# Research on Supplier Evaluation in a Green Supply Chain

Authors: Qiang Hou; Lei Xie
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

The supplier is considered to be attractive under green supply chain management; supplier evaluation is also confronted with a serious challenge. Based on this background, this paper studies green supplier evaluation and selection with weights unknown in hesitant fuzzy sets and presents evaluation indexes by the DEMATEL method. On the basis of the Euclidean distance, hesitancy degree is used to construct the improved type signed distance. At last, the paper calculates relative closeness to rank green suppliers by the improved TOPSIS method and compares the results of the traditional algorithm and SAW method. The result of the example shows that the algorithm has practicability and a dipartite degree.


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## Body

## 1. Introduction

With the increasing trend of environmental awareness and the change of competitive factor in enterprises, a green supply chain has developed rapidly under environmental management in recent years. The supplier is located upstream in a green supply chain, and it can realize transmission downstream. Meanwhile, the supplier has the characteristic of cost increase in posttreating. The green supplier which is located upstream can take into account both environment and benefit (Luo and Peng [1]; Humphreys (2003) [2]). Thus on the background of a green supply chain, supplier evaluation becomes an extremely important research subject.The current research situation of green supplier evaluation pays more attention to the evaluation index system and design method. It can be seen that the evaluation index system is mainly involved in factors of suppliers' coordination and matching of demanding. So many literatures have studied the evaluation index system. In fact, it has relevance highly among indexes. Because of the high correlation degree, some factors have been enhanced, and others have been weakened.From the point of view of the evaluation method, it mainly includes seven factors of the valuator, evaluation objective, evaluation object, evaluation index, evaluation criterion, index weight, and evaluation method. As the most important foundation of index attribute measure, the relationships of value are grey fuzzy. Actually, the index expression of supplier attribute is highly uncertain, and future expectation of experts is more uncertain, based on the limitation of information.Based on the above analysis, this paper is mainly to study the improvement method which includes correlation between indexes and hesitant fuzzy sets of index measurement. To be specific, the study proposes to measure the correlation of indexes by the DEMATEL method. On the other hand, it applies to solve the problem of attribute measurement by using a hesitant fuzzy set.The innovative points of this paper are the following: (1) according to the relations of indexes, the curtailment is more effective; (2) this paper projects the TOPSIS method to deal, but the hesitancy degree is higher when the deviation is bigger. It will affect the distance between hesitant fuzzy sets. Thus this paper makes innovations in hesitant distance.The organizational structure of the remainder of the paper is the following: Section2 presents the problem description and evaluation process. Section 3 shows the index selection method. Section 4 proposes the index evaluation and measurement in a hesitant fuzzy set. Section 5 carries out a case study. Section 6 discusses the analysis for the result. Finally, Section 7 draws a conclusion.


## 2. Problem Description and Evaluation Process

This paper discusses the existing problems of green supplier evaluation in the multiple attributes. And the hesitant fuzzy set can solve problems of the fuzzy information and the uncertain index system; the DEMATEL method can determine the evaluation index. Based on the above methods, the study shows the problem description and the evaluation process.


### 2.1. Problem Description

Under the background of a green supply chain, determining the alternative suppliers is necessary firstly. Then it can determine the attribute of evaluation index by the DEMATEL method and the structure decision matrix under a hesitant fuzzy set. Based on the above description, the paper defines the symbols and interpretation as follows.Definition 1.
A = A 1 , A 2 , ⋯ , A i ∣ i = 1,2 , ⋯ , n denotes the set of suppliers, and Ai is the ith green supplier.Definition 2.
O = O 1 , O 2 , ⋯ , O j ∣ j = 1,2 , ⋯ , m denotes the set of attributes, and Oj is the jth attribute.Definition 3.
ω = ω 1 , ω 2 , ⋯ , ω j ∣ j = 1,2 , ⋯ , m denotes the set of attribute weights, and ωjis the jth attribute weight, where ωj∈0,1 and ∑j=1mωj=1.Definition 4.
h i j is the hesitant fuzzy set, and it represents all evaluation values in Oj of Ai, and this paper structures a hesitant fuzzy set decision matrix H=hijn×m, and hij=γijλ∣λ=1,2,⋯,l, where γijλ is expert estimation.Definition 5.
Δ i j is the hesitancy degree in the scheme, and 0≤Δij≤1(i=1,2,⋯,n;j=1,2,⋯,m).Definition 6.
d H 1 , H 2 represents the Euclidean distance of two hesitant fuzzy sets.Definition 7.
d ∗ H i , H j represents the improvement distance of adding hesitancy degree.Definition 8.
c A i represents the relative closeness, and i=1,2,⋯,n.Definition 9.
B = b i j i × j is the direct effect matrix in the DEMATEL method, in which bij represents expert evaluation and is written by a 5-degree scoring standard. bij is given integer score of 0-4, representing “No influence(0),” “Low influence(1),” “General influence(2),” “High influence(3),” and “Very high influence(4).” bij represents correlation degree from index ith to index jth. And D is the standardized matrix of matrix B.Definition 10.
T is the total effect matrix in the DEMATEL method and T=tiji×j=DE-D-1.Definition 11.
f i represents effect degree, and fi=∑j=1ntij,(i=1,2,⋯,n).Definition 12.
e i represents affected degree and ei=∑j=1ntji(i=1,2,⋯,n).Definition 13.
m i represents centrality degree and mi=fi+ei(i=1,2,⋯,n).Definition 14.
n i represents cause degree and ni=fi-ei(i=1,2,⋯,n).


### 2.2. The Evaluation Process

The purpose of this paper is that using fuzzy language evaluates the green supplier; the process involves method selection, weight determination, and index selection. In order to solve the problems, this paper determines the evaluation index by the DEMATEL method, redefines the improved distance formula based on a hesitant fuzzy set, and extends a new evaluation method. Figure1 shows the system of supplier evaluation.Figure 1
The evaluation process of a green supplier.Based on the analysis, the algorithm of the evaluation system is summarized as follows.Step 1.
Formulate appropriate green suppliers and attributes by using the DEMATEL method and denote them byA=A1,A2,⋯,Ai∣i=1,2,⋯,n and O=O1,O2,⋯,Oj∣j=1,2,⋯,m, respectively.Step 2.
Construct hesitant fuzzy decision matrixH=(hij)n×m.Step 3.
Calculate positive ideal solutionA+ and negative ideal solution A-.Step 4.
Solve the attribute hesitancy degree of each supplier and redefine the new distance.Step 5.
Adopt the maximizing deviation method to determine attribute weightωj.Step 6.
Calculate distances between the green supplier and positive-negative ideal solution.Step 7.
Compute the relative closenesscAi.Step 8.
RankcAi(i=1,2,⋯,n) and select the optimal green supplier.


## 2.1. Problem Description

Under the background of a green supply chain, determining the alternative suppliers is necessary firstly. Then it can determine the attribute of evaluation index by the DEMATEL method and the structure decision matrix under a hesitant fuzzy set. Based on the above description, the paper defines the symbols and interpretation as follows.Definition 1.
A = A 1 , A 2 , ⋯ , A i ∣ i = 1,2 , ⋯ , n denotes the set of suppliers, and Ai is the ith green supplier.Definition 2.
O = O 1 , O 2 , ⋯ , O j ∣ j = 1,2 , ⋯ , m denotes the set of attributes, and Oj is the jth attribute.Definition 3.
ω = ω 1 , ω 2 , ⋯ , ω j ∣ j = 1,2 , ⋯ , m denotes the set of attribute weights, and ωjis the jth attribute weight, where ωj∈0,1 and ∑j=1mωj=1.Definition 4.
h i j is the hesitant fuzzy set, and it represents all evaluation values in Oj of Ai, and this paper structures a hesitant fuzzy set decision matrix H=hijn×m, and hij=γijλ∣λ=1,2,⋯,l, where γijλ is expert estimation.Definition 5.
Δ i j is the hesitancy degree in the scheme, and 0≤Δij≤1(i=1,2,⋯,n;j=1,2,⋯,m).Definition 6.
d H 1 , H 2 represents the Euclidean distance of two hesitant fuzzy sets.Definition 7.
d ∗ H i , H j represents the improvement distance of adding hesitancy degree.Definition 8.
c A i represents the relative closeness, and i=1,2,⋯,n.Definition 9.
B = b i j i × j is the direct effect matrix in the DEMATEL method, in which bij represents expert evaluation and is written by a 5-degree scoring standard. bij is given integer score of 0-4, representing “No influence(0),” “Low influence(1),” “General influence(2),” “High influence(3),” and “Very high influence(4).” bij represents correlation degree from index ith to index jth. And D is the standardized matrix of matrix B.Definition 10.
T is the total effect matrix in the DEMATEL method and T=tiji×j=DE-D-1.Definition 11.
f i represents effect degree, and fi=∑j=1ntij,(i=1,2,⋯,n).Definition 12.
e i represents affected degree and ei=∑j=1ntji(i=1,2,⋯,n).Definition 13.
m i represents centrality degree and mi=fi+ei(i=1,2,⋯,n).Definition 14.
n i represents cause degree and ni=fi-ei(i=1,2,⋯,n).


