Shu-Ping Wan^{*} and Yu Zheng
College of Information Technology, Jiangxi University of Finance and Economics, Nanchang 330013, China
Received: June 09, 2015 Accepted:June 25, 2015 Published: June 27, 2015
Citation: Wan SP, Zheng Y (2015) Supplier Selection of Foreign Trade Sourcing Company using ANP-VIKOR Method in Hesitant Fuzzy Environment. Ind Eng Manage 4:163. doi:10.4172/2169-0316.1000163
Copyright: © 2015 Wan SP, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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International supplier selection which includes different criteria can be regarded as a kind of multi-criteria decision making (MCDM) problems. By combining the analytic network process (ANP) with the Vlsekriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method in hesitant fuzzy (HF) environment, this paper proposes a HF-ANP-VIKOR method. First, a novel HF-ANP approach is presented to determine the weight of each criterion. In this approach, the preference relations between criteria are hesitant fuzzy preference relations (HFPRs) whose elements are hesitant fuzzy elements (HFEs). According to the distance between two HFPRs, a new compatibility measure for HFPRs is proposed to measure the compatibility degrees of HFPRs. If the HFPRs are acceptable compatibility, they are converted into fuzzy preference relations by which the weights of sub-criteria are determined. Subsequently, extending the classical VIKOR method into HF environment, a new HF-VIKOR method is put forward to rank the alternatives. Finally, a case of Nantong uasia import and export limited company is studied to illustrate the practicability and effectiveness of the HF-ANP-VIKOR method proposed in this paper.
Supplier selection; MCDM; ANP; VIKOR; HFS
With the development of globalization, lots of companies choose multinational operations to help them seek profits from development differences among nations. Foreign trade sourcing company provides foreign customers agency service, including labor-intensive, low valueadded primary products. According to customer demand, foreign trade sourcing company seeks sources, production, arrange transportation, export process and other value-added services. International sourcing process is quite complicated because it includes many procedures, such as selecting suppliers, confirming and sending the sample, examining the goods, booking cargo space, applying to the customs, exchange settlement and tax return. Figure 1 intuitively depicts these procedures.
In these procedures, the supplier selection may directly impact the purchase cost of enterprises. Usually the purchase cost reaches about 60% of the total cost in the enterprise, sometimes reaches more than 80%. It can be seen that an appropriate supplier can decrease the costs of the company. Therefore, supplier selection is a very important procedure. However, selecting an appropriate suppler is very complicated because many indicators have to be involved. Hence, supplier selection could be considered as a kind of multi-criteria decision making (MCDM) problems.
In general, existing research on selecting suppliers by using MCDM methods mainly focuses on two crucial issues: the evaluation criteria determination and the MCDM methods, which are briefly reviewed as follows.
Determination the criteria for supplier selection
To solve international supplier selection problems, first, the evaluation indices should be determined. Dickson [1] introduced price, quality, technology level and management indices. Eliram [2] suggested some hard targets and the soft targets. For example, product cost, quality and delivery are hard targets, whereas organization and management are soft targets. Choi [3] proposed price, technology status, financial and service indicators, and then used them to select the vendor of the US auto industry. Dowlatshahi [4] provided management, service and product development indictors. Sarkis and Talluri [5] evaluated suppliers using the quality, technology status, product cost and culture. The comparisons of the criteria studied in above works are shown in Table 1.
Criteria | Dickson[1] | Eliram[2] | Choi[3] | Dowlatshahi[4] | Sarkis and Talluri[5] |
---|---|---|---|---|---|
Price | √ | √ | √ | ||
Quality | √ | √ | √ | ||
Technology | √ | √ | √ | √ | |
Credit status | √ | ||||
Financial | √ | √ | |||
Product cost | √ | √ | |||
Management | √ | √ | |||
service | √ | √ | |||
Product development | √ | ||||
Culture | √ |
Table 1: The criteria selection in different works.
However, different kinds of companies will select diverse indicators according to their own scale and management tactics. Therefore, selecting criteria should be more cautious and evaluated by the authorities.
MCDM methods for supplier selection
The MCDM methods are mainly to rank the finite alternatives based on multiple criteria. Many researchers presented various MCDM methods for selecting supplier [1-24]. These methods mostly are divided into two categories.
