An Extensive Operational Law for Monotone Functions of LR Fuzzy Intervals with Applications to Fuzzy Optimization

The operational law put forward by Zhou et al. on strictly monotone functions with regard to regular LR fuzzy numbers makes a valuable push to the development of fuzzy set theory. However, its applicable conditions are confined to strictly monotone functions and regular LR fuzzy numbers, which restricts its application in practice to a certain degree. In this paper, we propose an extensive operational law that generalizes the one proposed by Zhou et al. to apply to monotone (but not necessarily strictly monotone) functions with regard to regular LR fuzzy intervals (LR-FIs), of which regular fuzzy numbers can be regarded as particular cases. By means of the extensive operational law, the inverse credibility distributions (ICDs) of monotone functions regarding regular LR-FIs can be calculated efficiently and effectively. Moreover, the extensive operational law has a wider range of applications, which can deal with the situations hard to be handled by the original operational law. Subsequently, based on the extensive operational law, the computational formulae for expected values (EVs) of LR-FIs and monotone functions with regard to regular LR-FIs are presented. Furthermore, the proposed operational law is also applied to dispose fuzzy optimization problems with regular LR-FIs, for which a solution strategy is provided, where the fuzzy programming is converted to a deterministic equivalent first and then a newly-devised solution algorithm is utilized. Finally, the proposed solution strategy is applied to a purchasing planning problem, whose performances are evaluated by comparing with the traditional fuzzy simulation-based genetic algorithm. Experimental results indicate that our method is much more efficient, yielding high-quality solutions within a short time.