# Rational Supplier Selection Based on Two-Phase Deep Analysis considering Fuzzy QFD and Game Theory

Authors: Feng Li; Senhao Luo; Yi Wang; Chia-Huei Wu
Journal: Mobile Information Systems
 (2021)

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## Abstract

Viewing from the perspective of a traditional manufacturing enterprise, the paper proposes a fuzzy comprehensive evaluation system model, which selects the most suitable supplier or suppliers to undertake the supply tasks and to achieve the purposes of reducing procurement costs and improving the supply quality and reliability, so as to fully meet customer demands. Based on decision-makers’ preferences and suppliers’ supply capabilities, subjective and objective weight of criteria are both developed for supplier selection with the corresponding weighting method, namely, quality function deployment (QFD) and entropy weight. Specially, the two-phase fuzzy QFD method combined with the trapezoidal fuzzy number (TFN) is creatively applied to realize the complicated conversion process. Furthermore, the game theory is employed to combine the advantages of the subjective and objective weighting method, by which the comprehensive weight can be determined. A numerical example is given to demonstrate the proposed modeling process. Through programming drawing images for comparative analysis and sensitivity analysis, study image trends, and draw conclusions, the proposed model effectively alleviates the weight deviation and has higher rationality and extensibility in supplier selection.


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## Body

## 1. Introduction

Supplier selection is of great significance to the manufacturing. Selecting the optimal supplier can effectively reduce the procurement costs and improve the quality and reliability of the supply, thereby improving the enterprise’s profit margin by minimizing the upstream costs of the supply chain [1]. At the same time, it can make contributions to improve the customers’ satisfaction. Therefore, the subjective and objective factors that affect the working efficiency of the supplier should be considered adequately by manufacturers. Multicriteria decision making (MCDM) methods that can effectively determine subjective and objective weights of evaluation criteria have been applied to various fields, such as several frequently used methods of obtaining weights include analytic network process (ANP), quality function deployment (QFD), best worst method (BWM), grey relational analysis (GRA), and entropy weight. However, both subjective and objective weights have their own limitations. In fact, the subjective weight only reflects the decision-makers’ preferences and ignores the actual efficiency of suppliers, which is greatly affected by human factors. On the contrary, the objective weight focuses on the objective supply capabilities of suppliers and ignores factors such as the environment of the manufacturing enterprise and the interest relationship of the upstream of the supply chain, which are closely related to the experience of decision makers.Game theory plays an important role in weighing the subjective and objective weight, which uses mathematical tools to study how the two sides with conflict of interest choose the optimal strategy, respectively [2]. In other words, game theory mainly studies the decision-making subjects interacting with each other, how to make decisions, and how to achieve equilibrium. Game theory was first used as a strategic bargain in economics [3], and then, it is widely used in the management, engineering, and many other fields [4]. Compared with other tools, game theory better deals with the common disequilibrium in reality, and more effectively makes individual and collective rationality converge under the condition of incomplete information and imperfect competition. In addition, game theory provides a replicable template for solving complex interactions between contradictory subjects in the process of complex modeling, which is conducive to finding the optimal solution for all parties. It works well when two or more players have a conflict of interest. Similarly, because the subjective weighting method is affected by human factors, while the objective weighting method is absolutely objective, there is a big difference in the weight coefficient determined by the two methods for the same evaluation index. Game theory seeks the consistency or compromise of its correlation weight values in fully considering the foundation of the characteristics of the subjective and objective weight method, so as to minimize the deviation of the subjective and objective weight. In the process, they can be regarded as the two sides with conflict of interest, while the comprehensive weight is regarded as the benefit combination of the game. Therefore, the comprehensive weight based on Nash equilibrium is obtained and applied to rational supplier selection. In this paper, a rational weighting model combining with game theory is proposed to evaluate and select the most suitable suppliers. Considering both decision-makers’ preferences and suppliers’ capabilities, the subjective and objective weight of supplier evaluation criteria are developed for supplier selection with QFD and entropy weight. Among them, a two-phase fuzzy QFD method combined with TFN is creatively used to realize the complicated conversion process. The most critical step, game theory, is employed to combine the advantages of the subjective and objective weight, by which the comprehensive weight can be determined. The structure of the proposed model is shown in Figure 1.Figure 1
The structure of the proposed model.The remainders of the paper are organized as follows. Section2 is literature review. Section 3 introduces the preliminaries involved in the paper. The proposed fusion model based on two-phase fuzzy QFD, entropy weight, and game theory is introduced in Section 4. Section 5 demonstrates a numerical example to illustrate the effectiveness of the proposed model. Section 6 makes comparative analysis and sensitivity analysis. The final research conclusions are presented in Section 7.