The first category is Single methods. The most popular MCDM method for selecting supplier is analytic hierarchy process (AHP) method [6]. The applications of AHP method in MCDM problem are briefly introduced as follows:
Hill and Nydick [7] employed AHP to rank alternatives by pairwise comparisons. Although this method is feasible, the workloads are very large. To simplify the calculation, Yahya and Kingsman [8] improved method [7] and only used AHP to determine the criterion weights. The alternatives are ranked based on comprehensive scores of alternatives which are obtained by weighting sum of the scores of alternatives on each criterion. Liu and Hai [9] proposed a voting-based AHP to select alternatives.
In addition, other single MCDM methods for supplier selection are often employed, such as the analytic network process (ANP) (Figure 2), the multiple attribute utility theory (MAUT) method, the outranking method, the technique for order preference by similarity to ideal solution (TOPSIS) method. Sarkis and Talluri [10] proposed an ANP model for strategic supplier selection. Min [11] presented a MAUTbased analytical approach to evaluate various international sourcing strategies under dynamically changing scenarios. De et al. [12] took an outranking approach as a suitable decision making tool for selecting supplier. Chen et al. [13] presented a closeness coefficient to determine the ranking order of all suppliers by calculating the distances between the alternatives and the fuzzy positive-ideal solution (FPIS) and fuzzy negative-ideal solution (FNIS).
The second category is hybrid methods. Jasmine [25] proposed an analytical approach combining quality function deployment (QFD) and ANP for guiding shipping companies’ design. Kaya and Kahraman [26] presented an integrated the VIKOR-AHP methodology, in which the weights of the selection criteria were derived by fuzzy AHP. Hsu et al. [27] discussed the recycled material vender selection problems and built a new MCDM model combining DEMATEL (decision-making trial and evaluation laboratory)-based ANP (DANP) with VIKOR. Nilashi et al. [28] used a DEMATEL-ANP based MCDM approach to evaluate the critical success factors in construction projects.
Although the above methods have advantages for supplier selection, there are some drawbacks which limit the applications of these methods. For example, these methods are unable to deal with the MCDM problems where several possible values may be supplied by decision makers (DMs) due to the ambiguity of human thinking. Hesitant fuzzy set (HFS) [29] is a proper tool to handle such a type of MCDM problems.
In this paper, a novel ANP-VIKOR method is proposed to solve MCDM problems in hesitant fuzzy (HF) environment. First, a HFANP approach is presented to determine the weights of criteria. A notable characteristic of this approach is that the elements of preference relation matrices are hesitant fuzzy elements (HFEs) which can more flexibly express the preferences of experts. When the values of criteria are in the form of HFEs, a new HF-VIKOR method is proposed and applied to rank the alternatives .The key features of the proposed method in this paper are listed as follows:
(1) Considering the interactions among criteria, we firstly extend ANP method into HF environment and propose HF-ANP to determine the weight of each criterion in supplier selection. Due to the fact that HFS permits the membership has a set of possible values, it is more suitable to use HFEs to describe the preferences of experts.
(2) In HF-ANP approach, we first propose a new measure to calculate the compatibility degree between hesitant fuzzy preference relations (HFPRs). If a HFPR is acceptable compatibility, we use the score function to convert the HFPRs to fuzzy preference relations (FPRs). Thus, using FPRs is much easier than using HFPRs to determine the weights of criteria.
(3) We propose a novel HF-VIKOR approach to rank alternatives. Concretely, we generalize the scope of the applications of the VIKOR method, which makes the VIKOR method solve more MCDM problems in different environments.
The paper is organized as follows. In Section 2, we briefly review the concepts, such as the score function, some operations of HFS and HFPR. In Section 3, we present the method of HF-ANP, HF-VIKOR and the HF-ANP-VIKOR method. In Section 4, a practical example of international supplier selection for foreign trade sourcing company with the ANP-VIKOR method in HFS environment. Finally, the conclusion is presented in Section 5.
In this section, we briefly review some basic concept and properties about HFS and HFPR.
Definition 1 [29,30]. Let X be a fixed set. It is defined as a HFS on X in terms of a function that when applied to X returns a subset of [0,1]
where is a set of some different values in [0,1] , denoting the possible membership degrees of the element x∈ X to the set E. For convenience, Xu and Xia [31] called HFEs denoted by where l(h) is the number of all elements in h. The elements in a HFE h are in an increasing order.
