Description Assignment Questions: (Marks 15) Read the above case study and answer the following Questions

Description\n \n\n\n\n\n\n\n\n\n Assignment Questions: (Marks 15)\n \n\n\n\n\n Read the above case study and answer the following Questions\n \n\n\n :\n \n\n\n\n\n\n\n\n\n\n\n\n Question 1: Explai\n \n n the decision-making approach discussed in this case study (250-300 words) (3-Marks).\n \n\n Question 2: Why supplier selection is a typical multi-criteria decision-making process involving subjectivity and vagueness? (250-300 words) (3-Marks).\n \n\n Question 3: Discuss the Sustainable supplier selection that is required for manufacturing companies. (250-300 words) (3-Marks).\n \n\n Question 4: Why Supplier selection decisions are important for most of the manufacturing firms? (250-300 words) (3-Marks).\n \n\n Question 5: What is your opinion about this study and how it is connected to course and beneficial for you? (250-300 words) (3-Marks).\n \n\n\n\nsustainability\nArticle\nA Rough Multi-Criteria Decision-Making Approach\nfor Sustainable Supplier Selection under\nVague Environment\nHuiyun Lu 1 , Shaojun Jiang 2 , Wenyan Song 1,3, * and Xinguo Ming 4\n1\n2\n3\n4\n*\nSchool of Economics and Management, Beihang University, Beijing 100191, China; lhybuaa@163.com\nSchool of Information Engineering, Handan University, Handan 056005, China; hh8582@163.com\nBeijing Key Laboratory of Emergency Support Simulation Technologies for City Operations,\nBeihang University, Beijing 100191, China\nSchool of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China;\nxgming@sjtu.edu.cn\nCorrespondence: songwenyan@buaa.edu.cn; Tel.: +86-010-8231-3693\nReceived: 13 June 2018; Accepted: 23 July 2018; Published: 26 July 2018\n

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\nAbstract: With the growing awareness of environmental and social issues, sustainable supply chain\nmanagement (SSCM) has received considerable attention both in academia and industry. Supplier\nselection plays an important role in the successful implementation of sustainable supply chain\nmanagement, because it can influence the performance of SSCM. Sustainable supplier selection is a\ntypical multi-criteria decision-making problem involving subjectivity and vagueness. Although some\nprevious researches of supplier selection use fuzzy approaches to deal with vague information, it has\nbeen criticized for requiring much priori information and inflexibility in manipulating vagueness.\nMoreover, the previous methods often omit the environmental and social evaluation criteria in the\nsupplier selection. To manipulate these problems, a new approach based on the rough set theory\nand ELECTRE (ELimination Et Choix Traduisant la REalité) is developed in this paper. The novel\napproach integrates the strength of rough set theory in handling vagueness without much priori\ninformation and the merit of ELECTRE in modeling multi-criteria decision-making problem. Finally,\na case study of sustainable supplier selection for solar air-conditioner manufacturer is provided to\ndemonstrate the application and potential of the approach.\nKeywords: sustainability; supplier selection; vague information; rough set theory; ELECTRE\n1. Introduction\nManufacturing companies today cannot ignore sustainability concerns in their business\nbecause of increased environmental awareness and ecological pressures from markets and various\nstakeholders [1–3]. Sustainable supplier selection is critical to enhance supply chain performance\nand competitive advantage [4]. This is because suppliers play an important role in implementing\nsustainable supply chain management (SSCM) practices and in achieving social, environmental and\neconomic goals [5]. In this respect, sustainable supplier selection based on the sustainability criteria\n(economic, environmental and social) is a critical strategic decision for SSCM [6,7] and it requires to be\nfurther explored methodically to help achieve sustainability of the whole supply chain.\nAlthough many researchers explore the topic of supplier selection, the study on the sustainable\nsupplier selection is still in the early stage. Most studies of sustainable supplier selection have only\nfocused on the economic and environmental aspects of sustainability. The social aspect of sustainability\nis often omitted in the decision–making for supplier selection. Besides, the problem of supplier\nselection is a typical multi-criteria decision-making (MCDM) problem. The decision makers always\nSustainability 2018, 10, 2622; doi:10.3390/su10082622\nwww.mdpi.com/journal/sustainability\nSustainability 2018, 10, 2622\n2 of 20\nneed to make trade-offs between conflicting criteria to select the most suitable supplier. It is difficult to\nobtain accurate judgments of decision makers in the process of supplier evaluation, because supplier\nselection involves large amount of linguistic information and subjective expert knowledge that are\nusually imprecise, vague or even inconsistent. To deal with this problem, fuzzy methods are often\nused to select suppliers. However, the fuzzy methods need much priori information (e.g., pre-set\nfuzzy membership function) which may increase the workload of decision makers [8,9]. The previous\napproaches also lack a flexible mechanism to deal with the subjective evaluations of experts [10,11].\nTherefore, to manipulate the above problems in sustainable supplier selection, this paper proposes\na novel integrated group decision method based on the ELECTRE (ELimination Et Choix Traduisant\nla REalité) approach and rough set theory in vague environments. Different with methods based on\nthe compensating accumulation principle (e.g., TOPSIS(Technique for Order Preference by Similarity\nto an Ideal Solution)), the ELECTRE method is based on a precedence relation and it can meet\ndifferent evaluation requirements by defining undifferentiated threshold, strict superior threshold\nand rejection threshold and thus, it has stronger flexibility in decision–making of supplier selection.\nFurthermore, the rough number originated from the rough set theory can flexibly reflect the uncertainty\nin decision–making process of supplier selection and it does not require much priori information.\nIn this respect, the proposed novel approach integrates the merit of ELECTRE in modeling multi-criteria\ndecision-making problem and the strength of rough set theory in handling vagueness without much\npriori information.\nThe paper is organized as follows: Section 2 presents a literature review of supplier selection,\nELECTRE method and rough set. Section 3 develops an integrated rough ELECTRE method for\nsustainable supplier selection. In the Section 4, a case study of sustainable supplier selection for solar\nair-conditioner manufacturer is used to validate the feasibility and effectiveness of the method and a\ncomparative analysis is also conducted in this section. In Section 5, conclusions and future research\ndirections are presented.\n2. Literature Review\n2.1. Evaluation Criteria for Sustainable Supplier Selection\nSupplier selection decisions are important for most of manufacturing firms, because a right\nsupplier can effectively improve the economic benefit of the manufacturing firm [12,13]. In the past,\neconomic criteria are usually used for supplier selection. The environment and social criteria are\noften overlooked. However, with the development of sustainable supply chain management (SSCM),\nboth the researchers and practitioners are paying more attention to environment criteria and social\ncriteria in supplier selection [14]. They find it is important to incorporating the social and environment\ncriteria into the supplier selection process [15,16]. This paper summarizes the sustainable supplier\nselection criteria from the economic, environment and social aspects. The details of the recognized\nsustainable supplier selection criteria with their sources and descriptions are summarized in Table 1.\nTable 1. Sustainable supplier selection criteria.\nSustainable Supplier\nSelection Criteria\nDescriptions\nEconomic criteria\nQuality [17,18]\nProduct quality and reliability level guaranteed by supplier.\nResponse [5]\nThe ability for timely response, completing orders on time and reliable delivery.\nCost [19]\nPurchasing cost, holding cost, ordering cost and supplier’s bidding price of\nthe product.\nSustainability 2018, 10, 2622\n3 of 20\nTable 1. Cont.\nSustainable Supplier\nSelection Criteria\nDescriptions\nEnvironmental criteria\nEnvironmental\nmanagement system\n(EMS) [20,21]\nA set of systematic processes and practices reducing environmental impacts.\nCarbon emission &\nresource\nconsumption [22,23]\nGreenhouse gas emissions in producing, transporting, using and recycling the\nproduct and the resource (e.g., energy, power and water) consumption of\nthe company.\nDesign for the\nenvironment [14,24]\nDesign reducing the overall impact of a product, process or service on human\nhealth and environment.\nGreen image [17]\nThe image of company in the green aspect, which can be improved by adopting\nenvironmental friendly products or implementing ‘green’ program. It can affect\nthe purchasing trend of customers, market share and the relationship\nwith stakeholders.\nSocial criteria\nProduct liability [25]\nBeing responsible for customer health and safety, providing products and\nservices with high quality and advertising based on real information.\nEmployee right and\nwelfare [26,27]\nTreating employee with dignity and respect and maintaining a culture of security,\nnondiscrimination and equality. Paying to employee shall comply with all\napplicable wage laws.\nSocial commitment [27]\nInvolving in local community, education, job creation, healthcare and\nsocial investment.\n2.2. The Methods of Sustainable Supplier Selection\nSelecting the right suppliers to set up optimal supplier networks can help to reduce purchasing\ncosts and increase the efficiency of the procurement logistics process [28]. Supplier selection is a\nmulti-criteria decision-making problem. There are some papers concerning sustainable (or green)\nsuppliers. Dai and Blackhurst (2012) integrate Analytical Hierarchy Process (AHP) with Quality\nFunction Deployment (QFD) for sustainable supplier selection [18]. The approach consists of four\nstages, that is, linking customer requirements with the firm’s sustainability strategy, determining the\nsustainable purchasing competitive priority, determining evaluation criteria of sustainable supplier\nand evaluating the sustainable suppliers. Hsu and Hu (2009) develop a method for selecting suppliers\nwith emphasis on issues of hazardous substance management based on Analytic Network Process\n(ANP) [29]. Liu and Hai (2005) provide a method called voting analytic hierarchy process for supplier\nselection [30]. Although AHP/ANP methods are more popular in the field of the supplier selection,\nthey are always used to determine the relative importance weightings of criteria and sub-factors merely.\nThey need to be integrated with other decision–making techniques. Besides, due to the number of\npairwise comparisons that need to be made, the number of supplier selections is practically limited in\nthe AHP/ANP-based supplier selection methods. Moreover, the conventional AHP/ANP methods do\nnot consider the vagueness of decision–making information.\nTo manipulate the increasing number of the suppliers, data envelopment analysis (DEA) is a\nprevalent approach used in supplier selection. This is because DEA can easily handle huge number of\nsuppliers with little managerial input and output required. Kuo et al. (2012) present a green supplier\nselection method using an analysis network process as well as data envelopment analysis (DEA) [31].\nANP which is able to consider the interdependency between criteria releases the constraint of DEA that\nthe users cannot set up criteria weight preferences. Wu and Blackhurst (2009) propose an augmented\nDEA approach for supplier evaluation and selection [32]. Sevkli et al. (2007) develop a new supplier\nselection method by embedding the DEA approach into AHP methodology [33]. They conclude that\nSustainability 2018, 10, 2622\n4 of 20\nthe integrated method outperforms the conventional AHP method for supplier selection. However,\nDEA-based supplier selection methods have some drawbacks. The practitioners may be confused with\ninput and output criteria. Besides, DEA is a linear programming to measure the relative efficiencies of\nhomogenous decision–making units (DMUs). An efficient supplier generating more outputs while\nrequiring less input may be not an effective supplier. Furthermore, the conventional DEA also does\nnot consider the subjectivity and vagueness in the decision–making process.\nBeside the multi-criteria decision–making method, some researchers use heuristic optimization\napproaches to select proper suppliers. Basnet and Leung (2005) develop an incapacitated mixed\nlinear integer programming which minimizes the aggregate purchasing, ordering and holding costs\nsubject to demand satisfaction [34]. They solve the problem with an enumerative search algorithm\nand a heuristic procedure. Veres et al. (2017) propose a heuristic method for optimizing supply\nchain including intelligent transportation systems (ITS) based vehicles for transportation operations\nproblems [35]. To solve the multi-product multi-period inventory lot sizing with supplier selection\nproblem, Cárdenas-Barrón et al. (2015) propose a heuristic algorithm based on reduce and optimize\napproach (ROA) and a new valid inequality [36]. Unfortunately, the heuristic optimization approaches\nomit the vagueness and subjectivity in the decision–making, which may lead to inaccurate results of\nsupplier selection.