## 2. Literature Review

Supplier selection is of great significance to the core competitiveness of manufacturing enterprises [5]. Choosing suitable suppliers is not only beneficial to reduce inventory costs and transaction costs but also improve the order fulfillment rate. At the same time, information technology and incentive mechanism are implemented to enable enterprises and suppliers to become a community of interests, and a rapid reaction system is established in order to shorten the procurement cycle and make an agile response to market demand. The common methods and related literature of supplier multicriteria evaluation are shown in Table 1.Table 1
The common methods and related literature.
AHP/ANPQFDBWMGRATOPSISEntropy weightGame theoryZavadskas et al.✓Dweiri et al.✓Ahmadi et al.✓Ayağ et al.✓Yazdani et al.✓Lima-Junior et al.✓Karsak et al.✓Gupta et al.✓✓Rezaei et al.✓Pitchipoo et al.✓✓Badi and Pamucar✓Govindan et al.✓Mousakhani et al.✓Blagojević et al.✓Wei et al.✓Wei et al.✓Lai et al.✓✓✓Liu et al.✓✓✓There are many studies on supplier evaluation and selection for manufacturing enterprises. Dweiri et al. put forward the decision support model of supplier selection in AHP and further carried out sensitivity analysis to test the robustness of supplier selection decision [6]. Starting from the concept of elasticity, Ahmadi et al. used AHP to calculate the weight of sustainable development criteria and interferon gamma release assay to rank suppliers and proposed a structured and integrated evaluation decision model for sustainable suppliers in telecommunications industry [7]. Zavadskas et al. used fuzzy AHP to evaluate and select raw material suppliers for pipeline production [8]. Ayağ and Samanlioglu comprehensively considered the influence of quantitative and qualitative factors on supplier selection and proposed an intelligent solution method for the supplier selection problem based on fuzzy ANP [9]. Yazdani et al. solved the relationship between customer demands with decision-making trial and evaluation laboratory (DEMATEL) and then determined the degree of relationship between each pair of supplier criteria and customer demands by the QFD [10]. Lima-Junior et al. used fuzzy QFD to weight each indicator and then evaluated the degree of difficulty of supply to obtain information for each indicator evaluation [11]. Karsak and Dursun proposed a fuzzy multicriteria group decision-making method for supplier selection based on QFD, fuzzy information fusion, and binary language representation model [12]. Gupta and Barua used BWM to rank selection criteria and then used fuzzy TOPSIS to rank the weight of supplier selection criteria, aiming to select suppliers based on the green innovation ability of small- and medium-sized enterprises [13]. Rezaei et al. combined with screening and BWM and developed a three-phase supplier selection method [14]. Pitchipoo et al. adopted grey relational analysis (GRA) as performance indicators to determine the optimal suppliers, then used the principal component analysis and entropy weight method to evaluate the corresponding weight values of each performance indicators, and proposed an optional decision-making model to evaluate the relative performance of suppliers with multiple outputs and inputs [15]. Badi and Pamucar applied the mixed grey theory and Marcos method in a steel company’s supplier selection decision to help it improve competitiveness [16]. Govindan and Sivakumar used the fuzzy technology of similarity ranking with fuzzy TOPSIS to evaluate and select potential suppliers and proposed a model to support the selection of the best green suppliers and the order allocation among potential suppliers [17]. Mousakhani et al. took into account the opinions of priority experts on the relative importance of criteria, calculated the weight of decision makers with the extended interval-2 fuzzy TOPSIS method, sorted the potential alternatives according to the interval-2 fuzzy hamming distance measure, and proposed a green supplier selection model based on the group decision method [18]. Blagojević et al. established a comprehensive entropy fuzzy pivot pairwise relative criteria importance assessment-data envelopment analysis (PIPRECIA-DEA) model to study how to determine the security state of B&H under particular uncertainty conditions [19]. Wei et al. obtained the optimal choice with the largest Euclidean and Hamming distance from NIS and designed a probabilistic uncertain language coding method based on sine entropy weight, which is applied in green supplier selection [20]. In order to obtain the optimal green supplier, Wei et al. provided an integrated model of entropy weight and multiattributive border approximation area comparison (MABAC) under uncertain probabilistic linguistic sets (UPLTSs). Information entropy is used to calculate the weight of criteria, and UPLTSs are used to get the final ranking results of green suppliers [21]. At the same time, there are many innovative methods for supplier evaluation and selection. Hosseini and Khaled discussed the elasticity criteria of supplier selection, combined binomial logistic regression, classification regression tree, and neural network, used the integrated method to predict the elasticity of a single supplier, and used different supplier selection models to rank [22]. Chakraborty et al. applied D-number to solve the uncertainty problem in the process of supplier selection [23]. Durmić used the full consistency method (FUCOM) to determine the importance of the criteria [24]. Xiao et al. used criteria importance through the intercriteria correlation (CRITIC) method to determine the weight of criteria and proposed a new intuitionistic fuzzy multicriteria group decision-making method. Finally, an application example of green supplier selection in iron and steel industry is given, and an objective weighting method is used to improve the effectiveness of evaluation [25]. Lei et al. proposed probabilistic uncertain linguistic QUALIFLEX (PUL-QUALIFLEX) with the CRITIC method to obtain the optimal solution and applied the method to an application example of green supplier selection [26]. In the research on the flood risk assessment model based on fuzzy comprehensive evaluation, Lai et al. used AHP and entropy weight method to calculate the subjective and the objective weight, respectively, and combined the two with game theory to reflect the intention of decision makers and the information content provided by each indicator, effectively correcting the one side of the single weight method [27]. Then, in order to overcome the deficiency that AHP cannot deal with the dependence and interaction between elements at different levels quantitatively, Liu et al. proposed an ANP method based on game theory, which effectively solved the problem of supplier selection under uncertain environment, and adjusted the index weight with DEMATEL, making the evaluation result more reasonable [1].However, the previous studies still have some drawbacks or challenges. QFD has the advantage of fully paying attention to customer demands, so many studies apply QFD to supplier selection. However, most of them that involve the QFD method only implement once or ignore the manufacturer criteria and directly implement the conversion from customer demands to supplier criteria, which makes the conversion process between criteria have gap, and the connection is not tight enough. Entropy weight has absolute objectivity. More importantly, both subjective and objective weights have limitations. In order to overcome these drawbacks, a fuzzy two-phase QFD method combined with TFN is creatively used to realize the complicated conversion process to select the most suitable supplier or suppliers to fully meet customer demands. On the basis of fully considering the characteristics of fuzzy QFD and entropy weight methods, the paper applies game theory to seek for the consistency or compromise of subjective and objective weight values from the perspective of a traditional manufacturing enterprise, so as to minimize the deviation of subjective and objective weights.


## 3. Materials and Methods

### 3.1. Quality Function Deployment

QFD, also known as house of quality (HOQ), is a customer-driven product development method. Starting from the perspective of quality assurance, customer demands can be obtained through certain market survey methods, converting customer demands into technical demands in the product development stage by the matrix graphic method, and identifying which parameter is the most important to customer satisfaction. These key parameters form the measurement index of design content and are finally decomposed into various stages and functional departments of product development. In order to ensure the quality of final products, it is necessary to coordinate the work of various departments to make the products designed and manufactured truly meet the requirements of customers.HOQ is the core tool for establishing the quality function deployment system. It realizes the quality function deployment process through a series of charts and matrices [28]. As shown in Figure 2(a), the HOQ looks like a house, with a row and column matrix filled with quantized values at the center of the house. The left wall represents the customer attributes, namely, customer demands (WHATs), while the ceiling is usually defined as product requirements (HOWs), i.e., how to design products to meet customer demands. The quantitative relationship between WHATs and HOWs is the core element of the QFD process. Other parts of the HOQ structure provide supporting information for the QFD. To be specific, the HOQ consists of the following parts: (A) represents the needs of customers; (B) represents engineering characteristics or methods; (C) represents the relationship matrix between WHATs and HOWs; (D) indicates the level of importance of expectations; (E) represents the customer benchmark; (F) represents the relationship among engineering characteristics.Figure 2
The graphics involved in preliminaries.QFD originated from the Japanese industry and gradually became a standard planning tool integrating collection, analysis, and optimization after the promotion by the Americans [29]. The main functions of QFD are reducing design changes, shorten development cycle, improve quality, improve customer satisfaction, and reduce design and manufacturing costs. QFD is applicable to both new product development and old product improvement, suitable for both general products and large complex products. It applies to both hardware and software products and service management, and it is an indispensable tool in the six sigma design process [30]. Toyota, General Motors, Volkswagen, Ford, and Chrysler all use QFD technology in the product planning process. For example, Toyota reduces the production cost of a commercial vehicle by 61% through QFD technology.


