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 decision- making 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 Decision- A 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” sub- criterion 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 2-5- 6-3-7-1-8-4 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. 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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=(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. 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