\nIn order to deal with the imprecise or vague nature of linguistic assessment in evaluation and\nselection of suppliers, fuzzy set theory is introduced into the conventional approaches. Considering\ntime pressure and lack of expertise in sustainable supplier selection, Büyüközkan and Çifçi (2011)\ndeveloped a method based on fuzzy analytic network process within group decision-making schema\nunder incomplete preference relations [37]. To manipulate the subjectivity of decision makers’\nevaluations, Amindoust et al. (2012) develop a new ranking method on the basis of fuzzy inference\nsystem (FIS) for sustainable supplier selection problem [6]. Azadnia et al. (2015) developed an\nintegrated method based on rule-based weighted fuzzy approach [38], fuzzy analytical hierarchy\nprocess and multi-objective mathematical programming for sustainable supplier selection and order\nallocation. Grisi et al. (2010) propose a fuzzy AHP method for green supplier selection using a\nseven-step approach [39]. Fuzzy logic is used to overcome uncertainty caused by human qualitative\njudgments. ELECTRE (ELimination Et Choix Traduisant la REalité) methods are able to make a\nsuccessful assessment of each alternative based on knowledge of the concordance and discordance\nsets for all pairs of alternatives. They are often used to select right suppliers [40]. Thus, Sevkli (2010)\nproposes a fuzzy ELECTRE for supplier selection [41]. Although the fuzzy methods can deal with\nthe imprecise or vague nature of linguistic assessment, it requires priori information (e.g., pre-set\nmembership function). Moreover, the fuzzy methods always convert linguistic variables into fuzzy\nnumbers with fixed intervals. Therefore, computation results usually do not exactly match initial\nlinguistic terms, which easily cause loss of information and lack of precision in the final results.\nAlthough these methods have brought great insights to supplier selection literature, most of them\nlack flexible mechanisms to handle the subjectivity and the vagueness of decision makers’ assessments.\nAlthough some fuzzy methods of supplier selection (e.g., fuzzy ELECTRE) consider the vagueness in\ndecision–making information, they require much priori information (e.g., pre-set fuzzy membership\nfunction) which consumes much time and effort of managers. Moreover, the previous fuzzy approaches\nuse fuzzy number with fixed interval to indicate the uncertainty, which cannot identify the changes in\ndecision makers’ judgments. For those reasons, there is a clear need for a new formal decision support\nmethodology for the sustainable supplier selection under vague environment.\n3. The Proposed Method\nThe main objective of this paper is to propose an integrated method for sustainable supplier\nselection based on rough set theory and ELECTRE. Besides, vagueness manipulation is also considered\nin the proposed approach. A flowchart of the proposed approach is shown in Figure 1.\nSustainability 2018, 10, 2622\nSustainability 2018, 10, x FOR PEER REVIEW\n5 of 20\n5 of 21\nFigure 1.\n1. The\nTheframework\nframework of\nof rough\nrough ELimination\nELimination Et\nEt Choix\nChoix Traduisant\nTraduisant la REalité (ELECTRE).\nFigure\n3.1. Determine\nDetermine the\nthe Supplier\nSupplier Evaluation\nEvaluation Criteria and Their Weights\nStep 1: determine the evaluation criteria of sustainable suppliers\nStep 1: determine the evaluation criteria of sustainable suppliers\nFirst of all, a panel of expert who are knowledgeable about supplier selection is established. The\nFirst of all, a panel of expert who are knowledgeable about supplier selection is established.\nD1,DD12, D\n,…,\nDkD)k )who\nThe group\nhas\nk decision-makers\nwhoare\nareresponsible\nresponsiblefor\nfor determining\ndetermining and\nand the\ngroup\nhas k\ndecision-makers\n(i.e.,(i.e.,\n2 , …,\nranking each criterion (i.e., C1 , C2 , …, Ck ). For the sustainable supplier selection, three aspects we\nC2,…,are\nCkeconomic\nranking\neachinto\ncriterion\n(i.e., C1,They\n). For thecriteria,\nsustainable\nsupplier selection,\nthree\naspects\nwe\nshould take\nconsideration.\nenvironmental\ncriteria and\nsocial\ncriteria.\nshould take into consideration. They are economic criteria, environmental criteria and social criteria.\nStep 2: determine the weights for the evaluation criteria of sustainable suppliers\nStep 2: determine the weights for the evaluation criteria of sustainable suppliers\nExperts have their own individual experience and knowledge. Therefore, they may have different\nExperts have their own individual experience and knowledge. Therefore, they may have\ncognitive vagueness for alternatives and criteria. Let us assume a judgment set P = { p1 , p2 , · · · , ph }\ndifferent cognitive vagueness for alternatives and criteria. Let us assume a judgment set\nwith h ordered judgments, in the manner of p1 ≤ p2 ≤ · · · ≤ ph . Let pi be a random judgment in the\np1 ≤ approximation\np2 ≤≤ ph . Let\nP =P and\np1, pd2is,\n, ph as\nwith\norderedof\njudgments,\nof lower\na\nset\ndefined\nthehdistance\nP, where din=the\nphmanner\n− p1 . The\nApr (ppii ) be\nand\nthe upper approximation Apr ( pi ) of the judgment pi can be identified as follows.\nrandom judgment in the set P and d is defined as the distance of P , where d = ph − p1 . The\nLower approximation set:\n{\n}\nlower approximation\nApr( pi ) and the \bupper approximation Apr\n
( pi ) of the judgment pi can\nApr ( pi ) = ∪ p j ∈ P p j ≤ pi , pi − p j ≤ d\nbe identified as follows\nLower\nUpper approximation\napproximation set:\nset:\n{\n
≤d\nApr\n( (ppi ))==∪∪\bpp j ∈\nj ≤p p\nj)d\nApr\n∈P\nP |pp ≥\n, i ,p( p−i −p p ≤\ni\nUpper approximation set:\nj\nj\ni\ni\nh\nRN ( pi ) = piL , pU\ni\n{\nj\ni\nApr ( pi ) = ∪ p j ∈ P | p j ≥ pi , ( p j − pi ) ≤ d\n(1)\n}\n}\n(1)\n(2)\n(3)\n(2)\nSustainability 2018, 10, 2622\n6 of 20\nq\n
\nWhere piL = m ∏ xij\nq\n
\nn\npU\n=\n∏ yij\ni\n(4)\n(5)\nwhere xij and yij are the elements of the lower approximation set Apr ( pi ) and the upper approximation\nset Apr ( pi ) of pi respectively and m and n are the number of elements in the two sets respectively.\nFor different criteria, experts might give different weights. Use wkj indicate the weight of jth\ncriterion with kth expert.\nWith the Formulas (1)–(5)\nn\no\nn\nd j = MAX wm\n(6)\nj − wj\n\n\no\n \nn\nm\nn\n≤ dj\n(7)\n= ∪ wnj ∈ P wnj ≤ wm\nApr wm\nj , wj − wj\nj\n\n\no\n \nn\nn\nm\n≤ dj\n(8)\nApr wm\n= ∪ wnj ∈ P wnj ≥ wm\nj , wj − wj\nj\nq\n \n
\nLim wkj = m ∏ x j\n(9)\n  q\n
\nLim wkj = n ∏ y j\n(10)\nwhere x j and y j are the elements of the lower approximation set Apr (wkj ) and the upper approximation\nset Apr (wkj ) of wkj respectively and m and n are the number of elements in the two sets respectively.\n  h\n \n i h\ni\nkU\nRN wkj = Lim wkj , Lim wkj\n= wkL\n,\nw\nj\nj\ns\nw jL =\ns\n(11)\ns\n∏ wkL\nj\n(12)\nk =1\ns\ns\ns\nwU\n∏ wkU\nj =\nj\n(13)\nk =1\nh\ni\nWe could get the weight of each criterion w j = w jL , wU\nj .\n3.2. Evaluate the Sustainable Suppliers with the Proposed Rough ELECTRE\nStep 1: Construct the rough decision matrix\nApart from the decision for the weight of criteria, the experts should give the assessment of the\nalternatives with consideration of all the criteria. Let’s use rijk to represent the kth expert scores on jth\ncriterion in ith alternative. The following is the scoring matrix. Aggregate all the scoring matrix.\n\nk\nr11\n k\n r21\nRk = \n ..\n .\nk\nrm1\nk\nr12\nk\nr22\n..\n.\nk\nrm2\n···\n···\n..\n.\n···\n\nk\nr1n\n\nk\nr2n\n\n.. \n\n. \nk\nrmn\n(14)\n\n\nrf\nrf\n· · · rf\n11\n12\n1n\n f f\n\n r21 r22 · · · rf\n2n \n\ne\nR= .\n.. \n..\n..\n\n.\n.\n. \n ..\nrf\nnm\nm2 · · · rg\nm1 rf\nn\no\nreij = rij1 , rij2 , · · · , rijh\n(15)\n(16)\nSustainability 2018, 10, 2622\n7 of 20\nDetermine the rough matrix with expert ratings.\n\n\nd = max rijm − rijn\n(17)\n\n\no\n \nn\nApr rijm = ∪ rijn ∈ P rijn ≤ rijm , rijm − rijn ≤ d\n\n\no\n \nn\nApr rijm = ∪ rijn ∈ P rijn ≥ rijm , rijn − rijm ≤ d\n(18)\nq\n \nLim rijk = m\n∏ xij\n(19)\n
\n(20)\n  q\n
\nLim rijk = n ∏ yij\n(21)\nwhere xij and yij are the elements of the lower approximation set Apr (rijk ) and the upper approximation\nset Apr (rijk ) of rijk respectively and m and n are the number of elements in the two sets respectively.\ni\n  \n h\nRN rijk = Lim, Lim = rijkL , rijkU\n(22)\ni h\ni\nh\nio\n
nh\nRN reij = rij1L , rij1U , rij2L , rij2U , · · · , rijsL , rijsU\ni\n
h\nRN reij = rijL , rijU\n(23)\ns\nrijL =\ns\ns\ns\n∏\n \n\nL , rU\nr11\n11\n \n\n\nL , rU\n r21\n21\nR=\n\n..\n\n.\n\n L U \nrm1 , rm1\ns\n∏ rijkU\nrijkL , rijU =\ns\n L U\nr12 , r12\n L U\nr22 , r22\n···\nk =1\n..\n.\n L U \nrm2 , rm2\n(24)\n(25)\nk =1\n···\n..\n.\n···\n L U\nr1n , r1n\n L U\nr2n , r2n\n..\n.\n L U \nrmn , rmn\n\n\n\n\n\n\n\n\n(26)\nThen, we normalize the rough decision matrix with the weight of criteria.\nh\ni h\ni\nL U\nsij = rij · w j = rijL w jL , rijU wU\nij = sij , sij\n(27)\n L U#\nh\ni\nsij sij\ntij =\n,\n= tijL , tU\nij\nCj Cj\nn o\nWhere Cj = MAX sU\nij\n \n\nL , tU\nt11\n11\n \n\n\nL , tU\n t21\n21\nT=\n\n..\n\n.\n\n L U \ntm1 , tm1\n L U\nt12 , t12\n L U\nt22 , t22\n···\n..\n.\n L U \ntm2 , tm2\n..\n···\n.\n···\n L U\nt1n , t1n\n L U\nt2n , t2n\n..\n.\n L U \ntmn , tmn\n(28)\n(29)\n\n\n\n\n\n\n\n\n(30)\nSustainability 2018, 10, 2622\n8 of 20\nStep 2: Construct the rough concordance matrix and discordance matrix\nIn this step, we construct some field for the comparison among all the alternatives. We compare\ndifferent alternatives in two aspects. One is the concordance and the other is the discordance. Construct\nthe concordance and discordance matrices.\n\b\nCS pq = Fj t pj ≥ tqj\n(31)\n\b\nDS pq = Fj t pj < tqj\n(32)\nCS pq represents the areas that alternative p is better than alternative q and DS pq represents the\nareas that alternative p is worse than alternative q.\nc pq =\n∑\nwj\n(33)\nFj ∈CS pq\n
\nmax Fj ∈ DS pq d t pj , tqj\n
\nd pq =\nmax Fj ∈ J d t pj , tqj\n\n L U\n L U  \n−\nc12 , c12\n· · · c1m\n, c1m\n \n \n\n\n\n\nL , cU\nL , cU\n−\n·\n·\n·\nc\n c21\n2m 2m \n21\n\nC=\n\n\n..\n..\n..\n..\n\n\n.\n.\n.\n.\n\n\n L U   L U \n···\n−\ncm1 , cm1\ncm2 , cm2\n\n\n− d12 · · · d1m\n\n− · · · d2m \n d21\n\n\nD= .\n..\n.. \n..\n\n.\n.\n.\n. \n .\ndm1 dm2 · · · −\n(34)\n(35)\n(36)\nBy means of the calculation, we could get the rough concordance matrix C and discordance\nmatrix D.\nStep 3: Determine the general Boolean matrix\nAfter we get the concordance matrix and discordance matrix, we should determine the threshold\nvalue. Using it to transform the matrix into Boolean matrix. First, we calculate the mean of the all\nfactors in matrix C and matrix D.\nm\nm\n∑\n∑ c pq\nc=\np=1,p6=q q=1,q6= p\nm\n∑\nd=\n(37)\nm ( m − 1)\nm\n∑\np=1,p6=q q=1,q6= p\nd pq\nm ( m − 1)\n(38)\nCompare the factors in matrix C with c and the factors in matrix D with d. According the result of\nthe comparison, we get the concordance Boolean matrix F and discordance Boolean matrix G.\n(\nf pq =\n(\ng pq =\n1\ni f : c pq ≥ c\n0\ni f : c pq < c\n1\ni f : d pq ≤ d\n0\ni f : d pq > d\n(39)\n(40)\nSustainability 2018, 10, 2622\n9 of 20\n \n \nF = f pq m×m , G = g pq m×m\n(41)\nThen we could construct the general Boolean matrix H.\nh pq = f pq · g pq\n(42)\n \nH = h pq m×m\n(43)\nAccording to the above calculations, we could get the general Boolean matrix. It is a basis for the\nranking of the alternatives. If h pq = 1, that means alternative p is better than alternative q.\nStep 4: Calculate the pure concordance index and discordance index\nBy the Boolean general matrix, we could get part relations between all alternatives. Since if\nh pq = 1, we know that alternative p is better than alternative q. But if h pq = 0 and we could not infer\nthe relationship of alternative p and alternative q from other alternatives, then we do not know which\nis better. In order to get a rank of all the alternatives, we bring into pure concordance index cˆi and\ndiscordance index d̂i .\nBefore calculating the pure index, we should transform rough interval into definite number.\nSong et al. (2017) has proposed this method. We use ∆−1 represents the calculation of changing rough\ninterval into definite number [14].\nThe calculation includes the following procedures.\n(1) Normalization\n\n\nzei L =\nzei U =\nziL − minziL /∆max\nmin\n(44)\ni\n\nL\nzU\ni − minzi\ni\n\n/∆max\nmin\n(45)\nU\nL\n∆max\nmin = maxzi − minzi\ni\n(46)\ni\nL\nwhere ziL and zU\ni are the lower limit and the upper limit of the rough number zei respectively; zei and\nzei U are the normalized form of ziL and zU\ni respectively.\n(2) Determine the total normalized definite value by\n\n\nzei L × 1 − zei L + zei U × zei U\nβi =\n(47)\n1 − zei L + zei U\n(3) Compute the final definite value form zei der for zei by\nzei der = minziL + β i ∆max\nmin\n(48)\ni\nTherefore, we can use this method to calculate the concordance index and discordance index.\nm\ncˆi =\n∑\nq=1,q6=i\n
\n∆−1 cf\niq −\nm\nd̂i =\n∑\nq=1,q6=i\nm\n∑\np=1,p6=i\n∆−1 cf\npi\n
\n(49)\nm\n
\ndiq −\n∑\nd pi\n
\n(50)\np=1,p6=i\nStep 5: Determine the final ranking\nAccording to the cˆi , we can get a priority in concordance. The bigger value of cˆi the higher place\nthe alternative would get. We use R1i for the ranking in concordance. The same we can get the priority\nSustainability 2018, 10, 2622\n10 of 20\nin discordance by d̂i . But on the contrary, the smaller value of d̂i the higher place the alternative would\nget. We use R2i for the ranking in discordance. The final ranking is calculated as follows:\nRi =\nR1i + R2i\n2\n(51)\nRi is the final rank of all the alternatives.\n4. Case Study\nIn this section, in order to validate the applicability and effectiveness of the proposed method, we\nuse an example to illustrate. We assume that there is a manufacturing company. For the purpose of\nchoosing a good supplier, they set up a panel of 4 experts. The experts come from various departments\nincluding purchasing, quality and production and planning who are involved in the supplier selection\nprocess. And there are 8 suppliers for selection.\n4.1. Implementation\n4.1.1. Determine the Supplier Evaluation Criteria and Their Weights\nStep 1: determine the evaluation criteria of sustainable suppliers\nFirst of all, the experts make a decision of the criteria. In addition to economic criteria,\nenvironmental criteria and social criteria should also be considered for the sustainable supplier\nselection. These criteria consist of three parts, we use C1~10 to represent these ten criteria. They are\nEconomic criteria including quality (C1), response (C2) and cost (C3); Environmental criteria including\nenvironmental management system (C4), carbon emission & resource consumption (C5), design for\nthe environment (C6), Green image (C7); Social criteria including product liability (C8), employee right\nand welfare (C9), social commitment (C10). The detailed introduction is shown in Table 1. We use\nA1~8 to represent alternatives, E1~4 to represent experts.\nStep 2: determine the weights for the evaluation criteria of sustainable suppliers\nAfter the decision of criteria, experts should evaluate the weight of each criterion. The experts\ngive their evaluation to the criteria in the Table 2. Firstly, we convert the grades which experts give to\ncriteria into rough number. Take criterion C1 for example.\nTable 2. The grade of each criterion.\nC1\nC2\nC3\nC4\nC5\nC6\nC7\nC8\nC9\nC10\nE1\nE2\nE3\nE4\n4\n3\n6\n5\n6\n6\n4\n4\n6\n7\n5\n6\n7\n5\n4\n6\n4\n3\n6\n4\n4\n4\n5\n5\n6\n5\n3\n2\n6\n5\n6\n4\n7\n6\n5\n5\n5\n4\n7\n4\nAccording to the Equations (6)–(13) in Section 3,\nd1 = 2\n
