Selection of a Sustainable Supplier by Using a Fuzzy MCDM Mathematical Modelling

Selection of a Sustainable Supplier by Using a Fuzzy MCDM Mathematical Modelling | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Selection of a Sustainable Supplier by Using a Fuzzy MCDM Mathematical Modelling Reema Agarwal, Ankur Agrawal, Nitendra Kumar, Samrat Ray, Liton Chandra Voumik This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-2517685/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract The primary objective of any business today is to remain competitive and sustainable so that it may continue operating profitably and efficiently. Therefore, in order to achieve the aforementioned goals, businesses must evaluate potential sustainable suppliers in light of the three pillars of sustainability (social responsibility, economic viability, and environmental friendliness). One example of an issue that requires multiple criterion decision-making (MCDM) is the challenge of finding sustainable suppliers. Fuzzy PROMETHEE, a method of multi-criteria decision making (MCDM) that utilizes triangular fuzzy numbers (TFNs) and linguistic concepts, is used in this study to establish the relative importance of three factors for choosing a sustainable supplier: social impact, economic viability, and environmental responsibility. Successful application of the fuzzy PROMETHEE method has allowed the organization's managers to arrive at the appropriate conclusion and implement the necessary solution. Fuzzy theory Multiple-Criteria Decision Making (MCDM) Fuzzy Preference Ranking Organization Method for Enrichment Evaluations (F-PROMETHEE) Sustainable Supplier (SS) 1. Introduction Each company relies heavily on their supply chain procedures to function smoothly and efficiently. In order for businesses to meet the needs of their customers, they rely on supply chain operations, which are the procedures by which raw materials and components are obtained from suppliers and refined into more complex products by in-house workers. Since the most effective sustainable supplier incorporates all three aspects of sustainable processes—social responsibility, economic viability, and environmental friendliness—it is essential that all businesses have access to the most suitable sustainable vendors. Vendor evaluation, which is part of MCDM, is crucial to the success of any business. When faced with a set of options that fail to satisfy a number of criteria, the Multi-Criteria Decision Making (MCDM) technique can be used to arrive at a final decision. The study's overarching goal is to apply the fuzzy PROMETHEE technique, a method of multi-criteria decision making (MCDM) that makes use of linguistic terms and involves TrFNs in order to deal with the issue of uncertainty, ambiguity, obscurity, vagueness, etc. in the process of sustainable supplier selection. On the basis of the views of the decision managers in an organization, this study identifies three decision makers, three suppliers, and three sustainable supplier selection criteria: social, economic, and environmentally friendly. An MCDM technique known as fuzzy PROMETHEE is applied in a fuzzy setting to first determine the relative importance of the identified sustainable supplier selection criteria, and then to choose the most suitable and optimal sustainable provider. Sustainable Supplier 1 (SS1) was found to be the most effective supplier after all, as found in this investigation. 2. Literature Review The term sustainability is considered the most significant aspect among the managers of supply chain processes to introduce a sustainable organization that contains the social, economic, and eco-friendly criteria for the betterment of the organization (Yazdani et al., 2021a [ 1 ], 2021b [ 2 ]). According to (Fallahpour et al. 2021[ 3 ]), the economic criteria for any manufacturing organization play a very significant role in any organization but in the present years, organization managers consider both the environmental and social needs concerning the people's judgments and executive orders (Dogan and Seker 2016 [ 4 ]; Dogan and Inglesi-Lotz 2017 [ 5 ]; Abdel-Baset et al. 2019 [ 6 ]. Therefore, the involvement of the triad areas of sustainability i.e. social, economic, and eco-friendly simultaneously generates, implements and introduces the hypothesis of the sustainable processes (Mojtahedi et al., 2021[ 7 ]). In any organization suppliers are the major step of the organizations and they greatly increase the production of the organizations by considering the concept of sustainability (Rashidi and Cullinane 2019 [ 8 ]; Liu et al. 2020 [ 9 ]). The companies require to evaluate their sustainable suppliers for everlasting relationships (Nezhadroshan et al. 2021 [ 10 ]). Identifying the best sustainable suppliers is the most important strategy for supply chain processes (Fallahpour et al. 2016 [ 11 ]; Kusi-Sarpong et al. 2019 [ 12 ]). Selecting the best sustainable suppliers helps the managers of the companies to get the satisfactory underdone substances at the punctual time, amount, and standard (Tavana et al. 2021[ 13 ]; Wang et al. 2021[ 14 ]). Up to now, there exist various investigations and publications in the process of choosing suppliers. This research study applies a MCDM technique by integrating PROMETHEE technique with fuzzy theory to assist the decision-makers with selection problems. Geldermann et al. (2000) [ 15 ] applied the F-PROMETHEE technique using TrFNs interval-valued in organizations making iron and steel components. Goumas and Lygerou in 2000 [ 16 ] implemented the fuzzy PROMETHEE II techniquein determining the alternative energy exploitation projects. Geldermann and Rentz (2001) [ 17 ] also applied the F-PROMETHEE technique for eco-friendly management system and constructed a sensitivity representation analysis. Bilsel et al. (2006) [ 18 ] applied the F-PROMETHEE technique in determining the ranking of the websites of Turkish hospitals. Chou et al. (2007) [ 19 ] applied the F-PROMETHEE technique to estimate the best environment-friendly selections for an actual construction area. Wang et al. (2008) [ 20 ] and Chen et al. (2011) [ 21 ] implemented a F-PROMETHEE technique for the information organizations outsourcing evaluation and assessment of providers. Moreira et al. (2009) [ 22 ] implemented both the PROMETHEE and F-PROMETHEE techniques in prioritizing the deterioration of the diagnostic of electrical power apparatus. Aloini et al. in 2009 [ 23 ] implemented an integrated F-PROMETHEE technique for choosing the logistic service providers. Zhou et al. (2009) [ 24 ] applied PROMETHEE II for presenting the pipe condition assessment problem. Behzadian et al. (2010) [ 25 ], Zhang et al. (2009) [ 26 ] applied the F-PROMETHEE technique to polluted areas based on a comparative risk procedure. Liu and Guan (2009) [ 27 ] developed the F-PROMETHEE technique for estimating the standard of passenger rail line. Giannopoulos and Founti in 2010 [ 28 ] studied the modified variety of the PROMETHEE technique integrated with a flexible ranking approach in a fuzzy environment. Li and Li (2010) [ 29 ] introduced a brand modified version of the improved PROMETHEE 2nd technique in terms of universal fuzzy values. Tuzkaya et al. (2010) [ 30 ] implemented a F-PROMETHEE technique for the selection of alternative material handling appliances. Shirinfar and Haleh (2011) [ 31 ] developed the fuzzy PROMETHEE technique for the selection of the suppliers. Yilmaz and Dagdeviren (2011) [ 32 ] applied the hybrid integrated F-PROMETHEE technique and GP technique for the choosing appliances. Shakhsi-Niaei et al. (2011) [ 33 ] presented the F-PROMETHEE technique in the form of a Monte Carlo simulation structure for estimating the ranking of a company's projects. Gupta et al. (2012) [ 34 ] applied the fuzzy PROMETHEE technique to choose the suppliers for logistics services in cement organizations. Tavakoli et al. (2013) [ 35 ] used the integration of F-PROMETHEE and the F-GP techniques to find the ranking of the vendors. Anojkumar et al. (2014) [ 36 ] introduced the involvement of tetrad MCDM techniques i.e. Fuzzy AHP-TOPSIS, F-AHP-VIKOR, F-AHP-ELECTRE, F-AHP-PROMTHEE for solving the problems of choosing the pipes stuff in sugar company and to select the best option among the numerous types of substances. In the present years, the researchers used the F-PROMETHEE technique to allow application novelty like squander disposal solution assessment (Lolli et al., 2016) [ 37 ] and methodological novelty like integration of interval-valued type-II fuzzy sets combined with PROMETHEE (Celik and Gumus, 2016 [ 38 ], Chen, 2015 [ 39 ], Chen, 2014 [ 40 ]) and introducing intuitionistic F-PROMETHEE technique (Liao and Xu, 2014 [ 41 ]). Chen and Pan (2016) [ 42 ] applied the variable fuzzy PROMETHEE technique in choosing the best low-carbon building extent. Krishankumar et al. (2017) [ 43 ] introduced an extended version of PROMETHEE outranking technique under an intuitionistic fuzzy logic surroundings for choosing the supplier based on linguistic terms. Wan et al. (2020) [ 44 ] utilized the fuzzy PROMETHEE technique for selecting the green vendors. Roy et al. (2020) [ 45 ] implemented the MCDM techniques named AHP-PROMETHEE for determining the methods for choosing the sustainable supplier by considering various types of transportation criteria in a ready-made clothing garments organization. 3. Fuzzy Promethee Technique Brans et al. in 1984 [ 46 ], Brans and Vincke in 1985 [ 47 ], Brans et al. in 1986 [ 48 ] proposed an MCDM technique named PROMETHEE (Preference Ranking Organization Method for Enrichment Evaluations) technique. This technique consists of simple philosophy and ideology and its methodology is also simple and easy in comparison to other techniques. This technique is an outranking technique and is classified into two techniques namely PROMETHEE I st and PROMETHEE II nd where PROMETHEE I st technique allows incomplete ranking of the set of choices and PROMETHEE II technique allows absolute or complete ranking of a set of choices (Ülengin et al., 2001) [ 49 ]. The application of PROMETHEE technique consists of real-life areas like environmental conservation and preservation, conservation and maintenance of water, business and financial capability, chemical science, conveyance management, constructing and gathering, conservation and preservation of energy, strategic planning in production, social management system, medical facilities, agricultural production, educational system, government policies, and sports fields, etc. The term "fuzzy PROMETHEE" was coined by Le Téno and Mareschal (1998) [ 50 ], who were the first to propose combining fuzzy theory and the PROMETHEE approach (F-PROMETHEE). Zadeh first proposed the idea of fuzziness in 1965 [ 51 ]. Using the collective wisdom of specialists, the fuzzy theory was created to provide a framework for resolving cases of ambiguity and uncertainty (Wang et al. in 2008 [ 20 ]). The mathematical formulation and the solutions to issues that are neither simple nor well-defined by mathematical modeling techniques can be constructed and analyzed using the principles of fuzzy logic theory (Kandel, 1986 [ 52 ]). The main advantages of the fuzzy PROMRTHEE technique are that it is I simple to learn and apply, (ii) transparent in its processes, and (iii) sensitive to issues of uncertainty, ambiguity, obscurity, vagueness, etc. in the decision-making setting. Additionally, the advantages of the PROMETHEE technique are supported by the expanded fuzzy version of the most popular PROMETHEE outranking technique. The aforementioned advantages make this method useful in a wide variety of practical settings. A unique aspect of the fuzzy PROMETHEE method is that it affords the opportunity to choose the types of preference words, so increasing the likelihood of obtaining a more workable definition for the many kinds of detected choice criteria. Therefore, this method has been validated as an efficient and effective method for use by decision managers in the supply chain processes. Under this technique, the decision managers of the organization firstly study the framework of the process of sustainable suppliers’ assessment secondly, they determine the criteria which are useful in selecting the sustainable supplier and thirdly they evaluate the relative weights of the identified criteria through linguistic terms expressed by TrFNs assessment by the judgment of the decision managers. Fourthly, the ratings of all the suppliers is also described through linguistic terms expressed by trapezoidal fuzzy numbers i.e. a quadruple denoted by \(({a}_{1},{a}_{2},{a}_{3},{a}_{4})\) assessment by the decision managers. Lastly, they evaluate the best sustainable supplier, and the ranking of suppliers is evaluated by calculating the fuzzy Hamming distances and are applied for making pair wise comparisons among the set of identified sustainable suppliers for each criterion. 