The additive consistency measure of fuzzy reciprocal preference relations
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ORIGINAL ARTICLE
The additive consistency measure of fuzzy reciprocal preference relations Yejun Xu1 · Xia Liu1 · Huimin Wang1,2
Received: 9 November 2015 / Accepted: 10 January 2017 © Springer-Verlag Berlin Heidelberg 2017
Abstract Fuzzy reciprocal preference relations (FPR) are one of the most common preference relations which decision makers (DMs) express their comparison information in decision making, and the consistency of preference relations is an important step for reasonable and reliable decision making. Based on the concept of deviation between two matrices, we develop some consistency measures for FPRs to ensure that the DMs are being neither random nor illogical. A consistency index (CI) and the threshold (CI) of FPRs are defined to measure whether a FPR is of acceptable consistency. For FPRs with unacceptable consistency, an optimization method and two iterative algorithms are presented to improve its consistency and the process terminates until the CI is controlled within the threshold CI. Furthermore, one of algorithms is extended to handle group decision making (GDM) of FPRs. Finally, two examples and comparative analysis are furnished to demonstrate the effectiveness of the developed methods. Keywords Fuzzy reciprocal preference relation · Additive consistency · Group decision making
* Yejun Xu [email protected] 1
Business School, Hohai University, Nanjing 211100, People’s Republic of China
2
State Key Laboratory of Hydrology‑Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, People’s Republic of China
1 Introduction Preference relation [6, 18, 35] is one of the most popular tools in decision processes, which uses pairwise comparison of alternatives to express the DMs’ preference information [18]. At present, there are different formats of preference relations: multiplicative preference relations (MPRs) [18, 26, 35], fuzzy reciprocal preference relations (FPRs) [8, 17, 21, 24], linguistic and extensions [11, 19, 38]. Due to the DMs may have different cultural and education backgrounds, personal habits and the vague nature of human judgment, the use of FPRs to express DMs’ preferences have received great attention in recently years [4, 10, 13, 16, 27, 34, 36]. One of the important problems of preference relations is the consistency of the information provided by the experts. Because consistency measures can be ensured that the DMs are being neither random nor illogical in his or her pairwise comparisons, and consistency in preference relations given by DMs has a direct impact on the ranking results of the finally decision [14]. If lack of consistency, it will lead to incredible result and the ranking result of alternatives is misleading [10, 20]. In practical situations, there are two main reasons that make consistency difficult to achieve: (1) the decision problem is complicated and the DMs lack knowledge or experience, and (2) the preference relations are based on pairwise comparisons of alternatives, instead of being handled as a whole set of alternatives
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