\n
\nApr w11 = {4, 4}, Apr w11 = {4, 5, 4, 6}\n
\n
\nApr w12 = {4, 5, 4}, Apr w12 = {5, 6}\nSustainability 2018, 10, 2622\n11 of 20\n
\n
\nApr w13 = {4, 4}, Apr w13 = {4, 5, 4, 6}\n
\n
\nApr w14 = {4, 5, 4, 6}, Apr w14 = {6}\n
√\n
√\nLim w11 = 2 4 × 4 = 4, Lim w11 = 4 4 × 5 × 4 × 6 = 4.68\n
√\n
√\nLim w12 = 3 4 × 5 × 4 = 4.31, Lim w12 = 2 5 × 6 = 5.48\n√\n√\n
\n
\nLim w13 = 2 4 × 4 = 4, Lim w13 = 4 4 × 5 × 4 × 6 = 4.68\n
√\n
\nLim w14 = 4 4 × 5 × 4 × 6 = 4.68, Lim w14 = 6\n√\n√\nw1L = 4 4 × 4.31 × 4 × 4.68 = 4.24, w1U = 4 4.68 × 5.48 × 4.68 × 6 = 5.18\nThe same as the other criteria, following the same procedure, we can get the importance degree of\nall the criteria in Table 3.\nTable 3. The importance of all the criteria.\nRough Importance\nW1\nW2\nW3\nW4\nW5\nW6\nW7\nW8\nW9\nW10\n[4.24, 5.18]\n[3.57, 4.77]\n[5.69, 6.70]\n[5.06, 5.42]\n[4.68, 5.70]\n[5.23, 5.73]\n[3.53, 4.37]\n[2.63, 3.67]\n[6.06, 6.42]\n[4.28, 5.60]\n4.1.2. Evaluate the Sustainable Suppliers with the Proposed Rough ELECTRE\nStep 1: Construct the rough decision matrix\nDifferent expert might hold different view for alternatives and criteria because of their personal\nexperience and knowledge. And the true information is just contained in the cognitive vagueness.\nAccording to the evaluation towards the alternatives from the experts, we could get the rough number\nof each alternative. We take the data for criterion 1 in Table 4 for example.\nTable 4. The evaluation for alternative under the criterion 1.\nC1\nA1\nA2\nA3\nA4\nA5\nA6\nA7\nA8\nE1\nE2\nE3\nE4\n6\n4\n5\n4\n3\n6\n7\n5\n4\n3\n4\n5\n5\n6\n6\n4\n6\n4\n6\n5\n3\n4\n5\n3\n5\n2\n3\n5\n4\n6\n7\n5\nAccording to the Equations (17)–(26), we use x cab for the cth expert’s evaluation towards alternative\nb in criterion a. We can get the rough matrix in Table 5.\nSustainability 2018, 10, 2622\n12 of 20\nTable 5. The rough matrix.\nA1\nA2\nA3\nA4\nA5\nA6\nA7\nA8\nC1\nC2\nC3\n…\nC10\n[4.68, 5.70]\n[2.63, 3.67]\n[3.65, 5.15]\n[4.53, 4.93]\n[3.23, 4.16]\n[5.02, 5.85]\n[5.69, 6.70]\n[3.66, 4.69]\n[5.23, 5.73]\n[3.66, 4.69]\n[2.22, 3.13]\n[5.54, 5.93]\n[4.54, 5.38]\n[5.69, 6.70]\n[4.68, 5.70]\n[4.06, 4.41]\n[4.24, 5.18]\n[5.11, 5.79]\n[4.68, 5.70]\n[3.23, 4.16]\n[4.24, 5.18]\n[6.06, 6.42]\n[5.23, 5.73]\n[3.53, 4.37]\n…\n…\n…\n…\n…\n…\n…\n…\n[3.66, 4.69]\n[4.68, 5.70]\n[4.67, 6.17]\n[4.68, 5.70]\n[3.96, 5.29]\n[4.68, 5.70]\n[5.02, 5.85]\n[4.24, 5.18]\nNote: not all of the data are provided in Table 5 due to the space limitation.\nThen, we normalize the rough matrix. According to the Equations (27)–(30). We can get the result\nin Table 6.\nTable 6. The normalized weighted decision matrix.\nA1\nA2\nA3\nA4\nA5\nA6\nA7\nA8\nC1\nC2\nC3\n…\nC10\n[0.57, 0.85]\n[0.32, 0.55]\n[0.45, 0.77]\n[0.55, 0.74]\n[0.39, 0.62]\n[0.61, 0.87]\n[0.69, 1.00]\n[0.45, 0.70]\n[0.58, 0.86]\n[0.41, 0.70]\n[0.25, 0.47]\n[0.62, 0.89]\n[0.51, 0.80]\n[0.64, 1.00]\n[0.52, 0.85]\n[0.45, 0.66]\n[0.56, 0.81]\n[0.68, 0.90]\n[0.62, 0.89]\n[0.43, 0.65]\n[0.56, 0.81]\n[0.80, 1.00]\n[0.69, 0.89]\n[0.47, 0.68]\n…\n…\n…\n…\n…\n…\n…\n…\n[0.45, 0.76]\n[0.58, 0.92]\n[0.58, 1.00]\n[0.58, 0.92]\n[0.49, 0.86]\n[0.58, 0.92]\n[0.62, 0.95]\n[0.53, 0.84]\nStep 2: Construct the rough concordance matrix and discordance matrix\nIn this step, we construct the concordance and discordance matrices according to the normalized\nrough decision matrix. For the construct of the concordance matrix, we take alternative1 and alternative\n2 for example. At the first, we should find in which criterion A1 performs better than A2, that means\nthe score in certain criterion, A1 is higher than A2.\nAccording to the Table 6, we could find in criterion 1, 2, 9, A1 performs better than A2. Add up all\nthese weights of the criteria. We could get the value of c12 = [13.87, 16.37] in the concordance matrix.\nAnd we can get the concordance matrix in Table 7 by repeat these procedures.\nTable 7. The concordance matrix.\nA1\nA2\nA3\nA4\nA5\nA6\nA7\nA8\nA1\nA2\nA3\n…\nA8\n[31.11, 37.19]\n[33.63, 39.23]\n[7.85, 10.37]\n[7.81, 9.97]\n[29.01, 35.71]\n[19.27, 22.89]\n[17.55, 21.08]\n[13.87, 16.37]\n[19.81, 22.93]\n[13.87, 16.37]\n[11.35, 14.32]\n[22.20, 26.74]\n[23.84, 28.67]\n[22.09, 26.44]\n[11.35, 14.32]\n[25.17, 30.63]\n[7.81, 9.95]\n[7.11, 9.14]\n[24.73, 30.11]\n[17.04, 21.03]\n[16.85, 20.25]\n…\n…\n…\n…\n…\n…\n…\n…\n[27.43, 32.48]\n[22.89, 27.12]\n[28.13, 33.30]\n[12.09, 15.55]\n[20.67, 25.25]\n[25.47, 31.34]\n[17.78, 22.25]\n-\nFor the construct of the discordance matrix. First of all, we find the criterion which A2 is better\nthan A1. And we could find that they are criterion 3, 4, 5, 6, 7, 8, 10. Then we find the biggest distance\nin these criteria. Using it divide the biggest distance between A1 and A2. We can get the value of\nd12 = 1. Repeating these procedures and we can get the discordance matrix in Table 8.\nSustainability 2018, 10, 2622\n13 of 20\nTable 8. The discordance matrix.\nA1\nA2\nA3\nA4\nA5\nA6\nA7\nA8\nA1\nA2\nA3\nA4\nA5\nA6\nA7\nA8\n0.85\n1.00\n1.00\n1.00\n0.46\n1.00\n0.93\n1.00\n0.64\n1.00\n1.00\n0.37\n0.71\n1.00\n0.53\n1.00\n0.73\n1.00\n0.43\n0.88\n0.91\n0.90\n0.49\n1.00\n1.00\n0.00\n0.37\n0.68\n0.20\n0.27\n0.64\n0.65\n0.22\n0.55\n0.32\n1.00\n1.00\n1.00\n1.00\n1.00\n1.00\n1.00\n0.88\n1.00\n1.00\n1.00\n1.00\n0.32\n1.00\n1.00\n0.97\n1.00\n1.00\n1.00\n0.27\n0.82\n-\nStep 3: Determine the general Boolean matrix\nBased on concordance and discordance matrix, we construct the concordance Boolean and\ndiscordance Boolean matrices. Calculate the concordance index and discordance index. Follow\nthe Equations (37)–(41).\nm\n∑\ncL =\nm\n∑\np=1,p6=q q=1,q6= p\nm\nc Lpq\nm ( m − 1)\n= 22.49, cU =\nm\n∑\nd=\nm\n∑\nm\n∑\np=1,p6=q q=1,q6= p\n∑\np=1,p6=q q=1,q6= p\ncU\npq\n= 26.78\nm ( m − 1)\nd pq\n= 0.79\nm ( m − 1)\nAnd we can get the concordance Boolean and discordance Boolean matrices in Tables 9 and 10.\nTable 9. The concordance Boolean matrix.\nA1\nA2\nA3\nA4\nA5\nA6\nA7\nA8\nA1\nA2\nA3\nA4\nA5\nA6\nA7\nA8\n1\n1\n0\n0\n1\n0\n0\n0\n0\n0\n0\n0\n1\n0\n0\n1\n0\n0\n1\n0\n0\n1\n1\n1\n1\n1\n1\n1\n1\n1\n1\n0\n1\n1\n1\n0\n1\n0\n0\n0\n0\n0\n1\n0\n1\n0\n0\n1\n1\n1\n1\n1\n0\n0\n1\n0\n-\nTable 10. The discordance Boolean matrix.\nA1\nA2\nA3\nA4\nA5\nA6\nA7\nA8\nA1\nA2\nA3\nA4\nA5\nA6\nA7\nA8\n0\n0\n0\n0\n1\n0\n0\n0\n1\n0\n0\n1\n1\n0\n1\n0\n1\n0\n1\n0\n0\n0\n1\n0\n0\n1\n1\n1\n1\n1\n1\n1\n1\n1\n1\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n1\n0\n0\n0\n0\n0\n0\n1\n0\n-\nAccording to the Equation (42), we could get the general matrix in Table 11.\nSustainability 2018, 10, 2622\n14 of 20\nTable 11. The general matrix.\nA1\nA2\nA3\nA4\nA5\nA6\nA7\nA8\nA1\nA2\nA3\nA4\nA5\nA6\nA7\nA8\n0\n0\n0\n0\n1\n0\n0\n0\n0\n0\n0\n0\n1\n0\n0\n0\n0\n0\n1\n0\n0\n0\n1\n0\n0\n1\n1\n1\n1\n1\n1\n0\n1\n1\n1\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n0\n1\n0\n0\n0\n0\n0\n0\n1\n0\n-\nAnd following\nthe PEER\ngeneral\nmatrix, we could draw the priority picture like Figure 2.\nSustainability\n2018, 10, x FOR\nREVIEW\n15 of 21\nFigure 2. The relations of alternatives in conventional ELECTRE.\nWe use\nuse ‘>’\n> A5;\nA2A2\n> {A4,\nA5};A5};\nA3 >A3\nA5;\n{A1,\nWe\n‘>’ indicating\nindicatingbetter,\nbetter,then\nthenwe\nwecould\ncouldfind\nfindthat\nthatA1\nA1\n> A5;\n> {A4,\n> A6\nA5;>A6\n>\nA2,\nA3,\nA4,\nA5,\nA7,\nA8};\nA7\n>\n{A2,\nA4,\nA5};\nA8\n>\n{A4,\nA5}.\nThat’s\nsome\nrelation\nbetween\nall\n{A1, A2, A3, A4, A5, A7, A8}; A7 > {A2, A4, A5}; A8 > {A4, A5}. That’s some relation between the\nall\nalternatives.\nthe\nalternatives.\nBut we\nwe cannot\ncannot have\nhave aa rank\nrank of\nof all\nall the\nthe alternatives\nalternatives just\njust though\nthough this\nthis figure.\nfigure. Like\nLike we\nwe do\ndo not\nnot know\nknow is\nis\nBut\nA4 better\nbetter than\nthan A5\nA5 or\nor A5\nA5 better\nbetter than\nthan A4\nA4 or\nor they\nthey are\nare the\nthe same.\nsame. So,\nSo, we\nwe bring\nbring in\nin the\nthe concept\nconcept of\nof the\nthe pure\npure\nA4\nconcordance\nindex\nand\ndiscordance\nindex.\nconcordance index and discordance index.\nStep 4: Calculate the pure concordance index and discordance index\nStep 4: Calculate the pure concordance index and discordance index\nBefore we calculate the pure concordance index and discordance index, we should convert the\nBefore we calculate the pure concordance index and discordance index, we should convert the\nrough concordance matrix into definite number matrix. According to the Equations (44)–(48). We can\nrough concordance matrix into definite number matrix. According to the Equations (44)–(48). We can\nget the result in Table 12.\nget the result in Table 12.\nTable 12. The definite number concordance matrix.\nA1\nA2\nA3\nA4\nA5\nA6\nA7\nA8\nA1\n35.87\n38.39\n8.04\n7.95\n33.91\n20.83\n18.89\nA2\n14.50\n21.58\n14.50\n11.78\n25.40\n27.62\n25.12\nA3\n12.16\n29.60\n8.05\n7.27\n29.01\n19.06\n18.51\nA4\n40.77\n33.42\n41.08\n27.71\n46.10\n29.21\n35.07\nA5\n42.17\n37.34\n43.11\n18.63\n41.68\n31.53\n25.65\nA6\n17.01\n26.20\n22.63\n4.35\n9.09\n15.08\n21.43\nA7\n28.50\n22.47\n30.90\n17.60\n16.55\n35.78\n29.69\nA8\n31.35\n25.39\n32.29\n12.58\n23.00\n29.64\n19.55\n-\nSustainability 2018, 10, 2622\n15 of 20\nTable 12. The definite number concordance matrix.\nA1\nA2\nA3\nA4\nA5\nA6\nA7\nA8\nA1\nA2\nA3\nA4\nA5\nA6\nA7\nA8\n35.87\n38.39\n8.04\n7.95\n33.91\n20.83\n18.89\n14.50\n21.58\n14.50\n11.78\n25.40\n27.62\n25.12\n12.16\n29.60\n8.05\n7.27\n29.01\n19.06\n18.51\n40.77\n33.42\n41.08\n27.71\n46.10\n29.21\n35.07\n42.17\n37.34\n43.11\n18.63\n41.68\n31.53\n25.65\n17.01\n26.20\n22.63\n4.35\n9.09\n15.08\n21.43\n28.50\n22.47\n30.90\n17.60\n16.55\n35.78\n29.69\n31.35\n25.39\n32.29\n12.58\n23.00\n29.64\n19.55\n-\nThen we could calculate the pure concordance index and discordance index of each alternative\nand the result is in Table 13.\nTable 13. The pure concordance index and discordance index of each supplier.\nA1\nA2\nA3\nA4\nA5\nA6\nA7\nA8\ncˆi\nd̂i\n22.58\n69.80\n106.31\n−169.60\n−136.74\n125.73\n−18.63\n0.57\n−0.75\n−0.15\n0.81\n1.93\n4.16\n−4.92\n−0.87\n−0.21\nStep 5: Determine the final ranking\nAccording to the pure concordance index and discordance index, we could get the ranking of\neach supplier in concordance and discordance aspects. With the Equation (51), we could get the final\nranking of all the suppliers in Table 14.\nTable 14. The final ranking of all the suppliers.\nA1\nA2\nA3\nA4\nA5\nA6\nA7\nA8\nR1i\nR2i\nRi\n4\n3\n2\n8\n7\n1\n6\n5\n3\n5\n6\n7\n8\n1\n2\n4\n2\n3\n3\n7\n7\n1\n3\n6\nFrom Table 14, we could see that priority is: A6 > A1 > {A2, A3, A7} > A8 > {A4, A5}.\n4.2. Comparisons and Discussion\nTo further validate the effectiveness and strengths of the approach proposed in this paper, we make\na comparison analysis.\nThe comparison is conducted between the modified ELECTRE method with rough number (the\nrough ELECTRE), fuzzy number (the fuzzy ELECTRE) and crisp number (the conventional ELECTRE).\nThe results are presented in Figure 3. From the Figure 3, we can see the rank of A2, A3 and A7 are\ndifferent with each other in the three methods. In the process of supplier selection, the top three\ncandidates are critical for the consideration. Different rankings will influence in the final performance\nof supply chain.\nSustainability 2018, 10, 2622\nSustainability 2018, 10, x FOR PEER REVIEW\nSustainability 2018, 10, x FOR PEER REVIEW\n16 of 20\n17 of 21\n17 of 21\nFigure 3.\n3. The\nThe rank\nrank of\nof different\ndifferent methods.\nmethods.\nFigure\nFigure 3. The rank of different methods.\nThe\nThe fuzzy\nfuzzy methods\nmethods of\nof supplier\nsupplier selection\nselection (e.g.,\n(e.g., methods\nmethods in\nin [39,41])\n[39,41]) often\noften use\nuse the\nthe fuzzy\nfuzzy number\nnumber\nwith\nfixed\ninterval\nto\ndeal\nwith\nthe\nuncertainty\nin\nsupplier\nselection,\nwhich\nwill\ncause\ninformation\nwith\nuncertainty\nin in\nsupplier\nselection,\nwhich\nwillwill\ncause\ninformation\nlost\nwith fixed\nfixedinterval\nintervaltotodeal\ndealwith\nwiththe\nthe\nuncertainty\nsupplier\nselection,\nwhich\ncause\ninformation\nlost\nin\nprocess.\nDifferent\nthe\nmethods,\nthe\napproach\nuses\nthe\nin\ndecision–making\nprocess.\nDifferent\nwith with\nthe\nmethods,\nthe proposed\napproach\nuses the\nrough\nlost\nin decision–making\ndecision–making\nprocess.\nDifferent\nwithfuzzy\nthe fuzzy\nfuzzy\nmethods,\nthe proposed\nproposed\napproach\nuses\nthe\nrough\nnumber\nwith\nflexible\ninterval\nto\ndescribe\nthe\nuncertainty\nand\nit\ndoes\nnot\nrequire\nto\nsubjectively\nnumber\nwith\nflexible\ninterval\nto\ndescribe\nthe\nuncertainty\nand\nit\ndoes\nnot\nrequire\nto\nsubjectively\nset\nrough number with flexible interval to describe the uncertainty and it does not require to subjectively\nset\nthe\nfuzzy\nmembership\nfunction\nin\nadvance.