## 2.2. The Evaluation Process

The purpose of this paper is that using fuzzy language evaluates the green supplier; the process involves method selection, weight determination, and index selection. In order to solve the problems, this paper determines the evaluation index by the DEMATEL method, redefines the improved distance formula based on a hesitant fuzzy set, and extends a new evaluation method. Figure1 shows the system of supplier evaluation.Figure 1
The evaluation process of a green supplier.Based on the analysis, the algorithm of the evaluation system is summarized as follows.Step 1.
Formulate appropriate green suppliers and attributes by using the DEMATEL method and denote them byA=A1,A2,⋯,Ai∣i=1,2,⋯,n and O=O1,O2,⋯,Oj∣j=1,2,⋯,m, respectively.Step 2.
Construct hesitant fuzzy decision matrixH=(hij)n×m.Step 3.
Calculate positive ideal solutionA+ and negative ideal solution A-.Step 4.
Solve the attribute hesitancy degree of each supplier and redefine the new distance.Step 5.
Adopt the maximizing deviation method to determine attribute weightωj.Step 6.
Calculate distances between the green supplier and positive-negative ideal solution.Step 7.
Compute the relative closenesscAi.Step 8.
RankcAi(i=1,2,⋯,n) and select the optimal green supplier.


## 3. Construct the Evaluation Index System

In Step1, the index system is the evaluation basis of the supplier; thus the index is very important. On the other hand, appropriate indexes play an important role in the evaluation green supplier.


### 3.1. Construct the Initial Index

The index is the important element to evaluate the supplier. Dickson (1996) [3] is the most influential in supplier evaluation; he evaluated the evaluation indexes of 23 suppliers and indicated that the important indexes are quality and historical performance, as well as delivery period. To face the current environmental situation, Handfield et al. (2002) [4], Shaw et al. (2012) [5], and Hsu et al. (2013) [6] added the green evaluation index, for instance, greenhouse gas emission and green purchasing. Based on the above, He et al. (2018) [7] put forward 18 indexes of qualitative and quantitative evaluation. It can be seen that evaluation indexes are various.Index selection should follow principles which include systematicness overall, science, stability, flexibility, and brevity. Based on the research and principles, this paper proposes the initial index by the Delphi method and literature method.The Delphi method can provide a more reliable index. Briefly, the Delphi method can make expert opinions in induction and arrangement scientifically and repeat verification until getting centralized indexes. On the one hand, the literature method can also analyze and epurate the evaluation index by collecting literatures. On the other hand, by combining the two methods, the initial index set is obtained completely.


### 3.2. Index Reduction: the DEMATEL Method

Obviously, the evaluation indexes have high relevance. The initial index is far from being refined. This paper is mainly for an enterprise which also seeks green and high efficiency. The DEMATEL (Decision-Making Trial and Evaluation Laboratory) (Gabus and Fontela [8]) model generally uses the knowledge and experience of an expert to select main indexes; particularly it is more useful to a system of an element uncertainty relation. So it can select the optimal supplier of a green industry enterprise by using the DEMATEL method. And the steps of DEMATEL are following.Firstly, through expert interview and literature retrieval, it aims at experts and enterprises with a questionnaire. And then it can finish indexes’ statistic by returning the questionnaire. Combining with statistical data and expert advice, the evaluation index can be divided into first-grade indexes and second-grade indexes.Then, it can establish the direct effect matrixB=biji×j. In order to establish the direct effect matrix, it is necessary to set up an assessment group; then the correlation degree can be defined by an analysis and discussion. And the interaction degree of the index is given integer score of 0-4. Then using formula (1) can get a standard matrix D:(1)D=1max1≤i≤n⁡∑j=1nbijB.b ij represents the correlation degree from index ith to index jth.Thirdly, according to matrix D, it can get the total influence matrix T, effect degreefi, and affected degree ei by formulas. Meanwhile centrality degree mi and cause degree ni can be calculated. And it can get cause-effect relationships.Finally, we can recognize the rank of indexes by cause-effect relationships. Then it can obtain the main indexes.


## 3.1. Construct the Initial Index

The index is the important element to evaluate the supplier. Dickson (1996) [3] is the most influential in supplier evaluation; he evaluated the evaluation indexes of 23 suppliers and indicated that the important indexes are quality and historical performance, as well as delivery period. To face the current environmental situation, Handfield et al. (2002) [4], Shaw et al. (2012) [5], and Hsu et al. (2013) [6] added the green evaluation index, for instance, greenhouse gas emission and green purchasing. Based on the above, He et al. (2018) [7] put forward 18 indexes of qualitative and quantitative evaluation. It can be seen that evaluation indexes are various.Index selection should follow principles which include systematicness overall, science, stability, flexibility, and brevity. Based on the research and principles, this paper proposes the initial index by the Delphi method and literature method.The Delphi method can provide a more reliable index. Briefly, the Delphi method can make expert opinions in induction and arrangement scientifically and repeat verification until getting centralized indexes. On the one hand, the literature method can also analyze and epurate the evaluation index by collecting literatures. On the other hand, by combining the two methods, the initial index set is obtained completely.


## 3.2. Index Reduction: the DEMATEL Method

Obviously, the evaluation indexes have high relevance. The initial index is far from being refined. This paper is mainly for an enterprise which also seeks green and high efficiency. The DEMATEL (Decision-Making Trial and Evaluation Laboratory) (Gabus and Fontela [8]) model generally uses the knowledge and experience of an expert to select main indexes; particularly it is more useful to a system of an element uncertainty relation. So it can select the optimal supplier of a green industry enterprise by using the DEMATEL method. And the steps of DEMATEL are following.Firstly, through expert interview and literature retrieval, it aims at experts and enterprises with a questionnaire. And then it can finish indexes’ statistic by returning the questionnaire. Combining with statistical data and expert advice, the evaluation index can be divided into first-grade indexes and second-grade indexes.Then, it can establish the direct effect matrixB=biji×j. In order to establish the direct effect matrix, it is necessary to set up an assessment group; then the correlation degree can be defined by an analysis and discussion. And the interaction degree of the index is given integer score of 0-4. Then using formula (1) can get a standard matrix D:(1)D=1max1≤i≤n⁡∑j=1nbijB.b ij represents the correlation degree from index ith to index jth.Thirdly, according to matrix D, it can get the total influence matrix T, effect degreefi, and affected degree ei by formulas. Meanwhile centrality degree mi and cause degree ni can be calculated. And it can get cause-effect relationships.Finally, we can recognize the rank of indexes by cause-effect relationships. Then it can obtain the main indexes.


## 4. Supplier Evaluation Method

This paper constructs the model from four aspects. First, in a previous paper, it could gain the main indexes according to the correlation degree among indexes by the DEMATEL method. Then, in multiple-attribute decision-making, the hesitancy degree of evaluation schemes is higher, and the distance is bigger. Thus this paper redefines the distance formula. Thirdly, in order to avoid the subjective effect, the method adopts the maximizing deviation method to determine weightsω. Finally, the improved traditional TOPSIS method can evaluate and select optimal suppliers.


### 4.1. Improvement Distance Metric

In a hesitant fuzzy set, the traditional Euclidean distance has its limitation. Particularly, the evaluation process shows a hesitancy degree. Thus, how to define the new distance metric is worth studying.


