A New Methodology of Multicriteria Decision-Making in Supplier Selection Based on Z-Numbers

Abstract

Supplier selection is a significant issue of multicriteria decision-making (MCDM), which has been heavily studied with classical fuzzy methodologies, but the reliability of the knowledge from domain experts is not efficiently taken into consideration. Z-number introduced by Zadeh has more power to describe the knowledge of human being with uncertain information considering both restraint and reliability. In this paper, a methodology for supplier selection using Z-numbers is proposed considering information transformation. It includes two parts: one solves the issue of how to convert Z-number to the classic fuzzy number according to the fuzzy expectation; the other solves the problem of how to get the optimal priority weight for supplier selection with genetic algorithm (GA), which is an efficient and flexible method for calculating the priority weight of the judgement matrix. Finally, an example for supplier selection is used to illustrate the effectiveness the proposed methodology.

[ProQuest: [...] denotes non US-ASCII text; see PDF]

Bingyi Kang 1 and Yong Hu 2 and Yong Deng 1,2,3 and Deyun Zhou 3

Academic Editor:Young Hae Lee

1, School of Computer and Information Sciences, Southwest University, Chongqing 400715, China
2, Big Data Decision Institute, Jinan University, Tianhe, Guangzhou 510632, China
3, School of Electronics and Information, Northwestern Polytechnical University, Xi'an, Shaanxi 710072, China

Received 14 October 2015; Revised 27 November 2015; Accepted 29 November 2015; 6 January 2016

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The selection of suppliers is a significant issue in the business management [1]. As the organization becomes more and more dependent on their suppliers, the direct and indirect consequences of poor decision-making will become more critical. Any deficiency in coordination of the supply chain will lead to excessive delays and poor customer service [2]. An effective methodology for supplier selection is the demand from the current business scenario.

Supplier selection can be modeled as a typical multicriteria decision-making problem. It depends on the broad comparison of suppliers using a common set of traditional criteria and measures. Several methodologies have been proposed for supplier selection. Some of the well-known examples of systematic analysis for domestic supplier selection include a categorical method, weighted point method [3], vendor profile analysis [4], and multiple objective programming [5-7]. Apart from the abovementioned techniques, the related works have been also proposed. Choy and Lee proposed a case-based supplier management tool (CBSMT) using the case-based reasoning (CBR) technique in the areas of intelligent supplier selection and management that will enhance performance as compared to using the traditional approach [8]. Liu et al. demonstrated the application of data envelopment analysis (DEA) in evaluating the overall performances of suppliers in a manufacturing firm [9]. Jiang et al. proposed a model to deal with supplier selection based on fuzzy set in the Dempster-Shafer framework [10]. Cho and Lee proposed a method to deal with supply chain with a seasonal demand process using three levels' information sharing [11]. Mari et al. proposed a network optimization model for a sustainable and resilient supply chain network [12]. Mari et al. also discussed the construction of a resilient supply chain growth algorithm based on a complex network theory for designing a resilient supply chain network [13]. Some other researches related to the issue of supply chain management can also be inspiring, such as [14-16]. Additionally, some models of uncertain information process are proposed to deal with decision-making under uncertain environment [17, 18]; these methods can be easily expanded to the issue of selection of suppliers.

The most common and available methodology applied for supplier selection is analytic hierarchy process (AHP). Wang et al. [19] used the advantages of AHP and preemptive goal programming to incorporate both quantitative and qualitative factor in supplier selection problem. Ghodsypour and O'Brien [20] proposed integration of an AHP and linear programming to consider both tangible and intangible factor in choosing the best suppliers and placing optimum order quantities among them. Amid et al. [21] used the AHP to determine the weights of criteria for fuzzy multiobjective supplier selection in a supply chain. Aydin and Kahraman [22] presented an AHP based analytical tool for decision support to establish an effective multicriteria supplier selection process in an air conditioner seller firm under fuzziness. Aydin and Kahraman [22] calculated the weight of each component by fuzzy AHP for balanced and defective supply chain problems. Wang and Yang [23] proposed a fuzzy model for supplier selection in quantity discount environment with method of AHP and fuzzy compromise programming. Yang et al. [24] established a model for vendor selection by integrated fuzzy MCDM techniques with independent and interdependent relationships. Sevkli [25] proposed an approach with AHP weighted fuzzy linear programming model for supplier selection. Chan et al. discussed the fuzzy based analytic hierarchy process (fuzzy AHP) to efficiently tackle both quantitative and qualitative decision factors involved in selection of global supplier in current business scenario and also applied the AHP methodology to select the supplier in the airline industry [26, 27]. Deng et al. proposed a methodology of DAHP to handle supplier selection based on [figure omitted; refer to PDF] -number considering the dependence of factors in the system of supplier selection [28]; similarly, Su et al. discussed the influence of the dependence of factors in the Dempster-Shafer framework in the view of AHP, which can be easily applied in the aspect of supplier selection [29]. Shahgholian et al. [30] proposed a multicriteria group decision-making approach based on fuzzy sets which can solve supplier selection problem that have much vagueness. In Shahgholian et al.'s method, linguistic variables were used to assess the importance weights of strategic and operational criterion. Some other works from the view of AHP are also demonstrated such as [31, 32].

Recently, It is noted that some fuzzy set theories such as [figure omitted; refer to PDF] -number theory and Grey system theory are applied in the supplier selection. [figure omitted; refer to PDF] -number theory takes the dependence of information into consideration, which can be useful in the framework with nonexclusive hypotheses [28, 33, 34]. Grey system theory can handle the sensitivity of the system efficiently for the interval number that was used to assess the importance weights of strategic and operational criterion [35, 36]. However, they cannot deal with the reliability of decision system efficiently.