\nThe\nrough\nnumber\ncan\nflexibly\nreflect\nthe\nchange\nof\nthe\nfuzzy\nmembership\nfunction\nin advance.\nTheThe\nrough\nnumber\ncan can\nflexibly\nreflect\nthe change\nof the\nset the\nfuzzy\nmembership\nfunction\nin advance.\nrough\nnumber\nflexibly\nreflect\nthe change\nof\nthe\nexperts’\npreference.\nFor\nexample,\nif\none\nexpert\nprovides\nthe\nscores\nof\n6,\n4,\n6,\n5.\nIt\nthen\ncan\nbe\nexperts’\npreference.\nFor example,\nif one if\nexpert\nprovides\nthe scores\n6, 4, 6,of5. 6,It 4,\nthen\ncanItbe\nconverted\nthe experts’\npreference.\nFor example,\none expert\nprovides\ntheofscores\n6, 5.\nthen\ncan be\nconverted\nto\n[4,\nwhich\nhave\nof\nto\nfuzzy intervals\nofintervals\n[5, 7], [3, of\n5],[5,\n[5,7],\n7] [3,\nand\n[4,[5,\n6],7]\nalland\nof which\nhave\ninterval\nof 2. interval\nBut\nthe proposed\nconverted\nto fuzzy\nfuzzy\nintervals\nof\n[5,\n7],\n[3, 5],\n5],\n[5,\n7]\nand\n[4, 6],\n6], all\nall of\noffixed\nwhich\nhave fixed\nfixed\ninterval\nof 2.\n2. But\nBut\nthe\nproposed\napproach\ntransforms\nthe\noriginal\nscores\ninto\nthe\nflexible\nrough\nintervals\nof\n[5.18,\n6],\napproach\ntransforms\nthetransforms\noriginal scores\ninto thescores\nflexible\nrough\nintervalsrough\nof [5.18,\n6], [4, of\n5.18],\n[5.18,\n6]\nthe proposed\napproach\nthe original\ninto\nthe flexible\nintervals\n[5.18,\n6], [4,\n[4,\n5.18],\n[5.18,\n6]\n[4.47,\n5.65],\nwhich\nare\n4.\nchange\nevaluations\nand\nwhich\nshown\nin Figure\n4. Ifin\ntheFigure\nexperts\nchange\ntheir evaluations\n3, 4, 6, 4,\n5.18],[4.47,\n[5.18,5.65],\n6] and\nand\n[4.47,are\n5.65],\nwhich\nare shown\nshown\nin\nFigure\n4. If\nIf the\nthe experts\nexperts\nchange their\ntheirinto\nevaluations\ninto\n3,\n4,\nwill\ninto\n[5,\n[3,\n5],\nthe\nELECTRE\nthe\nwill intervals\nchange\ninto\n[2,change\n4], [3, 5],\n[5,[2,\n7],4],\n[3,[3,\n5],5],\nwhile\nthe\nELECTRE\ntransform\nthe\nintofuzzy\n3, 4,\n4, 6,\n6,intervals\n4, the\nthe fuzzy\nfuzzy\nintervals\nwill\nchange\ninto\n[2,\n4],\n[3,\n5],\n[5, 7],\n7],\n[3,rough\n5], while\nwhile\nthe rough\nrough\nELECTRE\ntransform\nthe\noriginal\nscores\ninto\n[3,\n4.12],\n[3.63,\n4.58],\n[4.12,\n6]\nand\n[3.36,\n4.58].\nObviously,\nthe\noriginal\nscores\n[3, 4.12],\n[3.63,\n4.58],\nand [3.36,\nthe boundary\nof the fuzzy\ntransform\nthe into\noriginal\nscores\ninto\n[3, [4.12,\n4.12], 6][3.63,\n4.58], 4.58].\n[4.12,Obviously,\n6] and [3.36,\n4.58]. Obviously,\nthe\nboundary\nof\nthe\nfuzzy\ninterval\nhas\nno\nalteration\nwith\nthe\nchange\nof\nthe\nexperts’\nchange\nin\nthe\nfuzzy\ninterval\nhas\nno\nalteration\nwith\nthe\nchange\nof\nthe\nexperts’\nchange\nin\nthe\nfuzzy\nELECTRE.\nOn\nthe\nother\nboundary of the fuzzy interval has no alteration with the change of the experts’ change in the fuzzy\nELECTRE.\nOn\nother\nELECTRE\ncan\nidentify\nthe\nof\npreferences,\nhand,\nthe rough\nELECTRE\ncan the\nidentify\nthe\nchanges of\nexpert\npreferences,\nwhich\nwill make\nthe final\nELECTRE.\nOn the\nthe\nother hand,\nhand,\nthe rough\nrough\nELECTRE\ncan\nidentify\nthe changes\nchanges\nof expert\nexpert\npreferences,\nwhich\nwill\nmake\nthe\nfinal\nranking\nmore\naccurate\nand\nreasonable.\nranking\nmore\naccurate\nandranking\nreasonable.\nwhich will\nmake\nthe final\nmore accurate and reasonable.\nFigure 4. Different vagueness manipulations for judgements on alternative one of criterion one.\nFigure 4. Different vagueness manipulations for judgements on alternative one\none of\nof criterion\ncriterion one.\none.\nMoreover,\nMoreover, compared\ncompared with\nwith the\nthe traditional\ntraditional ELECTRE\nELECTRE method\nmethod (e.g.,\n(e.g., the\nthe ELECTRE\nELECTRE method\nmethod used\nused\nby\nby Bırgün\nBırgün and\nand Cıhan\nCıhan (2010)\n(2010) [40]),\n[40]), the\nthe proposed\nproposed method\nmethod provides\nprovides the\nthe rank\nrank of\nof all\nall the\nthe alternatives.\nalternatives. In\nIn\nthe\nthe traditional\ntraditional ELECTRE\nELECTRE method,\nmethod, we\nwe can\ncan only\nonly get\nget partial\npartial relationships\nrelationships among\namong alternatives.\nalternatives. This\nThis will\nwill\nSustainability 2018, 10, 2622\n17 of 20\nMoreover, compared with the traditional ELECTRE method (e.g., the ELECTRE method used by\nBırgün and Cıhan (2010) [40]), the proposed method provides the rank of all the alternatives. In the\ntraditional ELECTRE method, we can only get partial relationships among alternatives. This will\nhinder the managers to directly identify the best supplier. As shown in Figure 2, there is no direct or\nindirect relationship between A7 and A3, so we do not know whether A7 performs better than A3 or\nnot. However, we can get all the relationships in the proposed method based on the calculation of\npure concordance index and discordance index. All the ranks of the suppliers can be provided in the\nproposed approach. This is obviously more practical and reasonable than the conventional ELECTRE\nmethod (e.g., the ELECTRE method in [40]). Moreover, different with most of AHP/ANP-based\nmethods [29,30] and DEA approaches [31,32], the proposed rough ELECTRE method considers\nthe uncertainty in decision–making process, which makes the final ranking results of suppliers\nmore accurate.\nTheoretically, this study develops a rough multi-criteria decision-making approach for sustainable\nsupplier selection considering vagueness and subjectivity. The novel approach integrates the strength\nof rough set theory in handling vagueness without much priori information and the merit of ELECTRE\nin modeling multi-criteria decision-making problem. The comparisons between the proposed method,\nthe conventional ELECTRE and the fuzzy ELECTRE reveal that the rough ELECTRE performs better\nthan the conventional ELECTRE and the fuzzy ELECTRE in dealing with vague and imprecise\ninformation. Besides, this research contributes to modeling the problem of supplier selection based on\nthe economic, environmental and social aspects. The social aspects are often omitted in the previous\nsupplier selection methods. Practically, this method provides an effective method to identify the\nright suppliers to achieve the success of the sustainable supply chain management. It also provides a\nstandardized procedure for managers in sustainable supplier selection.\n5. Conclusions\nTo manipulate the vagueness in sustainable supplier selection, a new approach based on the rough\nset theory and ELECTRE is developed in this paper. The novel approach integrates both the strength\nof rough set theory in handling vagueness and the merit of ELECTRE in modeling multi-criteria\ndecision-making problem. A case study of sustainable supplier selection for solar air-conditioner\nmanufacturer is provided to demonstrate the application and potential of the approach. In sum, this\nproposed method has the following features:\nFirst, this study considers the social sustainability in the supplier selection, which is often omitted\nin the previous literature. This research contributes to modeling the problem of supplier selection\ndecision within the context of a sustainable supply chain management based on the Triple Bottom Line\n(TBL) concept (economic, environmental and social aspects). The sustainability criteria in this study\nare generic and can be used for sustainable supplier selection in different industries.\nSecond, the proposed rough ELECTRE method can flexibly reflect the uncertainty in\ndecision–making without much priori information. Different with the previous fuzzy methods,\nthe proposed approach utilizes the lower and upper approximations to describe uncertainty and\nit does not require the pre-set fuzzy membership function, which will reduce the decision–making\nburdens of managers.\nThird, the proposed approach can identify the preference changes of decision makers with flexible\nrough intervals. Due to the flexible uncertainty mechanism, the rough number is more sensitive than\nfuzzy number to the preference changes of decision makers, which makes the final ranking results\nmore accurate.\nFourth, different with the conventional ELECTRE revealing partial ranking orders, the proposed\nrough ELECTRE method can provide full ranking order of all the alternatives. This is especially useful\nfor managers to get a comprehensive view of suppliers and make reasonable decision–making in\nsupplier selection.\nSustainability 2018, 10, 2622\n18 of 20\nAlthough the proposed method has some merits in sustainable supplier selection, it also has\nseveral ameliorable aspects which may serve as implications for further study. To make the ranking\nresults more accurate, it would be favorable for future research to take decision makers’ weighs,\nobjective criteria weights and subjective criteria weights into consideration. To handle huge number\nof suppliers, the proposed rough ELECTRE method will be integrated with DEA method with little\nmanagerial input and output required. Moreover, a computerized tool based on the proposed approach\nwill be developed to reduce the computation burdens of managers. Besides, more testing work is\nnecessitated to gain external validity.\nAuthor Contributions: Conceptualization, S.J. and W.S.; Methodology, W.S.; Validation, H.L. and S.J.; Resources,\nX.M.; Writing-Original Draft Preparation, H.L.; Writing-Review & Editing, X.M.\nFunding: The work described in this paper was supported by the National Natural Science Foundation of China\n(Grant No. 71501006 and 71632003), the Technical Research Foundation (JSZL2016601A004), the Open Project of\nHenan Key Laboratory of Intelligent Manufacturing of Mechanical Equipment, Zhengzhou University of Light\nIndustry (No. IM201801), and the Fundamental Research Funds for the Central Universities.\nAcknowledgments: The authors would like to thank the editor and the anonymous reviewers for their helpful\ncomments and suggestions on the drafts of this paper.\nConflicts of Interest: The authors declare no conflict of interest.\nAbbreviations\nThe following abbreviations are used in this manuscript:\nDi\nCi\nEi\nAi\nP\npi\nApr ( pi )\nApr ( pi )\nd\nRN ( pi )\npiL\npU\ni\nxij\nyij\nwkj\n \nLim wkj\n \nLim wkj\nith decision-maker\nith criterion for supplier selection\nith expert\nith alternative\njudgement set\njudgement of ith expert\nlower approximation set of pi , which contains the elements that smaller than pi in set P\nupper approximation set of pi , which contains the elements that bigger than pi in set P\nmaximum distance of set P\nrough interval corresponding to pi\nlower approximation of pi\nupper approximation of pi\nelements of Apr ( pi )\nelements of Apr ( pi )\nweight of jth criterion with kth expert\nlower approximation of wkj\nupper approximation of wkj\nw jL\nlower approximation of the weight of jth criterion\nwU\nj\nupper approximation of the weight of jth criterion\nrijk\nRk\nreij\nkth expert’s judgement on jth criterion in ith alternative\nscoring matrix for kth expert\nSet of rijk\ntij\nT\nC\nD\nc\nd\nF\nG\nH\nrough interval corresponding to reij after normalized\nrough scoring matrix\nconcordance matrix\ndiscordance matrix\nconcordance index\ndiscordance index\nconcordance Boolean matrix\ndiscordance Boolean matrix\ngeneral Boolean matrix\nSustainability 2018, 10, 2622\ncˆi\nd̂i\nSSCM\nELECTRE\nMCDM\nEMS\nTOPSIS\nAHP\nANP\nDEA\nQFD\n19 of 20\npure concordance index\npure discordance index\nSustainable supply chain management\nELimination Et Choix Traduisant la Realité\nMulti-criteria decision-making\nEnvironmental management system\nTechnique for Order Preference by Similarity to an Ideal Solution\nAnalytical Hierarchy Process\nAnalytical Network Process\nData Envelopment Analysis\nQuality Function Deployment\nReferences\n1.\n2.\n3.\n4.\n5.\n6.\n7.\n8.\n9.\n10.\n11.\n12.\n13.\n14.\n15.\n16.\n17.\n18.\n19.\nMa, L.; Song, W.; Zhou, Y. 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[CrossRef]\nSevkli, M.; Lenny Koh, S.C.; Zaim, S.; Demirbag, M.; Tatoglu, E. An application of data envelopment analytic\nhierarchy process for supplier selection: A case study of BEKO in Turkey. Int. J. Prod. Res. 2007, 45, 1973–2003.\n[CrossRef]\nBasnet, C.; Leung, J.M. Inventory lot-sizing with supplier selection. Comput. Oper. Res. 2005, 32, 1–14.\n[CrossRef]\nVeres, P.; Bányai, T.; Illés, B. Intelligent transportation systems to support production logistics.\nVeh. Autom. Eng. 2017, 245–256. [CrossRef]\nCárdenas-Barrón, L.E.; González-Velarde, J.L.; Treviño-Garza, G. A new approach to solve the multi-product\nmulti-period inventory lot sizing with supplier selection problem. Comput. Oper. Res. 2015, 64, 225–232.\n[CrossRef]\nBüyüközkan, G.; Çifçi, G. A novel fuzzy multi-criteria decision framework for sustainable supplier selection\nwith incomplete information. Comput. Ind. 2011, 62, 164–174. [CrossRef]\nAzadnia, A.H.; Saman, M.Z.M.; Wong, K.Y. Sustainable supplier selection and order lot-sizing: An integrated\nmulti-objective decision-making process. Int. J. Prod. Res. 2015, 53, 383–408. [CrossRef]\nGrisi, R.M.; Guerra, L.; Naviglio, G. Supplier Performance Evaluation for Green Supply Chain Management;\nSpringer: Berlin/Heidelberg, Germany, 2010; pp. 149–163.\nBırgün, S.; Cıhan, E. Supplier selection process using ELECTRE method. In Proceedings of the 2010\nInternational Conference on Intelligent Systems and Knowledge Engineering (ISKE), Hangzhou, China,\n15–16 November 2010; pp. 634–639.\nSevkli, M. An application of the fuzzy ELECTRE method for supplier selection. Int. J. Prod. Res. 2010, 48,\n3393–3405. [CrossRef]\n© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access\narticle distributed under the terms and conditions of the Creative Commons Attribution\n(CC BY) license (http://creativecommons.org/licenses/by/4.0/).\nCollege of Administrative and Financial Sciences\nAssignment-2\nMGT425-Spreadsheet Decision Modeling\nCourse Name:\nSpreadsheet Decision Modeling\nStudent’s Name:\nCourse Code: MGT425\nStudent’s ID Number:\nSemester: 1\nCRN: 14264\nAcademic Year: First Term- 2022-2023 (1444 H)\nFor Instructor’s Use only\nInstructor’s Name: DR GHAZALA AZIZ\nStudents’ Grade:\nLevel of Marks:\nInstructions – PLEASE READ THEM CAREFULLY\n• The Assignment must be submitted on Blackboard (WORD format only) via\nallocated folder.