4. Steps In Fuzzy Promethee Technique Step 1 - Identification of alternatives and criteria- Firstly a group of \(K\) decision managers say \({s DM}_{K}\) where \(\left(k=\text{1,2},3,\dots K\right)\) then identifies a set of sustainable suppliers say \(m\) where ( \(i\) = 1, 2,…, \(m\) ) and various types of sustainable supplier selection criteria say \(n\) where \((j\) = 1, 2,…, \(n\) ) namely social, environmental, and economic and i.e. Soc., Env., Eco. Step 2 -Linguistic terms expressed for criteria and performance ratings of the options - The weights of identified criteria and the fuzzy performance ratings used for the choices are represented through linguistic terms described by TrFNs according to the opinions and assessment of the decision managers. Step 3 - Estimation of the weights of criteria - Fuzzy weights of each criterion are denoted by \({ w}_{j}\) Where $${ w}_{j}=\left({w}_{j}^{1},{w}_{j}^{2},{w}_{j}^{3},{w}_{j}^{4}\right)$$ 1 \({ w}_{j}^{1}=min\) ( \({w}_{j}^{1})\) (2) $${ w}_{j}^{2}=1/k\left({\Sigma }{w}_{j}^{2}\right)$$ 3 $${ w}_{j}^{3}=1/k\left({\Sigma }{w}_{j}^{3}\right)$$ 4 \({ w}_{j}^{4}=max\) ( \({w}_{j}^{4})\) (5) Step 4 - Developing a fuzzy decision matrix – This matrix says [ \(D\) ] is constructed as: $$\left[D\right]={\left[{x}_{ij}\right]}_{m*n}$$ 6 where \({x}_{ij}\) be the fuzzy performance rating of the \({i}^{th}\) choice for the \({j}^{th}\) criteria, \(i\) = 1,2,…, \(m\) ; \(m\) is the numeral of choices, \(j\) = 1,2,…, \(n\) ; \(n\) is the numeral of criteria. Step 5 - Construction of combined fuzzy decision matrix – This matrix is constructed by showing the fuzzy performance ratings of all the alternatives based on identified criteria through TrFNs and is as follows: $$\left[\begin{array}{ccc}{x}_{11}& \cdots & {x}_{1n}\\ ⋮& \ddots & ⋮\\ {x}_{m1}& \cdots & {x}_{mn}\end{array}\right]$$ 7 Where \({ x}_{ij}=\left({a}_{ij},{b}_{ij},{c}_{ij},{d}_{ij}\right)\) are the performance ratings in fuzzy numbers of all the choices through TrFNs. Where \({ a}_{ij}= min\) ( \({a}_{ij})\) (8) $${ b}_{ij}=\frac{1}{K}\left(\varSigma {b}_{ij}\right)$$ 9 \({c}_{ij}=\frac{1}{K}\left(\varSigma {c}_{ij}\right)\) (10) \({ d}_{ij}=\text{m}\text{a}\text{x}\left({d}_{ij}\right)\) (11) Step 5 – Developing a combined normalized fuzzy decision matrix- This matrix say \(\left[R\right]\) is constructed as: $$\left[R\right]=\left[{r}_{ij}\right]$$ 12 Normalization is done by applying the following two equations - In case of beneficial criteria i.e. social and environment criteria - $${ r}_{ij}=\left(\frac{{a}_{ij}}{{d}_{j}^{+}},\frac{{b}_{ij}}{{d}_{j}^{+}},\frac{{c}_{ij}}{{d}_{j}^{+}},\frac{{d}_{ij}}{{d}_{j}^{+}}\right)$$ 13 Where \({d}_{j}^{+}=\text{max}\left({d}_{ij}\right)\) (14) In the case of non-beneficial criteria i.e. economic criteria - $${r}_{ij}=\left(\frac{{a}_{j}^{-}}{{d}_{ij}},\frac{{a}_{j}^{-}}{{c}_{ij}},\frac{{a}_{j}^{-}}{{b}_{ij}},\frac{{a}_{j}^{-}}{{a}_{ij}}\right)$$ 15 Where \({a}_{j}^{-}=min{ a}_{ij }\) (16) Step 6 - Construction of weighted-normalized fuzzy decision matrix- This matrix say [ \(V\) ] is constructed by multiplying the values in the combined normalized matrix say \({r}_{ij}\) and the evaluated weights of the identified criteria say \({w}_{j}\) . $$V=\left[{r}_{ij}*{w}_{j}\right]$$ 17 Step 7 - Construction of preference functions - Preference functions among each alternative are constructed by evaluating Hamming distances which are used for comparing two choices say a and b according to each criterion. Firstly, we calculate the maximum number say c between the two fuzzy numbers. Then, finding the Hamming distance which is the sum of all values evaluated by taking the absolute value of the distinction between the maximum number and the first alternative and also with the second alternative. According to Hatami-Marbini & Tavanain 2011 [ 53 ], preference functions are determined by the following equations - $${ \text{V}}_{\text{a}\text{c}}\ge {\text{V}}_{\text{b}\text{c}} , \text{i}\text{f} \text{a}\text{n}\text{d} \text{o}\text{n}\text{l}\text{y} \text{i}\text{f} d\left\{\text{max}\left({V}_{ac} , {V}_{bc}\right), {V}_{bc}\right\}\ge d\left\{\text{max}\left({V}_{ac} , {V}_{bc}\right),{V}_{ac}\right\}$$ 20 $${ \text{V}}_{\text{a}\text{c}}<{\text{V}}_{\text{b}\text{c}} , \text{i}\text{f} \text{a}\text{n}\text{d} \text{o}\text{n}\text{l}\text{y} \text{i}\text{f} d\left\{\text{max}\left({V}_{ac} , {V}_{bc}\right), {V}_{bc}\right\}<d\left\{\text{max}\left({V}_{ac} , {V}_{bc}\right),{V}_{ac}\right\}$$ 21 $$\text{P}\left(\text{a},\text{b}\right)=d\left\{\text{max}\left({V}_{ac} , {V}_{bc}\right), {V}_{ac}\right\} \text{i}\text{f} {V}_{ac}<{V}_{bc} and d\left\{\text{max}\left({V}_{ac} , {V}_{bc}\right),{V}_{bc}\right\}\text{i}\text{f} {V}_{ac}\ge {V}_{bc }$$ 22 Step 8 - Construction of fuzzy preference index - Fuzzy preference index among each alternative is denoted by \(\pi \left(a,b\right)\) and estimated by using the following equation: $$\pi \left(a,b\right)={\Sigma }\left\{{w}_{j}*P\left(a,b\right)\right\}/{\Sigma }{w}_{j }$$ 23 Step 9 – Calculation of positive and negative flows - Positive and negative flows are evaluated by defuzzification of the trapezoidal fuzzy numbers used in the fuzzy preference index into crisp numbers (Giannopoulos & Founti 2010) [ 38 ]. According to Chen et al.'s, defuzzification of the TrFNs is determined by adding all the four fuzzy numbers and dividing by 4. Positive flows and negative flows are evaluated by defuzzification of the TrFNs in rows and columns respectively. Let \({x}_{A}\) be the defuzzified value of the \(\text{T}\text{r}\text{F}\text{N}\text{s} A\) where \(A=({a}_{1},{a}_{2},{a}_{3},{a}_{4})\) , then the defuzzification is calculated as: $${ x}_{A}=\frac{{a}_{1}+{a}_{2}+{a}_{3}+{a}_{4}}{4}$$ 24 Step 10 – Calculation of net flows \(\left({NF}^{{\prime }}\right)\) - Net flows are determined by subtracting the positive flows ( \(PF\) ) and negative flows (NF). $${ NF}^{{\prime }}=PF-NF$$ 25 Step 11 – Determining the ranking of each alternative – The Preference ranking of each choice is determined by the analysis of net flows. The ranking of the choices can be determined by considering the values of the net flows for each choice. If the value of the net flows is maximum then the rank of choice is one. The best choice is determined as: $${ A}^{*}=\{ {A}_{i} ;\text{max}(NF)\}$$ 26 5. Application Of The Fuzzy Promethee Technique Firstly, a team of three decision managers i.e. DM1, DM2, and DM3 are formed and they identified the three sustainable suppliers’ i.e.SS1, SS2, SS3, and three types of sustainable supplier selection criteria i.e. social, environmental, and economic i.e. Soc., Env., Eco. The linguistic terms for the weights of criteria and performance ratings of the sustainable suppliers, through TrFNs assessment by the opinions of all the decision managers, are shown in Tables 1 and 2 respectively. Table 1 Linguistic terms used for the evaluating weights of the criteria Linguistic terms Trapezoidal Fuzzy numbers Very low(VL) (0,0,0.1,0.2) Low(L) (0.1, 0.2, 0.2, 0.3) Medium low (ML) (0.2, 0.3, 0.4, 0.5) Medium (M) (0.4, 0.5, 0.5, 0.6) Medium high (MH) (0.5, 0.6, 0.7, 0.8) High (H) (0.7, 0.8, 0.8, 0.9) Very high (VH) (0.8, 0.9, 1, 1) Table 2 Linguistic terms used for the performance ratings of the suppliers Linguistic terms Trapezoidal Fuzzy numbers Very poor(VP) (0, 0, 1, 2) Poor (P) (1, 2, 2, 3) Medium poor(MP) (2, 3, 4, 5) Fair (F) (4, 5, 5, 6) Medium good (MG) (5, 6, 7, 8) Good (G) (7, 8, 8, 9) Very good(VG) (8, 9, 10, 10) The linguistic terms which are applied for evaluating the weights of the identified criteria assessment by the opinion of all the decision managers are shown in Table 3 . Table 3 Linguistic terms used for finding the weights of the criteria assessment by all the decision managers Soc. Env. Eco. DM1 VH M H DM2 H VH H DM3 M H VH The linguistic terms which are implemented for determining the performance ratings of all the sustainable suppliers’ assessments by the opinions of all the decision managers are shown in Tables 4 , 5 , 6 , and 7 . Table 4 Linguistic terms used for the performance ratings of the suppliers’ assessment by the first decision manager DM1 Soc. Env. Eco. SS1 VG MG VG SS2 G MG G SS3 MG G VG Table 5 Linguistic terms used for the performance ratings of the suppliers’ assessment by the second decision manager DM2 Soc. Env. Eco. SS1 G VG MG SS2 MG G VG SS3 VG MG G Table 6 Linguistic terms used for the performance ratings of the suppliers’ assessment by the third decision manager DM3 Soc. Env. Eco. SS1 G VG MG SS2 MG G MG SS3 VG VG G Table 7 Linguistic terms used for the performance ratings of the suppliers’ assessment by all the decision managers DM1,DM2,DM3 Soc. Env. Eco. SS1 VG, G, G MG, VG, VG VG, MG, MG SS2 G, MG, MG MG, G, G G, VG, MG SS3 MG, VG, VG G, MG, VG VG, G, G The fuzzy ratings which are implemented for estimating the weights of each identified criteria according to the opinions of all the decision managers through TrFNs are shown in Table 8 . Table 8 Fuzzy ratings used for finding the weights of the criteria assessment by all decision-makers in terms of trapezoidal fuzzy numbers Soc. Env. Eco. DM1 0.8,0.9,1,1 0.5,0.6,0.7,0.8 0.7,0.8,0.8,0.9 DM2 0.7,0.8,0.8,0.9 0.8,0.9,1,1 0.7,0.8,0.8,0.9 DM3 0.5,0.6,0.7,0.8 0.7,0.8,0.8,0.9 0.8,0.9,1,1 Now, determining the weights of all the criteria by using the equations (2)-(5) and Table 8 . Firstly, determining the weight of the Soc. criteria by using Table 8 . In Table 8 , \({w}_{j}^{1}\) = min (0.8, 0.7, 0.5) = 0.5, \({w}_{j}^{2}\) = 1/3(0.9 + 0.8+0.6) = 0.7667, \({w}_{j}^{3}\) = 1/3(1, 0.8, 0.7) = 0.8333 and \({w}_{j}^{4}\) = max (1, 0.9, 0.8) = 1. Thus, weights of the social criteria is (0.5, 0.7667, 0.8333,1). Similarly, the weights of the environment criteria is (0.5, 0.7667, 0.8333,1) and the weights of the economic criteria is (0.7, 0.8333, 0.8667, 1). The weights of all the identified criteria through TrFNs are shown in Table 9 . Table 9 Weights of all criteria in terms of TrFNs Criteria Weights Soc. (0.5, 0.7667, 0.8333, 1) Env. (0.5, 0.7667, 0.8333, 1) Eco. (0.7, 0.8333, 0.8667, 1) Now, constructing a decision matrix of all the green and sustainable suppliers for each criterion represented by Eq. ( 6 ) which shows the fuzzy performance ratings of all the green and sustainable suppliers by the opinions of all the decision managers in terms of TrFNs and is shown in Table 10 . Table 10 Fuzzy decision matrix of all the suppliers Soc. Env. Eco. SS1 (8,9,10,10), (7,8,8,9), (7,8,8,9) (5,6,7,8), (8,9,10,910, (8,9,10,10) (8,9,10,10), (5,6,7,8), (5,6,7,8) SS2 (7,8,8,9), (5,6,7,8), (5,6,7,8) (5,6,7,8), (7,8,8,9), (7,8,8,9) (7,8,8,9), (8,9,10,10), (5,6,7,8) SS3 (5,6,7,8), (8,9,10,10), (8,9,10,10) (7,8,8,9), (5,6,7,8), (8,9,10,10) (8,9,10,10), (7,8,8,9), (7,8,8,9) Now, constructing a combined decision matrix of all the sustainable suppliers for each criterion represented by Eq. ( 7 ) which depicts the combined fuzzy performance ratings of all the sustainable suppliers according to the opinions of all the decision managers in terms of TrFNs by using the equations (8)-(11) and Table 10 and is shown in Table 11 . In Table 11 , the numbers of the first cell is calculated as \({a}_{ij}\) = min(8,7,7) = 7, \({b}_{ij}\) = 1/3(9 + 8+8)=8.3, \({c}_{ij}\) = 1/3(10,8,8)=8.7, \({d}_{ij}=\) max(10,9,9)=10 by using the Table 10 . Similarly, we can find all the values of all cells. Table 11 Combined Fuzzy decision matrix of all the suppliers Soc. Env. Eco. SS1 7,8.3,8.7,10 5,8,9,10 5,7,8,10 SS2 5,6.7,7.3,9 5,7.3,7.7,9 5,7.7,8.3,10 SS3 5,8,9,10 5,7.7,8.3,10 7,8.3,8.7,10 Now, constructing a normalized decision matrix