Supplier Evaluation in Green Supply Chain: An Adaptive Weight D-S Theory Model Based on Fuzzy-Rough-Sets-AHP Method

1. Introduction

With the fast growth of economic globalization, the resources and environment are facing enormous pressure now. Under this background, green supply chain management (GSCM) appears very important [1]. Green supply chain (GSC) was put forward in 1996 by the Manufacturing Research Center (MRC) of Michigan State University in a research on environmentally responsible manufacturing [2,3]. GSCM contains many contents, such as green supplier evaluation (GSE), green product design (GPD), green production (GP), green marketing and waste recycling (GMWR). As the upstream in the whole supply chain, the role of supplier in protecting the environment and saving costs can be transmitted to every part of the downstream through the supply chain, so as to improve the compatibility of supply chain and environment [4]. Manufacturing enterprises begin to measure the green degree of their suppliers, and one of the key steps to measure the green degree of an enterprise is how to choose the best supplier as a long-term partner [5]. By choosing the suitable green supplier, enterprises can largely improve the resource recycling rate and reduce pollutant emissions, and provide green control and processing for raw materials supplied by suppliers. Thus, the whole supply chain will be green, and a green strategic partnership is established with the supplier to achieve sustainable development. In general, one of the keys to building a green supply chain is to choose a suitable supplier.

The rest of this paper is organized as follows. In Section 2, an overview of the existing researches on supplier evaluation in GSC is provided. In Section 3, an adaptive weight D-S evidence theory model based on fuzzy-rough-sets-AHP method is put forward for supplier evaluation in GSC. In Section 4, a bearing cage supplier evaluation case is given. At last, this paper is concluded in Section 5.

2. Literature Review

A lot of significant studies on supplier evaluation in GSC are seen in the existing literatures. The representative research mainly focuses on two aspects as follows.

One is application of single method to the supplier evaluation problem. Buyukozkan and Cifci [6] proposed a fuzzy analytic network process (ANP) method based on multi-person decision-making schema under incomplete preference relationships for vendor selection. Based on the application of rough set theory to study the relations among organizational properties, supplier development program involvement properties, and performance outcome properties, Bai and Sarkis [7] put forward a formal model for green supplier selection. Tseng and Chiu [8] determined the weights of criteria and alternatives according to both by qualitative and quantitative information and sorted alternative suppliers based on a grey relational analysis (GRA). To obtain the best green supplier for a plastic manufacturing company in Singapore, Kannan et al. [9] put forward a fuzzy axiomatic design (FAD) method. To evaluate environmental performance of suppliers, Awasthi et al. [[10] proposed a fuzzy multi-criteria method (FmCM). By adding green criteria into the criteria framework of supplier selection, Yeh and Chuang [11] proposed an optimum mathematical planning model (OMPM) for green partner selection. Wu et al. [12] proposed a fuzzy linguistic decision-making method to solve the problem of selecting green supplier.

The other is integrated application of two or more than two methods for supplier evaluation. Li and Zhao [13] built the assessment model by using threshold method and gray correlation analysis (GCA) for vehicle component supplier selecting. Yan [14] used genetic algorithm (GA) and AHP to realize the dynamic adjustment of index weights in green supplier selection. Kuo et al. [15] proposed a hybrid approach based on artificial neural network (ANN), data envelopment analysis (DEA), and ANP for green supplier selection. Kuo and Lin [16] put forward a supplier selection approach based on ANP and DEA with the consideration of green indicators due to environmental protection issues. Based on fuzzy decision-making trial and evaluation laboratory model (DEMATEL), ANP, and technique for order performance by similarity to ideal solution (TOPSIS), Buyukozkan and Cifci [17] proposed a hybrid fuzzy multi-criteria decision-making (MCDM) approach for green supplier evaluation. By combining AHP and TOPSIS, Luo and Peng [18] proposed an integrated model for both of evaluation and selection of green supplier.

The above two kinds of methods have theoretical basis and practical value, but they also have some limitations. The subjectivity of fuzzy AHP in determining the index weight is too large. Neural network calculation process is complex, redundant, which will result in lack of accurate calculation. TOPSIS method has the advantages of convenient calculation and strong applicability, but the evaluation process may be missing information and the results are not objective enough. Additionally, each evaluation expert is required to give personal subjective evaluation information when considering the same evaluation index set. When different evaluation experts compare multiple indicators on the same level, it is easy to appear contradictory or chaotic judgment and evaluation. Because of the limitations of evaluation experts’ understanding of supplier capabilities, the evaluation index value is often characterized by uncertainty and incompleteness. Moreover, the evaluation index weight is obviously variable when the demand has changed or the preference of the evaluation experts is different.

