Supplier selection is a significant issue of multicriteria decisionmaking (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.
The selection of suppliers is a significant issue in the business management [
Supplier selection can be modeled as a typical multicriteria decisionmaking 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 wellknown examples of systematic analysis for domestic supplier selection include a categorical method, weighted point method [
The most common and available methodology applied for supplier selection is analytic hierarchy process (AHP). Wang et al. [
Recently, It is noted that some fuzzy set theories such as
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.
Hence, how to transform
The remainder of the paper is organized as follows: Section
In 1965, the notion of fuzzy sets was firstly introduced by Zadeh [
A fuzzy set
For a finite set
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
A triangular fuzzy number
A triangular fuzzy number
A triangular fuzzy number.
Similarly, the trapezoidal fuzzy number can be defined as follows.
A trapezoidal fuzzy number
A trapezoidal fuzzy number
A trapezoidal fuzzy number.
Let
Membership function of the triangular fuzzy numbers.
Let a fuzzy set
Let fuzzy set
The distance of two fuzzy numbers
In the real world, uncertainty is a pervasive phenomenon. Much of the information on which decisions are based is uncertain [
A new concept,
A
Zadeh [
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.
Assuming
Eigenvector of
Consistency index (CI) [
Accordingly, the consistency ratio (
If the result of
In this section, we briefly introduce a typical FAHP method. For detailed information, please refer to [
In the first step, triangular fuzzy numbers are used for pairwise comparisons. Then, by using extent analysis method, the synthetic extent value
Let
The value of fuzzy synthetic extent with respect to the
The degree of possibility of
The degree of possibility for a convex fuzzy number to be greater than
The weight vector obtained in Step
In the following part, the methodology for supplier selection using
In the following subsection, a method of changing a
Assume a
A simple
Consider
The weighted
Consider
Consider
The converted regular fuzzy number is
Consider
Consider
The regular fuzzy number transformed from
Methodology for the weight of the pairwise reciprocal judging matrix with
Consider
From (
Here a simple numerical example is used to illustrate the proposed method of converting a
Assume that an expert gives his opinion as follows:
From the proof above, it can be concluded that the fuzzy expectation of
In the following subsection, the methodology for the optimal priority weight is proposed. At first, a reciprocal judging matrix using
In most situations, nonconsensus is a common phenomenon in the group decisionmaking. 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
Hence, when the reciprocal judging matrix
From (
Obviously, the solved priority weight should make the value of (
The above function (
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 [
The following example is used to illustrate the proposed method for the priority weight of the judging matrix using
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
Firstly, we should convert the linguistic variables to numerical
Secondly, according to the proposed method of converting
Suppose that the fuzzy variable is
According to the centroid method, the fuzzy priority can be defuzzified as
The normalized priority weight is
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; migrationdirection: forward; migrationfraction: 0.2; migrationinterval: 20; constraint parameterinitial penalty: 10; constraint parameterpenalty factor: 100; hybrid function: none; stopping criteria: 100, Inf, −Inf, 50, Inf,
Inconsistency is a critical problem that should be taken into consideration in the process of decisionmaking. Although the extent analysis method on fuzzy AHP proposed by Chang [
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
Then, Kwong’s method will be also applied to check the consistency of comparison matrix (Table
First, a triangular fuzzy number, denoted as
Hence, the crisp comparison matrix can be calculated according to formula (
Then, the consistency index (CI) and the consistency ratio (CR) for a comparison matrix can be computed with the use of the following formulas:
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 
Linguistic evaluation of criteria.


 


—  (H, VH)  (VH, H) 

(H^{−1}, VH)  —  (VH, VH) 

(VH^{−1}, H)  (VH^{−1}, VH)  — 
Numerical fuzzy evaluation of criteria.


 


((1, 1, 1), (1, 1, 1))  ((0.5, 0.75, 1), (0.75, 1, 1))  ((0.75, 1, 1), (0.5, 0.75, 1)) 

((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)) 

((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)) 
Regular fuzzy evaluation of criteria.


 


(1, 1, 1)  (0.48, 0.72, 0.96)  (0.65, 0.87, 0.87) 

(0.96, 1.27, 1.91)  (1, 1, 1)  (0.72, 0.96, 0.96) 

(0.87, 0.87, 1.15)  (0.96, 0.96, 1.27)  (1, 1, 1) 
Comparison matrix with triangular fuzzy number.