For any two HFEs h1 and h2, we extend the shorter one by adding the minimum element of it until both of HFEs have the same length. For example, let h_{1} = {0.2,0.3} and h_{2} = {0.3,0.4,0.5} We can extend h_{1} to
For three HFEs h , h_{1} and h_{2}, the following operations are defined as [29,30]:
1) Lower bound:
2) Upper bound:
3) Complement:
4)
5)
6)
7)
Let be a collection of HFEs. Liao et al. [32] generalized 6) and 7) to the following forms:
8)
9)
Definition 2 [33]. Let and
be two HFEs. The Manhattan distance between HFEs is defined as:
where
Example 1. Let h_{1} = {0.2,0.6} and h_{2} = {0.1,0.5,0.6} be two HFEs. First, extend the h_{1} to then the Manhattan distance between h1 and h2 is calculated as
It can be clear that the Manhattan distance between two HFEs h_{1} and h_{2} is a crisp number, Eq. (2) is a useful tool for defuzzying two HFEs into a crisp number.
Definition 3 [34]. Let be a HFE, where l(h) denotes the number of all elements in h. A score function S of the HFE h is defined as
For two HFEs h_{1} and h_{2} , if , then h_{2} > h_{2} ; if s(h_{1} ) = s(h_{2} ) , then h_{1}= h_{2}
Theorem 1 [34]. Let , and be three HFEs, then
1)
2)
3)
4)
5)
6)
Definition 4. and be two HFEs, Then the compatibility degree of h1 and h2 is defined as:
The compatibility degree is used to express the similarity degree between two HFEs. It is obvious that the larger the value of c(h1,h2), the greater the compatibility degree between h1 and h2.
Remark 2: In real-life decision making, all elements in h_{1} or h_{2} cannot be zero simultaneously. Therefore, we suppose that at most one HFE be zero in Definition 4.
Theorem 2. The compatibility degree c(h_{1},h_{1}) satisfies the properties:
i) ; if and only if
ii)
Proof. i) Since ,we get c(h_{1},h_{1})≥ 0. In the following, we only have to prove c(h_{1},h_{1}) ≤ 1.
Using Cauchy-Schwarz inequality, we have
≤
≤
Therefore,
If i.e., h_{1}=h_{2}, then c(h_{1},h_{2}) =1.
ii) From Eq. (4), we have
= c(h_{1},h_{2})
Namely, c(h_{1},h_{2})=c(h_{2},h_{1})
This proof is completed.
Definition 5 [32].Let be a fixed set. A HFPR H on X is presented by a matrix where is a HFE indicating all the possible degrees to which xi is preferred to x_{j }Moreover, h_{ij} should satisfy the following conditions: , i, j = 1,2,, n .
Definition 6. Let and be two HFPRs, where and Then the compatibility degree of H^{(1)} and H^{(2)} is defined as:
Theorem 3. The compatibility degree c(H^{(1)} ,H^{(2)} ) satisfies the properties:
i)
ii) if and only if
iii)
The proof procedure is similar to that of Theorem 3.
Definition 7. Let (k =1,2,,t) be t individual HFPRs, the HFPR is called the collective HFPR.
where the values of calculate as:
Definition 8. is called the compatibility measure of HFPR H^{(k)} . If
we call H^{(k)} perfect compatibility. If , we call H^{(k)} acceptable compatibility, where δ_{o} is the threshold value of acceptable compatibility. Usually, we take in real decision making.
Theorem 4. Each individual HFPR and the collective HFPR are perfectly compatible if and only if any two individual HFPRs are perfectly compatible. i.e.,
if and only if for all (k,l =1,2,,t) .
Proof. Sufficiency:
From Theorem 4, if for any k =1,2,,t , then we know that Therefore, for any k,l =1,2,,t , we have Hence,
Necessity:
If for any k,l =1,2,,t , then we have Therefore, for any i, j =1,2,,n , we acquire i.e.,
Thus, from Eq. (6), we get Hence, for any k =1,2,?,t ,Accordingly, we obtain
Thereby,
This completes the proof of the Theorem 5.
For a MCDM problem, suppose that G is the goal, C_{i}. (i =1,2,...,n) are the criteria and cijis the j-th sub-criterion of the i-th criterion, where j =1,2.... i_{q} and i_{q} is the number of the sub-criteria of the criterion C_{i}. m are the alternatives, indicate the k-th expert. The ratings of the alternative m regarding to the sub-criterion are give as , w is the weight vector of the subcriteria with respect to the goal.