Therefore, we proposed an adaptive weight D-S theory model in this paper to solve the uncertainty and incompleteness problems of index value of supplier evaluation in GSC. The adaptive weight of evaluation index is determined by our designed fuzzy-rough-sets-AHP method.

3. Adaptive Weight D-S Theory Model for Supplier Evaluation

3.1 Establishment of Index Framework

For supplier evaluation in green supply chain, to build a comprehensive index framework is of great significance. On the one hand, product attribute is the main ability embodiment of a supplier; on the other, comprehensive ability can give a strong support to the product attribute of a supplier. Here, the comprehensive ability mainly contains internal competitiveness, external competitiveness, and cooperation ability.

For internal competitiveness, it mainly can be divided into innovation ability, manufacturing capacity and agility. Because a supplier is not isolated in supply chain, it is unavoidably limited by its external competitiveness. For external competitiveness, it mainly can be divided into economic environment, geographical environment, social environment, and legal environment. For cooperation ability, it mainly can be divided into technical compatibility degree, cultural compatibility degree, information platform compatibility degree and reputation.

Therefore, the index framework of supplier evaluation in GSC is built as shown in Fig. 1. It can be represented as a criterion set {C₁, C₂,C₃,C₄}. Here, C₁ stands for product attribute, C₂ stands for internal competitiveness, C₃ stands for external competitiveness, and C₄ stands for cooperation ability. Among them, C₁={C₁₁, C₁₂, C₁₃, C₁₄}. In other words, C1 is divided into four indexes: C₁₁, C₁₂, C₁₃, and C₁₄. Here, C₁₁ stands for cost, C₁₂stands for quality, C₁₃ stands for service, and C₁₄ stands for flexibility.

Four criterions are divided into two types as follows. (1) Comprehensive qualitative type: C₁.(2) Quantitative type: C₂, C₃, and C₄. For comprehensive qualitative type criterion, its value is determined by its subordinate indexes. For quantitative type criterion, its value is obtained by expert score method. Similarly, four indexes of C₁ are divided into two types as follows. (1) Quantitative type: C₁₁ and C₁₂. (2) Direct qualitative type: C₁₃ and C₁₄.

3.2 Determination of the Adaptive Weight

The AHP method [19], which was put forward by Thomas L. Saaty, can not only make clear the hierarchical structure of the components of complex problem, but also verify the consistency of the results. Therefore, it has been widely applied in the weighting of multi-attribute decision-making problem [20-22]. The traditional AHP uses exact numbers to represent the relative importance between indexes. The evaluation of relative importance between indexes in supplier evaluation by experts depends on personal judgment and subjective experience, so using exact numbers to represent the relative importance between indexes is unjustified to some extent.

Fuzzy number can give expression to the connatural uncertainty of expert's preference. Additionally, the evaluation of relative importance between indexes by multiple experts is obviously indistinguishable when integrating the opinions of all experts. Instead of a membership function, rough boundary interval [21,23] can represent the indistinguishability as a set boundary area. It can better integrate the opinions of all evaluation experts. Accordingly, a fuzzy-rough-sets-AHP method is designed to solve the adaptive weight of evaluation index.

We use U to represent a domain which is actually a nonempty finite set of objects. Y is any object in U. In U, all objects are divided into n partitions: S₁,S₂,…,S_n. If these n partition has the order of [TeX:] $$S_{1}<S_{2}<\ldots<S_{n}$$, the upper and lower approximation sets of any partition S_i (1≤i≤n) can be defined as follows:

where U/R(Y) represents the partition of the indistinct relationship R(Y) in U.

According to the above definition, any ambiguous partition S_i in U can be represented by its rough boundary interval RN(S_i). RN(S_i) consists of its rough lower limit [TeX:] $$\underline{L}\left(S_{i}\right)$$ and rough upper limit [TeX:] $$\overline{L}\left(S_{i}\right)$$ which are defined as follows:

where [TeX:] $$\underline{N}\left(S_{i}\right)$$ is the number of objects in the upper approximation set of Si and [TeX:] $$\underline{N}\left(S_{i}\right)$$ is the number of objects in the upper approximation set of S_i.