### 3.2. Entropy Weight

Information entropy theory was originally introduced by Shannon from thermodynamics to information theory [31] and has been widely applied in decision science, social economy, and other fields. Information entropy is an uncertain measure of a system in a disordered state [32], and for the entropy weight, it can reflect the useful quantitative information of the evaluation indexes. Supposing that there are m different states in a system and the probability of each state is Pi (i = 1, 2, …, m), then the entropy of the system is(1)E=−1lnm∑i=1mpilnpi.The basic idea of the entropy weight method is to determine the objective weight according to the index variability [33]. Generally speaking, the smaller the information entropy E of an index is, the greater the variation degree of the index value is, the more information it can provide, the greater the role it can play in the comprehensive evaluation, and the greater its weight will be [19]. On the contrary, the higher the information entropy E of an index is, the smaller the variation degree of the index is, the less the information it provides, the smaller the role it plays in the comprehensive evaluation, and the smaller its weight is [34]. Entropy weighting steps are shown in detail in Section 4.2.


### 3.3. Game Theory

The study of game theory originated in the 18th century and developed in the work of Zermelo, Borel, and Von Neumann [35]. In 1944, Von Neumann and Morgenstern published The Theory of Games and Economic Behavior [36], which laid a solid foundation for the theories and methods of cooperative games. In 1951, Nash proposed the concept of Nash equilibrium [37, 38], from which game theory began to flourish and was applied in economics, psychology, politics, and other fields.Noncooperative static game with complete information, also known as a strategic game, is mainly composed of three elements: participants, strategy set, and utility function. In game theory, players are both rational and selfish. Each player has a set of all possible choices of strategies, and the specific combination of strategies of all players corresponds to a utility function. The utility function is an indicator to measure the benefit of participants from the game. It not only depends on the strategy choice of participants themselves but also is related to the strategies of other participants, reflecting the preferences of participants for the set of strategies. In the process of the game, when considering the opponent’s strategy choice, no participant has the motivation to deviate from the strategy he chose, and the Nash equilibrium is reached. Nash equilibrium is the best response of each participant to a given strategy from another participant.


### 3.4. TFN

Compared with traditional mathematics and real numbers, the fuzzy number is an important concept in fuzzy analysis. In 1976, Zadeh introduced the concept of fuzzy numbers. Since then, many scholars have carried out research on fuzzy numbers and proposed several fuzzy numbers of different backgrounds [39], such as L-R fuzzy number, triangular fuzzy number, and TFN. Now, the study of the fuzzy number has become more and more mature since TFN is an extension of the triangular fuzzy number and interval fuzzy number, which has a wider range of applications.In previous literature, most scholars used TFN to deal with problems. However, since TFN is represented by four parameters, which provide more detailed description, it can provide more detailed solutions than triangular fuzzy numbers. As shown in Figure2(b), TFN can be defined by four parameters (a1, a2, a3, and a4), and the expression for the piecewise function [40] is as follows:(2)Fx=x−a1a2−a1,ifx∈a1,a2,1,ifx∈a2,a3,a4−xa4−a3,ifx∈a3,a4,0,otherwise,wherea1≤a2≤a3≤a4.


## 3.1. Quality Function Deployment

QFD, also known as house of quality (HOQ), is a customer-driven product development method. Starting from the perspective of quality assurance, customer demands can be obtained through certain market survey methods, converting customer demands into technical demands in the product development stage by the matrix graphic method, and identifying which parameter is the most important to customer satisfaction. These key parameters form the measurement index of design content and are finally decomposed into various stages and functional departments of product development. In order to ensure the quality of final products, it is necessary to coordinate the work of various departments to make the products designed and manufactured truly meet the requirements of customers.HOQ is the core tool for establishing the quality function deployment system. It realizes the quality function deployment process through a series of charts and matrices [28]. As shown in Figure 2(a), the HOQ looks like a house, with a row and column matrix filled with quantized values at the center of the house. The left wall represents the customer attributes, namely, customer demands (WHATs), while the ceiling is usually defined as product requirements (HOWs), i.e., how to design products to meet customer demands. The quantitative relationship between WHATs and HOWs is the core element of the QFD process. Other parts of the HOQ structure provide supporting information for the QFD. To be specific, the HOQ consists of the following parts: (A) represents the needs of customers; (B) represents engineering characteristics or methods; (C) represents the relationship matrix between WHATs and HOWs; (D) indicates the level of importance of expectations; (E) represents the customer benchmark; (F) represents the relationship among engineering characteristics.Figure 2
The graphics involved in preliminaries.QFD originated from the Japanese industry and gradually became a standard planning tool integrating collection, analysis, and optimization after the promotion by the Americans [29]. The main functions of QFD are reducing design changes, shorten development cycle, improve quality, improve customer satisfaction, and reduce design and manufacturing costs. QFD is applicable to both new product development and old product improvement, suitable for both general products and large complex products. It applies to both hardware and software products and service management, and it is an indispensable tool in the six sigma design process [30]. Toyota, General Motors, Volkswagen, Ford, and Chrysler all use QFD technology in the product planning process. For example, Toyota reduces the production cost of a commercial vehicle by 61% through QFD technology.


## 3.2. Entropy Weight

Information entropy theory was originally introduced by Shannon from thermodynamics to information theory [31] and has been widely applied in decision science, social economy, and other fields. Information entropy is an uncertain measure of a system in a disordered state [32], and for the entropy weight, it can reflect the useful quantitative information of the evaluation indexes. Supposing that there are m different states in a system and the probability of each state is Pi (i = 1, 2, …, m), then the entropy of the system is(1)E=−1lnm∑i=1mpilnpi.The basic idea of the entropy weight method is to determine the objective weight according to the index variability [33]. Generally speaking, the smaller the information entropy E of an index is, the greater the variation degree of the index value is, the more information it can provide, the greater the role it can play in the comprehensive evaluation, and the greater its weight will be [19]. On the contrary, the higher the information entropy E of an index is, the smaller the variation degree of the index is, the less the information it provides, the smaller the role it plays in the comprehensive evaluation, and the smaller its weight is [34]. Entropy weighting steps are shown in detail in Section 4.2.