With reference to the past literatures, it can be observed that the discussion of the reliability of the domain experts for the supplier selection is little and limited. [figure omitted; refer to PDF] -number is a new notion proposed by Zadeh in 2011 which has a more power to describe the knowledge of human being [37]. It has a simple structure with constraint and reliability, which can easily represent and handle the reliability of uncertain knowledge. We will take the reliability of pairwise judgement into consideration using [figure omitted; refer to PDF] -number. Note that the theory about [figure omitted; refer to PDF] -number has not been figured out for its real application, yet the classic fuzzy set theory is relatively mature and has played much important role in the field of approximate reasoning [38, 39], fuzzy control [40, 41], group decision-making [42, 43], multiple criteria decision-making [44-46], and so forth.

Hence, how to transform [figure omitted; refer to PDF] -number to a classic fuzzy number is a rather significant issue for [figure omitted; refer to PDF] -number's application in supplier selection. In this paper, a method of converting [figure omitted; refer to PDF] -numbers to classic fuzzy numbers according to fuzzy expectation is introduced. The other problem is how to get the optimal priority weight of the supplier selection; in this paper a new methodology of handling the pairwise judgement with Genetic Algorithm (GA) is proposed, which can get the optimal priority weight. The proposed framework will be introduced in the following sections step by step.

The remainder of the paper is organized as follows: Section 2 discusses some definitions, concepts, and basic theory. In Section 3, the proposed methodology for supplier selection using [figure omitted; refer to PDF] -numbers is presented. It contains three parts: the first one is the method of converting [figure omitted; refer to PDF] -number to classic fuzzy number according to fuzzy expectation; the other is the methodology for the optimal priority weight with GA and the third part shows the comparison between the proposed method and the classic FAHP for priority weight. Section 4 shows an example in supplier selection to illustrate the proposed approach. Conclusions and future work are made in Section 5.

2. Preliminaries

2.1. Fuzzy Sets

In 1965, the notion of fuzzy sets was firstly introduced by Zadeh [47], providing a natural way of dealing with problems in which the source of imprecision is the absence of sharply defined criteria of class membership. The fuzzy set theory can be used in a wide range of domains. A brief introduction of fuzzy sets is given as follows.

Definition 1.

A fuzzy set [figure omitted; refer to PDF] that is defined on a universe [figure omitted; refer to PDF] may be given as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the membership function [figure omitted; refer to PDF] . The membership value [figure omitted; refer to PDF] describes the degree of belongingness of [figure omitted; refer to PDF] in [figure omitted; refer to PDF] .

For a finite set [figure omitted; refer to PDF] , the fuzzy set [figure omitted; refer to PDF] is often denoted by [figure omitted; refer to PDF] .

In the real application, the domain experts may give their opinions by fuzzy numbers. For example, in a new product price estimation, one expert may give his opinion as follows: the lowest price is 2 dollars, the most possible price of the product may be 3 dollars, and the highest price of this product will not be in excess of 4 dollars. Hence, we can use a triangular fuzzy number [figure omitted; refer to PDF] to represent the expert's opinion. The triangular fuzzy numbers can be defined as follows.

Definition 2.

A triangular fuzzy number [figure omitted; refer to PDF] can be defined by a triplet [figure omitted; refer to PDF] , where the membership can be determined as follows.

A triangular fuzzy number [figure omitted; refer to PDF] can be shown in Figure 1. Consider [figure omitted; refer to PDF]

Figure 1: A triangular fuzzy number.

[figure omitted; refer to PDF]

Similarly, the trapezoidal fuzzy number can be defined as follows.

Definition 3.

A trapezoidal fuzzy number [figure omitted; refer to PDF] can be defined by a triplet [figure omitted; refer to PDF] , where the membership can be determined as follows.

A trapezoidal fuzzy number [figure omitted; refer to PDF] can be shown in Figure 2. Consider [figure omitted; refer to PDF]

Figure 2: A trapezoidal fuzzy number.

[figure omitted; refer to PDF]

Let [figure omitted; refer to PDF] be the universe of discourse, which includes five linguistic variables describing the degree of security; [figure omitted; refer to PDF] , assuming that only two adjacent linguistic variables have the overlap of meanings. And let [figure omitted; refer to PDF] be a fuzzy set of the universe of discourse [figure omitted; refer to PDF] subjectively defined as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are the membership function of the fuzzy sets, which are shown in Figure 3.

Figure 3: Membership function of the triangular fuzzy numbers.

[figure omitted; refer to PDF]

Definition 4.

Let a fuzzy set [figure omitted; refer to PDF] be defined on a universe [figure omitted; refer to PDF] which may be given as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the membership function [figure omitted; refer to PDF] . The membership value [figure omitted; refer to PDF] describes the degree of belongingness of [figure omitted; refer to PDF] in [figure omitted; refer to PDF] . The expectation of a fuzzy number is denoted as follows: [figure omitted; refer to PDF] which is not the same as the meaning of the Expectation of Probability Space. It is as the Information Strength supporting the fuzzy set [figure omitted; refer to PDF] .

Example 5.

Let fuzzy set [figure omitted; refer to PDF] be defined as [figure omitted; refer to PDF] : [figure omitted; refer to PDF]

Definition 6.

The distance of two fuzzy numbers [figure omitted; refer to PDF] and [figure omitted; refer to PDF] described by [figure omitted; refer to PDF] -level set is defined as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are any two fuzzy numbers and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is the [figure omitted; refer to PDF] -level set of two fuzzy numbers [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

In the real world, uncertainty is a pervasive phenomenon. Much of the information on which decisions are based is uncertain [48-51]. Humans have a remarkable capability to make rational decisions based on information which is uncertain, imprecise, and/or incomplete. Formalization of this capability, at least to some degree, is a challenge that is hard to meet [52, 53]. Zadeh proposed a notion, namely, [figure omitted; refer to PDF] -number, which is an ordered pair of fuzzy numbers [figure omitted; refer to PDF] . The first component, [figure omitted; refer to PDF] , plays the role of a fuzzy restriction. And the second component, [figure omitted; refer to PDF] , is a reliability of the first component [37]. The definition of [figure omitted; refer to PDF] -number is shown below.