As can be seen, an ambiguous partition in the domain can be represented by a rough boundary interval containing a rough lower limit and a rough upper limit as follows:

We start from the bottom layer of index framework shown in Fig. 1. There are q experts. The index set is [TeX:] $$\left\{C_{11}, C_{12}, \ldots, C_{1 l}\right\} . \text { Here }, l=4$$

Step 1: According to the evaluation of expert [TeX:] $$k(k=1,2, \ldots, q) \text { on }\left\{C_{11}, C_{12}, \ldots, C_{1 l}\right\}$$, the fuzzy reciprocal judgment matrix E^k is constructed as follows:

where [TeX:] $$e_{i j}^{k}$$ represents the score of supplier j compared to supplier i evaluated by expert k, here i,j=1,2,...,l and [TeX:] $$i \neq j . \quad e_{i, j}^{k}=\left(a_{i, j}^{k}, b_{i, j}^{k}, c_{i, j}^{k}, d_{i, j}^{k}\right)$$ is a trapezoidal fuzzy number and [TeX:] $$a_{i, j}^{k}, b_{i, j}^{k}, c_{i, j}^{k} \quad \text { and } \quad d_{i, j}^{k}$$ [TeX:] $$\left(a_{i, j}^{k} \leq b_{i, j}^{k} \leq c_{i, j}^{k} \leq d_{i, j}^{k}\right)$$ are all positive real numbers. Then we verify the consistency of E^k. If it is qualified, do the next step; otherwise, redo this step.

Step 2: E^k is split into a^k, b^k, c^k, d^k. The expression of a^k is as follows:

Step 3: Based on a¹,a²,…, a^q, the rough group decision matrix is constructed as follows:

where [TeX:] $$a_{i, j}=\left\{a_{i, j}^{1}, a_{i, j}^{2}, \ldots, a_{i, j}^{q}\right\}$$, here [TeX:] $$i, j=1,2, \ldots, l \text { and } i \neq j$$.

The rough boundary interval of [TeX:] $$a_{i, j}^{k} \in a_{i, j}(k=1,2, \ldots, q)$$ is obtained as follows:

where [TeX:] $$a_{i, j}^{k,-}$$ is the rough lower limit of [TeX:] $$a_{i, j}^{k}$$ in set [TeX:] $$a_{i, j}^{k,+}$$ is the rough upper limit of [TeX:] $$a_{i, j}^{k}$$ in set [TeX:] $$a_{i, j}$$.

Therefore, the rough boundary interval of [TeX:] $$a_{i, j}$$ can be represented as follows:

Based on the operational rule of rough boundary interval, the average form of [TeX:] $$R N\left(a_{i, j}\right)$$ is obtained as follows:

where [TeX:] $$a_{i, j}^{-}$$ is the rough lower limit of set [TeX:] $$a_{i, j} \text { and } a_{i, j}^{+}$$ is the rough upper limit of set [TeX:] $$a_{i, j}$$.

Step 4: The rough judgement matrix is constructed as follows:

EA is divided into EA^- and EA⁺. Here, EA^- is the rough lower limit matrix and EA⁺ is the rough upper limit matrix. EA^- and EA⁺ are expressed as follows:

The eigenvectors corresponding to the maximum eigenvalues of EA^- and EA⁺ are obtained respectively as follows:

where [TeX:] $$v a_{i}^{-}$$ are the value of VA^- on the i (i=1,2,...,l) dimension and [TeX:] $$v a_{i}^{+}$$ are the value of VA⁺on the i (i=1,2,...,l) dimension.

Then, we can get that [TeX:] $$g a_{i}=\left(\left|v a_{i}^{-}\right|+\left|v a_{i}^{+}\right|\right) / 2$$, and a set [TeX:] $$G A=\left\{g a_{1}, g a_{2}, \ldots, g a_{l}\right\}$$ is obtained.

Step 5: We repeat steps 3 and 4, so [TeX:] $$G B_{t}=\left\{g b_{1}, g b_{2}, \ldots, g b_{l}\right\}, G C=\left\{g c_{1}, g c_{2}, \ldots, g c_{l}\right\} \text { and } G D=\left\{g d_{1}, g d_{2}, \ldots, g d_{l}\right\}$$ can be obtained. Then the adaptive weight of evaluation indexes [TeX:] $$C_{11}, C_{12}, \dots, C_{1 l}$$ with the trapezoidal fuzzy number form are [TeX:] $$z_{1}=\left(g a_{1}, g b_{1}, g c_{1}, g d_{1}\right), z_{2}=\left(g a_{2}, g b_{2}, g c_{2}, g d_{2}\right), \ldots, z_{l}=\left(g a_{l}, g b_{l}, g c_{l}, g d_{l}\right)$$. Here we use gravity model appoach to convert [TeX:] $$z_{i}=\left(g a_{i}, g b_{i}, g c_{i}, g d_{i}\right)(i=1,2, \ldots, l)$$ into real number ri as follows:

We normalize [TeX:] $$r_{1}, r_{2}, \ldots, r_{l}$$ and can obtain the adaptive weight of evaluation indexes [TeX:] $$C_{1 i}$$ as follows:

Step 6: For indexes [TeX:] $$C_{1}, C_{2}, C_{3 \mathrm{and}} \mathrm{C}_{4}$$, we repeat steps 1-5 and obtain the adaptive weights of them as [TeX:] $$\omega\left(C_{1}\right), \omega\left(C_{2}\right), \omega\left(C_{3}\right), \omega\left(C_{4}\right)$$.