## 3.3. Game Theory

The study of game theory originated in the 18th century and developed in the work of Zermelo, Borel, and Von Neumann [35]. In 1944, Von Neumann and Morgenstern published The Theory of Games and Economic Behavior [36], which laid a solid foundation for the theories and methods of cooperative games. In 1951, Nash proposed the concept of Nash equilibrium [37, 38], from which game theory began to flourish and was applied in economics, psychology, politics, and other fields.Noncooperative static game with complete information, also known as a strategic game, is mainly composed of three elements: participants, strategy set, and utility function. In game theory, players are both rational and selfish. Each player has a set of all possible choices of strategies, and the specific combination of strategies of all players corresponds to a utility function. The utility function is an indicator to measure the benefit of participants from the game. It not only depends on the strategy choice of participants themselves but also is related to the strategies of other participants, reflecting the preferences of participants for the set of strategies. In the process of the game, when considering the opponent’s strategy choice, no participant has the motivation to deviate from the strategy he chose, and the Nash equilibrium is reached. Nash equilibrium is the best response of each participant to a given strategy from another participant.


## 3.4. TFN

Compared with traditional mathematics and real numbers, the fuzzy number is an important concept in fuzzy analysis. In 1976, Zadeh introduced the concept of fuzzy numbers. Since then, many scholars have carried out research on fuzzy numbers and proposed several fuzzy numbers of different backgrounds [39], such as L-R fuzzy number, triangular fuzzy number, and TFN. Now, the study of the fuzzy number has become more and more mature since TFN is an extension of the triangular fuzzy number and interval fuzzy number, which has a wider range of applications.In previous literature, most scholars used TFN to deal with problems. However, since TFN is represented by four parameters, which provide more detailed description, it can provide more detailed solutions than triangular fuzzy numbers. As shown in Figure2(b), TFN can be defined by four parameters (a1, a2, a3, and a4), and the expression for the piecewise function [40] is as follows:(2)Fx=x−a1a2−a1,ifx∈a1,a2,1,ifx∈a2,a3,a4−xa4−a3,ifx∈a3,a4,0,otherwise,wherea1≤a2≤a3≤a4.


## 4. The Two-Phase Deep Analysis considering Fuzzy QFD and Game Theory

### 4.1. Applying Two-Phase Fuzzy QFD to Calculate Subjective Weight

By using the two-phase fuzzy QFD method, the complicated conversion process of customer demands to manufacturer techniques and to supplier capabilities is realized, and then, the subjective weight is calculated. The conversion process between criteria is shown in Figure2. In addition, in order to make the evaluation more practical, the QFD relational matrix uses TFN to provide more detailed description.According to the previous research, assuming that a group of TFN evaluation scale is represented byΩ=EL,ML,M,MH,EH, the specific meaning is shown in Figure 3.Figure 3
TFN evaluation scale.After completing the evaluation with TFN and calculating the subjective weight of supplier evaluation criteria based on the evaluation results, equations (3) and (4) are used to defuzzify and normalize the evaluation:(3)Qi=14∑a=14fa,i=1,2,…,m;a=1,2,3,and4,(4)NQi=Qi−Mini=1,2,…,mQiMaxi=1,2,…,mQi−Mini=1,2,…,mQi,where fa is the fuzzy evaluation value of the trapezoid vertex, a = 1, 2, 3, and 4, Qi defines the defuzzified weight of the ith supplier evaluation criterion, NQi is its normalized weight, i = 1, 2, …, m, and m is the number of supplier evaluation criteria.


### 4.2. Using Entropy Weight to Calculate Objective Weight

Step 1: normalizing the raw data matrix: there arem evaluation criteria and n evaluation objects, and the original data matrix B is as follows:(5)B=b11…b1n⋮⋱⋮bm1⋯bmn.After normalizing, the matrixC=cijm×n is obtained, where the content of symbol cij is the normalized value of the jth evaluation object on the ith evaluation criterion, j = 1, 2, …, n. The normalization formulas of positive and negative indicators are, respectively, as follows:(6)cij=bij−Minj=1,2,…,nbijMaxj=1,2,…,nbij−Minj=1,2,…,nbij,(7)cij=Maxj=1,2,…,nbij−bijMaxj=1,2,…,nbij−Minj=1,2,…,nbij.Step 2: defining entropy: based on the original data matrix in Step 1, the entropy of theith evaluation criterion is defined as the following equation:(8)Ei=−h∑j=1ndijlndij,i=1,2,…,m,where(9)h=1lnn,dij=cij∑j=1ncij.Step 3: calculating entropy weight: based on the entropy obtained in Step 2, the entropy weight of theith evaluation criterion is obtained through the following formulas:(10)WEi=1−Eim−∑i=1mEi,where(11)∑i=1mWEi=1,0≤WEi≤1.


### 4.3. Comprehensive Weight Based on Game Theory

Step 1: unifying vector definitions: use the following equation to unify the definition of each weight vector:(12)vt=ut,ut=s1,s2,…,sm,∑i=1msi=1,whereut represents the weight vector obtained by Section 4.2, t = 1, 2, …, r, r > 1, and m is the number of supplier evaluation criteria.Step 2: assuming weight coefficients: assume that the weight coefficient of each kind of weight isγt, and the comprehensive weight Z is expressed as follows:(13)Z=∑t=1γtvtT,γt>0.Step 3: determining weight coefficients: the principle of determiningγt is to minimize the deviation between the comprehensive weight and each weight:(14)MinZ−ut2=∑t=1γtvtT−utT2.According to the differential properties of the matrix [1], the condition for the optimal first derivative of equation (14) is as follows:(15)∑t=1γt×ut×vtT=ut×utT,equivalent to(16)u1⋅v1Tu1⋅v2T…u2⋅v1Tu2⋅v2T…⋮⋮⋮...γ1γ2⋮.=u1⋅u1Tu2⋅u2T⋮..By the above formula, the weight coefficientsγ1,γ2,… can be solved.Step 4: normalizing weight coefficients: weight coefficients are normalized by the following equation:(17)γt∗=γt∑t=1γt.So, the optimal weight coefficientsγt∗ are obtained. Meanwhile, the optimal comprehensive weight Z∗ is obtained through the following equation:(18)Z∗=∑t=1γt∗vtT.Step 5: scoring and ranking: according to the final scores and ranking, one or more suppliers participating in the evaluation with the highest score are selected.


## 4.1. Applying Two-Phase Fuzzy QFD to Calculate Subjective Weight

By using the two-phase fuzzy QFD method, the complicated conversion process of customer demands to manufacturer techniques and to supplier capabilities is realized, and then, the subjective weight is calculated. The conversion process between criteria is shown in Figure2. In addition, in order to make the evaluation more practical, the QFD relational matrix uses TFN to provide more detailed description.According to the previous research, assuming that a group of TFN evaluation scale is represented byΩ=EL,ML,M,MH,EH, the specific meaning is shown in Figure 3.Figure 3
TFN evaluation scale.After completing the evaluation with TFN and calculating the subjective weight of supplier evaluation criteria based on the evaluation results, equations (3) and (4) are used to defuzzify and normalize the evaluation:(3)Qi=14∑a=14fa,i=1,2,…,m;a=1,2,3,and4,(4)NQi=Qi−Mini=1,2,…,mQiMaxi=1,2,…,mQi−Mini=1,2,…,mQi,where fa is the fuzzy evaluation value of the trapezoid vertex, a = 1, 2, 3, and 4, Qi defines the defuzzified weight of the ith supplier evaluation criterion, NQi is its normalized weight, i = 1, 2, …, m, and m is the number of supplier evaluation criteria.


