belief degree when no samples are available to estimate a probability distribution. to deal with uncertainty in human belief, which is neither random nor fuzzy, liu (2009), (2015), (2016) introduced uncertainty theory. it deals with modeling of uncertainty, based on normality, monotonicity, self-duality, countable sub-additivityand product measure axioms. uncertain variable, uncertain set and uncertain measure are the basic tools to describe the uncertain phenomenon. mailto:prasantakumarghosh43@gmail.com mailto:jkdey1971@gmail.com ghosh and dey/decis. mak. appl. manag. eng. 3 (2) (2020) 1-18 2 expected value operator for the uncertain variable has become asignificant role in both theory and practice. the expected value of a monotone function of an uncertain variable is a lebesgue-stieltjes integral of the function concerning its uncertainty distribution. liu (2012) proposed the concept of expected value of uncertain variables to rank the variables. liu (2016) also verifiedthe linearity of the expected value operator. liu and ha (2010) derived a useful formula for calculating the expected values of strictly monotone functions of independent uncertain variables. liu (2016) founded uncertain programming involving uncertain variables, which has been used to model in practical view of the system reliability design, project scheduling problem, transportation problem (gao and kar (2017), majumder et al. (2018))portfolio selection problem (kar et al. (2017); qin et al. (2016);majumder et al. (2018)) and facility location problem ([liu et al. (2015), ke et al. (2015)). in financial mathematics, liu (2009), (2016) gave an uncertain stock model and european option price formula. zhou et al. (2014) studied a dynamic recruitment problem with enterprise performance in an uncertain environment and presented an optimal search strategy for the firms' employment decisions. chen et al. [3] analyses the pricing and effort decisions of a supply chain with a single manufacturer and single retailer considering the demand expansion effectiveness of sales effort under uncertainty. in most of the classical economic production quantity (epq) model, it is assumed that the production process is always in good condition and produces 100% perfect quality items. but this assumption may not true in a real production system. in most of the practical situations, the production process continuously deteriorates and produce a certain percentage of defective (imperfect) items. rosenblatt and lee (1986) studied the effect of production process deterioration on the epq model and considering the shifting of the process from ‘in-control’ state to ‘out-of-control’ state, which is exponentially distributed. the deteriorating production system is an imperfect production system that has a threshold level of defectiveness to separate the system into in-control and out-of-control states. khouja and meherez (1994), have considered the elapsed time until the production process shifts to an ‘out-ofcontrol’ state to be an exponentially distributed random variable and shown the weak and strong relationship between the rate of production and process quality. sana (2010) extended the model of khouja and meherez (1994), assuming the percentage of defective items varies not linearly with production rate and production run time. hariga and ben-daya(1998) extended the model of rosenbaltt and lee(1986) considering the general shift distribution and optimal production run time to be unique. yehet al. (2000), (2007) has considered different defective rates in in-control and out-of-control state for the imperfect production process and investigate production run length with warranty policy. chen and lo (2006) have developed an imperfect production process with allowable shortages and the products are sold with free minimal warranty. the probabilities of imperfect items in both states are different. again the epq models are derived under very modern heuristics and soft computing techniques, especially usingprobabilistic reasoning, fuzzy logic, roughfuzzy logic,uncertainty theory etc. chen et al. (2005) derived a fuzzy epq model with fuzzy opportunity cost. wang et al. (2009) investigate a model of the imperfect production process with fuzzy elapsed time. qin and kar (2013) investigate a newsboy model under uncertain environment. wang et al. (2015) contributed a paper is to provide a more general framework for single-period inventory problem by considering single-item and multiple items with a budget constraint under uncertain and random environment. the proposed models consider both uncertain imperfect production inventory model with uncertain elapsed time 3 and random behavior of the demands and cover not only the random instance but also the single-fold uncertain situation. in this paper, we investigate optimum production run time and optimum cost in an imperfect production process, where the rate of imperfect items are different in different states of the process. the process may be shifting from ‘in-control’ state to the ‘out-of-control’ state is an uncertain variable with certain uncertainty distribution proposed by liu (2009). the rest of the manuscript is organized as follows. some preliminary concepts related to our study are discussed in section 2. section 3, states the assumptions and notations of the model. section 4 and 5 are for the mathematical modeling and solution respectively. section 6 provides numerical examples and discuss the results. some sensitivity analyses are provided in section 7. the paper summarizes and concludes in section 8. 2. preliminaries before presenting the inventory model in an uncertain environment, in this section, we introduce some useful definitions and fundamental results of liu's uncertainty theory. uncertainty theory is an extremely important feature of the real world. the interpretation of uncertainty measure is the personal belief degree of an uncertain event. definition 1. let s be a non-empty set and  a  -algebra over s . each element a in s is called an event. a set function m from  to [0, 1] is called an uncertain measure if it satisfies the following axioms. axiom 1: (normality) { } 1m   for the universal set s . axiom 2: (duality) cm{a}+m{a }=1 for any event a in s . axiom 3: (subadditivity) for every countable sequence of events a1, a2..... . we have 11 { } { } i i ii m a m a      . the triplet ( , , )s m is called an uncertainty space. axiom 4: (product axiom) let ( , , ) k k k s m be uncertain spaces for k = 1, 2, …., n. then the product uncertain measure m is an uncertain measure on the product  algebra 1 2 ..... n     , satisfying k 1 1 { } m { } n k k k n k m a min a     . definition 2 (liu, 2015). a measurable function  from an uncertainty space ( , , )s m to the set of real numbersis defined as an uncertain variable. i.e. for any borel set b of real numbers, the set { } { / ( ) }b s b       is an event. definition 3 (liu, 2016). in practice, the uncertain variable is described by the concept of uncertainty distribution  . which is defined by ( ) { } x m r       . it is a monotone increasing function except ( ) 0 and ( ) 1.     definition 4 (liu, 2016). an uncertain variable  is said to have a first identification if, (i) ( )x is a nonnegative on r such that sup ( ) ( )) 1x y x y    (ii) and for any set b of real numbers, we have ghosh and dey/decis. mak. appl. manag. eng. 3 (2) (2020) 1-18 4 sup ( ) sup ( ) 0.5 { } 1 sup ( ) sup ( ) 0.5 c c x b x b x b x b x if x m b x if x              definition 5 (liu, 2016). an uncertain variable  is said to have a second identification  if, ( )x is a nonnegative and integrable function on r such that ( ) 1 r x dx  ; for any set b of real numbers, we have ( ) ( ) 0.5 { } 1 ( ) ( ) 0.5 0.5 otherwise c c b b b b x dx if x dx m b x dx if x dx                     definition 6 (liu, 2016). let be  an uncertain variable. then the expected value of  is denoted as [ ]e  and is defined by 0 0 [ ] { } { } e m d m d              , provided at least one of the two integrals is finite. theorem 1 (liu, 2016). let  be an uncertain variable with uncertainty distribution  such that lim ( ) 0 and lim ( ) 1 x x         . if n(  ) is a monotone function of x and if [ ( )]e n  exists, then 0 0 [ ( )] { ( ) ) { ( ) ) = ( ) ( )e n m n d m n d n d                    proof: let n(  ) is a monotone function with ( )n    and by the properties of uncertainty distribution ( ) , we have lim { } ( ) lim(1 ( )) ( ) 0m n n              and lim { } ( ) lim ( ) ( ) 0m n n              . assuming that expected value [ ( )]e n  is finite. let us consider two real number 1 2 and   such that 1 2 <0<   , then 1 12 2 2 2 11 ( ) ( )1 (0)0 0 (0) { ( ) } { ( )} = { } ( ) [ { } ( )] n n nn m n d m n d m u dn u m u n u                         1 1 2 2 1 1 ( ) ( ) 1 2 2 (0) (0) ( ) { } (1 ( ( )) ( ) ( ) n n n n n u dm u n n u d u                   taking as 2r   , it follows that 1 0 (0) { ( ) } ( ) ( ) n m n n u d u         . similarly, imperfect production inventory model with uncertain elapsed time 5 1 1 1 1 1 1 1 1 0 0 ( ) 1 (0) (0) 1 1 1 ( ) { ( ) } { ( )} { } ( ) ( ( ) ( ) ( ) n n n n m n d m n d m u dn u n n u d u                                   taking as 1 , it follows that   1 0 (0) { ( ) } ( ) ( ) n m n d n u d u           -1 -1 0 0 n (0) n (0) [ ( )] { ( ) } { ( ) } = ( ) ( ) ( ) ( ) = ( ) ( ) e n m n d m n d n u d u n u d u n u d u                           hence the theorem. note that the expected value of a monotone function nothing but the lebesguestieltjes integral of the function with respect to its uncertainty distribution. definition 7 (liu, 2016). to estimate the unknown parameter  of an uncertain distribution ( )x  , liu employed the principle of least squares, which minimizes the sum of squares of the distances of the expert’s experimental data to the uncertainty distribution. if the expert’s experimental data are 1 1 2 2 ( , ), ( , ).....( , ) n n x x x   and the vertical direction is accepted. then we have 2 1 min ( ( ) ) n i i i x       . the optimum solution ˆ of   is called the least squares estimate of . example 1: assume that an uncertainty distribution has a liner form with two unknown parameters  and  . we assume that the following are expert’s experimental data, (1,0.15),(2,0.45),(3,0.55),(4,0.85),(5,0.95). then the least squares uncertainty distribution is 0 ( ) ( ) / ( ) 1 x x x x x                  where  = 0.2273 and  = 4.7727. example 2: let ( , )l   be a linear uncertain variable. then its uncertainty distribution is 0 ( ) ( ) / ( ) 1 x x x x x                  and its inverse uncertainty distribution is 1 ( ) ( )        . the expected value can be attained. ghosh and dey/decis. mak. appl. manag. eng. 3 (2) (2020) 1-18 6 1 0 [ ] ( ( ) ) 2 e d              . example 3: let ( , , )z a b c  be a zigzag uncertain variable. then its uncertainty distribution is 0 ( ) / 2( ) ( ) ( 2 ) / 2( ) 1 x a x a b a a x b x x c b c b b x c x c                 and its inverse uncertainty distribution is 1 (1 2 ) 2 0.5 ( ) (2 2 ) (2 1) 0.5 a a b c                   and the expected value can be attained 0.5 0 1 0.5 [ ] ((1 2 ) 2 ) 2 ((2 2 ) (1 2 ) ) 4 e a b d a b c b c d                    3. assumptions& notations the following notations and assumptions are used in developing the model. 3.1. assumptions 1. the production process has two states ‘in-controlstate’ and ‘outofcontrol state’. at the beginning of each production process, the system is in ‘in-control-state’ and produces defective items at a rate 1 1 (0 1)   . during the production run, the process may be shifted from “in-control-state” to “out-of-control-state” at any uncertain time in production period and produces re-workable defective items at a rate 2 2 2 1 (0< <1 ) and ( )    . 2. the elapsed time until the production process shift is  assumed to be an uncertain variable with uncertainty distribution  . 3. the production rate and demand rate are constant and deterministic. 4. full (100%) inspection is considered at a certain cost. 5. the re-workable defective products are reworked at the end of the screening process with negligible reworked time. 6. the process is brought back to its initial conformable state ‘in-control-state’ for each setup so, incurred more setup cost including restoration cost which is fixed. 7. in real life situation for the competition market, shortages are not allowed. 8. the time horizon is infinite. 3.2. notations  ; uncertain variable (denote the shifting time from ‘inimperfect production inventory model with uncertain elapsed time 7 control’ state to ‘out-of-control’ state  ; uncertainty distribution p ; production quantity per unit time (deterministic and constant) d ; demand rate k ; setup cost 1  ; the rate of reworkable -defective items in ‘in-control’ state. 2  ; the rate of reworkabledefective items in ‘out-of-control’ state. t ; production up-time t ; production cycle length ( , ) rd ni t  ; the number of defective items in the production process. p c ; production cost/item/time s c ; screening cost/item/time h c ; holding cost/item/time r c : rework cost/item/time ( )ac t : average cost per unit time [ ( )]e ac t ; expected average cost per unit time 4. mathematical formulation of the proposed inventory model: 4.1. mathematical formulation in this proposed model, under the above assumptions, we consider an imperfect production process, in which the production process is in two states 'in-control' and 'out of control' state. we consider the production system has production uptime up to time t. in between production starting point and production uptime, the system shifted from 'in control' state to 'out of control' state at any uncertain time point having uncertainty distribution  . an inspection section separates the perfect and re-workable defective quality items through 100% screening process and the screening process finishes after production end. the perfect items are kept for satisfying customer demand and re-workable defective items are reworked with a cost after screening and stored in the main inventory. in the ‘in-control’ and 'out-ofcontrol' states, two types of items are produced among which the re-workable defective are produced at the rate 1 1 2 2 2 1 (0 1) and (0 1), where           respectively. so first we calculate the number of re-workable defective items throughout the production cycle. let ( , ) rd ni t  be the number of re-workable defective items in the production process, then 1 1 2 ; ( , ) ( ) ; rd pt t ni t p p t t               (1) the length of the production cycle is pt t d  . production cost= p c pt . rework cost= ( , ) r rd c ni t  . ghosh and dey/decis. mak. appl. manag. eng. 3 (2) (2020) 1-18 8 holding cost= 2 ( ) 2 h c t p p d d  .and screening cost= s c pt . 2 ( ) total cost ( , ) 2 h p s r rd c t p p d k c pt c pt c ni t d        (2) therefore the average cost per unit time ( ) ( , )kd ac(t)= ( ) pt 2 h r rd p s c p d t c d ni t c c d pt      (3) expected average cost per unit time is ( ) e[ ( , )]kd e[ac(t)]= +( ) pt 2 h r rd p s c p d t c d ni t c c d pt     (4) proposition 1. if  is a positive uncertain variable with uncertainty distribution with (0) 0 and ( ) 1.      then ( , ) rd ni t  is a positive uncertain variable and the expected value of ( , ) rd ni t  is 1 2 1 0 [ ( , )] ( ) ( ) t rd e ni t p t p x dx       (5) proof: let be  an uncertain variable. then the expected value of the uncertain variable  is denoted as [ ]e  and is defined by 0 0 [ ] { } { } e m d m d              . so, 0 0 [ ( , )] { ( , ) } { ( , ) } = ( , ) ( ) rd rd rd rd e ni t m ni t d m ni t d ni t u d u                  here ( , ) rd ni t  is positive valued, then 0 1 2 1 0 0 1 2 1 2 1 0 ( , ) ( ) ( , ) ( ) ( ) ( ) ( ) ( ) ( ) (0) (1 ( )) ( ) ( ) rd rd t t t t ni t u d u ni t u d u p ud u p t u d u p td u p t t p t p t t p x dx                                       (6) as (0) 0  , it follows that, 1 2 1 0 [ ( , )] ( ) ( ) t rd e ni t p t p x dx       (7) hence the result follows. corollary 1. if  be an uncertain variable with uncertainty distribution  whose support is ( , )a b ,then [ ( , )] rd e ni t  reduces to 1 2 1 [ ( , )] ( ) ( ) t rd a e ni t p t p x dx       proposition 2. the optimum production run length t  exists and is unique, which optimizes the function imperfect production inventory model with uncertain elapsed time 9 2 1 1 0 2 1 0 ( ) ( ) ( ) ( ) ( ) 2 ( ) ( ) ( ) t h r p s r t r c p d t c dkd h t c c c p d x dx pt t c d c t x dx t                    (8) where, 1 ( )kd c(t)= ( ) pt 2 h p s r c p d t c c c p d      (9) here c(t) represents the expected average cost per unit time for the classical epq model with constant defective rate and rework. proof: we have, 1 ( )kd c(t)= ( ) pt 2 h p s r c p d t c c c p d      and 2 ( ) kd c (t)= 0 2 pt h c p d    gives 2 ˆ ( ) h kd t c p p d   . again 3 2 ( ) 0 kd c t pt    ˆfor t t . so c(t) has a unique minimum at ˆ t t . here ( )c t is the cost function for the standard epq model with a constant defective rate of product throughout the production period and it is convex for all 0t  . let 2 1 0 ( ) ( ) ( ) t r c d g t x dx t     , from which it follows that 2 1 2 0 ( ) ( ) [ ( ) ( ) ] t r c d g t t t x dx t        . as 0 ( ) 1x   , ( )x is monotone non-decreasing and 0 ( ) ( ) t t t x dx   implies ( )g t  2 1 0 ( ) ( ) t r c d x dx t    is non-decreasing and 0 1 0 ( ) 1 t x dx t    implies 2 1 0 ( ) ( ) r g t c d     . as ( )h t is a sum of the convex function of c(t) and g(t), which is bounded and non-decreasing, there exist a unique 0t   such that t t  for which h(t) is minimum. if ( ) 0h t  is satisfied for t t   , then 2 1 2 20 ( )( ) { [ ( ) ( ) ] } 0 2 t hr c p dc d kd t t x dx t pt          from which we get 2 1 2 1 2 2 20 ( )( ) ( ) ( ) ] { ( ) } 2 t hr r c p dc d c d kd x dx t t t t pt            . and in this case 1 2 1 ( ) ( ) ( ) ( ) ( ) p s r h r h t c c c d c p d t dc t              (10) ghosh and dey/decis. mak. appl. manag. eng. 3 (2) (2020) 1-18 10 5. solution we cannot find out closed form solution of the optimum production run from the above objective function. using the search procedure along with the bisection method and bound of the optimum production time, a search procedure is appliedtofind out t  such that 2 ( ) ( ) 0h t t f t      , which gives 2 r 2 1 0 ( ) ( ) t +c d( )[ ( ) ( ) ] 2 t h c p d kd f t t t x dx p             (11) 5.1 algorithm of the search procedure step 1: set 0lt  and 2 ( ) u h kd t c p p d   where ( ) 0 l f t  and ( ) 0 u f t  , implies the existence of a value t  for which ( ) 0f t   . step 2: compute 2 l u o t t t   and the value of 0( )f t . step 3: if ( )of t , where( 0, error tolerance).control goes to step 5 otherwise goes to next step. step 4: if ( ) 0of t  , set u ot t , otherwise ( ) 0of t  , set l ot t , go to step 2. select the value of ot as optimum value t  . step5. terminate the search procedure. 6. numerical example the numerical examples are given for illustrative and verification of the real world problem. case 1: linear uncertain distribution consider the uncertain variable  as linear with support ( , )a b , where a=2.5 and b=10.0 and other parameters are k = 2000, 1  0.10, 2  0.20, hc  5.0, d = 50, p = 75, pc  50.0, sc  5.0., rc = 10.0. by the search procedure, the optimum production time is t  = 4.53537 and corresponding optimum cost = 3380.49. imperfect production inventory model with uncertain elapsed time 11 figure 1. expected average cost function is convex with respect to production time table 1: expected optimum production time and expected optimum cost/unit time with respect to the different change of the parameter. parameter change in parame ter optimum productio n time optimum expected average cost/unit time parame ter change in paramet er optimum productio n time optimum expected average cost/unit time p -20% -10% 0% +10% +20% 7.71744 5.64864 4.53537 3.81795 3.30973 3220.65 3315.25 3380.49 3429.20 3467.34 1  -20% -10% 0% +10% +20% 4.51952 4.52741 4.53537 4.54340 4.55750 3371.10 3375.79 3380.49 3385.19 3389.88 d -20% -10% 0% +10% +20% 3.46624 3.95285 4.53537 5.26842 6.25753 2851.74 3121.64 3380.49 3627.14 3859.37 2  -20% -10% 0% +10% +20% 4.56789 4.55150 4.53537 4.51952 4.50393 3379.36 3379.88 3380.49 3381.10 3381.69 k -20% -10% 0% +10% +20% 4.06436 4.30631 4.53537 4.75341 4.96138 3318.47 3350.33 3380.49 3409.20 3436.63 ghosh and dey/decis. mak. appl. manag. eng. 3 (2) (2020) 1-18 12 figure 2. production time with respect to the percentage change of the different parameter figure 3. optimum expected average cost/unit time with respect to the percentage change of the different parameter. case 2: zigzag uncertain distribution consider the uncertain variable  as zigzag z(a, b, c) with support ( , )a c , where a = 5.0 and b = 8.0 and c=10.0 and other parameters are k = 2000, 1   0.1, 2   0.20, h c  2.0, d=50,p=75, p c  25.0, sc  2.0, rc = 5.0. 0 1 2 3 4 5 6 7 8 9 -30% -20% -10% 0% 10% 20% 30% o p ti m u m p ro d u ct io n t im e change in parameter production rate demand rate setup cost rate of nonconforming items in 'in-control-state' rate of nonconforming items in 'out-of controlstate' 0 500 1000 1500 2000 2500 3000 3500 4000 4500 -30% -20% -10% 0% 10% 20% 30% o p ti m u m t o ta l co st / u n it t im e change in parameter production rate demand rate setup cost rate of nonconforming items in 'in-control-state' rate of nonconforming items in 'out-of controlstate' imperfect production inventory model with uncertain elapsed time 13 by the search procedure, the optimum production time is t  = 4.55557 and corresponding optimum cost = 3379.72. figure 4. expected cost function is convex with respect to production time table 2. expected optimum production time and expected optimum cost/unit time with respect to the different change of the parameter paramet er change in parameter optimu m product ion time optimum expected average cost/unit time para mete r chan ge in para meter optimu m product ion time optimum expected average cost/unit time p -20% -10% 0% 10% 20% 7.82140 5.68945 4.55557 3.82879 3.31550 3217.68 3313.77 3379.72 3428.82 3467.18 1  -20% -10% 0% 10% 20% 4.54340 4.54947 4.55557 4.56171 4.56789 3370.19 3374.96 3379.72 3384.49 3389.26 d -20% -10% 0% 10% 20% 3.47242 3.96434 4.55557 5.30377 6.32211 2851.56 3121.24 3379.72 3625.80 3857.09 2  -20% -10% 0% 10% 20% 4.58037 4.56789 4.55557 4.54340 4.53139 3378.79 3379.26 3379.72 3380.19 3380.64 k -20% -10% 0% 10% 20% 4.08052 4.32457 4.55557 4.77540 4.98556 3317.97 3349.69 3379.72 3408.30 3435.62 ghosh and dey/decis. mak. appl. manag. eng. 3 (2) (2020) 1-18 14 figure 5. production time with respect to the percentage change of different parameters figure 6. optimum cost/unit time with respect tothe percentage change of different parameters. 7. sensitivity analysis since the shifting time point from in–control state to out-of-control state is uncertain variable in between beginning and end of the production run and it 0 1 2 3 4 5 6 7 8 9 -30% -20% -10% 0% 10% 20% 30% e x p e ct e d o p ti m u m p ro d u ct io n t im e change in parameter production rate (p) demand rate(d) setup cost (k) rate of defective items in 'incontrol state' rate of defective items in 'out-ofcontrol state' 0 500 1000 1500 2000 2500 3000 3500 4000 4500 -30% -20% -10% 0% 10% 20% 30%e x p e ct e d o p ti m u m p ro d u ct io n t im e change in parameter production rate (p) demand rate(d) setup cost (k) rate of defective items in 'incontrol state' rate of defective items in 'out-ofcontrol state' imperfect production inventory model with uncertain elapsed time 15 depends on support set of uncertainty distribution, so, the optimum production time and optimum cost per unit time depending on that. from table-1, figure-2 and table-2, figure-5, for both the cases, it is observed that optimum production time is decreased with increase of production rate and rate of reworkable defective items in ‘out-of-control’ state and it increases with the increase of other parameter demand rate, setup cost and rate of reworkable defective items in ‘in-control’ state. similarly from table-1, figure-3 and table-2, figure-6, for both the cases, it follows that optimum cost increases with the increases of all parameters. optimum production time is sensitive to the change of parameters production rate (p) and demand rate (d) and optimum cost per unit time is sensitive to the change of parameter demand rate (d). optimum production time is moderately sensitive with the change of a parameter (p), slightly sensitive to the change of a parameter (d) and insensitive to the change of all other parameters. in the same way, the optimum cost/per unit time is slightly sensitive to the changes of the parameter (d) and insensitive to all other parameters. 8. conclusion in this article, we have discussed an imperfect production inventory model in an uncertain environment. it is assumed that an imperfect production process has two states ‘in-control’ state and ‘out-of-control’ state. here we also assumed that the elapsed time of the production process follow uncertain shift distribution, which is an uncertain variable follows an uncertainty distribution. the basic difference from the earlier research article is that our model is on the study of uncertain phenomena while the stochastic is about the study of stochastic phenomena. for the lack of historical data, the shifting of the production process is quantified by domain experts’ belief degree and by the principle of least square, the manager should follow a particular uncertainty distribution. the optimum production time and optimum cost depend on the type of uncertainty distribution along with the support set of that. two case examples justify the numerical verification of theorem and propositions in this proposed model on linear and zigzag uncertainty distribution with the same support. it follows that the expected optimum cost is near about approximately the same for both the uncertainty distribution, though optimum production time is slightly different. finally, for illustrating the procedure, an algorithm is designed to find out the optimum goal. inthe future, we would like to extend our modelfor the imperfect production process with random uncertain circumstances. moreover, the possible extension of different variants of imperfect production inventory problem like demand variability and trade credit policy to uncertain single/multi-objective models will also be the area of our research interest. author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research received no external funding. conflicts of interest: the authors declare no conflicts of interest. ghosh and dey/decis. mak. appl. manag. eng. 3 (2) (2020) 1-18 16 references chen, c. k., & lo, c. c. 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(2014). an uncertain search model for recruitment problem with enterprise performance. journal of intelligent manufacturing, 28(3), 695-704. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0318102022t * corresponding author. dineshkumartripathi1980@gmail.com (d.k. tripathi), nigamsantosh01@gmail.com (s.k. nigam), pratibha138@gmail.com (p. rani), raoofstat15@gmail.com (a.r. shah) new intuitionistic fuzzy parametric divergence measures and score function-based cocoso method for decision-making problems dinesh kumar tripathi1, santosh k. nigam1, pratibha rani2* and abdul raoof shah3 1 department of mathematics, government college satna, madhya pradesh, india 2 department of engineering mathematics, koneru lakshmaiah education foundation, vaddeswaram, andhra pradesh, india 3 department of statistics, government degree college, pulwama, jammu & kashmir, india received: 7 july 2022; accepted: 10 october 2022; available online: 17 october 2022. original scientific paper abstract: the present study introduces a decision-making approach with the combined compromise solution (cocoso) under intuitionistic fuzzy sets (ifss) named as the if-cocoso method based on proposed divergence measures and score function. the aim of the presented approach is to obtain an effective solution for multi-criteria decision-making problems on ifss context. in this line, a new procedure is presented to derive the criteria weights using generalized score function and parametric divergence measures of ifss. to compute the criteria weight, a generalized score function and parametric divergence measures are developed on ifss and discussed some interesting properties. further, the presented approach is applied to rank and evaluate the therapies for medical decision making problems, which demonstrates its applicability and feasibility. finally, comparative and sensitivity analyses are discussed for validating the developed method. key words: intuitionistic fuzzy sets, combined compromise solution, medical decision-making, divergence measure, score function. 1. introduction healthcare decision-making is a multifaceted procedure which involves data processing, assessment of evidence, and application of relevant knowledge in order mailto:dineshkumartripathi1980@gmail.com mailto:nigamsantosh01@gmail.com mailto:pratibha138@gmail.com mailto:raoofstat15@gmail.com tripathi et al./decis. mak. appl. manag. eng. (2022) 2 to choose the suitable interventions. decision-making in healthcare is challenging for experts because of the high level of ambiguity, complexity of decisions, a large number of tangible and intangible variables, and multiple objectives involved (basile et al., 2022). for such decisions, “multi-criteria decision-making (mcdm)” approaches are definitely appropriate. mcdm is defined as “an umbrella term to describe a collection of formal methods, which seek to take explicit account of numerous criteria in helping individuals or groups that explore decisions” (thokala et al., 2016). in the process of healthcare mcdm, lutz and bowers (2000) confirmed the capability of patients in making decisions in regard to what they want and need. on the other hand, patients’ skills and knowledge might be too insufficient to make a significant contribution to final clinical results in the group decision making procedure(lee and lin, 2010). if the patient’s circumstance and viewpoints are considered well, it will be clear that their judgments are essentially inaccurate, involving lots of uncertainties. thus, the present paper applies “intuitionistic fuzzy sets (ifss)” to capture inaccurate or ambiguous therapeutic information that may appear during medical decision-making analyses. the “fuzzy sets (fss)” established by zadeh (1965) is very popular in the decision-making field (bozanic et al., 2019; bhattacharya et al., 2022; torğul et al., 2022). atanassov (1986) modified the fss and developed the ifss that is known as having both “non-membership function (nf)” and “membership function (mf)”. the ifs has made available mathematical framework of a higher efficiency in managing the situations in which the “decision-expert (de)” is of two minds at the same moment whether to approve a certain decision or not. lots of topics concerning decision making with ifss have been taken into account (stanujkić and karabašević, 2018; kumari and mishra, 2020; son et al., 2020; kushwaha et al., 2020; rahman, 2022; hezam et al., 2022). “intuitionistic fuzzy numbers (ifns)” have been ranked in many situations through their conversion into representative crisp values(yager, 2004). in case of ifns, the representative crisp values are noted as score degrees and accuracy degrees. a score function of ifss was developed by chen and tan (1994) using the mf and nf. after that, hong and choi (2000) made an improvement on this function through adding an accuracy function to it. furthermore, in (wang and chen, 2018; zeng et al., 2019), authors have offerred other functions to rank ifns. scholars made use of the divergence measure for the purpose of assessing the discrimination degree between objects. pal (1993) was a pioneer in the fuzzy divergence measure. then, the idea of integrating divergence measure with ifss was introduced by vlachos and sergiadis (2007); they used it in segmentation of images, diagnosis of diseases, and recognition of patterns. in recent years, a numerous of divergence measures have been discussed by numerous researchers (montes et al., 2015; ansari et al., 2018; xiao, 2019; arora & naithani, 2022). although, the previous studies have focused on different divergence measures of ifss, they have not integrated de preferences with the measure. additionally, the extant measure is in a linear order; thus, it gives no precise nature of the alternative. as a result, the present study keeps the efficiency and flexibility criteria of ifss and proposes an innovative generalized parametric directed divergence measure that is capable to measure the fuzziness of a set. to this end, a divergence measure of order α and β is offered to give higher reliability and flexibility to des for various values of these parameters. the formulations of such measures are done by taking the convex linear combinations of mfs between two ifss. on the basis of such representations, a number of desirable properties of these measures are examined. it is analyzed that the existing divergence measures are distinct cases of the proposed measure; as a result, the proposed measure is with a higher suitability and generalizability. new intuitionistic fuzzy parametric divergence measures and score function-based… 3 recent years, numerous mcda approaches such as “technique for order of preference by similarity to ideal solution (topsis)”, “visekriterijumska optimizacija i kompromisno resenje (vikor)”, “complex proportional assessment (copras)”, “weighted aggregated sum product assessment (waspas)”, “evaluation based on distance from average solution (edas)” and others have been commenced to cope the realistic mcda problems. while employing these approaches in solving mcda problems, the ranking outcomes produced by the topsis, vikor, copras, waspas and edas may alter significantly in accordance with the variation of weight distributions of attributes (batool et al., 2021; rudnik et al., 2021, mishra et al., 2022b,c). alternatively, the dependability and permanence of the outcomes obtained by these approaches are inadequate (wen et al., 2019). to conquer this inadequacy, yazdani et al. (yazdani et al., 2019a) presented an mcda method, named as cocoso model, which unites the aggregation of compromise value with different models to find a “compromise solution (cs)”. the cocoso model is a combination of “weighted sum model (wsm)” and “exponentially weighted product model (ewpm)”. yazdani et al. (2019b) gave an integrated tool with the “decision making trial and evaluation laboratory (dematel)” and “best-worst method (bwm)” with cocoso method to choose suitable supplier. rani and mishra (2020) presented a cocoso model for the evaluation of “sustainable waste electrical and electronics equipment (sweee)” recycling partner with “single-valued neutrosophic sets (svnss)”. tavana et al. (2021) combined the fuzzy bwm and cocoso methods with bonferroni functions for assessing and prioritizing the suppliers in reverse supply chains. mishra and rani (2021) designed a model by combining the cocoso and critic approaches under the context of svnss to solve the 3prlps assessment problem. bai et al. (2022) discussed a new decision-making methodology based on the q-rung orthopair fuzzy swara and cocoso methods to assess the “sustainable circular supply chain (scsc)” risks in manufacturing firm. narang et al. (2022) studied an integrated decision support system based on generalized heronian operator and cocoso method, and applied for portfolio analysis. mishra et al. (2022a) combined the archimedean copula operator with the cocoso model for the assessment of smart cities to adopt “internet of things barriers (iotbs)” on “fermatean fuzzy sets (ffss)”. accordingly, the present study has a three-folded objective. first, a new method proposes to select, rank and evaluate the significant therapy for patients based on multiple criteria. second, a novel mcdm method is developed based on the cocoso and ifss named as if-cocoso approach. recently, yazdani et al. (2019a) introduced the novel cocoso approach with the aid of some aggregation strategies. the use of the cocoso is limited within the context of ifss. thus, we present the cocoso to handle the medical decision making issues based on ifss. in summary, the current work has the following contributions: • to propose an innovative “generalized score function (gsf)” and divergence measures to determine the criteria weights. • to introduce an innovative gsf by considering the “hesitancy degree (hd)” that exists between the mf and nf of ifns. • to present two new parametric divergence measures of order α and β represented as class of α and (α,β) for ifss. • to introduce an algorithm for if-cocoso method based the cocoso approach and ifss. • to make clear how applicable and reliable is the proposed if-cocoso approach, an application of therapy selection problem for patients is discussed. tripathi et al./decis. mak. appl. manag. eng. (2022) 4 the rest part of this study is presented in the following way: an extensive literature review on fuzzy mcdm approaches in healthcare sciences is provided in section 2. section 3 discusses the preliminaries related to the ifss, new gsf and parametric divergence measures for ifss. section 4 introduces the if-cocoso method with the gsf and parametric divergence measures. section 5 discusses a case study for medical decision making using ifss, which illustrates the feasibility of the proposed method. section 5 shows the comparative and sensitivity analyses to illustrate the utility of presented method. section 7 focuses on the conclusion, limitations and future recommendations. 2.application of fuzzy mcdm models in medical problems in the context of decision analysis, mcdm provides a systematic way for the assessment of options/cases/alternatives over a set of pre-determined selected criteria. ifss have gained much interest from authors in medical mcdm. szmidt & kacprzyk (2004) provided an application of medical mcdm based on the distance measures between ifss. vlachos & sergiadis (2007) attempted to extend the approach proposed by szmidt & kacprzyk (2001) through taking into consideration a new measure based on symmetric discrimination information. the two ifs-based applications developed in the biomedicine field are the classification of bacteria by khatibi & montazer (2009) and the medical image segmentation (chaira, 2014). nowadays, numerous mcdm methods are effectively employed to help medical mcdm in fss and ifss contexts. hsieh et al. (2018) integrated “analytic hierarchy process (ahp)” and fuzzy topsis to assess the important parameters of human errors in emergency sections. honarbakhsh et al. (2018) employed the ahp to assess the “respiratory protection program (rpp)” in teaching hospitals under fss. according to findings of the present study, the most proper option for ambulance location is road network. xiao (2018) proposed a fuzzy mcdm called d-number to assess the “healthcare waste treatment (hwt)” technologies. in addition various of previous studies used the classical mcdm method to evaluate the healthcare management topics, for example; a novel approach developed by malekpoor et al. (2022) was on the basis of topsis and “case based reasoning (cbr)” to optimize dose planning procedure for minimizing the concerning prostate cancer. bahadori et al. (2018) integrated the “grey relational analysis (gra)” and vikor models to assess the “quality control effectiveness (qce)” in hospitals. kirkire et al. (2018) introduced a model using the sem-topsis for prioritizing the risk factors of medical devices. rani et al. (2020) presented the pythagorean fuzzy-copras approach to treat the “pharmacological therapy selection” for “type 2 diabetes (t2d)” problem. liu et al. (2021) presented and ranked the “medical waste treatment technologies (mwtts)” through the cocoso method on “pythagorean fuzzy sets (pfss)”. 3. concepts related to the proposed approach this section firstly presents the basic concepts of ifss and then proposes generalized score function and divergence measure under intuitionistic fuzzy environment. new intuitionistic fuzzy parametric divergence measures and score function-based… 5 3.1. preliminaries this section concentrates on the demonstration of the decision information based on ifss. in the following step, this study focuses on the aggregation based on the ifcocoso method. in the fss doctrine, the mf of an element is represented based on the interval number of [0, 1], whereas the nf essentially is complement. though, in concern, this hypothesis does not meet with human opinions. hence, atanassov (1986) defined the ifss as follows: definition 1. atanassov (1986) defined the mathematical form of an ifs ‘s’ on  1 2, , ..., tw w w = as  , ( ), ( ) : ,k s k s k ks w w w w =  (1) wherein : [0, 1]s  → and : [0, 1]s  → show the mf and nf of kw to s in , respectively, with the condition ( ) ( ) ( ) ( ) 0 1, 0 1, 0 1, . s k s k s k s k k w w w w w          +    (2) the intuitionistic index of an element kw  to s is defined by ( ) ( ) ( ) ( )1 and 0 1, .s k s k s k s k kw w w w w   = − −     next, xu (2007) described this term ( ), ( )s k s kw w  as an “intuitionistic fuzzy number (ifn)”, denoted by ( ), ,   = which satisfies  , 0,1    and 0 1.   +  definition 2 (xu, 2007). consider ( ), ,k k k  = 1, 2, ...,j t= be the ifns. then ( ) ( ) ( ) ( )and .k k k k k k     = − = +s (3) are the score and accuracy functions, respectively. definition 3 (xu, 2007). let ( ), ,k k k  = 1, 2, ...,j t= be the ifns. then the “intuitionistic fuzzy weighted average (ifwa)” and “intuitionistic fuzzy weighted geometric (ifwg)” operators are given by ( ) ( )1 2 1 1 1 , ,..., 1 1 , , k k t tt w t k k k k k k k ifwa w         = = =   =  = − −      (4) ( ) ( )1 2 1 1 1 , ,..., , 1 1 , kk t tt t k k k k k k k ifwg w         = = =   =  = − −      (5) where ( )1 2, , ..., t k t    = is a weight vector of , 1, 2, ,k k t = with 1 1 t k k  = = and  0, 1 .k  the divergence measure of ifss is a tool for calculating the amount of difference between ifss. vlachos and sergiadis (2007) firstly gave the formula for if-divergence measure. further, montes et al. (2015) defined the new axiomatic definition of ifdivergence measure, which as tripathi et al./decis. mak. appl. manag. eng. (2022) 6 definition 4 (montes et al., 2015). let ( ), .s t ifss  then ( ) ( ):j ifss ifss   → r is a divergence measure, if it fulfills the following postulates: (d1). ( ) ( ), , ;j s t j t s= (d2). ( ), 0j s t = if and only if ;s t= (d3). ( ) ( ), , ,j s u t u j s t for every ( );u ifs  (d4). ( ) ( ), , ,j s u t u j s t for every ( ).u ifs  3.2. generalized score function and divergence measure under ifss context in the present section, a new generalized score function and divergence measures are introduced for ifss. 3.2.1. generalized score function (gsf) in this subsection, a new gsf is developed by taking the “hesitancy degree (hd)” between the mfs and nfs of ifns. definition 5. suppose ( ), ,k k k  = 1, 2, ,k t= be the ifns. a gsf of an ifn is given by ( ) ( )( )* 1 21 1 ,k k k k      = + + − − s (6) where 1 2 1 21, , 0   + =  denotes the attitudinal behaviors of the proposed function, showing the degree of weighted average of the hd between the mf and nf of ifns. when 1 1 2 2 , = = the gsf is decreased to the score function proposed by liu and wang (2007). theorem 1. let ( ), ,k k k  = 1, 2k = be two ifns and ( ) * .s be a gsf. then ( )* .s holds the properties as follows: (s1) for any ifn, ( )* [0,1]k s ; (s2) ( )( )* 0,1 0,=s ( )( )* 1, 0 1=s ; (s3) if ( ) ( )* *1 2 s s , then ( ) ( ) * * 1 2 c c  s s ; (s4) for a fuzzy subset ( ),1 ,k k k  = − ( ) * k k  =s ; (s5) if ( ) ( )1 1 2 2 0   − − −  and ( ) ( )1 1 2 2 0,   + − +  then ( ) ( )* *1 2 s s ; (s6) if ( ) ( )1 1 2 2 0   − − −  and ( ) ( )1 1 2 2 0,   + − +  then ( ) ( )* *1 2 . s s proof: the proof is omitted. figure 1 depicts the score value of ( )( )* 1 2 1 21, , 0k    + = s when ( ) ( ), 0.1, 0.7k k  = and ( ) ( ), 0.6, 0.2k k  = . the color of each point ( )1 2,  on the simplex illustrates the entropy of the fixed ifns. as the value of 1  and 2  become bigger, the value of ( )* ks becomes bigger. new intuitionistic fuzzy parametric divergence measures and score function-based… 7 figure 1. gsf ( )* ks with respect to parameters  1 2, 0,1 ,   (a) when ( ) ( ), 0.1, 0.7k k  = (b) when ( ) ( ), 0.6, 0.2k k  = 3.2.2. parametric divergence measures for ifss here, we present two novel generalizable parametric divergence measures of order α and β denoted as class of α and (α,β). a number of favored properties are also taken into account. corresponding to the divergence measure introduced by parkash and kumar (2011), we introduce a parametric divergence measure as follows: ( ) ( )( ) ( )( ) ( ) ( ) ( ) 2 ln ln 2 1 1 , 1 2 1 2 1 s k s k t k wt w w s k k ce s t w t             +  =  = − − −   (7) tripathi et al./decis. mak. appl. manag. eng. (2022) 8 ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 2 2 ln ln 1 2 1 2 1 , s k s k s k t k s k t k w w w w w w s k s k w w                     + +     + − + − −   where 1 2   and 1 2 .  since eq. (7) is not symmetric, we define the symmetric version as follows: ( ) ( ) ( )1 , || || .j s t ce s t ce t s = + (8) next, we introduce a biparametric if-divergence measure as ( ) ( )( ) ( ) 2 ( ) ( ) ( ) ln 1 2 2 2ln 2 1 1 || 1 2 ws k w ws k t k t s k k ce s t w t         + =  = −    −  −      (9) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 2 2 ln ln 1 1 2 2 2 2 1 , w ws k s k w w w ws k t k s k t k s k s k w w            + +              + − + − −   where   and .  here, eq. (9) is not symmetric. thus, the symmetric measure is given by ( ) ( ) ( )2 , || || .j s t ce s t ce t s     = + (10) theorem 2. let ( ), , .s t u ifss  then, the measure ( ), ; 1, 2,j s t  = shown in eqs (8) and (10), satisfies (p1) ( ) ( ), , ,j s t j t s = (p2) ( )0 , 1,j s t  ( ), 1 c j s s  = if and only if ( ) ,s p  where ( )p  is the set of all crisp sets, (p3) ( ), 0 iff ,j s t s t = = (p4) ( ) ( ), ,c cj s t j s t = and ( ) ( ), , , c c j s t j s t   = (p5) ( ) ( ), ,j s t j s u  and ( ) ( ), , ,j t u j s u  for ,s t u  (p6) ( ) ( ), , ,j s t s t j s t = (p7) ( ) ( ) ( ) ( ), , , , ,j s t u j s u j t u u ifs   +    (p8) ( ) ( ) ( ) ( ), , , , ,j s t u j s u j t u u ifs   +    (p9) ( ) ( ), ,j s u t u j s t  for every ( ) ,u ifs  (p10) ( ) ( ), ,j s u t u j s t  for every ( ).u ifs  proof: properties (p1)-(p4) and (p6)-(p8) are easily proved from the definition. hence, we omit the proof. (p5) let .s t u  then s t u    and .s t u    it implies , s t s t s t s u s u s u            − + − + −  − + − + − , t u t u t u s u s u s u            − + − + −  − + − + − therefore, ( ) ( ), ,j s t j s u  and ( ) ( ), , .j t u j s u  to prove (p9) and (p10), the fixed set  is partitioned into 8 subsets as follows: new intuitionistic fuzzy parametric divergence measures and score function-based… 9 ( ) ( ) ( )  ( ) ( ) ( ) | |k k k k k k k kw s w t w u w w s w u w t w =   =  =  ( ) ( ) ( )  ( ) ( ) ( ) | |k k k k k k k kw s w t w u w w s w u w t w      ( ) ( ) ( )  ( ) ( ) ( ) | |k k k k k k k kw t w s w u w w t w u w s w      ( ) ( ) ( )  ( ) ( ) ( ) | | ,k k k k k k k kw u w s w t w w u w t w s w      which are indicated by 1 2 8 , , ..., .   based on montes et al. (2015), for each ; 1, 2, ..., 8, j j = we have ( )( ) ( )( ) ( ) ( )k k k ks u w t u w s w t w−  − and ( )( ) ( )( ) ( ) ( ) .k k k ks u w t u w s w t w−  − thus, from (p5), we get ( ) ( ), ,j s u t u j s t  and ( ) ( ), ,j s u t u j s t  for every ( ).u ifs  remark 1: it is interesting to point out that if 1 2 , → then eq. (7) converts to the divergence measures of ifss proposed by wei and ye (2010). also, when 1 2 → and ( ) ( )0s k t kw w = = , the proposed divergence given by eq. (7) converts to measure given by shang and jiang (1997). similarly, the measure 2 ( || )ce s t transforms to measure proposed by wei and ye (2010) for  = and when ( ) ( )0s k t kw w = = and , = then the proposed divergence shown in eq. (9) transforms to the measure given by shang and jiang (1997). 4. proposed if-cocoso method the present section aims at the development of an extended cocoso approach for the purpose of handling the mcdm issues on ifss. the presented method extends the method given in yazdani et al. (2019a). in mcdm process, consider a discrete set of m alternatives/options  1 2, ,..., mh h h h= over a set of criteria/attributes  1 2, ,..., .np p p p= consider a group of “decision makers/experts (dms/des)”  1 2, ,...,d d d d= to make a suitable decision for given alternatives. the procedural steps of if-cocoso method is depicted in the following steps (see figure 2): step 1: create the “linguistic decision-matrix (ldm)”. owing to the vagueness of human's mind, lack of data and imprecise knowledge about the options, the des define the ldm to evaluate his/her decision on option hi concerning a criterion pj. then, we construct the performance evaluation matrix ( )r k ij m n y y   =   for each dm considering the criterion set. tripathi et al./decis. mak. appl. manag. eng. (2022) 10 ( ) ( ) ( ) ( ) ( ) 1 1 11 11 1 1 ( ) 1 1 . , , , . , , n r r nr nr r k ij m n m m r m r mnr mnr m n p p h y y r h               = =       (11) step 2: obtain the importance of des. the importance ratings of des are given as ( ), ,r r r  = 1, 2, , .r = for the aim of viewing their relative importance in the mcdm model, the crisp weights of des are expressed by eq. (12). ( ) ( ) 1 2 , 1, 2,..., . 2 r r r r r r r r r        = − − = = − −   (12) step 3: build an “aggregated intuitionistic fuzzy decision-matrix (a-if-dm)”. next, for aggregating all the single opinions and constructing the collective decision matrix, we need to form an a-if-dm by using ifwa operator. let d= [𝑦𝑖𝑗]𝑚×𝑛 be the a-if-dm, where ( ), ,ij ij ijy  = 1, 2, ..., ,i m= 1, 2, ..., ,j n= where ( ) ( ) 1 1 1 1 , . r r ij ijr ijr r r y     = = = − −  (13) step 4: create the “normalized aggregated intuitionistic fuzzy decision matrix (na-if-dm)”. the na-if-dm ij m n m    =   is evaluated and given by ( ) ( ) ( ) , , for benefit criterion, , , for cost criterion. ij ij ij ij c ij ij ij y y       =  =  = (14) step 5: compute criteria weights. when attribute weights are completely unknown, then if-divergence measurebased weight-determining procedure is used to derive the weights of criteria. thus, the attribute weight is determined as ( )( ) ( ) ( )( ) ( ) * 1 1, * 1 1 1, 1 1 , 1 , . 1 1 , 1 m m ij tj ij i t t i j n m m ij tj ij j i t t i j y y y m j j y y y m    = =  = = =    − +  −  =     − +   −        s s (15) step 6: calculate the wsm and ewpm. the wsm ( )(1)i value for each option is calculated using the ifwa operator as (1) 1 . n i j ij j   =  =  (16) new intuitionistic fuzzy parametric divergence measures and score function-based… 11 the ewpm ( )( 2)i value for each alternative is computed using the ifwg operator as ( 2) 1 . n i j ij j   =  =  (17) step 7: calculate the “balanced compromise scores (bcss)” of each option. here, the following procedures are applied to find the of alternatives, which are derived as ( ) ( ) ( ) ( ) ( )( ) * (1) * ( 2) 1 * (1) * ( 2) 1 , i i i m i i i q =  +  =  +  s s s s (18) ( ) ( ) ( ) ( ) ( ) * (1) * ( 2) 2 * (1) * ( 2) , min min i i i i i i i q   = +   s s s s (19) ( ) ( ) ( ) ( ) ( ) ( ) ( ) * (1) * ( 2) 3 * (1) * ( 2) 1 , max 1 max i i i i i ii q      + −  =  + −  s s s s (20) where  is a strategic coefficient and  0,1 .  generally, we take 0.5. = step 8: find the “overall compromise solution (ocs)” of alternatives. the ocs (𝑄𝑖) of each option is determined by ( ) ( ) ( )( ) ( ) ( ) ( )( ) 1 31 2 3 1 2 31 . 3 i i i i i i i q q q q q q q= + + + (21) to end, prioritize the options by arranging the ocs ( )iq in descending order. tripathi et al./decis. mak. appl. manag. eng. (2022) 12 figure 2. flowchart of the if-cocoso method 5. application of proposed method in medical decision-making in the healthcare system, mcdm regarding the patient healthcare structure is more complicated in compare to decision making procedure based on individual case because of multiple dms such as the patients, the families of patients and healthcare personnel. in the patient-centered healthcare system, there is a requirement to adopt group decision making techniques involving family opinions, patient inclinations and new intuitionistic fuzzy parametric divergence measures and score function-based… 13 professional judgments by medical personnel and team. the subjects and perspectives of group decision making are combined for evaluating the patient healthcare problems. the patient-centered care is a healthcare system which provides preferences and needs of patients as well as the autonomous of the patients to decide for care and treatment of themselves (pelzang, 2010; greene et al., 2012). therefore, the patient-centered care considers the relatives and patients are experienced to choose their own expectations and needs and they are eligible to take decisions and select their preferences. however, involving patients, relatives and professional medical decision-makers to provide the decision information is basically vague and contains numerous uncertainties. the case study was from the department of neurosurgery, new delhi in india. “multiple sclerosis (ms)” is more and more identified in india because of the increase in the number of practicing neurologists and affordable and easy availability of “magnetic resonance imaging (mri)”. an ms is an “autoimmune inflammatory demyelinating illness (aidi)” of the “central nervous system (cns)”. in india, the illness came to be identified in the 1960s. retrospective analysis with longitudinal follow-up of patients mentioned to a single tertiary healthcare center with neurology services in the neurosurgery department new delhi in india. for this study, we selected a 55-year-old male with his relatives. a physician evaluated the medical history of the patient and his current physical situation by providing three kinds of treatments, including rehabilitation (h1), corticosteroids (h2), and plasma exchange (plasmapheresis) (h3). to help the male patient and his relatives for understanding the disadvantages and advantages of each treatment option, the physician presents the related information based on various evaluation criteria, including related cost, probability of a recurrence, discomfort index of the treatment, survival rate, probability of a cure, number of days of hospitalization, selfcare capacity, severity of the complications and severity of the side effects. the physician requested to the patient and his relatives to discuss and evaluate the different kinds of treatment options thoroughly. in the following stage, the best treatment option is determined with the help of proposed if-cocoso approach. for doing so, three dms assessed the treatment options over the evaluative criteria, which are classified based on benefit/cost criteria. here, the criteria 1 ,p 3 p and 9 p are determined as the benefit types, and rest are determined as the cost types of criteria. based on the above discussion and current literature review of the medical decision making, several criteria have identified and provided in table 1. tripathi et al./decis. mak. appl. manag. eng. (2022) 14 table 1. important criteria for medical decision making in the literature review name of criteria symbol reference source survival rate p1 chen (2015); chen et al. (2013); hu et al. (2019a); chen (2017); chen (2013); hu et al. (2019b) severity of the side effects p2 chen (2015); ma et al. (2017); chen et al. (2013); chen (2017); huet al. (2019b); chen (2013); li et al. (2018) probability of a cure p3 chen (2015); chen et al. (2013); hu et al. (2019a); chen (2017); chen (2013); hu et al. (2019b); li et al. (2018) severity of the complications p4 chen (2015); chen et al. (2013); chen (2017); chen (2013); li et al. (2018) discomfort index of the treatment p5 chen (2015); hu et al. (2019b); chen et al. (2013); chen (2017); hu et al. (2019c); chen (2013); li et al. (2018) cost/expense p6 chen (2015); chen et al. (2013); ma et al. (2017); chen (2017); hu et al. (2019a); chen (2013); hu et al. (2019b); li et al. (2018) number of days of hospitalization p7 chen (2015); hu et al. (2019a); chen et al. (2013); chen (2017);hu et al. (2019c); chen (2013); li et al. (2018) probability of a recurrence p8 chen (2015); ma et al. (2017); chen et al. (2013); chen (2017); hu et al. (2019a); chen (2013); hu et al. (2019c); chen (2014); li et al. (2018) self-care capacity p9 chen (2015); chen (2017); chen (2013); hu et al. (2019c); hu et al. (2019b); li et al. (2018) the lvs and corresponding ifns for the significance of the alternative, des and criteria are presented in tables 2 and 3 adopted from rani et al., (2021) and hezam et al. (2022). according to the results of table 3 and eq. (12), the weights of des are estimated and shown in table 4. table 2. the lvs and corresponding ifns lvs ifns extremely unimportant (eu) (0.1, 0.8, 0.1) very unimportant (vu) (0.2, 0.7, 0.1) quite unimportant (qu) (0.3, 0.6, 0.1) unimportant (u) (0.4, 0.5, 0.1 ) neutral (n) (0.55, 0.4, 0.05) important(qi) (0.65, 0.3, 0.05) important (i) (0.75 , 0.2, 0.05) very important (vi) (0.9, 0.05, 0.05) extremely important (ei) (1, 0, 0) new intuitionistic fuzzy parametric divergence measures and score function-based… 15 table 3: the lvs for significance rating des’ lvs ifns outstanding (o) (1, 0, 0) exceeds expectations (ee) (0.9, 0.05, 0.05) meet expectations (me) (0.7,0.2, 0.1) moderate (m) (0.6,0.3, 0.1) needs improvements (ni) (0.3,0.6, 0.1) unacceptable (u) (0.1,0.8, 0.1) table 4: evaluated des’ weights next, tables 5-7 present the significance ratings of option over different attributes for each de. with the use of tables 5-7 and eq. (13), the a-if-dm is established and presented in table 8. by eq. (14) and table 8, the normalized a-ifdm can be showed in table 9. table 5. assessment given by de 1 d 1 d h1 h2 h3 p1 (0.8100, 0.0750) (0.7750, 0.1750) (0.7900, 0.1100) p2 (0.7250, 0.1750) (0.7250, 0.1850) (0.7250, 0.2350) p3 (0.4250, 0.3150) (0.5750, 0.3250) (0.5500, 0.3850) p4 (0.5750, 0.3750) (0.7100, 0.1750) (0.6900, 0.2100) p5 (0.3650, 0.1800) (0.5250, 0.4250) (0.5250, 0.3250) p6 (0.4950, 0.3500) (0.5400, 0.3250) (0.5400, 0.4250) p7 (0.7950, 0.2000) (0.7400, 0.1300) (0.7400, 0.2100) p8 (0.8100, 0.1250) (0.7750, 0.1650) (0.7750,0.1650) p9 (0.8100, 0.1600) (0.6550, 0.2250) (0.6550, 0.2250) table 6. assessment given by de 2 d 2 d h1 h2 h3 p1 (0.7300, 0.2250) (0.8850, 0.0300) (0.6950, 0.1250) p2 (0.6900, 0.2300) (0.8150, 0.1100) (0.8150, 0.1350) p3 (0.4000, 0.3100) (0.5250, 0.4200) (0.5000, 0.2300) p4 (0.4750, 0.3150) (0.6250, 0.2750) (0.7250, 0.2250) p5 (0.3750, 0.2250) (0.5500, 0.3500) (0.5000, 0.2250) p6 (0.4250, 0.3250) (0.5500, 0.3200) (0.5500, 0.3150) p7 (0.7250, 0.2250) (0.7500, 0.1600) (0.7500, 0.1600) p8 (0.8000, 0.1250) (0.7900, 0.1500) (0.7900, 0.1500) p9 (0.8250, 0.1100) (0.6800, 0.2250) (0.6800, 0.2250) decision expert (des) 1 d 2 d 3 d lvs me ee m ifns (0.7, 0.2, 0.1) (0.9, 0.05, 0.05) (0.6, 0.3, 0.1) weights 0.3242 0.3979 0.2779 tripathi et al./decis. mak. appl. manag. eng. (2022) 16 table 7. assessment given by de 3 d 3 d h1 h2 h3 p1 (0.8100, 0.1550) (0.7250, 0.2100) (0.6100, 0.3150) p2 (0.7750, 0.1500) (0.6900, 0.2350) (0.7250, 0.1750) p3 (0.4850, 0.3650) (0.5600, 0.3600) (0.5600, 0.3700) p4 (0.4500, 0.4250) (0.5250, 0.3600) (0.6250, 0.2250) p5 (0.3500, 0.2150) (0.6100, 0.2750) (0.4750, 0.3250) p6 (0.4750, 0.3250) (0.5000, 0.3400) (0.5000, 0.3600) p7 (0.8250, 0.1250) (0.6500, 0.2250) (0.6500, 0.2250) p8 (0.8500, 0.0300) (0.7500, 0.1250) (0.7500, 0.1250) p9 (0.8400, 0.1250) (0.6500, 0.2250) (0.6500, 0.2100) table 8. the a-if-dm ( ) 3 9 ij y  given by des h1 h2 h3 p1 (0.7910, 0.1169, 0.0921) (0.8011, 0.0857, 0.1132) (0.7312, 0.2130, 0.0558) p2 (0.7250, 0.1942, 0.0808) (0.7813, 0.1479, 0.0708) (0.7297, 0.1872, 0.0831) p3 (0.5237, 0.3372, 0.1390) (0.4803, 0.3220, 0.1978) (0.5370, 0.3644, 0.0987) p4 (0.6656, 0.2357, 0.0987) (0.6163, 0.2718, 0.1119) (0.5335, 0.3334, 0.1331) p5 (0.4781, 0.2986, 0.2233) (0.4846, 0.2682, 0.2472) (0.5001, 0.2660, 0.2339) p6 (0.5259, 0.3587, 0.1155) (0.5128, 0.3202, 0.1670) (0.4920, 0.3404, 0.1675) p7 (0.7593, 0.1708, 0.0699) (0.7422, 0.1787, 0.0791) (0.7204, 0.1860, 0.0936) p8 (0.7870, 0.1508, 0.0622) (0.7933, 0.1414, 0.0653) (0.7882, 0.0787, 0.1331) p9 (0.7157, 0.2015, 0.0829) (0.7369, 0.1784, 0.0847) (0.7284, 0.1824, 0.0891) table 9. normalized a-if-dm ( ) 3 9 ij   h1 h2 h3 p1 (0.7910, 0.1169, 0.0921) (0.8011, 0.0857, 0.1132) (0.7312, 0.2130, 0.0558) p2 (0.1942, 0.7250, 0.0808) (0.1479, 0.7813, 0.0708) (0.1872, 0.7297, 0.0831) p3 (0.5237, 0.3372, 0.1390) (0.4803, 0.3220, 0.1978) (0.5370, 0.3644, 0.0987) p4 (0.2357, 0.6656, 0.0987) (0.2718, 0.6163, 0.1119) (0.3334, 0.5335, 0.1331) p5 (0.2986, 0.4781, 0.2233) (0.2682, 0.4846, 0.2472) (0.2660, 0.5001, 0.2339) p6 (0.3587, 0.5259, 0.1155) (0.3202, 0.5128, 0.1670) (0.3404, 0.4920, 0.1675) new intuitionistic fuzzy parametric divergence measures and score function-based… 17 h1 h2 h3 p7 (0.1708, 0.7593, 0.0699) (0.1787, 0.7422, 0.0791) (0.1860, 0.7204, 0.0936) p8 (0.1508, 0.7870, 0.0622) (0.1414, 0.7933, 0.0653) (0.0787, 0.7882, 0.1331) p9 (0.7157, 0.2015, 0.0829) (0.7369, 0.1784, 0.0847) (0.7284, 0.1824, 0.0891) based on table 8 and eq. (15), the criteria weights using the proposed parametric divergence measure and gsf is derived and presented as )(0.1245, 0.1210, 0.0935, 0.1043, 0.0960, 0.0938, 0.1207, 0.1269, 0.1193 . t j  = (22) using table 9 and eqs (16) and (17), (1) i  and (2) i  with their score values ( )* (1)is and ( )( )2* is are determined and depicted in table 10. according to eqs (18)(20), the relative weights or balanced compromise scores ( )1 , i q ( )2 i q and ( ) ( ) 3 i q  (with 0.0, 0.2, 0.5, 0.8,1) = is computed and given in table 9, where the compromise decision mechanism coefficient  0,1  . further, the aggregated compromise index ( )iq  (with 0.5) = of the treatment choice is evaluated and presented in table 10. from table 10, h2 is the best treatment choice and h1 is the least favorable option. table 10. the ocs for each option (1) i  (2) i  ( )* (1)is ( ) * ( 2) i s ( )1 i q ( )2 i q ( ) ( ) 3 i q  ( )iq  h1 (0.4443, 0.4334) (0.3185, 0.5734) 0.4986 0.3529 0.3372 2.0678 0.9890 2.0073 h2 (0.4419, 0.4084) (0.3045, 0.5724) 0.5081 0.3420 0.3366 2.0557 0.9873 2.0148 h3 (0.4302, 0.4509) (0.2994, 0.5573) 0.4813 0.3423 0.3262 2.0009 0.9566 1.9492 also, we demonstrate a sensitivity analysis based on various decision-making coefficient values. the value of  = 0.5 is preferred to be analyzed. the variations of  aid us in the evaluation of the approach’s sensitivity to the movement from the weighted sum comparability sequence, power weight comparability sequence, balanced compromise scores and aggregating compromise index. the results are depicted in figure 3. hence, it is clearly recognizable that the presented method has high stability with various values of the parameter  (0.0, 0.2, 0.5, 0.8 and 1.0). thus, we can conclude that the presented combination results in enhancing the solidity of the developed method. tripathi et al./decis. mak. appl. manag. eng. (2022) 18 figure 3. ranking orders of options with strategic coefficient  6. comparative analyses here, a comparison is made between the outcomes achieved from the if-cocoso model and other existing approaches. to show the utility of the if-cocoso method, the if-topsis method (joshi and kumar (2014), if-vikor method (luo and wang, 2017) and if-waspas method (mishra et al., 2019) are used to handle the given case study. 6.1. if-waspas model the if-waspas framework (mishra et al., 2019) comprises the following procedures: steps 1-7: follow the steps of if-cocoso. step 8: estimate the waspas degree of each alternative as follows: ( )(1) (2)1 ,i i iq =  + −  (23) where signifies the strategic coefficient, where [0,1] . step 9: prioritize the option(s) based on the score values of . i q subsequently, the results of the if-waspas model are demonstrated in table 11. new intuitionistic fuzzy parametric divergence measures and score function-based… 19 table 11. the waspas degree of each option using if-waspas method alternative (1) i  (2) i  ( )* (1)is ( ) * ( 2) i s ( )iq ranking order h1 (0.4443, 0.4334) (0.3185, 0.5734) 0.4986 0.3529 0.4251 2 h2 (0.4419, 0.4084) (0.3045, 0.5724) 0.5081 0.3420 0.4257 1 h3 (0.4302, 0.4509) (0.2994, 0.5573) 0.4813 0.3423 0.4118 3 the prioritization of treatment choices is 2 1 3h h h and the alternative 2h has the maximum degree of suitability for treatment choice. next, a sensitivity analysis is presented, which is accompanied with five sets of various values of parameter ( ) . the value of = 0.5 is analyzed. the variations of aid us in the evaluation of the method’s sensitivity level to the variation from the wsm to the wpm. 6.2. if-topsis model the if-topsis method (joshi and kumar, 2014) is discussed as steps 1-5: similar as the presented method. step 6: compute the “intuitionistic fuzzy ideal solution (if-is)” and “intuitionistic fuzzy anti-ideal solution (if-ais)”. suppose pb and pn be the sets of benefit and non-benefit attributes, respectively. then the if-is + and the if-ais solution − can be given as , , max | , min | , min , max : 1, 2, ..., j ij b ij n i i ij b ij n i i p p p p p j j j j i m      + =     =        (24) ( ) . , min | , max | , max , min : 1, 2, ..., j ij b ij n i i ij b ij n ii p p p p p j j j j i m     −  =     = (25) step 7: calculate the divergences from if-is and if-ais. using eq. (8), we compute the weighted if-divergence ( ),ij h + between the options ,ih i and the if-is , +  and the divergence ( ),ij h − between the options ,ih i and the if-ais . −  tripathi et al./decis. mak. appl. manag. eng. (2022) 20 step 8: calculate the relative closeness coefficient (cc). the relative closeness coefficient of each option considering the intuitionistic fuzzy is is evaluated by ( ) ( ) ( ) , . , , , i i i i j h cc i j h j h    − − + = +     (26) step 9: choose the best option with maximum value of *cc among the values ; 1, 2, ..., .icc i m= from table 8, eq. (24) and eq. (25), the if-is and if-ais are estimated as follows:  = + {(0.8011, 0.0857, 0.1132), (0.1479, 0.7813, 0.0708), (0.5370, 0.3644, 0.0987), (0.2357, 0.6656, 0.0987), (0.2660, 0.5001, 0.2339), (0.3202, 0.5128, 0.1670), (0.1708, 0.7593, 0.0699), (0.0787, 0.7882, 0.1331), (0.7369, 0.1784, 0.0847)}, −  = {(0.7312, 0.2130, 0.0558), (0.1942, 0.7250, 0.0808), (0.4803, 0.3220, 0.1978) (0.3334, 0.5335, 0.1331), (0.2986, 0.4781, 0.2233), (0.3587, 0.5259, 0.1155) (0.1860, 0.7204, 0.0936), (0.1508, 0.7870, 0.0622), (0.7157, 0.2015, 0.0829)}. the results of the if-topsis model are presented in table 12. table 12. results of the if-topsis method alternative ( ),ij h +  ( ),ij h −  icc ranking order h1 0.0028 0.0031 0.5254 2 h2 0.0027 0.0040 0.5970 1 h3 0.0044 0.0032 0.4211 3 the prioritization of treatment choices is 2 1 3h h h and the alternative h2 has the higher degree of suitability for treatment choice than others. 6.3. if-vikor model the if-vikor model(luo and wang, 2017)consists of the following steps: steps 1-5: same as the presented method. step 6: derive the “group utility (gu)” and “individual regret (ir)” of each option. the key focus of the original vikor technique is to effectively rank and determine the compromise solution for a given problem that has some contradictory criteria. for the compromise solution, multiple measures are presented from the p l − metric that is applied as an aggregated value to a compromise solution model as follows: new intuitionistic fuzzy parametric divergence measures and score function-based… 21 ( ) ( ) 1 , 1 , , 1, 2, , p p n j ij p i j j j j j y l j     + + − =      = =         (27) wherein j  is the weight of ( )1, 2, ,jp j n= . with the divergence value-based p l -metric, the gu and ir are presented by eqs (28) and (29). ( ) ( ) 1, 1 , , , n j ij i i j j j j j y l j    + + − =    = =      g (28) ( ) ( ) , 1 , max . , j ij i i j j n j j j y l j    +  + −      = =      (29) step 7: compute of the “compromise solution (cs)” of each option. the vikor technique’s stimulus is to find the cs that is amidst an extreme gu for the least and majority of the ir for opponents. this study defines the cs , i i  as ( )1 ,i ii   + + − + − + −  −  = + − −  −  g g g g (30) where min , i i + =g g max , i i − =g g min , i i +  =  max i i −  =  and  is the weight or decision mechanism coefficient. with no generality loss, we take the value as 0.5. the smaller the value of ( )1, 2, , ,i i m = the better the option ( )1, 2, , .ih i m= the compromise solution can be chosen with “voting by majority ( )0.5  ” with “consensus ( )0.5 = ” and with “veto ( )0.5  ”. step 8: rank the alternative. the vikor approach involves ranking the alternative(s) , i h i which corresponds to the values of , i g i  and . i  the obtained result contains three ranking lists signified as , i g i  and . i  we suggest the most appropriate one (the smallest among i  values) as a cs with handling the given necessary conditions given in luo and wang (2017). here, the results of the if-vikor technique are illustrated in table 13. tripathi et al./decis. mak. appl. manag. eng. (2022) 22 table 13. outcomes of , i g i  and ( )  and the cs of each option h1 h2 h3 ranking compromise solution g 0.5875 0.2604 0.6369 2 1 3h h h 2h  0.1269 0.1070 0.1719 2 1 3 h h h 2 h ( )  0.3066 (0.0) 0.4191(0.2) 0.5877 (0.5) 0.7564 (0.8) 0.8688 (1.0) 0.0000 (0.0) 0.0000 (0.2) 0.0000 (0.5) 0.0000 (0.8) 0.0000 (1.0) 1.0000 (0.0) 1.0000 (0.2) 1.0000 (0.5) 1.0000 (0.8) 1.0000 (1.0) 2 1 3 h h h 2 h the prioritization of treatment option is 2 1 3 h h h and the alternativeh2has the higher degree of suitability for treatment choice. additionally, we elucidate a systematic comparison of the present if-cocoso with other mcdm approaches according to the several important standards applied in decision-making procedure (see table 14). it can be concluded from table 14 that the presented model is absolutely a novel contribution as it incorporates all major aspects of mcdm methods by comparing with the extant studies on mcdm approaches within ifss settings. table 14. a comparative discussion of the ranking orders with various methods tools standards criteria weights mcdm model assess ing hd expert weights ranking order opti mal choic e ivfelectr e (vahdani and hadipou r, 2011) intervalvalued fuzzy electre method assume d outran king model exclud ed not considered 2 1 h h 3 1 h h 2 3 ,h h iftopsis (joshi and kumar, 2014) distance measure based topsis method entropy measure method compr omisin g model exclud ed not considered 2 1 3 h h h 2h ifvikor (luo and wang, 2017) novel distance measurebased vikor method entropy measure method compr omisin g model include d considere d (entropy measure method) 2 1 3 h h h 2h new intuitionistic fuzzy parametric divergence measures and score function-based… 23 tools standards criteria weights mcdm model assess ing hd expert weights ranking order opti mal choic e ifwaspas method mishra et al. (2019) similarity measurebased waspas similarit y measure method scoring model (utility based metho d) include d not considered 2 1 3 h h h 2h propose d ifcocoso method proposed divergence measure and gsf-based cocoso method propose d divergen ce measure and gsf compr omisin g model include d considere d (score function model) 2 1 3 h h h 2h 7. conclusions to characterize uncertainty and fuzziness arguments in evaluating the medical decision-making problem, ifss were implemented and evaluative criteria were recognized, which contain various qualitative and quantitative influencing parameters from the physician to medical field. the present study has significant contributions to current knowledge on the decision-making methodology. first, an innovative gsf was proposed to mark out the most appropriate alternatives from the possible alternatives where the decision matrix, which is related to various criteria in regard to choosing the attributes, is characterized in ifss. in addition, it was examined from the developed score function that the enhanced score function introduced by liu & wang (2007) is obtained as a special case by taking ( ) ( )1 11 2 2 2, ,  = and thus, the gsf is of a higher profitability for obtaining the expert goals during their implementation in comparison with those that are currently used. second, two new parametric if-divergence measures of order α and (α,β) were developed through taking into consideration the mf, nf and hf and various attractive properties of the proposed measures that have been studied. third, a new compromising method, i.e., the extended cocoso method, was presented to handle medical mcdm problems with ifss with the proposed parametric divergence measure and gsf. we developed a method to evaluate criteria weights with ifns. finally, we implemented the if-cocoso method to rank and evaluate the therapies for medical mcdm problem. the effectiveness of the developed approach is justified by some comparative analyses. in the future, researchers can be extended the cocoso method in different uncertain environments such as “hesitant fuzzy sets (hfss)”, pfss, ffss, “intervalvalued fermatean fuzzy sets (ivffss)”. in addition, we will continue this study with expectation that the model could be considered more appropriate to other decisionmaking issues such as selection of disinfection facility for healthcare waste, low carbon suppliers assessment, blockchain technology adoption and so many others. author contributions: conceptualization, dinesh k. tripathi and abdul r. shah; methodology, pratibha rani; software, pratibha rani; validation, s. k. nigam and pratibha rani; formal analysis, pratibha rani; investigation, dinesh k. tripathi; tripathi et al./decis. mak. appl. manag. eng. (2022) 24 resources, s. k. nigam; data curation, pratibha rani; writing—original draft preparation, dinesh k. tripathi and abdul r. shah; writing—review and editing, pratibha rani; visualization, s. k. nigam and pratibha rani; supervision, s. k. nigam. all authors have read and agreed to the published version of the manuscript. funding: this research received no external funding. data availability statement: not applicable. acknowledgments: we would like to thank the editors and reviewers for their constructive comments and suggestions to improve the quality of the paper. conflicts of interest: the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. references ansari, m.d., mishra, a.r., & ansari, f.t. 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(2019). multiattribute decision making based on novel score function of intuitionistic fuzzy values and modified vikor method. information sciences, 488, 76-92. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 3, issue 1, 2020, pp. 1-21. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame2003001s * corresponding author. e-mail addresses: hareshshrm@gmail.com (h.k. sharma), kriti.kri89@gmail.com (k. kumari), samarjit.kar@maths.nitdgp.ac.in (s. kar) a rough set theory application in forecasting models haresh kumar sharma 1, kriti kumari 2 and samarjit kar 1* 1 department of mathematics, national institute of technology, durgapur, india 2 department of mathematics, banasthali university, jaipur, rajasthan, india received: 5 september 2019; accepted: 11 november 2019; available online: 15 november 2019. original scientific paper abstract. this paper introduces the performance of different forecasting methods for tourism demand, which can be employed as one of the statistical tools for time series forecasting. the holt-winters (hw), seasonal autoregressive integrated moving average (sarima) and grey model (gm (1, 1)) are three important statistical models in time-series forecasting. this paper analyzes and compare the performance of forecasting models using rough set methods, total roughness (tr), min-min roughness (mmr) and maximum dependency of attributes (mda). current research identifies the best time series forecasting model among the three studied time series forecasting models. comparative study shows that hw and sarima are superior models than gm (1, 1) for forecasting seasonal time series under tr, mmr and mda criteria. in addition, the authors of this study showed that gm (1, 1) grey model is unqualified for seasonal time series data. key words: forecasting, mean absolute percent error (mape), rough set, total roughness, maximum dependency degree 1. introduction the future planning has emerged as a key component for the success of a large number of entrepreneurs round the corners of the world. it certainly makes it easy for all the countries to formulate economic policies as well as act upon them efficiently. the perfection and accuracy are quite important in the process of highly accurate and reliable prediction. it helps the government in formulating development policies concerned with economics, infrastructure and many other sectors of the global as well as the domestic economy. it even improves the decision-making process. time series modeling and forecasting play a key role in accurate prediction. the current trend is mailto:kriti.kri89@gmail.com mailto:samarjit.kar@maths.nitdgp.ac.in sharma et al./decis. mak. appl. manag. eng. 3 (1) (2020) 1-21 2 incorporated for the future prediction; it thus becomes necessary to use highly consistent and precise forecasting tools. since the last couple of decades, a wide variety of forecasting models is available for the study of tourist arrivals and demand forecasting (chu, 1998, lim and mcaleer, 2002; wang, 2004). suggesting autoregressive integrated moving average (arima) model as a most suitable model for tourism demand forecast. arima was first brought out by box-jenkins (box and jenkins, 1976) and presently it is the most accepted model for forecasting univariate time series data. arima model is the combined result of autoregressive (ar) and moving average (ma) model. arima model develops an optimal univariate future prediction. moreover, the arima model has received worldwide confidence due to its ability to handle stationary and non-stationary series with seasonal and non-seasonal elements (pankratz, 1983). but, sarima is particularly designed for the time series data with trends and seasonal patterns. holtwinters has also gained more popularity to capture trend and seasonality (winters, 1960). the hw seasonal method consists of the forecast equation and smoothing equations for level, trend, and seasonality. later on, grey system theory developed by deng states that a system whose internal sources such as system characteristics operation mechanisms and architecture are completely clear, called a white system (deng, 1982). the added advantage of this system is that the theory cannot only estimate an uncertain system but sometimes it produces ideal results. for example, tseng et al. (2001) was reported the application of the grey model to forecast taiwan machinery industry and soft drinks time series data. however, nguyen et al. (2013) studied the forecasting of tourist arrivals in vietnam using gm (1, 1) grey model. from the last many years variety of forecasting criteria has been used to select the best time series models (lim and mcaleer, 2002; wang, 2004). modeling and forecasting consist of a large number of criteria. for instance, (chu, 1998) employed mape and u-statistics criteria to compare the holt winters and sarima models. however, (chen et al, 2009) applied mape criterion to evaluate the forecasting accuracy of holt-winters, sarima, and grey models. in earlier research accuracy of forecasting models have been evaluated using error based criteria (goh and law, 2002; law and au, 1999; law, 2000). sometimes it may be possible that one model may become a good one due to some set of criteria but at the same time some other model may turn out to be the best one due to some other set of criteria. moreover, these indicators have very much been exploited and only marginal improvements might be expected from their continued use. this research proposed a new approach that applies rough set theory to select the highly accurate models in time series forecasting. the rough set theory has been introduced to deal with vagueness, imprecision, and uncertainty. the original rough set theory depends on the equivalence relation (indiscernibility relation). this approach is taken into consideration the attributes in accordance with the normalized values (goh and law, 2003). rough set theory has been able to overcome one of its advantages in association with statistical analysis during the process of attribute selection using rough set indicators (hassanein and elmelegy, 2013; herawan, 2010). current research is an extended effort of chen et al. (2009) work where they have used the holt-winters (hw) model, seasonal autoregressive integrated moving average (sarima) model and grey model (gm (1, 1)). gm (1, 1) has been used to define monthly inbound air travel arrivals to taiwan and to distinguish the models based on their respective performance. mean absolute percent error (mape) has been used as an indicator to measure forecast accuracy. based on the results derived, they concluded that the hw and sarima models are better reliable models than gm (1, 1). a rough set theory application in forecasting models 3 the objective of this paper is to obtain the best forecasting models using tr, mmr, and mda rough set indicators. based on rough set information table, those techniques are used to calculate roughness of models. then, compare these three models in accordance with the roughness. the authors of this study showed that the gm (1, 1) is an inadequate model for forecasting with seasonality as compared to hw and sarima models. the rest of the research paper is organized as follows; section 2 contains literature review, section 3 briefly introduces the basic concepts rough set theory and some related properties. section 4, presents an algorithmic approach for the evaluation of rough data using mape indicator. in section 5, the experimental design and experimental results have been discussed. finally, the paper is concluded in section 6. 2. literature review in recent years, rough set theory have been employed in various literature to select the clustering attributes. for example, mazlack et al. (2000) proposed bi-clustering (bc) technique depend on balanced/unbalanced bi-valued attributes and total roughness (tr) technique based on the average accuracy of roughness (pawlak, 1982; pawlak & skowron, 2007). the tr technique is useful for selecting the clustering attributes in the data set, where the maximum tr is the maximum accuracy for selecting clustering attributes. three indicators i.e. tr, mmr, and mda of the rough set theory have been successfully used. for instance, parmar et al. (2007) developed a new method called min-min roughness (mmr) to develop bc technique for the information system with many valued attributes. in this technique, attributes for approximation are calculated using well-known corporate to the lower and upper approximations of a subset of the universe in the information system. herawan et al. (2010) developed a new technique known as maximum dependency of attributes (mda) to select clustering attributes. mda technique is based on the dependency of attributes using rough set theory in an information system. these three techniques tr, mmr and mda provide the same outcome in selecting the attributes. this makes the rough set criteria a very useful to select the different attributes. however, in previous literature, there is no any link of rough set theory with relationship time series modeling to select the best forecasting models. in time series analysis and forecasting, the selection of highly accurate model is very important to evaluate the best time series model. hence, this research proposed a rough set criterion for strong evidence in the selection of best suitable time series models that is different another traditional statistical indicator. rough set theory has been consistently employed in a variety of research areas for the extraction of decision rules (law & au 1998, 2000, goh & law, 2003; liou et al. 2016). celotto et al. (2012) applied rough set theory based forecasting model in data of tourist service demand.. moreover, li et al. (2011) predicted tourism in tangshan city of china using rough set model. golmohammadi and ghareneh (2011) analyze the importance of travel attributes by rough set approach. celotto et al. (2015) applied rough set theory to summarize tourist evaluations of a destination. , faustino et al. (2011) present a rough set analysis of electrical charge demand in the united states and the level of the sapucal river in brazil. liou (2016) used the rough set theory to study the airline service quality to taiwan. sharma et al. (2019) proposed hybrid rough set based forecasting model and applied on tourism demand of air transportation passenger data set in australia tourism demand. sharma et al./decis. mak. appl. manag. eng. 3 (1) (2020) 1-21 4 rough set theory use to alter the roughness of a data, which has been successfully applied to various real life decision making problems ( karavidić & projović, 2018; roy et al., 2018; vasiljević et al., 2018). moreover, the rough set concept can definitely be implemented to sets categorized by means of immaterial facts wherein statistical tools fail to provide fruitful outcomes (pawlak, 1991). pamučar et al. (2018) proposed interval rough number enabled ahp-mabac model for web pages evaluation. sharma et al. (2018) applied modified rough ahp-mabac method for prioritizing indian railway stations. 3. rough set theory the rough set theory was first introduced by pawlak (1982). the rough set concept is a new mathematical technique to tackle vagueness, imprecision, and uncertainty (pawlak, 1982; pawlak & skowron, 2007). it is a vital tool to examine the degree of dependencies and minimize the number of attributes within the dataset. its success is partly owed to the following properties: (1) analysis is performed on the hidden fact of the data; (2) supplementary information on data is not required like specialist awareness or thresholds; (3) equivalent relation is a basic idea of classical rough set theory. whereas, the attribute might be assign with both the values symbolic or real. pawlak proposed that the rough set theory is established on the assumption that with every member of the universe of discourse we relate some information. for example, symptoms of the disease develop a crucial part of information where objects are the patients suffering from the certain disease. the objects become indiscernible (similar) when characterized by the same information in view of the available information about them. the indiscernibility relation created in this way is the mathematical foundation of the rough set theory. the original concept of the rough set theory is the induction of approximation. the main aim of the rough set theory is the approximation of a set by a pair of two crisp sets called the lower and upper approximations of the sets. 3.1 indiscernibility relation let u be the non-empty finite set of all objects known as the universe and 𝐴 is the finite set of all attributes, then the couple 𝑆 = (𝑈, 𝐴) is known as an information system. for any non-empty subset 𝐵 of 𝐴 is associated with an equivalence relation inds(b) relation, 𝐼𝑁𝐷𝑆(𝐵) = { (𝑦𝑖 , 𝑦𝑗 ) ∈ 𝑈 × 𝑈 ∣∣ ∀ 𝑏 ∈ 𝐵, 𝑏(yi) = b(yj) } (1) where 𝑏(𝑦𝑖 ) represents the value of attribute 𝑏 for the element 𝑦𝑖 . 𝐼𝑁𝐷𝑆(𝐵) is called the indiscernibility relation on 𝑈. the notion [𝑦𝑖 ]𝐼𝑁𝐷𝑆(𝐵) represent the equivalence class of the indiscernibility relation. [𝑦𝑖 ]𝐼𝑁𝐷𝑆(𝐵) is also called as elementary set with respect to the attribute 𝐵. 3.2. lower and upper approximation lower approximation and upper approximation (pawlak, 1982; pawlak, 1991) of any set can be defined as follows: for an information system s = (u, a) given the set of attribute b ⊆ a, y ⊆u, the lower and upper approximation of y are defined as follows respectively, 𝑌𝐵 = ∪ {𝑦𝑖 |[𝑦𝑖 ]𝐼𝑁𝐷𝑆(𝐵) ⊆ 𝑌} (2) 𝑌𝐵 = ∪ {𝑦𝑖 |[𝑦𝑖 ]𝐼𝑁𝐷𝑆(𝐵) ∩ 𝑌 ≠ ∅} (3) a rough set theory application in forecasting models 5 clearly, lower approximation contains all members which certain objects of y and upper approximation consists all members which possible objects of y. the boundary region is the set of members that can possible member, but not surely, defined as follow: 𝐵𝑁𝐷𝐵 (𝑌) = 𝑌𝐵 − 𝑌𝐵 (4) the boundary region of an exact (crisp) set is an empty set like the lower approximation and upper approximation of exact set are similar. if the boundary region of a set is non-empty i.e. 𝐵𝑁𝐷𝐵 (𝑌) ≠ ∅, then the set y has been referred to as rough (vague). 3.3. roughness (r) inexactness of a category (set) is one of the reasons behind the existence of boundary line region. as the boundary line region of a category increases, the accuracy of the category decreases. to model such kind of imprecision the concept of accuracy of approximation (pawlak, 1991) is very much required. accuracy measure represented as follow: 𝛼𝐵(𝑌) = 𝑐𝑎𝑟𝑑 𝑌𝐵 𝑐𝑎𝑟𝑑 𝑌𝐵 the accuracy is intended to compute the degree of satisfaction of our knowledge about the category (set). obviously 0 ≤ 𝛼𝐵 (𝑌) ≤ 1. if 𝛼𝐵 (𝑌) =1, y is exact with respect to b, if 𝛼𝐵 (𝑌) < 1, y is rough with respect to b. assume that an attribute 𝑎𝑖 ∈ 𝐴 having k-distinct values, say 𝛼𝑘 , 𝑘 = 1,2, … . , 𝑚. suppose 𝑌(𝑎𝑖 = 𝛼𝑘 ),𝑤ℎ𝑒𝑟𝑒 𝑘 = 1,2, . . … , 𝑚 𝑖 a subset of the objects consists k-distinct values of attribute 𝑎𝑖 . the roughness of tr (mazlack, 2000) of the set(𝑎𝑖 = 𝛼𝑘 ), 𝑘 = 1, 2, … . , 𝑚, with respect to aj, where 𝑖 ≠ 𝑗, represented by 𝑅𝑎𝑗 (𝑌 ∣ 𝑎𝑖 = 𝛼𝑘 ) as is defined by 𝑅𝑎𝑗 ( 𝑌 ∣∣ 𝑎𝑖 = 𝛼𝑘 ) = |yaj (𝑎𝑖=𝛼𝑘)| |yaj (𝑎𝑖=𝛼𝑘)| ,𝑘 = 1, 2, … . , 𝑚 (5) 3.3.1. mean roughness (mr) the values of mean roughness of an attribute 𝑎𝑖 ∈ 𝐴 with respect to another attribute 𝑎𝑗 ∈ 𝐴, where, 𝑖 ≠ 𝑗, represented by the following formula 𝑅𝑜𝑢𝑔ℎ𝑎𝑗 (𝑎𝑖 )= ∑ 𝑅𝑎𝑗 (𝑌∣𝑎𝑖=𝛼𝑘) |𝑉(𝑎𝑖)| 𝑘=1 |𝑉(𝑎𝑖)| (6) where 𝑉(𝑎𝑖 ) is the set of all values of attribute 𝑎𝑖 ∈ a. 3.3.2. total roughness (tr) the total roughness of the attribute 𝑎𝑖 ∈ a with respect to the attribute 𝑎𝑗 ∈ 𝐴, where, 𝑖 ≠ 𝑗, represented by 𝑇𝑅 (𝑎𝑖 ), is defined by 𝑇𝑅(𝑎𝑖 ) = ∑ 𝑅𝑜𝑢𝑔ℎ𝑎𝑗 (𝑎𝑖) |𝐴| 𝑗=1 |𝐴|−1 (7) the maximum value of tr, the finest selection choice of clustering attributes. 3.3.3. minimum – minimum roughness (mmr) from the tr system, the mean roughness of attribute ai with respect to attribute aj, where, 𝑖 ≠ 𝑗 is define by sharma et al./decis. mak. appl. manag. eng. 3 (1) (2020) 1-21 6 𝑀𝑀𝑅𝑜𝑢𝑔ℎ𝑎𝑗 (𝑎𝑖 ) =1 − ∑ 𝑅𝑎𝑗 (𝑌∣𝑎𝑖=𝛼𝑘) |𝑉(𝑎𝑖)| 𝑘=1 |𝑉(𝑎𝑖)| 𝑀𝑀𝑅𝑜𝑢𝑔ℎ𝑎𝑗 (𝑎𝑖 ) = 1 – 𝑅𝑜𝑢𝑔ℎ𝑎𝑗 (𝑎𝑖 ) (8) 3.4. maximum dependency attribute (mda) [herawan et al. (2010)] suppose s = (u, a) is information system and let ai and aj be any subsets of a. dependency attribute ai on aj in a degree k (0<k<1), is denoted by 𝑎𝑗 ⟹ 𝑘𝑎𝑖 . the degree k is derived: 𝑘 = ∑ |𝑎𝑗(𝑌)|𝑌∈𝑈/𝑎𝑖 |𝑈| (9) if degree k =1, ai is fully depends on aj. otherwise ai partially depends on aj. for an assortment of clustering attributes a maximum degree of dependency of attributes is most appropriate. because as greater the degree of dependence of attribute as much accurate for the assortment of clustering attribute. 3.5. accuracy of tr, mmr and mda techniques section 4 involves the applications of tr, mmr and mda techniques, used for the selection of better forecasting models. the calculation of the accuracy consists of (i) tr which makes the use of the total average of mean roughness (ii) mmr which relies upon the minimum of mean roughness and (iii) mda depends on the maximum degree of dependency to select the forecasting attribute model. after determining the roughness and mean roughness of the attributes, we select the best model based on maximum tr, maximum mda values and lower values of minimum roughness. example 1. to demonstrate the degree of dependency of attributes, we consider the information system as shown in table 1. table 1. representation of food junction database with ten objects and five attributes customer food quality (𝐹1) menu variety (𝐹2) service (𝐹3) atmosphere (𝐹4) cleanliness (𝐹5) 𝑦1 good good poor good average 𝑦2 poor poor poor average good 𝑦3 poor good average good average 𝑦4 good average poor average average 𝑦5 good average poor good poor 𝑦6 average good average good poor 𝑦7 average good good good good 𝑦8 poor poor poor average good 𝑦9 good average poor average average 𝑦10 good good average average good table 1 shows the food junction data set in which 𝑈 = {𝑦1, 𝑦2 , . . , 𝑦10} (where 𝑦𝑗 represents a customer) and 𝐴 = (𝐹1, 𝐹2, 𝐹3, 𝐹4, 𝐹5), being 𝐹1, 𝐹2, 𝐹3, 𝐹4, and 𝐹5 attribute set.  in dataset 1, indiscernibility relations can described as follows: 𝑈/𝐹1 = {(𝑦2, 𝑦3, 𝑦8), (𝑦6, 𝑦7), (𝑦1, 𝑦4, 𝑦5, 𝑦9, 𝑦10)} since 𝐹1(𝑦1) = 𝐹1(𝑦4) = 𝐹1(𝑦5) = 𝐹1(𝑦9) = 𝐹1(𝑦10) = 𝑔𝑜𝑜𝑑 𝐹1(𝑦2) = 𝐹1(𝑦3) = 𝐹1(𝑦8) = 𝑝𝑜𝑜𝑟 a rough set theory application in forecasting models 7 𝐹1(𝑦6) = 𝐹1(𝑦7) = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒  𝑈/𝐹2 = {(𝑦2, 𝑦8), (𝑦4, 𝑦5, 𝑦9), (𝑦1, 𝑦3, 𝑦6, 𝑦7, 𝑦10),  𝑈/𝐹3 = {(𝑦1, 𝑦2, 𝑦4, 𝑦5, 𝑦8, 𝑦9), (𝑦3, 𝑦6, 𝑦10), (𝑦7),  𝑈/𝐹4 = {(𝑦2, 𝑦4, 𝑦8, 𝑦9, 𝑦10), (𝑦1, 𝑦3, 𝑦5, 𝑦6, 𝑦7)},  𝑈/𝐹5 = {(𝑦5, 𝑦6), (𝑦1, 𝑦3, 𝑦4, 𝑦9, ), (𝑦2, 𝑦7, 𝑦8, 𝑦10)}, lower and upper approximation of subsets of 𝑈 of attribute 𝐹1 with respect to attribute 𝐹2, 𝐹3, 𝐹4 and 𝐹5 are given below: (i) 𝐹1 with respect to 𝐹2 𝑋(𝐹1 = 𝑝𝑜𝑜𝑟) = {𝑦2, 𝑦8}, 𝑋(𝐹1 = 𝑝𝑜𝑜𝑟) = {𝑦1, 𝑦2, 𝑦3, 𝑦6, 𝑦7, 𝑦8, 𝑦10} 𝑋(𝐹1 = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒) = {∅}, 𝑋(𝐹1 = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒) = {𝑦1, 𝑦3, 𝑦6, 𝑦7, 𝑦10} 𝑋(𝐹1 = 𝑔𝑜𝑜𝑑) = {𝑦4, 𝑦5, 𝑦9}, 𝑋(𝐹1 = 𝑔𝑜𝑜𝑑) = {𝑦1, 𝑦3, 𝑦4, 𝑦5, 𝑦6, 𝑦7, 𝑦9, 𝑦10}. (ii) 𝐹1 with respect to 𝐹3 𝑋(𝐹1 = 𝑝𝑜𝑜𝑟) = {∅}, 𝑋(𝐹1 = 𝑝𝑜𝑜𝑟) = {𝑦1, 𝑦2, 𝑦3, 𝑦4, 𝑦5, 𝑦6, 𝑦8, 𝑦9, 𝑦10} 𝑋(𝐹1 = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒) = {𝑦7}, 𝑋(𝐹1 = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒) = {𝑦3, 𝑦6, 𝑦7, 𝑦10} 𝑋(𝐹1 = 𝑔𝑜𝑜𝑑) = {∅}, 𝑋(𝐹1 = 𝑔𝑜𝑜𝑑) = {𝑦1, 𝑦2, 𝑦3, 𝑦4, 𝑦5, 𝑦6, 𝑦8, 𝑦9, 𝑦10}. (iii) 𝐹1 with respect to 𝐹4 𝑋(𝐹1 = 𝑝𝑜𝑜𝑟) = {∅}, 𝑋(𝐹1 = 𝑝𝑜𝑜𝑟) = {𝑦1, 𝑦2, 𝑦3, 𝑦4, 𝑦5, 𝑦6, 𝑦7, 𝑦8, 𝑦9, 𝑦10} 𝑋(𝐹1 = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒) = {∅}, 𝑋(𝐹1 = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒) = {𝑦1, 𝑦3, 𝑦5, 𝑦6, 𝑦7} 𝑋(𝐹1 = 𝑔𝑜𝑜𝑑) = {∅}, 𝑋(𝐹1 = 𝑔𝑜𝑜𝑑) = {𝑦1, 𝑦2, 𝑦3, 𝑦4, 𝑦5, 𝑦6, 𝑦7, 𝑦8, 𝑦9, 𝑦10}. (iv) 𝐹1 with respect to 𝐹5 𝑋(𝐹1 = 𝑝𝑜𝑜𝑟) = {∅}, 𝑋(𝐹1 = 𝑝𝑜𝑜𝑟) = {𝑦1, 𝑦2, 𝑦3, 𝑦4, 𝑦7, 𝑦8, 𝑦9, 𝑦10} 𝑋(𝐹1 = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒) = {∅}, 𝑋(𝐹1 = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒) = {𝑦2, 𝑦5, 𝑦6, 𝑦7, 𝑦8, 𝑦10} 𝑋(𝐹1 = 𝑔𝑜𝑜𝑑) = {∅}, 𝑋(𝐹1 = 𝑔𝑜𝑜𝑑) = {𝑦1, 𝑦2, 𝑦3, 𝑦4, 𝑦5, 𝑦6, 𝑦7, 𝑦8, 𝑦9, 𝑦10} to explain in finding the degree of dependency of the attributes, we examine the information system as shown in table1. from table1, depend on each attribute there are five classes of 𝑈 induced by indiscernibility relation on each attribute. using (9), the degree of dependency of attribute 𝐹2 on attribute 𝐹1, denoted by, 𝐹1 ⟹𝑘 𝐹2can be computed as follows: 𝑘 = ∑ |𝐹1(𝑌)|𝑌∈𝑈/𝐹2 |𝑈| = |{𝑦6, 𝑦7}| |{𝑦1, 𝑦2, 𝑦3, 𝑦4, 𝑦5, 𝑦6, 𝑦7, 𝑦8, 𝑦9, 𝑦10}| = 0.2 sharma et al./decis. mak. appl. manag. eng. 3 (1) (2020) 1-21 8 similarly, we can compute 𝐹2 ⟹𝑘 𝐹3 as 𝑘 = ∑ |𝐹2(𝑌)|𝑌∈𝑈/𝐹3 |𝑈| = |{𝑦2, 𝑦4, 𝑦5, 𝑦8, 𝑦9}| |{𝑦1, 𝑦2, 𝑦3, 𝑦4, 𝑦5, 𝑦6, 𝑦7, 𝑦8, 𝑦9, 𝑦10}| = 0.5. the degree of dependency of all the attributes in table 1 can be outlined as in table 2. in mda performance, if the maximum degree of an attribute is equal with other attributes, then we check the next maximum value of the attribute. from the table 2 the first highest degree of the attribute, i.e., 0.5 comes out in attributes 𝐹1 and 𝐹3. then the second maximum degree of attribute 𝐹1 is 0.1 even as in attribute 𝐹3 is 0. hence, attribute 𝐹1 has been elected as a good attribute as compared with 𝐹2, 𝐹3, 𝐹4 and 𝐹5 attributes. therefore, food quality (𝐹1) is the most appropriate attribute for food junction data among rest of the attributes. table 2. degree of dependency using mda criteria 𝐹1 𝐹2 𝐹3 𝐹4 𝐹5 mda 𝐹1 ….. 0.5 0.1 0 0 0.5 0.1 𝐹2 0.2 ….. 0.4 0 0 0.4 0.2 𝐹3 0 0.5 ….. 0 0 0.5 0 𝐹4 0.2 0.2 0.1 ….. 0.2 0.2 𝐹5 0 0.2 0.1 0 ….. 0.2 0.1 4. data description 4.1. mean absolute percentage error (mape) the mean absolute percentage error (mape), as well renowned as mean absolute percentage deviation (mapd), is used as a measure of accuracy for constructing actual and fitted time series data. it usually denoted by the formula: 𝑀𝐴𝑃𝐸 = ∑ (|𝐴𝑐𝑡− 𝑃𝑟�̂�|)/ 𝐴𝑐𝑡 𝑛 𝑡=1 𝑛 (10) where 𝐴𝑐𝑡 (𝑡 = 1, 2, . … , 𝑛) is the actual value, prt ̂ (𝑡 = 1, 2, . … , 𝑛) represents the predicted values and 𝑛 is the total number of observations. 4.2. evaluation of rough set data according to lewis (1982), the range of mape is fixed for time series models. but there is no any strong evidence for others criteria. therefore mape is suitable criteria in the rough set analysis as compared to different indicators. for the measurement of the accuracy of roughness, we normalize the mape values in the following two ways. case1. mape range according to lewis (1982), the value of mape being less than 10% denotes a high degree of accuracy. moreover, when it lies between 10-20% predictions is good, 20-50% is reasonable and more than 50% depicts inaccuracy in prediction. a rough set theory application in forecasting models 9 case 2. the accuracy in the level of prediction relies on the minimum value of mape. henceforth in the paper, we have proposed our assumptions which offer us a valid and highly reliable result. in this sense value of mape, less than 5% provides highly accurate results. however, when it lies in the range of 5% to 10% the prediction is appreciably good and more than 10% shows inaccuracy in the prediction. the proposed procedures of modeling are: first normalized decision table is constructed in the range (0 to 1). then tr, mmr and mda are calculated to select best criteria for forecasting. also, the best criteria are selected on the basis of minimum mmr value and maximum tr and mda. all the steps also described in figure 1. figure 1. the stages of building best forecasting models. 4.3. data the study of chen et al. (2009) has been used to evaluate mape value for this research as shown table 3. in this research, the author analysed total arrivals of 12 countries objects, of japan, hong kong, and the us during the period 1996: 1-2007:12 and three attributes sarima, holt-winters and grey time series models. the entirety data series are separated into two groups, tourism, and non-tourism. 5. result analysis and discussions in this section, we begin with the analysis of the data and make a comparison of their precise results. the comparison of the performance is based on the accuracy of the models. we have considered two test cases to bring out the comparison and start normalize the mape of tourism, non-tourism and allpurpose objects into its equivalent rough set information system compute tr, mmr and mda of all the attributes of the data sets select the best attribute based on minimum mmr and maximum tr and mda stop sharma et al./decis. mak. appl. manag. eng. 3 (1) (2020) 1-21 10 evaluation of the accuracy of the three models using three different rough set criteria tr, mmr, and mda. 5.1. case 1: the rough set information system we have incorporated three models which are holt-winters, sarima, and grey model. all these three models have three normalized values, i.e. reasonable, good and high accuracy. these values are mention in table 4 for considered twelve objectives. table 3. mape of fitted models (%) (chen et al., 2009) objects holt-winters sarima model grey model tourism purpose japan 17.90 10.24 29.80 hong kong 19.27 15.97 24.61 us 5.23 5.01 16.37 total 16.56 7.03 31.16 non-tourism purpose japan 4.75 4.02 8.85 hong kong 4.03 4.87 19.57 us 3.54 3.24 7.55 total 2.69 2.94 4.25 all-purposes japan 10.54 8.50 23.12 hong kong 11.52 13.80 22.58 us 2.81 2.72 8.54 total 5.55 8.95 16.04 a rough set theory application in forecasting models 11 table 4. rough set information system of three models (case1) objects holt-winters sarima model grey model tourism purpose japan reasonable reasonable reasonable hong kong reasonable reasonable reasonable us good good reasonable total reasonable good reasonable non-tourism purpose japan high accuracy high accuracy good hong kong high accuracy high accuracy reasonable us high accuracy high accuracy good total high accuracy high accuracy high accuracy all-purposes reasonable japan reasonable good reasonable hong kong reasonable reasonable good us high accuracy high accuracy reasonable total good good reasonable to serve this purpose we obtain the equivalence classes determined by indiscernibility relation from equation (1). (i) 𝑌( 𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 = 𝑟𝑒𝑎𝑠𝑜𝑛𝑎𝑏𝑙𝑒 ) = { 1,2 4, 9, 10}, 𝑌(𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 = 𝑔𝑜𝑜𝑑) = {3,12}, 𝑌(𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 = 𝐻𝑖𝑔ℎ 𝑎𝑐𝑐𝑢𝑟𝑎𝑐𝑦) = {5,6,7,8,11}, 𝑈 / 𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 = {{1,2,4,9,10}, {3,12}, {5,6,7,8,11}} (ii) 𝑌 (𝑆𝐴𝑅𝐼𝑀𝐴 = 𝑟𝑒𝑎𝑠𝑜𝑛𝑎𝑏𝑙𝑒) = {1,2,10}, 𝑌(𝑆𝐴𝑅𝐼𝑀𝐴 = 𝑔𝑜𝑜𝑑) = {3,4,9,12}, (𝑆𝐴𝑅𝐼𝑀𝐴 = 𝐻𝑖𝑔ℎ 𝑎𝑐𝑐𝑢𝑟𝑎𝑐𝑦) = {5,6,7,8, 11}, 𝑈/𝑆𝐴𝑅𝐼𝑀𝐴 = {{1,2,10}, {3,4,9,12}, {5,6,7,8,11}} (iii) 𝑌 (𝐺𝑀(1,1) = 𝑟𝑒𝑎𝑠𝑜𝑛𝑎𝑏𝑙𝑒) = {1,2,3,4,6,9,10,12}, 𝑌(𝐺𝑀(1,1) = 𝑔𝑜𝑜𝑑) = {5,7,11}, 𝑌(𝐺𝑀(1,1) = 𝐻𝑖𝑔ℎ 𝑎𝑐𝑐𝑢𝑟𝑎𝑐𝑦) = {8}, 𝑈/ 𝐺𝑀(1,1) = {{1,2,3,4,6,9,10,12}, {5,7,11}, {8}}. in the next step, we finalize the lower and upper approximations of subset y of u based on holt-winters with respect to sarima and grey model using the formula in equation (2) and (3). sharma et al./decis. mak. appl. manag. eng. 3 (1) (2020) 1-21 12 (i) holt-winters with respect to sarima 𝑌 (𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 = 𝑟𝑒𝑎𝑠𝑜𝑛𝑎𝑏𝑙𝑒) = {1, 2, 10}, 𝑌 (𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 = 𝑟𝑒𝑎𝑠𝑜𝑛𝑎𝑏𝑙𝑒) = {1, 2, 3, 4, 9, 10, 12}, 𝑌 (𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 = 𝑔𝑜𝑜𝑑) = {∅} , 𝑌 (𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 = 𝑔𝑜𝑜𝑑) = {3, 4, 9, 12}, 𝑌 (𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 = 𝐻𝑖𝑔ℎ 𝑎𝑐𝑐𝑢𝑟𝑎𝑐𝑦) = {5, 6, 7, 8, 11}, 𝑌 (𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 = 𝐻𝑖𝑔ℎ 𝑎𝑐𝑐𝑢𝑟𝑎𝑐𝑦 ) = {5, 6, 7, 8, 11}. (ii) holt-winters with respect to grey 𝑌 (𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 = 𝑟𝑒𝑎𝑠𝑜𝑛𝑎𝑏𝑙𝑒) = {∅}, 𝑌 (𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 = 𝑟𝑒𝑎𝑠𝑜𝑛𝑎𝑏𝑙𝑒) = {1, 2, 3, 4, 6, 9, 10, 12}, 𝑌 (𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 = 𝑔𝑜𝑜𝑑) = {∅}, 𝑌 (𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 = 𝑔𝑜𝑜𝑑) = {1, 2, 3, 4, 6, 9, 10}, 𝑌 (𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 = 𝐻𝑖𝑔ℎ 𝑎𝑐𝑐𝑢𝑟𝑎𝑐𝑦) = {5, 6, 7, 8, 11}, 𝑌 (𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 = 𝐻𝑖𝑔ℎ 𝑎𝑐𝑐𝑢𝑟𝑎𝑐𝑦) = {5, 6, 7, 8, 11}. then, we obtain the roughness of all models, where tr makes the use of formula as in the equation (5). the roughness of subsets of u consist distinct value of holtwinters with respect to sarima and grey models are given below. (i) roughness(r) of holt winters with respect to sarima model 𝑅𝑟𝑒𝑎𝑠𝑜𝑛𝑎𝑏𝑙𝑒 = 0.42857, 𝑅𝑔𝑜𝑜𝑑 = 0 and 𝑅ℎ𝑖𝑔ℎ 𝑎𝑐𝑐𝑢𝑟𝑎𝑐𝑦 = 1 (ii) roughness of holt-winters with respect to grey model 𝑅𝑟𝑒𝑎𝑠𝑜𝑛𝑎𝑏𝑙𝑒 = 0, 𝑅𝑔𝑜𝑜𝑑 = 0 and 𝑅ℎ𝑖𝑔ℎ 𝑎𝑐𝑐𝑢𝑟𝑎𝑐𝑦 = 1. further, we obtain the mean roughness of all attributes model. where tr uses the formula in equation (6). (i) mean roughness(mr) of holt-winters with respect to sarima model 𝑀𝑅𝐻𝑊 = 0.47619. (ii) mean roughness of holt-winters with respect to grey model 𝑀𝑅𝐻𝑊 = 0.11. now finally, we obtain the total roughness of holt-winters with respect to the sarima and grey model. 