On the Extent to Which Certain Fixed-Charge Depot Location Problems Can be Solved by LP

TLDR: Computational results are reported to show that linear programming often produces integer solutions to uncapacitated problems as required, and it is suggested that this represents a practical solution approach.

Abstract: This paper considers a class of feasible set fixed-charge depot location problems which have been formulated as mixed-integer programmes. Computational results are reported to show that linear programming often produces integer solutions to uncapa- citated problems as required. It is suggested that this represents a practical solution approach. Computational evidence suggests this convenient property does not extend to capacitated problems. Discussion of reducing infinite set problems to such feasible set problems is included. This paper considers depot location-demand allocation problems where loca- tions are to be chosen from a finite set of candidate sites. This corresponds to the "feasible set approach", discussed by Rand.' The objective is to locate depots so that all customer demand is allocated among the depots while minimizing the sum of variable and fixed depot costs associated with satisfy- ing that demand. Demand is assumed known in each of a number of customer zones. The modelling intent is to answer the four fundamental questions listed by Rand as: How many depots should there be? Where should they be? Which customers should they serve? How big should they be? A mixed integer programming model-sometimes called the fixed-charge plant loca- tion model is used for which efficient special purpose algorithms exist (see Elshafei2 and Geoffrion and Graves3). However, it is a nontrivial task to develop the necessary computer programs. The purpose of this paper is to provide computational results which indicate that ordinary linear program- ming typically produces integer solutions to uncapacitated problems. It is suggested that this represents a practical solution approach if the problem size is not too large, but computational results show that this approach seems inappropriate for capacitated problems. Discussion of reducing certain infinite set problems to feasible set problems is presented in an Appendix.