A hybrid multistart heuristic for the uncapacitated facility location problem

Abstract

We present a multistart heuristic for the uncapacitated facility location problem, based on a very successful method we originally developed for the p-median problem. We show extensive empirical evidence to the effectiveness of our algorithm in practice. For most benchmarks instances in the literature, we obtain solutions that are either optimal or a fraction of a percentage point away from it. Even for pathological instances (created with the sole purpose of being hard to tackle), our algorithm can get very close to optimality if given enough time. It consistently outperforms other heuristics in the literature.

Introduction

Consider a set F of potential facilities, each with a setup cost c(f), and let U be a set of users (or customers) that must be served by these facilities. The cost of serving user u with facility f is given by the distance d(u, f) between them (often referred to as service cost or connection cost as well). The facility location problem consists in determining a set S ⊆ F of facilities to open so as to minimize the total cost (including setup and service) of covering all customers:

$$\mathrm{cost}(S)=\sum _{f\in S}c(f)+\sum _{u\in U}{\min }_{f\in S}d(u\text{,}f)\text{.}$$

Note that we assume that each user is allocated to the closest open facility, and that this is the uncapacitated version of the problem: there is no limit to the number of users a facility can serve. Even with this assumption, the problem is NP-hard [8].

This is perhaps the most common location problem, having been widely studied in the literature, both in theory and in practice.

Exact algorithms for this problem do exist (some examples are [7], [25]), but its NP-hard nature makes heuristics the natural choice for larger instances.

Ideally, one would like to find heuristics with good performance guarantees. Indeed, much progress has been made in terms of approximation algorithms for the metric version of this problem (in which all distances obey the triangle inequality). In 1997, Shmoys et al. [37] presented the first polynomial-time algorithm with a constant approximation factor (roughly 3.16). Several improved algorithms have been developed since then, with some of the latest [21], [22], [29] being able to find solutions within a factor of around 1.5 from the optimum. Unfortunately, there is not much room for improvement in this area. Guha and Khuller [16] have established a lower bound of 1.463 for the approximation factor, under some widely believed assumptions.

In practice, however, these algorithms tend to be much closer to optimality for non-pathological instances. The best algorithm proposed by Jain et al. in [21], for example, has a performance guarantee of only 1.61, but was always within 2% of optimality in their experimental evaluation.

Although interesting in theory, approximation algorithms are often outperformed in practice by more straightforward heuristics with no particular performance guarantees. Constructive algorithms and local search methods for this problem have been used for decades, starting from the pioneering work of Kuehn and Hamburger [27]. Since then, more sophisticated metaheuristics have been applied, such as simulated annealing [2], genetic algorithms [26], tabu search [13], [31], [38], [39], and the so-called “complete local search with memory” [13]. Dual-based methods, such as Erlenkotter’s dual ascent [10], Guignard’s Lagragean dual ascent [17], and Barahona and Chudak’s volume algorithm [3] have also shown promising results.

An experimental comparison of some state-of-the-art heuristics is presented by Hoefer in [20] (slightly more detailed results are presented in [18]). Five algorithms are tested: JMS, an approximation algorithm presented by Jain et al. in [22]; MYZ, also an approximation algorithm, this one by Mahdian et al. [29]; swap-based local search; Michel and Van Hentenryck’s tabu search [31]; and the volume algorithm [3]. Hoefer’s conclusion, based on experimental evidence, is that tabu search finds the best solutions within reasonable time, and recommends this method for practitioners.

In this paper, we provide an alternative that can be even better in practice. It is a hybrid multistart heuristic akin to the one we developed for the p-median problem in [36]. A series of minor adaptations is enough to build a very robust algorithm for the facility location problem, capable of obtaining near-optimal solutions for a wide variety of instances of the facility location problem.

The remainder of the paper is organized as follows. In Section 2, we describe our algorithm and its constituent parts. Section 3 presents empirical evidence to the effectiveness of our method, including a comparison with Michel and Van Hentenryck’s tabu search. Final remarks are made in Section 4.