𝑇𝑅(𝑎𝑖 ) = (0.47 + 0.11) /2= 0.2931, where ai is the numbers of attributes model, 𝑖 = 1, 2. now, from table 5, we observe that the tr value 0.2931 of holt-winters is much higher than tr value 0.1944 of grey model. thus, the holt-winters consider as a better model than the grey model. similarly, tr value of sarima is greater than that of the grey model. hence, the sarima model serves us as a better model in view of the grey model. a rough set theory application in forecasting models 13 table 5. the total roughness of three model using tr criteria attribute holt-winters sarima model grey model total roughness holt winters ……… 0.47619 0.11 0.2931 sarima model 0.42857 ……….. 0.11 0.26929 grey model 0.19444 0.19444 ………….. 0.19444 fourthly, we find the mean roughness of the three attributes using the principle of the mmr technique as in the equation (8) which differs from that in the tr technique. mean roughness of the model holt-winters with respect to sarima and grey model is as below. (i) mmr of holt winters with respect to sarima model 𝑀𝑀𝑅𝐻𝑊 = 1 − 𝑀𝑅𝐻𝑊 = 1 − 0.47619 = 0.52381 (ii) mmr of holt winters with respect to grey model 𝑀𝑀𝑅𝐻𝑊 = 1 − 𝑀𝑅𝐻𝑊 = 1 − 0.11 = 0.89 consequently, the mmr of the entire models is depicted in table 6. we perceive that the mmr of the holt-winters and sarima have attained their minimum values respectively i.e., 0.52381 and 0.57143. these values are lower than that of the mmr of grey model i.e., 0.80556. thus, holt-winters and sarima are preferred as the good models. also, the results of the experiment were summarized in fig. 2 and fig. 3 shows the accuracy of three models with tr, mmr, and mda values. however, as far as mda techniques are considered, holt-winters and sarima have the maximum degree of dependency in table 7 in contrast with the grey model. as a result, the holt-winters and sarima are preferred as the good models. table 6. the minimum-minimum roughness of three models using mmr criteria models holtwinters sarima model grey model mmr holt winters …….. 0.52381 0.89 0.52381 sarima model 0.57143 ……….. 0.89 0.57143 grey model 0.80556 0.80556 ……….. 0.80556 sharma et al./decis. mak. appl. manag. eng. 3 (1) (2020) 1-21 14 table 7. degree of dependency using mda criteria of three models figure 2. the roughness of hw, sarima and gm (1, 1) models using tr criterion for case 1 models holt-winters sarima model grey model mda holt winters ….. 0.6667 0.333 0.6667 sarima model 0.583 ….. 0.333 0.583 0.333 grey model 0.583 0.583 ….. 0.583 0.2931 0.26929 0.19444 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 hw sarima gm(1,1) r o u g h n e ss m e a su re m e n t forecasting models a rough set theory application in forecasting models 15 figure 3. the roughness of hw, sarima and gm (1, 1) models using mmr and mda criteria for case 1 5.3. case 2: the rough set information system table 8 shows in the information system of rough set data of case2. in this case, the analysis to evaluate tr, mmr, and mda of each method is similar as in case1. with tr techniques holt-winters model is considered an as good model because this attribute has the highest total roughness in table 9 i.e. 0.5625 as compared to sarima and grey model i.e. 0.29165 and 0.2065. thus, we select holt-winters model is good as compare to sarima and grey model. the results of mmr and mda are given in table 10 and table 11. according to mmr and mda reports, holt-winters models are chosen to select good model as compared to sarima and grey model. also, fig. 4 and fig. 5 illustrate the accuracy of three models with tr, mmr and mda values which are involving the datasets of case 2. from the above considerations, we can see that case 1 gives better results of three models in terms of rough accuracy compare to case 2 results. therefore, case 1 is an appropriate range of mape for applying the rough set approach in the field of time series modeling. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r o u g h n e ss models mmr mda sharma et al./decis. mak. appl. manag. eng. 3 (1) (2020) 1-21 16 table 8. rough set information system of three models (case 2) objects holt-winters sarima model grey model tourism purpose japan good good reasonable hong kong good good reasonable us high accuracy high accuracy good total good high accuracy reasonable non-tourism purpose japan high accuracy high accuracy high accuracy hong kong high accuracy high accuracy good us high accuracy high accuracy high accuracy total high accuracy high accuracy high accuracy all-purposes japan good high accuracy reasonable hong kong good good reasonable us high accuracy high accuracy high accuracy total high accuracy high accuracy good table 9. the total roughness of three models using tr criteria attribute holt-winters sarima model grey model total roughness holt winters ……… 0.125 1 0.5625 sarima model 0.29165 ……….. 0.29165 0.29165 grey model 0.33 0.083 ………….. 0.2065 table 10. the minimum-minimum roughness of three models using mmr criteria attribute holtwinters sarima model grey model mmr holt winters …….. 0.875 0 0 0.875 sarima model 0.70835 ……….. 0.70835 0.70835 grey model 0.67 0.917 ……….. 0.67 0.917 a rough set theory application in forecasting models 17 table 11. degree of dependency using mda criteria attribute holt-winters sarima model grey model mda holt winters ….. 0.25 1 1 sarima model 0.5833 ….. 0.5833 0.5833 grey model 0.4167 0.25 ….. 0.4167 figure 4. the roughness of hw, sarima and gm (1, 1) models based on tr criterion for case 2 figure 5. the accuracy of hw, sarima and gm (1,1) models based on mmr and mda criteria for case 2 0 0.1 0.2 0.3 0.4 0.5 0.6 hw sarima gm(1,1) r o u g h n e ss models 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 hw-sarima sarima-hw gm(1,1)-hw gm(1,1)-sarima r o u g h n e ss models mmr mda sharma et al./decis. mak. appl. manag. eng. 3 (1) (2020) 1-21 18 6. conclusions current research proposes a new technique to select the best forecasting model using rough set approach. this technique is based on the rough set criteria i.e., tr, mmr and mda. in the present study, the problems concerning the selection of good forecasting model using rough set methods, such as normalized mape values. two test cases are considered to implement rough set and three models i.e. holt winters, sarima, and grey models are employed. at first, we normalize mape values into the rough set information system and then we calculate the accuracy of each model. a close analysis of all the results furnished by the respective models, it reports that holtwinters and sarima models are far better models when compared to grey model. according to the analysis reports, holt-winters and sarima are good models in consideration of seasonal time series. also, grey model is an unqualified model to forecast seasonal time series under tr, mmr, and mda rough set criteria. in this research, we would recommend that the proposed research is feasible and it offers a powerful statistical evidence for rough set methods. . herewith these concepts, we suppose that various 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(2009). an interval type-2 fuzzy rough set model forattribute reduction, ieee transactions on fuzzy systems, 17(2), 301-315. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 1, number 1, 2018, pp. 51-66 issn: 2560-6018 doi: https://doi.org/10.31181/dmame180151b * corresponding author. e-mail addresses: dragan.bojanic@mod.gov.rs (d. bojanic), marbojanic@gmail.com (m. bojanic), vladimir.ristic@mod.gov.rs (v. ristic) multi-criteria decision-making in a defensive operation of the guided anti-tank missile battery: an example of the hybrid model fuzzy ahp mabac dragan bojanic1*, mitar kovač2, marina bojanic3 and vladimir ristic4 1 strategic research institute, university of defense, belgrade, serbia 2 university of edukons, faculty of project and innovation management, belgrade, serbia 3 electrical engineering school „nikola tesla“, pančevo, serbia 4 university of defense, military academy, belgrade, serbia received: 3 january 2018; accepted: 31 january 2018; published: 15 march 2018. original scientific paper abstract: the military decision-making process is a proven analytical process for designing operations, troops’ movements, logistics or air defense planning. the hybrid fahp-mabac model is tested for obtaining/selecting the results for an optimal firing position of the guided anti-tank missile battery (gamb).this study provides a multi-criteria decision-making (mcdm) model so that the confidence interval of fuzzy numbers describes the comparison in pairs whose degree is not determined before the comparison. by using mathematical expressions the confidence interval is brought into direct connection with the degree of certainty of decision-makers/expert of the comparison performed. in the group decision-making, the confidence intervals differ depending on the decision-maker/expert’s opinion. finally the sensitivity analysis is used to determine how sensitive a decision model is. the suggested model is expected to contribute to the development of the science of military-operations as well as to prove itself useful to the actors related to defense. key words: fuzzy ahp, mabac, mcdm, sensitivity analysis. 1. introduction mcdm is a systematic way of problem solving for any scientific research area. the military decision-making process is a proven analytical process for designing operations, troops’ movements, logistics or air defense planning. in most cases, the decision-making process in the military organization implies not having relevant information, which is characterized by a high degree of mailto:dragan.bojanic@mod.gov.rs mailto:marbojanic@gmail.com mailto:vladimir.ristic@mod.gov.rs bojanić et al./decis. mak. appl. manag. eng. 1 (1) (2018) 51-66 52 uncertainty, subjectivity and ambiguity. generally, in the decision-making process for these problems, a large number of key and different criteria are involved. therefore, it is advisable to use tools for their resolution such as mcdm processes that are nowadays widely used in the military (kewley & embrechts, 2002; de leeneer & pastijn, 2002; zanjirani & asgarib, 2007; kose et al., 2013; gyarmati, 2015; goztepe & kahraman, 2015; andersson et al., 2015; boccia et al., 2017), as well as in other research disciplines (sánchez-lozano et al., 2013). fuzzy logic proves to be an appropriate tool (sánchez-lozano et al., 2015) to address the described uncertainties and ambiguities. in that way, fuzzy logic enables the exploitation of tolerance that exists in imprecision, ambiguity and partial truth of the results obtained through given research. anti-tank is defined as a combat against enemy’s tanks and other armored equipment and vehicles. its associated tasks include the closing of endangered directions, flanks, and junctions (gordic et al., 2013). anti-tank is a part of the combined combat arms directed against the tanks grouped for an attack or already directly attacking, as well as the armored equipment detached to their battle formation. for a successful combat, the organization of a solid anti-armor system is required, which includes anti-tank units as the basic element. the aim of anti-tank operations in a defensive operation is to prevent accidental penetration of the enemy's armored forces into our elements of combat disposition (slavkovic et al., 2013; jotić & slavković,2016). the new operational environment imposes the need to upgrade the defense science in the field of the preparation and conduct of combat operations (knežević & slavković, 2012). in the serbian army many decisions are made in the processes of planning, organization and preparation for the execution of missions and tasks. the useful tools which support the decision-making process are the methods of multi criteria decision-making. the fundamental issue of multi-criteria group decision-making is finding procedures for choosing decisions that correspond to the desired solution, with the option of selecting and allocating the most acceptable alternative. the complexity of group decision-making is reflected in the fact that there are a number of criteria and alternatives with different levels of significance for the decision-makers/experts involved in decision-making. this paper presents a hybrid model, using the fuzzificated saaty’s scale and the mabac method (multi-attributive border approximation area comparison) (božanić et al., 2015; božanić et al., 2016a; božanić et al., 2016b; pamučar et al., 2016b). the paper is focused on the demonstration of a new way of fuzzification of the saaty’s scale used for comparison in pairs while varying a confidence interval depending on the comparison. the scale is used for obtaining criteria weight coefficients, while the mabac method is used for the final ranking. this model is illustrated by an example of decision-making during the selection of firing position (pop) of the gamb in the army's defensive operation. the example presents only one segment from a series of decisions that decision-makers face in the preparation and execution of (military) operations. the sensitivity analysis is used to determine how sensitive a decision model is. 2. description of the methods used in the hybrid ahp-mabac model the described mcdm model is based on the knowledge of several decision making methods (areas), fuzzy logic, the ahp method (saaty's scale) and the mabac https://www.sciencedirect.com/science/article/pii/s0377221701003721#! https://www.sciencedirect.com/science/article/pii/s0377221705008775#! http://www.emeraldinsight.com/author/kose%2c+erkan https://www.researchgate.net/profile/kerim_goztepe https://www.researchgate.net/profile/cengiz_kahraman multi-criteria decision-making in a defensive operation of the guided anti-tank... 53 method. fuzzy logic successfully covers vagueness and uncertainty that are often present in decision-supporting models. the saaty's scale, which is an indispensable part of the ahp method, shows good results in defining the criteria weight coefficients, and it is increasingly applied with other methods (knežević et al., 2015; zhu et al., 1999). the mabac method provides stable (consistent) solutions and it represents a reliable tool for rational decision-making (pamučar & ćirović, 2015). 3. fuzzy logic and fuzzy sets in fuzzy logic, an element’s belonging to the specific set is not precisely defined the element can be more or less part of the set; therefore, it is closer to human perception than conventional logic (pamučar et al., 2016b). fuzzy logic allows quantification of seemingly imprecise information, which is a very common situation when describing social phenomena. fuzzy logic uses the experience of human expert in the form of linguistic if-then rules, while approximate reasoning mechanism uses managing action for the individual case. in this paper, approximate reasoning algorithm is used to display the influence of the entry criteria on decision preference in choosing the most appropriate firing gamb position. the first step in designing fuzzy sets is to define the degree of the membership of an element x (xx) in the set a. this is described with membership function a (x), which in the classic theory has a value of 0 (does not belong) or 1 (belongs), while in a fuzzy set the membership function can have any value between 0 and 1. so, it can be said that the closer а (x) is to 1, the greater belonging of x to a is, and vice versa. a fuzzy set a is defined as a set of ordered pairs      a aa x, x x x, 0 x 1     (1) where:  x is a universal set or a set of considerations based on which fuzzy set a is defined;  a (x) function of element x belonging to set a. in this paper, triangular fuzzy numbers will be used. they will be presented in the form t = (t1, t2, t3), where fig. 1:  t2 is where the membership function of a fuzzy number has a value of 1;  t1 is the left distribution of the confidence interval of fuzzy number t, and,  t3 is the right distribution of the confidence interval of fuzzy number t. t1 t2 t3 1  t x    2 1, t x x t     1 1 2 2 1 , t x x t t x t t t         3 2 3 3 2 , t x t x t x t t t         1 0, t x x t     3 0, t x x t    0 αt1 αt2 bojanić et al./decis. mak. appl. manag. eng. 1 (1) (2018) 51-66 54 figure 1. triangular fuzzy number t (pamučar et al., 2016a) the membership function of fuzzy number t is defined in the following way:   1 1 1 2 2 1 2t 3 2 3 3 2 3 0, x t x t , t x t t t x 1, x t t x , t x t t t 0, x t                     (2) for its final purpose, fuzzy number t= (t1, t2, t3) is converted into a real number. 4. fuzzification of the saaty’s scale one of the key phases in the application of this method is the development of the comparison matrix by pairs, corresponding to every level of the hierarchy. a pairwise comparison is performed according to the data collected by measuring them on the basis of beliefs, estimates or experiences of those who carry out the assessment (čupić & suknović 2010). the shown fuzzification of the saaty’s scale is presented in (božanić et al., 2015; božanić et al., 2016a; božanić et al., 2016b; pamučar et al., 2016b; božanić, 2017). the definition of this new fuzzificated saaty’s scale (table 1) starts from the assumption that the decision-makers and analysts have a different degree of certainty ji, concerning the accuracy of comparisons in pairs (božanić et al., 2016a; pamučar et al., 2016b). this degree of certainty differs from one comparative pair to another. the value of the degree of certainty belongs to interval ji0,1. in the cases when ji=0, it is considered that the decision-maker/analyst has no data about this relationship; hence it should not be used in the decision-making process because it points to absolute ignorance of the decision-making subject. the value of the degree of certainty where ji=1 describes the absolute certainty of the decisionmakers/analysts in the given comparison. the lower the certainty in the performed comparison is, the lower the element ji. table 1. fuzzified saaty's scale for comparison in pairs (božanić, 2017). definition standard value fuzzy number inverse values of the fuzzy number same importance 1 (1, 1, 1) (1, 1, 1) low dominance 3   ji ji3 , 3, 2 3    ji ji1 2 3,1/ 3,1 3  high dominance 5   ji ji5 , 5, 2 5    ji ji1 2 5,1/ 5,1 5  very high dominance 7   ji ji7 , 7, 2 7   ji ji1 (2 )7 ,1 / 7,1 7  absolute dominance 9   ji ji9 , 9, 2 9   ji ji1 (2 )9 ,1 / 9,1 9  multi-criteria decision-making in a defensive operation of the guided anti-tank... 55 intermediate values 2, 4, 6, 8   ji jix , x, 2 x  x 2, 4, 6, 8   ji ji1 2 x ,1/ x,1 x  x = 2, 4, 6, 8 by defining different values of parameter ji, the left and the right distribution of fuzzy numbers change from one comparison to another, according to the expression:           1 2 1 2 1 2 1 2 3 2 2 2 3 2 3 2 2 3 t t , t t , t , t 1 / 9, 9 t t , t , t t t , t 1 / 9, 9 t 2 t , t t , t , t 1 / 9, 9                    (3) the value of t2 represents the value of linguistic expressions from the classic saaty’s scale, which in a fuzzy number has a maximum membership t2 = 1 fuzzy number     1 2 3t t , t , t x , x, 2 x    ,  x 1,9 is defined by the expressions: 1 x , 1 x x t x 1, x 1             (4)  2t x, x 1,9   (5)    3t 2 x, x 1,9    (6) inverse fuzzy number     1 2 3t t , t , t x , x, 2 x    ,  x 1, 9 is defined as follows:          3 1 2 x , 1 2 x 1 1/ t 1 2 x , x 1, 9 1, 1 2 x 1                (7) 21 / t 1 / x, 1 / x 1, 9   (8)  11/ t 1 x , 1/ x 1,9   (9) the defined scale is further used in standard steps of the ahp method, which is described in a number of papers (čupić & suknović2010), (inđić et al., 2014) and others. based on the pre-defined scale, the decision-makers and analysts fill in the new, modified matrix: 1 2 n 1 11 11 12 12 1n 1n 2 21 21 22 22 2n 2n n n1 n1 n 2 n 2 nn nn c c c c a ; a ; a ; a c a ; a ; a ; c a ; a ; a ;                       (10) as can be seen, the matrix is extended with the degree of certainty in the comparison made, whereby ji=ij, where ji0,1. after the calculation is finished, the defuzzification can be performed using one of well-known methods. some of the well-known expressions for defuzzification (seiford, 1996) are the following ones: bojanić et al./decis. mak. appl. manag. eng. 1 (1) (2018) 51-66 56     13 1 2 1 1defazzy s= t t t t 3 t        (11)   13 2 1defazzy s= t t 1 t 2        (12) where  represents the degree of optimism. the defined scale is also suitable for group decision-making, which is nowadays becoming more and more popular. involving experts greatly improves the quality of decisions made because knowledge and experience are collected and consolidated into a single unit. 5. mabac method the mabac method was developed by pamučar and ćirović (pamučar & ćirović, 2015). the basic setting of the mabac method is reflected in the definition of the distance of the criterion function of each of the observed alternatives from the approximate border area. the text that follows shows the procedure of implementation of the mabac method in six steps, its mathematical formulation being: step 1. creation of initial decision matrix (x). in the first step, the evaluation of m alternatives by n criteria is carried out. the alternatives are presented with vectors ai = (xi1,xi2, ……xin), where xij.is the value of alternative i according to criteria j (i = 1,2,..., m; j = 1,2,..., n). 1 2 n 1 11 12 1n 2 11 22 2n m 1m 2m mn c c ... c a x x ... x a x x x x ... ... ... ... ... a x x ... x             (13) where m indicates the number of alternatives, and n indicates the total number of criteria. step 2. normalization of initial matrix (x) elements. 1 2 n 1 11 12 1n 2 11 22 2n m 1m 2m mn c c ... c a t t ... t a t t t n ... ... ... ... ... a t t ... t             (14) the elements of normalized matrix (n) are obtained using the following expressions: a) for the "benefit" type criteria ij i ij i i x x t x x       (15) b) for the "cost" type criteria multi-criteria decision-making in a defensive operation of the guided anti-tank... 57 ij i ij i i x x t x x       (16) where xij, xi+and xi− are the components of initial decision matrix (x), where xi+ and xi− are defined as: xi+= max (x1, x2,..., xm) and represents the maximum value of the observed criteria by alternatives, xi+= min(x1, x2,..., xm) and represents the minimum value of the observed criteria by alternatives. step 3. calculation of weighted matrix elements (v). the elements of weighted matrix (v) are calculated on the basis of expression (17): ij i ij i v w t w  (17) where tij are the elements of normalized matrix (n), and w represents the weight coefficient of criteria. by applying expression (17), we get weighted matrix v that otherwise can be written as: 11 12 1n 1 11 1 2 12 2 n 1n n 21 22 2n 1 21 1 2 22 2 n 2n n m1 m 2 mn 1 m1 1 2 m 2 2 n mn n v v ... v w t w w t w ... w t w v v ... v w t w w t w ... w t w v ... ... ... ... ... ... ... ... v v ... v w t w w t w ... w t w                                 (18) where n is the total number of criteria, and m is the total number of alternatives. step 4. determination of approximate border area (g) matrix. the border approximate area (baa) for each criterion is determined by expression (19) 1/ m m i ij j 1 g v          (19) where vij are weighted matrix elements (v) and m represents the total number of alternatives. after determining value gi according to the criteria, we form the matrix of approximate border areas g (20) size nx1 (n is the total number of criteria by which the election of the offered alternatives is made).   1 2 n 1 2 n c c ... c g g g ... g (20) step 5. calculation of the matrix elements distance from border approximate area (q) 11 12 1n 21 22 2n m1 m 2 mn q q ... q q q q q ... ... ... ... q q ... q             (21) the distance of the alternatives from border approximate area (qij) is defined as the difference between weighted matrix elements (v) and the values of border approximate areas (g). bojanić et al./decis. mak. appl. manag. eng. 1 (1) (2018) 51-66 58 q v g  (22) which otherwise can be written as: 11 1 12 2 1n n 21 1 22 2 2n n m1 1 m 2 2 mn n v g v g ... v g v g v g ... v g q ... ... ... ... v g v g ... v g                    (23) where gi represents the border approximate area for criterion ci, vij. is weighted matrix elements (v), n represents the number of criteria, and m represents the number of alternatives. alternative ai may belong to border approximate area (g), upper approximate area (g+) or lower approximate area (g¯), respectively ai⋲{g¯˅g+v g¯}. upper approximate area (g+) is an area in which the ideal alternative is found (a+), while lower approximate area (g¯) is an area in which the anti-ideal alternative (a¯) is found (fig. 2). g  g  a  a  1 a 3 a 4 a 2 a 5 a 6 a 7 a g upper approximation area lower approksimation area bordere approksimation area 0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 c ri te ri o n f u n c ti o n s figure 2. presentation of upper (g+), lower (g¯) and border (g ) approximation areas (pamučar, ćirović, 2015) belonging of alternative ai to approximate area (g ,g+ or g¯) is determined on the basis of expression (24) ij i ij ij g if q 0 a g if q 0 g if q 0          (24) in order for alternative at to be chosen as the best from the set, it is necessary that, according to as many criteria as possible, it belongs to upper approximate area (g+). step 6. ranking alternatives. calculation of the criteria function values by alternatives is obtained as the sum of the distances of the alternatives from border approximate areas (qi). summing the elements of matrix q by rows gives the final values of the criteria function alternatives multi-criteria decision-making in a defensive operation of the guided anti-tank... 59 n i ij j 1 s q , j 1, 2,..., n, i 1, 2,..., m     (25) where n represents the number of criteria, and m represents the number of alternatives. 6. selection of the optimal firing position of the guided anti-tank missile battery in a defense operation 6.1. identification of the criteria and calculation of the weight coefficients of the criteria a guided anti-tank missile battery performs its tasks from the firing position (pop). it is composed of self-propelled launchers lrspm83. the (pop) is a part of the land in the area of the operation, prepared and occupied or intended to be occupied by the artillery units for the execution of firing support (the military lexicon, 1981). “in order to ensure the effectiveness of fire control and fire coordination, the distance in length is 100–300m between self-propelled launchers, and 300–400m between platoons. the dimensions of deployment areas can extend to 1,4-2 km in width and 1 km in length for batteries, 1km in width and 500m in length for platoons depending on the combat situation, terrain, and the number of assets involved.” (fig. 3) (rule book self-propelled anti-tank battery – platoon, 2016). figure 3. battery deployment on pop decision-makers usually have to select an (pop) area relying on the acquired theoretical knowledge, experience and assessment in the specific situation. a number of criteria that influence the ranking and selection of alternatives indicate a bojanić et al./decis. mak. appl. manag. eng. 1 (1) (2018) 51-66 60 possibility of applying multiple criteria methods. founded on the available literature, the ranking criteria that are the basis for selection-making are also defined: c1– „distance to the targets area“the distance at which the guided armorpiercing missiles will neutralize the enemy armored vehicle in the direction of the attack. the range of guided armor-piercing missiles is determined by their maximum launching distance and the distance of the fire line – influenced by the terrain in front of their firing positions; c2– „speed of occupancy pop“ represents the estimate of time for which launching devices from the expected region will make an arrival at pop; c3– „maneuver capability “ -the maneuver capability of guided anti-tank missile units is determined by their mobility (movements to firing positions, during maneuvers to deployment areas), and the time required for occupation and leaving of the deployment areas; c4 – „fortification conditions“ terrain features that allow successful fortification of artillery to enhance force protection; c5 – „quality of access roads“ road characteristics that appreciate the possibility of fast and successful settling and abandonment of pop; c6 – „masking conditions “ terrain features that enable successful masking of gamb and movement of gamb of parts as well as masking of the effects of the rocket launching. the values of criteria c1 and c2 are described numerically and the values of criteria c3, c4, c5i c6 are described with fuzzy linguistic descriptors, fig. 4 (very poor vl, poor l, medium s, good d and great o). 1 0.8 0.6 0.4 0.2 1 32 54 0 vl l s d o figure 4. graphic display of fuzzy linguistic descriptors (božanić et al., 2016b; pamučar et al., 2016b; božanić, 2017) the first step in defining the weight coefficients is to define the square comparison matrix. two elements of the hierarchy (models) are compared using the saaty’s classic scale and by defining the degree of certainty of a given claim according to expression (10). the degree of inconsistency of the given matrix is 0.0352. table 2. comparison matrix in pairs crit. c1 c2 c3 c4 c5 c6 c1 1;1 1.4;0.7 1.2;0.6 4.5;0.9 2.5;0.9 3.1;0.9 c2 0.7;07 1;1 1.1;0.1 1.3;0.9 1.8;0.6 2.6;0.4 multi-criteria decision-making in a defensive operation of the guided anti-tank... 61 c3 0.8;0.6 0.9;0.1 1;1 2.0;0.5 2.0;0.7 4.1;0.9 c4 0.2;0.9 0.8;0.9 0.5;0.5 1;1 2.4;0.5 2.5;0.9 c5 0.4;0.9 0.5;0.6 0.5;0.7 0.4;0.5 1;1 1.2;0.5 c6 0.3;0.9 0.4;0.4 0.2;0.9 0.4;0.9 0.8;0.5 1;1 the values of the table 2 matrix are converted into fuzzy numbers by applying the fuzzificated saaty’s scale (table 1), so that a new matrix is shown in table 3. table 3. comparison matrix in pairs after fuzzification crit. c1 c2 c3 c4 c5 c6 c1 1;1;1 1.0;1.4;1.8 1.0;1.3;1.8 4.1;4.5;4.9 2.3;2.5;2.8 2.8;3.1;3.4 c2 0.6;0.7;1.0 1;1;1 1.0;1.1;2.1 1.2;1.3;1.4 1.1;1.9;2.6 1.0;2.6;4.2 c3 0.6;0.8;1.0 0.5;0.9;1.0 1;1;1 1.0;2.0;3.0 1.4;2.0;2.6 3.7;4.1;4.5 c4 0.2;0.2;0.2 0.7;0.8;0.8 0.3;0.5;1.0 1;1;1 1.2;2.4;3.7 2.3;2.5;2.8 c5 0.4;0.4;0.4 0.4;0.5;0.9 0.4;0.5;0.7 0.3;0.4;0.8 1;1;1 1.0;1.2;1.8 c6 0.3;0.3;0.4 0.2;0.4;0.9 0.2;0.2;0.3 0.4;0.4;0.4 0.5;0.8;1.0 1;1;1 the weight vector w of every criterion of the table 3 matrix is the sum of the linguistic expressions that describe the criteria in the same row of the table 3 matrix, which is divided by the sum of all linguistic expressions that describe the criteria of the table 3 matrix. after the calculation, the weight vectors of the criteria are obtained (table 4). the label "l" represents the left distribution of the fuzzy number; "d" represents the right distribution of the fuzzy number and "s" a place where the level of the membership of the fuzzy number has a value one. table 4. fuzzy weight vectors of the criteria criteria "l" "s" "d" c1 0.203 0.293 0.424 c2 0.113 0.191 0.339 c3 0.126 0.217 0.327 c4 0.085 0.139 0.234 c5 0.059 0.092 0.159 c6 0.045 0.068 0.119 finally, by applying expression (11) the defuzzification of the weight coefficients of the evaluation criteria is performed. the final values of the weight coefficients of the criteria are given in table 5. table 5. final values of the weight coefficients criteria weight coefficients (w) c1 0.307 c2 0.214 c3 0.223 c4 0.152 c5 0.103 c6 0.077 bojanić et al./decis. mak. appl. manag. eng. 1 (1) (2018) 51-66 62 6.2. application of the mabac method for ranking alternatives in order to apply the mabac method, six alternatives have been selected (from a1 to a6) (table 6). the alternatives represent a land area that the artillery means will be deployed on. the initial decision matrix is presented in table 6. since evaluation criteria c3, c4, c5 and c6 have a qualitative character, for the evaluation of alternatives by criteria, the fuzzy linguistic descriptors scale is used. table 6. initial decision matrix alternative c1 (min) c2 (min) c3 (max) c4 (max) c5 (max) c6 (max) a1 1300 6.00 s o d s a2 2200 8.00 o vl s o a3 1950 11.00 l s l s a4 1400 10.00 vl l s o a5 2500 5.00 s vl l s a6 1700 9.00 o d d l defuzzification is performed by applying expression (12). the next step is the normalization of the matrix elements and the final values of the normalized matrix are presented in table 7. table 7. normalized matrix alternative c1 (min) c2 (min) c3 (max) c4 (max) c5 (max) c6 (max) a1 1.000 0.833 0.500 1.000 1.000 0.375 a2 0.250 0.500 1.000 0.000 0.500 1.000 a3 0.458 0.000 0.201 0.500 0.000 0.375 a4 0.917 0.167 0.000 0.201 0.500 1.000 a5 0.000 1.000 0.500 0.000 0.000 0.375 a6 0.667 0.333 1.000 0.799 1.000 0.000 using the next steps of the mabac method, the distance values of the alternatives from the border approximate area are obtained and their ranking is shown in table 8. table 8. ranking of the alternatives alternative si rang a1 0.309 1 a2 0.055 3 a3 -0.200 6 a4 0.032 4 a5 -0.119 5 a6 0.229 2 based on the obtained distances from the ideal alternative, it can be concluded that alternative (a1) is the most appropriate alternative while alternative (a3) is the least favorable one. multi-criteria decision-making in a defensive operation of the guided anti-tank... 63 7. sensitivity analysis of the output results it is recommended as a means of checking the stability of the results against the subjectivity of decision-makers. the sensitivity analysis of the results is carried out by changing the initial weight coefficients of the evaluation criteria. table 9 gives the scenarios of change of weight coefficients (seven scenarios), based on which the ranking of the already presented alternatives is performed. table 9. scenarios with different weights of the criteria criteria s1 s2 s3 s4 s5 s6 s7 c1 0.2 0.6 0.1 0.1 0.1 0.1 0.1 c2 0.2 0.1 0.6 0.1 0.1 0.1 0.1 c3 0.2 0.1 0.1 0.6 0.1 0.1 0.1 c4 0.2 0.1 0.1 0.1 0.6 0.1 0.1 c5 0.2 0.1 0.1 0.1 0.1 0.6 0.1 c6 0.2 0.1 0.1 0.1 0.1 0.1 0.6 the ranking of the alternatives after the application of the scenario is given in table 10. by analyzing the obtained results, it can be concluded that there is significant stability of the output results in most of the scenarios. this is supported by the fact that a1 and a4 are most often ranked as the first or the second, which is expected for a stable system. table 10. ranks of alternatives by applying different situations alternative rank from table 8 s1 s2 s3 s4 s5 s6 s7 a1 1 1 1 1 3 1 1 3 a2 3 3 4 3 2 5 3 1 a3 6 6 5 6 6 3 6 6 a4 4 4 2 5 5 4 2 2 a5 5 5 6 2 4 6 5 5 a6 2 2 3 4 1 2 4 4 8. conclusions the application of the (mcdm) model is successfully presented in the paper while the sensitivity analysis indicates the potential of the application of the created models to support decision-making during the planning process of the land forces operation. in a similar way, the application would be conducted in group decisionmaking. a practical example of the fuzzy scale demonstrated a possibility of using the hybrid fuzzy ahp-mabac model, its performance in ranking the offered alternatives. the analysis of the output results sensitivity has shown that the 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accepted: 10 march 2020; available online: 14 march 2020. original scientific paper abstract: third-party logistics (3pl) has involved a significant response among researchers and practitioners in the recent decade. in the global competitive scenario, multinational companies (mncs) not only improve the quality of the service and increase efficiency, but they also decrease costs by means of 3pl. however, the assessment and selection of 3pl is a very critical decision to make, comprising intricacy due to the existence of various imprecisely based criteria. also, uncertainty is an unavoidable part of information in the decision-making process and its importance in the selection process is relatively high and needs to be carefully considered. consequently, incomplete and inadequate data or information may occur among other various selection criteria, which can be termed as a multi-criteria decisionmaking (mcdm) problem. rough numbers are very flexible to model this type of uncertainty occurring in mcdm problems. in this paper, the hamy mean (hm) operator and dombi operations are expanded by rough numbers (rns) so as to propose the rough number dombi-hamy mean (rndhm) operator. then, the multi-attribute decision-making (madm) model is designed with the rndhm operator. finally, the rndhm is employed to achieve the final ranking of the 3pl providers. key words: madm; rough numbers; rough dombi-hamy mean operator; third-party logistics. 1. introduction the number of transport requests for the relocation of this specific type of goods is inevitably increasing. the basic difference in relation to the transport of the classical types of goods is that, in the transport of dangerous materials, participants are faced with additional safety requirements. this type of transport is specific because of the mailto:marko.vasiljevic@sf.ues.rs.ba an evaluation of a third-party logistics provider: the application of the rough dombi-hamy... 93 risks that present themselves during the implementation of this logistics process (tanackov et al., 2018), which may endanger people’s life and health, the environment, and material resources due to the dangerous properties of the transported matter and potential reactions (di fazio et al., 2016). that is the reason why, at both the national and international levels, the hazardous substance domain is one of the most regulated areas in terms of the set directives tailored for each type of transport. the companies that produce or use hazardous materials are frequently unable to provide full transport services as they have the specific and varied capacities that must be used in the provision of complete transport services, which is why they often turn to 3pl services. also, numerous extra security requirements specific to dangerous materials must be met in organizing transport. when systematically viewed in the economic context, there are companies which find it more economically viable to outsource transport to third-party logistics providers (3pl), rather than possess and use their own fleets. when assessing the logistics sector, a number of companies will abolish or reduce their own fleet and hire 3pl service providers to handle transport, with the goal of reducing total costs and increasing profits (azadi & saen, 2013). in certain cases in practice, this solution has proven to be correct (lien & day, 2017), which is especially interesting as being relevant to the dangerous goods transport field, where the cost is not the only criterion. the evaluation of 3pl providers is a critical step for any manufacturer seeking to select a suitable 3pl provider as a business partner (jung, 2017). today, the evaluation of 3pl providers is most often performed by using multi-criteria decision-making methods. to deal with uncertainties and determine the values of qualitative attributes, the majority of authors use fuzzy sets (zadeh, 1965) or various extensions of fuzzy theory, such as: interval-valued fuzzy sets (sizong and tao, 2016; zywica, 2016), intuitionistic fuzzy sets (atanassov, 1986), interval intuitionistic fuzzy sets (nguyen, 2016), hesitant fuzzy sets (ngan, 2017), as well as other extensions. fuzzy sets are a very powerful and commonly used tool for dealing with imprecision. however, when selecting an appropriate membership function for fuzzy sets, subjectivity may affect the final decision, for which reason particular care needs to be taken with that respect (pamucar et al., 2017; pamucar et al., 2018). in addition to fuzzy theory, rough set theory, originally introduced by pawlak (1991), is another suitable tool for treating imprecision. in recent years, rough set theory has been successfully implemented in various fields of human activities. its application can be said to be adequate and usually irreplaceable when handling uncertainty and inaccuracy analyses. knowing the advantages of rough set theory (pawlak 1991), the application of rough sets and rough numbers is fully justified in today’s modern practice in the decision-making process when it includes imprecision in data. although rns have been effectively used in some fields, all the existing approaches are unsuitable to depict the interrelationships among any number of rns assigned by a variable vector. the hamy mean (hm) operator (li et al., 2018) is a famous operator able to show interrelationships among any number of arguments assigned by a variable vector. therefore, the hm operator can assign a robust and flexible mechanism to solve an information fusion in madm problems. thus, the hm operator is proposed in this paper in order to overcome these limits. how to aggregate rns by applying traditional hm operators while simultaneously using the dombi t-norms and co-norms (sremac et al., 2018) is an interesting issue. so, the purpose of this paper is to propose the hm operator in order to solve a madm problem of the evaluation of managers in the human resource department with rns. in order to do that, the rest of the paper is organized as follows: in section 2, irns are introduced; in section 3, the hm operator with rns is developed based on the dombi operations; in section 4, an sinani et al./decis-mak. appl. manag. eng. 3 (1) (2020) 92-107 94 example of the evaluation of managers in a human resource department with rns is presented, and finally in section 5, comments are given at the end of the paper. 2. rough numbers and operations rough numbers (rns), consisting of the upper, lower and boundary intervals, respectively, determine the intervals of multiple experts’ evaluations without requiring any additional information, only relying on original data (stevic et al., 2018). hence, the obtained experts’ preferences objectively represent and improve the decision-making process. the definition of rns according to song et al. (2013) is given below. let u be a universe containing all the objects and let x be a random object from u. then, a set of k classes is assumed to exist, which represents the dm’s preferences, 1 2( , ,..., )kr j j j , with the condition 1 2 ,..., kj j j   . then, for every , , 1qx u j r q k    , the lower approximation ( )qapr j , the upper approximation ( )qapr j and the boundary interval ( )qbnd j are determined as follows:  ( ) / ( )q qapr j x u r x j   (1)  ( ) / ( )q qapr j x u r x j   (2)      ( ) / ( ) / ( ) / ( )q q q qbnd j x u r x j x u r x j x u r x j        (3) the object can be represented by a rough number with the lower limit ( )qlim j and the upper limit ( )qlim j in eqs. (4)-(5). 1 ( ) ( ) ( )q q l lim j r x x apr j m   (4) 1 ( ) ( ) ( )q q u lim j r x x apr j m   (5) where lm and um represent the sum of the objects given in the lower and upper object approximations of qj , respectively. for the object qj , the rough boundary interval ( ( )qirbnd j ) is the interval between the lower and upper limits (pamucar et al., 2018). the rough boundary interval is a measure of uncertainty. a bigger ( )qirbnd j value shows that there are variations in experts’ preferences, while smaller values show that experts’ opinions do not considerably differ. all the objects between the lower limit ( )qlim j and the upper limit ( )qlim j of the rough number ( )qrn j are included in ( )qirbnd j . since rns belong to a group of interval numbers, arithmetic operations applied to interval numbers are also appropriate for rns. an evaluation of a third-party logistics provider: the application of the rough dombi-hamy... 95 3. the dombi operations of rns and the dombi-hamy mean operator with rough numbers definition 1. let p and q be any two real numbers. then, the dombi t-norm and tconorm between p and q are defined as follows (dombi, 1982): 1/ 1 ( , ) 1 1 1 do p q p q p q                        (6) 1/ 1 ( , ) 1 1 1 1 c do p q p q p q                         (7) where 0  and  ( , ) 0,1p q  . in accordance with the dombi t-norm and the t-conorm, dombi operations are defined by rough numbers. definition 2. assuming that 1 1 1( ) [ ( ), ( )]rn lim lim   i 2 2 2( ) [ ( ), ( )]rn lim lim   are two rough numbers, , 0   and let       1 1 ( ) ( ) ( ) ( ) , ( ) , ( ) ( ) i i i i i n n i i i i lim lim f rn f lim f lim lim lim                       be a rough function, then some operational laws of rough numbers based on the dombi t-norm and t-conorm can be defined as follows: (1) addition “+”                 2 1 1/ 1 2 1 2 2 1 1 2 1 2 ( )2 , 1 ( ) ( ) 1 1 ( ) 1 ( ) 1 2 ( )2 1 ( ) ( ) 1 1 ( ) 1 ( ) ( ) ( ) ( ) ( ) jj jj lim j j f lim f lim f lim f lim lim j j f lim f lim f lim f lim lim rn rn lim                                                                         1/                            (8) (2) multiplication “×”                 2 1 1/ 1 2 1 2 2 1 1/ 1 2 1 2 ( ) , 1 ( ) 1 ( ) 1 ( ) ( ) 1 2 ( ) 1 ( ) 1 ( ) 1 ( ) ( ) ( ) ( ) jj jj lim f lim f lim f lim f lim lim f lim f lim f lim f lim rn rn                                                                                            (9) (3) scalar multiplication, where 0  sinani et al./decis-mak. appl. manag. eng. 3 (1) (2020) 92-107 96 1 1/ 1 1 1 1/ 1 1 ( ) ,1 ( ) 1 1 ( ) 1 ( ) 1 ( ) 1 1 ( ) ( ) ( ) ( ) lim lim lim lim lim lim lim rn lim                                                           (10) (4) power, where 0    1 11/ 1/ 1 1 1 1 ( ) ( ) ,1 1 ( ) 1 ( ) 1 1 ( ) ( ) ( ) lim lim lim lim lim lim rn                                                        (11) the hamy mean (hm) (hara et al., 1998) is used to aggregate values, simultaneously including mutual correlations among multiple arguments, and is defined in the following way: definition 3 (hara et al., 1998). assume that ( 1, 2,..., )ix i n represents a set of nonnegative real numbers and the parameter 1, 2,...,k n , then the hm is defined as: 1 1/ ( ) 1 2 1 ... 1 1 ( , ,..., ) j k k k k n i i i n j hm x x x x n k                    (12) where 1 2( , ,..., )ki i i includes all the k-tuple combinations of (1, 2,..., )n and n k       represents a binomial coefficient calculated as: ! !( )! n n k k n k       (13) based on the rn operators (8)-(11), the rn dombi-hamy mean (rndhm) operator is proposed. theorem 1. let ( ) [ ( ), ( )]j j jrn lim lim   ;  1, 2,...,j n , the collection of rns in r, then the rndhm operator is defined as follows: an evaluation of a third-party logistics provider: the application of the rough dombi-hamy... 97       1 1 1/ 1 ...1 1 1/ ,( ) 1 2 1 ... 1 ( ) , 1 1 1 ( ) ( ) 1 ( ), ( ),.., ( ) ( ) ( ) ( j k n i i i i nk k i j j i j k k kk n i i i n j n lim i i n k k f lim f lim i rndhm rn rn rn rn n k lim lim                                                                                1 1/ 1 ...1 1 ( ) 1 1 1 ( ) ( ) ) n i i i i nk k i j j i j n lim i n k k f lim f lim                                                                                        (14) where       1 1 ( ) ( ) ( ) ; ( ) ( ) ( ) ( ). n i i i i i n i i i i f lim lim lim f rn f lim lim lim                    represents a rough function. the proof. based on the dombi operations of the rn, the following is obtained: a)         1/ 1/ 1 1 ( ) ( ) , 1 ( ) 1 ( )1 1 1 ( ) ( ) ( ) i ij j j k k i ij j j ji ij j k lim lim i f lim f limj f lim f lim rn                                                                             thus,         1/ 1/ 1 1 1/ ( ) ( ) , 1 ( ) 1 ( )1 1 1 1 1 ( ) ( ) ( ) i ij j j k k i ij j j ji ij j k k lim lim i f lim f limj k kf lim f lim rn                                                                                   sinani et al./decis-mak. appl. manag. eng. 3 (1) (2020) 92-107 98 b) thereafter,         1/ 1 ...1 1 1 1 ...1 1 ( ) , 1 1 ( ) 1/ ( ) ( ) 1 ... 1 1 1 ( ) ( ) ( ) ( ) ( ) i j j i i nk k i j j i j j i j k j i i k i j j i j lim i k f lim k f limk i lim i i n j i k f lim f lim lim rn lim                                                                                1/ nk                                                              c) therefore,       1 1 1/ 1 ...1 1 1/ ,( ) 1 2 1 ... 1 ( ) , 1 1 1 ( ) ( ) 1 ( ), ( ),.., ( ) ( ) ( ) ( j k n i i i i nk k i j j i j k k kk n i i i n j n lim i i n k k f lim f lim i rndhm rn rn rn rn n k lim lim                                                                                1 1/ 1 ...1 1 ( ) 1 1 1 ( ) ( ) ) n i i i i nk k i j j i j n lim i n k k f lim f lim                                                                                        besides, since an evaluation of a third-party logistics provider: the application of the rough dombi-hamy... 99         1 1/ 1 ...1 1 1 ( ) 1 1 1 ( ) ( ) ( ) 1 1 1 ( ) ( ) 0 ( ) ( ), ( ) ( ) n i i i i nk k i j j i j n i i i j i j n lim i rnghm i n k k f lim f lim n lim rnghm i i n k k f lim f lim lim lim lim lim                                                                             1/ 1 ...1 1 ( ) i i nk k j rnghmlim                                 , then         1 1/ 1 ...1 1 1 1 ...1 1 ( ) , 1 1 1 ( ) ( ) ( ) 1 1 1 ( ) ( ) ( ) ( ) n i i i i nk k i j j i j n i i i i nk k i j j i j n lim i i n k k f lim f lim n lim i i n k k f lim f lim lim lim                                                                                    1/                                                             represents the rn, so that theorem 1 has been proven and equation (14) is correct. the rndhm operator also contains the following properties: idempotency, boundedness and monotonicity. theorem 2. (idempotency). if ( ) ( ) [ ( ), ( )]jrn rn lim lim     for all  1, 2,...,j n then  ( ), ( ),.., ( ) ( ) [ ( ), ( )]rnhm rn rn rn rn lim lim       . the proof. since ( ) [ ( ), ( )] ( )j j jrn lim lim rn     ;  1, 2,...,j n , then the following calculations are obtained by using theorem 1: sinani et al./decis-mak. appl. manag. eng. 3 (1) (2020) 92-107 100       1 1/ 1 ...1 1 1 ( ) ( ) , 1 1 1 ( ) ( ) ( ) 1 1 1 ( ), ( ),.., ( ) ( ), ( ) ( ) ( ) n i i i i nk k i j j i j n i i k n lim i i n k k f lim f lim n lim i i n k k f l rndhm rn rn rn lim lim lim lim                                                                                  1/ 1 ...1 1 1 1 1/ ( ) ( ) ( ) ( ) , 1 11 ( ) 1 ( ) ( ) i i nk k i j j i j n n i i i i i j im f lim n lim n lim i i n n i ilim k k lim lim                                                                                                               1/ ( ) ( ), ( ) ( ) i j lim lim lim rn                              theorem 3. (commutativity). let the rn rough set ' ' '1 2( ( ), ( ),.., ( )nrn rn rn   be any permutation of 1 2( ( ), ( ),.., ( ))nrn rn rn   . then, there is    ' ' '1 2 1 2( ), ( ),.., ( ) ( ), ( ),.., ( )n nrndhm rn rn rn rndhm rn rn rn      . the proof. based on definition 2, the conclusion is obvious:         ' ' 1 1 1/ ' '1 ...1 ' 1 ( ) ' ' ' 1 2 ( ) ( ) ' , 1 1 1 1 1 ( ) 1 ( ) ( ) ( ), ( ),.., ( ) ( ) ( ) n n i i i i i i nk k i ij j j i j k n n lim n lim i i i i n nk k k k f lim f lim f lim rndhm rn rn rn lim lim                                                                     1/ 1 ...1 ' 1 ( ) i i nk k j i j f lim                                                                    an evaluation of a third-party logistics provider: the application of the rough dombi-hamy... 101         1 1 1/ 1 ... 1 ...1 1 1 1 ( ) ( ) , 1 1 1 1 1 ( ) 1 ( ) ( ) ( ) ( ) ( ) n n i i i i i i n i i nk kk k i ij j j ji ij j n lim n lim i i i i n nk k k k f lim f lim f lim f lim lim lim                                                                                     1/ ( ) ( ), ( ),.., ( ) j j j k i i irndhm rn rn rn                                                        theorem 4. (boundedness). let ( ) [ ( ), ( )]j j jrn lim lim   ;  1, 2,...,j n , tte collection of rns in r, let    ( ) min ( ) , min ( )j j jrn lim lim       and    ( ) max ( ) , max ( )j j jrn lim lim       , then  1 2( ) ( ), ( ),.., ( ) ( )j n jrn rnhm rn rn rn rn         . the proof. let  1 2( ) min ( ), ( ),..., ( ) ( ), ( )j n j jrn rn rn rn lim lim             and  1 2( ) max ( ), ( ),..., ( ) ( ), ( )j n j jrn rn rn rn lim lim             . then,  ( ) min ( )j j j lim lim    ,  ( ) min ( )j j j lim lim    ,  ( ) max ( )j j j lim lim    and  ( ) max ( )j j j lim lim    . based on that, it follows:         1 1 1/ 1 ... 1 ...1 1 1 1 ( ) ( ) 1 1 1 1 1 ( ) 1 ( ) ( ) ( ) ( ) ( ) n n i i i i i i n i ik kk k i ij j j ji ij j n lim n lim i i i i n nk k k k f lim f lim f lim f lim lim lim                                                                                     1/ n                             1 1/ 1 ...1 1 ( ) 1 1 1 ( ) ( ) ( ) n i i i i n kk i j j i j n lim i i n k k f lim f lim lim                                                    sinani et al./decis-mak. appl. manag. eng. 3 (1) (2020) 92-107 102         1 1 1/ 1 ... 1 ...1 1 1 1 ( ) ( ) 1 1 1 1 1 ( ) 1 ( ) ( ) ( ) ( ) ( ) n n i i i i i i n i ik kk k i ij j j ji ij j n lim n lim i i i i n nk k k k f lim f lim f lim f lim lim lim                                                                                     1/ n                             1 1/ 1 ...1 1 ( ) 1 1 1 ( ) ( ) ( ) n i i i i n kk i j j i j n lim i i n k k f lim f lim lim                                                     according to the foregoing inequalities, it can be concluded that  1 2( ) ( ), ( ),.., ( ) ( )j n jrn rnhm rn rn rn rn         . theorem 5. (monotonicity). let ( ) [ ( ), ( )]j j jrn lim lim   , ( ) [ ( ), ( )]j j jrn lim lim   ;  1, 2,...,j n , be the two collections of rns, if ( ) ( )j jlim lim  , ( ) ( )j jlim lim  for all j , then    1 2 1 2( ), ( ),.., ( ) ( ), ( ),.., ( )n nrnhm rn rn rn rnhm rn rn rn      . the proof. since 0 ( ) ( )j jlim lim   , 0 ( ) ( )j jlim lim   , then, based on theorem 1, the following is obtained:         1 1 1/ 1 ... 1 ...1 1 1 1 ( ) ( ) 1 1 1 1 1 ( ) 1 ( ) ( ) ( ) ( ) ( ) n n i i i i i i n i i nk kk k i ij j j ji ij j n lim n lim i i i i n nk k k k f lim f lim f lim f lim lim lim                                                                                    1/                             1 1 1/ 1 ... 1 ...1 1 1 1 ( ) ( ) 1 1 1 1 1 ( ) 1 ( ) ( ) ( ) ( ) ( ) n n i i i i i i n i i nk kk k i ij j j ji ij j n lim n lim i i i i n nk k k k f lim f lim f lim f lim lim lim                                                                                    1/                     let  1 2( ) ( ), ( ),.., ( )nrn rnhm rn rn rn    and  1 2( ) ( ), ( ),.., ( )nrn rnhm rn rn rn    be two rns, then: (1) if ( ) ( )lim lim  and ( ) ( )lim lim  , then the following is obtained:    1 2 1 2( ), ( ),.., ( ) ( ), ( ),.., ( )n nrnhm rn rn rn rnhm rn rn rn      ; an evaluation of a third-party logistics provider: the application of the rough dombi-hamy... 103 (2) if ( ) ( )lim lim  and ( ) ( )lim lim  , then it can be concluded that there are the following equalities:         1 1 1/ 1 ... 1 ...1 1 1 1 ( ) ( ) 1 1 1 1 1 ( ) 1 ( ) ( ) ( ) ( ) ( ) n n i i i i i i n i i nk kk k i ij j j ji ij j n lim n lim i i i i n nk k k k f lim f lim f lim f lim lim lim                                                                                    1/                             1 1 1/ 1 ... 1 ...1 1 1 1 ( ) ( ) 1 1 1 1 1 ( ) 1 ( ) ( ) ( ) ( ) ( ) n n i i i i i i n i i nk kk k i ij j j ji ij j n lim n lim i i i i n nk k k k f lim f lim f lim f lim lim lim                                                                                    1/                     finally, it can be concluded that there is the following inequality    1 2 1 2( ), ( ),.., ( ) ( ), ( ),.., ( )n nrnhm rn rn rn rnhm rn rn rn      . 4. a numerical example: an evaluation of a third-party logistics provider the practical application of the rndhm aggregator is a logical conclusive phase of the subject-matter research. the multi-criteria model consists of a total of the five logistics providers who represent the alternatives having been evaluated based on the four criteria presented in detail in table 1. the research study involved five experts, who evaluated the significance of the criteria, i.e. the alternatives according to the given criteria. the experts used a 1-9 scale for the evaluation of 3pl. table 1. the criteria based on which the logistics provider was evaluated no criteria description of the criteria 1. financial stability the logistics provider delivers a financial report on positive business practices and a certificate of the settlement of tax liabilities, i.e. a certificate issued by the tax authorities, which confirms no extant debts in connection with the payment of taxes, fees, etc. 2. the professionalization of drivers the logistics provider employs the drivers who have a certificate of approval for dangerous goods transport. the ongoing education of drivers is organized through seminars, additional and specialist training programs, safe and eco-friendly driving programs, etc. 3. the cost of transport covers all costs incurred by the carrier during the performance of services. 4. the application of risk mitigation measures the logistics provider has one or more designated security advisors for the transport of dangerous sinani et al./decis-mak. appl. manag. eng. 3 (1) (2020) 92-107 104 no criteria description of the criteria goods. at points of loading/unloading, the property of the contracting authority or third parties is treated with special care. the carrier acquaints his workers with the rules of conduct, movement, safety, fire protection, and all other measures and procedures. the five logistics providers ai (i=1, 2, 3, 4, 5) were evaluated: 1 2 ( ) 3 4 5 1 2 3 4 4, 5, 6, 8 8, 8, 9, 8 8, 8, 9, 7 8, 8, 9, 9 5, 5, 5, 5 7,8, 6, 9 8, 8, 8, 7 9, 9, 6, 9 7, 5, 8, 4 9, 9, 9, 7 7, 7,8, 9 6, 6, 7, 6 6, 5, 8, 3 7, 5, 6, 8 5, 5, 5, 6 9, 9, 8, 7 5, 5, 6, 4 4, 4, 5, 8 9, 9, 9, 8 6, 6, 8, 6 e c c c c a a ax a a       ;1 e m             where m (in our case m=4) is the number of experts. by applying the rules for the transformation of crisp numbers into rough numbers, the elements of the matrix ( )ex are transformed into rns and presented in the matrix ( )e rn x (table 2). table 2. the expert rough decision matrix ( ( )e rn x ) alt./crit. c1 c2 a1 [4.0,5.75];[4.5,6.33];[5,7];[5.75,8] [8,8.25];[8,8.25];[8.25,9];[8,8.25] a2 [5.0,5.0];[5.0,5.0];[5.0,5.0];[5.0,5.0] [6.5,8.0];[7,8.5];[6.0,7.5];[7.5,9.0] a3 [5.33,7.5];[4.5,6.67];[6,8];[4.0,6.0] [8.5,9.0];[8.5,9.0];[8.5,9.0];[7.0,8.5] a4 [4.67,7];[4,6.33];[5.5,8];[3,5.5] [6.0,7.5];[5.0,6.5];[5.5,7.0];[6.5,8.0] a5 [4.67,5.33];[4.67,5.33];[5,6];[4,5] [4,5.25];[4,5.25];[4.33,6.5];[5.25,8] alt./crit. c3 c4 a1 [8,8.25];[8,8.25];[8.25,9];[8,8.25] [8,8.5];[8,8.5];[8.5,9];[8.5,9] a2 [7.75,8];[7.75,8];[7.75,8];[7,7.75] [8.25,9];[8.25,9];[6,8.25];[8.25,9] a3 [7,7.75];[7,7.75];[7.33,8.5];[7.75,9] [6,6.25];[6,6.25];[6.25,7];[6,6.25] a4 [5,5.25];[5,5.25];[5,5.25];[5.25,6] [8.25,9];[8.25,9];[7.5,8.67];[7,8.25] a5 [8.75,9];[8.75,9];[8.75,9];[8,8.75] [6.0,6.5];[6,6.5];[6.5,8.0];[6.0,6.5] the aggregated initial rough decision matrix ( rnx ) is obtained by using the rndhm aggregator. thus, based on eq. (14), the element 11( )rn x in the matrix ( ) 5 4 ( ) e ijrnx rn x      is obtained as follows:              1 1 1 1 1 1 1 1 1 1 1 1 (1) (2) (3) (4) 11 11 11 11 19.25 11 2 1 1 1 1 1 1 1 4 3.81 3.28 3.81 2.85 3.81 2.35 3.28 2.85 3.28 2.35 2.85 2.35 2 4,5.75 ; 4.5, 6.33 ; 5, 7 ; 5.( ), ( ), ( ), ( ) ( ) 19.25 75,8rndhma rn x rn x rn x rn x lim x                                  1/1 1/1 1 1 1 1 1 1 1 1 1 1 1 1 4.76 27.08 6.7111 2 1 1 1 1 1 1 1 4 3.71 3.28 3.71 2.87 3.71 2.39 3.28 2.87 3.28 2.39 2.87 2.39 2 ( ) 27.08lim x                                                     an evaluation of a third-party logistics provider: the application of the rough dombi-hamy... 105 where the values  ( )11( )ef rn x ; 1 4e  represent the rough functions. in a similar way, the other aggregated elements of the matrix rnx are obtained.                             1 2 3 4 5 4.76; 6.71 8.06;8.43 8.06;8.43 8.25;8.75 5.00; 5.00 6.72;8.23 7.55; 7.94 7.62;8.81 4.89; 7.00 8.10;8.87 7.26;8.23 6.06; 6.43 4.18; 6.64 5.72; 7.23 5. 1 2 3 4 rn k k k k a a ax a a              06; 5.43 7.73;8.72 4.57; 5.4 4.36; 6.14 8.56;8.94 6.12; 6.84                 in the following steps, the final values of the criteria functions of the alternatives were obtained and ranked by applying additive normalization and weighting the normalized matrix using the following equation   1 n i ij ji q a x w    , where xij represents the elements of the normalized matrix and wj = (0.2, 0.22, 0.28, 0.30)t represents the criteria weights. thus, the alternative score functions: q(a1)=[0.2;0.25], q(a2)=[0.18;0.23], q(a3)=[0.18;0.24], q(a4)=[0.15;0.22] and q(a5)=[0.16;0.21] were obtained. it is desirable for an alternative to have the highest possible value of its criteria function, so a1 > a2 > a3 > a5 > a4. 5. conclusion dangerous goods transport and its potential consequences arouse the attention of the public because of the detrimental effect of these materials on the environment, as well as people, and possible accidents, too. in order to meet the complex demands of today’s market, the hazardous substance increased consumption trend has appeared, thereby increasing the volume of the production and transport of these goods. transport is a mandatory logistic activity for supplying users with these materials, regardless of whether they are in a raw, semi-processed, or fully-processed form. in this paper, madm problems with rns were subjected to investigation, which was only followed by the utilization of the hm operator and the dombi operations in order to design an hm operator with rns, i.e. the rndhm operator. after that, the rndhm was used so as to propose a model for madm problems with rns. finally, a real example of the evaluation of 3pl was used in order to show the developed approach. the procedure for selecting 3pl carriers is particularly specific in the field of dangerous substances due to high risks and additional security requirements. this reflects in the criteria defined for the evaluation of the logistics providers. one of the benefits of using 3pl carriers’ services is that they have a wider range of operations and are able to satisfy the clients who have a high frequency of transport needs on a weekly basis. specialized carriers better understand the market and customers’ needs. they also have well-designed strategies and business models to continue improving their offers, regional coverage, and specialization in all industry sectors. a direction for a further research study implies supporting the environmental sustainability of 3pl dangerous material providers, particularly in the areas marked as the five topical areas: influencing factors, green actions, an impact on performance, information and communication technology (ict) tools supporting green actions, energy efficiency in road freight transport, and shippers’ perspective and sinani et al./decis-mak. appl. manag. eng. 3 (1) (2020) 92-107 106 collaboration. in subsequent studies, the extension and application of rns needs to be studied in many other uncertain environments and other applications. author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research received no external funding. conflicts of interest: the authors declare no conflicts of interest. references atanassov, k. 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(2016). an algorithmic study of relative cardinalities for interval-valued fuzzy sets. fuzzy sets and systems, 294, 105–124. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 1, number 1, 2018, pp. 82-96 issn: 2560-6018 doi: https://doi.org/10.31181/dmame180182v * corresponding author. e-mail addresses: zeljkostevic88@yahoo.com (ž. stević) ** an earlier version of this paper was presented at the 1st international conference on management, engineering and environment – “icmnee 2017” (fazlollahtabar et al., 2017). a rough multicriteria approach for evaluation of the supplier criteria in automotive industry marko vasiljević1, hamed fazlollahtabar2, željko stević1*, slavko vesković3 1 university of east sarajevo, faculty of transport and traffic engineering doboj, bosnia and herzegovina 2 department of industrial engineering, school of engineering, damghan univeristy, damghan , iran 3 university of belgrade, faculty of transport and traffic engineering, serbia received: 25 december 2018; accepted: 29 january 2018; published: 15 march 2018. original scientific paper abstract. ensuring costs reduction and increasing competitiveness and satisfaction of end users are the goals of each participant in the supply chain. taking into account these goals, the paper proposes methodology for defining the most important criteria for suppliers’ evaluation. from a set of twenty established criteria, i.e. four sets of criteria: finances, logistics, quality and communication and business including its sub-criteria, we have allocated the most important ones for supplier selection. analytic hierarchy process (ahp) based on rough numbers is presented to determine the weight of each evaluation criterion. for the criteria evaluation we have used knowledge from the expert in this field. the efficacy of the proposed evaluation methodology is demonstrated through its application to the company producing metal washers for the automotive industry. next a sensitivity analysis is carried out in order to show the stability of the model. for checking stability the ahp method in conventional form is used in combination with fuzzy logic. key words: rough ahp, supplier criteria, fuzzy ahp, logistics, quality. 1. introduction one of the most important strategic issues in logistics procurement according to stević, (2017c) is a correct and optimal supplier selection, which enables an increase of market competitiveness. the importance of an adequate supplier selection was recognized at the beginning of the last decade of the 20th century when davis (1993) emphasized that the failure of suppliers to fulfill the promises and expectations regarding delivery is one of the three main sources of uncertainty plaguing the mailto:veskolukovac@yahoo.com a rough multicriteria approach for evaluation of the supplier criteria in automotive industry 83 supply chain. kagnicioglu (2016) considers that the supplier selection is a critical procurement activity in the supply chain management because of the crucial role that the supplier characteristics play regarding price, quality, delivery and service in achieving the objectives of the supply chain. van weele (2009) points out that a healthy relationship with the suppliers can improve the financial position in a short term and so can a competitive strategy over a long period of time. today the company must strive to enlarge the quality of product itself so that the end user is satisfied with provided services, which would make him a loyal user. due to the above mentioned, it is necessary, during the first phase of logistics, i.e. purchasing logistics, to commit good evaluation and choice of the supplier, which can largely influence the forming of the product’s final price; thus, it can, in this way, accomplish a significant effect in the complete supply chain. it is possible to accomplish the above mentioned if the evaluation is done on the basis of a multicriteria decision-making that includes a large number of criteria as well as an expert’s estimation of their relative significance (stević et al. 2016). this paper is structured as follows. section 2 shows the fundamentals of a rough set theory, operations with rough numbers and rough analytic hierarchy process. section 3 makes up the body of the paper: it gives a practical example besides showing results of the proposed model. section 4 presents a sensitivity analysis. this section also gives discussion and model stability. section 5 presents conclusions before the paper ends with a list of references. 2. methods 2.1. a rough set theory due to the complexity and uncertainty of numerous real indicators in the process of multi-criteria decision-making, as well as the occurrence of the ambiguity of human thinking, there are difficulties in presenting information about the attributes of decisions through accurate (precise) numerical values. these uncertainties and ambiguities are commonly exploited through application of rough numbers (song et al., 2014; zhu et al., 2015). in addition to the fuzzy theory, a very suitable tool for the treatment of uncertainty without any impact of subjectivism is a rough set theory, which was first introduced in (pawlak, 1982). from the beginning until today, the theory of rough sets has evolved through solving many problems by using rough sets (khoo & chai, 2001; chai & liu, 2014; nauman et al. 2016; liang et al. 2017; pamučar et al. 2017a) and through the use of rough numbers as in (tiwari et al., 2016; shidpour et al., 2016; stević et al., 2017b; 2017d; 2017e). in the theory of rough sets only the internal knowledge is used, i.e. operational data, and there is no need to rely on the models of assumptions. in other words, in the application of rough sets, instead of various additional/external parameters, we use exclusively the structure of the data provided (duntsch et al., 1997). in rough sets the measurement of uncertainty is based on the uncertainty that is already contained in the data (khoo & chai, 2001). this leads to objective indicators that are contained in the data. in addition, the theory of rough sets is suitable for application in the sets that are characterized by a small number of data, and for which statistical methods are not suitable (pawlak, 1991). vasiljević et al./decis. mak. appl. manag. eng. 1 (1) (2018) 82-96 84 2.2. operations with rough numbers in the rough set theory, any vague concept can be represented as a pair of precise concepts based on the lower and upper approximations (pawlak, 1991) as shown in figure 1. figure 1. basic notions of the rough set theory (stević et al., 2017a) let’s u be a universe containing all objects and x be a random object from u . then we assume that there exists set build with k classes representing dms preferences, ( ) q apr j 1 2 ( , ,..., ) k r j j j with condition 1 2 ,..., k j j j   . then, , , 1 q x u j r q k     lower approximation , upper approximation ( ) q apr j and boundary interval ( ) q bnd j are determined, respectively, as follows:  ( ) / ( )q qapr j x u r x j   (1)  ( ) / ( )q qapr j x u r x j   (2)      ( ) / ( ) / ( ) / ( )q q q qbnd j x u r x j x u r x j x u r x j        (3) the object can be presented with rough number (rn) defined with lower limit ( ) q lim j and upper limit ( ) q lim j , respectively: 1 ( ) ( ) ( ) q q l lim j r x x apr j m   (4) 1 ( ) ( ) ( ) q q u lim j r x x apr j m   (5) where l m and u m represent the sum of objects contained in the lower and upper object approximation of q j , respectively. obviously, the lower limit and upper limit denote the mean value of elements included in the lower approximation and upper approximation, respectively. their difference is defined as rough boundary interval (𝐼𝑅𝐵𝑛𝑑(𝐺𝑞)): 𝐼𝑅𝐵𝑛𝑑(𝐺𝑞) = 𝐿𝑖𝑚(𝐺𝑞) − 𝐿𝑖𝑚(𝐺𝑞) (7) operation for two rough numbers 𝑅𝑁(𝛼) = [𝐿𝑖𝑚(𝛼), 𝐿𝑖𝑚(𝛼)] and 𝑅𝑁(𝛽) = [𝐿𝑖𝑚(𝛽), 𝐿𝑖𝑚(𝛽)] according to (zhai et al., 2009) are: a rough multicriteria approach for evaluation of the supplier criteria in automotive industry 85 addition (+) of two rough numbers (𝛼) and (𝛽) 𝑅𝑁(𝛼) + 𝑅𝑁(𝛽) = [𝐿𝑖𝑚(𝛼) + 𝐿𝑖𝑚(𝛽), 𝐿𝑖𝑚(𝛼) + 𝐿𝑖𝑚(𝛽)] (8) subtraction (-) of two rough numbers (𝛼) and (𝛽) 𝑅𝑁(𝛼) − 𝑅𝑁(𝛽) = [𝐿𝑖𝑚(𝛼) − 𝐿𝑖𝑚(𝛽), 𝐿𝑖𝑚(𝛼) − 𝐿𝑖𝑚(𝛽)] (9) multiplication (×) of two rough numbers (𝛼) and (𝛽) 𝑅𝑁(𝛼) × 𝑅𝑁(𝛽) = [𝐿𝑖𝑚(𝛼) × 𝐿𝑖𝑚(𝛽), 𝐿𝑖𝑚(𝛼) × 𝐿𝑖𝑚(𝛽)] (10) division (÷) of two rough numbers 𝑅𝑁(𝑎) and 𝑅𝑁(𝑏) 𝑅𝑁(𝛼) ÷ 𝑅𝑁(𝛽) = [𝐿𝑖𝑚(𝛼) ÷ 𝐿𝑖𝑚(𝛽), 𝐿𝑖𝑚(𝛼) ÷ 𝐿𝑖𝑚(𝛽)] (11) scalar multiplication of rough number 𝑅𝑁(𝛼), where 𝜇 is a nonzero constant 𝜇 × 𝑅𝑁(𝛼) = [𝜇 × 𝐿𝑖𝑚(𝛼), 𝜇 × 𝐿𝑖𝑚(𝛼)] (12) 2.3. rough analytic hierarchy process the procedure of the rough ahp is described as follows (zhu et al., 2015): step 1: identify the evaluation objective, criteria and alternatives. construct a hierarchical structure with the evaluation objective at the top layer, criteria in the middle and alternatives at the bottom. step 2: conduct ahp survey and construct a group of pair-wise comparison matrices. the pair-wise comparison matrix of the eth expert is described as: 𝐵𝑒 = [ 1 𝑥21 𝑒 ⋮ 𝑥𝑚1 𝑒 𝑥12 𝑒 1 ⋮ 𝑥𝑚2 𝑒 ⋯ ⋯ ⋱ ⋯ 𝑥1𝑚 𝑒 𝑥2𝑚 𝑒 ⋮ 1 ] (13) where 𝑥𝑔ℎ 𝑒 (1 ≤ 𝑔 ≤ 𝑚, 1 ≤ ℎ ≤ 𝑚, 1 ≤ 𝑒 ≤ 𝑠)is the relative importance of criterion g on criterion h given by expert e, m is the number of criteria, s is the number of experts. calculate maximum eigenvalue 𝜆𝑚𝑎𝑥 𝑒 of be, then compute consistency index 𝐶𝐼 = (𝜆𝑚𝑎𝑥 𝑒 − 𝑛)/(𝑛 − 1). determine random consistency index (ri) in table 1 according to n. compute consistency ratio cr=ci/ri. table 1. value of random index depending on the rank of matrix (saaty & vargas, 2012) n 1 2 3 4 5 6 7 8 9 10 ri 0.00 0.00 0.52 0.89 1.11 1.25 1.35 1.40 1.45 1.49 conduct consistency test. if cr<0.1, the comparison matrix is acceptable. otherwise, experts’ judgments should be adjusted until cr < 0.1 then, integrated comparison matrix �̃� is built as: �̃� = [ 1 �̃�21 𝑒 ⋮ �̃�𝑚1 𝑒 �̃�12 𝑒 1 ⋮ �̃�𝑚2 𝑒 ⋯ ⋯ ⋱ ⋯ �̃�1𝑚 𝑒 �̃�2𝑚 𝑒 ⋮ 1 ] (14) vasiljević et al./decis. mak. appl. manag. eng. 1 (1) (2018) 82-96 86 where �̃�𝑔ℎ{𝑥𝑔ℎ 1 , 𝑥𝑔ℎ 2 , … . , 𝑥𝑔ℎ 𝑠 }, �̃�𝑔ℎ is the sequence of relative importance of criterion g on criterion h. step 3: construct a rough comparison matrix. translate element 𝑥𝑔ℎ 𝑒 in �̃� into rough number 𝑅𝑁(𝑥𝑔ℎ 𝑒 ) using eqs. (1) (6): 𝑅𝑁(𝑥𝑔ℎ 𝑒 ) = [𝑥𝑔ℎ 𝑒𝐿 , 𝑥𝑔ℎ 𝑒𝑈] (15) where 𝑥𝑔ℎ 𝑒𝐿 is the lower limit of 𝑅𝑁(𝑥𝑔ℎ 𝑒 ) while 𝑥𝑔ℎ 𝑒𝑈 is the upper limit. then rough sequence 𝑅𝑁(�̃�𝑔ℎ) is represented as: 𝑅𝑁(�̃�𝑔ℎ) = {[𝑥𝑔ℎ 1𝐿, 𝑥𝑔ℎ 1𝑈], [𝑥𝑔ℎ 2𝐿, 𝑥𝑔ℎ 2𝑈], … , [𝑥𝑔ℎ 𝑠𝐿 , 𝑥𝑔ℎ 𝑠𝑈]} (16) it is further translated into an average rough number 𝑅𝑁(𝑥𝑔ℎ) by rough arithmetic eqs. (8) (12): 𝑅𝑁(𝑥𝑔ℎ) = [𝑥𝑔ℎ 𝐿 , 𝑥𝑔ℎ 𝑈 ] (17) 𝑥𝑔ℎ 𝐿 = 𝑥𝑔ℎ 1𝐿 +𝑥𝑔ℎ 2𝐿 +⋯+𝑥𝑔ℎ 𝑠𝐿 𝑆 (18) 𝑥𝑔ℎ 𝑈 = 𝑥𝑔ℎ 1𝑈 +𝑥𝑔ℎ 2𝑈 +⋯+𝑥𝑔ℎ 𝑠𝑈 𝑆 (19) where 𝑥𝑔ℎ 𝐿 is the lower limit of 𝑅𝑁(𝑥𝑔ℎ) and 𝑥𝑔ℎ 𝑈 is the upper limit. then rough comparison matrix m is formed as: 𝑀 = [ [1,1] [𝑥21 𝐿 , 𝑥21 𝑈 ] ⋮ [𝑥𝑚1 𝐿 , 𝑥𝑚1 𝑈 ] [𝑥12 𝐿 , 𝑥12 𝑈 ] [1,1] ⋮ [𝑥𝑚2 𝐿 , 𝑥𝑚2 𝑈 ] ⋯ ⋯ ⋱ ⋯ [𝑥1𝑚 𝐿 , 𝑥1𝑚 𝑈 ] [𝑥2𝑚 𝐿 , 𝑥2𝑚 𝑈 ] ⋮ [1,1] ] (20) step 4: calculate rough weight wg of each criterion: 𝑤𝑔 = [ √∏ 𝑥𝑔ℎ 𝐿 ,𝑚ℎ=1 𝑚 √∏ 𝑥𝑔ℎ 𝑈 ,𝑚ℎ=1 𝑚 ] (21) 𝑤𝑔 ′ = 𝑤𝑔/max (𝑤𝑔 𝑈) (22) where 𝑤𝑔 ′ is the normalization form. finally, the criteria weights are obtained. 3. numerical example the main activity of the company which is the subject of research is the production of metal washers for the automotive industry. its product range covers up over 3000 types of metal washers, and the largest part of it is used for mechanical transmissions in heavy machinery, cranes, trucks, and the like. the company is focused on the production and sale of flat and elastic washers. the ability of production is over 3500 tons of finished products. the aim of this paper is to determine the most important criteria for suppliers’ evaluation in the mentioned company. figure 2 presents criteria finance, logistics, quality and communications and business, and each of these criteria contains five subcriteria which are also shown in the figure below each criterion. a review of the given criteria for suppliers’ evaluation through literature is presented in the paper (stević, 2017c). a rough multicriteria approach for evaluation of the supplier criteria in automotive industry 87 figure 2. criteria for supplier selection (stević, 2017b) collect individual judgments and construct a group of pairwise comparison matrices. take the consistency examination until all the comparison matrices can pass through. integrate individual comparison matrices to generate an integrated comparison matrix. the individual pair-wise comparison matrices are as follows: 𝐵1 = [ 1 1/4 1/3 4 1 3 3 1/3 1 1/3 1/5 1/4 3 5 4 1 ] , 𝐶𝑅 = 0,068 < 0,10 𝐵2 = [ 1/ 1 4 1 4 1 1/4 1 4 1 1/3 3 1/3 3 1/3 4 1 ] , 𝐶𝑅 = 0,047 < 0,10 𝐵3 = [ 1 4 1/4 1 1 4 1/4 3 1 4 1/4 1/3 1 5 1/5 1 ] , 𝐶𝑅 = 0,049 < 0,10 obviously cre < 0.1 (e= 1, 2, 3), all the comparison matrices are acceptable. then integrated comparison matrix �̃� is generated by combining with the above three individual comparison matrices. �̃� = [ 1,1,1 1/4,4,4 4,1/4, 1/4 1,1,1 1/3,1,1 3,3,4 3,1/4,1/4 5,1/3,3 3,1,1 1/3,4,4 1/3,1/4,1/4 1/5,3,1/3 1,1,1 4,4,5 1/4,1/3,1/5 1,1,1 ] translate the elements in �̃� into rough numbers and correspondingly original integrated comparison matrix �̃� is converted into a rough comparison matrix. vasiljević et al./decis. mak. appl. manag. eng. 1 (1) (2018) 82-96 88 take as an example x̃24 = {5,1/3,3} 𝐿𝑖𝑚 ( 1 3 ) = 1 3 , 𝐿𝑖𝑚 ( 1 3 ) = 1 3 (5 + 1 3 + 3) = 2,78 𝐿𝑖𝑚(3) = 1 2 ( 1 3 + 3) = 1,67, 𝐿𝑖𝑚(3) = 1 2 (3 + 5) = 4 𝐿𝑖𝑚(5) = 1 3 (5 + 1 3 + 3) = 2,78, 𝐿𝑖𝑚(5) = 5 thus, 𝑥24 𝑒 can be expressed in rough number: 𝑅𝑁(𝑥24 1 ) = 𝑅𝑁(5) = [2,78; 5] 𝑅𝑁(𝑥24 2 ) = 𝑅𝑁 ( 1 3 ) = [0,33; 2,78] 𝑅𝑁(𝑥24 3 ) = 𝑅𝑁(3) = [1,67; 4] according to eqs. (17) (19) 𝑥24 𝐿 = 𝑥24 1 + 𝑥24 2 + 𝑥24 𝑠 𝑆 = 2,78 + 0,33 + 1,67 3 = 1,59 𝑥24 𝑈 = 𝑥24 1 + 𝑥24 2 + 𝑥24 𝑠 𝑆 = 5 + 2,78 + 4 3 = 3,93 thus rough sequence �̃�24 in �̃� is transformed into rough number 𝑅𝑁(𝑥24) = [1,59; 3,93]. the transformation of other elements in �̃� is implemented in the same way. then, the rough comparison matrix is obtained: 𝑀 = [ [1; 1] [0,67; 2,33] [1,22; 2,11] [0,28; 0,32] [1,92; 3,58] [1; 1] [1,96; 3,59] [0,57; 1,95] [0,63; 0,93] [0,56; 1,78] [1; 1] [0,22; 0,24] [3,11; 3,55] [1,59; 3,93] [4,11; 4,55] [1; 1] ] calculate rough weights of the criteria using eqs. (21) and (22). 𝑤 = {[1,39; 1,85]; [0,88; 2,01]; [1,77; 2,42]; [0,43; 0,62]} 𝑤′ = {[0,57; 0,76]; [0,36; 0,83]; [0,73; 1]; [0,18; 0,26]} according to the obtained results, the third criterion quality is the most important in the target company. observing the obtained values of the upper and lower limits of the rough number in all, except for the second criterion, shows that they have relatively approximate values. the cause of a large difference between the lower and the upper limit of logistics criteria are the different attitudes of decision-makers when this criterion is concerned. that is why the decision-makers gave different assessments of their preferences, for example, 1/3 and 5, etc. a rough multicriteria approach for evaluation of the supplier criteria in automotive industry 89 figure 3. comparison of rough numbers (zhai et al., 2008) figure 3 shows a comparison of rough numbers on the basis of which the criteria or alternatives are ranked. a comparison of the two rough numbers is clearly defined, depending on the lower and upper limits. after obtaining the values that mark the weight of the criteria in the same way it is necessary to make calculation for sub-criterion; so, the following is an example of the calculation for the subcriteria that belong to the logistics. the individual pair-wise comparison matrices are as follows: 𝐵1 = [ 1 1/4 1/4 4 4 4 1 1/3 5 5 4 3 1 5 5 1/4 1/5 1/5 1 1 1/4 1/5 1/5 1 1 ] , 𝐶𝑅 = 0,088 < 0,10 𝐵2 = [ 1 1/4 3 1 3 4 1 5 4 5 1/3 1/5 1 1/3 1 1 1/4 3 1 3 1/3 1/5 1 1/3 1 ] , 𝐶𝑅 = 0,028 < 0,10 𝐵3 = [ 1 3 1/4 1/5 1/7 1/3 1 1/5 1/7 1/9 4 5 1 1/3 1/4 5 7 3 1 1/3 7 9 4 3 1 ] , 𝐶𝑅 = 0,062 < 0,10 obviously cre < 0.1 (e= 1, 2, 3), all the comparison matrices are acceptable. then integrated comparison matrix �̃� is generated by combining with the above three individual comparison matrices. �̃� = [ 1,1,1 4,4,1/3 1/4,1/4, 3 1,1,1 4,1/3,4 1/4,1,5 1/4,1/3,7 3,1/5,5 1/5,1/4,7 1/5,1/5, 9 1/4,3,1/4 1/3,5,1/5 4,1,1/5 5,4,1/7 4,3,1/7 5,5,1/9 1,1,1 1/5,3,3 1/5,1,4, 5,1/3,1/3 1,1,1 1,1/3,3 5,1,1/4 1,3,1/3 1,1,1 ] translate the elements in �̃� into rough numbers and correspondingly original integrated comparison matrix �̃� is converted into a rough comparison matrix. take as an example �̃�45 = {1,1/3,3} vasiljević et al./decis. mak. appl. manag. eng. 1 (1) (2018) 82-96 90 𝐿𝑖𝑚 ( 1 3 ) = 1 3 , 𝐿𝑖𝑚 ( 1 3 ) = 1 3 (1 + 1 3 + 3) = 1,44 𝐿𝑖𝑚(1) = 1 2 (1 + 1 3 ) = 0,66, 𝐿𝑖𝑚(1) = 1 2 (1 + 3) = 2 𝐿𝑖𝑚(3) = 1 3 (1 + 1 3 + 3) = 1,44, 𝐿𝑖𝑚(3) = 3 thus, 𝑥45 𝑒 can be expressed in rough number: 𝑅𝑁(𝑥45 1 ) = 𝑅𝑁(1) = [0,66; 2] 𝑅𝑁(𝑥45 2 ) = 𝑅𝑁 ( 1 3 ) = [0,33; 1,44] 𝑅𝑁(𝑥45 3 ) = 𝑅𝑁(3) = [1,44; 3] according to eqs. (17) (19) 𝑥45 𝐿 = 𝑥45 1 + 𝑥45 2 + 𝑥45 𝑠 𝑆 = 0,66 + 0,33 + 1,44 3 = 0,81 𝑥45 𝑈 = 𝑥45 1 + 𝑥45 2 + 𝑥45 𝑠 𝑆 = 2 + 1,44 + 3 3 = 2,15 thus rough sequence �̃�245 in �̃� is transformed into rough number 𝑅𝑁(𝑥45) = [0,81; 2,15]. the transformation of other elements in �̃� is implemented in the same way. then, the rough comparison matrix is obtained: 𝑀 = [ [1; 1] [0,56; 1,78] [0,56; 1,78] [0,84; 2,74] [1,36; 3,29] [1,96; 3,59] [1; 1] [0,77; 3,17] [1,75; 4,18] [2,28; 4,46] [1,96; 3,59] [1,51; 3,91] [1; 1] [0,85; 2,93] [0,98; 3,36] [0,98; 3,36] [0,97; 4,37] [1,45; 2,69] [1; 1] [0,81; 2,15] [1,02; 4,40] [1,18; 5,09] [0,84; 2,74] [0,81; 2,15] [1; 1] ] calculate rough weights of the criteria using eqs. (21) and (22). 𝑤 = {[1,31; 2,86]; [0,99; 2,74]; [0,88; 2,11]; [1,00; 2,35]; [1,20; 2,54]} 𝑤′ = {[0,46; 1,00]; [0,35; 0,96]; [0,31; 0,74]; [0,35; 0,82]; [0,42; 0,89]} in order to obtain the final values of the subcriteria belonging to the logistics group, the following values are needed: 𝑤′ = {[0,46; 1,00]; [0,35; 0,96]; [0,31; 0,74]; [0,35; 0,82]; [0,42; 0,89]} multiplying with the values of the main criterion-logistics [0,36; 0,83] gives the following values: 𝑤′′ = {[0,17; 0,83]; [0,13; 0,80]; [0,11; 0,61]; [0,13; 0,68]; [0,15; 0,74]} the most important logistics sub-criteria are delivery and reliability, which in the overall ranking occupy high positions, which can be seen in table 2. following the above described methodology, the values for all the twenty criteria are obtained and shown in figure 4. a rough multicriteria approach for evaluation of the supplier criteria in automotive industry 91 figure 4 values of all criteria in rough numbers figure 4 shows that the certification of products which is used in (birgün barla, 2003; jamil et al., 2013; ting & cho, 2008; uygun et al., 2013) and quality (fallahpour et al., 2017; kilic, 2013; özbek, 2015; stević et al., 2016; wang et al., 2017) are of utmost importance in the company which is the subject of our research. these two criteria are very important because the company exports its products to the international market. the third place is taken by the criterion of volume discounts (jamil et al., 2013; wang, 2010) because the company is located on the territory of bosnia and herzegovina which is a very poor country; thus, additional discounts in business are very popular. the next most important criterion is that of delivery time (chan & kumar, 2007; sawik, 2010; yücenur et al., 2011; rezaei et al., 2014) and reliability (gencer & gürpinar, 2007; muralidharan et al., 2002; büyüközkan & göçer, 2017). 4. comparison and discussion once the results are obtained, a sensitivity analysis including comparison of the values of the criteria using three different forms of the ahp method is carried out. figure 5 presents the values of the main criteria obtained using conventional ahp, fuzzy ahp and rough ahp, while in table 2 presented are all the results of the sensitivity analysis including all the twenty criteria. the sensitivity analysis is very important for all types of research; a very studious example of the sensitivity analysis in the multicriteria decision-making can found in the paper (pamučar et al., 2017b) in which the authors use different methods for ranking solution. vasiljević et al./decis. mak. appl. manag. eng. 1 (1) (2018) 82-96 92 figure 5. values of main criteria using ahp, fahp and rahp certification of products is the most important criterion using ahp and rough ahp, while quality is the most important one using fuzzy ahp. of equal rank by all the methods is volume discount which thus occupies the third position. the results show that the rough ahp has more similarity with the conventional ahp for this research. table 2. results of sensitivity analysis criteria ahp fahp rahp values rank values rank values rank price of material 0,084 4 0,068 4 (0,28;0,53) 7 financial stability 0,047 5 0,052 6 (0,16;0,58) 8 method of payment 0,025 14 0,039 13 (0,08;0,19) 14 price of transport 0,025 14 0,026 18 (0,08;0,17) 15 volume discounts 0,122 3 0,074 3 (0,42;0,76) 3 delivery time 0,047 5 0,056 5 (0,17;0,83) 4 reliability 0,033 10 0,043 11 (0,13;0,80) 5 flexibility 0,034 9 0,042 12 (0,11;0,61) 9 logistics capacity 0,039 7 0,050 7 (0,13;0,68) 10 the percentage of correct realization of delivery 0,043 6 0,047 10 (0,15;0,74) 6 quality of material 0,149 2 0,114 1 (0,46;0,90) 2 warranty period 0,032 11 0,037 14 (0,09;0,20) 13 certification of products 0,164 1 0,102 2 (0,50;1,00) 1 reputation 0,037 8 0,033 15 (0,10;0,21) 12 awards and honors 0,017 16 0,021 19 (0,04;0,09) 18 communication system 0,012 17 0,032 16 (0,04;0,10) 17 speed of response to requirements 0,029 12 0,048 8 (0,10;0,26) 11 reactions to reclamation 0,027 13 0,044 9 (0,08;0,24) 11 information technology 0,021 15 0,043 11 (0,08;0,24) 11 clean of business 0,011 18 0,031 17 (0,04;0,14) 16 ranking all criteria from the first to the twentieth place is also shown in figure 6. a rough multicriteria approach for evaluation of the supplier criteria in automotive industry 93 figure 6. ranking criteria by the three forms of the ahp method the consequence of different results using different methods is reflected in different scales for evaluating criteria, which according to mukhametzyanov & pamučar (2017) is one of the five main reasons that influence the obtaining of results and their ranking. rough ahp according to (roy et al., 2016) enables us to measure consistency of preferences, manipulate multiple decision-makers and calculate relative importance for each criterion. the rough ahp according to (song et al., 2013) combines the strength of rough sets in handling subjectivity and the advantage of ahp in hierarchy evaluation. 5. conclusion this study proposes a rough group ahp approach to the evaluation supplier criteria in the company for producing metal washers for the automotive industry. according to the methodology applied in this paper the conclusion is that decisionmaking based on the rough ahp can be very helpful in production companies. the proposed models allow the evaluation of alternatives despite the imprecision and lack of quantitative information in the decision-making process. future research related to this work based on the most important criteria represents the application of some of the multicriteria methods based on the rough theory, for example the rough topsis for suppliers evaluation and their ranking. references birgün barla, s. 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(2015). an integrated ahp and vikor for design concept evaluation based on rough number. advanced engineering informatics, 29(3), 408-418. plane thermoelastic waves in infinite half-space caused decision-making: applications in management and engineering vol. 3, issue 1, 2020, pp. 60-78. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame2003078m * corresponding author. e-mail addresses: dmitri_muravev@sjtu.edu.cn (d. muravev), nemanja.mijic@gmail.com (n. mijic) a novel integrated provider selection multicriteria model: the bwm-mabac model dmitri muravev 1,2* and nemanja mijic3 1 school of naval architecture, ocean and civil engineering and state key laboratory of ocean engineering, shanghai jiao tong university, shanghai, china 2 nosov magnitogorsk state technical university, department of logistics and transportation systems management, mining engineering and transport institute, magnitogorsk, russia 3 university of defence in belgrade, department of logistics, belgrade, serbia received: 12 december 2019; accepted: 4 march 2020; available online: 14 march 2020. original scientific paper abstract: the supply chain is a very complex area aimed at obtaining the optimum from the point of view of all participants. in order to achieve the overall optimum and satisfaction of all participants, it is necessary to make an adequate evaluation and selection of providers at the initial stage. in this paper, the selection of providers is based on a new approach in the field of multicriteria decision-making. the weight coefficients were determined using the best-worst method (bwm), whereas provider evaluation and selection were performed using the multi-attributive border approximation area comparison (mabac) method, which is one of more recent methods in this field. in order to determine the stability of the model and the applicability of the proposed hybrid bwm-mabac model, the results were compared with the mairca and vikor models, and the results of the comparative analysis are presented herein. in addition, a total of 18 different scenarios were formed in the sensitivity analysis, in which the criteria change their original value. at the end of the sensitivity analysis, the statistical dependence of the results was determined using spearman's correlation coefficient, which confirmed the applicability of the proposed multicriteria approaches. key words: multicriteria decision-making, bwm, mabac, mairca, vikor, supply chain. mailto:dmitri_muravev@sjtu.edu.cn mailto:nemanja.mijic@gmail.com a novel integrated provider selection multicriteria model: the bwm-mabac model 61 1. introduction when looking at the efficiency of the entire supply chain, it is impossible not to notice that it largely depends on an adequate selection of providers, because this process is one of the most important factors that directly affect the performance of a company. with a proper valuation and selection of the right provider, this logistics subsystem can efficiently carry out tasks related to supplying a company, since the right providers can meet the requirements and needs posed in the procurement subsystem, which on their part relate to the quality, price and quantity of goods, delivery times and other deadlines, flexibility, reliability, and so on. a search for providers in order to fulfill this is a permanent and primary task. in order to enable the former, it is necessary to continuously collect and process providers’ data, establish and maintain adequate links with them. according to soheilirad et al. (2017), provider selection is an important item when speaking about management decisions, which considers several qualitative and quantitative criteria. the importance of this process in organizations reflects in the formation of the final product price, since the price of raw materials plays a significant role in the price of the final product (bai and sarkis 2010; ramanathan 2007). provider selection is one of more important items in the supply chain management (zhong and yao, 2017), while managing and developing provider relationships is a critical issue for achieving competitive advantage (bai and sarkis, 2011). considering the fact that provider selection in the supply chain is multicriteria group decision-making, according to zolfani et al. (2012), it is necessary for managers to know the most appropriate method for them to use so as to choose the right provider. this is especially true when we know that modern supply chains require that stringent requirements should be met, for which reason managers are faced with a difficult task of properly evaluating potential providers that rarely price the products that affect the company’s competitiveness in the market (cox and ireland, 2002). every day, a large number of decisions are made on the basis of certain criteria, so it can be safely said that multicriteria decision-making (mcdm) plays a significant role in real-life problems, including logistical problems. particularly important is the role that multicriteria decision-making plays in the decision-making process that affects the business system or the environment. therefore, mcdm is an efficient systematic and quantitative way to solve vital logistical problems, including supply chain management. the increasing use of multicriteria decision-making methods has contributed to the increasing popularity of this field on a daily basis (zavadskas et al., 2014a). this paper presents a hybrid mcdm vendor evaluation model, which is based on the application of the two models, i.e. the best-worst method (bwm) and the multiattributive border approximation area comparison (mabac) model. the bwm model (rezaei, 2015) was used to determine criteria weights, while the mabac model was used to evaluate the providers. the bwm and mabac models had been opted for because of the many advantages that recommend them for use in this field. in addition to this, no application of the hybrid bwm-mabac model has been reported in the literature yet, which enriches the methodology for provider evaluation and selection. the paper is structured into several chapters. in the second section of the paper, a literature analysis is performed through an overview of the existing multicriteria decision-making methods in the field of supply chains. the analysis of the applied methods, as well as the criteria for the work done in the field of the selection of transport service providers is carried out. based on the data obtained from the analysis of the work done, a new multicriteria decision-making model is proposed, as muravev and mijic./dec-mak. appl. manag. eng. 3 (1) (2020) 60-78 62 well as the criteria that will be used to select the right transport service provider. in the third section, the mathematical foundations of the hybrid bwm-mabac model are presented. in the fourth section, the testing of the proposed model is performed on a real-life example, in which the evaluation of providers at the ministry of defense of the republic of serbia is conducted. in the fourth section of the paper, the results are validated in three phases. the first phase involves comparing the results of the bwmmabac model with those obtained by applying other multicriteria decision-making methods. in the second phase, the validation is performed in a dynamic environment by applying dynamic initial decision matrices. the third phase of the validation includes a sensitivity analysis of the change in the weights of the criteria coefficients. the fifth section contains the conclusive considerations, where the presented conclusions are derived from the conducted research and the suggestions for further research. 2. literature review according to a large number of authors, provider selection is one of the most challenging management problems in logistics (stojicic et al., 2019). as a result, a number of methods for the evaluation of transport service providers have been developed to date. in the literature (fallahpour et al., 2017), the author uses the fuzzy modifications of the analytic network process (anp) method and the technique for order preference by similarity to ideal solution (topsis). govindan et al. (2013) used the fuzzy topsis method to rank the sustainable performance of a transport service provider. in order to make a choice of a logistics provider from the sustainability perspective while keeping an eye on the company’s goals, dai & blackhurst (2012) introduced a new integrated approach based on the ahp and the quality function deployment (qfd) methods, with four hierarchical stages. rezaei et al. (2016) introduced a new approach to selecting a transport service provider, which consists of three phases, the essential one being the phase in which the implementation of the best-worst method (bwm) is demonstrated. this approach can benefit companies looking for new markets. azadnia et al. (2013) propose an integrated approach to choosing a logistics provider, which, in addition to the application of the fuzzy ahp method, is also based on multi-objective mathematical programming, as well as the rule-based weighted fuzzy method. luthra et al. (2017) introduced an integrated approach to selecting a transport service provider from the sustainability perspective. the approach was implemented through a combination of the ahp and vikor methods based on 22 criteria. barata et al. (2014) demonstrated the application of mcdm methods in the evaluation of the degree of the organizational sustainability of a company. hsu et al. (2014) presented an approach based on several mcdm methods in order to select a transport service provider from the environmental point of view, i.e. with respect to carbon emissions. also, validi et al. (2014) ranked logistics providers and traffic routes based on co2 emissions by using the topsis method. the evaluation of the performance of logistics providers in the electronics industry is the topic of the research conducted in the paper (chatterjee et al., 2018) from the environmental point of view as well. in this paper, the rough decision-making trial and evaluation laboratory (dematel) model is used in combination with the rough multi-attribute ideal real comparative analysis (mairca) method. a quantitative assessment of the performance of transport service providers is presented in the paper from the sustainability perspective (erol et al., 2011). in addition to mcdm methods, fuzzy a novel integrated provider selection multicriteria model: the bwm-mabac model 63 techniques are used in this paper because of the presence of indeterminacy. more specifically, the fuzzy entropy and fuzzy multi-attribute utility theory (maut) methods are used. kusi-sarpong et al. (2018) presented a framework for the ranking and selection of sustainable innovation in logistics, given the fact that innovation plays a very significant role in sustainability. this framework is based on the bwm method, which has been tested and applied by several companies in india. managing the evaluation of providers from the sustainability perspective is significant for many industries, and for logistics as well. therefore, it is an increasingly common research topic. table 1 presents an analysis of the papers dedicated to this topic that have been published in the last few years. table 1. the application of mcdm methods in supply chains the problem solved by applying mcdm methods applied methods literature sustainable provider selection fpp, fuzzy topsis, ahp, vikor, fuzzy electre, fuzzy ahp, fuzzy multiplicative ahp, fuzzy smart, fuzzy vikor, fuzzy maut, bwm, ahpqfd, delphi, fuzzy dematel dai et al. (2012); erol et al. (2011); fallahpour et al. (2017); govindan et al. (2013); kusisarpong et al. (2018); luthra et al. (2017); luthra et al. (2018); padhi et al. (2018) provider evaluation from the environmental point of view fuzzy entropy-topsis, electre tri barata et al. (2014); zhao et al. (2014) assessing provider performance in supply chains dematel, anp, mairca, fuzzy delphi, danp, vikor, topsis chatterjee et al. (2018); hsu et al. (2014); validi et al. (2014) suggesting an innovative provider selection methodology bwm, danp, dematel, vikor, multimoora, ahp, fuzzy topsis das & shaw (2017); entezaminia et al. (2016); kuo et al. (2015); liu et al. (2018); rezaei et al. (2016) an integrated approach to identifying and analyzing criteria and alternatives under uncertain conditions grey-dematel, fahp azadnia et al. (2015); su et al. (2016) also, mcdm methods have been applied in solving a broad range of problems in the logistics field. depending on a specific problem in the logistics field, various mcdm methods have been used, such as: ahp, topsis, vikor, mairca, electre, fuzzy ahp, fuzzy topsis and dematel (alikhani et al., 2019; ahmadi et al., 2019; buyukozkan & gocer, 2017). according to the table, it is possible to see that the ahp, topsis and fuzzy topsis methods have been applied to the greatest extent in the logistics field. the bwm and fuzzy preferences programming (fpp) methods have most commonly been used to determine weight coefficients. this literature review allows us to see that the bwm and mabac models have not yet been applied in providers’ supply chain. due to the aforementioned fact, as well as the numerous advantages of the bwm and mabac models (pamucar and cirovic, 2015; gigovic et al., 2017; pamucar et al., 2018a, 2018b), a hybrid bwm-mabac model is proposed in this paper. based on the search muravev and mijic./dec-mak. appl. manag. eng. 3 (1) (2020) 60-78 64 of the most important index databases of international journals, table 2 presents an analysis of the criteria that have been applied in the mcdm methods for optimizing the selection of providers in supply chains. t a b le 2 . a p p li e d c ri te ri a i n m c d m m e th o d s in t h e f ie ld o f tr a n sp o rt s e rv ic e p ro v id e r se le ct io n : a re a l it e ra tu re c 1 c 2 c 3 c 4 c 5 c 6 c 7 c 8 c 9 s u st a in a b le p ro v id e r se le ct io n i n s u p p ly ch a in s a li k h a n i e t a l (2 0 1 9 ) f a ll a h p o u r e t a l (2 0 1 7 ) m e m a ri e t a l (2 0 1 9 ) y u e t a l (2 0 1 9 ) f a sh o to e t a l (2 0 1 6 ) + + + + + + + + + + + + + + + + x x + x x x x + x x x x x x x + x x x x + x x + x x + x x p ro v id e r e v a lu a ti o n fr o m a n e n v ir o n m e n ta l p o in t o f v ie w r a z a e i a n d h a e ri ( 2 0 1 9 ) p a rk o u h i e t a l (2 0 1 9 ) d o b o s a n d v o ro sm a rt y ( 2 0 1 9 ) l in e t a l (2 0 1 1 ) + + + + + + + + + + + + + + + x + + x + + x x x + + x x + + x x + x x x a n i n te g ra te d p ro v id e r e v a lu a ti o n a p p ro a ch f u e t a l (2 0 1 9 ) a h m a d i a n d a m in ( 2 0 1 9 ) r o u y e n d e g h a n d s a p u tr o (2 0 1 4 ) w u e t a l (2 0 1 9 ) l iu e t a l (2 0 1 9 ) + + + + + + + + + + + + + + + x x x x x x + + x + + + x + x + x x x x x x x x x x x x x x p ro v id e r s e le ct io n a p p li ca ti o n o f f u z z y t h e o ry i n m c d m m o d e ls b u y u k o z k a n a n d g o ce r (2 0 1 7 ) s a rk a r e t a l (2 0 1 7 ) z h a n g e t a l (2 0 1 5 ) l im a -j u n io r a n d c a rp in e tt i (2 0 1 6 ) + + + + + + + + + + + + x x x x x x x x x x + x x x x x x + + x x x x x p ro v id e r s e le ct io n a n p a p p ro a ch d a rg i e t a l (2 0 1 4 ) g u p ta e t a l (2 0 1 5 ) + + + + + x x x + x + x x + x x x x a novel integrated provider selection multicriteria model: the bwm-mabac model 65 based on the literature review presented, the following criteria are identified in order to address the provider evaluation issues in supply chains: c1 – price (min.), c2 – quality (max.), c3 – service and delivery (max.), c4 – flexibility (max.), c5 – technological capabilities (max.), c6 – trust (max.), c7 – relationship (max.), c8 risk (min.) and c9 – innovation (max.). based on table 2, a conclusion can be drawn that the criteria c1, c2 and c3, i.e. the price, the quality and the service and delivery, respectively, are indispensable when choosing a provider, regardless of the specific situation. another important criterion is c5, i.e. technological capabilities, which was applied in eight papers, only to be followed by the criteria c6 and c8, i.e. trust and risk, respectively, which were applied in six papers, and c4 and c7, i.e. flexibility and relationship (trust), respectively, which were applied in five papers. as a criterion, innovation (c9) is used least in solving the transport service provider selection problem, having only been used in two papers. in this paper, all the analyzed criteria will be included for the purpose of a comprehensive consideration of the problem. 