3.3 D-S Theory Decision Regulations

By D-S theory, we can deal with the multi-criteria decision problems with uncertainty and incompleteness [24,25]. In the existing researches, it has been certified that a content result can be obtained and the uncertainty of decision can be decreased based on D-S theory [26,29]. According to D-S theory, we define the suppliers to be evaluated [TeX:] $$x_{1}, x_{2}, \dots, x_{i}, \dots, x_{N}$$ as the D-S identification framework [TeX:] $$\Theta=\left\{x_{1}, x_{2}, \ldots, x_{i}, \ldots, x_{N}\right\}$$. All possible subsets of can be expressed by power set 2. When all elements in are incompatible and independent with each other, the number of elements in 2 is 2^N. Then, a set function [TeX:] $$m : 2^{\Theta} \rightarrow[0,1]$$, which satisfies [TeX:] $$m(\phi)=0 \text { and } \sum_{A \subset \Theta} m(A)=1$$ , is defined. Here, m is known as the basic probability allocation (BPA) function on and A is a supplier to be evaluated. m(A), which is the BPA value of A, represents the trust degree in A. Any supplier to be evaluated satisfying the condition m(A)>0 is called a focal element.

For [TeX:] $$A \subseteq \Theta$$, the fusion rule of finite BPA functions on is as follows:

K is the normalization constant and is expressed as follows:

The overall trust degree of A on can be represented as a belief function [TeX:] $$\operatorname{Bel}(A)=\sum_{B \subseteq A} m(B)$$ where [TeX:] $$B \subset \Theta$$, and the uncertainty degree of A on can be represented as a plausible function [TeX:] $$P l(A)=\sum_{B \cap A \neq \phi} m(B) \text { where } B \subset \Theta$$.

For a supplier A on , Bel(A) shows the sum of the possibility estimate of all its subsets, and Pl(A) shows the sum of the uncertainty estimate of all its subsets. For A, the degree of confirmation can be expressed by the trust interval [Bel(A), Pl(A)].

According to the above analysis, the sum of credibility which the evidences support A is shown by Bel(A) and the sum of credibility which the evidences does not negative A is shown by Pl(A). Thus, the trust interval is formed as [Bel(A), Pl(A)]. The support degree to a supplier of belief function and plausible function can be reflected by [Bel(A), Pl(A)] comprehensively.

According to references [30,31], to evaluate the suppliers by trust interval approach is more reasonable than max-belief-function decision-making approach or max-plausible-function decision-making approach. The D-S theory decision regulations based on trust interval for supplier evaluation in GSC are as follows.

(i) It is assumed that supplier A_i is better than supplier A_j with a degree of [TeX:] $$P\left(A_{i}>A_{j}\right)$$. The trust interval of A_i is [Bel(A_i), Pl(A_i)], and the trust interval of A_j is [Bel(A_j), Pl(A_j)]. [TeX:] $$P\left(A_{i}>A_{j}\right)$$ is obtained as follows:

where [TeX:] $$P\left(A_{i}>A_{j}\right) \in[0,1]$$.

(ii) The partial order relationship: (a) When [TeX:] $$P\left(A_{i}>A_{j}\right)>0.5$$, A_i is better than A_j, which is expressed as A_i A_j ; (b) When [TeX:] $$P\left(A_{i}>A_{j}\right)>0.5 \text { and } P\left(A_{j}>A_{k}\right)>0.5, A_{i}$$ is better than A_k, which is expressed as A_i A_j A_k.

3.4 Supplier Evaluation Based on D-S Theory

The weighted BPA value of focal element [TeX:] $$A_{i}\left(i<2^{N}\right)$$ under index [TeX:] $$t(t \in I F)$$, which is [TeX:] $$\tilde{m}_{t}\left(A_{i}\right)$$, is introduced into the D-S theory model as evidence input. The calculating and processing approaches for weighted BPA value of each focal element are as follows.