2.2. [figure omitted; refer to PDF] -Numbers

A new concept, [figure omitted; refer to PDF] -numbers, is proposed by Zadeh to model uncertain information [37]. A [figure omitted; refer to PDF] -number can be defined as an ordered pair of fuzzy numbers as follows.

Definition 7.

A [figure omitted; refer to PDF] -number is an ordered pair of fuzzy numbers denoted as [figure omitted; refer to PDF] . The first component [figure omitted; refer to PDF] , a restriction on the values, is a real-valued uncertain variable [figure omitted; refer to PDF] . The second component [figure omitted; refer to PDF] is a measure of reliability for the first component.

Zadeh [37] points out that [figure omitted; refer to PDF] is a restriction on the probability measure of [figure omitted; refer to PDF] rather than on the probability of [figure omitted; refer to PDF] . Conversely, if [figure omitted; refer to PDF] is a restriction on the probability of [figure omitted; refer to PDF] rather than on the probability measure of [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] is not a [figure omitted; refer to PDF] -number. It means that [figure omitted; refer to PDF] measures the sureness, confidence, and reliability of measurement of restriction of [figure omitted; refer to PDF] .

[figure omitted; refer to PDF] -numbers can be used to model uncertain information in the real world. For example, in risk analysis, the loss of severity of the fifth component is very low, with a confidence of very likely, which can be written as a [figure omitted; refer to PDF] -number as follows: [figure omitted; refer to PDF] . For simplicity, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are assumed as triangular fuzzy numbers as defined earlier in (4) and Figure 3.

Then the notion of AHP will be introduced as follows. The first step of AHP is to establish a hierarchical structure of the problem. In each hierarchical level, a nominal scale is used to construct pairwise comparison judgement matrix.

Definition 8.

Assuming [figure omitted; refer to PDF] are [figure omitted; refer to PDF] decision elements, the pairwise comparison judgement matrix is denoted as [figure omitted; refer to PDF] , which satisfies [figure omitted; refer to PDF] where each element [figure omitted; refer to PDF] represents the judgement concerning the relative importance of decision element [figure omitted; refer to PDF] over [figure omitted; refer to PDF] .

Definition 9.

Eigenvector of [figure omitted; refer to PDF] pairwise comparison judgement matrix can be denoted as [figure omitted; refer to PDF] , which is calculated as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the maximum eigenvalue in the eigenvector [figure omitted; refer to PDF] of matrix [figure omitted; refer to PDF] .

Definition 10.

Consistency index (CI) [54] is used to measure the inconsistency within each pairwise comparison judgement matrix, which is formulated as follows: [figure omitted; refer to PDF]

Accordingly, the consistency ratio ( [figure omitted; refer to PDF] ) can be calculated by using the following equation: [figure omitted; refer to PDF] where RI is the random consistency index. The value of RI is related to the dimension of the matrix, which is listed in Table 2.

If the result of [figure omitted; refer to PDF] is less than 0.1, the consistency of the pairwise comparison matrix [figure omitted; refer to PDF] is acceptable. Moreover, the eigenvector of pairwise comparison judgement matrix can be normalized as final weights of decision elements. Otherwise, the consistency is not passed and the elements in the matrix should be revised.

2.3. FAHP

In this section, we briefly introduce a typical FAHP method. For detailed information, please refer to [2, 55, 56].

In the first step, triangular fuzzy numbers are used for pairwise comparisons. Then, by using extent analysis method, the synthetic extent value [figure omitted; refer to PDF] of the pairwise comparison is introduced and by applying the principle of the comparison of fuzzy numbers, the weight vectors with respect to each element under a certain criterion are calculated. The details of the methodology are presented in the following steps.

Let [figure omitted; refer to PDF] be an object set and let [figure omitted; refer to PDF] be a goal set. According to the method of Chang's extent analysis, each object is taken and an extent analysis for each goal, [figure omitted; refer to PDF] , is performed. Therefore, [figure omitted; refer to PDF] extent analysis values for each object can be obtained, with the following signs: [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , where all [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) are TFNs.

Step 1.

The value of fuzzy synthetic extent with respect to the [figure omitted; refer to PDF] th object is defined as [figure omitted; refer to PDF] In order to obtain [figure omitted; refer to PDF] , perform the fuzzy addition operation of [figure omitted; refer to PDF] extent analysis values for a particular matrix such that [figure omitted; refer to PDF] To obtain [figure omitted; refer to PDF] , perform the fuzzy addition operation of [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) values such that [figure omitted; refer to PDF] and then compute the inverse of the vector.

Step 2.

The degree of possibility of [figure omitted; refer to PDF] is expressed as follows: [figure omitted; refer to PDF] To compare [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , both [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are required.

Step 3.

The degree of possibility for a convex fuzzy number to be greater than [figure omitted; refer to PDF] convex fuzzy numbers [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) can be defined as [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] . Then the weight vector is given by [figure omitted; refer to PDF]

Step 4.

The weight vector obtained in Step 3 is normalized to get the normalized weights.

In the following part, the methodology for supplier selection using [figure omitted; refer to PDF] -number is proposed, which includes three parts: the first is how to convert [figure omitted; refer to PDF] -number to a classic fuzzy number; the second is how to get the optimal priority weight; the process is shown in Figure 7. The first part corresponds to [...] shown in Figure 7 and the second part corresponds to [...], [...], and [...] in Figure 7; the third one is a comparison between the proposed method for the optimal priority weight and the classic fuzzy AHP.