3. the bwm-mabac hybrid provider evaluation model in this paper, a new hybrid bwm-mabac model (figure 1) for provider evaluation is presented. the model includes the application of the two methods: (1) the bwm method, which is used to determine the weight criteria, and (2) the mabac method, which is used to evaluate the ranking of the alternatives. experts’ opinions determining b and w criteria no yes comparison in criteria pairs: forming of bo and ow comparison vectors determining level of consistency for bwm – if cr satisfied? aggregation of bo and ow vectors of experts aggregated bo and ow vectors evaluation of optimum value of coefficient criteria experts’ evaluation of alternatives aggregation of experts’ opinion into matrix xk normalization of interval rough home matrix of decision making determining elements of weighted matrix determining the matrix of baa ranking of alternatives using mabac method sensitivity analysis – modification of wj through scenarios rank reversal problem spearman coefficient of correlation of ranks selection of best alternative p h a se 1 : in te rv a l ro u g h b w m p h a se 2 : m a b a c m e th o d optimal weights coefficients figure 1. the bwm-mabac model muravev and mijic./dec-mak. appl. manag. eng. 3 (1) (2020) 60-78 66 the model is implemented through three phases. in the first phase, the criteria are evaluated by using the bwm method. based on a questionnaire and the experts’ evaluation, the ranking of the criteria and the comparison of the ranking criteria are made. the values of the weight coefficients of the criteria are obtained as the output from the bwm method. the output data from the bwm are further processed through the mabac method algorithm. in the second phase, the alternatives are ranked by applying the mabac method. in the third phase, the results are validated. 3.1. the bwm algorithm like the ahp model, the bwm is one of the models of a more recent data based on comparison principles in criteria pairs and the validation of results through a deviation from the maximum consistency (pamucar et al., 2018). the bwm is a model which eliminates the drawbacks of the ahp model to some extent. the advantages of the implementation of the bwm are a small number of comparisons in criteria pairs, the ability to validate results by defining the consistency deviation (cr) of the comparisons, and taking into consideration transitivity during comparisons in criteria pairs. also, the bwm methodological procedure eliminates the problem of a comparison redundancy in criteria pairs, which is present in some subjective criteria weight determination models. suppose that there are the n evaluation criteria in the multicriteria model that are denoted as wj, j = 1,2, ..., n and that their weight coefficients need to be determined. subjective weighting models based on comparisons in criteria pairs require that the decision-maker should determine the degree of the influence of the criterion i on the criterion j as well. in accordance with the defined settings, the following section introduces the bwm algorithm (pamucar et al., 2018). algorithm: bwm input: the experts’ comparison in criteria pairs output: the optimal values of criteria/sub-criteria weight coefficients step 1: the experts’ ranking of the criteria/sub-criteria. step 2: the determination of the bo and ow vectors of the comparative significance of the evaluation criteria. step 4: defining a model for the determination of the final values of the weighting coefficients of the evaluation criteria: 1 min . . ; ; 1; 0; 1, 2,..., jb bj jw j w n j jj s t ww a a w w w w j n                 step 5: the calculation of the final values of the evaluation criteria/sub-criteria  1 2, ,..., t n w w w 3.2. the mabac model algorithm the mabac method is one of more recent methods (pamucar and cirovic, 2015). to date, it has been broadly applied in solving numerous problems in the multicriteria decision-making field. the basic assumption of the mabac method is that the distance a novel integrated provider selection multicriteria model: the bwm-mabac model 67 of the criterion function of the observed alternative from the boundary approximate area should be defined. the following section introduces the mabac method algorithm (pamucar and cirovic, 2015). algorithm: mabac method input: bwm weights and the initial decision matrix output: the ranking of the alternatives step 1: the formation of the initial decision matrix (x). step 2: the normalization of the elements of the initial matrix. step 3: the calculation of the elements of the weighted matrix (v). step 4: the determination of the matrix of the boundary approximate regions (g). the matrix of the boundary approximation domains g (12) format 1n x :   1 2 1 2 ... ... n n c c c g g g g , where 1/ 1 m m i ij j g v          ( ijv represents the elements of the weighted matrix). step 5: the calculation of the elements of the distance matrix of the alternatives from the boundary approximation region (q). the distance of the alternatives from the boundary approximation region (qij) is determined as the difference between the elements of the constrained matrix (v) and the value of the boundary approximation regions (g). 11 1 12 2 1 11 12 1 21 1 22 2 2 21 22 2 1 1 2 2 1 2 ... ... ... ... ... ... ... ... ... ... ... ... ... n n n n n n m m mn n m m mn v g v g v g q q q v g v g v g q q q q v g v g v g q q q                                 step 6: the ranking of the alternatives. the calculation of the values of the criteria functions is obtained as the sum of the distances of the alternatives from the boundary approximation regions ( iq ). 4. the application of the bwm-mabac model the model has been tested on a real issue, which includes the evaluation of the providers of spare parts for transport vehicles at the ministry of defense of the republic of serbia. a total of eight providers were considered, whose names were not included in this survey because of the confidentiality of the tender documents. the study involved four experts with at least 10 years of experience in supply chain evaluation. in the first phase of the implementation of the bwm-mabac model, it is necessary to define the weight coefficients of the criteria by using the bwm. the first phase of the bwm involves the experts’ ranking of the criteria. based on the significance of the criteria presented in the bo and ow vectors, a nonlinear model was formed for the calculation of the optimal values of the weight coefficients. a total of four models were formed, one for each expert. the following section provides the model for the calculation of the optimal values of the weighting coefficients for the first expert. muravev and mijic./dec-mak. appl. manag. eng. 3 (1) (2020) 60-78 68 1 1 1 2 3 7 3 74 2 2 2 7 1 1 min . . 7 ; 2 ;....; 3 ; 6.00 ; 3 ;;...; 2 ; 1; 0; 1, 2,..., 7j j j expert s t w w w w w w w ww w w w w w j                                   by solving the nonlinear models, the optimal values of the weight coefficients for each expert were defined, as in table 4. table 4. the criteria weight coefficients experts e1 e2 e3 e4 medium c ri te ri a w e ig h t co e ff ic ie n ts c1 0.311 0.100 0.117 0.030 0.1395 c2 0.207 0.120 0.180 0.149 0.1640 c3 0.076 0.299 0.269 0.149 0.1982 c4 0.104 0.067 0.096 0.149 0.1040 c5 0.060 0.086 0.054 0.075 0.0687 c6 0.065 0.120 0.077 0.075 0.0842 c7 0.052 0.050 0.045 0.149 0.0740 c8 0.086 0.075 0.128 0.149 0.1095 c9 0.039 0.054 0.034 0.075 0.0505 by averaging the obtained values, the optimal values of the weight coefficients of the criteria were defined, which were further used to evaluate providers by applying the mabac method. the paper evaluates a total of eight providers, designated a1 through a8. based on the provider data, the mabac model was implemented through the following six steps: step 1. in the mabac model, the initial decision matrix (x) was the starting point: 3 5 6 7 8 91 2 4 max max max max min maxmin max max 1 65 23 56 53 54 95 53 59 62 2 45 29 50 49 49 87 63 7344 3 56 43 70 57 59 52 5941 41 4 70 35 82 43 91 93 38 6641 5 82 68 63 95 35 81 79 39 49 6 90 56 71 80 62 71 91 23 81 7 48 39 63 74 25 66 66 72 52 8 76 56 59 61 53 6 f f f f f ff f f a a a x a a a a a  7 59 46 77                             step 2. using linear normalization, the elements of the matrix x were normalized, thus obtaining the normalized matrix n: a novel integrated provider selection multicriteria model: the bwm-mabac model 69 3 5 6 7 8 91 2 4 max max max max min maxmin max max 1 0.556 0 0.188 0.192 0.439 0.283 0.265 0.4061 2 0.133 0 0.115 0.364 0 0.925 0.184 0.7501 3 0.756 0.444 0.625 0.269 0.252 0.294 0.057 0.408 0.313 4 0.444 0.267 0 0.961 0 0.61 1 5 6 7 8 f f f f f ff f f a a a n a a a a a  33 0.531 0.111 0.406 0.152 0.725 0.774 0.673 01 1 0 0.733 0.656 0.712 0.561 0.529 1 1 1 0.933 0.356 0.406 0.596 0 0.431 0.528 0 0.094 0.311 0.733 0.281 0.346 0.424 0.451 0.396 0.531 0.875                             step 3. by multiplying the optimal values of the weighting coefficients obtained by applying the bwm by the elements of the normalized matrix n a weighted normalized matrix v was obtained. 3 5 6 7 8 91 2 4 max max max max min maxmin max max 1 0.217 0.164 0.235 0.124 0.099 0.168 0.095 0.139 0.071 2 0.279 0.186 0.198 0.116 0.094 0.084 0.142 0.130 0.088 3 0.245 0.237 0.322 0.132 0.085 0.109 0.078 0.154 0.066 4 0. 5 6 7 8 f f f f f ff f f a a a v a a a a a  202 0.208 0.396 0.104 0.137 0.165 0.074 0.179 0.077 0.155 0.328 0.279 0.208 0.079 0.145 0.131 0.183 0.051 0.140 0.284 0.328 0.178 0.107 0.129 0.148 0.219 0.101 0.270 0.222 0.279 0.166 0.069 0.121 0.113 0.110 0.055 0.183 0.284 0.254 0.140 0.098 0.122 0.103 0.168 0.095                             step 4. in step 4, the defined matrix of the boundary approximate regions (g) was approached. the boundary approximation area (gao) for each criterion was determined by geometrically averaging the values of the matrix v. 3 5 6 7 8 91 2 4 0.2055 0.2335 0.2807 0.1425 0.0942 0.1276 0.1074 0.1568 0.0736 c c c c c cc c c g        step 5. in this step, the distance of the elements v of the matrix from the matrix g was calculated. thus, the matrix q, which represents the distance of the alternatives from the gao, was obtained. 3 5 6 7 8 91 2 4 max max max max min maxmin max max 1 0.011 0.070 0.045 0.018 0.005 0.041 0.012 0.018 0.003 2 0.073 0.048 0.083 0.026 0.001 0.043 0.035 0.027 0.015 3 0.039 0.003 0.041 0.010 0.009 0.019 0 4 5 6 7 8 f f f f f ff f f a a a q a a a a a                  .029 0.003 0.007 0.004 0.026 0.116 0.038 0.043 0.037 0.033 0.022 0.004 0.051 0.094 0.002 0.066 0.015 0.018 0.024 0.026 0.023 0.066 0.051 0.048 0.036 0.013 0.001 0.041 0.062 0.027 0.064 0.011 0.002 0.024 0.025 0.007 0.006                0.047 0.018 0.023 0.051 0.027 0.0020 0.004 0.005 0.004 0.011 0.021                                 muravev and mijic./dec-mak. appl. manag. eng. 3 (1) (2020) 60-78 70 step 6. the ranking of the alternatives was performed based on the value of the alternative score functions. the criteria functions of the alternatives were obtained by summing up the elements of the matrix q by rows. thus, the values of the criteria functions of the alternatives were obtained for each provider: s2 = -0.104 s3 = 0.007 s4 = 0.120 s5 = 0.137 s6 = 0.212 s7 = -0.018 s8 = 0.025 based on the values of the criteria functions, the final ranking of the alternatives was defined as: a6 > a5 > a4 > a8 > a3 > a7 > a2 > a1. before making a decision, it is necessary to evaluate the reliability of the results obtained. the validation of the results of the bwm-mabac model was carried out through three phases. in the first phase, the initial ranking of the alternatives was compared with that of the other mcdm models, as in figure 2. since the mabac method uses linear normalization, the multi attributive ideal-real comparative analysis (mairca) method and the multicriteria compromise ranking (vikor) method also have linear normalization. figure 2. the ranks of the alternatives according to the presented methods, the ranking of the alternatives shows that the alternative a6 remained the first-ranked by all the methods. the same rank was obtained by the mairca method as it was by the mabac method, whereas the ranking changed with the vikor method (the alternative a1 replaced its rank with the alternative a2, and the alternative a3 replaced the rank with te alternative a8). in order to determine the statistical significance between the rankings obtained by the bwm-mabac model and the other approaches, the spearman correlation coefficient (scc) was used. this correlation coefficient is a simple linear correlation coefficient between the ranks. the spearman rank correlation coefficient is a non-parametric method for the estimation of the strength of the association that is applied when data for at least one variable are given as ordinal data or a rank, when at least one variable has no normal distribution and when the relationship between the variables is not linear. the results of the ranking comparisons by using the scc are given in table 5. 0 1 2 3 4 5 6 7 8 9 а1 а2 а3 а4 а5 а6 а7 а8 mabac mairca vikor a novel integrated provider selection multicriteria model: the bwm-mabac model 71 table 5. the rank correlation of the tested methods method mcdm mabac mairca vikor scc 1.000 1.000 0.952 table 5 allows us to see that the mabac and mairca methods are in a complete correlation. also, the vikor method shows a high correlation compared to the mabac method. given the fact that, in this particular case, all the scc values are significantly higher than 0.9 (exceptional correlation) and the mean value is 0.976, it can be concluded that there is a very large correlation (closeness) between the proposed model and the other mcdm methods. in doing so, we can conclude that the proposed ranking is validated and credible. in the second stage of the results validation, a performance analysis of the proposed model was conducted under the conditions of the dynamic initial matrix. in the dynamic starting matrix for each scenario, the number of the alternatives was changed and the obtained ranks were analyzed. scenarios are formed for situations where one inferior alternative is removed from subsequent considerations, while the remaining dominant alternatives are ranked according to a newly-acquired initial decision matrix. in this study, the initial solution a6> a5> a4> a8> a3> a7> a2> a1 was obtained. clearly, the alternative a1 is the worst option. in the first scenario, the alternative a1 was eliminated from the list of the alternatives and a new decision matrix with a total of seven alternatives was obtained. the new decision matrix was re-solved by using the bwm-mabac model. in the following scenario, the next worst alternative was eliminated and the remaining alternatives were ranked. thus, a total of seven scenarios were formed, which are shown in table 6. table 6. the ranks of the alternatives within the dynamic decision matrix scenario rang s1 a6>a5>a4>a8>a3>a7>a2>a1 s2 a6>a5>a4>a8>a3>a7>a2 s3 a6>a5>a4>a8>a3>a7 s4 a6>a5>a4>a8>a3 s5 a6>a5>a4>a8 s6 a6>a5>a4 s7 a6>a5 it is clear from table 6 that, when the worst-case alternative is eliminated, there is no change in the best-ranked alternative in the rearranged matrix. based on this, it can be concluded that the bwm-mabac model does not lead to a rank reversal among the alternatives. the alternative a6 remained the best-ranked across all the scenarios, thus confirming the robustness and accuracy of the resulting rankings of the alternatives in the dynamic environment. since the results of multicriteria decision-making depend on the values of the weighting coefficients of the evaluation criteria, an analysis of the sensitivity of the results to a change in the criteria weights was performed. the analysis of the sensitivity of the rankings of the alternatives to changes in the weight coefficients of the criteria was conducted through the 18 scenarios given in table 7. muravev and mijic./dec-mak. appl. manag. eng. 3 (1) (2020) 60-78 72 table 7. the sensitivity analysis scenarios scenario criteria weights scenario criteria weights s1 wc1=1.25× wc11(ov); wci=0.25× wci(ov) s10 wc1=1.55× wc11(ov); wci=0.55× wci(ov) s2 wc2=1.25× wc11(ov); wci=0.25× wci(ov) s11 wc2=1.55× wc11(ov); wci=0.55× wci(ov) s3 wc3=1.25× wc11(ov); wci=0.25× wci(ov) s12 wc3=1.55× wc11(ov); wci=0.55× wci(ov) s4 wc4=1.25× wc11(ov); wci=0.25× wci(ov) s13 wc4=1.55× wc11(ov); wci=0.55× wci(ov) s5 wc5=1.25× wc11(ov); wci=0.25× wci(ov) s14 wc5=1.55× wc11(ov); wci=0.55× wci(ov) s6 wc6=1.25× wc11(ov); wci=0.25× wci(ov) s15 wc6=1.55× wc11(ov); wci=0.55× wci(ov) s7 wc7=1.25× wc11(ov); wci=0.25× wci(ov) s16 wc7=1.55× wc11(ov); wci=0.55× wci(ov) s8 wc8=1.25× wc11(ov); wci=0.25× wci(ov) s17 wc8=1.55× wc11(ov); wci=0.55× wci(ov) s9 wc9=1.25× wc11(ov); wci=0.25× wci(ov) s18 wc9=1.55× wc11(ov); wci=0.55× wci(ov) *ov (old value) the sensitivity analysis scenarios for the change in the criteria weights are grouped into two groups. within each group, the weighting coefficients of the criteria were increased by 25% and 55%, respectively. in each of the 18 scenarios, one criterion was favored within the two groups, by which the weight coefficient increased by the indicated values. in the same scenario, the weighting coefficients were reduced by 75% (s1-s9) and 45% (s10-s18), respectively. the changes in the ranking of the alternatives across the 18 scenarios are shown in table 8. table 8. the changes in the ranking due to the changes in the criteria weights scenario rank s1 a7>a2>a3>a4>a6>a1>a5>a8 s2 a5>a6>a8>a3>a4>a7>a2>a1 s3 a4>a6>a3>a5>a7>a8>a1>a2 s4 a5>a6>a7>a8>a4>a3>a1>a2 s5 a4>a6>a5>a8>a3>a1>a2>a7 s6 a4>a6>a5>a1>a8>a7>a3>a2 s7 a6>a5>a2>a8>a7>a4>a3>a1 s8 a6>a5>a4>a8>a3>a1>a7>a2 s9 a6>a4>a8>a5>a3>a2>a7>a1 s10 a4>a7>a6>a3>a5>a2>a8>a1 s11 a5>a6>a8>a4>a3>a7>a2>a1 s12 a4>a6>a5>a3>a7>a8>a1>a2 s13 a6>a5>a4>a7>a8>a3>a1>a2 a novel integrated provider selection multicriteria model: the bwm-mabac model 73 scenario rank s14 a6>a4>a5>a8>a3>a7>a1>a2 s15 a6>a4>a5>a8>a3>a7>a1>a2 s16 a6>a5>a4>a8>a7>a2>a3>a1 s17 a6>a5>a4>a8>a3>a7>a1>a2 s18 a6>a4>a5>a8>a3>a7>a2>a1 the results show that assigning different criteria weights across the 18 scenarios presented does not lead to a significant change in the ranking of the alternatives. in the scenarios s2-s9 and s11-s18, the fact that the alternative a6 ranks either first or second is noticed. in the scenarios s1 and s10, the alternative a6 is not among the top two alternatives, due to the high value of the a6 provider engagement price, whereas in the scenarios s1 and s10, the impact of the weighting factor on the final decision only increased. the results (table 8) show that assigning different weights to the criteria across the scenarios leads to a change in the rank of the alternatives, thus confirming that the model is sensitive to changes in weight coefficients. the following section compares the ranks in table 8 with the initial ranks. the scc values are shown in figure 3. based on figure 3, it can be concluded that there is a high rank correlation in the 12 scenarios, since the scc value is greater than 0.80, whereas in the four scenarios, that correlation is satisfactory, i.e. it is greater than 0.50. in the two scenarios, the scc value is below 0.50. however, the mean scc across the scenarios is 0.778, which shows a satisfactory average correlation. based on this, it can be concluded that there is a satisfactory closeness of the ranks and the proposed ranking is validated and credible. -0,400 -0,200 0,000 0,200 0,400 0,600 0,800 1,000 s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11 s12 s13 s14 s15 s16 s17 s18 figure 3. the correlation of the ranks muravev and mijic./dec-mak. appl. manag. eng. 3 (1) (2020) 60-78 74 5. conclusion the multicriteria model presented in this paper is an integration of the bwm and mabac methods, where the bwm was used to calculate the values of the criteria weights, while the mabac method was applied for provider evaluation and selection. the model was verified through the provider selection process in a real system, based on the nine criteria. the results show that provider 6 is the best solution in all the scenarios including different criteria values, except in the two scenarios in which the price criterion was favored. the analysis of the results has shown that the obtained ranks of the alternatives of the bwm-mabac model completely correlate with the ranks of the other multicriteria models which they were compared with. one of the contributions of this paper is the bwm-mabac model that provides us with an objective aggregation of experts’ decisions. in addition, another contribution of the paper reflects in the improvement of the methodology of provider evaluation and selection through a new hybrid multicriteria model. no use of this or a similar approach in the selection of providers has been seen in the literature analysis. by applying the developed approach, it is possible to approach multicriteria decision-making in an easy way and evaluate and select those providers who have a significant impact on the achievement of the efficiency of the whole of the supply chain. the four-stage model may also be applied so as to make other decisions. it is applicable in the provider evaluation process in all areas and may be particularly suitable for manufacturing companies. the flexibility of the model reflects in the fact that it can be verified by integrating any multicriteria decision-making methods. further research related to this paper pertains to the application of uncertainty theories (fuzzy, rough, neutrosophic, etc.) together with this and other multicriteria methods in this field. author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research received no external funding. conflicts of interest: the authors declare no conflicts of interest. references ahmadi, s., & amin, s. h. 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(2012). a hybrid mcdm model encompassing ahp and copras-g methods for selecting company provider in iran. technological and economic development of economy, 18(3), 529-543. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 3, issue 2, 2020, pp. 19-36. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame2003019d * corresponding author. e-mail addresses: dbozanic@yahoo.com (d. božanić), tesic.dusko@yahoo.com (d. tešić), milickm5@gmail.com (a. milić). multicriteria decision making model with znumbers based on fucom and mabac model darko božanić 1*, duško tešić1 and aleksandar milić1 1 university of defense in belgrade, military academy, belgrade, serbia received: 5 january 2020; accepted: 6 april 2020; available online: 12 april 2020. original scientific paper abstract. in the paper is presented a model for selecting a location for a brigade command post during combat operations. considering that this is a very complex model, which can be approached from several aspects, this paper is limited only to the criteria related to the construction or arrangement of the command post, respectively, the engineering aspect. the selection process is conducted using hybrid fucom – z-number – mabac model. the fucom method is used to define the weight coefficients of criteria based on which the selection is made. the mabac method, modified by applying z-number, is used to rank alternatives. the end results indicate that the application of z-number in decision making includes broader set of uncertainties than standard fuzzy numbers, which is very important for deciding in combat situations. key words: fucom, mabac, z-number, fuzzy number, brigade command post. 1. introduction – problem description the serbian army performs various combat and non-combat operations. through the implementation of these, commanders and leaders (from the highest to the lowest level of command) are often in situations in which they have to make more decisions. most often the end result is a decision made on the basis of previously acquired, mostly theoretical knowledge and on the experience gained by officers during their military service. one example of such an issue is the selection of a brigade command post in a defense operation. like others, this problem can be solved on the basis of experience and knowledge, but it is much better when the decision is followed by adequate mathematical decision-making model, used as an aid or tool for decisionmakers. mailto:dbozanic@yahoo.com mailto:tesic.dusko@yahoo.com mailto:milickm5@gmail.com božanić et al./decis. mak. appl. manag. eng. 3 (2) (2020) 19-36 20 “a command post presents an area, premises or technical means (ship, tank, conveyor, aircraft) in the area of operation of a unit, in which the command is placed with appropriate means during the preparation and conduct of a combat.“ (military lexicon, 1981). there are numerous factors influencing the selection of a command post. these factors (criteria or conditions) can in principle be divided into two groups: 1) the criteria related to the construction of a command post, respectively, performance of works, and 2) the criteria related to the functionality of a command post. the first set of criteria defines the criteria primarily related to fortifying and partly to masking, while in the second set of criteria would be included the criteria related to successful command during combat operations. throughout this paper, the authors focused on the first set of criteria, namely, the development of a model to support decision-making when selecting a brigade command post from the perspective of the ability to perform works, respectively, fortifying and masking. the decision-making support model is based on two methods: 1) the fucom method for defining criteria weights, and 2) the mabac method, which is fuzzified using standard fuzzy numbers and z numbers for ranking alternatives. the fucom method was first presented in 2018 (pamučar et al., 2018). due to its simple application and reliable results, this method has quickly begun to be applied in other papers (prentkovskis et al., 2018; badi & abdulshahed, 2019; puška et al., 2019; cao et al., 2019; durmić et al., 2019; stević et al., 2019; ibrahimović et al., 2019). the most common application of the fucom method is found in the process of defining weight coefficients of criteria. the mabac method was firstly described in the paper made by pamučar and ćirović (2015). after the first publication, large number of authors applied the method (božanić et al., 2016a; peng & yang, 2016; božanić et al., 2016b; chatterjee et al., 2017; gigović et al., 2017; majchrzycka & poniszewska, 2018; ji et al., 2018; hondro, 2018, ibrahimović et al., 2019; luo & xing, 2019; wei et. al, 2019). very soon after the first appearance, the method was applied in fuzzy environment (roy et al., 2016; xue et al., 2016; yu et al., 2017; sun et al., 2018; hu et al., 2019; božanić et al., 2018, bobar et al., 2020), neutrosophic environment (peng & dai, 2018; pamučar & božaić, 2019), as well as with the application of rough numbers (roy et al., 2017; sharma et al., 2018). 2. methods applied in the paper in the following part of the paper, the description of the methods used in the paper is provided. 2.1. fucom method considering that basic version of the fucom method is used, which is presented in pamučar et al. (2018), in the further part of the paper, only the steps of the method are listed. more detailed review with the examples is available at pamučar et al (2018). the fucom method consists of three steps: step 1. in the first step, the criteria from the predefined set of the evaluation criteria  1 2 nc c , c ,..., c are ranked. the ranking is performed according to the significance of the criteria, i.e. starting from the criterion which is expected to have the highest weight coefficient to the criterion of the least significance. step 2. in the second step, a comparison of the ranked criteria is carried out and the comparative priority ( k / ( k 1)  , k 1, 2,..., n , where k represents the rank of the multicriteria decision making model with z-numbers based on fucom and mabac model 21 criteria) of the evaluation criteria is determined. the comparative priority of the evaluation criteria ( k / ( k 1)   ) is an advantage of the criterion of the j( k ) c rank compared to the criterion of the j( k 1) c  rank. step 3. in the third step, the final values of the weight coefficients of the evaluation criteria   t 1 2 n w , w ,..., w are calculated. the final values of the weight coefficients should satisfy the two conditions. after the verification of the fulfillment of conditions, the weight coefficients of criteria are defined by using the expression (1): j( k ) k / ( k 1) j( k 1) j( k ) k / ( k 1) ( k 1) / ( k 2) j( k 2) n j j 1 j min s.t. w , j w w , j w w 1, j w 0, j                          (1) 2. 2. z number мавас method the mabac method is developed by (pamučar & ćirović, 2015). it is developed as the method providing crisp values. in this paper is used fuzzified mabac method by applying z-numbers. the fuzzification is performed using triangular fuzzy numbers. a general form of triangular fuzzy number is given in the figure 1. 0 t1 µ(x) x. t2 t3 1 figure 1. triangular fuzzy number (pamučar et al., 2012) triangular fuzzy numbers have the form 1 2 3 t (t , t , t ) t1 the left distribution of the confidence interval of fuzzy number t, t2 fuzzy number membership function has the maximum value equal to 1, and t3 the right distribution of the confidence interval of fuzzy number t (pamučar et al., 2012). z–number presents an extension of classic fuzzy number and provides wider opportunities for considering additional uncertainties following decision making. the concept of z-number was proposed by zadeh (2011). in 2012 already kang et al. (2012a, 2012b) shown in detail the application of z-numbers in uncertain božanić et al./decis. mak. appl. manag. eng. 3 (2) (2020) 19-36 22 environment. later authors consider the application of z-numbers with different methods of multi-criteria decision making. sahrom & dom (2015) present the use of z-numbers in the hybrid ahp-z-number-dea method. azadeh & kokabi (2016) use znumbers with the dea method. azadeh et al. (2013) with the ahp, yaakob & gegov (2015) with the topsis method, aboutorab et al. (2018) with the best worst method. salari et al. (2014) elaborate a novel earned value management model using znumber. z-number represents an ordered pair of fuzzy numbers that appear as z=( a , b ) (zadeh, 2011). the first component, fuzzy number a , represents the fuzzy limit of a particular variable x, while the second component fuzzy number b represents, the reliability of the first component ( a ). the appearance of the z-number with triangular fuzzy numbers is shown in figure 2 (zadeh, 2011). a1 a3a2 ã(x) 1 x ã b1 b3b2  (x) 1 x bb ,z a b figure 2. a-simple z-number (kang et al., 2012a) the general record of triangular z-numbers can be displayed as   1 2 3 1 2 3 baz a , a , a ; w , (b , b , b ; w ) (2) where the values a w i b w represent weight factors of fuzzy numbers a referring to b , which for the initial z-number the majority of authors defines as ba w w 1  ,   ba w , w 0,1 ( a w is the height of the generalized fuzzy number and a 0 w 1  ) (chutia et al., 2013). the transformation of the z-number into the classical fuzzy number, with the presented evidence, is shown in kang et al. (2012b). this transformation consists of three steps: 1) convert the second part ( b ) into a crisp number using the centered method (kang et al., 2012b): 1 2 3 a a a 3     (3) 2) add the weight of the second part ( b ) to the first part ( a ). the weighted z-number can be denoted as kang et al. (2012b)  aa az x, (x) (x) (x)         (4) which can be denoted by the figure 3a. this can be written as (azadeh et al., 2013): multicriteria decision making model with z-numbers based on fucom and mabac model 23 1 2 3 z (a , a , a ; )   (5) a1 a3a2 ã(x) 1 x ã  a a1 a3a2 ã(x) 1 x ã  a ã(x) 1 x `z 1  a 2  a 3  a a) b) figure 3. z-number after multiplying the reliability (a) and the regular fuzzy number transformed from z-number (b) 3) convert the weighted z-number into a regular fuzzy number. the regular fuzzy set can be denoted as kang et al. (2012b) ‚ ‚ ‚ az z x z x, (x) (x) ( )             (6) ‚ 1 2 3 z *a ( *a , *a , *a )     (7) and it can be present as figure 3b (kang et al., 2012b). after describing z-numbers it is necessary to explain their application in a particular model. these numbers present more comprehensive treatment of uncertainty because when the value of an alternative by a criterion in the form of a standard fuzzy number ( a ) is shown, the degree of certainty of the decision maker or expert ( b ) is also presented. by the above expressions (2-7) is made a transformation of the above fuzzy numbers into a unique fuzzy number. standard fuzzy mabac method is further applied. the degree of certainty of the decision makers in the values provided for the evaluation of alternatives by criteria is defined by the expressions presented on the fuzzy linguistic scale, as in the figure 4. 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 very small small medium high very high 0 figure 4. fuzzy linguistic descriptors for evaluating the degree of conviction of experts (bobar et al., 2020) božanić et al./decis. mak. appl. manag. eng. 3 (2) (2020) 19-36 24 hybrid model z number – mabac is taken from bobar et al. (2020). the fuzzy mabac method consists of 7 steps (božanić et al., 2018, bobar et al., 2020): step 1. forming of the initial decision matrix ( x ). matrix is formed with a grade of alternatives based on criteria  1 2, ...,i i i ina x x x , where ijx represents the value of ith alternative (i=1,2,...m), based on j-th criteria (j=1,2,...,n) 1 2 1 11 12 1 2 11 22 2 1 2 ... ... ... ... ... ... ... ... n n n m m m mn k k k a x x x a x x x x a x x x             (8) step 2. converting z-numbers to regular fuzzy number. this process is performed by applying the expressions (2) to (7). the output provides new initial fuzzy decisionmaking matrix ( p ) step 3. normalization of new initial decision-making matrix ( p ). the calculation of the elements of normalized matrix ( n ) depends on the type of criteria. for beneficial criteria this calculation is executed according to the expression: ij i ij i i x x t x x       (9) for detriment criteria the calculation is executed according to the expression: ij i ij i i x x t x x       (10) values ij x , i x  , i x  represent elements of the initial matrix of decision-making ( x ). the values i x  , i x  are defined as explained bellow 1 2 max( , ,..., ) i r r mr x x x x   represent maximal values of the right distribution of fuzzy numbers of the observed criteria alternatives 1 2 min( , ,..., ) i l l ml x x x x   represent minimal values of the left distribution of fuzzy numbers of the observed criteria alternatives. consequently, the normalized matrix ( )n is calculated 1 2 1 11 12 1 2 11 22 2 1 2 ... ... ... ... ... ... ... ... n n n m m m mn k k k a t t t a t t t n a t t t             (11) step 4. calculation of the weighted matrix ( v ) elements. elements of this matrix are calculated based on the following expression:   ij i ij i v w t w (12) multicriteria decision making model with z-numbers based on fucom and mabac model 25 in the previous expression ij t represents elements of the normalized matrix( n ), whereas i w presents weight coefficients of the criteria. weighted matrix ( v ) is visualized in the following way 11 12 1 21 22 2 1 2 ... ... ... ... ... ... ... n n m m mn v v v v v v v v v v             (13) step 5. determination of the approximate border area matrix ( g ). the border approximate area for each criteria is determined based on the expression: 1/ 1 m m i ij j g v          (14) the matrix of approximate area ( )g has a format n x 1, where n presents overall sum of criteria number and is represented in the following way   1 2 1 2 ... ... n n k k k g g g g (15) step 6. calculation of the matrix elements of alternatives distance from the border approximate area ( q ). the distance of alternatives from the border approximate area ( ijq ) is defined with the expression:  q v g (16) afterwards the matrix is calculated q 11 12 1 21 22 2 1 2 ... ... ... ... ... ...             n n m m mn q q q q q q q q q q (17) step 7. ranking of alternatives. the value estimation of criteria functions of alternatives is gained from the sum of the distance of alternatives from the border approximate areas ( i q ). the ultimate values of criteria functions of alternatives are gained from the sum of elements of the matrix q in rows: 1 , 1, 2,..., , 1, 2,..., n i ij j s q j n i m     (18) by defuzzification of the values obtained, final rank of the alternatives is obtained. defuzzification can be performed by applying the expressions (seiford, 1996; liou and wang, 1992): 3 1 2 1 1 a ((t t ) (t t )) / 3 t     (19)  3 2 1a t t 1 t / 2       (20) božanić et al./decis. mak. appl. manag. eng. 3 (2) (2020) 19-36 26 3. description of criteria and calculation of weight coefficients the selection of a location for a command post is made on the basis of five criteria, obtained by analyzing available literature. basic criteria for selecting a location of a brigade command post are shown from the most significant (c1) to the least significant (c5), respectively, c1> c2> c3> c4> c5. the criteria on which depends the location of a command post are as follows:  c1 time required for engineering works. this criterion implies the total time required for preparatory, main and final works on the engineering arrangement of a command post. (hristov, 1978). through this criterion, various elements such as the influence of land to the selection of the type of object to be constructed, geological composition of the soil, etc., are indirectly evaluated.  c2 deposits of building materials. various materials are used in the construction of fortification structures, such as: timber, steel and concrete elements and stone. through this criterion, the existence of material deposits in the vicinity of the area of works, the quantities and types of materials, as well as the possibility of its incorporation into facilities in its existing form or after processing are evaluated.  c3 masking conditions. masking conditions include the possibility of concealing preparations for the execution of works, centralized processing of certain elements (timber, reinforced concrete elements, etc.) and direct works on the fortification.  c4 influence of the enemy. this criterion implies the ability of the enemy to detect the preparation and execution of works and the possibility of direct action from the ground and from the air. (šećković, 1972).  c5 possibilities of use of workshops, technical means and tools. in the areas of a potential command posts, it is desirable to have the possibility of using local plants (workshops, quarries, sawmills, etc.), tools (pickaxes, shovels, crowbars, etc.) and technical means suitable for fortification (dozers, loaders, diggers, etc.), in order to economize the forces, resources and time required to perform the works. the set of criteria from c1 to c5 consists of two subsets:  the "c +" is a set of criteria of the benefit type, which means that the higher value of criteria is more favorable (the criteria c2, c3 and c5), and  the"c -" is a set of criteria of the cost type, which means that the lower value of criteria is more favorable (the criteria c1 and c4). the criterion c1 is presented as numerical, while the other criteria are presented as linguistic. the weight coefficients of the criteria are obtained using the fucom method. criteria ranks are calculated based on the data on their mutual comparison, as in the table 1. table 1. importance of criteria criteria c1 c2 c3 c4 c5 importance ( ( )j kc  ) 1 2 3.5 5 6 the values of the calculated weight coefficients are provided in the table 2. multicriteria decision making model with z-numbers based on fucom and mabac model 27 table 2. weight coefficient of criteria criteria c1 c2 c3 c4 c5 wj 0.465 0.232 0.133 0.093 0.077 4. model testing ten alternatives were defined to test the model. prior to the process of selecting the best alternative from the set of offered ones, a scale for evaluating linguistic criteria had been defined, as in the figure 5 1 0.8 0.6 0.4 0.2 1 32 54 0 vs s m l vl figure 5. graphic display of fuzzy linguistic descriptors (božanić et al., 2016b) linguistic criterion can be described with five values: very small (vs), small (s), medium (m), large (l), very large (vl). the initial decision-making matrix is shown in the table 3. table 3. initial decision making matrix alternative index c1 c2 c3 c4 c5 a b a b a b a b a b a1 (3,4,6) m vs vs vs h l h vl s a2 (2,3,4) vs s vh m m vl vs m m a3 (4,5,7) h l vs s vh vs m l h a4 (3,6,7) s m m vs vs m s vl vs a5 (4,8,8) vh vl s s h vl vh vs vh a6 (3,5,6) vs l h vl m l m s m a7 (4,6,7) m vs s l vs vs vs l s a8 (5,8,9) s m h l s vl h m vs a9 (6,6,8) h s vh m vh s s s h a10 (4,6,9) vh vl m vl s m vh vs vh božanić et al./decis. mak. appl. manag. eng. 3 (2) (2020) 19-36 28 in the next step, the quantification of linguistic descriptors is performed, as shown in the table 4. table 4. quantification of linguistic descriptors alternative index, c1 c2 c5 a b a b ... a b a1 (3,4,6) (0.8,1,1) (1,1,2) (0,0,0.2) ... (4,5,5) (0.1,0.25,0.4) a2 (2,3,4) (0,0,0.2) (1,2,3) (0.8,1,1) ... (2,3,4) (0.3,0.5,0.7) a3 (4,5,7) (0.55,0.75,0.95) (3,4,5) (0,0,0.2) ... (3,4,5) (0.55,0.75,0.95) a4 (3,6,7) (0.1,0.25,0.4) (2,3,4) (0.3,0.5,0.7) ... (4,5,5) (0,0,0.2) a5 (4,8,8) (0.8,1,1) (4,5,5) (0.1,0.25,0.4) ... (1,1,2) (0.8,1,1) a6 (3,5,6) (0,0,0.2) (3,4,5) (0.55,0.75,0.95) ... (1,2,3) (0.3,0.5,0.7) a7 (4,6,7) (0.55,0.75,0.95) (1,1,2) (0.1,0.25,0.4) ... (3,4,5) (0.1,0.25,0.4) a8 (5,8,9) (0.1,0.25,0.4) (2,3,4) (0.55,0.75,0.95) ... (2,3,4) (0,0,0.2) a9 (6,6,8) (0.55,0.75,0.95) (1,2,3) (0.8,1,1) ... (1,2,3) (0.55,0.75,0.95) a10 (4,6,9) (0.8,1,1) (4,5,5) (0.3,0.5,0.7) ... (1,1,2) (0.8,1,1) in the second step, z-numbers were converted to regular fuzzy numbers. after converting into a regular fuzzy number, new initial decision-making matrix was obtained, as in the table 5. table 5. new initial decision-making matrix alternative index c1 c2 ... c5 a1 (2.12,2.83,4.24) (0.26.0.26,0.52) ... (2,2.50,2,50) a2 (0.52,0.77,1.03) (0.97,1.93,2.90) ... (1.41,2.12,2.83) a3 (3.46,4.33,6.06) (0.77,1.03,1.29) ... (2.6,3.46,4.33) a4 (1.5,3,3.5) (1.41,2.12,2.83) ... (1.03,1.29,1.29) a5 (3.86,7.73,7.73) (2,2.5,2.5) ... (0.97,0.97,1.93) a6 (0.77,1.29,1.55) (2.6,3.46,4.33) ... (0.71,1.41,2.12) a7 (2.83,4.24,4.95) (0.5,0.5,1) ... (1.5,2,2.5) a8 (2.5,4,4.5) (1.73,2.6,3.46) ... (0.52,0.77,1.03) a9 (5.2,5.2,6.93) (0.97,1.93,2.9) ... (0.87,1.73,2.6) a10 (3.86,5.8,8.69) (2.83,3.54,3.54) ... (0.97,0.97,1.93) in the third step, the normalization of the new initial decision-making matrix was performed, using the expressions 9 and 10 respectively, as in the table 6. table 6. normalized initial decision-making matrix alternative index c1 c2 ... c5 a1 (0.54,0.72,0.8) (0,0,0.06) ... (0.39,0.52,0.52) a2 (0.94,0.97,1) (0.17,0.41,0.65) ... (0.24,0.42,0.61) a3 (0.32,0.53,0.64) (0.13,0.19,0.25) ... (0.55,0.77,1) a4 (0.64,0.7,0.88) (0.28,0.46,0.63) ... (0.14,0.2,0.2) a5 (0.12,0.12,0.59) (0.43,0.55,0.55) ... (0.12,0.12,0.37) a6 (0.87,0.91,0.97) (0.57,0.79,1) ... (0.05,0.24,0.42) multicriteria decision making model with z-numbers based on fucom and mabac model 29 alternative index c1 c2 ... c5 a7 (0.46,0.54,0.72) (0.06,0.06,0.18) ... (0.26,0.39,0.52) a8 (0.51,0.57,0.76) (0.36,0.57,0.79) ... (0,0.7,0.14) a9 (0.22,0.43,0.43) (0.17,0.41,0.65) ... (0.09,0.32,0.55) a10 (0,0.35,0.59) (0.63,0.8,0.8) ... (0.12,0.12,0.37) by applying the expression (12) in the following step is obtained the weighted matrix ( v ), as in the table 7. table 7. weighted matrix alternative index c1 c2 ... c5 a1 (0.72,0.8,0.84) (0.23,0.23,0.25) ... (0.11,0.12,0.12) a2 (0.4,0.92,0.93) (0.27,0.33,0.38) ... (0.1,0.22,0.12) a3 (0.61,0.71,0.76) (0.26,0.28,0.29) ... (0.12,0.14,0.15) a4 (0.76,0.79,0.87) (0.30,0.34,0.38) ... (0.09,0.09,0.09) a5 (0.52,0.52,0.74) (0.33,0.36,0.36) ... (0.09,0.09,0.11) a6 (0.87,0.89,0.92) (0.37,0.41,0.46) ... (0.08,0.1,0.11) a7 (0.68,0.72,0.8) (0.06,0.06,0.18) ... (0.1,0.11,0.12) a8 (0.7,0.73,0.82) (0.25,0.25,0.27) ... (0.08,0.08,0.09) a9 (0.57,0.66,0.66) (0.32,0.37,0.41) ... (0.08,0.1,0.12) a10 (0.47,0.63,0.74) (0.27,0.33,0.38) ... (0.09,0.09,0.11) in the fifth step is obtained the approximate border area matrix ( g ), by applying the expression (14), as in the table 8. table 8. approximate border area matrix alternative index c1 c2 ... c5 a1 (0.67,0.73,0.8) (0.29,0.32,0.35) ... (0.09,0.1,0.11) in the sixth step, using the expression (16), the distance of the alternatives from the border approximate area was obtained, as in the table 9. table 9. matrix of the distance of alternatives from border approximate areas alternative index c1 c2 ... c5 a1 (-0.09,0.07,0.07) (-0.12,-0.09,-0.05) ... (0,0.02,0.03) a2 (0.1,0.19,0.26) (-0.08,0,0.09) ... (-0.02,0.01,0.03) a3 (-0.19,-0.01,0.1) (-0.09,-0.05,0) ... (0.01,0.04,0.06) a4 (-0.04,0.06,0.21) (-0.06,0.01,0.08) ... (-0.02,-0.01,0) a5 (-0.28,-0.21,0.07) (-0.02,0.04,0.07) ... (-0.03,-0.01,0.01) a6 (0.07,0.16,0.25) (0.01,0.09,0.17) ... (-0.03,-0.01,0.02) a7 (-0.13,-0.01,0.13) (-0.11,-0.08,-0.02) ... (-0.02,0.01,0.03) božanić et al./decis. mak. appl. manag. eng. 3 (2) (2020) 19-36 30 alternative index c1 c2 ... c5 a8 (-0.1,0,0.15) (-0.04,0.04,0.12) ... (-0.03,-0.02,0) a9 (-0.24,-0.06,0) (-0.08,0,0.09) ... (-0.03,0,0.03) a10 (-0.34,-0.1,0.07) (0.02,0.09,0.13) ... (-0.03,-0.01,0.01) the final values of the criteria functions with the rank of alternatives are provided in the table 10. table 10. ranking of alternatives alternative index z-number mabac method fuzzy mabac method classic mabac method i s si rank is si rank si rank a1 (-0.28,-0.06,0.14) -0.07 9 (-0.38,-0.02,0.3) -0.03 7 0.04 5 a2 (0.01,0.24,0.44) 0.23 2 (-0.26,0.11,0.45) 0.1 3 0.19 3 a3 (-0.28,0.01,0.22) -0.01 5 (-0.28,0.17,0.49) 0.13 2 0.2 1 a4 (-0.17,0.03,0.27) 0.04 3 (-0.37,-0.01,0.44) 0.02 5 -0.01 7 a5 (-0.41,-0.24,0.14) -0.17 10 (-0.4,-0.12,0.36) -0.05 8 -0.17 9 a6 (0.07,0.3,0.5) 0.29 1 (-0.23,0.17,0.54) 0.16 1 0.19 2 a7 (-0.24,-0.06,0.15) -0.05 8 (-0.33,0,0.38) 0.02 6 0 6 a8 (-0.22,0,0.26) 0.01 4 (-0.5,-0.13,0.34) -0.09 9 -0.18 10 a9 (-0.32,0.01,0.22) -0.03 6 (-0.5,-0.04,0.24) -0.1 10 0.04 8 a10 (-0.36,0,0.24) -0.04 7 (-0.34,0.16,0.47) 0.1 4 0.16 4 in addition to the rank of alternatives obtained by applying z-number mabac model, in the table 10 are also provided the ranks of alternatives obtained by applying classic mabac method and by applying fuzzy mabac method (excluding znumber). comparative ranking of alternatives provides significant differences in ranking. the alternative a6, the second-ranked in the application of classic mabac method, appears as the first-ranked in the rest of the cases. the alternative a3, which is the first-ranked when applied the mabac method, is the second-ranked when fuzzy mabac method is applied, and even the fifth-ranked when z –number mabac model is applied. this difference clearly indicates the need to mathematically examine rank correlation. considering that these are different models, rank differences can be expected, but they should not be significantly different. in this sense, rank correlation control is performed using the spearman’s coefficient n 2 i i 1 2 6 d s 1 n(n 1)     (21) where is:  s the value of the spearman’s coefficient,  di the difference in the rank of the given element in the vector w and the rank of the correspondent element in the reference vector,  n number of ranked elements. multicriteria decision making model with z-numbers based on fucom and mabac model 31 spearman's coefficient takes values from the interval -1,1. when the ranks of the elements completely coincide, the spearman’s coefficient is 1 ("ideal positive correlation"). when the ranks are completely opposite, the spearman’s coefficient is 1 ("ideal negative correlation"), that is, when s = 0 the ranks are unregulated. the rank correlation of alternatives using spearman’s coefficient is provided in the table 11. table 11. spearman’s coefficient values using different models z-number mabac method fuzzy mabac method classic mabac method fuzzy z number mabac method 1 0.923 0.895 fuzzy mabac method 1 0.984 classic mabac method 1 table 11 shows that the rank correlation is extremely high, suggesting that new model is performing well, considering two new uncertainties which are not considered by classic mabac method (uncertainty about the evaluation of alternatives by criteria, as well as the degree of certainty in assigned values of alternatives by criteria). 5. sensitivity analysis logically, the last step in model evaluation is sensitivity analysis. sensitivity analysis is performed by applying different scenarios changing the weight coefficients of criteria, where different criterion was favored in each scenario. (pamučar et. al. 2017). the display of weight coefficients according to the scenarios is given in the table 12. table 12. weight coefficient in different scenario criterion s-0 s-1 s-2 s-3 s-4 s-5 c1 0.465 0.4 0.15 0.15 0.15 0.15 c2 0.232 0.15 0.4 0.15 0.15 0.15 c3 0.133 0.15 0.15 0.4 0.15 0.15 c4 0.093 0.15 0.15 0.15 0.4 0.15 c5 0.077 0.15 0.15 0.15 0.15 0.4 in the table 13 is provided the rank of alternatives using different scenarios. božanić et al./decis. mak. appl. manag. eng. 3 (2) (2020) 19-36 32 table 13. ranking of alternatives by applying different scenarios alternative index s-1 s-2 s-3 s-4 s-5 si rank si rank si rank si rank si rank a1 -0.05 8 -0.17 10 -0.12 10 -0.13 8 -0.04 7 a2 0.23 2 0.13 2 0.15 2 0.20 1 0.16 2 a3 0.06 3 0.03 5 0.09 4 0.16 2 0.19 1 a4 0.02 4 0.00 6 -0.12 8 0.03 6 -0.05 8 a5 -0.19 10 -0.09 9 -0.12 9 -0.23 10 -0.14 10 a6 0.23 1 0.24 1 0.25 1 0.12 4 0.13 3 a7 0.00 6 -0.07 8 -0.05 7 0.12 5 0.02 5 a8 -0.04 7 -0.01 7 -0.04 6 -0.14 9 -0.11 9 a9 0.01 5 0.07 4 0.14 3 0.14 3 0.06 4 a10 -0.06 9 0.09 3 0.04 5 -0.02 7 -0.02 6 table 13 shows different ranks of alternatives for different scenarios, indicating that the model produced is sensitive to changes in the criteria weights. regardless of the different ranks, it is noted that the alternatives a6 and a2 are at the top in all scenarios, while the alternatives a1 and a5 are ranked as the worst in most scenarios. the next step in sensitivity analysis is the application of the spearman’s coefficient, to establish and analyze rank correlations when applying different scenarios, as in the table 14. table 14. spearman’s coefficient values obtained using different sensitivity analysis scenarios s-0 s-1 s-2 s-3 s-4 s-5 s-0 1 0.976 0.960 0.954 0.927 0.911 s-1 1 0.945 0.956 0.972 0.960 s-2 1 0.988 0.945 0.943 s-3 1 0.960 0.962 s-4 1 0.990 s-5 1 as observed from the table 14, the values of the spearman’s coefficient are extremely high and very close to the ideal positive correlation. this indicates a stable and sensitive enough model 6. conclusions by the fucom – z-number – mabac model presented have successfully been evaluated the locations for a command post selection in military combat operations. with a parallel presentation of the application of classic mabac method and its modifications by the use of fuzzy numbers, respectively, z-number, it can be noted that the modification of the mabac method using z-number provides broader range of possibilities for considering uncertainty. it is difficult to cover large number of uncertainties following combat operations through conventional multi-criteria decision-making methods, which is why it is important to include at least a part of those uncertainties in the decision-making process. fuzzy mabac model includes multicriteria decision making model with z-numbers based on fucom and mabac model 33 some uncertainties, while z-number – mabac model increases the number of uncertainties treated. this indicates that the application of z-number is extremely useful in the processes in which is not possible to predict all the elements that influence decision making, because, contrary to uncertainty, it can include other factors that are not fully measurable but can influence the final outcome. the introduction of a model for the selection of a command post location in combat operations significantly advances this process: it helps decision makers understand the factors that influence the selection more comprehensively, and provides less experienced decision makers with the support in decision making based on their predecessor's experience. considering that there is a number of decisions made during combat operations and followed by a high degree of uncertainty, undoubtedly, the model presented can significantly facilitate making decision on selection of a command post. author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research received no external funding. conflicts of interest: the authors declare no conflicts of interest. references aboutorab, h., saberi, m., asadabadi, m.r., hussain, o. & chang, e. 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(2011). a note on z-number. information sciences, 181, 2923-2932. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 3, issue 1, 2020, pp. 146-164. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame2003134p * corresponding author. e-mail addresses: ivanpetrovic1977@gmail.com (i. petrovic), kankaras.milan@outlook.com (m. kankaras) a hybridized it2fs-dematel-ahp-topsis multicriteria decision making approach: case study of selection and evaluation of criteria for determination of air traffic control radar position ivan petrovic 1* and milan kankaras 1 1 university of defence, military academy, belgrade, serbia received: 5 january 2020; accepted: 11 march 2020; available online: 14 march 2020. original scientific paper abstract: in this paper the criteria for selection of air traffic control (atc) radar position that provide successfully fulfilled role of radar in air traffic management are determined and evaluated. using the questionnaire, experts determined the initial criteria for selecting the radar position. furthermore, the hybridized dematel-ahp-topsis model was modified by using the interval type-2 fuzzy sets (it2fs). less important criteria were eliminated by using the it2fs-dematel method, the prioritization of the final criteria was carried out by using the it2fs-ahp method and a multi-criteria decision making model was proposed. of the four atc radar positions offered, the optimal position was selected by using the it2fs-topsis method. validation of model was carried out by using fuzzy and the it2fs modified methods: topsis, copras and mabac. a sensitivity analysis was carried out through 36 scenarios of changes in the criteria’s weights. key words: ahp, air traffic control radar position, dematel, interval type2 fuzzy sets, topsis. 1. introduction the complexity of air traffic arises from the fact that it takes place in the third dimension of space (an air). furthermore, air traffic’s intensity and internationality, complexity of airspace routes and corridors, an organization’s complexity and various types of aircrafts with visual flight’s rules or instrumental flight’s rules have a significant impact on air traffic flow management (carey, 2019; fleischer, 2019). all over the world, the crucial role in air traffic flow management has an air traffic control, whose functioning is impossible without logistical support in the form of atc radars. mailto:ivanpetrovic1977@gmail.com mailto:kankaras.milan@outlook.com a hybridized it2fs-dematel-ahp-topsis multi-criteria decision making approach… 147 in addition to the numerous advantages of this type of traffic, unfortunately, modern age also brings some new security risks for the air traffic (for example 9/11), which has become increasingly vulnerable to asymmetric threats. some of the potential forms of air traffic violations are the hijacking of aircraft or terrorist attack from the airspace or from the ground (petrović et al, 2015). in addition, the air traffic control has a very important role in preventing aircraft’s accidents related with human mistakes or technical defects in the aircraft. the modern age is characterized by the possibility of using micro unmanned aerial vehicles (drones) whose purpose is to endanger air traffic (bergen and tiedemann, 2010). the technological development of modern multifunctional primary-secondary radars based on active phased array antenna system, as well as the active electronically scanned antenna system, provide high frequency radar agility and quick scanning of the airspace. furthermore, these types of radars have a channel for weather forecast and modern modes of the moving target detector system and the sweeping of ground and airspace clutter due to bad meteorological conditions (zhao and yue, 2014). this is the consequence of modern technological solutions (on the radar and on the telecommunication system) based on which radar is collected and sent data, technical staff’s abilities, and the selection of atc radar position on the terrain. the maximum utilization of all technical performance of the radar system, as well as the minimization of the possibility of attacking asymmetric threats from the airspace, should be ensured by the selection of the optimal radar position. the selection of radar position is especially significant from the aspect of reducing the possibility of using unmanned aerial vehicles for endangering the safety of air traffic, because they mostly fly at low altitudes and have a small radar cross section. determination of the radar position implies researching all possible criteria that have an impact on the work of individual radars, as well as entire radar network. due to the lack of adequate literature, the experts determined the initial criteria for the selection of the atc radar position. the initial criteria were as follows: к1 the quality of providing of continuous radar coverage in accordance with the requirements of air traffic flow management; к2 the quality of providing the detection of small radar cross section aircraft at the maximum range of observation; к3 reflection coefficient of the terrain of the radar position; к4 terrain configuration (the existence of natural obstacles that reduce the range of radar observation); к5 the influence of forests on the interference of electromagnetic wave signals; к6 the influence of meteorological conditions on the formation of radar beam; к7 the accessibility of the radar position from the aspect of realization of logistics functions (supplying spare parts and maintenance of radar system equipment, ensuring optimal conditions for the work of technical personnel, providing a continuous and secure communication system between correspondents on all modes); к8 the position in relation to airspace routes and prohibited, restricted and dangerous area. taking into consideration the lack of adequate literature, as well as the fact that the small number of experts were participated in the research, for the purpose of multi criteria decision making, it2fs are applied (zhang, 2018). this type of fuzzy sets provides valid results in conditions of significant uncertainty, which represented the basic feature of this research (deveci et al, 2018). traditional methods of multi-criteria decision making already had significant application in the realization of the research. the dematel (decision-making trial and evaluation laboratory) method is often muravev and mijic./decis. mak. appl. manag. eng. 3 (1) (2020) 146-164 148 applied to prioritize criteria (stević et al, 2017; kaya and yet, 2019; wang, 2019). in this research the dematel was used to eliminate less important criteria (petrović and kankaraš, 2018). the prioritization of the final criteria was carried out using the ahp (analytic hierarchy process) method as one of the most appropriate method of subjective determination of the criteria’s weights (kahraman et al, 2014; singh and prasher, 2019). the application of this method ensured a significant validity of the research's results, including the application of the topsis (technique for order preference by similarity to ideal solution) method that was used to test the proposed model (chen, 2019). in addition, it should be noted that during the literature analysis, no papers were found which had the methodological approach applied in this paper (the it2fs-dematel-ahp-topsis approach). thus, validation of the hybridized model and results was carried out using other multi-criteria decision making methods modified with fuzzy (topsis, mabac and copras) and it2f sets (mabac and copras). 2. methods the research was carried out in accordance with the algorithm shown in figure 1. hybridized it2fs dematel-ahp-topsis model was carried out in three phases: in the first phase, the less-important criteria in relation to other criteria are eliminated using it2fs-dematel; in the second phase, the prioritization of the final criteria was carried out by the it2fs-ahp method; in the third phase, the optimal alternative of the four offered alternatives was selected using the it2fs-topsis method. figure 1. algorithm of a multi-criteria selection of the atc radar position 2.1. background of interval type-2 fuzzy sets the application of it2fs, as a special type of type-2 fuzzy sets (t2fs) (milošević et al, 2019; haghighi et al, 2019) was caused by the lack of valid research in these a hybridized it2fs-dematel-ahp-topsis multi-criteria decision making approach… 149 fields. unlike t2fs, which represent an extended type of t1fs, the it2fs are easier for calculation and they ensure the validity of the results in the conditions of a high level of uncertainty of the subjective opinion of the experts (liang et al, 2019). the t2fs а ~ in the universe of discourse x can be presented by the following membership functions:          1,0,1,0,,,,~ ~~  uxjuxxuxuxа axa  (1) or,    uxuxа ajuxx x ,, ~ ~ ,   (2)  1,0 x j and  presents union of all x and u. type t2fs а ~ for which it is for   1, ~ ~  uxа a  presents it2fs, which based on functions 1 and 2 can be represented as (figure 2):        1,,1,0,1,,~ ~  uxjuxxuxа ax  (3) alternatively:  uxа xjuxx ,1 ~ ,  ,  1,0xj (4) bearing in mind that are upper and lower membership functions of the it2fs are type-1 membership functions, trapezoidal it2fs iа ~ can be presented in the next form (kahraman, et al, 2014):            l i l i l i l i l i l i u i u i u i u i u i u i l i u ii ahahaaaaahahaaaaaaа 214321214321 ,;,,,,,;,,,, ~  (5)  u ij ah is the membership value of the element   u ji a 1 in the upper trapezoidal function of membership 21,  ja u i . and the same is for  lij ah in the lower trapezoidal function of membership 21,  ja l i ,                1,0,1,0,1,0,1,0 2121  l i l i u i u i ahahahah , ni 1 . figure. 2. the form of trapezoidal interval type-2 fuzzy sets if are given two it2fs:            lllllluuuuuulu ahahaaaaahahaaaaaaа 121114131211121114131211111 ,;,,,,,;,,,, ~             lllllluuuuuulu ahahaaaaahahaaaaaaа 222124232221222124232221222 ,;,,,,,;,,,, ~  muravev and mijic./decis. mak. appl. manag. eng. 3 (1) (2020) 146-164 150 their addition operations are defined as follows:                      llllllllllll uuuuuuuuuuuu ahahahahaaaaaaaa ahahahahaaaaaaaaaа 221221112414231322122111 22122111241423132212211121 ,min,,min;,,, ,,min,,min;,,, ~~   (6) their subtraction operations are defined as follows:                      llllllllllll uuuuuuuuuuuu ahahahahaaaaaaaa ahahahahaaaaaaaaaа 221221112114221323122411 22122111211422132312241121 ,min,,min;,,, ,,min,,min;,,, ~~   (7) their multiplication operations are defined as follows:                      llllllllllll uuuuuuuuuuuu ahahahahaaaaaaaa ahahahahaaaaaaaaaа 221221112414231322122111 22122111241423132212211121 ,min,,min;,,, ,,min,,min;,,, ~~   (8) in accordance with calculation rules with fuzzy sets, the division operations between two trapezoidal it2fs are follows (kahraman et al, 2014):                     ,,min,,min;,,, ,,min,,min;,,,~ ~ 22122111 21 14 22 13 23 12 24 11 22122111 21 14 22 13 23 12 24 11 2 1              llll l l l l l l l l uuuu u u u u u u u u ahahahah a a a a a a a a ahahahah a a a a a a a a a a (9) the multiplication and division operations between the trapezoidal it2fs and scalar k are defined as follows:            l i l i l i l i l i l i u i u i u i u i u i u i l i u ii ahahakakakak ahahakakakakaakаk 214321 2114321 ,;,,, ,,;,,,, ~   (10)                     l i l i l i l i l i l iu i u i u i u i u i u ii ahah k a k a k a k a ahah k a k a k a k a k a 21 4321 211 4321 ,;,,,,,;,,, ~ (11) the reciprocal of the trapezoidal it2fs are defined as:                     l i l il i l i l i l i u i u iu i u i u i u ii ahah aaaa ahah aaaaa 21 1234 211 1234 ,; 1 , 1 , 1 , 1 ,,; 1 , 1 , 1 , 1 ~ 1 (12) for any trapezoidal it2fs iа ~ , m ia ~ is defined as:           l i l i m l i m l i m l i m l i u i u i m u i m u i m u i m u i m i ahahaaaaahahaaaaa 2143212114321 ,;,,,,,;,,, ~  (13) the ranking value of the trapezoidal it2fs iа ~ is calculated as follows (baykasoğlu et al, 2017):                                       l i u i l i u i l i u i l i u i l i u i l i u i l i u i l i u i l i u ii ahahahah asasasasasasasas amamamamamamаrank 2211 44332211 3321211 4 1 ~    (14)      31, 2 1     p aa am j pi j ipj ip (15)  j iq as the standard deviation of the elements j iq a and   j qi a 1 :   31, 1 2 1 2 1 2 1                qas q qk q qk j ik j ik aa j iq (16)  j i as 4 the standard deviation of the elements j ip a , 41  q : a hybridized it2fs-dematel-ahp-topsis multi-criteria decision making approach… 151              4 1 2 4 1 4 4 1 4 1 k k j ik j ik j i aaas (17) according to kahraman et al (2014), defuzzification of the trapezoidal it2fs iа ~ is calculated as follows:                              l i l i l i l i l i l i l i l i l i u i u i u i u i u i u i u i u i u i i aaaahaaahaa aaaahaaahaa аdtrat 113212114 113212114 4/ ~~ 4/ ~~ 2 1~ (18) these equations are necessary for calculating the dematel, the ahp and the topsis procedures with the trapezoidal it2fs. 2.2. background of interval type-2 fuzzy sets-ahp method the determination of the final criteria’s weights was carried out by the it2fs-ahp method (celik and akyuz, 2018). the initial values were gathered by experts in the form of linguistic it2fs and suited by questionnaire of satty. the average matrix of pairwise comparisons was obtained using the equations (6), (11) and (12)               1 ~ ~ 1 ~ 1 ~ ~~ ij ij nnij a a aa            l ij l ij l ij l ij l ij l ij u ij u ij u ij u ij u ij u ij l ij u ijij ahahaaaaahahaaaaaaа 214321214321 ,;,,,,,;,,,, ~           l ij l ij u ij u ij ahahahah 2121 ,;1,1,1,1,,;1,1,1,11 ~  using equation (18) defuzzification of the it2fs elements of the average matrix of pairwise comparisons were carried out and the crisp values for determination of the consistency ratio were calculated. the consistency ratio was calculated as follows: ri ci cr  (23) 1 max    n n ci  (24)    n i i n 1 max 1  (25) i i i w b  (26)                                nnnnn n n n w w w aaa aaa aaa b b b 2 1 21 22221 11211 2 1 (27) ri random index, which depends on the number of rows – columns. if 10.0cr then the result is consistent. the prioritization of criteria was carried out by geometric mean method applied to the it2fs. if         ij l ij l ij u ij u ij aahahahah ~ , 2121  then the it2fs values of the criteria weights i w~ were derived from the following equations: muravev and mijic./decis. mak. appl. manag. eng. 3 (1) (2020) 146-164 152                                                                                                                                                                                                      n i l ij l ij nn j l ij nn j l ij nn j l ij nn j l ij u ij u ij nn j u ij nn j u ij nn j u ij nn j u ij l ij l ij nn j l ij nn j l ij nn j l ij nn j l ij u ij u ij nn j u ij nn j u ij nn j u ij nn j u ij i ahahaaaa ahahaaaa ahahaaaa ahahaaaa w 1 21 1 1 4 1 1 3 1 1 2 1 1 1 21 1 1 4 1 1 3 1 1 2 1 1 1 21 1 1 4 1 1 3 1 1 2 1 1 1 21 1 1 4 1 1 3 1 1 2 1 1 1 ,;,,, ,,;,,, ,;,,, ,,;,,, ~ (28) according to equation (9):                                                                                                                                                                                                                                                        l ij l ij n i nn j l ij nn j l ij n i nn j l ij nn j l ij n i nn j l ij nn j l ij n i nn j l ij nn j l ij u ij u ij n i nn j u ij nn j u ij n i nn j u ij nn j u ij n i nn j u ij nn j u ij n i nn j u ij nn j u ij i ahah a a a a a a a a ahah a a a a a a a a w 21 1 1 1 1 1 1 4 1 1 1 2 1 1 3 1 1 1 3 1 1 2 1 1 1 4 1 1 1 21 1 1 1 1 1 1 4 1 1 1 2 1 1 3 1 1 1 3 1 1 2 1 1 1 4 1 1 1 ,;,,, ,,;,,, ~ (29) using equation (18), the values of niwdtra i ,...,1, ~  were calculated, whose aggregations were obtained the weights i w . 2.3. background of interval type-2 fuzzy sets-topsis method after the criteria's weights were determined, the optimal radar position was selected by the it2fs-topsis method (deveci, 2018). this method is based on the ranking of alternatives in relation to the ideal and negative ideal solution. in the first step, individual k it2fs decision matrices was formed from data gathered by six experts ( 6k ). the average it2fs decision matrix was derived from the individual it2fs decision matrices using equation (6) and (11):   mjniff ij  1,1, ~~ (30) n number of criteria, m number of alternatives. the normalized it2fs decision matrix was calculated as follows:   mjni f f rr m j ij ij ij                   1,1, ~ ~ ~~ 1 2 , or: a hybridized it2fs-dematel-ahp-topsis multi-criteria decision making approach… 153 if         ij l ij l ij u ij u ij fafhfhfhfh ~ , 2121  then according to equation (8), (9) and (13) is:                         , ,;,,, ,,;,,, ~ 11 1 2 1 4 1 2 2 3 1 2 3 2 1 2 4 1 11 1 2 1 4 1 2 2 3 1 2 3 2 1 2 4 1                                            l ij l ij m j l ij l ij m j l ij l ij m j l ij l ij m j l ij l ij u ij u ij m j u ij u ij m j u ij u ij m j u ij u ij m j u ij u ij ij fhfh f f f f f f f f fhfh f f f f f f f f r (31) in the next step, the weighted it2fs decision matrix was constructed using equation (32): iijij wrv ~~  (32) ranking values iijij wrv ~~  were calculated by equations (14)-(17), and a new matrix was obtained:   ij vrankv ~  in the next step, the positive and negative ideal solutions are respectively calculated using equation (33) and (34):         nijijj vrankvrankvrankgivrankgivrankv ~ ,.. ~ , ~ , ~ min,, ~ max 21 (33)         nijijj vrankvrankvrankgivrankgivrankv ~ ,.. ~ , ~ , ~ max,, ~ min 21 (34)  g benefit criteria (criteria that are maximized);  g cost criteria (criteria that are minimized). in the next step, the distance between each alternative, positive, and negative ideal solution was calculated as follows (hwang and yoon, 2012):      n i iijj vrankvranks 1 2~~ , mj 1 (35)      n i iijj vrankvranks 1 2~~ mj 1 (36) in the next step, for each alternative value of the relative degree of closeness to ideal solutions was calculated as follows (hwang and yoon, 2012):      jj j j ss s q * , 10 *  j q (37) finally, the alternatives are ranked. the optimal alternative is the one that has the largest value of * j q (hwang and yoon, 2012). muravev and mijic./decis. mak. appl. manag. eng. 3 (1) (2020) 146-164 154 3. results the elimination of less important initial criteria was carried out using the it2fsdematel method. at first, six experts carried out the pairwise comparisons of influence between initial criteria. the influence that one criterion can have on other criteria, as well as the influence that same criterion can receive from other criteria is the following: no influence (n), low influence (l), medium influence (m), high influence (h) and very high influence (vh). the linguistic variables of influence expressed by the trapezoidal it2fs values are shown in table 1. table 1. dematel linguistic variables for causal influence among criteria linguistic variable of influence trapezoidal it2fs no (n) ((0,0,0,0;1,1),(0,0,0,0;0.8,0.8)) low (l) ((0,0.2,0.2,0.4;1,1),(0,0.1,0.1,0.3;0.8,0.8)) medium (m) ((0.2,0.4,0.4,0.6;1,1),(0.1,0.3,0.3,0.5;0.8,0.8)) high (h) ((0.4,0.6,0.6,0.8;1,1),(0.3,0.5,0.5,0.7;0.8,0.8)) very high (vh) ((0.6,0.8,0.8,1;1,1),(0.5,0.7,0.7,0.9;0.8,0.8)) after the average it2fs matrix of the influence between initial criteria and the normalized direct-relation matrix was calculated, the total relation matrix т ~ was obtained using the formulas 19, 20 and 21. this matrix was defuzzificated by formula 18 (table 2). table 2. defuzzificated the total relation matrix tdtra ~ k k1 k2 k3 k4 k5 k6 k7 k8 k1 0.081 0.180 0.190 0.178 0.179 0.171 0.174 0.175 k2 0.140 0.077 0.177 0.156 0.169 0.137 0.140 0.165 k3 0.056 0.066 0.036 0.062 0.055 0.051 0.056 0.061 k4 0.090 0.097 0.124 0.048 0.094 0.066 0.091 0.096 k5 0.066 0.069 0.069 0.069 0.035 0.062 0.059 0.060 k6 0.115 0.120 0.110 0.084 0.073 0.046 0.070 0.083 k7 0.080 0.084 0.125 0.106 0.105 0.114 0.045 0.080 k8 0.121 0.125 0.085 0.102 0.103 0.086 0.123 0.055 the threshold value 099.0 was obtained using equation (22). by subtracting the threshold value from the value of the elements of tdtra ~ was obtained the matrix determines the significance of the criteria. table 3. comparison of the elements of the defuzzificated total relation matrix with the threshold values k k1 k2 k3 k4 k5 k6 k7 k8 k1 -0.018 0.081 0.091 0.079 0.080 0.072 0.075 0.076 k2 0.041 -0.022 0.078 0.057 0.070 0.038 0.041 0.066 k3* -0.043 -0.033 -0.063 -0.037 -0.044 -0.048 -0.043 -0.038 k4 -0.009 -0.002 0.025 -0.051 -0.005 -0.033 -0.008 -0.003 a hybridized it2fs-dematel-ahp-topsis multi-criteria decision making approach… 155 k k1 k2 k3 k4 k5 k6 k7 k8 k5* -0.033 -0.030 -0.030 -0.030 -0.064 -0.037 -0.040 -0.039 k6 0.016 0.021 0.011 -0.015 -0.026 -0.053 -0.029 -0.016 k7 -0.019 -0.015 0.026 0.007 0.006 0.015 -0.054 -0.019 k8 0.022 0.026 -0.014 0.003 0.004 -0.013 0.024 -0.044 * non-significant criteria based on table 3 it can be noted that all values of the criteria k3 and k5 of tdtra ~ are lower than the threshold value. these two criteria were eliminated. the final criteria are the following: с1 the quality of providing of continuous radar coverage in accordance with the requirements of air traffic flow management; с2 the quality of providing the detection of small radar cross section aircraft at the maximum range of observation; с3 terrain configuration (the existence of natural obstacles that reduce the range of radar observation); с4 the influence of meteorological conditions on the formation of radar beam; с5 the accessibility of the radar position from the aspect of realization of logistics functions (supplying spare parts and maintenance of radar system equipment, ensuring optimal conditions for the work of technical personnel, providing a continuous and secure communication system between correspondents on all modes); с6 the position in relation to airspace routes and prohibited, restricted and dangerous area. the linguistic variables and their it2fs values applied for the pairwise comparisons of criteria in accordance with the procedures of the ahp method are shown in table 4. table 4. ahp linguistic variables of criteria pairwise comparison linguistic variables trapezoidal it2fs absolutely strong (as) ((7,8,9,9;1,1),(6,7,8,8;0.8,0.8)) very strong (vs) ((5,6,7,8;1,1),(4,5,5,6;0.8,0.8)) fairly strong (fs) ((3,4,4,5;1,1),(2,3,3,4;0.8,0.8)) slightly strong (ss) ((1,2,3,3;1,1),(1,1,2,2;0.8,0.8)) equal (e) ((1,1,1,1;1,1),(1,1,1,1;1,1)) slightly weak (sw) ((0.333,0.333,0.5,1;1,1),(0.5,0.5,1,1;0.8,0.8)) fairly weak (fw) ((0.2,0.25,0.25,0.333;1,1),(0.25,0.333,0.333,0.5;0.8,0.8)) very weak (vw) ((0.125,0.143,0.167,0.2;1,1),(0.167,0.2,0.2,0.25;0.8,0.8)) absolutely weak (aw) ((0.111,0.111,0.125,0.143;1,1),(0.125,0.125,0.143,0.167; 0.8,0.8)) after the average it2fs pairwise comparisons matrix was constructed, the it2fs values of the criteria weights were obtained using formula 8, 9, 11, 12, 13, 28 and 29. defuzzification of the it2fs was carried out by formula 18. by the aggregation of i wdtra ~ , the final criteria’s weights were obtained i w ~ ,  6,...,1i (table 5). muravev and mijic./decis. mak. appl. manag. eng. 3 (1) (2020) 146-164 156 table 5. criteria’s weights iw ~ c trapezoidal it2fs i wdtra ~ iw ~ c1 (0.325,0.44,0.514,0.69;1,1),(0.292,0.393,0.481,0.64;0.8,0.8) 0.45 0.452 c2 (0.17,0.235,0.278,0.382;1,1),(0.166,0.234,0.281,0.389;0.8,0.8) 0.254 0.255 c3 (0.021,0.027,0.032,0.046;1,1),(0.028,0.035,0.046,0.06;0.8,0.8) 0.035 0.035 c4 (0.05,0.071,0.091,0.128;1,1),(0.059,0.077,0.105,0.14;0.8,0.8) 0.086 0.086 c5 (0.029,0.037,0.051,0.077;1,1),(0.04,0.048,0.07,0.091;0.8,0.8) 0.053 0.053 c6 (0.07,0.102,0.131,0.178;1,1),(0.078,0.1,0.145,0.189;0.8,0.8) 0.118 0.119 the consistency ratio is obtained as follows: 1) defuzzification of the average it2fs pairwise comparisons matrix was carried out by formula 18 and иј аdtra ~ values were obtained. 2) using the formula 23-27 on the elements of иј аdtra ~ matrix and i w (the values of the initial weights determined by the elements of иј аdtra ~ ), the value of 25.16,011.0 1  rincr was obtained. 3) using the formula 23-27 on the elements of иј аdtra ~ matrix and i w ~ the value of 25.16,019.0 1  rincr was obtained. 4) bearing in mind that both values of the consistency ratio are less than 0.1, the ahp method was valid for determining the criteria’s weights. based on the obtained results, the diagram of the criteria’s weights was shown in figure 4. figure. 4. diagram of the criteria weights proposed model of criteria’s weights was tested by the it2fs-topsis method. using the it2fs-topsis method, the optimal atc radar position was selected based on the criteria’s weights and the it2fs value of the linguistic variables of alternatives for each criterion, shown in table 6. 0.000 0.100 0.200 0.300 0.400 0.500 c1 c2 c3 c4 c5 c6 a hybridized it2fs-dematel-ahp-topsis multi-criteria decision making approach… 157 table 6. linguistic variables for ranking of alternatives by topsis linguistic variables for ranking of alternatives trapezoidal it2fs very poor (vp) ((0,0.1,0.1,0.2;1,1),(0,0,0.1,0.1;0.8,0.8)) poor (p) ((0.1,0.2,0.2,0.4;1,1),(0,0.1,0.1,0.2;0.8,0.8)) medium (m) ((0.2,0.4,0.4,0.6;1,1),(0.1,0.3,0.3,0.5;0.8,0.8)) good (g) ((0.3,0.5,0.6,0.8;1,1),(0.2,0.4,0.4,0.6;0.8,0.8)) very good (vg) ((0.4,0.6,0.8,1;1,1),(0.3,0.5,0.6,0.8;1,1)) after the average it2fs decision matrix was constructed (formula 30), the normalized it2fs decision matrix and the weighted it2fs decision matrix was calculated by formula 31 and formula 32. based on formula 14-17, ranking values of the weighted it2fs decision matrix were calculated (table 7). table 7. ranking values of the weighted it2fs decision matrix rank a1 a2 a3 a4 c1 4,937 4,717 4,498 3,817 c2 4,006 4,006 4,006 4,006 c3 3,411 3,363 3,363 3,378 c4 3,652 3,557 3,384 3,384 c5 3,523 3,412 3,376 3,412 c6 3,717 3,602 3,423 3,660 where: c1, c2, c5 and c6 are benefit criteria  g ; c3 and c4 are cost criteria  g . according to formula 33 and 34 were respectively calculated positive and negative ideal solutions.  717.3,523.3,384.3,363.3,006.4,937.4 j v ,  423.3,376.3,652.3,411.3,006.4,817.3 j v according to 35 and 36 the distance between each alternative, positive, and negative ideal solution were calculated using formula 35 and 36. table 8. ranks of alternatives alternatives a1 a2 a3 a4 s+ 0.272 0.322 0.548 1.127 s1.167 0.925 0.733 0.361 q* 0.811 0.742 0.572 0.242 rank 1 2 3 4 the ranks of the alternatives (table 8), which depend on values of the relative degree of closeness q*, were calculated using equation (37) (the higher value is the value of the optimal alternative). muravev and mijic./decis. mak. appl. manag. eng. 3 (1) (2020) 146-164 158 4. discussion by literature analysis, it is not possible to determine the criteria for the selection of atc radar positions, which required the engagement of experts for the formation of the initial criteria. after determination the initial criteria, based on the obtained results by the it2fs-dematel method, the criterion k3 (reflection coefficient of the terrain of the radar position) and criterion k5 (the influence of forests on the interference of electromagnetic wave signals) were eliminated. based on the obtained results it can be concluded that both criteria were already included in other criteria by experts’ opinion. namely, if reflection coefficient of the terrain of the radar position is low and if the influence of forests on the interference of electromagnetic wave signals is high, it is impossible to provide a quality assurance of continuous radar beam at all flight levels and the high probability of detection of the aircraft of small radar cross section. in the second phase of the research, using the it2fs-ahp method, the final criteria were evaluated. based on table 5 and figure 3, it was concluded that the significance of criterion c1 is the highest (the highest weight’s value). furthermore, according to table 5 and figure 3, it can be noted that the criteria c2 and c6 have significantly higher weights than the criteria c3, c4 and c5. similarly, it can be concluded from the values obtained in tables 2 and 3 (the dematel method is often applied to prioritize the criteria (stević et al, 2017). bearing in mind that the basic purpose of atc radars is to ensure the smooth functioning of air traffic, the significance of criterion c1 (the quality of providing of continuous radar coverage in accordance with the requirements of air traffic flow management) could not be specifically explained. namely, the reliability of the operation of the area control centre, approach control unit and tower control unit on all air routes and corridors depends on the quality of the radar beam's continuity. in the conditions of existence of asymmetric threats, as well as increasingly frequent use of micro unmanned aerial vehicles for various purposes, there is no doubt the possibility of detecting aircraft of low radar cross section is extremely significant. despite being used for useful purposes (detecting and monitoring major fires or nuclear-chemical accidents, monitoring the situation on the terrain after industrial or other accidents, etc.), the unmanned aerial vehicles can often be used to perform spy or terrorist activities, as well as other forms of airspace violation (islam et al, 2018; card, 2018). therefore, their quick detection is extremely significant for the safety of the functioning of air traffic. considering aforementioned the criterion c2 is very significant. in the case that the radar position has a great possibility of detecting low cross section aircraft, especially at low altitudes, the safety of air traffic at lower flight levels, as well as below transition level and transition altitude is very appropriate. the significance of the criterion c6 is a consequence of the fact that the radar position must maximize the number of the air routes and the flight levels covered by the radar. this criterion is also significant because of fast detection the airspace violations if the aircraft is in prohibited, restricted or dangerous areas. based on the obtained results of the weights, the other criteria are less significant. weight of the criterion c4 is higher than for the criterion c3 (according to experts, and this criterion is the integral part of other criteria) and for the criterion c5. namely, despite the fact that radar technology is developing exponentially, low ceilings and weather disturbances, such as heavy rain or snow, storms, strong winds, large hail, can still affect the atc radar coverage. furthermore, the weather precipitation and low cloudiness have a major influence on the interference of the electromagnetic waves, causing significant clutters reducing radar visibility. the criterion c5 is less significant because of exponential a hybridized it2fs-dematel-ahp-topsis multi-criteria decision making approach… 159 development of radar's maintenance technology, ensuring optimal conditions for the work of technical personnel and continuous and safe communication system between correspondents. aforementioned is something what is relatively easy to regulate even in extreme conditions. based on the test results obtained by the it2fs-topsis method, the optimal atc radar position ensures: the high quality of providing of continuous radar coverage in accordance with the requirements of air traffic flow management; the detection of small radar cross section aircraft at the maximum range of observation; the very good position of covering of airspace routes and prohibited, restricted and dangerous area. validation of the hybridized multi-criteria decision making approach was carried out using: topsis, copras and mabac methods, because of the reliability of the results obtained using these methods (pamučar et al, 2018a; pamučar et al, 2018b; garg, 2019). these methods were modified using fuzzy (trapezoidal fuzzy sets) and it2fs (except topsis). at the same time, the validation of the method was used to evaluate the reliability of the obtained results of the hybridized model (ghorabaee et al, 2015). the ranks of alternatives according to modified topsis, copras and mabac methods are shown in table 9. table 9. comparison of the ranks of alternatives according to modified methods alt. it2fstopsis fuzzytopsis it2fscopras fuzzycopras it2fsmabac fuzzymabac a1 1 1 1 1 1 2 a2 2 2 2 2 2 1 a3 3 3 3 3 3 3 a4 4 4 4 4 4 4 based on the results in the table, it can be noted that the rank of alternatives was changed only for fuzzy-mabac method. using this method, alternatives a1 and a2 replaced ranks. the correlation of results was tested using spearman's correlation coefficient of ranks. this statistical technique is extremely useful for ranking a small number of variables (pamučar et al, 2018; ghorabaee et al, 2015). using spearman's correlation coefficient of ranks, it was found that the correlation is less than 1 only in the case of the fuzzy-mabac method (spearman’s correlation coefficient is 0.9). the average value of the correlation is 0.98. based on the average value of spearman's correlation coefficient of ranks, it can be concluded that the application of the hybridized model is extremely reliable under conditions of the uncertainties. a sensitivity analysis was carried out through changes in the criteria’s weights. the sensitivity analysis carried out through 36 scenarios. in each scenario, the weight of one criterion is increased (reduced) by 25%, 50% and 75%, respectively. the weights of the other criteria are increased (decreased) due to the following condition 1 1 n i i w   (table 10). the results in the table show that the ranking of alternatives changed through five scenarios. in other scenarios, the ranking of alternatives did not change. based on spearman’s correlation coefficient, in 31 scenarios, values of correlation is one, while in five scenarios values of correlation is 0.9. thus, it can be concluded that there is very high correlation (closeness) of ranks through the scenarios and that the results obtained using hybridized it2fs-dematel-ahp-topsis approach are credible. muravev and mijic./decis. mak. appl. manag. eng. 3 (1) (2020) 146-164 160 t a b le 1 0 . t h e s e n si ti v it y a n a ly si s o f re su lt s a hybridized it2fs-dematel-ahp-topsis multi-criteria decision making approach… 161 5. conclusion in the paper, the criteria for selection of the optimal atc radar position, which will ensure observation of air traffic at all flight levels (including flights below the altitude transition), were determined and evaluated. furthermore, radar positions are ranked from the aspect of influence of the radar ability to detect potential air traffic violations, as well as flying through prohibited, restricted or dangerous areas. in the research, special attention is devoted to significance of radar positions in the detection of unmanned aerial vehicles that could be used for endangering safety from the airspace. the it2fs, which were used in the research, enabled valid decision making in conditions of high level of uncertainty, when partially reliable data (as a consequence of the lack of appropriate literature) was gathered by a small number of experts. bearing in mind aforementioned, future research could be focused on: 1) the application of other traditional objective and subjective methods of multicriteria decision making in combination with it2fs in the determination and evaluation of criteria for the selection of the radar position and for solving other poorly structured problems (for example: critic, best-worst, anp, electra, copras, mairca, vikor, mabac, etc.). 2) the application of other types of tools that accept uncertainty in decisionmaking. such as type 2 fuzzy sets, interval (type 1 or type 2) valued fuzzy sets, (interval valued) intuitionistic fuzzy sets. these types and forms of fuzzy sets can be applied by themselves or in construction with some numbers such as rough numbers or grey theory. the application of the proposed model in this paper with the geographic information system, which can provide a practical purpose of this model in the selection of the optimal radar position that, ensures maximization of the technical characteristics’ utilization of the atc radar author contributions: each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content. funding: this research received no external funding. conflicts of interest: the authors declare no conflicts of interest. reference baykasoğlu, a., & gölcük, i̇. 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(2014). research on the conformal phased-array antenna. applied mechanics and materials, 685, 324-327. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). muravev and mijic./decis. mak. appl. manag. eng. 3 (1) (2020) 146-164 164 appendix table a1. the average linguistic variables matrix of the influence between initial criteria (dematel) k k1 k2 k3 k4 k5 k6 k7 k8 k1 n vh vh vh vh vh vh vh k2 h n vh 1.167×h vh h h vh k3 1.17×l 1.5×l n 1.33×l l l 1.17×l 1.33×l k4 m 1.085×m h n m l m 1.085×m k5 1.5×l 1.5×l 1.33×l 1.5×l n 1.33×l 1.17×l 1.17×l k6 h h 1.167×m 1.5×l l n l 1.5×l k7 1.5×l 1.5×l h 1.25×m 1.25×m h n 1.5×l k8 h h l m m 1.5×l h n table a2. the average linguistic variables pairwise comparisons matrix (ahp) c c1 c2 c3 c4 c5 c6 c1 e fs as vs (vs+as)/2 (fs+vs)/2 c2 1/fs e as (4×fs+2×vs) vs (ss+fs)/2 c3 1/as 1/as e 1/fs e 2/(fs+vs) c4 1/vs 1/(4×fs+2×vs) fs e e e c5 2/(vs+as) 1/vs e e e 1/(4×ss+2×fs) c6 2/(fs+vs) 2/(ss+fs) (fs+vs)/2 e (4×ss+2×fs) e table a3. the average it2fs decision matrix (topsis) c/a a1 a2 a3 a4 c1 g (m+g)/2 m p c2 (m+g)/2 (p+m)/2 (p+m)/2 (p+m)/2 c3 (g+vg)/2 m m (p+m)/2 c4 (g+vg)/2 (p+m)/2 p p c5 (4×m+2×g)/6 (4×p+2×m)/6 p (4×p+2×m)/6 c6 g m p (p+m)/2 plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 1, number 1, 2018, pp. 165-184 issn: 2560-6018 doi: https://doi.org/10.31181/dmame1801165k * corresponding author. e-mail addresses: nkomazec@gmail.com (n. komazec), mladenovicmilica21@yahoo.com (m. mladenović), cacabezbfco@yahoo.com (s. dabižljević) etiology of the notion of event in terms of decision-making and determination of organizational system risk conditions nenad komazec1*, milica mladenović2, slavica dabižljević3 1 university of defence, military academy, belgrade, serbia 2 s4 glosec global security, belgrade, serbia 3 regional security and crisis management asociation, belgrade, serbia received: 18 january 2018; accepted: 23 february 2018; published: 15 march 2018. original scientific paper abstract: the problem of functioning of organizational systems in a dynamic environment implies permanent influences from the environment. the tendency of these influences, since these are in connection with the functioning of other systems in terms of meeting their needs, is the creation of entropy of organizational systems. the causes of the impact are found in various occurrences in the environment, which are most often of a situational character. an impact can be made by one phenomenon, two or more. the interaction between phenomena usually contributes to an increase of the impact by intensity, time and number of exposed protected values. management of an organizational system in such conditions depends on risk management quality, that is, on the quality of decision-making process in terms of risk. by understanding, identifying and responding to such events, it is possible to determine the risk to organizational systems elements and to make a decision on future actions. the problems of identifying optimal solution, that is, optimization tasks, are met and analyzed in all phases of an organizational system existence. the process of decision-making and the choice of the "best" alternative is mostly based on more than one criterion and various limits. this paper presents an approach to the analysis of elements of organizational system environment, which generate events that influence on the behavior of organizational systems. deciding quality depends on the quality and availability of information about events in organizational system environment, which can be determined through different multicriteria decision-making models. the result of the research is a qualitatively new definition of the notions of event and the extraordinary event viewed through the risk function as immanent characteristic of all events in the environment.). key words: organizational system, decision-making, risk, event, hazard. mailto:nkomazec@gmail.com mailto:mladenovicmilica21@yahoo.com mailto:cacabezbfco@yahoo.com komazec et al./decis. mak. appl. manag. eng. 1 (1) (2018) 165-184 166 1. introduction a daily functioning of organizational systems consists of a series of activities in space and time during which different risks and individual decisions create different sensitivity modalities of the system (alexander, 1996). the resulting modalities are conditioned by the characteristics of people, organizational systems, nature and social phenomena. a significant feature of organizational systems, from the aspect of various environmental influences, is sensitivity to these influences. organizational systems which do not have the capacity to identify influences and take appropriate measures to protect themselves are considered sensitive. sensitivity decreases with the establishment of these capacities. the environment variability in which organizational systems function implies the influence of the resulting conditions on system elements functioning. a situational approach to the study of system functioning, from the aspect of generating a set of circumstances which characterizes the resulting situation, is fundamental question what are the elements of the environment and in what way they influence an organizational system. namely, from every new situation (a set of circumstances) a new spectrum of influences is generated, by the analysis of which can be assessed the risk on system functioning. hence, the need arises for observation of several criteria in the decision-making process on future conditions of organizational systems. in the problems of multi-criteria optimization in a decision-making process, a decision-maker in organizational systems implicitly strives to find a solution that meets the above criteria to the greatest extent possible, without breaking existing limitations. unfortunately, such problems do not have a single and global solution, that is, there is no optimal solution applicable to all criteria simultaneously. it often happens that some criteria, partly or completely, are mutually opposed. in addition, the criteria considered can by their very nature be very heterogeneous and expressed in different measuring units, from monetary units, through physical quantities, up to probability or subjective estimations determined on the basis of a scale formed for a particular problem. all this indicates that a final, single solution can not be determined without the involvement of a decision-maker. the importance of a decision-maker in organizational systems is especially evident when risk conditions are concerned, that is, when the experience and the ability of a decision-maker to identify and determine new conditions that imply the risk plays significant role. the aim of the paper is to show the possibility of determining stochastic elements in organizational systems environment in conditions of risk and uncertainty based on the modeling of multi-criteria determination. the research problem is based on the need for determination of elements of phenomena and events in system environment, in order to start with an experiment, so as to create quality information for a decision-maker. in the first part of the paper, the problem of researching the stochasticity of organizational systems elements is presented. on the basis of the existing knowledge through the analysis of the content and preliminary analysis of a hazard to system elements, the problem of generating the impact to organizational systems values is presented and the relation between the environment elements and system elements. it is shown the possibility of application of different models of multi-criteria analysis, from the aspect of presence of a number of impact elements from organizational system environment. by applying these models, it is possible to experiment with different types of influences to organizational systems in different circumstances. in the second part, it is developed a study of the interaction between different sets of circumstances influencing the elements of organizational systems. the conditions of the system with the elements of exposure and resistance are defined. the result of the etiology of the notion of event in terms of decision-making and determination of ... 167 research is a quality and functionally new definition of the notion of event and extraordinary event, from the aspect of dependence of temporary and spatial dimension of environment influences and risks. in the third part, the results of the research are commented, with the emphasis on new definition of events and extraordinary events in terms of risk and uncertainty of space and time of their occurrence. also, it is emphasized the significance of the methods and models of the experimental work by which are created the assumptions on how to improve the quality of the information necessary for decision-making. 