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 PROMETHEE- II, 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 up- to-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 pair- wise 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 pair- wise 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 here- in-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-BWM- MOORA, 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. 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