of all the sustainable suppliers for all criteria represented by Eq. ( 12 ) and by using Table 11 . For beneficial criteria, namely social and environmental criteria, equations ( 13 ) and (14)are used and for non-beneficial criteria, namely economic criteria, equations ( 15 ) and (16) are used. A normalized decision matrix of all the sustainable suppliers based on each criterion is shown in Table 12 . In beneficial criteria, the value of the first cell is calculated as (7/10,8.3/10,8.7/10,10/10) = (0.7,0.83,0.87,1) while in the case of non-beneficial criteria, the value of the cell is calculated as (5/10,5/8,5/7,5/5) = (0.5,0.62,0.71,1). Similarly, we can find all the values of all cells. Table 12 Normalized Decision matrix of all the suppliers wrt each criterion Soc. Env. Eco. SS1 0.7,0.83,0.87,1 0.5,0.8,0.9,1 0.5,0.62,0.71,1 SS2 0.5,0.67,0.73,0.9 0.5,0.73,0.77,0.9 0.5,0.60,0.64,1 SS3 0.5,0.8,0.9,1 0.5,0.77,0.83,1 0.5,0.57,0.60,0.71 Now, constructing a weighted normalized decision matrix of all the sustainable suppliers for each criterion represented as the Eq. ( 17 ) by using Table 12 and is shown in Table 13 . By using the Table 12 , the value of the first cell is calculated as: (0.7*0.5 + 0.83*0.7667 + 0.87*0.8333 + 1*1)=(0.35,0.6363,0.7249,1). Similarly, we can find all the values of all cells. Table 13 Weighted Normalized Decision matrix of all the suppliers wrt each criterion Soc. Env. Eco. SS1 0.35,0.6363,0.7249,1 0.25,0.6133,0.7499,1 0.35,0.5166,0.6153,1 SS2 0.25,0.5136,0.6083,0.9 0.25,0.5596,0.6416,0.9 0.35,0.4999,0.5546,1 SS3 0.25,0.6133,0.7499,1 0.25,0.5903,0.6916,1 0.35,0.4749,0.5200,1 Now, determining the preference functions among the sustainable suppliers for each criterion by using the equations ( 20 )-( 22 ) and Table 13 . Firstly, determining the preference function between sustainable suppliers SS1 and SS2 for social criteria is shown in Table 14 . Table 14 Preference function between supplier 1 and supplier 2 Alternative a 0.35 0.6363 0.7249 1 Alternative b 0.25 0.5136 0.6083 0.9 The maximum number between a and b say c 0.35 0.6363 0.7249 1 distance between a and c 0 0 0 0 distance between b and c 0.1 0.1227 0.1166 0.1 \(d\left\{\text{max}\left({V}_{ac} ,{V}_{bc}\right),{V}_{ac}\right\}=0\) and \(d\left\{\text{max}\left({V}_{ac} ,{V}_{bc}\right),{V}_{bc}\right\}=\) 0.1 + 0.1227+0.1166+0.1= 0.4393 then \({V}_{ac}\) ≥ \({V}_{bc}\) condition is satisfied. Therefore, the preference function will be 0.4343. Similarly, determining the preference functions among all the sustainable suppliers for each criterion is shown in Table 15 . Table 15 Preference functions of each supplier wrt each criterion Soc. Env. Eco. P(SS1,SS2) 0.4343 0.262 0.0774 P(SS1,SS3) 0.123 0.0813 0.137 P(SS2,SS1) 0 0 0 P(SS2,SS3) 0.3413 0.1807 0.0596 P(SS3,SS1) 0.025 0 0 P(SS3,SS2) 0 0 0 Now determining the fuzzy preference index among the sustainable suppliers for each criterion by multiplying the weights of criteria and the preference functions of each supplier using the Eq. ( 23 ) and Table 15 . Firstly, determining the fuzzy preference index of suppliers SS1 and SS2 as: \(\left\{{w}_{j}*P\left(a,b\right)\right\}\) =(0.5*0.4343,0.7667*0.4343,0.8333*0.4343,1*0.4343)= (0.2171,0.3329,0.3619,0.4343) (0.5*0.262, 0.7667*0.262,0.8333*0.262,1*0.262)= (0.131, 0.2008, 0.2183, 0.262) (0.7*0.0774, 0.8333*0.0774,0.8667*0.0.774,1*0.0.774)= (0.0541,0.0644,0.0670,0.0774) Now finding \({\Sigma }\left\{{w}_{j}*P\left(a,b\right)\right\}\) =( 0.2171 + 0.131 + 0.0541, 0.3329 + 0.2008 + 0.0644, 0.3619 + 0.2183 + 0.0670, 0.4343 + 0.262 + 0.0774) = (0.4022,0.5891,0.6472,0.7737). Now finding \({\Sigma }{w}_{j }=\) (0.5 + 0.5 + 0.7,0.7667 + 0.7667 + 0.8333,0.8333 + 0.8333 + 0.8667,1 + 1 + 1)=(1.7,2.3667,2.5333,3) Now finding \({\Sigma }\left\{{w}_{j}*P\left(a,b\right)\right\}/{\Sigma }{w}_{j }=\) (0.4022/3, 0.5891/2.5333, 0.6472/2.3667, 0.7737/1.7)= (0.1340,0.2325,0.2734,0.4551). Similarly, determining the fuzzy preference index of all sustainable suppliers for each identified criteria is shown in Table 16 . Table 16 Fuzzy preference index of all the suppliers M1 M2 M3 M1 0,0,0,0 0.1340,0.2325,0.2734,0.4551 0.066,0.1068,0.3796,0.2007 M2 0,0,0,0 0,0,0,0 0.2260,0.1775,0.2055,0.3421 M3 0.0041,0.0075,0.0008,0.0147 0,0,0,0 0,0,0,0 Now, determining the positive flows after applying defuzzification in rows by using the Eq. ( 24 ) in Table 16 i.e. (0.1340 + 0.066,0.2325 + 0.1068,0.2734 + 0.3796,0.4551 + 0.2007)/4 = 0.2987 Similarly, determining the negative flows after applying defuzzification in columns by using the Eq. ( 24 ) in Table 16 i.e. (0.0041 + 0.0075 + 0.0008 + 0.0147)/4 = 0.0067 Similarly, determining the positive and negative flows of all the sustainable suppliers. Now, determining the net flows by using the Eq. ( 25 ).And finally, determining the ranking of the sustainable suppliers by considering the values of the net flows for each choice. If the values of the net flows are maximum then the rank of the sustainable supplier is one. The best sustainable supplier is determined by using the Eq. ( 26 ). The positive flows, negative flows, net flows and the ranking of the sustainable suppliers, are shown in Table 17 . Table 17 Ranking of all the suppliers Positive flows Negative flows Net flows Ranking SS1 0.2987 0.0067 0.292 1 SS2 0.2377 0.2737 -0.036 2 SS3 0.0067 0.4260 -0.4193 3 Hence, supplier1 is the best sustainable supplier. Ranking of all the sustainable suppliers - SS1 > SS2 > SS3 6. Conclusions, Drawbacks, And Future Outlook Of Study- To increase the quality standards and competitive strategies, outsourcing is unavoidable. The process of evaluating and selecting the best and optimal sustainable suppliers is a complex feature for any organization. The methods considered in suppliers’ assessment care only for the basic needs of the isolated system but do not consider the whole supply chain processes. Maintaining the relationships among the sustainable suppliers and buyers strongly in the processes of a supply chain is very essential to their active participation. Therefore, the organizations prefer to collaborate with a finite numeral of suppliers or consider only a single supplier to attain and maintain high performance in the organizations. Due to the plan significance of the process of suppliers’ assessment, large-scale research is developing to cope with the problems related to the MCDM problems. This paper applied a PROMETHEE technique under the fuzzy environment for solving the problem of sustainable suppliers’ assessment by considering the sustainable supplier selection criteria. After applying this technique we achieve that the supplier 1 is the best sustainable supplier. The main goal of this research paper is to decide the appropriate, best, and optimal sustainable supplier. The two major benefits of this technique are - (i) it is easily accessible by the users and (ii) it pays attention to the problems of uncertainties, ambiguities, obscurities, vagueness, etc. indecision-making problems. Hence, this technique is considered to be very fruitful and is applied by the decision managers in the supply chain processes. While considering the future study and research, this technique can also be easily implemented to any other assessment problem in various types of real fields, particularly in manufacturing and service organizations. 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Introduction","content":"\u003cp\u003eEach company relies heavily on their supply chain procedures to function smoothly and efficiently. In order for businesses to meet the needs of their customers, they rely on supply chain operations, which are the procedures by which raw materials and components are obtained from suppliers and refined into more complex products by in-house workers. Since the most effective sustainable supplier incorporates all three aspects of sustainable processes—social responsibility, economic viability, and environmental friendliness—it is essential that all businesses have access to the most suitable sustainable vendors. Vendor evaluation, which is part of MCDM, is crucial to the success of any business. When faced with a set of options that fail to satisfy a number of criteria, the Multi-Criteria Decision Making (MCDM) technique can be used to arrive at a final decision.\u003c/p\u003e \u003cp\u003eThe study's overarching goal is to apply the fuzzy PROMETHEE technique, a method of multi-criteria decision making (MCDM) that makes use of linguistic terms and involves TrFNs in order to deal with the issue of uncertainty, ambiguity, obscurity, vagueness, etc. in the process of sustainable supplier selection. On the basis of the views of the decision managers in an organization, this study identifies three decision makers, three suppliers, and three sustainable supplier selection criteria: social, economic, and environmentally friendly. An MCDM technique known as fuzzy PROMETHEE is applied in a fuzzy setting to first determine the relative importance of the identified sustainable supplier selection criteria, and then to choose the most suitable and optimal sustainable provider. Sustainable Supplier 1 (SS1) was found to be the most effective supplier after all, as found in this investigation.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e\u003c/p\u003e "},{"header":"2. Literature Review","content":"\u003cp\u003eThe term sustainability is considered the most significant aspect among the managers of supply chain processes to introduce a sustainable organization that contains the social, economic, and eco-friendly criteria for the betterment of the organization (Yazdani et al., 2021a [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e], 2021b [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]). According to (Fallahpour et al. 2021[\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e]), the economic criteria for any manufacturing organization play a very significant role in any organization but in the present years, organization managers consider both the environmental and social needs concerning the people's judgments and executive orders (Dogan and Seker 2016 [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]; Dogan and Inglesi-Lotz 2017 [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]; Abdel-Baset et al. 2019 [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e]. Therefore, the involvement of the triad areas of sustainability i.e. social, economic, and eco-friendly simultaneously generates, implements and introduces the hypothesis of the sustainable processes (Mojtahedi et al., 2021[\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e]).\u003c/p\u003e\u003cp\u003eIn any organization suppliers are the major step of the organizations and they greatly increase the production of the organizations by considering the concept of sustainability (Rashidi and Cullinane 2019 [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e]; Liu et al. 2020 [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]). The companies require to evaluate their sustainable suppliers for everlasting relationships (Nezhadroshan et al. 2021 [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]). Identifying the best sustainable suppliers is the most important strategy for supply chain processes (Fallahpour et al. 2016 [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e]; Kusi-Sarpong et al. 2019 [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e]). Selecting the best sustainable suppliers helps the managers of the companies to get the satisfactory underdone substances at the punctual time, amount, and standard (Tavana et al. 2021[\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e]; Wang et al. 2021[\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e]).\u003c/p\u003e\u003cp\u003eUp to now, there exist various investigations and publications in the process of choosing suppliers. This research study applies a MCDM technique by integrating PROMETHEE technique with fuzzy theory to assist the decision-makers with selection problems. Geldermann et al. (2000) [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e] applied the F-PROMETHEE technique using TrFNs interval-valued in organizations making iron and steel components. Goumas and Lygerou in 2000 [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e] implemented the fuzzy PROMETHEE II techniquein determining the alternative energy exploitation projects. Geldermann and Rentz (2001) [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e] also applied the F-PROMETHEE technique for eco-friendly management system and constructed a sensitivity representation analysis. Bilsel et al. (2006) [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e] applied the F-PROMETHEE technique in determining the ranking of the websites of Turkish hospitals.