Intelligent-guided adaptive search for the maximum covering location problem

doi:10.1016/j.cor.2016.08.018

Computers & Operations Research

Volume 78, February 2017, Pages 129-137

https://doi.org/10.1016/j.cor.2016.08.018 Get rights and content

Highlights

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IGAS has a learning stage based on the mapping of solutions into a GNG network.
•
IGAS provides an efficient solution to the maximum covering facility location problem.
•
IGAS was better than the solutions obtained by popular methods from the literature.

Abstract

Computational intelligence techniques are part of the search process in several recent heuristics. One of their main benefits is the use of an adaptive memory to guide the search towards regions with promising solutions. This paper follows this approach proposing a variation of a well-known iteration independent metaheuristic. This variation adds a learning stage to the search process, which can improve the quality of the solutions found. The proposed metaheuristic, named Intelligent-Guided Adaptive Search (IGAS), provides an efficient solution to the maximum covering facility location problem. Computational experiments conducted by the authors showed that the solutions found by IGAS were better than the solutions obtained by popular methods found in the literature.

Introduction

Facility location is a special class of problems whose goal is to locate a limited number of facilities that fulfill particular constraints. These constraints are primarily dictated by practical necessities, such as a reasonable public coverage. One of these problems, the maximum covering location problem (MCLP) [1], has been fairly studied in the literature [1], [2], [3]. The MCLP addresses the identification of the best sites to locate a pre-defined number of facilities, covering the maximum possible number of customers. Since each facility has a coverage area, each customer must be solely addressed by the facilities that cover its area. Garey and Johnson [4] proved that this problem is NP-hard.

The MCLP can deal with several applications, ranging from environmental monitoring to the management of transport networks [5], [6]. Many variants of the MCLP have been investigated for different applications. For example, Lee and Lee [7] developed a tabu search-based heuristic to solve a hierarchical covering location problem. This adaptation of the MCLP to meet more than one type (level) of related service (facility) was introduced in [8] for a health care system design. The facilities can be hospitals and clinics. In the hierarchy, clinics belong to a level inferior to hospitals, since they cover a restricted set of patients, due to their more specialized services. Thus, each clinic can attend a specific set of patient demands. As a result, the different types of facilities should be located in regions that guarantee the maximum coverage of the services, taking into account the availability of resources.

The interest in the classical MCLP has steadily reduced in the last decades. The main reason is the development of exact solvers [9] able to solve various instances of this problem. Nevertheless, for large-scale applications, MCLP remains a challenge. For these applications, the most recent version of a particular solver, the CPLEX package [9], has problems when dealing with large search trees. To the best of the authors' knowledge, only a small number of attempts have successfully solved large-scale instances of the MCLP [2], [3].

This paper proposes and investigates a novel Intelligent-Guided Adaptive Search metaheuristic (IGAS) to solve large-scale instances of the MCLP. The proposed metaheuristic is based on the Greedy Randomized Search Procedure (GRASP) [10], [11], [12]. GRASP is an iterative strategy with two main steps: a semi-greedy construction step and a local search step. Nonetheless, different from GRASP, the construction phase of IGAS is based on an artificial neural network, the Growing Neural Gas (GNG) network [13]. GNG was chosen because it is easy to integrate with GRASP, since it can conveniently map solutions in its structure [14], [15]. This mapping allows the creation of a memory not present in the independent iterations of GRASP, which can be used in future iterations.

To evaluate the performance of IGAS, two sets of experiments were carried out. Both experiments compare IGAS with the GRASP proposed in [12] and CPLEX v.12.6, within the time limit of 20,000 s. For the first set of experiments, new instances, larger than the existing benchmark instances, based on real data, were generated. The second set of experiments used the same instances used in the evaluation of most heuristics found in the literature. To assess the quality of the solutions in the second set of experiments, the gaps between the heuristic solutions and their linear relaxations are reported. The experimental results show that the value obtained by IGAS was competitive and smaller than the average gap obtained by GRASP and CPLEX.

The remaining of this paper is organized as follows. Section 2 briefly presents related studies. Section 3 introduces a mathematical formulation for the MCLP, proposed in [1], which is adopted in this paper. Section 4 describes how the MCLP was approached with the proposed method, the IGAS metaheuristic. Section 5 presents the computational experiments, details the behavior of IGAS and analyzes the quality of the solutions obtained. Finally, Section 6 summarizes the main contributions from this work and presents the final remarks.

