α-cut fuzzy arithmetic:: Simplifying rules and a fuzzy function optimization with a decision variable

The problems of a-cut fuzzy arithmetic have been shown, like in interval arithmetic, that distinct states of fuzzy parameters (or fuzzy variable values) may be chosen and produce an overestimated fuzziness. Meanwhile, local extrema of a function may exist inside the support of fuzzy parameters and cause an underestimation of fuzziness and an illegal fuzzy number's result. Previous approaches to overcoming these problems have appeared in literature. Yet, the computational burden of these approaches became even heavier. Thus, this paper is based on the vertex method in literature and extensively proposes newly devised rules observed greatly useful for simplifying the vertex method. These rules are devised through a function partitioned into subfunctions, distinguishing the types of fuzzy parameter/variable occurrences, and types of subfunctions or functions with the various observations. The improved efficiency has been found able to significantly reduce the combination (vertex) test of the vertex method for the fuzzy parameters' alpha-cut endpoints possibly to only a few fuzzy parameters' endpoint combinations. Also as related, a procedure for the fuzzy optimization of fuzzy functions with a fuzzy blurred argument (a single variable) is examined with the vertex method as well. A proper and useful preliminary algorithm is proposed. Numerical examples with results are provided.