Based on investigating the actual status of each supplier, the expert gives the initial value of indexes which belongs to quantitative type or direct qualitative type. Here, the indexes which belong to definite quantitative type or direct qualitative type are assigned exact values, the indexes which are relatively fuzzy quantitative type are assigned value intervals, and the indexes which are completely unknown are assigned null value. By membership approach, we can calculate the tendency degree of the initial index value.

To an index, five levels of expert’s remark are given as: {G₁,G₂,G₃,G₄,G₅}={very bad, bad, middle, good,very good}. Here, G₁ is the remark level corresponding to the minimum initial index value D₁, and G₅ is the remark level corresponding to the maximal initial index value G₅. However, to cost-based index C₁₁, G₁ is the remark level corresponding to the maximal initial index value D₁, and G₅ is the remark level corresponding to the minimum initial index value D₅.

We assume that the corresponding exact numbers of the five levels of expert’s remark are: [TeX:] $$E\left(G_{1}\right)=0,E\left(G_{2}\right)=0.25, E\left(G_{3}\right)=0.5, E\left(G_{4}\right)=0.75, \text { and } E\left(G_{5}\right)=1$$. The membership degree of the initial index value to G_i is defined as _i. On t, the tendency degree of A_i is expressed as [TeX:] $$P_{t}\left(A_{i}\right)$$, and the calculation of [TeX:] $$P_{t}\left(A_{i}\right)$$ is divided into two circumstances as follows:

(1) Index t belongs to quantitative type.

In this circumstance, the initial index value of A_i on t is a point-value a or an value-interval [a,b].

If [TeX:] $$D_{i} \leq a \leq D_{i+1} \text { or } D_{i} \leq a \leq b \leq D_{i+1}, P_{t}\left(A_{i}\right)=\beta_{i} E\left(G_{i}\right)+\beta_{i+1} E\left(G_{i+1}\right)$$.

If [TeX:] $$D_{i} \leq a \leq D_{i+1} \text { and } D_{i+1} \leq b \leq D_{i+2}, P_{t}\left(A_{i}\right)=\beta_{i} E\left(G_{i}\right)+\beta_{i+1} E\left(G_{i+1}\right)+\beta_{i+2} E\left(G_{i+2}\right)$$.

If [TeX:] $$D_{i} \leq a \leq D_{i+1} \text { and } D_{j} \leq b \leq D_{j+1}, P_{t}\left(A_{i}\right)=\beta_{i} E\left(G_{i}\right)+\beta_{i+1} E\left(G_{i+1}\right)+\ldots+\beta_{j} E\left(G_{j}\right)+\beta_{j+1} E\left(G_{j+1}\right)$$.

(2) Index t belongs to direct qualitative type.

In this circumstance, the tendency degree of A_i on t is equal to the number corresponding to the remark level.

By the above approach, the tendency degree of each focal element except under any index can be solved. Here, the expert’s uncertainty is indicated by . Without the consideration of the influence of , the supplier evaluation problem is a simple probability allocation problem. However, the advantages of D-S theory in solving multiple indexes decision problem are not reflected. Simultaneously, the trust degree of expert on any index is different and the uncertainty of an index is expressed by the probability allocation of

Thus, the probability allocation value of on different indexes should also be considered differently. For example, in the supplier evaluation problem, the weight of evaluation index is obviously variable in different requirements. If cost reduction is needed, C₁₁ will be more important and its trust degree should be larger. The BPA value of on C₁₁ should be smaller. Accordingly, we introduce the adaptive weight (determined in Section 3.2) to regulate the preference of each index and solve the probability allocation problem of on all indexes, then the weighted BPA value of every focal element on any index is calculated as [TeX:] $$\tilde{m}_{t}\left(A_{i}\right)$$.

We assume that the adaptive weight of t is [TeX:] $$\omega_{t}\left(\omega_{t} \in(0,1)\right)$$. The bigger w_t is, the higher the trust degree of expert to t, the lower the uncertainty of t is, and vice versa. Therefore, a weighted normalization processing of the BPA values of all focal elements is taken as follows:

According to Fig. 1, the evidences of C₁₁,C₁₂,C₁₃ and C₁₄ of C₁ are fused and processed, and then the weighted BPA value of C₁ is calculated as [TeX:] $$\tilde{m}_{1}\left(A_{i}\right)$$. After that, we execute a secondary fusion which includes the weighted BPA value of C₁,C₂,C₃ and C₄, and the evaluation result of suppliers can be obtained.

4. Case Study

As an important part of modern mechanical equipment, the main function of bearing is to sustain the mechanical revolving body, depress the friction in movement and ensure the rotary precision. A bearing manufacturing enterprise has three candidate bearing-cage suppliers. To select the optimal bearing-cage supplier, supplier evaluation should be executed.