## 4.2. Using Entropy Weight to Calculate Objective Weight

Step 1: normalizing the raw data matrix: there arem evaluation criteria and n evaluation objects, and the original data matrix B is as follows:(5)B=b11…b1n⋮⋱⋮bm1⋯bmn.After normalizing, the matrixC=cijm×n is obtained, where the content of symbol cij is the normalized value of the jth evaluation object on the ith evaluation criterion, j = 1, 2, …, n. The normalization formulas of positive and negative indicators are, respectively, as follows:(6)cij=bij−Minj=1,2,…,nbijMaxj=1,2,…,nbij−Minj=1,2,…,nbij,(7)cij=Maxj=1,2,…,nbij−bijMaxj=1,2,…,nbij−Minj=1,2,…,nbij.Step 2: defining entropy: based on the original data matrix in Step 1, the entropy of theith evaluation criterion is defined as the following equation:(8)Ei=−h∑j=1ndijlndij,i=1,2,…,m,where(9)h=1lnn,dij=cij∑j=1ncij.Step 3: calculating entropy weight: based on the entropy obtained in Step 2, the entropy weight of theith evaluation criterion is obtained through the following formulas:(10)WEi=1−Eim−∑i=1mEi,where(11)∑i=1mWEi=1,0≤WEi≤1.


## 4.3. Comprehensive Weight Based on Game Theory

Step 1: unifying vector definitions: use the following equation to unify the definition of each weight vector:(12)vt=ut,ut=s1,s2,…,sm,∑i=1msi=1,whereut represents the weight vector obtained by Section 4.2, t = 1, 2, …, r, r > 1, and m is the number of supplier evaluation criteria.Step 2: assuming weight coefficients: assume that the weight coefficient of each kind of weight isγt, and the comprehensive weight Z is expressed as follows:(13)Z=∑t=1γtvtT,γt>0.Step 3: determining weight coefficients: the principle of determiningγt is to minimize the deviation between the comprehensive weight and each weight:(14)MinZ−ut2=∑t=1γtvtT−utT2.According to the differential properties of the matrix [1], the condition for the optimal first derivative of equation (14) is as follows:(15)∑t=1γt×ut×vtT=ut×utT,equivalent to(16)u1⋅v1Tu1⋅v2T…u2⋅v1Tu2⋅v2T…⋮⋮⋮...γ1γ2⋮.=u1⋅u1Tu2⋅u2T⋮..By the above formula, the weight coefficientsγ1,γ2,… can be solved.Step 4: normalizing weight coefficients: weight coefficients are normalized by the following equation:(17)γt∗=γt∑t=1γt.So, the optimal weight coefficientsγt∗ are obtained. Meanwhile, the optimal comprehensive weight Z∗ is obtained through the following equation:(18)Z∗=∑t=1γt∗vtT.Step 5: scoring and ranking: according to the final scores and ranking, one or more suppliers participating in the evaluation with the highest score are selected.


## 5. Numerical Experiment

Considering the subjective preferences of decision makers and the objective supply capacities of suppliers, the proposed model has high extensibility, which is generally applicable to different criteria for manufacturers or suppliers. Therefore, assuming that there are suppliersA, B, C, D, E, F, and G, whose supply capacities are shown in Table 2, the supply capacities of suppliers are on time delivery rate (SC1), defective rate of raw materials (SC2), fast response time (SC3), resistance to interruption risk rate (SC4), and recycling rate (SC5).Table 2
The supply capacities of suppliers.
SupplierSC1SC2SC3SC4SC5A95.31.201.5069.355.0B94.51.401.8561.061.2C87.21.001.3078.265.7D84.61.301.8075.960.4E93.51.301.7562.062.2F86.21.001.2079.266.7G82.01.502.2073.555.0


### 5.1. Evaluation Criteria Architecture

Different literature have different considerations and interpretations on the setting of customer and manufacturer evaluation criteria. After consulting and contrasting, we set the following evaluation criteria. Customer demand criteria are price (CD1), quality (CD2), service (CD3), uniqueness (CD4), and brand (CD5). For the manufacturer technique criteria, process optimization (MT1) is an operation method that is superior to existing processes in order to improve operational efficiency, reduce the production cost, strictly control the process procedures, and reorganize or improve the original process. The low defective rate (MT2) is an important index to measure the technological level and the basic guarantee of product quality. Flexibility in production (MT3) is a new requirement put forward by manufacturing enterprises in the face of rapidly changing market demand. Customer organization (MT4) refers to the ability of manufacturing enterprises to attract customers and maintain customer loyalty. Buffer stock (MT5) refers to the inventory quantity higher than the average demand, which can avoid the expected demand rise or imbalance between production stages. It can effectively reduce the risk of production interruption resulting in the failure of raw materials to be delivered on time in a short period of time due to extreme circumstances. Enterprise reputation (MT6) can help enterprises establish a good public image and then improve the market competitiveness of enterprises.


