On minimum-risk problems in fuzzy random decision systems

In a decision-making process, we may face a hybrid environment where linguistic and frequent imprecision nature coexists. The problem of frequent imprecision can be solved by probability theory, while the problem of linguistic imprecision can be tackled by possibility theory. Therefore, to solve this hybrid decision-making problem, it is necessary to combine both theories effectively. In this paper, we restrict our attention to this hybrid decision-making problem, where the input data are imprecise and described by fuzzy random variables. Fuzzy random variable is a mapping from a probability space to a collection of fuzzy variables, it is an appropriate tool to deal with twofold uncertainty with fuzziness and randomness in an optimization framework. The purpose of this paper is to present reasonable chances of a fuzzy random event characterized by fuzzy random variables so that they can connect with the expected value operators of a fuzzy random variable via Choquet integrals, just like the relation between the probability of a random event and the mathematical expectation of a random variable, and that between the credibility of a fuzzy event and the expected value operator of a fuzzy variable. Toward that end, we take fuzzy measure and fuzzy integral theory as our research tool, and present three kinds of mean chances of a fuzzy random event via Choquet integrals. After discussing the duality of the mean chances, we use the mean chances to define the expected value operators of a fuzzy random variable via Choquet integrals. To show the reasonableness of the mean chance approach, we prove the expected value operators defined in this paper coincide with those presented in our previous work. Using the mean chances, we present a new class of fuzzy random minimum-risk problems, where the objective and the constraints are all defined by the mean chances. To solve general fuzzy random minimum-risk optimization problems, a hybrid intelligent algorithm, which integrates fuzzy random simulations, genetic algorithm and neural network, is designed, and its feasibility and effectiveness are illustrated by numerical examples.