\u003c/p\u003e\u003cp\u003eChou et al. (2007) [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e] applied the F-PROMETHEE technique to estimate the best environment-friendly selections for an actual construction area. Wang et al. (2008) [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e] and Chen et al. (2011) [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e] implemented a F-PROMETHEE technique for the information organizations outsourcing evaluation and assessment of providers. Moreira et al. (2009) [\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e] implemented both the PROMETHEE and F-PROMETHEE techniques in prioritizing the deterioration of the diagnostic of electrical power apparatus. Aloini et al. in 2009 [\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e] implemented an integrated F-PROMETHEE technique for choosing the logistic service providers. Zhou et al. (2009) [\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e] applied PROMETHEE II for presenting the pipe condition assessment problem. Behzadian et al. (2010) [\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e], Zhang et al. (2009) [\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e] applied the F-PROMETHEE technique to polluted areas based on a comparative risk procedure. Liu and Guan (2009) [\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e] developed the F-PROMETHEE technique for estimating the standard of passenger rail line.\u003c/p\u003e\u003cp\u003eGiannopoulos and Founti in 2010 [\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e] studied the modified variety of the PROMETHEE technique integrated with a flexible ranking approach in a fuzzy environment. Li and Li (2010) [\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e] introduced a brand modified version of the improved PROMETHEE 2nd technique in terms of universal fuzzy values. Tuzkaya et al. (2010) [\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e] implemented a F-PROMETHEE technique for the selection of alternative material handling appliances. Shirinfar and Haleh (2011) [\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e] developed the fuzzy PROMETHEE technique for the selection of the suppliers. Yilmaz and Dagdeviren (2011) [\u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e] applied the hybrid integrated F-PROMETHEE technique and GP technique for the choosing appliances. Shakhsi-Niaei et al. (2011) [\u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e] presented the F-PROMETHEE technique in the form of a Monte Carlo simulation structure for estimating the ranking of a company's projects.\u003c/p\u003e\u003cp\u003eGupta et al. (2012) [\u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e34\u003c/span\u003e] applied the fuzzy PROMETHEE technique to choose the suppliers for logistics services in cement organizations. Tavakoli et al. (2013) [\u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e] used the integration of F-PROMETHEE and the F-GP techniques to find the ranking of the vendors. Anojkumar et al. (2014) [\u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e] introduced the involvement of tetrad MCDM techniques i.e. Fuzzy AHP-TOPSIS, F-AHP-VIKOR, F-AHP-ELECTRE, F-AHP-PROMTHEE for solving the problems of choosing the pipes stuff in sugar company and to select the best option among the numerous types of substances. In the present years, the researchers used the F-PROMETHEE technique to allow application novelty like squander disposal solution assessment (Lolli et al., 2016) [\u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e] and methodological novelty like integration of interval-valued type-II fuzzy sets combined with PROMETHEE (Celik and Gumus, 2016 [\u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e38\u003c/span\u003e], Chen, 2015 [\u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e39\u003c/span\u003e], Chen, 2014 [\u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e]) and introducing intuitionistic F-PROMETHEE technique (Liao and Xu, 2014 [\u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e41\u003c/span\u003e]).\u003c/p\u003e\u003cp\u003eChen and Pan (2016) [\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e] applied the variable fuzzy PROMETHEE technique in choosing the best low-carbon building extent. Krishankumar et al. (2017) [\u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e43\u003c/span\u003e] introduced an extended version of PROMETHEE outranking technique under an intuitionistic fuzzy logic surroundings for choosing the supplier based on linguistic terms. Wan et al. (2020) [\u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e44\u003c/span\u003e] utilized the fuzzy PROMETHEE technique for selecting the green vendors. Roy et al. (2020) [\u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e45\u003c/span\u003e] implemented the MCDM techniques named AHP-PROMETHEE for determining the methods for choosing the sustainable supplier by considering various types of transportation criteria in a ready-made clothing garments organization.\u003c/p\u003e"},{"header":"3. Fuzzy Promethee Technique","content":"\u003cp\u003eBrans et al. in 1984 [\u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e46\u003c/span\u003e], Brans and Vincke in 1985 [\u003cspan citationid=\"CR47\" class=\"CitationRef\"\u003e47\u003c/span\u003e], Brans et al. in 1986 [\u003cspan citationid=\"CR48\" class=\"CitationRef\"\u003e48\u003c/span\u003e] proposed an MCDM technique named PROMETHEE (Preference Ranking Organization Method for Enrichment Evaluations) technique. This technique consists of simple philosophy and ideology and its methodology is also simple and easy in comparison to other techniques. This technique is an outranking technique and is classified into two techniques namely PROMETHEE I\u003csup\u003est\u003c/sup\u003e and PROMETHEE II\u003csup\u003end\u003c/sup\u003e where PROMETHEE I\u003csup\u003est\u003c/sup\u003e technique allows incomplete ranking of the set of choices and PROMETHEE II technique allows absolute or complete ranking of a set of choices (\u0026Uuml;lengin et al., 2001) [\u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e49\u003c/span\u003e]. The application of PROMETHEE technique consists of real-life areas like environmental conservation and preservation, conservation and maintenance of water, business and financial capability, chemical science, conveyance management, constructing and gathering, conservation and preservation of energy, strategic planning in production, social management system, medical facilities, agricultural production, educational system, government policies, and sports fields, etc.\u003c/p\u003e \u003cp\u003eThe term \"fuzzy PROMETHEE\" was coined by Le T\u0026eacute;no and Mareschal (1998) [\u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e50\u003c/span\u003e], who were the first to propose combining fuzzy theory and the PROMETHEE approach (F-PROMETHEE). Zadeh first proposed the idea of fuzziness in 1965 [\u003cspan citationid=\"CR51\" class=\"CitationRef\"\u003e51\u003c/span\u003e]. Using the collective wisdom of specialists, the fuzzy theory was created to provide a framework for resolving cases of ambiguity and uncertainty (Wang et al. in 2008 [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e]). The mathematical formulation and the solutions to issues that are neither simple nor well-defined by mathematical modeling techniques can be constructed and analyzed using the principles of fuzzy logic theory (Kandel, 1986 [\u003cspan citationid=\"CR52\" class=\"CitationRef\"\u003e52\u003c/span\u003e]).\u003c/p\u003e \u003cp\u003eThe main advantages of the fuzzy PROMRTHEE technique are that it is I simple to learn and apply, (ii) transparent in its processes, and (iii) sensitive to issues of uncertainty, ambiguity, obscurity, vagueness, etc. in the decision-making setting. Additionally, the advantages of the PROMETHEE technique are supported by the expanded fuzzy version of the most popular PROMETHEE outranking technique. The aforementioned advantages make this method useful in a wide variety of practical settings. A unique aspect of the fuzzy PROMETHEE method is that it affords the opportunity to choose the types of preference words, so increasing the likelihood of obtaining a more workable definition for the many kinds of detected choice criteria. Therefore, this method has been validated as an efficient and effective method for use by decision managers in the supply chain processes.\u003c/p\u003e \u003cp\u003eUnder this technique, the decision managers of the organization firstly study the framework of the process of sustainable suppliers\u0026rsquo; assessment secondly, they determine the criteria which are useful in selecting the sustainable supplier and thirdly they evaluate the relative weights of the identified criteria through linguistic terms expressed by TrFNs assessment by the judgment of the decision managers. Fourthly, the ratings of all the suppliers is also described through linguistic terms expressed by trapezoidal fuzzy numbers i.e. a quadruple denoted by\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(({a}_{1},{a}_{2},{a}_{3},{a}_{4})\\)\u003c/span\u003e\u003c/span\u003eassessment by the decision managers. Lastly, they evaluate the best sustainable supplier, and the ranking of suppliers is evaluated by calculating the fuzzy Hamming distances and are applied for making pair wise comparisons among the set of identified sustainable suppliers for each criterion.\u003c/p\u003e"},{"header":"4. Steps In Fuzzy Promethee Technique","content":"\u003cp\u003e \u003cb\u003eStep 1 - Identification of alternatives and criteria-\u003c/b\u003e Firstly a group of\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(K\\)\u003c/span\u003e\u003c/span\u003edecision managers say\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({s DM}_{K}\\)\u003c/span\u003e\u003c/span\u003ewhere\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left(k=\\text{1,2},3,\\dots K\\right)\\)\u003c/span\u003e\u003c/span\u003ethen identifies a set of sustainable suppliers say \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(m\\)\u003c/span\u003e\u003c/span\u003e where (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(i\\)\u003c/span\u003e\u003c/span\u003e = 1, 2,\u0026hellip;,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(m\\)\u003c/span\u003e\u003c/span\u003e) and various types of sustainable supplier selection criteria say \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(n\\)\u003c/span\u003e\u003c/span\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((j\\)\u003c/span\u003e\u003c/span\u003e = 1, 2,\u0026hellip;, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(n\\)\u003c/span\u003e\u003c/span\u003e) namely social, environmental, and economic and i.e. Soc., Env., Eco.\u003c/p\u003e \u003cp\u003e \u003cb\u003eStep 2 -Linguistic terms expressed for criteria and performance ratings of the options\u003c/b\u003e- The weights of identified criteria and the fuzzy performance ratings used for the choices are represented through linguistic terms described by TrFNs according to the opinions and assessment of the decision managers.