A k-product uncapacitated facility location problem

Abstract

A k-product uncapacitated facility location problem can be described as follows. There is a set of demand points where clients are located and a set of potential sites where facilities of unlimited capacities can be set up. There are k different kinds of products. Each client needs to be supplied with k kinds of products by a set of k different facilities and each facility can be set up to supply only a distinct product with a non-negative fixed cost determined by the product it intends to supply. There is a non-negative cost of shipping goods between each pair of locations. These costs are assumed to be symmetric and satisfy the triangle inequality. The problem is to select a set of facilities to be set up and their designated products and to find an assignment for each client to a set of k facilities so that the sum of the setup costs and the shipping costs is minimized. In this paper, an approximation algorithm within a factor of $2k+1$ of the optimum cost is presented. Assuming that fixed setup costs are zero, we give a $2k-1$ approximation algorithm for the problem. In addition we show that for the case $k=2$, the problem is NP-complete when the cost structure is general and there is a 2-approximation algorithm when the costs are symmetric and satisfy the triangle inequality. The algorithm is shown to produce an optimal solution if the 2-product uncapacitated facility location problem with no fixed costs happens to fall on a tree graph.

Introduction

In the classical simple plant location or 1-product uncapacitated facility location problem, we have to select a set of facilities to be set up and a set of clients for each facility to service so as to minimize the total cost of setting up the facilities and servicing the clients. In the last few years, a number of constant factor approximation algorithms have been proposed for this problem when the service cost is assumed to be in the metric space by Aardal et al., 1999, Charikar et al., 1999, Charikar and Guha, 1999, Guha and Khuller, 1999, Jain and Vazirani, 1999, Shmoys et al., 1997. That is, the service cost is assumed to be symmetric and satisfy the triangle inequality.

The first heuristic algorithm with a performance guarantee of $\frac{3}{1-e^{-3}}$ was given by Shmoys et al. (1997), which is based on a linear program (LP) rounding algorithm extended from the filter technique of Lin and Vitter, 1992a, Lin and Vitter, 1992b. Coupling with a local search phase with the LP rounding, Guha and Khuller (1999) improved the factor to be 2.408. Later Chudak and Shmoys (2003) further strengthened the LP rounding approach to obtain a $1+\frac{2}{e}$-approximation algorithm. This idea has since been extended by Aardal et al. (1999) for the k-level facility location problem. Another interesting and elegant approach to obtain a constant factor approximation is via the primal–dual algorithm proposed by Jain and Vazirani (1999). This idea is extended to the two-level facility location problem where a performance guarantee of 1.77 was obtained by Zhang (2004). Jain et al. (2002) gave a simple greedy algorithm for the 1-product facility location problem with a performance guarantee of 1.61. This ratio was improved to 1.52 by Mahdian et al. (2002), which is close to the lower bound 1.463 proved by Guha and Khuller (1999).

In practice, a client usually requires more than one product to be supplied by several sources. This problem can be referred as multiproduct uncapacitated facility location problem or k-product uncapacitated facility location problem (k-PUFLP). It was mentioned as a special case of the submodular set function by Fisher et al. (1978). Various optimal and heuristic algorithms for the k-PUFLP have previously been considered in the general space by Klincewicz et al., 1986, Klincewicz and Luss, 1987.

For the multiproduct capacitated facility location problem, it has been studied by Lee, 1991, Lee, 1993, Mazzola and Neebe, 1999.

In this paper we consider the k-PUFLP and present heuristic algorithms for a variety of k-PUFLPs in the metric space. In Section 2, we formulate the k-PUFLP. In Section 3, we consider the k-product uncapacitated facility location problem with no fixed costs (k-PUFLPN). We show that the 2-PUFLPN is NP-complete when the cost structure is assumed to be general without the metric space property and a heuristic algorithm with a performance guarantee of 2 is presented when the costs are assumed to satisfy the metric space property. It is shown that the algorithm produces an optimal solution if the 2-PUFLPN happens to fall on a tree graph. For the k-PUFLPN ($k⩾3$), we present a heuristic algorithm with a performance guarantee of $2k-1$. In Section 4, we give a heuristic algorithm with a performance guarantee of $2k+1$ for the k-PUFLP when k is fixed. In Section 5 we conclude the paper by suggesting some questions for future research.