Convergence of optimal solutions about approximation scheme for fuzzy programming with minimum-risk criteria

This paper presents a class of two-stage fuzzy programming with minimum-risk criteria in the sense of Value-at-Risk (VaR). Since the proposed two-stage fuzzy minimum-risk problem (FMRP) often includes fuzzy variable coefficients defined through possibility distributions with infinite supports, it is inherently an infinite-dimensional optimization problem that can rarely be solved directly. Thus, algorithm procedures for solving such an optimization problem must rely on soft computing and approximation schemes, which result in a finite-dimensional optimization problem. In this paper, we develop an approximation method to compute the objective function of the two-stage FMRP, and discuss the convergent results about the use of the approximation method in FMRP, including the convergence of the objective value, optimal value, and the optimal solutions. To apply the convergent results about the approximation method, we consider a two-stage fuzzy facility location-allocation (FLA) problem with VaR objective, and solve the problem indirectly by solving its approximating problem. Since the approximating fuzzy FLA problem is neither linear nor convex, conventional optimization algorithms cannot be used to solve it. In this paper, we design a hybrid particle swarm optimization (PSO) algorithm to solve the approximating fuzzy FLA problem. One numerical example with five facilities and ten customers is also presented to demonstrate the effectiveness of the designed algorithm.