Determining the criterion weights by HF-ANP approach in HF environment
In this sub-section, we firstly introduce the classical ANP method, and then extend the ANP into the HF environment.
The classical ANP method: The ANP method [35] is generalized from AHP method. The AHP method claims that criteria in the same layer are mutually independent. However, the ANP method allows that the criteria in the same layer are interactive. The procedure of ANP is introduced step by step as follows:
(1) The network construction. The problem should be stated clearly and be decomposed into a network structure. An example of the network is shown in Figure 2. The network model is composed of three levels, including goal level, criterion level and sub-criterion level.
(2) Weighting matrix determination. Similar to the comparisons performed in the AHP, the weighting matrix expresses the degrees of interaction between criteria in the network, which means the relative importance of each C_{i} (i=1,2,…,n) with respect to the goal. To obtain the weighting matrix A we can calculate the eigenvalues and eigenvectors for pair-wise matrices.
where column vectors in A are the weight vectors expressing the influence degree of C_{i} on C_{j} (i,j=1,2,…,n) under the goal.
(3) Super-matrix formation. The super-matrix concept is similar to the Markov chain process [35]. It denotes the degree of mutual effect between sub-criteria under the criterion level. In other words, it expresses the degree to which one sub-criterion is preferred to the other. The calculation process for super-matrix W is the same as that of weighting matrix.
where column vectors in W are the weight vectors expressing the impact degrees of the sub-criteria in C_{i}. on the sub-criteria in C_{j}.
(4) Weighting super-matrix. The degree of mutual influence among the criteria in ANP can be expressed by weighting super-matrix W , which is computed as
(5) Determine the sub-criteria weights. Calculate the limit of weighting super-matrix. The elements in the each row of weighting super-matrix will tend to the same value which denotes the corresponding weight of each sub-criterion over the goal.
The HF-ANP approach: In most situations, crisp data are insufficient to model real-life situations. Since human judgments or preferences are often vague and cannot evaluate his preferences with real numbers. Due to the fact that the HFS can express the degree of fuzzy clearly, it is more suitable to use HFSs to deal with the human judgments or preferences.
The HF-ANP is composed of six steps:
Step 1: For criterion C_{k} (k =1,2,...,n) , expert construct the following HFPR by comparing criteria.
Step 2: Measure the compatibility degree of HFPR H^{(k)} (k =1,2,....,n) by Definition 7 and Definition 8.
Step 3: Determine weighting matrix
First, using Eq. (3), HFPR is converted into a FPR
The weights of FPR can be calculated as:
Then, the weighting matrix can be determined, where and
As the similar way in determining A, we can determine the super-matrix w.
Step 4: By Eq. (7), the weighted super-matrix W is derived.
Step 5: Utilizing Eq. (8), the limit of weighted super-matrix can be calculated to obtain the weight of sub-criteria over alternatives. Then the weight vector w^{T} is acquired.
A HF-VIKOR approach to ranking alternatives
VIKOR is a compromise MCDM technique proposed by Opricovic and Tzeng [36]. This method determines compromise solutions to rank alternatives and select best one(s). The compromise solution is a feasible solution which is the closest to the ideal solution, and the “compromise” means an agreement established by mutual concessions [37].
The basic measure for compromise ranking is developed from the L_{p}-metric function which is used as an aggregation function in the compromise programming (Yu 1973) [38].
The compromise ranking procedure of the VIKOR method can be set up as follows:
(i) Determine the ideal solution and negative solution as
where F_{1} and F_{2} are respectively the sets of benefit criteria and cost criteria.
Then the form of Lp-metric distance measure over the alternatives in compromise programming was developed as:
where w_{o} is the weight vector of the criteria.
(ii) Compute the group utility and individual regret values of the alternatives as:
(iii) Calculate the value of Qi as follows:
where and
(iv) Rank the by the values of S_{i} , R_{i} and Q_{i}
(v) Determine a compromise solution the alternatives A′ which is the best ranked by the measure Q (minimum) if the following two conditions should be satisfied:
C1. Acceptable advantage:
where is the alternative with second position in the ranking list by Q; DQ = 1/ (m−1) ; m is the number of alternatives.
C2. Acceptable stability in decision making: The alternative A′ must also be the best ranked by S or/and R. This compromise solution could be “voting by majority rule” (when v>0.5 is needed), or “by consensus” v ≈ 0.5 ,or “with veto” (v<0.5). Here, v is the weight of decision making strategy “the majority of criteria” (or “the maximum group utility”).