### 5.2. The Application of Two-Phase Fuzzy QFD

Based on the evaluation scale in Figure3, the decision makers evaluate customer demands and provide importance levels for them, as shown in Table 3. The weight values are expressed in awl, l = 1, 2, …, L.Table 3
Importance levels of customer demands.
DemandImportance levelawlCD1EH(7,8,9,9)CD2EH(7,8,9,9)CD3EH(7,8,9,9)CD4MH(5,6,7,8)CD5MH(5,6,7,8)In order to reflect the effect of manufacturer techniques on customer demands, the linguistic approach is used again to establish an interrelation matrix. The result of which the manufacturer technique is contributing more to customer demand is easily observed in Table4.Table 4
Interrelation between customer demands and manufacturer techniques.
CD/MTMT1MT2MT3MT4MT5MT6CD1EHMHEHEHEHMHCD2EHEHELEHELEHCD3ELELEHEHEHEHCD4MLELMLMHELEHCD5MLEHMLMHMHEHThe weight calculation of manufacturer techniques is processed, and TFNbwm is used to express the result, as shown in Table 5:(19)cwn=1M⊗bw1⊗c1n⊕bw2⊗c2n⊕⋯⊕bwM⊗cMn,where the array of row l and column m of the matrix is indicated by blm, m = 1, 2, …, M.Table 5
The weight calculation of manufacturer techniques.
CD/MTMT1MT2MT3MT4MT5MT6awlCD1(7,8,9,9)(5,6,7,8)(7,8,9,9)(7,8,9,9)(7,8,9,9)(5,6,7,8)(7,8,9,9)CD2(7,8,9,9)(7,8,9,9)(0,0,1,2)(7,8,9,9)(0,0,1,2)(7,8,9,9)(7,8,9,9)CD3(0,0,1,2)(0,0,1,2)(7,8,9,9)(7,8,9,9)(7,8,9,9)(7,8,9,9)(7,8,9,9)CD4(1,2,3,4)(0,0,1,2)(1,2,3,4)(5,6,7,8)(0,0,1,2)(7,8,9,9)(5,6,7,8)CD5(1,2,3,4)(7,8,9,9)(1,2,3,4)(5,6,7,8)(5,6,7,8)(7,8,9,9)(5,6,7,8)bwmbw1bw2bw3bw4bw5bw6(21.600, 30.400, 42.600, 48.800)(23.800, 32.000, 44.600, 51.800)(21.600, 30.400, 42.600, 48.800)(39.400, 52.800, 68.200, 74.200)(24.600, 32.800, 45.400, 52.000)(40.600, 54.400, 70.200, 75.600)So far, the first phase of the proposed two-phase fuzzy QFD method is completed. Then, the matrix reflecting the effect of supplier capacities on manufacturer techniques is built at the beginning of the second phase and shown in Table6.Table 6
Interrelation between manufacturer techniques and supplier capacities.
MT/SCSC1SC2SC3SC4SC5MT1ELEHMLELELMT2ELEHELMLELMT3EHELMHEHELMT4MMHMMHEHMT5EHELMHEHELMT6MHMHEHMHEHThe weight calculation of supplier capacities is processed, and the result is represented by TFNcwn, as shown in Table 7:(20)cwn=1M⊗bw1⊗c1n⊕bw2⊗c2n⊕⋯⊕bwM⊗cMn,where cmn is the array of row m and column n of the matrix, n = 1, 2, …, N.Table 7
The weight calculation of supplier capacities.
MT/SCSC1SC2SC3SC4SC5bwmMT1(0,0,1,2)(7,8,9,9)(1,2,3,4)(0,0,1,2)(0,0,1,2)(21.600, 30.400, 42.600, 48.800)MT2(0,0,1,2)(7,8,9,9)(0,0,1,2)(1,2,3,4)(0,0,1,2)(23.800, 32.000, 44.600, 51.800)MT3(7,8,9,9)(0,0,1,2)(5,6,7,8)(7,8,9,9)(0,0,1,2)(21.600, 30.400, 42.600, 48.800)MT4(3,4,5,6)(5,6,7,8)(3,4,5,6)(5,6,7,8)(7,8,9,9)(39.400, 52.800, 68.200, 74.200)MT5(7,8,9,9)(0,0,1,2)(5,6,7,8)(7,8,9,9)(0,0,1,2)(24.600, 32.800, 45.400, 52.000)MT6(5,6,7,8)(5,6,7,8)(7,8,9,9)(5,6,7,8)(7,8,9,9)(40.600, 54.400, 70.200, 75.600)cwncw1cw2cw3cw4cw5(107.433, 173.867, 285.267, 359.733)(119.633, 190.400, 306.933, 384.233)(109.167, 181.067, 293.533, 371.800)(124.533, 202.133, 322.867, 401.733)(93.333, 142.933, 236.800, 291.833)The fuzzy weight is defuzzified by equation (3), and the result is shown in Table 8.Table 8
Defuzzification and normalization.
Supplier criteriaDefuzzification weightNormalization weightSC1231.5750.197SC2250.3000.213SC3238.8920.203SC4262.8170.224SC5191.2250.163Until now, the weight calculation of the proposed two-phase fuzzy QFD method is completed.


### 5.3. The Application of Entropy Weight

On the basis of Table2, the original data can be constructed with matrix B as follows:(21)B=0.9530.9450.8720.8460.9350.8620.8200.0120.0140.0100.0130.0130.0100.0151.5001.8501.3001.8001.7501.2002.2000.6930.6100.7820.7590.6200.7920.7350.5500.6120.6570.6040.6220.6670.550.The matrixA is normalized by equations (6) and (7) to get matrix C:(22)C=1.0000.9400.3910.1950.8650.31600.6000.2001.0000.4000.4001.00000.7000.3500.9000.4000.4501.00000.45600.9450.8190.0551.0000.68700.5300.9150.4620.6151.0000.The matrixC is processed by equations (8) and (9) to get entropy E1–E5.E1 = 0.844, E2 = 0.853, E3 = 0.880, E4 = 0.836, and E5 = 0.804.After the entropy,E1–E5 are obtained, and the entropy weight is calculated by equations (10) and (11) as follows:WE1 = 0.199, WE2 = 0.188, WE3 = 0.153, WE4 = 0.209, and WE5 = 0.250.Until now, the weight calculation of entropy weight is completed.


### 5.4. The Combination with Game Theory

Each weight vector processed by equation (12) and unification results are as follows:v1 = (0.197, 0.213, 0.203, 0.224, 0.163) and v2 = (0.199, 0.188, 0.153, 0.209, 0.250)The matrix equation is calculated by equations (13)–(16) as follows:With the MATLAB R2019a to assist the calculation, the weight coefficients’ vectorα can be obtained:(23)γ=γ1,γ2=0.398,0.615.Equation (17) is applied to normalize the weight coefficients vector α, and the optimal comprehensive weight W∗ is obtained by equation (18) as follows:(24)γ∗=γ1∗,γ2∗=0.393,0.607,Z∗=0.198,0.198,0.173,0.215,0.216.


### 5.5. Suppliers Evaluation and Ranking

On the basis of the original data matrixB and the optimal comprehensive weight Z∗, the final scores and ranking of suppliers participating in the evaluation are calculated, as shown in Table 9.Table 9
Score and rank.
SupplierScoreRankA0.7185B0.7733C0.7096D0.7752E0.7584F0.6947G0.8221Therefore, the comprehensive ranking of the suppliers is G > D > B > E > A > C > F.