3. The Proposed Methodology for Supplier Selection Using [figure omitted; refer to PDF] -Numbers

In the following subsection, a method of changing a [figure omitted; refer to PDF] -number to a classic fuzzy number is used [57] according to the fuzzy expectation. Then the pairwise reciprocal judging matrices are converted to classic reciprocal judging matrices.

3.1. The Method of Changing a [figure omitted; refer to PDF] -Number to a Classic Fuzzy Number

Assume a [figure omitted; refer to PDF] -number [figure omitted; refer to PDF] , which is shown in Figure 4. The left is the part of restriction, and the right is the part of reliability. Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Assume [figure omitted; refer to PDF] is a trapezoidal membership function and [figure omitted; refer to PDF] is a triangular membership function.

Figure 4: A simple [figure omitted; refer to PDF] -number.

[figure omitted; refer to PDF]

3.1.1. Convert the Second Part (Reliability) into a Crisp Number with Centroid Method

Consider [figure omitted; refer to PDF] where [figure omitted; refer to PDF] denotes an algebraic integration.

3.1.2. Add the Weight of the Second Part (Reliability) to the First Part (Restriction)

The weighted [figure omitted; refer to PDF] -number can be denoted as [figure omitted; refer to PDF] .

Theorem 11.

Consider [figure omitted; refer to PDF]

Proof.

Consider [figure omitted; refer to PDF] which can be shown in Figure 5.

Figure 5: [figure omitted; refer to PDF] -number after multiplying the confidence.

[figure omitted; refer to PDF]

3.1.3. Convert the Irregular Fuzzy Number (Weighted Restriction) to Regular Fuzzy Number

The converted regular fuzzy number is [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

Theorem 12.

Consider [figure omitted; refer to PDF]

Proof.

Consider [figure omitted; refer to PDF] which can be shown in Figure 6.

Figure 6: The regular fuzzy number transformed from [figure omitted; refer to PDF] -number.

[figure omitted; refer to PDF]

Figure 7: Methodology for the weight of the pairwise reciprocal judging matrix with [figure omitted; refer to PDF] -numbers in supplier selection. [...] A method of converting [figure omitted; refer to PDF] -number to classic fuzzy number; [...] an objective function according to the min-bias to the consistent weight; [...] the genetic algorithm is applied to solve the optimal fuzzy weight; [...] a centroid method is used to convert the fuzzy weight to crisp priority weight.

[figure omitted; refer to PDF]

Theorem 13.

Consider [figure omitted; refer to PDF]

Proof.

From (21) and (24), [figure omitted; refer to PDF]

Here a simple numerical example is used to illustrate the proposed method of converting a [figure omitted; refer to PDF] -number to a classical fuzzy number.

Example 14.

Assume that an expert gives his opinion as follows: [figure omitted; refer to PDF] and his reliability, confidence, sureness, strength of belief, and so forth are [figure omitted; refer to PDF] Hence the expert's knowledge can be expressed with [figure omitted; refer to PDF] -number as [figure omitted; refer to PDF] At first, we should convert expert's reliability into crisp number according to (19): [figure omitted; refer to PDF] Second, add the weight of reliability [figure omitted; refer to PDF] to the constraint [figure omitted; refer to PDF] : [figure omitted; refer to PDF] Third, convert the weighted [figure omitted; refer to PDF] -number to regular fuzzy number according to the proposed approach: [figure omitted; refer to PDF]

From the proof above, it can be concluded that the fuzzy expectation of [figure omitted; refer to PDF] and the fuzzy expectation of [figure omitted; refer to PDF] are equal.

In the following subsection, the methodology for the optimal priority weight is proposed. At first, a reciprocal judging matrix using [figure omitted; refer to PDF] -numbers is shown. Then the method of converting [figure omitted; refer to PDF] -number to classic fuzzy number shown as the subsection above is applied. Note that the inconsistency is a common phenomenon in the decision-making; the genetic algorithm is used to figure out the optimal fuzzy priority weights. An illustration will be used to prove its advantage with the most popular method proposed by Chang [55]. At last, the centroid method is used to get the crisp priority weight. The procedure can be shown in Figure 7.

3.2. The Methodology for the Optimal Priority Weight

In most situations, nonconsensus is a common phenomenon in the group decision-making. Although the fuzzy set has been applied to soften the conflicts among the different opinions from experts, how to get the optimal priority weight is a critical problem and open issue. In this part, a biased function is defined to establish the objective function. Then the problem is converted to solve the optimal issue under some constraint. In the following, the methodology of searching the optimal priority weight based on GA is proposed.

A reciprocal judging matrix [figure omitted; refer to PDF] is consistent if and only if [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is the eigenvector of the matrix [figure omitted; refer to PDF] . However, the consistency of the reciprocal judging matrix [figure omitted; refer to PDF] is always not satisfied due to the complexity of the real application, and the equation [figure omitted; refer to PDF] is not satisfied.

