Consistency-index-driven group decision making under the environment of triangular fuzzy numbers
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METHODOLOGIES AND APPLICATION
Consistency-index-driven group decision making under the environment of triangular fuzzy numbers Fang Liu1
· Caixia Huang1 · Tong Liu1
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Group decision making (GDM) under a fuzzy environment is one of the research focuses recently. Triangular fuzzy number can be used as an effective tool to capture the vagueness encountered by decision makers (DMs). In this study, a novel GDM model is proposed when triangular fuzzy multiplicative reciprocal matrices (TFMRMs) are adopted to express the opinions of DMs. A generalized consistency index is constructed to quantify the inconsistency degree of TFMRMs, which reflects the basic idea of fuzzy set theory that everything has some elasticity. The interesting properties of the new consistency index are studied, and acceptable consistency of TFMRMs is discussed. Then, based on the proposed consistency index, an operator is proposed to aggregate the individual TFMRMs. The properties of the collective TFMRM are further investigated. Finally, a new algorithm for solving a GDM problem with TFMRMs is elaborated on. Numerical results are reported to illustrate the advantages and novelty of the proposed consistency-index-driven GDM model. Keywords Group decision making (GDM) · Triangular fuzzy multiplicative reciprocal matrix (TFMRM) · Generalized consistency index · Acceptable consistency · Aggregation operator
1 Introduction The analytic hierarchy process (AHP) is one of the qualitative and quantitative decision-making methods, and it has been studied and extended comprehensively (Saaty 1980; Golden et al. 1989; Vaidya and Kumar 2006; Bernasconi et al. 2010; Saaty 2013; Brunelli 2015; Rezaei 2015; Liu et al. 2020a). Due to the vagueness and uncertainty involved in the realworld decision-making problems, the precise number-based decision-making models such as the typical AHP are worth to be modified.
1.1 Uncertain decision environments In order to cope with the vagueness and uncertainty of human-originated information, many mathematical tools have been proposed such as fuzzy sets (Zadeh 1965), intuitionistic fuzzy sets (Atanassov 2012), rough sets (Pawlak Communicated by V. Loia.
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1982) and others (Liu 2010). Here of much interest is the fuzzy environment, where the opinions of DMs could be evaluated by using a fuzzy number (Dubois 2011; Kacprzak 2019; Qin and Liu 2019; De et al. 2020; Garg and Chen 2020). It is noted that interval numbers are based on the idea that the distribution of DMs’ judgments is uniform. A triangular fuzzy number means that there is a centroid in the opinions of DMs. A trapezoidal fuzzy number is a generalized case of interval numbers and triangular fuzzy numbers (Wu et al. 2020; Zhou et al. 2017). A Gaussian fuzzy number implies that the distribution of DMs’ opinions in an interval is a normal function (Chen et al. 2019). One can see that different fuzzy numbers describe the different situations of the judgments of DMs. In the present study, it is suppo
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