Random Fuzzy Multiple Objective Decision Making

Since the fuzzy set was initialized by Zadeh [356], it has been applied to many fields. Later, many scholars proposed the concept of two-fold uncertain variables, combined fuzzy variables, and random variables. Two different definitions from different perspectives have been proposed. The first comes from Kwakernaak [180], who coined the term “fuzzy random variable” and who regarded that a fuzzy random variable is a random variable whose value is not real, but fuzzy number. Since Kruse and Meyer [176] worked on an expanded version of a similar model, they are often mentioned with as Kwakernaak. The second comes from Puri and Ralescu [256] who regarded fuzzy random variables as random fuzzy sets. Once again, because of Klement et al. [162], and other collaborations, these three authors often are jointly credited with the second definition. In this chapter, we mainly take Kwakernaak’s point and regard it as a random fuzzy variable (Abbr. Ra-Fu variable) which has been renamed by some scholars [32, 209, 236, 341] in order to avoid the confusion. It has been widely extended to many fields. Xu and Liu [341] discussed a class of supply chain networks optimal problems with random fuzzy shipping costs and customer demand, and proposed a random fuzzy multi-objective mixed-integer non-linear programming model to gain an optimal strategy. Xu and He [339] provided an auxiliary programming model of the random fuzzy programming, converted it to a deterministic mixed 0–1 integer programming model, of which the solutions are proved to exist and showed its efficiency by application to a supply chain problem. Zhou and Xu [363] discussed a class of integrated logistics network models under random fuzzy environments and applied them to a Chinese beer company.