\u003c/p\u003e \u003cp\u003e \u003cb\u003eStep 3 - Estimation of the weights of criteria -\u003c/b\u003e \u003c/p\u003e \u003cp\u003eFuzzy weights of each criterion are denoted by\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({ w}_{j}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003cp\u003eWhere\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$${ w}_{j}=\\left({w}_{j}^{1},{w}_{j}^{2},{w}_{j}^{3},{w}_{j}^{4}\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({ w}_{j}^{1}=min\\)\u003c/span\u003e \u003c/span\u003e(\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({w}_{j}^{1})\\)\u003c/span\u003e\u003c/span\u003e (2)\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$${ w}_{j}^{2}=1/k\\left({\\Sigma }{w}_{j}^{2}\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$${ w}_{j}^{3}=1/k\\left({\\Sigma }{w}_{j}^{3}\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({ w}_{j}^{4}=max\\)\u003c/span\u003e \u003c/span\u003e(\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({w}_{j}^{4})\\)\u003c/span\u003e\u003c/span\u003e (5)\u003c/p\u003e \u003cp\u003e \u003cb\u003eStep 4 - Developing a fuzzy decision matrix \u0026ndash;\u003c/b\u003e This matrix says [\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(D\\)\u003c/span\u003e\u003c/span\u003e] is constructed as:\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$\\left[D\\right]={\\left[{x}_{ij}\\right]}_{m*n}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{ij}\\)\u003c/span\u003e\u003c/span\u003e be the fuzzy performance rating of the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({i}^{th}\\)\u003c/span\u003e\u003c/span\u003e choice for the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({j}^{th}\\)\u003c/span\u003e\u003c/span\u003e criteria, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(i\\)\u003c/span\u003e\u003c/span\u003e= 1,2,\u0026hellip;,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(m\\)\u003c/span\u003e\u003c/span\u003e ; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(m\\)\u003c/span\u003e\u003c/span\u003e is the numeral of choices, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(j\\)\u003c/span\u003e\u003c/span\u003e= 1,2,\u0026hellip;,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(n\\)\u003c/span\u003e\u003c/span\u003e ; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(n\\)\u003c/span\u003e\u003c/span\u003eis the numeral of criteria.\u003c/p\u003e \u003cp\u003e \u003cb\u003eStep 5 - Construction of combined fuzzy decision matrix\u003c/b\u003e \u0026ndash; This matrix is constructed by showing the fuzzy performance ratings of all the alternatives based on identified criteria through TrFNs and is as follows:\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$\\left[\\begin{array}{ccc}{x}_{11}\u0026amp; \\cdots \u0026amp; {x}_{1n}\\\\ ⋮\u0026amp; \\ddots \u0026amp; ⋮\\\\ {x}_{m1}\u0026amp; \\cdots \u0026amp; {x}_{mn}\\end{array}\\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({ x}_{ij}=\\left({a}_{ij},{b}_{ij},{c}_{ij},{d}_{ij}\\right)\\)\u003c/span\u003e\u003c/span\u003eare the performance ratings in fuzzy numbers of all the choices through TrFNs.\u003c/p\u003e \u003cp\u003eWhere\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({ a}_{ij}= min\\)\u003c/span\u003e \u003c/span\u003e(\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({a}_{ij})\\)\u003c/span\u003e\u003c/span\u003e (8)\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$${ b}_{ij}=\\frac{1}{K}\\left(\\varSigma {b}_{ij}\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({c}_{ij}=\\frac{1}{K}\\left(\\varSigma {c}_{ij}\\right)\\)\u003c/span\u003e \u003c/span\u003e(10) \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({ d}_{ij}=\\text{m}\\text{a}\\text{x}\\left({d}_{ij}\\right)\\)\u003c/span\u003e\u003c/span\u003e (11)\u003c/p\u003e \u003cp\u003e \u003cb\u003eStep 5 \u0026ndash; Developing a combined normalized fuzzy decision matrix-\u003c/b\u003e This matrix say \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left[R\\right]\\)\u003c/span\u003e\u003c/span\u003eis constructed as:\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$$\\left[R\\right]=\\left[{r}_{ij}\\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e12\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eNormalization is done by applying the following two equations -\u003c/p\u003e \u003cp\u003eIn case of beneficial criteria i.e. social and environment criteria -\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e\n$${ r}_{ij}=\\left(\\frac{{a}_{ij}}{{d}_{j}^{+}},\\frac{{b}_{ij}}{{d}_{j}^{+}},\\frac{{c}_{ij}}{{d}_{j}^{+}},\\frac{{d}_{ij}}{{d}_{j}^{+}}\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e13\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({d}_{j}^{+}=\\text{max}\\left({d}_{ij}\\right)\\)\u003c/span\u003e\u003c/span\u003e(14)\u003c/p\u003e \u003cp\u003eIn the case of non-beneficial criteria i.e. economic criteria -\u003cdiv id=\"Equ9\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ9\" name=\"EquationSource\"\u003e\n$${r}_{ij}=\\left(\\frac{{a}_{j}^{-}}{{d}_{ij}},\\frac{{a}_{j}^{-}}{{c}_{ij}},\\frac{{a}_{j}^{-}}{{b}_{ij}},\\frac{{a}_{j}^{-}}{{a}_{ij}}\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e15\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({a}_{j}^{-}=min{ a}_{ij }\\)\u003c/span\u003e\u003c/span\u003e(16)\u003c/p\u003e \u003cp\u003e \u003cb\u003eStep 6 - Construction of weighted-normalized fuzzy decision matrix-\u003c/b\u003e This matrix say [\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(V\\)\u003c/span\u003e\u003c/span\u003e] is constructed by multiplying the values in the combined normalized matrix say \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({r}_{ij}\\)\u003c/span\u003e\u003c/span\u003e and the evaluated weights of the identified criteria say\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({w}_{j}\\)\u003c/span\u003e\u003c/span\u003e.\u003cdiv id=\"Equ10\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ10\" name=\"EquationSource\"\u003e\n$$V=\\left[{r}_{ij}*{w}_{j}\\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e17\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003cb\u003eStep 7 - Construction of preference functions -\u003c/b\u003e Preference functions among each alternative are constructed by evaluating Hamming distances which are used for comparing two choices say a and b according to each criterion. Firstly, we calculate the maximum number say c between the two fuzzy numbers. Then, finding the Hamming distance which is the sum of all values evaluated by taking the absolute value of the distinction between the maximum number and the first alternative and also with the second alternative. According to Hatami-Marbini \u0026amp; Tavanain 2011 [\u003cspan citationid=\"CR53\" class=\"CitationRef\"\u003e53\u003c/span\u003e], preference functions are determined by the following equations -\u003cdiv id=\"Equ11\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ11\" name=\"EquationSource\"\u003e\n$${ \\text{V}}_{\\text{a}\\text{c}}\\ge {\\text{V}}_{\\text{b}\\text{c}} , \\text{i}\\text{f} \\text{a}\\text{n}\\text{d} \\text{o}\\text{n}\\text{l}\\text{y} \\text{i}\\text{f} d\\left\\{\\text{max}\\left({V}_{ac} , {V}_{bc}\\right), {V}_{bc}\\right\\}\\ge d\\left\\{\\text{max}\\left({V}_{ac} , {V}_{bc}\\right),{V}_{ac}\\right\\}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e20\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ12\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ12\" name=\"EquationSource\"\u003e\n$${ \\text{V}}_{\\text{a}\\text{c}}\u0026lt;{\\text{V}}_{\\text{b}\\text{c}} , \\text{i}\\text{f} \\text{a}\\text{n}\\text{d} \\text{o}\\text{n}\\text{l}\\text{y} \\text{i}\\text{f} d\\left\\{\\text{max}\\left({V}_{ac} , {V}_{bc}\\right), {V}_{bc}\\right\\}\u0026lt;d\\left\\{\\text{max}\\left({V}_{ac} , {V}_{bc}\\right),{V}_{ac}\\right\\}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e21\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ13\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ13\" name=\"EquationSource\"\u003e\n$$\\text{P}\\left(\\text{a},\\text{b}\\right)=d\\left\\{\\text{max}\\left({V}_{ac} , {V}_{bc}\\right), {V}_{ac}\\right\\} \\text{i}\\text{f} {V}_{ac}\u0026lt;{V}_{bc} and d\\left\\{\\text{max}\\left({V}_{ac} , {V}_{bc}\\right),{V}_{bc}\\right\\}\\text{i}\\text{f} {V}_{ac}\\ge {V}_{bc }$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e22\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003cb\u003eStep 8 - Construction of fuzzy preference index -\u003c/b\u003e Fuzzy preference index among each alternative is denoted by \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\pi \\left(a,b\\right)\\)\u003c/span\u003e\u003c/span\u003e and estimated by using the following equation:\u003cdiv id=\"Equ14\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ14\" name=\"EquationSource\"\u003e\n$$\\pi \\left(a,b\\right)={\\Sigma }\\left\\{{w}_{j}*P\\left(a,b\\right)\\right\\}/{\\Sigma }{w}_{j }$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e23\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003cb\u003eStep 9 \u0026ndash; Calculation of positive and negative flows -\u003c/b\u003e Positive and negative flows are evaluated by defuzzification of the trapezoidal fuzzy numbers used in the fuzzy preference index into crisp numbers (Giannopoulos \u0026amp; Founti 2010) [\u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e38\u003c/span\u003e]. According to Chen et al.'s, defuzzification of the TrFNs is determined by adding all the four fuzzy numbers and dividing by 4. Positive flows and negative flows are evaluated by defuzzification of the TrFNs in rows and columns respectively.\u003c/p\u003e \u003cp\u003eLet \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{A}\\)\u003c/span\u003e\u003c/span\u003e be the defuzzified value of the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{T}\\text{r}\\text{F}\\text{N}\\text{s} A\\)\u003c/span\u003e\u003c/span\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(A=({a}_{1},{a}_{2},{a}_{3},{a}_{4})\\)\u003c/span\u003e\u003c/span\u003e, then the defuzzification is calculated as:\u003cdiv id=\"Equ15\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ15\" name=\"EquationSource\"\u003e\n$${ x}_{A}=\\frac{{a}_{1}+{a}_{2}+{a}_{3}+{a}_{4}}{4}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e24\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003cb\u003eStep 10 \u0026ndash; Calculation of net flows\u003c/b\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\left({NF}^{{\\prime }}\\right)\\)\u003c/span\u003e \u003c/span\u003e \u003cb\u003e-\u003c/b\u003e Net flows are determined by subtracting the positive flows (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(PF\\)\u003c/span\u003e\u003c/span\u003e) and negative flows (NF).\u003cdiv id=\"Equ16\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ16\" name=\"EquationSource\"\u003e\n$${ NF}^{{\\prime }}=PF-NF$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e25\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003cb\u003eStep 11 \u0026ndash; Determining the ranking of each alternative \u0026ndash;\u003c/b\u003e The Preference ranking of each choice is determined by the analysis of net flows. The ranking of the choices can be determined by considering the values of the net flows for each choice. If the value of the net flows is maximum then the rank of choice is one. The best choice is determined as:\u003cdiv id=\"Equ17\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ17\" name=\"EquationSource\"\u003e\n$${ A}^{*}=\\{ {A}_{i} ;\\text{max}(NF)\\}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e26\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e"},{"header":"5. Application Of The Fuzzy Promethee Technique","content":"\u003cp\u003eFirstly, a team of three decision managers i.e. DM1, DM2, and DM3 are formed and they identified the three sustainable suppliers\u0026rsquo; i.e.SS1, SS2, SS3, and three types of sustainable supplier selection criteria i.e. social, environmental, and economic i.e. Soc., Env., Eco.\u003c/p\u003e \u003cp\u003eThe linguistic terms for the weights of criteria and performance ratings of the sustainable suppliers, through TrFNs assessment by the opinions of all the decision managers, are shown in Tables\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e and \u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e respectively.