Ranking Alternatives Using a Fuzzy Preference Relation-Based Fuzzy VIKOR Method

Abstract

The process of evaluating and ranking alternatives, including the aggregation of various qualitative and quantitative criteria and weights of criteria, can be recognized as a fuzzy multiple criteria decision-making (MCDM) problem. In fuzzy MCDM problems, qualitative criteria and criteria weights are usually indicated in linguistic values expressed in terms of fuzzy numbers, and values under quantitative criteria are usually crisp numbers. How to properly aggregate them for evaluating and selecting alternatives has been an important research issue. To help decision-makers make the most suitable selection, this paper proposes a fuzzy preference relation-based fuzzy VIKOR method. VIKOR is a compromise ranking method to solve discrete MCDM problems in complex systems. In this study, the F-preference relation is applied to compare fuzzy numbers with their means to produce a single index of a dominance level while still maintaining fuzzy meaning of the original linguistic values. The inverse function is applied to obtain the defuzzification values of Beta 1–4 to assist in the completion of the proposed method, and formulas can be clearly derived to facilitate the ranking procedure. Introducing fuzzy preference relation into fuzzy VIKOR can simplify the calculation procedure for more efficient decision-making. The proposed method is new and has never been seen before. A numerical example and a comparison of the proposed method are conducted to show and verify its expedience and advantage.

1. Introduction

VIKOR is a compromise ranking method to clarify discrete multiple criteria decision-making (MCDM) problems, where the criteria can be incompatible and incomparable. This method has been proved to be an effective MCDM tool, especially where the decision-makers are not in the right positions to reveal their preferences at the first phase of the decision-making process [1,2,3,4]. Due to the reason that qualitative criteria and criteria weights are usually indicated by linguistic values, which can be expressed in terms of fuzzy numbers, this led to the development of fuzzy VIKOR [5,6]. Many extensions and applications of fuzzy VIKOR have been investigated. Most of the existing studies applying fuzzy VIKOR method, such as [4,7,8], used the approximation for the multiplication result of two positive triangular fuzzy numbers, $\tilde{M}=(m_1,m_2,m_3)$ and $\tilde{N}=(n_1,n_2,n_3)$ as $\tilde{M}\otimes \tilde{N}=\left(m_1{n}_1,m_2{n}_2,m_3{n}_3\right)$, which is still a linear triangular fuzzy number. However, according to [9], the multiplication of $\tilde{M}\otimes \tilde{N}$ is a nonlinear fuzzy number. Usually, a defuzzification method can be adopted to avoid the limitation related to the complicated multiplication process between two fuzzy numbers. In this study, fuzzy preference relation is used to obtain a fuzzy preference degree, which can be presented by crisp numbers for a better comparison of fuzzy numbers.

Although fuzzy preference relation ranking method is considered as more complex than defuzzification methods, which may lose fuzzy message and information, it maintains the fuzzy meaning [10,11,12]. Each fuzzy preference relation method has its own merits and demerits. This paper applies Li’s [13] F-preference relation method to compare fuzzy numbers, in which a single index is obtained by measuring the preference degree of one fuzzy number over another. By comparing the fuzzy numbers with their mean, the number of pairwise comparisons becomes smaller. Moreover, the inverse function [14] is applied to obtain the four areas, including ${\beta }_1$, ${\beta }_2$, ${\beta }_3$, and ${\beta }_4$, to facilitate the completion of the proposed method. To the best of our knowledge, research on introducing fuzzy preference relation to fuzzy VIKOR is yet to be investigated. To fill this gap, this paper proposes a fuzzy preference relation-based fuzzy VIKOR method for ranking alternatives. A numerical example and comparison of the proposed method are constructed to prove and validate its expedience and advantage.

The rest of this study is organized as follows. Section 2 demonstrates the literature review, including fuzzy VIKOR and fuzzy preference relation. Section 3 presents the basic concepts of fuzzy set theory. Section 4 suggests the fuzzy preference relation-based fuzzy VIKOR model; meanwhile, Section 5 displays a numerical example and a comparison to show the feasibility and advantage of the proposed method. Finally, Section 6 contains the study conclusions.

2. Literature Review

2.1. Fuzzy VIKOR

Multiple criteria decision-making (MCDM) is regarded as a complex and dynamic process [3]. MCDM methods can be considered as methodological and analytic tools, which can support the decision-making process to obtain the optimal alternative by which different criteria and involved expectations can be evaluated [15]. VIKOR is an effective MCDM and comprehensive analysis tool to rank and select the most suitable compromise solution from a set of alternatives based on contradictory criteria [6]. The name “VIseKriterijumska Optimizacija I Kompromisno Resenje” (VIKOR) is in the Serbian language, which means “Multicriteria Optimization and Compromise Solution” in English. Taking the compromise ranking approach as a foundation, Opricovic [16] built a multicriteria decision-making procedure for the assessment of alternatives, criteria, and criteria weights. According to [1,2,3,4], VIKOR was developed as a method for the multicriteria optimization of complex systems, and it establishes the compromise solution from a ranking set of alternatives, which also depends on the weight stability intervals. The compromise solution is feasibly the closest one to the ideal solution, in a situation where there is a mutual agreement among the decision-makers as shown by [3,4]. Based on this idea, the solution generated by VIKOR is considered to be an easily accepted one among the decision-makers; therefore, it can serve as a mutual ground for conflict settlement.

The merit of this compromise ranking was developed from the L_p-metric in compromising programming [17]. A L_p-metric in compromise programming was introduced to find a feasible solution that is the closest to the ideal one [18], which was based on the statement that the closer a solution to the ideal, the more preferable it becomes [19]. L₁ (p = 1) is the sum of all individual regrets, or can also be referred to as “disutility”; L_∞ (p = ∞) is the maximal possible regret that an individual could have. According to [1,2,3,4,20], within the VIKOR method, L_1j and L_∞j was adopted as an aggregating function to calculate S_j and R_j, respectively, to formulate ranking. The compromise solution is acquired by a minimum value of S_j, which represents a maximum group utility for majority rule, and a minimum value of R_j, which represents a minimum individual regret of the opponent [1,2,3,4,20]. Therefore, the ranking index of VIKOR can be considered as an aggregation of all criteria, and a combination of a balance between total and individual satisfaction.

Qualitative criteria and the fuzzy weights can be determined in terms of linguistic values to deal with inconsistent and uncertain environments [21]. Fuzzy set theory can transform the linguistic values to fuzzy numbers to effectively complete the calculation procedure of the fuzzy VIKOR model [22]. Therefore, fuzzy VIKOR has been extensively examined [8]. In the existing literature, the fuzzy VIKOR ranking results must depend on a defuzzification step to translate fuzzy values into crisp numbers [6].