Hence, when the reciprocal judging matrix [figure omitted; refer to PDF] with triangular fuzzy number is inconsistent, the biased function is defined as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is the [figure omitted; refer to PDF] -set cut of fuzzy number [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is the [figure omitted; refer to PDF] -set cut of fuzzy number [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is the [figure omitted; refer to PDF] -set cut of fuzzy number [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

From (8), the biased function can be denoted as follows: [figure omitted; refer to PDF]

Obviously, the solved priority weight should make the value of (35) to get the minimum. Hence the objective function for the reciprocal judging matrix using triangular fuzzy numbers can be denoted as follows: [figure omitted; refer to PDF]

The above function (36) equals the following equation denoted as [figure omitted; refer to PDF]

Such nonlinear programming models can be easily implemented by using existing optimization packages such as LINGO software package or MATLAB optimization tool box. Note that genetic algorithm is an available tool to solve the optimal issue, which has an excellent power to search the global optimal solution with complex constraints [58-60]; it is applied to figure out the optimal fuzzy priority weight [figure omitted; refer to PDF] . In this paper, the parameters of experimental GA are some default values (population size: 20; scaling function: rank; selection function: stochastic uniform; elite count: 2; crossover fraction: 0.8; mutation function: constraint dependent; crossover function: scattered; migration-direction: forward; migration-fraction: 0.2; migration-interval: 20; constraint parameter-initial penalty: 10; constraint parameter-penalty factor: 100; hybrid function: none; stopping criteria: 100, Inf, -Inf, 50, Inf, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ). Then the centroid method is applied to convert the optimal fuzzy priority weight to crisp weight.

The following example is used to illustrate the proposed method for the priority weight of the judging matrix using [figure omitted; refer to PDF] -numbers.

Example 15.

This example is used to calculate the weight of three different criteria through fuzzy evaluation matrix; the fuzzy evaluation of the criteria is constructed by the pairwise comparison of the different criterion relevant to the overall object using [figure omitted; refer to PDF] -numbers, which is shown in Table 2. The linguistic variable is applied to represent the experts' opinion, which is defined like Figure 3.

Firstly, we should convert the linguistic variables to numerical [figure omitted; refer to PDF] -numbers denoted by Table 3.

Secondly, according to the proposed method of converting [figure omitted; refer to PDF] -number to regular fuzzy number, we can get the regular triangular fuzzy number matrix shown in Table 4.

Suppose that the fuzzy variable is [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ; then the objective function according to (36) can be established. After the procedure of selection, mating, crossover, and mutation of genetic algorithm, the optimal fuzzy weight can be figured out as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and the minimum of the objective function is 0.0277. The detailed result is shown in Figure 8.

According to the centroid method, the fuzzy priority can be defuzzified as [figure omitted; refer to PDF]

The normalized priority weight is [figure omitted; refer to PDF]

Figure 8: Optimal fuzzy priority weight (upper right) with genetic algorithm. Parameters of GA are as follows: population size: 20; scaling function: rank; selection function: stochastic uniform; elite count: 2; crossover fraction: 0.8; mutation function: constraint dependent; crossover function: scattered; migration-direction: forward; migration-fraction: 0.2; migration-interval: 20; constraint parameter-initial penalty: 10; constraint parameter-penalty factor: 100; hybrid function: none; stopping criteria: 100, Inf, -Inf, 50, Inf, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

3.3. The Methodology for the Optimal Weight Comparing with the Classic FAHP

Inconsistency is a critical problem that should be taken into consideration in the process of decision-making. Although the extent analysis method on fuzzy AHP proposed by Chang [55] has been applied to soften the conflicts from different experts/commanders, it is still lack of capability to handle this issue. Sometimes, Chang's method gets an unreasonable result in spite of the fact that the decision-matrix is consistent according to the calculation method of consistency index and consistency ratio presented by Kwong [61]. Kwong's method has been used to check the consistency of pairwise judgement of each comparison matrix in the application of global supplier development [2].

Now, a designed example is used to denote the shortcoming of the extent analysis method on fuzzy AHP and to illustrate the efficiency and advantage of our proposed method.

Suppose that a designed comparison matrix with three criteria [figure omitted; refer to PDF] is constructed according to the pairwise judgement, which is shown in Table 5.

Then, Kwong's method will be also applied to check the consistency of comparison matrix (Table 5). The detailed steps are shown as follows.

First, a triangular fuzzy number, denoted as [figure omitted; refer to PDF] , can be defuzzified to a crisp number as follows: [figure omitted; refer to PDF]

Hence, the crisp comparison matrix can be calculated according to formula (41); the crisp comparison matrix is shown in Table 6.

Then, the consistency index (CI) and the consistency ratio (CR) for a comparison matrix can be computed with the use of the following formulas: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the largest eigenvalue of the comparison matrix, [figure omitted; refer to PDF] is the dimension of the matrix, and [figure omitted; refer to PDF] is a random index, which depends on [figure omitted; refer to PDF] , as shown in Table 1.

Table 1: The value of RI (random consistency index).

Dimension	1	2	3	4	5	6	7	8	9	10
RI	0	0	0.52	0.89	1.12	1.26	1.36	1.41	1.46	1.49

Table 2: Linguistic evaluation of criteria.

	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	-	(H, VH)	(VH, H)
[figure omitted; refer to PDF]	(H-1 , VH)	-	(VH, VH)
[figure omitted; refer to PDF]	(VH-1 , H)	(VH-1 , VH)	-

Table 3: Numerical fuzzy evaluation of criteria.

	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	((1, 1, 1), (1, 1, 1))	((0.5, 0.75, 1), (0.75, 1, 1))	((0.75, 1, 1), (0.5, 0.75, 1))
[figure omitted; refer to PDF]	((1, 1/0.75, 1/0.5), (0.75, 1, 1))	((1, 1, 1), (1, 1, 1))	((0.75, 1, 1), (0.75, 1, 1))
[figure omitted; refer to PDF]	((1, 1, 1/0.75), (0.5, 0.75, 1))	((1, 1, 1/0.75), (0.75, 1, 1))	((1, 1, 1), (1, 1, 1))

Table 4: Regular fuzzy evaluation of criteria.

	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	(1, 1, 1)	(0.48, 0.72, 0.96)	(0.65, 0.87, 0.87)
[figure omitted; refer to PDF]	(0.96, 1.27, 1.91)	(1, 1, 1)	(0.72, 0.96, 0.96)
[figure omitted; refer to PDF]	(0.87, 0.87, 1.15)	(0.96, 0.96, 1.27)	(1, 1, 1)

Table 5: Comparison matrix with triangular fuzzy number.