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eLinguistic terms used for the evaluating weights of the criteria\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eLinguistic terms\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eTrapezoidal Fuzzy numbers\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eVery low(VL)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e(0,0,0.1,0.2)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eLow(L)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e(0.1, 0.2, 0.2, 0.3)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMedium low (ML)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e(0.2, 0.3, 0.4, 0.5)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMedium (M)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e(0.4, 0.5, 0.5, 0.6)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMedium high (MH)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e(0.5, 0.6, 0.7, 0.8)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eHigh (H)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e(0.7, 0.8, 0.8, 0.9)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eVery high (VH)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e(0.8, 0.9, 1, 1)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eLinguistic terms used for the performance ratings of the suppliers\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eLinguistic terms\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eTrapezoidal Fuzzy numbers\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eVery poor(VP)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e(0, 0, 1, 2)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePoor (P)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e(1, 2, 2, 3)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMedium poor(MP)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e(2, 3, 4, 5)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFair (F)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e(4, 5, 5, 6)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMedium good (MG)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e(5, 6, 7, 8)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGood (G)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e(7, 8, 8, 9)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eVery good(VG)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e(8, 9, 10, 10)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe linguistic terms which are applied for evaluating the weights of the identified criteria assessment by the opinion of all the decision managers are shown in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eLinguistic terms used for finding the weights of the criteria assessment by all the decision managers\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSoc.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eEnv.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eEco.\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDM1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eVH\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eM\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eH\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDM2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eH\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eVH\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eH\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDM3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eM\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eH\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eVH\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe linguistic terms which are implemented for determining the performance ratings of all the sustainable suppliers\u0026rsquo; assessments by the opinions of all the decision managers are shown in Tables\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e, \u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e5\u003c/span\u003e, \u003cspan refid=\"Tab6\" class=\"InternalRef\"\u003e6\u003c/span\u003e, and \u003cspan refid=\"Tab7\" class=\"InternalRef\"\u003e7\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eLinguistic terms used for the performance ratings of the suppliers\u0026rsquo; assessment by the first decision manager\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDM1\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSoc.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eEnv.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eEco.\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSS1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eVG\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eMG\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eVG\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSS2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eG\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eMG\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eG\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSS3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMG\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eG\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eVG\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab5\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 5\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eLinguistic terms used for the performance ratings of the suppliers\u0026rsquo; assessment by the second decision manager\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDM2\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSoc.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eEnv.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eEco.\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSS1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eG\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eVG\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eMG\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSS2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMG\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eG\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eVG\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSS3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eVG\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eMG\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eG\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab6\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 6\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eLinguistic terms used for the performance ratings of the suppliers\u0026rsquo; assessment by the third decision manager\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDM3\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSoc.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eEnv.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eEco.\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSS1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eG\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eVG\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eMG\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSS2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMG\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eG\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eMG\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSS3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eVG\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eVG\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eG\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab7\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 7\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eLinguistic terms used for the performance ratings of the suppliers\u0026rsquo; assessment by all the decision managers\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDM1,DM2,DM3\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSoc.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eEnv.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eEco.\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSS1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eVG, G, G\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eMG, VG, VG\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eVG, MG, MG\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSS2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eG, MG, MG\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eMG, G, G\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eG, VG, MG\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSS3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMG, VG, VG\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eG, MG, VG\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eVG, G, G\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe fuzzy ratings which are implemented for estimating the weights of each identified criteria according to the opinions of all the decision managers through TrFNs are shown in Table\u0026nbsp;\u003cspan refid=\"Tab8\" class=\"InternalRef\"\u003e8\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab8\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 8\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eFuzzy ratings used for finding the weights of the criteria assessment by all decision-makers in terms of trapezoidal fuzzy numbers\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSoc.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eEnv.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eEco.\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDM1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.8,0.9,1,1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.5,0.6,0.7,0.8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.7,0.8,0.8,0.9\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDM2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.7,0.8,0.8,0.9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.8,0.9,1,1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.7,0.8,0.8,0.9\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDM3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.5,0.6,0.7,0.8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.7,0.8,0.8,0.9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.8,0.9,1,1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eNow, determining the weights of all the criteria by using the equations (2)-(5) and Table\u0026nbsp;\u003cspan refid=\"Tab8\" class=\"InternalRef\"\u003e8\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eFirstly, determining the weight of the Soc. criteria by using Table\u0026nbsp;\u003cspan refid=\"Tab8\" class=\"InternalRef\"\u003e8\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eIn Table\u0026nbsp;\u003cspan refid=\"Tab8\" class=\"InternalRef\"\u003e8\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({w}_{j}^{1}\\)\u003c/span\u003e\u003c/span\u003e = min (0.8, 0.7, 0.5) = 0.5, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({w}_{j}^{2}\\)\u003c/span\u003e\u003c/span\u003e = 1/3(0.9\u0026thinsp;+\u0026thinsp;0.8+0.6) = 0.7667, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({w}_{j}^{3}\\)\u003c/span\u003e\u003c/span\u003e = 1/3(1, 0.8, 0.7) = 0.8333 and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({w}_{j}^{4}\\)\u003c/span\u003e\u003c/span\u003e= max (1, 0.9, 0.8) = 1.\u003c/p\u003e \u003cp\u003eThus, weights of the social criteria is (0.5, 0.7667, 0.8333,1). Similarly, the weights of the environment criteria is (0.5, 0.7667, 0.8333,1) and the weights of the economic criteria is (0.7, 0.8333, 0.8667, 1).\u003c/p\u003e \u003cp\u003eThe weights of all the identified criteria through TrFNs are shown in Table\u0026nbsp;\u003cspan refid=\"Tab9\" class=\"InternalRef\"\u003e9\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab9\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 9\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eWeights of all criteria in terms of TrFNs\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCriteria\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eWeights\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSoc.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e(0.5, 0.7667, 0.8333, 1)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eEnv.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e(0.5, 0.7667, 0.8333, 1)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eEco.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e(0.7, 0.8333, 0.8667, 1)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eNow, constructing a decision matrix of all the green and sustainable suppliers for each criterion represented by Eq.