Fuzzy VIKOR has been employed to a wide range of applications in decision-making problems, such as post-earthquake sustainable construction [20], water resource planning [4], supplier selection [5,7], material selection [23], healthcare quality assessment [21], healthcare supplier selection [24], production management [25], employee selection [6], and risk management [8,26]. There are a number of previous studies that explored the extension or hybrid combinations of fuzzy VIKOR with other MCDM methods, such as an AHP-fuzzy VIKOR model for evaluating integrated management systems [27], an integration of fuzzy AHP-ELECTRE-VIKOR to select a catering company [28], a fuzzy DEMATEl-fuzzy VIKOR for machine tool selection [29] (Li et al., 2020), an application of fuzzy AHP-fuzzy VIKOR model in renewable energy systems [30] and urban waterlogging prevention systems [31]. However, fuzzy preference relation-based fuzzy VIKOR has not been explored before.

2.2. Fuzzy Preference Relation

Based on Zadeh’s [22] fuzzy sets, Orlovsky [32] developed a concept of fuzzy preference relations; the corresponding fuzzy equivalence and preference relations were defined. Kołodziejczyk [33] analyzed Orlovsky’s [32] concept of decision-making with a fuzzy preference relation and formulated the new fuzzy preference relation properties. Nakamura [34] applied extended minimum operator and Hamming distance to define a fuzzy preference relation between two fuzzy sets. Tanino [35] proposed the application of fuzzy preference orderings as a fuzzy binary relation in group decision-making problems. Later, Yuan [10] reviewed Nakamura’s [34] method and suggested an improved method that compared the subtraction of two fuzzy numbers with the real number zero, and then presented the properties of the ranking method based on fuzzy preference relations. Li’s study [13] introduced a method that was based on fuzzy preference relation to measure the degree of preference of one fuzzy number over another with a smaller number of pairwise comparisons, by comparing the fuzzy numbers with their mean. Lee [11] presented a method based on Li’s [13] fuzzy preference relation and added a comparable property. Hipel et al. [36] overviewed the literature related to fuzzy preference relation to solve the multi-participant decision-making problem regarding the export of water in bulk quantities. Wang [12] proposed the revised method, which is a relative preference relation method with the membership function expressing the preference degrees of fuzzy numbers over their average. Liu et al. [37] defined the heterogeneous preference relation with self-confidence. Based on Li’s [13] method, Sadiq et al. [38] applied a combination of AHP and α-level-weighted fuzzy preference relation to identify the requirements of the software used in this method. Roldán López de Hierro et al. [39] developed a fuzzy binary relation from Li’s [13] algorithm for the production of two fuzzy numbers.

According to [10,11,12], fuzzy ranking methods can be classified into two main categories. The first one is based on defuzzification, and the second one implements preference relation to compare fuzzy numbers. Although the defuzzification method is determined to be simpler and easier, it loses the fuzzy messages and information by defuzzifying the fuzzy numbers into crisp numbers [11]. Although the fuzzy preference ranking method is more complicated, it is able to maintain the fuzzy meaning. By representing the preference degree, it establishes a fuzzy relation among fuzzy numbers for further pairwise comparisons [12]. A fuzzy MCDM method using inverse function-based total utility approach on maximizing set and minimizing set [40] was suggested by Chu and Yeh [41] to rank fuzzy numbers, in which a complicated procedure was used.

Due to the mentioned advantage of fuzzy preference relation, this study considers embedding Li’s [13] fuzzy preference relation with fuzzy VIKOR model for efficiently evaluating and selecting alternatives, where formulas of membership functions of the four areas, including ${\beta }_1,{\beta }_2,{\beta }_3,{\beta }_4$, and the fuzzy weighted ratings can be clearly developed. In addition, the concept of inverse function from Liou an Wang [14] is suggested to obtain the integral areas of ${\beta }_1,{\beta }_2,{\beta }_3,{\beta }_4$, in which formulas can be derived to help complete the proposed fuzzy preference relation-based fuzzy VIKOR method.

3. Fuzzy Set Theory

3.1. Fuzzy Set and Fuzzy Numbers

A fuzzy set M can be described as $M=\left\{\left(x,f_M\left(x\right)\right)|x\in U\right\}$, where ${\mu }_M\left(x\right)$ is the membership function of A at x in the universe of discourse U with the interval [0, 1] [9]. A fuzzy number M with membership function ${\mu }_M$ can be characterized as in [42].

A membership function ${\mu }_M\left(x\right)$ of a triangular fuzzy number $M=(m_1,m_2,m_3)$ can be denoted by the following [42]:

$${\mu }_M(x)=\{\begin{array}{l}\frac{x-m_1}{m_2-m_1},m_1\le x\le m_2\\ \frac{m_3-x}{m_3-m_2},m_2\le x\le m_3\\ 0,otherwise\end{array}$$

(1)

3.2. Operations on Fuzzy Numbers

Given two fuzzy numbers $M$ and $N$, $M, N\in R^{+}$, $M^{\alpha }=\left[M_l^{\alpha },M_u^{\alpha }\right]$ and $N^{\alpha }=\left[N_l^{\alpha },N_u^{\alpha }\right]$. $M^{\alpha }=\left[M_l^{\alpha },M_u^{\alpha }\right]=\left\{x|f_A\left(x\right)\ge \alpha \right\} , \alpha \in \left[0, 1\right]$ is the definition α-cuts of a fuzzy number $M$ where $M_l^{\alpha }$ and $M_u^{\alpha }$ are its lower and upper bounds [9] (Kaufmann and Gupta, 1991). The main operations of $M$ and $N$ can be expressed as follows [9]:

$${\left(M\oplus N\right)}^{\alpha }=\left[M_l^{\alpha }+N_l^{\alpha },{ M}_u^{\alpha }+N_u^{\alpha }\right]$$

(2)

$${\left(M㊀N\right)}^{\alpha }=\left[M_l^{\alpha }-N_u^{\alpha }, M_u^{\alpha }-N_l^{\alpha }\right]$$

(3)

$${\left(M\otimes N\right)}^{\alpha }=\left[M_l^{\alpha }\cdot N_l^{\alpha }, M_u^{\alpha }\cdot N_u^{\alpha }\right]$$

(4)

$${\left(M\otimes r\right)}^{\alpha }=\left[M_l^{\alpha }\cdot r, M_u^{\alpha }\cdot r\right] , r\in R^{+}$$

(5)

3.3. Linguistic Values

Linguistic values are demonstrated in linguistic terms. They are beneficial, especially in the situations where the variables are vague or too complicated to be prudently expressed with a traditional quantitative interpretation [22]. The linguistic values for “importance” can be interpreted using triangular fuzzy numbers as VL (Very Low) = (0,0.1,0.3), L (Low) = (0.1,0.3,0.5), M (Medium) = (0.3,0.5,0.7), H (High) = (0.5,0.7,0.9), and VH (Very High) = (0.7,0.9,1). The linguistic ratings of qualitative criteria can be assessed using triangular fuzzy numbers as VP (Very Poor) = (0,0.1,0.3), P (Poor) = (0.1,0.3,0.5), M (Moderate) = (0.3,0.5,0.7), G (Good) = (0.5,0.7,0.9), and VG = (Very Good) = (0.7,0.9,1).

4. Model Establishment

A fuzzy preference relation-based fuzzy VIKOR method is established using the following steps.

• Step 1. Establish the alternatives and criteria.