\n• Assignments submitted through email will not be accepted.\n• Students are advised to make their work clear and well presented, marks may be\nreduced for poor presentation. This includes filling your information on the cover\npage.\n• Students must mention question number clearly in their answer.\n• Late submission will NOT be accepted.\n• Avoid plagiarism, the work should be in your own words, copying from students or\nother resources without proper referencing will result in ZERO marks. No exceptions.\n• All answered must be typed using Times New Roman (size 12, double-spaced) font.\nNo pictures containing text will be accepted and will be considered plagiarism).\n• Submissions without this cover page will NOT be accepted.\nCourse Learning Outcomes-Covered\nAligned (PLOs)\nMGT.K.1\n(1.1)\nMGT.K.3\n(1.2)\nMGT.S.1\n(2.1)\nMGT.V.1\n(3.1)\nCourse Learning Outcomes (CLOs)\nQuestion\nFind some structured ways of dealing with complex managerial\ndecision problems.\nExplain simple decision models and management science ideas\nthat provide powerful and (often surprising) qualitative insight\nabout large spectrum of managerial problems.\nDemonstrate the tools for deciding when and which decision\nmodels to use for specific problems.\nBuild an understanding of the kind of problems that is tackled\nusing Spreadsheet Modeling and decision analysis.\nQuestion- 2.\nQuestion- 1, 3\nQuestion-4\nQuestion-5\nAssignment Instructions:\n• Log in to Saudi Digital Library (SDL) via University’s website\n• On first page of SDL, choose “English Databases”\n• From the list find and click on EBSCO database.\n• In the Search Bar of EBSCO find the following article:\nTitle: A Rough Multi-Criteria Decision-Making Approach for Sustainable Supplier Selection\nunder Vague Environment: A Case Study.\nAuthor: Huiyun Lu , Shaojun Jiang , Wenyan Song, Xinguo Ming\nDate: 26 July 2018\nAssignment Questions: (Marks 15)\nRead the above case study and answer the following Questions:\nQuestion 1: Explain the decision-making approach discussed in this case study (250-300 words) (3Marks).\nQuestion 2: Why supplier selection is a typical multi-criteria decision-making process involving\nsubjectivity and vagueness? (250-300 words) (3-Marks).\nQuestion 3: Discuss the Sustainable supplier selection that is required for manufacturing companies.\n(250-300 words) (3-Marks).\nQuestion 4: Why Supplier selection decisions are important for most of the manufacturing firms?\n(250-300 words) (3-Marks).\nQuestion 5: What is your opinion about this study and how it is connected to course and beneficial\nfor you? (250-300 words) (3-Marks).\nAnswers:\n1\n2\n3\n4\n5\n\nPurchase answer to see full\nattachment

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Unformatted previewDescription Assignment Questions: (Marks 15) Read the above case study and answer the following Questions : Question 1: Explai n the decision-making approach discussed in this case study (250-300 words) (3-Marks). Question 2: Why supplier selection is a typical multi-criteria decision-making process involving subjectivity and vagueness? (250-300 words) (3-Marks). Question 3: Discuss the Sustainable supplier selection that is required for manufacturing companies. (250-300 words) (3-Marks). Question 4: Why Supplier selection decisions are important for most of the manufacturing firms? (250-300 words) (3-Marks). Question 5: What is your opinion about this study and how it is connected to course and beneficial for you? (250-300 words) (3-Marks). sustainabilityArticleA Rough Multi-Criteria Decision-Making Approachfor Sustainable Supplier Selection underVague EnvironmentHuiyun Lu 1 , Shaojun Jiang 2 , Wenyan Song 1,3, * and Xinguo Ming 41234*School of Economics and Management, Beihang University, Beijing 100191, China; lhybuaa@163.comSchool of Information Engineering, Handan University, Handan 056005, China; hh8582@163.comBeijing Key Laboratory of Emergency Support Simulation Technologies for City Operations,Beihang University, Beijing 100191, ChinaSchool of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China;xgming@sjtu.edu.cnCorrespondence: songwenyan@buaa.edu.cn; Tel.: +86-010-8231-3693Received: 13 June 2018; Accepted: 23 July 2018; Published: 26 July 2018

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Abstract: With the growing awareness of environmental and social issues, sustainable supply chainmanagement (SSCM) has received considerable attention both in academia and industry. Supplierselection plays an important role in the successful implementation of sustainable supply chainmanagement, because it can influence the performance of SSCM. Sustainable supplier selection is atypical multi-criteria decision-making problem involving subjectivity and vagueness. Although someprevious researches of supplier selection use fuzzy approaches to deal with vague information, it hasbeen criticized for requiring much priori information and inflexibility in manipulating vagueness.Moreover, the previous methods often omit the environmental and social evaluation criteria in thesupplier selection. To manipulate these problems, a new approach based on the rough set theoryand ELECTRE (ELimination Et Choix Traduisant la REalité) is developed in this paper. The novelapproach integrates the strength of rough set theory in handling vagueness without much prioriinformation and the merit of ELECTRE in modeling multi-criteria decision-making problem. Finally,a case study of sustainable supplier selection for solar air-conditioner manufacturer is provided todemonstrate the application and potential of the approach.Keywords: sustainability; supplier selection; vague information; rough set theory; ELECTRE1. IntroductionManufacturing companies today cannot ignore sustainability concerns in their businessbecause of increased environmental awareness and ecological pressures from markets and variousstakeholders [1–3]. Sustainable supplier selection is critical to enhance supply chain performanceand competitive advantage [4]. This is because suppliers play an important role in implementingsustainable supply chain management (SSCM) practices and in achieving social, environmental andeconomic goals [5]. In this respect, sustainable supplier selection based on the sustainability criteria(economic, environmental and social) is a critical strategic decision for SSCM [6,7] and it requires to befurther explored methodically to help achieve sustainability of the whole supply chain.Although many researchers explore the topic of supplier selection, the study on the sustainablesupplier selection is still in the early stage. Most studies of sustainable supplier selection have onlyfocused on the economic and environmental aspects of sustainability. The social aspect of sustainabilityis often omitted in the decision–making for supplier selection. Besides, the problem of supplierselection is a typical multi-criteria decision-making (MCDM) problem. The decision makers alwaysSustainability 2018, 10, 2622; doi:10.3390/su10082622www.mdpi.com/journal/sustainabilitySustainability 2018, 10, 26222 of 20need to make trade-offs between conflicting criteria to select the most suitable supplier. It is difficult toobtain accurate judgments of decision makers in the process of supplier evaluation, because supplierselection involves large amount of linguistic information and subjective expert knowledge that areusually imprecise, vague or even inconsistent. To deal with this problem, fuzzy methods are oftenused to select suppliers. However, the fuzzy methods need much priori information (e.g., pre-setfuzzy membership function) which may increase the workload of decision makers [8,9]. The previousapproaches also lack a flexible mechanism to deal with the subjective evaluations of experts [10,11].Therefore, to manipulate the above problems in sustainable supplier selection, this paper proposesa novel integrated group decision method based on the ELECTRE (ELimination Et Choix Traduisantla REalité) approach and rough set theory in vague environments. Different with methods based onthe compensating accumulation principle (e.g., TOPSIS(Technique for Order Preference by Similarityto an Ideal Solution)), the ELECTRE method is based on a precedence relation and it can meetdifferent evaluation requirements by defining undifferentiated threshold, strict superior thresholdand rejection threshold and thus, it has stronger flexibility in decision–making of supplier selection.Furthermore, the rough number originated from the rough set theory can flexibly reflect the uncertaintyin decision–making process of supplier selection and it does not require much priori information.In this respect, the proposed novel approach integrates the merit of ELECTRE in modeling multi-criteriadecision-making problem and the strength of rough set theory in handling vagueness without muchpriori information.The paper is organized as follows: Section 2 presents a literature review of supplier selection,ELECTRE method and rough set. Section 3 develops an integrated rough ELECTRE method forsustainable supplier selection. In the Section 4, a case study of sustainable supplier selection for solarair-conditioner manufacturer is used to validate the feasibility and effectiveness of the method and acomparative analysis is also conducted in this section. In Section 5, conclusions and future researchdirections are presented.2. Literature Review2.1. Evaluation Criteria for Sustainable Supplier SelectionSupplier selection decisions are important for most of manufacturing firms, because a rightsupplier can effectively improve the economic benefit of the manufacturing firm [12,13]. In the past,economic criteria are usually used for supplier selection. The environment and social criteria areoften overlooked. However, with the development of sustainable supply chain management (SSCM),both the researchers and practitioners are paying more attention to environment criteria and socialcriteria in supplier selection [14]. They find it is important to incorporating the social and environmentcriteria into the supplier selection process [15,16]. This paper summarizes the sustainable supplierselection criteria from the economic, environment and social aspects. The details of the recognizedsustainable supplier selection criteria with their sources and descriptions are summarized in Table 1.Table 1. Sustainable supplier selection criteria.Sustainable SupplierSelection CriteriaDescriptionsEconomic criteriaQuality [17,18]Product quality and reliability level guaranteed by supplier.Response [5]The ability for timely response, completing orders on time and reliable delivery.Cost [19]Purchasing cost, holding cost, ordering cost and supplier’s bidding price ofthe product.Sustainability 2018, 10, 26223 of 20Table 1. Cont.Sustainable SupplierSelection CriteriaDescriptionsEnvironmental criteriaEnvironmentalmanagement system(EMS) [20,21]A set of systematic processes and practices reducing environmental impacts.Carbon emission &resourceconsumption [22,23]Greenhouse gas emissions in producing, transporting, using and recycling theproduct and the resource (e.g., energy, power and water) consumption ofthe company.Design for theenvironment [14,24]Design reducing the overall impact of a product, process or service on humanhealth and environment.Green image [17]The image of company in the green aspect, which can be improved by adoptingenvironmental friendly products or implementing ‘green’ program. It can affectthe purchasing trend of customers, market share and the relationshipwith stakeholders.Social criteriaProduct liability [25]Being responsible for customer health and safety, providing products andservices with high quality and advertising based on real information.Employee right andwelfare [26,27]Treating employee with dignity and respect and maintaining a culture of security,nondiscrimination and equality. Paying to employee shall comply with allapplicable wage laws.Social commitment [27]Involving in local community, education, job creation, healthcare andsocial investment.2.2. The Methods of Sustainable Supplier SelectionSelecting the right suppliers to set up optimal supplier networks can help to reduce purchasingcosts and increase the efficiency of the procurement logistics process [28]. Supplier selection is amulti-criteria decision-making problem. There are some papers concerning sustainable (or green)suppliers. Dai and Blackhurst (2012) integrate Analytical Hierarchy Process (AHP) with QualityFunction Deployment (QFD) for sustainable supplier selection [18]. The approach consists of fourstages, that is, linking customer requirements with the firm’s sustainability strategy, determining thesustainable purchasing competitive priority, determining evaluation criteria of sustainable supplierand evaluating the sustainable suppliers. Hsu and Hu (2009) develop a method for selecting supplierswith emphasis on issues of hazardous substance management based on Analytic Network Process(ANP) [29]. Liu and Hai (2005) provide a method called voting analytic hierarchy process for supplierselection [30]. Although AHP/ANP methods are more popular in the field of the supplier selection,they are always used to determine the relative importance weightings of criteria and sub-factors merely.They need to be integrated with other decision–making techniques. Besides, due to the number ofpairwise comparisons that need to be made, the number of supplier selections is practically limited inthe AHP/ANP-based supplier selection methods. Moreover, the conventional AHP/ANP methods donot consider the vagueness of decision–making information.To manipulate the increasing number of the suppliers, data envelopment analysis (DEA) is aprevalent approach used in supplier selection. This is because DEA can easily handle huge number ofsuppliers with little managerial input and output required. Kuo et al. (2012) present a green supplierselection method