\u0026nbsp;(\u003cspan refid=\"Equ4\" class=\"InternalRef\"\u003e6\u003c/span\u003e) which shows the fuzzy performance ratings of all the green and sustainable suppliers by the opinions of all the decision managers in terms of TrFNs and is shown in Table\u0026nbsp;\u003cspan refid=\"Tab10\" class=\"InternalRef\"\u003e10\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab10\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 10\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eFuzzy decision matrix of all the suppliers\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSoc.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eEnv.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eEco.\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSS1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e(8,9,10,10), (7,8,8,9), (7,8,8,9)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e(5,6,7,8), (8,9,10,910, (8,9,10,10)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e(8,9,10,10), (5,6,7,8), (5,6,7,8)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSS2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e(7,8,8,9), (5,6,7,8), (5,6,7,8)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e(5,6,7,8), (7,8,8,9), (7,8,8,9)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e(7,8,8,9), (8,9,10,10), (5,6,7,8)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSS3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e(5,6,7,8), (8,9,10,10), (8,9,10,10)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e(7,8,8,9), (5,6,7,8), (8,9,10,10)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e(8,9,10,10), (7,8,8,9), (7,8,8,9)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eNow, constructing a combined decision matrix of all the sustainable suppliers for each criterion represented by Eq.\u0026nbsp;(\u003cspan refid=\"Equ5\" class=\"InternalRef\"\u003e7\u003c/span\u003e) which depicts the combined fuzzy performance ratings of all the sustainable suppliers according to the opinions of all the decision managers in terms of TrFNs by using the equations (8)-(11) and Table\u0026nbsp;\u003cspan refid=\"Tab10\" class=\"InternalRef\"\u003e10\u003c/span\u003e and is shown in Table\u0026nbsp;\u003cspan refid=\"Tab11\" class=\"InternalRef\"\u003e11\u003c/span\u003e. In Table\u0026nbsp;\u003cspan refid=\"Tab11\" class=\"InternalRef\"\u003e11\u003c/span\u003e, the numbers of the first cell is calculated as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({a}_{ij}\\)\u003c/span\u003e\u003c/span\u003e = min(8,7,7) = 7, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({b}_{ij}\\)\u003c/span\u003e\u003c/span\u003e= 1/3(9\u0026thinsp;+\u0026thinsp;8+8)=8.3, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({c}_{ij}\\)\u003c/span\u003e\u003c/span\u003e = 1/3(10,8,8)=8.7, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({d}_{ij}=\\)\u003c/span\u003e\u003c/span\u003emax(10,9,9)=10 by using the Table\u0026nbsp;\u003cspan refid=\"Tab10\" class=\"InternalRef\"\u003e10\u003c/span\u003e. Similarly, we can find all the values of all cells.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab11\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 11\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eCombined Fuzzy decision matrix of all the suppliers\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSoc.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eEnv.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eEco.\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSS1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e7,8.3,8.7,10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e5,8,9,10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e5,7,8,10\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSS2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e5,6.7,7.3,9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e5,7.3,7.7,9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e5,7.7,8.3,10\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSS3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e5,8,9,10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e5,7.7,8.3,10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e7,8.3,8.7,10\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eNow, constructing a normalized decision matrix of all the sustainable suppliers for all criteria represented by Eq.\u0026nbsp;(\u003cspan refid=\"Equ7\" class=\"InternalRef\"\u003e12\u003c/span\u003e) and by using Table\u0026nbsp;\u003cspan refid=\"Tab11\" class=\"InternalRef\"\u003e11\u003c/span\u003e. For beneficial criteria, namely social and environmental criteria, equations (\u003cspan refid=\"Equ8\" class=\"InternalRef\"\u003e13\u003c/span\u003e) and (14)are used and for non-beneficial criteria, namely economic criteria, equations (\u003cspan refid=\"Equ9\" class=\"InternalRef\"\u003e15\u003c/span\u003e) and (16) are used. A normalized decision matrix of all the sustainable suppliers based on each criterion is shown in Table\u0026nbsp;\u003cspan refid=\"Tab12\" class=\"InternalRef\"\u003e12\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eIn beneficial criteria, the value of the first cell is calculated as (7/10,8.3/10,8.7/10,10/10) = (0.7,0.83,0.87,1) while in the case of non-beneficial criteria, the value of the cell is calculated as (5/10,5/8,5/7,5/5) = (0.5,0.62,0.71,1). Similarly, we can find all the values of all cells.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab12\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 12\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eNormalized Decision matrix of all the suppliers wrt each criterion\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSoc.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eEnv.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eEco.\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSS1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.7,0.83,0.87,1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.5,0.8,0.9,1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.5,0.62,0.71,1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSS2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.5,0.67,0.73,0.9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.5,0.73,0.77,0.9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.5,0.60,0.64,1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSS3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.5,0.8,0.9,1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.5,0.77,0.83,1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.5,0.57,0.60,0.71\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eNow, constructing a weighted normalized decision matrix of all the sustainable suppliers for each criterion represented as the Eq.\u0026nbsp;(\u003cspan refid=\"Equ10\" class=\"InternalRef\"\u003e17\u003c/span\u003e) by using Table\u0026nbsp;\u003cspan refid=\"Tab12\" class=\"InternalRef\"\u003e12\u003c/span\u003e and is shown in Table\u0026nbsp;\u003cspan refid=\"Tab13\" class=\"InternalRef\"\u003e13\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eBy using the Table\u0026nbsp;\u003cspan refid=\"Tab12\" class=\"InternalRef\"\u003e12\u003c/span\u003e, the value of the first cell is calculated as:\u003c/p\u003e \u003cp\u003e(0.7*0.5\u0026thinsp;+\u0026thinsp;0.83*0.7667\u0026thinsp;+\u0026thinsp;0.87*0.8333\u0026thinsp;+\u0026thinsp;1*1)=(0.35,0.6363,0.7249,1).\u003c/p\u003e \u003cp\u003eSimilarly, we can find all the values of all cells.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab13\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 13\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eWeighted Normalized Decision matrix of all the suppliers wrt each criterion\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSoc.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eEnv.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eEco.\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSS1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.35,0.6363,0.7249,1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.25,0.6133,0.7499,1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.35,0.5166,0.6153,1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSS2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.25,0.5136,0.6083,0.9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.25,0.5596,0.6416,0.9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.35,0.4999,0.5546,1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSS3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.25,0.6133,0.7499,1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.25,0.5903,0.6916,1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.35,0.4749,0.5200,1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eNow, determining the preference functions among the sustainable suppliers for each criterion by using the equations (\u003cspan refid=\"Equ11\" class=\"InternalRef\"\u003e20\u003c/span\u003e)-(\u003cspan refid=\"Equ13\" class=\"InternalRef\"\u003e22\u003c/span\u003e) and Table\u0026nbsp;\u003cspan refid=\"Tab13\" class=\"InternalRef\"\u003e13\u003c/span\u003e. Firstly, determining the preference function between sustainable suppliers SS1 and SS2 for social criteria is shown in Table\u0026nbsp;\u003cspan refid=\"Tab14\" class=\"InternalRef\"\u003e14\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab14\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 14\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003ePreference function between supplier 1 and supplier 2\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAlternative a\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.35\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.6363\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.7249\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAlternative b\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.25\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.5136\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.6083\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.9\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eThe maximum number between a and b say c\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.35\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.6363\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.7249\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003edistance between a and c\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003edistance between b and c\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.1227\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.1166\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(d\\left\\{\\text{max}\\left({V}_{ac} ,{V}_{bc}\\right),{V}_{ac}\\right\\}=0\\)\u003c/span\u003e \u003c/span\u003eand\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(d\\left\\{\\text{max}\\left({V}_{ac} ,{V}_{bc}\\right),{V}_{bc}\\right\\}=\\)\u003c/span\u003e\u003c/span\u003e0.1\u0026thinsp;+\u0026thinsp;0.1227+0.1166+0.1= 0.4393 then\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({V}_{ac}\\)\u003c/span\u003e \u003c/span\u003e \u0026ge; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({V}_{bc}\\)\u003c/span\u003e\u003c/span\u003econdition is satisfied. Therefore, the preference function will be 0.4343.\u003c/p\u003e \u003cp\u003eSimilarly, determining the preference functions among all the sustainable suppliers for each criterion is shown in Table\u0026nbsp;\u003cspan refid=\"Tab15\" class=\"InternalRef\"\u003e15\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab15\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 15\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003ePreference functions of each supplier wrt each criterion\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSoc.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eEnv.\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eEco.