A committee of k decision-makers ($D_t$, t = 1~k) is accountable to consider and evaluate m alternatives ($A_i$, i = 1~m) under n criteria ($C_j$, j = 1~n), which include both qualitative and quantitative ones. Quantitative criteria can be divided into benefit $(j\in B)$ and cost criteria $(j\in C)$. Furthermore, the performance ratings of alternatives versus qualitative criteria are in linguistic values, which can also be expressed by positive triangular fuzzy numbers.

• Step 2. Determine the average ratings of alternatives versus qualitative criteria.

Let $x_{ijt}=\left(a_{ijt},b_{ijt},c_{ijt}\right)$, $x_{ijt}\in R^{+}$, which is the rating specified by decision-maker $D_t$ to alternative $A_i$ for criterion $C_j$.

$$x_{ij}=\left(a_{ij},b_{ij},c_{ij}\right) \mathrm{where} a_{ij}={\displaystyle \sum _{t=1}^k\frac{{}_{{{\displaystyle a}}_{ijt}}}{k}}, b_{ij}={\displaystyle \sum _{t=1}^k\frac{{}_{{{\displaystyle b}}_{ijt}}}{k}}, c_{ij}={\displaystyle \sum _{t=1}^k\frac{{}_{{{\displaystyle c}}_{ijt}}}{k}}$$

(6)

• Step 3. Normalize the values under quantitative criteria.

Quantitative criteria can be categorized into benefit criteria, which features the nature, characterized as the larger the better, and cost criteria, characterized as the smaller the better. For quantitative criteria, the values may be either crisp or fuzzy with different units. Therefore, they must be normalized into a comparable scale for calculation rationale. Chu and Charnsethikul [43] ’s approach is adapted for the normalization process, and it is capable of maintaining the ranges of normalized triangular fuzzy numbers in [0, 1].

Assuming that $x_{ij}=(e_{ij},f_{ij},g_{ij})$ is the performance value of alternative $A_i$ versus criteria $C_j$, the normalization of $r_{ij}$ is as follows:

$$r_{ij}=\left(\frac{e_{ij}-\underset{i}{\min }{e}_j}{\underset{i}{\max }{g}_{ij}-\underset{i}{\min }{e}_j},\frac{f_{ij}-\underset{i}{\min }{e}_j}{\underset{i}{\max }{g}_{ij}-\underset{i}{\min }{e}_j},\frac{g_{ij}-\underset{i}{\min }{e}_j}{\underset{i}{\max }{g}_{ij}-\underset{i}{\min }{e}_j}\right), j\in B$$

(7)

$$r_{ij}=\left(\frac{\underset{i}{\max }{g}_{ij}-g_{ij}}{\underset{i}{\max }{g}_{ij}-\underset{i}{\min }{e}_j},\frac{\underset{i}{\max }{g}_{ij}-f_{ij}}{\underset{i}{\max }{g}_{ij}-\underset{i}{\min }{e}_j},\frac{\underset{i}{\max }{g}_{ij}-e_{ij}}{\underset{i}{\max }{g}_{ij}-\underset{i}{\min }{e}_j}\right), j\in C$$

(8)

• Step 4. Determine the fuzzy best values using the extended fuzzy preference relation.

Using the Li’s [13] method, fuzzy numbers are compared with their mean instead of comparing every fuzzy numbers pairwise. The Li’s [13] extended fuzzy preference relation is applied to rank and determine best and worst fuzzy values as follows.

First, the mean value of the alternatives is calculated as follows:

$${\overline{x}}_j=\left({\overline{a}}_j,{\overline{b}}_j,{\overline{c}}_j\right)={\displaystyle \sum _{i=1}^m\frac{x_{ij}}{m}}$$

(9)

The preference relation among two fuzzy values $x_{ij}$ and ${\overline{x}}_j$ is given below:

$$\begin{array}{l}{\mu }_R(x_{ij},{\overline{x}}_j)=\{\begin{array}{l}\frac{\left({\beta }_1+{\beta }_2\right)}{\beta },\beta \ne 0\\ 0.5,\beta =0\end{array}\\ \beta ={\beta }_1+{\beta }_2+{\beta }_3+{\beta }_4\end{array}$$

(10)

According to Liou and Wang [14], with a fuzzy number $A=\left[a,b,c,d\right]$, the left inverse function of the left membership function $f_A^L\left(x\right),a\le x\le b,$ is $g_A^L\left(y\right)$, and the right inverse function of the right membership function $f_A^R\left(x\right),c\le x\le d,$ is $g_A^R\left(y\right)$. Both $g_A^L\left(y\right)$ and $g_A^R\left(y\right)$ are integrable, $0\le y\le 1$.

Suppose in Equation (10), $x_{ij}-{\overline{x}}_j$ is represented by a triangular fuzzy number $A=\left(a,b,c\right)$, then, the Liou and Wang‘s [14] inverse function is used to develop ${\beta }_1,{\beta }_2,{\beta }_3,{\beta }_4$ as follows:

$$\begin{array}{l}{\beta }_1={\displaystyle {\int }_{g_A^R\left(y\right)>0}{g}_A^R\left(y\right)dy}\\ {\beta }_2={\displaystyle {\int }_{g_A^L\left(y\right)>0}{g}_A^L\left(y\right)dy}\\ {\beta }_3={\displaystyle {\int }_{g_A^R\left(y\right)<0}-g_A^R\left(y\right)dy}\\ {\beta }_4={\displaystyle {\int }_{g_A^L\left(y\right)<0}-g_A^L\left(y\right)dy}\end{array}$$

(11)

Situation 1: if $a,b\ge 0;c>0$, then ${\beta }_3={\beta }_4=0$, and therefore,

$${\mu }_{R\left(x_{ij},{\overline{x}}_j\right)}={\mu }_{R(A)}=1$$

(12)

Situation 2: if $a<0;b\ge 0;c>0$, then, ${\beta }_1+{\beta }_2=I_A^R(y)$; ${\beta }_2-{\beta }_4=I_A^L(y)$; ${\beta }_3=0$; and ${\beta }_2={\displaystyle \underset{\raisebox{1ex}{$a$}\!\left/ \!\raisebox{-1ex}{$(a-b)$}\right.}{\overset{1}{\int }}{g}_A^L\left(y\right)dy=\frac{b^2}{2(b-a)}}$, and therefore,

$${\mu }_{R\left(x_{ij},\overline{x}\right)}={\mu }_{R(A)}=\frac{I_A^R(y)}{I_A^R(y)-I_A^L\left(y\right)+{\beta }_2}=\frac{b^2+bc-ab-ac}{a^2+b^2+bc-ab-ac}$$

(13)

Situation 3: if $a,b<0;c\ge 0$, then, ${\beta }_1-{\beta }_3=I_A^R(y)$; ${\beta }_3+{\beta }_4=I_A^L(y)$; ${\beta }_2=0$; and ${\beta }_1={\displaystyle \underset{0}{\overset{c/(c-b)}{\int }}{g}_A^R\left(y\right)dy=\frac{c^2}{2(c-b)}}$, and therefore,

$${\mu }_{R\left(x_{ij},\overline{x}\right)}={\mu }_{R(A)}=\frac{{\beta }_1}{{\beta }_1-I_A^L(y)}=\frac{c^2}{c^2+b^2+ab-bc-ac}$$

(14)

Situation 4: if $a,b,c<0$, then, ${\beta }_1={\beta }_2=0$, and therefore,

$${\mu }_{R\left(x_{ij},\overline{x}\right)}={\mu }_{R(A)}=0$$

(15)

Based on the values of ${\mu }_R(x_{ij},{\overline{x}}_j)$, we can rank and find out the fuzzy best value, $f_j^{\ast }$.

