A method to analyze the rank reversal problem in the ELECTRE II method. (July 2021) - Liu, Xianliang
Highlights: Some properties regarding the optimal alternatives are obtained in the situation that there is only one type of outranking relations. When deleting a non-optimal alternative does not affect the denominator of the discordance index, two the methods are proposed to analyze and precisely predict the occurrence of rank reversal if there is only one type of outranking relations. If there are two types of outranking relations, two combination relation digraphs are proposed. We prove that the rankings of the alternatives obtained from the combination relation digraph are the same as those of the alternatives obtained from the two relation digraphs of the strongly and weakly outranking relations via the ascending distillation. There has the same conclusion in the descending distillation. When deleting a non-optimal alternative does not affect the denominator of the discordance index, a necessary and sufficient condition to decide whether the rank reversal occurs or not is obtained in the combination relation digraph, if there are two types of outranking relations. When the denominator of the discordance index changes after a nonoptimal alternative is deleted, an efficient method is proposed to decide whether the optimal alternative set changes or not. Abstract: In multi-criteria decision-making (MCDM) methods, the rank reversal is a highly important problem for analyzing the reliability of MCDM methods. The main objective of this paper is to analyze and precisely predict the occurrence of rank reversal in the ELECTRE II method. First, some useful properties of the optimal alternatives are obtained in the descending and ascending distillation processes. Second, two new concepts named combination relation digraph and numerator matrix are proposed to precisely analyze and greatly simplify the rank reversal in the ELECTRE II method. Third, all the reasons for the rank reversal in the ELECTRE II method are revealed more completely. Fourth, the necessary and sufficient conditions are proposed for determining whether or not the rank reversal occurs in the ELECTRE II method. Additionally, the proposed methods in this paper are valid for analyzing the rank reversal in the ELECTRE method with discriminating thresholds. Finally, five examples are used to evaluate the correctness and effectiveness of the methods that are proposed in this paper.