2. stochastic elements of phenomena in organizational systems environment existing knowledge and methods of research the manifastations of diverse organization in human society, which meet certain objevtives in real world, are complex phenomena (beck, 2011). such phenomena, due to their complexity, meaningfulness and connectivity, are called a system (kljajic, 1994). starting from the view according to which there are important elements common to all areas of reality, it can be considered that there are principles for the functioning of all systems in real world, and therefore in society (bertalanfy, 1968). knowledge of the interaction between the system and the environment has led to the creation of a general theory of a system, which changed the perception and intellectual perspective from which reality has been observed (bertalanfy, 1968). although different theoreticians have tried (and succeeded) to prove certain regularities in the origin, development and disappearance of a system, the practice has been reminding them of a scope of circumstances that have random character. such a character is caused by various factors, but as a rule, it has certain influence on the system. a system unifying various elements into a functional entity, with the aim of pursuing common goal, is called an organization, and such systems are called organizational systems (figure 1). figure 1. elements of organizational systems (stevanović & subošić, 2007) an organizational effect has shown positive signs of overcoming individual problems in the struggle for survival and progress of an organization. new values have been created, which have been above individual values (daft, 2004). the new ways of organizing have implied certain enhanced effects, such as: 1. use of common resources to achieve common goals; 2. increase in efficiency of operation; 3. creating new ideas about problem solving; 4. use of new technologies; 5. adapting to changes in the environment and 6. generating new values. organizational system structure function envirnoment komazec et al./decis. mak. appl. manag. eng. 1 (1) (2018) 165-184 168 newly created values are not only characterized by positive effects on communities, on the contrary, a significant number of negative effects also appears. a number of new phenomena are not expected. the complexity of organizational systems causes series of interactions between elements of deterministic and stochastic character. the existence of elements of stochastic character leads to the emergence of phenomena and events over which subjects of the organizational system must exercise certain influences, in order to keep them within planned frameworks. system regulation of organizational systems enables preliminary identification of stochastic phenomena and events and taking effective measures to control them. however, in addition to all indicators of progress, a part of the events has still been out of control, with the elements of uncertainty and negative consequences. sets of circumstances are being created, which are not expected, whose causes are not familiar, whose effects can not be predicted and, ultimately, against which it is not possible to protect. the result of generating different circumstances in new phenomena is the emergence of various events that can have the capacity without affecting the process, can affect the process in the form of anomalies, or make certain changes in the process. the diversity and multidimensionality of environmental impacts requires from a decision-makers to apply different methods of multi-criteria decision-making in order to find the best alternative. the question arises whether it is possible to anticipate such events, or take preventive measures in terms of their removal, mitigation and reduction. by analyzing the contents of existing literature and by preliminary analysis of the influence of the environment to organizational system elements, the etiology of the occurrence of events in the environment of organizational systems is investigated. 3. models of multi-criteria decision-making in the process of determination of risk condition in organizational systems multi-criteria decision-making models (mcdm) containing qualitative or quantitative attribute values have wide application in the fields of operational research, management science, urban planning, natural sciences and military affairs. the mcdm problem usually is solved in a two-phase process: (1) the rating, that is, the aggregation of the values of criteria for each alternative and (2) the ranking or ordering between the alternatives, with respect to the global consensual degree of satisfaction. the step-by-step sequence of the problem of multi-criteria decisionmaking is defined as follows (mukhametzyanov and meshalkin, 2014; pamučar et al., 2017a): (1) choice of alternatives ( ; 1, 2,..., i a i m ); (2) choice of evaluating criteria ( ; 1, 2,..., j c j n ); (3) acceptance of scales of an estimation of alternatives on each criterion; (4) determination of priorities (weights) of criteria ( ; 1, 2,.., j w j n ); (5) determination of evaluation matrix i.e. decision matrix ij m n x a      ; (6) choosing a method for ranking alternatives. clasic methods, such as saw (stević et al., 2017; kaklauskas et al., 2006), moora (brauers and zavadskas, 2006; kalibatas & turskis, 2008; brauers, 2008), vikor etiology of the notion of event in terms of decision-making and determination of ... 169 (opricović & tzeng, 2004), copras (viteikiene & zavadskas, 2007), topsis (pamučar et al., 2017), mabac (pamučar & ćirović, 2015), are mostly used in solving problems of multi-criteria optimization. mentioned models imply that weight coefficients of criteria are determined by some other model, such as the ahp method (božanić et al., 2016; pamučar et al., 2016), the dematel method (pamučar et al., 2017b; gigović et al. 2016) and the best-worst method (stević et al., 2017; pamučar et al., 2018). basic settings of the most commonly used traditional models of multi-criteria optimization are presented in the following part of the paper. 3.1. multi-criteria compromise ranking (vikor method) the vikor method represents an often used method for multi-criteria ranking, suitable for solving different decision-making problems. it is especially suitable for situations where criteria of quantitative nature are prevalent. the vikor method was developed based on the elements of compromise programming. the method starts from the “border” forms of p l metrics (opricović & tzeng, 2004). it seeks the solution that is the closest to the ideal. in order to find the distance from the ideal point it uses the following function:     1/ * * 1 , ,1 p n p p j j j l f f f f x p               (1) this function represents the distance between the ideal point *f and the point  f x in space of criteria functions. the essence of vikor method is that for every action it finds the value of i q , and then it chooses the action which has the lowest listed value (the smallest distance from the “ideal” point). the measurement for multi-criteria ranking of the i -th action ( i q ) is calculated from the equation:  1i i iq v qs v qr    (2) where * * i i s s qs s s     (3) * * i i r r qr r r     (4) where * min j s s , max j s s   , * min j r r i max j r r   , while js represents pessimistic solution, and j r the expected solution. by calculating the values of i qs , i qr and i q for every action, three independent ranking lists can be formed. the size of i qs represents the measurement of deviation through which the demand for maximal group benefit (the first ranking list) is expressed. the value of i qr represents the measurement of deviation through which the demand for minimization of distance of some action from the “ideal” action (second ranking list) is expressed. the value of i q represents the forming of compromise ranking list which ties together the values of i qs and i qr (the third ranking list). by choosing the smaller or the greater value for v (the strategic weight of satisfying the majority of criteria), a decision-maker can factor the impact of the value of i qs or the value of i qr in the compromise ranking list. as the reliable komazec et al./decis. mak. appl. manag. eng. 1 (1) (2018) 165-184 170 ranking list by the vikor method, we take the compromise ranking list with the value of 0.5v  . 3.2. technique for order of preference by similarity to ideal solution (topsis) method the basic principle of the topsis method is that the best alternative should have the shortest distance from the ideal solution and the farthest distance from the antiideal solution. a relative distance of each alternative from the ideal and anti-ideal solution is obtained as (pamučar et al., 2017b) , 1,...,i i i i s q i n s s       (5) where i s  and i s  are separation measures of alternative i from the ideal and anti-ideal solution, respectively; i q is the relative distance of alternative i from the ideal solution, and  0,1iq  . the largest value of the criterion i q correlates with the best alternative. the best ranked, or the most preferable, alternative * tps a can be determined using the following formula (pamučar et al., 2018):  * maxtps i i i a a q the separation measures of each alternative, from the ideal and anti-ideal solution, are computed using following formulae (pamučar et al., 2018):   1/ 2 2 1 n j ij i j s w r r              (6)   1/ 2 2 1 n j ij i j s w r r              (7) where element ij r represents the performance of alternative i a in relation to criterion j c . for m criteria ( 1, 2, ..., m c c c ) and n alternatives ( 1, 2, ..., n a a a ), the matrix r has the shape ij nxm r r    . the values ( 1, 2 ,..., mw w w ) represent weight values of criteria that satisfy the condition 1 n ii w  . the ideal a  and the anti-ideal a  solution in the topsis method can be determined using the formula (8) and (9), respectively.    ' 1 2(max | ), (min , ), 1,.., , ,...,ij ij ma v j g v j g i n v v v          (8)    ' 1 2(min | ), (max , ), 1,.., , ,...,ij ij ma v j g v j g i n v v v          (9) it can be seen from the formula (6) and (7) that the ordinary topsis method is based on the euclidean distance (gigović et al., 2016, 2017). 3.3. multi-attributive border approximation area comparison (mabac) method basic setting of the mabac method is represented in defining distance of the criteria function of every observed alternative from the border approximate area (pamučar & ćirović, 2015). after forming the initial decision-making matrix ( x ), in etiology of the notion of event in terms of decision-making and determination of ... 171 the first step are evaluated m alternative by n criteria. the alternatives are presented with vectors  1 2, ,...,i i i ina x x x , where ijx is the value of -th alternative by j -th criteria ( 1, 2,..., ; 1, 2,...,i m j n  ). in the next step it is performed the normalization of the initial matrix elements ( x ) by applying linear normalization (pamučar & ćirović, 2015). after weighting normalized matrix, it is determined the matrix of border approximate areas ( g ) 1/ 1 m m i ij j g v          (10) where ij v present weighted matrix elements, ..present total number of alternatives. after the calculation of the value i g by criteria, it is formed the matrix of border approximate areas g in 1n  form and it is determined the distance of alternatives from the border approximate area (božanić et al., 2016). the alternative i a can belong to the border approximate area ( g ), upper approximate area ( g  ) or lower approximate area ( g  ), i.e.  ia g g g      . the upper approximate area ( g  ) present the area in which the ideal alternative is located ( a  ), while lower approximate area ( g  ) present the area in which the antiideal alternative is located ( a  ). belonging of the alternative i a to the approximate area ( g , g  or g  ) is determined based on the expression (11) ij i i ij i ij i g if q g a g if q g g if q g          (11) in order the alternative i a to be chosen as the best from the set, it is necessary to belong to the upper approximate area ( g  ) by as many criteria as possible. for example, if the alternative i a by 5 criteria (out of total of 6 criteria) belongs to the upper approximate area, and by one criterion belongs to the lower approximate area ( g  ), this means that by 5 criteria the alternative is close or equal to the ideal alternative, while by one criterion it is close or equal to the anti-ideal alternative. in case the value is 0 ij q  , i.e. ij q g   , the alternative i a then is close or equal to the ideal alternative. the value 0 ij q  , i.e. ij q g   shows that the alternative i a is close or equal to the anti-ideal alternative. ranking alternatives. the calculation of the values of criteria functions by alternatives is obtained as the sum of distances of alternatives from border approximate areas. 3.4. complex proportional assessment (copras) method ranking alternatives by the copras method assumes direct and proportional dependence of significance and priority of investigated alternatives on a system of criteria (ustinovichius et al., 2007).the selection of significance and priorities of alternatives, by using copras method, can be expressed concisely using four stages (viteikiene & zavadskas, 2007). komazec et al./decis. mak. appl. manag. eng. 1 (1) (2018) 165-184 172 for normalization in the copras method, the following formula is used (viteikiene & zavadskas, 2007): 1 ij ij m ij i a n a    (12) where xij is the performance of the i-th alternative with respect to the j-th criterion, ij a is its normalized value, and m is the number of alternatives. in the copras method, each alternative is described with the sum of maximizing attributes s+i. in order to simplify calculation of +is and is in the decision-making matrix, the columns maximizing criteria are placed first, followed by the minimizing criteria. in such cases, +i s and i s  are calculated as follows (viteikiene & zavadskas, 2007): 1 k +i ij j j= s = n q (13) 1 n i ij j j=k + s = n q   (14) in formulas (2) and (3), k is the number of maximizing criteria; n is total number of criteria; and qj is significance of the j-th criterion. the relative weight i q of i-th alternative is calculated as follows: 1 1 1 m i i= i +i m i i= i s q = s + s s      (15) the priority order of compared alternatives is determined on the basis of their relative weight (the higher relative weight, the higher priority/rank). the methods presented form part of the corpus of methods applicable in the study of the influence of environmental elements on organizational systems functioning. the application depends on the conditions and time for experimentation. the problem of organizational systems functioning refers to the influence of various factors, of permanent or situational character. the complexity of methods and models is inversely proportional to the time of event generation in organizational system environment. 4. theoretical and functional concept of event and extraordinary event the environment of organizational systems represents a set of different phenomena and interactions between them. individual or cumulative action of a phenomenon or a set of phenomena is determined as an event. the analysis of the content of certain references presents different interpretations of the term event, which have certain common characteristics (table 1). in law lexicon (1964), an event presents a circumstance that occurs against the will of the organization subjects, and to which it is objectively related the occurrence, the cessation or the change of condition. an event is often qualified as force majeure. etiology of the notion of event in terms of decision-making and determination of ... 173 the flow of time is an event of great importance for the acquisition and loss of subjective rights (law lexicon, 1964). table 1. the most significant characteristics of different interpretations of the concept of an event (komazec, 2017) source most frequent elements of the concept of an event common characteristics law lexicon (1964) a circumstance ocurring against the will of the subject. force mejaure. accidental occurence familiar or unfamiliar cause series of circumstances arises undefined in space and time no influence of the subject little encyclopedia (1978) a subset of the set of possible results of an experiment. dictionary of the croatian or serbian language (1903) what is happening, with familiar or unfamiliar cause. a chance, case, intention. new larousse encyclopedia (1999) a realized circumstance. a fact, an act. dictionary of serbocroatian literary language (1967) what happened, in a particular place. an occasion, opportunity. iso guide 73:2009 risk management appearance of series of circumstances. standard srps a.l2.003:2017 appearance or change of particular set of circumstances. according to little encyclopedia (1978), an event is also a subset of the set of all possible outcomes of an experiment (little encyclopedia, 1978). in dictionary of croatian or serbian language (1903), an event (m. eventus, casus) is defined as something that is happening. in general, it refers to what is happening, whether good or bad, with familiar or unfamiliar cause; what can happen or is thought to be possible to happen; something special that can happen, where it is required or it is said what is to be done; chance, case, intention, when the cause of what is happening is not familiar, and it is thought to occur with no cause. occasio, opportunitas as a chance, an experience; what can lure a man to do something. happening an act of happening and what is happening (dictionary of the croatian or serbian language, 1903). according to the larousse encyclopedia, an event means (lat. evenire) to happen, to befall 1. what happens, what comes or acts, a fact, circumstance. 2. significant, striking act. 3. in statistics, a coincidence that occurs, in a particular place. a set of significant facts that occurred (new larousse encyclopedia, 1999). in dictionary of serbo-croatian literary language (1967), an event is presented as 1. what happened at a certain place; 2. an occasion, opportunity; 3. an important phenomenon, a peculiar thing (serbo-croatian literary language, 1967). according to the international standard iso guide 73: 2009 risk management vocabulary, an event is the emergence of a certain set of circumstances. an event is occurrence or change of a particular set of circumstances (standard srps a.l2.003: 2017). the same standard in the explanation provides the following interpretations: 1. an event may consist of one or more occurrences and may have several causes; 2. an event may consist of something that has not happened; 3. an event can sometimes refer to an "incident" or an "accident" and komazec et al./decis. mak. appl. manag. eng. 1 (1) (2018) 165-184 174 4. an event without consequences can also be considered as an event that is "barely avoided", "just about to happen" or "almost happened". the serbian standard srps a.l2.003: 2017 states that an event is characterized by a consequence, as an outcome that affects the objectives. an event gets important for an organizational system in the moment when it acquires capacity, or when a set of circumstances is such that it can result in negative consequences on system values. therefore, an event may pose a threat to system values, whether it familiar or not. in case it is familiar, an event can be studied, analyzed and monitored. in case when it is not familiar, an event is hypothetically observed, through the development of potential scenarios. bearing in mind mentioned characteristics and the results of the content analysis of available references, an event can be defined as every accidental result of a set of circumstances, which ocured in a particular place and in a particular time, against the will of the subject which is directly or indirectly influenced by it. 4.1. influence of the event condition to the existence of hazard to system elements the term related to events in organizational system environment, which is in significant relation with the condition of the organizational system, is a hazard. from the aspect of the existence of a hazard, an accumulated set of circumstances due to which there is a risk, uncertainty or certainty that it will result in negative consequences on system values, becomes significant for system management and the subject of its monitoring and analysis. thus, a set of circumstances caused by various occurrences in the environment, with or without a negative impact on organizational systems values, obtains the form of an event. the declaration of the resulting set of circumstances an event in the management of the system arises at the moment when competent authorities assess that the resulting set of circumstances is significant from the aspect of the impact on planned functioning of organizational systems (pavličić, 2010). mentioned significance of an event, from the aspect of the impact on organizational systems functioning, is related to the emergence of a hazard to system values. any disruption in organizational system functioning implies a threat to the protected values of a system known or unknown (komazec et al., 2015). the term "hazard" comes from the french word ‘hasard’ and the arabic word ‘azzahr’, meaning "chance" or "opportunity" (benson, 1981). a hazard is defined as "potentially harmful physical event, phenomenon or human activity that can cause loss of life or injury, property damage, social and economic disorders or environmental degradation. this event has the probability of occurrence within a specific time period in a given area, with certain intensity ", (un international strategy for disaster reduction, www.unisdr.org (2009). different authors also defne a hazard as (шойгу et al., 2004): 1. a possibility of causing injury, material, physical or moral damage to a person, society or state; 2. an accompanying phenomenon or probability of occurrence of a potentially destructive phenomenon in a specific period of time and in a particular region; 3. a situation in which processes and phenomena are possible which can lead to injuries of people, causing material damage, destructive action on the environment; 4. a process, property or state of the environment, in the event of occurrence of conditions which can lead to one or several negative consequences to human health, the state of the environment, which cause material or social damage etiology of the notion of event in terms of decision-making and determination of ... 175 with a deterioration of living and working conditions and the process of normal economic activity or deterioration of the environment quality. an event that is preliminarily recognized as a "hazard" is a source of possible damage (standard srps a.l2.003: 2017) and a hazard can be a source of risk. the term "possible" refers to its potentiality. a potential hazard relates to the fact that a set of circumstances is recognized as potentially dangerous to system values. the degree of danger is determined by analyzing the risk of occurrence of an event with negative impact, based on the available knowledge about phenomena that form the resulting set of circumstances. one of the approaches to defining the notion of a hazard, from the aspect of natural disasters appearance (thywissen, 2006): a hazard is an extreme geographical event that leads to a natural disaster. in this case, extreme means significant deviation in positive or negative direction from what is considered normal. basics for determining hazards are place, time, scope and frequency. many hazardous phenomena occur and their locations can be predicted. natural hazards can be defined as extreme events that occur in the biosphere, lithosphere, hydrosphere and atmosphere. based on the approach to defining the concept of hazard from the point of view of the causes in natural and social systems, (thywissen, 2006): a hazard is a product of combination of natural and social systems. a hazard is the result of the interaction of nature and man. would it be treated as completely climatic, geological, political or economic, important components that need to be considered when seeking the right solution for them would be nissed. thenature is neutral, however, the environment becomes dangerous only when it interacts with a man. a certain event turns into a natural disaster when: 1. it is extreme in scope; 2. the population is extremely high and 3. the systems used by people are extremely sensitive. the determination of the concept of a hazard from the aspect of general impact on system values, (thywissen, 2006): in the broadest terms, a hazard is a threat to people, to valuable inanimate nature. hazards can happen, but they also do not have to. however, when they occur they imply real impact on people and other values. hazards arise from the interaction of social, technological and natural systems. a hazard is a follow-up event or the possibility of its occurring at a certain time in a particular place (iso guide 73: 2009). it implies a potential threat to people, as well as a real event that affects them. there are many ways to characterize a hazard, for example, natural, technical, created by human factor, nuclear, ecological. the categories are probably as diverse as the disciplines and sectors of social life being covered. but what they have in common is the potential to cause serious, harmful effects which root in any incident, accident, and disaster. a hazard can be individual or general. in tha case, it is a specific hazard scenario. an important feature of a hazard is that it gives an impression of the likelihood, or the possibility to happen. a hazard is a threat, not an event itself, at the initial stage (smith, 2013). any hazard can manifest itself through a real harmful event. in other words, if a hazard can be measured in the units of real damage, then a hazard is no longer a hazard, it becomes an event, an accident, or a disaster (thywissen, 2006). based on the above, it can be concluded that a hazard may imply direct or indirect impact on the values of organizational systems. organizational systems, in relation to events that can potentially pose a threat to the environment, can have two states: exposure and vulnerability. the degree of presentation of both states depends on komazec et al./decis. mak. appl. manag. eng. 1 (1) (2018) 165-184 176 important characteristics of the system: persistence, resistance and sensitivity (komazec et al., (2016). 4.2. risk elements of the system an organizational system functions in its environment. environmental variability implies the variability of the conditions in which a certain impact of the event on organizational system values is realized (adigees, 2004). the level of exposure of the organizational system elements and their sensitivity (vulnerability) to events are basic characteristics of the existence of a risk to an organizational system (louis, 2009). exposure of organizational systems values to the impact of an event from the environment is a very important feature of possibility of occurrence of a hazard. exposure is the degree to which an organization and/or interested party is susceptible to the impact of an event (standard srps a.l2.003: 2017). exposure means number of people and/or other elements of the system (values) at risk that may be affected by the effects of a particular event. together with vulnerability and hazard, exposure is another precondition of the risk and negative impact of events on organizational systems values. the exposure of organizational systems is very low, if the system is inactive or out of function. thywissen cites an interesting relationship, stating that exposure determines the severity of the event impact on elements at risk, and vulnerability determines final damage level. therefore, in its economic dimension, vulnerability is shown through the projection according to which at a given event organizational systems will suffer damage in certain percentage. which parts of the system and what level of damage is shown through exposure (thywissen, 2006). based on everything mentioned, it can be concluded that exposure is not a risk element, but it directly affects the possibility, manner and intensity of the risk event on organizational systems values. vulnerability, according to srps standard a.l2.003: 2017, is characteristic feature of organizational systems values that results in sensitivity to the source of risk, which can allow the influence of the event with consequences (standard srps a.l2.003: 2017). further, in the same standard it is stated: 1. vulnerability can be considered a measure of quality of the existing protection conditions; 2. vulnerability can be defined as the degree to which an organization and/or interested party is susceptible to the impact of an event due to its exposure; 3. if damage scope is defined by the duration of harmful effects on protected values, then vulnerability includes also resistance. this conclusion stems from the assumption that vulnerability implies susceptibility of an event, or sensitivity of an organization to an event. besides exposure, another precondition of a negative event is vulnerability. vulnerability is a dynamic, characteristic feature of every system (household, region, state, infrastructure, or other risk element) that contains many components. the importance is determined by seriousness of an event. vulnerability points to the potential of damage and represents a forward-looking variable. vulnerability should include an anticipatory feature of imagination, of what could happen to a particular system in terms of certain risk and hazard (institute of management accountants, 2007). determining vulnerability means questioning what would happen if a particular event (events) affected certain elements at risk. vulnerability is an inherent feature of a system that is always present even in a peaceful period between the events. it does not appear or disappear depending on the event appearance or disappearance, but it is a constant and dynamic feature that exhibits in a certain etiology of the notion of event in terms of decision-making and determination of ... 177 amount during the event, depending on the severity of the harmful event. this means that vulnerability can often be measured only indirectly, and for this indirect measurement as a benchmark is taken the resulting damage (bukov, porfiriev 2005). what is usually seen in the aftermath of a negative event is not vulnerability itself, but the damage occurred. by examining the form of damage of a particular society without knowing the magnitude of the event, does not allow the conclusion about the vulnerability of that society. in this sense, the strength damage relationship reflects the vulnerability of an endangered system element (thywissen, 2006). poverty is also a measure of sensitivity, i.e., potential generation of disorders in a particular system. poverty (vulnerability) indicates existing state of subject's protection, that is, the sensitivity of the subject to potential hazards (standard srps a.l2.003: 2017). under sensitivity are considered the characteristics of the system, territory, community and the conditions in which they are located. these conditions affect the ability of organizational systems to anticipate, resist, fight and recover from the consequences of risky events from the environment. the degree of sensitivity also represents the difference between existing and necessary protection measures of organizational systems values. the greater the difference, the greater the degree of sensitivity, that is, the community is more vulnerable to potential hazards. knowledge of the difference between existing and required state is measure of knowledge of organizational system sensitivity (nocera, 2009). 4.3. resistance of organizational systems to the influences of risky events according to thywissen, in the life cycle of organizational systems resulting damage does not only depend on hazard, vulnerability and exposure, but also on persistence and resistance of elements at risk. in the literature, most of considerations indicate major overlap between persistence and toughness, which are often used as synonyms. these two dimensions of a harmful event are very difficult to separate (thywissen, 2006). persistence represents strategies and measures that directly affect damage during events, by alleviating, reducing pressure or reducing effects, and flexible strategies that change behavior or activities to avoid adverse effects. resistance represents persistence enhanced with the ability to maintain functionality of organizational systems during an event and to ensure complete recovery (bozanic et al., 2016). the notion of toughness is used to characterize the ability of the system to return to the reference level after the operation disorders and to maintain certain structure and functions. the toughness of the system is often represented by flexibility of the system itself, i.e., how many changes and obstacles it can tolerate, while retaining the desired level of functioning (un international strategy for disaster reduction, www.uisdr.org, 2009). adaptibility, flexibility or elasticity are characteristics of the ability to absorb the influence of an event. resistance is represented by various elements, such as: organization, competence, types of constructions, barriers, land composition, geography, atomic shelters, locations, and so on. as resistance increases, the ability to protect the system, society and the environment increases also. resistance is inversely proportional to vulnerability (un international strategy for disaster reduction, www.uisdr.org, 2009). the ability of organizational systems, community or society exposed to hazards is the capacity to adapt with resistance and changes in order to achieve an acceptable level of functionality. this is determined by the ability degree of the system to organize itself and to increase learning ability from past events, as well as to improve risk reduction measures. resistence is the ability of the organization to absorb the komazec et al./decis. mak. appl. manag. eng. 1 (1) (2018) 165-184 178 consequences of business cessation, and to maintain the level of services to a minimum (un international strategy for disaster reduction, www.uisdr.org, 2009). the capability or capacity of the organization is the ability to maintain basic functions during and after the event with consequences to the protected values as soon as possible, and with as little harmful effects as possible (standard srps a.l2.003: 2017). the following explanations are provided in the standard: 1. resistance also means an organization's ability to absorb negative effects from the environment, or to adapt and recover from the event with consequences to the protected values; 2. resistance includes strategies and measures that mitigate or suppress harmful effects, as well as adaptive measures to avoid adverse effects. in this way, resilience also implies the ability of the organization to maintain its functionality during the event, as well as to recover from the event occurred. 3. resistance is a characteristic of an organization that is inverse to vulnerability. it should be considered that in conditions of acting of the events with negative effects and the exposure of system values and their vulnerability, it is necessary to connect and analyze the notion of sensitivity of system elements in order to examine the process of acting of events with negative impact. 4.4. changes of system conditions due to the influence of hazards disorders in organizational systems functioning represent a state created by the impact of risky events and the degree of sensitivity of organizational systems elements. there can be no disorders if there is a hazrd, and there is no sensitivity. if there is no sensitivity, the system element that is exposed to hazard is not vulnerable, because protection measures had been taken. on the other hand, the system element may be sensitive to the hazard, but there could be no set of circumstances for the emergence of the hazard. therefore, the existence of a risk of occurrence of a certain hazard does not necessarily imply negative effects on system elements, unless there is sensitivity (alexander, 1996). it can be concluded that events are everyday occurrences that represent a set of circumstances and interactions in the real world (table 2). events, as a state, have neutral value from the aspect of a hazard to the system, up to the moment of their identification or materialization. organizational systems are daily exposed to different phenomena, with varying intensity and method of acting on system elements. the sensitivity of system elements is proportional to the degree of awareness of management of the need for risk management, on one hand, and concrete measures taken to reduce the degree of negative impact, on the other. the events in the process of organizational systems functioning can appear as regular (planned, expected) events and extraordinary (unplanned, unexpected) events (karović & komazec, 2015). all the events that imply a hazard to organizational systems values conditionally represent extraordinary events. in order for a particular event to obtain a legal form of an extraordinary event, it must be verified in a lawful manner by competent authority. the term "extraordinary" event refers to an unplanned phenomenon, unexpected action, or deviation from a regular one. according to alexander, the notion extraordinary means beyond usual order, which is not regular, distinctive, unusual, exqusite (expresses the state), (alexander, 1996). etiology of the notion of event in terms of decision-making and determination of ... 179 in rečnik srpskohrvatskoga književnog jezika a syntagm extraordinary situation is presented as a situation in which, due to the occurrence of extraordinary circumstances, it is departed from the application of a certain number of legal norms, and other norms foreseen for such a case are applied instead. table 2. the most important characteristics of different interpretations of the concept of extraordinary event (komazec, 2017) source most frequent elements of the concept of event mutual characteristics alexander (1996) beyond usual order, which is not regular not regular deviates from the application of existing legal norms use of specially developed norms endangers values not defined in time and space need of extraordinary forces to react potencially dangerous dictionary of the croatian or serbian language (1903) because of the emergence of an exceptional circumstance, it deviates from the application of legal norms, but others are applied, in accordance with the situation mlađan (2014) the capabilities of regular forces to go beyond the needs of the endangered system komazec et al., (2015) all events in the armed forces, as well as those who are directly or indirectly related to it, resulting in the endangering of life and combat readiness the notion of extraordinary event from the aspect of the criteria of needs and possibilities, according to law lexicon (1964), is interpreted as the possibility of regular forces to respond, satisfy and overcome the needs of an endangered system. every extraordinary event represents a unique case in itself (mladjan, 2014). besides extraordinary events, there are also everyday events (regular, immanent to the system), for the elimination of which an organizational system engages minimum forces and resources within regular activity, whereby these can successfully and efficiently simultaneously eliminate more of these events (mladjan, 2014; komazec et al., 2014). extraordinary events refer to all unwanted sets of circumstances, phenomena or interactions that provoke negative consequences on human life and health, material and cultural property, combat readiness of organizational systems, order and discipline, business and reputation (mučibabić, 1995). kasagić states that all incidents in the armed forces, as well as those who are directly or indirectly connected with it, are considered to be extraordinary events, which result in endangering the lives of members of the army, affecting combat readiness of the unit and causing material damage to the army (komazec et al., 2015). from the above analysis of the content of different definitions of the notion of event and extraordinary event, the following general definition of event and extraordinary event can be made: "an event is a state created by a disruptive action of a set of circumstances unexpected in time and space, due to which are possible to occur or have occurred negative consequences to organizational system values. the consequences of the event can be eliminated or reduced by regular actions (measures), forces and resources" (komazec, 2017). komazec et al./decis. mak. appl. manag. eng. 1 (1) (2018) 165-184 180 "an extraordinary event is a state created by a negative action of a set of circumstances unexpected in time and space, resulting in negative consequences or unacceptable risk to organizational system values and verified by competent authority. the consequences of such event can not be eliminated or reduced by the application of regular procedures (measures), forces and resources, but it is also necessary to engage additional capacity of organizational systems over a longer period of time" (komazec, 2017). an extraordinary event can overcome the dimension of an event and pass into a higher state of endangering organizational systems values. more events occuring at a given moment represent a situation. the development of the negative capacity of the event, through the extraordinary event situation, can lead to an emergency situation, that is, a crisis situation (komazec et al., 2016). 5. comments on the result of the research an analysis of existing literature, which deals with the definition of the concept of risk through the etiology of the occurrence of events that affect organizational systems functioning, proves the possibility of investigating the effects from the aspect of riskiness of events. namely, a new set of circumstances is stochastic, in most cases, therefore, the level of uncertainty is proportionally higher. under such conditions, the management of organizational systems is difficult. management of organizational systems is responsible for identifying potential hazards, taking measures to prevent events with negative consequences, as well as measures to improve the capacity for identifying and remediating risky events. deciding on future management moves depends on the degree of danger of new circumstances. it is precisely in this segment that a key difference in the perception of a new set of circumstances arises, and it is classified as a potential "event" or as a potential "extraordinary event". the research provided a qualitatively new definition of two terms, of "event" and "extraordinary event". both concepts are observed from the perspective of risk existence to the organizational system as an immanent property, rather than physical occurrence of events. with this approach, hypothetical observation of an event is also possible, through creating different scenarios for the development of future situations. basic characteristic of a scenario is the use of past and present information in real-time for the purpose of designing future state of organizational systems. the generation of scenarios can be improved by applying multi-criteria decision-making methods and different simulations. the sensitivity of the results of this research can show the limits of decision-making under the conditions of uncertainty and risk, but also to show to decision-makers how to determine the boundary between a harmless state, an event and an extraordinary event. a deciding based on the results of the application of different decision-making models provides certain degree of security in the quality of information about events or extraordinary events. for a decision-maker, reaching a level of security in information is a key moment in the decision-making process. namely, by knowing whether a new set of circumstances is a hazard to an organizational system, a decision-maker makes a decision to take measures to protect organizational system values. the measures undertaken may vary in structure, feasibility, impact, possibility and needs for financing, etc. it is certain that making a decision on system reaction to the occurrence of an "event" has less implications on the weight of the decision on riskiness of the measure taken. while, on the other hand, the adoption of measures in case of an "extraordinary event" may require urgency, intensity and etiology of the notion of event in terms of decision-making and determination of ... 181 extent of taking measures, which further implies errors in the decision-making process. 6. conclusion by exploring the conditions of the creation of a "set of circumstances" that has an impact on organizational systems elements and generates new states of organizational systems, qualitatively and functionally new definitions of the state of events and extraordinary events have emerged. namely, the management of an organizational system requires permanent decision-making, which will direct the system towards some future desired situations. a changeable environment generates events that have different implications on organizational systems elements. the deviation of any element from the planned behavior leads to disorders in the system operation. there are events that have permanent and predictable influence, and there are also those with stochastic character. stochastic events have predispositions of risky events, or the possibility of occurrence with negative consequences. also, the study of stochastic events opens the questions of possibilities of exploring the influence on organizational systems values. methods of multi-criteria decisionmaking can be used to experiment with the influence of different factors in different conditions. namely, the system of rules and limits set up through the applied multicriteria decision-making models provides a study of the impacts in controlled conditions. by applying different simulations, it is possible to investigate the sensitivity of the influence of different factors and the emergence of new states of 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(2004). учeбник спaсaтeљa. издaтeљствo: сoв кубањ. plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 5, issue 2, 2022, pp. 219-240. issn: 2560-6018 eissn: 2620-0104 doi:_ https://doi.org/10.31181/dmame0307102022p * corresponding author. e-mail addresses: kumarpaulvivek@gmail.com (v.k. paul), santonabchakraborty@gmail.com (santonab chakraborty), s_chakraborty00@yahoo.co.in (shankar chakraborty) an integrated irn-bwm-edas method for supplier selection in a textile industry vivek kumar paul1, santonab chakraborty2 and shankar chakraborty1* 1 department of production engineering, jadavpur university, kolkata, west bengal, india 2 national institute of industrial engineering, mumbai, india received: 6 june 2022; accepted: 11 september 2022; available online: 7 october 2022. original scientific paper abstract: like all other manufacturing industries, supplier selection also plays a pivotal role in a textile industry with respect to timely and cost-effective delivery of raw materials (cotton, yarn or fabric), chemicals and dyes, machineries, spare parts and other auxiliary parts/items. an appropriately selected supplier would help the textile industry in seamless production of final or semi-finished products leading to effective deployment of supply chain management concept. due to involvement of many competing suppliers and a set of conflicting criteria, supplier selection is often treated as a typical multi-criteria decision making problem. the process of choosing the right supplier for a given item often becomes more difficult due to presence of both quantitative and qualitative evaluation criteria. in this paper, based on six most significant criteria, an attempt is put forward to integrate interval rough number (irn) with best worst method (bwm) and evaluation based on distance from average solution (edas) method to solve a supplier selection problem for a textile industry. the application of irn helps in expressing opinions of the decision makers with respect to relative importance of the considered criteria and performance of the suppliers against each of the criteria using rough boundary intervals under group decision making environment. later, the criteria weights are determined using irn-bwm and the alternative suppliers are ranked from the best to the worst employing irn-edas method. an irn dombi weighted geometric averaging (irndwga) technique is considered to aggregate the opinions of the decision makers. this integrated approach identifies alternative 3 as the most apposite supplier for the textile industry under consideration. key words: supplier selection, textile industry, rough numbers, bwm, edas, mcdm, ranking. 1. introduction in today’s highly competitive global market, supply chain management has emerged out as a major decisive process of efficiently organizing all the activities from the placement of customers’ orders to the timely and cost-effective delivery of end products. it emphasizes on seamless integration of suppliers, producers, distributors, retailers and customers for achieving their goals through transformation of raw materials into quality products (tayyab & sarkar, 2021). the basic objective of supply chain management is focused on producing the right product for the right customer in the right amount and at the right time. supplier evaluation and selection appears to be one of the key determinants for the success of supply chain, influencing the long-term commitment and performance of any manufacturing organization. suppliers have varying strengths and weaknesses which require careful appraisal before they are ranked based on some specified evaluation criteria. supplier selection thus deals with shortlisting a set of mailto:kumarpaulvivek@gmail.com mailto:santonabchakraborty@gmail.com mailto:s_chakraborty00@yahoo.co.in paul et al./decis. mak. appl. manag. eng. 5 (2) (2022) 219-240 220 competent suppliers having the highest potential to consistently fulfill the manufacturing organization’s needs with an acceptable overall performance. an efficient supplier selection process would reduce purchasing risks, ensure uninterrupted production, maximize overall value for the buyers, develop proximity and long-term relationships between buyers and suppliers, and maximize benefits by improving the organization’s performance. an improper supplier selection decision may have severe detrimental effects, like shortage of raw material inventory, undue interruption in the production process etc. (amindoust & saghafinia, 2016; acar et al., 2016). in india, textile industry plays an increasingly important role in the national economy, not only by meeting the growing and diverse requirements of the people, but also generating huge job opportunities, making a major contribution in promoting economic development. the indian textile industry contributes 5% to the country’s gdp, 7% to the industry output in value terms, 12% to the country’s export earnings, and 5% to the global trade in textiles and apparel. the growth rate of indian textile industry was estimated to be 8.7% during 2015-2020, increased from about 7% from 2010-2015. india ranks as the world’s sixth largest textile and clothing exporter, and is also the major cotton and jute producer. it is also the second largest silk producer and 95% of the world’s hand-woven fabric comes from india. the indian technical textiles sector is estimated at usd 16 billion, approximately 6% of the global market. the textile and apparel industry in india is the second largest employer in the country providing direct employment to 45 million people and 100 million people in allied industries. the domestic technical textile market for synthetic polymer was valued at usd 7.1 billion in 2020 and is projected to reach usd 11.6 billion by 2027, growing at a cagr of 7.2%, while the technical textile market for woven fabrics is expected to grow at a cagr of 7.4% to usd 15.7 billion by 2027, up from usd 9.5 billion in 2020. investment in the indian textile industry has witnessed an erroneous growth of almost 69%, increasing from usd 1.41 million in 2010 to usd 2.38 billion in 2019. by 2029, the indian textiles market is expected to be worth more than usd 209 billion. like all other manufacturing industries, evaluation and selection of a set of competent suppliers also plays a key role in timely and cost-effective delivery of raw materials (fiber, yarn or fabric), chemicals and dyes, machineries, spare parts and other auxiliary parts/items in a textile industry. those suppliers should provide the items that are matched to the textile industry’s needs and requirements. thus, it has now become critical to clearly identify the industry’s needs and what it actually wants to procure before selecting a supplier. selection of suppliers from a large number of candidate suppl iers having varying potentialities and capabilities is a complex task due to involvement of several qualitative and quantitative evaluation criteria (nong & ho, 2019). conflicting nature of the criteria also makes the supplier selection problem more complicated. a supplier supposed to be the best with respect to a particular criterion may poorly perform against another criterion. the supplier selection problem having a set of equally compatible suppliers and conflicting evaluation criteria can be treated as a typical mcdm problem (chakraborty & chakraborty, 2022; chakraborty et al., 2023). in this direction, the past researchers have attempted the applications of several mcdm tools in identifying the most apposite suppliers for textile industries involved in production of varieties of end products (yıldız & yayla, 2015; manucharyan, 2021). in earlier days, evaluation of the suppliers and selection of the best one usually depend on the opinion on a single decision maker associated with the purchasing department of the organization. although it is a simple, straightforward and less computational intensive task, it may include individual biasness in the decision making process. nowadays, in order to make this process more scientific and unbiased, decisions from a group of participating experts (from various departments having valued experience) are sought. at the later stage of the evaluation process, judgments of the experts are weighted aggregated to derive a single collective decision. an organization would strive on both individual and group decision making approaches to be successful in the present-day competitive market. keeping in mind the basic objective of supplier selection, this paper first identifies six pivotal criteria, and attempts to express the opinions of four experts with respect to the relative significance of the considered criteria and performance of each supplier against each of the criteria using irns. the weights of the six evaluation criteria are determined using irn-bwm approach and the competing suppliers are ranked from the best to the worst based on irn-edas method. this integrated approach (irn-bwm-edas) appears to be a useful tool for supplier selection in a given textile industry engaged in procurement of raw materials in the form of cotton bales. this paper is structured as follows: section 2 provides a concise literature review of different mcdm methods employed for solving supplier selection problems in textile industries. the mathematical details of irn, irn-bwm and irn-edas are presented in section 3. a demonstrative example consisting of four suppliers is solved in section 4 using the proposed approach and conclusions are drawn in section 5. an integrated irn-bwm-edas method for supplier selection in a textile industry 221 2. literature review it has already been mentioned that the supplier selection process can often be treated as an mcdm problem with an aim to select the most apposite supplier fulfilling the requirements of a textile industry. table 1 presents a concise review of supplier selection problems in textile industries taking into account the number of suppliers, evaluation criteria, mcdm tool(s) applied and integration of mcdm techniques with other methods. it can be interestingly noticed that ahp has been mainly employed for criteria weight measurement, followed by anp. unlike ahp, anp considers inter-dependencies between the criteria and it has not a strictly hierarchical structure. on the other hand, topsis, moora, waspas and vikor have been the other popular tools used for evaluation and ranking of the suppliers. fuzzy theory and grey theory have been integrated with the mcdm tools to evaluate relative importance of the criteria under uncertain decision making environment. in the similar direction, dea has been applied for shortlisting the efficient suppliers through an initial screening process, pca has been adopted for criteria weight measurement and data dimensionality reduction, and dematel has been employed to segregate the evaluation criteria into cause and effect groups with development of the corresponding causal diagrams. ali et al. (2020) developed a fuzzy-ahp-topsis-based decision support system for solving a cotton supplier selection problem in a pakistani textile industry. the weights of five evaluation criteria, i.e. cost, quality, service, delivery and payment terms were first estimated using fuzzy-ahp method and topsis was later applied to rank the candidate suppliers. utama et al. (2021a) integrated ahp method with moora to solve a green supplier selection problem in a textile industry. the weights of eight evaluation criteria were estimated using ahp and the considered suppliers were ranked based on moora appraisal scores. product price was identified as the most important criterion affecting the supplier selection decision. while assessing the performance of apparel retailers, sarıçam and yilmaz (2021) presented the combined application of dea, ahp and topsis methods. ahp was employed to determine the criteria and sub-criteria weights and the apparel retailers were initially ranked using topsis method. a set of feasible and most efficient retailers was finally identified based on the application of dea. celik et al. (2021) first estimated weights of the considered evaluation criteria using bwm and interval type-2 fuzzy numbers, and later ranked the green suppliers for a textile industry based on todim and interval type-2 fuzzy numbers. product design and pattern suitability, purchase cost, dye and print quality, profit, and variation in price were identified as the most significant sub-criteria. based on this literature review of the applications of different mcdm techniques in solving textile supplier selection problems, it can be noticed that the past researchers have endeavored to mainly integrate fuzzy theory and grey theory with different mcdm tools to rank the suppliers from the best to the worst under uncertain decision making environment. this paper proposes an integrated approach combining irn, bwm and edas methods for solving a supplier selection problem in an indian textile mill. to the best of the authors’ knowledge, till date, there has been no application of irn-bwm-edas method for solving supplier selection problems in textile industries. table 1: literature review on mcdm-based supplier selection in textile industries author(s) no. of suppliers criteria mcdm tool(s) other tool(s) hlyal et al. (2015) 5 cost, quality, logistics efficiency, production capacity, social climate, versatility ahp sasi and digalwar (2015) 2 quality, labor and pollution rules, product variety, transportation facility, raw material cost, labor cost, counterpart flexibility, research background, export cost, degree of specialization, international relation, flexibility in production, number of production centers, dependency on import ahp, topsis kara et al. (2016) 3 basic requirements, performance requirement, attractive service requirement anp shukla (2016) 3 cost, quality, reliability, delivery, flexibility ahp ayvaz and kuşakcı (2017) 4 cost, delivery performance, customer relationships, payment options, technical capability topsis fuzzy theory table 1: contd. r&d rate, productivity, gross profit rate, paul et al./decis. mak. appl. manag. eng. 5 (2) (2022) 219-240 222 jing (2018) 12 quantity discount, inventory turnover ratio, return rate, discount rate, operating expense rate topsis fuzzy theory, dea bakhat and rajaa (2019) 7 quality, cost, technological capability, technical support, delivery, flexibility, supplier reputation, discount opportunities ahp, waspas grey theory guarnieria and trojan (2019) 10 ability to fulfil customers’ requirements, quality, on-time delivery, technological capacity, accordance with the law, continuous improvement, environmental impact, managing hazardous waste, environmental management electre copeland method, ahp burney and ali (2019) 4 cost, quality, service, delivery, payment terms ahp fuzzy theory wang et al. (2020) 10 reliability, responsiveness, flexibility, cost, assets prometheeii, ahp fuzzy theory karami et al. (2020) 12 quality, price, location, lead time, monetary position, financial position, on-time delivery, ability to product change, support and service, technical capacity vikor pca, dea ersoy and dogan (2020) 16 price, quality, delivery, reliability, inventory availability, flexibility, pollution rate of the raw material ahp fuzzy theory, dea ali et al. (2020) 5 cost, quality, service, delivery, payment terms ahp, topsis fuzzy theory mondragon et al. (2021) 1 technology used by the suppliers, technology used by the customers, automation, rapid manufacturing, capacity, reduced cycle time, cost, roi, supply chain performance, on-time delivery, skill, environmental impact ahp fuzzy theory utama et al. (2021a) 8 company profile, quality, cost, delivery, service, environment moora ahp sarıçam and yilmaz (2021) 4 management and organization, usage of upto-date technology and equipment, quality system and certification, geographical location, product price, seamless production, product quality, follow up, lead time, technical capability, accuracy, reliability ahp, topsis dea celik et al. (2021) 3 environmental, social, quality, risk, cost/price, capability, business structure bwm, todim interval type-2 fuzzy number utama et al. (2021b) 3 price, quality, conformance to specifications, on-time delivery, appropriateness of quantities, replacement of damaged goods, performance history, flexibility, eco-friendly material, permittance, delivery cost, mode of transportation, capability, environmental certificate, payment method anp dematel this paper 4 cost, quality, delivery, technical support, payment terms, flexibility edas irn, bwm 3. methods 3.1 irn let us assume a supplier selection problem involving k experts specifying their preferences in the form of a decision matrix x = [xijk]m×n using a predefined scale, where m and n are the numbers of alternative suppliers and criteria respectively, and xijk represents the preference of kth expert for ith an integrated irn-bwm-edas method for supplier selection in a textile industry 223 alternative against jth criterion. the preference of kth expert is expressed in the form of rns as  .,  kij k ij k ij xxx thus, the initial decision matrix evaluating m alternatives against n criterion by k th decision maker (1 ≤ e ≤ k) can be expressed as below:                   ),(...),(),( ............ ),(...),(),( ),(...),(),( 2211 2222222121 1112121111 e m n e m n e m e m e m e m e n e n eeee e n e n eeee e xxxxxx xxxxxx xxxxxx x (1) there is a set of k classes of expert’s preferences },...,,{ 21   kxxxx satisfying the condition }....{ 21   kxxx there is also another set of k classes of expert’s preferences }.,...,,{ 21   kxxxx now, an interval can be defined in each class ,,;1;];,[ rxxmixxxxx u i l i u i l i u i l ii   where l ix and u ix represent the lower and upper boundaries of ith class respectively. suppose that x is a universe containing all objects and x is an arbitrary object in x. if the lower and upper interval limits are sequenced as follows: u k uul l ll xxxxxx  ...,;..., 2121 (1 ≤ l, k ≤ m), the above sequences can then be denoted as two sets: a) a set of lower classes },,...,,{ 21 l i lll xxxx  and a set of upper classes },...,,{ 21 u i uuu xxxx  ).1,and1,( kixxlixx uu i ll i  the lower and upper approximations of l ix and u ix can be described as follows (chattopadhyay et al., 2022; ghosh et al., 2022). a) lower approximation:  li ll i xxxxxxapr  )(/)( (2)  ui uu i xxxxxxapr  )(/)( (3) b) upper approximation:  li ll i xxxxxxapr  )(/)( (4)  ui uu i xxxxxxapr  )(/)( (5) now, the lower and upper limits of l ix and u ix can be defined as below: a) lower limit:    ln b l i bl i bl i l l i xaprxx n xlim 1 )( 1 )( (6)    * 1* )( 1 )( ln b u i bu i bu i l u i xaprxx n xlim (7) b) upper limit:    un b l i bl i bl i u l i xaprxx n xlim 1 )( 1 )( (8)    * 1* )( 1 )( un b u i bu i bu i u u i xaprxx n xlim (9) where nl and nl* are the numbers of objects contained in lower approximations of the classes of objects l ix and u ix respectively, and where nu and nu * are the numbers of objects contained in upper approximations of the classes of objects l ix and u ix respectively. then, the corresponding irn can be defined using the following expression (pamučar et al., 2017): paul et al./decis. mak. appl. manag. eng. 5 (2) (2022) 219-240 224     ),(),,())(),(()),(),(( )(),()( u i l i u i l i u i u i l i l i u i l ii xxxxxlxlxlxl xrnxrnxirn    (10) thus, irns can effectively represent both uncertainty and imprecision in a decision making process. to illustrate its numerical formulations, let us assume a group decision making situation where three experts require to qualitatively evaluate a specific criterion (attribute) based on a 1-5 scale. suppose, expert e1 assigns a score 3-4, expert e2 appraises the importance of that criterion with a score of 4-5 and expert e3 assigns a value of 4 to that criterion. thus, two of the experts (e1 and e2) are not sure of their opinions, whereas, the other expert (e3) perfectly judges the importance of the considered criterion. these experts’ preferences on criterion importance can now be represented as: p(e1) = (3, 4), p(e2) = (4, 5) and p(e3) = (4, 4). based on the formulations of irns, two classes of objects xi´ and xi are formed as: xi´ = (3, 4, 4) and xi = (4, 5, 4). these object classes are now converted into two rough sequences,  ui l i xx  , and  ui l i xx , . thus, for the first class of objects: )4,7.3()4(4)4( ,7.3)443( 3 1 )4(),7.3,3()3(7.3)443( 3 1 )3(,3)3(     i u i l ii u i l i xx xxxx similarly, for the second class of objects: ),4,7.3()4(4)3(,7.3)443( 3 1 )4(  i u i l i xxx )5,3.4()5(5)5( ,3.4)544( 3 1 )5(),33.4,4()4(3.4)544( 3 1 )4(,4)4(   i u i l ii u i l i xx xxxx thus, the rns expressing the judgments of the three experts are converted into the following irns: irn(e1) = [(3, 3.7), (4, 4.3)], irn(e2) = [(3.7, 4), (4.3, 5)], irn(e3) = [(3.7, 4), (4, 4.3)] application of irns relieves involvement of the decision makers while abstracting complex problems and qualitatively evaluating them based on knowledge and common sense. use of additional intervals minimizes chances of losing information and provides greater scope to the decision makers to express their judgments more preciously without making biased decisions (yazdani et al., 2020). 3.2 irn-bwm the bwm, proposed by rezaei (2015), is an mcdm technique for criteria weight measurement, where the decision maker first identifies the best and the worst criteria, and subsequently develops two pairwise comparison vectors for the best and the worst criteria. the best criterion is considered to have the most important role in the decision making process, whereas, the worst criterion has the reverse role. using a pre-defined scale (e.g. 1-9), the decision maker evaluates the performance of the best criterion over all other criteria and the performance of all other criteria over the worst criterion. these two pairwise comparison vectors (i.e. bo and ow) are treated as the inputs to a linear programming model, which is finally solved to determine the optimal criteria weight values. as this method is based on only the best and the worst criteria for pair-wise comparisons, it requires fewer computational steps, while providing a clear understanding of the evaluation process, and more consistent and unbiased results (sadjadi and karimi, 2018; pamučar et al., 2020; khan et al., 2021; rodríguez-gutiérrez et al., 2021; hasan et al., 2022; srdjevic et al., 2022). in this paper, bwm is combined with irns to deal with uncertainty and ambiguity present while assigning the relative importance (weight) to the considered supplier selection criteria in a group decision making environment. integration of irns with bwm protects quality of the existing data by realistically describing expert’s preferences with respect to two matrixes, i.e. aggregated best-to-other (bo) and other-to-worst (ow). to take advantages of bwm, it has already been combined with different uncertainty theories in the literature, like fuzzy bwm (guo & zhao, 2017), intuitionistic fuzzy multiplicative bwm (mou et al., 2016), intuitionistic multiplicative preference bwm (you et al., 2016), intuitionistic preferences relation bwm (yang et al., 2016), interval-valued fuzzy-rough bwm (pamučar et al., 2018) and rough bwm (stević et al., 2017a; badi & ballem, 2018). the application of the proposed irn-bwm is illustrated using the following steps: an integrated irn-bwm-edas method for supplier selection in a textile industry 225 step 1: define a set of criteria for evaluating the alternatives. suppose there is a group of e experts in the decision making process, who defines the set of criteria c = {c1, c2,...,cn} (where n is the total number of criteria). step 2: define the best (b) and worst (w) criteria from the set c. the experts arbitrarily choose the b and w criteria. step 3: define the irnbo vector in which the experts represent their preferences comparing b criterion to the criteria in the set c = {c1, c2,...,cn}. the comparison of criterion b with other criterion in c is expressed through the advantage of criterion b over criterion j (j = 1,2,...,n), i.e. ).1(),( keaaa ue bj el bj e bj   as a result of this comparison, a vector )(bo e ba is obtained, where ),,;,;,( 2211 ue bn el bn ue b el b ue b el b e b aaaaaaa   ue bj el bj aa  and represent the advantage of criterion b over criterion j, .1and1  ue bb el bb aa so, for each e th expert, a bo matrix k b e bbb aaaa ,...,,...,, 21 is formed. these individual expert bo matrixes would be utilized to obtain an aggregated irnbo matrix (in step 5). step 4: define the irnow vector. each expert compares jth criterion to w criterion, whereby the advantage of jth criterion over criterion w is represented as ).1(),( keaaa ue jw el jw e jw   thus, a vector )(o e waw is obtained for e th expert, where ),,;,;,( 2211 ue nw el nw ue w el w ue w el w e w aaaaaaa   ue jw el jw aa  and denote the advantage of jth criterion over criterion w, .1and1  ue w w el w w aa thus, for each expert, a ow matrix k w e www aaaa ,...,,...,, 21 is framed. similar to the previous step, the individual ow matrixes are employed to derive an aggregated irnow matrix (in step 6). step 5: define the aggregated irnbo matrix of the expert’s opinions. based on individual expert’s bo matrix   ,, 1 n le bj el bj e b aaa    two separate matrixes el ba * and le ba * are formed in which the expert decisions are aggregated:   n kl bn l bn l bn kl b l b l b kl b l b l b el b aaaaaaaaaa  1 21 2 2 2 1 21 2 1 1 1 * ,...,,;,...,,;,...,, (11)   n uk bn u bn u bn uk b u b u b uk b u b u b ue b aaaaaaaaaa    1 21 2 2 2 1 21 2 1 1 1 * ,...,,;,...,,;,...,, (12) where },...,,{ 21 kl bj l bj l bj el bj aaaa  and },...,,{ 21 uk bj u bj u bj ue bj aaaa   represent advantage of criterion b over criterion j. after forming el ba * and ue ba * matrixes, each pair of sequences el bja and ue bja  is transformed into the corresponding irns, using eqs. (2)-(10),  ))((),((()),((),((()(  eubjelbjeubjelbjebj alalalalairn where )( el bjal and )( el bjal represent lower limits, and )( eu bjal and) )( eu bjal denote upper limits of )( e bjairn respectively. so for each sequence ),( e bjairn the corresponding bo matrixes )1(,...,...,, 21 keaaaa k b e bbb  are formed. now, by applying the irndwga operator, the average irn sequence is obtained. the aggregated irnbo matrix is expressed in eq. (13):   nbnbbb airnairnairna   121 )(),...,(),( (13) where      ubjlbjubjlbjbj aaaaairn ,,,)( presents average irns obtained using the following equation: paul et al./decis. mak. appl. manag. eng. 5 (2) (2022) 219-240 226 𝐼𝑅𝑁𝐷𝑊𝐺𝐴{𝐼𝑅𝑁(𝜑1), . . . , 𝐼𝑅𝑁(𝜑𝑛)} = [ { ∑ ∅𝑗 𝐿−𝑛 𝑗=1 1+{∑ 𝑤𝑗 𝑛 𝑗=1 { 1−𝑓(∅ 𝑗 𝐿−) 𝑓(∅ 𝑗 𝐿−) } 𝜌 } 1 𝜌⁄ , ∑ ∅𝑗 𝑈−𝑛 𝑗=1 1+{∑ 𝑤𝑗 𝑛 𝑗=1 { 1−𝑓(∅ 𝑗 𝑈−) 𝑓(∅ 𝑗 𝑈−) } 𝜌 } 1 𝜌⁄ } { ∑ ∅𝑗 𝐿+𝑛 𝑗=1 1+{∑ 𝑤𝑗 𝑛 𝑗=1 { 1−𝑓(∅ 𝑗 𝐿+) 𝑓(∅ 𝑗 𝐿+) } 𝜌 } 1 𝜌⁄ , ∑ ∅𝑗 𝑈+𝑛 𝑗=1 1+{∑ 𝑤𝑗 𝑛 𝑗=1 { 1−𝑓(∅ 𝑗 𝑈+) 𝑓(∅ 𝑗 𝑈+) } 𝜌 } 1 𝜌⁄ } ] (14) step 6: define the aggregated irnow matrix of the expert’s opinions. similar to step (5), two separate matrixes el wa * and ue wa * are formed on the basis of individual expert’s ow matrixes   ., 1 n ue jw el jw e w aaa      n ml wn l wn l wn ml w l w l w ml w l w l w el w aaaaaaaaaa  1 21 2 2 2 1 21 2 1 1 1 * ,...,,;,...,,;,...,, (15)   n um wn u wn u wn um w u w u w um w u w u w ue w aaaaaaaaaa    1 21 2 2 2 1 21 2 1 1 1 * ,...,,;,...,,;,...,, (16) where },...,,{ 21 ml nw l jw l jw el iw aaaa  and },...,,{ 21 um nw u jw u jw ue jw aaaa   denote advantage of criterion j over criterion w. by applying eqs. (2)-(10), each pair of sequences el jwa and ue jwa  is transformed into:  ))((),((()),((),((()(  euiweliweuiweliwejw alalalalairn sequence, where )( eljwal and )( eljwal represent lower limits, while )( eu jwal and )( eu jwal represent upper limits of )( e jwairn sequence, respectively. so, for each )( e jwairn sequence, the ow matrixes )1(,...,,...,, 21 keaaaa k w e www  are obtained. as in the previous step, applying irndwga operator, the following aggregated irn sequences are achieved:   nnwwww airnairnairna   121 )(),...,(),( (17) where     _,,,)( ujwljwujwljwjw aaaaairn  is the average irns obtained using irndwga operator. now, based on the aggregate values of irnbo and irnow matrixes, a nonlinear model for calculating optimal values of the weight coefficients is formed, as presented in the next step. the irndwga operator is chosen in this paper due to its minimum number of operational parameters and flexibility against changing values of those parameters. step 7: calculate the optimal values of criteria weights. by solving the following set of equations, the irn values of criteria weights are derived (rezaei, 2015). min ξ subject to ;    u bju j l b a w w ;    l bjl j u b a w w ;    u bju j l b a w w ;    l bjl j u b a w w ;    u jwu w l j a w w ;    l jwl jw u j a w w ;    u jwu w l j a w w     l jwl w u j a w w (18) ,1 1   n j l jw ,1 1   n j l jw ,1 1   n j u jw ,1 1   n j u jw ,   u j u j l j l j wwww njwwww u j u j l j l j ,...,2,1,0,,,   where  ),(),,()(  ujljujljj wwwwwirn represents the optimal value of weight coefficient,      ujwljwujwljwjw aaaaairn ,,,)( and      ubjlbjubjlbjbj aaaaairn ,,,)( are the values from irnow and irnbo matrixes respectively. an integrated irn-bwm-edas method for supplier selection in a textile industry 227 step 8: check the level of consistency for irn-bwm method-based weight coefficients. since the expert’s comparisons captured by irnbo and irnow matrixes are adopted to define the above model, a check is required for consistency of the comparisons. it also represents validation of the criteria weight coefficients. an expression can be defined to represent minimum consistency in the irn-bwm model. since there is a requirement that ,   u bw u bw l bw l bw aaaa the advantage of the best criterion over the worst criterion cannot be greater than . u bwa thus, the upper limit u b wa can be considered to fix the value of consistency index (ci) and all the variables related to )( bwairn can employ ci to calculate the consistency ratio (cr). thus, it can be concluded that the ci which corresponds to u b wa would take the maximum value in the interval [ l b wa , u b wa ]. based on this assumption, eq. (19) can be framed to determine the ci value. 0)()21( 2   u bw u bw u bw aaa  (19) now, the cr can be expressed using the following equation: ci cr *   (20) where cr [0, 1] and ξ* is the optimal consistency index. 3.3 irn-edas the edas method (ghorabaee et al., 2015) belongs to the group of mcdm techniques overcoming some of the drawbacks of the traditional topsis method. in topsis method, the best alternative should be positioned nearest to the ideal solution and farthest from the anti-ideal solution. identifying the ideal and anti-ideal solutions in a given decision making problem appears to be quite difficult as there may be no alternative having all of its best beneficial criteria and worst non-beneficial criteria. on the other hand, the desirability of an alternative in edas method is estimated based on its distance from the average solution which is the arithmetic mean of criteria values for the considered alternatives. this method has excellent efficiency, requiring fewer computational steps as compared to other mcdm techniques. in a short time, it has become a popular technique in solving both engineering and managerial decision making problems, like machine selection (ulutaş, 2017), materials selection (chatterjee et al., 2018; dhanalakshmi et al., 2022), evaluation of the performance of steam boilers (kundakcı, 2019), selection of cotton fabrics (mitra, 2022)), grading of jute fibres (mitra, 2021), industrial robot selection (rashid et al., 2021), parametric optimization of a wire electrical discharge machining process (okponyia and oke, 2021), evaluation of alternative facility locations (el-araby et al., 2022) etc. it has also a large number of extensions, like fuzzy edas (ghorabaee et al., 2016), interval grey edas (stanujkic et al., 2017), picture fuzzy edas (zhang et al., 2019), rough edas (stević et al., 2017b), interval-valued pythagorean fuzzy edas (yanmaz et al., 2020) etc. the procedural steps of irn-edas method are presented as below: step 1: develop the initial decision matrixes based on the judgments of k experts appraising the performance of m alternatives against n criteria in the form of irns. step 2: transform the individual decision matrixes into a group irn matrix.              )(...)()( ............ )(...)()( )(...)()( )( 21 22221 11211 mnmm n n ij xirnxirnxirn xirnxirnxirn xirnxirnxirn xirn (21) step 3: calculate an average solution by forming an irn(avj).   nm u j l j u j l jj avavavavavirn    ),(),,()( (22) the values of irn(avj) can be determined by applying the following equation: paul et al./decis. mak. appl. manag. eng. 5 (2) (2022) 219-240 228                     m xirn m xirn m xirn m xirn m xirn u ij l ijm i u ij l ijm i ij )( , )( , )( , )()( 11 (23) step 4: calculate the positive distance irn(pdaij) and negative distance irn(ndaij) matrixes in relation to the average solution irn(avj) for all criteria.     nm u j l j u j l jij pdapdapdapdapdairn    ,,,)( (24)     nm u j l j u j l jij ndandandandandairn    ,,,)( (25) to obtain elements of these matrixes, it is necessary to take into account the type of criterion (beneficial or non-beneficial) in the supplier selection problem.                                          l ij u ij u ij l ij l ij u ij u ij l ij l ij u ij u ij l ij l ij u ij u ij l iju ij l ij u ij l ijij av c av c av c av c av b av b av b av b pdapdapdapdapdairn ,,,or,,,,,,)( (26)         ljuijujlijljuijujlijuijlijuijlijij avxavxavxavxbbbbbirn   ,,,,0max,,,)( (27)         lijujuijljlijujuijljuijlijuijlijij xavxavxavxavcccccirn   ,,,,0max,,,)( (28)                                           l ij u ij u ij l ij l ij u ij u ij l ij l ij u ij u ij l ij l ij u ij u ij l iju ij l ij u ij l ijij av c av c av c av c av b av b av b av b ndandandandandairn ,,,or,,,,,,)( (29)         lijujuijljlijujuijljuijlijuijlijij xavxavxavxavbbbbbirn   ,,,,0max,,,)( (30)         ljuijujlijljuijujlijuijlijuijlijij avxavxavxavxcccccirn   ,,,,0max,,,)( (31) where ijb belongs to the set of beneficial criteria and ijc belongs to the set of non-beneficial criteria. step 5: multiply the irn matrixes irn(pdaij) and irn(ndaij) by the corresponding criteria weights.      ujuijljlijujuijljlijnm u j l j u j l jij wpdawpdawpdawpdavpvpvpvpvpirn     ,,,,,,)( (32)      ujuijljlijujuijljlijnm u j l j u j l jij wndawndawndawndavnvnvnvnvnirn     ,,,,,,)( (33) step 6: calculate sums of the weighted irn matrixes,     )(,,,)( 1    n i ij u i l i u i l ii vpirnspspspspspirn (34)     )(,,,)( 1    n i ij u i l i u i l ii vnirnsnsnsnsnsnirn (35) step 7: calculate the normalized values for the matrixes.                     l i u i u i l i l i u i u i l i i iu ij l ij u ij l iji sp sp sp sp sp sp sp sp irn(sp spirn nspnspnspnspnspirn max , max , max , max )max )( ,,,)( (36)                      l i u i u i l i l i u i u i l i i iu ij l ij u ij l iji sn sn sn sn sn sn sn sn irn(sn snirn nsnnsnnsnnsnnsnirn max , max , max , max 1 )max )( 1,,,)( (37) step 8: calculate the appraisal scores irn(asi) of all the alternatives.              2 )()( ,,,)( ii u i l i u i l ii nsnirnnspirn asasasasasirn (38) an integrated irn-bwm-edas method for supplier selection in a textile industry 229 step 9: rank the considered alternatives based on the converted crisp values of irn(asi). any two irns, i.e.  ],[],,[)( ui l i u i l i xxxxirn   and  ],[],,[)( ui l i u i l i xxxxirn   can be ranked using their points of intersection i(α) and i(β), while satisfying the following two conditions: (a) if i(α) < i(β), then irn(α) < irn(β) (b) if i(α) > i(β), then irn(α) > irn(β) for a decision making problem considering four alternatives, the corresponding intersection points can be obtained using the following equations: l i u ili l i u iui liui ui xxrbxxrb rbrb rb     )(;)(; )()( )(     (39) l i u ili l i u iui liui ui xxrbxxrb rbrb rb     )(;)(; )()( )(     (40) l i u ili l i u iui liui ui xxrbxxrb rbrb rb     )(;)(; )()( )(     (41) l i u ili l i u iui liui ui xxrbxxrb rbrb rb     )(;)(; )()( )(     (42) u i l i xxi   )1()(   (43) u i l i xxi   )1()(   (44) u i l i xxi   )1()(   (45) u i l i xxi   )1()(   (46) 4. irn-bwm-edas-based supplier selection for an indian textile industry this section demonstrates the application of the proposed integrated methodology for selecting the most apposite supplier engaged in providing cotton bales in an indian textile mill. in this supplier selection process under group decision making environment, involvement of four experts is considered. they are respectively engaged in the purchasing (12 years industrial experience having master’s in business administration degree), blowroom (20 years experience with a bachelor’s degree in textile technology), spinning (carding, speed frame and ring frame) (10 years experience possessing a bachelor’s degree in textile technology) and quality control (8 years of experience with master’s degree in textile technology) departments of the said textile mill. the supplier selection problem is solved herein-under using irn-bwm-edas approach through the adoption of the following steps: step 1: identify the relevant evaluation criteria. based on the literature review (table 1) and valued opinions of the participating experts, six evaluation criteria, as provided in table 2, are considered for solving this supplier selection problem. table 2: evaluation criteria for supplier selection in a textile mill criteria symbol description cost c1 it is the net price offered by a supplier. the procurement decision is usually made based on the minimum price for a particular item. quality c2 it can be defined as the ability of a supplier to consistently meet and maintain the quality specifications. any deviation in the specified quality level may adversely affect the production processes leading to loss of goodwill of the organization. delivery c3 it is the ability of a supplier to meet the specified delivery schedule. strict adherence to the delivery schedule is highly recommended to paul et al./decis. mak. appl. manag. eng. 5 (2) (2022) 219-240 230 maintain proper inventory level in order to streamline all the production processes. technical support c4 it can be described as the capability of a supplier to upkeep itself with the advanced technologies to support the procuring organization. the supplier must be aware of all the cutting edge technologies, products and services to meet the ever-changing requirements of the organizations. payment terms c5 it deals with different payment-related terms, like payment in advance, consequences of late payment and delivery, payment disputes etc., to be taken into consideration when a purchase order is placed to a supplier. it also takes into account the ability of a supplier to manage the letter of credit, collection of documents, opening of accounts etc. flexibility c6 it refers to the capability of a supplier to quickly respond to the changing demands of the buying organization with respect to delivery, volume and product design. it can be treated as a tool to cope with the environmental uncertainties. besides providing the actual items, a flexible supplier may also be capable to deal with supplying/processing other items. step 2: identification of the best and the worst criteria. after defining the most important evaluation criteria for this problem, all the four experts (e1, e2, e3 and e4) unanimously decide criterion c1 (cost) and criterion c5 (payment terms) as the best (b) and the worst (w) criteria respectively. if there are discrepancies in opinions among the experts with respect to identification of the best and the worst criteria, separate bo and ow vectors would be formed leading to different weight information of the considered criteria. these varying weights expressed in the form of irns would later be aggregated together using a suitable operator to derive a common criteria weight set for its subsequent application. step 3: formation of the bo and ow vectors for each of the experts. based on the identified best and the worst criteria, each of the experts now appraises the relative importance of the remaining criteria with respect to the best and the worst criteria, leading to the formation of bo and ow vectors, as exhibited in table 3. these judgments are initially expressed in terms of rns based on a 1-9 scale to resolve the uncertainty and ambiguity present in the group decision making environment. it is worthwhile to mention here that in this problem, equal importance is assigned to each of the experts. table 3: bo and ow vectors criteria evaluation criteria evaluation best: c1 e1 e2 e3 e4 worst: c5 e1 e2 e3 e4 c2 (3,4) (3, 5) (2, 3) (4, 5) c1 (5, 6) (5, 7) (4, 5) (3, 4) c3 (7, 9) (5, 7) (6, 7) (8, 9) c2 (8, 9) (7, 8) (5, 8) (7, 9) c4 (5, 6) (5, 7) (4, 5) (3, 4) c3 (6, 7) (6, 9) (5, 6) (8, 9) c5 (6, 7) (6, 9) (5, 6) (8, 9) c4 (3, 4) (3, 5) (2, 3) (4, 5) c6 (8, 9) (7, 8) (5, 8) (7, 9) c6 (7, 9) (5, 7) (6, 7) (8, 9) step 4: based on the mathematical steps, as mentioned in sub-section 3.1, the decisions of the four experts with respect to bo and ow vectors are now transformed into corresponding irnbo and irnow vectors, as depicted in tables 4 and 5 respectively. for example, in bo vector for criterion c3, p(e1) = (7, 9), p(e2) = (5, 7), p(e3) = (6, 7) and p(e4) = (8, 9), which lead to the formation of two classes of objects xi´ and xi as: xi´ = (7,5,6,8) and xi = (9,7,7,9). thus, for the first class of objects: )8,5.6()8(8)8(,65.6)8765( 4 1 )8( )5.7,6()7(5.7)87( 2 1 )7(,6)576( 3 1 )7( ),7,5.5()6(7)876( 3 1 )6(,5.5)65( 2 1 )6( ),5.6,5()5(5.6)8765( 4 1 )5(,5)5(         i u i l i i u i l i i u i l i i u i l i xxx xxx xxx xxx similarly, for the second class of objects: an integrated irn-bwm-edas method for supplier selection in a textile industry 231 ),8,7()7(8)9977( 4 1 )7(,7)77( 2 1 )7(  i u i l i xxx ).9,8()9(9)99( 2 1 )9(,8)9977( 4 1 )9(  i u i l i xxx thus, irn(e1) = [(6,7.5), (8,9)], irn(e2) = [(5,6.5), (7,8)], irn(e3) = [(5.5,7), (7,8)] and irn(e4) = [(6.5,8), (8,9)]. table 4: bo vector in terms of irns best : c1 e1 e2 e3 e4 c2 [(2.67,3.33), (3.50,4.67)] [(2.67,3.33), (4.5,5.00)] [(2.00,3.00), (3.00,4.25)] [(3.00,4.00), (4.5,5.00)] c3 [(6.00,7.50), (8.00, 9.00)] [(5.00,6.50), (7.00,8.00)] [(5.50,7.00), (7.00,8.00)] [(6.50,8.00), (8.00,9.00)] c4 [(4.25,5.00), ( 5.00,6.50)] [(4.25,5.00), (5.50,7.00)] [(3.50,4.67), (4.50,6.00)] [(3.00,4.25), (4.00,5.50)] c5 [(5.67,6.67), (6.50,8.33)] [(5.67,6.67), (7.75,9.00)] [(5.00,6.25), (6.00,7.75)] [(6.25,8.00), (7.75,9.00)] c6 [(6.75,8.00), (8.50,9.00)] [(6.33,7.33), (8.00,8.50)] [(5.00,6.75), (8.00,8.50)] [(6.33,7.33), (8.50,9.00)] table 5: ow vector in terms of irns worst : c5 e1 e2 e3 e4 c1 [(4.25,5.00), (5.00,6.50)] [(4.25,5.00), (5.50,7.00)] [(3.50,4.67), (4.50,6.00)] [(3.00,4.25), (4.00,5.50)] c2 [(6.75,8.00), (8.50,9.00)] [(6.33,7.33), (8.00,8.50)] [(5.00,6.75), (8.00,8.50)] [(6.33,7.33), (8.50,9.00)] c3 [(5.67,6.67), (6.50,8.33)] [(5.67,6.67), (7.75,9.00)] [(5.00,6.25), (6.00,7.75)] [(6.25,8.00), (7.75,9.00)] c4 [(2.67,3.33), (3.50,4.67)] [(2.67,3.33), (4.5,5.00)] [(2.00,3.00), (3.00,4.25)] [(3.00,4.00), (4.5,5.00)] c6 [(6.00,7.50), (8.00,9.00)] [(5.00,6.50), (7.00,8.00)] [(5.50,7.00), (7.00,8.00)] [(6.50,8.00), (8.00,9.00)] step 5: development of the aggregated irnbo and irnow vectors. using the irndwga operator of eq. (14), the irnbo and irnow vectors are aggregated into unique irn vectors considering equal importance to all the four experts, as shown in table 6. the calculation steps to convert the irns for criterion c3 in the bo vector of table 4 into the corresponding aggregated irns are presented as below:                                                                                                                                    26.9 26.0 26.01 25.0... 28.0 28.01 25.01 34 12.7 24.0 24.01 25.0... 24.0 24.01 25.01 30 59.6 24.0 24.01 25.0... 22.0 22.01 25.01 29 04.6 26.0 26.01 25.0... 26.0 26.01 25.01 23 )( 31 31 31 31 31 u l u l x x x x xirndwga table 6: aggregated irn bo and ow vectors best : c1 irn bo worst: c5 irn ow paul et al./decis. mak. appl. manag. eng. 5 (2) (2022) 219-240 232 c2 [(2.51,3.59),(3.21,5.37)] c1 [(4.01,5.28),(4.54,5.33)] c3 [(6.04,6.59),(7.12,9.26)] c2 [(6.48,7.30),(7.55,8.95)] c4 [(4.01,5.28),(4.54,5.33)] c3 [(5.47,7.11),(6.22,9.43)] c5 [(5.47,7.11),(6.22,9.43)] c4 [(2.51,3.59),(3.21,5.37)] c6 [(6.48,7.30),(7.55,8.95)] c6 [(6.04,6.59),(7.12,9.26)] step 6: determine the optimal values of criteria weights. based on the aggregated irnbo and irnow vectors, the following optimization problem is framed, which is subsequently solved using lindo 19 software to estimate the optimal criteria weights. the derived irn-based criteria weights are provided in table 7. while solving this problem, the ξ* value is attained as 0.18 and the corresponding ci for n = 6 is 3.00 (rezaei, 2015). thus, the cr value becomes 0.18/3.00 = 0.06 symbolizing excellent consistency in the derived criteria weights. min ξ subject to ;12.7;26.9;51.2;59.3;21.3;37.5 332222      l u b u l b l u b u l b l u b u l b w w w w w w w w w w w w  ;01.4;28.5;54.4;33.5;04.6;59.6 444433      l u b u l b l u b u l b l u b u l b w w w w w w w w w w w w ;55.7;95.8;47.5;11.7;22.6;43.9 665555      l u b u l b l u b u l b l u b u l b w w w w w w w w w w w w ;01.4;28.5;54.4;33.5;48.6;30.7 1111 66      l w u u w l l w u u w l l u b u l b w w w w w w w w w w w w ;22.6;43.9;48.6;30.7;55.7;95.8 332222      l w u u w l l w u u w l l w u u w l w w w w w w w w w w w w ;51.2;59.3;21.3;37.5;47.5;11.7 444433      l w u u w l l w u u w l l w u u w l w w w w w w w w w w w w ;04.6;59.6;12.7;26.9 6666     l w u u w l l w u u w l w w w w w w w w ;1;1;1;1 6 1 6 1 6 1 6 1       j u j j u j j l j j l j wwww ; u j u j l j l j wwww   6,...,2,1,0,,,   jwwww u j u j l j l j table 7: optimal criteria weights criteria irn weights c1 [(0.280, 0.365), (0.220, 0.342)] c2 [(0.142, 0.180), (0.140, 0.168)] c3 [(0.038, 0.065), (0.028, 0.061)] c4 [(0.221, 0.210), (0.202, 0.150)] c5 [(0.025, 0.050), (0.015, 0.030)] c6 [(0.112, 0.131), (0.110, 0.122)] step 7: appraisal of the relative performance of the competing suppliers with respect to the considered evaluation criteria by each of the experts. as the initial step of irn-edas method, all the four experts now evaluate the performance of the suppliers against each criterion in terms of rns, as provided in table 8. these rn-based evaluation scores are later converted into irn-based scores, as shown in table 9. table 8: individual expert’s responses while evaluating the suppliers an integrated irn-bwm-edas method for supplier selection in a textile industry 233 e1 supplier criteria c1 c2 c3 c4 c5 c6 s1 (3, 4) (2, 5) (6, 7) (4, 6) (8, 9) (5, 6) s2 (6, 7) (3, 6) (5, 6) (8, 9) (2, 5) (2, 4) s3 (4, 7) (5, 7) (7, 8) (3, 4) (6, 7) (1, 2) s4 (3, 5) (6, 7) (2, 5) (4, 7) (3, 4) (4, 6) e2 supplier criteria c1 c2 c3 c4 c5 c6 s1 (4, 7) (5, 7) (7, 8) (3, 4) (6, 7) (1, 2) s2 (3, 5) (6, 7) (2, 5) (4, 7) (3, 4) (4, 6) s3 (6, 7) (3, 6) (5, 6) (8, 9) (2, 5) (2, 4) s4 (3, 4) (2, 5) (6, 7) (4, 6) (8, 9) (5, 6) e3 supplier criteria c1 c2 c3 c4 c5 c6 s1 (3, 5) (6, 7) (2, 5) (4, 7) (3, 4) (4, 6) s2 (3, 4) (2, 5) (6, 7) (4, 6) (8, 9) (5, 6) s3 (4, 7) (5, 7) (7, 8) (3, 4) (6, 7) (1, 2) s4 (6, 7) (3, 6) (5, 6) (8, 9) (2, 5) (2, 4) e4 supplier criteria c1 c2 c3 c4 c5 c6 s1 (3, 4) (2, 5) (6, 7) (4, 6) (8, 9) (5, 6) s2 (6, 7) (3, 6) (5, 6) (8, 9) (2, 5) (2, 4) s3 (3, 5) (6, 7) (2, 5) (4, 7) (3, 4) (4, 6) s4 (4, 7) (5, 7) (7, 8) (3, 4) (6, 7) (1, 2) step 8: formation of the aggregated irn-edas matrix using irndwga operator. the individual decision matrixes for the four participating experts in terms of irns are now aggregated using irndwga operator to form the corresponding irn matrix, as shown in table 10. table 9: irn matrix for irn-edas method e1 supplier criteria c1 c2 c3 c4 c5 c6 s1 [2.50,5.20], [4.00,6.16] [2.00,4.67], [3.50,6.60] [4.00,7.00], [5.60,8.00] [3.00,5.75], [5.25,7.00] [4.67,8.00], [6.16,9.00] [3.50,6.33], [5.25,7.00] s2 [3.60,7.00], [5.60,8.00] [2.33,5.50], [5.25,7.00] [3.00,6.33], [5.25,7.00] [4.33,8.00], [6.16,9.00] [2.00,4.33], [3.50,6.60] [2.00,4.33], [4.00,6.16] s3 [2.67,5.50], [5.40,7.25] [3.25,6.00], [5.40,7.25] [4.33,7.00], [5.83,8.00] [2.00,5.00], [3.00,6.60] [3.80,6.50], [5.40,7.25] [1.00,4.33], [2.00,5.83] s4 [2.67,4.00], [4.67,6.00] [3.67,6.00], [5.67,7.00] [2.00,3.67], [4.67,6.00] [3.20,4.67], [5.67,7.00] [2.67,4.00], [4.00,5.67] [3.20,4.67], [5.00,6.67] e2 supplier criteria c1 c2 c3 c4 c5 c6 s1 [2.67,5.50], [5.40,7.25] [3.25,6.00], [5.40,7.25] [4.33,7.00], [5.83,8.00] [2.00,5.00], [3.00,6.60] [3.80,6.50], [5.40,7.25] [1.00,4.33], [2.00,5.83] s2 [2.67,4.00], [4.67,6.00] [3.67,6.00], [5.67,7.00] [2.00,3.67], [4.67,6.00] [3.20,4.67], [5.67,7.00] [2.67,4.00], [4.00,5.67] [3.20,4.67], [5.00,6.67] s3 [3.60,7.00], [2.33,5.50], [3.00,6.33], [4.33,8.00], [2.00,4.33], [2.00,4.33], paul et al./decis. mak. appl. manag. eng. 5 (2) (2022) 219-240 234 [5.60,8.00] [5.25,7.00] [5.25,7.00] [6.16,9.00] [3.50,6.60] [4.00,6.16] s4 [2.50,5.20], [4.00,6.16] [2.00,4.67], [3.50,6.60] [4.00,7.00], [5.60,8.00] [3.00,5.75], [5.25,7.00] [4.67,8.00], [6.16,9.00] [3.50,6.33], [5.25,7.00] e3 supplier criteria c1 c2 c3 c4 c5 c6 s1 [2.67,4.00], [4.67,6.00] [3.67,6.00], [5.67,7.00] [2.00,3.67], [4.67,6.00] [3.20,4.67], [5.67,7.00] [2.67,4.00], [4.00,5.67] [3.20,4.67], [5.00,6.67] s2 [2.50,5.20], [4.00,6.16] [2.00,4.67], [3.50,6.60] [4.00,7.00], [5.60,8.00] [3.00,5.75], [5.25,7.00] [4.67,8.00], [6.16,9.00] [3.50,6.33], [5.25,7.00] s3 [2.67,5.50], [5.40,7.25] [3.25,6.00], [5.40,7.25] [4.33,7.00], [5.83,8.00] [2.00,5.00], [3.00,6.60] [3.80,6.50], [5.40,7.25] [1.00,4.33], [2.00,5.83] s4 [3.60,7.00], [5.60,8.00] [2.33,5.50], [5.25,7.00] [3.00,6.33], [5.25,7.00] [4.33,8.00], [6.16,9.00] [2.00,4.33], [3.50,6.60] [2.00,4.33], [4.00,6.16] e4 supplier criteria c1 c2 c3 c4 c5 c6 s1 [2.50,5.20], [4.00,6.16] [2.00,4.67], [3.50,6.60] [4.00,7.00], [5.60,8.00] [3.00,5.75], [5.25,7.00] [4.67,8.00], [6.16,9.00] [3.50,6.33], [5.25,7.00] s2 [3.60,7.00], [5.60,8.00] [2.33,5.50], [5.25,7.00] [3.00,6.33], [5.25,7.00] [4.33,8.00], [6.16,9.00] [2.00,4.33], [3.50,6.60] [2.00,4.33], [4.00,6.16] s3 [2.67,4.00], [4.67,6.00] [3.67,6.00], [5.67,7.00] [2.00,3.67], [4.67,6.00] [3.20,4.67], [5.67,7.00] [2.67,4.00], [4.00,5.67] [3.20,4.67], [5.00,6.67] s4 [2.67,5.50], [5.40,7.25] [3.25,6.00], [5.40,7.25] [4.33,7.00], [5.83,8.00] [2.00,5.00], [3.00,6.60] [3.80,6.50], [5.40,7.25] [1.00,4.33], [2.00,5.83] table 10: irn matrix for irn-edas method supplier criteria c1 c2 c3 c4 c5 c6 s1 [2.49,5.54], [4.26,6.15] [2.26,6.08], [5.30,5.67] [3.94,6.98], [3.73,7.47] [2.98,4.15], [4.99,7.36] [4.63,6.41], [3.71,9.05] [3.26,2.94], [4.48,7.71] s2 [3.58,4.79], [4.19,8.15] [2.57,6.26], [4.01,6.87] [3.07,4.35], [6.14,7.16] [4.23,5.57], [5.01,9.10] [2.32,4.44], [6.30,5.70] [2.29,5.21], [5.47,5.57] s3 [2.87,5.43], [5.21,5.97] [3.18,5.27], [5.53,7.49] [3.93,5.78], [6.21,5.08] [2.23,7.79], [3.45,7.40] [3.61,4.19], [5.39,5.57] [1.31,4.74], [2.37,7.96] s4 [2.47,4.79], [5.95,7.00] [3.16,4.38], [4.77,7.59] [2.36,6.76], [5.19,8.41] [2.91,5.80], [6.55,5.21] [2.57,7.65], [3.53,7.90] [2.71,6.22], [3.69,3.84] step 9: calculate the average solution by forming the irn(avj) matrix. based on the mathematical steps, as highlighted in sub-section 3.3, the average solutions are computed leading to the following matrix:                      ]27.6,00.4[],78.4,39.2[ ]06.7,73.4[],68.5,28.3[ ]27.7,00.5[],83.5,09.3[ ]03.7,32.5[],97.5,32.3[ ]91.6,90.4[],50.5,79.2[ ]82.6,90.4[],39.5,85.2[ )( javirn the calculations steps of the average solution for criterion 𝐶6 are shown as below: an integrated irn-bwm-edas method for supplier selection in a textile industry 235                               27.6 4 ]84.396.757.571.7[ 00.4 4 ]69.337.247.548.4[ 78.4 4 ]22.674.421.594.2[ 39.2 4 ]71.231.129.226.3[ )( 1 m i ij m xirn step 10: formulate the positive distance matrix irn(pdaij) and negative distance matrix irn(ndaij) in relation to the average solution irn(avij) for all the criteria. an example of calculation of these matrixes for element irn(pda46) = [0.00, 0.55], [0.00, 0.60] is provided as below:                            39.2 45.1 , 78.4 00.0 , 00.4 22.2 , 27.6 00.0 ,,,)( 46 46 46 46 46 46 46 46 46 l u u l l u u l av b av b av b av b pdairn where ]45.1,00.0[],22.2,00.0[],[],,[)( 4646464646   ulul bbbbbirn = max (0, [2.71 – 6.27, 6.22 – 4.00], [3.69 – 4.78, 3.84 – 2.39]) similarly, an example of calculation of these matrixes for element irn(nda46) = [0.39, 0.08], [0.00, 0.00] is shown as below:                            39.2 00.0 , 78.4 00.0 , 00.4 31.0 , 27.6 43.2 ,,,)( 46 46 46 46 46 46 46 46 46 l u u l l u u l av b av b av b av b ndairn where ]00.0,00.0[],31.0,43.2[],[],,[)( 4646464646   ulul bbbbbirn = max (0, [6.27 – 3.84,4.00 – 3.69 ], [4.78 – 6.22, 2.39 – 3.84]) step 11: develop the weighted positive distance and negative distance matrixes. here, irn(pdaij) and irn(ndaij) matrixes are multiplied by the corresponding criteria weights. an example of the corresponding calculation steps is provided as below: irn(vp46) = [0.00, 0.07], [0.00, 0.07] = [0.00×0.112, 0.55×0.131], [0.00×0.110, 0.60×0.122] irn(vn46) = [0.04, 0.01], [0.00, 0.00] = [0.39×0.112, 0.08×0.131], [0.00×0.110, 0.00×0.122] step 12: compute the sums of the weighted irn matrixes. an example of these calculation steps is as follows:                   26.107.004.010.026.029.050.0 05.000.000.003.000.000.002.0 19.007.003.004.005.000.000.0 12.000.000.007.000.000.005.0 )()( 6 1 46 j ijvpirnspirn                   60.000.003.001.004.000.052.0 02.000.000.001.000.001.000.0 07.001.003.000.000.003.000.0 11.005.000.006.000.000.000.0 )()( 6 1 46 j ijvnirnsnirn step 13: normalize the above matrixes. an example of these calculation steps is exhibited as below:              12.0 26.1 , 28.0 05.0 , 05.0 19.0 , 55.1 12.0 ]50.10,17.0[],80.3,08.0[)( 4nspirn              11.0 60.0 , 23.0 02.0 , 08.0 07.0 , 60.0 11.0 1]45.4,92.0[],125.0,82.0[)( 4nsnirn step 14: estimate irn(asi) values of all the alternative suppliers. the irn-edas method-based calculation of irn(asi) value for the fourth supplier is shown as follows: paul et al./decis. mak. appl. manag. eng. 5 (2) (2022) 219-240 236                2 45.450.10 , 2 92.017.0 , 2 125.080.3 , 2 82.008.0 ]03.3,54.0[],96.1,45.0[)( 4asirn the irn(asi) values of all the four competing suppliers are provided in table 11. using eqs. (39)-(46), these irn(asi) values are now converted into their corresponding crisp values which would lead to developing the condition as i(γ) > i(δ) > i(α) > i(β). this analysis reveals that for supplying cotton bales to the considered indian textile mill, supplier 3 is the most suitable choice, followed by supplier 4. in order to validate the performance of this integrated approach, the derived rank order of the considered suppliers is compared with that of other popular mcdm methods, like irn-bwm-waspas, irn-bwmmoora, irn-bwm-topsis and irn-bwm-vikor. it can be interestingly noticed that in all the considered integrated approaches, supplier 3 appears to be the best choice, while there are alternations in the positions of the remaining suppliers in the derived ranking lists. table 11: appraisal scores of the alternative suppliers supplier irn(asi) crisp value rank s1 [0.52, 1.63], [0.41, 3.32] 1.29 3 s2 [0.49, 0.73], [0.37, 4.82] 0.71 4 s3 [0.45, 2.57], [0.44, 3.06] 1.62 1 s4 [0.45, 1.96], [0.54, 3.03] 1.42 2 5. conclusions this paper proposes an integrated approach combining irn, bwm and edas methods for solving a supplier selection problem for an indian textile mill. for this purpose, six evaluation criteria, i.e. cost, quality, delivery, technical support, payment terms and flexibility, four alternative suppliers and four experts engaged in the purchasing, blowroom, spinning and quality control departments of the said mill are considered. at first, the relative importance assigned to different criteria by the experts is expressed in terms of irns which are aggregated together to estimate the corresponding optimal criteria weights using bwm. similarly, the performance of each of the competing suppliers with respect to the considered evaluation criteria is also expressed using irns. the aggregated irns for supplier performance evaluation are the inputs to edas method which would finally help in ranking those suppliers. based on this integrated approach, supplier 3 emerges out as the most apposite choice, followed by supplier 4. although it is a computationally extensive method, but it leads to more accurate and reliable solution while providing unbiased decision reducing the chances of losing information. one main limitation of this paper is that it does not consider effects of the changing values of different operational parameters in irndwga operator on the final solutions. the accuracy of the derived ranking results may be contrasted against other existing integrated mcdm approaches, like rough bwm-mairca, rough-mabac-doe, irn-swara-mabac etc. to ease out the computational steps involved in the approach, a decision support framework may be developed as a future scope of this paper. list of abbreviations ahp analytic hierarchy process anp analytic network process bwm best worst method cagr compound annual growth rate dea data envelopment analysis dematel decision making trial and evaluation laboratory doe design of experiments edas evaluation based on distance from average solution electre elimination et choice translating reality irn interval rough number irndwga irn dombi weighted geometric averaging mabac multi-attributive border approximation area comparison mairca multi-attributive real-ideal comparative analysis mcdm multi-criteria decision making moora multi-objective optimization on the basis of ratio analysis pca principal component analysis promethee preference ranking organization method for enrichment evaluation rn rough number roi return on investment swara step-wise weight assessment ratio analysis an integrated irn-bwm-edas method for supplier selection in a textile industry 237 todim tomada de decisao interativa multicriterio topsis technique for order of preference by similarity to ideal solution vikor viekriterijumsko kompromisno rangiranje waspas weighted aggregated sum product assessment author contributions: conceptualization, v.k.p. and santonab chakraborty; data collection and analysis, v.k.p; draft and calculations, santonab chakraborty; literature review and editing, shankar chakraborty funding: this research received no external funding. data availability statement: this paper has not used any data repository. conflicts of interest: the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. references acar, a.z., önden, i̇., & gürel, ö. 