• Step 5. Determine the dominance of $f_j^{\ast }$ over $x_{ij}$.

The dominance of $f_j^{\ast }$ over $x_{ij}$ is obtained by calculating the fuzzy preference relation using Equation (9)–(15):

$${\mu }_R(f_j^{\ast },x_{ij})=\{\begin{array}{l}\frac{\left({\beta }_1+{\beta }_2\right)}{\beta },\beta \ne 0\\ 0.5,\beta =0\end{array}$$

(16)

• Step 6. Determine the importance weights of criteria.

Fuzzy weights of each alternative, $w_{ij}=(w_{a_{ij}},w_b{}_{{}_{ij}},w_{c_{ij}})$, are assigned by decision-maker $D_t$ for criterion $C_j$. The aggregated fuzzy weights $w_j$ of each criterion can be calculated as shown in Equation (17).

$$w_j=(w_{a_j},w_{b_j},w_{c_j})\mathrm{where} w_{a_j}={\displaystyle \sum _{t=1}^k\frac{w_{a_{ij}}}{k}}, w_{b_j}={\displaystyle \sum _{t=1}^k\frac{w_b{}_{{}_{ij}}}{k}}, w_{c_j}={\displaystyle \sum _{t=1}^k\frac{w_{c_{ij}}}{k}}$$

(17)

• Step 7. Compute the separation measures of $S_i$ and $R_i$.

The value of $S_i=\left(S_i^a,S_i^b,S_i^c\right)$ is calculated as follows:

$$S_i={\displaystyle \sum _{j=1}^n\left(w_j\otimes {\mu }_R(f_j^{\ast },x_{ij})\right)}$$

(18)

The value of $R_i=\left(R_i^a,R_i^b,R_i^c\right)$ is calculated as follows:

$$R_i=\underset{j}{\max }\left(w_j\otimes {\mu }_R(f_j^{\ast },x_{ij})\right)$$

(19)

The worst value of $S_i$, which is $S^{\ast }$, and the worst value of $R_i$, which is $R^{\ast }$, can then be obtained by the applied fuzzy preference relation using Equation (9)–(15).

• Step 8. Compute the separation measures of $Q_i$.

$$Q_i=\upsilon \times {\mu }_R(S^{\ast },S_i)+\left(1-\upsilon \right){\mu }_R\left(R^{\ast },R_i\right)$$

(20)

Here, $\upsilon$ is modified as $\upsilon =\left(n+1\right)/2n$ [4]. Herein, the inverse function based total integral value method [14] is used to rank $Q_i$.

• Step 9. Propose a compromise solution by ranking the values of $Q_i$ in the ascending order. The conditions that should be satisfied are as follows:

-

Condition 1: acceptable advantage. The following conditions should be satisfied: $Q(A_2)-Q(A_1)\ge DQ$, where $DQ=1/\left(m-1\right)$.

-

Condition 2: acceptance stability in decision-making. The best alternative $A_i$ must also be the best ranked by $S_i$ or/and $R_i$. This result should be stable within a decision-making process.

5. Numerical Example and Comparison

5.1. A Numerical Example

• Step 1. Determine the alternatives and criteria.

Suppose a manufacturing company is looking for a new green supplier with a suitable packaging line. A committee of three decision-makers ($D_1$, $D_2$, and $D_3$) is formed, who are in the management board and are crucial stakeholders of a company. Furthermore, suppose there are four potential suppliers for evaluation, $A_1$, $A_2$, $A_3$, and $A_4$. Moreover, assume that decision-makers have reached a consensus after a discussion to determine the following twelve criteria of evaluation, with two quantitative ones, such as environmentally friendly investment $\left(C_1\right)$ and environmental cost $\left(C_2\right)$, and ten qualitative criteria, such as green images $\left(C_3\right)$, management competency $\left(C_4\right)$, employees’ competency $\left(C_5\right)$, green cooperation $\left(C_6\right)$, service and delivery quality $\left(C_7\right)$, environmentally friendly applications $\left(C_8\right)$, green/sustainable technology $\left(C_9\right)$, green/sustainable manufacturing system $\left(C_{10}\right)$, green/sustainable procedures $\left(C_{11}\right)$, and environmentally friendly management and control $\left(C_{12}\right)$.

• Step 2. Determine the average ratings of alternatives versus qualitative criteria.

Assume that the ratings of alternatives versus qualitative criteria are evaluated by decision-makers based on professional perceptions and experience, as shown in

Table 1. The ratings of alternatives versus qualitative criteria.

Criteria	Alternatives	Decision-Makers
		D1	D2	D3
$C_3$	$A_1$	M	P	M
	$A_2$	G	G	G
	$A_3$	VG	VG	G
	$A_4$	M	G	M
$C_4$	$A_1$	P	M	VP
	$A_2$	G	M	G
	$A_3$	G	G	G
	$A_4$	P	M	M
$C_5$	$A_1$	VP	P	M
	$A_2$	G	VG	VG
	$A_3$	VG	VG	VG
	$A_4$	G	G	G
$C_6$	$A_1$	M	P	VP
	$A_2$	G	M	M
	$A_3$	VG	G	G
	$A_4$	G	P	M
$C_7$	$A_1$	M	M	G
	$A_2$	G	VG	VG
	$A_3$	G	G	G
	$A_4$	VG	G	G
$C_8$	$A_1$	M	M	P
	$A_2$	G	VG	G
	$A_3$	VG	G	G
	$A_4$	VP	P	VP
$C_9$	$A_1$	M	P	P
	$A_2$	VG	G	G
	$A_3$	VG	VG	VG
	$A_4$	M	G	M
$C_{10}$	$A_1$	P	P	VP
	$A_2$	G	M	G
	$A_3$	G	VG	VG
	$A_4$	M	G	M
$C_{11}$	$A_1$	M	P	P
	$A_2$	M	VG	G
	$A_3$	G	VG	M
	$A_4$	G	P	M
$C_{12}$	$A_1$	P	P	M
	$A_2$	G	G	VG
	$A_3$	G	VG	VG
	$A_4$	M	M	M

. The average ratings of alternatives versus qualitative criteria can be obtained using Equation (6), as shown in

Table 2. Average ratings of alternatives versus qualitative criteria.