#### 4.1.1. Measure Index: Hesitant Fuzzy Set

In 1965, Professor Zadeh [9] brought forward the concept of a fuzzy set; he thought a set includes two states: support and opposition; in other words, a fuzzy set is characterized by a membership degree and nonmembership degree. Thereafter, scholars have paid more attention to fuzzy sets. Moreover, hesitant fuzzy sets are multidimensional observers which has been considered first by Molaei (2009) [10] as an extension of a one-dimensional observer (Molaei (2004) [11]). Atanassov (1986) [12] proposed the intuitionistic fuzzy set theory; the new theory added hesitancy degree which respects the neutralizing attitude for judging. Torra and Narukawa (2009) [13] suggested a hesitant fuzzy set that could describe the possibility of each element. In Step 2, actually it often shows information which makes investors hesitate when they are making decisions; thus Xia and Xu (2011) [14] defined the mathematical expression of a hesitant fuzzy set.Definition 15.
It hypothesizes thatx=x1,x2,⋯,xl is a nonempty set, and a hesitant fuzzy set is denoted in (2)H=x,hx∣x∈X.
h x is the hesitant fuzzy element, it denotes the degree that the scheme satisfies the attribute, and hx∈0,1. Then the hesitant fuzzy decision-making matrix is shown in (3)H=h11h12…h1mh21h22…h2m⋮⋮⋱⋮hn1hn2…hnm.
The membership degree of a hesitant fuzzy set reflects the hesitancy degree. The greater the deviation among elements, the higher the hesitancy degree. Based on this, in Step4, Zhang and Xu (2015) [15] defined the hesitancy degree in a hesitant fuzzy set.Definition 16.
For a hesitant fuzzy setHij=γijλ∣λ=1,2,⋯,l;i=1,2,⋯,n;j=1,2,⋯,m, γijλ is λth of Hij and the hesitancy degree of Hijis defined in (4)Δ=1Cl∑λ>σ=1lγλ-γσ,ifl>10,ifl=1,
whereCl=1/2l(l-1), γλ∈0,1, and γσ∈0,1 are the smallest λth and σth in Hij.
Moreover, distance measurement is widely applied in many fields. Xu and Xia (2011) [16] indicated the distance of two hesitant fuzzy sets based on the Euclidean distance.Definition 17.
For two hesitant fuzzy setsH1 and H2, dH1,H2 is the distance of two hesitant fuzzy sets:(5)dH1,H2=1l∑q=1lH1λq-H2λq2,
whereH1λq∈0,1 and H2λq∈0,1 are the qth value in H1 and H2, respectively, and they possess the following properties:(1)
0≤dH1,H2≤1(2)
dH1,H2=dH2,H1(3)
dH1,H2=0, if and only if H1=H2


#### 4.1.2. New Distance Metric

Based on the fuzziness of information, this paper employs a hesitant fuzzy set to deal with information. Some literatures have adopted hesitant fuzzy sets with various ways (Shi and Xiao (2018) [17]; Wu and Cao (2012) [18]). And some have improved the hesitant fuzzy sets. Additionally, the distance from the ideal point of TOPSIS is always improved, such as in Ran (2018) [19] and Lin et al. (2018) [20]. Actually, the hesitancy degree is higher when the deviation is bigger; then the distance of two hesitant fuzzy sets becomes larger accordingly. And it can be aware of limitation in distance formula (5). This paper redefines the distance formula in Step 4, as following in (6)d∗Hi,Hj=dHi,Hj1-Δi1-Δj=1/l∑q=1lHiλq-Hjλq21-Δi1-Δj.Δ i and Δj are the hesitancy degrees in schemes i and j, respectively, and the value of the hesitancy degree in the positive ideal solution and negative ideal solution is 0. dHi,Hj is the Euclidean distance of two hesitant fuzzy sets Hi and Hj. And formula (6) should satisfy the following properties:(1)
d∗Hi,Hj≥dHi,Hj(2)
d∗Hi,Hj=d∗Hj,Hi(3)
d∗Hi,Hj=0, if and only if Hi=HjProof.
(1)0≤Δi≤1, d∗Hi,Hj=dHi,Hj/1-Δi(1-Δj)≥dHi,Hj
(2) BecausedHi,Hj=dHj,Hi, therefore d∗Hi,Hj=dHi,Hj/1-Δi(1-Δj)=dHj,Hi/1-Δi(1-Δj)=d∗Hj,Hi
(3) Ifd∗Hi,Hj=0, H1λq=H2λq, thus Hi=Hj, while if Hi=Hj, d∗Hi,Hj=0. And d∗Hi,Hj=0, if and only if Hi=Hj.


### 4.2. A Method of Determining Attribute Weight

Weight occupies an important position in evaluation. Referencing Xu and Zhang (2013) [21], we adopt the maximizing deviation method to determine attribute weight. In Step 5, deviation formulas of what Ai is related to other schemes are following in (7)dij∗ω=∑k=1nd∗hij,hkjωj,i=1,2,…,n;j=1,2,…,m,whered∗hij,hkj=1/l∑q=1lhijλq-hkjλq2/1-Δij(1-Δkj) denotes the Euclidean distance of hij and hkj.Thus we can construct the deviations of all schemes under the conditionOj∈O, in (8)dj∗ω=∑i=1ndij∗ω=∑i=1n∑k=1nωj1/l∑q=1lhijλq-hkjλq21-Δij1-Δkj,j=1,2,…,m.Based on this analysis, we propose a nonlinear programming model that could determine weight vectorω and maximum deviation. The model is following in (9)maxd∗ω=∑j=1m∑i=1n∑k=1nωjd∗hij,hkjs.t.ωj≥0,j=1,2,…,m,∑j=1mωj=1.In order to solve the model, let(10)fω,φ=∑j=1m∑i=1n∑k=1nωjd∗hij,hkj+φ2∑j=1mωj-1.Formula (10) denotes the Lagrangian function of constraining the optimization problem, and φ is a real number and denotes the Lagrangian multiplier. Thus we calculate the partial differential.(11)∂f∂ωj=∑i=1n∑k=1nd∗hij,hkj+φωj=0(12)∂f∂φ=12∑j=1mωj2-1=0.ω j is prepared from formula (11):(13)ωj=-∑i=1n∑k=1nd∗hij,hkjφ,j=1,2,…,m.Bringing formula (13) into formula (12), it can be obtained in (14)φ=-∑j=1m∑i=1n∑k=1nd∗hij,hkj2.Obviouslyφ<0; ∑i=1n∑k=1nd∗hij,hkj denotes the deviations of all schemes in attribute j.Then combining with (13) and (14) gives(15)ωj=∑i=1n∑k=1nd∗hij,hkj∑j=1m∑i=1n∑k=1nd∗hij,hkj2.For the sake of brevity, we writeDj=∑i=1n∑k=1nd∗hij,hkj,j=1,2,⋯,m, and simplify (15): (16)ωj=Dj∑j=1mDj2,j=1,2,…,m,ωj∈0,1j=1,2,…,m,whereωjj=1,2,⋯,m is positive number, and the model has a unique solution. Finally, we make ωj standardized, as followed in (17)ωj∗=ωj∑j=1mωj,j=1,2,…,m.By solving the model, optimal solutionω=ω1,ω2,⋯,ωmT can be drawn.


