Preference implication-based approach to ranking fuzzy numbers

Dombi and Baczy'{n}ski presented a new approach to the problem of implication operation by introducing the preference implication, which has very advantageous properties. In this paper, it is presented how the preference implication is connected with soft inequalities and with sigmoid functions. Utilizing this connection the preference implication-based preference measure for two fuzzy numbers is introduced and its key properties, including the reciprocity, are described. Then, the exact expression for computing the new preference measure for trapezoidal fuzzy numbers is presented. Here, using the new preference measure, two crisp relations over trapezoidal fuzzy numbers are introduced. It is shown that one of them is a strict (but not a total) order relation, and the other one is an equivalence relation. The strict order relation can be used to rank comparable fuzzy numbers, while the equivalence relation, which we call the indifference relation, expresses that the order of some fuzzy numbers is indifferent. These two crisp relations can be used to rank a collection of trapezoidal fuzzy numbers. Lastly, the proposed ranking method is compared with some well-known existing fuzzy number ranking methods.