	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	(1, 1, 1)	(1, 2, 3)	(4, 5, 6)
[figure omitted; refer to PDF]	(1/3, 1/2, 1)	(1, 1, 1)	(2, 3, 4)
[figure omitted; refer to PDF]	(1/6, 1/5, 1/4)	(1/4, 1/3, 1/2)	(1, 1, 1)

Table 6: Comparison matrix with triangular fuzzy number.

	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	1	2	5
[figure omitted; refer to PDF]	0.56	1	3
[figure omitted; refer to PDF]	0.2	0.35	1

As seen in formulas (42), the consistency ratio is less than 0.1, and the pairwise judgement of comparison matrix is consistent.

Now, we will use the extent analysis method on fuzzy AHP proposed by Chang [55] to calculate the weight of the comparison matrix (see Table 5).

First Step . By applying formula (13), we have [figure omitted; refer to PDF]

The relation among [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is shown in Figure 9.

Figure 9: Relation of the fuzzy synthetic extent [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Using formulas (16), [figure omitted; refer to PDF]

Finally, by using formula (17), we obtain [figure omitted; refer to PDF]

Therefore, [figure omitted; refer to PDF]

Via normalization, we obtain the weight vectors with respect to the decision criteria [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] : [figure omitted; refer to PDF]

Then we will use the proposed methodology (GAFAHP) to calculate the optimal priority weight of the comparison matrix (see Table 5).

According to the proposed methodology for the optimal priority weight, after the 50 times of iterations, the changing trend of the fitness of population and the final optimal weight are shown in Figure 10. At last, we obtain the optimal fuzzy weight: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

Figure 10: Optimal fuzzy priority weight (upper right) with Genetic Algorithm. Parameters of GA are as follows: population size: 20; scaling function: rank; selection function: stochastic uniform; elite count: 2; crossover fraction: 0.8; mutation function: constraint dependent; crossover function: scattered; migration-direction: forward; migration-fraction: 0.2; migration-interval: 20; constraint parameter-initial penalty: 10; constraint parameter-penalty factor: 100; hybrid function: none; stopping criteria: 100, Inf, -Inf, 50, Inf, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Then, the centroid method is used to convert the optimal fuzzy weight to crisp weight: [figure omitted; refer to PDF]

Via normalization, the final priority weight is [figure omitted; refer to PDF]

From Table 7, the conclusion can be made that our proposed method is a more efficient method to obtain the priority weight. The reason that classic FAHP is not applicable is that the value of fuzzy synthetic extent has no intersection sometimes, which can be shown in Figure 9, and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] have no intersection. Hence, zero is assigned to the weight of criteria [figure omitted; refer to PDF] , which implies that [figure omitted; refer to PDF] is of no use at all. It is extremely unreasonable. Our proposed method can overcome the shortcoming and can get the optimal priority weight all the way.

Table 7: Proposed method versus classic FAHP.

	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Classic FAHP [2, 55]	0.68	0.32	0.00
The proposed method	0.59	0.29	0.12

4. An Application of Supplier Selection

Decision-making is widely used in supplier management and selection. In this section, a numerical example originated from [2] is presented to illustrate the procedure of the proposed model.

Owing to the large number of factors affecting the supplier selection decision, an orderly sequence of steps should be required to tackle it. The problem taken here has four levels of hierarchy, and the different decision criterion, attributes, and the decision alternatives will be further discussed. The main objective here is the selection of best global supplier for a manufacturing firm. Application of common criteria to all suppliers makes objective comparisons possible. The criteria which are considered here in selection of the global supplier are as follows:

(a) [figure omitted; refer to PDF] ) Overall cost of the product.

(b) ( [figure omitted; refer to PDF] ) Quality of the product.

(c) ( [figure omitted; refer to PDF] ) Service performance of supplier.

(d) ( [figure omitted; refer to PDF] ) Supplier profile.

(e) ( [figure omitted; refer to PDF] ) Risk factor.

The hierarchy model of supplier selection can be constructed as shown in Figure 11.

Figure 11: Hierarchy for the global supplier selection.

[figure omitted; refer to PDF]

As can be seen from Figure 11, the overall cost of the product ( [figure omitted; refer to PDF] ) has three factors (attributes): product price ( [figure omitted; refer to PDF] ), freight cost ( [figure omitted; refer to PDF] ), and tariff and custom duties ( [figure omitted; refer to PDF] ). The quality of the product ( [figure omitted; refer to PDF] ) has four factors: rejection rate of the product ( [figure omitted; refer to PDF] ), increased lead time ( [figure omitted; refer to PDF] ), quality assessment ( [figure omitted; refer to PDF] ), and remedy for quality problems ( [figure omitted; refer to PDF] ). The service performance ( [figure omitted; refer to PDF] ) has four attributes: delivery schedule ( [figure omitted; refer to PDF] ), technological and R&D support ( [figure omitted; refer to PDF] ), response to changes ( [figure omitted; refer to PDF] ), and ease of communication ( [figure omitted; refer to PDF] ). The suppliers profile ( [figure omitted; refer to PDF] ) has four attributes: financial status ( [figure omitted; refer to PDF] ), customer base ( [figure omitted; refer to PDF] ), performance history ( [figure omitted; refer to PDF] ), and production facility and capacity ( [figure omitted; refer to PDF] ). The risk factor ( [figure omitted; refer to PDF] ) has four attributes: geographical location ( [figure omitted; refer to PDF] ), political stability ( [figure omitted; refer to PDF] ), economy ( [figure omitted; refer to PDF] ), and terrorism ( [figure omitted; refer to PDF] ). Refer to [2] for more detailed information about the attributes mentioned above.

