Ranking trapezoidal fuzzy numbers using a parametric relation pair

Abstract

Huynh et al. introduced a probability-based fuzzy relation for comparing fuzzy numbers (see V. Huynh, Y. Nakamori and J. Lawry (2008) [40]), but they did not detail how to compute it. Here, we will consider this fuzzy relation as a probability-based preference intensity index and present closed formulas for the integrals needed to compute this index for fuzzy sets that have trapezoidal membership functions. Also, we will propose an algorithm to compute this index and a numerical method to approximate it. The comparison of two fuzzy numbers should also be able to capture the situation where the order of the fuzzy numbers cannot be judged; and so, their order may be considered as being indifferent. Here, using the probability-based preference intensity index, we will introduce two crisp relations, which have a common parameter, over a collection of fuzzy sets that have trapezoidal membership functions. Next, we will show that - depending on the parameter value - one of them is a strict order relation and the other one may be interpreted as a relation that expresses the order indifference of fuzzy numbers. We will call this latter one the order indifference relation. Lastly, we will demonstrate how these two relations can be utilized to rank a collection of fuzzy sets that have trapezoidal membership functions.

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