Conditional median as a robust solution concept for uncapacitated location problems

Abstract

While making location decisions, one intends to increase effects (reduce distances) for the service recipients (clients). A conventional optimization approach to location problems considers only the optimality of locational decisions for specific clients data. Real-world applications inevitably involve errors and uncertainties in the operating conditions, and thereby the resulting performance may be lower than expected. In particular, a distribution system design is very sensitive to the varying demands for goods, and the demand changes may deteriorate drastically the system efficiency when optimized for different demand structure. Several approaches have been developed to deal with uncertain or imprecise data. The approaches focused on the quality or on the variation (stability) of the solution for some data domains are considered robust. Frequently, uncertainty is represented by limits (intervals) on possible values of demand weights varying independently rather than by scenarios for all the weights simultaneously. In this paper we show that a solution concept of the conditional median can be used to optimize effectively such robust location problems. The conditional median is a generalization of the minimax solution concept extended to take into account the number of services (the portion of demand) related to the worst performances. Namely, for a specified portion of demand, we take into account the corresponding portion of the maximum results, and we consider their average as the worst conditional mean to be minimized. Similar to the standard minimax approach, the minimization of the worst conditional mean can be defined by a linear objective and a number of auxiliary linear inequalities.