Criteria	Alternatives
	A1			A2			A3			A4
	${\mathit{a}}_{\mathit{i}\mathit{j}}$	${\mathit{b}}_{\mathit{i}\mathit{j}}$	${\mathit{c}}_{\mathit{i}\mathit{j}}$	${\mathit{a}}_{\mathit{i}\mathit{j}}$	${\mathit{b}}_{\mathit{i}\mathit{j}}$	${\mathit{c}}_{\mathit{i}\mathit{j}}$	${\mathit{a}}_{\mathit{i}\mathit{j}}$	${\mathit{b}}_{\mathit{i}\mathit{j}}$	${\mathit{c}}_{\mathit{i}\mathit{j}}$	${\mathit{a}}_{\mathit{i}\mathit{j}}$	${\mathit{b}}_{\mathit{i}\mathit{j}}$	${\mathit{c}}_{\mathit{i}\mathit{j}}$
$C_3$	0.300	0.483	0.667	0.600	0.750	0.900	0.733	0.850	0.967	0.433	0.617	0.800
$C_4$	0.217	0.367	0.517	0.517	0.683	0.850	0.600	0.750	0.900	0.300	0.483	0.667
$C_5$	0.217	0.367	0.517	0.800	0.800	0.917	0.800	0.867	0.967	0.600	0.700	0.850
$C_6$	0.217	0.367	0.517	0.433	0.617	0.800	0.667	0.800	0.933	0.383	0.550	0.717
$C_7$	0.433	0.617	0.800	0.733	0.850	0.967	0.600	0.750	0.900	0.667	0.800	0.933
$C_8$	0.300	0.483	0.667	0.667	0.800	0.933	0.667	0.800	0.933	0.133	0.250	0.367
$C_9$	0.250	0.417	0.583	0.667	0.800	0.933	0.800	0.900	1.000	0.433	0.617	0.800
$C_{10}$	0.167	0.300	0.433	0.517	0.683	0.850	0.733	0.850	0.967	0.433	0.617	0.800
$C_{11}$	0.250	0.417	0.583	0.583	0.733	0.883	0.583	0.733	0.883	0.383	0.550	0.717
$C_{12}$	0.217	0.367	0.517	0.517	0.683	0.850	0.600	0.750	0.900	0.300	0.483	0.667

.

• Step 3. Normalize the values under quantitative criteria.

The ratings of alternatives versus quantitative criteria are shown in

Table 3. Ratings of alternatives versus quantitative criteria.

Criteria	Alternatives
	A1			A2			A3			A4
	${\mathit{e}}_{\mathit{i}\mathit{j}}$	${\mathit{f}}_{\mathit{i}\mathit{j}}$	${\mathit{g}}_{\mathit{i}\mathit{j}}$	${\mathit{e}}_{\mathit{i}\mathit{j}}$	${\mathit{f}}_{\mathit{i}\mathit{j}}$	${\mathit{g}}_{\mathit{i}\mathit{j}}$	${\mathit{e}}_{\mathit{i}\mathit{j}}$	${\mathit{f}}_{\mathit{i}\mathit{j}}$	${\mathit{g}}_{\mathit{i}\mathit{j}}$	${\mathit{e}}_{\mathit{i}\mathit{j}}$	${\mathit{f}}_{\mathit{i}\mathit{j}}$	${\mathit{g}}_{\mathit{i}\mathit{j}}$
$C_1$	12	14	16	20	21	22	17	19	21	17	19	21
$C_2$	5	7	10	11	12	13	6	7	9	8	10	12

, and the normalized values of quantitative criteria can be obtained using Equations (7) and (8), as shown in

Table 4. Normalized ratings of alternatives versus quantitative criteria.

Criteria	Alternatives
	A1			A2			A3			A4
	${\mathit{e}}_{\mathit{i}\mathit{j}}$	${\mathit{f}}_{\mathit{i}\mathit{j}}$	${\mathit{g}}_{\mathit{i}\mathit{j}}$	${\mathit{e}}_{\mathit{i}\mathit{j}}$	${\mathit{f}}_{\mathit{i}\mathit{j}}$	${\mathit{g}}_{\mathit{i}\mathit{j}}$	${\mathit{e}}_{\mathit{i}\mathit{j}}$	${\mathit{f}}_{\mathit{i}\mathit{j}}$	${\mathit{g}}_{\mathit{i}\mathit{j}}$	${\mathit{e}}_{\mathit{i}\mathit{j}}$	${\mathit{f}}_{\mathit{i}\mathit{j}}$	${\mathit{g}}_{\mathit{i}\mathit{j}}$
$C_1$	0.000	0.200	0.400	0.800	0.900	1.000	0.500	0.700	0.900	0.500	0.700	0.900
$C_2$	0.375	0.750	1.000	0.000	0.125	0.250	0.500	0.750	0.875	0.125	0.375	0.625

.

• Step 4. Determine the fuzzy best value using the extended fuzzy preference relation.

The mean values of the alternatives ${\overline{x}}_j$ can be obtained using Equation (9) as shown in

Table 5. Mean values of the alternatives.

$\overline{\mathit{x}}$j	$\overline{\mathit{a}}$j	$\overline{\mathit{b}}$j	$\overline{\mathit{c}}$j
$\overline{x}$1	0.450	0.625	0.800
$\overline{x}$2	0.250	0.500	0.688
$\overline{x}$3	0.517	0.675	0.833
$\overline{x}$4	0.408	0.571	0.733
$\overline{x}$5	0.604	0.683	0.813
$\overline{x}$6	0.425	0.583	0.742
$\overline{x}$7	0.608	0.754	0.900
$\overline{x}$8	0.442	0.583	0.725
$\overline{x}$9	0.538	0.683	0.829
$\overline{x}$10	0.463	0.613	0.763
$\overline{x}$11	0.450	0.608	0.767
$\overline{x}$12	0.500	0.654	0.808

. The preference relation between $x_{ij}$ and ${\overline{x}}_j$ can be obtained by Equations (10)–(15) as shown in

Table 6. Fuzzy preference relation between $x_{ij}$ and ${\overline{x}}_j$.

${\mathit{\mu }}_{\mathit{R}}({\mathit{x}}_{\mathit{i}\mathit{j}},{\overline{\mathit{x}}}_{\mathit{j}})$	$\mathit{i}=1$	$\mathit{i}=2$	$\mathit{i}=3$	$\mathit{i}=4$
$j=1$	0.000	1.000	0.686	0.686
$j=2$	0.852	0.000	0.916	0.290
$j=3$	0.083	0.722	0.945	0.339
$j=4$	0.050	0.795	0.922	0.270
$j=5$	0.000	0.997	0.999	0.571
$j=6$	0.035	0.595	0.974	0.401
$j=7$	0.156	0.811	0.486	0.655
$j=8$	0.229	0.983	0.983	0.000
$j=9$	0.008	0.844	0.995	0.312
$j=10$	0.000	0.706	0.996	0.512
$j=11$	0.072	0.837	0.837	0.331
$j=12$	0.026	0.892	0.970	0.239

. Based on Table 6, we can obtain the fuzzy best value $f_j^{\ast }$, as shown in

Table 7. Fuzzy best values.