### 4.3. The Evaluation Method: the TOPSIS Method

The multiattribute decision-making method is involved in standardization of the decision matrix, choice of attribute weight, and alternative ranking. In the above, weight has already been redefined. Different methods of standardizing the matrix lead to different results. Moreover the traditional methods are SAW (Simple Additive Weighting) [22], ELECTRE [23], and others. The methods of green supplier evaluation and selection are various at present. Briefly, DEA (Saen (2010) [24]; Azadeh (2010) [25]), AHP (Yang (2003) [26]; Diego (2012) [27]), fuzzy AHP (Shaw (2012) [5]; Mangla (2017) [28]), ANP (Saaty (1996) [29]; Vinodh (2010) [30]; Bakeshlou (2017) [31]), and TOPSIS (Hwang et al. (1981) [32]; Büyüközkan (2012) [33]; Fallahpour (2017) [34]) are most widely used in supplier selection. But the TOPSIS method can make full use of original data, and the result of TOPSIS precisely reflects the gap among suppliers. And green supplier selection is decision-making of fuzzy multiple attributes. Generally, the above methods have a certain limitation, and evaluation mechanisms have some defects. Because of the uncertain and fuzzy information in the environment and the cognition difference, in Steps 6 and 7, this paper will hold green supplier evaluation by the TOPSIS method under a hesitant fuzzy set.Attribute weights have been already determined. In order to overcome the drawback of losing integration information easily, this paper redefines the improved type signed distance formula. Then we sequence the relative closeness between each green supplier and ideal solution. Finally, the optimal supplier can be selected.Under a hesitant fuzzy environment, in Step3, A+ and A- are the positive ideal solution and negative ideal solution, as follows:(18)A+=xj,maxi⁡hijλq∣j=1,2,…,m=x1,h11+,h12+,…,h1l+,x2,h21+,h22+,…,h2l+,…,xm,hm1+,hm2+,…,hml+(19)A-=xj,mini⁡hijλq∣j=1,2,…,m=x1,h11-,h12-,…,h1l-,x2,h21-,h22-,…,h2l-,…,xm,hm1-,hm2-,…,hml-.According to (6), we calculate distance di∗+ and di∗- between each green supplier and ideal solution:(20)di∗+=∑j=1md∗hij,hj+ωj=∑j=1mωj1/l∑q=1lhijλq-hjλq+21-Δi1-∆+,i=1,2,…,n(21)di∗-=∑j=1md∗hij,hj-ωj=∑j=1mωj1/l∑q=1lhijλq-hjλq-21-Δi1-∆-,i=1,2,…,n.Then we can solve relative closeness between supplierAi and positive ideal solution A+:(22)cAi=di∗-di∗++di∗-,i=1,2,…,n,where0<cAi<1,i=1,2,⋯,n⁡。 Based on the magnitude of closeness degree, it can select the optimal green supplier.


## 4.1. Improvement Distance Metric

In a hesitant fuzzy set, the traditional Euclidean distance has its limitation. Particularly, the evaluation process shows a hesitancy degree. Thus, how to define the new distance metric is worth studying.


### 4.1.1. Measure Index: Hesitant Fuzzy Set

In 1965, Professor Zadeh [9] brought forward the concept of a fuzzy set; he thought a set includes two states: support and opposition; in other words, a fuzzy set is characterized by a membership degree and nonmembership degree. Thereafter, scholars have paid more attention to fuzzy sets. Moreover, hesitant fuzzy sets are multidimensional observers which has been considered first by Molaei (2009) [10] as an extension of a one-dimensional observer (Molaei (2004) [11]). Atanassov (1986) [12] proposed the intuitionistic fuzzy set theory; the new theory added hesitancy degree which respects the neutralizing attitude for judging. Torra and Narukawa (2009) [13] suggested a hesitant fuzzy set that could describe the possibility of each element. In Step 2, actually it often shows information which makes investors hesitate when they are making decisions; thus Xia and Xu (2011) [14] defined the mathematical expression of a hesitant fuzzy set.Definition 15.
It hypothesizes thatx=x1,x2,⋯,xl is a nonempty set, and a hesitant fuzzy set is denoted in (2)H=x,hx∣x∈X.
h x is the hesitant fuzzy element, it denotes the degree that the scheme satisfies the attribute, and hx∈0,1. Then the hesitant fuzzy decision-making matrix is shown in (3)H=h11h12…h1mh21h22…h2m⋮⋮⋱⋮hn1hn2…hnm.
The membership degree of a hesitant fuzzy set reflects the hesitancy degree. The greater the deviation among elements, the higher the hesitancy degree. Based on this, in Step4, Zhang and Xu (2015) [15] defined the hesitancy degree in a hesitant fuzzy set.Definition 16.
For a hesitant fuzzy setHij=γijλ∣λ=1,2,⋯,l;i=1,2,⋯,n;j=1,2,⋯,m, γijλ is λth of Hij and the hesitancy degree of Hijis defined in (4)Δ=1Cl∑λ>σ=1lγλ-γσ,ifl>10,ifl=1,
whereCl=1/2l(l-1), γλ∈0,1, and γσ∈0,1 are the smallest λth and σth in Hij.
Moreover, distance measurement is widely applied in many fields. Xu and Xia (2011) [16] indicated the distance of two hesitant fuzzy sets based on the Euclidean distance.Definition 17.
For two hesitant fuzzy setsH1 and H2, dH1,H2 is the distance of two hesitant fuzzy sets:(5)dH1,H2=1l∑q=1lH1λq-H2λq2,
whereH1λq∈0,1 and H2λq∈0,1 are the qth value in H1 and H2, respectively, and they possess the following properties:(1)
0≤dH1,H2≤1(2)
dH1,H2=dH2,H1(3)
dH1,H2=0, if and only if H1=H2


### 4.1.2. New Distance Metric

Based on the fuzziness of information, this paper employs a hesitant fuzzy set to deal with information. Some literatures have adopted hesitant fuzzy sets with various ways (Shi and Xiao (2018) [17]; Wu and Cao (2012) [18]). And some have improved the hesitant fuzzy sets. Additionally, the distance from the ideal point of TOPSIS is always improved, such as in Ran (2018) [19] and Lin et al. (2018) [20]. Actually, the hesitancy degree is higher when the deviation is bigger; then the distance of two hesitant fuzzy sets becomes larger accordingly. And it can be aware of limitation in distance formula (5). This paper redefines the distance formula in Step 4, as following in (6)d∗Hi,Hj=dHi,Hj1-Δi1-Δj=1/l∑q=1lHiλq-Hjλq21-Δi1-Δj.Δ i and Δj are the hesitancy degrees in schemes i and j, respectively, and the value of the hesitancy degree in the positive ideal solution and negative ideal solution is 0. dHi,Hj is the Euclidean distance of two hesitant fuzzy sets Hi and Hj. And formula (6) should satisfy the following properties:(1)
d∗Hi,Hj≥dHi,Hj(2)
d∗Hi,Hj=d∗Hj,Hi(3)
d∗Hi,Hj=0, if and only if Hi=HjProof.
(1)0≤Δi≤1, d∗Hi,Hj=dHi,Hj/1-Δi(1-Δj)≥dHi,Hj
(2) BecausedHi,Hj=dHj,Hi, therefore d∗Hi,Hj=dHi,Hj/1-Δi(1-Δj)=dHj,Hi/1-Δi(1-Δj)=d∗Hj,Hi
(3) Ifd∗Hi,Hj=0, H1λq=H2λq, thus Hi=Hj, while if Hi=Hj, d∗Hi,Hj=0. And d∗Hi,Hj=0, if and only if Hi=Hj.


## 4.1.1. Measure Index: Hesitant Fuzzy Set

In 1965, Professor Zadeh [9] brought forward the concept of a fuzzy set; he thought a set includes two states: support and opposition; in other words, a fuzzy set is characterized by a membership degree and nonmembership degree. Thereafter, scholars have paid more attention to fuzzy sets. Moreover, hesitant fuzzy sets are multidimensional observers which has been considered first by Molaei (2009) [10] as an extension of a one-dimensional observer (Molaei (2004) [11]). Atanassov (1986) [12] proposed the intuitionistic fuzzy set theory; the new theory added hesitancy degree which respects the neutralizing attitude for judging. Torra and Narukawa (2009) [13] suggested a hesitant fuzzy set that could describe the possibility of each element. In Step 2, actually it often shows information which makes investors hesitate when they are making decisions; thus Xia and Xu (2011) [14] defined the mathematical expression of a hesitant fuzzy set.Definition 15.
It hypothesizes thatx=x1,x2,⋯,xl is a nonempty set, and a hesitant fuzzy set is denoted in (2)H=x,hx∣x∈X.
h x is the hesitant fuzzy element, it denotes the degree that the scheme satisfies the attribute, and hx∈0,1. Then the hesitant fuzzy decision-making matrix is shown in (3)H=h11h12…h1mh21h22…h2m⋮⋮⋱⋮hn1hn2…hnm.
The membership degree of a hesitant fuzzy set reflects the hesitancy degree. The greater the deviation among elements, the higher the hesitancy degree. Based on this, in Step4, Zhang and Xu (2015) [15] defined the hesitancy degree in a hesitant fuzzy set.Definition 16.
For a hesitant fuzzy setHij=γijλ∣λ=1,2,⋯,l;i=1,2,⋯,n;j=1,2,⋯,m, γijλ is λth of Hij and the hesitancy degree of Hijis defined in (4)Δ=1Cl∑λ>σ=1lγλ-γσ,ifl>10,ifl=1,
whereCl=1/2l(l-1), γλ∈0,1, and γσ∈0,1 are the smallest λth and σth in Hij.
Moreover, distance measurement is widely applied in many fields. Xu and Xia (2011) [16] indicated the distance of two hesitant fuzzy sets based on the Euclidean distance.Definition 17.
For two hesitant fuzzy setsH1 and H2, dH1,H2 is the distance of two hesitant fuzzy sets:(5)dH1,H2=1l∑q=1lH1λq-H2λq2,
whereH1λq∈0,1 and H2λq∈0,1 are the qth value in H1 and H2, respectively, and they possess the following properties:(1)
0≤dH1,H2≤1(2)
dH1,H2=dH2,H1(3)
dH1,H2=0, if and only if H1=H2


