On solving large instances of the capacitated facility location problem

Abstract

We present two sets of results pertaining to the solution of capacitated facility location problems that are large, especially with regard to the number of customers. One set of results relates to customer aggregation, while another set of results concerns the judicious selection of variable-upper-bounding (VUB) constraints to include in the initial integer-programming formulation.

In many real-world instances of facility location problems, cities and towns define ‘customers’ and their ‘demands’. Such problems typically feature large metropolises that have numerous satellite townships whose total population is exceeded (often, greatly) by that of the associated metropolis. We argue that both sets of our results would be relevant in solving such problems. We discuss our computational experiences with reference to a real-world variant of the classical capacitated facility location problem that spurred the results reported here.

Introduction

In the capacitated facility location problem (CFLP), we have a set of customers with known demands of a certain commodity as well as a set of potential sites where facilities that satisfy the demands of the customers can be located. The capacity of the facility at each potential site is known. The problem is to determine the set of facilities to open as well as the manner in which the demand of each customer is satisfied without violating the capacity restriction of any facility, so that the total cost is minimized.

The polyhedral structures of the CFLP (and/or variants thereof) have been studied by Aardal et al., 1995, Aardal et al., 1996, Chen and Guignard, 1998, Deng and Simchi-Levi, 1993, Leung and Magnanti, 1989. Aardal (1998) provided separation algorithms and tested a strong fractional cutting-plane algorithm for the problem.

The present study has been motivated by a real-world exercise on capacitated facility location that the author has conducted; Sankaran and Raghavan (1997) reported an earlier stage of the study, in which the sites and customers respectively numbered 18 and 449.

Customer aggregation is a common device for managing the size of IP formulations for solving the CFLP and its variants (see for instance, Simchi-Levi et al., 2003); nevertheless, it necessitates an approximation of the true cost of servicing customer demand. For two continuous location models, the n-center and the n-median location problems introduced by Hakimi, 1964, Hakimi, 1965, Francis et al., 1996, Rayco et al., 1997 respectively developed aggregation schemes that exploited the underlying structures of the models. The CFLP is, in contrast, a discrete location model.

The disaggregate IP formulation of the CFLP, which contains variable-upper-bounding (VUB) constraints, is much tighter than the aggregate formulation. However, the VUB constraints are numerous, the more so for progressively large instances of the CFLP. In terms of applying a polyhedral approach for solving very large, real-world instances of the CFLP, a small subset (or even none) of the numerous VUB constraints could be included initially. The rest of the VUB constraints could be included subsequently on-the-fly, as and when needed, possibly along with any other strong fractional cutting-planes which might be yielded through separation algorithms, such as those featured in MINTO (Savelsbergh et al., 1994) and in Aardal (1998).

Two questions emerge from the above discussion. Given a large instance of the CFLP,

• How should one aggregate customers to compress the instance to a pre-specified size while controlling the extent to which the cost function is approximated?

• Which manageably small subset of the VUB constraints should one include at the outset of the solution of the instance of the CFLP so that the resulting duality gap is as small as possible?

Note the two questions speak to divergent means of pursuing the common goal of tractability. The first issue addresses the minimization of the error that customer aggregation introduces, while the second issue addresses the suspension of a large subset of VUB constraints that are not very critical to the closure of the duality gap.

In this note, we establish theoretical results that speak to the two issues listed above (proofs of all the results are presented in Appendix A). We begin by presenting the notation and the formulation. Subsequently, we present theoretical results, as also a heuristic for customer aggregation. Both sets of results, namely those pertaining to customer aggregation and those pertaining to the suspension of VUB constraints, are relevant in any real-world instance of facility location problems in which cities and towns define ‘customers’ and their ‘demands’. We present computational results concerning the first set of results with reference to a real-world variant of the CFLP.