Firstly, the adaptive weight of evaluation index is determined by our designed fuzzy-rough-sets-AHP method. Four experts (expert 1, expert 2, expert 3, and expert 4) participate in the judgment on C₁₁,C₁₂,C₁₃ and C₁₄. Using the proportional scale method of trapezoidal fuzzy number [21], the trapezoidalfuzzy- number reciprocal judgment matrices E₁, E₂, E₃, and E₄ are shown as follows.

Taking E¹ for example, it can be converted to the following form:

After consistency check, E¹, E², E³ and E⁴ are all qualified. Then they are split, and a¹ is as follows:

Based on a¹, a², a³ and a⁴, the rough group-decision matrix is obtained as follows:

In element [TeX:] $$a_{1,4}=\{3 / 2,1,1,7 / 3\}$$, the upper approximation set of partition '3/2' is {3/2,7/3} and the lower approximation set of partition '3/2' is {3/2,1,1}, so [TeX:] $$\underline{L} \quad\left(3 / 2^{\prime}\right)=(3 / 2+1+1) / 3=1.17$$, [TeX:] $$\overline{L}\left(3 / 2^{\prime}\right)=(3 / 2+7 / 3) / 2=1.92 \text { and } R N\left(3 / 2^{\prime}\right)=[1.17,1.92]$$. Similarly, RN('1')=[1,1.46], RN('7/3')=[1.46,2.33] and [TeX:] $$R N\left(a_{1,4}\right)=\{[1.17,1.92],[1,1.46],[1,1.46],[1.46,2.33]\}$$.

Thus, [TeX:] $$A v g_{-} R N\left(a_{1,4}\right)=[1.16,1.79]$$. The rough boundary intervals in average form of other elements of a can be also calculated. The rough judgment matrix is constructed as follows:

Then EA is split into EA^- and EA⁺. The eigenvector corresponding to the maximum eigen-value of EA^- is [TeX:] $$V A^{-}=[0.71,0.44,0.45,0.30]^{\mathrm{T}}$$ and the eigenvector corresponding to the maximum eigen-value of EA⁺ is [TeX:] $$ V A^{+}=[0.65,0.49,0.47,0.34]^{\mathrm{T}}$$. So GA={0.68,0.47,0.46,0.32}. Similarly, GB={0.73,0.51,0.66,0.58}, GC={0.82,0.67,0.73,0.69} and GD={0.95,0.77,0.83,0.75}. Then the adaptive weight of indexes C₁₁,C₁₂,C₁₃ and C₁₄ with the trapezoidal fuzzy number form are [TeX:] $$z_{1}=(0.68,0.73,0.82,0.95), z_{2}=(0.47,0.51,0.67,0.77), z_{3}=(0.46,0.66,0.73,0.83) \text { and } z_{4}=(0.32,0.58,0.69,0.75)$$. After gravity model approach processing and normalization processing, we obtain the adaptive weight of evaluation indexes C₁₁,C₁₂,C₁₃ and C₁₄: [TeX:] $$\omega\left(C_{11}\right)=0.30, \omega\left(C_{12}\right)=0.23, \omega\left(C_{13}\right)=0.25, \text { and } \omega\left(C_{14}\right)=0.22$$. Similarly, the adaptive weights of criterions C₁,C₂,C₃ and C₄ are [TeX:] $$\omega\left(C_{1}\right)=0.57, \omega\left(C_{2}\right)=0.18, \omega\left(C_{3}\right)=0.26, \text { and } \omega\left(C_{4}\right)=0.09$$.

Secondly, we use the proposed adaptive weight D-S theory model to deal with the decision of supplier evaluation problem. For the three candidate suppliers, their initial index values are shown in Table 1.

Corresponding to the remark level [TeX:] $$\left\{G_{1}, G_{2}, G_{3}, G_{4}, G_{5}\right\}$$, the reference values of the index belonging to quantitative type are as follows: [TeX:] $$G\left(C_{11}\right)=\left\{10^{5}, 10^{4}, 10^{3}, 10^{2}, 10^{1}\right\}, G\left(C_{12}\right)=\{0.05,0.04,0.03,0.02,0.01\}$$, [TeX:] $$G\left(C_{2}\right)=\{17,13,9,5,1\}, G\left(C_{3}\right)=\{1,3,5,7,9\} \text { and } G\left(C_{4}\right)=\{0,0.25,0.5,0.75,1\}$$.

Then, the membership degree of initial index value to every remark level is obtained. The data in Table 1 is translated into the membership degree form corresponding to remark grade. As shown in Table 2, the tendency degree form of initial index value is obtained.

