A consensus-based approach for multi-criteria decision making with probabilistic hesitant fuzzy information
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METHODOLOGIES AND APPLICATION
A consensus-based approach for multi-criteria decision making with probabilistic hesitant fuzzy information Jian Li1 • Li-li Niu2 • Qiongxia Chen1 • Guang Wu2
Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract As a generalized fuzzy number, probabilistic hesitant fuzzy element (PHFE) improves the flexibility for decision makers in expressing hesitant information, and it has been receiving increased attention. This study develops a multi-criteria decisionmaking (MCDM) approach that considers consensus reaching among decision makers with probabilistic hesitant fuzzy information. To obtain this aim, first, a new approach to derive normalized PHFE (NPHFE) is proposed to overcome the shortcomings in previous studies. Subsequently, a new Euclidean distance and some operations related to PHFEs are developed based on the new proposed NPHFEs. At the same time, the effectiveness and rationality of the new proposed approaches are discussed. Second, a consensus index of group with PHFEs is presented, which based on the proposed Euclidean distance of decision-makers’ evaluation information on all the criteria. Third, if the consensus level of the group does not reach the expect threshold value, an iteration algorithm is designed to improve its consensus level. Moreover, the proof of the convergence of the proposed algorithm is provided to verify its effectiveness, and a MCDM approach based on group consensus is proposed. Finally, the most comprehensive candidate selection problems are provided to demonstrate the effectiveness of the proposed MCDM approach. And a comparative study with other methods is conducted with the same illustrative example. Keywords Multi-criteria decision making Euclidean distance Group consensus Probabilistic hesitant fuzzy element
1 Introduction The consensus-reaching process plays an essential role in multi-criteria decision-making (MCDM) problems (Wu et al. 2019). Since in MCDM process, decision makers may come from varying professional backgrounds, and the opinions provided from these decision makers may very different from each other. What is more, there are maybe noncooperative decision makers (Dong et al. 2016b), dishonest decision makers (Dong et al. 2018a) and the decision makers are continuously updating their opinions in decision-making process (Dong et al. 2016a, 2017). These Communicated by V. Loia. & Qiongxia Chen [email protected] 1
School of Logistics Management and Engineering, Nanning Normal University, Nanning 530001, People’s Republic of China
2
Guangxi University Xingjian College of Science and Liberal Arts, Nanning 530005, People’s Republic of China
may adversely affect the overall efficiency of the consensus-reaching process. For this reason, the consensusreaching algorithms for handling MCDM problems have been studied deeply (Triantaphyllou et al. 2020; Ding et al. 2020; Li et al. 2019a; Wu et al. 2020; Tian et al. 2019). Such as, Uren˜a et al. (2019) developed consensus-
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