\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eP(SS1,SS2)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.4343\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.262\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.0774\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eP(SS1,SS3)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.123\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.0813\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.137\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eP(SS2,SS1)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eP(SS2,SS3)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.3413\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.1807\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.0596\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eP(SS3,SS1)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.025\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eP(SS3,SS2)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eNow determining the fuzzy preference index among the sustainable suppliers for each criterion by multiplying the weights of criteria and the preference functions of each supplier using the Eq.\u0026nbsp;(\u003cspan refid=\"Equ14\" class=\"InternalRef\"\u003e23\u003c/span\u003e) and Table\u0026nbsp;\u003cspan refid=\"Tab15\" class=\"InternalRef\"\u003e15\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eFirstly, determining the fuzzy preference index of suppliers SS1 and SS2 as:\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\left\\{{w}_{j}*P\\left(a,b\\right)\\right\\}\\)\u003c/span\u003e \u003c/span\u003e=(0.5*0.4343,0.7667*0.4343,0.8333*0.4343,1*0.4343)= (0.2171,0.3329,0.3619,0.4343)\u003c/p\u003e \u003cp\u003e \u003col\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003e(0.5*0.262, 0.7667*0.262,0.8333*0.262,1*0.262)= (0.131, 0.2008, 0.2183, 0.262)\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003e(0.7*0.0774, 0.8333*0.0774,0.8667*0.0.774,1*0.0.774)= (0.0541,0.0644,0.0670,0.0774)\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003c/ol\u003e \u003c/p\u003e \u003cp\u003eNow finding \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\Sigma }\\left\\{{w}_{j}*P\\left(a,b\\right)\\right\\}\\)\u003c/span\u003e\u003c/span\u003e=( 0.2171\u0026thinsp;+\u0026thinsp;0.131\u0026thinsp;+\u0026thinsp;0.0541, 0.3329\u0026thinsp;+\u0026thinsp;0.2008\u0026thinsp;+\u0026thinsp;0.0644, 0.3619\u0026thinsp;+\u0026thinsp;0.2183\u0026thinsp;+\u0026thinsp;0.0670, 0.4343\u0026thinsp;+\u0026thinsp;0.262\u0026thinsp;+\u0026thinsp;0.0774) = (0.4022,0.5891,0.6472,0.7737).\u003c/p\u003e \u003cp\u003eNow finding \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\Sigma }{w}_{j }=\\)\u003c/span\u003e\u003c/span\u003e (0.5\u0026thinsp;+\u0026thinsp;0.5\u0026thinsp;+\u0026thinsp;0.7,0.7667\u0026thinsp;+\u0026thinsp;0.7667\u0026thinsp;+\u0026thinsp;0.8333,0.8333\u0026thinsp;+\u0026thinsp;0.8333\u0026thinsp;+\u0026thinsp;0.8667,1\u0026thinsp;+\u0026thinsp;1\u0026thinsp;+\u0026thinsp;1)=(1.7,2.3667,2.5333,3)\u003c/p\u003e \u003cp\u003eNow finding \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\Sigma }\\left\\{{w}_{j}*P\\left(a,b\\right)\\right\\}/{\\Sigma }{w}_{j }=\\)\u003c/span\u003e\u003c/span\u003e (0.4022/3, 0.5891/2.5333, 0.6472/2.3667, 0.7737/1.7)= (0.1340,0.2325,0.2734,0.4551).\u003c/p\u003e \u003cp\u003eSimilarly, determining the fuzzy preference index of all sustainable suppliers for each identified criteria is shown in Table\u0026nbsp;\u003cspan refid=\"Tab16\" class=\"InternalRef\"\u003e16\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab16\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 16\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eFuzzy preference index of all the suppliers\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eM1\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eM2\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eM3\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eM1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0,0,0,0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.1340,0.2325,0.2734,0.4551\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.066,0.1068,0.3796,0.2007\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eM2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0,0,0,0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0,0,0,0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.2260,0.1775,0.2055,0.3421\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eM3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.0041,0.0075,0.0008,0.0147\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0,0,0,0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0,0,0,0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eNow, determining the positive flows after applying defuzzification in rows by using the Eq.\u0026nbsp;(\u003cspan refid=\"Equ15\" class=\"InternalRef\"\u003e24\u003c/span\u003e) in Table\u0026nbsp;\u003cspan refid=\"Tab16\" class=\"InternalRef\"\u003e16\u003c/span\u003ei.e. (0.1340\u0026thinsp;+\u0026thinsp;0.066,0.2325\u0026thinsp;+\u0026thinsp;0.1068,0.2734\u0026thinsp;+\u0026thinsp;0.3796,0.4551\u0026thinsp;+\u0026thinsp;0.2007)/4\u0026thinsp;=\u0026thinsp;0.2987\u003c/p\u003e \u003cp\u003eSimilarly, determining the negative flows after applying defuzzification in columns by using the Eq.\u0026nbsp;(\u003cspan refid=\"Equ15\" class=\"InternalRef\"\u003e24\u003c/span\u003e) in Table\u0026nbsp;\u003cspan refid=\"Tab16\" class=\"InternalRef\"\u003e16\u003c/span\u003ei.e. (0.0041\u0026thinsp;+\u0026thinsp;0.0075\u0026thinsp;+\u0026thinsp;0.0008\u0026thinsp;+\u0026thinsp;0.0147)/4\u0026thinsp;=\u0026thinsp;0.0067\u003c/p\u003e \u003cp\u003eSimilarly, determining the positive and negative flows of all the sustainable suppliers.\u003c/p\u003e \u003cp\u003eNow, determining the net flows by using the Eq.\u0026nbsp;(\u003cspan refid=\"Equ16\" class=\"InternalRef\"\u003e25\u003c/span\u003e).And finally, determining the ranking of the sustainable suppliers by considering the values of the net flows for each choice. If the values of the net flows are maximum then the rank of the sustainable supplier is one. The best sustainable supplier is determined by using the Eq.\u0026nbsp;(\u003cspan refid=\"Equ17\" class=\"InternalRef\"\u003e26\u003c/span\u003e). The positive flows, negative flows, net flows and the ranking of the sustainable suppliers, are shown in Table\u0026nbsp;\u003cspan refid=\"Tab17\" class=\"InternalRef\"\u003e17\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab17\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 17\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eRanking of all the suppliers\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePositive flows\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eNegative flows\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eNet flows\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eRanking\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSS1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.2987\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.0067\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.292\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSS2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.2377\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.2737\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-0.036\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSS3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.0067\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.4260\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-0.4193\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eHence, supplier1 is the best sustainable supplier.\u003c/p\u003e \u003cp\u003eRanking of all the sustainable suppliers - SS1\u0026thinsp;\u0026gt;\u0026thinsp;SS2\u0026thinsp;\u0026gt;\u0026thinsp;SS3\u003c/p\u003e"},{"header":"6. Conclusions, Drawbacks, And Future Outlook Of Study-","content":"\u003cp\u003eTo increase the quality standards and competitive strategies, outsourcing is unavoidable. The process of evaluating and selecting the best and optimal sustainable suppliers is a complex feature for any organization. The methods considered in suppliers\u0026rsquo; assessment care only for the basic needs of the isolated system but do not consider the whole supply chain processes. Maintaining the relationships among the sustainable suppliers and buyers strongly in the processes of a supply chain is very essential to their active participation. Therefore, the organizations prefer to collaborate with a finite numeral of suppliers or consider only a single supplier to attain and maintain high performance in the organizations. Due to the plan significance of the process of suppliers\u0026rsquo; assessment, large-scale research is developing to cope with the problems related to the MCDM problems. This paper applied a PROMETHEE technique under the fuzzy environment for solving the problem of sustainable suppliers\u0026rsquo; assessment by considering the sustainable supplier selection criteria. After applying this technique we achieve that the supplier 1 is the best sustainable supplier.\u003c/p\u003e \u003cp\u003eThe main goal of this research paper is to decide the appropriate, best, and optimal sustainable supplier. The two major benefits of this technique are - (i) it is easily accessible by the users and (ii) it pays attention to the problems of uncertainties, ambiguities, obscurities, vagueness, etc. indecision-making problems. Hence, this technique is considered to be very fruitful and is applied by the decision managers in the supply chain processes. While considering the future study and research, this technique can also be easily implemented to any other assessment problem in various types of real fields, particularly in manufacturing and service organizations.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eYazdani, M., Kabirifar, K., Frimpong, B.E., Shariati, M., Mirmozaffari, M. and Boskabadi, A., 2021a. Improving construction and demolition waste collection service in an urban area using a simheuristic approach: A case study in Sydney, Australia. \u003cem\u003eJournal of Cleaner Production\u003c/em\u003e, \u003cem\u003e280\u003c/em\u003e, p.124138.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eYazdani, M., Mojtahedi, M., Loosemore, M., Sanderson, D. and Dixit, V., 2021b. 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Omega, \u003cem\u003e39\u003c/em\u003e(4), pp.373\u0026ndash;386.\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Fuzzy theory, Multiple-Criteria Decision Making (MCDM), Fuzzy Preference Ranking Organization Method for Enrichment Evaluations (F-PROMETHEE), Sustainable Supplier (SS)","lastPublishedDoi":"10.21203/rs.3.rs-2517685/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-2517685/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe primary objective of any business today is to remain competitive and sustainable so that it may continue operating profitably and efficiently. Therefore, in order to achieve the aforementioned goals, businesses must evaluate potential sustainable suppliers in light of the three pillars of sustainability (social responsibility, economic viability, and environmental friendliness). One example of an issue that requires multiple criterion decision-making (MCDM) is the challenge of finding sustainable suppliers. Fuzzy PROMETHEE, a method of multi-criteria decision making (MCDM) that utilizes triangular fuzzy numbers (TFNs) and linguistic concepts, is used in this study to establish the relative importance of three factors for choosing a sustainable supplier: social impact, economic viability, and environmental responsibility. Successful application of the fuzzy PROMETHEE method has allowed the organization's managers to arrive at the appropriate conclusion and implement the necessary solution.\u003c/p\u003e","manuscriptTitle":"Selection of a Sustainable Supplier by Using a Fuzzy MCDM Mathematical Modelling","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2023-01-30 20:01:24","doi":"10.21203/rs.3.rs-2517685/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"d7852f37-41b4-4c12-9453-34d6af220d1e","owner":[],"postedDate":"January 30th, 2023","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2023-01-30T20:01:24+00:00","versionOfRecord":[],"versionCreatedAt":"2023-01-30 20:01:24","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-2517685","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-2517685","identity":"rs-2517685","version":["v1"]},"buildId":"FbvkV6FR0MCFSLy54lSbu","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}