If one of the conditions is not satisfied, then a set of compromise solution is proposed, which consist of:
• Alternatives A′ and A′ if only condition C2 is not satisfied, or
• Alternatives if condition C1 is not satisfied; A^{(M)} is determined by the relation for maximum M (the positions of these alternatives are “in closeness”).
A novel ANP-VIKOR method to MCDM in HF environment
In this section, we extend the ANP-VIKOR in the HF environment to solve the MCDM problem.
The HF-ANP-VIKOR method is composed of eight major steps:
Step 1: Specify the criteria, sub-criteria and alternatives.
Step 2: Construct a network structure according to the relations among criteria.
Step 3: Construct the HFPRs H^{(k)}(k =1,2,.....,n) and decision matrix.
Step 4: Measure the compatibility degree of HFPR H^{(k) }
(k = 1,2,...,n) by Definition 7 and Definition 8.
Step 5: Use Eqs. (9)-(10), determine weighting matrix and super-matrix W .
Step 6: By Eqs. (7)-(8), calculate the weights of sub-criteria over total goal.
Step 7: Convert the to a FPR matrix.
Step 8: Determine the ideal solution and negative ideal solution using Equations (11)-(12).
Step 9: Calculate the group utility and individual regret values of S_{i}, R_{i} and the value of Q_{i}1 by Equations (14)-(16).
Step 10: Rank alternatives according to the conditions C1 and C2.
An Application of the HF-ANP-VIKOR for Supplier Selection
In this section, we gave an application of the HF-ANP-VIKOR for supplier selection.
Nantong uasia import and export limited company intends to select a supplier for artificial flavors. In order to select the best supplier, four potential suppliers ( A_{1} , A_{2} , A_{3} , A_{4} ) are required assessment. The object indicators, which are confirmed by the procurement department in the company, are shown in Table 2.
Police risk e_{11} | Due to the changes of political to bring the possibility of economic losses. | |
Risk C_{1} | Tariff e_{12} | Import and export commodities after the declaration of tax levied. |
Credit risk e_{13} | Borrowers, issuers for various reasons are unwill or unable to fulfill the contract conditions. | |
product price ratio e_{21} | The average price ratio of products (and services) between the supplier and other suppliers, in order to evaluate the price level between suppliers. | |
Price C_{2} | exchange rate e_{22} | The currency exchange rate of another currency, a currency that the price is based on another currency. |
payment e_{23} | Payment for the performance of debt instruments adopted specific practices. | |
product quality pass rate e_{31} | The ratio the number of quality products to that of total products. | |
Quality C_{3} | quality certification e_{32} | The authority to prove that product complies with the standards and the technical requirements through the issuance of the certificates or certification marks. |
technical level e_{33} | Technical level and updated equipment usage. |
Table 2: The criteria and sub-criteria.
Step 1: The network construction. The network construct by DMs for the MCDM problems is showed in Figure 3.
Step 2: Determine the weights of sub-criteria.
First, by comparing criteria, the HFPRs matrices are constructed as Tables 3-5.
C1 | C1 | C2 | C3 |
C1 | {0.5,0.5,0.5} | {0.2,0.3,0.5} | {0.3,0.3,0.4} |
C2 | {0.5,0.7,0.8} | {0.5,0.5,0.5} | {0.1,0.3,0.7} |
C3 | {0.6,0.7,0.7} | {0.3,0.7,0.9} | {0.5,0.5,0.5} |
Table 3: Pair-wise comparison matrix under C_{1}.
C_{1} | C_{1} | C_{2} | C_{3} |
C_{1} | {0.5,0.5,0.5} | {0.1,0.4,0.5} | {0.3,0.4,0.7} |
C_{2} | {0.5,0.7,0.9} | {0.5,0.5,0.5} | {0.2,0.4,0.6} |
C_{3} | {0.3,0.6,0.7} | {0.4,0.6,0.8} | {0.5,0.5,0.5} |
Table 4: Pair-wise comparison matrix under C_{2}.
C_{3} | C_{1} | C_{2} | C_{3} |
C_{1} | {0.5,0.5,0.5} | {0.5,0.7,0.9} | {0.3,0.3,0.6} |
C_{2} | {0.1,0.3,0.5} | {0.5,0.5,0.5} | {0.3,0.5,0.7} |
C_{3} | {0.4,0.7,0.7} | {0.3,0.5,0.7} | {0.5,0.5,0.5} |
Table 5: Pair-wise comparison matrix under C_{3}.