We define the set of candidate suppliers as the D-S theory identification framework: [TeX:] $$\Theta=\left\{x_{1}, x_{2}, x_{3}\right\}$$, Here, x₁,x₂ and x₃ represent bearing-cage suppliers 1, 2, and 3, respectively.

For four indexes C₁₁,C₁₂,C₁₃ and C₁₄ and three criterions C₂,C₃, and C₄, the weighted BPA values of all focal elements are obtained according to the tendency degree shown in Table 2 and the weight vectors [TeX:] $$\left(\omega\left(C_{11}\right), \omega\left(C_{12}\right), \omega\left(C_{13}\right), \omega\left(C_{14}\right)\right)=(0.30,0.23,0.25,0.22), \text { and }\left(\omega\left(C_{2}\right), \omega\left(C_{3}\right), \omega\left(C_{4}\right)\right)=(0.18,0.26,09)$$. The calculation result is as follows:

(1) C₁₁: [TeX:] $$\tilde{m}_{11}\left(x_{1}\right)=0.1672, \quad \tilde{m}_{11}\left(x_{2}\right)=0.0934, \quad \tilde{m}_{11}\left(x_{3}\right)=0.0393, \quad \tilde{m}_{11}(\Theta)=0.7000$$

(2) C₁₂: [TeX:] $$\tilde{m}_{12}\left(x_{1}\right)=0.0920, \quad \tilde{m}_{12}\left(x_{2}\right)=0.0920, \quad \tilde{m}_{12}\left(x_{3}\right)=0.0460, \quad \tilde{m}_{12}(\Theta)=0.7700$$

(3) C₁₃: [TeX:] $$\tilde{m}_{13}\left(x_{1}\right)=0.1071, \tilde{m}_{13}\left(x_{2}\right)=0.1429, \quad \tilde{m}_{13}\left(x_{3}\right)=0, \quad \tilde{m}_{13}(\Theta)=0.7500$$

(4) C₁₄: [TeX:] $$\tilde{m}_{14}\left(x_{1}\right)=0, \quad \tilde{m}_{14}\left(x_{2}\right)=0.1100, \tilde{m}_{14}\left(x_{3}\right)=0.1100, \tilde{m}_{14}(\Theta)=0.7800$$

(5) C₂: [TeX:] $$\tilde{m}_{2}\left(x_{1}\right)=0.0739, \quad \tilde{m}_{2}\left(x_{2}\right)=0.0744, \quad \tilde{m}_{2}\left(x_{3}\right)=0.0317, \quad \tilde{m}_{2}(\Theta)=0.8200$$

(6) C₃: [TeX:] $$\tilde{m}_{3}\left(x_{1}\right)=0.1300, \quad \tilde{m}_{3}\left(x_{2}\right)=0.1300, \quad \tilde{m}_{3}(\Theta)=0.7400$$

(7) C₄: [TeX:] $$\tilde{m}_{4}\left(x_{1}\right)=0.0297, \tilde{m}_{4}\left(x_{2}\right)=0.0302, \tilde{m}_{4}\left(x_{3}\right)=0.0302, \tilde{m}_{4}(\Theta)=0.9100$$

After that, we take [TeX:] $$\tilde{m}_{11}\left(x_{i}\right), \tilde{m}_{12}\left(x_{i}\right), \tilde{m}_{13}\left(x_{i}\right), \text { and } \tilde{m}_{14}\left(x_{i}\right)$$ as the evidence input and implement the first evidence fusion. The BPA values of all focal elements are obtained as follows: [TeX:] $$m_{1}\left(x_{1}\right)=0.1001,m_{1}\left(x_{2}\right)=0.7815, m_{1}\left(x_{3}\right)=0.0772, m_{1}\left(x_{1}, x_{2}\right)=0.0102, m_{1}\left(x_{2}, x_{3}\right)=0.0201, m_{1}\left(x_{1}, x_{3}\right)=0.0098, \text { and } m_{1}(\Theta)=0.0011.$$

We normalize BPA values [TeX:] $$m_{1}\left(A_{i}\right)$$ of the suppliers to be evaluated and on index C₁. With the consideration of [TeX:] $$\omega\left(C_{1}\right)$$, the weighted BPA values are obtained as follows: [TeX:] $$\tilde{m}_{1}\left(x_{1}\right)=0.0571, \tilde{m}_{1}\left(x_{2}\right)=0.4455, \tilde{m}_{1}\left(x_{3}\right)=0.0440, \quad \tilde{m}_{1}\left(x_{1}, x_{2}\right) 0.0058, \quad \tilde{m}_{1}\left(x_{2}, x_{3}\right)=0.0115, \tilde{m}_{1}\left(x_{1}, x_{3}\right)=0.0056, \text { and } \tilde{m}_{1}(\Theta)=0.0006$$.