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(2019). edas method for multiple criteria group decision making with picture fuzzy information and its application to green suppliers selections. technological and economic development of economy, 25(6), 1123-1138. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 3, issue 1, 2020, pp. 126-145. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame2003114d * corresponding author. i.naric@yahoo.com, irena.djalic@sf.ues.rs.ba (i. đalić), zeljkostevic88@yahoo.com, zeljko.stevic@sf.ues.rs.ba (ž. stević), ckaramasa@hotmail.com (c. karamasa), adispuska@yahoo.com (a. puška) a novel integrated fuzzy piprecia–interval rough saw model: green supplier selection irena đalić 1, željko stević 1*, caglar karamasa 2 and adis puška 3 1 university of east srajevo, faculty of transport and traffic engineering doboj, bosnia and herzegovina 2 anadolu university, faculty of business administration, turkey 3 institute for scientific research and development, brčko district, bosnia and herzegovina received: 5 december 2019; accepted: 2 march 2020; available online: 14 march 2020. original scientific paper abstract: a novel integrated fuzzy–rough multi-criteria decision-making (mcdm) model based on integration fuzzy and interval rough set theories is presented. the model integrates the fuzzy pivot pairwise relative criteria importance assessment fuzzy piprecia and interval rough simple additive weighting (saw) methods. an illustrative example of the model demonstration is proposed, representing the evaluation and supplier selection based on nine environmental criteria. the fuzzy piprecia method is used to determine the significance of the following seven criteria: c1 – the environmental image, c2 – recycling, c3 – pollution control, c4 – the environmental management system, c5 – environmentally friendly products, c6 – resource consumption, and c7 – green competencies. the interval rough saw method is applied so as to evaluate four alternatives. the results show that the third criterion is most important, whereas the fourth alternative is the best solution. key words: fuzzy piprecia, interval rough saw method, supplier selection, environment. 1. introduction green supplier selection is one of the most important tasks for the functioning of the whole supply chain, especially for production companies. in this paper, an innovative integrated fuzzy–rough mcdm model is proposed for the evaluation of suppliers, based on environmental criteria. mcdm is an important and powerful tool for solving such problems, as is confirmed by stević et al. (2020): multi-criteria decision-making is quite an applicable tool for analyzing complex real problems mailto:i.naric@yahoo.com mailto:irena.djalic@sf.ues.rs.ba mailto:zeljkostevic88@yahoo.com mailto:zeljko.stevic@sf.ues.rs.ba mailto:ckaramasa@hotmail.com mailto:adispuska@yahoo.com a novel integrated fuzzy piprecia – interval rough saw model: green supplier selection 127 because of its ability to evaluate different alternatives by using certain criteria. there are a certain number of research studies of green supplier selection by using various mcdm methods. büyüközkan & çifçi (2012) used a combination of mcdm methods in order to evaluate green suppliers. qin et al. (2017) solved the problem of making a decision on green supplier selection by using a combination of mcdm methods. considering various environmental performance requirements and criteria, yazdani et al. (2017) introduced a new model, i.e. an integrated approach to green supplier selection. green supplier selection is carried out in various business areas. zhao & guo (2014) made the green supplier selection for a supplier of thermal power equipment by using mcdm methods. banaeian et al. (2018) made green supplier selection in the agri-food industry, while tsui & tzeng used the mcdm approach to improve the performance of green suppliers in the tft-lcd industry. uppala et al. (2017) used the mcdm approach to green supplier selection in an electronics company, whereas yu & hou (2016) conducted green supplier selection in the automotive manufacturing industry. from the economic and environmental aspects, chen et al. (2016) used the fuzzy mcdm approach to green supplier selection. the paper is aimed at taking the advantages of the implemented approaches and allowing for more accurate and balanced decision-making through their integration. the rest of the paper is structured as follows: in the second section, the applied methods are presented, i.e. the fuzzy piprecia and interval rough saw methods, and some basic operations with interval rough numbers are also shown; in the third section, the results obtained are demonstrated in detail, and the section is divided into two subsections; in the fourth section, the conclusion of the paper is given, inclusive of an emphasis on the advantages offered by the proposed integrated model. 2. methods 2.1. fuzzy pivot pairwise relative criteria importance assessment – the fuzzy piprecia method the main advantage of the piprecia (stanujkić et al. 2017) method is that it allows the evaluation of criteria without sorting them first by significance, which is not the case with the swara method (keršuliene et al. 2010; vesković et al. 2018). today, the largest number of multi-criteria decision-making problems are solved by applying group decision-making. in such cases, especially as the number of decision-makers involved in the fuzzy piprecia model increases, achieves its benefits. the fuzzy piprecia method was developed by stević et al. (2018). it consists of the 11 steps shown below. step 1. forming the required benchmarking set of criteria and forming a team of decision-makers. sorting the criteria according to the marks from the first to the last, which means they need to be sorted unclassified. therefore, their significance is irrelevant in this step. step 2. in order to determine the relative importance of the criteria, each decisionmaker individually evaluates the presorted criteria by starting from the second criterion, equation (1). 1 1 1 1 1 1 j j r j j j j j if c c s if c c if c c             (1) đalić et al./decis-mak. appl. manag. eng. 3 (1) (2020) 126-145 128 where r j s denotes the evaluation of the criteria by the decision-maker r. in order to obtain the matrix j s , it is necessary to perform the averaging of the matrix r j s by using the geometric mean. the decision-makers evaluate the criteria by applying the defined scales shown in tables 1 and 2. the second and third steps of the developed method are closely interdependence, and new fuzzy scales are defined so as to meet the second and third steps of the fuzzy piprecia method. if the fact that the nature of fuzzy number operations and the fact that, in the third step, the values are subtracted from two, it is then required that these scales should be define. it is important to note that, by defining these scales, the appearance of the number two is avoided, which might cause difficulties and wrong results when the calculation is concerned. therefore, no other previously used fuzzy scales could be used. only the scales defined in this paper are applicable. table 1. the 1-2 scale for the assessment of the criteria scale 1-2 l m u dfv an almost equal value 1 1.000 1.000 1.050 1.008 slightly more significant 2 1.100 1.150 1.200 1.150 moderately more significant 3 1.200 1.300 1.350 1.292 more significant 4 1.300 1.450 1.500 1.433 much more significant 5 1.400 1.600 1.650 1.575 dominantly more significant 6 1.500 1.750 1.800 1.717 absolutely more significant 7 1.600 1.900 1.950 1.858 when the criterion is of greater importance in relation to the previous one, an assessment is made by using the above-mentioned scale in table 1. in order to make it easier for the decision-makers to evaluate the criteria, the table shows the defuzzified value (dfv) for each comparison. table 2. the 0-1 scale for the assessment of the criteria scale 0-1 l m u dfv 0.667 1.000 1.000 0.944 weakly less significant 0.500 0.667 1.000 0.694 moderately less significant 0.400 0.500 0.667 0.511 less significant 0.333 0.400 0.500 0.406 really less significant 0.286 0.333 0.400 0.337 much less significant 0.250 0.286 0.333 0.288 dominantly less significant 0.222 0.250 0.286 0.251 absolutely less significant when the criterion is of lesser importance compared to the previous one, an assessment is made by using the above-mentioned scale in table 2. step 3. determining the coefficient jk 1 1 2 1 j j if j k s if j       . (2) j s a novel integrated fuzzy piprecia – interval rough saw model: green supplier selection 129 step 4. determining the fuzzy weight jq 1 1 1 1 jj j if j qq if j k          . (3) step 5. determining the relative weight of the criterion j w 1 j j n j j q w q    . (4) in the following steps, it is necessary to apply the inverse methodology of the fuzzy piprecia method. step 6. the evaluation of the applicable scale defined above, this time starting from the penultimate criterion. 1 1 1 1 ' 1 1 j j r j j j j j if c c s if c c if c c             . (5) ' r j s denotes the evaluation of the criteria by the decision-maker r. it is again necessary to average the matrix r j s by applying the geometric mean. step 7. determining the coefficient 'jk 1 ' 2 ' j j if j n k s if j n       . (6) n denotes a total number of the criteria. specifically, in this case, it means that the value of the last criterion is equal to the fuzzy number one. step 8. determining the fuzzy weight 'jq 1 1 '' ' jj j if j n qq if j n k          . (7) step 9. determining the relative weight of the criterion 'jw đalić et al./decis-mak. appl. manag. eng. 3 (1) (2020) 126-145 130 1 ' ' ' j j n j j q w q    . (8) step 10. in order to determine the final weights of the criteria, it is first necessary to perform the defuzzification of the fuzzy values jw and 'jw 1'' ( ') 2 j j j w w w  . (9) step 11. checking the results obtained by applying the spearman and pearson correlation coefficients. 2.2. interval rough numbers the process of group decision-making is accompanied by a large amount of uncertainty and subjectivity, so decision-makers often have dilemmas when assigning certain values to decision attributes. in this paper, a new approach in rough sets theory based on interval rough numbers (irn) is applied so as to process uncertainty contained in data in group decision-making. suppose that one decision attribute should be assigned a value represented by a qualitative scale, whose values range from 1 to 5. the first decision-maker (dm) may consider that the decision attribute should have a value ranging between 3 and 4, the second dm may consider that a value between 4 and 5 should be assigned, whereas the third dm has no dilemma about the value of the decision attribute and assigns it the value 4. the presented dilemmas are extremely common in the group decision-making process. in such situations, one of the solutions is to geometrically average two values, which individual decision-makers are in doubt which one to assign. in such situations, however, the uncertainty (ambiguity) that prevailed in the decision-making process would be lost, and a further calculation would be reduced to crisp values. on the other hand, the use of fuzzy or grey techniques would entail predicting the existence of uncertainty and subjectively defining the interval which such uncertainty is exploited by. subjectively defined intervals in further data processing may significantly influence the final decision (duntsch et al., 1997), which should definitely be avoided if impartial decision-making is aimed at. on the contrary, the approach based on interval rough numbers includes the exploitation of the uncertainty contained in the obtained data. by applying the arithmetic operations explained in the following section, the values of the attributes that fully describe the specified uncertainties without subjectively affecting their values are obtained. thus, the uncertainties of the first dm can be described by an interval rough number irn = [(3,3.67), (4,4.33)], of the second dm by irn = [(3.67,4), (4.33,5)], while those of the third dm can be described by irn = [(3.67,4), (4,4.33)]. the detailed procedure for the determination of an irn is explained in the following section. suppose that there is a set of k classes representing the dm’s preferences, 1 2 ( , ,..., ) k r j j j , provided that they belong to the sequence that satisfies the condition 1 2 ,..., k j j j   , and another set of m classes, which also represents a novel integrated fuzzy piprecia – interval rough saw model: green supplier selection 131 the dm’s preferences, * 1 2 ( , ,..., ) k r i i i . all objects are defined in the universe and related to the dm’s preferences. in * r , each class of objects is presented in an interval  ,i li uii i i , where the conditions that li uii i (1 i m  ) ,li uii i r , too, are satisfied. then, li i is the lower limit of the interval, while ui i is the upper limit of the interval of the i th class of the objects. if both limits of the object classes (the upper and the lower limits) are arranged in such a way that * * * * * * 1 2 1 2 ,..., , ,..., l l lj u u uk i i i i i i      ( 1 ,j k m  ), respectively, then the two new sets that contain the lower object class * * * * 1 2 ( , ,..., ) l l l lj r i i i and upper object class * * * * 1 2 ( , ,..., ) u u u uk r i i i , respectively, can be defined. then, for any class of objects * li i r ( 1 i j  ) and * ui i r (1 i k  ), it is possible to define the lower approximation of * li i and * ui i as follows:  * * *( ) / ( )li l liapr i y u r y i   (10)  * * *( ) / ( )ui u uiapr i y u r y i   (11) the upper approximations of * li i and * ui i are defined by applying the following equations:  * * *( ) / ( )li l liapr i y u r y i   (12)  * * *( ) / ( )ui u uiapr i y u r y i   (13) both classes of objects (the upper and the lower classes of the objects * li i and * ui i ) are defined by their lower limits * ( ) li lim i and * ( ) ui lim i , and their upper limits * ( ) li lim i and * ( ) ui lim i , respectively, * * *1 ( ) ( ) ( ) li l li l lim i r y y apr i m   (14) * * * * 1 ( ) ( ) ( ) ui u ui l lim i r y y apr i m   (15) where l m and * l m represent the total number of the objects contained in the lower approximation of the classes of the objects * li i and * ui i , respectively. the upper limits * ( ) li lim i and * ( ) ui lim i are defined by applying the equations (16) and (17), as follows: * * *1 ( ) ( ) ( ) li l li u lim i r y y apr i m   (16) đalić et al./decis-mak. appl. manag. eng. 3 (1) (2020) 126-145 132 * * * * 1 ( ) ( ) ( ) ui u ui u lim i r y y apr i m   (17) where u m and * u m represent a total number of the objects contained in the upper approximation of the classes of the objects * li i and * ui i , respectively. for the lower class of the objects, a rough boundary interval of * li i is presented as * ( ) li rb i , denoting the interval between the lower and the upper limits: * * * ( ) ( ) ( ) li li li rb i lim i lim i  (18) whereas for the upper class of the objects the rough boundary interval of * ui i is obtained as * * * ( ) ( ) ( ) ui ui ui rb i lim i lim i  (19) the uncertain class of the objects * li i and * ui i can be presented by using their lower and upper limits * * * ( ) ( ), ( ) li li li rn i lim i lim i    (20) * * * ( ) ( ), ( ) ui ui ui rn i lim i lim i    (21) as can be seen, each class of the objects is defined by its lower and upper limits, which represent the interval rough number defined as * * * ( ) ( ), ( ) i li ui irn i rn i rn i    . (22) the irn determination procedure will be explained by the example of the determination of the weight coefficient of the criterion wi, which is participated in by four experts. the experts evaluated the criteria by using the scale that includes integer values, ranging within the following 1-5 intervals: 1 – a very small impact, 2 – a small impact, 3 – a medium impact, 4 – a large impact, and 5 – a very large impact. the experts’ evaluations are shown in table 3. table 3. the experts’ evaluation of the criterion i w criterion experts e1 e2 e3 e4 wi (2;3) (3;4) (4;5) (5;5) the experts’ evaluations in table 3 are presented in the form of ordered pairs (ai;bi), where ai and bi are the values assigned by the experts to the criteria from the 1-5 scale. the experts who cannot confidently opt for one of the values in the scale enter both values they have a dilemma of (e1, e2 and e3). in our example, only the expert e4 had no dilemma and chose a unique value from the scale. these uncertainties can be represented by trapezoidal fuzzy numbers of the form a=(a1, a2, a3, a4), where a2 and a3 represent the values in which the membership function reaches its maximum value, whereas a1 and a4 represent the left and the right limits of a fuzzy set, respectively. in our example (table 3), the four trapezoidal fuzzy numbers a (e1) = (1,2,3,4), a (e2) = (2,3,4,5), a (e3) = (3. 4,5,5) and a (e4) = (4,5,5,5) were obtained. the trapezoidal fuzzy numbers are graphically shown in a novel integrated fuzzy piprecia – interval rough saw model: green supplier selection 133 figure 1, where the darker nuance indicates the values in which the membership function reaches its maximum value (a2 and a3), whereas the light nuance indicates the elements of the set more or less belonging to the fuzzy set (a1 and a4). 1 2 3 4 5 3.5 4.5 2.5 1.5 trapezoidal fuzzy numbers interval rough numbers 4.25 figure 1. the criterion evaluation – the interval rough and fuzzy evaluations in addition to the fuzzy approach, the uncertainties described can also be presented by interval rough numbers, since it was defined in the previous section (the equations (10)-(21)) that an irn consists of two rough sequences and the two classes of the objects wi and w'i:  2;3; 4;5iw  and  ' 3;4;5;5iw  were defined. by applying the equations (10)-(17), the rough sequences (20) and (21) are formed for each class of the objects. for the first class of the objects, the following was obtained: (2) 2lim  , 1 (2) (2 3 4 5) 3.5 4 lim      ; (2) [2, 3.5]rn  1 (3) (2 3) 2.5 2 lim    , 1 (3) (3 4 5) 4 3 lim     ; (3) [2.5, 4]rn  1 (4) (2 3 4) 3 3 lim     , 1 (4) (4 5) 4.5 2 lim    ; (4) [3, 4.5]rn  1 (5) (2 3 4 5) 3.5 4 lim      , (5) 5lim  ; (4) [3.5, 5]rn  for the second class of the objects, the following was obtained: (3) 3lim  , 1 (3) (3 4 5 5) 4.25 4 lim      ; (2) [3, 4.5]rn  1 (4) (3 4) 3.5 2 lim    , 1 (4) (4 5 5) 4.67 3 lim     ; (3) [2.5, 4]rn  1 (5) (3 4 5 5) 4.25 4 lim      , (5) 5lim  ; (5) [4.25, 5]rn  đalić et al./decis-mak. appl. manag. eng. 3 (1) (2020) 126-145 134 based on the rough sequences, the following interval rough numbers:     ( 1) 2, 3.5 , 3, 4.25irn e  ,     ( 2) 2.5, 4 , 3.5, 4.67irn e  ,     ( 3) 3, 4.5 , 4.25, 5irn e  and     ( 4) 3.5, 5 , 4.25, 5irn e  were obtained. rationally reasoning, without applying the rough and fuzzy sets, it can be concluded that the values of the criterion wi should range between the values 3.5 and 4.25. these values are obtained by the geometrical averaging of the classes of the objects  2;3; 4;5iw  and  ' 3;4;5;5iw  . in figure 3, the rational (expected) values 3.5 and 4.25 are shown by the dashed line. figure 3 allows us to notice that the expected values (3.5 and 4.25) are completely within the range of all the irns. on the other hand, the fuzzy numbers only partially cover the expected values. the affiliation function of the fuzzy numbers a(e2) and a(e3) with the maximum affiliation only partially covers the expected values, whereas the fuzzy numbers a(e1) and a(e4) cover the expected values, with an affiliation degree of 0.5. on the other hand, all the irns fully cover the expected values (3.5 and 4.25) by their intervals. interval rough numbers are characterized by specific arithmetic operations, which are different from the arithmetic operations with classical rough numbers. arithmetic operations between two interval rough numbers     1 2 3 4( ) , , ,irn a a a a a and     1 2 3 4( ) , , ,irn b b b b b are performed by using the following equations (23), (24), (25), (26) and (27): (1) the addition of interval rough numbers, "+",               1 2 3 4 1 2 3 4 1 1 2 2 3 3 4 4( ) ( ) , , , , , , , , ,irn a irn b a a a a b b b b a b a b a b a b        (23) (2) the subtraction of interval rough numbers, "-",               1 2 3 4 1 2 3 4 1 4 2 3 3 2 4 1( ) ( ) , , , , , , , , ,irn a irn b a a a a b b b b a b a b a b a b        (24) (3) the multiplication of interval rough numbers, "×",               1 2 3 4 1 2 3 4 1 1 2 2 3 3 4 4( ) ( ) , , , , , , , , ,irn a irn b a a a a b b b b a b a b a b a b        (25) (4) the division of interval rough numbers, "/",               1 2 3 4 1 2 3 4 1 4 2 3 3 2 4 1( ) / ( ) , , , / , , , / , / , / , /irn a irn b a a a a b b b b a b a b a b a b  (26), and (5) the scalar multiplication of interval rough numbers where 0k           1 2 3 4 1 2 3 4( ) , , , , , ,k irn a k a a a a k a k a k a k a        (27) any two interval rough numbers  ' '( ) , , ,l u l uirn             and  ' '( ) , , ,l u l uirn             are ranked according to the following rules: (1) if the interval of the interval rough number is not strictly bounded by another interval, then: (a) ff the condition that { ' 'u u   and l l   } or { ' 'u u   i l l   } is met, then ( ) ( )irn irn  , figure 2a; (b) if the condition that { ' 'u u   and l l   } is met, then ( ) ( )irn irn  , figure 2b. a novel integrated fuzzy piprecia – interval rough saw model: green supplier selection 135 (2) if the intervals of the interval rough numbers ( )irn  and ( )irn  are strictly bounded, then it is necessary to determine the intersection points ( )i  and ( )i  of the interval rough numbers ( )irn  and ( )irn  . if the condition that ' 'u u   and l l   is met, then (a) ff the condition that ( ) ( )i i  is met, then ( ) ( )irn irn  , figures 2c and 2d; (c) if the condition that ( ) ( )i i  is met, then ( ) ( )irn irn  , figure 2e. the intersection points of the interval rough numbers are obtained in the following manner: ' '( ) ; ( ) ; ( ) ( ) ( ) u l u lui ui li ui li rb rb rb rb rb                  (28) ' '( ) ; ( ) ; ( ) ( ) ( ) u l u lui ui li ui li rb rb rb rb rb                  (29) ( ) ( )irn irn  ( ) ( )irn irn  'u  l  ( )i  ( )i  ( ) ( )irn irn ( ) ( )irn irn  ( ) ( )irn irn  l  'u  a) b) c) d) e) ( )i  ( )i  ( )i  ( )i  l  l  l  l  l  l  l  l  'u  'u  'u  'u  'u  'u  'u  'u  figure 2. ranking interval rough numbers ' ( ) (1 ) l u i            (30) ' ( ) (1 ) l u i            (31) 2.3. interval rough saw method the saw method is a simple and easily applicable multi-criteria decision-making method. using only crisp numbers, however, it is impossible to obtain the results that treat uncertainty and objectivity in an adequate way (stević et al. 2017). the rough saw method was developed two years ago and presented in the study (stević et al., 2017). the interval rough saw method consists of the following steps (stević et al., 2019): step 1: forming a multi-criteria decision-making model which consists of m alternatives and n criteria. step 2: forming a team of r experts, who will make an assessment of alternatives according to all the criteria and sub-criteria. đalić et al./decis-mak. appl. manag. eng. 3 (1) (2020) 126-145 136 step 3: the transformation of individual matrices into a group interval rough matrix. in this step, it is necessary to transform each individual matrix of the experts r1,r2,...,rn into an interval rough group matrix by using the equations (10)-(21): 1 2 1 11 12 1 2 21 22 2 1 2 ... ( ) ( ) ... ( ) ( ) ( ) ( ) ... ... ... ... ... ( ) ( ) ... ( ) n n n m l l ln m n c c c a irn x irn x irn x a irn x irn x irn x y a irn x irn x irn x              (32) where m denotes the number of alternatives and n denotes the number of criteria. step 4: the normalization of the initial interval rough group matrix (33) by using the equations (34) and (35): 1 2 1 11 12 1 2 21 22 2 1 2 ... ( ) ( ) ... ( ) ( ) ( ) ( ) ... ... ... ... ... ( ) ( ) ... ( ) n n n m l l ln m n c c c a irn n irn n irn n a irn n irn n irn n y a irn n irn n irn n              (33) if the criterion belongs to the benefit group, then equation (34) is used for the normalization process:       ' ' ' ' ' ' , , , ( ) , , , max , , , l u l u ij ij ij ijl u l u ij ij ij ij ij l u l u ij ij ij ij n n n n irn n n n n n n n n n             , (34) whereas for the criteria belonging to the cost group, equation (35) is applied:       ' ' ' ' ' ' min , , , ( ) , , , , , , l u l u ij ij ij ijl u l u ij ij ij ij ij l u l u ij ij ij ij n n n n irn n n n n n n n n n             (35) equations (25) and (26) are further broken down into equations (36) and (37):   ' ' ' ' ' ' , , , , , , max max max max l u l u ij ij ij ijl u l u ij ij ij ij nu l u l ij ij ij ij n n n n n n n n if c b n n n n                (36)   ' ' ' ' ' ' min min min min , , , , , , l u l u ij ij ij ijl u l u ij ij ij ij nu l u l ij ij ij ij n n n n n n n n if c c n n n n                (37) step 5: weighting the previously normalized matrix:    ' ' ' ' ' '( ) , , , w , w , w , wl u l u l l u u l l u uij ij ij ij ij ij ij ij ij ij ij ij ij m n irn v v v v v n n n n              (38) step 6: summing up all of the values of the obtained alternatives (summing up by rows):  ' ' 1 ( ) , , , l u l u i i i i i m irn s s s s s      (39) step 7: ranking the alternatives in descending order, i.e. the highest value is the best alternative. in order to rank the potential solutions more easily, a rough number can be converted into a crisp number by using the average value. a novel integrated fuzzy piprecia – interval rough saw model: green supplier selection 137 3. results the selection of a green supplier depends on the precise determination and selection of adequate criteria and their evaluation. a novel integrated mcdm model is modified from (stević et al 2019) where supplier selection carried out 21 sustainable criteria. in this example we left only environmental criteria and made decision. the criteria for selecting a sustainable supplier are as follows: c1 – environmental image, c2 – recycling, c3 – pollution control, c4 – environmental management system, c5 – environmentally friendly products, c6 – resource consumption and c7 – green competencies. 3.1. determining criteria weights by using the fuzzy piprecia method the evaluation of the criteria was performed by using the linguistic scale that involves quantification into fuzzy triangle numbers. table 4 shows the evaluation of the criteria for fuzzy piprecia and inverse fuzzy piprecia carried out by the decision-makers. based on the evaluation of the criteria and equation (1), the matrix sj was formed. 𝑠𝑗 = [ ⋯ 1.100 1.150 1.200 1.050 1.075 1.125 0.310 0.367 0.450 1.150 1.225 1.275 0.310 0.367 0.450 1.050 1.075 1.125] applying equation (2), these values were subtracted from two. following the rules of operations with the fuzzy numbers of the matrix kj 𝑘𝑗 = [ 1.000 1.000 1.000 0.800 0.850 0.900 0.875 0.925 0.950 1.550 1.633 1.690 0.725 0.775 0.850 1.550 1.633 1.690 0.875 0.925 0.950] the following was obtained: according to equation (2), the value 1 (1.000, 1.000, 1.000)k  𝑘2̅̅ ̅ = (2 − 1.200, 2 − 1.150, 2 − 1.100) = (0.800, 0.850, 0.900) applying equation (3), the values qj 𝑞𝑗 = [ 1.000 1.000 1.000 1.111 1.176 1.250 1.170 1.272 1.429 0.692 0.779 0.922 0.814 1.005 1.271 0.482 0.615 0.820 0.507 0.665 0.937] đalić et al./decis-mak. appl. manag. eng. 3 (1) (2020) 126-145 138 were obtained as follows: 1 (1.000,1.000,1.000)q  𝑞2̅̅ ̅ = ( 1.000 0.900 , 1.000 0.850 , 1.000 0.800 ) = (1.111, 1.176, 1.250) applying equation (4), relative weights were calculated: 𝑤1̅̅̅̅ = ( 1.000 7.629 , 1.000 6.512 , 1.000 5.775 ) = (0.131, 0.154, 0.173) in order to determine the final weights of the criteria, it was necessary to apply equations (5)-(9), or the methodology of the inverse fuzzy piprecia method. based on the evaluation performed by the decision-makers, the matrix sj' was obtained as follows: 𝑠𝑗′ = [ ⋯ 0.517 0.708 0.917 0.583 0.833 1.000 1.350 1.525 1.575 0.450 0.583 0.833 1.350 1.525 1.575 0.583 0.833 1.000] applying equation (6), the values of the matrix kj' were obtained as follows: 𝑘𝑗′ = [ 1.083 1.292 1.483 1.000 1.167 1.417 0.425 0.475 0.650 1.167 1.417 1.550 0.425 0.475 0.650 1.000 1.167 1.417 1.000 1.000 1.000] 𝑘7̅̅ ̅ ′ = (1.000, 1.000, 1.000) 𝑘3̅̅ ̅ ′ = (2 − 1.575, 2 − 1.525, 2 − 1.350) = (0.425, 0.475, 0.650) a novel integrated fuzzy piprecia – interval rough saw model: green supplier selection 139 t a b le 4 . e v a lu a ti o n o f th e c ri te ri a f o r fu z z y p ip r e c ia a n d i n v e rs e f u z z y p ip r e c ia p ip r . c 1 c 2 c 3 c 4 c 5 c 6 c 7 d m 1 1 .1 0 0 1 .1 5 0 1 .2 0 0 1 .0 0 0 1 .0 0 0 1 .0 5 0 0 .3 3 3 0 .4 0 0 0 .5 0 0 1 .1 0 0 1 .1 5 0 1 .2 0 0 0 .3 3 3 0 .4 0 0 0 .5 0 0 1 .0 0 0 1 .0 0 0 1 .0 5 0 d m 2 1 .2 0 0 1 .3 0 0 1 .3 5 0 1 .1 0 0 1 .1 5 0 1 .2 0 0 0 .2 8 6 0 .3 3 3 0 .4 0 0 1 .2 0 0 1 .3 0 0 1 .3 5 0 0 .2 8 6 0 .3 3 3 0 .4 0 0 1 .1 0 0 1 .1 5 0 1 .2 0 0 d m 3 1 .1 0 0 1 .1 5 0 1 .2 0 0 1 .0 0 0 1 .0 0 0 1 .0 5 0 0 .3 3 3 0 .4 0 0 0 .5 0 0 1 .1 0 0 1 .1 5 0 1 .2 0 0 0 .3 3 3 0 .4 0 0 0 .5 0 0 1 .0 0 0 1 .0 0 0 1 .0 5 0 d m 4 1 .0 0 0 1 .0 0 0 1 .0 5 0 1 .1 0 0 1 .1 5 0 1 .2 0 0 0 .2 8 6 0 .3 3 3 0 .4 0 0 1 .2 0 0 1 .3 0 0 1 .3 5 0 0 .2 8 6 0 .3 3 3 0 .4 0 0 1 .1 0 0 1 .1 5 0 1 .2 0 0 a v 1 .1 0 0 1 .1 5 0 1 .2 0 0 1 .0 5 0 1 .0 7 5 1 .1 2 5 0 .3 1 0 0 .3 6 7 0 .4 5 0 1 .1 5 0 1 .2 2 5 1 .2 7 5 0 .3 1 0 0 .3 6 7 0 .4 5 0 1 .0 5 0 1 .0 7 5 1 .1 2 5 p ip r i c 7 c 6 c 5 c 4 c 3 c 2 c 1 d m 1 0 .6 6 7 1 .0 0 0 1 .0 0 0 1 .3 0 0 1 .4 5 0 1 .5 0 0 0 .5 0 0 0 .6 6 7 1 .0 0 0 1 .3 0 0 1 .4 5 0 1 .5 0 0 0 .6 6 7 1 .0 0 0 1 .0 0 0 0 .5 0 0 0 .6 6 7 1 .0 0 0 d m 2 0 .5 0 0 0 .6 6 7 1 .0 0 0 1 .4 0 0 1 .6 0 0 1 .6 5 0 0 .4 0 0 0 .5 0 0 0 .6 6 7 1 .4 0 0 1 .6 0 0 1 .6 5 0 0 .5 0 0 0 .6 6 7 1 .0 0 0 0 .4 0 0 0 .5 0 0 0 .6 6 7 d m 3 0 .6 6 7 1 .0 0 0 1 .0 0 0 1 .3 0 0 1 .4 5 0 1 .5 0 0 0 .5 0 0 0 .6 6 7 1 .0 0 0 1 .3 0 0 1 .4 5 0 1 .5 0 0 0 .6 6 7 1 .0 0 0 1 .0 0 0 0 .5 0 0 0 .6 6 7 1 .0 0 0 d m 4 0 .5 0 0 0 .6 6 7 1 .0 0 0 1 .4 0 0 1 .6 0 0 1 .6 5 0 0 .4 0 0 0 .5 0 0 0 .6 6 7 1 .4 0 0 1 .6 0 0 1 .6 5 0 0 .5 0 0 0 .6 6 7 1 .0 0 0 0 .6 6 7 1 .0 0 0 1 .0 0 0 a v 0 .5 8 3 0 .8 3 3 1 .0 0 0 1 .3 5 0 1 .5 2 5 1 .5 7 5 0 .4 5 0 0 .5 8 3 0 .8 3 3 1 .3 5 0 1 .5 2 5 1 .5 7 5 0 .5 8 3 0 .8 3 3 1 .0 0 0 0 .5 1 7 0 .7 0 8 0 .9 1 7 đalić et al./decis-mak. appl. manag. eng. 3 (1) (2020) 126-145 140 applying equation (7), the following values were obtained: 𝑞𝑗′ = [ 0.513 1.780 4.380 0.761 2.299 4.745 1.078 2.682 4.745 0.701 1.274 2.017 1.086 1.805 2.353 0.706 0.857 1.000 1.000 1.000 1.000] 𝑞7̅̅ ̅ ′ = (1.000, 1.000, 1.000) 𝑞3′ ̅̅̅̅ = ( 0.701 0.650 , 1.274 0.475 , 2.017 0.425 ) = (1.078, 2.682, 4.745) after that, it was necessary to apply equation (8) in order to obtain the relative weights for the fuzzy inverse piprecia method. 𝑤4′ ̅̅ ̅̅ = ( 0.701 20.241 , 1.274 11.695 , 2.017 5.844 ) = (0.035, 0.109, 0.345) the results of the applied methodology are presented in table 5. using equation (9), the final weights of the criteria were obtained. before applying this equation, it was necessary to defuzzify the values of the criteria obtained by applying the equations (1)-(9). table 5 shows the complete previous calculation, and the last column shows the defuzzified values of the relative weights of the criteria. the spearman (erceg et al., 2019) correlation coefficient for the obtained ranks is 0.964, which means that there is a minimum difference in these ranks. the first and the fifth criteria are replaced in the third and the fourth place, respectively. the pearson (stevic et al., 2018) correlation coefficient for the criterion weights (0.977) was also calculated. table 6 presents the final weight results obtained by using the fuzzy piprecia method. in table 6, the criteria are ranked by significance. the most significant criterion is c3 – pollution control. the function value of this criterion is 0.247. the least significant criterion is c6 – resource consumption. the function value of this criterion is 0.090. table 6. the final weight results obtained by applying the fuzzy piprecia method i ii wj c1 0.153 0.231 0.192 3 c2 0.181 0.273 0.227 2 c3 0.197 0.297 0.247 1 c4 0.121 0.136 0.129 5 c5 0.157 0.179 0.168 4 c6 0.097 0.083 0.090 7 c7 0.106 0.094 0.100 6 a novel integrated fuzzy piprecia – interval rough saw model: green supplier selection 141 t a b le 5 . c ri te ri a w e ig h ts o b ta in e d u si n g f u z z y p ip r e c ia m e th o d p s j k j q j w j d e fa z i w j r a n g c 1 1 ,0 0 0 1 ,0 0 0 1 ,0 0 0 1 ,0 0 0 1 ,0 0 0 1 ,0 0 0 0 .1 3 1 0 .1 5 4 0 .1 7 3 0 .1 5 3 0 .1 9 2 3 c 2 1 .1 0 0 1 .1 5 0 1 .2 0 0 0 .8 0 0 0 .8 5 0 0 .9 0 0 1 .1 1 1 1 .1 7 6 1 .2 5 0 0 .1 4 6 0 .1 8 1 0 .2 1 6 0 .1 8 1 0 .2 2 7 2 c 3 1 .0 5 0 1 .0 7 5 1 .1 2 5 0 .8 7 5 0 .9 2 5 0 .9 5 0 1 .1 7 0 1 .2 7 2 1 .4 2 9 0 .1 5 3 0 .1 9 5 0 .2 4 7 0 .1 9 7 0 .2 4 7 1 c 4 0 .3 1 0 0 .3 6 7 0 .4 5 0 1 .5 5 0 1 .6 3 3 1 .6 9 0 0 .6 9 2 0 .7 7 9 0 .9 2 2 0 .0 9 1 0 .1 2 0 0 .1 6 0 0 .1 2 1 0 .1 2 9 5 c 5 1 .1 5 0 1 .2 2 5 1 .2 7 5 0 .7 2 5 0 .7 7 5 0 .8 5 0 0 .8 1 4 1 .0 0 5 1 .2 7 1 0 .1 0 7 0 .1 5 4 0 .2 2 0 0 .1 5 7 0 .1 6 8 4 c 6 0 .3 1 0 0 .3 6 7 0 .4 5 0 1 .5 5 0 1 .6 3 3 1 .6 9 0 0 .4 8 2 0 .6 1 5 0 .8 2 0 0 .0 6 3 0 .0 9 4 0 .1 4 2 0 .0 9 7 0 .0 9 0 7 c 7 1 .0 5 0 1 .0 7 5 1 .1 2 5 0 .8 7 5 0 .9 2 5 0 .9 5 0 0 .5 0 7 0 .6 6 5 0 .9 3 7 0 .0 6 6 0 .1 0 2 0 .1 6 2 0 .1 0 6 0 .1 0 0 6 p -i sj k j q j w j c 1 0 .5 1 7 0 .7 0 8 0 .9 1 7 1 .0 8 3 1 .2 9 2 1 .4 8 3 0 .5 1 3 1 .7 8 0 4 .3 8 0 0 .0 2 5 0 .1 5 2 0 .7 5 0 0 .2 3 1 c 2 0 .5 8 3 0 .8 3 3 1 .0 0 0 1 .0 0 0 1 .1 6 7 1 .4 1 7 0 .7 6 1 2 .2 9 9 4 .7 4 5 0 .0 3 8 0 .1 9 7 0 .8 1 2 0 .2 7 3 c 3 1 .3 5 0 1 .5 2 5 1 .5 7 5 0 .4 2 5 0 .4 7 5 0 .6 5 0 1 .0 7 8 2 .6 8 2 4 .7 4 5 0 .0 5 3 0 .2 2 9 0 .8 1 2 0 .2 9 7 c 4 0 .4 5 0 0 .5 8 3 0 .8 3 3 1 .1 6 7 1 .4 1 7 1 .5 5 0 0 .7 0 1 1 .2 7 4 2 .0 1 7 0 .0 3 5 0 .1 0 9 0 .3 4 5 0 .1 3 6 c 5 1 .3 5 0 1 .5 2 5 1 .5 7 5 0 .4 2 5 0 .4 7 5 0 .6 5 0 1 .0 8 6 1 .8 0 5 2 .3 5 3 0 .0 5 4 0 .1 5 4 0 .4 0 3 0 .1 7 9 c 6 0 .5 8 3 0 .8 3 3 1 .0 0 0 1 .0 0 0 1 .1 6 7 1 .4 1 7 0 .7 0 6 0 .8 5 7 1 .0 0 0 0 .0 3 5 0 .0 7 3 0 .1 7 1 0 .0 8 3 c 7 1 .0 0 0 1 .0 0 0 1 .0 0 0 1 .0 0 0 1 .0 0 0 1 .0 0 0 0 .0 4 9 0 .0 8 6 0 .1 7 1 0 .0 9 4 đalić et al./decis-mak. appl. manag. eng. 3 (1) (2020) 126-145 142 3.2. the evaluation of the alternatives by applying the interval rough saw method table 7. the initial interval rough matrix a1 a2 a3 a4 c1 [4.11, 4.55]; [4.5, 5.5] [4.11, 4.55]; [4.45, 4.89] [4.45, 4.89]; [5.45, 5.89] [5.45, 5.89]; [6, 6] c2 [5.45, 5.89]; [6, 6] [5, 5]; [5.45, 5.89] [5.45, 5.89]; [6.45, 6.89] [4.89, 5.78]; [5.45, 5.89] c3 [4.22, 5.11]; [4.22, 5.11] [5, 5]; [5.45, 5.89] [4.5, 5.5]; [4.5, 5.5] [4.5, 5.5]; [5.45, 5.89] c4 [3.11, 3.55]; [3.45, 3.89] [4, 4]; [4.11, 4.55] [3.28, 5.28]; [3.89, 5.39] [3.89, 5.39]; [4.5, 5.5] c5 [4.11, 4.55]; [5.11, 5.55] [3.45, 3.89]; [4, 4] [3.89, 5.39]; [4.5, 5.5] [4.45, 4.89]; [5, 5] c6 [4.45, 4.89]; [4.45, 4.89] [4.45, 4.89]; [5, 5] [4.5, 5.5]; [5.11, 5.55] [3.11, 3.55]; [4, 4] c7 [3.11, 3.55]; [4, 4] [3.45, 3.89]; [4.11, 4.55] [3.5, 4.5]; [4.45, 4.89] [4.45, 4.89]; [4.5, 5.5] then, equations (33)-(37) need to be applied in order to normalize the initial interval rough matrix. criterion c6 belongs to the cost group, while the other criteria need to be maximized, i.e. they belong to the beneficial criteria group. table 8 shows the normalized interval rough matrix. table 8. the normalized interval rough matrix a1 a2 a3 a4 c1 [0.69, 0.76], [0.76, 1.01] [0.69, 0.76], [0.76, 0.9] [0.74, 0.82], [0.93, 1.08] [0.91, 0.98], [1.02, 1.1] c2 [0.79, 0.91], [1.02, 1.1] [0.73, 0.78], [0.93, 1.08] [0.79, 0.91], [1.1, 1.26] [0.71, 0.9], [0.93, 1.08] c3 [0.72, 0.94], [0.77, 1.02] [0.85, 0.92], [0.99, 1.18] [0.76, 1.01], [0.82, 1.1] [0.76, 1.01], [0.99, 1.18] c4 [0.57, 0.79], [0.64, 0.97] [0.73, 0.89], [0.76, 1.14] [0.6, 1.17], [0.72, 1.35] [0.71, 1.2], [0.83, 1.38] c5 [0.74, 0.89], [0.95, 1.25] [0.62, 0.76], [0.74, 0.9] [0.7, 1.05], [0.83, 1.24] [0.8, 0.96], [0.93, 1.12] c6 [0.64, 0.8], [0.82, 0.9] [0.62, 0.71], [0.82, 0.9] [0.56, 0.69], [0.73, 0.89] [0.78, 0.89], [1.13, 1.29] c7 [0.57, 0.79], [0.82, 0.9] [0.63, 0.86], [0.84, 1.02] [0.64, 1], [0.91, 1.1] [0.81, 1.09], [0.92, 1.24] an example of the calculation of the normalized matrix for the criteria belonging to the cost group is: 𝐼𝑅𝑁 (𝑛16) = ([0.636, 0.798, 0.818, 0.899]) = ([ 3.110 4.890 , 3.550 4.450 , 4.000 4.890 , 4.000 4.450 ]) for the alternative a1. an example of the calculation of the normalized matrix for the criteria belonging to the benefit group is: 𝐼𝑅𝑁 (𝑛27) = ([0.527, 0.864, 0.840, 1.022]) = ([ 3.450 5.500 , 3.890 4.500 , 4.110 4.890 , 4.550 4.450 ]) for the alternative a2. subsequently, the normalized interval rough matrix was weighted by the criterion values obtained by applying the fuzzy piprecia method. the weighting was performed by applying equation (38), while the summing up of the values for the alternatives by rows was performed by applying equation (39). table 9 shows the weighted normalized interval rough matrix. a novel integrated fuzzy piprecia – interval rough saw model: green supplier selection 143 table 9. the weighted normalized interval rough matrix a1 a2 a3 a4 c1 [0.13, 0.14], [0.16, 0.19] [0.13, 0.14], [0.14, 0.17] [0.14, 0.16], [0.18, 0.21] [0.17, 0.19], [0.19, 0.21] c2 [0.18, 0.21], [0.23, 0.25] [0.16, 0.18], [0.21, 0.24] [0.18, 0.21], [0.25, 0.29] [0.16, 0.2], [0.21, 0.24] c3 [0.18, 0.23], [0.19, 0.25] [0.21, 0.23], [0.24, 0.29] [0.19, 0.25], [0.2, 0.27] [0.19, 0.25], [0.24, 0.29] c4 [0.07, 0.1], [0.08, 0.12] [0.09, 0.11], [0.1, 0.15] [0.08, 0.15], [0.09, 0.17] [0.09, 0.15], [0.11, 0.18] c5 [0.12, 0.15], [0.16, 0.21] [0.1, 0.13], [0.12, 0.15] [0.12, 0.18], [0.14, 0.21] [0.13, 0.16], [0.16, 0.19] c6 [0.06, 0.07], [0.07, 0.08] [0.06, 0.06], [0.07, 0.08] [0.05, 0.06], [0.07, 0.08] [0.07, 0.08], [0.1, 0.12] c7 [0.06, 0.08], [0.08, 0.09] [0.06, 0.09], [0.08, 0.1] [0.06, 0.1], [0.09, 0.11] [0.08, 0.11], [0.09, 0.12] table 10 shows the final results of the integrated fuzzy piprecia-interval rough saw approach. table 10. the results of supplier selection by applying the integrated fuzzy piprecia-interval rough saw approach si av rank 0.799 0.986 0.964 1.201 0.988 3 0.823 0.941 0.980 1.189 0.983 4 0.819 1.104 1.018 1.337 1.069 2 0.901 1.144 1.107 1.353 1.126 1 the ranking was performed in descending order, which means that the highest value was the best and the lowest value was the worst solution. the alternative 4 is the most acceptable solution according to the results obtained. 4. conclusion in this paper, the evaluation of green suppliers was carried out by applying an innovative fuzzy-rough mcdm model. the advantages of fuzzy piprecia, which was used to determine the criteria weights, and the interval rough saw method, applied for supplier evaluation, are demonstrated throughout the paper. the fuzzy piprecia method allows for the evaluation of criteria without first sorting them by significance. group decision-making is also an advantage of this method. today, the largest number of multi-criteria decision-making problems are solved by applying group decisionmaking. in such cases, especially given the fact that the number of decision-makers involved in the fuzzy piprecia model increases, benefits are achieved from it. the saw method is a simple and easily applicable multi-criteria decision-making method. using only crisp numbers, however, it is impossible to obtain the results that treat uncertainty and objectivity in an adequate manner. for that reason, the interval rough saw method was implemented for supplier selection based on the environmental criteria. the obtained results show that the fourth supplier is the best solution. author contributions: each 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(2014). selecting green supplier of thermal power equipment by using a hybrid mcdm method for sustainability. sustainability, 6(1), 217-235. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). decision making: applications in management and engineering vol. 5, issue 2, 2022, pp. 176-200. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0311072022b * corresponding author. broumisaid78@gmail.com, s.broumi@flbenmsik.ma e-mail addresses: sundareswaranr@ssn.edu.in (r. sundareswaran), shanmugapriyma@ssn.edu.in (m. shanmugapriya), giorgio.nordo@unime.it (g. nordo), taleamohamed@yahoo.fr (m. talea), assiabakali@yahoo.fr (a. bakali), fsmarandache@gmail.com (f. smarandache) intervalvalued fermatean neutrosophic graphs said broumi1*, raman sundareswaran2, marayanagaraj shanmugapriya3, giorgio nordo4, mohamed talea1, assia bakali5, and florentin smarandache6 1 laboratory of information processing, faculty of science ben m’sik, university of hassan ii, casablanca, morocco & regional center for the professions of education and training(c.r.m.e.f), casablanca,morocco. 2,3department of mathematics, sri sivasubramaniya nadar college of engineering, india 4dipartimento di scienze matematiche e informatiche, scienze fisiche e scienze della terra dell'università degli studi di messina viale ferdinando stagno d'alcontres, italy 5 ecole royale navale-boulevard sour jdid, morocco 6department of mathematics, university of new mexico, usa received: 2 may 2022; accepted: 29 june 2022; available online: 12 july 2022. original scientific paper abstract: in this work, we define interval-valued fermatean neutrosophic graphs (ivfns) and present some operations on interval-valued fermatean neutrosophic graphs. further, we introduce the concepts of regular intervalvalued fermatean neutrosophic graphs, strong interval-valued fermatean neutrosophic graphs, cartesian, composition, lexicographic product of interval-valued fermatean neutrosophic graphs. finally, we give the applications of interval-valued fermatean neutrosophic graphs. key words: interval-valued fermatean fuzzy sets, interval-valued fermatean neutrosophic sets, interval-valued fermatean neutrosophic graphs mailto:s.broumi@flbenmsik.ma mailto:shanmugapriyma@ssn.edu.in intervalvalued fermatean neutrosophic graphs 177 1. introduction the concept of neutrosophic set theory was proposed by jun (2017). the idea of neutrosophic set which is a generalization of the fuzzy set (zadeh, 1965), intuitionistic fuzzy set (atanassov, 1986). the neutrosophic sets are characterized by a truth function (t), an indeterminate function (i) and a false function (f) independently. smarandache (2019) introduced the concept of spherical neutrosophic oversets as generalization of spherical fuzzy sets. by bending the concept of single valued neutrosophic set and graph theory, different classes of neutrosophic graphs is discussed by broumi (2016) and many works available in the literature (broumi et al., 2016a, 2016b, 2016c, 2016d, 2022). nagarajan et al. (2019) investigated the intervalvalued neutrosophic graphs and its applications. recently, ajay et al. (2020, 2021) extended the concept of pythagorean neutrosophic sets to graphs and called it pythagorean neutrosophic graph (png) and investigated some of their properties. the same authors presented the idea of labelling of pythagorean neutrosophic fuzzy graphs and investigate their properties. ajay et al. (2022) studied the concept of regularity in png and introduced the ideas of regular, full edge regular, edge regular, and partially edge regular pythagorean neutrosophic graphs. in addition, a new mcdm method has been introduced using the pythagorean neutrosophic graphs with an illustrative example. by integrating the concepts pythagorean neutrosophic fuzzy graph and dombi operator. furthermore, ajay et al. (2021) proposed a new extension of neutrosophic graph called pythagorean neutrosophic dombi fuzzy graphs (pndfg) and suggested some basic operations of pndfg and computed the degree and total degree of a vertex of pndfg. akalyadevi et al. (2022) introduced the concept of spherical neutrosophic graph coloring and discussed some of their important properties also they suggested the chromatic number of spherical neutrosophic graph as a crisp number. duleba et al. (2021) applied the concept of interval-valued spherical fuzzy ahp method to public transportation problem. aydın et al. (2021) proposed a novel fuzzy multimoora method based on interval-valued spherical fuzzy sets to evaluate companies that are using industry 4.0 technologies. lathamaheswari et al. (2021) proposed the concept of interval valued spherical fuzzy aggregation operators and applied it for solving a decision-making problem. kutlu gündoğdu et al. (2021) extended spherical fuzzy analytic hierarchy process to interval-valued spherical fuzzy ahp (ivsf-ahp) method and applied it to compare the service performances of several hospitals. kutlu gündoğdu et al. (2019) presented the idea of spherical fuzzy sets (sfs) as an integration of pythagorean fuzzy sets and neutrosophic sets. smarandache (2017) proposed the concept of spherical neutrosophic numbers. senapati et al. (2019) defined basic operators over the ffss. on the other hand, division, and subtraction operations on ffss were proposed. donghai liu et al. (2019) focused on a distance measure for fermatean fuzzy linguistic term sets. ganie et al. (2022) proposed some novel distance measures for fermatean fuzzy sets using t-conorms. on the other hand, jeevaraj et al. (2021) proposed the concept of interval-valued fermatean fuzzy sets (ivffss) and establishes some mathematical operations on the class of ivffss. a new total ordering principle on the class of ivffns is introduced. they implemented the interval-valued fermatean fuzzy topsis (ivfftopsis) method for solving multi-criteria decision-making problems. based on neutrosophic pythagorean sets, stephen et al. (2021) introduced the concept of interval-valued neutrosophic pythagorean sets with dependent interval valued pythagorean components and discussed some of its properties. recently, lakhwani et al. (2022) introduced a novel concept of dombi neutrosophic graph and presented some kinds of dombi neutrosophic graph such as a regular dombi neutrosophic graph, said broumi et al./decis. mak. appl. manag. eng. 5 (2) (2022) 176-200 178 strong dombi neutrosophic graph, complete dombi neutrosophic graph, and complement dombi neutrosophic graph and described some of their properties, also, and discussed some operations on dombi neutrosophic graphs are defined. in this paper, we present the concept of interval-valued fermatean neutrosophic graphs (ivfng) and the concepts of regular interval-valued fermatean neutrosophic graphs, strong interval-valued fermatean neutrosophic graphs, cartesian, composition, lexicographic product of interval-valued fermatean neutrosophic graphs. we also introduce some theorems and examples on ivfng’s finally, we give the applications of interval-valued fermatean neutrosophic graphs. 2. preliminaries the extension of crisp set theory with membership degree is known as fuzzy set theory in which each element of the set gets a real number between 0 and 1. but in many real time situations, it is not always possible to give an exact degree of membership because of lack of knowledge, vague information, and so forth. to overcome this problem, we can use interval-valued fuzzy sets, which assign to each element a closed interval which approximates the “real,” but unknown, membership degree. the length of this interval is a measure for the uncertainty about the membership degree. an interval number i is an interval [𝑐 −, 𝑐+] with 0 ≤ 𝑐 − ≤ 𝑐+ ≤ 1. the interval [c, 𝑐] is identified with the number 𝑐 ∈ [0, 1]. let 𝐼[0, 1] be the set of all closed subintervals of [0, 1]. an extension of fuzzy sets by zadeh (1965), intervalvalued fuzzy sets which stated that the values of the membership degrees are intervals of numbers instead of the numbers. it provides a more sufficient information about uncertainty than traditional fuzzy sets. in this section, we provide all the basic definitions of interval valued sets and its corresponding graphs. table 1 depicts the types of sets and graphs for interval-valued fuzzy and neutrosophic environments. table 1. different types of interval-valued sets and its graphs type of sets definition type of graphs definition interval-valued fuzzy set (ivfs) -zadeh, 1975 𝐴 = {(𝑥, [𝜇𝐴 −(𝑥), 𝜇𝐴 +(𝑥)]): 𝑥 ∈ 𝑉} intervalvalued fuzzy graph (ivfg) muhammad akram, wieslaw a. dudek, 2011. g = (a, b), where a = [𝜇𝐴 −(𝑥), 𝜇𝐴 +(𝑥)] is an interval-valued fuzzy set on v and b = [𝜇𝐵 −(𝑥), 𝜇𝐵 +(𝑥)]is an interval-valued fuzzy relation on e. interval-valued intuitionistic fuzzy set (ivifs) atanassov, k., gargov, g., 1989 𝐴 = {(𝑥, [𝑇𝐴 −(𝑥), 𝑇𝐴 +(𝑥)]): ∈ 𝑉}; 𝐵 = {(𝑥, [𝐹𝐴 −(𝑥), 𝐹𝐴 +(𝑥)]): 𝑥 ∈ 𝑉} such that 0 ≤ 𝑇𝐴 +(𝑥) + 𝐹𝐴 +(𝑥) ≤ 1 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥𝜖 𝑋 intervalvalued intuitionistic fuzzy graph (ivifg) s. n. mishra and a. pal, 2013  𝜇𝐴: 𝑉 → 𝐷[0,1]; 𝜂𝐴: 𝑉 → 𝐷[0,1] such that 0 ≤ 𝜇𝐴(𝑥) + 𝜂𝐴(𝑥) ≤ 1 , ∀ 𝑥 ∈ 𝑉  𝜇𝐵 : 𝐸 ⊆ 𝑉 × 𝑉 → 𝐷[0,1] ; 𝜂𝐵 : 𝐸 ⊆ 𝑉 × 𝑉 → 𝐷[0,1] 𝜇𝐵 −((𝑥, 𝑦)) ≤ min( 𝜇𝐴 −(𝑥), 𝜇𝐴 −(𝑦)); 𝜂𝐵 −((𝑥, 𝑦)) ≥ min( 𝜂𝐴 −(𝑥), 𝜂𝐴 −(𝑦)) 𝜇𝐵 +((𝑥, 𝑦)) ≤ min( µ𝐴 +(𝑥), µ𝐴 +(𝑦)); 𝜂𝐵 +((𝑥, 𝑦)) ≥ min( 𝜂𝐴 +(𝑥), 𝜂𝐴 +(𝑦)) such that 0 ≤ 𝜇𝐵 +((𝑥, 𝑦)) + 𝜂𝐵 +((𝑥, 𝑦)) ≤ 1 , ∀(𝑥, 𝑦) ∈ 𝐸 interval-valued neutrosophic set (ivns) said broumi , mohamed talea , assia bakali , florentin for each point 𝑥 ∈ 𝑋, we have that 𝑇𝐴(𝑥) = [𝑇𝐴 −(𝑥), 𝑇𝐴 +(𝑥)], 𝐼𝐴(𝑥) = [𝐼𝐴 −(𝑥), 𝐼𝐴 +(𝑥)], 𝐹𝐴(𝑥) = [𝐹𝐴 −(𝑥), 𝐹𝐴 +(𝑥)] ⊆ [0, 1] and 0 ≤ 𝑇𝐴(𝑥) + 𝐼𝐴(𝑥) + 𝐹𝐴(𝑥) ≤ 3. intervalvalued neutrosophic graph (ivng) said broumi, mohamed talea, assia bakali, 𝐺 = (𝐴, 𝐵), where 𝐴 =< [𝑇𝐴 −, 𝑇𝐴 +], [𝐼𝐴 −, 𝐼𝐴 +], [𝐹𝐴 −, 𝐹𝐴 +] > is an intervalvalued neutrosophic set on v; and 𝐵 = < [𝑇𝐵 −, 𝑇𝐵 +], [𝐼𝐵 −, 𝐼𝐵 +], [𝐹𝐵 −, 𝐹𝐵 +] > 𝑇𝐵 −: 𝑉 × 𝑉 → [0, 1], 𝑇𝐵 +: 𝑉 × 𝑉 → [0, 1], 𝐼𝐵 −: 𝑉 × 𝑉 → [0, 1], 𝐼𝐵 +: 𝑉 × 𝑉 → [0, 1] and 𝐹𝐵 −: 𝑉 × 𝑉 → [0,1], 𝐹𝐵 +: 𝑉 × 𝑉 → [0, 1] are such that 𝑇𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≤ min[𝑇𝐴 −(𝑣𝑖 ), 𝑇𝐴 −(𝑣𝑗 )], intervalvalued fermatean neutrosophic graphs 179 definition 2.1 (akram et al., 2013) the interval-valued fuzzy set (ivfs) 𝐴 in 𝑉 is defined by = {(𝑥, { µ𝐴 − (𝑥), µ𝐴 + (𝑥))} ∶ 𝑥 ∈ 𝑉 } , where µ𝐴 − (𝑥) and µ𝐴 + (𝑥) are fuzzy subsets of 𝑉 such that µ𝐴 − (𝑥) ≤ µa + (x) for all 𝑥 ∈ 𝑉. for any two interval-valued sets 𝐴 = [µ𝐴 − (𝑥), µ𝐴 + (𝑥)] and 𝐵 = [µ𝐵 − (𝑥), µ𝐵 + (𝑥)] in v. define: • 𝐴 ⋃ 𝐵 = {(𝑥, 𝑚𝑎𝑥(µ𝐴 − (𝑥), µ𝐵 − (𝑥)), 𝑚𝑎𝑥(µ𝐴 + (𝑥), µ𝐵 + (𝑥))) ∶ 𝑥 ∈ 𝑉}, • 𝐴 ⋂ 𝐵 = {(𝑥, 𝑚𝑖𝑛(µ𝐴 − (𝑥), µ𝐵 − (𝑥)), 𝑚𝑖𝑛(µ𝐴 + (𝑥), µ𝐵 + (𝑥))) ∶ 𝑥 ∈ 𝑉}. smarandache, (2016) florentin smarandache, 2016 𝑇𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≤ min[𝑇𝐴 +(𝑣𝑖 ), 𝑇𝐴 +(𝑣𝑗 )], 𝐼𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐼𝐵 − (𝑣𝑖 ), 𝐼𝐵 −(𝑣𝑗 )], 𝐼𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐼𝐵 +(𝑣𝑖 ), 𝐼𝐵 +(𝑣𝑗 )], 𝐹𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐹𝐵 −(𝑣𝑖), 𝐹𝐵 −(𝑣𝑗 )], 𝐹𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≥ 𝑚𝑎𝑥[𝐹𝐵 +(𝑣𝑖 ), 𝐹𝐵 +(𝑣𝑗 )] interval-valued pythagorean fuzzy set (ivpfs) f. teng, z. liu, and p. liu, (2018). for each point 𝑥 ∈ 𝑋, we have that 𝑇𝐴(𝑥) = [𝑇𝐴 −(𝑥), 𝑇𝐴 +(𝑥)], 𝐹𝐴(𝑥) = [𝐹𝐴 −(𝑥), 𝐹𝐴 +(𝑥)] ⊆ [0, 1] and 0 ≤ 𝑇𝐴 +(𝑥)2 + 𝐹𝐴 +(𝑥)2 ≤ 1. intervalvalued pythagorean fuzzy graph (ivpfg) mohamed s.y., ali a.m.,2018 𝐺 = (𝐴, 𝐵), where 𝐴 =< [𝑇𝐴 −, 𝑇𝐴 +], [𝐹𝐴 −, 𝐹𝐴 +] > is an interval-valued neutrosophic set on v; and 𝐵 = < [𝑇𝐵 −, 𝑇𝐵 +], [𝐹𝐵 −, 𝐹𝐵 +] > 𝑇𝐵 −: 𝑉 × 𝑉 → [0, 1], 𝑇𝐵 +: 𝑉 × 𝑉 → [0, 1], and 𝐹𝐵 −: 𝑉 × 𝑉 → [0,1], 𝐹𝐵 +: 𝑉 × 𝑉 → [0, 1] are such that 𝑇𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≤ min[𝑇𝐴 −(𝑣𝑖 ), 𝑇𝐴 −(𝑣𝑗 )], 𝑇𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≤ min[𝑇𝐴 +(𝑣𝑖 ), 𝑇𝐴 +(𝑣𝑗 )], 𝐹𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐹𝐵 −(𝑣𝑖), 𝐹𝐵 −(𝑣𝑗 )], 𝐹𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐹𝐵 +(𝑣𝑖), 𝐹𝐵 +(𝑣𝑗 )] such that 0 ≤ 𝑇𝐴 +(𝑥)2 + 𝐹𝐴 +(𝑥)2 ≤ 1. interval-valued fermatean fuzzy set (ivffs) jeevaraj s, (2021) for each point 𝑥 ∈ 𝑋, we have that 𝑇𝐴(𝑥) = [𝑇𝐴 −(𝑥), 𝑇𝐴 +(𝑥)], 𝐹𝐴(𝑥) = [𝐹𝐴 −(𝑥), 𝐹𝐴 +(𝑥)] ⊆ [0, 1] and 0 ≤ 𝑇𝐴 +(𝑥)3 + 𝐹𝐴 +(𝑥)3 ≤ 1. intervalvalued fermatean fuzzy graph (ivffg) 𝐺 = (𝐴, 𝐵), where 𝐴 =< [𝑇𝐴 −, 𝑇𝐴 +], [𝐹𝐴 −, 𝐹𝐴 +] > is an interval-valued neutrosophic set on v; and 𝐵 = < [𝑇𝐵 −, 𝑇𝐵 +], [𝐹𝐵 −, 𝐹𝐵 +] > 𝑇𝐵 −: 𝑉 × 𝑉 → [0, 1], 𝑇𝐵 +: 𝑉 × 𝑉 → [0, 1], and 𝐹𝐵 −: 𝑉 × 𝑉 → [0,1], 𝐹𝐵 +: 𝑉 × 𝑉 → [0, 1] are such that 𝑇𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≤ min[𝑇𝐴 −(𝑣𝑖 ), 𝑇𝐴 −(𝑣𝑗 )], 𝑇𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≤ min[𝑇𝐴 +(𝑣𝑖 ), 𝑇𝐴 +(𝑣𝑗 )], 𝐹𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐹𝐵 −(𝑣𝑖), 𝐹𝐵 −(𝑣𝑗 )], 𝐹𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≥ 𝑚𝑎𝑥[𝐹𝐵 +(𝑣𝑖 ), 𝐹𝐵 +(𝑣𝑗 )] such that 0 ≤ 𝑇𝐴 +(𝑥)3 + 𝐹𝐴 +(𝑥)3 ≤ 1. interval-valued fermatean neutrosophic set (ivfns) – said broumi, raman sundareswaran, marayanagaraj shanmugapriya, giorgio nordo mohamed talea, assia bakali, and florentin smarandache, (2022) a={〈𝑥, 𝑇𝐴(𝑥), ia(𝑥), fa(𝑥)〉| 𝑥 ∈ x } where 𝑇𝐴(𝑥) = [ta −(𝑥) , 𝑇𝐴 +(𝑥)], ia(𝑥)=[ia −(𝑥) , ia +(x)] and fa(𝑥) = [𝐹𝐴 −(𝑥), fa +(𝑥)], 𝑇𝐴(𝑥): 𝑋 → 𝐷[0,1] ia(𝑥): 𝑋 → 𝐷[0,1], 𝐹𝐴(𝑥): 𝑋 → 𝐷[0,1] and 0 ≤ (𝑇𝐴(𝑥)) 𝟑 +(𝐹𝐴(𝑥)) 𝟑 ≤1 and 0 ≤ (𝐼𝐴 (𝑥)) 𝟑 ≤ 1 0 ≤ (𝑇𝐴(𝑥)) 𝟑 +(𝐹𝐴(𝑥)) 𝟑 +(𝐼𝐴(𝑥)) 𝟑 ≤ 2 means 0 ≤ (𝑇𝐴 +(x))𝟑+(𝐼𝐴 +(𝑥))𝟑+(𝐹𝐴 +(𝑥))𝟑 ≤ 2 ∀ 𝑥 ∈ x intervalvalued fermatean neutrosophic graph (ivfng) said broumi, et al., 2022 𝐺 = (𝐴, 𝐵), where 𝐴 =< [𝑇𝐴 −, 𝑇𝐴 +], [𝐼𝐴 −, 𝐼𝐴 +], [𝐹𝐴 −, 𝐹𝐴 +] > is an intervalvalued fermatean neutrosophic set on v; and 𝐵 = < [𝑇𝐵 −, 𝑇𝐵 +], [𝐼𝐵 −, 𝐼𝐵 +], [𝐹𝐵 −, 𝐹𝐵 +] > 𝐸 satisfying the following condition: 𝑇𝐴 − ∶ 𝑉 → [0, 1], 𝑇𝐴 + ∶ 𝑉 → [0, 1], 𝐼𝐴 − ∶ 𝑉 → [0, 1], 𝐼𝐴 +: 𝑉 → [0, 1]𝑎𝑛𝑑 𝐹𝐴 −: 𝑉 → [0, 1], 𝐹𝐴 +: 𝑉 → [0, 1], and 0 ≤ 𝑇𝐴(𝑣𝑖) + 𝐼𝐴(𝑣𝑖 ) + 𝐹𝐴(𝑣𝑖 ) ≤ 3, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣𝑖 ∈ 𝑉 (𝑖 = 1, 2, … , 𝑛 the functions 𝑇𝐵 −: 𝑉 × 𝑉 → [0, 1], 𝑇𝐵 +: 𝑉 × 𝑉 → [0, 1], 𝐼𝐵 −: 𝑉 × 𝑉 → [0, 1], 𝐼𝐵 +: 𝑉 × 𝑉 → [0, 1] and 𝐹𝐵 −: 𝑉 × 𝑉 → [0,1], 𝐹𝐵 +: 𝑉 × 𝑉 → [0, 1] are such that 𝑇𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≤ min[𝑇𝐴 −(𝑣𝑖 ), 𝑇𝐴 −(𝑣𝑗 )], 𝑇𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≤ min[𝑇𝐴 +(𝑣𝑖 ), 𝑇𝐴 +(𝑣𝑗 )], 𝐼𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐼𝐵 − (𝑣𝑖 ), 𝐼𝐵 −(𝑣𝑗 )], 𝐼𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐼𝐵 +(𝑣𝑖 ), 𝐼𝐵 +(𝑣𝑗 )], 𝐹𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐹𝐵 −(𝑣𝑖), 𝐹𝐵 −(𝑣𝑗 )], 𝐹𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐹𝐵 +(𝑣𝑖), 𝐹𝐵 +(𝑣𝑗 )], denoting the degree of truth-membership, indeterminacy-membership and falsitymembership of the edge (𝑣𝑖 , 𝑣𝑗 ) ∈ 𝐸 respectively, where 0 ≤ 𝑇𝐵 ({𝑣𝑖 , 𝑣𝑗 }) 3 + 𝐼𝐵({𝑣𝑖 , 𝑣𝑗 }) 3 + 𝐹𝐵 ({𝑣𝑖 , 𝑣𝑗 }) 3 ≤ 2 for all {𝑣𝑖 , 𝑣𝑗 } ∈ 𝐸 (𝑖, 𝑗 = 1, 2, … , 𝑛) means 0 ≤ (𝑇𝐵 +(𝑣𝑖 , 𝑣𝑗 )) 𝟑 +(𝐼𝐵 +(𝑣𝑖 , 𝑣𝑗 )) 𝟑 +(𝐹𝐵 +(𝑣𝑖 , 𝑣𝑗 )) 𝟑 ≤ 2 ∀ 𝑥 ∈ x. said broumi et al./decis. mak. appl. manag. eng. 5 (2) (2022) 176-200 180 definition 2.2 (akram et al., 2013) if 𝐺 ∗ = (𝑉, 𝐸) is a graph, then by an interval-valued fuzzy relation (ivfr) 𝐵 on a set 𝐸 we mean an interval-valued fuzzy set such that µ𝐵 − (𝑥𝑦) ≤ 𝑚𝑖𝑛(µ𝐴 − (𝑥), µ𝐴 − (𝑦)), µ𝐵 + (𝑥𝑦) ≤ 𝑚𝑖𝑛(µ𝐴 +(𝑥), µ𝐴 + (𝑦)) for all 𝑥𝑦 ∈ 𝐸. definition 2.3 (akram et al., 2013) by an interval-valued fuzzy graph (ivfg) of a graph 𝐺 ∗ = (𝑉, 𝐸) we mean a pair 𝐺 = (𝐴, 𝐵), where 𝐴 = [µ𝐴 − , µ𝐴 + ] is an interval-valued fuzzy set on 𝑉 and 𝐵 = [µ𝐵 − , µ𝐵 + ] is an interval-valued fuzzy relation on 𝐸. example 2.4 (akram et al., 2013) consider a graph 𝐺 ∗ = (𝑉, 𝐸) such that 𝑉 = {𝑥, 𝑦, 𝑧}, 𝐸 = {𝑥𝑦, 𝑦𝑧, 𝑧𝑥}. let a be an interval-valued fuzzy set of v and b be an interval-valued fuzzy set of 𝐸 ⊆ 𝑉 × 𝑉 defined by 𝐴 = 〈( 𝑥 0.2 , 𝑦 0.3 , 𝑧 0.4 ) , ( 𝑥 0.4 , 𝑦 0.5 , 𝑧 0.5 ) 〉 , 𝐵 = 〈( 𝑥 0.2 , 𝑦 0.3 , 𝑧 0.4 ) , ( 𝑥𝑦 0.3 , 𝑦𝑧 0.4 , 𝑧𝑥 0.4 ) 〉 figure 1. interval-valued fuzzy graph g akram et al. (2013) introduced certain types of interval-valued fuzzy graphs including balanced interval-valued fuzzy graphs, neighbourly irregular intervalvalued fuzzy graphs, neighbourly total irregular interval-valued fuzzy graphs, highly irregular interval-valued fuzzy graphs, and highly total irregular interval-valued fuzzy graph. hossein et al. (2013) define three new operations on interval-valued fuzzy graphs; namely direct product, semi strong product and strong product. definition 2.5 (mishra et al., 2013; ismayil et al., 2014) an interval-valued intuitionistic fuzzy set (ivifs) 𝐴 in 𝑋, is given by 𝐴 = { 〈𝑥, 𝜇𝐴(𝑥), 𝜂𝐴(𝑥)〉/ 𝑥𝜖 𝑋} where 𝜇𝐴: 𝑋 → [0, 1], 𝜂𝐴: 𝑋 → 𝐷[0, 1]. the intervals 𝜇𝐴(𝑥) and 𝜂𝐴(𝑥) denote the degree of membership and the degree of non-membership of the element 𝑥 to the set, where 𝜇𝐴(𝑥) = [ 𝜇𝐴 −(𝑥), 𝜇𝐴 +(𝑥)] and 𝜂𝐴(𝑥) = [𝜂𝐴 −(𝑥), 𝜂𝐴 +(𝑥)] with the condition 0 ≤ 𝜇𝐴 +(𝑥) + 𝜂𝐴 +(𝑥) ≤ 1 for all 𝑥𝜖 𝑋. definition 2.6 (mishra et al., 2013; ismayil et al., 2014) an interval-valued intuitionistic fuzzy graph (ivifg) with underlying set v is defined to be a pair 𝐺 = (𝐴, 𝐵) where  the functions 𝜇𝐴: 𝑉 → 𝐷[0,1]; 𝜂𝐴: 𝑉 → 𝐷[0,1] denote the degree of membership and non-membership of the element 𝑥 ∈ 𝑉 respectively, such that 0 ≤ 𝜇𝐴(𝑥) + 𝜂𝐴(𝑥) ≤ 1 , ∀ 𝑥 ∈ 𝑉 intervalvalued fermatean neutrosophic graphs 181  the functions 𝜇𝐵 : 𝐸 ⊆ 𝑉 × 𝑉 → 𝐷[0,1] ; 𝜂𝐵 : 𝐸 ⊆ 𝑉 × 𝑉 → 𝐷[0,1] are defined by 𝜇𝐵 −((𝑥, 𝑦)) ≤ min( 𝜇𝐴 −(𝑥), 𝜇𝐴 −(𝑦)) ; 𝜂𝐵 −((𝑥, 𝑦)) ≥ min( 𝜂𝐴 −(𝑥), 𝜂𝐴 −(𝑦)) 𝜇𝐵 +((𝑥, 𝑦)) ≤ min( µ 𝐴 +(𝑥), µ 𝐴 +(𝑦)) ; 𝜂𝐵 +((𝑥, 𝑦)) ≥ min( 𝜂 𝐴 +(𝑥), 𝜂 𝐴 +(𝑦)) such that 0 ≤ 𝜇𝐵 +((𝑥, 𝑦)) + 𝜂𝐵 +((𝑥, 𝑦)) ≤ 1 , ∀(𝑥, 𝑦) ∈ 𝐸 example 2.7 𝐺 = (𝐴, 𝐵) defined on a graph 𝐺 ∗ = (𝑉, 𝐸) such that 𝑉 = {𝑥, 𝑦, 𝑧}, 𝐸 = {𝑥𝑦, 𝑦𝑧, 𝑧𝑥}, a is an interval valued intuitionistic fuzzy set of 𝑉and let 𝐵 is an intervalvalued intuitionistic fuzzy set of 𝐸 ⊆ 𝑉 𝑋 𝑉. here 𝐴 = {〈𝑥, [0.5,0.7], [0.1,0.3]〉, 〈𝑦, [0.6,0.7], [0.1,0.3]〉 , 〈𝑧, [0.4,0.6], [0.2,0.4]〉 } 𝐵 = {〈𝑥𝑦, [0.3,0.6], [0.2,0.4]〉, 〈𝑦𝑧, [0.3,0.5], [0.3,0.4]〉 , 〈𝑥𝑧, [0.3,0.5], [0.2,0.4]〉} figure 2. interval-valued intuitionistic fuzzy graph g mishra et al. (2013) introduced product of ivifg and ismayil et al. (2014) defined on strong interval-valued intuitionistic fuzzy graph. akram et al. (2013) studied the certain types of interval-valued fuzzy graphs. peng xu et al. (2022) studied the concept of certain interval-valued intuitionistic fuzzy graphs and its applications. xindong et al. (2016) introduced the concept of interval-valued pythagorean fuzzy set. mohamed et. al. (2018) introduced and studied interval-valued pythagorean fuzzy graphs. definition 2.8 (mohamed et. al., 2018) an intervalvalued pythagorean fuzzy set (ivpfs) a defined in a finite universe of discourse 𝑋 is given by 𝐴 = {〈𝑥, 𝜇𝐴(𝑥) = [ 𝜇𝐴 −(𝑥), 𝜇𝐴 +(𝑥)], 𝜂𝐴(𝑥) = [𝜂𝐴 −(𝑥), 𝜂𝐴 +(𝑥)]〉 / 𝑥𝜖 𝑋} where 𝜇𝐴 −(𝑥), 𝜇𝐴 +(𝑥) ∶ 𝑋  [0,1] and 𝜂𝐴 −(𝑥), 𝜂𝐴 +(𝑥): 𝑋  [0,1] and 0 ≤ (𝜇𝐴 +(𝑥))2 + (𝜂𝐴 +(𝑥))2 ≤ 1 . here 𝜇𝐴(𝑥) and 𝜂𝐴(𝑥) denote the degree of membership and degree of non-membership of 𝑥 ∈ 𝑋 in 𝐴. definition 2.9 (mohamed et. al., 2018) an pythagorean fuzzy graph (pfg) with underlying set 𝑉 defined to be a pair 𝐺 = (𝐴, 𝐵)where  the functions 𝜇𝐴: 𝑉 → 𝐷[0,1]; 𝜂𝐴: 𝑉 → 𝐷[0,1] denote the degree of membership and non-membership of the element 𝑥 ∈ 𝑉 respectively, such that 0 ≤ 𝜇𝐴(𝑥) + 𝜂𝐴(𝑥) ≤ 1 , ∀ 𝑥 ∈ 𝑉  the functions 𝜇𝐵 : 𝐸 ⊆ 𝑉 × 𝑉 → 𝐷[0,1] ; 𝜂𝐵 : 𝐸 ⊆ 𝑉 × 𝑉 → 𝐷[0,1] are defined by 𝜇𝐵 −((𝑥, 𝑦)) ≤ min( 𝜇𝐴 −(𝑥), 𝜇𝐴 −(𝑦)) ; 𝜂𝐵 −((𝑥, 𝑦)) ≥ min( 𝜂𝐴 −(𝑥), 𝜂𝐴 −(𝑦)) 𝜇𝐵 +((𝑥, 𝑦)) ≤ min( µ 𝐴 +(𝑥), µ 𝐴 +(𝑦)) ; 𝜂𝐵 +((𝑥, 𝑦)) ≥ min( 𝜂 𝐴 +(𝑥), 𝜂 𝐴 +(𝑦)) such that 0 ≤ 𝜇𝐵 +((𝑥, 𝑦)) 2 + 𝜂𝐵 +((𝑥, 𝑦)) 2 ≤ 1 , ∀(𝑥, 𝑦) ∈ 𝐸 said broumi et al./decis. mak. appl. manag. eng. 5 (2) (2022) 176-200 182 yahya et al. (2018) defined the strong interval-valued pythagorean fuzzy graph and cartesian product, composition and join of two strong interval-valued pythagorean fuzzy graph are studied. definition 2.10 (broumi et al., 2016d) an interval-valued neutrosophic set (ivns) 𝐴 in 𝑋 is characterized by truthmembership function 𝑇𝐴(𝑥) , indeterminacy-membership function 𝐼𝐴 (𝑥) and falsitymembership function 𝐹𝐴(𝑥). for each point 𝑥 ∈ 𝑋, we have that 𝑇𝐴(𝑥) = [𝑇𝐴 −(𝑥), 𝑇𝐴 +(𝑥)], 𝐼𝐴(𝑥) = [𝐼𝐴 −(𝑥), 𝐼𝐴 +(𝑥)], 𝐹𝐴(𝑥) = [𝐹𝐴 −(𝑥), 𝐹𝐴 +(𝑥)] ⊆ [0, 1] and 0 ≤ 𝑇𝐴 (𝑥) + 𝐼𝐴(𝑥) + 𝐹𝐴(𝑥) ≤ 3. definition 2.11 (broumi et al., 2016d) an intervalvalued neutrosophic graph (ivng) of a graph 𝐺 ∗ = (𝑉, 𝐸) we mean a pair 𝐺 = (𝐴, 𝐵), where 𝐴 =< [𝑇𝐴 −, 𝑇𝐴 +], [𝐼𝐴 −, 𝐼𝐴 +], [𝐹𝐴 −, 𝐹𝐴 +] > is an intervalvalued neutrosophic set on v; and 𝐵 = 〈[𝑇𝐵 −, 𝑇𝐵 +], [𝐼𝐵 −, 𝐼𝐵 +], [𝐹𝐵 −, 𝐹𝐵 +]〉 is an interval valued neutrosophic relation on 𝐸 satisfying the following condition: i. 𝑉 = { 𝑣1 , 𝑣2 , … , 𝑣𝑛 }, such that 𝑇𝐴 − ∶ 𝑉 → [0, 1], 𝑇𝐴 + ∶ 𝑉 → [0, 1], 𝐼𝐴 − ∶ 𝑉 → [0, 1], 𝐼𝐴 +: 𝑉 → [0, 1] and 𝐹𝐴 −: 𝑉 → [0, 1], 𝐹𝐴 +: 𝑉 → [0, 1] denote the degree of truth-membership, the degree of indeterminacy-membership and falsitymembership of the element 𝑦 ∈ 𝑉, respectively, and 0 ≤ 𝑇𝐴(𝑣𝑖 ) + 𝐼𝐴 (𝑣𝑖 ) + 𝐹𝐴(𝑣𝑖 ) ≤ 3, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣𝑖 ∈ 𝑉 (𝑖 = 1, 2, … , 𝑛) ii. the functions 𝑇𝐵 −: 𝑉 × 𝑉 → [0, 1], 𝑇𝐵 +: 𝑉 × 𝑉 → [0, 1], 𝐼𝐵 −: 𝑉 × 𝑉 → [0, 1], 𝐼𝐵 +: 𝑉 × 𝑉 → [0, 1] and 𝐹𝐵 −: 𝑉 × 𝑉 → [0,1], 𝐹𝐵 +: 𝑉 × 𝑉 → [0, 1] are such that 𝑇𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≤ min[𝑇𝐴 −(𝑣𝑖 ), 𝑇𝐴 −(𝑣𝑗 )], 𝑇𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≤ min[𝑇𝐴 +(𝑣𝑖 ), 𝑇𝐴 +(𝑣𝑗 )], 𝐼𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐼𝐵 − (𝑣𝑖 ), 𝐼𝐵 −(𝑣𝑗 )], 𝐼𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐼𝐵 +(𝑣𝑖 ), 𝐼𝐵 +(𝑣𝑗 )], 𝐹𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐹𝐵 −(𝑣𝑖 ), 𝐹𝐵 −(𝑣𝑗 )], 𝐹𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≥ 𝑚𝑎𝑥[𝐹𝐵 +(𝑣𝑖 ), 𝐹𝐵 +(𝑣𝑗 )], denoting the degree of truth-membership, indeterminacy-membership and falsitymembership of the edge (𝑣𝑖 , 𝑣𝑗 ) ∈ 𝐸 respectively, where 0 ≤ 𝑇𝐵 ({𝑣𝑖 , 𝑣𝑗 }) + 𝐼𝐵 ({𝑣𝑖 , 𝑣𝑗 }) + 𝐹𝐵 ({𝑣𝑖 , 𝑣𝑗 }) ≤ 3 for all {𝑣𝑖 , 𝑣𝑗 } ∈ 𝐸 (𝑖, 𝑗 = 1, 2, … , 𝑛). 3. interval-valued fermatean neutrosophic graphs fuzzy sets, intuitionistic fuzzy sets, neutrosophic sets are the generalization of the classical set and which are also the most popular mathematical tools in the study uncertainty. later, researchers combined these sets with graph structures and studied its properties in literature. these combinations, fuzzy graphs, intuitionistic fuzzy graphs and neutrosophic graphs are useful in decision making problems. in an administrative setup, electing a leader among a group of people through the voting process, a judgement may give based on a candidate satisfies his expectations with a possibility of 0.80 and this candidate dissatisfies the expectations with a possibility of 0.95 and neutrally give 0.85 but their sum is 2.80 (>2) and their square sum is 2.265 (>2) and the sum of the cubes is equal to 1.9835 (<2). it is impossible to give an exact degree of membership in every instant, because the lack of knowledge, vague intervalvalued fermatean neutrosophic graphs 183 information, and so forth may produce higher values to the membership values. to overcome this problem, we can use interval-valued fuzzy sets, which assign to each element a closed interval which approximates the “real,” but unknown, membership degree. in this series, we are adding one more class of graphs namely, interval-valued fermatean neutrosophic graphs and certain types of interval-valued fermatean neutrosophic graphs are introduced and discussed in this section. definition 3.1 an interval-valued fermatean neutrosophic set (ivfns) 𝐴 on the universe of discourse 𝑋 is of the structure: 𝐴 = {〈𝑥, 𝑇𝐴(𝑥), ia(𝑥), fa(𝑥)〉| 𝑥 ∈ x }, where 𝑇𝐴 (𝑥) = [ta −(𝑥) , 𝑇𝐴 +(𝑥)], ia(𝑥) = [ia −(𝑥) , ia +(x)] and fa(𝑥) = [𝐹𝐴 −(𝑥), fa +(𝑥)] represents the truth-membership degree, indeterminacy-membership degree and falsity-membership degree, respectively. consider the mapping 𝑇𝐴 (𝑥): 𝑋 → 𝐷[0,1] , ia(𝑥): 𝑋 → 𝐷[0,1], 𝐹𝐴(𝑥): 𝑋 → 𝐷[0,1] and 0 ≤ (𝑇𝐴 (𝑥)) 𝟑 +(𝐹𝐴(𝑥)) 𝟑 ≤1 and 0 ≤ (𝐼𝐴 (𝑥)) 𝟑 ≤ 1 0 ≤ (𝑇𝐴 (𝑥)) 𝟑 +(𝐹𝐴(𝑥)) 𝟑 +(𝐼𝐴 (𝑥)) 𝟑 ≤ 2 means 0 ≤ (𝑇𝐴 +(x))𝟑+(𝐼𝐴 +(𝑥))𝟑+(𝐹𝐴 +(𝑥))𝟑 ≤ 2 ∀ 𝑥 ∈ x definition 3.2 an interval-valued fermatean neutrosophic graph (ivfng) of a graph 𝐺 ∗ = (𝑉, 𝐸) we mean a pair 𝐺 = (𝐴, 𝐵), where 𝐴 = 〈[𝑇𝐴 −, 𝑇𝐴 +], [𝐼𝐴 −, 𝐼𝐴 +], [𝐹𝐴 −, 𝐹𝐴 +]〉 is an interval-valued fermatean neutrosophic set on v; and 𝐵 = 〈[𝑇𝐵 −, 𝑇𝐵 +], [𝐼𝐵 −, 𝐼𝐵 +], [𝐹𝐵 −, 𝐹𝐵 +〉] is an interval valued fermatean neutrosophic relation on 𝐸 satisfying the following condition: i. 𝑉 = { 𝑣1 , 𝑣2 , … , 𝑣𝑛 }, such that 𝑇𝐴 − ∶ 𝑉 → [0, 1], 𝑇𝐴 + ∶ 𝑉 → [0, 1], 𝐼𝐴 − ∶ 𝑉 → [0, 1], 𝐼𝐴 +: 𝑉 → [0, 1] and 𝐹𝐴 −: 𝑉 → [0, 1], 𝐹𝐴 +: 𝑉 → [0, 1]denote the degree of truth -membership, the degree of indeterminacy-membership and falsitymembership of the element 𝑦 ∈ 𝑉, respectively, and 0 ≤ 𝑇𝐴(𝑣𝑖 ) + 𝐼𝐴(𝑣𝑖 ) + 𝐹𝐴(𝑣𝑖 ) ≤ 3, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣𝑖 ∈ 𝑉 (𝑖 = 1, 2, … , 𝑛). ii. the functions 𝑇𝐵 −: 𝑉 × 𝑉 → [0, 1], 𝑇𝐵 +: 𝑉 × 𝑉 → [0, 1], 𝐼𝐵 −: 𝑉 × 𝑉 → [0, 1], 𝐼𝐵 +: 𝑉 × 𝑉 → [0, 1] and 𝐹𝐵 −: 𝑉 × 𝑉 → [0,1], 𝐹𝐵 +: 𝑉 × 𝑉 → [0, 1] are such that 𝑇𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≤ min[𝑇𝐴 −(𝑣𝑖 ), 𝑇𝐴 −(𝑣𝑗 )] , 𝑇𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≤ min[𝑇𝐴 +(𝑣𝑖 ), 𝑇𝐴 +(𝑣𝑗 )], 𝐼𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐼𝐵 − (𝑣𝑖 ), 𝐼𝐵 −(𝑣𝑗 )], 𝐼𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐼𝐵 +(𝑣𝑖 ), 𝐼𝐵 +(𝑣𝑗 )], 𝐹𝐵 −({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐹𝐵 −(𝑣𝑖 ), 𝐹𝐵 −(𝑣𝑗 )], 𝐹𝐵 +({𝑣𝑖 , 𝑣𝑗 }) ≥ max[𝐹𝐵 +(𝑣𝑖 ), 𝐹𝐵 +(𝑣𝑗 )] denoting the degree of truth-membership, indeterminacy-membership and falsitymembership of the edge (𝑣𝑖 , 𝑣𝑗 ) ∈ 𝐸 respectively, where 0 ≤ 𝑇𝐵 ({𝑣𝑖 , 𝑣𝑗 }) 3 + 𝐼𝐵 ({𝑣𝑖 , 𝑣𝑗 }) 3 + 𝐹𝐵 ({𝑣𝑖 , 𝑣𝑗 }) 3 ≤ 2 for all {𝑣𝑖 , 𝑣𝑗 } ∈ 𝐸 (𝑖, 𝑗 = 1, 2, … , 𝑛) means 0 ≤ (𝑇𝐵 +(𝑣𝑖 , 𝑣𝑗 )) 𝟑 +(𝐼𝐵 +(𝑣𝑖 , 𝑣𝑗 )) 𝟑 +(𝐹𝐵 +(𝑣𝑖 , 𝑣𝑗 )) 𝟑 ≤ 2 ∀ 𝑥 ∈ x. example 3.3 consider a graph 𝐺 ∗, such that 𝑉 = {𝑥1, 𝑥2, 𝑥3}, 𝐸 = {𝑥1𝑥2, 𝑥2𝑥3, 𝑥3𝑥4, 𝑥4𝑥1}. let 𝐴 be an interval valued fermatean neutrosophic subset of 𝑉 and 𝐵 be an interval valued fermatean neutrosophic subset of 𝐸, denoted by 𝐴 = { 〈𝑥1[0.85,0.95], [0.90,0.95], [0.85,0.85]〉, 〈𝑥2, [0.85,0.90], [0.90,0.95], [0.85,0.90]〉, 〈𝑥3, [0.85,0.95], [0.95,0.95], [0.85,0.95]〉 } 𝐵 = { 〈𝑥1𝑥2, [0.80,0.90], [0.90,0.95], [0.80,0.85]〉, 〈𝑥2𝑥3, [0.85,0.90], [0.90,0.95], [0.85,0.85]〉, 〈𝑥3𝑥1, [0.85,0.95], [0.90,0.95], [0.85,0.85]〉 } said broumi et al./decis. mak. appl. manag. eng. 5 (2) (2022) 176-200 184 figure 3. interval-valued fermatean neutrosophic graph g definition 3.4. let 𝐺 = (𝐴, 𝐵) be an ivfng. 𝐺 is an interval valued regular fermatean neutrosophic graph if it satisfies the following conditions: ∑ 𝑇𝐵 −(𝑣1, 𝑣2) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑣1≠𝑣2 ; ∑ 𝑇𝐵 +(𝑣1, 𝑣2) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑣1≠𝑣2 ∑ 𝐼𝐵 −(𝑣1, 𝑣2) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑣1≠𝑣2 ; ∑ 𝐼𝐵 +(𝑣1, 𝑣2) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑣1≠𝑣2 ∑ 𝐹𝐵 −(𝑣1, 𝑣2) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑣1≠𝑣2 ; ∑ 𝐹𝐵 +(𝑣1, 𝑣2) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑣1≠𝑣2 definition 3.5. let g = (a, b) be an ivfng. g is an interval valued regular strong neutrosophic graph if it satisfies the following conditions 𝑇𝐵 −(𝑣1, 𝑣2) = min(𝑇𝐴 −(𝑣1), 𝑇𝐴 −(𝑣2)) ; ∑ 𝑇𝐵 −(𝑣1, 𝑣2) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑣1≠𝑣2 𝑇𝐵 +(𝑣1, 𝑣2) = min(𝑇𝐴 +(𝑣1), 𝑇𝐴 +(𝑣2)) ; ∑ 𝑇𝐵 +(𝑣1, 𝑣2) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑣1≠𝑣2 𝐼𝐵 −(𝑣1, 𝑣2) = max(𝐼𝐴 −(𝑣1), 𝐼𝐴 −(𝑣2)) ; ∑ 𝐼𝐵 −(𝑣1, 𝑣2) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑣1≠𝑣2 𝐼𝐵 +(𝑣1, 𝑣2) = max(𝐼𝐴 +(𝑣1), 𝐼𝐴 +(𝑣2)) ; ∑ 𝐼𝐵 +(𝑣1, 𝑣2) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑣1≠𝑣2 𝐹𝐵 −(𝑣1, 𝑣2) = max(𝐹𝐴 −(𝑣1), 𝐹𝐴 −(𝑣2)) ; ∑ 𝐹𝐵 −(𝑣1, 𝑣2) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑣1≠𝑣2 𝐹𝐵 +(𝑣1, 𝑣2) = max(𝐹𝐴 +(𝑣1), 𝐹𝐴 +(𝑣2)) ; ∑ 𝐹𝐵 +(𝑣1, 𝑣2) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑣1≠𝑣2 definition 3.6. let g = (a, b) be an ivfng. g is a strong interval valued regular strong neutrosophic graph if it satisfies the following conditions: 𝑇𝐵 −(𝑣1, 𝑣2) = min(𝑇𝐴 −(𝑣1), 𝑇𝐴 −(𝑣2)) ; 𝐼𝐵 −(𝑣1, 𝑣2) = max(𝐼𝐴 −(𝑣1), 𝐼𝐴 −(𝑣2)) ; 𝐹𝐵 −(𝑣1, 𝑣2) = max(𝐹𝐴 −(𝑣1), 𝐹𝐴 −(𝑣2)); 𝑇𝐵 +(𝑣1, 𝑣2) = min(𝑇𝐴 +(𝑣1), 𝑇𝐴 +(𝑣2)) ; 𝐼𝐵 +(𝑣1, 𝑣2) = max(𝐼𝐴 +(𝑣1), 𝐼𝐴 +(𝑣2)) ; 𝐹𝐵 +(𝑣1, 𝑣2) = max(𝐹𝐴 +(𝑣1), 𝐹𝐴 +(𝑣2)) ; intervalvalued fermatean neutrosophic graphs 185 such that 0≤ 𝑇𝐵 +(𝑣1, 𝑣2 )) + i𝐵 + (𝑣1, 𝑣2)) + 𝐹𝐵 +(𝑣1, 𝑣2)) ≤ 3, for all 𝑣1, 𝑣2 ∈ 𝐸 and 0 ≤ (𝑇𝐵 +(𝑣𝑖 , 𝑣𝑗)) 𝟑 +(𝐼𝐵 +(𝑣𝑖 , 𝑣𝑗)) 𝟑 +(𝐹𝐵 +(𝑣𝑖 , 𝑣𝑗)) 𝟑 ≤ 2 ∀ 𝑥 ∈ x example 3.7. let 𝐺 = (𝐴, 𝐵)be an interval-valued fermatean neutrosophic graph with𝑉 = {𝑥1, 𝑥2, 𝑥3}. 𝐴 = { 〈𝑥1, [0.85,0.95], [0.90,0.95], [0.85,0.85]〉, 〈𝑥2, [0.85,0.90], [0.90,0.95], [0.85,0.90]〉, 〈𝑥3, [0.85,0.95], [0.95,0.95], [0.85,0.95]〉, }, 𝐵 = { 〈𝑥1𝑥2, [0.85,0.90], [0.90,0.95], [0.80,0.90]〉, 〈𝑥2𝑥3, [0.85,0.90], [0.95,0.95], [0.85,0.95]〉, 〈𝑥1𝑥3, [0.85,0.95], [0.95,0.95], [0.85,0.95]〉, }, figure 4. strong interval-valued fermatean neutrosophic graph g definition 3.8. let 𝐴1 and 𝐴2 be interval-valued neutrosophic subsets of 𝑉1 and 𝑉2 respectively. let 𝐵1 and 𝐵2 interval-valued neutrosophic subsets of 𝐸1 and 𝐸2 respectively. the cartesian product of two ivfngs 𝐺1 and 𝐺2 is denoted by 𝐺1 × 𝐺2 = (𝐴1 × 𝐴2 , 𝐵1 × 𝐵2) and is defined as follows: i. (𝑇𝐴1 − × 𝑇𝐴2 − )(𝑥1, 𝑥2) = min (𝑇𝐴1 − (𝑥1), 𝑇𝐴2 − (𝑥2)) (𝑇𝐴1 + × 𝑇𝐴2 + )(𝑥1, 𝑥2) = min (𝑇𝐴1 + (𝑥1), 𝑇𝐴2 + (𝑥2)) (𝐼𝐴1 − × 𝐼𝐴2 − )(𝑥1, 𝑥2) = max (𝐼𝐴1 − (𝑥1), 𝐼𝐴2 − (𝑥2)) (𝐼𝐴1 + × 𝐼𝐴2 + )(𝑥1 , 𝑥2) = max (𝐼𝐴1 + (𝑥1), 𝐼𝐴2 + (𝑥2)) (𝐹𝐴1 − × 𝐹𝐴2 − )(𝑥1, 𝑥2) = max (𝐹𝐴1 − (𝑥1 ), 𝐹𝐴2 − (𝑥2)) (𝐹𝐴1 + × 𝐹𝐴2 + )(𝑥1, 𝑥2) = max (𝐹𝐴1 + (𝑥1), 𝐹𝐴2 + (𝑥2)) 𝑓𝑜𝑟 𝑎𝑙𝑙 ( 𝑥1, 𝑥2) ∈ 𝑉 ii. (𝑇𝐵1 − × 𝑇𝐵2 − )((𝑥, 𝑥2)(𝑥, 𝑦2)) = min (𝑇𝐴1 − (𝑥), 𝑇𝐵1 − (𝑥2𝑦2 )) (𝑇𝐵1 + × 𝑇𝐵2 + )((𝑥, 𝑥2)(𝑥, 𝑦2)) = min (𝑇𝐴1 + (𝑥), 𝑇𝐵1 + (𝑥2𝑦2)) (𝐼𝐵1 − × 𝐼𝐵2 − )((𝑥, 𝑥2)(𝑥, 𝑦2)) = max (𝐼𝐴1 − (𝑥), 𝐼𝐵2 − (𝑥2𝑦2)) (𝐼𝐵1 + × 𝐼𝐵2 + )((𝑥, 𝑥2)(𝑥, 𝑦2)) = max (𝐼𝐴1 + (𝑥), 𝐼𝐵2 + (𝑥2𝑦2)) (𝐹𝐵1 − × 𝐹𝐵2 − )((𝑥, 𝑥2)(𝑥, 𝑦2 )) = max (𝐹𝐴1 − (𝑥), 𝐹𝐵2 − (𝑥2𝑦2)) (𝐹𝐵1 + × 𝐹𝐵2 + )((𝑥, 𝑥2 )(𝑥, 𝑦2 )) = max (𝐹𝐴1 + (𝑥), 𝐹𝐵2 + (𝑥2𝑦2 )) ∀ 𝑥 ∈ 𝑉1 𝑎𝑛𝑑 ∀ 𝑥2𝑦2 ∈ 𝐸2 iii. (𝑇𝐵1 − × 𝑇𝐵2 − ) ((𝑥1, 𝑧) (𝑦1, 𝑧)) = min (𝑇𝐵1 − (𝑥1𝑦1), 𝑇𝐴2 − (𝑧)) said broumi et al./decis. mak. appl. manag. eng. 5 (2) (2022) 176-200 186 (𝑇𝐵1 + × 𝑇𝐵2 + ) ((𝑥1, 𝑧) (𝑦1, 𝑧)) = min (𝑇𝐵1 + (𝑥1𝑦1), 𝑇𝐴2 + (𝑧)) (𝐼𝐵1 − × 𝐼𝐵2 − )((𝑥1, 𝑧) (𝑦1, 𝑧)) = max (𝐼𝐵1 − (𝑥1𝑦1), 𝐼𝐴2 − (𝑧)) (𝐼𝐵1 + × 𝐼𝐵2 + )((𝑥1, 𝑧) (𝑦1, 𝑧)) = max (𝐼𝐵1 + (𝑥1𝑦1), 𝐼𝐴2 + (𝑧)) (𝐹𝐵1 − × 𝐹𝐵2 − )((𝑥1, 𝑧) (𝑦1 , 𝑧)) = max (𝐹𝐵1 − (𝑥1𝑦1 ), 𝐹𝐴2 − (𝑧)) (𝐹𝐵1 + × 𝐹𝐵2 + )((𝑥1, 𝑧)(𝑦1, 𝑧)) = max (𝐹𝐵1 + (𝑥1𝑦1 ), 𝐹𝐴2 + (𝑧)) ∀ 𝑧 ∈ 𝑉2 𝑎𝑛𝑑 ∀ 𝑥1𝑦1 ∈ 𝐸1 example 3.9. let 𝐺1 ∗ = (𝐴1, 𝐵1)and 𝐺2 ∗ = (𝐴2, 𝐵2 ) be two graphs where 𝑉1 = {𝑢1, 𝑢2}, 𝑉2 = {𝑣1, 𝑣2}. consider two interval valued fermatean neutrosophic graphs: 𝐴1 = {〈u1, [0.85,0.95], [0.95,0.95], [0.95,0.95]〉, 〈𝑢2, [0.90,0.90], [0.95,0.95], [0.85,0.85]〉, }, 𝐵1 = {〈𝑢1𝑢2, [0.85,0.90], [0.95,0.95], [0.95,0.95]〉} ; 𝐴2 = {〈𝑣1, [0.80,0.90], [0.85,0.95], [0.95,0.85]〉, 〈𝑣2, [0.95,0.90], [0.95,0.95], [0.80,0.85]〉, }, 𝐵2 = {〈𝑣1𝑣2, [0.80,0.90], [0.95,0.95], [0.95,0.85]〉}. figure 6. cartesian product of two ivfngs 𝐺1 × g2 definition 3.10. let 𝐺 $ = 𝐺1 $ × 𝐺2 $ = (𝑉1 × 𝑉2, 𝐸) be the composition of two graphs where 𝐸 = {(𝑥, 𝑥2) (𝑥, 𝑦2 ) /𝑥 ∈ 𝑉1, 𝑥2𝑦2 ∈ 𝐸2} ∪ {(𝑥1, 𝑧) (𝑦1, 𝑧) /𝑧 ∈ 𝑉2, 𝑥1𝑦1 ∈ 𝐸1} ∪ {( 𝑥1, 𝑥2) ( 𝑦1 , 𝑦2) |𝑥1𝑦1 ∈ 𝐸1, 𝑥2 ≠ 𝑦2 }, then the composition of interval valued fermatean neutrosophic graphs 𝐺1[ 𝐺2] = (𝐴1 ∘ 𝐴2, 𝐵1 ∘ 𝐵2) is an interval valued fermatean neutrosophic graphs defined by: i. (𝑇𝐴1 − ∘ 𝑇𝐴2 − ) (𝑥1, 𝑥2) = min (𝑇𝐴1 − (𝑥1), 𝑇𝐴2 − (𝑥2)) (𝑇𝐴1 + ∘ 𝑇𝐴1 + ) (𝑥1, 𝑥2) = min (𝑇𝐴1 + (𝑥1), 𝑇𝐴1 + (𝑥2)) (𝐼𝐴1 − ∘ 𝐼𝐴2 − )(𝑥1, 𝑥2) = max (𝐼𝐴1 − (𝑥1), 𝐼𝐴2 − (𝑥2)) (𝐼𝐴1 + ∘ 𝐼𝐴2 + )(𝑥1, 𝑥2) = max (𝐼𝐴1 + (𝑥1), 𝐼𝐴2 + (𝑥2)) (𝐹𝐴1 − ∘ 𝐹𝐴2 − )(𝑥1, 𝑥2) = max (𝐹𝐴1 − (𝑥1), 𝐹𝐴2 − (𝑥2)) (𝐹𝐴1 + ∘ 𝐹𝐴2 + ) (𝑥1, 𝑥2) = max (𝐹𝐴1 + (𝑥1), 𝐹𝐴2 + (𝑥2)) ∀ 𝑥1 ∈ 𝑉1, 𝑥2 ∈ 𝑉2 ii. (𝑇𝐴1 − ∘ 𝑇𝐴2 − )((𝑥, 𝑥2)(𝑥, 𝑦2)) = min (𝑇𝐴1 − (𝑥), 𝑇𝐵2 − (𝑥2𝑦2 )) figure 5. interval − valued fermatean neutrosophic graphs g1 , g2 intervalvalued fermatean neutrosophic graphs 187 (𝑇𝐴1 + ∘ 𝑇𝐴1 + )((𝑥, 𝑥2)(𝑥, 𝑦2)) = min (𝑇𝐴1 + (𝑥), 𝑇𝐵2 + (𝑥2𝑦2)) (𝐼𝐴1 − ∘ 𝐼𝐴2 − ) ((𝑥, 𝑥2)(𝑥, 𝑦2)) = max (𝐼𝐴1 − (𝑥), 𝐼𝐵2 − (𝑥2𝑦2)) (𝐼𝐴1 + ∘ 𝐼𝐴2 + ) ((𝑥, 𝑥2)(𝑥, 𝑦2 )) = max (𝐼𝐴1 + (𝑥), 𝐼𝐵2 + (𝑥2𝑦2)) (𝐹𝐴1 − ∘ 𝐹𝐴2 − )((𝑥, 𝑥2)(𝑥, 𝑦2)) = max (𝐹𝐴1 − (𝑥), 𝐹𝐵2 − (𝑥2𝑦2)) (𝐹𝐴1 + ∘ 𝐹𝐴2 + ) ((𝑥, 𝑥2)(𝑥, 𝑦2)) = max (𝐹𝐴1 + (𝑥), 𝐹𝐵2 + (𝑥2𝑦2)) ∀ 𝑥 ∈ 𝑉1, ∀𝑥2 𝑦2 ∈ 𝐸2 iii. (𝑇𝐵1 − ∘ 𝑇𝐵2 − ) ((𝑥1, 𝑧) (𝑦1, 𝑧)) = min (𝑇𝐵1 − (𝑥1𝑦1), 𝑇𝐴 2 −(𝑧)) (𝑇𝐵1 + ∘ 𝑇𝐵2 + ) ((𝑥1, 𝑧) (𝑦1, 𝑧)) = min (𝑇𝐵1 + (𝑥1𝑦1 ), 𝑇𝐴 2 +(𝑧)) (𝐼𝐵1 − ∘ 𝐼𝐵2 − ) ((𝑥1, 𝑧) (𝑦1, 𝑧)) = max (𝐼𝐵1 − (𝑥1𝑦1), 𝐼𝐴2 − (𝑧)) (𝐼𝐵1 + ∘ 𝐼𝐵2 + ) ((𝑥1, 𝑧) (𝑦1, 𝑧)) = max (𝐼𝐵1 + (𝑥1𝑦1), 𝐼𝐴2 + (𝑧)) (𝐹𝐵1 − ∘ 𝐹𝐵2 − ) ((𝑥1, 𝑧) (𝑦1, 𝑧)) = max (𝐹𝐵1 − (𝑥1𝑦1 ), 𝐹𝐴2 − (𝑧)) (𝐹𝐵1 + ∘ 𝐹𝐵2 + ) ((𝑥1, 𝑧) (𝑦1, 𝑧)) = max (𝐹𝐵1 + (𝑥1𝑦1), 𝐹𝐴2 + (𝑧)) ∀ 𝑧 ∈ 𝑉2, ∀𝑥1 𝑦1 ∈ 𝐸1 ; iv. (𝑇𝐵1 − ∘ 𝑇𝐵2 − ) ((𝑥1, 𝑥2) (𝑦1, 𝑦2)) = min (𝑇𝐴2 − (𝑥2), 𝑇𝐴2 − (𝑦2), 𝑇𝐵1 − (𝑥1𝑦1)) (𝑇𝐵1 + ∘ 𝑇𝐵2 + )((𝑥1, 𝑥2)(𝑦1 , 𝑦2)) = min (𝑇𝐴2 + (𝑥2), 𝑇𝐴2 + (𝑦2), 𝑇𝐵1 + (𝑥1𝑦1)) (𝐼𝐵1 − ∘ 𝐼𝐵2 − )((𝑥1, 𝑥2)(𝑦1, 𝑦2)) = max (𝐼𝐴2 − (𝑥2), 𝐼𝐴2 − (𝑦2), 𝐼𝐵1 − (𝑥1𝑦1)) (𝐼𝐵1 + ∘ 𝐼𝐵2 + )((𝑥1, 𝑥2)(𝑦1, 𝑦2)) = max (𝐼𝐴2 + (𝑥2), 𝐼𝐴2 + (𝑦2), 𝐼𝐵1 + (𝑥1𝑦1)) (𝐹𝐵1 − ∘ 𝐹𝐵2 − )((𝑥1, 𝑥2)(𝑦1, 𝑦2 )) = max (𝐹𝐴2 − (𝑥2), 𝐹𝐴2 − (𝑦2), 𝐹𝐵1 − (𝑥1𝑦1)) ( 𝐹𝐵1 + ∘ 𝐹𝐵2 + )((𝑥1, 𝑥2)(𝑦1 , 𝑦2)) = max ( 𝐹𝐴2 + (𝑥2 ), 𝐹𝐴2 + (𝑦2 ), 𝐹𝐵1 + (𝑥1𝑦1 )), ∀ (𝑥1, 𝑥2)( 𝑦1, 𝑦2) ∈ 𝐸 0 − 𝐸, 𝑤ℎ𝑒𝑟𝑒 𝐸0 = 𝐸 ∪ {( 𝑥1, 𝑥2) ( 𝑦1, 𝑦2) |𝑥1 𝑦1 ∈ 𝐸1, 𝑥2 ≠ 𝑦2 }. example 3.11. let 𝐺1 ∗ = (𝐴1, 𝐵1)and 𝐺2 ∗ = (𝐴2, 𝐵2 ) be two graphs where 𝑉1 = {𝑢1, 𝑢2}, 𝑉2 = {𝑣1, 𝑣2}. consider two interval valued fermatean neutrosophic graphs: 𝐴1 = {〈u1, [0.85,0.95], [0.95,0.95], [0.95,0.95]〉, 〈u2, [0.90,0.90], [0.95,0.95], [0.85,0.85]〉}, 𝐵1 = {〈𝑢1𝑢2, [0.85,0.90], [0.95,0.95], [0.95,0.95]〉 } ; 𝐴2 = {〈𝑣1, [0.80,0.90], [0.85,0.95], [0.95,0.85]〉, 〈𝑣2, [0.95,0.90], [0.95,0.95], [0.80,0.85]〉}, 𝐵2 = {〈𝑣1𝑣2, [0.80,0.90], [0.95,0.95], [0.95,0.85]〉}. figure 7. interval − valued fermatean neutrosophic graphs g1 , g2 said broumi et al./decis. mak. appl. manag. eng. 5 (2) (2022) 176-200 188 figure 8. composition of interval valued fermatean neutrosophic graphs 𝐺1[ 𝐺2] definition 3.12. the union 𝐺1 ∪ 𝐺2 = (𝐴1 ∪ 𝐴2, 𝐵1 ∪ 𝐵2) of two interval valued fermatean neutrosophic graphs of the graphs 𝐺1 ∗ and 𝐺2 ∗ is an interval-valued fermatean neutrosophic graph of 𝐺1 ∗ ∪ 𝐺2 ∗ .  (𝑇𝐴1 − ∪ 𝑇𝐴2 − )(𝑥) = { 𝑇𝐴1 − (𝑥), 𝑖𝑓 𝑥 ∈ 𝑉1 𝑎𝑛𝑑 𝑥 ∉ 𝑉2 𝑇𝐴2 − (𝑥)𝑖𝑓 𝑥 ∉ 𝑉1 𝑎𝑛𝑑 𝑥 ∈ 𝑉2 min (𝑇𝐴1 − (𝑥), 𝑇𝐴2 − (𝑥)) if 𝑥 ∈ v1 ∩ v2, ,  (𝑇𝐴1 + ∪ 𝑇𝐴2 + )(𝑥) = { 𝑇𝐴1 + (𝑥), 𝑖𝑓 𝑥 ∈ 𝑉1 𝑎𝑛𝑑 𝑥 ∉ 𝑉2 𝑇𝐴2 + (𝑥)𝑖𝑓 𝑥 ∉ 𝑉1 𝑎𝑛𝑑 𝑥 ∈ 𝑉2 min (𝑇𝐴1 + (𝑥), 𝑇𝐴2 + (𝑥)) if 𝑥 ∈ v1 ∩ v2,  (𝐼𝐴1 − ∪ 𝐼𝐴2 − )(𝑥) = { 𝐼𝐴1 − (𝑥), 𝑖𝑓 𝑥 ∈ 𝑉1 𝑎𝑛𝑑 𝑥 ∉ 𝑉2 𝐼𝐴2 − (𝑥)𝑖𝑓 𝑥 ∉ 𝑉1 𝑎𝑛𝑑 𝑥 ∈ 𝑉2 max (𝐼𝐴1 − (𝑥), 𝐼𝐴2 − (𝑥)) if 𝑥 ∈ v1 ∩ v2,  (𝐼𝐴1 + ∪ 𝐼𝐴2 + )(𝑥) = { 𝐼𝐴1 + (𝑥), 𝑖𝑓 𝑥 ∈ 𝑉1 𝑎𝑛𝑑 𝑥 ∉ 𝑉2 𝐼𝐴2 + (𝑥)𝑖𝑓 𝑥 ∉ 𝑉1 𝑎𝑛𝑑 𝑥 ∈ 𝑉2 max (𝐼𝐴1 + (𝑥), 𝐼𝐴2 + (𝑥)) if 𝑥 ∈ v1 ∩ v2,  (𝐹𝐴1 − ∪ 𝐹𝐴2 − )(𝑥) = { 𝐹𝐴1 − (𝑥), 𝑖𝑓 𝑥 ∈ 𝑉1 𝑎𝑛𝑑 𝑥 ∉ 𝑉2 𝐹𝐴2 − (𝑥)𝑖𝑓 𝑥 ∉ 𝑉1 𝑎𝑛𝑑 𝑥 ∈ 𝑉2 max (𝐹𝐴1 − (𝑥), 𝐹𝐴2 − (𝑥)) if 𝑥 ∈ v1 ∩ v2,  (𝐹𝐴1 + ∪ 𝐹𝐴2 + )(𝑥) = { 𝐹𝐴1 + (𝑥), 𝑖𝑓 𝑥 ∈ 𝑉1 𝑎𝑛𝑑 𝑥 ∉ 𝑉2 𝐹𝐴2 + (𝑥)𝑖𝑓 𝑥 ∉ 𝑉1 𝑎𝑛𝑑 𝑥 ∈ 𝑉2 max (𝐹𝐴1 + (𝑥), 𝐹𝐴2 + (𝑥)) if 𝑥 ∈ v1 ∩ v2,  (𝑇𝐵1 − ∪ 𝑇𝐵2 − )(𝑥𝑦) = { 𝑇𝐵1 − (𝑥𝑦), 𝑖𝑓 𝑥𝑦 ∈ 𝐸1 𝑎𝑛𝑑 𝑥𝑦 ∉ 𝐸2 𝑇𝐵2 − (𝑥𝑦) 𝑖𝑓 𝑥𝑦 ∉ 𝐸1 𝑎𝑛𝑑 𝑥𝑦 ∈ 𝐸2 min (𝑇𝐵1 − (𝑥𝑦), 𝑇𝐵2 − (𝑥𝑦) ) if 𝑥𝑦 ∈ e1 ∩ 𝐸2,  (𝑇𝐵1 + ∪ 𝑇𝐵2 + )(𝑥𝑦) = { 𝑇𝐵1 + (𝑥𝑦), 𝑖𝑓 𝑥𝑦 ∈ 𝐸1 𝑎𝑛𝑑 𝑥𝑦 ∉ 𝐸2 𝑇𝐵2 + (𝑥𝑦) 𝑖𝑓 𝑥𝑦 ∉ 𝐸1 𝑎𝑛𝑑 𝑥𝑦 ∈ 𝐸2 min (𝑇𝐵1 + (𝑥𝑦), 𝑇𝐵2 + (𝑥𝑦) ) if 𝑥𝑦 ∈ e1 ∩ 𝐸2, intervalvalued fermatean neutrosophic graphs 189  (𝐼𝐵1 − ∪ 𝐼𝐵2 − )(𝑥𝑦) = { 𝐼𝐵1 − (𝑥𝑦), 𝑖𝑓 𝑥𝑦 ∈ 𝐸1 𝑎𝑛𝑑 𝑥𝑦 ∉ 𝐸2 𝐼𝐵2 − (𝑥𝑦) 𝑖𝑓 𝑥𝑦 ∉ 𝐸1 𝑎𝑛𝑑 𝑥𝑦 ∈ 𝐸2 min (𝐼𝐵1 − (𝑥𝑦), 𝐼𝐵2 − (𝑥𝑦) ) if 𝑥𝑦 ∈ e1 ∩ 𝐸2,  (𝐼𝐵1 + ∪ 𝐼𝐵2 + )(𝑥𝑦) = { 𝐼𝐵1 + (𝑥𝑦), 𝑖𝑓 𝑥𝑦 ∈ 𝐸1 𝑎𝑛𝑑 𝑥𝑦 ∉ 𝐸2 𝐼𝐵2 + (𝑥𝑦) 𝑖𝑓 𝑥𝑦 ∉ 𝐸1 𝑎𝑛𝑑 𝑥𝑦 ∈ 𝐸2 max (𝐼𝐵1 + (𝑥𝑦), 𝐼𝐵2 + (𝑥𝑦) ) if 𝑥𝑦 ∈ e1 ∩ 𝐸2,  𝐹𝐵1 − ∪ 𝐹𝐵2 − )(𝑥𝑦) = { 𝐹𝐵1 − (𝑥𝑦), 𝑖𝑓 𝑥𝑦 ∈ 𝐸1 𝑎𝑛𝑑 𝑥𝑦 ∉ 𝐸2 𝐹𝐵2 − (𝑥𝑦) 𝑖𝑓 𝑥𝑦 ∉ 𝐸1 𝑎𝑛𝑑 𝑥𝑦 ∈ 𝐸2 max (𝐹𝐵1 − (𝑥𝑦), 𝐹𝐵2 − (𝑥𝑦) ) if 𝑥𝑦 ∈ e1 ∩ 𝐸2,  (𝐹𝐵1 + ∪ 𝐹𝐵2 + )(𝑥𝑦) = { 𝐹𝐵1 + (𝑥𝑦), 𝑖𝑓 𝑥𝑦 ∈ 𝐸1 𝑎𝑛𝑑 𝑥𝑦 ∉ 𝐸2 𝐹𝐵2 + (𝑥𝑦) 𝑖𝑓 𝑥𝑦 ∉ 𝐸1 𝑎𝑛𝑑 𝑥𝑦 ∈ 𝐸2 max (𝐹𝐵1 + (𝑥𝑦), 𝐹𝐵2 + (𝑥𝑦) ) if 𝑥𝑦 ∈ e1 ∩ 𝐸2, definition 3.13. the join of 𝐺1 + 𝐺2 = (𝐴1 + 𝐴2 , 𝐵1 + 𝐵2 ) interval valued neutrosophic graphs 𝐺1 and 𝐺2 of the graphs 𝐺1 ∗ and 𝐺2 ∗ is defined as follows:  (𝑇𝐴1 − + 𝑇𝐴2 − )(𝑥) = { 𝑇𝐴1 − (𝑥) 𝑖𝑓 𝑥 ∈ 𝑉1 𝑇𝐴2 − (𝑥) 𝑖𝑓 𝑥 ∈ 𝑉2 min(𝑇𝐴1 − , 𝑇𝐴2 − )(𝑥) if 𝑥 ∈ v1 ∪ v2, ,  (𝑇𝐴1 + + 𝑇𝐴2 + )(𝑥) = { 𝑇𝐴1 + (𝑥) 𝑖𝑓 𝑥 ∈ 𝑉1 𝑇𝐴2 + (𝑥) 𝑖𝑓 𝑥 ∈ 𝑉2 min(𝑇𝐴1 + , 𝑇𝐴2 + )(𝑥) if 𝑥 ∈ v1 ∪ v2,  (𝐼𝐴1 − + 𝐼𝐴2 − )(𝑥) = { 𝐼𝐴1 − (𝑥) 𝑖𝑓 𝑥 ∈ 𝑉1 𝐼𝐴2 − (𝑥) 𝑖𝑓 𝑥 ∈ 𝑉2 max(𝐼𝐴1 − , 𝐼𝐴2 − )(𝑥) if 𝑥 ∈ v1 ∪ v2,  (𝐼𝐴1 + + 𝐼𝐴2 + )(𝑥) = { 𝐼𝐴1 + (𝑥) 𝑖𝑓 𝑥 ∈ 𝑉1 𝐼𝐴2 + (𝑥) 𝑖𝑓 𝑥 ∈ 𝑉2 max(𝐼𝐴1 + , 𝐼𝐴2 + )(𝑥) if 𝑥 ∈ v1 ∪ v2,  (𝐹𝐴1 − + 𝐹𝐴2 − )(𝑥) = { 𝐹𝐴1 − (𝑥) 𝑖𝑓 𝑥 ∈ 𝑉1 𝐹𝐴2 − (𝑥) 𝑖𝑓 𝑥 ∈ 𝑉2 max(𝐹𝐴1 − , 𝐹𝐴2 − )(𝑥) if 𝑥 ∈ v1 ∪ v2,  (𝐹𝐴1 + + 𝐹𝐴2 + )(𝑥) = { 𝐹𝐴1 + (𝑥) 𝑖𝑓 𝑥 ∈ 𝑉1 𝐹𝐴2 + (𝑥) 𝑖𝑓 𝑥 ∈ 𝑉2 max(𝐹𝐴1 + , 𝐹𝐴2 + )(𝑥) if 𝑥 ∈ v1 ∪ v2,  (𝑇𝐵1 − + 𝑇𝐵2 − )(𝑥𝑦) = { 𝑇𝐵1 − (𝑥𝑦), 𝑖𝑓 𝑥𝑦 ∈ 𝐸1 𝑇𝐵2 − (𝑥𝑦) 𝑖𝑓 𝑥𝑦 ∈ 𝐸2 min (𝑇𝐵1 − (𝑥𝑦), 𝑇𝐵2 − (𝑥𝑦) ) if 𝑥𝑦 ∈ e1 ∪ 𝐸2, said broumi et al./decis. mak. appl. manag. eng. 5 (2) (2022) 176-200 190  (𝑇𝐵1 + + 𝑇𝐵2 + )(𝑥𝑦) = { 𝑇𝐵1 + (𝑥𝑦), 𝑖𝑓 𝑥𝑦 ∈ 𝐸1 𝑇𝐵2 + (𝑥𝑦) 𝑖𝑓 𝑥𝑦 ∈ 𝐸2 min (𝑇𝐵1 + (𝑥𝑦), 𝑇𝐵2 + (𝑥𝑦) ) if 𝑥𝑦 ∈ e1 ∪ 𝐸2,  𝐼𝐵1 − + 𝐼𝐵2 − )(𝑥𝑦) = { 𝐼𝐵1 − (𝑥𝑦), 𝑖𝑓 𝑥𝑦 ∈ 𝐸1 𝐼𝐵2 − (𝑥𝑦) 𝑖𝑓 𝑥𝑦 ∈ 𝐸2 max(𝐼𝐵1 − (𝑥𝑦), 𝐼𝐵2 − (𝑥𝑦) ) if 𝑥𝑦 ∈ e1 ∪ 𝐸2,  (𝐼𝐵1 + + 𝐼𝐵2 + )(𝑥𝑦) = { 𝐼𝐵1 + (𝑥𝑦), 𝑖𝑓 𝑥𝑦 ∈ 𝐸1 𝐼𝐵2 + (𝑥𝑦) 𝑖𝑓 𝑥𝑦 ∈ 𝐸2 max(𝐼𝐵1 + (𝑥𝑦), 𝐼𝐵2 + (𝑥𝑦) ) if xy ∈ e1 ∪ 𝐸2,  𝐹𝐵1 − + 𝐹𝐵2 − )(𝑥𝑦) = { 𝐹𝐵1 − (𝑥𝑦), 𝑖𝑓 𝑥𝑦 ∈ 𝐸1 𝐹𝐵2 − (𝑥𝑦) 𝑖𝑓 𝑥𝑦 ∈ 𝐸2 max(𝐹𝐵1 − (𝑥𝑦), 𝐹𝐵2 − (𝑥𝑦) ) if 𝑥𝑦 ∈ e1 ∪ 𝐸2,  (𝐹𝐵1 + + 𝐹𝐵2 + )(𝑥𝑦) = { 𝐹𝐵1 + (𝑥𝑦), 𝑖𝑓 𝑥𝑦 ∈ 𝐸1 𝐹𝐵2 + (𝑥𝑦) 𝑖𝑓 𝑥𝑦 ∈ 𝐸2 max(𝐹𝐵1 + (𝑥𝑦), 𝐹𝐵2 + (𝑥𝑦) ) if 𝑥𝑦 ∈ e1 ∪ 𝐸2,  (𝑇𝐵1 − + 𝑇𝐵2 − ) (𝑥 𝑦) = min (𝑇𝐵1 − (𝑥), 𝑇𝐵2 − (𝑥))  (𝑇𝐵1 + + 𝑇𝐵2 + ) (𝑥𝑦) = min (𝑇𝐵1 + (𝑥), 𝑇𝐵2 + (𝑥))  (𝐼𝐵1 − + 𝐼𝐵2 − ) (𝑥𝑦) = max (𝐼𝐵1 − (𝑥), 𝐼𝐵2 − (𝑥))  (𝐼𝐵1 + + 𝐼𝐵2 + ) (𝑥 𝑦) = max (𝐼𝐵1 + (𝑥), 𝐼𝐵2 + (𝑥)  (𝐹𝐵1 − + 𝐹𝐵2 − ) (𝑥𝑦) = max (𝐹𝐵1 − (𝑥), 𝐹𝐵2 − (𝑥))  (𝐹𝐵1 + + 𝐹𝐵2 + ) (𝑥 𝑦) = max (𝐹𝐵1 + (𝑥), 𝐹𝐵1 + (𝑥))𝑖𝑓𝑥𝑦 ∈ 𝐸′ , where 𝐸′is the set of all edges joining the nodes of 𝑉1 and 𝑉2, assuming 𝑉1 ∩ 𝑉2 = ∅. example 3.14. let 𝐺1 ∗ = (𝐴1, 𝐵1)and 𝐺2 ∗ = (𝐴2, 𝐵2 ) be two graphs where𝑉1 = {𝑢1, 𝑢2, 𝑢3,𝑢4}, 𝑉2 = {𝑣1, 𝑣2, 𝑣3}. consider two interval valued fermatean neutrosophic graphs: 𝐴1 = { 〈𝑢1, [0.85,0.95], [0.95,0.95], [0.95,0.95]〉, 〈𝑢2, [0.90,0.90], [0.95,0.95], [0.85,0.85]〉, 〈𝑢3, [0.90,0.95], [0.85,0.95], [0.85,0.85]〉, 〈𝑢4, [0.90,0.95], [0.95,0.90], [0.80,0.85]〉 }, 𝐵1 = { 〈𝑢1𝑢2, [0.85,0.90], [0.95,0.95], [0.95,0.95]〉, 〈𝑢2𝑢3, [0.90,0.90], [0.95,0.95], [0.85,0.85]〉, 〈𝑢3𝑢4, [0.90,0.95], [0.95,0.95], [0.85,0.85]〉, 〈𝑢1𝑢4, [0.85,0.95], [0.95,0.95], [0.95,0.95]〉, 〈𝑢1𝑢3, [0.85,0.95], [0.95,0.95], [0.95,0.95]〉 } ; 𝐴2 = { 〈𝑢1, [0.80,0.90], [0.85,0.95], [0.95,0.85]〉, 〈𝑢2, [0.95,0.90], [0.95,0.95], [0.80,0.85]〉, 〈𝑢3, [0.90,0.90], [0.95,0.95], [0.80,0.80]〉 }, 𝐵2 = { 〈𝑢1𝑢2, [0.80,0.90], [0.95,0.95], [0.95,0.85]〉 , 〈𝑢2𝑢3, [0.90,0.90], [0.95,0.95], [0.80,0.85]〉, 〈𝑢1𝑢3, [0.80,0.90], [0.95,0.95], [0.95,0.85]〉 }. intervalvalued fermatean neutrosophic graphs 191 figure 9. interval − valued fermatean neutrosophic graph g1 figure 10. interval − valued fermatean neutrosophic graph g2 figure 11. union two interval − valued fermatean neutrosophic graphs 𝐺1 ∪ 𝐺2 { 〈𝑢1𝑢2, [0.80,0.90], [0.95,0.95], [0.95,0.95]〉, 〈𝑢2𝑢3, [0.90,0.90], [0.95,0.95], [0.85,0.85]〉, 〈𝑢3𝑢4, [0.90,0.95], [0.95,0.95], [0.85,0.85]〉, 〈𝑢1𝑢4, [0.85,0.95], [0.95,0.95], [0.95,0.95]〉, 〈𝑢1𝑢3, [0.80,0.90], [0.95,0.95], [0.95,0.95]〉 } example 3.15 let 𝐺1 ∗ = (𝐴1, 𝐵1)and 𝐺2 ∗ = (𝐴2, 𝐵2 ) be two graphs where 𝑉1 = {𝑥1, 𝑥2, 𝑥3}, 𝑉2 = {𝑦1, 𝑦2, 𝑦3}. consider two interval valued fermatean neutrosophic graphs : 𝐴1 = { 〈𝑥1, [0.85,0.95], [0.95,0.95], [0.95,0.95]〉, 〈𝑥2, [0.90,0.90], [0.95,0.95], [0.85,0.85]〉, 〈𝑥3, [0.90,0.95], [0.85,0.95], [0.85,0.85]〉 }, 𝐵1 = {〈𝑥1𝑥2, [0.85,0.90], [0.95,0.95], [0.95,0.95]〉, 〈𝑥2𝑥3, [0.90,0.90], [0.95,0.95], [0.85,0.85]〉 } 𝐴2 = { 〈𝑦1, [0.85,0.85], [0.95,0.95], [0.90,0.90]〉, 〈𝑦2, [0.95,0.90], [0.90,0.95], [0.80,0.85]〉, 〈𝑦3, [0.95,0.95], [0.85,0.85], [0.85,0.85]〉 }, 𝐵2 = {〈𝑦1𝑦2, [0.85,0.85], [0.95,0.95], [0.90,0.90]〉, 〈𝑦2𝑦3, [0.95,0.90], [0.90,0.95], [0.85,0.85]〉} said broumi et al./decis. mak. appl. manag. eng. 5 (2) (2022) 176-200 192 figure 12. interval − valued fermatean neutrosophic graph g1 figure 13. interval − valued fermatean neutrosophic graph g2 figure 14. join of interval − valued fermatean neutrosophic graphs 𝐺1 + 𝐺2 𝐸(𝐺1 + 𝐺2) : < 𝑥1𝑥2, [0.85,0.90], [0.95,0.95], [0.95,0.95] >, < 𝑥2𝑥3, [0.90,0.90], [0.95,0.95], [0.85,0.85] > < 𝑦1𝑦2, [0.85,0.85], [0.95,0.95], [0.90,0.90] >, < 𝑦2𝑦3, [0.95,0.90], [0.90,0.95], [0.85,0.85] > < 𝑥1𝑦1, [0.85,0.85], [0.95,0.95], [0.95,0.95] >, < 𝑥1𝑦2, [0.85,0.90], [0.95,0.95], [0.95,0.95] >, < 𝑥1𝑦3, [0.85,0.95], [0.95,0.95], [0.95,0.95] > < 𝑥2𝑦1, [0.85,0.90], [0.95,0.95], [0.90,0.90] >, < 𝑥2𝑦2, [0.90,0.90], [0.95,0.95], [0.85,0.85] >, < 𝑥2𝑦3, [0.90,0.90], [0.95,0.95], [0.85,0.85] > < 𝑥3𝑦1, [0.85,0.85], [0.95,0.95], [0.90,0.90] >, < 𝑥3𝑦2, [0.90,0.90], [0.90,0.95], [0.85,0.85] >, < 𝑥3𝑦3, [0.90,0.95], [0.85,0.95], [0.85,0.85] > intervalvalued fermatean neutrosophic graphs 193 definition 3.16. an interval valued fermatean neutrosophic graph g = (a, b) is called complete if 𝑇𝐵 −({𝑣𝑖 , 𝑣𝑗}) = min[𝑇𝐴 −(𝑣𝑖 ), 𝑇𝐴 −(𝑣𝑗 )] , 𝑇𝐵 +({𝑣𝑖 , 𝑣𝑗 }) = min[𝑇𝐴 +(𝑣𝑖 ), 𝑇𝐴 +(𝑣𝑗 )] 𝐼𝐵 −({𝑣𝑖 , 𝑣𝑗 }) = max[𝐼𝐵 − (𝑣𝑖 ), 𝐼𝐵 −(𝑣𝑗 )] , 𝐼𝐵 +({𝑣𝑖 , 𝑣𝑗 }) = max[𝐼𝐵 +(𝑣𝑖 ), 𝐼𝐵 +(𝑣𝑗 )] 𝐹𝐵 −({𝑣𝑖 , 𝑣𝑗 }) = max[𝐹𝐵 −(𝑣𝑖 ), 𝐹𝐵 −(𝑣𝑗 )], 𝐹𝐵 +({𝑣𝑖 , 𝑣𝑗 }) = max[𝐹𝐵 +(𝑣𝑖 ), 𝐹𝐵 +(𝑣𝑗 )] definition 3.17. let g = (a,b) be an interval-valued fermatean neutrosophic graph where 𝐴 = 〈[𝑇𝐴 −, 𝑇𝐴 +], [𝐼𝐴 −, 𝐼𝐴 +], [𝐹𝐴 −, 𝐹𝐴 +]〉 is an interval-valued fermatean neutrosophic set on v; and 𝐵 = 〈[𝑇𝐵 −, 𝑇𝐵 +], [𝐼𝐵 −, 𝐼𝐵 +], [𝐹𝐵 −, 𝐹𝐵 +]〉 is an interval valued fermatean neutrosophic relation on 𝐸 satisfying 𝑉 = { 𝑣1 , 𝑣2 , … , 𝑣𝑛 }, such that 𝑇𝐴 − ∶ 𝑉 → [0, 1], 𝑇𝐴 + ∶ 𝑉 → [0, 1], 𝐼𝐴 − ∶ 𝑉 → [0, 1], 𝐼𝐴 +: 𝑉 → [0, 1] and 𝐹𝐴 −: 𝑉 → [0, 1], 𝐹𝐴 +: 𝑉 → [0, 1] denote the degree of truth-membership, the degree of indeterminacy-membership and falsitymembership of the element 𝑦 ∈ 𝑉, respectively. the positive degree of a vertex 𝑢 ∈ 𝑉(𝐺) is 𝑇+(𝑢) = ∑ [𝑇𝐴 +]𝑢𝑣∈𝐸(𝐺) ; 𝐼 +(𝑢) = ∑ [𝐼𝐴 +]𝑢𝑣∈𝐸(𝐺) ; 𝐹 +(𝑢) = ∑ [𝐹𝐴 +]𝑢𝑣∈𝐸(𝐺) and 𝑑 +(𝑢) = (𝑇𝐴 +, 𝐼𝐴 +, 𝐹𝐴 +). 𝑇−(𝑢) = ∑ [𝑇𝐴 −]𝑢𝑣∈𝐸(𝐺) ; 𝐼 −(𝑢) = ∑ [𝐼𝐴 −]𝑢𝑣∈𝐸(𝐺) ; 𝐹 −(𝑢) = ∑ [𝐹𝐴 −]𝑢𝑣∈𝐸(𝐺) and 𝑑 −(𝑢) = (𝑇𝐴 −, 𝐼𝐴 −, 𝐹𝐴 −). the degree of a vertex 𝑢 is 𝑑(𝑢) = [𝑑+ (𝑢), 𝑑− (𝑢)]. if 𝑑+ (𝑢) = 𝑘1 , 𝑑 − (𝑢) = 𝑘2 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑢 ∈ 𝑉, 𝑘1, 𝑘2 are two real numbers, then the graph is called [𝑘1 , 𝑘2] -regular interval valued fermatean neutrosophic graph. example 3.18. we consider an interval-valued fermatean neutrosophic graph. figure 15. intervalvalued fermatean neutrosophic graph g 𝑑(x1) = ([ 1.65,1.80,1.65], [1.85,1.90,1.70]); d(x2) = ([1.65,1.8,1.65], [1.8,1.9,1.7]); d(x3) = ([1.7,1.8,1.7], [1.85,1.9,1.7]). 4. proposed ivfng framework for mcdm problem the most of real life problems deal with uncertain domain. recently, researchers (sriganesh et al. 2021; sundareswaran et al. 2022) have been studied the assessment of structural cracks in buildings using single-valued neutrosophic dematel model and graph theoretical approach. the new concepts of ivfng are employed to find the best materials that are used for making dental implants in the case of smokers. there are many researchers developed and studied different types uncertainty sets and their application in multi-criteria decisionmaking (mcdm) (duran et al., 2021; ejegwa et al. 2022; mohanta et al., 2020; li et al., 2022; smarandache, 2020; smarandache, 2022; wang et al., 2022; zhang et al., 2022). mahesh et al. (2022), made a comparative study said broumi et al./decis. mak. appl. manag. eng. 5 (2) (2022) 176-200 194 of dental implant materials using digraph techniques. dental implants are the most popular option to replace missing teeth. they create direct contact with the bone which mimics the root of the tooth, upon which dental prosthesis can be fitted. these implants are designed in such a way that they can last for a long time without any failure. they get adhered to the bone without intervening in any connective tissue and this phenomenon is known as osseointegration. titanium is considered the gold standard as it is the most commonly used dental implant material in use since the 1960s zirconia is a non-metallic alternative to metal dental implants like 𝑇𝑖 𝑎𝑙𝑙𝑜𝑦 (𝑇𝑖 − 6𝐴𝑙 − 4𝑉) and 𝑇𝑖 alloys. figure 16. fishbone diagram with the various factors and subfactors in this section, the concept of interval-valued fermatean neutrosophic graphtheoretic approach has been used to selection of material. the condition of osseointegration in smokers is taken into consideration to compare the different material dental implants namely 𝑇𝑖 𝑎𝑙𝑙𝑜𝑦 (𝑇𝑖 − 6𝐴𝑙 − 4𝑉), 𝑇𝑖 alloy, and zirconia. the material to be chosen should exhibit certain properties to satisfy the purpose. while designing a dental implant, many factors come into consideration such as materials, dimensions, shape, etc. material selection is the most important property for a dental implant to serve the required function. the material of the implant must be affordable and available. following are the factors that are important for the selection of the material. biocompatibility (b): a biocompatible material does not invoke an immune response and does not release any toxic substances. the major subfactors of biocompatibility are corrosion, inflammation, and allergy. surface properties (s): surface properties refer to macroscopic and microscopic features of the implant surface and it plays a major role in determining the level of osseointegration between the implant and the bone. the major subfactors of surface properties are surface tension and surface energy, surface roughness, porosity. mechanical properties(m): the implant biomaterial should possess a high degree of modulus of elasticity, to withstand the forces applied to the implant, thus preventing its deformation. it also ensures uniform stress distribution, thus reducing the implant movement concerning the bone. cost (c): dental implants in india range from 30,000-50,000 rupees. the price depends on many factors like the type of tooth implant, material, and design of the implant, etc. titanium is more expensive than stainless steel. the cost of titanium is slightly lower than zirconia. intervalvalued fermatean neutrosophic graphs 195 titanium (𝑀1) and titanium alloys (𝑀2): titanium is an excellent corrosion– resistant material due to the formation of 𝑇𝑖 𝑎𝑙𝑙𝑜𝑦 (𝑇𝑖 − 6𝐴𝑙 − 4𝑉) when 𝑇𝑖 atoms react with water molecules and oxygen. they show excellent biocompatibility properties and support osseointegration. titanium-based dental implants are strong and resist fracture. the cost of titanium is slightly lower than the zirconia. however, titanium implants are less aesthetically pleasing than zirconia and hence they are not preferable to use in the case of front teeth implant placement. zirconia could be preferred in this case due to its ivory color. zirconia (𝑀3): zirconia is a non-metallic alternative to metal dental implants like ti. an advantage of zirconia over titanium is its ivory color. its low modulus of elasticity and thermal conductivity, low affinity to plaque, and high biocompatibility, in addition to its white color, have made zirconia ceramics a very attractive alternative to titanium. it is highly corrosion resistant and does not involve any release of ions hence no cytotoxicity. figure 17. types of dental implants in the process of applying ivfng in identifying the best material. ivfng can be represented as a matrix whose rows and columns are the sub-factors. 𝑉 = { 𝑀1, 𝑀2, 𝑀3} be the three different material under the selection on the basis of wishing param eters or attributes set 𝐴 = {𝐵 , 𝑆}. figure 18. ivfng based on biocompatibility & surface properties we construct the adjacency matrix for 𝑀(𝐵), 𝑀(𝑆) listed below: 𝑴(𝑩) = ( < [0, 0 ], [0, 0 ], [0, 0 ] > < [0.85,0.95], [0.95,0.95], [0.85,0.85] > < [0, 0 ], [0, 0 ], [0, 0 ] > < [0.85,0.95], [0.95,0.95], [0.85,0.85] > < [0, 0 ], [0, 0 ], [0, 0 ] > < [0.85,0.85], [0.95,0.95], [0.85,0.95] > < [0, 0 ], [0, 0 ], [0, 0 ] > < [0.85,0.85], [0.95,0.95], [0.85,0.95] > < [0, 0 ], [0, 0 ], [0, 0 ] > ) 𝑴(𝑺) = ( < [0, 0 ], [0, 0 ], [0, 0 ] > < [0.85,0.95], [0.90,0.95], [0.85,0.85] > < [0.85,0.90], [0.95,0.95], [0.95,0.85] > < [0.85,0.95], [0.90,0.95], [0.85,0.85] > < [0, 0 ], [0, 0 ], [0, 0 ] > < [0.85,0.90], [0.95,0.95], [0.95,0.85] > < [0.85,0.90], [0.95,0.95], [0.95,0.85] > < [0.85,0.90], [0.95,0.95], [0.95,0.85] > < [0, 0 ], [0, 0 ], [0, 0 ] > ) said broumi et al./decis. mak. appl. manag. eng. 5 (2) (2022) 176-200 196 we obtain the resultant interval valued fermatean neutrosophic graph g by performing some operation (and or or). the incidence matrix of resultant interval fermatean neutrosophic graph is 𝑴(𝑩) = ( < [0, 0 ], [0, 0 ], [0, 0 ] > < [0.85,0.95], [0.95,0.95], [0.85,0.85] > < [0, 0 ], [0, 0 ], [0, 0 ] > < [0.85,0.95], [0.95,0.95], [0.85,0.85] > < [0, 0 ], [0, 0 ], [0, 0 ] > < [0.85,0.85], [0.95,0.95], [0.95,0.95] > < [0, 0 ], [0, 0 ], [0, 0 ] > < [0.85,0.85], [0.95,0.95], [0.95,0.95] > < [0, 0 ], [0, 0 ], [0, 0 ] > ) sahin (2015) defined the average possible membership degree of element x to interval valued neutrosophic set 𝐴 = 〈[𝑇𝐴 − (𝑥), 𝑇𝐴 +(𝑥)], [𝐼𝐴 − (𝑥), 𝐼𝐴 +(𝑥)], [𝐹𝐴 − (𝑥), 𝐹𝐴 +(𝑥)]〉 as follows: 𝑆𝑘 (𝑥) = 𝑇𝐴 − (𝑥) + 𝑇𝐴 +(𝑥) + 4 − 𝐼𝐴 − (𝑥) − 𝐼𝐴 +(𝑥) − 𝐹𝐴 − (𝑥) − 𝐹𝐴 +(𝑥) 6 based on 𝑆𝑘 (𝑥), table 2 depicted the score value of adjacency matrix of resultant interval valued fermatean neutrosophic graph g with 𝑆𝑘 and choice value for both materials. table 2. score value of adjacency matrix materials 𝑴𝟏 𝑴𝟐 𝑴𝟑 overall 𝑴𝟏 0 0.383 0 0.383 𝑴𝟐 0.383 0 0.317 0.7 𝑴𝟑 0 0.317 0 0.317 further, it is noticed from table 2, 𝑇𝑖 𝑎𝑙𝑙𝑜𝑦 (𝑇𝑖 − 6𝐴𝑙 − 4𝑉) has higher level of osseointegration in smokers followed by 𝑇𝑖 and zirconia. therefore, we may claim that ivfng is a new way to tackle the uncertainty in fermatean neutrosophic environment. 5. conclusion the concept of uncertainty plays a vital role in all science and engineering problems. especially, fuzzy theory, intuitionistic fuzzy theory and then neutrosophic theory are the most valuable tools to find the optimum solution in mutli-criteria decision making problems. in this work, we include one more concept called intervalvalued fermatean neutrosophic graphs in the list which has pythagorean neutrosophic, single valued neutrosophic, bipolar neutrosophic graphs. we have discussed various types of interval-valued fermatean neutrosophic graphs and the other types of these graphs in this paper. we also apply this new type of graph in a decision making problem. we are extending our research on this new concept to introduce interval-valued fermatean neutrosophic number and interval-valued fermatean triangle and trapezoidal neutrosophic number and its applications in our future work. interval-valued fermatean neutrosophic graph has many advantages in mcdm problems such as mobile networking, supply chain management system, bio-medical applications, e-waste management and networking, etc. in future, one may determine the optimum alternatives in mcdm problems using ivfng based score and accuracy functions. author contributions: conceptualization, s.b and f.s; methodology, s.b., m.t., and a.b; software, r.s.; validation, r.s. and m.s.; formal analysis, r.s.; investigation, s.b.; resources, s.b., r.s., and m.s.; writing—original draft preparation, r.s., and m.s.; writing—review and editing, r.s., and m.s.; visualization, r.s., and m.s.; supervision, intervalvalued fermatean neutrosophic graphs 197 s.b and g.n. all authors have read and agreed to the published version of the manuscript. funding: this research is not financially supported by any funding agencies. data availability statement: not applicable. acknowledgments: the authors would like to thank the management of sri sivasubramaniya nadar college of engineering for giving the needed facilities to complete this research work. conflicts of interest: authors declares that no competing interests for this research article. references ajay, d., & chellamani, p. 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(2022), neutrosophic management evaluation of insurance companies by a hybrid todim-bsc method: a case study in private insurance companies, management decision, vol. ahead-of-print no. ahead-of-print. https://doi.org/10.1108/md-01-2022-0120 © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1016/j.matpr.2022.04.156 plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 1, number 1, 2018, pp. 121-142 issn: 2560-6018 doi: https://doi.org/10.31181/dmame1801121r * corresponding author. e-mail addresses: jaga.math23@gmail.com (j. roy), dpamucar@gmail.com (d. pamučar), kar_s_k@yahoo.com (s. kar), krish.math23@gmail.com (k. adhikary) a rough strength relational dematel model for analysing the key success factors of hospital service quality jagannath roy1*, krishnedu adhikary1, samarjit kar1, dragan pamučar2, 1 department of mathematics, national institute of technology durgapur, india 2 university of defence in belgrade, military academy, department of logistics, belgrade, serbia received: 18 january 2018; accepted: 6 february 2018; published: 15 march 2018. original scientific paper abstract. successful management of hospital service quality (hsq) is increasingly becoming strategic perspective of hospitals to excel in medical care within reasonable prices which is among the customers’/patients’ primary needs. several key success factors (ksfs) can control the proper hsq management and make it a complex problem. for these reasons, it is vital to set hospitals’ goals and detect ksfs via customers’ feedback and viewpoints. the preceding researches discussed not much about the effect of internal strength impact on the interdependencies between ksfs of hsq management. to resolve these issues, a rough strength relationaldecision making and trial evaluation laboratory (rsr-dematel) model is developed to analyse the individual priorities of ksf of hospital’s performance measures. the rsrdematel method reflects completely the internal as well as external total influences between the ksfs. additionally, the proposed model has intelligence and flexibility in manipulating the inherent uncertainty due to the subjective ang vague information in ksf analysis for managing hsq. the validity and efficiency of the rsr-dematel model are examined by applying it to a hospital data available in the literature. the result analysis shows that “medical staff with professional abilities” (ksf5) is the most significant ksf in hsq management, i.e., recruiting more skilled doctors and nurses will eventually increase the performance of hospital medical services. finally, a comparative analysis is conducted to cross check the obtained results with other two methods from literature. key words: strength-relation analysis, rough numbers, dematel, hsq, ksfs. mailto:jaga.math23@gmail.com mailto:dpamucar@gmail.com mailto:kar_s_k@yahoo.com mailto:krish.math23@gmail.com roy et al./decis. mak. appl. manag. eng. 1 (1) (2018) 121-142 122 1. introduction the hospital service quality management (hsqm) in todays’ competitive marketplace is a complex decision making problem where organizations, people, information, and resources take part in achieving some predetermined objectives. although in the past few decades hsqm referred to merely limited to medical services, nowadays, the success of a hospital broadly depends on hsqm and its business process management (bpm) (shieh et al., 2010). to achieve the preset goals, keeping the hospital competitive, sustaining its growth rate, and raising profits not only at a local but also a global level, the hospital authority must implement bpm and simultaneously improve hsq. although bpm is known as business concept for decades, its strategic and operational roles with in organizations is still an important issue requiring investigation from various perspectives such as operations and information management (bai & sarkis, 2013). however, bpm in hsq can be a risky intent with the possibility of huge investments and uncertain consequences. many authors (bai & sarkis, 2013; abdolvand et al., 2008; bandara et al., 2005) warned and suggested about the failure rate of bpm. thus, in order to implement bpm in hsq successfully, and it is necessary to know the resulting initiatives and identify key success factors (ksfs) in hsq incorporating bpm. several papers sought to identify ksfs of hsqm. most of these papers focused conceptual elements or performed qualitative analyses. many authors (bowers et al., 1993; babakus & mangold, 1992; koerner, 2000; andaleeb, 2001; youssef et al., 1995; parasuraman et al., 1985) have studied development and implementation of bpm after maintaining hsq strategies, but only a few used robust methodologies to conduct systematic evaluation of hsqm key/critical factors as well as bpm employment. hospital managers should push its boundaries and implement a well-organized bpm strategy that facilitates identifying and analyzing ksfs in hsqm. hence, bpm to improve hsq can be well accomplished as multiple criteria decision making (mcdm) problems that consider several complex and usually conflicting or interacting factors. as mentioned earlier hsqm includes factors from organizations, people, information, and resources, it is necessary to narrow down the factors’/criteria set. such minimal collection of ksfs in hsqm will reduce the complexity of decision making process. this is why, hospitals should pay more attention these ksfs while implementing bpm. with this, hospital managers will be able to understand better of bpm in hsqm and regulate the corresponding ksfs to successful management of hsq. now, in order to assess the interactions between the ksfs, a well-organized method for quantitative analysis may provide valuable insights of cause and effect relationships. it is difficult to make actionable strategies directly from opinions given by experts and managers since they lack clear visions about the cause/effect relationships amongst the ksfs, which play vital roles in the hsqm. hence, a systematic exploration tool, decision making trial and evaluation laboratory (dematel) method can be applied to depict complex cause/effect relations through matrices (shieh et al. 2010). additionally, this particular tool makes use of cognitive maps to draw digraphs which portray the inter-relationships between ksfs. the dematel method is beneficial in illuminating the relations amongst ksfs and ordering them depending on the cause/effect relations and prominence of their influences on other factors. but it becomes difficult to describe those relations if uncertainty exists in the data to be used for decision making process. in response to the uncertainty, rough set theory (rst) is an excellent choice to manipulate subjective and vague data involved in the analysis of ksfs of hsqm. rst has a rich theoretical background in analyzing vague information and incomplete a rough strength relational dematel model for analysing the key success factors of hospital... 