Second, measure the compatibility of each preference relation matrix by Definition 7 and Definition 8 (Table 6).
Cd | Cd | Cd | Cd | Cd | |||||
---|---|---|---|---|---|---|---|---|---|
C_{1} | 0.96 | e^{3}_{11} | 0.96 | e^{3}_{21} | 0.98 | e^{2}_{31} | 0.98 | e^{1}_{11} | 0.99 |
C_{2} | 0.96 | e^{3}_{12} | 0.96 | e^{3}_{22} | 0.99 | e^{2}_{32} | 0.97 | e^{1}_{12} | 0.97 |
C_{3} | 0.96 | e^{3}_{13} | 0.96 | e^{3}_{23} | 0.98 | e^{2}_{33} | 0.99 | e^{1}_{13} | 0.97 |
e^{2}_{11} | 0.96 | e^{3}_{21} | 0.97 | e^{1}_{31} | 0.99 | e^{2}_{21} | 0.99 | e^{1}_{31} | 0.99 |
e^{1}_{12} | 0.95 | e^{3}_{22} | 0.98 | e^{2}_{21} | 0.99 | e^{2}_{22} | 0.96 | e^{2}_{32} | 0.97 |
e^{2}_{13} | 0.99 | e^{3}_{23} | 0.98 | e133 | 0.95 | e223 | 1 | e^{3}_{33} | 0.99 |
Table 6: Compatible degree (Cd).
Third, determine the weighting matrix and super-matrix.
Using Eq. (2), the HFPRs are converted to FPRs (Tables 7-9). By Eqs. (9)-(10), the weights of FPRs are computed. These weights compose the following weighting matrix.
C_{1} | C_{1} | C_{2} | C_{3} |
C_{1} | 0.5 | 0.3833 | 0.35 |
C_{2} | 0.6167 | 0.5 | 0.467 |
C_{3} | 0.65 | 0.533 | 0.5 |
Table 7: FPR matrix of C_{1}.
C_{2} | C_{1} | C_{2} | C_{3} |
C_{1} | 0.5 | 0.367 | 0.533 |
C_{2} | 0.633 | 0.5 | 0.467 |
C_{3} | 0.467 | 0.533 | 0.5 |
Table 8: FPR matrix of C_{2}.
C_{3} | C_{1} | C_{2} | C_{3} |
C_{1} | 0.5 | 0.767 | 0.45 |
C_{2} | 0.233 | 0.5 | 0.567 |
C_{3} | 0.55 | 0.433 | 0.5 |
Table 9: FPR matrix of C_{3}.
Similarly, the super-matrix is calculated as
According to Eqs. (7)-(8), we compute the limit matrix as follows:
From the limit matrix, the weights of sub-criteria are obtained as w = (0.1,0.106,0.116,0.093,0.119,0.117,0.102,0.121,0.126)^{T} .
Step 3: Elicit the hesitant fuzzy decision matrix, see Table 10.
e_{11} | e_{12} | e_{13} | e_{21} | e_{22} | e_{23} | e_{31} | e_{32} | e_{33} | |
A_{1} | {0.1,0.2,0.3} | {0.2,0.2,0.3} | {0.3,0.3,0.4} | {0.1,0.1,0.2} | {0.2,0.4,0.6} | {0.2,0.3,0.4} | {0.3,0.5,0.7} | {0.2,0.2,0.3} | {0.1,0.3,0.5} |
A_{2} | {0.2,0.2,0.5} | {0.2,0.3,0.4} | {0.1,0.3,0.5} | {0.3,0.3,0.5} | {0.2,0.4,0.6} | {0.1,0.2,0.3} | {0.1,0.1,0.3} | {0.3,0.4,0.5} | {0.2,0.4,0.6} |
A_{3} | {0.2,0.4,0.6} | {0.3,0.4,0.5} | {0.1,0.3,0.5} | {0.3,0.7,0.9} | {0.4,0.5,0.6} | {0.2,0.4,0.7} | {0.3,0.4,0.5} | {0.1,0.2,0.3} | {0.1,0.2,0.3} |
A_{4} | {0.3,0.5,0.7} | {0.2,0.4,0.6} | {0.1,0.2,0.4} | {0.2,0.4,0.6} | {0.2,0.2,0.3} | {0.2,0.5,0.7} | {0.1,0.1,0.3} | {0.3,0.5,0.7} | {0.3,0.3,0.4} |
Table 10: Hesitant fuzzy decision matrix.