Then, we take [TeX:] $$\tilde{m}_{1}\left(A_{i}\right), \tilde{m}_{2}\left(A_{i}\right), \tilde{m}_{3}\left(A_{i}\right) \text { and } \tilde{m}_{4}\left(A_{i}\right)$$ as the evidence input and implement the second evidence fusion. The comprehensive BPA values of all focal elements are obtained as follows: [TeX:] $$m\left(x_{1}\right)=0.1255, m\left(x_{2}\right)=0.7088, m\left(x_{3}\right)=0.0102, m\left(x_{1}, x_{2}\right)=0.0999, m\left(x_{2}, x_{3}\right)=0.0032, m\left(x_{1}, x_{3}\right)=0.0506\ and\ m(\Theta)=0.0018_{\circ}$$

[TeX:] $$\operatorname{Bel}\left(A_{i}\right) \text { and } \operatorname{Pl}\left(A_{i}\right)$$ of all suppliers are calculated. Then the trust intervals of all suppliers are obtained as follows:

(1) [TeX:] $$x_{1} :[0.1255,0.2778]$$

(2) [TeX:] $$x_{2} :[0.7088,0.8137]$$

(3) [TeX:] $$x_{3} :[0.0102,0.0658]$$

On the basis of the D-S theory decision regulations, the result is as follows:

(1) [TeX:] $$P\left(x_{1}>x_{2}\right)=0,$$ so x₁ x₂.

(2) [TeX:] $$P\left(x_{1}>x_{3}\right)=1,$$ so x₃ x₁

Therefore, the evaluation result of three suppliers is x₃ x₁ x₂ and supplier 2 is the optimal bearingcage supplier. Thus, the proposed adaptive weight D-S theory model can solve the supplier evaluation problem in GSC even the initial index value is uncertain and incomplete (See in Table 1, the initial values of x₂ and x₃ on index C₂ are interval values and the initial value of x₃ on index C₃ is missing).

To verify the effectiveness of the proposed adaptive weight D-S theory model, we use traditional TOPSIS method [18,21] to make a comparison. Because the traditional TOPSIS method can only solve the evaluation problem with certain and complete index value, we replace the interval with its mid-value and ignore the index with missing index value (The initial values of x₂ and x₃ on index C₂ are 15.5 and 7, respectively. Index C3 is ignored). The tendency degree method is still used to process the initial index value. Then, processing of the adaptive weights is executed on the basis of the hierarchical structure shown in Fig. 1 and the final weight vector of C₁₁,C₁₂,C₁₃,C₁₄,C₂ and C₄ is [TeX:] $$\omega=(0.15,0.11,0.12,0.09,0.18,0.09 )$$ in which index C₃ is ignored. In Table 3, the weighted index value matrix is obtained.

From Table 3, the positive and negative ideal points are (0.1275, 0.1100, 0.1200, 0.0675, 0.1586, 0.0900) and (0.0300, 0.0330, 0, 0, 0.0675, 0.0886), respectively. So we can obtain the close degree to positive ideal point of each supplier is as follows:

(1) x₁:0.7065.

(2) x₂:0.7686.

(3) x₃:0.2569.

Therefore, the evaluation result of three candidate suppliers by traditional TOPSIS method is x₃ x₁ x₂ and the bearing manufacturing enterprise should choose supplier 2 as the optimal bearing cage supplier. The evaluation results of the proposed adaptive weight D-S theory model and traditional TOPSIS method are consistent. This shows that the proposed adaptive weight D-S theory model is feasible and effective.

5. Conclusion

In this paper, an adaptive weight D-S theory model is proposed for the evaluation problem characterized by uncertainty and incompleteness and variable index weight in GSC. In addition, a fuzzyrough- sets-AHP approach is designed to obtain the adaptive index weight. The index framework is established considering of the main factors affecting the supplier evaluation in GSC, which can improve the scientific nature and rationality. The case study and the comparison with TOPSIS show that the optimal supplier of manufacturing enterprise can be correctly selected by the proposed adaptive weight D-S theory model.

Acknowledgement

This paper is supported by Key Scientific Research Projects in 2017 at North Minzu University (No. 2017KJ22), the Third Batch of Ningxia Youth Talents Supporting Program (No. TJGC2018048), Natural Science Foundation of Ningxia Province (No. NZ17113), and Ningxia First-class Discipline and Scientific Research Project: Electronic Science and Technology (No. NXYLXK2017A07).