using an analysis network process as well as data envelopment analysis (DEA) [31].ANP which is able to consider the interdependency between criteria releases the constraint of DEA thatthe users cannot set up criteria weight preferences. Wu and Blackhurst (2009) propose an augmentedDEA approach for supplier evaluation and selection [32]. Sevkli et al. (2007) develop a new supplierselection method by embedding the DEA approach into AHP methodology [33]. They conclude thatSustainability 2018, 10, 26224 of 20the integrated method outperforms the conventional AHP method for supplier selection. However,DEA-based supplier selection methods have some drawbacks. The practitioners may be confused withinput and output criteria. Besides, DEA is a linear programming to measure the relative efficiencies ofhomogenous decision–making units (DMUs). An efficient supplier generating more outputs whilerequiring less input may be not an effective supplier. Furthermore, the conventional DEA also doesnot consider the subjectivity and vagueness in the decision–making process.Beside the multi-criteria decision–making method, some researchers use heuristic optimizationapproaches to select proper suppliers. Basnet and Leung (2005) develop an incapacitated mixedlinear integer programming which minimizes the aggregate purchasing, ordering and holding costssubject to demand satisfaction [34]. They solve the problem with an enumerative search algorithmand a heuristic procedure. Veres et al. (2017) propose a heuristic method for optimizing supplychain including intelligent transportation systems (ITS) based vehicles for transportation operationsproblems [35]. To solve the multi-product multi-period inventory lot sizing with supplier selectionproblem, Cárdenas-Barrón et al. (2015) propose a heuristic algorithm based on reduce and optimizeapproach (ROA) and a new valid inequality [36]. Unfortunately, the heuristic optimization approachesomit the vagueness and subjectivity in the decision–making, which may lead to inaccurate results ofsupplier selection.In order to deal with the imprecise or vague nature of linguistic assessment in evaluation andselection of suppliers, fuzzy set theory is introduced into the conventional approaches. Consideringtime pressure and lack of expertise in sustainable supplier selection, Büyüközkan and Çifçi (2011)developed a method based on fuzzy analytic network process within group decision-making schemaunder incomplete preference relations [37]. To manipulate the subjectivity of decision makers’evaluations, Amindoust et al. (2012) develop a new ranking method on the basis of fuzzy inferencesystem (FIS) for sustainable supplier selection problem [6]. Azadnia et al. (2015) developed anintegrated method based on rule-based weighted fuzzy approach [38], fuzzy analytical hierarchyprocess and multi-objective mathematical programming for sustainable supplier selection and orderallocation. Grisi et al. (2010) propose a fuzzy AHP method for green supplier selection using aseven-step approach [39]. Fuzzy logic is used to overcome uncertainty caused by human qualitativejudgments. ELECTRE (ELimination Et Choix Traduisant la REalité) methods are able to make asuccessful assessment of each alternative based on knowledge of the concordance and discordancesets for all pairs of alternatives. They are often used to select right suppliers [40]. Thus, Sevkli (2010)proposes a fuzzy ELECTRE for supplier selection [41]. Although the fuzzy methods can deal withthe imprecise or vague nature of linguistic assessment, it requires priori information (e.g., pre-setmembership function). Moreover, the fuzzy methods always convert linguistic variables into fuzzynumbers with fixed intervals. Therefore, computation results usually do not exactly match initiallinguistic terms, which easily cause loss of information and lack of precision in the final results.Although these methods have brought great insights to supplier selection literature, most of themlack flexible mechanisms to handle the subjectivity and the vagueness of decision makers’ assessments.Although some fuzzy methods of supplier selection (e.g., fuzzy ELECTRE) consider the vagueness indecision–making information, they require much priori information (e.g., pre-set fuzzy membershipfunction) which consumes much time and effort of managers. Moreover, the previous fuzzy approachesuse fuzzy number with fixed interval to indicate the uncertainty, which cannot identify the changes indecision makers’ judgments. For those reasons, there is a clear need for a new formal decision supportmethodology for the sustainable supplier selection under vague environment.3. The Proposed MethodThe main objective of this paper is to propose an integrated method for sustainable supplierselection based on rough set theory and ELECTRE. Besides, vagueness manipulation is also consideredin the proposed approach. A flowchart of the proposed approach is shown in Figure 1.Sustainability 2018, 10, 2622Sustainability 2018, 10, x FOR PEER REVIEW5 of 205 of 21Figure 1.1. TheTheframeworkframework ofof roughrough ELiminationELimination EtEt ChoixChoix TraduisantTraduisant la REalité (ELECTRE).Figure3.1. DetermineDetermine thethe SupplierSupplier EvaluationEvaluation Criteria and Their WeightsStep 1: determine the evaluation criteria of sustainable suppliersStep 1: determine the evaluation criteria of sustainable suppliersFirst of all, a panel of expert who are knowledgeable about supplier selection is established. TheFirst of all, a panel of expert who are knowledgeable about supplier selection is established.D1,DD12, D,…,DkD)k )whoThe grouphask decision-makerswhoareareresponsibleresponsibleforfor determiningdetermining andand thegrouphas kdecision-makers(i.e.,(i.e.,2 , …,ranking each criterion (i.e., C1 , C2 , …, Ck ). For the sustainable supplier selection, three aspects weC2,…,areCkeconomicrankingeachintocriterion(i.e., C1,They). For thecriteria,sustainablesupplier selection,threeaspectsweshould takeconsideration.environmentalcriteria andsocialcriteria.should take into consideration. They are economic criteria, environmental criteria and social criteria.Step 2: determine the weights for the evaluation criteria of sustainable suppliersStep 2: determine the weights for the evaluation criteria of sustainable suppliersExperts have their own individual experience and knowledge. Therefore, they may have differentExperts have their own individual experience and knowledge. Therefore, they may havecognitive vagueness for alternatives and criteria. Let us assume a judgment set P = { p1 , p2 , · · · , ph }different cognitive vagueness for alternatives and criteria. Let us assume a judgment setwith h ordered judgments, in the manner of p1 ≤ p2 ≤ · · · ≤ ph . Let pi be a random judgment in thep1 ≤ approximationp2 ≤≤ ph . LetP =P andp1, pd2is,, ph aswithorderedofjudgments,of lowerasetdefinedthehdistanceP, where din=thephmanner− p1 . TheApr (ppii ) beandthe upper approximation Apr ( pi ) of the judgment pi can be identified as follows.random judgment in the set P and d is defined as the distance of P , where d = ph − p1 . TheLower approximation set:{}lower approximationApr( pi ) and the \bupper approximation Apr
( pi ) of the judgment pi canApr ( pi ) = ∪ p j ∈ P p j ≤ pi , pi − p j ≤ dbe identified as followsLowerUpper approximationapproximation set:set:{
≤dApr( (ppi ))==∪∪\bpp j ∈j ≤p pj)dApr∈PP |pp ≥, i ,p( p−i −p p ≤iUpper approximation set:jjiihRN ( pi ) = piL , pUi{jiApr ( pi ) = ∪ p j ∈ P | p j ≥ pi , ( p j − pi ) ≤ d(1)}}(1)(2)(3)(2)Sustainability 2018, 10, 26226 of 20q
Where piL = m ∏ xijq
npU=∏ yiji(4)(5)where xij and yij are the elements of the lower approximation set Apr ( pi ) and the upper approximationset Apr ( pi ) of pi respectively and m and n are the number of elements in the two sets respectively.For different criteria, experts might give different weights. Use wkj indicate the weight of jthcriterion with kth expert.With the Formulas (1)–(5)nond j = MAX wm(6)j − wjo nmn≤ dj(7)= ∪ wnj ∈ P wnj ≤ wmApr wmj , wj − wjjo nnm≤ dj(8)Apr wm= ∪ wnj ∈ P wnj ≥ wmj , wj − wjjq 
Lim wkj = m ∏ x j(9)  q
Lim wkj = n ∏ y j(10)where x j and y j are the elements of the lower approximation set Apr (wkj ) and the upper approximationset Apr (wkj ) of wkj respectively and m and n are the number of elements in the two sets respectively.  h  i hikURN wkj = Lim wkj , Lim wkj= wkL,wjjsw jL =s(11)s∏ wkLj(12)k =1ssswU∏ wkUj =j(13)k =1hiWe could get the weight of each criterion w j = w jL , wUj .3.2. Evaluate the Sustainable Suppliers with the Proposed Rough ELECTREStep 1: Construct the rough decision matrixApart from the decision for the weight of criteria, the experts should give the assessment of thealternatives with consideration of all the criteria. Let’s use rijk to represent the kth expert scores on jthcriterion in ith alternative. The following is the scoring matrix. Aggregate all the scoring matrix.kr11 k r21Rk =  .. .krm1kr12kr22...krm2······...···kr1nkr2n.. . krmn(14)rfrf· · · rf11121n f f r21 r22 · · · rf2n eR= ... .......  ..rfnmm2 · · · rgm1 rfnoreij = rij1 , rij2 , · · · , rijh(15)(16)Sustainability 2018, 10, 26227 of 20Determine the rough matrix with expert ratings.d = max rijm − rijn(17)o nApr rijm = ∪ rijn ∈ P rijn ≤ rijm , rijm − rijn ≤ do nApr rijm = ∪ rijn ∈ P rijn ≥ rijm , rijn − rijm ≤ d(18)q Lim rijk = m∏ xij(19)
(20)  q
Lim rijk = n ∏ yij(21)where xij and yij are the elements of the lower approximation set Apr (rijk ) and the upper approximationset Apr (rijk ) of rijk respectively and m and n are the number of elements in the two sets respectively.i   hRN rijk = Lim, Lim = rijkL , rijkU(22)i hihio
nhRN reij = rij1L , rij1U , rij2L , rij2U , · · · , rijsL , rijsUi
hRN reij = rijL , rijU(23)srijL =sss∏ L , rUr1111 L , rU r2121R=... L U rm1 , rm1s∏ rijkUrijkL , rijU =s L Ur12 , r12 L Ur22 , r22···k =1... L U rm2 , rm2(24)(25)k =1···...··· L Ur1n , r1n L Ur2n , r2n... L U rmn , rmn(26)Then, we normalize the rough decision matrix with the weight of criteria.hi hiL Usij = rij · w j = rijL w jL , rijU wUij = sij , sij(27) L U#hisij sijtij =,= tijL , tUijCj Cjn oWhere Cj = MAX sUij L , tUt1111 L , tU t2121T=... L U tm1 , tm1 L Ut12 , t12 L Ut22 , t22···... L U tm2 , tm2..···.··· L Ut1n , t1n L Ut2n , t2n... L U tmn , tmn(28)(29)(30)Sustainability 2018, 10, 26228 of 20Step 2: Construct the rough concordance matrix and discordance matrixIn this step, we construct some field for the comparison among all the alternatives. We comparedifferent alternatives in two aspects. One is the concordance and the other is the discordance. Constructthe concordance and discordance matrices.\bCS pq = Fj t pj ≥ tqj(31)\bDS pq = Fj t pj < tqj(32)CS pq represents the areas that alternative p is better than alternative q and DS pq represents theareas that alternative p is worse than alternative q.c pq =∑wj(33)Fj ∈CS pq
max Fj ∈ DS pq d t pj , tqj
d pq =max Fj ∈ J d t pj , tqj L U L U  −c12 , c12· · · c1m, c1m  L , cUL , cU−···c c212m 2m 21C=............ L U   L U ···−cm1 , cm1cm2 , cm2− d12 · · · d1m− · · · d2m  d21D= ..... ......  .dm1 dm2 · · · −(34)(35)(36)By means of the calculation, we could get the rough concordance matrix C and discordancematrix D.Step 3: Determine the general Boolean matrixAfter we get the concordance matrix and discordance matrix, we should determine the thresholdvalue. Using it to transform the matrix into Boolean matrix. First, we calculate the mean of the allfactors in matrix C and matrix D.mm∑∑ c pqc=p=1,p6=q q=1,q6= pm∑d=(37)m ( m − 1)m∑p=1,p6=q q=1,q6= pd pqm ( m − 1)(38)Compare the factors in matrix C with c and the factors in matrix D with d. According the result ofthe comparison, we get the concordance Boolean matrix F and discordance Boolean matrix G.(f pq =(g pq =1i f : c pq ≥ c0i f : c pq < c1i f : d pq ≤ d0i f : d pq > d(39)(40)Sustainability 2018, 10, 26229 of 20  F = f pq m×m , G = g pq m×m(41)Then we could construct the general Boolean matrix H.h pq = f pq · g pq(42) H = h pq m×m(43)According to the above calculations, we could get the general Boolean matrix. It is a basis for theranking of the alternatives. If h pq = 1, that means alternative p is better than alternative q.Step 4: Calculate the pure concordance index and discordance indexBy the Boolean general matrix, we could get part relations between all alternatives. Since ifh pq = 1, we know that alternative p is better than alternative q. But if h pq = 0 and we could not inferthe relationship of alternative p and alternative q from other alternatives, then we do not know whichis better. In order to get a rank of all the alternatives, we bring into pure concordance index cˆi anddiscordance index d̂i .Before calculating the pure index, we should transform rough interval into definite number.Song et al. (2017) has proposed this method. We use ∆−1 represents the calculation of changing roughinterval into definite number [14].The calculation includes the following procedures.(1) Normalizationzei L =zei U =ziL − minziL /∆maxmin(44)iLzUi − minzii/∆maxmin(45)UL∆maxmin = maxzi − minzii(46)iLwhere ziL and zUi are the lower limit and the upper limit of the rough number zei respectively; zei andzei U are the normalized form of ziL and zUi respectively.(2) Determine the total normalized definite value byzei L × 1 − zei L + zei U × zei Uβi =(47)1 − zei L + zei U(3) Compute the final definite value form zei der for zei byzei der = minziL + β i ∆maxmin(48)iTherefore, we can use this method to calculate the concordance index and discordance index.mcˆi =∑q=1,q6=i
∆−1 cfiq −md̂i =∑q=1,q6=im∑p=1,p6=i∆−1 cfpi
(49)m
diq −∑d pi