123 data. thus, rough strength relational decision making trial and evaluation laboratory (rsr-dematel) method can efficiently handle uncertainty due to subjectivity and vagueness. rsr-dematel uses to rough numbers to manipulate the strengths and inter-relationships among the ksfs of hsqm under uncertainty caused by decision makers’ knowledge based linguistic (qualitative) assessment. this paper aims to assess ksfs of hsqm, with the objectives of this study being: (1) to apply a flexible and unique method that appraises ksfs of hsqm and obtain the structure of complicated causal relationships and the influence level of these factors; and (2) to help hospital managers having better control of ksfs while implementing bpm in hsq and examine several capacities of bpm implementation practices. to meet the abovementioned goals, this paper is organized as follows: we briefly introduce the basic elements of rough numbers and rough arithmetic in section 2. in section 3 we provide the step-wise description of the rsr-dematel model while section 4 deals with an empirical case example. the major implications of our study are articulated in section 5. the final section concludes the paper and tells about the limitations and future research directions. 2. rough numbers and its operations in group decision making problems, the priorities are defined on multi-expert’s aggregated decision and process subjective evaluation of expert’s decisions. rough numbers consisting of upper, lower and boundary interval respectively, determine intervals of their evaluations without requiring additional information by relying only on original data (zhai et al., 2008). hence, obtained expert decision makers (dms) perceptions objectively present and improve their decision making process. according to zhai et al. (2009), the definition of rough number is shown below. let’s u be a universe containing all objects and x be a random object from u . then we assume that there exists set build with k classes representing dms preferences, 1 2 k r (j , j ,..., j ) with condition 1 2 k j j ,..., j   . then, q x u, j r, 1 q k     lower approximation q apr(j ) , upper approximation q apr(j ) and boundary interval q bnd(j ) are determined, respectively, as follows:  q qapr(j ) x u / r(x) j   (1)  q qapr(j ) x u / r(x) j   (2)       q q q q bnd(j ) x u / r(x) j x u / r(x) j x u / r(x) j         (3) the object can be presented with rough number (rn) defined with lower limit q lim(j ) and upper limit q lim(j ) , respectively: q q l 1 lim(j ) r(x) x apr(j ) m   (4) q q u 1 lim(j ) r(x) x apr(j ) m   (5) roy et al./decis. mak. appl. manag. eng. 1 (1) (2018) 121-142 124 where l m and u m represent the sum of objects contained in the lower and upper object approximation of q j , respectively. for object q j , rough boundary interval  qirbnd(j ) presents interval between lower and upper limit as: q q q irbnd(j ) lim(j ) lim(j )  (6) rough boundary interval presents measure of uncertainty. the bigger q irbnd(j ) value shows that variations in experts’ preferences exist, while smaller values show that experts had harmonized opinions without major deviations. in q irbnd(j ) are comprised all objects between lower limit q lim(j ) and upper limit q lim(j ) of rough number q rn(j ) . that means that q rn(j ) can be presented using q lim(j ) and q lim(j ) . q q q rn(j ) lim(j ), lim(j )    (7) since rough numbers belong to the group of interval numbers, arithmetic operations applied in interval numbers is also appropriate for rough numbers. since rough numbers belong to the group of interval numbers, arithmetic operations applied in interval numbers is also appropriate for rough numbers. if a and b presents two rough numbers rn(a) lim(a), lim(a)    and rn(b) lim(b), lim(b)    , k denotes constant, k 0 , then the arithmetic operations with rn(a) , rn(b) and k are as follows: (1) addition of rough numbers "+" rn(a) rn(b) lim(a), lim(a) lim(b), lim(b) lim(a) lim(b), lim(a) lim(b)                 (8) (2) subtraction of rough numbers "-" rn(a) rn(b) lim(a), lim(a) lim(b), lim(b) lim(a) lim(b), lim(a) lim(b)                 (9) (3) multiplication of rough numbers "×" rn(a) rn(b) lim(a), lim(a) lim(b), lim(b) lim(a) lim(b), lim(a) lim(b)                 (10) (4) dividing of rough numbers "/" rn(a) / rn(b) lim(a), lim(a) / lim(b), lim(b) lim(a) / lim(b), lim(a) / lim(b)              (11) (5) scalar multiplication of rough numbers, where k 0 a rough strength relational dematel model for analysing the key success factors of hospital... 125 k rn(a) k lim(a), lim(a) k lim(a), k lim(a)             (12) ranking rule of rough numbers: any two rough numbers, rn(a) lim(a), lim(a)    and rn(b) lim(b), lim(b)    , where lim(a) and lim(b) , and lim(a) , lim(b) represent their lower and upper limits, respectively, are ranked by the use of the following rules: if the rough boundary interval of a rough number is not strictly bound by another, then the ranking order is easily determined, i.e. (a) if lim(a) lim(b) and lim(a) lim(b) lim(a) lim(b) and lim(a) lim(b)       then rn(a) rn(b) (b) if lim(a) lim(b) and lim(a) lim(b) , then rn(a) rn(b) . if the rough boundary interval of a rough number is strictly bound by another, then ranking becomes awkward and medians m(a) and m(b) of rn(a) and rn(b) respectively, are used in ranking. (a) if lim(b) lim(a) and lim(b) lim(a) then if m(a) m(b) then rn(a) rn(b) if m(a) m(b) then rn(a) rn(b)      (b) similar rules can be derived if lim(a) lim(b) and lim(a) lim(b) . 3. the rough strength-relation dematel method the dematel method is a comprehensive method used in both the design and analysisof structural method characterized by the causal relations between complex factors (fontela & gabus, 1976). the method is based on graph theory, which enables visual planning and problem solving so thatall relevant factors can be classified into causal and consequential factors,for better understanding of their interrelations. this method makes it possible to better understand the complex structure of a problem and define the relations between factors (gigović et al., 2017). for the purpose of accepting the imprecision in the collective decision making process, this paper modifies the dematel method by applying rough numbers strength-relation analysis (rsr-dematel). the application of rough numbers eliminates the necessity for additional information for defining uncertain number intervals. in such a way, the quality of the existing data in the collective decision making process can be retained, as well as the experts’ perception, which is expressed through the aggregation matrix. the text below shows the steps governing the rsrdematel method, which was used in the group decision making process. step 1. evaluate internal strength of factors with linguistic scale. when considering the interactions between two factors, the interaction not only depends on the intensity of influencing but also on the strength of the factors that exerts (song et al. 2017). the decision maker (expert) can evaluate the internal strength of all factors using the 5-point verbal scale 0 –no strength (ns); 1 –low strength (ls); 2 –medium strength (ms); 3 –high strength (hs); 4 –very high strength (vhs). assuming that there are m experts in the research and n observed factors, each expert should determine the degree of intrenal strength of all factors. the evaluation roy et al./decis. mak. appl. manag. eng. 1 (1) (2018) 121-142 126 matrix of k (1 ≤ k ≤ m) expert is presented as a non-negative matrix of n×1 rank, and each element of the k matrix in equation yk=[yki]n×1 denotes a non-negative number ykij, where 1 ≤ k ≤ m. k 1 k k 2 k n nx1 y y y ;1 i n; 1 k m y                  (13) where k i y represent linguistic variable taken from the preliminary defined linguistic scale used by expert k for the purpose of intrenal strength evaluation. in accordance with this, y1, y2, …,ym matrices are evaluation matrices of each of m experts. step 2. determination of experts’ weight coefficients  iw . experts’ weight coefficients are determined using three parameters: an objective expert’s evaluation ( o w ) which is determined on the basis of experience that an expert possesses in the field of research; mutual expert’s evaluation ( u w ) which is determined on the basis of mutual assessment of the experts participating in the study; and subjective expert’s evaluation ( s w ) which is determined on the basis of expert’s assessment of their own competence for participation in the study. reviews on all three weight parameters  o u sw , w , w are awarded on the basis of pre-defined linguistic scale: 0 – no influence (ni); 1 –low influence (li); 2 –medium influence (mi); 3 –high influence (hi); 4 –very high influence (vhi). the obtained weighting coefficient ( i w ) is calculated from the score representing the sum of individual assessment parameters ( o w , u w and s w ). since the requirement m ii 1 w 1   needs to be satisfied, the final iw values are calculated using equation (14) where i i ii o u s w w w w   is the weighting coefficient of expert m (i=1,2,...,m). i i m i i 1 w w w    (14) step 3. determine the aggregated internal strength of factors. on the basis of the step 1, we receive y1, y2, …,ym matrices of each of m experts 1 2 m 1 1 1 1 2 m k 2 2 2 1 2 m n n n nx1 y , y ,..., y y , y ,..., y y y , y ,..., y              (15) where  1 2 mi i i iy y , y ,..., y denote the sequences used to describe the intrenal strength of the factor i. by applying equations (1) through (7), each sequence k i x is converted to rough sequences  k ki i k i lim( )rn y y y, lim( )     , where k i lim(y ) and k i lim(y ) represent the lower limit and upper limit of rough sequence  kirn y , respectively. a rough strength relational dematel model for analysing the key success factors of hospital... 127 thus we obtain rough matrices y1, y2, …,ym, where m denotes the number of experts. therefore for each rough matrix (y1, y2, …,ym) in position (i,1) we obtain rough sequence    i1 i1 i1 i1 i1 i1 i11 1 2 2 m mlim( ), lim(y ) , lim(y ), lim(y ) ,..., lim(y ), lim(r y )n y y           . by applying equation (16) the rough aggregated internal strength of factor i is obtained m e e ii e 1 i i i m e e ii e 1 i i lim(y ) lim( ) w rn(y ) lim(y ), lim(y ) lim(y ) li y ym( ) w                 (16) where i lim(y ) and i lim(y ) represent the lower limit and upper limit of the rough aggregated internal strength i rn(y ) , respectively. thus, the agregated matrix of internal strengths y is obtained 1 1 1 2 2 2 n n nnx1 nx1 rn(y ) [lim(y ), lim(y )] rn(y ) lim(y ), lim(y ) y rn(y ) lim(y ), lim(y )                              (17) step 4. analysis ofexpert’s response matrix for the factors. assuming that there are m experts in the research and n observed factors (criteria), each expert should determine the degree to which criterion i affects criterion j. comparative analysis of the ith and jth pairwise by k expert is denoted as xije, where: i=1,...,n; j=1,...,n. the value of each xijk pair is an integer, where: 0 –no influence (ni); 1 –low influence (li); 2 – medium influence (mi); 3 –high influence (hi); 4 –very high influence (vhi). the judgment of k expert is presented as a non-negative matrix of n×n rank, and each element of the k matrix in equation xe=[xkij]n×n denotes a non-negative number xeij, where 1 ≤ k ≤ m. k k 12 1n k k k 21 2n k k n1 n 2 nxn 0 x x x 0 x x ; 1 i, j n; 1 k m x x 0                  (18) where e ij x arepresent linguistic variable taken from the preliminary defined linguistic scale. in accordance with this, x1, x2, …,xm matrices are judgment matrices of each of m experts. the diagonal elements of the judgment matrix are all set to zero since the same factors do not influence each other. step 5. determine the aggregated rough direct-relation matrix. based on response matrices xk=[xkij]n×n obtained from each m expert, we built the integrated rough direct relation matrix *x . roy et al./decis. mak. appl. manag. eng. 1 (1) (2018) 121-142 128 1 2 k 1 2 k 1 2 k 11 11 11 12 12 12 1n 1n 1n 1 2 k 1 2 k 1 2 k * 21 21 21 22 22 22 2n 2n 2n 1 2 k 1 2 k 1 2 k n1 n1 n1 n 2 n 2 n 2 nn nn nn x , x , , x x x ; x , , x x , , x x , x , , x x x ; x , , x x , ; ; ; ; ; ; ; , x x x , x , , x x x ; x , , x x ,; x; ,                          (19) where  1 2 kij ij ij ijx x , x , , x  denote the sequence used to describe the relative importance of criterion i in relation to criterion j. by applying equations (1) through (7), sequence k ij x is converted to rough number  ij ij i k j k k lim( ), limr ( )n x x x     , where i k j xlim( ) and i k j xlim( ) represent the lower limit and upper limit of rough number  kijrn x , respectively. thus we obtain x1, x2, …, xm rough matrices (where m denotes the number of experts). by applying equation (20) the aggregated rough element ijrn(x ) of the aggregated rough direct-relation matrix is obtained i m e e ij i e 1 ij ij ij i m e e ij i e 1 lim(x ) lim(x ) w rn(x ) lim(x ), lim(x ) lim(x ) lim(x ) w                 (20) where ijlim(x ) and ijlim(x ) represent the lower limit and upper limit of the rough aggregated rough element ijrn(x ) , respectively. finally, we get the agregated rough direct-relation matrix,  ij ij ij n nn n x x lim(x ), lim(x )rn           12 1n 21 2n n1 n 2 12 12 1n 1n 21 21 2n 2n n1 n1 n 2 0 rn(x ) rn(x ) rn(x ) 0 rn(x ) x rn(x ) rn(x ) 0 [0, 0] lim(x ), lim(x ) lim(x ), lim(x ) lim(x ), lim(x ) [0, 0] lim(x ), lim(x ) [lim(x ), lim(x )] lim(x                                n 2), lim(x ) [0, 0]                    (21) step 6. construct the group direct strength-relation matrix. the rough numbers representing the strength of factors, eq. (17), are inserted into the principal diagonal of the group direct-relation matrix (21). the group direct strength-relation matrix d is obtained as a rough strength relational dematel model for analysing the key success factors of hospital... 129 12 1n 21 2n n1 n 2 0 rn(d ) rn(d ) rn(d ) 0 rn(d ) d rn(d ) rn(d ) 0             (22) where ij ij ij rn(d ) lim(d ), lim(d )     m m e e e e ij i ii e 1 e 1 m m e e e e ij i ij i i e 1 1 i i e i if i j then rn(d ) rn(y ) lim(x ) w , lim( ) w if i j then rn(d ) rn(x ) lim(x ) w , lim(x w y )                                 (23) matrix d shows the first effects that particular factor causes, as well as the initial effects one gets from other factors. the sum of each i-th matrix row d represents total direct effects which factor i has caused to other factors, and the sum of each j-th column of matrix d represents total direct effects which factor j has received from other factors. step 7. normalize the group direct strength-relation matrix. based on matrix z, a normalized initial direct-relation matrix ij n n z irn(z )      is obtained, equation (24). by normalization, each element in matrix z is assigned a value between zero and one. the z matrix is obtained when each element ij rn(d ) of matrix d is divided by number s, as shown in equations (25) and (26) 11 12 1n 21 22 2n n1 n 2 nn rn(z ) rn(z ) rn(z ) rn(z ) rn(z ) rn(z ) z rn(z ) rn(z ) rn(z )             (24) where ij rn(z ) is obtained by applying equation (25) ij ij ij ij rn(d ) lim(d ) lim(d ) rn(z ) rn , s s s          (25) where         n n n ij ij ijj 1 j 1 j 1 n n ij ijj 1 j 1 s max rn(d ) max lim(d ), lim(d ) max max lim(d ) , max lim(d )                  (26) step 8. determine the total strength-relation matrix. by applying equations (27) and (28), the totalstrength-relation matrix ij n n t rn(t )      of rank n×n is calculated, where i denotes the identity matrix of the nxn rank. the element ij rn(t ) denotes a direct influence of factor i on factor j, while t matrix denotes total strengthrelations among each pair of factors. roy et al./decis. mak. appl. manag. eng. 1 (1) (2018) 121-142 130 since each rough number is composed of two sequences (upper and lower approximation), then the normalized matrix of average perception ij n n z rn(z )      can be divided into two sub-matrices, i.e. l u z z , z    , where l ij n n z lim(z )      and u ij n n z lim(z )      . moreover,   m l m lim z o   and   m u m lim z o   , where o denotes a zero matrix.         1 l 2l ml l m 1 u 2 u mu u m lim i z z z i z lim i z z and z i z                      (27) therefore, the matrix of the total influences t will be obtained by calculating of the following elements         l l ij 1 l 2l ml l m 1 u 2u mu u m n n u u ij n n t lim(t )lim i z z z i z lim i z and t limi z tz ( )z                               (28) where l ij n n z lim(z )      and u ij n n z lim(z )      . sub-matrices lt and u t together represent the rough matrix of the total influences  l ut t , t . based on equations (27) and (28), a total strength-relation matrix is defined: 11 12 1n 21 22 2n n1 n 2 nn rn(t ) rn(t ) rn(t ) rn(t ) rn(t ) rn(t ) t rn(t ) rn(t ) rn(t )             (29) where ij ij ij rn(t ) lim(t ), lim(t )    is the overall influence rating of the decision maker for each factor i on factor j , thus reflecting mutual dependence of each factor pair. step 9. calculating the sum of rows and columns of total strength-relation matrix t. in matrix t, the sum of rows and sum of columns are denoted as vectors r and c, rank n×1: n i ij j 1 n 1 rn(r ) rn(t )           (30) n i ij i 1 1 n rn(c ) rn(t )           (31) the value ri denotes the sum of the i-th row of matrix t and shows the total direct and indirect effects that criterion i delivers to other factors. similarly, the value ci is the sum of the j-th column of matrix t, and represents the total direct and indirect a rough strength relational dematel model for analysing the key success factors of hospital... 131 effects that factor j receives from other factors. in cases where i=j, equation (ri+ci) indicates the impact of the factors and equation (ri-ci) indicates the intensity of the factors compared to others (pamučar & ćirović, 2015). to effectively determine the “prominence” and the “relation”, the sum of rows ri to the sum of columns ci in the total strength-relation matrix t need to be converted into the crisp forms crisp i r and crisp i c by applying equations (32)-(34)             i i i i i i ii i i i i i i i i i ii lim(r ) min lim(r ) lim(r ) max lim(r ) min lim(r ) rn(r ) lim(r ), lim(r ) lim(r ) min lim(r ) lim(r ) max lim(r ) min lim(r )              (32) where i lim(r ) and i lim(r ) represent the lower limit and upper limit of the rough number i rn(r ) , respectively; ilim(r ) and ilim(r ) are the normalized forms of i lim(r ) and i lim(r ) . after normalization we obtain a total normalized crisp value  i i i i i i i lim(r ) 1 lim(r ) lim(r ) lim(r ) 1 lim(r ) lim(r )         (33) finally crisp form crisp i r for i rn(r ) is obtained by applaying eq. (34)      crispi i i i i i ii r min lim(r ) max lim(r ) min lim(r )        (34) the final crisp form crisp i c for i rn(c ) can be obtained similarly. step 10. calculate “prominence”/“relation” and prioritize factors. the vector i p named “prominence” is made by adding crisp i r to crisp i c . the vector i f named “relation” is made by subtracting crisp i r to crisp i c . crisp crisp i i i p r c  (35) crisp crisp i i i f r c  (36) the vector i p combines the interrelations of both directions (the horizontally exerted and the vertically received influence) of the factor i and therefore is interpreted as an overall influence intensity of that factor. it reveals how much importance the factor has. the larger value of i p the greater overall importance/influence of factor i in terms of overall relationships with other factors. all the factors can then be prioritized based on the i p (song et al., 2017). the vector i f shows the difference between the exerted and received influence, and it is a basis for classification of the factors. when the value i f is positive, the factor i belongs to the cause group. the factor i is a net cause for other factors. if the value i f is negative, the factor i belongs to the effect group. roy et al./decis. mak. appl. manag. eng. 1 (1) (2018) 121-142 132 step 11. determine the cause and effect relationships between factors. based on the i p and i f the cause-and-effect diagram (ced) can be acquired by mapping the dataset of the ( i p , i f ). in the ced the prominence axis shows how important a criterion relative to the available set of factors, whereas the relation axis will divide the factors into cause and effect groups (song et al., 2017). the construction of a ced visualizes the complex interrelationship and provides information in order to determine the most important factors and how they influence the affected factors. factors with a value higher than threshold value α are selected and shown in the ced. n n iji 1 j 1 rn(t ) n           (37) where n denotes the number of matrix elements (29). the rough numbers in the matrix (29) should be converted into crisp numbers. using eqs. (32)-(34) the crisp total strength-relation matrix * crisp ij n n t t      can be obtained. elements of matrix *t with values higher than the threshold α arese lected and shown in the diagram where the x–axis represents i p , and the y – axis i f and they are used to denote the relationship between two factors. when presenting the factor relationships, the arrow of the cause-and-effect relationship will be directed from the factor with the value lower than threshold value α to the element with the value higher than threshold value α. 4. an emperical case study 4.1. case background to validate the feasibility and efficiency of the proposed rsr-dematel method for analyzing the key success factors (ksfs) of hospital service quality (hsq), the method is applied to the case example from shieh et al. (2010). due to the increased competition, health care institutions/hospitals face huge challenge to attract and retain patients, fulfill their several needs, like high quality medical care in reasonable price, proper health insurance etc. successful management of hospital service quality is the only way to meet those goals. in order to do that the hospital authority should identify key criteria and their importance by conducting a survey from the viewpoints of patients and or their families. for further analysis, the authority needs feedbacks from several departmental personnel (we call them decision makers (dms)) in the hospital. it’s quite evident that the dms would possess different opinions about the importance of the key success factors. some of them also may have conflicts with others regarding the interrelationships and the mutual influences between the key success factors, which shall help mitigating the priorities of these factors. thus, in this case study, the proposed method is utilized for evaluating and analyzing the critical success factors a successful hospital as well as examining their interrelationships. twenty one managerial personnel (dms) having knowledge in networking with medical services from diverse occupations in the hospital are invited. more details on the dms can be found in shieh et al. (2010). in the data collection phase, the key success factors are deducted from the operational process (asking patients’/their family’s responses) in the hospital and literature review. the research team then organized a focused group discussion a rough strength relational dematel model for analysing the key success factors of hospital... 133 lasting an hour to understand and validate the key success factors identified from the literature. the decision panned finalize all the seven key success factors (see table 1) which are significant for their work, and thus decide to provide the necessary inputs to be used in this research based on the seven key success factors in table 1. table 1. the internal strength of factors evaluated by decision makers (experts) and expert weights experts factors expert weights c1 c2 c3 c4 c5 c6 c7 dm1 hs ms ms ms ms hs hs 0.059 dm2 hs ms ms hs ms vhs hs 0.049 dm3 vhs vhs vhs vhs hs ms ms 0.053 dm4 vhs hs hs ms hs vhs vhs 0.044 dm5 vhs vhs vhs vhs vhs vhs vhs 0.048 dm6 ms ms ms ls ms hs ls 0.058 dm7 ms ms ls ls ms vhs hs 0.057 dm8 hs ms ms ls ls vhs hs 0.058 dm9 hs ls ms hs ms ms ms 0.052 dm10 ms ms ls ls ls vhs ms 0.058 dm11 hs vhs hs hs ms vhs ms 0.044 dm12 vhs ms ms hs hs ms hs 0.057 dm13 ms ms ls ms ms ms ms 0.054 dm14 vhs vhs vhs vhs hs vhs hs 0.048 dm15 vhs vhs vhs vhs vhs vhs vhs 0.048 dm16 hs ms ms ls ms ms ms 0.046 dm17 hs ls ms hs ms ms ms 0.058 dm18 ms ms ls ls ls vhs ms 0.051 dm19 hs vhs hs hs ms vhs ms 0.058 4.2. implementation step 1 to 2. internal strength of each ksfs of hsq is assessed with verbal language. in this stage, the nineteen dms are requested to appraise the internal strength of different ksfs of hsq according to pre-defined linguistic scale: 0 –no strength (ns); 1 –low strength (ls); 2 –medium strength (ms); 3 –high strength (hs); 4 –very high strength (vhs). all the internal strength of ksfs are provided in form of linguistic scales in table 1. the evaluation set of the ksf1 (well-equipped medical equipment) can be denoted as ksf1 y ={hs, hs, vhs, vhs, vhs, ms, ms, hs. hs, ms, hs, vhs, ms, hs, vhs, hs, hs, ms, hs }={3, 3, 4, 4, 4, 2, 2, 3, 3, 2, 3, 4, 2, 4, 4, 3, 3, 2, 3}. for manipulating the impreciseness, subjectivity and vagueness due to the decision makers’ verbal information in the internal strength of ksf1, ksf1 y is transformed into the rough interval number according to eqs. (1)(7) as follows: 2 2 2 2 2 3 3 3 3 3 3 3 3 lim(3) 2.62 13               , 3 3 3 3 3 3 3 3 4 4 4 4 4 4 lim(3) 3.43 14                . similarly, lim(4) 3.05 , lim(4) 4.00 , lim(2) 2.00 , lim(2) 3.05 thus, ksf1 y can then be transformed into a set of rough intervals as roy et al./decis. mak. appl. manag. eng. 1 (1) (2018) 121-142 134 ksf1 y = {[2.62, 3.43], [2.62, 3.43], …, [2.00, 3.05], [2.62, 3.43]} the other internal strengths of ksfs can be obtained similarly in terms of rough interval numbers. step 3. allowing for different background of the dms, different weights from table 1 are assigned to them for calculating the rough aggregated internal strength of factor i according to eq. (16). the rough aggregated internal strength of ksfs are shown in table 2. table 2. the rough internal strength of factors factor internal strength of factor c1 [2.575, 3.494] c2 [1.967, 3.159] c3 [1.698, 3.023] c4 [1.648, 2.875] c5 [1.746, 2.771] c6 [2.739, 3.746] c7 [2.078, 3.067] step 4 to 5. evaluate influence between ksfs to construct direct-relation matrix. the nineteen decision experts evaluate the direct impacts among the seven ksfs with the help of the vocal ratings: 0 –no influence (ni); 1 –low influence (li); 2 –medium influence (mi); 3 –high influence (hi); 4 –very high influence (vhi). based on these ratings, the influence evaluations in table 3 can be transformed into non-negative integers from 0 to 4. all the direct-relation matrices xk (k=1, 2, …, 19) of ksfs of hsq could be obtained according to eq. (18) and then the individual direct-relation matrices are then blended consecutively to generate a group direct-relation matrix (see table 3). table 3. the verbal scores of direct-relations between factors c1 c2 c3 ... c7 c1 0;0;0;0;0;0;0;0...0;0 1;0;1;2;2;2;1;1...2;1 2;2;2;2;2;3;1;3...3;2 ... 1;0;2;2;0;2;1;2...2;1 c2 2;2;0;2;2;2;1;2...0;2 0;0;0;0;0;0;0;0...0;0 2;1;1;2;3;2;3;3...3;2 1;1;1;2;3;2;2;3...1;1 c3 2;1;1;2;3;2;1;3...2;2 2;0;1;1;3;2;2;3...3;2 0;0;0;0;0;0;0;0...0;0 2;1;2;2;2;2;2;3...3;2 c4 1;1;2;2;2;0;1;2...1;1 1;3;2;2;3;2;2;3...2;1 2;1;1;2;3;2;3;3...2;2 1;1;1;2;2;2;1;2...2;1 c5 3;1;1;2;2;0;2;1...1;3 2;0;1;2;3;1;2;3...2;2 3;2;2;2;2;2;2;3...2;3 1;1;1;2;1;1;2;2...0;1 c6 3;2;2;2;2;0;1;2...2;3 2;0;1;1;3;1;2;3...2;2 3;2;2;2;3;2;2;3...1;3 1;0;1;2;3;2;2;2...1;1 c7 1;0;0;1;1;0;1;1...1;1 2;0;1;1;3;0;2;3...2;2 2;1;1;2;2;1;2;3...3;2 0;0;0;0;0;0;0;0...0;0 rough numbers are utilized for manipulating the imprecision and subjectivity in data inputs from dms. according to eqs. (20)-(21), the aggregated rough directrelation matrix ( x̂ ) (shown in table 4) of different expert can be obtained. a rough strength relational dematel model for analysing the key success factors of hospital... 135 table 4. the group direct-relation matrix in the rough interval form c1 c2 c3 c4 ... c7 c1 [0.00, 0.00] [1.13, 1.90] [1.69, 2.54] [1.35, 2.42] ... [0.78, 1.98] c2 [1.03, 2.11] [0.00, 0.00] [1.68, 2.45] [2.02, 2.61] [1.13, 1.92] c3 [1.55, 2.35] [1.51, 2.38] [0.00, 0.00] [1.73, 2.37] [1.62, 2.28] c4 [0.79, 1.92] [1.75, 2.53] [1.65, 2.45] [0.00, 0.00] [1.26, 1.89] c5 [1.28, 2.50] [1.51, 2.45] [1.88, 2.54] [1.33, 2.13] [1.04, 1.80] c6 [1.62, 2.55] [1.42, 2.36] [1.79, 2.55] [1.50, 2.28] [1.19, 2.06] c7 [0.63, 1.63] [1.15, 2.31] [1.41, 2.30] [1.09, 2.08] [0.00, 0.00] step 6. the aggregated group direct strength-relation matrix (d) (shown in table 5) is obtained using eqs. (22)-(23). in this stage, the rough intervals denoting the strength of ksfs of hsq (calculated in step 3, table 2) are implanted into the main diagonal of the aggregated group direct-relation matrix (attained in step 5). table 5. the group direct strength-relation matrix c1 c2 c3 c4 ... c7 c1 [2.58, 3.49] [1.13, 1.90] [1.69, 2.54] [1.35, 2.42] ... [0.78, 1.98] c2 [1.03, 2.11] [1.97, 3.16] [1.68, 2.45] [2.02, 2.61] [1.13, 1.92] c3 [1.55, 2.35] [1.51, 2.38] [1.70, 3.02] [1.73, 2.37] [1.62, 2.28] c4 [0.79, 1.92] [1.75, 2.53] [1.65, 2.45] [1.65, 2.87] [1.26, 1.89] c5 [1.28, 2.50] [1.51, 2.45] [1.88, 2.54] [1.33, 2.13] [1.04, 1.80] c6 [1.62, 2.55] [1.42, 2.36] [1.79, 2.55] [1.50, 2.28] [1.19, 2.06] c7 [0.63, 1.63] [1.15, 2.31] [1.41, 2.30] [1.09, 2.08] [2.08, 3.07] step 7. to convert the interface scales of ksfs of hsq into equivalent scales and confirm the existence of the total strength-relation matrix t, the group direct-relation matrix ( x̂ ) is normalized according to eqs. (24)-(26). the normalized rough directrelation matrix (z) is shown in table 6. table 6. normalized the group direct strength-relation matrix c1 c2 c3 c4 ... c7 c1 [0.02, 0.03] [0.01, 0.02] [0.01, 0.02] [0.01, 0.02] ... [0.01, 0.02] c2 [0.01, 0.02] [0.02, 0.03] [0.01, 0.02] [0.02, 0.02] [0.01, 0.02] c3 [0.01, 0.02] [0.01, 0.02] [0.01, 0.03] [0.01, 0.02] [0.01, 0.02] c4 [0.01, 0.02] [0.01, 0.02] [0.01, 0.02] [0.01, 0.02] [0.01, 0.02] c5 [0.01, 0.02] [0.01, 0.02] [0.02, 0.02] [0.01, 0.02] [0.01, 0.01] c6 [0.01, 0.02] [0.01, 0.02] [0.01, 0.02] [0.01, 0.02] [0.01, 0.02] c7 [0.01, 0.01] [0.01, 0.02] [0.01, 0.02] [0.01, 0.02] [0.02, 0.03] step 8. the total strength-relation matrix t, shown in table 7, can be computed by applying eqs. (27)-(29). the components in table 7 specify the overall influence grades of decision makers for the ksf(i) against the ksf(j) bearing in mind their internal strengths. table 7. the total strength-relation matrix roy et al./decis. mak. appl. manag. eng. 1 (1) (2018) 121-142 136 c1 c2 c3 c4 ... c7 c1 [0.02, 0.03] [0.01, 0.02] [0.02, 0.02] [0.01, 0.02] ... [0.01, 0.02] c2 [0.01, 0.02] [0.02, 0.03] [0.02, 0.02] [0.02, 0.03] [0.01, 0.02] c3 [0.01, 0.02] [0.01, 0.02] [0.02, 0.03] [0.02, 0.02] [0.01, 0.02] c4 [0.01, 0.02] [0.02, 0.02] [0.02, 0.02] [0.02, 0.03] [0.01, 0.02] c5 [0.01, 0.02] [0.01, 0.02] [0.02, 0.02] [0.01, 0.02] [0.01, 0.02] c6 [0.01, 0.02] [0.01, 0.02] [0.02, 0.02] [0.01, 0.02] [0.01, 0.02] c7 [0.01, 0.02] [0.01, 0.02] [0.01, 0.02] [0.01, 0.02] [0.02, 0.03] step 9. the sum of rows ( i rn(r ) and columns ( j rn(c ) ) of rough total strengthrelation matrix t are calculated using eqs. (30)-(31) and presented in table 8. in order to effectively rank the ksfs of hsq and investigate the cause-effect relations between them, it is necessary to remove roughness from data according to eqs. (32)(34). the final crisp form ( crisp i r ) and ( crisp j c ), which are also provided in table 8. table 8. the sum of rows, sum of columns, “prominence” and “relation” fac tor rn(ri) rn(ci) ricrisp cicrisp pi fi rank cause /effect c1 [0.097, 0.169] [0.087, 0.162] 0.135 0.122 0.257 0.014 6 cause c2 [0.105, 0.170] [0.097, 0.166] 0.141 0.131 0.273 0.010 4 cause c3 [0.113, 0.175] [0.109, 0.172] 0.151 0.144 0.294 0.007 2 cause c4 [0.101, 0.162] [0.097, 0.163] 0.133 0.130 0.263 0.004 5 cause c5 [0.098, 0.163] [0.109, 0.179] 0.132 0.149 0.281 -0.017 3 effect c6 [0.109, 0.176] [0.122, 0.181] 0.149 0.159 0.308 -0.011 1 effect c7 [0.079, 0.150] [0.081, 0.144] 0.109 0.106 0.215 0.003 7 cause step 10. the “prominence” (pi) and the “relation” (fi) vectors are computed via eqs. (35) and (36), respectively, and shown in table 8. now, depending on “prominence” and “relation” vectors, the impact-relation map of ksfs of hsq can be accomplished by plotting the dataset of (pi, fi) in fig. 1. in this figure, the prominence axis tells about the relative importance of a ksf of hsq compared to the other ksfs of hsq under consideration, whereas the relation axis divides the ksfs of hsq into cause and effect groups. a ksf of hsq will be given top most priority if it’s has the highest prominence value (visibility/importance/influence) in terms of overall relationships with other ksfs. from table 8, we observe that c6 (medical staff with professional abilities) is the most important ksf of hsq followed by c3 (trusted medical staff with professional competence of health care), c5 (detailed description of the patient’s condition by the medical doctor), c2 (service personnel with good communication skills), c4 (service personnel with immediate problem-solving abilities), c1 (wellequipped medical equipment) and c7 (pharmacist’s advices on taking medicine). based on the “relation” vector from table 8, all the ksfs of hsq can be categorized into cause group and effect group, as shown in fig. 1. fig. 1 depicts that “relations” of five ksfs are positive. they are c3 (trusted medical staff with professional competence of health care), c2 (service personnel with good communication skills), c4 (service personnel with immediate problem-solving abilities), c1 (well-equipped medical equipment) and c7 (pharmacist’s advices on taking medicine). these factors belong to the cause group and have net cause for other ksfs. the “relations” of the rest of ksfs (c6, c5) are negative, and they belong to the effect group which are reliant on the change of cause ksfs of hsq. a rough strength relational dematel model for analysing the key success factors of hospital... 137 step 11. finally, it remains to explore the comprehensive interactions between ksfs of hsq. to do so we need to plot a relationship digraph to recognize essential influencing relationships of ksfs depending upon the rough total strength-relation matrix (table 7). the rough intervals in table 7 are transformed to crisp numbers to form crisp total strength-relation matrix (table 9) using eqs. (32)-(34). table 9. the crisp total strength-relation matrix of factors c1 c2 c3 c4 c5 c6 c7 c1 0.030 0.013 0.020 0.017 0.022 0.018 0.011 c2 0.015 0.025 0.019 0.023 0.023 0.020 0.014 c3 0.019 0.019 0.023 0.020 0.024 0.024 0.018 c4 0.012 0.020 0.019 0.022 0.022 0.021 0.014 c5 0.018 0.018 0.021 0.016 0.022 0.021 0.013 c6 0.019 0.018 0.021 0.018 0.021 0.032 0.015 c7 0.009 0.015 0.016 0.014 0.014 0.014 0.025 the bold numbers indicates the relationships that exceed the threshold α=0.0189 a threshold value (  ) of total strength relation can be computed according to eq. (37) for drawing the interpretational diagraph to graphically describe the interrelationship maps between the ksfs of hsq. particular relationships that exceed the threshold 0.0189 (note the bold numbers in table 9) are encompassed in the concluding interacting maps in fig. 1. d1 ci+ri ci-ri 0.150 0.200 0.250 0.300 0.350 -0.020 -0.015 -0.010 0.000 0.010 0.015 0.020 d2 d3 d4d7 d6 d5 figure 1. cause and effect relationships between factors 4.3. comparisons and discussion to endorse the efficiency and powers of the rough strength relational dematel model for analyzing ksfs of hsq in this paper, a comparative exploration is accompanied to analyze the same problem. traditional dematel (shieh et al. 2010) and fuzzy dematel (pamučar and ćirović, 2015) are well-known in the literature. the rank priorities of the seven ksfs derived from these two methods are shown in table 10 along with the ranking produced by rough dematel model. fig. 2 is a pictorial demonstration and relationship of the rank orders according to all those methods. roy et al./decis. mak. appl. manag. eng. 1 (1) (2018) 121-142 138 figure 2. comparison analysis of ranking of the ksfs using different methods firstly, the ranking results from the traditional dematel and rough dematel method are different except for c3, c5 and c6. also, the major interrelations among the ksfs differ in traditional dematel and the rough dematel method. the impact of c2 on c4 (ksf2→ksf4) is reflected as one of the most precarious relations in the rough dematel (fig. 3c). but this is missing in case of the crisp dematel model (fig. 3a). figure 3. comparison of the top ten impact relations in different methods this is possibly due to rough dematel ponders strengthimpacts of the ksfs c2 and c4 ([1.967, 3.159] and [1.648, 2.875]) on the relation ksf2→ksf4. note that traditional dematel does not consider the strengths of c2 and c4 in examining the interactions between ksfs. the rough dematel method is also able to manipulate uncertainty in the ksfs analysis based on decision makers’ opinions. the key success a rough strength relational dematel model for analysing the key success factors of hospital... 139 factors input data from dms is transformed into rough number that reflects the uncertainty in the decision making process due to the linguistic assessments of dms. for example, nineteen dms opined on the direct-relation between c1 (ksf1) and c5 (ksf5) as {hi, ni, li, mi, hi, hi, li, mi, hi, mi, hi, li, hi, mi, mi, hi, hi, hi, hi}. then the proposed method transforms such linguistic ratings into a sequence of rough interval numbers as {[2.26, 3], [0, 2.26], [0.75, 2.39], [1.30, 2.67], …, [2.26, 3], [2.26, 3]} which deliberates the vague information in decision making problem. the traditional dematel only characterizes the linguistic assessment set, {hi, ni, li, mi, hi, hi, li, mi, hi, mi, hi, li, hi, mi, mi, hi, hi, hi, hi} into crisp score set, {3, 0, 1, 2, 3, 3, 1, 2, 3, 2, 3, 1, 3, 2, 2, 3, 3, 3, 3}. thus, to analyze the ksfs for maintaining hsq, the rough dematel can deliver more appreciated suggestion than the crisp dematel method. the second comparative analysis is accompanied with the outcome from the fuzzy dematel method. the attained ranking grades according to fuzzy dematel model are accessible from table 10. there is notable similarity between the major interrelations among the ksfs produced by the fuzzy dematel and the rough dematel. all of them are exactly same except the relations c2→c3, c2→c4 and c3→c4. it is because both the rough and fuzzy approaches consider subjectivity and vagueness while making decisions. while fuzzy dematel operates conventional symmetric triangular fuzzy numbers (stfns), the rough dematel makes use of rough numbers. the rough numbers can flexibly manipulate uncertainty to the highest extent when it is caused by subjective and vague information (zhu et al., 2015). as before we consider the direct impact relation between c1 and c5 as crisp rating set {3, 0, 1, 2, 3, 3, 1, 2, 3, 2, 3, 1, 3, 2, 2, 3, 3, 3, 3}. table 10. comparison analysis of the ranking results of factors factors crisp dematel fuzzy dematel rsr dematel pi ranking pi ranking pi ranking c1 16.30 6 4.97 6 0.26 6 c2 17.64 4 5.37 4 0.27 4 c3 18.93 1 5.71 1 0.29 2 c4 17.48 5 5.31 5 0.26 5 c5 18.52 3 5.61 2 0.28 3 c6 18.57 2 5.58 3 0.31 1 c7 14.80 7 4.65 7 0.22 7 the rough dematel changes this decision set into {[2.26, 3], [0, 2.26], [0.75, 2.39], [1.30, 2.67], …, [2.26, 3], [2.26, 3]} and combines these intervals into [1.69, 2.77]. on the other hand, the fuzzy dematel adapts the decision set as {[2, 4], [0, 1], [0, 2], [1, 3], …, [2, 4], [2, 4]} and combines the these stfns into [1.32, 3.32] with fixed interval of 2. now, on changing the original direct impact ratings between c1 and c5 (ksf1→ksf5) are changed to {2, 2, 1, 2, 2, 2, 2, 1, 3, 3, 2, 3, 1, 3, 1, 2, 2, 3, 1, 2}, then rough aggregation will produce the combined rough rating as [1.57, 2.57]. on contrary the fuzzy aggregation yields the combined stfn rating as [1.00, 2.93] with fixed interval of 2 which does not replicate the changes in dms’ judgments. this is due to the predetermined fuzzy membership function in fuzzy dematel method (song et al. 2017). hence, the rough dematel possesses higher flexibility and more rational than the fuzzy dematel. roy et al./decis. mak. appl. manag. eng. 1 (1) (2018) 121-142 140 5. major implications the result analysis of ksfs of hsq reflects important comprehensions in the theoretical and practical perspective, hence contributes to the hospital service quality management. depending on these outcomes, hospital management can take unambiguous actions to measure, regulate and alleviate the acknowledged ksfs of hsq. this paper, theoretically, advances a framework that helps in identifying the key success factors of hsq as well as the strength and impact between them. this study fills the gap of identifying ksfs of hsq management and their strength impact interrelationships under subjectivity and vagueness. in real world problems, many dms/administrators focus little on the interrelationships of ksfs of hsq. the proposed model for analyzing ksfs in hsq management may help to apprehend the structure of interacting relations among these factors. a hospital can develop truly proactive services with such management and decision-making tool which delivers provision in scheduling the path of managing hsq by regulating the influences of ksfs on each other. the rough dematel method also describes the inter-dependencies among the ksfs systematically, since it deliberates the strength effect of ksfs on their interdependencies, which is discussed by no previous researcher. the rough dematel model aids the managers of hsq to ensure the customers’/patients’ needs are fulfilled and the hospital performs well even though stakes are high due to today’s competitive market. this paper offers a practical impact in the literature hospital service quality management. the proposed model also facilitates the consciousness of ksfs in hsq management. the decision panel includes essentially executives from several departments to establish a comprehensive deliberation of ksfs and direct impact relations in detailed analysis and prioritization of them. numerous useful suggestions can also be deliberated as follows. first, the most important ksf is “medical staff with professional abilities” (c6) in hsq management, i.e., engaging more skilled doctors and nurses will help the achieving primary goals, like, attracting and retaining the patients. failure to select skilled doctors and nurses can affect handling “well-equipped medical equipment” (c1) and “detailed description of the patient’s condition by the medical staff” (c5), because trained medical staff with professional abilities plays a vital role (ksf) for a hospital to be successful. hence, the hospital authority should pay more attention in recruiting accomplished medical professionals to serve better medical treatments in order to govern its ksfs and improve overall performance of hospital services. second, “well-equipped medical equipment” (c1), “trusted medical staff with professional competence of health care” (c3), “service personnel with good communication skills” (c2), “service personnel with immediate problem-solving abilities” (c4), and “pharmacist’s advices on taking medicine” (c7) are among influential ksfs since it they belong to the cause group. improvement of wellequipped medical apparatus would reflect impacts in c5 and c3, even though “wellequipped medical equipment” (c1) is ranked sixth in the final list. on contrary, “medical staff with professional abilities” (c6) is the most essential criterion and conjointly have mutual impacts with other two top ksfs-c5 and c3. this means that the managers of hsq should focus on the interaction between medical staff and patients as this is far more important. a better interaction will help to grow higher satisfaction in the patients (shieh et al. 2010). this is how a hospital can retain its customers who are satisfied with their care. finally, if the hospital wants to accomplish high performance in hospital services, it should control the “cause ksfs” (c1, c2, c3, c4 and c7 beforehand if it is willing to a rough strength relational dematel model for analysing the key success factors of hospital... 141 take care of the “effect ksfs” (c5, c6). if the hospital authority thinks to control the ksfs of “detailed description of the patient’s condition by the medical staff” (c5) and medical staff with professional abilities” (c6), it will be essential to pay more attention to the ksfs of c2, c3 and c4. this is because the “medical staff with professional abilities” (c6) and “detailed description of the patient’s condition by the medical staff” (c5) are the influenced ksf and can be improved, while the “trusted medical staff with professional competence of health care” (c3) and “service personnel with good communication skills” (c2) are the influencing ksfs and can dispatch influences. hospital managing board must be aware of such relationships to control and diminish the risk ksfs in hsq management. 6. conclusions to categorize the ksfs of hospital service quality, a systematic research framework grounded on rough numbers and the dematel technique are proposed in this study. the theoritical and real-world importance of this paper can be listed below: the proposed model can concurrently analyse the internal strength and external impacts of ksfs in hsq management. this speciality serves better information for imposing key/critical decision and provides more accurate ranking orders in ksfs. the rogh dematel model is also very effective in manipulating the vagueness and subjectivity in data since the rough numbers intervals flexiblely specifies the uncertain information in experts’ knowledge based decisions. unlike the fuzzy approaches, the rough dematel needs no auxiliary data (e.g., robust fuzzy membership value, data distribution) in real-world decision problems, which keeps simplier for managers to adopt it in practice. the proposed model helps practitioners to apprehend the inter-relationships among ksfs of hsq to produce valuable perceptions and actionable trials. it can also help hospital management to concentrate on the major evolving issues of ksfs which might boost the overall performance of the hospital. although the proposed model serves good in both theoretical and practical perspectives, it has still some margins. the ksf internal strength analyses are totally based on final verdicts of decision experts (dms), which can make the decision making process more difficult. it is one of the limits of our proposed model. so, for future works, we will consider probability theory to implement the internal strengths of ksfs in hsq management and to measure their impacts on hospital’s performances more accurately. finaaly, the different 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(2015). an integrated ahp and vikor for design concept evaluation based on rough number. advanced engineering informatics, 29, 408–418. plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 5, issue 2, 2022, pp. 140-175. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0306102022b * corresponding author. e-mail addresses: sb.16ms1302@phd.nitdgp.ac.in (s. biswas), gautam.bandyopadhyay@dms.nitdgp.ac.in (g. bandyopadhyay), jnmukhopadhyay@gmail.com (j.n. mukhopadhyaya) a multi-criteria framework for comparing dividend pay capabilities: evidence from indian fmcg and consumer durable sector sanjib biswas1*, gautam bandyopadhyay1 and jayanta nath mukhopadhyaya2 1 department of management studies, national institute of technology, india 2 finance area, army institute of management, india received: 16 july 2022; accepted: 29 september 2022; available online: 6 october 2022. original scientific paper abstract: in this paper, we aim to carry out a comparative analysis of the dividend pay capabilities (dpc) of the selected organizations belonging to the fast moving consumer goods (fmcg) and consumer durables (cd) sectors listed in bse, india during the period fy 2013-14 to fy 2019-20. we select top 25 companies from fmcg group and top 5 firms from the cd sector on the basis of average market capitalization. for comparison purpose, we have considered six aspects (grounded on the extant theories on dividend policy) such as ownership, size, profitability, growth, liquidity and risk. we have used a new integrated logarithmic percentage change-driven objective weighting (lopcow) and evaluation based on distance from average solutions (edas) framework for our analysis. the result shows that companies do not show consistent performance over the years. we further have noticed that fmcg organizations show comparatively better capabilities that cd firms vis-à-vis dividend payment. since, there are considerable variations in the ranking, we apply aggregation methods like borda count (bc), copeland method (cm) and simple additive weighting (saw). we use two other popular multicriteria decision making (mcdm) methods like multi-attributive border approximation area comparison (mabac) and the complex proportional assessment (copras) for comparison with our framework to ascertain the reliability of our result. key words: dividend payment, investment decision making, lopcow, edas, borda count, copeland method. mailto:sb.16ms1302@phd.nitdgp.ac.in mailto:gautam.bandyopadhyay@dms.nitdgp.ac.in mailto:jnmukhopadhyay@gmail.com a multi-criteria framework for comparing dividend pay capabilities: evidence from indian … 141 1. introduction dividend is a part of the profit distributed to the investors recognizing their stake in the business and cooperation. the remaining part of the profit (after paying the dividend) is retained by the firms for reinvestment in the ongoing and future activities. dividends are paid primarily to allure the investors who perceive the same as a sign of company’s growth and steady income out of their investment (khan et al., 2019). however, the decisions on dividend payment is a complex one that stands on conflicting perspectives. while a higher payment toward dividend is an indication of potential for monetary growth and a means for income for the investors, a lower dividend pay-out (dpo) enables the firms to use the surplus for future expansion of the business to provide a higher gain in future against the capital investment. the discussions on formulation of the policy decisions for determining the dpo keeping in mind two contradictory objectives such as providing income opportunities to investors for attracting them for further investment and retaining earnings for future expansions and growth of the continuing business have been progressing over many decades. the researchers have been able to put forth several theories in this regard. literature on the agency problem has advanced hypotheses on the relation between free cash flow and business performance. research papers have also included variables reflecting the agency problem in their explanation for dpo. as dpo reduces the free cash flow available to companies, it is expected that this will reduce the incidence of the agency problem. the general argument that is advanced is that managers keep free cash and invest them in growth of companies to consolidate their position as a larger company has more activities and requires more people and more supervision. it is possible that these investments may not be justified and is against the interest of the shareholders. in this regard, it is also emphasized that expansion through debt is desirable as there is better monitoring by lenders and acts as a disciplinary tool for managers. the dividend discount model of share price determination states that higher the dpo and its expected growth rate, higher is the value of the share of that company. that is dpo reflects income generation and helps in expectation formation for future growth. in other words, by declaring the dividends the companies provide an indication or signal to investors about the performance vis-à-vis income generation and prospects (brigham and houston, 2001). the purpose of declaration of dividends is to minimize the degree of asymmetry in information available to the internal parties (i.e., managers) and external (i.e., investors) shareholders (lin et al., 2017; hardy and andestiana, 2019). a higher dividend transmits a positive signal to the investors while a lesser cash dividend payment provides a negative signal (affandi et al., 2018). it is true that companies that skip dividends or lowers the rate of dividend, are penalized by the market. this approach, therefore, relates dpo to expected future growth and does not focus on the agency problem per se. the agency problems stem from the agency cost which is defined as sum of the expenditures related to monitoring (due to governance of the activities of the agents by the principal) and bonding (to ensure that the interests of the principals are met by the agents) and residual loss in form of the opportunity cost due to the difference in the interests of the agents and the principals (jensen and meckling, 1976). payment of dividend also means less retained earnings for reinvestment purposes and can signify that the company does not have any expansion plans in the near future. thus dividend payment can give conflicting signals. rozeff (1982) made three propositions. first, the companies resort to paying lower dividends for reinforcing their investment plans and safeguarding from costly external finance. secondly, in case of meeting biswas et al./decis. mak. appl. manag. eng. 5 (2) (2022) 140-175 142 short term obligations and fixed charges, companies with higher debt-equity ratio tend to lower the dividend payment. finally, to lower the agency cost companies prefer to have lower dpo if there is a higher external shareholding. the problem of agency cost gets escalated when there is a conflict of interest between the managers (i.e., agents) and the shareholders (i.e., owners) where the principal’s expectations are not reflected in the actions of the agents (affandi et al., 2018). to this end, the agency cost can be minimized by striking a balance of the conflicting objectives of the agents and the principals. dividend payment is one of the ways to reduce the agency conflict (kilincarslan, 2021). in this connection, easterbrook (1984) explained that although companies find dividend payments obvious, this is all cost and no benefit to them. dividends are taxed at a higher rate than capital gains which would result from investments of the retained earnings. further, in the presence of dividend payments, external finance for investments would add cost to the company. however, companies paying dividends and simultaneously raising funds from the market is very common. this he states is a way shareholders reduce the monitoring costs of the managers. as a single shareholder is not in a position to monitor the activities of the managers, they rely on external fund providers to do the job for them. paying dividends and raising external funds leads to a check on the nature of investments undertaken by the managers, while keeping the leverage unaltered. this is further elaborated in jensen (1986). according to him, companies that are involved in new activities are the ones that have not been yet subject to disciplinary forces of the market and hence generate higher free cash flow. such companies may move into riskier ventures or unrelated diversification. debt as a substitute for dividends can control this agency problem. baker et al. (2002) mentioned about four explanations behind dpo such as signalling, tax-preference (i.e., transactional), agency problem and bird-in-hand. the theory of bird-in-hand relies on short-term gain in terms of payment of dividends rather waiting for long-term capital gains under uncertainty (widiyanti et al., 2019). to sum up, it is evidenced that disclosure of dividends and building the capabilities to pay the dividends can converge the theories. investors look for a consistent and increasing dpo to get confidence about the appropriate utilizations of the funds invested in the company (chaniago and ekadjaja, 2022). therefore, we spot that there have been different schools of thoughts in explaining the motive and basis for taking dividend policies by the organizations. further, it is an established fact that dpo has distinguished effect on firms’ valuation at the market place vis-à-vis investors’ behaviours and performance of the organizations. however, the evidence of a sizeable number of work continuously carried out over many decades in past suggest that the stated field has not been exhaustedly explored yet. this motivates us to undertake the current study that aims to find answers to the following research questions: rq1. how can a model be formulated to compare a group of companies on the basis of several influencing factors of dpo? rq2. to what extent do the firms differ from each other in terms of their capabilities to pay dividends subject to the influence of multiple indicative variables concerning the dpo? in this paper we intend to carry out a comparative analysis of the dividend pay capabilities (dpc) of the fmcg and cd organizations listed in bse, india over a period of fy 2013-14 to fy 2019-20. dpc of a particular company is defined as the final appraisal score which is obtained considering the performance of the company with respect to the criteria (i.e., the factors that influence the dividend payment). a company with higher appraisal score is considered as having more capability (with a multi-criteria framework for comparing dividend pay capabilities: evidence from indian … 143 respect to the criteria) in paying dividend as compared with the other available alternative options (i.e., companies). since, we consider multiple factors grounded on theoretical foundations of dividend policy that affect the decisions of dpo, our work aims to build a mcdm model for the comparative analysis. mcdm models are particularly useful when a set of alternative choices are compared subject to the influence of a number of conflicting attributes or features or criteria to select the best possible choice(s) (pamucar et al., 2021; laha and biswas, 2019). for the purpose of such kind of analysis we use a very recently developed mcdm algorithm such as lopcow (ecer and pamucar, 2022) for calculating of criteria weights and edas method (keshavarz ghorabaee et al., 2015) for final ranking. the remaining part of the present paper is structured as follows. section 2 is devoted for summarizing the observations and findings of some of the past work related to effect of the dpo on firm performance and determining factors for dividend payment. in section 3 we include a brief description of the data and methodology while section 4 provides the summary of the findings of the current work. section 5 sheds light on the inferences and implications of the results through a brief discussion and in section 6 we make the concluding remarks alongside some scopes for future work. 2. related work a plethora of research spanning over last several decades have been carried out by the researchers and the practitioners in the field of dividend policy and its effect on firm’s performance (financial and market), value, shareholders’ sentiments vis-àvis the disclosure of the dividends and the underlying factors that influence the decision on dpo. the principal objective of corporate financial management is to maximize the market value of equity shares. the key question of interest is: what is the relationship between dividend policy and market price of equity shares? the jury is still out on this unresolved issue in corporate finance. according to the traditional position enunciated by graham and dodd (1934), the stock market places considerably more weight on dividends than on retained earnings. the gordon model (gordon, 1959) has shown that for firms, where the rate of return generated by the firm is greater than the rate of return required by shareholders, the price per share increases as the dividend payout ratio decreases and vice versa. miller and modigliani (1961) expounded that the value of a firm depends solely on its earnings power and is not influenced by the manner in which its earnings are split between dividends and retained earnings. according to them dividends matter because of the uncertainty characterizing the future, the imperfections in the capital market and the existence of taxes. in real life different investors hold different views about future prospects and managers are better informed about future prospects than investors. consequently, the information or signaling content of such dividend announcements. muth's paper (muth, 1961) says that what matters in economics is not what actually happens but the difference between what actually happens and what was supposed or expected to happen. consequently, only surprises in policy would have the kind of effect the policy maker is striving to achieve. what happens if the dividend announced is higher than what was expected by the market? in such a situation the market revises its assessment of future earnings and would lead to an upward price movement in the share and vice versa. the academic thinking is that the price changes that occur look like responses to dividends themselves, though they are caused by an underlying revision of the earnings potential. mathematical models like the walter model (walter, 1963) have shown that the optimal payout ratio for a biswas et al./decis. mak. appl. manag. eng. 5 (2) (2022) 140-175 144 growth firm is nil. clearly this leads to an extreme course of action which makes limited sense in the real world. in most countries dividends are taxed more heavily than capital gains. hence it can be argued that firms should pay little dividend so that investors earn more by way of capital gains. the tax laws in all countries favor capital gains in one more way. taxes on dividends are payable immediately but taxes on capital gains are payable only when shares are sold. consequently, the effective tax rate on capital gains diminishes as the period of holding increases. brennan (1971) attempted to provide a connectivity between the gordon’s model and miller and modigliani framework. lintner (1956) made some very important observations. mature firms with significant stable earnings have higher payout ratios, whereas fast growing firms have low payout ratios. large fmcg companies may fall under this category. dividends tend to follow earnings, but dividends follow a smoother path than earnings. transitory changes in earnings are not likely to have an impact on dividend payment. moreover, dividends are sticky in nature as managers are reluctant to have a dividend payout that may have to be reversed. a subsequent study by fama and babiak (1968) supported the lintner model. in the subsequent studies, the authors (black and scholes, 1974; asquith and mullins, 1983; lease et al., 1999) extended the explanations on how dividend yield and policy influence the stock price movements and impact of dividend intimations on stock price hike at the market place. there were a number of early contributions to discern the influence of industry, managers’ views and other subsequent factors on dividend policy (michel, 1979; baker et al., 1985; miller and rock, 1985; baker and powell, 1999; baker et al., 2001) in the following sub-sections, we present a summary of some of the recently published research available in the extant literature where the first one discusses how are the dividend policy and dpo relevant and important for firms’ market performances and valuations while in the second sub-section we enfold the findings of the past work to explore various determinants of the dpo. 2.1. effect of dividend policy and dpo on firm performance there has been a number of past research that attempted to establish the impact of dividend payment not only to bring new investments but also the enhance the firm’s value and performance. for instance, jiang et al. (2019) conducted an analysis over 210 stocks listed in shanghai and shenzhen 300 index, china and noted that the drop in the share prices is lower on the days of dividend payment. in another study, taofeek et al. (2019) focused on dividend management on stock price movement in long run as well as short run. five variables were used namely stock price volatility, dividend pay-out ratio, dividend yield, earnings volatility and firm size. this research considered non-financial sectors listed in nigerian stock exchange. in this study the researcher highlighted that low dividend pay-out ratio serves as good signal to investors for expectation of return which increases the firm value. the work of pandey and narayani (2019) focused to explore the impact of dpo on the share price in auto sector of india for a longitudinal spectrum of 12 years ranging from 2004 to 2016 encompassing the recession period 2008-09. ten auto companies listed in nse were considered and six variables were contemplated namely market share price as dependent variable and dividend yield, dividend pay-out ratio, earning retention ratio, earning per share and leverage. the researchers found out that dividend yield and dpo have a significant effect on share price in given time period. odum et al. (2019) attempted to find out the impact of dpo on firm’s value. the research employed panel ordinary least square regression techniques on 11 beverages and breweries companies listed on nigeria stock exchange covering ten years from 2007. a multi-criteria framework for comparing dividend pay capabilities: evidence from indian … 145 as a manifestation of the values of the firm, five indicating variables such as profit after tax, cash holding, leverage ratio, dividend pay-out ratio and firm size were considered. based on the findings form the study author recommended into order to increase the value of the firm, manager must ensure to increase pat and leverage ratio. puspitaningtyas (2019) tried to determine the effect of dividend announcement on stock return on the period 2017 in indonesia stock market. 53 companies were considered in this research which were listed in indonesia stock market who have announced cash dividend in consecutive during 2016-2017 and do not conduct corporate action other than announcement of dividend. four criteria were considered namely actual return, expected return, abnormal return and average abnormal return for three time periods such as pre event i.e. 5 days before the event, event day and post event i.e. 5 days after the event. researcher found that the market reacts to the announcement of dividend which is indicated by the existence of abnormal return value which is directly proportion to increase and decrease in dividends which strengths the perspective of signalling theory. the work of omar and echchabi (2019) examined the potential role of dividend pay-out plays in influencing the fund managers in selecting and recommending a stock. semi-structured interview method was conducted with six malaysian investment manager and the results indicates that other factors coupled with dividend pay-out pays a catalyst for investors and fund managers to select a stock in their portfolio. in the context of signalling theory, salman (2019) worked on investigating the influence of shareholder preference and dividend signalling on the dividend policy of the corporations in pakistan. through a structured questionnaire based survey of 61 executives, the study reported that there are significant positive relationships between dividend policy and shareholder’s preferences and dividend signalling. in a recent work, yin and nie (2021) attempted to predict the returns of the stocks listed in chinese market using raw and multiple adjusted dividend pay-out ratios (dpr). the research showed that stock returns can be positively predicted by dpr during the study period (2002-2018). in a different study, the researchers observed a moderating effect of the dividend policy on the causal relationship between profitability and value of the firm (setyabudi, 2021). in the context of nigeria, ifeanyichukwu and yusuf (2021) worked on examining the effect of the share dividends and cash dividend on the market price of the share during the time period 2014-2018. the authors observed a positive effect of cash dividend on share price at the market and also recommended the organizations to work for increasing the priceearnings ratio. paying dividends to shareholders not adds to increase their wealth, but also helps to paying organizations to achieve sustainability in the long run (sami and abdallah, 2021). a policy with higher dividend payment increases the corporate value significantly (dang et al., 2021; gupta & arora, 2021). dividend payment is associated with investors’ sentiments that enhances the demand of the investors and eventually escalates the market return (kumar et al., 2022). seth and mahenthiran (2022) further extended the growing volume of the literature to establish the relevance of the signals of csr disclosure and dpo for maintaining long-term relationship with the shareholders that eventually enables the firms to become sustainable in future. 2.2. determinants of dpo over the years the researchers from various countries have conducted several studies from various perspectives to find out the determinants of the dpo. in amidu and abor (2006), the authors used financial statements for six consecutive years to find out the factors that affect the dpo decision for the organizations listed on the biswas et al./decis. mak. appl. manag. eng. 5 (2) (2022) 140-175 146 ghana stock exchange. the authors considered two perspectives such as agency cost and opportunity for investments. it is evidenced in their work that profitability and cash flow hold positive associations with dpo while risk maintains the inverse relationship. in a later work, hamill and al-shattarat (2012) tested the hypothesis of agency cost to discern the effect of ownership structure, free cash flow and firm size on dpr and observed significant influence. mui and mustapha (2016) had worked on public organizations in malaysia and used multiple regression to conclude that investment opportunity, liquidity and size of the firm bear significant effect on dpo. the study of khan et al. (2017) on pakistani firms advocated for taxes and cash flow in addition to profitability as enablers of dividend policy. the authors conducted the study during 2003-2012 using panel regression. based on their analysis over chinese state controlled and non-state-controlled firms during a period of 10 years, lin et al. (2017) realized that information asymmetry lowers the dpo. however, the authors observed evidence that for state controlled firms, higher information asymmetry leads to higher dpo. continuing in the same direction, malik and sattar (2018) applied the ordinary least square (ols) method to figure out notable influence of governance related variables such as size of the board, ceo duality, ownership structure, size of the firm and operating cash flow on dpo for the companies in pakistan. while working on 19 companies from indonesian stock exchange during 2013-2015, tumiwa and mamuaya (2019) noted significant impact of firm size, profitability, and leverage on the dpo and stock price. the work of le et al. (2019) supported the growing strand of work and revealed that profitability is positively related with dpo. however, the authors did not notice any notable influence of firm size, free cash flow, financial leverage and liquidity on dividend payment. nidar et al. (2019) found positive influence of ownership structure and presence of independence in the board on dpr while they noticed insignificant and negative effect of board size. budiarso (2019) extended the literature with their work on indonesian consumer durable firms to investigate the footprints of profitability (variable: return on asset), efficiency (variable: growth of asset), risk (variable: debt ratio) and non-discretionary accruals and discretionary accruals on dividend policy using logistics regression over a period of 2010-2017. the author reported the consequential role of profitability for deciding the dpo. the extant literature further evidenced with the work of lloren-alcantara (2020) on philippine-listed organizations during 2014-2018. the study pointed out the affirmative effects of profitability, liquidity and firm size but negative impact of the insider ownership on dividend payment. the authors argued for further work in this regard. in the recent works (setyabudi, 2021; salim and aulia, 2021) the authors reflected in tune with past work and noted the significant associations of profitability, liquidity and leverage with dpo. yakubu et al. (2021) reported a positive causal association among working capital management through cash conversion cycle and days inventory outstanding with dpo based on a study made on a group of non-financial firms in ghana during 2007 to 2016. bakri et al. (2021) conducted a two period comparison of the determinants of dpo with respect to formal corporate governance mechanisms in malaysian context and noted that profitability, lagged of dividends and firm size remain as a constant factor. al sawalqa (2021) worked on life cycle theory of dpo on selected jordanian non-financial firms and noted the importance of asset value and shareholders’ equity on determination of the dividend policy. the study of taher and al-shboul (2022) has focused on delving into the relationship of liquidity and dividend policy and found that an increase in liquidity decreases the dpo. chaniago and ekadjaja (2022) figured out positive and significant impact of return on equity and ownership structure on dpr while they discovered a multi-criteria framework for comparing dividend pay capabilities: evidence from indian … 147 insignificant and positive effect of cash ratio for the indonesian firms. novia and marlina (2022) provided a contrasting result as they observed no effect of leverage and liquidity and negative impact of profitability on the dividend policy. in indian context, there have been a number of work in sync with the research at global platform. for example, labhane and das (2015) investigated for the trend and determinants of dividend policy for 239 firms listed on the national stock exchange (nse), india over a period of 20 years. the authors put forth some interesting observations. first, the authors observed a decline in the number of companies that pay dividend while there had been a rise in the total amount paid in form of dividends over the study period. secondly, the pattern of dpo varies across the industries. finally, the authors concluded that given a conditions of higher free cash flow, better investment opportunities, larger size, age and profitability, and lower leverage, the firms tend to pay more dividends. in a later work (singla and samanta, 2018), it was found that profitability, life cycle and size lead to increase in dividend payment while cash flow exhibits negative relationship with dpo due to the presence of agency problem. thakur and kannadhasan (2018) applied quantile regression model to establish the differences in dividend payments by the companies due to changes in the profitability, growth, and size. in the work of labhane and mahakud (2019), 781 indian organizations listed in nse were examined for a period of 1995 to 2015 based on 14 variables related to profitability, efficiency, risk, liquidity, size, market capitalization and nature of business. the results highlighted the notable effect of the business group and profitability on dpo. garg and bhargaw (2019) diverted the stream of ongoing work by using lintner’s model and noted the effect of current year’s earning on dividend payment for the indian firms listed in the bombay stock exchange (bse). katakwar et al. (2021) pointed out the positive impact of return on equity on dpo while they found risk and tax rate negative influence the dividend payments for nse listed firms. 2.3. motivations and contributions of the research from the literature review we make out that the subject area is not an unknown one. there has been a continuous effort in introspecting the motives behind formulating the dividend policy and its impact on financial and market performance of the stocks and investors’ behaviours. a steady growth in the volume of the literature is observed that deal with unveiling the determinants of the dividend payment in the context of leading indices of the global stock market while considering different types of the industries and firms. however, there is a scantiness in the work that considers multiple perspectives and provide a comprehensive multi-criteria based evaluation of a number of organizations to enfold the competitive positions with respect to their relative capabilities for paying dividends. it is evidenced in the extant literature that most of the past research have utilized time series based predictive models and frameworks to detect the causal associations. in this regard, the current work adds value to the growing literature in two ways. firstly, in indian context the present paper may be considered as a work of its kind that provides a multi-period, multi-criteria based comparison of fmcg and cd companies with respect to the features rooted through the theoretical base of the dividend policy and findings of the previous work. secondly, we present a new integrated framework of lopcow-edas methods for carrying out mcdm based analysis wherein we utilize the multiple aggregation methods. lopcow has not been explored for variety of applications yet. biswas et al./decis. mak. appl. manag. eng. 5 (2) (2022) 140-175 148 3. data and methodology in this paper, we aim to carry out a comparative analysis of the dpc of the selected organizations belonging to the fmcg and cd sectors listed in bse, india. the present section discusses the selection of the sample, description of the criteria and methods used in the paper. figure 1 depicts the flow of the steps followed in the current study. 3.1. sample in the present paper, we consider the fmcg and cd companies listed in bse. we apply two filtrations. first, we discard all companies which are not listed in bse during april 1, 2013 to march 31, 2020 (our study period is fy 2013-14 to fy 201920). second, we calculate the average market capitalizations (over the study period) of the companies shortlisted at the first stage by using geometric mean. we select top 25 companies from fmcg group and top 5 firms from the cd sector. therefore, our final sample consists of total 30 organizations (see table 1). these 30 organizations are the alternative options in our paper. in the present paper we have adopted convenience sampling. we have considered the fmcg and cd sectors. fmcg aka consumer packaged products are regularly bought by the consumers and consumed by households in daily use. fmcg sector is characterized by a huge variety of household products with higher consumption and variable price range (lowest may be below inr 10), a large number of consumers (both from urban and rural markets), a diverse distribution network, lower penetration level (that lowers the entry and exit barriers), and a higher level of competition with presence of many domestic as well as multinational firms alongside unorganized players (dhingra et al., 2018). in the decade the sector has undergone a transformational change because of technological progress, e-commerce, enhanced penetration to rural markets, covid-19, and changing nature of the consumer behaviours which have posited promises for potential future growth and challenges for the organizations to design and deliver unique value propositions (toi report, 2022). according to the recent report by indian brand equity foundation (ibef, 2022a) the estimated market potential for fmcg is usd 220 billion by 2025 with a cagr of 14.9% while the projected value of the packaged food market in india is usd 70 billion. the observed rural spending is around 50 percent of the total spending in fmcg products. the fdi inflow in the last two years has been usd 20.11 billion. on the other hand, cd refers to a group of products consumed by the household over a period of time such as kitchen appliances, electronic gadgets, home furnishing and leisure items etc. the products are classified under three broad categories: white goods, brown goods and consumer electronics. the sector is also characterized by wide variety, a higher level of technology dependency, a mix of several domestic and multinational firms in addition to numerous unorganized and/or organized support firms and higher level of competition on brands. given the developments in the software and hardware technology and enhanced disposable income, cd sector has emerged as one of dynamic and happening industry having a widespread awareness. with government initiatives (e.g., rural electrification and affordable housing schemes), cd products have a notable rural penetration too. in recent time, the sector has witnessed a fdi inflow of usd 3.19 billion (ibef, 2022b; sarangi, 2019). considering the growth potential, familiarity to the households, variety of products, higher level of competition within the industry, promising amount of fdi, and increased level of use at all levels of the society, have made the fmcg and cd sectors the sectors of interest for the investment decision analysis. a multi-criteria framework for comparing dividend pay capabilities: evidence from indian … 149 in the fmcg and cd sectors there are 72 and 10 listed companies respectively. since, our study period starts from april 01, 2013, at the first level of filtration, we discard the companies that do not appear in the listing throughout the study period, i.e., got enlisted after april 01, 2013 and/or got discontinued before march 31, 2020. after the first level of filtration we obtain 60 companies from the fmcg and 09 firms from the cd sector. now, we calculate the average market capitalization for all companies qualified at the first stage (i.e., 69 companies). we use geometric mean (gm) for calculating the average as gm is acceptable than the arithmetic mean in presence of outliers, if any. since any of the 69 companies did not have any missing and/or zero value for the market capitalization, gm is also justified in use. after obtaining the average market capitalization, we select top 25 organizations from the fmcg (out of 60) and top 5 companies from the cd (out of 9) sectors. here, our final sample consists of more than 30 percent of the total elements available in the population. the total size of the final sample is 30. the extant literature has advocated for 30 as a minimum standard size of the sample in sync with the central limit theorem, n-hat and n-omega methods (for example, roscoe, 1975; luanglath and rewtrakunphaiboon, 2013; louangrath, 2014; luanglath, 2014; agresti and kateri, 2021). hence, the sample size used in this paper satisfies the minimum requirement. 3.2. criteria description in line with past work, we select the criteria for carrying out the comparative analysis of dpc of the sample organizations. for example, the extant literature shows that institutional ownership (io) plays a momentous role in corporate governance. the distribution pattern of io is one of the significant enablers for supporting the organizations in maintaining the optimum cash holding vis-à-vis agency cost issue to safeguard the interest of the investors. with the cash holding by the organizations. in this context, a higher % of non-promoter ownership reduces the cash holdings and thereby support the objective of “efficient monitoring hypothesis (emh)” as observed by gupta and bedi (2020). the size of the organization has a positive impact on the profitability of the firm (hirdinis, 2019). profitability indicates the earnings prospect of the firms that favours the dividend pay-out (dewasiri et al., 2019). however, earning is supported by the growth. a growing organization has a better prospect of earnings in future. hence, growth is an important enabler of dividend pay-out. liquidity in terms of free cash flow (fcf) on the other hand has a positive effect on the dividend policy (rochmah and ardianto, 2020; pattiruhu and paais, 2020). according to the signalling theory, dividend is an indicator of the potential earnings in future. however, the uncertainties due to business risk blur the future earning prospect. therefore, risk negatively influences the dpo (hamill and alshattarat, 2012). therefore, leverage as a measure of risk undermines dpo. the criteria that are used in the current work for comparing the fmcg and cd organizations are summarized in table 2. biswas et al./decis. mak. appl. manag. eng. 5 (2) (2022) 140-175 150 table 1. list of companies under comparison s/l company category s/l company category a1 avanti feeds ltd. fmcg a16 i t c ltd. fmcg a2 bajaj consumer care ltd. fmcg a17 jyothy labs ltd. fmcg a3 bombay burmah trdg. corpn. ltd. fmcg a18 k r b l ltd. fmcg a4 britannia industries ltd. fmcg a19 marico ltd. fmcg a5 c c l products (india) ltd. fmcg a20 nestle india ltd. fmcg a6 colgate-palmolive (india) ltd. fmcg a21 procter & gamble hygiene & health care ltd. fmcg a7 dabur india ltd. fmcg a22 radico khaitan ltd. fmcg a8 e i d-parry (india) ltd. fmcg a23 tata consumer products ltd. fmcg a9 emami ltd. fmcg a24 united breweries ltd. fmcg a10 future consumer ltd. fmcg a25 zydus wellness ltd. fmcg a11 gillette india ltd. fmcg a26 rajesh exports ltd. cd a12 godfrey phillips india ltd. fmcg a27 symphony ltd. cd a13 godrej consumer products ltd. fmcg a28 titan company ltd. cd a14 hatsun agro products ltd. fmcg a29 voltas ltd. cd a15 hindustan unilever ltd. fmcg a30 whirlpool of india ltd. cd figure 1. research framework a multi-criteria framework for comparing dividend pay capabilities: evidence from indian … 151 it is evident from the literature review that for better governance and utilization of the surplus earned from the business operations, the organizations need to be under independent vigilance. an increase in the percentage of shareholding by the non-promoters may reduce the excessive cash holding and the possibility of misuse by the agents (i.e., managers) and hence address the agency cost problem, if any. hence, in this study we take % ownership by non-promoters as a proxy of io. for an effective management of cash and earnings, io should be maximized. a company with greater amount of total assets is likely to operate with freedom. it is also an indication of company’s financial wellbeing and future prospect. hence, size which is a natural log of total assets is treated as beneficial for building dpc. it is amply evident from the past work that a more profitable firm is likely to be capable for enhancing dividend payout. therefore, all profitability indicators are considered as of maximizing nature with respect to dpc. the same explanations hold true for the growth variables for having a better dpc. hence, all growth indicators are mentioned in the maximizing direction. if an organizations are having greater liquidity, the short run obligations can be made. further, liquidity also indicates efficiency in business operations in generating cash. therefore, ncf is considered in the maximizing direction. finally, a firm can operate with stability for long run growth, if the debt is lower than the profit. to this end, leverage (considered as a proxy indicator of risk) is considered as a non-beneficial criterion with respect to dpc in this paper for which we set minimizing objective. table 2. list of criteria dimension criteria definition code effect direction uom ownership institutional ownership (io) % ownership by nonpromoters c1 maximize % size size of the firm (s) natural log of total assets c2 maximize value profitability net profit margin (npm) (net profit/ revenue)*100% c3 maximize % return on capital employed (roce) (pbit/capital employed)*100% c4 maximize % growth sales growth (sg) natural log of (sales at t / sales at (t-1)) c5 maximize value market cap/ enterprise value (mcev) market capitalization/ enterprise value c6 maximize times liquidity net cash flow (from operating activities) (ncf) net amount of money being generated from regular business operations c7 maximize rs. million risk leverage (l) debt/ pbitda c8 minimize times 3.3. data the total spectrum for study has been selected as 10 years, i.e., fy 2012-13 to fy 2021-22. however, fy 2012-13 has been considered as a base year for the calculation of the year on year growth attributes (for example, sales growth). further, fy 202021 and 2021-22 have been the periods affected by the “black swan” event, covid-19 which impacted the stock market unprecedentedly and yet we believe may not be suitable to be considered for a stable analysis. hence, the study period is effectively selected as fy 2013-14 to fy 2019-20. the data for finding out various indicating criteria for the companies under study have been collected from cmie prowess iq (version 1.96). accordingly, the decision matrices for the financial years (i.e., fy 2013-14 to fy 2020-21) have been constructed using the definitions as mentioned in the table 2. the decision matrices for the various fys are given in the appendix a. the biswas et al./decis. mak. appl. manag. eng. 5 (2) (2022) 140-175 152 study period spans over fy 2013-14 to fy 2019-20. in our paper we have not considered the period fy 2020-21 as the same is characterized by an unprecedented disruption because of the rapid spread of covid-19. during this ‘black swan’ period there has been a massive impact on socio-cultural and economic environment across the globe. hence, for a deeper understanding of the comparative dpc of the companies under study, we have selected a considerably uninterrupted period. 3.4. criteria weight calculation: lopcow method the lopcow method calculates the criteria weights based on objective information (ecer and pamucar, 2022). it provides the following advantages the criteria weights are comparatively even in distribution negative performance values of the alternatives can be used in deriving the criteria weights. this is a useful feature in many complex real-life scenarios such as stock returns. ability to work efficiently with a large number of criteria and alternatives let, ij m n x x      be the decision-matrix where, m is the number of alternatives (i.e., companies under comparison; 30m ) and n is the number of criteria (in our case, 8n  ). in what follows are the computational steps (ecer and pamucar, 2022) step 1. normalization of the decision-matrix using the linear max-min type of normalization, we obtain the normalized decision matrix as given by ij m n r r      where, min max min j ij ij j j x x r x x    (when j j   , desired effect: maximizing) (1) max max min j ij ij j j x x r x x    (when j j   , desired effect: minimizing) (2) step 2. derive the percentage value (pv) for the criteria the pv for each criterion is given by the natural log of the mean square value as a proportion of the standard deviation expressed in percentage. this step helps to reduce the uneven distribution of the weights. accordingly, pv is calculated as 2 1 ln .100 m rij i m pj                   (3)  denotes the standard deviation step 3. computation of criteria weights the weight for the th j criterion is given by a multi-criteria framework for comparing dividend pay capabilities: evidence from indian … 153 1 ij j n ij j p w p    (4) where, 1 1 n j j w   (i.e., sum of the weights of all criteria = 1) 3.5. edas method edas considers average solution as a yardstick for figuring out the suitability of the alternatives. in this method, two distances used such as pda (positive distance from the average) and nda (negative distance from the average) are calculated subject to the desired effect of the corresponding criterion, i.e. maximizing and minimizing. the alternative, which has higher pda and/or lower nda, is considered as the best alternative among the others (keshavarz ghorabaee et al., 2015). edas has been applied in various real-life problems concerned with selection of best possible alternatives subject to influence of a set of criteria, for example, performance based selection of mutual funds (karmakar et al., 2018), carpenter manufacturer selection (stević et al., 2018), resource selection under dynamic environment for crowd computing for smartphones (pramanik et al., 2021), green supplier selection (wei et al., 2021), strategic decision for international market selection (zolfani et al., 2021), 3d printer selection in digital manufacturing (lei et al., 2022), and green financing (su et al., 2022) among others. in what follows are the advantages of the edas method: edas is useful method in the situations with fluctuations in the performance values it does not consider the extreme solutions for benchmarking and therefore, it works fine in realistic situations provides stable and reliable solutions (even with larger alternative and criteria sets) that are free from the rank reversal issues. the procedural steps of the algorithm are as under. step 1. formation of the decision matrix the decision matrix is represented as 11 1 1 n ij m n m mn x x x x x x                 (5) where, is the number of alternatives and is the number of criteria. ij x is the performance value of the alternative subject to the criterion. step 2. derive the average solution the average solution is derived as biswas et al./decis. mak. appl. manag. eng. 5 (2) (2022) 140-175 154 ; 1, 2, ... 1 j n m xij i x j m     (6) step 3. derive the pda and nda the pda and nda are calculated by using the following expressions pda: (0,( )) ; (max ) (0,( )) ; (min ) max x xij j j j imizing x j ij max x xj ij j j imizing x j d                  (7) nda: (0,( )) ; (max ) (0,( )) ; (min ) max x xj ij j j imizing x j ij max x xij j j j imizing x j d                  (8) step 4. calculation of the weighted sum of pda and nda values for all alternatives subject to the criteria the weighted sums are calculated as 1 n s w ji ij j d     (9) 1 n s w ji ij j d     (10) here, w j is the weight of the criterion. step 5. normalization of the weighted sum of pda and nda values the normalization is done as under for weighted sum of pdas: ( ) si ns i max si i     (11) for weighted sum of ndas: 1 ( ) si ns i max si i      (12) a multi-criteria framework for comparing dividend pay capabilities: evidence from indian … 155 step 6. computation of the appraisal score of the alternatives the appraisal score of the alternative is computed as 1 ( ) 2 s ns ns ai i i     (13) here, 0 1s ai   step 7. ranking of the alternatives the alternatives are ranked as per their appraisal scores. higher is the score, more preferred is the corresponding alternative. 3.6. aggregation of the mcdm results in many real-life mcdm applications arriving at a consensus decision is a critical issue (biswas, 2020a). the problem arises when a group of opinion makers or a set of different mcdm algorithms are involved in selection of a best possible alternative. to aggregate the outcomes of different decision making frameworks, the researchers have developed a number of algorithms. in this section, we discuss some of the approaches. 