## 4.1.2. New Distance Metric

Based on the fuzziness of information, this paper employs a hesitant fuzzy set to deal with information. Some literatures have adopted hesitant fuzzy sets with various ways (Shi and Xiao (2018) [17]; Wu and Cao (2012) [18]). And some have improved the hesitant fuzzy sets. Additionally, the distance from the ideal point of TOPSIS is always improved, such as in Ran (2018) [19] and Lin et al. (2018) [20]. Actually, the hesitancy degree is higher when the deviation is bigger; then the distance of two hesitant fuzzy sets becomes larger accordingly. And it can be aware of limitation in distance formula (5). This paper redefines the distance formula in Step 4, as following in (6)d∗Hi,Hj=dHi,Hj1-Δi1-Δj=1/l∑q=1lHiλq-Hjλq21-Δi1-Δj.Δ i and Δj are the hesitancy degrees in schemes i and j, respectively, and the value of the hesitancy degree in the positive ideal solution and negative ideal solution is 0. dHi,Hj is the Euclidean distance of two hesitant fuzzy sets Hi and Hj. And formula (6) should satisfy the following properties:(1)
d∗Hi,Hj≥dHi,Hj(2)
d∗Hi,Hj=d∗Hj,Hi(3)
d∗Hi,Hj=0, if and only if Hi=HjProof.
(1)0≤Δi≤1, d∗Hi,Hj=dHi,Hj/1-Δi(1-Δj)≥dHi,Hj
(2) BecausedHi,Hj=dHj,Hi, therefore d∗Hi,Hj=dHi,Hj/1-Δi(1-Δj)=dHj,Hi/1-Δi(1-Δj)=d∗Hj,Hi
(3) Ifd∗Hi,Hj=0, H1λq=H2λq, thus Hi=Hj, while if Hi=Hj, d∗Hi,Hj=0. And d∗Hi,Hj=0, if and only if Hi=Hj.


## 4.2. A Method of Determining Attribute Weight

Weight occupies an important position in evaluation. Referencing Xu and Zhang (2013) [21], we adopt the maximizing deviation method to determine attribute weight. In Step 5, deviation formulas of what Ai is related to other schemes are following in (7)dij∗ω=∑k=1nd∗hij,hkjωj,i=1,2,…,n;j=1,2,…,m,whered∗hij,hkj=1/l∑q=1lhijλq-hkjλq2/1-Δij(1-Δkj) denotes the Euclidean distance of hij and hkj.Thus we can construct the deviations of all schemes under the conditionOj∈O, in (8)dj∗ω=∑i=1ndij∗ω=∑i=1n∑k=1nωj1/l∑q=1lhijλq-hkjλq21-Δij1-Δkj,j=1,2,…,m.Based on this analysis, we propose a nonlinear programming model that could determine weight vectorω and maximum deviation. The model is following in (9)maxd∗ω=∑j=1m∑i=1n∑k=1nωjd∗hij,hkjs.t.ωj≥0,j=1,2,…,m,∑j=1mωj=1.In order to solve the model, let(10)fω,φ=∑j=1m∑i=1n∑k=1nωjd∗hij,hkj+φ2∑j=1mωj-1.Formula (10) denotes the Lagrangian function of constraining the optimization problem, and φ is a real number and denotes the Lagrangian multiplier. Thus we calculate the partial differential.(11)∂f∂ωj=∑i=1n∑k=1nd∗hij,hkj+φωj=0(12)∂f∂φ=12∑j=1mωj2-1=0.ω j is prepared from formula (11):(13)ωj=-∑i=1n∑k=1nd∗hij,hkjφ,j=1,2,…,m.Bringing formula (13) into formula (12), it can be obtained in (14)φ=-∑j=1m∑i=1n∑k=1nd∗hij,hkj2.Obviouslyφ<0; ∑i=1n∑k=1nd∗hij,hkj denotes the deviations of all schemes in attribute j.Then combining with (13) and (14) gives(15)ωj=∑i=1n∑k=1nd∗hij,hkj∑j=1m∑i=1n∑k=1nd∗hij,hkj2.For the sake of brevity, we writeDj=∑i=1n∑k=1nd∗hij,hkj,j=1,2,⋯,m, and simplify (15): (16)ωj=Dj∑j=1mDj2,j=1,2,…,m,ωj∈0,1j=1,2,…,m,whereωjj=1,2,⋯,m is positive number, and the model has a unique solution. Finally, we make ωj standardized, as followed in (17)ωj∗=ωj∑j=1mωj,j=1,2,…,m.By solving the model, optimal solutionω=ω1,ω2,⋯,ωmT can be drawn.


## 4.3. The Evaluation Method: the TOPSIS Method

The multiattribute decision-making method is involved in standardization of the decision matrix, choice of attribute weight, and alternative ranking. In the above, weight has already been redefined. Different methods of standardizing the matrix lead to different results. Moreover the traditional methods are SAW (Simple Additive Weighting) [22], ELECTRE [23], and others. The methods of green supplier evaluation and selection are various at present. Briefly, DEA (Saen (2010) [24]; Azadeh (2010) [25]), AHP (Yang (2003) [26]; Diego (2012) [27]), fuzzy AHP (Shaw (2012) [5]; Mangla (2017) [28]), ANP (Saaty (1996) [29]; Vinodh (2010) [30]; Bakeshlou (2017) [31]), and TOPSIS (Hwang et al. (1981) [32]; Büyüközkan (2012) [33]; Fallahpour (2017) [34]) are most widely used in supplier selection. But the TOPSIS method can make full use of original data, and the result of TOPSIS precisely reflects the gap among suppliers. And green supplier selection is decision-making of fuzzy multiple attributes. Generally, the above methods have a certain limitation, and evaluation mechanisms have some defects. Because of the uncertain and fuzzy information in the environment and the cognition difference, in Steps 6 and 7, this paper will hold green supplier evaluation by the TOPSIS method under a hesitant fuzzy set.Attribute weights have been already determined. In order to overcome the drawback of losing integration information easily, this paper redefines the improved type signed distance formula. Then we sequence the relative closeness between each green supplier and ideal solution. Finally, the optimal supplier can be selected.Under a hesitant fuzzy environment, in Step3, A+ and A- are the positive ideal solution and negative ideal solution, as follows:(18)A+=xj,maxi⁡hijλq∣j=1,2,…,m=x1,h11+,h12+,…,h1l+,x2,h21+,h22+,…,h2l+,…,xm,hm1+,hm2+,…,hml+(19)A-=xj,mini⁡hijλq∣j=1,2,…,m=x1,h11-,h12-,…,h1l-,x2,h21-,h22-,…,h2l-,…,xm,hm1-,hm2-,…,hml-.According to (6), we calculate distance di∗+ and di∗- between each green supplier and ideal solution:(20)di∗+=∑j=1md∗hij,hj+ωj=∑j=1mωj1/l∑q=1lhijλq-hjλq+21-Δi1-∆+,i=1,2,…,n(21)di∗-=∑j=1md∗hij,hj-ωj=∑j=1mωj1/l∑q=1lhijλq-hjλq-21-Δi1-∆-,i=1,2,…,n.Then we can solve relative closeness between supplierAi and positive ideal solution A+:(22)cAi=di∗-di∗++di∗-,i=1,2,…,n,where0<cAi<1,i=1,2,⋯,n⁡。 Based on the magnitude of closeness degree, it can select the optimal green supplier.


