Fuzzy Arithmetic for LR Fuzzy Numbers with Applications to Fuzzy Programming

LR fuzzy numbers are quite common in real world applications involving the triangular fuzzy number, the Gaussian fuzzy number, and the Cauchy fuzzy number as special cases. In this paper, we first prove that a fuzzy number is an LR fuzzy number if and only if its credibility distribution is strictly increasing. After that, based on credibility measure, an operational law on independent LR fuzzy numbers is proposed for fuzzy arithmetic, providing a novel approach to analytically and exactly calculating the inverse credibility distribution of some specific arithmetical operations. As an application of the operational law, an equivalent form of the expected value operator as well as a theorem for computing the expected value of strictly monotone functions is suggested. Finally, we utilize the operational law to construct a solution framework of fuzzy programming with parameters of LR fuzzy numbers, and such type of fuzzy programming problems can be handled by the operational law as the classic deterministic programming without any particular solving techniques.