A branch-and-cut algorithm for the discrete (r∣p)-centroid problem

Abstract

The environment of the (r∣p)-centroid problem is composed of two noncooperative firms, a leader and a follower, competing to serve the demand of customers from a given market. The demand of each customer is totally served by a facility of the leader or follower according to a customer choice rule. The goal of both the leader and the follower is to maximize its own market share. The (r∣p)-centroid problem consists of deciding where the leader should place p facilities knowing that the follower will react by placing r facilities. The discrete version of the problem is a ${\sum }_2^p$-hard one, where both the applicant facilities and the customers are nodes on a graph. In spite of it, we present an integer programming formulation with polynomially many variables and exponentially many constraints. Moreover, we report several experiments with different number of customers and applicant facilities and different values of p and r. Our results show that our method requires less computational time than the two exact algorithms found in the literature, being able to optimally solve 29 previously open instances with up to 100 customers, 100 applicant facilities and p = r = 15.

Introduction

In competitive location models, two or more noncooperative firms compete to provide customers from a given market. Each customer is partially or totally served by facilities placed by the firms according to a customer choice rule. Different objective functions can be pursued by each firm when it decides where to place its facilities. For example, they might aim to maximize their own market share, their own profit, their own number of customers served, among others. When the decision is made sequentially, the model is called sequential competitive location model. A review of competitive location models and of sequential ones can be found in Eiselt and Laporte, 1989, Friesz et al., 1988, Eiselt et al., 1993, Eiselt and Laporte, 1997, Kress and Pesch, 2012.

In this paper, we focus on the discrete (r∣p)-centroid problem which is a sequential competitive facility location problem where the demand of each customer is totally served by the closest facility placed by either the first firm to enter the market (leader) or the second one (follower). The leader places p facilities on a graph knowing that the follower will react by placing r facilities. The goal of both the leader and the follower is to maximize its own market share. The problem of deciding the leader’s strategy is called discrete (r∣p)-centroid problem (Hakimi, 1983).

In Voting theory, there is an interesting problem that can be modeled as a discrete (p∣p)-centroid problem. This problem consists of finding a p-simpson solution in a graph, i.e., a set S of p nodes that minimizes the maximum number of nodes closer to another set of p nodes than to S. For more details, see Campos Rodrı´guez and Moreno Perez (2008).

Noltemeier et al. (2007) proved that the discrete (r∣p)-centroid problem is ${\sum }_2^{\mathrm{p}}$-hard. Hence, that problem turns out to be harder than any optimization problem whose decision version is in NP. The hardness of that problem comes from the fact that evaluating a single leader’s strategy requires to solve an NP-hard problem to optimize the follower’s strategy. Fortunately, optimizing the follower’s strategy often spends less than one second for instances with 100 customers and 100 applicant facilities by solving an integer programming (IP) model. Besides, the ${\sum }_2^p$-hardness is also known for the absolute (non-discrete) variant as shown in Spoerhase (2009).

Some heuristics can be found in the literature for the discrete (r∣p)-centroid problem. Benati et al. (1994) proposed a Tabu Search algorithm for the problem. A hybrid memetic algorithm was proposed by Alekseeva et al. (2009). Recently, Davydov (2012) also proposed a tabu search algorithm for the problem and tested it on several instances from the benchmark library Discrete Location Problems with up to 100 customers, 100 applicant facilities and p = r = 20.

Few researchers have proposed exact methods for the discrete (r∣p)-centroid problem. Polynomial time algorithms can be found for the discrete (1∣1)-centroid problem (Campos Rodriguez and Moreno Perez, 2003) and the discrete (r∣p)-centroid on paths and discrete (1∣p)-centroid on trees (Spoerhase and Wirth, 2009). For the problem on general graphs, we found two similar exact procedures based on an elimination process in a candidate list proposed by Rodrı´guez et al., 2010, Alekseeva et al., 2010. In the instances proposed by Rodrı´guez et al. (2010), the customers and facility coordinates are randomly distributed on a (50 × 50) grid graph G_50,50. The authors report that the average computational time consumed to solve the instances with 20 applicant facilities, 30 customers, p = 4 and r = 2 is about 50 min, while larger instances consumed more than 2 h on average. The exact procedure proposed by Alekseeva et al. (2010) optimally solved instances from the benchmark library Discrete Location Problems with up to 100 customers, 100 applicant facilities and p = r = 5. Although most instances were solved in a few hours, the most difficult one required more than two days of computation. Moreover, the authors could not solve the instances with p = r = 10.

In this paper, we propose an IP formulation for the discrete (r∣p)-centroid problem with polynomially many variables and exponentially many constraints. One should note that, since the problem is ${\sum }_2^{\mathrm{p}}$-hard, neither a polynomial formulation nor a formulation where all constraints can be separated in polynomial time is possible unless $\mathit{NP}={\sum }_2^{\mathrm{p}}$. Hence, the constraints are separated during the optimization either using a greedy heuristic or exactly solving an IP model. We test our method on two groups of instances. The first one is composed of instances randomly generated in the same way as in Rodrı´guez et al. (2010). The second group is composed of 60 instances from the benchmark library Discrete Location Problems. Those instances have 100 customers and 100 applicant facilities and different values of p and r as defined in Alekseeva et al., 2010, Davydov, 2012. Our exact method is faster than the one proposed by Alekseeva et al. (2010) for 19 out of 20 instances with p = r = 5, and allows for optimally solving 20 open instances with p = r = 10 and 9 open instances with p = r = 15.

This paper is divided as follows. In Section 2, we define the discrete (r∣p)-centroid problem and present some examples. In Section 3, we describe an IP model for the problem and propose strengthened inequalities that improves its continuous relaxation. In Section 4, we define the separation problem for the only exponential family of constraints used in our formulation, and present an IP model and a greedy heuristic for it. In Section 5, we report our experiments and compare our method to the two exact ones found in the literature. Finally, in Section 6, we summarize our conclusions.