	$\mathbf{Fuzzy} \mathbf{Best} \mathbf{Value} {\mathit{f}}_{\mathit{j}}^{\ast }$
	${\mathit{a}}_{\mathit{j}}^{\ast }$	${\mathit{b}}_{\mathit{j}}^{\ast }$	${\mathit{c}}_{\mathit{j}}^{\ast }$
$j=1$	0.800	0.900	1.000
$j=2$	0.500	0.750	0.875
$j=3$	0.733	0.850	0.967
$j=4$	0.600	0.750	0.900
$j=5$	0.800	0.867	0.967
$j=6$	0.667	0.800	0.933
$j=7$	0.733	0.850	0.967
$j=8$	0.667	0.800	0.933
$j=9$	0.800	0.900	1.000
$j=10$	0.733	0.850	0.967
$j=11$	0.583	0.733	0.883
$j=12$	0.733	0.850	0.967

.

• Step 5. Determine the dominance of $f_j^{\ast }$ over $x_{ij}$.

The dominance of $f_j^{\ast }$ over $x_{ij}$ can be obtained using Equation (16) as shown in

Table 8. Fuzzy preference relation between $f_j^{\ast }$ and $x_{ij}$.

${\mathit{\mu }}_{\mathit{R}}({\mathit{f}}_{\mathit{j}}^{\ast },{\mathit{x}}_{\mathit{i}\mathit{j}})$	$\mathit{i}=1$	$\mathit{i}=2$	$\mathit{i}=3$	$\mathit{i}=4$
$j=1$	1.000	0.500	0.955	0.955
$j=2$	0.500	1.000	0.500	0.973
$j=3$	1.000	0.818	0.500	0.981
$j=4$	1.000	0.695	0.500	0.985
$j=5$	1.000	0.759	0.500	0.979
$j=6$	1.000	0.924	0.500	0.990
$j=7$	0.981	0.500	0.818	0.686
$j=8$	1.000	0.500	0.500	1.000
$j=9$	1.000	0.850	0.500	1.000
$j=10$	1.000	0.928	0.500	0.981
$j=11$	1.000	0.500	0.500	0.924
$j=12$	1.000	0.686	0.500	0.999

.

• Step 6. Determine the importance weights of criteria.

Assume that the importance weights of criteria are determined in linguistic values by decision-makers based on their professional perceptions and experience, as shown in

Table 9. Importance weights of criteria in linguistic values.

	Importance Weights of Criteria
	D1	D2	D3
$C_1$	VI	VI	VI
$C_2$	MI	IM	MI
$C_3$	MI	VI	IM
$C_4$	VI	VI	IM
$C_5$	MI	VI	MI
$C_6$	MI	IM	IM
$C_7$	UI	MI	IM
$C_8$	MI	IM	MI
$C_9$	VI	VI	IM
$C_{10}$	UI	MI	IM
$C_{11}$	VI	VI	VI
$C_{12}$	MI	IM	MI

. Then, the aggregated fuzzy weights can be obtained by Equation (17) as shown in

Table 10. Aggregated importance weights.

	Average Important Weight
	${\mathit{w}}_{{\mathit{a}}_{\mathit{j}}}$	${\mathit{w}}_{{\mathit{b}}_{\mathit{j}}}$	${\mathit{w}}_{{\mathit{c}}_{\mathit{j}}}$
$C_1$	0.775	0.875	1.000
$C_2$	0.517	0.675	0.825
$C_3$	0.583	0.725	0.867
$C_4$	0.650	0.775	0.908
$C_5$	0.642	0.775	0.917
$C_6$	0.458	0.625	0.775
$C_7$	0.367	0.517	0.650
$C_8$	0.517	0.675	0.825
$C_9$	0.650	0.775	0.908
$C_{10}$	0.367	0.517	0.650
$C_{11}$	0.775	0.875	1.000
$C_{12}$	0.517	0.675	0.825

.

• Step 7. Compute the separation measures of $S_i$ and $R_i$.

The values of $S_i$ and $R_i$ can be obtained by Equations (18) and (19) and are shown in

Table 11. The values of $S_i$.

$S_i$	$S_i^a$	$S_i^b$	$S_i^c$
$S_1$	6.551	8.136	9.725
$S_2$	4.820	6.044	7.256
$S_3$	3.877	4.803	5.736
$S_i$	6.547	8.138	9.732

and

Table 12. The values of $R_i$.

$R_i$	$R_i^a$	$R_i^b$	$R_i^c$
$R_1$	0.775	0.875	1.000
$R_2$	0.553	0.659	0.773
$R_3$	0.740	0.835	0.955
$R_4$	0.740	0.835	0.955

, respectively.

• Step 8. Compute the separation measures of $Q_i$.

The separation measures $Q_i$ can be obtained by Equation (20) as shown in

Table 13. The values of $Q_i$.

$Q_1$	1.000
$Q_2$	0.729
$Q_3$	0.725
$Q_4$	0.996

. According to the values in Table 13, we obtain the following ranking result: $A_3\to A_2\to A_4\to A_1$. Therefore, $A_3$ is the best supplier, while $A_1$ is the worst supplier for the manufacturing company.

5.2. A Comparison with Fuzzy VIKOR Method [4]

Since VIKOR was first fully described by [16], later, fuzzy VIKOR was applied by [4] for water resources planning. Since then, the model used in this section has been applied and developed in various fields as can be observed from the fuzzy VIKOR studies conducted later.

• Step a. Determine the alternatives and criteria—same as Step 1 above.

• Step b. Aggregate the ratings of the alternatives against criteria.

The equation $x_{ij}=\left(a_{ij},b_{ij},c_{ij}\right)$, where $a_{ij}={\min }_k\left\{a_{ijt}\right\}$, $b_{ij}={\displaystyle \sum _{t=1}^k\frac{{}_{{{\displaystyle b}}_{ijt}}}{k}}$, and $c_{ij}={\max }_k\left\{c_{ijt}\right\}$, is used to obtain the aggregated ratings of criteria versus criteria, as shown in

Table 14. Aggregate ratings of alternatives versus criteria.

