Solving fuzzy (stochastic) linear programming problems using superiority and inferiority measures

An inexact optimization model for regional energy systems planning in the mixed stochastic and fuzzy environment

“…According to Zimmerman [80] and Van Hop [74], letT be a family of triangular fuzzy numbers, which can be defined as follows:…”

“…Therefore, the total superiority ofQ overP is defined as the area ofQ larger thanP. Mathematically, this area is presented as follows [74]:…”

“…where F 0 ðRÞ andX a $ stand for the set of fuzzy numbers with compact supports and the a-cut level of the fuzzy setX $ , respectively.~ X is a fuzzyrandom variable if and only if given w 2 O,X a $ is a random interval and a 2 ð0; 1 [66,74]. This also represents that~ X exists in the mixed stochastic and fuzzy environment.…”

“…According to Van Hop [74], for two triangular fuzzy-random variables~ Pp~ Q, the superiority of fuzzy-random variable~ Q over~ P is…”

“…Moreover, they could not handle dual uncertainties such as the mixture of fuzziness and randomness, resulting in the loss of valuable information. In order to address the above limitations, Van Hop [74] proposed a superiority-inferiority-based fuzzystochastic programming (SI-FSP) approach as an extension of the existing FMP methods. Through this method, uncertainties expressed as fuzzyrandom variables in objective function and/or system constraints can be effectively handled; onerous computational efforts as required by the conventional FMP methods can be greatly reduced.…”