## 5. A Case Study for Green Suppliers Evaluation

This paper takes green suppliers for example. There are 4 suppliers provided (Ai(i=1,2,3,4)) in a green supply chain. Through expert interview and literature retrieval and combining with statistical data and expert advice, the evaluation index can be divided into 4 first-grade indexes and 20 second-grade indexes, as shown in Table 1.Table 1
The evaluation index system of the green supplier.
Evaluation objective Level indicators Secondary indicators Evaluation index system of green supplier selection Product level Quality management system (O1) Relative price level (O2) Equipment (O3) Product qualification ratio (O4) Logistics cost (O5) Service level On-time delivery rate (O6) Order completion rate (O7) Services response speed (O8) Service satisfaction (O9) Development level Information level (O10) Profit growth rate (O11) R&D input (O12) Corporate reputation (O13) Staff quality (O14) Financial situation (O15) Environmental competitiveness Exhaust gas emission (O16) Energy consumption (O17) Cleaner production level (O18) “Three-waste” recycling rate (O19) Green enterprise image (O20)Then, the direct effect matrix B is calculated by expert evaluation. The matrix can make information put in order as shown in Table2.Table 2
The direct effect matrix B of the green supplier.
Index O 1 O 2 O 3 O 4 O 5 O 6 O 7 O 8 O 9 O 10 O 11 O 12 O 13 O 14 O 15 O 16 O 17 O 18 O 19 O 20 O 1 0 2 4 4 0 0 0 0 1 0 1 1 2 1 0 2 2 0 1 0 O 2 3 0 1 0 1 0 0 0 0 0 4 3 0 0 4 0 1 0 1 0 O 3 2 4 0 4 0 2 3 2 0 2 2 4 0 0 4 2 2 3 1 2 O 4 4 3 4 0 2 2 3 0 4 0 3 1 4 3 2 0 0 1 0 0 O 5 1 0 0 1 0 1 0 0 0 1 4 0 0 0 3 0 0 0 0 0 O 6 0 0 2 2 2 0 4 0 4 2 1 0 3 0 2 0 0 0 0 0 O 7 1 0 4 1 0 4 0 0 4 2 3 0 3 0 3 0 0 0 0 0 O 8 4 0 3 1 0 2 3 0 4 0 3 0 2 3 1 0 0 0 0 0 O 9 4 0 2 3 0 4 4 0 0 0 4 0 4 4 0 2 0 0 0 0 O 10 0 0 2 0 1 0 0 2 0 0 2 1 0 0 0 0 1 1 0 1 O 11 3 4 4 1 3 0 4 0 0 0 0 2 0 0 4 0 0 1 1 0 O 12 0 2 4 0 1 0 0 0 0 3 3 0 0 0 4 1 0 3 1 3 O 13 3 0 1 3 0 0 0 2 3 1 2 0 0 2 0 1 1 1 0 3 O 14 0 0 0 1 0 0 1 2 4 0 0 0 1 0 0 0 0 0 0 0 O 15 0 3 4 1 1 1 0 0 0 2 3 3 0 0 0 0 0 1 0 0 O 16 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 3 2 4 4 O 17 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 3 0 2 4 3 O 18 0 0 2 0 0 0 1 0 0 0 1 1 0 0 2 4 4 0 3 4 O 19 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 4 3 0 4 O 20 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 4 4 0 4 0Thus we can acquire the total influence matrix T. The computing formulas of the matrix result in the cause-effect relationship which is presented in Table3.Table 3
The cause-effect relationship of the green supplier.
Evaluation index Effect Affected Centrality Cause degree (fi) degree (ei) degree (mi) degree (ni) O1 Quality management system 1.3020 1.4770 2.7790 -0.1750 O2 Relative price level 1.0294 1.2872 2.3166 -0.2578 O3 Equipment 2.1965 2.2783 4.4748 -0.0818 O4 Product qualification ratio 2.1557 1.3253 3.4810 0.8304 O5 Logistics cost 0.6581 0.6563 1.3144 0.0018 O6 On-time delivery rate 1.3786 0.9162 2.2948 0.4624 O7 Order completion rate 1.5961 1.3306 2.9267 0.2655 O8 Services response speed 1.6362 0.4482 2.0844 1.1880 O9 Service satisfaction 1.8683 1.1995 3.0678 0.6688 O10 Information level 0.6564 0.7670 1.4234 -0.1106 O11 Profit growth rate 1.5873 1.9612 3.5485 -0.3739 O12 R&D input 1.3670 1.0711 2.4381 0.2959 O13 Corporate reputation 1.3510 1.2392 2.5902 0.1118 O14 Staff quality 0.6213 0.6765 1.2978 -0.0552 O15 Financial situation 1.1584 1.7655 2.9239 -0.6071 O16 Exhaust gas emission 0.7622 1.1726 1.9348 -0.4104 O17 Energy consumption 0.6145 1.2672 1.8817 -0.6527 O18 Cleaner production level 1.0914 1.0870 2.1784 0.0044 O19 “Three-waste” recycling rate 0.7153 1.1618 1.8771 -0.4465 O20 Green enterprise image 0.6856 1.3436 2.0292 -0.6580From Table3, it is significant that the equipment, product qualification ratio, and service satisfaction have high effect degrees. Among them, equipment is highest, because equipment is related to not only product quality and qualified rate, but also environmental pollution. Meanwhile, the indexes have a lower centrality degree and cause degree including the information level, staff quality, energy consumption, and “three-waste” recycling rate. After eliminating the secondary indexes, this paper would select 16 main evaluation indexes of the green supplier.Index selection should follow principles which include systematicness overall, science, stability, flexibility, and brevity. Based on the research and principles, this paper proposes the index system of the green supplier. 16 evaluation indexes are shown in Figure2, respectively.Figure 2
The evaluation index system of a green supplier.And suppliers’ indexes can be scored by experts after obtaining the index system. Evaluation information is shown in Table4.Table 4
The hesitant fuzzy decision-making matrix.
O 1 O 2 O 3 O 4 O 5 O 6 O 7 O 8 A 1 0.5,0.8,0.7,0.5 0.9,0.4,0.3,0.3 0.4,0.5,0.2,0.1 0.6,0.5,0.3,0.9 0.7,0.6,0.3,0.3 0.8,0.6,0.5,0.2 0.7,0.6,0.9,0.6 0.8,0.1,0.2,0.1 A 2 0.9,0.7,0.6,0.6 0.7,0.5,0.3,0.3 0.8,0.6,0.3,0.3 0.9,0.7,0.6,0.6 0.5,0.2,0.3,0.2 0.4,0.5,0.6,0.4 0.6,0.8,0.9,0.6 0.4,0.5,0.8,0.9 A 3 0.1,0.6,0.4,0.3 0.4,0.2,0.5,0.2 0.2,0.3,0.6,0.2 0.4,0.6,0.5,0.2 0.3,0.2,0.4,0.2 0.3,0.5,0.6,0.1 0.4,0.5,0.6,0.4 0.2,0.3,0.5,0.2 A 4 0.3,0.1,0.5,0.2 0.1,0.3,0.6,0.1 0.4,0.3,0.6,0.3 0.4,0.1,0.5,0.6 0.6,0.2,0.1,0.1 0.3,0.2,0.6,0.2 0.5,0.4,0.5,0.2 0.5,0.1,0.4,0.1  O 9 O 10 O 11 O 12 O 13 O 14 O 15 O 16  A 1 0.9,0.8,0.6,0.5 0.3,0.4,0.5,0.4 0.3,0.6,0.5,0.3 0.1,0.3,0.5,0.1 0.5,0.5,0.2,0.4 0.2,0.6,0.1,0.5 0.5,0.3,0.4,0.2 0.3,0.4,0.1,0.8 A 2 0.2,0.4,0.7,0.5 0.6,0.2,0.1,0.4 0.2,0.5,0.8,0.6 0.6,0.5,0.2,0.7 0.4,0.5,0.2,0.6 0.5,0.3,0.4,0.1 0.4,0.3,0.1,0.6 0.6,0.4,0.3,0.4 A 3 0.6,0.3,0.1,0.4 0.2,0.3,0.5,0.2 0.5,0.6,0.4,0.2 0.3,0.5,0.2,0.1 0.4,0.6,0.7,0.9 0.1,0.3,0.6,0.7 0.3,0.4,0.2,0.7 0.4,0.3,0.6,0.1 A 4 0.2,0.6,0.5,0.4 0.4,0.2,0.3,0.9 0.5,0.3,0.4,0.7 0.5,0.6,0.1,0.5 0.5,0.1,0.6,0.2 0.2,0.4,0.8,0.2 0.4,0.7,0.6,0.3 0.5,0.6,0.2,0.9Then the proposed approach of the improved type signed distance in a hesitant fuzzy set is the following.In Step1, suppliers and indexes are denoted by A=A1,A2,A3,A4 and O=O1,O2,⋯,O16∣j=1,2,⋯,16, respectively.In Step2, we denote the hesitant fuzzy decision-making matrix H=hij4×16 as in Table 4.In Step3, we calculate the positive ideal solution A+ and negative ideal solution A-, respectively:(23)A+=x1,0.6,0.6,0.5,0.4,x2,0.5,0.6,0.8,0.9,x3,0.6,0.5,0.5,0.7,x4,0.5,0.6,0.7,0.9,x5,0.5,0.6,0.6,0.7,x6,0.2,0.8,0.8,0.5,x7,0.6,0.9,0.9,0.8,x8,0.5,0.7,0.6,0.7,x9,0.6,0.6,0.6,0.9,x10,0.6,0.6,0.5,0.4,x11,0.5,0.6,0.8,0.9,x12,0.6,0.5,0.5,0.7,x13,0.5,0.6,0.7,0.9,x14,0.5,0.6,0.6,0.7,x15,0.5,0.7,0.6,0.7,x16,0.6,0.6,0.6,0.9;A-=x1,0.2,0.3,0.1,0.2,x2,0.2,0.2,0.3,0.2,x3,0.1,0.3,0.2,0.1,x4,0.4,0.5,0.1,0.4,x5,0.1,0.1,0.1,0.1,x6,0.1,0.4,0.3,0.2,x7,0.1,0.1,0.2,0.2,x8,0.3,0.3,0.1,0.2,x9,0.3,0.3,0.1,0.1,x10,0.2,0.2,0.1,0.2,x11,0.2,0.2,0.3,0.2,x12,0.1,0.3,0.2,0.1,x13,0.4,0.5,0.1,0.4,x14,0.1,0.1,0.1,0.1,x15,0.3,0.3,0.1,0.2,x16,0.3,0.3,0.1,0.1.In Steps4 and 5, there are the results of attribute hesitancy degree and the improved type signed distance in Table 5 and Table 6, respectively.Table 5
Attribute hesitancy degree of supplier.
O 1 O 2 O 3 O 4 O 5 O 6 O 7 O 8 A 1 0.1833 0.3167 0.2333 0.3167 0.25 0.3167 0.1667 0.3667 A 2 0.1667 0.2333 0.3000 0.1667 0.1667 0.1167 0.1833 0.3000 A 3 0.2667 0.1833 0.2167 0.2167 0.1167 0.2833 0.1167 0.1667 A 4 0.2167 0.2833 0.1667 0.2667 0.2667 0.2167 0.1667 0.2500  O 9 O 10 O 11 O 12 O 13 O 14 O 15 O 16  A 1 0.2333 0.1000 0.1833 0.2333 0.1667 0.3000 0.1667 0.3667 A 2 0.2667 0.2833 0.3167 0.2667 0.2167 0.2167 0.2667 0.1500 A 3 0.2667 0.1667 0.2167 0.2167 0.2667 0.3333 0.2667 0.2667 A 4 0.2833 0.2167 0.3667 0.2167 0.2500 0.3000 0.2333 0.3667Table 6
The improved type signed distance.
O 1 O 2 O 3 O 4 O 5 O 6 O 7 O 8 A 1 1.4348 1.6376 1.2367 1.7734 1.2612 1.5341 0.9454 2.3452 A 2 1.7654 1.2589 1.6130 1.6354 0.7189 0.9087 1.0320 2.9301 A 3 1.6868 1.2014 1.2638 1.7710 0.9284 1.0717 0.8589 1.6761 A 4 1.8422 1.8601 1.0545 1.7748 0.9914 1.1574 1.1494 1.6274  O 9 O 10 O 11 O 12 O 13 O 14 O 15 O 16  A 1 2.2699 0.7393 1.3220 1.7144 0.8998 2.0262 1.2613 1.8329 A 2 1.5705 1.5438 1.7981 1.6100 0.8564 1.5685 1.2320 1.4020 A 3 2.0515 0.9487 1.6218 1.4917 1.7668 2.0106 1.2394 2.3445 A 4 1.7027 1.0518 2.3468 1.4318 0.9706 1.8417 1.5723 1.9326According to the distance, the weight vector can be calculated:(24)ω=0.0702,0.0621,0.0539,0.0725,0.0407,0.0487,0.0416,0.0894,0.0792,0.0447,0.0739,0.0651,0.0468,0.0776,0.0553,0.0783T.In Step6, we compute the distances di∗+ and di∗- from each supplier Ai to the positive-negative ideal solution by using Eqs. (25) and (26):(25)d1∗+=0.3652,d2∗+=0.2999,d3∗+=0.4589,d4∗+=0.4695;d1∗-=0.4284,d2∗-=0.4427,d3∗-=0.3110,d4∗-=0.3368.In Steps7 and 8, we calculate the relative closeness cAi values and rank them:(26)cA1=0.5398;cA2=0.5962,cA3=0.4039,cA4=0.4177.A 2 ≻ A 1 ≻ A 4 ≻ A 3, so A2 is the ideal supplier.