Criteria	Alternatives
	A1			A2			A3			A4
	${\mathit{a}}_{\mathit{i}\mathit{j}}$	${\mathit{b}}_{\mathit{i}\mathit{j}}$	${\mathit{c}}_{\mathit{i}\mathit{j}}$	${\mathit{a}}_{\mathit{i}\mathit{j}}$	${\mathit{b}}_{\mathit{i}\mathit{j}}$	${\mathit{c}}_{\mathit{i}\mathit{j}}$	${\mathit{a}}_{\mathit{i}\mathit{j}}$	${\mathit{b}}_{\mathit{i}\mathit{j}}$	${\mathit{c}}_{\mathit{i}\mathit{j}}$	${\mathit{a}}_{\mathit{i}\mathit{j}}$	${\mathit{b}}_{\mathit{i}\mathit{j}}$	${\mathit{c}}_{\mathit{i}\mathit{j}}$
$C_1$	0.000	0.200	0.400	0.800	0.900	1.000	0.500	0.700	0.900	0.500	0.700	0.900
$C_2$	0.375	0.750	1.000	0.000	0.125	0.250	0.500	0.750	0.875	0.125	0.375	0.625
$C_3$	0.000	0.483	0.750	0.600	0.750	0.900	0.600	0.850	1.000	0.350	0.617	0.900
$C_4$	0.000	0.367	0.750	0.350	0.683	0.900	0.350	0.683	0.900	0.200	0.483	0.750
$C_5$	0.000	0.367	0.750	0.600	0.850	1.000	0.800	0.900	1.000	0.600	0.750	0.900
$C_6$	0.000	0.367	0.750	0.350	0.617	0.900	0.600	0.800	1.000	0.200	0.550	0.900
$C_7$	0.000	0.617	0.900	0.600	0.850	1.000	0.600	0.750	0.900	0.600	0.800	1.000
$C_8$	0.000	0.483	0.750	0.600	0.800	1.000	0.600	0.800	1.000	0.200	0.350	0.500
$C_9$	0.000	0.417	0.750	0.600	0.800	1.000	0.800	0.900	1.000	0.350	0.617	0.900
$C_{10}$	0.000	0.300	0.500	0.350	0.683	0.900	0.600	0.850	1.000	0.350	0.617	0.900
$C_{11}$	0.000	0.417	0.750	0.350	0.733	1.000	0.350	0.733	1.000	0.200	0.550	0.900
$C_{12}$	0.000	0.417	0.750	0.600	0.800	1.000	0.600	0.850	1.000	0.350	0.550	0.750

.

• Step c. Aggregate the weighted weightings.

The $w_{jt}=(w_{a_{jt}},w_{b_{jt}},w_{c_{jt}})$ is the weighted rating assigned by decision-maker $D_t$ for criterion $C_j$. The values in Table 9 are used.

The aggregated fuzzy weight $w_j$ of each criterion can be obtained by $w_j=(w_{a_j},w_{b_j},w_{c_j})$, where $w_{a_j}={\min }_k\left\{w_{a_{jt}}\right\}, w_{b_j}={\displaystyle \sum _{t=1}^k\frac{w_{b_{jt}}}{k}}, w_{c_j}={\max }_k\left\{w_{c_{jt}}\right\}$, as shown in

Table 15. Aggregated importance weights of criteria.

	Average Important Weight
	${\mathit{d}}_{\mathit{j}}$	${\mathit{e}}_{\mathit{j}}$	${\mathit{f}}_{\mathit{j}}$
$C_1$	0.775	0.875	1.000
$C_2$	0.400	0.675	0.875
$C_3$	0.400	0.725	0.725
$C_4$	0.400	0.775	0.725
$C_5$	0.575	0.775	0.875
$C_6$	0.400	0.625	0.725
$C_7$	0.125	0.517	0.725
$C_8$	0.400	0.675	0.875
$C_9$	0.400	0.775	0.725
$C_{10}$	0.125	0.517	0.725
$C_{11}$	0.775	0.875	1.000
$C_{12}$	0.400	0.675	0.875

.

• Step d. Determine the fuzzy best value and the fuzzy worst value.

The fuzzy best value $f_j^{\ast }=(a_j^{\ast }, b_j^{\ast }, c_j^{\ast })$ and fuzzy worst value $f_j^{°}=(a_j^{°},b_j^{°},c_j^{°})$, in which $f_j^{\ast }=\underset{i}{\max }{x}_{ij}, f_j^{°}=\underset{i}{\min }{x}_{ij}$ if $C_j$ is a benefit criterion, and $f_j^{\ast }=\min x_{ij}, f_j^{°}=\max x_{ij}$ if $C_j$ is a cost criterion, can be obtained as shown in

Table 16. Fuzzy best values.

	$\mathbf{Fuzzy} \mathbf{Best} \mathbf{Value} {\mathit{f}}_{\mathit{j}}^{\ast }$
	${\mathit{a}}_{\mathit{j}}^{\ast }$	${\mathit{b}}_{\mathit{j}}^{\ast }$	${\mathit{c}}_{\mathit{j}}^{\ast }$
$C_1$	0.800	0.900	1.000
$C_2$	0.500	0.750	1.000
$C_3$	0.600	0.850	1.000
$C_4$	0.350	0.683	0.900
$C_5$	0.800	0.900	1.000
$C_6$	0.600	0.800	1.000
$C_7$	0.600	0.850	1.000
$C_8$	0.600	0.800	1.000
$C_9$	0.800	0.900	1.000
$C_{10}$	0.600	0.850	1.000
$C_{11}$	0.350	0.733	1.000
$C_{12}$	0.600	0.850	1.000

and

Table 17. Fuzzy worst values.

	$\mathbf{Fuzzy} \mathbf{Worst} \mathbf{Value} {\mathit{f}}_{\mathit{j}}^{°}$
	${\mathit{a}}_{\mathit{j}}^{°}$	${\mathit{b}}_{\mathit{j}}^{°}$	${\mathit{c}}_{\mathit{j}}^{°}$
$C_1$	0.000	0.200	0.400
$C_2$	0.000	0.125	0.250
$C_3$	0.000	0.483	0.750
$C_4$	0.000	0.367	0.750
$C_5$	0.000	0.367	0.750
$C_6$	0.000	0.367	0.750
$C_7$	0.000	0.617	0.900
$C_8$	0.000	0.350	0.500
$C_9$	0.000	0.417	0.750
$C_{10}$	0.000	0.300	0.500
$C_{11}$	0.000	0.417	0.750
$C_{12}$	0.000	0.417	0.750

, respectively.

• Step e. Compute the fuzzy difference $d_{ij}$.

$d_{ij}=(f_j^{\ast }-x_{ij})/(c_j^{\ast }-a_j^{°})$ if $C_j$ is a benefit criterion, and $d_{ij}=(x_{ij}-f_j^{\ast })/(c_j^{°}-a_j^{\ast })$ if $C_j$ is a cost criterion, as shown in

Table 18. Fuzzy difference $d_{ij}$.