3.6.1. borda count (bc) bc is an age old established preference based aggregation method (borda, 1784) that has been applied for consolidation of the ranking results of various mcdm algorithms (lansdowne and woodward, 1996; wu, 2011; pourjavad and shirouyehzad, 2011; gandhi et al., 2018; barak and mokfi, 2019; ecer, 2021). in what follows are the steps for this aggregation method. step 1. the ranking of the alternatives (subject to the influence of the criteria) is made by each opinion maker or method. step 2. suppose, there are alternative options. each alternative is given a point equal to the number of options succeeding the considered one. hence, the most preferred or best alternative receives points while the second best alternative gets points and so on. step 3. calculation of the sum of the points obtained by each alternative option step 4. ranking of the alternatives based on the total points. the alternative which obtains the highest points would be ranked first and so on. 3.6.2. copeland method (cm) the cm is the extended and modified form of bc. the cm starts after the bc. this method puts emphasis on the number of other alternative options subordinated to the given alternative (lestari et al., 2018; dortaj et al., 2020; ecer, 2021). the procedural steps are as follows. step 1. computation of the win score for each alternative (vis-à-vis the others) step 2. computation of the loss score (after subtracting of the score obtained in the first stage from majority wins’ score) step 3. calculation of the final score which is the difference between the win and loss scores. the alternative that obtains the highest overall score will be ranked first and so on. biswas et al./decis. mak. appl. manag. eng. 5 (2) (2022) 140-175 156 3.6.3. grade average method (gam) this is a simple method of aggregation of the ranks by various models. according to this method, the alternatives are ranked using the different methods. then, for each alternative, an average of ranks or grades (obtained by using various models) is calculated. the alternative that scores least grade average, overall is the first preferred one (dortaj et al., 2020). in the present paper, for calculation purpose, we have used ms office (2016) and spss (version 25) software tools on a computer with intel(r) core(tm) i3-1005g1 cpu @ 1.20ghz 1.19 ghz, 8gb ram. 4. results in this section, we briefly highlight the findings step by step. first, we carry out the calculations for year wise criteria weights using the procedural steps of the lopcow method (see expressions (1) to (4), section 3.4). table 3 provides the normalized decision matrix while table 4 exhibits the calculations of the criteria weights using lopcow for fy 2013-14. the calculations of the criteria weights (using lopcow method) for other fys are given in appendix b. table 3. normalized decision matrix (for lopcow method) for fy 2013-14 company criteria c1 c2 c3 c4 c5 c6 c7 c8 a1 0.4344 0.0577 0.1501 0.2905 1.0000 0.2794 0.2799 0.9185 a2 0.0000 0.1219 0.5079 0.2556 0.4407 0.2912 0.2874 0.9878 a3 0.1216 0.0990 0.0193 0.0045 0.4562 0.2843 0.2774 0.3190 a4 0.3245 0.3660 0.1313 0.3296 0.4598 0.2892 0.3377 0.9632 a5 0.4075 0.1086 0.2871 0.1359 0.3695 0.2735 0.2836 0.9028 a6 0.3211 0.4183 0.2979 0.7412 0.4771 0.2892 0.3218 0.9554 a7 0.0851 0.5086 0.3227 0.3064 0.4549 0.2892 0.3484 0.9019 a8 0.3979 0.5389 0.0000 0.0000 0.1891 1.0000 0.2893 0.2103 a9 0.0302 0.3151 0.4217 0.2989 0.3670 0.2931 0.3136 0.9695 a10 0.4530 0.2476 0.1680 0.0293 0.4402 0.2902 0.2639 0.0000 a11 0.0000 0.2481 0.0613 0.0555 0.4896 0.2902 0.2842 0.6220 a12 0.0528 0.3728 0.1069 0.1140 0.4908 0.2912 0.3041 0.8387 a13 0.1564 0.5793 0.3124 0.1506 0.4711 0.2873 0.3523 0.7436 a14 0.0003 0.1994 0.0757 0.1078 0.4892 0.2735 0.2950 0.8825 a15 0.1037 0.7721 0.2792 1.0000 0.4151 0.2902 0.6624 0.9976 a16 1.0000 1.0000 0.4348 0.3062 0.4525 0.2912 1.0000 0.9688 a17 0.1100 0.3675 0.1883 0.0664 0.5628 0.2775 0.2895 0.6293 a18 0.2188 0.4402 0.2255 0.1121 0.6764 0.2353 0.2521 0.7968 a19 0.2048 0.5085 0.3594 0.1907 0.4128 0.3078 0.3009 0.8562 a20 0.1638 0.6208 0.2833 0.2935 0.4254 0.2873 0.4567 0.9738 a21 0.0583 0.3215 0.3401 0.2890 0.5574 0.2892 0.3091 0.9539 a22 0.4621 0.3827 0.0504 0.0332 0.5637 0.2569 0.2884 0.6911 a23 0.5116 0.5671 0.1995 0.0704 0.4926 0.3039 0.2895 0.7106 a24 0.0024 0.5193 0.0752 0.0653 0.4458 0.2824 0.3044 0.7568 a25 0.0329 0.0609 1.0000 0.2661 0.2452 0.2961 0.2833 0.8777 a26 0.2940 0.7343 0.0265 0.0365 0.0000 0.0000 0.0000 0.1537 a27 0.0000 0.0000 0.5091 0.4015 0.7822 0.2931 0.2827 1.0000 a28 0.2937 0.6119 0.1668 0.2441 0.4027 0.2873 0.2166 0.8155 a29 0.5981 0.5341 0.0802 0.0842 0.2180 0.2961 0.3089 0.3508 a30 0.0000 0.3300 0.0938 0.1544 0.3287 0.2980 0.2954 0.7640 a multi-criteria framework for comparing dividend pay capabilities: evidence from indian … 157 table 4. criteria weights (fy 2013-14) – lopcow method c1 c2 c3 c4 c5 c6 c7 c8 mean square 0.1042 0.2119 0.0976 0.0904 0.2351 0.1100 0.1308 0.6358 sd 0.2325 0.2344 0.2044 0.2143 0.1771 0.1427 0.1593 0.2742 pv 32.8284 67.4634 42.4134 33.8469 100.7452 84.3134 81.9621 106.7331 wj 0.0597 0.1226 0.0771 0.0615 0.1831 0.1532 0.1489 0.1940 table 5 provides the summary of the criteria weights for all years. the orders of the criteria (as per their relative importance) for different years are given in table 7. it is noticed that leverage (i.e, risk) obtains the higher priority while liquidity in most of the cases holds the less weight. we now use these weights to compare and rank the companies under study using edas method. table 5. year wise criteria weights – summary fy criteria weights c1 c2 c3 c4 c5 c6 c7 c8 2013-14 0.0597 0.1226 0.0771 0.0615 0.1831 0.1532 0.1489 0.1940 2014-15 0.0986 0.1891 0.2256 0.1537 0.1479 0.1272 0.0394 0.0186 2015-16 0.0596 0.1148 0.1515 0.0984 0.1533 0.1779 0.0165 0.2279 2016-17 0.0749 0.1216 0.1643 0.1411 0.1908 0.0074 0.0241 0.2759 2017-18 0.0601 0.1060 0.1818 0.1124 0.1553 0.0019 0.0990 0.2835 2018-19 0.0564 0.0952 0.1225 0.0765 0.2130 0.2299 0.0199 0.1867 2019-20 0.0751 0.1072 0.1788 0.0840 0.0919 0.2119 0.0157 0.2355 2020-21 0.0459 0.0624 0.1604 0.0734 0.1653 0.1774 0.1359 0.1792 table 7 exhibits the average solution for the criteria for the fy 2013-14 (using the expression (6)). table 8 provides the calculation of the appraisal scores of the alternatives (using the expressions (7) to (13)) for the fy 2013-14. the ranking order of the alternatives are also included in table 8. in the similar way we find the ranking order of the alternatives for all other financial years (see appendix c). table 6. year wise criteria weights – priority order fy priority order 2013-14 2014-15 2015-16 2016-17 2017-18 2018-19 2019-20 biswas et al./decis. mak. appl. manag. eng. 5 (2) (2022) 140-175 158 table 7. average solution (fy 2013-14) criteria c1 c2 c3 c4 c5 c6 c7 c8 avg. sol. 42.0386 9.9983 9.8703 25.081 0.1114 1.1347 4970.837 6.488 table 8. ranking of alternatives (fy 2013-14) company s+ sns+ nssi rank a1 0.8600 0.2105 0.3817 0.9113 0.6465 3 a2 0.2170 0.1798 0.0963 0.9242 0.5103 12 a3 0.0049 0.5360 0.0022 0.7740 0.3881 27 a4 0.1920 0.0527 0.0852 0.9778 0.5315 10 a5 0.1101 0.3097 0.0489 0.8695 0.4592 22 a6 0.3161 0.0274 0.1403 0.9885 0.5644 6 a7 0.2028 0.0306 0.0900 0.9871 0.5385 8 a8 0.9898 0.8649 0.4393 0.6354 0.5374 9 a9 0.1923 0.1861 0.0853 0.9215 0.5034 13 a10 0.0239 0.6651 0.0106 0.7196 0.3651 29 a11 0.0496 0.3346 0.0220 0.8590 0.4405 24 a12 0.0948 0.1677 0.0421 0.9293 0.4857 16 a13 0.1351 0.0475 0.0600 0.9800 0.5200 11 a14 0.1144 0.2460 0.0508 0.8963 0.4735 20 a15 1.3503 0.0775 0.5994 0.9673 0.7834 2 a16 2.2528 0.0128 1.0000 0.9946 0.9973 1 a17 0.1476 0.2699 0.0655 0.8862 0.4759 19 a18 0.3251 0.3370 0.1443 0.8580 0.5011 14 a19 0.1083 0.1346 0.0481 0.9432 0.4957 15 a20 0.5365 0.0613 0.2382 0.9742 0.6062 5 a21 0.2954 0.0865 0.1311 0.9635 0.5473 7 a22 0.1737 0.3120 0.0771 0.8685 0.4728 21 a23 0.0985 0.1788 0.0437 0.9246 0.4842 17 a24 0.0103 0.2165 0.0046 0.9087 0.4567 23 a25 0.3267 0.4470 0.1450 0.8116 0.4783 18 a26 0.0268 2.3722 0.0119 0.0000 0.0059 30 a27 0.7069 0.1821 0.3138 0.9232 0.6185 4 a28 0.0602 0.4227 0.0267 0.8218 0.4243 26 a29 0.0473 0.6561 0.0210 0.7234 0.3722 28 a30 0.0067 0.3502 0.0030 0.8524 0.4277 25 table 9 provides the summary of the year wise rankings which reflect that there have been some considerable irregularities in the ranking orders of the alternatives. a multi-criteria framework for comparing dividend pay capabilities: evidence from indian … 159 table 9. summary of year wise ranking of the companies company rank 2013-14 2014-15 2015-16 2016-17 2017-18 2018-19 2019-20 a1 3 8 2 7 2 26 9 a2 12 7 11 6 11 10 8 a3 27 29 30 30 30 25 29 a4 10 14 7 9 14 15 13 a5 22 17 21 18 12 27 10 a6 6 4 10 10 19 19 11 a7 8 11 17 19 18 12 14 a8 9 1 9 27 28 29 28 a9 13 5 12 14 21 22 21 a10 29 30 27 29 5 13 1 a11 24 22 23 17 23 16 25 a12 16 25 26 12 25 30 7 a13 11 15 19 20 17 9 16 a14 20 26 5 11 26 20 24 a15 2 3 4 4 3 5 5 a16 1 2 3 2 1 6 3 a17 19 19 20 21 27 23 27 a18 14 20 16 25 22 3 15 a19 15 9 13 16 9 11 18 a20 5 13 29 5 10 8 6 a21 7 12 14 8 15 2 12 a22 21 27 1 23 6 1 19 a23 17 21 22 26 20 24 4 a24 23 24 6 24 8 17 26 a25 18 6 8 3 4 14 30 a26 30 28 25 28 29 4 20 a27 4 10 18 1 13 28 2 a28 26 16 28 15 7 7 22 a29 28 23 24 22 24 18 23 a30 25 18 15 13 16 21 17 we notice that the alternatives do not hold consistent positions over the years of the study period. to set the overall preferential order, it is necessary to arrive at a consensus. to meet this objective, we apply the widely used aggregation of voting technique such as bc as described in section 3.6.1. we also use another popular method like cm to carry out the aggregation to validate the result of bc. further, we formulate a new decision matrix using the appraisal scores of the alternatives each year. in this newly formed decision matrix all the years are assigned same weights (i.e., equal importance). table 10 presents the decision matrix used for obtaining the overall ranking of the alternatives. we apply simple additive weighting (saw) method for deriving the overall ranks as followed in many past research (for instance, biswas, 2020b, pramanik et al., 2021). biswas et al./decis. mak. appl. manag. eng. 5 (2) (2022) 140-175 160 table 10. decision matrix for overall ranking of the alternatives weight 0.1429 0.1429 0.1429 0.1429 0.1429 0.1429 0.1429 models score values 2013-14 2014-15 2015-16 2016-17 2017-18 2018-19 2019-20 a1 0.6465 0.5059 0.7904 0.6126 0.7993 0.5276 0.6147 a2 0.5103 0.5195 0.6576 0.6204 0.6235 0.7946 0.6173 a3 0.3881 0.0466 0.0000 0.1170 0.0000 0.5295 0.1992 a4 0.5315 0.4417 0.7217 0.6001 0.6042 0.7130 0.5810 a5 0.4592 0.3752 0.5353 0.5023 0.6065 0.5057 0.6134 a6 0.5644 0.5633 0.6684 0.5807 0.5506 0.6584 0.6098 a7 0.5385 0.4802 0.5905 0.4953 0.5676 0.7643 0.5799 a8 0.5374 0.8558 0.6924 0.2756 0.3544 0.3277 0.2807 a9 0.5034 0.5423 0.6307 0.5360 0.5237 0.5667 0.5116 a10 0.3651 -0.0342 0.4138 0.1395 0.7495 0.7582 0.8963 a11 0.4405 0.3029 0.4669 0.5188 0.5183 0.6890 0.4955 a12 0.4857 0.2362 0.4211 0.5457 0.4995 0.0642 0.6229 a13 0.5200 0.4384 0.5813 0.4922 0.5681 0.8194 0.5449 a14 0.4735 0.2026 0.7413 0.5656 0.4755 0.5892 0.5021 a15 0.7834 0.6031 0.7545 0.6856 0.7962 0.8985 0.7209 a16 0.9973 0.6479 0.7768 0.7339 0.8843 0.8878 0.7911 a17 0.4759 0.3611 0.5734 0.4655 0.4666 0.5407 0.4430 a18 0.5011 0.3575 0.5928 0.3601 0.5196 0.9232 0.5477 a19 0.4957 0.4949 0.6180 0.5234 0.6356 0.7791 0.5423 a20 0.6062 0.4680 0.3949 0.6206 0.6275 0.8371 0.6969 a21 0.5473 0.4698 0.5965 0.6028 0.6001 0.9278 0.5950 a22 0.4728 0.1913 0.9223 0.3799 0.6752 0.9285 0.5319 a23 0.4842 0.3254 0.5146 0.3512 0.5290 0.5406 0.7687 a24 0.4567 0.2692 0.7303 0.3696 0.6473 0.6781 0.4616 a25 0.4783 0.5383 0.7043 0.6883 0.7915 0.7196 0.0006 a26 0.0059 0.1620 0.4382 0.1747 0.2546 0.9225 0.5190 a27 0.6185 0.4912 0.5825 0.9695 0.6060 0.4094 0.8245 a28 0.4243 0.3918 0.4030 0.5352 0.6501 0.8469 0.5095 a29 0.3722 0.3025 0.4642 0.4323 0.5067 0.6764 0.5030 a30 0.4277 0.3700 0.5934 0.5360 0.5913 0.5780 0.5428 table 11 exhibits the overall ranking of the alternatives using the bc method. we proceed to calculate the win score and loss score for each alternatives using the findings presented in table 9 and the steps described in section 3.6.2 (cm) to derive the copeland score and accordingly, rank the alternatives. a multi-criteria framework for comparing dividend pay capabilities: evidence from indian … 161 table 11. overall rank (using borda count) company borda count final rank_ borda company borda count final rank_ borda a1 153 3 a16 192 1 a2 145 4 a17 54 27 a3 10 30 a18 95 16 a4 128 9 a19 119 11 a5 83 19 a20 134 6 a6 131 8 a21 140 5 a7 111 13 a22 112 12 a8 79 21 a23 76 24 a9 102 15 a24 82 20 a10 76 23 a25 127 10 a11 60 26 a26 46 29 a12 69 25 a27 134 7 a13 103 14 a28 89 17 a14 78 22 a29 48 28 a15 184 2 a30 85 18 table 12 provides the findings of the cm. we apply the regular procedural steps of the saw method (simanaviciene and ustinovichius, 2010) and obtain the overall (after aggregating) ranks of the alternatives (refer table 13). table 12. overall rank (using copeland approach) company wins losses final score final rank company wins losses final score final rank a1 153 2892 -2739 3 a16 192 2853 -2661 1 a2 145 2900 -2755 4 a17 54 2991 -2937 27 a3 10 3035 -3025 30 a18 95 2950 -2855 16 a4 128 2917 -2789 9 a19 119 2926 -2807 11 a5 83 2962 -2879 19 a20 134 2911 -2777 6 a6 131 2914 -2783 8 a21 140 2905 -2765 5 a7 111 2934 -2823 13 a22 112 2933 -2821 12 a8 79 2966 -2887 21 a23 76 2969 -2893 24 a9 102 2943 -2841 15 a24 82 2963 -2881 20 a10 76 2969 -2893 23 a25 127 2918 -2791 10 a11 60 2985 -2925 26 a26 46 2999 -2953 29 a12 69 2976 -2907 25 a27 134 2911 -2777 7 a13 103 2942 -2839 14 a28 89 2956 -2867 17 a14 78 2967 -2889 22 a29 48 2997 -2949 28 a15 184 2861 -2677 2 a30 85 2960 -2875 18 table 13. overall rank (using saw method) company final rank_ saw company final rank_ saw a1 4 a16 1 a2 5 a17 24 a3 30 a18 16 a4 8 a19 11 a5 20 a20 7 a6 9 a21 6 a7 12 a22 10 a8 25 a23 22 biswas et al./decis. mak. appl. manag. eng. 5 (2) (2022) 140-175 162 company final rank_ saw company final rank_ saw a9 15 a24 19 a10 26 a25 14 a11 23 a26 29 a12 28 a27 3 a13 13 a28 17 a14 21 a29 27 a15 2 a30 18 figure 2 pictorially represents the comparison of the overall ranking of the alternatives using bc, cm and saw methods which reflects a consensus. we also calculate the correlations among the overall ranking by using bc method and others (see table 14) which indicates the consistency of bc method with others. table 14. correlation test among the rankings by bc, cm and saw methods final_rank_copeland final_rank_saw final_rank_ borda spearman's rho 1.000** .977** sig. (2-tailed) 0.000 0.000 ** correlation is significant at the 0.01 level (2-tailed). figure 2. comparison of overall ranks by bc, cm and saw methods we further test the consistency of the year wise ranking of the alternatives (obtained by using edas method) and the overall ranking (obtained by using bc method) as given in table 15. we note that the correlation is statistically significant. further, it is evident that fy 2013-14, fy 2016-17 and fy 2017-18 show higher consistency, fy 2015-16 and fy 2019-20 are moderately consistent and fy 2018-19 exhibits low consistency with the final ranking. a multi-criteria framework for comparing dividend pay capabilities: evidence from indian … 163 table 15. correlation test among the year wise rankings and the overall ranking rank_ 13_14 rank_ 14_15 rank_ 15_16 rank_ 16_17 rank_ 17_18 rank_ 18_19 rank_ 19_20 overall rank spearman's rho .816** .744** .588** .780** .720** .370* .507** sig. (2-tailed) 0.000 0.000 0.001 0.000 0.000 0.044 0.004 ** correlation is significant at the 0.01 level (2-tailed). * correlation is significant at the 0.05 level (2-tailed). mcdm methods are dependent on the given conditions such as selection of alternative and criteria sets, effects of the criteria on the alternatives, criteria weights, computational steps of the algorithms and so on. therefore, it is imperative to examine whether the result obtained by using a specific mcdm method is reliable or not (biswas et al., 2019; gupta et al., 2019; gupta et al., 2022; biswas et al., 2022a). the extant literature shows several instances (e.g., biswas and pamucar, 2021; biswas et al., 2021; biswas et al., 2022b; biswas and anand, 2020) wherein the authors use a group of widely used methods to compare with the selected framework for the given problem. in our paper, we rank the alternatives using two other popular and extensively used mcdm models such as multi-attributive border approximation area comparison (mabac) (pamučar and ćirović, 2015) and the complex proportional assessment (copras) method (zavadskas et al., 1994) for all years. next, we use the bc method to derive the final ranks for both mabac and copras method. then, we examine the correlations among the rankings (year wise and overall) provided by our framework (using edas), mabac and copras (see tables 16-18). the tables 16-18 suggest that our edas based ranking is comparable and in sync with the other methods. hence, there is a reason to consider our result as a reliable one. table 16. rank correlation test (year wise) between edas and copras edas_ 13_14 edas_ 14_15 edas_ 15_16 edas_ 16_17 edas_ 17_18 edas_ 18_19 edas_ 19_20 copras_13_14 spearman's rho .993** sig. (2-tailed) 0.000 copras_14_15 spearman's rho .892** sig. (2-tailed) 0.000 copras_15_16 spearman's rho .928** sig. (2-tailed) 0.000 copras_16_17 spearman's rho .957** sig. (2-tailed) 0.000 copras_17_18 spearman's rho .964** sig. (2-tailed) 0.000 copras_18_19 spearman's rho .960** sig. (2-tailed) 0.000 copras_19_20 spearman's rho .821** sig. (2-tailed) 0.000 ** correlation is significant at the 0.01 level (2-tailed). biswas et al./decis. mak. appl. manag. eng. 5 (2) (2022) 140-175 164 table 17. rank correlation test (year wise) between edas and mabac edas_ 13_14 edas_ 14_15 edas_ 15_16 edas_ 16_17 edas_ 17_18 edas_ 18_19 edas_ 19_20 mabac_13_14 spearman's rho .905** sig. (2-tailed) 0.000 mabac_14_15 spearman's rho .683** sig. (2-tailed) 0.000 mabac_15_16 spearman's rho .453* sig. (2-tailed) 0.012 mabac_16_17 spearman's rho .916** sig. (2-tailed) 0.000 mabac_17_18 spearman's rho .731** sig. (2-tailed) 0.000 mabac_18_19 spearman's rho .663** sig. (2-tailed) 0.000 mabac_19_20 spearman's rho .749** sig. (2-tailed) 0.000 ** correlation is significant at the 0.01 level (2-tailed). * correlation is significant at the 0.05 level (2-tailed). table 18. rank correlations among the final results by edas, mabac and copras mabac_final copras_final edas_final spearman's rho .782** .972** sig. (2-tailed) 0.000 0.000 ** correlation is significant at the 0.01 level (2-tailed). we further conduct a non-parametric statistical test such as kruskal wallis test (kwt) to examine whether the distribution functions of edas, copras and mabac are significantly different. we find the value of the asymp. sig. as 1.00 which strongly supports the null hypothesis that the distribution functions of all methods are equal. hence, the result obtained by edas method is further validated statistically. 5. discussion the current study reveals some interesting observations. firstly, we see that when multiple criteria (ownership, size, profitability, growth, liquidity and risk) are considered for comparing the companies (i.e., fmcg and cd), the rankings are not consistent over the years. we observe that as per overall aggregated ranking, top two organizations such as a16 (itc limited) and a15 (hindustan unilever ltd) hold their positions more or less consistent. the same behavior is noticed for the bottom three organizations such as a3 (bombay burmah trdg. corpn. ltd), a26 (rajesh exports ltd.) and a29 (voltas ltd). looking at the nature of these organizations, we find that market capitalization (see table d1 in appendix d) for the top two capable organizations are higher than others. in addition, both a16 and a15 are having multiproduct portfolio with strong global presence. further, it is seen that fmcg organizations are in the top bracket as far as dpc is concerned. the companies in the bottom bracket are mostly cd firms. the result is an indication that organizations producing luxury goods may tend to be less capable in paying the dividends. further, a multi-criteria framework for comparing dividend pay capabilities: evidence from indian … 165 we figure out that fy 2018-19 shows considerably higher variations in the usual rankings of the companies. it may be an indirect effect of the declaration of the gst bill in india and demonetization initiative by the government of india. the present study has some significant implications. from the theoretical point of view, the current study is a distinguished work for comparing the firms on the basis of their dpc. so far, studies have been made to explore the effect of dividend payment on the firms’ performance and their values and to enfold the determinants of the dividend policy. but, the question arises, are the firms capable enough to pay dividends? therefore, this paper provides a holistic multi-perspective analysis framework to gauge the capabilities of the firms beforehand. we have noticed that dpc varies significantly over the years. hence, although the firms may realize the importance of paying dividend as a positive signal to the investors, they may not be equally capable of the providing the same over the years. hence, there is a need of striking a balance between the principal’s interest and manager’s decisions. further, the present paper has its importance from the perspectives of behavioral finance also. despite the tax disadvantage of dividends (typically dividend income is taxed more than capital gains) and issuance cost associated with new equity, firms pay dividends and investors generally regard such dividend payments positively. information signaling, clientele effect, agency costs are some important reasons. in addition, investors have preference for dividends due to behavioral reasons. lack of self-control and aversion for regret could be important reasons. consequently, dividends and capital receipts are not perfectly substitutable. the experts of the behavioral finance field (e.g., kahneman and tversky, 1979; thaler and shefrin, 1981; shefrin and statman, 1984) regarded the internal conflict as one of the major reasons behind such mismatch. the individual wishes to deny himself a present indulgence, yet simultaneously finds that he yields to the temptation. in the area of personal finance, individuals would like to protect their principal from their wasteful spending tendencies. a simple way to do this is to limit their spending to the dividend income so that the capital amount is preserved. such behavioral nature explains a preference for dividend by those who otherwise have difficulty in exercising self-control. the individuals who set aside funds for their children's college education at one interest rate, yet borrow to finance their consumer goods at a higher interest rate, are not acting as standard utility maximizers. yet the underlying rationale seems quite straightforward. similarly, it implies that an individual may be better off by allowing current consumption to be determined by the dividend payout from his stock portfolio. in other words, this individual may wish to follow a rule stipulating that portfolio capital is not to be consumed, only dividends. empirical evidence suggests that most investors feel more regret when they sell their stock to generate income compared to using the dividend income. regret shall be more, if stock prices rise subsequently. therefore, despite all arguments and counter discussions related to dividend policy, dpo holds its importance in attracting the investors over the years. hence, this paper puts forth a notable extension to the growing strand of work that renders a new direction to the individual investors and policy makers. 6. conclusion the present study has been designed to add a new dimension to the ongoing strand of literature on dividend policy and dpo. the current work has provided a multi-period, multi-criteria based framework to compare the dpc of 25 fmcg and 5 cd organizations (listed in bse, india) for the period fy 2013-14 to fy 2019-20. for comparison purpose, we have considered six aspects (grounded on the extant biswas et al./decis. mak. appl. manag. eng. 5 (2) (2022) 140-175 166 theories on dividend policy) such as ownership, size, profitability, growth, liquidity and risk. we have used a new integrated lopcow-edas framework for our analysis. the result shows that companies do not show consistent performance over the years. however, the aggregate overall performance is in sync with the market capitalization for most of the organizations. we further have noticed that fmcg organizations show comparatively better capabilities that cd firms vis-à-vis dividend payment. for aggregation (of the ranks for different years) we have used widely used techniques such as bc, cm in addition to saw. the aggregated overall ranking shows consistency with the same obtained for individual years. as per the aggregated ranking the companies like a16: itc limited; a15: hindustan unilever ltd. a1: avanti feeds ltd. a2: bajaj consumer care ltd. a21: procter & gamble hygiene & health care ltd. hold the top positions while a3: bombay burmah trdg. corpn. ltd. a26: rajesh exports ltd. a29: voltas ltd. a17: jyothy labs ltd. a11: gillette india ltd. fall into the lower bracket. however, the present work posits a number of further scope of research. firstly, in this paper we have not considered subjective opinions of the investors in deriving the criteria weights. the criteria weights, though have been found by using objective information, but the susceptible to abrupt variations in the performance values of the alternatives. one general drawback of the opinion-based decision making is subjective bias. hence, one future study may also take opinions of some seasoned investors and experts to derive the criteria weights which shall be aggregated with the weights found by using objective values. then the same weights may be used to compare the companies for different years during the study period. secondly, in our study we observe considerable variations in ranking for different years. one future work may attempt to gauge the impact of macroeconomic events during each fy and shall draw a causal association with the variations in the ranking. thirdly, it shall be an interesting future work to examine whether dpc has any positive association with the stock market performance of the organizations under study here. further, the present study may be extended to test whether sales and operational performance, innovativeness, financial stability and economic sustainability have any positive influence on dpc or not. fourthly, in this paper we have not examined the impact of covid-19 on dpc. a near future research may be designed in this regard by considering the fy 2020-21 and fy 2021-22. fifthly, it may also be a notable work if an investigation may be made to find out the association of dpc and dpr and dividend yield. sixthly, the current work focuses on fmcg and cd sectors. the same framework as used in this paper may be modified/extended for assessing the comparative dpc of the constituent firms belong to other sectors. seventh, from the technical point of view, lopcow is very recently introduced. the method may be tested in other complex scenarios, especially under uncertainty wherein future research shall extend the model to work with imprecise information. eighth, a future work may be done to compare a group of companies at the different phases of the business cycle and are managed differently (e.g., by professionals, promoter dominated, multinational governance etc.) with our model to obtain their dpc and subsequently relate to their dpo. ninth, there is a possibility to examine the performance of the companies from behavioural perspectives vis-à-vis dpc. tenth, there may be other measures of risk, for instance, cost of the capital employed, degree of operating leverage among others that may also be considered in further analysis. the edas method has some limitations. edas method is more appropriate for risk neutral situations as it considers the average solution point as a benchmark. the average solution may not always portray the true picture in all real-life scenarios. a multi-criteria framework for comparing dividend pay capabilities: evidence from indian … 167 further, in some occasions, it is seen that the nda or pda values for some alternatives equal to zero. in those cases, the weighted sum values become undefined. nevertheless, the above-mentioned scopes, we hope, will not lessens the value and potential of the present work. we believe that the current work shall contribute new dimensions and perspectives for the policy makers in business organizations and government and help the investors in investment decision making. author contributions: conceptualization: sb, gb, dp, jnm; methodology: sb, gb, dp; software: sb; validation: sb, gb, dp; formal analysis: sb.; investigation: sb, gb; data curation: sb; writing—original draft preparation: sb; writing—review and editing: jnm.; supervision: gb. all authors have read and agreed to the published version of the manuscript funding: this research received no external funding. data availability statement: necessary data is provided. acknowledgments: the authors express their sincere thanks to all anonymous referees whose valuable comments helped to improve the quality of this paper. conflicts of interest: the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. references affandi, f., sunarko, b., & yunanto, a. 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(2021). international market selection: a maba based edas analysis framework. oeconomia copernicana, 12(1), 99-124. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). decision making: applications in management and engineering vol. 2, issue 2, 2019, pp. 138-158. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame2003079b * corresponding author. e-mail addresses: sanjibb@acm.org (s. biswas), gautam.bandyopadhyay@dms.nitdgp.ac.in (g. bandyopadhyay), banhi.guha@gmail.com (b. guha), malay.bhattacharjee100@gmail.com (m. bhattacharjee), an ensemble approach for portfolio selection in a multi-criteria decision making framework sanjib biswas 1*, gautam bandyopadhyay 2, banhi guha 3 and malay bhattacharjee 2* 1 calcutta business school, diamond harbour road, bishnupur, west bengal, india 2 department of management studies, national institute of technology, durgapur, west bengal, india 3 amity university, kolkata, west bengal, india received: 2 february 2019; accepted: 10 august 2019; available online: 23 august 2019. original scientific paper abstract: investment in mutual funds (mf) has generated increasing interest among the investors over last few decades as it provides an opportunity for flexible and transparent choice of funds to diversify risk while having return potential. mf are essentially a portfolio wherein investors’ funds are invested in the securities traded in the capital market while sharing a common objective. however, selection and management of different asset classes pertaining to a particular mf are done by an active fund manager under regulatory supervision. hence, for an individual investor, it is important to assess the performances of the mf before investment. performances of mf depend on several criteria based on risk-return measures. hence, selection of mf is subject to satisfying multiple criteria. in this paper, we have adopted an ensemble approach based on a two-stage framework. our sample consists of the open ended equity large cap funds (direct plan) in india. in the first stage, the efficiencies of the funds are analyzed using dea for primary selection of the funds. in order to rank the funds based on risk and return parameters for investment portfolio formulation, we have used mabac approach in the second stage wherein criteria weights have been calculated using the entropy method. keywords: mutual fund, portfolio selection, multi-criteria decision making, data envelopment analysis (dea), entropy, multi-attribute border approximation area comparisons (mabac). mailto:sanjibb@acm.org mailto:gautam.bandyopadhyay@dms.nitdgp.ac.in mailto:banhi.guha@gmail.com mailto:malay.bhattacharjee100@gmail.com an ensemble approach for portfolio selection in a multi-criteria decision making framework 139 1. introduction the aftermath of the economic liberalization, indian capital market (icm) has witnessed significant changes in investment pattern. with the development of information and communication technology (ict), there is no dearth of information regarding available investment opportunities. moreover, with increasing efforts from the govt. of india (goi), mf have emerged as a preferred choice for common investors because of many reasons. first, it provides reportedly high return as compared to other popular investment options like fixed deposits (fd), national savings certificates (nsc), public provident funds (ppf), and other postal savings. alongside, with increasing awareness and available information, risk can also be brought down to an affordable level by carefully evaluating the performances of the mf for the prudent selection of funds to invest. the concept of mf resembles the selection of portfolio wherein the active fund managers invest the total amount invested in other asset classes such as stocks. the objective is to generate sizeable return out of the investment made in the portfolio while minimizing the risk through diversification i.e. by appropriate selection of the securities and allocating optimum weights dynamically (gupta et al., 2018). mf in india have a long stint with icm since the inception of unit trust of india (uti) in 1963. recent reports(amfi, india, 2018)indicate a significant growth in the asset base of the indian mf industry (imi) from inr 5.05 trillion (31st march, 2008) to inr 22.20 trillion (february, 2018)despite the event of bankruptcy of the renowned us bank lehman brothers in september, 2008. therefore, a large number of investors have been attracted by the promising nature of imi. however, in order to ensure the possibility of considerable return at an affordable risk, one has to select the funds apt to his/her risk appetite and financial goal. with this preamble, in this study, we have focused on open ended equity mf(direct plans) in india belonging to the large cap segment. the equity mf segment accounts for around 50.7% share of the total asset base of the industry in february, 2018 as reported by amfi. unlike the close ended funds (cf), open ended funds (of) allow the investors to buy and sell the units of the funds on a continuous basis, which means the new investors are allowed to enter at convenience and so as the existing investors can exit whenever needed. moreover, for cf the unit capital is fixed and also there is a limit on sales. in other words, of allow greater flexibility for the investors than cf. large cap funds show relatively stable movements in the return as compared to the mid-cap and small cap funds. we have considered direct plans only since, unlike regular plans, they impose less pressure on expense ratio as no intermediate commission is involved. in essence, we like to perceive the performance of mfs from a common investors’ point of view without imposing significant burden on net assets value (nav). in our approach, we have used non-parametric methods (npm). literatures manifest comparatively less evidence of such methods than their traditional parametric counterparts (babalos et al., 2011). selection of mf needs to satisfy the objectives pertaining to several criteria based on risk and return and time horizon, etc. therefore, among npm, dea has been a popular method, though, moderate evidences of the use of mcdm techniques have been found in the state-of-the art. the reason lies in the fundamental use of dea to assess and differentiate between the efficient and nonefficient decision making units (dmu). mcdm techniques allow to rank the dmu based on a number of criteria. hence, for identifying efficient dmu or mf while ranking them based on performance parameters, we propose a two-stage assessment framework. the rest of the paper proceeds as follows. section 2 highlights some of the related work while in the section 3 we discuss the research methodology. section 4 biwas et al/decis. mak. appl. manag. eng. 2 (2) (2019) 138-158 140 summarizes the results and put forward necessary discussions on the same. finally, section 5 concludes the paper and posits some future scope of research. 2. related work a plethora of research has been conducted on mf for understanding the nature and setting performance measurement framework with an objective to inflate the expected utility while reducing the risk level. in one of the seminal works in the stated field, markowitz postulated the mean-variance model related to efficiently diversified portfolio (markowitz, 1952). in the following works, the researchers (tobin, 1958; markowitz, 1959; sharpe, 1966; jensen, 1968; treynor, 1965) further explained and extended the framework by introducing new risk measures such as semi-variance and risk adjusted performance metrics such as reward to volatility ratio, alpha and reward to variability ratio based on the capital assets pricing model (capm). the objective was to assess portfolio performance with respect to the benchmark with an objective to minimize the systematic risk which is represented by beta. the authors exhibited that unsystematic risk can be reduced through diversification of the portfolio. based on these measures, several researchers and practitioners worked on evaluating the performances of the mf (kacperczyk et al., 2005; pedersen &rudholm­alfvin, 2003; eling & schuhmacher, 2007; plantinga & de groot, 2002; redman et al., 2000). in the indian context also, across different periods, many researchers (barua&verma, 1991; jaydev, 1996; gupta, 2000; sehgal&jhanwar, 2008; tripathy, 2004; anand & murugaiah, 2006; anitha et al., 2011; arora, 2015; kundu, 2009) worked on selection of funds based the criteria like sharpe ratio, jensen ratio, and sortino ratio, alpha, beta, nav, timing to market, and selectivity skill following traditional statistical approaches. over the years apart from the traditional parametric approaches, applied operations research techniques have also been adopted by the researchers. the authors (pendaraki et al., 2004; sharma & sharma, 2006) have applied goal programming to evaluate the performances of mf for formulating the portfolio. dea has been a widely accepted method by the researchers and practitioners (murthi& choi, 2001; murthi et al., 1997; anderson et al., 2004; sengupta, 2003; daraio & simar, 2006; babalos et al., 2015; mcmullen & strong, 1998; wilkens & zhu, 2001; tarim& karan, 2001; galagedera & silvapulle, 2002; chang, 2004; carlos matallín et al., 2014; nguyen-thi-thanh, 2006; chu et al., 2010; tsolas, 2011; morey & morey, 1999; basso & funari, 2001; briec et al., 2004; zhao et al., 2011; jaro & na, 2006; kooli et al., 2005; haslem & scheraga, 2003)among the non-parametric applied operations research techniques. the researchers have considered the variables like standard deviation, expense ratio, loads, turnover, beta, costs, fund size, variance, percentage of periods with negative return, lower semi-variance, sales charges, operating expenses, cash percentage, p/e ratio, p/b ratio, total assets, lower mean, lower semi-skewness, and excess kurtosis as input while considering the variables like return, deviations from median return, capital flow, skewness, sharpe ratio, upper semi-variance, upper semiskewness, and jensen’s α as output in assessing the performances of the funds under study. there has been another string of the literature in which mcdm techniques are applied for selection of mf (pendaraki et al., 2005; lin et al., 2007; gladish et al., 2007; chang et al., 2010; babalos et al., 2011; alptekin, 2009; karmakar et al., 2018; pendaraki & zopounidis, 2003; sielska, 2010). attribute based classification approaches like utadis (utilités additives discriminantes) (pendaraki et al., 2005; lin et al., 2007), fuzzy mcdm techniques (gladish et al., 2007), distance based mcdm methods like topsis (lin et al., 2007; chang et al., 2010; alptekin, 2009; karmakar an ensemble approach for portfolio selection in a multi-criteria decision making framework 141 et al., 2018; sielska, 2010) and edas (karmakar et al., 2018), outranking methods such as promethee (sielska, 2010) and promethee ii (pendaraki&zopounidis, 2003) have been selected for portfolio selection issue based on the parameters like sharpe ratio, sortinoratio, treynorratio, jensen’s α, aum, beta, standard deviation, nav, annualized return, average return, information ratio, and r-squared. in this context, in paper (babalos et al., 2011) the authors used stochastic multi-criteria acceptability analysis (smaa-2) framework for assessing performances of mf. predominantly, objective weight method using euclidean distance has been applied for calculating criteria weight. however, some authors like chang et al. (2010) experimented with different distance measures for examining performances of the mf. 3. data and methodology 3.1. development of the framework in this study, we have followed a non-parametric two stage framework wherein we have ranked the funds under study. in the first stage, we have applied dea to appraise the efficiencies of the funds for a primary level selection. for a refined selection in a common setting for investment choice among the relatively efficient portfolios, at stage two the mcdm technique like mabac has been used. the entropy method has been employed for calculation of criteria weight in this regard. figure 1 depicts the framework followed in this study. we have started our analysis with total 48 number of open ended equity large cap funds under direct plan. in the seminal works of markowitz (1952, 1959), the underline assumption considered first two central moments of the utility function of the return which is treated as a normal distribution. however, the studies (lau et al., 1990; cambell & hentsche, 1992) arguably reported that portfolio returns are always not normally distributed in practice. hence, it is imperative to consider higher moments. average investors prefer lower value of kurtosis (scott & horvath, 1980) as it entails a higher degree of sensitivity of the funds with respect to non-favorable market condition. in view of this, for filling the gap in the literature in indian context, this study considers kurtosis as one of the inputs. expense ratio puts load on the profitability as it covers the management fees and operating expense. the third quartile return has a typical significance in the sense that it indicates relative closeness to the highest value than the average return. the ratio psv to nsv signifies inclination of the deviation of the return from the mean towards the higher side which essentially acts as a favorable proposition for the investors. we have used standard variable returns to scale (vrs) model as it resembles real life situation compared to constant returns to scale (crs) which reflects the proportionate change in the output with respect to the input (ali & seiford, 1990). further, we have calculated super efficiency in order to discriminate the funds to a considerable extent. biwas et al/decis. mak. appl. manag. eng. 2 (2) (2019) 138-158 142 figure 1. research framework the higher value of net assets of a fund indicates the greater possibility of a good return. the sharpe ratio, α, and sortino ratio specify the excess return with respect to risk, i.e. risk-adjusted performance while beta captures the systematic risk. to what extent, the portfolio return with respect to the benchmark, i.e. index, adjusts the volatility of the return is reflected in the value of information ratio which stands beneficial for the investors. the value of r-squared manifests the nature of diversification of the portfolio, which leads to reduction of unsystematic risk. 3.2. sample in our study, we focus on open ended and large cap equity mutual funds under direct plan. in this context, we have excluded the fixed maturity plan, and plans suspended for sales which otherwise leaves lesser chance to gain more at calculated risk. for spotting the funds, we have referred the value research online data base and subsequently for collecting information on the performance criteria. appendix 1 shows the descriptive statistics of the total 48 funds selected initially for the study. returns of the last 12 quarters (i.e. sep 2015 to jun 2018) have been considered for calculating distribution based parameters used in dea. 3.3. data envelopment analysis (dea) dea evaluates a set of peer entities (homogenous) i.e. decision making units (dmus) having multiple inputs or outputs. as introduced by charnes et al. (1978), this technique has gained extensive importance by the researchers in measuring performance efficiency in terms of the frontiers or envelop rather than the central tendency as in the case of fitting a regression model. in dea, the efficiency of the homogenous entities is calculated by using the linear programming method. calculations for input oriented, constant return to scale (crs) are as follows: min 𝜃 subject to: ∑ xij n j=1 λj ≤ θxit i = 1,2, … , m; input constraint (1) ∑ yrj n j=1 λj ≥ yrt r = 1,2, … , s; output constraint (2) where, 𝜆𝑗 ≥ 0 ∀ 𝑖, 𝑗, 𝑟 for variable return to scale (vrs) the sets of equations are: min 𝜃 stage 1 dea input/ output variables input: kurtosis; expense ratio output: third quartile return; ratio between positive semi-variance (psv) to negative semi-variance (nsv) stage 2 mabac efficiency based selection final ranking portfolio selection criteria net assets, 5-year annualized return, sharpe ratio, sortino ratio, information ratio, alpha, r-squared, and beta. entropy method for criteria weight funds under study an ensemble approach for portfolio selection in a multi-criteria decision making framework 143 subject to: ∑ xij n j=1 λj ≤ θxit i = 1,2, … , m; input constraint (3) ∑ yrj n j=1 λj ≥ yrt r = 1,2, … , s; output constraint (4) where, ∑ λj n j=1 = 1 ; λj ≥ 0 ∀ i, j, r if two or more dmus are found to be efficient (i.e. θ = 1 or 100%) then the superefficiency value is calculated to discriminate them. for vrs the super efficiency is calculated as below: min 𝜃 subject to ∑ xij n j=1 λj ≤ θxit i = 1,2, … , m; input constraint (5) ∑ yrj n j=1 λj ≥ yrt r = 1,2, … , s; output constraint (6) where, ∑ λj n j=1 = 1; λj ≥ 0 ∀ i, j, r ; j ≠ t. 3.4. entropy method it is an objective method for calculating criteria weights based on relative information content wherein, higher value of the entropy (degree of disorder) indicates more information content for the respective criterion (shannon, 1948). the steps are as given below where, 𝑎𝑖 : i th alternative where i = 1,2,3,…..m; 𝑐𝑗 : j th criterion where j = 1,2,3,…..n; 𝑥𝑖𝑗 : j th criterion value for the ith alternative; step1. normalization of the criteria. for this purpose, we have used the enhanced accuracy method of normalization zeng et al. (2013) as mentioned in (jahan & edwards, 2015). accordingly, the normalized matrix r = [𝑟𝑖𝑗 ]𝑚 ×𝑛 is given by: 𝑟𝑖𝑗 = 1 𝑥𝑗 𝑚𝑎𝑥 −𝑥𝑖𝑗 ∑ (𝑥𝑗 𝑚𝑎𝑥−𝑥𝑖𝑗) 𝑚 𝑖=1 (beneficial criteria) (7) 𝑟𝑖𝑗 = 1 𝑥𝑖𝑗−𝑥𝑗 𝑚𝑖𝑛 ∑ (𝑥𝑖𝑗−𝑥𝑗 𝑚𝑖𝑛)𝑚𝑖=1 (non-beneficial criteria) (8) step 2. entropy calculation for the criterion entropy of the jth criterion is given by: 𝐻𝑗 = ∑ 𝑓𝑖𝑗 ln 𝑓𝑖𝑗 𝑚 𝑖=1 ln 𝑚 , 𝑖 = 1,2, … 𝑚; 𝑗 = 1,2, . . 𝑛 (9) where, 𝑓𝑖𝑗 = 𝑟𝑖𝑗 ∑ 𝑟𝑖𝑗 𝑚 𝑖 , 𝑖 = 1, 2, … 𝑚; 𝑗 = 1,2, … (10) step 3. calculation of the entropy weight for the criterion entropy weight of the jth criterion is determined by: 𝑤𝑗 = 1−𝐻𝑗 𝑛−∑ 𝐻𝑗 𝑛 𝑗=1 , where ∑ 𝑤𝑗 𝑛 𝑗=1 = 1 (11) biwas et al/decis. mak. appl. manag. eng. 2 (2) (2019) 138-158 144 3.5. multi-attribute border approximation area comparisons (mabac) since its proposal (pamučar & ćirović, 2015), this method has drawn significant attention from the researchers for its inherent computational ease and stability. unlike topsis, this method classifies the performances of the criteria into two areas such as upper approximation area (uaa) for ideal solutions and lower approximation area (laa) for non-ideal solutions instead of calculating the distance of any solution from the ideal and non-ideal solutions. in a sense, this method examines the relative strength and weakness of each alternative with respect to the others pertaining to each criterion (roy et al., 2016). this method has been widely applied by the researchers for solving multi-criteria decision making problems such as railway management (sharma et al., 2018; vesković et al., 2018)medical tourism site selection (roy et al., 2018) and selection of hotels (yu et al., 2017). let, d is the initial decision matrix represented by 𝑎𝑖 : i th alternative where i = 1,2,3,…..m; 𝑐𝑗 : j th criterion where j = 1,2,3,…..n; 𝑥𝑖𝑗 : j th criterion value for the ith alternative; the broad steps under this method are given below. step 1. normalization of the criteria values 𝑟𝑖𝑗 = (𝑥𝑖𝑗− 𝑥𝑖 −) (𝑥𝑖 +− 𝑥𝑖 − ) ; for beneficial criteria (12) 𝑟𝑖𝑗 = (𝑥𝑖𝑗− 𝑥𝑖 + ) (𝑥𝑖 −− 𝑥𝑖 + ) ; for non-beneficial criteria (13) where, 𝑥𝑖 + and 𝑥𝑖 − are the maximum and minimum criteria values respectively. step 2. construction of weighted normalization matrix (y) elements of y are given by: 𝑦𝑖𝑗 = 𝑤𝑗 (𝑟𝑖𝑗 + 1) ; where, 𝑤𝑗 are the criteria weight. (14) step 3. determination of the border approximation area (baa) the elements of the border approximation area (baa) t is given by: 𝑡𝑗 = (∏ 𝑦𝑖𝑗 𝑚 𝑖=1 ) 1/𝑚 (15) where, m is the total number of alternatives &𝑡𝑗 corresponds to each criterion. step 4. calculation of the matrix q related to the separation of the alternatives from baa q = y-t (16) a particular alternative 𝑎𝑖 is said to be belonging to the upper approximation area (uaa) i.e. 𝑇+ if 𝑞𝑖𝑗 > 0 or lower approximation area (laa) i.e. 𝑇 − if 𝑞𝑖𝑗 < 0or baa i.e. t if 𝑞𝑖𝑗 = 0. the alternative 𝑎𝑖 is considered to be the best among the others if more numbers of criteria pertaining to it possibly belong to 𝑇+. step 5. ranking of the alternatives it is done according to the final values of the criterion functions as given by si = ∑ qij n j=1 for j = 1,2, … n and i = 1,2, … m (17) higher the value, the better is the rank. in this study for carrying out dea, we have used lingo (version 11) software while for mcdm related calculations, microsoft office excel (version 2010) is utilized. an ensemble approach for portfolio selection in a multi-criteria decision making framework 145 4. results and discussions we have considered total 48 funds initially. however, before analyzing them using dea, we have checked whether the condition of the required number of dmus is satisfied or not. in this regard, though there are several studies, we have followed one of the widely accepted study conducted by banker et al. (1989). according to the study, the rule of the thumb is n >= max {p × q, 3(p + q)} where, p be the number of inputs and q is the number of outputs used in the analysis, and n is the number of dmus to be considered. our model has two inputs and two outputs. hence, it satisfies the condition. the top 20 funds (i.e. dmus) based on the result of dea considering vrs model is given in the table 1 while the details is included in the appendix 2. the performance criteria values of the above funds (table 1) are given in the appendix 1. we then have used mcdm model for ranking the above mentioned funds. for calculating criteria weights, we have used a modified entropy method. the results are listed in tables 2-3. after calculating the criteria weights, we then proceed to the stage 2 i.e. ranking of the funds (primary selection through dea) using the mabac technique. the results are given in tables 4-6. from the result, it is evident that the top five funds (i.e. a27, a19, a38, a22 and a25; the names are given in appendix 1) are rated 4-star and 5-star by value research and except one of them, and their risk grades are above average or more. on the other hand, bottom five funds (i.e. a40, a42, a21, a3 and a5; the names are given in appendix 1) are rated 2-star or below and having an overall average risk grade. hence, this study also conforms to the market based rating of the funds by value research. in order to check the dependability of the result obtained from mabac, we have also ranked the funds selected from dea result using topsis technique. in line with the method suggested by hwang and yoon (1981), we have obtained the rankings as given in the table 7. for checking consistency with mabac based ranking, we have performed spearman’s rank correlation test using ibm spss 22, which is 0.952 (significant at 0.01 level). hence, the result obtained from mabac is acceptable. biwas et al/decis. mak. appl. manag. eng. 2 (2) (2019) 138-158 146 t a b le 1 . d e a r e su lt ( t o p 2 0 f u n d s) f u n d s u n d e r st u d y k u rt o si s k u rt o si s (n o rm a li z e d )( in p u t) e x p e n se r a ti o (i n p u t) p s v / n s v (o u tp u t) q 3 ( o u tp u t) v r s r e su lt r a n k v r s d m u 3 -0 .9 9 4 2 1 8 2 9 7 0 .2 2 8 3 5 7 7 8 5 0 .5 1 0 .6 9 2 5 2 4 2 0 6 6 .3 2 5 0 .2 7 7 2 0 d m u 5 0 .4 3 0 0 5 2 7 9 6 1 1 .0 5 1 .1 9 0 7 5 9 4 5 3 7 .5 1 2 5 0 .4 1 9 6 1 7 d m u 1 0 0 .0 6 8 7 4 5 8 7 5 0 .8 0 4 2 5 0 9 7 8 2 .0 9 1 .7 1 2 9 8 8 5 7 3 6 .0 6 5 1 7 d m u 1 3 -1 .3 9 2 0 7 2 4 5 1 0 .0 1 2 8 0 8 1 8 8 0 .1 5 1 .0 2 4 9 4 6 6 6 7 7 .3 4 7 5 1 .5 2 7 2 2 d m u 1 4 -1 .0 6 9 7 3 6 0 7 3 0 .1 8 7 4 4 3 7 3 2 0 .1 5 0 .6 9 7 8 2 1 4 7 9 6 .6 5 7 5 0 .8 3 4 2 1 1 d m u 1 5 -0 .9 8 6 2 8 6 2 2 9 0 .2 3 2 6 5 5 2 2 5 1 .2 6 1 .6 5 8 3 9 9 1 9 8 .5 3 7 5 1 7 d m u 1 9 -0 .8 6 7 6 8 1 9 8 1 0 .2 9 6 9 1 2 6 8 6 0 .4 4 1 .1 6 7 1 8 5 8 9 2 9 .8 0 2 5 1 .4 6 3 d m u 2 1 -1 .0 4 7 2 4 5 0 8 9 0 .1 9 9 6 2 8 9 0 7 0 .4 3 0 .7 0 8 5 7 5 8 5 1 6 .5 2 2 5 0 .3 2 7 6 1 8 d m u 2 2 -0 .9 1 4 6 4 0 3 5 9 0 .2 7 1 4 7 1 5 5 5 0 .5 6 1 .1 7 7 3 4 2 7 8 9 9 .7 4 2 5 1 .0 2 6 d m u 2 4 -1 .0 2 0 2 0 4 2 8 6 0 .2 1 4 2 7 9 0 8 5 0 .1 7 0 .7 0 2 1 0 1 3 0 9 6 .5 9 2 5 0 .7 3 4 9 1 3 d m u 2 5 -0 .7 5 6 9 4 1 7 1 1 0 .3 5 6 9 0 9 5 9 8 0 .7 5 0 .7 0 3 9 5 1 6 5 9 8 .0 9 0 .3 1 6 9 1 9 d m u 2 7 -1 .3 4 0 7 0 3 7 1 5 0 .0 4 0 6 3 8 7 6 4 2 .9 4 0 .7 0 3 0 1 8 8 2 5 1 2 .0 2 2 5 1 7 d m u 3 2 -1 .4 1 5 7 1 3 3 7 1 0 1 .1 5 1 .0 3 8 4 4 5 7 3 5 7 .1 8 1 7 d m u 3 6 -0 .9 6 8 7 9 4 4 0 .2 4 2 1 3 1 9 5 5 0 .2 9 0 .7 0 6 5 9 5 0 6 6 6 .6 1 0 .4 5 8 6 1 6 d m u 3 7 -1 .4 1 4 5 6 6 8 3 5 0 .0 0 0 6 2 1 1 7 1 0 .2 9 1 .0 2 7 5 7 9 3 9 7 .0 4 5 3 .0 3 4 3 1 d m u 3 8 -0 .9 5 0 0 8 2 1 3 3 0 .2 5 2 2 6 9 8 9 5 1 .3 2 1 .5 2 8 3 5 8 0 1 3 7 .0 7 0 .7 8 1 9 1 2 d m u 4 0 -1 .0 4 6 7 9 3 8 1 7 0 .1 9 9 8 7 3 3 9 7 0 .2 9 0 .7 0 3 3 5 6 9 4 5 6 .6 0 7 5 0 .4 6 8 9 1 5 d m u 4 2 -1 .0 6 3 8 9 2 4 5 6 0 .1 9 0 6 0 9 6 8 9 0 .2 1 0 .7 0 2 1 4 8 7 6 8 6 .6 8 5 0 .6 2 6 4 1 4 d m u 4 3 -1 .4 0 3 7 3 3 1 3 2 0 .0 0 6 4 9 0 6 5 9 0 .2 2 1 .0 3 3 5 2 9 5 3 2 7 .2 4 1 .4 1 0 3 4 d m u 4 8 -1 .0 7 2 4 5 1 5 0 9 0 .1 8 5 9 7 2 5 6 2 0 .1 2 0 .6 9 1 0 7 3 6 2 9 6 .5 8 7 5 1 .2 5 5 an ensemble approach for portfolio selection in a multi-criteria decision making framework 147 table 2. normalization table alt. c1 c2 c3 c4 c5 c6 c7 c8 a3 0.9450 0.9338 0.9315 0.9315 0.8427 0.9350 1.0000 0.9600 a5 0.9477 0.9561 0.8910 0.8883 0.9688 0.9165 0.8951 0.9850 a10 0.9450 0.9446 0.9688 0.9694 0.9866 0.9611 0.9161 1.0000 a13 0.9453 0.9381 0.9564 0.9514 0.9839 0.9499 0.9930 0.9600 a14 0.9460 0.9406 0.9470 0.9477 0.9304 0.9453 1.0000 0.9600 a15 1.0000 0.9627 0.9315 0.9333 0.9831 0.9350 0.9510 0.8650 a19 0.9454 0.9834 0.9844 1.0000 0.9914 0.9815 0.7832 0.9800 a21 0.9453 0.9359 0.9377 0.9369 0.8585 0.9389 1.0000 0.9550 a22 0.9446 0.9809 0.9782 0.9928 0.9902 0.9784 0.7762 0.9850 a24 0.9449 0.9400 0.9470 0.9459 0.9346 0.9446 1.0000 0.9550 a25 0.9459 0.9509 0.9751 0.9712 0.9917 0.9655 0.9231 0.9550 a27 0.9446 1.0000 1.0000 0.9946 1.0000 1.0000 0.8531 0.7850 a32 0.9445 0.9314 0.9315 0.9279 0.9634 0.9363 0.9930 0.9600 a36 0.9449 0.9383 0.9377 0.9405 0.9144 0.9399 1.0000 0.9550 a37 0.9445 0.9333 0.9470 0.9441 0.9756 0.9452 0.9930 0.9700 a38 0.9841 0.9808 0.9502 0.9423 0.9848 0.9491 0.9301 0.9400 a40 0.9456 0.9371 0.9439 0.9459 0.8882 0.9435 1.0000 0.9550 a42 0.9445 0.9375 0.9439 0.9441 0.8936 0.9426 1.0000 0.9550 a43 0.9445 0.9345 0.9470 0.9441 0.9774 0.9457 0.9930 0.9600 a48 0.9478 0.9401 0.9502 0.9477 0.9405 0.9459 1.0000 0.9600 table 3. hj values and criteria weights hj 0.748 0.748 0.748 0.748 0.747 0.748 0.747 0.747 wj 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125 table 4. normalization result alt. c1 c2 c3 c4 c5 c6 c7 c8 a3 0.009 0.035 0.371 0.387 0.000 0.221 1.000 0.814 a5 0.058 0.359 0.000 0.000 0.802 0.000 0.531 0.930 a10 0.009 0.193 0.714 0.726 0.915 0.534 0.625 1.000 a13 0.016 0.098 0.600 0.565 0.898 0.399 0.969 0.814 a14 0.028 0.134 0.514 0.532 0.558 0.345 1.000 0.814 a15 1.000 0.456 0.371 0.403 0.892 0.221 0.781 0.372 a19 0.017 0.759 0.857 1.000 0.945 0.779 0.031 0.907 a21 0.014 0.065 0.429 0.435 0.100 0.268 1.000 0.791 a22 0.003 0.722 0.800 0.935 0.938 0.741 0.000 0.930 a24 0.007 0.125 0.514 0.516 0.584 0.336 1.000 0.791 a25 0.026 0.284 0.771 0.742 0.947 0.587 0.656 0.791 a27 0.002 1.000 1.000 0.952 1.000 1.000 0.344 0.000 a32 0.001 0.000 0.371 0.355 0.767 0.237 0.969 0.814 a36 0.008 0.100 0.429 0.468 0.456 0.280 1.000 0.791 a37 0.000 0.028 0.514 0.500 0.845 0.344 0.969 0.860 a38 0.714 0.720 0.543 0.484 0.904 0.390 0.688 0.721 biwas et al/decis. mak. appl. manag. eng. 2 (2) (2019) 138-158 148 alt. c1 c2 c3 c4 c5 c6 c7 c8 a40 0.021 0.083 0.486 0.516 0.289 0.324 1.000 0.791 a42 0.000 0.088 0.486 0.500 0.323 0.313 1.000 0.791 a43 0.000 0.045 0.514 0.500 0.856 0.350 0.969 0.814 a48 0.061 0.126 0.543 0.532 0.622 0.351 1.000 0.814 table 5. border approximation area matrix baa 0.1347 0.1552 0.1907 0.1919 0.2066 0.1730 0.2181 0.2189 table 6. ranking of the funds fund a3 a5 a10 a13 a14 a15 a19 a21 a22 a24 sum (si) -0.134 -0.154 0.101 0.056 0.002 0.073 0.173 -0.101 0.144 -0.005 rank 19 20 6 8 12 7 2 18 4 13 fund a25 a27 a32 a36 a37 a38 a40 a42 a43 a48 sum (si) 0.111 0.173 -0.050 -0.048 0.019 0.156 -0.050 -0.051 0.017 0.017 rank 5 1 15 14 9 3 16 17 11 10 table 7. topsis ranking fund a3 a5 a10 a13 a14 a15 a19 a21 a22 a24 topsis rank 20 15 7 8 12 3 4 19 5 13 fund a25 a27 a32 a36 a37 a38 a40 a42 a43 a48 topsis rank 6 1 14 16 10 2 17 18 9 11 5. conclusion we have made an attempt to assess the funds from two perspectives such as efficiency and performance. accordingly, we have filtrated the funds through a two stage process using dea at stage 1 and mabac at stage 2. the rationale behind this study lies in the selection of the funds to form an investment portfolio based on their return distribution and performance parameters encompassing risk-return tradeoff. in effect, this study not only has adjudged the funds on efficiency dimension, but also sets out to establish a ranking based on risk-return criteria. our results conform to the market based rating of the funds. a combination of dea-entropy-mabac turns this study considerably different from the existing contributions in the indian context as far as the approach is concerned. the results of this study provide the investors a broader perspective for selection of the portfolio. however, future research shall be required to focus more clinical approach by considering fundamental parameters and stock level analysis. it is important to analyze the stocks on which the fund managers invest the amount invested by the investors both on fundamental dimension and organizational dimensions to ascertain the decision and establish a causal relationship among the stock performance and mf performance. further, the efficiencies of the fund houses need to be examined. there are requirements to investigate the relationship between investors’ sentiments and market 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(2011). mutual funds performance evaluation based on endogenous benchmarks. expert systems with applications, 38 (4), 3663–3670. © 2018 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). biwas et al/decis. mak. appl. manag. eng. 2 (2) (2019) 138-158 154 a p p e n d ix 1 . d e sc ri p ti v e s ta ti st ic s s / l f u n d r a t in g r is k g ra d e n e t a ss e ts (c r) (c 1 ) 5 -y e a r a n n u a li ze d r e t (c 2 ) s h a r p e r a ti o (c 3 ) s o rt i n o r a ti o (c 4 ) in fo rm a t io n r a ti o (c 5 ) a lp h a (c 6 ) r s q u a r e d (c 7 ) b e ta (c 8 ) s d e x p e n s e r a ti o a 1 a d it y a b ir la s u n l if e f o cu se d e q u it y f u n d d ir e ct p la n * * * * b e lo w a v e ra g e 4 2 3 9 .3 9 2 1 .9 8 0 .3 6 0 .5 5 -0 .1 6 0 .1 2 0 .9 3 0 .9 2 1 3 .2 2 1 .2 6 a 2 a d it y a b ir la s u n l if e f ro n tl in e e q u it y f u n d d ir e ct p la n * * * * b e lo w a v e ra g e 2 1 3 8 0 . 4 2 2 1 .3 9 0 .3 8 0 .6 0 -0 .3 8 0 .1 3 0 .9 4 0 .9 3 1 3 .3 2 1 .3 1 a 3 a d it y a b ir la s u n l if e i n d e x f u n d d ir e ct p la n * a b o v e a v e ra g e 1 4 2 .9 1 1 6 .1 9 0 .3 1 0 .5 0 -4 .7 5 1 .0 9 1 .0 0 0 .9 9 1 3 .8 4 0 .5 1 a 4 a x is b lu e ch ip f u n d d ir e ct p la n * * * * * l o w 2 5 6 8 .1 0 2 0 .7 7 0 .5 9 0 .7 8 0 .2 4 3 .0 6 0 .8 7 0 .8 7 1 3 .0 0 0 .9 4 a 5 b n p p a ri b a s l a rg e c a p f u n d d ir e ct p la n * * a v e ra g e 8 9 2 .5 2 1 9 .7 1 0 .1 8 0 .2 6 -0 .5 1 2 .5 2 0 .8 5 0 .9 4 1 4 .2 2 1 .0 5 a 6 c a n a ra r o b e co b lu e ch ip e q u it y f u n d d ir e ct p la n * * * b e lo w a v e ra g e 1 3 1 .6 6 1 8 .7 2 0 .4 0 0 .6 1 0 .0 9 0 .4 9 0 .9 2 0 .9 7 1 4 .0 7 1 .4 8 a 7 d h f l p ra m e ri ca l a rg e c a p f u n d d ir e ct p la n * * * b e lo w a v e ra g e 4 1 4 .4 9 1 9 .8 5 0 .3 5 0 .6 0 -0 .3 4 0 .4 4 0 .9 7 0 .9 3 1 3 .1 8 1 .5 1 a 8 d s p t o p 1 0 0 e q u it y f u n d d ir e ct p la n * h ig h 3 0 0 7 .8 0 1 7 .4 0 0 .2 5 0 .4 1 -0 .3 3 1 .6 9 0 .9 0 1 .0 2 1 4 .9 9 1 .4 7 a 9 e d e lw e is s l a rg e c a p f u n d d ir e ct p la n * * * a v e ra g e 1 4 0 .8 1 2 0 .3 3 0 .3 8 0 .5 9 0 .0 0 0 .0 2 0 .9 4 1 .0 0 1 4 .2 8 0 .5 8 a 1 0 e ss e l l a rg e c a p e q u it y f u n d d ir e ct p la n * * * b e lo w a v e ra g e 1 5 0 .6 6 1 7 .9 0 0 .4 3 0 .7 1 0 .0 9 0 .9 3 0 .8 8 0 .9 1 1 3 .4 7 2 .0 9 a 1 1 f ra n k li n i n d ia b lu e ch ip f u n d d ir e ct p la n * * a v e ra g e 8 1 0 7 .9 4 1 8 .8 4 0 .2 3 0 .4 3 -0 .7 9 1 .8 7 0 .9 5 0 .9 0 1 2 .8 4 1 .1 6 a 1 2 f ra n k li n i n d ia i n d e x f u n d n s e n if ty p la n d ir e ct p la n * * a v e ra g e 2 5 3 .2 1 1 6 .6 4 0 .3 2 0 .5 2 -3 .6 8 0 .8 6 1 .0 0 0 .9 9 1 3 .7 7 0 .6 4 an ensemble approach for portfolio selection in a multi-criteria decision making framework 155 a 1 3 h d f c i n d e x f u n d s e n se x p la n d ir e ct p la n * * * b e lo w a v e ra g e 2 4 4 .0 4 1 6 .8 7 0 .3 9 0 .6 1 0 .0 0 0 .0 6 0 .9 9 0 .9 9 1 3 .8 4 0 .1 5 a 1 4 h d f c i n d e x f u n d n if ty 5 0 p la n d ir e ct p la n * * * a v e ra g e 4 3 5 .8 9 1 7 .2 6 0 .3 6 0 .5 9 -1 .8 0 -0 .2 9 1 .0 0 0 .9 9 1 3 .8 4 0 .1 5 a 1 5 h d f c t o p 1 0 0 f u n d d ir e ct p la n * * * h ig h 1 5 2 6 0 .7 9 2 0 .7 6 0 .3 1 0 .5 1 -0 .0 3 -1 .0 9 0 .9 3 1 .1 8 1 6 .9 9 1 .2 6 a 1 6 h s b c l a rg e c a p e q u it y f u n d d ir e ct p la n * * * a v e ra g e 7 2 9 .8 2 1 8 .9 0 0 .4 1 0 .6 6 0 .2 1 0 .4 9 0 .9 5 1 .0 4 1 4 .8 5 1 .5 1 a 1 7 ic ic i p ru d e n ti a l b lu e ch ip f u n d d ir e ct p la n * * * * l o w 1 8 7 4 7 .2 8 2 0 .6 2 0 .4 7 0 .7 3 0 .2 5 1 .2 1 0 .9 4 0 .9 3 1 3 .2 7 1 .1 7 a 1 8 ic ic i p ru d e n ti a l n if ty i n d e x f u n d d ir e ct p la n * * a v e ra g e 3 5 7 .3 4 1 7 .2 0 0 .3 4 0 .5 6 -3 .5 5 -0 .6 2 1 .0 0 0 .9 9 1 3 .8 4 0 .5 7 a 1 9 ic ic i p ru d e n ti a l n if ty n e x t 5 0 i n d e x f u n d d ir e ct p la n * * * * * a b o v e a v e ra g e 2 6 5 .3 6 2 4 .0 4 0 .4 8 0 .8 8 0 .2 5 2 .5 1 0 .6 9 0 .9 5 1 5 .8 5 0 .4 4 a 2 0 id b i in d ia t o p 1 0 0 e q u it y f u n d d ir e ct p la n * * a v e ra g e 4 1 7 .4 9 1 8 .9 4 0 .1 6 0 .2 8 -0 .7 1 -2 .6 7 0 .8 9 0 .8 9 1 3 .1 3 1 .0 6 a 2 1 id b i n if ty i n d e x f u n d d ir e ct p la n * * a b o v e a v e ra g e 2 2 4 .6 4 1 6 .5 2 0 .3 3 0 .5 3 -4 .2 2 -0 .7 9 1 .0 0 1 .0 0 1 3 .8 7 0 .4 3 a 2 2 id b i n if ty j u n io r in d e x f u n d d ir e ct p la n * * * * * a b o v e a v e ra g e 5 5 .9 9 2 3 .6 4 0 .4 6 0 .8 4 0 .2 1 2 .2 7 0 .6 8 0 .9 4 1 5 .7 9 0 .5 6 a 2 3 id f c l a rg e c a p f u n d d ir e ct p la n * * * b e lo w a v e ra g e 3 7 3 .2 5 1 7 .3 0 0 .4 4 0 .6 5 0 .1 7 0 .9 2 0 .9 5 0 .9 3 1 3 .2 6 1 .8 3 a 2 4 id f c n if ty f u n d d ir e ct p la n * * * a v e ra g e 1 1 7 .7 0 1 7 .1 7 0 .3 6 0 .5 8 -1 .6 6 -0 .3 5 1 .0 0 1 .0 0 1 3 .8 5 0 .1 7 a 2 5 in d ia b u ll s b lu e ch ip f u n d d ir e ct p la n * * * * a v e ra g e 4 0 1 .7 5 1 8 .8 9 0 .4 5 0 .7 2 0 .2 6 1 .2 7 0 .8 9 1 .0 0 1 4 .7 0 0 .7 5 a 2 6 in v e sc o i n d ia l a rg e ca p f u n d d ir e ct p la n * * * * l o w 1 5 2 .2 6 2 1 .0 5 0 .4 7 0 .7 2 0 .2 8 1 .2 5 0 .9 6 0 .9 2 1 2 .9 9 0 .9 6 a 2 7 jm c o re 1 1 f u n d d ir e ct p la n * * * * * h ig h 3 6 .2 2 2 6 .6 6 0 .5 3 0 .8 5 0 .5 4 3 .9 4 0 .7 9 1 .3 4 2 0 .8 9 2 .9 4 a 2 8 jm l a rg e c a p f u n d d ir e ct p la n * b e lo w a v e ra g e 2 8 1 8 .0 1 1 7 .2 5 0 .0 6 0 .0 9 -1 .3 1 -3 .5 7 0 .9 6 0 .8 0 1 1 .3 6 1 .6 1 a 2 9 k o ta k b lu e ch ip f u n d d ir e ct p la n * * * b e lo w a v e ra g e 1 4 2 3 .6 0 2 0 .4 2 0 .3 5 0 .5 6 -0 .2 3 -0 .4 1 0 .9 6 0 .9 5 1 3 .5 2 1 .1 5 biwas et al/decis. mak. appl. manag. eng. 2 (2) (2019) 138-158 156 a 3 0 l & t i n d ia l a rg e c a p f u n d d ir e ct p la n * * a v e ra g e 4 2 3 .8 5 1 8 .7 4 0 .2 1 0 .3 8 -0 .6 2 -2 .1 6 0 .9 2 0 .9 5 1 3 . 7 7 2 .0 4 a 3 1 l ic m f i n d e x -n if ty p la n d ir e ct p la n * a b o v e a v e ra g e 2 3 .7 7 1 6 .3 9 0 .3 0 0 .4 9 -3 .4 9 -1 .1 7 1 .0 0 1 .0 0 1 3 . 9 6 0 .6 9 a 3 2 l ic m f i n d e x -s e n se x p la n d ir e ct p la n * a v e ra g e 2 0 .6 9 1 5 .8 1 0 .3 1 0 .4 8 -0 .6 9 -0 .9 9 0 .9 9 0 .9 9 1 3 . 8 6 1 .1 5 a 3 3 l ic m f l a rg e c a p f u n d d ir e ct p la n * * b e lo w a v e ra g e 2 4 5 .0 9 1 8 .0 9 0 .1 9 0 .3 2 -0 .7 7 -2 .3 5 0 .9 3 0 .9 0 1 3 . 0 2 1 .5 2 a 3 4 m o ti la l o sw a l f o cu se d 2 5 f u n d d ir e ct p la n * * * * l o w 1 1 7 0 . 5 3 2 1 .9 4 0 .3 8 0 .5 5 -0 .0 6 0 .7 2 0 .7 2 0 .7 8 1 2 . 8 3 1 .1 5 a 3 5 p ri n ci p a l n if ty 1 0 0 e q u a l w e ig h t f u n d d ir e ct p la n * a b o v e a v e ra g e 1 8 .1 7 1 5 .8 4 0 .2 3 0 .3 9 -1 .2 6 -2 .1 6 0 .9 8 0 .9 9 1 3 . 9 2 0 .5 0 a 3 6 r e li a n ce i n d e x f u n d n if ty p la n d ir e ct p la n * * a b o v e a v e ra g e 1 3 2 .9 9 1 6 .9 0 0 .3 3 0 .5 5 -2 .3 4 -0 .7 1 1 .0 0 1 .0 0 1 3 . 9 7 0 .2 9 a 3 7 r e li a n ce i n d e x f u n d s e n se x p la n d ir e ct p la n * * a v e ra g e 7 .6 3 1 6 .1 1 0 .3 6 0 .5 7 -0 .2 8 -0 .3 0 0 .9 9 0 .9 7 1 3 . 5 5 0 .2 9 a 3 8 r e li a n ce l a rg e c a p f u n d d ir e ct p la n * * * * a b o v e a v e ra g e 1 0 8 9 7 .8 2 2 3 .6 2 0 .3 7 0 .5 6 0 .0 3 0 .0 0 0 .9 0 1 .0 3 1 5 . 0 6 1 .3 2 a 3 9 s b i b lu e ch ip f u n d d ir e ct p la n * * * * * l o w 2 0 2 8 3 .9 2 2 2 .6 1 0 .4 1 0 .6 2 -0 .0 3 0 .5 3 0 .9 0 0 .8 7 1 2 . 8 3 1 .1 8 a 4 0 s b i n if ty i n d e x f u n d d ir e ct p la n * * a v e ra g e 3 2 0 .9 9 1 6 .7 1 0 .3 5 0 .5 8 -3 .2 2 -0 .4 3 1 .0 0 1 .0 0 1 3 . 9 3 0 .2 9 a 4 1 s u n d a ra m s e le ct f o cu s f u n d d ir e ct p la n * * * * b e lo w a v e ra g e 8 3 5 .7 0 1 8 .6 2 0 .4 3 0 .6 6 0 .0 5 0 .6 9 0 .9 5 0 .9 0 1 2 . 8 2 0 .5 5 a 4 2 t a ta i n d e x n if ty f u n d d ir e ct p la n * * a v e ra g e 1 2 .0 6 1 6 .7 7 0 .3 5 0 .5 7 -3 .0 4 -0 .5 0 1 .0 0 1 .0 0 1 3 . 8 7 0 .2 1 a 4 3 t a ta i n d e x s e n se x f u n d d ir e ct p la n * * * a v e ra g e 5 .8 7 1 6 .3 0 0 .3 6 0 .5 7 -0 .2 2 -0 .2 6 0 .9 9 0 .9 9 1 3 . 8 1 0 .2 2 a 4 4 t a ta l a rg e c a p f u n d d ir e ct p la n * * * b e lo w a v e ra g e 8 0 3 .2 9 1 8 .5 0 0 .2 9 0 .4 7 -0 .4 5 -1 .1 5 0 .9 5 0 .9 5 1 3 . 5 3 0 .7 4 a 4 5 t a u ru s l a rg e ca p e q u it y f u n d d ir e ct p la n * h ig h 3 9 .3 8 1 7 .0 4 0 .0 0 0 .0 0 -1 .2 7 -5 .3 5 0 .9 1 0 .9 9 1 4 . 4 1 2 .1 5 a 4 6 t a u ru s n if ty i n d e x f u n d d ir e ct p la n * * * a v e ra g e 1 9 .0 6 1 7 .1 1 0 .3 7 0 .6 0 -0 .2 6 -0 .2 0 0 .9 9 0 .9 8 1 3 . 6 3 1 .2 2 a 4 7 u t i m a st e rs h a re f u n d d ir e ct p la n * * * b e lo w a v e ra g e 5 5 3 0 . 6 6 1 9 .5 7 0 .3 0 0 .5 3 -0 .4 9 -0 .9 7 0 .9 6 0 .9 1 1 2 . 9 6 1 .4 3 a 4 8 u t i n if ty i n d e x f u n d d ir e ct p la n * * * a v e ra g e 9 3 5 .9 4 1 7 .1 8 0 .3 7 0 .5 9 -1 .4 6 -0 .2 5 1 .0 0 0 .9 9 1 3 . 7 9 0 .1 2 an ensemble approach for portfolio selection in a multi-criteria decision making framework 157 appendix 2. calculation of efficiency using dea funds under study input output vrs result rank vrs kurtosis kurtosis (normalized) expense ratio psv/ nsv q3 dmu1 -0.4904 0.5013 1.2600 1.0549 6.6575 16.07% 31 dmu2 -0.7490 0.3612 1.3100 1.0393 7.1600 13.37% 36 dmu3 -0.9942 0.2284 0.5100 0.6925 6.3250 27.70% 20 dmu4 -0.5445 0.4720 0.9400 0.5358 7.4000 16.62% 30 dmu5 0.4301 1.0000 1.0500 1.1908 7.5125 41.96% 17 dmu6 0.2381 0.8960 1.4800 0.7074 5.9725 9.31% 43 dmu7 -0.0014 0.7662 1.5100 0.7634 5.5975 9.27% 44 dmu8 -0.9583 0.2478 1.4700 0.6868 6.0125 10.06% 41 dmu9 -0.5300 0.4798 0.5800 0.6661 6.8850 22.95% 25 dmu10 0.0687 0.8043 2.0900 1.7130 6.0650 100% 7 dmu11 -0.5220 0.4842 1.1600 0.7371 4.7950 12.24% 38 dmu12 -1.0308 0.2085 0.6400 0.7048 6.3925 22.51% 26 dmu13 -1.3921 0.0128 0.1500 1.0249 7.3475 152.72% 2 dmu14 -1.0697 0.1874 0.1500 0.6978 6.6575 83.42% 11 dmu15 -0.9863 0.2327 1.2600 1.6584 8.5375 100% 7 dmu16 -0.5595 0.4639 1.5100 0.7604 6.7950 9.57% 42 dmu17 -1.2878 0.0693 1.1700 0.9866 6.7800 15.06% 32 dmu18 -1.0226 0.2130 0.5700 0.7052 6.5200 25.08% 22 dmu19 -0.8677 0.2969 0.4400 1.1672 9.8025 146.00% 3 dmu20 -0.0496 0.7401 1.0600 1.0686 6.1950 21.37% 27 dmu21 -1.0472 0.1996 0.4300 0.7086 6.5225 32.76% 18 dmu22 -0.9146 0.2715 0.5600 1.1773 9.7425 102.00% 6 dmu23 -1.0924 0.1751 1.8300 0.5602 6.7950 8.18% 46 dmu24 -1.0202 0.2143 0.1700 0.7021 6.5925 73.49% 13 dmu25 -0.7569 0.3569 0.7500 0.7040 8.0900 31.69% 19 dmu26 -0.7229 0.3754 0.9600 0.6534 6.3100 14.85% 33 dmu27 -1.3407 0.0406 2.9400 0.7030 12.0225 100% 7 dmu28 -0.6121 0.4354 1.6100 0.7645 3.8550 9.03% 45 dmu29 -0.5916 0.4465 1.1500 1.0974 5.9675 24.09% 24 dmu30 -0.3986 0.5510 2.0400 0.7255 6.5800 7.13% 47 dmu31 -1.0855 0.1789 0.6900 0.6874 6.5025 21.11% 28 dmu32 -1.4157 0.0000 1.1500 1.0384 7.1800 100% 7 dmu33 -1.3228 0.0504 1.5200 0.5927 5.8700 14.05% 35 dmu34 -0.8733 0.2939 1.1500 0.6444 7.5100 14.71% 34 dmu35 -0.7201 0.3768 0.5000 0.7821 4.9900 26.93% 21 dmu36 -0.9688 0.2421 0.2900 0.7066 6.6100 45.86% 16 dmu37 -1.4146 0.0006 0.2900 1.0276 7.0450 303.43% 1 dmu38 -0.9501 0.2523 1.3200 1.5284 7.0700 78.19% 12 dmu39 -0.6039 0.4398 1.1800 0.8370 7.2125 12.26% 37 dmu40 -1.0468 0.1999 0.2900 0.7034 6.6075 46.89% 15 dmu41 -0.1160 0.7042 0.5500 0.8305 5.9050 24.10% 23 dmu42 -1.0639 0.1906 0.2100 0.7021 6.6850 62.64% 14 biwas et al/decis. mak. appl. manag. eng. 2 (2) (2019) 138-158 158 funds under study input output vrs result rank vrs kurtosis kurtosis (normalized) expense ratio psv/ nsv q3 dmu43 -1.4037 0.0065 0.2200 1.0335 7.2400 141.03% 4 dmu44 -0.8221 0.3216 0.7400 0.6315 6.6900 19.13% 29 dmu45 -1.0498 0.1983 2.1500 0.4023 6.5025 6.97% 48 dmu46 -0.6233 0.4293 1.2200 0.7896 6.4950 11.76% 39 dmu47 -0.8883 0.2857 1.4300 0.7594 5.8050 10.29% 40 dmu48 -1.0725 0.1860 0.1200 0.6911 6.5875 125.00% 5 plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering vol. 5, issue 2, 2022, pp. 1-29. issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0305102022t * corresponding author. e-mail addresses: magdalena.tutak@polsl.pl (m. tutak), jaroslaw.brodny@polsl.pl (j. brodny) evaluating differences in the level of working conditions between the european union member states using topsis and k-means methods magdalena tutak1*, and jarosław brodny1 1 silesian university of technology, gliwice, poland received: 5 september 2022; accepted: 5 october 2022; available online: 5 october 2022. original scientific paper abstract: work, which is a conscious activity of man, plays an immensely important role in their life and is the basis for the development of civilization. the work process is closely related to the conditions in which work is performed. these conditions include a number of social, technical, environmental as well as economic and organizational factors necessary to perform work safely in accordance with the applicable legal conditions. the role and importance of working conditions is appreciated by all organizations, countries and their groups taking action to improve them, including formal order. given the importance and topicality of this issue, research has been carried out, the main goal of which was to assess the level of working conditions in the european union (eu) countries according to the adopted criteria. the research was based on data from the european foundation for the improvement of living and working conditions (eurofound). accordingly, eight main criteria were adopted, which were characterized by 64 subindicators. such a broad approach to describing individual areas related to working conditions made it possible to analyze many factors influencing them. the research covered the 27 eu member states by determining indicators for working conditions criteria and an indicator for general (overall) working conditions. on this basis, their ranking and the level of working conditions in these countries were specified. the topsis method was applied to this part of the research. with the use of partial levels of working conditions evaluation criteria and the k-means method, the authors identified countries similar in terms of the level of studied working conditions criteria. based on the spearman's rho and kendall's tau correlation coefficients, relationships were examined between the working conditions and the level of economic development and indicators characterizing the area of health and safety at work in the countries under study, which is very important from the point of view of working conditions. the results showed significant differences in working conditions between the eu-27. they were found to be definitely worse in the economically less developed countries (mainly the so-called "new" eu) than in the economically stronger states (the so-called "old" eu countries). the tutak et al./decis. mak. appl. manag. eng. 5 (2) (2022) 1-29 2 assessment and groups of similar countries in terms of working conditions should be used to develop strategies to improve these conditions in the eu-27. this is particularly significant in the context of dynamic technological, social and geopolitical changes across europe, which have a significant impact on the labor market. key words: working conditions, work-life balance, sustainability, labor market, eu-27, health and safety at work, mcdm method. 1. introduction one of the most important factors that determines the quality of human life in the modern world is gainful employment (clark, 2001; haller & hadler, 2006). it is the basis of existence, providing the means of subsistence, and thus determines the quality and sense of life as well as provides a clear time structure, a sense of identity, social status and integration and opportunities for personal development (grote & guest, 2017; mosadeghrad et al., 2011; taylor-gooby, 2008). therefore, it can be assumed that the necessity of work in a social and individual sense is fundamental for human existence. it is also obvious that performing work, especially paid work, occupies a significant part of a person's life. on average, a worker in the eu-27 works 1513 hours per year. the lowest workload is reported in germany (1332 hours), denmark (1336), the netherlands (1399) and austria (1400), while the highest in malta (1827), croatia (1834), romania (1795), and poland (1766) (oecd.stat). given the role and importance of work for individual and social development and the amount of time spent on it, it is reasonable to address the issue of conditions in which work is performed. it is mainly about providing employees with safety against loss of life and health and favorable circumstances for carrying out the work process. more often than not, work conditions are defined as a set of factors present in the work environment, resulting from the realized process and factors related to work performance, which affect the mental and physical well-being of employees (feldman et al., 2002; matthews et al., 2010; tutak et al., 2020). important factors affecting the quality of work are also the elements related to working time (working hours, rest, etc.) and remuneration (davidescu et al., 2020). it is obvious that providing favorable working conditions has a positive effect on the quality and efficiency of work performed, both from the point of view of the employer and the employee. however, performing work in unfavorable conditions has a negative impact on the work process and may translate into unsatisfactory results. the risk of physical and mental fatigue resulting from the high quantitative and qualitative demands to which employees may be exposed can cause a threat to their health and disturb their work-life balance. in the long run, this situation can have very negative economic and social consequences for the workers themselves, the employer and society. therefore, working conditions and professional life associated with them have a significant impact on the physical and psychological state of the employee, including in particular motivation and commitment at work. published research results indicate that appropriate working conditions and work organization are crucial for the ability to perform work, health, well-being and skills of workers (davidescu et al., 2020; dorenbosch, 2014; de wind et al., 2016). safe and hygienic working conditions significantly affect the quality of both work and life outside work (greubel et al., 2016; lunau et al., 2014). the role and importance of proper working conditions, which are the responsibility of the employer, is appreciated in many countries and regions of the world. this also applies to the evaluating differences in the level of working conditions between the european union… 3 european union (eu), where ensuring safe and employee-friendly working conditions is one of the basic assumptions of the social and economic policy. this is confirmed by numerous pieces of legislation, as well as the european parliament resolution of 10 march 2022 on a new eu strategic framework for health and safety at work after 2020 (including better protection of workers from exposure to harmful substances, work stress and repetitive motion injuries). according to article 151 of the eu treaty, member states should actively work to improve living and working conditions. at the same time, individual countries may set stricter standards than those laid down in eu directives. as a result, working conditions vary considerably across the eu member states. this is due to the wealth of countries, changes in demographics and employment structure resulting from the ongoing processes of digitalization, development of new technologies and increasing labor market flexibility and fragmentation of work. all these factors are increasingly affecting working conditions in the eu member states. therefore, given the role and importance of working conditions for the economic development of the eu, as well as the dynamic changes taking place in the labor market in the recent period, it is fully justified to conduct research on the assessment of the level of working conditions in the eu-27 countries. the main purpose of this research was to determine, according to the adopted criteria, the level of working conditions in the eu-27 countries, to compare this level and to find out whether this level is related to the economic development of these countries. an important objective of the research was also to check for similarities between these countries in the level of criteria for evaluating working conditions. the evaluation of the level of working conditions in the eu-27 countries was based on 8 criteria, including: physical environment, work intensity, working time, social environment, skills, discretion and other cognitive factors, prospects, job and company context, and working life perspectives. these criteria were characterized, in total, by 64 diagnostic variables (sub-indicators). for this part of the study, the topsis approach, from the mcdm methods group, was used. similarities between the studied countries were determined based on the k-means method. non-parametric tests such as the kendall's tau and spearman's rank correlation coefficients were used to examine the relationship between the level of working conditions and the economic development of individual countries and indicators describing the area of health and safety at work. the research presented in this paper, conducted for the first time in such a scope, fills the existing research gap resulting from the lack of such analyses for the eu market and makes a new contribution to the existing literature in several aspects. for the first time, an assessment of the level of working conditions in the eu-27 was carried out using the topsis method (from the mcdm group of methods), dedicated to this type of analysis. the application of this method made it possible to assess objectively the level of working conditions, considering eight evaluation criteria which cover the most important aspects of this issue. their characteristics include as many as 64 sub-indicators, which proves its comprehensiveness and transparency. it also made it possible to determine similarities between the studied countries in terms of the level of particular criteria of this evaluation, which is also a new approach to studying working conditions in the eu-27. this is particularly important from the point of view of economic diversification of these countries and the assessment of the impact on working conditions in these countries. another important and new element presented in this paper is the research on the relationship between the level of working conditions in the eu-27 countries and their economic development and the tutak et al./decis. mak. appl. manag. eng. 5 (2) (2022) 1-29 4 values of indicators characterizing the area of health and safety at work, which is crucial from the point of view of occupational safety. the novelty of the presented research as well as the importance and significance of the problem of working conditions for the economic development of the eu in the dynamically changing labor market and for the employees and employers themselves fully justifies the advisability of undertaking this research. the application of a new analytical approach to the study of this topic guarantees new knowledge regarding the characteristics of the eu-27 countries in terms of the level of working conditions in these countries. 2. literature review the presented literature review focuses on issues related to the influence of working conditions on productivity and effectiveness of work and workers' health condition (physical and mental). the number of publications related to this subject has been growing recently, which proves its importance and topicality. published research has most often concerned the effect of working conditions on employee satisfaction (agbozo et al., 2017), the effect of presenteeism, working conditions and absenteeism on work performance (stromberg et al., 2017), as well as the effect of personality on employee performance (mustafa & ali, 2019). recently, an increasing number of studies have also examined motivation as a determinant of job performance (yuen et al., 2018) and the effect of working conditions on employees' physical and psychological well-being (arenas et al., 2015; jeong et al., 2020; schütte et al., 2014). a number of research results indicate the importance of working conditions that enable employees to properly perform their duties and use their full potential. this mainly affects work performance and employee satisfaction (masadeh et al., 2016). work performance is often taken up in studies, the results of which indicate its strict dependence on working conditions. the fewer factors limiting an employee, the more effectively the employee is able to perform his or her tasks (guan & frenkel, 2019; marshall et al., 2015; matsuo, 2019; yozgat et al., 2013). on the other hand, inadequate working conditions disrupt the work process, often causing employees to increase their energy expenditure while performing assigned tasks (jimenez et al., 2017). as shown by one study (mustafa & ali, 2019; rossberg & friss, 2004), poor working conditions are one of the main reasons for high employee turnover in companies and low satisfaction and productivity. currently, much research in the scientific literature also addresses the issue of working conditions and satisfaction with work-life balance (gragnano et al., 2020; isaacs., 2016; rich et al., 2016). the literature is also rich in articles on physical and ergonomic factors affecting working conditions for specific companies or occupational groups. this includes teachers (amin, 2015), members of the professional guard (nelson et al., 2015), as well as shearer loader assembly line workers (saurin & ferreira, 2009) and standing sewing machine operators (nagaraj et al., 2019). in the latter case, the body parts where musculoskeletal symptoms are most common during such work (knee, foot, thigh, shin, and lower back) were identified. similar studies have been conducted for workers who perform hand sewing of shoes (diant & salimi, 2014). the results of these studies clearly indicate that work experience, daily working hours, working time without breaks, feeling pressure to perform tasks, and posture during work have a significant impact on the health of workers. evaluating differences in the level of working conditions between the european union… 5 the analysis of the literature also shows a lack of works on the formation of working conditions in different countries, including both developed and developing countries. this is because such a comprehensive approach provides an opportunity for a broader assessment of the processes related to working conditions in individual countries and regions. in this regard, it is possible to only mention studies on working conditions at the macro level, in different countries, including the eu countries, and relating to the issue of satisfaction with work-life balance (greubel et al., 2016; lunau et al., 2014; wepfer et al, 2015). however, the existing literature lacks studies on working conditions in different groups of countries, including the eu countries. this is important because nowadays, especially in europe, major changes can be observed in the labor market related to the ongoing industrial revolution, coronavirus pandemic and geopolitical turmoil. these phenomena have a tremendous impact on the labor market, for which the conditions of work performance can be a serious asset in attracting new employees with appropriate competencies. given the current state of the literature, it is reasonable to fill the arising research gap by carrying out comprehensive studies on the assessment of the level of working conditions in the eu-27 countries. an approach based on mcdm methods will be used to evaluate working conditions. this methodology includes a set of methods, for multi-criteria decision support. most often, the purpose of their application – through analysis and synthesis of discrete multi-criteria problems – is to evaluate a set of alternatives in terms of multiple, often conflicting, decision criteria that characterize a phenomenon or process (zavadskas & turskis, 2011; zavadskas et al., 2014). thus, having a set of alternatives (objects) and a number of evaluation criteria, the goal of applying the mcdm method/methods is to create a ranking of alternatives, e.g., from best to worst. a number of different methodologies for the application of mcdm methods have been presented in the literature, along with their characteristics, level of complexity and varied range of applications (božanić et al., 2021; kizielewicz et al., 2021; opricovic & tzeng, 2004; peng et al., 2011; stojčić et al., 2019; yorulmaz et al., 2021). studies to date have used mcdm methods, for example, to evaluate the development of renewable energy sources (kumar et al., 2017), energy security (tutak & brodny, 2022), sustainable development (stanujkic et al., 2020), selection of materials in engineering applications (emovon & oghenenyerovwho, 2020), evaluation of public transportation (kundu et al., 2014), and many other phenomena and processes. by contrast, for the evaluation of working conditions, these methods have not been used so far, which can undoubtedly be considered a research gap in this area. on the other hand, the multifaceted nature of the issues related to the labor market and its conditions fully predestines them for such application. 3. material and methods this section introduces the study area and the source from which the data were obtained, which are then discussed in section 3.2. sections 3.3, 3.4 and 3.5 characterize the research methods used. 3.1. area of research the european union, the countries of which were enrolled in the study, has 27 member states (as of 2020) (figure 1.) and occupies a combined area of 4.4 million km2 on the european continent. it is inhabited by approximately 437 million people. tutak et al./decis. mak. appl. manag. eng. 5 (2) (2022) 1-29 6 the promotion of employment, better living and working conditions, adequate social protection, dialogue between management and employees, the development of personnel resources to ensure sustainable and high levels of employment and the prevention of social exclusion are the common objectives of the eu and the member states in the field of social and employment policy. these tasks, together with the free movement of workers, constitute one of the main pillars of the eu, listed in article 151 treaty on the functioning of the european union (treaty on the functioning of the european union, 2012). it should also be emphasized that the eu-27 is the only economic and political union of its kind in the world, bringing together as many as 27 countries with diverse energy structures, employment, economic levels and wealth of societies. the origins of the idea to create the eu go back to the period after world war ii. the first stages of integration of european countries consisted of improving economic cooperation according to the principle that countries that trade with each other benefit from it, avoiding conflicts. the result of this premise was the creation of the european economic community (eec) in 1958. initially, economic cooperation involved six countries: belgium, france, germany, italy, luxembourg, and the netherlands. since then, 22 more members have joined the eu and a huge single and common market for the free movement of labor, capital and goods has been created, which is constantly developing its potential. it should also be noted that on january 31, 2020, the united kingdom became the first country to withdraw from the eu. figure 1. the eu-27 countries area (own elaboration) 3.2. data data from the european foundation for the improvement of living and working conditions (eurofound) database were used for the study (eurofound). eurofound is a tripartite eu agency that conducts activities to improve living and working conditions. as part of its activities since 2005, it has been conducting "european working conditions surveys" in 36 european countries, including 27 eu member states. from 2005 to 2015, the survey was conducted on a 5-year cycle basis. evaluating differences in the level of working conditions between the european union… 7 the next survey was conducted not in 2020, but in 2021, due to the ongoing pandemic of the sars coronavirus cov-2. the working conditions survey was conducted on a sample of over 44,000 respondents using the cati method (computer-assisted telephone interview). respondents were selected by random direct calls to cell phone numbers. sample sizes ranged from 1000 to 4200 interviews per country. this sample size for each country enabled very robust estimates to be made at the european level and allowed information to be collected and analyses to be performed on working conditions in the surveyed countries and the eu as a whole. the working conditions survey covered a total of 8 evaluation criteria: physical environment, work intensity, working time, social environment, skills, discretion and cognitive factors, prospects, job and company context, and working life perspectives. a total of 64 indicators were used for the eight criteria for evaluating working conditions (table 1). table 1. list of criteria and indicators used to evaluate working conditions criteria to evaluate working conditions evaluation indicators physical environment (12) exposure to high temperature exposure to low temperature exposure to smoke, vapors, dust or particles exposure to inhalation of vapors exposure to chemical products or substances exposure to tobacco smoke from other people exposure to materials which may be infectious work requires strenuous or painful positions work requires lifting or carrying people work requires carrying or moving heavy loads work requires repetitive hand or arm movements work requires wearing personal protective equipment work intensity (6) work at very high speed work requires meeting tight deadlines adequate time to perform one's duties frequent interruptions that disrupt work rhythm number of factors influencing the pace of work work requires participating in situations that cause emotional discomfort working time (6) number of hours per week spent on main paid work performing more than 10 hours of work per month doing night work doing weekend work doing shift work doing work according to a set start and end time social environment (5) exposure to undesirable social behavior support and assistance by co-workers support and assistance by a manager/head fair treatment at work experience of discrimination at work in the past 12 months skills, discretion work requires solving unforeseen problems independently work requires performing complex tasks tutak et al./decis. mak. appl. manag. eng. 5 (2) (2022) 1-29 8 criteria to evaluate working conditions evaluation indicators and other cognitive factors (15) work requires learning new things work requires using computers, laptops, smartphones, etc. while performing work, it is possible to select tasks to be performed or to change the order of performing these tasks while performing work, it is possible to select or change the speed or pace of work while performing work, it is possible to choose or change the manner of work possibility to have a say in selecting co-workers possibility to participate in improving the work organization or work processes in a department or organization possibility to influence decisions that are important to the job participation in training paid for by an employer (or by oneself if self-employed) within the last 12 months participation in on-the-job training in the past 12 months work involves monotonous activities possibility to take a break at any time prospects (3) job provides good prospects for further career development workplace restructuring or reorganization that has had a significant impact on your job in the past three years possibility to find a new job with similar pay after losing current job or leaving voluntarily job and company context (7) change in hours worked per week in the last 12 months change in salary or income in the last 12 months knowledge of health and safety risks associated with the job exposure to health or safety risks because of the job trade union/works council or similar organizations representing workers in the company health and safety delegate or committee within an organization regular meetings at which workers can express their views on the situation in a company working life perspectives (10) sense of work being well done well done work gives a sense of well-done duty sense of work being useful feeling adequately rewarded for work work makes a difference to health feeling that current job or a similar job will be possible until age 60 number of hours per week spent doing paid and unpaid work worrying about work-related issues outside of work hours in the past 12 months feeling tired after work, which prevents from doing housework that needed to be done (in the past 12 months) devoting an inadequate amount of time to family because of work (in the past 12 months) evaluating differences in the level of working conditions between the european union… 9 3.3. the topsis methods the technique for order preference by similarity to ideal solution (topsis) method was used for a multivariate analysis aimed at assessing the level of working conditions in the eu-27 countries. this method consists in comparing the vector of values of decision (evaluation) criteria for a given object (eu country) with vectors of ideal and anti-ideal solution. the vector of ideal solution is the vector of values selected as the best from the set of values available for each of the indicators in the whole set of considered objects. similarly, the negatively ideal vector is the vector of the worst values. in order to assess a given object (eu country) and compare it with others, it is necessary to determine the distance in euclidean space between the vector of values of a given object and vectors: ideal and anti-ideal. the best object is the one of the objects (eu countries), for which its value vector has simultaneously the smallest distance from the ideal vector and the largest from the negatively ideal vector. the idea of topsis method is to determine the distance of considered objects (eu countries) from ideal (pattern) and anti-ideal (anti-pattern) solution. the result of the analysis is a synthetic index that creates a ranking of the studied countries. the best country is considered to be the one with the smallest distance from the ideal solution and at the same time the largest distance from the anti-ideal solution (chakraborty, 2022). the algorithm of the research procedure using the topsis method consists of the following steps:  to construct a new decision matrix: 𝑋 = [𝑥𝑖𝑗 ]𝑚×𝑛 = [ 𝑥11 ⋯ 𝑥1𝑛 ⋮ ⋱ ⋮ 𝑥𝑚1 ⋯ 𝑥𝑚𝑛 ] (1) where: where n is the number of alternatives and m is the number of criteria  to calculate a normalized decision matrix: 𝑥𝑖𝑗 = 𝑥𝑖𝑗 √∑ 𝑥𝑖𝑗 2𝑚 𝑖=1 ; ∀𝑖, 𝑗 (2)  to determine the ideal s+ and anti-ideal salternative: 𝑆 + = (𝑥1 +, 𝑥2 +, 𝑥3 +, … . 𝑥𝑛 + ) = {(𝑚𝑎𝑥𝑖 𝑥𝑖𝑗 |𝑗 ∈ 𝐵|), (𝑚𝑖𝑛𝑖 , 𝑥𝑖𝑗 |𝑗 ∈ 𝐶|)} (3) 𝑆 − = (𝑥1 −, 𝑥2 −, 𝑥3 −, … . 𝑥𝑛 − ) = {(𝑚𝑖𝑛𝑖 𝑥𝑖𝑗 |𝑗 ∈ 𝐵|), (𝑚𝑎𝑥𝑖 , 𝑥𝑖𝑗 |𝑗 ∈ 𝐶|)} (4)  to determine the euclidean distance of a given alternative from the ideal alternative s+ and anti-ideal alternative s: 𝑑𝑖 + = √∑ (𝑥𝑖𝑗 − 𝑥𝑗 +) 2𝑛 𝑗=1 (5) 𝑑𝑖 − = √∑ (𝑥𝑖𝑗 − 𝑥𝑗 −) 2𝑛 𝑗=1 (6)  to determine the coefficient of relative closeness of alternatives si to the ideal alternative s+ (indicator of working conditions criteria): 𝑃𝑖 = 𝑑𝑖 − 𝑑𝑖 ++𝑑𝑖 − (7) tutak et al./decis. mak. appl. manag. eng. 5 (2) (2022) 1-29 10 the values of the indicator of relative closeness of objects to the ideal object in the topsis method are within the range from 0 to 1, and the higher the value of the indicator, the higher position a country achieves in the ranking. in the study of the evaluation of the level of working conditions, the topsis index was determined for each of the eight evaluation criteria, i.e., physical environment, work intensity, working time, social environment, skills, discretion and cognitive factors, prospects, job and company context, and working life perspectives. on the other hand, the final value of the topsis index is the sum of the indices determined for the eight evaluation criteria: 𝑃𝑖 ∗ = ∑ 𝑃𝑖𝑗 𝑖 𝑗=8 (8) based on the determined values of the topsis index (𝑃𝑖𝑗 )) for each criterion of working conditions evaluation and based on the final value of this index 𝑃𝑖 ∗, the levels for each criterion and the overall level of working conditions in the eu-27 countries were determined. a division was made into four classes of working conditions level and is as follows: class 1 – very high level 𝑃𝑖 ∗ ≥ 𝑃𝑖 ∗̅̅ ̅ + 𝑠𝑃𝑖 ∗ (overall evaluation) (9) 𝑃𝑖𝑗 ≥ 𝑃𝑖𝑗̅̅ ̅ + 𝑠𝑃𝑖𝑗 (criterion evaluation) (10) class 2 –high level 𝑃𝑖 ∗̅̅ ̅ + 𝑠𝑃𝑖 ∗ > 𝑃𝑖 ∗ ≥ 𝑃𝑖 ∗̅̅ ̅ (overall evaluation) (11) 𝑃𝑖𝑗̅̅ ̅ + 𝑠𝑃𝑖𝑗 > 𝑃𝑖𝑗 ≥ 𝑃𝑖𝑗 ̅̅ ̅ (criterion evaluation) (12) class 3 – acceptable level 𝑃�̅� > 𝑃𝑖 ∗ ≥ 𝑃�̅� − 𝑠𝑃𝑖 (overall evaluation) (13) 𝑃𝑖𝑗̅̅ ̅ > 𝑃𝑖𝑗 ≥ 𝑃𝑖𝑗̅̅ ̅ − 𝑠𝑃𝑖𝑗 (criterion evaluation) (14) class 4– low level 𝑃𝑖 ∗ < 𝑃�̅� − 𝑠𝑃𝑖 (overall evaluation) (15) 𝑃𝑖𝑗 < 𝑃𝑖𝑗̅̅ ̅ − 𝑠𝑃𝑖𝑗 (criterion evaluation) (16) where: 𝑃𝑖 ∗ is the working conditions indicator of a given country; pij is indicator of working conditions criterion of a given country; 𝑃𝑖 ∗̅̅ ̅ is the average value of the 𝑃𝑖 ∗ indicator for the population of countries under study; 𝑃𝑖𝑗̅̅ ̅ is the average value of the 𝑃𝑖𝑗 indicator for the population of countries under study; 𝑠𝑃𝑖 is the standard deviation of the mean value of the 𝑃𝑖 ∗ indicator for the population of countries under study and 𝑠𝑃𝑖𝑗 is the standard deviation from the mean value of the 𝑃𝑖𝑗 indicator the population of countries under study. based on the values of the criteria for evaluating working conditions 𝑃𝑖𝑗 , groups of similar countries were identified. 3.4. the k-menas methods the k-means clustering method belongs to non-hierarchical clustering methods. the analysis is based on finding and separating groups of similar objects (clusters). according to the algorithm of this method, k (given a priori) different, possibly dissimilar clusters are created. in the next stage, the objects are moved from cluster to cluster until the intra-cluster variation becomes optimized. clusters created in accordance with the algorithm are characterized by the greatest similarity between countries contained in them. individual clusters, on the other hand, should differ from one another as much as possible. one of the most important stages of grouping evaluating differences in the level of working conditions between the european union… 11 countries by the k-means method is to determine cluster centers and the euclidean distance, which determines the distance of the tested object (country) from the center of gravity of the cluster (17): 𝑑(𝑥, 𝑦) = √∑ (𝑥𝑖 − 𝑦𝑖 ) 2𝑝 𝑖=1 (17) where: d(x,y) is euclidean distance; p is number of objects (eu-27 countries). on the basis of the determined euclidean distances, the process of assigning the examined objects (eu-27 countries) to groups of similar countries/clusters was carried out. 3.5. method for analyzing the relationship between the level of working conditions and economic development of individual countries two non-parametric tests in the form of the kendall's tau and spearman's rank correlation coefficients were used to determine a relationship between the level of working conditions in the eu-27 countries and economic development and indicators characterizing health and safety at work. the kendall's tau correlation coefficient provides a measure of the monotonic relationship between two random variables (x, y). the kendall's tau correlation coefficient takes values between -1 and +1 inclusive. this coefficient shows both the direction and the strength of a given relationship. it is determined from the following equation: 𝜏(𝑋, 𝑌) = 2𝑃[(𝑋1 − 𝑋2)(𝑌1 − 𝑌2) > 0] − 1 (18) the second test used in this study was the spearman's rho coefficient. it is also one of the non-parametric measures of monotonic statistical relationship between random variables (x, y). it is used to analyze the interdependence of objects in terms of a twodimensional feature (x, y). the correlation coefficient is used in analyses of the interdependence of objects with respect to two-dimensional feature (x, y). the spearman's rho correlation coefficient can take values from -1 to +1 inclusive and is calculated from the following equation: 𝜌𝑠 (𝑋, 𝑌) = 3[𝑃𝑟 [(𝑋 − 𝑋1) ∙ (𝑌 − 𝑌2) > 0] − 𝑃𝑟 [(𝑋 − 𝑋1) ∙ (𝑌 − 𝑌2) < 0]] (19) a diagram showing the research procedure is shown in figure 2. tutak et al./decis. mak. appl. manag. eng. 5 (2) (2022) 1-29 12 figure 2. diagram of the research procedure 4. results based on the data obtained from the eurofound database and the methods discussed, research methodology was developed, and the study was conducted, the results of which are presented in this section. according to the adopted methodology, these results were divided into preliminary and fundamental. the preliminary research (statistical analysis of data) involved determining values for selected indicators characterizing working conditions in the eu-27 countries. the fundamental research involved evaluating the level of working conditions in particular countries and determining their relation to economic development of these countries and the area of health and safety at work, as well as determining similarities between the examined countries. 4.1. preliminary statistical analysis the indicators (diagnostic variables) characterizing the working conditions used in the study were subjected to preliminary processing and their basic statistical parameters were determined. the results for the selected group of indicators are presented in table 2. evaluating differences in the level of working conditions between the european union… 13 table 2. basic descriptive statistics for selected indicators related to working conditions indicators mean median min max standard deviation coefficient of variation skewness kurtosis exposure to chemical products or substances more than 1/4 of the work time 15.84 16.00 8.90 22.00 3.33 20.99 -0.04 -0.54 less than 1/4 of the work time 84.16 84.00 78.00 91.10 3.33 3.95 0.04 -0.54 exposure to materials that may be infectious more than 1/4 of the work time 12.41 12.20 6.70 20.30 3.18 25.60 0.39 0.16 less than 1/4 of the work time 87.59 87.80 79.70 93.30 3.18 3.63 -0.39 0.16 work requires wearing personal protective equipment no 59.68 59.50 44.90 75.40 8.05 13.49 0.02 -0.45 yes 40.32 36.90 21.80 51.30 7.37 20.08 0.10 0.04 work requires doing tasks at high speed more than 1/4 of the work time 60.93 60.90 34.20 89.60 13.18 21.64 0.08 0.19 never 39.07 39.10 10.50 65.80 13.18 33.73 -0.08 0.19 work requires meeting tight deadlines more than 1/4 of the work time 62.53 62.20 49.80 80.50 7.68 12.28 0.44 -0.02 never 37.47 37.80 19.50 50.20 7.68 20.48 -0.44 -0.02 number of factors influencing the pace of work 0-2 66.61 68.10 49.00 78.00 6.90 10.36 -0.90 0.83 3 do 5 33.38 31.90 22.00 51.10 6.91 20.70 0.90 0.83 number of hours per week spent on main paid work up to 40 hours 75.28 77.60 56.90 85.40 7.31 9.72 -0.92 0.36 over 40 hours 24.71 22.30 14.60 43.10 7.31 29.57 0.92 0.37 doing night work never 80.28 81.60 73.70 87.00 3.75 4.67 -0.34 -1.07 at least once 19.72 18.40 13.00 26.30 3.75 19.02 0.34 -1.07 exposure to undesirable social behavior yes 75.28 77.60 56.90 85.40 7.31 9.72 -0.92 0.36 no 24.71 22.30 14.60 43.10 7.31 29.57 0.92 0.37 experience of discrimination at work in the past 12 months yes 21.19 21.00 10.40 40.20 6.39 30.18 0.84 1.88 no 78.81 79.00 59.80 89.60 6.39 8.11 -0.84 1.88 work requires performing complex tasks yes 62.36 65.80 38.80 80.00 11.16 17.89 -0.53 -0.38 no 37.64 34.20 20.00 61.20 11.16 29.64 0.53 -0.38 participation in training paid for by the employer (or by oneself if self-employed) within the last 12 months yes 36.47 37.90 8.60 53.90 12.13 33.27 -0.55 -0.51 no 63.53 62.10 46.10 91.40 12.13 19.10 0.55 -0.51 job provides good prospects for further career development yes 40.00 41.60 26.70 52.80 6.78 16.96 -0.30 -0.64 no 36.40 37.10 24.20 46.70 6.10 16.75 -0.08 -0.81 tutak et al./decis. mak. appl. manag. eng. 5 (2) (2022) 1-29 14 indicators mean median min max standard deviation coefficient of variation skewness kurtosis knowledge of the health and safety risks associated with the job very good 90.33 91.60 78.90 96.40 4.88 5.40 -0.85 0.14 not the best 9.67 8.40 3.60 21.10 4.88 50.42 0.85 0.14 exposure to health or safety risks because of the job yes 25.34 25.30 13.10 46.80 7.70 30.38 0.70 0.99 no 74.66 74.70 53.20 86.90 7.70 10.31 -0.70 0.99 feeling adequately rewarded for work yes 50.46 48.20 32.20 66.90 9.07 17.97 0.10 -0.71 no 49.54 51.80 33.10 67.80 9.07 18.31 -0.10 -0.71 worrying about work-related issues outside of work hours in the past 12 months always, often 14.75 13.40 4.70 26.10 5.89 39.94 0.38 -0.90 sometimes, never 85.25 86.60 73.90 95.30 5.89 6.91 -0.38 -0.90 when looking at the results, on average, nearly 16% of workers in the eu-27 are exposed to chemical products or substances for more than a quarter of their working time, with the highest proportion in hungary (22%) and the lowest in the netherlands (8.9%). on average, nearly 40% of workers in the eu-27 are required to use personal protective equipment at their workplace, with the highest proportion in slovenia (51.3%) and the lowest in greece (21.8%). eu-27 workers who spend up to 40 hours a week on paid work account for just over 75%, with the highest proportion in the netherlands (85.4%) and the lowest in greece (56.9%). an average of 75% of workers in the eu-27 are exposed to undesirable social behavior at work, with the highest proportion in the netherlands (85.5%) and the lowest in greece (56.9%). across the eu-27, on average, only just over 50% of workers feel adequately rewarded for their work. the best situation in this respect is found in denmark, where 66.9% of workers feel that they are well paid for their work, and the worst situation is reported in greece, where only just over 32% of workers confirm that they are well paid for their work. as regards upskilling, only 37% of the eu-27 workforce had participated in an employer-paid training course in the 12 months prior to the survey. the worst situation in this respect was reported in greece, where less than 9% of employees had participated in training, and the best situation in finland (more than 50% of employees). 