The Scalar Mean Chance and Expected Value of Regular Bifuzzy Variables

Abstract

As a natural extension of the fuzzy variable, a bifuzzy variable is defined as a mapping from a credibility space to the collection of fuzzy variables, which is an appropriate tool to model the two-fold fuzzy phenomena. In order to enrich its theoretical foundation, this paper explores some important measures for regular bifuzzy variables, the most commonly used type of bifuzzy variables. Firstly, we introduce the regular bifuzzy variables’ mean chance measure and some properties, including self-duality and its calculation formulas. Furthermore, we also investigate the mean chance distribution for strictly monotone functions of regular bifuzzy variables based on the proposed operational law. Finally, we present the expected value operator as well as equivalent analytical formulas of the expected value of regular bifuzzy variables and their strictly monotone functions.

Fuzzy Rough Programming Models: An Expected Value Perspective

“…Then, its expected value can be derived as E[ ζ] = (a + 2b + c)/4 from Definitions 2 and 3. In addition to the special case b − a = c − b, we call ζ a symmetric fuzzy number [5] (note that the symmetric fuzzy number in [5] is defined as a fuzzy variable with not only the same left and right shape functions but same left and right spreads). Definition 4 (Liu [30]).…”

“…Whereafter, relying on the classical mathematical programming methods or the hybrid algorithms, these models can be well settled. However, in many practical scenarios, it is difficult to offer the coefficients any accurate values due to the reality that some of the related data are incomplete, inexistent, or unavailable [4,5]. From the optimization point of view, this opens a new field of research called "uncertain programming".…”