## 6. Result Analysis

Based on the proposed algorithm, this part compares the results of the improved TOPSIS and traditional TOPSIS methods and studies the different multiattribute decision-making methods of the improved TOPSIS and SAW methods. By comparing the results among the three methods, the same and different points and advantages can be found.


### 6.1. The Result of the Traditional TOPSIS Method

The hesitant fuzzy set reflects the hesitancy degree of differences. And deviations among elements are greater; hesitancy degree is higher. Meanwhile Xu and Zhang (2013) [21] applied the ideal point method and obtained attribute weight:(27)ω=0.0778,0.0600,0.0557,0.0736,0.0453,0.0491,0.0520,0.0815,0.0769,0.0494,0.0666,0.0678,0.0488,0.0691,0.0577,0.0687T.By using the attribute weight, it can get the closeness degree as follows:(28)d1∗+=0.2595,d2∗+=0.2078,d3∗+=0.3843,d4∗+=0.4083;d1∗-=0.3693,d2∗-=0.3830,d3∗-=0.2092,d4∗-=0.1773.Through the closeness degree, the result can be drawn:(29)cA1=0.5873,cA2=0.6483,cA3=0.3525,cA4=0.3028.Then the ranking isA2≻A1≻A3≻A4, and A2 is the ideal supplier.


### 6.2. The Result of the SAW Method

SAW is the simple multiple-attribute decision-making method and is the basis of decision-making analysis. The process of SAW involves the standardization of the decision matrix, weight determination, and alternative ranking. it is a little different from the TOPSIS method.Firstly, we standardize the decision-making matrix. According to Zhu and Xu [35], all attribute values can be transformed into the benefit attribute.(30)hij′=hij,ifOjisthebenefitattributehijC,ifOjisthecostattribute,wherehijC is the complement of hij, as follows: hijC=1-hij.Meanwhile, it can calculate the attribute weightω by using the maximizing deviation method:(31)ω=0.0778,0.0600,0.0557,0.0736,0.0453,0.0491,0.0520,0.0815,0.0769,0.0494,0.0666,0.0678,0.0488,0.0691,0.0577,0.0687T.Then we calculate the Euclidean distance of the decision-making matrix by using (5), as in Table 7.Table 7
The Euclidean distance of suppliers.
O 1 O 2 O 3 O 4 O 5 O 6 O 7 O 8 A 1 0.7274 0.5361 0.4212 0.5326 0.6673 0.5507 0.5954 0.7472 A 2 0.8744 0.4244 0.5472 0.7771 0.3436 0.4894 0.6404 1.1386 A 3 0.7344 0.4444 0.3669 0.5104 0.4059 0.4387 0.5959 0.6272 A 4 0.8674 0.4924 0.3505 0.5677 0.4823 0.5302 0.7634 0.6444  O 9 O 10 O 11 O 12 O 13 O 14 O 15 O 16  A 1 0.9364 0.3591 0.3947 0.5530 0.4012 0.1573 0.3288 0.4081 A 2 0.4416 0.3591 0.4306 0.6003 0.3498 0.2850 0.2995 0.4555 A 3 0.5036 0.3591 0.3590 0.4898 0.7335 0.2532 0.2932 0.4055 A 4 0.4645 0.3591 0.4817 0.5157 0.3971 0.1573 0.4387 0.5639Finally, it can rank green suppliers by(32)Ci=∑j=1nωjdij,whereCi represents the evaluation value of the nth supplier.Based on calculation, the evaluation values are as follows:(33)C1=0.5315,C2=0.5533,C3=0.4755,C4=0.5132.Then the ranking isA2≻A1≻A4≻A3, and A2 is the ideal supplier.


