The relationship between the minimum-variance and minimax disparity RIM quantifier problems

Recently, Liu and Lou [On the equivalence of some approaches to the OWA operator and RIM quantifier determination, Fuzzy Sets and Systems 159 (2007) 1673–1688] investigated the equivalence of solutions to the minimum-variance and minimax disparity RIM quantifier problems. However, their proofs are very sensitive to the assumption, and some are mathematically incomplete. In this regard, this paper provides a counterexample of the minimax disparity RIM quantifier problem for the case in which generating functions are continuous. The paper also provides a correct proof of the minimax disparity RIM quantifier problem for the case in which generating functions are absolutely continuous and a generalized result for the minimum-variance RIM quantifier problem for the case in which generating functions are Lebesgue integrable. Based on the results, the paper provides a correct relationship between the minimum-variance and minimax disparity RIM quantifier problems.