(50)p=1,p6=iStep 5: Determine the final rankingAccording to the cˆi , we can get a priority in concordance. The bigger value of cˆi the higher placethe alternative would get. We use R1i for the ranking in concordance. The same we can get the prioritySustainability 2018, 10, 262210 of 20in discordance by d̂i . But on the contrary, the smaller value of d̂i the higher place the alternative wouldget. We use R2i for the ranking in discordance. The final ranking is calculated as follows:Ri =R1i + R2i2(51)Ri is the final rank of all the alternatives.4. Case StudyIn this section, in order to validate the applicability and effectiveness of the proposed method, weuse an example to illustrate. We assume that there is a manufacturing company. For the purpose ofchoosing a good supplier, they set up a panel of 4 experts. The experts come from various departmentsincluding purchasing, quality and production and planning who are involved in the supplier selectionprocess. And there are 8 suppliers for selection.4.1. Implementation4.1.1. Determine the Supplier Evaluation Criteria and Their WeightsStep 1: determine the evaluation criteria of sustainable suppliersFirst of all, the experts make a decision of the criteria. In addition to economic criteria,environmental criteria and social criteria should also be considered for the sustainable supplierselection. These criteria consist of three parts, we use C1~10 to represent these ten criteria. They areEconomic criteria including quality (C1), response (C2) and cost (C3); Environmental criteria includingenvironmental management system (C4), carbon emission & resource consumption (C5), design forthe environment (C6), Green image (C7); Social criteria including product liability (C8), employee rightand welfare (C9), social commitment (C10). The detailed introduction is shown in Table 1. We useA1~8 to represent alternatives, E1~4 to represent experts.Step 2: determine the weights for the evaluation criteria of sustainable suppliersAfter the decision of criteria, experts should evaluate the weight of each criterion. The expertsgive their evaluation to the criteria in the Table 2. Firstly, we convert the grades which experts give tocriteria into rough number. Take criterion C1 for example.Table 2. The grade of each criterion.C1C2C3C4C5C6C7C8C9C10E1E2E3E44365664467567546436444556532656476555474According to the Equations (6)–(13) in Section 3,d1 = 2

Apr w11 = {4, 4}, Apr w11 = {4, 5, 4, 6}

Apr w12 = {4, 5, 4}, Apr w12 = {5, 6}Sustainability 2018, 10, 262211 of 20

Apr w13 = {4, 4}, Apr w13 = {4, 5, 4, 6}

Apr w14 = {4, 5, 4, 6}, Apr w14 = {6}
√
√Lim w11 = 2 4 × 4 = 4, Lim w11 = 4 4 × 5 × 4 × 6 = 4.68
√
√Lim w12 = 3 4 × 5 × 4 = 4.31, Lim w12 = 2 5 × 6 = 5.48√√

Lim w13 = 2 4 × 4 = 4, Lim w13 = 4 4 × 5 × 4 × 6 = 4.68
√
Lim w14 = 4 4 × 5 × 4 × 6 = 4.68, Lim w14 = 6√√w1L = 4 4 × 4.31 × 4 × 4.68 = 4.24, w1U = 4 4.68 × 5.48 × 4.68 × 6 = 5.18The same as the other criteria, following the same procedure, we can get the importance degree ofall the criteria in Table 3.Table 3. The importance of all the criteria.Rough ImportanceW1W2W3W4W5W6W7W8W9W10[4.24, 5.18][3.57, 4.77][5.69, 6.70][5.06, 5.42][4.68, 5.70][5.23, 5.73][3.53, 4.37][2.63, 3.67][6.06, 6.42][4.28, 5.60]4.1.2. Evaluate the Sustainable Suppliers with the Proposed Rough ELECTREStep 1: Construct the rough decision matrixDifferent expert might hold different view for alternatives and criteria because of their personalexperience and knowledge. And the true information is just contained in the cognitive vagueness.According to the evaluation towards the alternatives from the experts, we could get the rough numberof each alternative. We take the data for criterion 1 in Table 4 for example.Table 4. The evaluation for alternative under the criterion 1.C1A1A2A3A4A5A6A7A8E1E2E3E464543675434556646465345352354675According to the Equations (17)–(26), we use x cab for the cth expert’s evaluation towards alternativeb in criterion a. We can get the rough matrix in Table 5.Sustainability 2018, 10, 262212 of 20Table 5. The rough matrix.A1A2A3A4A5A6A7A8C1C2C3…C10[4.68, 5.70][2.63, 3.67][3.65, 5.15][4.53, 4.93][3.23, 4.16][5.02, 5.85][5.69, 6.70][3.66, 4.69][5.23, 5.73][3.66, 4.69][2.22, 3.13][5.54, 5.93][4.54, 5.38][5.69, 6.70][4.68, 5.70][4.06, 4.41][4.24, 5.18][5.11, 5.79][4.68, 5.70][3.23, 4.16][4.24, 5.18][6.06, 6.42][5.23, 5.73][3.53, 4.37]……………………[3.66, 4.69][4.68, 5.70][4.67, 6.17][4.68, 5.70][3.96, 5.29][4.68, 5.70][5.02, 5.85][4.24, 5.18]Note: not all of the data are provided in Table 5 due to the space limitation.Then, we normalize the rough matrix. According to the Equations (27)–(30). We can get the resultin Table 6.Table 6. The normalized weighted decision matrix.A1A2A3A4A5A6A7A8C1C2C3…C10[0.57, 0.85][0.32, 0.55][0.45, 0.77][0.55, 0.74][0.39, 0.62][0.61, 0.87][0.69, 1.00][0.45, 0.70][0.58, 0.86][0.41, 0.70][0.25, 0.47][0.62, 0.89][0.51, 0.80][0.64, 1.00][0.52, 0.85][0.45, 0.66][0.56, 0.81][0.68, 0.90][0.62, 0.89][0.43, 0.65][0.56, 0.81][0.80, 1.00][0.69, 0.89][0.47, 0.68]……………………[0.45, 0.76][0.58, 0.92][0.58, 1.00][0.58, 0.92][0.49, 0.86][0.58, 0.92][0.62, 0.95][0.53, 0.84]Step 2: Construct the rough concordance matrix and discordance matrixIn this step, we construct the concordance and discordance matrices according to the normalizedrough decision matrix. For the construct of the concordance matrix, we take alternative1 and alternative2 for example. At the first, we should find in which criterion A1 performs better than A2, that meansthe score in certain criterion, A1 is higher than A2.According to the Table 6, we could find in criterion 1, 2, 9, A1 performs better than A2. Add up allthese weights of the criteria. We could get the value of c12 = [13.87, 16.37] in the concordance matrix.And we can get the concordance matrix in Table 7 by repeat these procedures.Table 7. The concordance matrix.A1A2A3A4A5A6A7A8A1A2A3…A8[31.11, 37.19][33.63, 39.23][7.85, 10.37][7.81, 9.97][29.01, 35.71][19.27, 22.89][17.55, 21.08][13.87, 16.37][19.81, 22.93][13.87, 16.37][11.35, 14.32][22.20, 26.74][23.84, 28.67][22.09, 26.44][11.35, 14.32][25.17, 30.63][7.81, 9.95][7.11, 9.14][24.73, 30.11][17.04, 21.03][16.85, 20.25]……………………[27.43, 32.48][22.89, 27.12][28.13, 33.30][12.09, 15.55][20.67, 25.25][25.47, 31.34][17.78, 22.25]-For the construct of the discordance matrix. First of all, we find the criterion which A2 is betterthan A1. And we could find that they are criterion 3, 4, 5, 6, 7, 8, 10. Then we find the biggest distancein these criteria. Using it divide the biggest distance between A1 and A2. We can get the value ofd12 = 1. Repeating these procedures and we can get the discordance matrix in Table 8.Sustainability 2018, 10, 262213 of 20Table 8. The discordance matrix.A1A2A3A4A5A6A7A8A1A2A3A4A5A6A7A80.851.001.001.000.461.000.931.000.641.001.000.370.711.000.531.000.731.000.430.880.910.900.491.001.000.000.370.680.200.270.640.650.220.550.321.001.001.001.001.001.001.000.881.001.001.001.000.321.001.000.971.001.001.000.270.82-Step 3: Determine the general Boolean matrixBased on concordance and discordance matrix, we construct the concordance Boolean anddiscordance Boolean matrices. Calculate the concordance index and discordance index. Followthe Equations (37)–(41).m∑cL =m∑p=1,p6=q q=1,q6= pmc Lpqm ( m − 1)= 22.49, cU =m∑d=m∑m∑p=1,p6=q q=1,q6= p∑p=1,p6=q q=1,q6= pcUpq= 26.78m ( m − 1)d pq= 0.79m ( m − 1)And we can get the concordance Boolean and discordance Boolean matrices in Tables 9 and 10.Table 9. The concordance Boolean matrix.A1A2A3A4A5A6A7A8A1A2A3A4A5A6A7A811001000000010010010011111111110111010000010100111110010-Table 10. The discordance Boolean matrix.A1A2A3A4A5A6A7A8A1A2A3A4A5A6A7A800001000100110101010001001111111111000000000000100000010-According to the Equation (42), we could get the general matrix in Table 11.Sustainability 2018, 10, 262214 of 20Table 11. The general matrix.A1A2A3A4A5A6A7A8A1A2A3A4A5A6A7A800001000000010000010001001111110111000000000000100000010-And followingthe PEERgeneralmatrix, we could draw the priority picture like Figure 2.Sustainability2018, 10, x FORREVIEW15 of 21Figure 2. The relations of alternatives in conventional ELECTRE.We useuse ‘>’> A5;A2A2> {A4,A5};A5};A3 >A3A5;{A1,We‘>’ indicatingindicatingbetter,better,thenthenwewecouldcouldfindfindthatthatA1A1> A5;> {A4,> A6A5;>A6>A2,A3,A4,A5,A7,A8};A7>{A2,A4,A5};A8>{A4,A5}.That’ssomerelationbetweenall{A1, A2, A3, A4, A5, A7, A8}; A7 > {A2, A4, A5}; A8 > {A4, A5}. That’s some relation between theallalternatives.thealternatives.But wewe cannotcannot havehave aa rankrank ofof allall thethe alternativesalternatives justjust thoughthough thisthis figure.figure. LikeLike wewe dodo notnot knowknow isisButA4 betterbetter thanthan A5A5 oror A5A5 betterbetter thanthan A4A4 oror theythey areare thethe same.same. So,So, wewe bringbring inin thethe conceptconcept ofof thethe purepureA4concordanceindexanddiscordanceindex.concordance index and discordance index.Step 4: Calculate the pure concordance index and discordance indexStep 4: Calculate the pure concordance index and discordance indexBefore we calculate the pure concordance index and discordance index, we should convert theBefore we calculate the pure concordance index and discordance index, we should convert therough concordance matrix into definite number matrix. According to the Equations (44)–(48). We canrough concordance matrix into definite number matrix. According to the Equations (44)–(48). We canget the result in Table 12.get the result in Table 12.Table 12. The definite number concordance matrix.A1A2A3A4A5A6A7A8A135.8738.398.047.9533.9120.8318.89A214.5021.5814.5011.7825.4027.6225.12A312.1629.608.057.2729.0119.0618.51A440.7733.4241.0827.7146.1029.2135.07A542.1737.3443.1118.6341.6831.5325.65A617.0126.2022.634.359.0915.0821.43A728.5022.4730.9017.6016.5535.7829.69A831.3525.3932.2912.5823.0029.6419.55-Sustainability 2018, 10, 262215 of 20Table 12. The definite number concordance matrix.A1A2A3A4A5A6A7A8A1A2A3A4A5A6A7A835.8738.398.047.9533.9120.8318.8914.5021.5814.5011.7825.4027.6225.1212.1629.608.057.2729.0119.0618.5140.7733.4241.0827.7146.1029.2135.0742.1737.3443.1118.6341.6831.5325.6517.0126.2022.634.359.0915.0821.4328.5022.4730.9017.6016.5535.7829.6931.3525.3932.2912.5823.0029.6419.55-Then we could calculate the pure concordance index and discordance index of each alternativeand the result is in Table 13.Table 13. The pure concordance index and discordance index of each supplier.A1A2A3A4A5A6A7A8cˆid̂i22.5869.80106.31−169.60−136.74125.73−18.630.57−0.75−0.150.811.934.16−4.92−0.87−0.21Step 5: Determine the final rankingAccording to the pure concordance index and discordance index, we could get the ranking ofeach supplier in concordance and discordance aspects. With the Equation (51), we could get the finalranking of all the suppliers in Table 14.Table 14. The final ranking of all the suppliers.A1A2A3A4A5A6A7A8R1iR2iRi432871653567812423377136From Table 14, we could see that priority is: A6 > A1 > {A2, A3, A7} > A8 > {A4, A5}.4.2. Comparisons and DiscussionTo further validate the effectiveness and strengths of the approach proposed in this paper, we makea comparison analysis.The comparison is conducted between the modified ELECTRE method with rough number (therough ELECTRE), fuzzy number (the fuzzy ELECTRE) and crisp number (the conventional ELECTRE).The results are presented in Figure 3. From the Figure 3, we can see the rank of A2, A3 and A7 aredifferent with each other in the three methods. In the process of supplier selection, the top threecandidates are critical for the consideration. Different rankings will influence in the final performanceof supply chain.Sustainability 2018, 10, 2622Sustainability 2018, 10, x FOR PEER REVIEWSustainability 2018, 10, x FOR PEER REVIEW16 of 2017 of 2117 of 21Figure 3.3. TheThe rankrank ofof differentdifferent methods.methods.FigureFigure 3. The rank of different methods.TheThe fuzzyfuzzy methodsmethods ofof suppliersupplier selectionselection (e.g.,(e.g., methodsmethods inin [39,41])[39,41]) oftenoften useuse thethe fuzzyfuzzy numbernumberwithfixedintervaltodealwiththeuncertaintyinsupplierselection,whichwillcauseinformationwithuncertaintyin insupplierselection,whichwillwillcauseinformationlostwith fixedfixedintervalintervaltotodealdealwithwiththetheuncertaintysupplierselection,whichcauseinformationlostinprocess.Differentthemethods,theapproachusestheindecision–makingprocess.Differentwith withthemethods,the proposedapproachuses theroughlostin decision–makingdecision–makingprocess.Differentwithfuzzythe fuzzyfuzzymethods,the proposedproposedapproachusestheroughnumberwithflexibleintervaltodescribetheuncertaintyanditdoesnotrequiretosubjectivelynumberwithflexibleintervaltodescribetheuncertaintyanditdoesnotrequiretosubjectivelysetrough number with flexible interval to describe the uncertainty and it does not require to subjectivelysetthefuzzymembershipfunctioninadvance.Theroughnumbercanflexiblyreflectthechangeofthefuzzymembershipfunctionin advance.TheTheroughnumbercan canflexiblyreflectthe changeof theset thefuzzymembershipfunctionin advance.roughnumberflexiblyreflectthe changeoftheexperts’preference.Forexample,ifoneexpertprovidesthescoresof6,4,6,5.Itthencanbeexperts’preference.For example,if one ifexpertprovidesthe scores6, 4, 6,of5. 6,It 4,thencanItbeconvertedthe experts’preference.For example,one expertprovidestheofscores6, 5.thencan beconvertedto[4,whichhaveoftofuzzy intervalsofintervals[5, 7], [3, of5],[5,[5,7],7] [3,and[4,[5,6],7]allandof whichhaveintervalof 2. intervalButthe proposedconvertedto fuzzyfuzzyintervalsof[5,7],[3, 5],5],[5,7]and[4, 6],6], allall ofoffixedwhichhave fixedfixedintervalof 2.2. ButButtheproposedapproachtransformstheoriginalscoresintotheflexibleroughintervalsof[5.18,6],approachtransformsthetransformsoriginal scoresinto thescoresflexibleroughintervalsroughof [5.18,6], [4, of5.18],[5.18,6]the proposedapproachthe originalintothe flexibleintervals[5.18,6], [4,[4,5.18],[5.18,6][4.47,5.65],whichare4.changeevaluationsandwhichshownin Figure4. IfintheFigureexpertschangetheir evaluations3, 4, 6, 4,5.18],[4.47,[5.18,5.65],6] andand[4.47,are5.65],whichare shownshowninFigure4. IfIf thethe expertsexpertschange theirtheirintoevaluationsinto3,4,willinto[5,[3,5],theELECTREthewill intervalschangeinto[2,change4], [3, 5],[5,[2,7],4],[3,[3,5],5],whiletheELECTREtransformtheintofuzzy3, 4,4, 6,6,intervals4, thethe fuzzyfuzzyintervalswillchangeinto[2,4],[3,5],[5, 7],7],[3,rough5], whilewhilethe roughroughELECTREtransformtheoriginalscoresinto[3,4.12],[3.63,4.58],[4.12,6]and[3.36,4.58].Obviously,theoriginalscores[3, 4.12],[3.63,4.58],and [3.36,the boundaryof the fuzzytransformthe intooriginalscoresinto[3, [4.12,4.12], 6][3.63,4.58], 4.58].