## 5.1. Evaluation Criteria Architecture

Different literature have different considerations and interpretations on the setting of customer and manufacturer evaluation criteria. After consulting and contrasting, we set the following evaluation criteria. Customer demand criteria are price (CD1), quality (CD2), service (CD3), uniqueness (CD4), and brand (CD5). For the manufacturer technique criteria, process optimization (MT1) is an operation method that is superior to existing processes in order to improve operational efficiency, reduce the production cost, strictly control the process procedures, and reorganize or improve the original process. The low defective rate (MT2) is an important index to measure the technological level and the basic guarantee of product quality. Flexibility in production (MT3) is a new requirement put forward by manufacturing enterprises in the face of rapidly changing market demand. Customer organization (MT4) refers to the ability of manufacturing enterprises to attract customers and maintain customer loyalty. Buffer stock (MT5) refers to the inventory quantity higher than the average demand, which can avoid the expected demand rise or imbalance between production stages. It can effectively reduce the risk of production interruption resulting in the failure of raw materials to be delivered on time in a short period of time due to extreme circumstances. Enterprise reputation (MT6) can help enterprises establish a good public image and then improve the market competitiveness of enterprises.


## 5.2. The Application of Two-Phase Fuzzy QFD

Based on the evaluation scale in Figure3, the decision makers evaluate customer demands and provide importance levels for them, as shown in Table 3. The weight values are expressed in awl, l = 1, 2, …, L.Table 3
Importance levels of customer demands.
DemandImportance levelawlCD1EH(7,8,9,9)CD2EH(7,8,9,9)CD3EH(7,8,9,9)CD4MH(5,6,7,8)CD5MH(5,6,7,8)In order to reflect the effect of manufacturer techniques on customer demands, the linguistic approach is used again to establish an interrelation matrix. The result of which the manufacturer technique is contributing more to customer demand is easily observed in Table4.Table 4
Interrelation between customer demands and manufacturer techniques.
CD/MTMT1MT2MT3MT4MT5MT6CD1EHMHEHEHEHMHCD2EHEHELEHELEHCD3ELELEHEHEHEHCD4MLELMLMHELEHCD5MLEHMLMHMHEHThe weight calculation of manufacturer techniques is processed, and TFNbwm is used to express the result, as shown in Table 5:(19)cwn=1M⊗bw1⊗c1n⊕bw2⊗c2n⊕⋯⊕bwM⊗cMn,where the array of row l and column m of the matrix is indicated by blm, m = 1, 2, …, M.Table 5
The weight calculation of manufacturer techniques.
CD/MTMT1MT2MT3MT4MT5MT6awlCD1(7,8,9,9)(5,6,7,8)(7,8,9,9)(7,8,9,9)(7,8,9,9)(5,6,7,8)(7,8,9,9)CD2(7,8,9,9)(7,8,9,9)(0,0,1,2)(7,8,9,9)(0,0,1,2)(7,8,9,9)(7,8,9,9)CD3(0,0,1,2)(0,0,1,2)(7,8,9,9)(7,8,9,9)(7,8,9,9)(7,8,9,9)(7,8,9,9)CD4(1,2,3,4)(0,0,1,2)(1,2,3,4)(5,6,7,8)(0,0,1,2)(7,8,9,9)(5,6,7,8)CD5(1,2,3,4)(7,8,9,9)(1,2,3,4)(5,6,7,8)(5,6,7,8)(7,8,9,9)(5,6,7,8)bwmbw1bw2bw3bw4bw5bw6(21.600, 30.400, 42.600, 48.800)(23.800, 32.000, 44.600, 51.800)(21.600, 30.400, 42.600, 48.800)(39.400, 52.800, 68.200, 74.200)(24.600, 32.800, 45.400, 52.000)(40.600, 54.400, 70.200, 75.600)So far, the first phase of the proposed two-phase fuzzy QFD method is completed. Then, the matrix reflecting the effect of supplier capacities on manufacturer techniques is built at the beginning of the second phase and shown in Table6.Table 6
Interrelation between manufacturer techniques and supplier capacities.
MT/SCSC1SC2SC3SC4SC5MT1ELEHMLELELMT2ELEHELMLELMT3EHELMHEHELMT4MMHMMHEHMT5EHELMHEHELMT6MHMHEHMHEHThe weight calculation of supplier capacities is processed, and the result is represented by TFNcwn, as shown in Table 7:(20)cwn=1M⊗bw1⊗c1n⊕bw2⊗c2n⊕⋯⊕bwM⊗cMn,where cmn is the array of row m and column n of the matrix, n = 1, 2, …, N.Table 7
The weight calculation of supplier capacities.
MT/SCSC1SC2SC3SC4SC5bwmMT1(0,0,1,2)(7,8,9,9)(1,2,3,4)(0,0,1,2)(0,0,1,2)(21.600, 30.400, 42.600, 48.800)MT2(0,0,1,2)(7,8,9,9)(0,0,1,2)(1,2,3,4)(0,0,1,2)(23.800, 32.000, 44.600, 51.800)MT3(7,8,9,9)(0,0,1,2)(5,6,7,8)(7,8,9,9)(0,0,1,2)(21.600, 30.400, 42.600, 48.800)MT4(3,4,5,6)(5,6,7,8)(3,4,5,6)(5,6,7,8)(7,8,9,9)(39.400, 52.800, 68.200, 74.200)MT5(7,8,9,9)(0,0,1,2)(5,6,7,8)(7,8,9,9)(0,0,1,2)(24.600, 32.800, 45.400, 52.000)MT6(5,6,7,8)(5,6,7,8)(7,8,9,9)(5,6,7,8)(7,8,9,9)(40.600, 54.400, 70.200, 75.600)cwncw1cw2cw3cw4cw5(107.433, 173.867, 285.267, 359.733)(119.633, 190.400, 306.933, 384.233)(109.167, 181.067, 293.533, 371.800)(124.533, 202.133, 322.867, 401.733)(93.333, 142.933, 236.800, 291.833)The fuzzy weight is defuzzified by equation (3), and the result is shown in Table 8.Table 8
Defuzzification and normalization.
Supplier criteriaDefuzzification weightNormalization weightSC1231.5750.197SC2250.3000.213SC3238.8920.203SC4262.8170.224SC5191.2250.163Until now, the weight calculation of the proposed two-phase fuzzy QFD method is completed.


## 5.3. The Application of Entropy Weight

On the basis of Table2, the original data can be constructed with matrix B as follows:(21)B=0.9530.9450.8720.8460.9350.8620.8200.0120.0140.0100.0130.0130.0100.0151.5001.8501.3001.8001.7501.2002.2000.6930.6100.7820.7590.6200.7920.7350.5500.6120.6570.6040.6220.6670.550.The matrixA is normalized by equations (6) and (7) to get matrix C:(22)C=1.0000.9400.3910.1950.8650.31600.6000.2001.0000.4000.4001.00000.7000.3500.9000.4000.4501.00000.45600.9450.8190.0551.0000.68700.5300.9150.4620.6151.0000.The matrixC is processed by equations (8) and (9) to get entropy E1–E5.E1 = 0.844, E2 = 0.853, E3 = 0.880, E4 = 0.836, and E5 = 0.804.After the entropy,E1–E5 are obtained, and the entropy weight is calculated by equations (10) and (11) as follows:WE1 = 0.199, WE2 = 0.188, WE3 = 0.153, WE4 = 0.209, and WE5 = 0.250.Until now, the weight calculation of entropy weight is completed.