### 6.3. Results Comparison

It is showed that even though the comparison result of the three methods is thatA2 is always the ideal supplier, the new algorithm pays more attention to the environmental index, and A4 changes greatly. It is the reason that the hesitancy degree of A4 is higher. Then a higher hesitancy degree makes distance bigger between A4 and the negative ideal point. In this case, A4 in the improved type signed distance is closer to the ideal point. So the improved distance has a more dipartite degree.It can be found that the improved TOPSIS method is simple and highly sensitive. For example, we change the value of attributeO8 of supplier Α2; that is to say, h28=0.4,0.2,0.4,0.1, and it can get the changed result by using the improved TOPSIS and SAW method: A1≻A2≻A4≻A3; obviously A1 is the ideal supplier. Moreover it can get the unchanged result by the TOPSIS method: A2≻A1≻A3≻A4, so it has the same order as before, and A2 is still the ideal supplier. Therefore, the result responds to the subtle change due to the high sensitivity of the improved TOPSIS method. On the other hand, it reflects that the improved TOPSIS method is used widely and responds to changes sensitively.


### 6.4. Algorithm Advantage

By comparison, the advantages of the algorithm are the following:(i) This paper proposes the improved type signed distance which adds the hesitancy degree. And the new method has a dipartite degree(ii) The method of determining weight is the maximizing deviation method which is more objective. On the other hand, deviation of adding hesitancy degree is more reliable(iii) The hesitancy degree pays more attention to the hesitancy level of the decision-maker, and the ideal solution method truthfully reflects distance. So it not only affects alternatives’ ranking, but also affects the selection of green suppliers


## 6.1. The Result of the Traditional TOPSIS Method

The hesitant fuzzy set reflects the hesitancy degree of differences. And deviations among elements are greater; hesitancy degree is higher. Meanwhile Xu and Zhang (2013) [21] applied the ideal point method and obtained attribute weight:(27)ω=0.0778,0.0600,0.0557,0.0736,0.0453,0.0491,0.0520,0.0815,0.0769,0.0494,0.0666,0.0678,0.0488,0.0691,0.0577,0.0687T.By using the attribute weight, it can get the closeness degree as follows:(28)d1∗+=0.2595,d2∗+=0.2078,d3∗+=0.3843,d4∗+=0.4083;d1∗-=0.3693,d2∗-=0.3830,d3∗-=0.2092,d4∗-=0.1773.Through the closeness degree, the result can be drawn:(29)cA1=0.5873,cA2=0.6483,cA3=0.3525,cA4=0.3028.Then the ranking isA2≻A1≻A3≻A4, and A2 is the ideal supplier.


## 6.2. The Result of the SAW Method

SAW is the simple multiple-attribute decision-making method and is the basis of decision-making analysis. The process of SAW involves the standardization of the decision matrix, weight determination, and alternative ranking. it is a little different from the TOPSIS method.Firstly, we standardize the decision-making matrix. According to Zhu and Xu [35], all attribute values can be transformed into the benefit attribute.(30)hij′=hij,ifOjisthebenefitattributehijC,ifOjisthecostattribute,wherehijC is the complement of hij, as follows: hijC=1-hij.Meanwhile, it can calculate the attribute weightω by using the maximizing deviation method:(31)ω=0.0778,0.0600,0.0557,0.0736,0.0453,0.0491,0.0520,0.0815,0.0769,0.0494,0.0666,0.0678,0.0488,0.0691,0.0577,0.0687T.Then we calculate the Euclidean distance of the decision-making matrix by using (5), as in Table 7.Table 7
The Euclidean distance of suppliers.
O 1 O 2 O 3 O 4 O 5 O 6 O 7 O 8 A 1 0.7274 0.5361 0.4212 0.5326 0.6673 0.5507 0.5954 0.7472 A 2 0.8744 0.4244 0.5472 0.7771 0.3436 0.4894 0.6404 1.1386 A 3 0.7344 0.4444 0.3669 0.5104 0.4059 0.4387 0.5959 0.6272 A 4 0.8674 0.4924 0.3505 0.5677 0.4823 0.5302 0.7634 0.6444  O 9 O 10 O 11 O 12 O 13 O 14 O 15 O 16  A 1 0.9364 0.3591 0.3947 0.5530 0.4012 0.1573 0.3288 0.4081 A 2 0.4416 0.3591 0.4306 0.6003 0.3498 0.2850 0.2995 0.4555 A 3 0.5036 0.3591 0.3590 0.4898 0.7335 0.2532 0.2932 0.4055 A 4 0.4645 0.3591 0.4817 0.5157 0.3971 0.1573 0.4387 0.5639Finally, it can rank green suppliers by(32)Ci=∑j=1nωjdij,whereCi represents the evaluation value of the nth supplier.Based on calculation, the evaluation values are as follows:(33)C1=0.5315,C2=0.5533,C3=0.4755,C4=0.5132.Then the ranking isA2≻A1≻A4≻A3, and A2 is the ideal supplier.


## 6.3. Results Comparison

It is showed that even though the comparison result of the three methods is thatA2 is always the ideal supplier, the new algorithm pays more attention to the environmental index, and A4 changes greatly. It is the reason that the hesitancy degree of A4 is higher. Then a higher hesitancy degree makes distance bigger between A4 and the negative ideal point. In this case, A4 in the improved type signed distance is closer to the ideal point. So the improved distance has a more dipartite degree.It can be found that the improved TOPSIS method is simple and highly sensitive. For example, we change the value of attributeO8 of supplier Α2; that is to say, h28=0.4,0.2,0.4,0.1, and it can get the changed result by using the improved TOPSIS and SAW method: A1≻A2≻A4≻A3; obviously A1 is the ideal supplier. Moreover it can get the unchanged result by the TOPSIS method: A2≻A1≻A3≻A4, so it has the same order as before, and A2 is still the ideal supplier. Therefore, the result responds to the subtle change due to the high sensitivity of the improved TOPSIS method. On the other hand, it reflects that the improved TOPSIS method is used widely and responds to changes sensitively.


## 6.4. Algorithm Advantage

By comparison, the advantages of the algorithm are the following:(i) This paper proposes the improved type signed distance which adds the hesitancy degree. And the new method has a dipartite degree(ii) The method of determining weight is the maximizing deviation method which is more objective. On the other hand, deviation of adding hesitancy degree is more reliable(iii) The hesitancy degree pays more attention to the hesitancy level of the decision-maker, and the ideal solution method truthfully reflects distance. So it not only affects alternatives’ ranking, but also affects the selection of green suppliers


## 7. Conclusion

The green industry has developed into a pillar industry in the economy development. Meanwhile, suppliers occupy an important position in a supply chain; particularly supplier evaluation involves the optimization problem of a green supply chain. Thus this paper proposes the new type algorithms and indicators which have an important value of practical significance.This paper determines the index system of a green supplier firstly by using the DEMATEL method and then gets the decision-making matrix by a decision-maker and hesitancy degree. Moreover, the method of maximizing deviation can help in obtaining attribute weight. Based on calculation, it can get relative closeness. Finally, suppliers can be ranked by relative closeness. Subsequently, a case proves the effectiveness of the algorithm.The example proves that the improved type signed distance of TOPSIS can distinguish greatly for green suppliers. Due to space limitation, the study only lists the limiting factors. In the future, research would be devoted to algorithm improvement and study on the influence factors.



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*Source: 2601301-2019-03-27.xml*