After the construction of the decision hierarchy of supplier selection, the fuzzy evaluation matrix of the criterion is constructed by the pairwise comparison of the different criterion relevant to the overall objective using [figure omitted; refer to PDF] -numbers.

The fuzzy evaluation of criteria with respect to the overall objective can be listed in Table 8. The final weights of each criteria can be determined by the proposed method. The detailed calculation process is given in Sections 3 and 4. The results are listed in right side of Table 8.

Table 8: Fuzzy evaluation of criteria with respect to the overall objective.

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[(1, 1, 1), (1, 1, 1)]	[(3/2, 2, 5/2), VH]	[(3/2, 2, 5/2), VH]	[(5/2, 3, 7/2), H]	[(5/2, 3, 7/2), VH]	0.49
[figure omitted; refer to PDF]	[(2/5, 1/2, 2/3), VH]	[(1, 1, 1), (1, 1, 1)]	[(3/2, 2, 5/2), M]	[(5/2, 3, 7/2), VH]	[(5/2, 3, 7/2), VH]	0.19
[figure omitted; refer to PDF]	[(2/5, 1/2, 2/3), VH]	[(2/5, 1/2, 2/3), M]	[(1, 1, 1), (1, 1, 1)]	[(3/2, 2, 5/2), VH]	[(3/2, 2, 5/2), VH]	0.15
[figure omitted; refer to PDF]	[(2/7, 1/3, 2/5), H]	[(2/7, 1/3, 2/5), VH]	[(2/5, 1/2, 2/3), VH]	[(1, 1, 1), (1, 1, 1)]	[(3/2, 2, 5/2), M]	0.11
[figure omitted; refer to PDF]	[(2/7, 1/3, 2/5), VH]	[(2/7, 1/3, 2/5), VH]	[(2/5, 1/2, 2/3), VH]	[(2/5, 1/2, 2/3), M]	[(1, 1, 1), (1, 1, 1)]	0.05

In a similar way, the fuzzy evaluation of the attributes with respect to criteria [figure omitted; refer to PDF] to [figure omitted; refer to PDF] can be given by domain experts and the corresponding results based on the proposed method are listed in Tables 9-13, respectively.

Table 9: Fuzzy evaluation of the attributes with respect to criterion [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[(1, 1, 1), (1, 1, 1)]	[(3/2, 2, 5/2), VH]	[(3/2, 2, 5/2), H]	0.47
[figure omitted; refer to PDF]	[(2/5, 1/2, 2/3), VH]	[(1, 1, 1), (1, 1, 1)]	[(3/2, 2, 5/2), VH]	0.30
[figure omitted; refer to PDF]	[(2/5, 1/2, 2/3), H]	[(2/5, 1/2, 2/3), VH]	[(1, 1, 1), (1, 1, 1)]	0.24

Table 10: Fuzzy evaluation of the attributes with respect to criterion [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[(1, 1, 1), (1, 1, 1)]	[(3/2, 2, 5/2), VH]	[(2/3, 1, 3/2), M]	[(5/2, 3, 7/2), VH]	0.32
[figure omitted; refer to PDF]	[(2/5, 1/2, 2/3), VH]	[(1, 1, 1), (1, 1, 1)]	[(2/3, 1, 3/2), VH]	[(3/2, 2, 5/2), H]	0.23
[figure omitted; refer to PDF]	[(2/3, 1, 3/2), M]	[(2/3, 1, 3/2), VH]	[(1, 1, 1), (1, 1, 1)]	[(3/2, 2, 5/2), VH]	0.32
[figure omitted; refer to PDF]	[(2/7, 1/3, 2/5), VH]	[(2/5, 1/2, 2/3), H]	[(2/5, 1/2, 2/3), VH]	[(1, 1, 1), (1, 1, 1)]	0.14

Table 11: Fuzzy evaluation of the attributes with respect to criterion [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[(1, 1, 1), (1, 1, 1)]	[(3/2, 2, 5/2), M]	[(5/2, 3, 7/2), VH]	[(7/2, 4, 9/2), H]	0.43
[figure omitted; refer to PDF]	[(2/5, 1/2, 2/3), M]	[(1, 1, 1), (1, 1, 1)]	[(5/2, 3, 7/2), H]	[(5/2, 3, 7/2), VH]	0.28
[figure omitted; refer to PDF]	[(2/7, 1/3, 2/5), VH]	[(2/7, 1/3, 2/5), H]	[(1, 1, 1), (1, 1, 1)]	[(3/2, 2, 5/2), VH]	0.17
[figure omitted; refer to PDF]	[(2/9, 1/4, 2/7), H]	[(2/7, 1/3, 2/5), VH]	[(2/5, 1/2, 2/3), VH]	[(1, 1, 1), (1, 1, 1)]	0.11

Table 12: Fuzzy evaluation of the attributes with respect to criterion [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[(1, 1, 1), (1, 1, 1)]	[(3/2, 2, 5/2), VH]	[(3/2, 2, 5/2), VH]	[(7/2, 4, 9/2), H]	0.44
[figure omitted; refer to PDF]	[(2/5, 1/2, 2/3), VH]	[(1, 1, 1), (1, 1, 1)]	[(2/5, 1/2, 2/3), VH]	[(3/2, 2, 5/2), H]	0.21
[figure omitted; refer to PDF]	[(2/5, 1/2, 2/3), VH]	[(2/7, 1/3, 2/5), VH]	[(1, 1, 1), (1, 1, 1)]	[(3/2, 2, 5/2), VH]	0.21
[figure omitted; refer to PDF]	[(2/9, 1/4, 2/7), H]	[(2/5, 1/2, 2/3), H]	[(2/5, 1/2, 2/3), VH]	[(1, 1, 1), (1, 1, 1)]	0.13