4.2. analysis and evaluation of working conditions in the first stage of fundamental research, using the topsis method, an index of working conditions criteria was determined for the eu countries and their rankings were made in terms of the value of these indices. the results are presented in table 3. then, based on the values of the working conditions criteria indices pi, the level of these conditions was assessed for each of the eight criteria: physical environment, work intensity, working time, social environment, skills, discretion and other cognitive factors, prospects, job and company context, and working life perspectives. to assess this level, a 4-point scale was adopted (according to equations 9-16), using the arithmetic mean and standard deviation calculated from the values of indices of evaluating differences in the level of working conditions between the european union… 15 working conditions pi (equation 7). the results of the assessment conducted are shown in table 4. table 3. indices and rankings of the eu countries for particular criteria of working conditions evaluation physical environment work intensity working time social environment skills, discretion and other cognitive factors working life perspectives job and company context prospects pi ra nk pi ra nk pi ra nk pi ra nk pi ra nk pi ra nk pi ra nk pi ra nk at 0.71 10 0.58 14 0.79 10 0.67 6 0.67 12 0.38 11 0.56 10 0.91 1 be 0.85 4 0.50 17 0.81 7 0.64 7 0.73 8 0.42 10 0.48 13 0.72 8 bg 0.79 7 0.94 2 0.80 9 0.68 5 0.20 26 0.19 18 0.46 15 0.81 5 cy 0.47 20 0.15 27 0.93 1 0.82 1 0.21 25 0.30 12 0.24 24 0.65 13 cz 0.69 11 0.78 7 0.34 26 0.63 8 0.41 15 0.16 20 0.70 6 0.61 15 de 0.84 5 0.56 15 0.86 3 0.58 14 0.51 13 0.25 16 0.74 3 0.90 2 dk 0.78 8 0.30 25 0.56 15 0.42 20 0.89 2 1.00 1 0.73 5 0.89 3 ee 0.65 13 0.81 6 0.52 17 0.26 25 0.83 4 0.71 7 0.62 7 0.67 12 el 0.43 21 0.35 22 0.37 22 0.70 4 0.15 27 0.05 25 0.09 27 0.30 26 es 0.41 23 0.40 20 0.72 12 0.52 15 0.45 14 0.16 19 0.22 25 0.43 24 fi 0.49 19 0.31 24 0.37 21 0.24 27 0.92 1 0.91 3 0.58 8 0.52 17 fr 0.36 25 0.40 21 0.59 14 0.43 17 0.68 9 0.56 9 0.42 18 0.44 21 hr 0.53 17 0.74 8 0.34 25 0.37 22 0.35 18 0.07 23 0.30 22 0.44 22 hu 0.42 22 0.69 10 0.72 11 0.62 10 0.29 20 0.15 21 0.43 17 0.43 23 ie 0.71 9 0.62 13 0.33 27 0.29 24 0.68 11 0.74 6 0.57 9 0.79 6 it 0.84 6 0.82 4 0.86 4 0.74 3 0.29 20 0.04 27 0.20 26 0.68 9 lt 0.34 26 0.68 11 0.84 5 0.61 11 0.31 19 0.11 22 0.41 19 0.48 19 lu 0.86 3 0.55 16 0.84 6 0.59 13 0.86 3 0.64 8 0.56 11 0.63 14 lv 0.53 18 0.95 1 0.81 8 0.63 9 0.21 24 0.28 14 0.44 16 0.45 20 mt 0.63 14 0.27 26 0.53 16 0.42 19 0.81 6 0.90 4 0.46 14 0.67 10 nl 0.86 2 0.65 12 0.67 13 0.49 16 0.82 5 0.75 5 0.53 12 0.89 3 pl 0.38 24 0.81 5 0.50 18 0.43 18 0.38 16 0.27 15 0.41 20 0.32 25 pt 0.92 1 0.88 3 0.91 2 0.81 2 0.25 23 0.07 24 0.27 23 0.77 7 ro 0.02 5 27 0.45 19 0.44 19 0.60 12 0.29 22 0.29 13 0.78 2 0.51 18 se 0.60 15 0.31 23 0.36 23 0.25 26 0.80 7 0.91 2 0.85 1 0.59 16 si 0.66 12 0.48 18 0.36 24 0.33 23 0.68 9 0.23 17 0.36 21 0.67 10 sk 0.55 16 0.69 9 0.38 20 0.38 21 0.36 17 0.04 26 0.74 4 0.30 27 table 4. levels of evaluating working conditions in individual eu-27 countries for the adopted criteria (according to equations 9-16) tutak et al./decis. mak. appl. manag. eng. 5 (2) (2022) 1-29 16 c o u n tr ie s criterion (dimension) physical environm ent work intensity working time social environm ent skills, discretion and other cognitive factors, prospects working life perspectiv es job and company context prospects at high acceptable high high high acceptable high very high be very high acceptable high high high high acceptable high bg high very high high high low acceptable acceptable very high cy acceptable low very high very high low acceptable low high cz high high low high acceptable acceptable very high high de very high acceptable very high high acceptable acceptable very high very high dk high low acceptable acceptable very high very high very high very high ee high very high acceptable low very high very high high high el acceptable low low very high low low low low es acceptable acceptable high acceptable acceptable acceptable low acceptable fi acceptable low low low very high very high high acceptable fr low acceptable acceptable acceptable high high acceptable acceptable hr acceptable high low acceptable acceptable low acceptable acceptable hu acceptable high high high acceptable acceptable acceptable acceptable ie high high low low high very high high high it very high very high very high very high acceptable negative low high lt low high very high high acceptable acceptable acceptable acceptable lu very high acceptable very high high very high high high high lv acceptable very high high high low acceptable acceptable acceptable mt high low acceptable acceptable very high very high acceptable high nl very high high high acceptable very high very high high very high pl low very high acceptable acceptable acceptable acceptable acceptable low pt very high very high very high very high low low low high ro low acceptable acceptable high acceptable acceptable very high acceptable se acceptable low low low very high very high very high acceptable si high acceptable low low high high acceptable high sk acceptable high low acceptable acceptable low very high low the analysis showed that for the criterion of 'physical environment', taking into account the indicators characterizing this criterion in the study, a very good result was obtained by denmark, belgium, portugal, the netherlands, luxembourg, and italy. all these countries belong to the developed eu countries (the so-called "old" eu countries). on the other hand, a low score for this criterion was obtained by romania, poland, lithuania, and france. among these countries as many as 3 belong to developing countries, i.e., the countries of the so-called "new" eu. in the case of assessing the criterion "work intensity", the highest scores were reported for estonia, italy, latvia, poland and portugal, and the lowest scores for denmark, finland, sweden, as well as cyprus, greece, and malta. a very important criterion, which is now of increasing importance to society, is the balance between work and private life, which is contained in the criterion ‘working life perspectives.’ very high scores for this criterion were obtained by denmark, estonia, finland, ireland, malta, the netherlands, and sweden. such a high rating for this criterion for the scandinavian countries is due to the fact that employees from these countries report high satisfaction with their working time and work-life balance. this is consistent with the social welfare system of these countries, where a great deal of attention is paid to the problem of reconciling work and private life. the lowest scores for this criterion were obtained by greece, croatia, portugal, and slovakia. this is due to the fact that in these countries little action is taken to facilitate work-life balance (matilla-santander et al., 2019). all in all, when analyzing the results presented in tables 3 and 4, it can be seen that within the assessed criteria, the eu-27 countries are characterized by considerable diversity. this makes it relatively difficult to distinguish groups of similar countries. therefore, in order to identify countries similar in terms of the level of sub-criteria for assessing working conditions, they were grouped using the k-means method. on this basis, the eu countries were divided into four clusters. the compositions of the formed clusters and distances from their centers (cluster centers) are presented in evaluating differences in the level of working conditions between the european union… 17 table 5. the greater the distance of the eu country from the center of the cluster in which it is located, the greater its differentiation from countries whose distance from the center of the cluster is smaller. table 5. elements of clusters with distances form centers cluster 1 distances from center of cluster 1 cluster 2 distances from center of cluster 2 cluster 3 distances from center of cluster 3 cluster 4 distances from center of cluster 4 at 0.074 cy 0.253 bg 0.071 dk 0.148 be 0.059 cz 0.178 it 0.072 ee 0.143 de 0.134 el 0.192 lv 0.137 fi 0.103 lu 0.099 es 0.118 pt 0.084 fr 0.175 nl 0.123 hr 0.128 ie 0.116 hu 0.086 mt 0.091 lt 0.124 se 0.112 pl 0.119 ro 0.210 si 0.192 sk 0.167 the results indicate that cluster 2 contains the largest number of countries (11 countries), and cluster 3 – the lowest number of countries (4 countries). no homogeneity of any of the clusters was found. countries from cluster 1 show the greatest similarity in terms of physical environment, work intensity, working time, social environment and prospects. as many as 11 countries from cluster 2 show the highest similarity in terms of social environment, prospects and skills, discretion and other cognitive factors. cluster 3 consists of countries that are similar in terms of physical environment, work intensity, working time, social environment, job and company context, prospects and skills, discretion and other cognitive factors. countries in cluster 4 are characterized by the highest similarity in terms of working life perspectives, working time, skills, discretion and other cognitive factors, social environment. for the clusters formed by very different countries, it is reasonable to use the mean value to determine the average rating of the individual criteria used to assess working conditions. a summary of the mean values of the working conditions evaluation criteria for each cluster is presented in figure 3. tutak et al./decis. mak. appl. manag. eng. 5 (2) (2022) 1-29 18 figure 3. average values of working conditions evaluation criteria for particular clusters of similar eu-27 countries when analyzing the results, it can be seen that countries in cluster 1 are characterized by the highest average score for the criteria physical environment and prospects. at the same time, for the criteria work intensity, working time, social environment, skills, discretion and other cognitive factors, the scores are slightly lower than for the countries in cluster 3, and for the criteria job and company context and working life perspectives, the scores are lower than for the countries in cluster 2. countries in cluster 2 were reported the worst in terms of the average evaluation of partial criteria for working conditions. countries from this cluster perform poorly in comparison with other countries in terms of physical environment and prospects, and also quite poorly in terms of work intensity, working time and social environment (only countries from cluster 4 have worse scores for these criteria) as well as job and company context, working life perspectives and skills, discretion and other cognitive factors (only countries from cluster 3 have worse scores). in the next stage of the research on the evaluation of working conditions, a total value of index of working conditions (equation 11) was determined for each eu-27 country, taking into account partial evaluation results for individual evaluation criteria (table 3) and the level of working conditions (fig. 4). based on the analyses, the final levels of working conditions in the eu-27 countries were determined. this breakdown for different levels is as follows (fig. 5): very high level of working conditions: denmark, germany, the netherlands, luxembourg, austria high level of working conditions: belgium, estonia, bulgaria, finland, ireland, malta, portugal, sweden, italy acceptable level of working conditions: czech republic, latvia, france, slovenia, lithuania, cyprus, hungary and poland evaluating differences in the level of working conditions between the european union… 19 low level of working conditions: greece, croatia, spain, romania, slovakia. figure 4. summary values of the working conditions index in the eu-27 figure 5. levels of working conditions in the eu countries on the basis of the conducted research, it can be concluded that definitely better working conditions can be found in the developed countries than in the developing countries of the eu. the exceptions in this regard are greece and spain (countries of the "old" eu), where the level of working conditions was reported to be low. the overall assessment of working conditions shows that in as many as eleven (out of fourteen) countries of the so-called "old" eu, the level of working conditions is either tutak et al./decis. mak. appl. manag. eng. 5 (2) (2022) 1-29 20 very high or high; in one of them, it is acceptable (france), and in two of them – low (spain and greece). among the countries of the so-called "new" eu, i.e., developing countries, a high level of working conditions can be found only in malta, estonia and bulgaria, and acceptable conditions in cyprus, hungary, lithuania, latvia, poland, slovenia and the czech republic. in two countries (romania and croatia), a low level of working conditions was found. unfortunately, the results show that in developed countries ("old" eu), there are much better working conditions than in the countries of the socalled "new" eu. the next stage of the research was to check if and what relations exist between the working conditions index 𝑃𝑖 ∗ (determined using the topsis method) and the level of economic development and indicators characterizing the area of health and safety at work in the eu-27 countries. for this purpose, the spearman's rho and kendall's tau correlation coefficients were adopted. the level of economic development of the eu countries was characterized by the values of gdp, gdp per capita, as well as gross domestic expenditure on r&d (% of gdp). the indicators describing the area of health and safety at work were: accidents at work (as percentage of persons employed), persons reporting exposure to risk factors that can adversely affect mental well-being (as percentage of total employment) and persons reporting a work-related health problem (as percentage of persons employed). the results of the calculations are summarized in table 6. table 6. the spearman's rank correlation coefficient and the kendall's tau correlation coefficient between the value of the work conditions index p_i^* and the economic development and health and safety at work in the eu-27 countries tested parameters kendall’s tau correlation coefficient spearman rho work conditions index p work conditions index p gdp, million euro 0.111 0.416 0.190 0.341 gdp per capita, million euro 0.521 0.000 0.673 0.000 gross domestic expenditure on r&d, % of gdp 0.308 0.024 0.451 0.018 accidents at work, percentage of persons employed 0.136 0.320 0.243 0.223 persons reporting exposure to risk factors that can adversely affect mental well-being, percentage of total employment 0.094 0.491 0.133 0.508 persons reporting a work-related health problem, percentage of persons employed 0.275 0.044 0.389 0.045 note: statistically significant values are marked in bold based on the results, it can be concluded that statistically significant positive relationships exist between the working conditions index and gdp per capita and gross domestic expenditure on r&d and between the working conditions index and persons reporting a work-related health problem, but in this case, the strength of the correlation is relatively weak. the strength of the relationship between the examined parameters and the working conditions index is higher for the spearman's rho correlation coefficients than for the kendall's tau correlation coefficient. no relationship was found between the gdp of a country and the working conditions evaluating differences in the level of working conditions between the european union… 21 index, nor between the index and occupational accidents and persons reporting exposure to risk factors that can adversely affect mental well-being. that is why the results obtained can be taken as statistical evidence that a relatively high economic level measured by gdp per capita is also associated with better working conditions in the eu-27. it is also very important to emphasize that business enterprise expenditure on r&d is statistically significant for the level of working conditions (i.e., has a positive impact on these conditions). this means that developed countries are characterized by significantly better working conditions than developing countries. 5. discussion free movement of workers and their access to work under the same conditions as nationals is one of the fundamental principles of the eu. this principle plays an important role in achieving the sustainable development goals and the 2030 sustainable europe strategy for economic growth and reducing social disparities. since many of the eu's policy priorities include increasing the level of employment, prolonging labor market participation and raising labor productivity, the issue of creating better working conditions appears to be an absolute necessity and even of fundamental importance for the success of the eu's economic and social strategies. since work is the basis for the development of the eu-27, which currently employs some 195.7 million people (eurostat, 2022), the problem of ensuring appropriate working conditions becomes extremely important and up to date. also, changes in the structure of employment associated with the digitalization of the economy, the aging of the european population and the ongoing processes of immigration, as well as the recent geopolitical turmoil, make the issue of access and working conditions of key importance for individual countries and the eu as a whole (cirillo et al., 2021; cristea et al., 2020; frennert, 2019; nica, 2015). of particular importance are the working conditions, which, in addition to economic aspects, are a very important social factor determining the efficiency and success of an enterprise (organization) and the sustainable development of the economy and society (davidescu, 2020; matilla-santander et al., 2019). working conditions affect the commitment and willingness to improve skills by employees, their satisfaction and productivity. they also have a major impact on work-life balance. therefore, it is important to assess these conditions, both at the level of an individual company and at the national and regional levels. a broader approach to assessing these conditions provides opportunities to compare them, identify similarities and differences, and identify shortcomings that can be improved through good practice or cooperation between groups of countries. in the case of the eu, which is building and promoting a common social policy and an open labor market, solutions to this issue are of particular importance. therefore, for the first time, such a wide range of research has been conducted on the evaluation of working conditions in the eu-27. the results made it possible to achieve the main objective of the paper, which was the overall evaluation of the level of working conditions in the eu-27 and a partial evaluation of the criteria influencing these conditions as a whole. the relationships between working conditions and the economic development of individual countries were also determined, as well as indicators characterizing an extremely important area, from the point of view of working conditions, i.e., safety and health at work. tutak et al./decis. mak. appl. manag. eng. 5 (2) (2022) 1-29 22 the research carried out and its results show that the eu-27 countries are characterized by great diversity in the level of working conditions. a comprehensive evaluation of working conditions, including eight evaluation criteria and as many as 64 diagnostic variables, showed that very good working conditions are found in denmark, germany, the netherlands, luxembourg, austria, and good conditions in belgium, estonia, bulgaria, finland, ireland, malta, portugal, sweden, and italy. this means that the most favorable working conditions occur mainly in developed countries, conventionally classified as countries of the "old" eu. only in three developing countries (so-called "new" eu), namely estonia, bulgaria and malta, a high level of working conditions was found. these results therefore indicate that in developing countries, which joined the eu after 2004, apart from estonia, bulgaria and malta, working conditions were reported to be acceptable and for two countries – croatia and romania – negative. at the same time, only in two countries of the "old" eu (spain and greece), working conditions were reported to be at a low level. a number of factors contribute to these differences, but the main reasons include insufficiently active labor policies and lower macroeconomic indicators (gdp per capita), which are ultimately important for working conditions in the evaluated countries (table 6). indeed, the results obtained showed that the level of working conditions is related to the economic development of a country, which is also in line with the results of giordano and kostova (2002). their results reflect that in most post-socialist countries, working conditions are often much worse than in western countries ("old" eu). this, in turn, indirectly affects the issues of migration in search for better working conditions to western countries from post-communist countries, such as poland (cieslik, 2011). apart from political and security aspects, the issues of working conditions related to economics have the greatest impact on migration decisions (bygnes & bivand erdal, 2017). an important factor that also affects working conditions is the level of technological development. most of the countries belonging to the "old" eu are much more technologically developed (brodny & tutak, 2021). the processes of digitalization, automation and robotization related to the activities of enterprises make it possible, for example, to reduce the contact of the employee with aggressive environments, which significantly improves the comfort of work. this process, often associated with the need to increase competence by the employee, generally improves his/her well-being and results in greater commitment (marenco & seidl, 2021). in the case of developing countries (the "new" eu), these processes do not occur so dynamically, which makes working conditions less favorable (grigorescu et al., 2021). by and large, the assessment of working conditions should be approached much more broadly than just by looking at the hazards in the work environment and health and safety issues. also, social factors including relations with co-workers and other work partners (subcontractors, customers or users) are also crucial. in this respect, very high and high levels of working conditions were found in cyprus, greece, italy and portugal, austria, belgium, bulgaria, the czech republic, germany, hungary, lithuania, luxembourg, latvia and romania, i.e. in more than half of the eu member states. it is somewhat surprising to note that in countries such as finland and sweden, employees rated these conditions negatively. this is due to, among other things, relatively poor ratings compared to other eu countries for exposure to undesirable social behavior, and issues of help and support from co-workers and from the manager/head. working conditions should also help to satisfy the need to develop professional activity. employees need a feeling of recognition, a sense of meaningfulness in their evaluating differences in the level of working conditions between the european union… 23 work, improvement of skills and a sense of accomplishment and purpose. from the point of view of the working life perspectives criterion, a very high level of working conditions was found in denmark, finland, sweden, ireland, the netherlands (eu-14), malta, and estonia (eu-13), which is due to the social policies of these countries. in a general sense, favorable working conditions influence the perception of a country as a suitable place to work and live. from an enterprise perspective, the provision of the best possible working conditions improves productivity and is a strong determinant of its success. inadequate working conditions, on the other hand, pose risks, cause economic losses and lead to negative social impacts. and adverse physical factors that have a harmful effect on the human body can promote the emergence of occupational diseases and deterioration of workers' health. therefore, good and comfortable working conditions, both in the physical and psychosocial sense, increase the productivity of enterprises and increase their value contributing to the economic growth of a country. the results indicate that working conditions need to be improved as soon as possible in four eu countries: greece, spain, croatia, and romania. improvements are also needed in the czech republic, latvia, france, slovenia, lithuania, cyprus, hungary, and poland. the improvement of working conditions in these countries concerns practically each of the evaluation criteria. the improvement of these criteria, and thus of working conditions in general, should be based on a well-prepared strategy at both national and company levels. a very important element of such a process should be responsible preparation and convincing employees to such changes. cooperation with employees and their engagement in the process of improving working conditions can be a critical factor for success in this area. the conducted research also indicates groups of similar countries and leaders in terms of working conditions. it is fully justified to take advantage of experiences and good practices of the leading countries in this area and use them by countries with less success in this field. it is clear that sustainable economic development of individual countries and the entire eu-27 must be linked to improvements in working conditions and, as far as possible, their equalization across the eu. the differences in this area shown in this paper result not only in external migration, but also internal migration, which adversely affects the sustainable development of the entire eu-27. solidarity between countries and mutual assistance should result in improved working conditions throughout the eu and in building a highly attractive labor market. the results should therefore provide valuable insights to support employment policy making and related improvements in working conditions in the eu-27. 6. conclusions the paper presents the results of the assessment of the level of working conditions in the eu-27. the topsis method, which belongs to the mcdm group of methods, and the k-means method were used for the assessment. the evaluation of the level of working conditions was carried out by means of a set of 8 criteria, which took into account a total of as many as 64 indicators (diagnostic variables) characterizing these criteria. the evaluation of the level of working conditions was also supplemented by studies aimed at indicating whether they are related to the basic economic parameters of the economy and indicators of the area of health and safety at work in the countries studied. tutak et al./decis. mak. appl. manag. eng. 5 (2) (2022) 1-29 24 the results of the research on the assessment of working conditions showed the following:  very good working conditions were found in denmark, germany, the netherlands, luxembourg, and austria.  good working conditions were found in belgium, estonia, bulgaria, finland, ireland, malta, portugal, sweden, and italy.  acceptable working conditions were found in the czech republic, latvia, france, slovenia, lithuania, cyprus, hungary, and poland.  negative working conditions were found in greece, croatia, spain, romania, and slovakia. thus, on the basis of these results, it can be concluded that definitely better working conditions are found in developed countries, which are part of the so-called "old eu-14" than in developing countries (the group of countries of the so-called "new eu-13"). on the other hand, the second part of the study conducted using non-parametric tests, such as spearman's rho and kendall's tau correlation coefficient showed that working conditions are associated with a country's economic development characterized by the value of gdp per capita and gross domestic expenditure on r&d and are also related to the parameter of persons reporting a work-related health problem. the results clearly indicate that working conditions are also significantly better in countries that are more prosperous and have higher levels of economic development, as well as higher expenditures on r&d. therefore, the findings provide new knowledge in assessing the level of working conditions found in the eu-27 countries and the factors that influence these conditions. it should also be emphasized that the developed and applied research methodology can also be successfully used to study working conditions at the regional level and in individual groups of enterprises. the results of such research would undoubtedly complement those presented in the paper. 7. future directions and limitations the developed methodology, conducted research and its results make it possible, as in the case of most of this type of analysis, to formulate their limitations and future research directions. in terms of limitations that may have affected the findings, an issue that should first be mentioned is related to the data obtained from the eurofound database. all data collected in this database concern subjective evaluation of particular issues (table 1), which is done by respondents. in this context, the concept of working conditions and their assessment by respondents may be relative. another limitation, which at the same time indicates the direction of further research, is related to the fact that this study presents a general evaluation of working conditions in the eu-27. it would be reasonable to conduct such an analysis taking into account different economic sectors, as well as age groups, gender and education, and the form of employment of workers and size of enterprises. such research would provide an opportunity for a very broad analysis of working conditions, which could result in more targeted recommendations for their improvement or harmonization. systematic monitoring of working conditions would also provide an opportunity to observe changes and indicate areas for improvement. evaluating differences in the level of working conditions between the european union… 25 author contributions: conceptualization, m.t. and j.b.; methodology, j.b. and m.t.; software, m.t. and j.b.; formal analysis, j.b. and m.t.; investigation, j.b. and m.t.; resources, m.t. and j.b.; data curation, m.t. and j.b.; writing—original draft preparation, m.t. and j.b.; writing—review and editing, j.b. and m.t.; visualization, m.t.; supervision, m.t. and j.b.; project administration, m.t. and j.b.; funding acquisition, m.t. all authors have read and agreed to the published version of the manuscript. funding: this publication was funded by the statutory research performed at silesian university of technology, department of production engineering, faculty of organization and management and department of safety engineering, faculty of mining, safety engineering and industrial automation. data availability statement: not applicable. acknowledgments: the authors would like to express their gratitude to the editors and anonymous referees for their informative, helpful remarks and suggestions to improve this paper as well as the important guiding significance to our researches. conflicts of interest: the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. references agbozo, g. k., owusu, i. s., hoedoafia, m. a., & atakorah, y. b. 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(2014). state of art surveys of overviews on mcdm/madm. technological and economic development of economy, 20, 165-179. © 2022 by the authors. submitted for possible open access publication under the terms and conditions of the creative commons attribution (cc by) license (http://creativecommons.org/licenses/by/4.0/). plane thermoelastic waves in infinite half-space caused decision making: applications in management and engineering issn: 2560-6018 eissn: 2620-0104 doi: https://doi.org/10.31181/dmame0329102022a * corresponding author. e-mail addresses: anushagogineni76@kluniversity.in (g. anusha), paladuguvenkataramana@gmail.com (p.v. ramana), rs4magt@gmail.com (r. sarkar) hybridizations of archimedean copula and generalized msm operators and their applications in interactive decision-making with q-rung probabilistic dual hesitant fuzzy environment gogineni anusha1*, paladugu venkata ramana2 and rupak sarkar3 1 department of engg. mathematics, college of engg., koneru lakshmaiah education foundation, vaddeswaram-522032, andhra pradesh, india 2 department of basic science & humanities, vignan’s lara institute of technology and sciences, vadlamudi-522213, andhra pradesh, india 3 department of mathematics, women’s polytechnic, hapania, tripura, india received: 17 august 2022; accepted: 16 october 2022; available online: 3 november 2022. original scientific paper abstract: the q-rung probabilistic dual hesitant fuzzy sets (qrpdhfss), which outperform dual hesitant fuzzy sets, probabilistic dual hesitant fuzzy sets, and probabilistic dual hesitant pythagorean fuzzy sets, are used in this research to develop an interactive group decision-making approach. we first suggest the archimedean copula-based operations on q-rung probabilistic dual hesitant fuzzy (qrpdhf) components and investigate their key features before constructing the approach. we then create some new aggregation operators (aos) in light of these operations, including the qrpdhf generalized maclaurin symmetric mean (msm) operator, qrpdhf geometric generalized msm operator, qrpdhf weighted generalized msm operator, and qrpdhf weighted generalized geometric generalized msm operator. these aggregation operators are better than current operators on qrpdhf because they can take into account the interactions between a large number of criteria and probability distributions. the evaluation findings are distorted since the present methodologies do not take expert involvement into account in order to achieve the requisite consistency level. we employ the idea of interaction, consistency, resemblance, and consensus-building among the decision-makers in our method to get around this. we create an optimization model based on the cross-entropy of the qrpdhf components to estimate the weights of the criterion. we provide a contextual research on the choice of open-source software lms in order to demonstrate the relevance of the anusha et al./decis. mak. appl. manag. eng. (2022) 2 recommended aos. likewise, we ran a sensitivity test on the weights of the criterion to make sure that our model is consistent. the comparison investigation has demonstrated that the suggested approach can overcome the challenges of previous works. key words: q-rung probabilistic dual hesitant fuzzy set, archimedean copula, generalized maclaurin symmetric mean operators, group decision-making. 1. introduction finding the best option(s) from a pool of readily available possibilities based on several features, both quantitative and qualitative, is the main goal of the multiple criteria decision-making (mcdm) technique. uncovering the numerical values of qualities can occasionally be a difficult process for an expert. in light of recent scientific and technological developments, uncertainty now dominates decisionmaking (dm) analyses. zadeh (1965) proposed the idea of fuzzy sets (fss) to deal with the data's ambiguity. the hesitant fuzzy set (hfs) (torra, 2010) is an extension of fss that permits membership degrees (mds) to assume a limited number of likely entries as opposed to only one. by include hesitant non-membership degrees (nmds) together with mds in the inquiry, zhu et al. (2012) created the concept of dual hfss (dhfss) and named its fundamental component dual hesitant fuzzy elements (dhfes). the case about the presence of the probabilities of the components in the dhfss and hfss is not resolved even if dhfss and hfss are successfully used to many dm situations. let's look at an illustration to better comprehend this: consider a professional who indicates his propensity for anything negative as a hesitant fuzzy element (hfe) of 0.4, 0.6, or 0.7. he mentioned throughout the tests that the comfort level associated to 0.6 is the greatest when compared to others, while the comfort level related to 0.7 is the worst. thus, under such a situation, the hfe {0.4, 0.6, 0.7} isn't reasonable to depict the data. additionally, think about a rating of an expert to assess the quality of an item as the dhfe <{0.4, 0.6, 0.7}, {0.2, 0.3, 0.4}>. during the appraisals, the expert accepts that his solace level toward the item evaluating 0.4, 0.7 is twofold than 0.6 in mds, while triple toward the 0.4 in the nmds concerning the others. hence, the dhfe <{0.4, 0.6, 0.7}, {0.2, 0.3, 0.4}> isn't reasonable to portray the information. to handle such issues, the idea of the probabilistic hfs (phfs) and probabilistic hfes (phfes) were presented by xu and zhou (2017) and were extended to probabilistic dhfs (pdhfs) and probabilistic dhfes (pdhfes) respectively by hao et al. (2017). the pdhfe gives a more exact depiction compare to phfe, hfe, and dhfe and can effectively portray the data in the above-expressed example. the todim approach with pdhfss was utilised by ren et al. (2017) for enterprise strategy evaluation. new correlation coefficients were put up by garg and kaur (2018) and used to solve problems with pdhfss and multi-criteria decisionmaking (mcdm). a strategy selection problem was handled by ren et al. (2019) utilising an integrated vikor and ahp technique using pdhf information. pdhfss is constrained in that the total grade for both membership and non-membership should not be more than 1. ji et al. (2021) developed the idea of probabilistic dual-hesitant pythagorean fuzzy sets (pdhpfss), which adhere to the requirement that the sum of the squares of md and nmd should not be more than 1. the q-rung probabilistic dual hesitant fuzzy sets (qrpdhfss) sets, introduced by li et al. (2020) hold the constraint that the addition of qth power of the md and the nmd must accomplish the value in [0, 1]. the qrpdhfs reduces to pdhfs when q =1 and pdhpfs when q =2, which means hybridizations of archimedean copula and generalized msm operators and their … 3 that the qrpdhfss are extended versions of pdhfss and pdhpfss. thus qrpdhfss are more powerful than pdhfss and pdhpfss. in the past few decades, interactive technology has made a series of developments. sakawa took the lead in considering the interaction between group decision makers (experts) to resolve inconsistencies (sakawa & yano, 1985). some studies have shown that interactive dm gradually and dynamically learns about the personal preference structure under the continuous communication and interaction between experts, and finally obtains the most satisfactory results (bashiri & badri, 2011; reverberi & talamo, 1999; shi & xia, 1997). watson et al. (1991) believed that the interactive mode within the group is a key variable that affects the rationality of the dm results. xu and chen (2007) believe that experts modify their preference information through interaction during the dm process, which can make the decision result more reasonable, and they use a hybrid weighted average operator to aggregate decision information in a fuzzy environment. cheng et al. (2018) considered the consistency of evaluation results and attribute weights through the interaction between venture capital providers and between venture capital providers and entrepreneurs. gou el al. (2019) introduced a consistency index to judge the linguistic preference relation of acceptable consistency. thus, in the literature, there is a significant gap regarding the consideration of interactive dm problems with qrung probabilistic dual hesitant fuzzy (qrpdhf) information. in any mcdm method, the primary concern is how to fuse the assessment data of various criterias’ for alternatives, and afterward to get the fittest one. two different ways are there to pick the most suitable alternative. one is the conventional assessment tools, and the other is the information aggregation operators (aos). the conventional assessment tools can only generate the preference order of alternatives, while information aos not only generate the preference order of alternatives effectively yet additionally provides comprehensive assessment value of each alternative. as a result, the information aos can tackle mcdm issues in a more feasible way compares to the conventional assessment tools. recently, the study of pdhf aggregation operators and their extensions has drawn significant attention to researchers. the pdhf weighted averaging (pdhfwa) operator was created by hao et al. (2017) and used for risk assessment. to address the problem of decisionmaking, garg and kaur (2018) designed various pdhfs-based einstein aos with certain information metrics. the pdhf fuzzy power weighted hamy mean (pdhpfpwhm) operator was created by ji et. al. (2021) and utilised to address the mcdm issues. for the purpose of resolving mcdm issues, li et al. (2020) suggested the q-rung pdhf power weighted muirhead mean (q-rpdhfpwmm) operator. a) objectives of research real-world multi-criteria group decision-making (mcgdm) situations allow for the observation of the relationships between a variety of factors. in this situation, it is crucial to take into account how the various criteria interact in order to arrive at a more logical conclusion. to date, the pdhf einstein weighted averaging operator (garg & kaur, 2018) and the pdhf weighted averaging ao (hao et al., 2017) have been used to average data incorporating pdhf information. additionally, the pdhpfpwhm operator is capable of recording the relationship between characteristics. however, they miss out on the linkages between several input criteria. the q-rpdhfpwmm operator (li et al., 2020) can manage multi-input dependence between criteria, but it is unable to handle probability distributions, leading to information loss during the aggregation phase. the q-rpdhfpwmm operator (li et al., 2020) was also solely used to address mcdm issues with variable anusha et al./decis. mak. appl. manag. eng. (2022) 4 weightings of criteria. in order to prevent information loss during aggregation, an ao that addresses the link between multiple input attributes and probability distributions in the context of an mcgdm setup with qrpdhf information is required. b) research gaps and motivations a pdhfs reduces to a phfs if the nmds alongside their associated probabilities are ignored. also, a pdhfs turns into a dhfs if the hesitant mds and nmds are equally probable. hence, pdhfss are generalized versions of the phfss and dhfss. but, pdhfss cannot fully express the real decision information because sum of md and nmd must not exceed 1. extending this restriction to their square sums, we get pdhpfss. but pdhfss and pdhpfss are special cases of qrpdhfss, since they require that the sum of the qth power of md and qth power of nmd should not surpass 1.thus, qrpdhfss can express the criteria values with higher flexibility. since hao et al.’s method (2017) and garg and kaur’s method (2018) are based on pdhfss and ji et’s method (2021) is based on pdhpfss, so li et al’s (li et al., 2020) method based on qrpdhfss is more effective compare to hao et al.’s method (2017), and garg and kaur’s method (2018)and ji et’s method (2021) for solving real decision-making problems. but li et al’s method has certain drawbacks too. to analyze the shortcomings of li et al.’s method (2020), we consider the following two counter examples: example 1: suppose, an institute is interested to choose an oss-lms package among three oss-lms packages, namelysakai (a1), efront (a2), and moodle (a3). these oss-lmss are to be assessed by three experts (e1, e2 and e3) depending on three attributes. the details of these options and attributes are presented in table 1 and table 2, respectively. to choose the best alternative among these five oss-lms, a team is formed involving of three experts. their initial evaluation results are presented in terms of pdhfes. table 1. description of the oss-lms alternatives oss-lms description sakai (a1) sakai is an oss-lms scheme that offers a flexible and versatile context for teaching, training, analysis, and other associations. sakai constantly grows based on the requirements of the faculty, learners, and corporations (https://sakaiproject.org/). efront (a2) efront lms extends the finest open source resolutions through the most useful of e-learning. the structure is adaptable, commanding, efficient, and completely functional (http://www.efrontlearning.net/). moodle (a3) this is the most prevalent open-source lms to provide teachers, administrators, and students with one robust, secure, and combined system for training atmospheres (moodle.org). hybridizations of archimedean copula and generalized msm operators and their … 5 table 2. criteria details criteria description functionality (c1) functionality is the strength of the software to accommodate functions that match the user’s specifications when utilizing the software under particular conditions. functionality is utilized to estimate the level in which an lms meets the functional specifications of an establishment. reliability (c2) reliability is the capacity of the software package to work consistently without falling under specific situations. reliability is practiced to evaluate the level of fault tolerance of the software packages. security and privacy (c3) security and privacy standards are required to authenticate the efficacy of a structure to safeguard private data and safeguard information from attacks and exposure on a user’s computer. the initial assessment matrices are: ( ) ( ) ijd d m n m p       ( ) ( ) ( ) ( ) 3 3 { ( )}, { ( )} a a b b ijd ijd a b p p            ( 1(1)3, 1(1)3, 1(1)3)d j i   are given in the form of table 3. suppose that the consensus coefficient among experts should be above 0.97 (i.e; * 0.97  ). table 3. initial assessment matrix c1 c2 c3 e1 a1 <{0.1(0.3), 0.3(0.2), 0.4(0.5)}, {0.6(0.4), 0.2(0.3), 0.3(0.3)}> <{0.6(0.4), 0.8(0.2), 0.3(0.3), 0.2(0.1)}, {0.3(0.4), 0.2(0.2), 0.7(0.3), 0.9(0.1)}> <{0.4(0.2), 0.1(0.4), 0.5(0.1), 0.7(0.3)}, {0.1(0.1), 0.5(0.3), 0.4(0.3), 0.3(0.3)}> a2 <{0.3(0.1), 0.7(0.5), 0.8(0.2), 0.5(0.2)}, {0.9(0.5), 0.6(0.1), 0.5(0.4)}> <{0.2(0.5), 0.5(0.2), 0.6(0.1), 0.3(0.2)}, {0.8(0.4), 0.6(0.2), 0.4(0.4)}> <{0.5(0.1), 0.8(0.6), 0.4(0.3)}, {0.5(0.2), 0.3(0.4), 0.1(0.4)}> a3 <{0.6(0.5), 0.2(0.2), 0.3(0.3)}, {0.1(0.3), 0.3(0.2), 0.4(0.5)}> <{0.3(0.2), 0.2(0.1), 0.7(0.1), 0.9(0.6)}, {0.6(0.5), 0.8(0.1), 0.3(0.2), 0.2(0.2)}> <{0.1(0.2), 0.5(0.4), 0.4(0.3), 0.3(0.1)}, {0.4(0.1), 0.1(0.3), 0.5(0.5), 0.7(0.1)}> e2 a1 <{0.4(0.3), 0.6(0.2), 0.5(0.2), 0.3(0.3)}, {0.3(0.3), 0.7(0.1), 0.8(0.3), 0.5(0.3)}> <{0.7(0.4), 0.8(0.2), 0.6(0.3), 0.4(0.1)}, {0.2(0.4), 0.5(0.2), 0.6(0.3), 0.3(0.1)}> <{0.5(0.2), 0.3(0.4), 0.1(0.1), 0.4(0.3)}, {0.5(0.1), 0.8(0.6), 0.4(0.3)}> a2 <{0.1(0.1), 0.3(0.5), 0.4(0.4)}, {0.6(0.6), 0.2(0.2), 0.3(0.2)}> <{0.6(0.5), 0.8(0.2), 0.3(0.1), 0.2(0.2)}, {0.3(0.3), 0.2(0.1), 0.7(0.2), 0.9(0.4)}> <{0.4(0.1), 0.1(0.1), 0.5(0.5), 0.7(0.3)}, {0.1(0.1), 0.5(0.4), 0.4(0.4), 0.3(0.1)}> a3 <{0.9(0.4), 0.6(0.1), 0.5(0.5)}, {0.9(0.3), 0.6(0.2), 0.5(0.5)}> <{0.8(0.3), 00.6(0.1), 0.4(0.6)}, {0.8(0.6), 0.6(0.2), 0.4(0.2)}> <{0.5(0.3), 0.3(0.4), 0.1(0.3)}, {0.3(0.3), 0.1(0.5), 0.5(0.2)}> e3 a1 <{0.6(0.5), 0.2(0.2), 0.3(0.3)}, {0.1(0.3), 0.3(0.1), 0.4(0.6)}> <{0.3(0.4), 0.2(0.2), 0.7(0.3), 0.9(0.1)}, {0.6(0.4), 0.8(0.2), 0.3(0.3), 0.2(0.1)}> <{0.1(0.2), 0.5(0.4), 0.4(0.1), 0.3(0.3)}, {0.4(0.1), 0.1(0.3), 0.5(0.3), 0.7(0.3)}> a2 <{0.9(0.1), 0.6(0.5), 0.5(0.4)}, {0.3(0.5), 0.7(0.1), 0.8(0.2), 0.5(0.2)}> <{0.8(0.7), 0.6(0.1), 0.4(0.2)}, {0.2(0.3), 0.5(0.1), 0.6(0.2), 0.3(0.4)}> <{0.5(0.4), 0.3(0.1), 0.1(0.5)}, {0.5(0.1), 0.8(0.8), 0.4(0.1)}> anusha et al./decis. mak. appl. manag. eng. (2022) 6 c1 c2 c3 a3 <{0.3(0.4), 0.7(0.1), 0.8(0.2), 0.5(0.3)}, {0.6(0.5), 0.2(0.1), 0.3(0.4)}> <{0.2(0.2), 0.5(0.1), 0.6(0.1), 0.3(0.6)}, {0.3(0.5), 0.2(0.1), 0.7(0.2), 0.9(0.2)}> <{0.5(0.2), 0.8(0.7), 0.4(0.1)}, {0.1(0.1), 0.5(0.3), 0.4(0.5), 0.3(0.1)}> li et al.’s method (li et al., 2020) has the limitation that it fails to generate any ranking order in the mcgdm problem described above. example 2: (li et al., 2020) “after preliminary analysis, four possible investment alternatives are taken into account; they are denoted by a1, a2, a3, a4. in this paper, we consider three commonly used attributes in investment evaluation decision: (1) g1 the quality of product and service; (2) g2 social and environmental impacts; (3) g3 economic benefits. the weight vector of the attributes is w = (0.3, 0.2, 0.5)t. the experts need to assess the four alternatives’ performance from three aspects respectively”. the initial assessment matrix is presented in table 4”. table 4. initial assessment matrix c1 c2 c3 a1 <{0.7(0.2),0.6(0.2), 0.5(0.6)},{0.2(1)}> <{0.7(1)},{0.2(1)}> <{0.2(1)},{0.2(1)}> a2 <{0.1(1)},{0.4(1)}> <{0.3(1)},{0.7(1)}> <{0.7(1)},{0.3(0.5), 0.2(0.5)}> a3 <{0.3(1)},{0.5(1)}> <{0.6(1)},{0.2(1)}> <{0.1(1)},{0.7(1)}> a4 <{0.05(0.7),0.2(0.3)}, {0.5(1)}> <{0.3(1)}, {0.6(0.5), 0.4(0.5)}> <{0.8(1)},{0.1(1)}> using li et al.’s method (li et al., 2020) with qrpdhfpwmm operator, the scores of the alternatives are respectively 0.2574, 0.1535, -0.2095 and 0.3497. thus, the ranking order is 4 1 2 3a a a a . as we can see from example 2, li et al’s method (li et al., 2020) is capable of producing the ranking order for any mcdm problem, but it has certain drawbacks mentioned below: 1. in li et al’s method (li et al., 2020), experts’ assessments were carried out separately, making it challenging to draw a consistent conclusion. specifically, it fails to depict the ambiguity of articulating information with the collaboration among experts. so, the assessment outcomes get distorted. 2. sometimes, the information related to criteria weights is not known or partially unknown due to lack of data, and the expert’s limited proficiency. these criteria weights can be determined by experts’ personnel inclinations. in the dm technique (li et al., 2020), due to arbitrary assignment of weights of criteria’ for the final aggregation procedure, the preference ranking obtained gets affected. moreover, the method (li et al., 2020) leads to information loss as it doesn’t not consider any information measure. 3. the q-rpdhfpwmm operator (li et al., 2020) is capable of capturing the dependency among multiple criteria, but it cannot deal with probability distributions during the aggregation process. as a result, the ranking order obtained is not reasonable. c) contributions the following contributions are included in this paper: 1. we have put out a paradigm for aggregation based on interactions and qrpdhf data. the consistency harmonious weight index (chwi) and expert assessment hybridizations of archimedean copula and generalized msm operators and their … 7 similarity ideas have been used in this framework to examine the expert's subjective and objective weights. the final expert weights are then calculated by taking into account a combination of these factors. finally, until the required consensus value is obtained, the coefficient of consensus is computed again with expert participation. 2. the cross-entropy measure takes into account the weight of each criterion to address the amount of unclear information. taking use of this, an optimization model is developed in this work to determine the weights of the criterion. 3. copulas are functions that link several marginal distributions, which can indicate the correlation among variables and also prevent information loss during aggregation, according to a number of scholars (bacigal et al., 2015; beliakov et al., 2007; grabisch et al., 2011; han et al., 2020; nather, 2010; nelsen, 2013; tao et al., 2018). the generalised maclaurin symmetric mean (gmsm) (wang et al., 2018) operator generalises the bonferroni mean, hamy mean, and maclaurin symmetric mean operators by changing the parameter values. the gmsm operator takes into account the connections between several criteria. therefore, the gmsm operator has been expanded to include qrpdhfgmsm operators with their weighted forms employing archimedean copula operations on qrpdhf elements (qrpdhfes). in section 2, we concisely discuss some essential concepts namely qrpdhfs, gmsm operator and archimedean copula. section 3 investigates the shortcomings of li et. al’s method (2017). in section 4, we present the archimedean copula based operations between qrpdhfes and the associated gmsm operators. this section also puts forward the qrpdhfgmsm, qrpdhfggmsm, qrpdhfwgmsm and qrpdhfwggmsm operators along their characteristics. in section 5, we provide a group dm methodology using the developed aos. a case study of open-source software lms assessment is considered in section 6 to express the applicability of the developed approach. section 7 deals with impact of parameters, and comparison study. the last section is the conclusions. 2. preliminaries some basic notions are presented here that are relevant to our research. 2.1 q-rung probabilistic dual hesitant fuzzy set (qrpdhfs) definition 1 (li et al., 2020): a qrpdhfs ( )p on a universe set 1 2 { , ,..., } n u u u u is described as: ( ) { , ( )( ), ( )( ) : } i i i i p u p u p u u u         where ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) { ( )}, and ( )( ) { ( )} (0 , 1; 1; 1, 0 ( ) ( ) 1) i i i i a a b b i u i u a b a b a b a q b q u u a b p u p p u p p p p p                   (for each a and b where 1q  ) express the membership and non-membership degrees, respectively of iu u and the related probabilities are ( )a p and ( )b p respectively. ( )p transforms into a qrpdhf element (qrpdhfe) if it is singleton. it is expressed as: anusha et al./decis. mak. appl. manag. eng. (2022) 8 ( ) ( ) ( ) ( ) ( ) { ( )}, { ( )} a a b b a b p p p     where ( ) ( ) ( ) ( )0 , 1; 1; 1 a b a b a b p p p p     and ( ) ( ) 0 ( ) ( ) 1 a q b q      for each a and b). motivated by the score value, deviation degree and ranking rules of pdhfes (hao et al., 2017), we define the followings: definition 2: the score value of ( ) ( ) ( ) ( ) ( ) { ( )}, { ( )} a a b b a b p p p     is given by ( ) ( ) ( ) ( ) ( ( )) (( ) ) (( ) ) a q a b q b a b sc p p p        . sometimes qrpdhfs cannot be compared if their score values become identical. to address this issue, their deviation degrees can be used. definition 3: the deviation degree of ( ) ( ) ( ) ( ) ( ) { ( )}, { ( )} a a b b a b p p p     is given by: ( ) 2 ( ) ( ) 2 ( ) ( ( )) (( ) ( ( ( ))) ) (( ) ( ( ( ))) ) a q a b q b a b d p sc p p sc p p            thus, deviation degree of a qrpdhfe reflects describes the distance from the average value. definition 4: for the qrpdhfes (1) (2) ( ) and ( )p p  , a ranking rule is defined as: a. if (1) (2)( ( )) ( ( ))sc p sc p   , then (1) (2) ( ) ( )p p  b. if (1) (2)( ( )) ( ( ))sc p sc p   , then (a) if (1) (2)( ( )) ( ( ))d p d p   , then (1) (2) ( ) ( )p p  (b) if (1) (2)( ( )) ( ( ))d p d p   , then (1) (2) ( ) ( )p p   2.2 generalized maclaurin symmetric mean operator (gmsm) definition 5 (wang et al., 2018): the gmsm operator is defined by: 1 2 1 2 1 2 1 ..... ; , ,....., 1 2 1 ... 1 1 ( , ,..., ) ( ) q q j j q q t t t q t t t t n pn p p p n jq gmsm c                            where 1 2 , ,....., 0 q t t t  , q is a parameter, and 1 2 ( , ,..., ) q p p p denotes a q-tuple combination of (1, 2,..., )n . a few specific cases of the gmsm operator are as follows: case-i: when q=2 and 1 2 t t t  , we get the bonferroni mean (bm) operator (bonferroni, 1950) given below: 1 2 1 1 22 2 1 2 1 , 1( )12 1 1 ( , ,..., ) ( ) ( ) ( 1) j j t n t t t t n p i jn p p n i j i jj bm n nc                              case-ii: when 1 2 ..... 1 q t t t    , we get the msm operator (maclaurin, 1729) given below: hybridizations of archimedean copula and generalized msm operators and their … 9 1 2 1 2 1 1 1 1 ..... 1 1 2 1 ... 1 ...1 1 1 1 ( , ,..., ) j j q q q q q n p pn n p p p n p p p nj jq q msm c c                                                 case-iii: when 1 2 1 ..... q t t t q     , we get the hamy mean (hm) operator (hara et al., 1998) given below: 1 2 1 2 1 1 1 1 1 1 ..... 1 2 1 ... 1 ...1 1 1 1 ( , ,..., ) ( ) j j q q q q q q q q q n p pn n p p p n p p p nj jq q hm c c                                         2.3 archimedean copula definition 6 (sklar, 1959): a function : [0,1] [0,1] [0,1]g   is termed as a copula if: (i) ( , 0) (0, ) 0, ( ,1) (1, ) [0,1]g u g u g u g u u u      (ii) 1 1 2 2 2 1 1 2 1 1 2 2 1 2 1 2 ( , ) ( , ) ( , ) ( , ) 0, for , , , [0,1] with and .g u r g u r g u r g u r u r u r u u r r       definition 7 (sklar, 1959): an archimedean copula is a mapping : [0,1] [0,1] [0,1]g   given by ( , ) ( ( ) ( ))g u r u r    where  is a strictly decreasing function and : [0, ) [0,1]   is given as: 1 ( ), [0, (0)] ( ) 0, [ (0), ] t t t t           further, if g is strictly increasing and  coincides with  , then g is called strict archimedean copula and we write: 1 ( , ) ( ( ) ( ))g u r u r      . 3. operations between qrpdhfes and associated gmsm operators let us take two qrpdhfes (1) (2) ( ) and ( )p p  and suppose that the probabilities and the fuzzy values presented in (1) (2) ( ) and ( )p p  are different. then, multiplying the fuzzy values with their corresponding probabilities, we may get some unreasonable results. to avoid this, the probabilities can be adjusted in the following manner. example 3: suppose (1) ( ) {0.4(0.8), 0.3(0.2)},{0.7(0.5), 0.8(0.5)}p   and (2) ( ) {0.5(1)},{0.6(0.4),p  0.8(0.6)}  . then their corresponding adjusted qrpdhfes are: (1) ( ) {0.4(0.8), 0.3(0.2)},p  {0.7(0.4), 0.7(0.1), 0.8(0.5)}  and (2) ( ) {0.5(0.8), 0.5(0.2)},{0.6(0.4), 0.8(0.1), 0.8(0.5)}p   . now, based on the adjusted qrpdhfes, we propose archimedean copula (ac) operations and develop the corresponding gmsm operators. 3.1 operations between qrpdhfes based on archimedean copula definition 8: let ( ) ( ) ( ) ( ) ( ) ( ) { ( )}, { ( )} ( 1, 2) j a a b b j j a b p p p j      be two adjusted qrpdhfes. then, for any 1 2 , , 0    , we define: anusha et al./decis. mak. appl. manag. eng. (2022) 10 1 ( ) ( ) ( ) 1 2 (1) ( 2) 1 ( ) ( ) ( ) 1 2 { 1 ( (1 ( ) ) (1 ( ) )) ( )}, 1. ( ) ( ) { ( (( ) ) (( ) )) ( )} a q a q aq a b q b q bq b p p p p                      1 ( ) ( ) ( ) 1 2 (1) ( 2) 1 ( ) ( ) ( ) 1 2 { ( (( ) ) (( ) )) ( )}, 2. ( ) ( ) { 1 ( (1 ( ) ) (1 ( ) )) ( )} a q a q aq a b q b q bq b p p                        (1) 1 ( ) ( ) 1 ( ) ( ) 1 1 3. ( ) { 1 ( (1 ( ) )) ( )}, { ( (( ) )) ( )} a q a b q bq q a b p p p               (1) 1 ( ) ( ) 1 ( ) ( ) 1 1 4. ( ( )) { ( (( ) ) ) ( )}, { 1 ( (1 ( ) )) ( )} a q a b q bq q a b p p p                theorem 1: let ( ) ( ) ( ) ( ) ( ) ( ) { ( )}, { ( )} ( 1, 2) j a a b b j j a b p p p j      be two adjusted qrpdhfes. then for 1 2, , 0    , we have, 1. (1) (2) (2) (1) ( ) ( ) ( ) ( )p p p p     2. (1) (2) (2) (1) ( ) ( ) ( ) ( )p p p p     3. (1) (2) (1) (2) ( ( ) ( )) ( ) ( )p p p p        4. (1) (2) (1) (2) ( ( ) ( )) ( ( )) ( ( ))p p p p          5. (1) (1) (1) 1 2 1 2 ( ) ( ) ( ) ( )p p p         6. 1 2 1 2 (1) (1) (1) ( ) ( ( )) ( ( ))p p p          proof: (1)-(2) straight forward. 3. we have, (1) (2) ( ) ( )p p  1 ( ) ( ) ( ) 1 2 1 ( ) ( ) ( ) 1 2 { 1 ( (1 ( ) ) (1 ( ) )) ( )}, { ( (( ) ) (( ) )) ( )} a q a q aq a b q b q bq b p p                   therefore, (1) (2) ( ( ) ( ))p p   1 ( ) ( ) ( ) 1 2 1 ( ) ( ) ( ) 1 2 { 1 ( (1 ( ) ) (1 ( ) )) ( )}, { ( (( ) ) (( ) )) ( )} a q a q aq a b q b q bq b p p                       on the other hand, (1) (2) ( ) ( )p p    hybridizations of archimedean copula and generalized msm operators and their … 11 1 ( ) ( ) 1 ( ) ( ) 1 1 1 ( ) ( ) 1 ( ) ( ) 2 2 1 ( ) ( ) ( ) 1 2 1 ( ) ( ) 1 2 { 1 ( (1 ( )) ) ( )}, { ( (( )) ) ( )} { 1 ( (1 ( )) ) ( )}, { ( (( ) )) ( )} { 1 ( (1 ( ) ) (1 ( )) ) ( )}, { ( (( ) ) (( a q a b q bq q a b a q a b q bq q a b a q a q aq a b q b p p p p p                                                 ( ) ) )) ( )} q bq b p hence, (1) (2) (1) (2)( ( ) ( )) ( ) ( )p p p p        . (4)-(6) proof is similar to (3). 3.2 archimedean copula based gmsm operators based on the ac operational laws for qrpdhfes, we firstly propose qrpdhfgmsm operator and study it’s properties. definition 9: let ( ) ( ) ( ) ( ) ( ) ( ) { ( )}, { ( )} ( 1(1) ) j a a b b j j a b p p p j n      be a collection of adjusted qrpdhfes. then the ac based gmsm operator on qrpdhfes is denoted by qrpdhfgmsm and is defined by: 1 2 1 2 1 2 ; , ,....., (1) ( 2) ( ) 1 ..... ( ) 1 ... 1 ( ( ), ( ),..., ( )) 1 ( ( )) r r j j r r t t t n t t tr p t n p p p n j r qrpdhfgmsm p p p p c                         where 1 2 , ,....., 0 r t t t  , r is a parameter, and 1 2 ( , ,..., ) r p p p denotes a r-tuple combination of (1, 2,..., )n . theorem 2: 1 2 ; , ,....., (1) (2) ( ) ( ( ), ( ),..., ( ))r r t t t n qrpdhfgmsm p p p   is also a qrpdhfe and  1 2 1 2 ; , ,....., (1) ( 2 ) ( ) 1 1 1 ( ) ( ) 1 ... 11 2 1 1 2 ( ( ), ( ),..., ( )) 1 1 1 1 (( ) ) ( ) ..... 1 1 ..... r j r r t t t n r a q a q j pn p p p n ja r r r qrpdhfgmsm p p p t p t t t c t t t                                                        1 2 1 1 ( ) ( ) 1 ... 1 1 1 1 (1 ( ) ) ( ) j r r b q b q j pn p p p n jb r t p c                                        proof: we have, ( ) ( ( )) j jp t p    1 ( ) ( ) 1 ( ) ( )( (( ) )) ( ) , 1 ( (1 ( ) )) ( )j j a q a b q bq q j p j p a b t p t p            . therefore, ( ) 1 ( ( )) j j r p t j p    1 ( ) ( ) 1 ( ) ( ) 1 1 (( ) ) ( ) , 1 (1 ( ) ) ( ) j j r r a q a b q b q q j p j p j ja b t p t p                                           anusha et al./decis. mak. appl. manag. eng. (2022) 12 then, 1 2 ( ) 1 ... 1 ( ( )) j j r r p t p p p n j p               1 2 1 2 1 1 ( ) ( ) 1 ... 1 1 1 ( ) ( ) 1 ... 1 1 1 (( ) ) ( ) , 1 (1 ( ) ) ( ) j r j r r a q a q j p p p p n ja r b q b q j p p p p n jb t p t p                                                                          now 1 2 ( ) 1 ... 1 1 ( ( )) j j r r p t n p p p n j r p c                 1 2 1 2 1 1 ( ) ( ) 1 ... 1 1 1 ( ) ( ) 1 ... 1 1 1 1 (( ) ) ( ) , 1 1 (1 ( ) ) ( ) j r j r r a q a q j pn p p p n ja r r b q b q j pn p p p n jb r t p c t p c                                                                  hence, 1 2 ; , ,....., (1) (2) ( ) ( ( ), ( ),..., ( ))r r t t t n qrpdhfgmsm p p p   1 2 1 2 1 ..... ( ) 1 ... 1 1 ( ( )) r j j r t t tr p t n p p p n j r p c                       1 2 1 1 1 ( ) ( ) 1 ... 11 2 1 1 1 ( ) 11 2 1 1 1 1 (( ) ) ( ) ..... 1 1 1 1 1 (1 ( ) ) ..... j r j r a q a q j pn p p p n ja r r r b q j pn jr r t p t t t c t t t t c                                                                         1 2 ( ) 1 ... ( ) r b q p p p nb p                      in the following, some vital properties of the qrpdhfgmsm operator are presented. theorem 3: (idempotency) if ( ) ( ) ( ) ( ) j l p p j    (l being a fixed natural number), then 1 2 ; , ,....., (1) (2) ( ) ( ) ( ( ), ( ),..., ( )) ( )r r t t t n l qrpdhfgmsm p p p p     . theorem 4: (monotonicity) let ( ) ( ) ( ) ( ) ( )( ) { ( )}, { ( )} ( 1(1) )j a a b b j j a b p p p j n        be another collection of adjusted qrpdhfes such that j, ( ) ( )a a j j    and ( ) ( )b b j j    . then, 1 2 1 2; , ,..., ; , ,...,(1) (2) ( ) ( )( ( ), ( ),..., ( )) ( ))r r r t t t r t t tn n qrpdhfgmsm p p p qrpdhfgmsm p    1 2; , ,....., (1) (2) ( )( ( ), ( ),..., ( ))r r t t t n qrpdhfgmsm p p p     . hybridizations of archimedean copula and generalized msm operators and their … 13 theorem 5: (boundedness) if ( ) ( ) ( ) ( ) ( ) ( ) {min ( )},{max ( )} j a a b b j j a b p p p       and ( ) ( ) ( ) ( ) {max ( )}, j a a j a p p     ( ) ( ) {min ( )} , b b j b p  then 1 2; , ,.....,( ) (1) (2)( ) ( ( ), ( ),r r t t tj p qrpdhfgmsm p p     ( ) ( ) ..., ( )) ( ) n j p p     . next, based on the archimedean copula operational laws for qrpdhfes, we propose qrpdhf geometric gmsm operator and study it’s properties. definition 10: the ac based geometric gmsm operator on qrpdhfes is denoted by qrpdhfggmsm and is defined by:   1 2 1 2 ; , ,..., (1) ( 2) ( ) 1 ( ) 1 ... 1 1 2 ( ( ), ( ),..., ( )) 1 ( ) ..... r n r j r r t t t n r cp j p p p n j r qrpdhfggmsm p p p t p t t t                         where 1 2 , ,....., 0 r t t t  , r is a parameter, and 1 2 ( , ,..., ) r p p p denotes a r-tuple combination of (1, 2,..., )n . theorem 6: 1 2 ; , ,....., (1) (2) ( ) ( ( ), ( ),..., ( ))r r t t t n qrpdhfggmsm p p p   is also a qrpdhfe and 1 2 1 2 ; , ,....., (1) ( 2) ( ) 1 1 1 ( ) ( ) 1 ... 1 1 1 ( ( ), ( ),..., ( )) 1 1 1 1 1 (1 ( ) ) ( ) , 1 1 1 1 r j r r t t t n r a q a q j pn p p p n ja r r r n r r r qrpdhfggmsm p p p t p t c t c                                                          1 2 1 ( ) ( ) 1 ... 1 (( ) ) ( ) j r r b q b q j p p p p n jb t p                                    proof: similar to theorem 2. in the following, some vital properties of the qrpdhfggmsm operator are presented. theorem 7: (idempotency) if ( ) ( ) ( ) ( ) j l p p j    (l being a fixed natural number), then 1 2 ; , ,....., (1) (2) ( ) ( ) ( ( ), ( ),..., ( )) ( )r r t t t n l qrpdhfggmsm p p p p     . theorem 8: (monotonicity) let ( ) ( ) ( ) ( ) ( ) ( ) { ( )}, { ( )} ( 1(1) ) j a a b b j j a b p p p j n        be another collection of adjusted qrpdhfes such that j, ( ) ( )a a j j    and ( ) ( )b b j j    . then, 1 2 1 2; , ,..., ; , ,...,(1) (2) ( ) (1)( ( ), ( ),..., ( )) ( ( ),r r r t t t r t t tn qrpdhfggmsm p p p qrpdhfggmsm p    (2) ( ) ( ),..., ( )) n p p   . theorem 9: (boundedness) if ( ) ( ) ( ) ( ) ( ) ( ) {min ( )},{max ( )} j a a b b j j a b p p p       and ( ) ( ) ( ) ( ) {max ( )}, j a a j a p p     ( ) ( ) {min ( )} , b b j b p  then 1 2; , ,.....,( ) (1) (2)( ) ( ( ), ( ),r r t t tj p qrpdhfggmsm p p     ( ) ( ) ..., ( )) ( ) n j p p     . anusha et al./decis. mak. appl. manag. eng. (2022) 14 3.3 archimedean copula based weighted gmsm operator although the qrpdhfgmsm operator can tackle the interrelationship among multiple input criteria, it does not take into account the self importance of the qrpdhfes. to overcome this problem, we propose qrpdhf weighted gmsm operator (qrpdhfwgmsm operator) based on archimedean copula. definition 11: the ac based qrpdhfwgmsm operator on qrpdhfes is defined by: 1 2 1 2 1 2 ; , ,....., (1) ( 2) ( ) 1 ..... ( ) 1 ... 1 ( ( ), ( ),..., ( )) 1 ( ( )) r r j j j r r t t t n t t tr p t pn p p p n j r qrpdhfwgmsm p p p w p c                         where 1 2 , ,....., 0 r t t t  , r is a parameter, 1 2 ( , ,..., ) r p p p denotes a r-tuple combination of (1, 2,..., )n and j w denotes the weight of ( ) ( ) j p with 0 j w  and 1 j j w  . theorem 10: 1 2 ; , ,....., (1) (2) ( ) ( ( ), ( ),..., ( ))r r t t t n qrpdhfwgmsm p p p   is also a qrpdhfe and  1 2 1 2 r; , ,....., (1) ( 2) ( ) 1 1 1 1 ( ) ( ) 1 ... 1 1 ( ( ), ( ),..., ( )) 1 1 1 1 (1 ( (1 ( ) ))) ( ) , 1 1 1 r j j r t t t n r a q a q j p pn p p p n ja r r r r r qrpdhfwgmsm p p p t w p t c t                                                            1 2 1 1 1 ( ) ( ) 1 ... 1 1 1 (1 ( (( ) ))) ( ) j j r r b q b q j p pn p p p n jb r t w p c                                         proof: we have, ( ) ( ) j j p p w p    1 ( ) ( ) 1 ( ) ( )1 ( (1 ( ) )) ( ) , ( (( ) )) ( )j j j j a q a b q bq q p p p p a b w p w p            . then, ( ) ( ( )) j j j p t p w p    1 1 ( ) ( ) 1 1 ( ) ( ) ( (1 ( (1 ( ) )))) ( ) , 1 ( (1 ( (( ) )))) ( ) j j j j a q aq j p p a b q bq j p p b t w p t w p                         so, ( ) 1 ( ( )) j j j r p t p j w p    1 1 ( ) ( ) 1 1 1 ( ) ( ) 1 (1 ( (1 ( ) ))) ( ) , 1 (1 ( (( ) ))) ( ) j j j j r a q a q j p p ja r b q b q j p p jb t w p t w p                                              hybridizations of archimedean copula and generalized msm operators and their … 15 therefore, 1 2 ( ) 1 ... 1 ( ( )) j j j r r p t p p p p n j w p                 1 2 1 2 1 1 1 ( ) ( ) 1 ... 1 1 1 1 ( ) ( ) 1 ... 1 1 1 (1 ( (1 ( ) ))) ( ) , 1 (1 ( (( ) ))) ( ) j j r j j r q a q a q j p p p p p n ja q b q b q j p p p p p n jb t w p t w p                                                                    then, 1 2 ( ) 1 ... 1 1 ( ( )) j j j r r p t pn p p p n j r w p c                1 2 1 2 1 1 1 ( ) ( ) 1 ... 1 1 1 1 ( ) ( 1 ... 1 1 1 1 (1 ( (1 ( ) ))) ( ) , 1 1 (1 ( (( ) ))) ( j j r j j r r a q a q j p pn p p p n ja r r b q b q j p pn p p p n jb r t w p c t w p c                                                                       ) ) hence, 1 2 ; , ,....., (1) (2) ( ) ( ( ), ( ),..., ( ))r r t t t n qrpdhfwgmsm p p p    1 2 1 1 1 1 ( ) ( ) 1 ... 1 1 1 1 1 ( ) 1 1 1 1 1 (1 ( (1 ( ) ))) ( ) , 1 1 1 1 1 (1 ( (( ) ))) j j r j j r a q a q j p pn p p p n ja r r r r b q j p pn jr r r t w p t c t w t c                                                                              1 2 ( ) 1 ... ( ) r b q p p p nb p                      i n the following, some vital properties of the qrpdhfwgmsm operator are presented. theorem 11: (idempotency) if ( ) ( ) ( ) ( ) j l p p j    (l being a fixed natural number), then 1 2 ; , ,....., (1) (2) ( ) ( ) ( ( ), ( ),..., ( )) ( )r r t t t n l qrpdhfwgmsm p p p p     . theorem 12: (monotonicity) let ( ) ( ) ( ) ( ) ( ) ( ) { ( )}, { ( )} ( 1(1) ) j a a b b j j a b p p p j n        be another collection of adjusted qrpdhfes such that j, ( ) ( )a a j j    and ( ) ( )b b j j    . then 1 2 1 2; , ,..., ; , ,...,(1) (2) ( ) (1)( ( ), ( ),..., ( )) ( ( ),r r r t t t r t t tn qrpdhfwgmsm p p p qrpdhfwgmsm p    (2) ( ) ( ),..., ( )) n p p   . theorem 13: (boundedness) if ( ) ( ) ( ) ( ) ( ) ( ) {min ( )},{max ( )} j a a b b j j a b p p p       and ( ) ( ) ( ) ( ) {max ( )}, j a a j a p p     ( ) ( ) {min ( )} , b b j b p  then 1 2; , ,.....,( ) (1) (2)( ) ( ( ), ( ),r r t t tj p qrpdhfwgmsm p p     ( ) ( ) ..., ( )) ( ) n j p p     . the qrpdhfgmsm operator considers the interrelationship among multiple criteria. but, it does not deal with the self priority of the qrpdhfes. to overcome this problem, we propose qrpdhf weighted geometric gmsm operator (qrpdhfwggmsm operator) based on archimedean copula. anusha et al./decis. mak. appl. manag. eng. (2022) 16 definition 12: the ac based qrpdhfwggmsm operator on qrpdhfes is defined by:    1 2 1 2 ; , ,....., (1) ( 2) ( ) 1 ( ) 1 ... 1 ( ( ), ( ),..., ( )) 1 ( ) r n p rj j r r t t t n r w cp j p p p n j r r qrpdhfwggmsm p p p t p t                      where 1 2 , ,....., 0 r t t t  , r is a parameter, 1 2 ( , ,..., ) r p p p denotes a r-tuple combination of (1, 2,..., )n and j w denotes the weight of ( ) ( ) j p with 0 j w  and 1 j j w  . theorem 14: 1 2 ; , ,....., (1) (2) ( ) ( ( ), ( ),..., ( ))r r t t t n qrpdhfwggmsm p p p   is also a qrpdhfe and  1 2 1 2 ; , ,....., (1) ( 2) ( ) 1 1 1 1 ( ) ( ) 1 ... 1 1 ( ( ), ( ),..., ( )) 1 1 1 1 1 (1 ( (( ) ))) ( ) , 1 1 r j j r r t t t n r a q a q j p pn p p p n ja r r r r r qrpdhfwggmsm p p p t w p t c t                                                            1 2 1 1 1 ( ) ( ) 1 ... 1 1 1 (1 ( (1 ( ) ))) ( ) j j r r b q b q j p pn p p p n jb r t w p c                                         p roof: similar to theorem 10. a few crucial properties of the qrpdhfwggmsm operator are demonstrated below. theorem 15: (idempotency) if ( ) ( ) ( ) ( ) j l p p j    (l being a fixed natural number), then 1 2 ; , ,....., (1) (2) ( ) ( ) ( ( ), ( ),..., ( )) ( )r r t t t n l qrpdhfwggmsm p p p p     . theorem 16: (monotonicity) let ( ) ( ) ( ) ( ) ( ) ( ) { ( )}, { ( )} ( 1(1) ) j a a b b j j a b p p p j n        be another collection of adjusted qrpdhfes such that j, ( ) ( )a a j j    and ( ) ( )b b j j    . then, 1 2 1 2; , ,....., ; , ,.....,(1) (2) ( )( ( ), ( ),..., ( ))r r r t t t r t t tn qrpdhfwggmsm p p p qrpdhfwggmsm   (1) (2) ( ) ( ( ), ( ),..., ( )) n p p p     . theorem 17: (boundedness) for ( ) ( ) ( ) ( ) ( )( ) {min ( )},{max ( )}j a a b b j j a b p p p       and ( ) ( ) ( ) ( ) {max ( )}, j a a j a p p     ( ) ( ){min ( )} ,b b j b p  1 2; , ,.....,( ) ( ) r r t t tj p qrpdhfwggmsm   (1) (2) ( ) ( ) ( ( ), ( ),..., ( )) ( ) n j p p p p       . 4. group decision-making methodology with interaction of experts suppose there are m number of options ( 1, 2,..., ) i a i m and n number of criteria ( 1, 2,..., ) j c j n connected with a decision-making issue with ‘d’ number of experts hybridizations of archimedean copula and generalized msm operators and their … 17 ( 1, 2,..., ) d e d l under qrpdhf setting. consider the qrpdhf matrices d m  ( ) ( ) ( ) ( ) ( ) ( ) { ( )}, { ( )} ijd a a b b ijd ijd m n a b m n p p p                as initial assessments of experts. then the developed method has the steps mentioned below: step 1: derive the adjusted qrpdhf matrices ( ) ( ) ( ) ( ) ( ) ( ) { ( )}, { ( )} ijd a a b b d ijd ijd m n a b m n m p p p                 ( 1, 2,..., )d l where each ( ) ( ) ijd p represents an adjusted qrpdhfe. step 2: calculate the weights of the experts. suppose d = weight of the expert ,de d = subjective weight of the expert ,de d = objective weight of the expert de . as pointed out by cheng et al. (2018), d gives the chwi (consistency harmonious weight index defined by ( ) ( ) 1 1 1 ( ) ( ( )) ( ( )) ( 1, 2,..., ) n m m ijd tji d j i t chwi e n p p d l                (1) where ( ) ( ) ( ) ( ) ( )( ( )) .ijd a a b b ijd ijd a b p p p        if ( )dchwi e =1, dm is consistent (han & li, 1994). the subjective weight d is calculated as: 1 ( ) ( ) ( 1(1) ) l d d d d chwi e chwi e d l     (2) in the event that the closeness between d m and ( ) s m d s is high, the impact of d m is more important. let ds  be the angle between d m and ( ) s m d s . then ds  can be computed by ( ), ( ) ( , 1(1) ) ( ) ( ) d s ds d s v m v m d s l v m v m     (3) where ( ) d v m =derived vector of d m  (11 ) ( 21 ) ( 1 ) (12 ) ( 22 ) ( 2 ) (1 )( ), ( ),..., ( ) , ( ), ( ),...., ( ),..., ( ),d d m d d d m d ndp p p p p p p        ( 2 ) ( )( ),..., ( )nd mndp p  and ( ) ( ) 1 1 ( ), ( ) ( ( ( )) ( ( ))) m n ijd ijs d s i j v m v m p p       . obviously, 0 1 ( , 1(1) ) ds d s l   . suppose 1, l d ds s s d       (4) then d  expresses the closeness between d m and ( ) s m d s . we normalize d  using eq. (5) to obtain the objective weight of each criterion. anusha et al./decis. mak. appl. manag. eng. (2022) 18 1 d d l d d        ( 1(1) )d l (5) utilizing a mix of ( 1(1) )d d l   and ( 1(1) )d d l  , experts’ final weights can be calculated by: (1 )d d d       ( 1(1) )d l (6) in eq. (6),  sorts out which weight is dominating. the experts lean towards the subjective weights if  is high; and in case if  is low, experts favor the objective weights. the parameter  [0, 1] is termed as the risk attitude of experts. the greater the value of  , the more inclination of the expert towards risk. also, he/she turns into a risk adverse person who regards the trustworthiness of the evaluation group. step 3: obtain the consensus coefficient. in an innovative mcgdm procedure, the weights of experts will change if the information provided changes. to get a more reasonable decision result, experts should interrelate with each other and lastly take a decision on assessing information. we utilize the symbol * to express the consensus coefficient. the judgment matrix of the dth expert obtained from the  round interaction is defined by ( ) d v m  =  (11 )( ) ( 21 )( )( ), ( ),d dp p   ( 1 )( ) (12 )( ) ( 22 )( ) ( 2 )( ) (1 )( ) ( 2 )( ) ( )( )..., ( ), ( ), ( ),...., ( ),..., ( ), ( ),..., ( ) ( 1(1) ).m d d d m d nd nd mndp p p p p p p d l              the consensus coefficient ( )  which is computed by the  th adjustment can be defined by ( ) 1 1 ( ) ( 1) l l ds d s d s l l          (7) especially ( )  =1 if and only if ( ) ( ) d s v m v m    . in that case the opinions of the experts are fully unified, then ( ) 0 1     . thus, throughout the dm procedure, one expert should provide the weights to others based on the assessment information and estimate the consensus coefficient in the  round after he/she has obtained a consensus value * in advance, and then check whether meets ( ) *     . if ( ) *     , experts need to carry out interactive communication and then recalculate ( )  until ( ) *     . step 4: aggregate the qrpdhf matrices using the proposed aos. here qrpdhfwgmsm (or qrpdhfwggmsm) operator is utilized to obtain the aggregated qrpdhf matrix ( ) ( ) ( ) ( ) ( ) ( ) { ( )}, { ( )} ij a a b b ij ij m n a b p p p          . ( ) ( 1) ( 2) ( ) ( ) ( ( ), ( ),..., ( ) ) ij ij ij ijl pdhfwgmsp qr m p p p     (8) or ( ) ( 1) ( 2) ( ) ( ) ( ( ), ( ),..., ( ) ) ij ij ij ijl pdhfwggmp qr p psm p     (9) step 5: construct the normalized aggregated qrpdhf matrix ( ) ( ) ( 1(1) , 1(1) ) ij m n p i m j n         . hybridizations of archimedean copula and generalized msm operators and their … 19 here ( ) ( ) ij p   = ( ) ( ) ( ) ( ){ ( )}, { ( )}a a b b ij ij a b p p    (or ( ) ( ) ( ) ( ){ ( )}, { ( )}b b a a ij ij b a p p    ) if j c is a benefit criteria (or cost criteria). step 6: calculate the criteria weights. the weights of criteria’ play an important role on the final outcome. consider an expert de , and qrpdhf data under jc . then the following divergence measure can be used to describe how the alternative ia differs from other alternatives. ( ) ( ) 1 1 ( ( ), ( )) 1 m d ijd pjd ij p div c p p m       (10) where ( ) ( ) ( ( ), ( )) ijd pjd c p p  stands for cross-entropy measure between the qrpdhfes ( ) ( ) ( ) and ( ) ijd pjd p p  . we define it by: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( ), ( )) 2 ( ) 1 ( ) ln ( ) ( ) 2 ( ) 1 ( ) ln ( ) ( ) ijd pjd a a ijd a a a ijd a a a aa ijd pjd a a a a pjd a a a pjd a a a aa ijd pjd a a c p p p p a p p p p a p p                                                               ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 ( ) 1 1 ( ) ln 1 1 ( ) ( ) 2 1 1 ( ) 1 1 ( ) ln 1 1 ( ) ( ) 2 a a ijd a a a ijd a a a a a ijd pjd a a a a pjd a a a pjd a a a a a ijd pjd a p a p a p p a p a p a p p a                                                         ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 ( ) 1 ( ) ln ( ) ( ) 2 ( ) 1 ( ) ln ( ) ( ) a b b ijd b b b ijd b b b bb ijd pjd b b b b pjd b b b pjd b b b bb ijd pjd b b p p b p p p p b p p                                                                           anusha et al./decis. mak. appl. manag. eng. (2022) 20 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 ( ) 1 1 ( ) ln 1 1 ( ) ( ) 2 1 1 ( ) 1 1 ( ) ln 1 1 ( ) ( ) 2 b b ijd b b b ijd b b b b b ijd pjd b b b b pjd b b b pjd b b b b b ijd pjd b p b p b p p a p b p b p p a                                                         b                 (11) the total divergence for the criterion j c is: ( ) ( ) 1 1 1 ( ( ), ( )) 1 m m d ijd pjd j i p div c p p m        (12) considering the importance of all experts, we can determine the overall divergences for each option over the given criterion j c . ( ) ( ) 1 1 1 1 1 ( ( ), ( )) 1 l l m m d ijd pjd j d j d d d i p div div c p p m               (13) from the above discussions, it is clear that the following optimization model can be used to calculate weights of criteria. ( ) ( ) 1 1 1 1 1 1 ( ( ), ( )) 1 subject to , 1, 0 n l m m ijd pjd j d j d i p j n j j j max w c p p m w w w j                          (14) where  is the set of partial information’s about criteria weights. step 7: applying the idea of adjusted probabilities, we create the adjusted aggregated qrpdhf matrix: ( ) ( ) ( ) ( ) ( ) ( ) { ( )}, { ( )} ( 1(1) , 1(1) ) ij a a b b ij ij m n a b m n p p p i m j n                       . to develop this final adjusted aggregated qrpdhf matrix ( ) ( ) i m n p       , the qrpdhfwgmsm operator (or qrpdhfwggmsm operator) is used. ( ) ( 1)* ( 2)* ( )* ( ) ( ( ) , ( ) ,........, ( ) ) i i i in dp p pqrp hfwgmsm p          (15) or ( ) ( 1)* ( 2)* ( )* ( ) ( ( ) , ( ) ,........, ( ) ) i i i in hp pqrpd fwggmsm p p          (16) step 8: generate the preference the options ( 1(1) ) i a i m using the scores of ( ) ( ) ( 1(1) ) i p i m    and select the optimal option. hybridizations of archimedean copula and generalized msm operators and their … 21 5. case study and solution the proposed method is employed in example-1 in order to assess the oss-lms alternatives with qrpdhf information. step 1: the initial assessments of experts are shown in table-5. table 5. adjusted assessment matrix c1 c2 c3 e1 a1 <{0.1(0.3), 0.3(0.2), 0.4(0.2), 0.4(0.3)}, {0.6(0.3), 0.6(0.1), 0.2(0.3), 0.3(0.3)}> <{0.6(0.4), 0.8(0.2), 0.3(0.3), 0.2(0.1)}, {0.3(0.4), 0.2(0.2), 0.7(0.3), 0.9(0.1)}> <{0.4(0.2), 0.1(0.4), 0.5(0.1), 0.7(0.3)}, {0.1(0.1), 0.5(0.3), 0.4(0.3), 0.3(0.3)}> a2 <{0.3(0.1), 0.7(0.5), 0.8(0.2), 0.5(0.2)}, {0.9(0.5), 0.6(0.1), 0.5(0.2), 0.5(0.2)}> <{0.2(0.5), 0.5(0.2), 0.6(0.1), 0.3(0.2)}, {0.8(0.3), 0.8(0.1), 0.6(0.2), 0.4(0.4)}> <{0.5(0.1), 0.8(0.1), 0.8(0.5), 0.4(0.3)}, {0.5(0.1), 0.3(0.4), 0.1(0.4), 0.5(0.1)}> a3 <{0.6(0.4), 0.6(0.1), 0.2(0.2), 0.3(0.3)}, {0.1(0.3), 0.3(0.2), 0.4(0.1), 0.4(0.4)}> <{0.3(0.2), 0.2(0.1), 0.7(0.1), 0.9(0.6)}, {0.6(0.5), 0.8(0.1), 0.3(0.2), 0.2(0.2)}> <{0.1(0.2), 0.5(0.4), 0.4(0.3), 0.3(0.1)}, {0.4(0.1), 0.1(0.3), 0.5(0.5), 0.7(0.1)}> e2 a1 <{0.4(0.3), 0.6(0.2), 0.5(0.2), 0.3(0.3)}, {0.3(0.3), 0.7(0.1), 0.8(0.3), 0.5(0.3)}> <{0.7(0.4), 0.8(0.2), 0.6(0.3), 0.4(0.1)}, {0.2(0.4), 0.5(0.2), 0.6(0.3), 0.3(0.1)}> <{0.5(0.2), 0.3(0.4), 0.1(0.1), 0.4(0.3)}, {0.5(0.1), 0.8(0.3), 0.8(0.3), 0.4(0.3)}> a2 <{0.1(0.1), 0.3(0.5), 0.4(0.2), 0.4(0.2)}, {0.6(0.5), 0.6(0.1), 0.2(0.2), 0.3(0.2)}> <{0.6(0.5), 0.8(0.2), 0.3(0.1), 0.2(0.2)}, {0.3(0.3), 0.2(0.1), 0.7(0.2), 0.9(0.4)}> <{0.4(0.1), 0.1(0.1), 0.5(0.5), 0.7(0.3)}, {0.1(0.1), 0.5(0.4), 0.4(0.4), 0.3(0.1)}> a3 <{0.9(0.4), 0.6(0.1), 0.5(0.2), 0.5(0.3)}, {0.9(0.3), 0.6(0.2), 0.5(0.1), 0.5(0.4)}> <{0.8(0.2), 0.8(0.1), 0.6(0.1), 0.4(0.6)}, {0.8(0.5), 0.8(0.1), 0.6(0.2), 0.4(0.2)}> <{0.5(0.2), 0.3(0.4), 0.1(0.3), 0.5(0.1)}, {0.5(0.1), 0.3(0.3), 0.1(0.5), 0.5(0.1)}> e3 a1 <{0.6(0.3), 0.6(0.2), 0.2(0.2), 0.3(0.3)}, {0.1(0.3), 0.3(0.1), 0.4(0.3), 0.4(0.3)}> <{0.3(0.4), 0.2(0.2), 0.7(0.3), 0.9(0.1)}, {0.6(0.4), 0.8(0.2), 0.3(0.3), 0.2(0.1)}> <{0.1(0.2), 0.5(0.4), 0.4(0.1), 0.3(0.3)}, {0.4(0.1), 0.1(0.3), 0.5(0.3), 0.7(0.3)}> a2 <{0.9(0.1), 0.6(0.5), 0.5(0.2), 0.5(0.2)}, {0.3(0.5), 0.7(0.1), 0.8(0.2), 0.5(0.2)}> <{0.8(0.5), 0.8(0.2), 0.6(0.1), 0.4(0.2)}, {0.2(0.3), 0.5(0.1), 0.6(0.2), 0.3(0.4)}> <{0.5(0.1), 0.3(0.1), 0.1(0.5), 0.5(0.3)}, {0.5(0.1), 0.8(0.4), 0.8(0.4), 0.4(0.1)}> a3 <{0.3(0.4), 0.7(0.1), 0.8(0.2), 0.5(0.3)}, {0.6(0.3), 0.6(0.2), 0.2(0.1), 0.3(0.4)}> <{0.2(0.2), 0.5(0.1), 0.6(0.1), 0.3(0.6)}, {0.3(0.5), 0.2(0.1), 0.7(0.2), 0.9(0.2)}> <{0.5(0.2), 0.8(0.4), 0.8(0.3), 0.4(0.1)}, {0.1(0.1), 0.5(0.3), 0.4(0.5), 0.3(0.1)}> step 2: we compute the subjective weights of experts with the use of eq. (1) and eq. (2). the connections between the experts’ opinions are gained using eq. (3) and eq.(4) and are shown in table-6. the objective weights of are calculated using eq. (5). their overall weight of experts’ are calculated by eq. (6) (choosing  =0.5) and are given in table-7. anusha et al./decis. mak. appl. manag. eng. (2022) 22 table 6. similarity between experts’ judgments d1 d2 d3 d1 1 0.94892766 0.975803776 d2 0.94892766 1 0.955188852 d3 0.975803776 0.955188852 1 table 7. weights of experts subjective objective final d1 1  = 0.355950765 1 = 0.334164012 1 = 0.3451 d2 2 = 0.307840016 2 =0.330584933 2 = 0.3192 d3 3  = 0.336209218 3 = 0.335251055 3 = 0.3357 step 3: the consensus coefficient value is obtained using eq. (7) and is found to be equal to 0.960680. this means that consensus level is not reached since ( ) 0.9606 7* 080 .9       . according to table-6, low level of connections is found among the 2nd and other experts. this signifies a biased decision of 2nd expert. therefore, the evaluation data of 2nd expert should get modified. the revised assessment information of the expert e2 is presented in table-8. table 8. revised adjusted evaluation matrix for expert e2 c1 c2 c3 a1 <{0.4(0.3), 0.6(0.2), 0.5(0.2), 0.3(0.3)}, {0.3(0.3), 0.7(0.1), 0.8(0.3), 0.5(0.3)}> <{0.7(0.4), 0.8(0.2), 0.6(0.3), 0.4(0.1)}, {0.2(0.4), 0.5(0.2), 0.6(0.3), 0.3(0.1)}> <{0.5(0.2), 0.3(0.4), 0.1(0.1), 0.4(0.3)}, {0.5(0.1), 0.8(0.3), 0.8(0.3), 0.4(0.3)}> a2 <{0.9(0.1), 0.3(0.5), 0.4(0.2), 0.4(0.2)}, {0.6(0.5), 0.6(0.1), 0.2(0.2), 0.3(0.2)}> <{0.6(0.5), 0.8(0.2), 0.3(0.1), 0.3(0.2)}, {0.3(0.3), 0.2(0.1), 0.7(0.2), 0.9(0.4)}> <{0.4(0.1), 0.9(0.1), 0.5(0.5), 0.7(0.3)}, {0.1(0.1), 0.5(0.4), 0.4(0.4), 0.3(0.1)}> a3 <{0.9(0.4), 0.6(0.1), 0.5(0.2), 0.5(0.3)}, {0.9(0.3), 0.6(0.2), 0.5(0.1), 0.5(0.4)}> <{0.8(0.2), 0.8(0.1), 0.6(0.1), 0.4(0.6)}, {0.8(0.5), 0.8(0.1), 0.6(0.2), 0.4(0.2)}> <{0.5(0.2), 0.3(0.4), 0.9(0.3), 0.5(0.1)}, {0.5(0.1), 0.3(0.3), 0.1(0.5), 0.5(0.1)}> the updated similarity between the experts is shown in table-9. the revised subjective weights, objective weights and final weights of experts are given in table10. table 9. similarity between experts’ judgments d1 d2 d3 d1 1 0.960562903 0.975803776 d2 0.960562903 1 0.973687808 d3 0.975803776 0.973687808 1 table 10. weights of experts subjective objective final d1 1  = 0.36673953 1 = 0.332702822 1 = 0.3497 d2 2 = 0.286860847 2 = 0.33233926 2 =0.3096 d3 3  = 0.346399623 3 = 0.334957918 3 = 0.3407 hybridizations of archimedean copula and generalized msm operators and their … 23 we again calculate the consensus coefficient and we get ( ) 0.970018 * 0.97       . this means that the required consensus level has been achieved. step 4: utilizing eq. (8) and taking r=2, q=3, 1 2 2t t  , and 1 ( ) 1 ( (0,1])x x x     , the aggregated qrpdhf matrix is obtained (table-11). table 11. aggregated qrpdhf matrix c1 c2 c3 a1 <{0.227895306 (0.3), 0.34146202 (0.2), 0.238279964 (0.2), 0.22633804 (0.3)}, {0.463528684 (0.3), 0.743639962 (0.1), 0.611580768 (0.3), 0.548554159 (0.3)}> <{0.366855654 (0.4), 0.473175705 (0.2), 0.363325668 (0.3), 0.257291879 (0.1)}, {0.309685763 (0.4), 0.62809717 (0.2), 0.741543607 (0.3), 0.487899542 (0.1)}> <{0.220229689(0.2), 0.174306177 (0.4), 0.221593385 (0.1), 0.282222658 (0.3)}, {0.518936043 (0.1), 0.693934888 (0.3), 0.740074614 (0.3), 0.531694149 (0.3)}> a2 <{0.61529503 (0.1), 0.367099559 (0.5), 0.366606178 (0.2), 0.326855285 (0.2)}, {0.805611894 (0.5), 0.743422141 (0.1), 0.328076273 (0.2), 0.455493106 (0.2)}> <{0.361526112 (0.5), 0.527071257 (0.2), 0.343395786 (0.1), 0.226706449 (0.2)}, {0.48300963 (0.3), 0.333296918 (0.1), 0.783092523 (0.2), 0.65499119 (0.4)}> <{0.326855285 (0.1), 0.53612698 (0.1), 0.296510226 (0.5), 0.361337545 (0.3)}, {0.168341209 (0.1), 0.63591727 (0.4), 0.517516245 (0.4), 0.451420463 (0.1)}> a3 <{0.402114899 (0.4), 0.463087898 (0.1), 0.307250966 (0.2), 0.294264417 (0.3)}, {0.797044611 (0.3), 0.679474704 (0.2), 0.535805538 (0.1), 0.55446316 (0.4)}> <{0.211311653 (0.2), 0.313720714 (0.1), 0.462607172 (0.1), 0.287719327 (0.6)}, {0.778036281 (0.5), 0.869439288 (0.1), 0.705266928 (0.2), 0.526905632 (0.2)}> <{0.25019694 (0.2), 0.336742161 (0.4), 0.548149113 (0.3), 0.269852656 (0.1)}, {0.527350985 (0.1), 0.387209963 (0.3), 0.168263919 (0.5), 0.677942932 (0.1)}> step 5: normalization is not required as no cost type criterion is considered. step 6: assume that 1 2 3 {0.10 w 0.30, 0.15 w 0.40, 0.20 w 0.35}        then the following optimization model is obtained: 1 2 3 1 2 3 1 2 3 1 2 3 0.018622 0.015832 0.025745 subject to 0.10 w 0.30, 0.15 w 0.40, 0.20 w 0.35, 1, , , 0 max w w w w w w w w w                 solving the above optimization model, we get, 1 2 3 0.30, 0.35, 0.35w w w   and 0.0201max   . step 7: based on table-11 and the criteria weights ( 1 2 3 0.30, 0.35, 0.35w w w   ), the final adjusted aggregated qrpdhf matrix ( ) 3 1 ( ) i         (table-12) is constructed using on eq. 15 by taking 1 ( ) 1 ( (0,1])x x x     . anusha et al./decis. mak. appl. manag. eng. (2022) 24 table 12. final adjusted aggregated qrpdhf matrix aggregated qrpdhfes a1 <{ 0.174194042 (0.2), 0.157802072 (0.1), 0.192021822 (0.1), 0.204677709 (0.1), 0.165325088 (0.1), 0.177843433 (0.1), 0.193274562 (0.2), 0.175371303 (0.1)}, {0.607032006 (0.1), 0.673758863 (0.2), 0.808611568 (0.1), 0.821571833 (0.2), 0.850950169 (0.1), 0.781393058 (0.2), 0.695389123 (0.1)}> a2 <{0.278236296 (0.1), 0.281989064 (0.1), 0.236699614 (0.3), 0.258450061 (0.1), 0.258286059 (0.1), 0.24977418 (0.1), 0.204800915 (0.2)}, {0.684475187 (0.1), 0.82164947 (0.2), 0.803535681 (0.1), 0.886928017 (0.1), 0.856991069 (0.1), 0.685299996 (0.2), 0.717825489 (0.1), 0.695270867 (0.1)}> a3 <{0.183693202 (0.2), 0.242842796 (0.1), 0.275604637 (0.1), 0.243090499 (0.1), 0.215248104 (0.1), 0.235707269 (0.1), 0.231045195 (0.2), 0.197489731 (0.1)}, {0.873194635 (0.1), 0.865508021 (0.2), 0.821881338 (0.1), 0.813389698 (0.1), 0.756819281 (0.1), 0.719803034 (0.1), 0.719803034 (0.1), 0.638226621 (0.1), 0.759939279 (0.1)}> step 8: the score values of the aggregated qrpdhfes are: (1) ( ( ))sc p   = -0.431971, ( 2) ( ( ))sc p   = -0.447465, (3) ( ( ))sc p   = -0.481656. hence, the priority order is: 1 2 3a a a . therefore, the optimal option is 1.a 6. discussion 6.1 impact of parameters upon priority order to signify the effects of the parameters 1 2 andt t (taking r=2, q=3) upon score values, the operators qrpdhfwgmsm and qrpdhfwggmsm are used choosing 1 2 , {1, 2,...,10}t t  . (1) suppose the value of 2 t is fixed (say, 2 t =2). to assess the parameter's effect 1 t upon ranking order, we employ a range of parameter values  1t with qrpdhfwgmsm operator. the related scores of alternatives are depicted in fig. 1. as demonstrated by fig.1, the ranking order is 2 3 1 a a a and 1 2 3 a a a for 1 1 2t  and 1 2 10t  respectively and thus the best alternative is 1a or 2a when qrpdhfwgmsm operator is used. hybridizations of archimedean copula and generalized msm operators and their … 25 figure 1. scores of alternatives when 1 t =1(1)10 using qrpdhfwgmsm operator (2) next, suppose the value of 1 t is fixed (say, 1 t =4). we utilize diverse parameter  2t values to illustrate its impact upon priority order using the qrpdhfwgmsm operator. the related score values of alternatives are depicted in fig. 2. as indicated by fig.2, the ranking order is 1 2 3 a a a and 2 3 1 a a a for 1 1 7t  and 1 7 10t  respectively and thus the best alternative is 1a or 2a when qrpdhfwgmsm operator is used. figure 2. scores of alternatives when 2 t =1(1)10 using qrpdhfwgmsm operator (3) to assess the parameter's effect 1 t upon ranking order, we employ a range of parameter values  1t with qrpdhfwggmsm operator. we first take a fixed value of 2 t , say, 2 t =2. the related scores of alternatives are depicted in fig. 3. as anusha et al./decis. mak. appl. manag. eng. (2022) 26 demonstrated by fig.3, the ranking order is 2 3 1 a a a and 3 2 1 a a a for 1 1 6t  and 1 6 10t  respectively and thus the best alternative is 3a or 2a when qrpdhfwggmsm operator is used. figure 3. scores of alternatives when 1 t =1(1)10 using qrpdhfwggmsm operator (4) lastly, suppose the value of 1 t is fixed (say, 1 t =4). we utilize diverse parameter  2t values to illustrate its impact upon priority order using the qrpdhfwggmsm operator. the related score values of alternatives are depicted in fig. 4. as demonstrated by fig.4, the priority is 3 2 1 a a a and 2 3 1 a a a for 1 1 2t  and 1 2 10t  respectively and thus the best alternative is 3a or 2a when qrpdhfwggmsm operator is used. figure 4. scores of alternatives when 2 t =1(1)10 using qrpdhfwggmsm operator hybridizations of archimedean copula and generalized msm operators and their … 27 6.2 comparative study a research comparing our suggested approach with li et al.’s method (2020) is offered to assess its efficacy. it is based on the qrpdhfwgmsm and qrpdhfwggmsm operators. table 13 lists the characteristics that set them apart from one another. in section 3's example 1, we use the methodology. because it is based on several mcdm approaches, li et al.’s method (2020) fails to produce any preference order of options, as shown in table 13. . but, our proposed method gives the priority order 2 1 3a a a (without interaction) and 1 2 3a a a (with interaction). as a result, the approach created using the qrpdhfwgmsm (or qrpdhfwggmsm) operator is successful. table 13. comparative investigation aspects proposed li et al. (2020) information type qrpdhf qrpdhf decision-making type group dm individual dm hesitation in preferences considered considered probabilistic information considered considered adjustment of probabilities considered no determination of experts’ weights subjective and objective weights with interaction among experts not applicable criteria’ weights selection a cross entropy based optimization model direct aggregation operators archimedean copula based weighted generalized maclaurin symmetric mean aos qrpdhf power weighted muirhead mean (qrpdhfpwmm) operator flexibility of the aos very high high whether consider dependency among multi-input criteria yes no determination of consensus coefficient considered not considered ranking of alternatives (without interaction of experts) with qrpdhfwgmsm operator taking r=2, q=3, and 1 2 2t t  2 1 3a a a can’t be determined ranking of alternatives (with interaction of experts) with qrpdhfwgmsm operator taking r=2, q=3, and 1 2 2t t  1 2 3a a a can’t be determined in the last sub-section, we have seen that li et al.’s method (2020) fail to generate any preference order of alternatives as we had taken a group decision-making problem. so, in order to compare our method with the methods developed by li et al. anusha et al./decis. mak. appl. manag. eng. (2022) 28 (2020), one case study related to mcdm (example 2 in section 3) is considered here. in this case, by the developed method, we obtain the scores -0.52571, -0.39996, 0.14094, -0.12571 respectively and the ranking order is 4 3 2 1a a a a .thus, our proposed method and li et al.’s method (2020) generate different preference order, but the best alternative remains the same in both cases which means that our method is effective. the advantages of our method in comparison to existing approach are as follows: 1. li et al.’s technique (2020) is based on the power ao and can therefore lessen the influence of an expert's bias on the results. however, these techniques can't handle mcgdm issues and reduce the accuracy of the results of decisions.however, our approach can handle mcgdm issues. our approach is therefore far more trustworthy and efficient. 2. because li et al.’s technique (2020) is based on the muirhead mean operator, it can take into account the relationships among several criteria, but it is limited in that it can only link one marginal distribution. but our proposed qrpdhf generalized msm operators’ aggregate qrpdhf data with higher flexibility. it also takes into account the relationships between many input criteria. these operators may link more than one marginal distribution and can thus prevent the information loss that results from the aggregation process since they were designed based on the archimedean copula operations on qrpdhfes. 3. it is clear that approach (li et al., 2020) falls short in its attempt to reduce the impact of highly inflated attribute values from a few unreliable experts who have different biases. this unspoken problem has a negative impact on how decisions are made in any mcgdm process. by permitting the expert engagement that is lacking in the present study, this issue has been remedied utilising the suggested methodology (li et al., 2020). 4. the preference ranking produced in the dm approach (li et al., 2020) is impacted by the random distribution of weights of criteria during the final aggregation step. additionally, the current approach (li et al., 2020) loses information because it doesn't take any information measures into account. our technique computes criteria weights using an optimization model based on the cross entropy measure. by highlighting the importance of each criterion, this optimization approach quantifies the amount of ambiguous data. 7. conclusion the qrpdhfss can effectively portray the dubiousness and uncertainty in reality due to the inclusion of the mds and nmds with their corresponding probabilities. the joint occurrence of the stochastic and the non-stochastic ambiguity make the qrpdhfss more realistic and superior. a comprehensive study on the usefulness of archimedean copulas under qrpdhf setting is demonstrated in our study. new operations for qrpdhf elements are formed via archimedean copulas. the existing aos (hao et al., 2017; garg & kaur, 2018) for aggregation pdhf data are limited to algebraic, and einstein operators. so, they are not capable of considering dependency between multiple attributes. on the other hand, although the aos (ji et al., 2021; li et al., 2020) based on hamy mean operator and muirhead mean operator respectively can consider dependency among criteria, cannot connect more than one marginal distribution. these facts motivated us to develop the archimedean copula based gmsm operators with their weighted forms under qrpdhf setting. some pivotal hybridizations of archimedean copula and generalized msm operators and their … 29 qualities like idempotency, boundedness, and monotonicity, and of proposed aos are introduced. subsequently, a mcgdm procedure is exhibited to track down the best option in qrpdhf setting. here, the weights of criteria are determined using an optimization model and experts weights are figured utilizing the linear combination of objective and subjective weights and interaction among experts. to give a superior comprehension of our technique, we have incorporated a case study including osslms selection. the robustness of our method has been demonstrated through sensitivity analysis of weights of criteria. the comparative study suggests that the proposed methodology can be adequately utilized in mcgdm issues containing correlated criteria’s in the pdhf setting. the only limitation of the proposed method is that in absence of partial weights information of criteria the proposed method fails. in such a scenario, other objective methods like critic, merec, entropy measure, etc can be utilized for determination of criteria weights. in further research, other aggregation operators (saha et al., 2022a; saha et al., 2021a; senapati, 2021; senapati et al., 2022; saha et al., 2022b; saha et al., 2021b) can also be extended to tackle the dependency among attributes with qrpdhf information and the proposed weight determination technique. our model can be used to provide a realistic solution to well-known problems, such as sustainable supplier selection (mishra et al., 2022a), warehouse site selection (saha et al., 2023), bio-energy production technology selection (hezam et al., 2023), solid waste disposal method selection (mishra et al., 2022b), renewable energy source selection (mishra et al., 2022c), low carbon tourism assessment (mishra et al., 2022d), biomass feedstock selection (saha et al., 2021c), cloud vendor selection (krishankumar et al., 2022), and food waste treatment technology selection (rani et al., 2021) as it can effectively avoid distorting evaluation information and handle the relationships between multiple criteria. author contributions: conceptualization: g. anusha and p.v. ramana; methodology, p.v. ramana; software: r. sarkar; validation, g. anusha, p.v. ramana and r. sarkar; formal analysis: g. anusha; investigation, g. anusha, p.v. ramana; resources: g. anusha; writing—original draft preparation: g. anusha, p.v. ramana and r. sarkar; writing—review and editing: r. sarkar. all authors have read and agreed to the published version of the manuscript.” funding: this research received no external funding. conflicts of interest: the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. anusha et al./decis. mak. appl. manag. eng. 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