[4.12,Obviously,6] and [3.36,4.58]. Obviously,theboundaryofthefuzzyintervalhasnoalterationwiththechangeoftheexperts’changeinthefuzzyintervalhasnoalterationwiththechangeoftheexperts’changeinthefuzzyELECTRE.Ontheotherboundary of the fuzzy interval has no alteration with the change of the experts’ change in the fuzzyELECTRE.OnotherELECTREcanidentifytheofpreferences,hand,the roughELECTREcan theidentifythechanges ofexpertpreferences,whichwill makethe finalELECTRE.On thetheother hand,hand,the roughroughELECTREcanidentifythe changeschangesof expertexpertpreferences,whichwillmakethefinalrankingmoreaccurateandreasonable.rankingmoreaccurateandrankingreasonable.which willmakethe finalmore accurate and reasonable.Figure 4. Different vagueness manipulations for judgements on alternative one of criterion one.Figure 4. Different vagueness manipulations for judgements on alternative oneone ofof criterioncriterion one.one.Moreover,Moreover, comparedcompared withwith thethe traditionaltraditional ELECTREELECTRE methodmethod (e.g.,(e.g., thethe ELECTREELECTRE methodmethod usedusedbyby BırgünBırgün andand CıhanCıhan (2010)(2010) [40]),[40]), thethe proposedproposed methodmethod providesprovides thethe rankrank ofof allall thethe alternatives.alternatives. InInthethe traditionaltraditional ELECTREELECTRE method,method, wewe cancan onlyonly getget partialpartial relationshipsrelationships amongamong alternatives.alternatives. ThisThis willwillSustainability 2018, 10, 262217 of 20Moreover, compared with the traditional ELECTRE method (e.g., the ELECTRE method used byBırgün and Cıhan (2010) [40]), the proposed method provides the rank of all the alternatives. In thetraditional ELECTRE method, we can only get partial relationships among alternatives. This willhinder the managers to directly identify the best supplier. As shown in Figure 2, there is no direct orindirect relationship between A7 and A3, so we do not know whether A7 performs better than A3 ornot. However, we can get all the relationships in the proposed method based on the calculation ofpure concordance index and discordance index. All the ranks of the suppliers can be provided in theproposed approach. This is obviously more practical and reasonable than the conventional ELECTREmethod (e.g., the ELECTRE method in [40]). Moreover, different with most of AHP/ANP-basedmethods [29,30] and DEA approaches [31,32], the proposed rough ELECTRE method considersthe uncertainty in decision–making process, which makes the final ranking results of suppliersmore accurate.Theoretically, this study develops a rough multi-criteria decision-making approach for sustainablesupplier selection considering vagueness and subjectivity. The novel approach integrates the strengthof rough set theory in handling vagueness without much priori information and the merit of ELECTREin modeling multi-criteria decision-making problem. The comparisons between the proposed method,the conventional ELECTRE and the fuzzy ELECTRE reveal that the rough ELECTRE performs betterthan the conventional ELECTRE and the fuzzy ELECTRE in dealing with vague and impreciseinformation. Besides, this research contributes to modeling the problem of supplier selection based onthe economic, environmental and social aspects. The social aspects are often omitted in the previoussupplier selection methods. Practically, this method provides an effective method to identify theright suppliers to achieve the success of the sustainable supply chain management. It also provides astandardized procedure for managers in sustainable supplier selection.5. ConclusionsTo manipulate the vagueness in sustainable supplier selection, a new approach based on the roughset theory and ELECTRE is developed in this paper. The novel approach integrates both the strengthof rough set theory in handling vagueness and the merit of ELECTRE in modeling multi-criteriadecision-making problem. A case study of sustainable supplier selection for solar air-conditionermanufacturer is provided to demonstrate the application and potential of the approach. In sum, thisproposed method has the following features:First, this study considers the social sustainability in the supplier selection, which is often omittedin the previous literature. This research contributes to modeling the problem of supplier selectiondecision within the context of a sustainable supply chain management based on the Triple Bottom Line(TBL) concept (economic, environmental and social aspects). The sustainability criteria in this studyare generic and can be used for sustainable supplier selection in different industries.Second, the proposed rough ELECTRE method can flexibly reflect the uncertainty indecision–making without much priori information. Different with the previous fuzzy methods,the proposed approach utilizes the lower and upper approximations to describe uncertainty andit does not require the pre-set fuzzy membership function, which will reduce the decision–makingburdens of managers.Third, the proposed approach can identify the preference changes of decision makers with flexiblerough intervals. Due to the flexible uncertainty mechanism, the rough number is more sensitive thanfuzzy number to the preference changes of decision makers, which makes the final ranking resultsmore accurate.Fourth, different with the conventional ELECTRE revealing partial ranking orders, the proposedrough ELECTRE method can provide full ranking order of all the alternatives. This is especially usefulfor managers to get a comprehensive view of suppliers and make reasonable decision–making insupplier selection.Sustainability 2018, 10, 262218 of 20Although the proposed method has some merits in sustainable supplier selection, it also hasseveral ameliorable aspects which may serve as implications for further study. To make the rankingresults more accurate, it would be favorable for future research to take decision makers’ weighs,objective criteria weights and subjective criteria weights into consideration. To handle huge numberof suppliers, the proposed rough ELECTRE method will be integrated with DEA method with littlemanagerial input and output required. Moreover, a computerized tool based on the proposed approachwill be developed to reduce the computation burdens of managers. Besides, more testing work isnecessitated to gain external validity.Author Contributions: Conceptualization, S.J. and W.S.; Methodology, W.S.; Validation, H.L. and S.J.; Resources,X.M.; Writing-Original Draft Preparation, H.L.; Writing-Review & Editing, X.M.Funding: The work described in this paper was supported by the National Natural Science Foundation of China(Grant No. 71501006 and 71632003), the Technical Research Foundation (JSZL2016601A004), the Open Project ofHenan Key Laboratory of Intelligent Manufacturing of Mechanical Equipment, Zhengzhou University of LightIndustry (No. IM201801), and the Fundamental Research Funds for the Central Universities.Acknowledgments: The authors would like to thank the editor and the anonymous reviewers for their helpfulcomments and suggestions on the drafts of this paper.Conflicts of Interest: The authors declare no conflict of interest.AbbreviationsThe following abbreviations are used in this manuscript:DiCiEiAiPpiApr ( pi )Apr ( pi )dRN ( pi )piLpUixijyijwkj Lim wkj Lim wkjith decision-makerith criterion for supplier selectionith expertith alternativejudgement setjudgement of ith expertlower approximation set of pi , which contains the elements that smaller than pi in set Pupper approximation set of pi , which contains the elements that bigger than pi in set Pmaximum distance of set Prough interval corresponding to pilower approximation of piupper approximation of pielements of Apr ( pi )elements of Apr ( pi )weight of jth criterion with kth expertlower approximation of wkjupper approximation of wkjw jLlower approximation of the weight of jth criterionwUjupper approximation of the weight of jth criterionrijkRkreijkth expert’s judgement on jth criterion in ith alternativescoring matrix for kth expertSet of rijktijTCDcdFGHrough interval corresponding to reij after normalizedrough scoring matrixconcordance matrixdiscordance matrixconcordance indexdiscordance indexconcordance Boolean matrixdiscordance Boolean matrixgeneral Boolean matrixSustainability 2018, 10, 2622cˆid̂iSSCMELECTREMCDMEMSTOPSISAHPANPDEAQFD19 of 20pure concordance indexpure discordance indexSustainable supply chain managementELimination Et Choix Traduisant la RealitéMulti-criteria decision-makingEnvironmental management systemTechnique for Order Preference by Similarity to an Ideal SolutionAnalytical Hierarchy ProcessAnalytical Network ProcessData Envelopment AnalysisQuality Function DeploymentReferences1.2.3.4.5.6.7.8.9.10.11.12.13.14.15.16.17.18.19.Ma, L.; Song, W.; Zhou, Y. 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This article is an open accessarticle distributed under the terms and conditions of the Creative Commons Attribution(CC BY) license (http://creativecommons.org/licenses/by/4.0/).College of Administrative and Financial SciencesAssignment-2MGT425-Spreadsheet Decision ModelingCourse Name:Spreadsheet Decision ModelingStudent’s Name:Course Code: MGT425Student’s ID Number:Semester: 1CRN: 14264Academic Year: First Term- 2022-2023 (1444 H)For Instructor’s Use onlyInstructor’s Name: DR GHAZALA AZIZStudents’ Grade:Level of Marks:Instructions – PLEASE READ THEM CAREFULLY• The Assignment must be submitted on Blackboard (WORD format only) viaallocated folder.• Assignments submitted through email will not be accepted.• Students are advised to make their work clear and well presented, marks may bereduced for poor presentation. This includes filling your information on the coverpage.• Students must mention question number clearly in their answer.• Late submission will NOT be accepted.• Avoid plagiarism, the work should be in your own words, copying from students orother resources without proper referencing will result in ZERO marks. No exceptions.• All answered must be typed using Times New Roman (size 12, double-spaced) font.No pictures containing text will be accepted and will be considered plagiarism).• Submissions without this cover page will NOT be accepted.Course Learning Outcomes-CoveredAligned (PLOs)MGT.K.1(1.1)MGT.K.3(1.2)MGT.S.1(2.1)MGT.V.1(3.1)Course Learning Outcomes (CLOs)QuestionFind some structured ways of dealing with complex managerialdecision problems.Explain simple decision models and management science ideasthat provide powerful and (often surprising) qualitative insightabout large spectrum of managerial problems.Demonstrate the tools for deciding when and which decisionmodels to use for specific problems.Build an understanding of the kind of problems that is tackledusing Spreadsheet Modeling and decision analysis.Question- 2.Question- 1, 3Question-4Question-5Assignment Instructions:• Log in to Saudi Digital Library (SDL) via University’s website• On first page of SDL, choose “English Databases”• From the list find and click on EBSCO database.• In the Search Bar of EBSCO find the following article:Title: A Rough Multi-Criteria Decision-Making Approach for Sustainable Supplier Selectionunder Vague Environment: A Case Study.Author: Huiyun Lu , Shaojun Jiang , Wenyan Song, Xinguo MingDate: 26 July 2018Assignment Questions: (Marks 15)Read the above case study and answer the following Questions:Question 1: Explain the decision-making approach discussed in this case study (250-300 words) (3Marks).Question 2: Why supplier selection is a typical multi-criteria decision-making process involvingsubjectivity and vagueness? (250-300 words) (3-Marks).Question 3: Discuss the Sustainable supplier selection that is required for manufacturing companies.(250-300 words) (3-Marks).Question 4: Why Supplier selection decisions are important for most of the manufacturing firms?(250-300 words) (3-Marks).Question 5: What is your opinion about this study and how it is connected to course and beneficialfor you? (250-300 words) (3-Marks).Answers:12345Purchase answer to see fullattachment APA paper format The American Psychological Association (APA) format is a widely used style for writing academic papers in the social sciences. The APA format provides specific guidelines for formatting papers, including margins, font size and type, spacing, and the use of headings. These guidelines ensure that papers written in the APA format are visually consistent and easy to read. In the APA format, papers are typically double-spaced and written in 12-point Times New Roman font. The margins should be 1 inch on all sides, and the text should be left-aligned. Headings are used to organize the paper into sections, with different levels of headings used to indicate the hierarchy of information. In-text citations are an essential aspect of the APA format, and they must be included whenever information from an outside source is used in the paper. The reference page is also an important component of an APA paper, as it lists all of the sources used in the paper. The reference page should be formatted according to the APA guidelines, including the use of a hanging indent for each reference and the use of italics for book titles. It is important to note that the APA format is not just a matter of style, but it is also a way of communicating research findings and ideas. The use of the APA format helps to ensure that the information presented in a paper is clear, concise, and easy to understand. In conclusion, the APA format is a widely used style for writing academic papers in the social sciences. It provides specific guidelines for formatting papers, including margins, font size and type, spacing, and the use of headings. By following the APA format, students can ensure that their papers are visually consistent, easy to read, and meet academic standards.

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