Table 13: Fuzzy evaluation of the attributes with respect to criterion [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[(1, 1, 1), (1, 1, 1)]	[(2/3, 1, 3/2), H]	[(2/3, 1, 3/2), VH]	[(2/3, 1, 3/2), VH]	0.28
[figure omitted; refer to PDF]	[(2/3, 1, 3/2), H]	[(1, 1, 1), (1, 1, 1)]	[(3/2, 2, 5/2), VH]	[(3/2, 2, 5/2), M]	0.31
[figure omitted; refer to PDF]	[(2/3, 1, 3/2), VH]	[(2/5, 1/2, 2/3), VH]	[(1, 1, 1), (1, 1, 1)]	[(3/2, 2, 5/2), VH]	0.24
[figure omitted; refer to PDF]	[(2/3, 1, 3/2), VH]	[(2/5, 1/2, 2/3), M]	[(2/5, 1/2, 2/3), VH]	[(1, 1, 1), (1, 1, 1)]	0.17

For the criterion [figure omitted; refer to PDF] , the summary combination of priority weights can be listed in Table 14. Also, the other summary combinations of priority weights of [figure omitted; refer to PDF] to [figure omitted; refer to PDF] are shown in Tables 15-18.

Table 14: Summary combination of priority weights: attributes of criterion [figure omitted; refer to PDF] .

Weight	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	Priority weight
0.47	0.30	0.24
Alternatives
[figure omitted; refer to PDF]	0.71	0.44	0.69	0.63
[figure omitted; refer to PDF]	0.13	0.36	0.08	0.19
[figure omitted; refer to PDF]	0.16	0.20	0.23	0.19

Table 15: Summary combination of priority weights: attributes of criterion [figure omitted; refer to PDF] .

Weight	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	Priority weight
0.32	0.23	0.32	0.14
Alternatives
[figure omitted; refer to PDF]	0.51	0.51	0.69	0.87	0.62
[figure omitted; refer to PDF]	0.23	0.23	0.08	0.00	0.15
[figure omitted; refer to PDF]	0.26	0.26	0.23	0.13	0.23

Table 16: Summary combination of priority weights: attributes of criterion [figure omitted; refer to PDF] .

Weight	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	Priority weight
0.43	0.28	0.17	0.11
Alternatives
[figure omitted; refer to PDF]	0.27	0.69	0.05	0.49	0.38
[figure omitted; refer to PDF]	0.18	0.08	0.64	0.32	0.25
[figure omitted; refer to PDF]	0.55	0.23	0.31	0.19	0.38

Table 17: Summary combination of priority weights: attributes of criterion [figure omitted; refer to PDF] .

Weight	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	Priority weight
0.44	0.21	0.21	0.13
Alternatives
[figure omitted; refer to PDF]	0.83	0.45	0.69	0.33	0.66
[figure omitted; refer to PDF]	0.17	0.45	0.08	0.33	0.51
[figure omitted; refer to PDF]	0.00	0.10	0.23	0.34	0.18

Table 18: Summary combination of priority weights: attributes of criterion [figure omitted; refer to PDF] .

Weight	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	Priority weight
0.28	0.31	0.24	0.17
Alternatives
[figure omitted; refer to PDF]	0.72	0.49	0.83	0.27	0.60
[figure omitted; refer to PDF]	0.00	0.32	0.17	0.18	0.17
[figure omitted; refer to PDF]	0.28	0.19	0.00	0.55	0.23

The Fuzzy evaluation of criterion with respect to the overall objective can be shown in Table 19. As can be seen from Table 19 and Figure 12, the best supplier is [figure omitted; refer to PDF] .

Table 19: Summary combination of priority weights: main criterion of the overall objective.

Weight	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	Priority weight
0.49	0.19	0.15	0.11	0.05
Alternatives
[figure omitted; refer to PDF]	0.63	0.62	0.38	0.66	0.60	0.59
[figure omitted; refer to PDF]	0.19	0.15	0.25	0.51	0.17	0.22
[figure omitted; refer to PDF]	0.19	0.23	0.38	0.18	0.23	0.23

Figure 12: Final priority weights of each supplier.

[figure omitted; refer to PDF]

5. Conclusions and Future Work

The supplier selection is a significant issue of multicriteria decision-making (MCDM), which has been researched for decades. However, the reliability of the knowledge from experts/commanders is not efficiently taken into consideration. After the notion of [figure omitted; refer to PDF] -number was introduced by Zadeh in 2011, some attention has been paid to this concept. [figure omitted; refer to PDF] -number has more capability to describe uncertain information with both restraint and reliability. In this paper, a methodology for supplier selection using [figure omitted; refer to PDF] -numbers is proposed, which includes two parts: one solves the issue of how to convert [figure omitted; refer to PDF] -number to classic fuzzy number; the other solves the problem of how to get the optimal priority weight for supplier selection with GA, which is more efficient comparing with the classic FAHP. At last, a designed example for supplier selection is used to illustrate the proposed methodology.

[figure omitted; refer to PDF] -number is a new notion which has a flexible presentation of uncertain information to simulate the knowledge or behavior of human being. The comparison of different [figure omitted; refer to PDF] -numbers is an important and open issue, which is also the fundamental issue to expand the application of [figure omitted; refer to PDF] -number. This is the future work we will concern.

Acknowledgments

The work is partially supported by National High Technology Research and Development Program of China (863 Program) (Grant no. 2013AA013801), National Natural Science Foundation of China (Grant nos. 61174022, 61573290, and 61503237), and China State Key Laboratory of Virtual Reality Technology and Systems, Beihang University (Grant no. BUAA-VR-14KF-02).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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