$d_{ij}$
Alternatives	A1			A2			A3			A4
Criteria	${\mathit{a}}_{\mathit{i}\mathit{j}}$	${\mathit{b}}_{\mathit{i}\mathit{j}}$	${\mathit{c}}_{\mathit{i}\mathit{j}}$	${\mathit{a}}_{\mathit{i}\mathit{j}}$	${\mathit{a}}_{\mathit{i}\mathit{j}}$	${\mathit{c}}_{\mathit{i}\mathit{j}}$	${\mathit{a}}_{\mathit{i}\mathit{j}}$	${\mathit{b}}_{\mathit{i}\mathit{j}}$	${\mathit{c}}_{\mathit{i}\mathit{j}}$	${\mathit{a}}_{\mathit{i}\mathit{j}}$	${\mathit{b}}_{\mathit{i}\mathit{j}}$	${\mathit{c}}_{\mathit{i}\mathit{j}}$
$C_1$	0.400	0.700	1.000	−0.200	0.000	0.000	−0.100	0.200	0.500	−0.100	−0.556	0.100
$C_2$	−0.500	0.000	0.625	0.250	0.625	0.750	−0.375	0.000	0.500	−0.125	−0.333	0.375
$C_3$	−0.150	0.367	1.000	−0.300	0.100	0.100	−0.400	0.000	0.400	−0.300	−0.157	0.100
$C_4$	−0.444	0.352	1.000	−0.611	0.000	0.000	−0.611	0.000	0.611	−0.444	−0.171	0.167
$C_5$	0.050	0.533	1.000	−0.200	0.050	0.000	−0.200	0.000	0.200	−0.100	−0.426	0.100
$C_6$	−0.150	0.433	1.000	−0.300	0.183	0.100	−0.400	0.000	0.400	−0.300	−0.229	0.100
$C_7$	−0.300	0.233	1.000	−0.400	0.000	0.000	−0.300	0.100	0.400	−0.400	−0.216	0.000
$C_8$	−0.150	0.317	1.000	−0.400	0.000	0.000	−0.400	0.000	0.400	0.100	0.000	0.500
$C_9$	0.050	0.483	1.000	−0.200	0.100	0.000	−0.200	0.000	0.200	−0.100	−0.222	0.100
$C_{10}$	0.100	0.550	1.000	−0.300	0.167	0.100	−0.400	0.000	0.400	−0.300	−0.373	0.100
$C_{11}$	−0.400	0.317	1.000	−0.650	0.000	0.000	−0.650	0.000	0.650	−0.550	−0.182	0.100
$C_{12}$	−0.150	0.433	1.000	−0.400	0.050	0.000	−0.400	0.000	0.400	−0.150	−0.157	0.250

.

• Step f. Compute the separation measures.

The $S_i=\left(S_i^a,S_i^b,S_i^c\right)$ derived from $S_i={\displaystyle \sum _{i=1}^n\left(w_j\otimes d_{ij}\right)}$ and $R_i=\left(R_i^a,R_i^b,R_i^c\right)$ derived from $R_i=\underset{j}{\max }\left(w_j\otimes d_{ij}\right)$ can be obtained as shown in

Table 19. The values of $S_i$.

$S_i$	$S_i^a$	$S_i^b$	$S_i^c$
$S_1$	−0.594	3.398	9.522
$S_2$	−1.646	0.845	0.874
$S_3$	−1.898	0.227	4.211
$S_4$	−1.177	−2.172	1.683

and

Table 20. The values of $R_i$.

$R_i$	$R_i^a$	$R_i^b$	$R_i^c$
$R_1$	0.310	0.613	1.000
$R_2$	0.100	0.422	0.656
$R_3$	−0.038	0.175	0.650
$R_4$	0.040	0.000	0.438

, respectively.

• Step g. Compute the value of $Q_i$.

The equation for $Q_i$ is $Q_i=\frac{\upsilon \left(S_i-S^{\ast }\right)}{\left(S^{°}{}^c-S^{\ast }{}_{}^a\right)}\oplus \frac{\left(1-\upsilon \right)\left(R_i-R^{\ast }\right)}{\left(R^{°}{}_{}^c-R^{\ast }{}_{}^a\right)}$, where $S^{°}=\underset{j}{\max }{S}_i$, $S^{\ast }=\underset{j}{\min }{S}_i$, $R^{\ast }=\underset{j}{\min }{R}_i$, and $R^{°}=\underset{j}{\max }{R}_i$. Here, $\upsilon$ is modified as $\upsilon =\left(n+1\right)/2n$ [4]. Through this equation, the values of $Q_i$ can be obtained, as shown in

Table 21. The values of $Q_i$.

$Q_i$	$Q_i^a$	$Q_i^b$	$Q_i^c$
$Q_1$	−0.126	0.539	1.000
$Q_2$	−0.273	0.335	0.456
$Q_3$	−0.350	0.189	0.498
$Q_4$	−0.281	0.000	0.386

.

• Step h. Defuzzify the values of $S_i$, $R_i$, and $Q_i$.

The values of $S_i$, $R_i$, and $Q_i$ are defuzzified by using the second-weighted mean [3,4]. The defuzzified value of a fuzzy number ${\tilde{A}}_i=(a_i,b_i,c_i)$ is $crisp\left({\tilde{A}}_i\right)=\frac{a_i+2b_i+c_i}{4}$. The result is shown in

Table 22. Defuzzified values.

i	1	2	3	4
$S_i$	3.931	0.230	0.691	−0.959
$R_i$	0.634	0.400	0.241	0.119
$Q_i$	0.488	0.213	0.132	0.026

.

• Step i. Propose a compromise solution by ranking the defuzzified values of $Q_i$, $S_i$, and $R_i$ in the ascending order. The following conditions need to be satisfied:

-

Condition 1: acceptable advantage. The following conditions should be satisfied: $Q(A_2)-Q(A_1)\ge DQ$, where $DQ=1/\left(m-1\right)$

-

Condition 2: acceptance stability in decision-making. The alternative A_i must also be the best ranked by S or/and R. This result should be stable within a decision-making process.

Herein, the ranking result of $A_4\to A_3\to A_2\to A_1$ is obtained. Therefore, $A_4$ is the best supplier, while $A_1$ is the worst supplier for the manufacturing company. The ranking result obtained by the Opricovic’s [4] fuzzy VIKOR method is inconsistent with the one produced by the proposed method. The nature of the two ranking methods might lead to different ranking results. The fuzzy VIKOR method applies the approximation multiplication calculation of two fuzzy numbers. As the multiplication of two fuzzy numbers should be considered as nonlinear result, the defuzzification is needed to further convert the fuzzy values into crisp numbers. The proposed fuzzy preference relation-based fuzzy VIKOR method can overcome this limitation, in which the dominance of the fuzzy preference relation among two fuzzy numbers is used to simplify the multiplication operation on fuzzy numbers and strengthen the ranking result.

6. Conclusions

VIKOR provides a compromise ranking method, which has been proved to be an applicable technique to deal with MCDM problems. The obtained solution is based on the closeness to an ideal solution, which can serve as a mutual ground for further negotiation. Since the introduction of fuzzy VIKOR, many extensions and applications of this method have been investigated. However, there are some limitations of fuzzy VIKOR that may lead to an incorrect ranking result. To avoid this limitation, a fuzzy preference relation-based fuzzy VIKOR method has been proposed. The Li’s [13] fuzzy preference relation was used to compare two fuzzy numbers, in which a single index is obtained by measuring the degree of preference of one fuzzy number over another with a smaller number of pairwise comparisons, by comparing the fuzzy numbers with their mean. The Liou and Wang’s [14] inverse function was applied to obtain the four areas ${\beta }_1,{\beta }_2,{\beta }_3,{\beta }_4$ in the Li’s [13] method, in which formulas can be derived for better executing the calculation. A numerical example has shown the feasibility of the proposed method. Moreover, a comparison with a fuzzy VIKOR method [4] has demonstrated the effectiveness of the proposed method.

While VIKOR is a proven efficient compromise ranking method in fuzzy MCDM, many previous studies depend mostly on the approximation multiplication of two fuzzy numbers and the defuzzification process to convert the fuzzy values into crisp numbers. This study proposed a fuzzy preference relation-based fuzzy VIKOR method, which adopts the dominance of one fuzzy number over another to not only easily rank fuzzy numbers but also maintain the fuzzy meaning.