The relationship between the maximum entropy and minimax ratio RIM quantifier problems

Liu and Lou [Fuzzy Sets and Systems 159 (2007) 1673–1688] investigated the equivalence of solutions to the maximum entropy and minimax ratio RIM quantifier problems. The paper improves Liu and Louʹs theorems from a mathematical perspective. Because their theorems are not suitable for absolutely continuous generating functions, the paper provides a correct proof of the minimax ratio RIM quantifier problem for the case in which generating functions are absolutely continuous and a generalized solution to the maximum entropy 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 maximum entropy and minimax ratio RIM quantifier problems.