The convergent results about approximating fuzzy random minimum risk problems

Based on mean chance theory, this paper presents a new class of two-stage fuzzy random minimum risk problem (FRMRP). In order to solve the two-stage FRMRP, we use a sequence of finitely supported primitive fuzzy random variables to approximate a continuous fuzzy random variable, which result in a finite-dimensional two-stage FRMRP. We also discuss the convergence of the approximation method, including the convergence of the objective value, optimal value and optimal solutions. To illustrate the modeling idea of this paper, we apply the proposed two-stage FRMRP to the capacitated location–allocation problem with fuzzy random demands. Finally, we integrate the approximation method, neural network (NN) and particle swarm optimization (PSO) to design a hybrid PSO algorithm to solve the two-stage capacitated location–allocation problem, and provide a numerical example with five facilities and ten customers to demonstrate the effectiveness of the algorithm.