## 5.4. The Combination with Game Theory

Each weight vector processed by equation (12) and unification results are as follows:v1 = (0.197, 0.213, 0.203, 0.224, 0.163) and v2 = (0.199, 0.188, 0.153, 0.209, 0.250)The matrix equation is calculated by equations (13)–(16) as follows:With the MATLAB R2019a to assist the calculation, the weight coefficients’ vectorα can be obtained:(23)γ=γ1,γ2=0.398,0.615.Equation (17) is applied to normalize the weight coefficients vector α, and the optimal comprehensive weight W∗ is obtained by equation (18) as follows:(24)γ∗=γ1∗,γ2∗=0.393,0.607,Z∗=0.198,0.198,0.173,0.215,0.216.


## 5.5. Suppliers Evaluation and Ranking

On the basis of the original data matrixB and the optimal comprehensive weight Z∗, the final scores and ranking of suppliers participating in the evaluation are calculated, as shown in Table 9.Table 9
Score and rank.
SupplierScoreRankA0.7185B0.7733C0.7096D0.7752E0.7584F0.6947G0.8221Therefore, the comprehensive ranking of the suppliers is G > D > B > E > A > C > F.


## 6. Comparative Analysis and Sensitivity Analysis

In this chapter, through comparative analysis and sensitivity analysis, the rationality of the proposed method and the ranking result of the suppliers are, respectively, processed to prove the robustness.


### 6.1. Comparative Analysis

In this section, subjective and objective weights are calculated by fuzzy QFD and entropy weights, respectively. These two types of weights and the comprehensive weight calculated by the proposed method are drawn in the same coordinate system, and the broken line diagram is shown in Figure4.Figure 4
Supplier capacity criteria weights under three methods.From the analysis of the figure, it can be seen that the weight value obtained by the single weight method is not convincing because there is no reference. Although subjective and objective methods have their own applicability, the weight values obtained by different methods vary greatly. The proposed method combined with game theory fully refers the existing weight values and solves each weight proportion problem through the combination of weight coefficients, which effectively alleviates the weight deviation. Under the same criterion, the comprehensive weight value is always in the range of subjective and objective weight values. Therefore, the comprehensive weight is the optimal equilibrium of subjective and objective weights, and the proposed method combined with game theory is more rational than the single method.


### 6.2. Sensitivity Analysis

This section studies the score and ranking of suppliersA, B, C, D, E, F, and G when the normalized subjective weight coefficient γ1∗ varies continuously in the interval (0, 1) and are drawn in Figure 5.Figure 5
The supplier ranking with the variation of the subjective weight coefficient.Analysis of the figure shows that, with the increase of the subjective weight coefficient, the scores of all suppliers are increasing. Suppliers A and G are more sensitive to the subjective weight coefficient, and their scores grow faster. When there are no decision makers participating in the evaluation, the suppliers’ scores are completely determined by their own supply capacities. When the decision makers participate in the evaluation moderately, the gap between suppliers A and C becomes significant. In other words, as the participation of decision makers increases, the gap among suppliers will become more obvious. The effective participation of decision makers can make it easier to select the most suitable suppliers.


## 6.1. Comparative Analysis

In this section, subjective and objective weights are calculated by fuzzy QFD and entropy weights, respectively. These two types of weights and the comprehensive weight calculated by the proposed method are drawn in the same coordinate system, and the broken line diagram is shown in Figure4.Figure 4
Supplier capacity criteria weights under three methods.From the analysis of the figure, it can be seen that the weight value obtained by the single weight method is not convincing because there is no reference. Although subjective and objective methods have their own applicability, the weight values obtained by different methods vary greatly. The proposed method combined with game theory fully refers the existing weight values and solves each weight proportion problem through the combination of weight coefficients, which effectively alleviates the weight deviation. Under the same criterion, the comprehensive weight value is always in the range of subjective and objective weight values. Therefore, the comprehensive weight is the optimal equilibrium of subjective and objective weights, and the proposed method combined with game theory is more rational than the single method.


## 6.2. Sensitivity Analysis

This section studies the score and ranking of suppliersA, B, C, D, E, F, and G when the normalized subjective weight coefficient γ1∗ varies continuously in the interval (0, 1) and are drawn in Figure 5.Figure 5
The supplier ranking with the variation of the subjective weight coefficient.Analysis of the figure shows that, with the increase of the subjective weight coefficient, the scores of all suppliers are increasing. Suppliers A and G are more sensitive to the subjective weight coefficient, and their scores grow faster. When there are no decision makers participating in the evaluation, the suppliers’ scores are completely determined by their own supply capacities. When the decision makers participate in the evaluation moderately, the gap between suppliers A and C becomes significant. In other words, as the participation of decision makers increases, the gap among suppliers will become more obvious. The effective participation of decision makers can make it easier to select the most suitable suppliers.


## 7. Conclusions

Supplier selection is of great significance to the core competitiveness of manufacturing enterprises. The paper proposes a fuzzy comprehensive supplier evaluation system model to improve the supply quality and reliability, so as to fully meet customer demands. Based on decision makers’ preferences and suppliers’ supply capabilities, subjective and objective weights of criteria are both developed for supplier selection. Specially, the two-phase fuzzy QFD method is creatively applied to realize the complicated conversion process. Furthermore, the game theory is employed to combine the advantages of the subjective and objective weighting method, by which the comprehensive weight can be determined. Although the objective evaluation method can alleviate the excessive subjectivity of decision-making to a certain extent, the subjective decision-making ability of decision makers can still affect the decision-making process. Green supplier selection, computer aided evaluation, and artificial intelligence simulation decision are the future research direction [41–43]. Through comparative analysis and sensitivity analysis, some revelations are obtained:(1)
The proposed method combined with game theory fully refers the existing weight values and solves each weight proportion problem through the combination of weight coefficients, which effectively alleviates the weight deviation. Therefore, the proposed method is more rational than the single method.(2)
The proposed model has high extensibility, in which the subjective and objective weight can be replaced by other methods which are more suitable for solving the practical problem.(3)
When the participation of decision makers increases, the gap among suppliers will become more obvious. The effective participation of decision makers can make it easier to select the most suitable suppliers.In addition, the limitations of the proposed method are that the subjective evaluation and analysis process of suppliers requires decision makers to have sufficient experience and decision-making ability. Computer-aided decision-making can be used as a new combination direction.



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*Source: 9979017-2021-07-26.xml*