Heuristics for the probabilistic capacitated multi-facility weber problem

Abstract

In this study, we consider the probabilistic capacitated multi-facilityWeber problem with general distance functions and probability density functions where customer locations are assumed to be random variables. The aim of the problem is to find supply points and allocations by minimizing the expected sum of demand weighted distances. Since customer locations are random, it is not always possible to obtain analytical expressions for expected distances between facilities and customers. First, we used general distance functions and probability distributions for our problem. Afterwards, we specialized the problem using two distance functions to examine different instances: Euclidean and rectilinear distances. For the Euclidean distance, we assume that customer locations follow symmetric bivariate normal and symmetric bivariate exponential probability distributions. In addition to these two probability distributions, symmetric bivariate uniform probability distribution is also assumed for the rectilinear distance. Solving this problem to optimality with known optimization techniques is very hard since the objective function of the problem is neither convex nor concave. In order to solve the problem, we have developed four heuristic methods. The first one is the probabilistic alternating location-allocation heuristic and the other three of them depend on discrete approximations. Furthermore, we have also developed three approximation methods for calculating the expected distances for different situations. These new approaches are implemented and computational results based on extensive experiments are also provided.