Ranking triangular interval-valued fuzzy numbers based on the relative preference relation

In this paper, we first use a fuzzy preference relation with a membership function representing preference degree for comparing two interval-valued fuzzy numbers and then utilize a relative preference relation improved from the fuzzy preference relation to rank a set of interval-valued fuzzy numbers. Since the fuzzy preference relation is a total ordering relation that satisfies reciprocal and transitive laws on interval-valued fuzzy numbers, the relative preference relation is also a total ordering relation. Practically, the fuzzy preference relation is more reasonable on ranking interval-valued fuzzy numbers than defuzzification because defuzzification does not present preference degree between fuzzy numbers and loses messages. However, fuzzy pair-wise comparison for the fuzzy preference relation is more complex and difficult than defuzzification. To resolve fuzzy pair-wise comparison tie, the relative preference relation takes the strengths of defuzzification and the fuzzy preference relation into consideration. The relative preference relation expresses preference degrees of interval-valued fuzzy numbers over average as the fuzzy preference relation does, and ranks fuzzy numbers by relative crisp values as defuzzification does. In fact, the application of relative preference relation was shown in traditional fuzzy numbers, such as triangular and trapezoidal fuzzy numbers, for previous approaches. In this paper, we extend and utilize the relative preference relation on interval-valued fuzzy numbers, especially for triangular interval-valued fuzzy numbers. Obviously, interval-valued fuzzy numbers based on the relative preference relation are easily and quickly ranked, and able to reserve fuzzy information.