An extensive operational law for monotone functions of LR fuzzy intervals with applications to fuzzy optimization

Abstract

The operational law proposed by Zhou et al. (J Intell Fuzzy Syst 30(1): 71–87, 2016) contributes to developing fuzzy arithmetic, while its applicable conditions are confined to strictly monotone functions and regular LR fuzzy numbers, which are hindering their operational law from dealing with more general cases, such as problems formulated as monotone functions and problems with fuzzy variables represented as fuzzy intervals (e.g., trapezoidal fuzzy numbers). In order to handle such cases we generalize the operational law of Zhou et al. in both the monotonicity of function and fuzzy variables in this paper and then apply the extensive operational law to the cases with monotone (but not necessarily strictly monotone) functions with regard to regular LR fuzzy intervals (LR-FIs) (of which regular LR fuzzy numbers are special cases). Specifically, we derive the computational formulae for expected values (EVs) of LR-FIs and monotone functions with regard to regular LR-FIs, respectively. On the other hand, we develop a solution scheme to dispose of fuzzy optimization problems with regular LR-FIs, in which a fuzzy programming is converted to a deterministic equivalent one and a newly devised solution algorithm is utilized to get the deterministic programming solved. The numerical experiments are conducted using our proposed solution scheme and the traditional fuzzy simulation-based genetic algorithm in the context of a purchasing planning problem. Computational results show that our method is much more efficient, yielding high-quality solutions.