Step 4: By Eqs. (11)-(12), the ideal and negative ideal solutions are seen in Table 11.
e_{11} | e_{12} | e_{13} | e_{21} | e_{22} | e_{23} | e_{31} | e_{32} | e_{33} | |
h_{+} | {0.1,0.2,0.3} | {0.2,0.2,0.3} | {0.1,0.2,0.4} | {0.1,0.1,0.2} | {0.2,0.2,0.3} | {0.1,0.2,0.3} | {0.3,0.5,0.7} | {0.3,0.5,0.7} | {0.2,0.4,0.6} |
h_{-} | {0.3,0.5,0.7} | {0.2,0.4,0.6} | {0.3,0.3,0.4} | {0.3,0.7,0.9} | {0.4,0.5,0.6} | {0.2,0.5,0.7} | {0.1,0.1,0.3} | {0.1,0.2,0.3} | {0.1,0.2,0.3} |
Table 11: The values of ideal and negative ideal solutions.
Step 5: Calculateand (Table 12).
d_{10} | e_{11} | e_{12} | e_{13} | e_{21} | e_{22} | e_{23} | e_{31} | e_{32} | e_{33} |
A_{1} | 0 | 0 | 0.1 | 0 | 0.167 | 0.1 | 0 | 0.267 | 0.1 |
A_{2} | 0.1 | 0.067 | 0 | 0.233 | 0.167 | 0 | 0.333 | 0.1 | 0 |
A_{3} | 0.2 | 0.167 | 0.067 | 0.5 | 0.267 | 0.233 | 0.1 | 0.3 | 0.2 |
A_{4} | 0.3 | 0.167 | 0 | 0.267 | 0 | 0.267 | 0.333 | 0 | 0.067 |
Table 12: The values of d(h_{0} ,h_{10} )
The values of over each sub-criterion are show as:
={0.3,0.167,0.1,0.5,0.267,0.267,0.333,0.3,0.2}
Step 6: By Eqs. (14)-(15), calculate the group utility Si and individual regret values Ri, which are shown as Table 13.
S_{1} | R_{1} | Q_{1} | Rank | |
A_{1} | 0.405 | 0.116 | 0.373 | 3 |
A_{2} | 0.335 | 0.102 | 0 | 4 |
A_{3} | 0.841 | 0.126 | 1 | 1 |
A_{4} | 0.516 | 0.117 | 0.499 | 2 |
Table 13: The values of S_{1} , R_{1} and l Q_{1}.
Step 7: Take v = 0.5 in Eq. (16), the Q_{i} are computed and listed in Table 14.
C | C_{1} | C_{2} | C_{3} |
C_{1} | {0.5,0.5,0.5} | {0.27,0.43,0.63} | {0.3,0.33,0.57} |
C_{2} | {0.43,0.57,0.73} | {0.5,0.5,0.5} | {0.2,0.4,0.67} |
C_{3} | {0.43,0.67,0.7} | {0.33,0.6,0.8} | {0.5,0.5,0.5} |
Table 14: The collective HFPR matrix.
Step 8: The rank by HF-ANP-VIKOR is Therefore, the best supplier is A_{3}.
In this paper, we proposed an ANP-VIKOR method for solving supplier selection problems with HFSs. First, we developed a HF-ANP approach and applied to calculate the weights of sub-criteria. In this calculation process, we not only presented a new measure to calculate the compatibility degree, but also converted the HFPRs to FPRs by the score function, which can save the calculations. Then, we presented a HF-VIKOR approach to rank alternatives. The main idea of this approach is to determine the values of group utility and individual regret over the alternatives, integrate the maximum group utility and the minimum individual regret to rank the alternatives. Finally, the numeral analysis indicated that the HF-ANP-VIKOR method is practicable and valid to solve the MCDM problem with HFEs.
This research was supported by the National Natural Science Foundation of China (Nos. 71061006, 61263018 and 11461030), the Humanities Social Science Programming Project of Ministry of Education of China (No. 09YGC630107), the Natural Science Foundation of Jiangxi Province of China (Nos. 20114BAB201012 and 20142BAB201011), “Twelve five” Programming Project of Jiangxi province Social Science (2013) (No. 13GL17) and the Excellent Young Academic Talent Support Program of Jiangxi University of Finance and Economics.
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