Fuzzy VIKOR and Fuzzy Preference Relation: Comparison
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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. This research proposes a fuzzy preference relation‑based fuzzy VIKOR method to help decision‑makers make the most suitable selection.

  • fuzzy preference relation
  • fuzzy VIKOR
  • inverse function

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, 𝑀̃̃=(𝑚1,𝑚2,𝑚3) and 𝑁̃=(𝑛1,𝑛2,𝑛3) as 𝑀̃⊗𝑁̃=(𝑚1𝑛1,𝑚2𝑛2,𝑚3𝑛3), which is still a linear triangular fuzzy number. However, according to [9], the multiplication of 𝑀⊗𝑁̃ 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 research, 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.

2. 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 [13]. 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 [14] 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 Lp-metric in compromising programming [15]. A Lp-metric in compromise programming was introduced to find a feasible solution that is the closest to the ideal one [16], which was based on the statement that the closer a solution to the ideal, the more preferable it becomes [17]. 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][18], within the VIKOR method, L1j and L∞j was adopted as an aggregating function to calculate Sj and Rj, respectively, to formulate ranking. The compromise solution is acquired by a minimum value of Sj, which represents a maximum group utility for majority rule, and a minimum value of Rj, which represents a minimum individual regret of the opponent [1][2][3][4][18]. 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 [19]. Fuzzy set theory can transform the linguistic values to fuzzy numbers to effectively complete the calculation procedure of the fuzzy VIKOR model [20]. 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 [18], water resource planning [4], supplier selection [5][7], material selection [21], healthcare quality assessment [19], healthcare supplier selection [22], production management [23], employee selection [6], and risk management [8][24]. 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 [25], an integration of fuzzy AHP-ELECTRE-VIKOR to select a catering company [26], a fuzzy DEMATEl-fuzzy VIKOR for machine tool selection [27] (Li et al., 2020), an application of fuzzy AHP-fuzzy VIKOR model in renewable energy systems [28] and urban waterlogging prevention systems [29]. However, fuzzy preference relation-based fuzzy VIKOR has not been explored before.

3. Fuzzy Preference Relation

Based on Zadeh’s [20] fuzzy sets, Orlovsky [30] developed a concept of fuzzy preference relations; the corresponding fuzzy equivalence and preference relations were defined. Kołodziejczyk [31] analyzed Orlovsky’s [30] concept of decision-making with a fuzzy preference relation and formulated the new fuzzy preference relation properties. Nakamura [32] applied extended minimum operator and Hamming distance to define a fuzzy preference relation between two fuzzy sets. Tanino [33] proposed the application of fuzzy preference orderings as a fuzzy binary relation in group decision-making problems. Later, Yuan [10] reviewed Nakamura’s [32] 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 [34] 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 [34] fuzzy preference relation and added a comparable property. Hipel et al. [35] 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. [36] defined the heterogeneous preference relation with self-confidence. Based on Li’s [34] method, Sadiq et al. [37] 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. [38] developed a fuzzy binary relation from Li’s [34] 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 [39] was suggested by Chu and Yeh [40] to rank fuzzy numbers, in which a complicated procedure was used.

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