 


(1, 1, 1)  (1, 2, 3)  (4, 5, 6) 

(1/3, 1/2, 1)  (1, 1, 1)  (2, 3, 4) 

(1/6, 1/5, 1/4)  (1/4, 1/3, 1/2)  (1, 1, 1) 
Comparison matrix with triangular fuzzy number.


 


1  2  5 

0.56  1  3 

0.2  0.35  1 
As seen in formulas (
Now, we will use the extent analysis method on fuzzy AHP proposed by Chang [
The relation among
Relation of the fuzzy synthetic extent
Using formulas (
Finally, by using formula (
Therefore,
Via normalization, we obtain the weight vectors with respect to the decision criteria
Then we will use the proposed methodology (GAFAHP) to calculate the optimal priority weight of the comparison matrix (see Table
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
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; migrationdirection: forward; migrationfraction: 0.2; migrationinterval: 20; constraint parameterinitial penalty: 10; constraint parameterpenalty factor: 100; hybrid function: none; stopping criteria: 100, Inf, −Inf, 50, Inf,
Then, the centroid method is used to convert the optimal fuzzy weight to crisp weight:
Via normalization, the final priority weight is
From Table
Proposed method versus classic FAHP.


 

Classic FAHP [ 
0.68  0.32 

The proposed method  0.59  0.29  0.12 
Decisionmaking is widely used in supplier management and selection. In this section, a numerical example originated from [
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:
(
(
(
(
The hierarchy model of supplier selection can be constructed as shown in Figure
Hierarchy for the global supplier selection.
As can be seen from Figure
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
The fuzzy evaluation of criteria with respect to the overall objective can be listed in Table
Fuzzy evaluation of criteria with respect to the overall objective.









[(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 

[(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 

[(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 

[(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 

[(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
Fuzzy evaluation of the attributes with respect to criterion







[(1, 1, 1), (1, 1, 1)]  [(3/2, 2, 5/2), VH]  [(3/2, 2, 5/2), H]  0.47 

[(2/5, 1/2, 2/3), VH]  [(1, 1, 1), (1, 1, 1)]  [(3/2, 2, 5/2), VH]  0.30 

[(2/5, 1/2, 2/3), H]  [(2/5, 1/2, 2/3), VH]  [(1, 1, 1), (1, 1, 1)]  0.24 
Fuzzy evaluation of the attributes with respect to criterion








[(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 

[(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 

[(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 

[(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 
Fuzzy evaluation of the attributes with respect to criterion








[(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 

[(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 

[(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 

[(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 
Fuzzy evaluation of the attributes with respect to criterion








[(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 

[(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 

[(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 

[(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 
Fuzzy evaluation of the attributes with respect to criterion








[(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 

[(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 

[(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 

[(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
Summary combination of priority weights: attributes of criterion
Weight 



Priority weight 

0.47  0.30  0.24  
Alternatives  

0.71  0.44  0.69  0.63 

0.13  0.36  0.08  0.19 

0.16  0.20  0.23  0.19 
Summary combination of priority weights: attributes of criterion
Weight 




Priority weight 

0.32  0.23  0.32  0.14  
Alternatives  

0.51  0.51  0.69  0.87  0.62 

0.23  0.23  0.08  0.00  0.15 

0.26  0.26  0.23  0.13  0.23 
Summary combination of priority weights: attributes of criterion
Weight 




Priority weight 

0.43  0.28  0.17  0.11  
Alternatives  

0.27  0.69  0.05  0.49  0.38 

0.18  0.08  0.64  0.32  0.25 

0.55  0.23  0.31  0.19  0.38 
Summary combination of priority weights: attributes of criterion
Weight 




Priority weight 

0.44  0.21  0.21  0.13  
Alternatives  

0.83  0.45  0.69  0.33  0.66 

0.17  0.45  0.08  0.33  0.51 

0.00  0.10  0.23  0.34  0.18 
Summary combination of priority weights: attributes of criterion
Weight 




Priority weight 

0.28  0.31  0.24  0.17  
Alternatives  

0.72  0.49  0.83  0.27  0.60 

0.00  0.32  0.17  0.18  0.17 

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





Priority weight 

0.49  0.19  0.15  0.11  0.05  
Alternatives  

0.63  0.62  0.38  0.66  0.60  0.59 

0.19  0.15  0.25  0.51  0.17  0.22 

0.19  0.23  0.38  0.18  0.23  0.23 
Final priority weights of each supplier.
The supplier selection is a significant issue of multicriteria decisionmaking (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
The authors declare that there is no conflict of interests regarding the publication of this paper.
Bingyi Kang and Yong Hu contributed equally to this work.
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. BUAAVR14KF02).