Section snippets

Related works

The MCLP emerged in the late 1970s, motivated by the necessity of designing better models to address the location of facilities in the public sector. Different from the private services, where the maximum profit is usually the main goal of the decision makers, the priorities behind the decision of locating public facilities lie on public service reasons, such as the minimization of the total distance between the facilities and the users. More importantly, the location of the facilities must

The MCLP formulation

Among the possible formulations for the MCLP, this paper presents its original form, proposed by Church and ReVelle [1]: $Maximize \sum_{c_{i} \in N} k_{i} x_{i}$ subject to: $\sum_{j \in S_{i}} y_{j} \geq x_{i} \forall c_{i} \in N$ $\sum_{l_{j} \in M} y_{j} = p$ $x_{i} \in {0, 1} \forall c_{i} \in N$ $y_{j} \in {0, 1} \forall l_{j} \in M$ where:

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$N = {c_{1}, c_{2}, \dots, c_{n}}$ is the set of customers (demand points) to be addressed.
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$M = {l_{1}, l_{2}, \dots, l_{m}}$ is the set of possible facility sites.
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p is the number of facilities to be located ( $p \leq m)$ .
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k_i is the demand of customer c_i.
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x_i is a binary variable that indicates whether or not the demand of the customer c_i is

Proposed solution method

This paper proposes a variant of GRASP, named Intelligent Guided Adaptive Search (IGAS), to solve the MCLP for large-sized instances. Greedy Randomized Adaptive Search Procedure (GRASP), an iterative metaheuristic, was first proposed in [10], and has been widely employed to solve optimization problems [27], [28]. It has virtually two phases in each of its iterations: a construction phase and a local search phase. The former is a semi-greedy algorithm that seeks for feasible solutions. The

Computational experiments

The performance of IGAS was evaluated by two sets of experiments. These experiments compared IGAS with GRASP and CPLEX using instances with more than 2700 nodes. Both IGAS and GRASP were programmed in Java. The experiments were run on a cluster with 104 nodes, each node with 2 Intel Xeon E5-2680v2 processors of 2.8 GHz, 10 cores and 128 GB DDR3 1866 MHz of RAM memory.

In the first set of experiments, 24 instances using the same methodology as in [36] were created. The data, available at //www.census.gov

Final remarks

The main contribution of this paper is the proposal and investigation of a novel metaheuristic, variation of GRASP, named IGAS. The introduced solution method incorporates a GNG network to GRASP to further explore promising regions in the solution space. The key difference from GRASP is that IGAS includes an adaptive memory able to learn from solutions found in past iterations, to influence the solutions to be found in future iterations.

To evaluate the performance of the proposed strategy, this

Acknowledgments

The authors thank Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) (Proc. 2015/21660-4, 2010/20231-9) for the financial support. In the experiments carried out for this research, the authors used computational resources from the Center of Applied Mathematics to Industry (CeMEAI) supported by FAPESP (Proc. 2013/07375-0). They also thank the reviewers' valuable contribution, which helped to improve the quality of the paper.

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Díaz et al. [22] developed a heuristic algorithm by integrating a greedy randomized adaptive search procedure and a Tabu search algorithm to solve MCLPs with customer preference ordering. Máximo et al. [23] presented an intelligent-guided adaptive search algorithm for MCLPs. Cordeau et al. [24] addressed an effective Benders decomposition algorithm to solve large-scale MCLPs based on the reformulation of branch and Benders cuts.
It is essential to design a product service system with the adequate location of maintenance facilities for guaranteeing the fast and dependable response to failures of extremely busy agricultural machinery during harvest time. This paper aims to optimize where to locate/relocate two hierarchical levels of facilities respectively composed of maintenance stations and mobile service fleets over multiple periods of a harvest season to support the safe and stable operation of agricultural machinery. We formulate a multi-objective dynamic covering location problem for these hierarchical facilities that seeks to maximize the total demand covered within some response radii while minimizing the modification to facility locations between the periods. The augmented $ɛ$ -constraint method based on lexicographic optimization is then developed to obtain a set of non-inferior solutions that allow us to evaluate the trade-off between multiple decision objectives. Lastly, a real-life problem is handled to illustrate the effectiveness of our multi-objective decision model and solution method. Computational results recommend the optimal decision of facility locations for implementation, examine the sensitivity analysis of selected parameters, and compare the solution obtained from the proposed model with the period-specific solutions that take no account of the changing locations of facilities between the periods.
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The rest of the article is organized as follows: the related works are discussed in Section 2. The proposed problem is formally defined along with its mathematical formulation in Section 3, where it is shown how the formulation mentioned in this article is different from the existing formulation of the traditional single objective MCLP mentioned in Atta et al. (2018), Church and ReVelle (1974), Fazel Zarandi, Davari, and Haddad Sisakht (2011) and Máximo, Nascimento, and Carvalho (2017). In Section 4, a brief description of the harmony search algorithm (HSA) is given.
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A critical characteristic of the MCLP is that the number of covering resources is fixed and is given in advance. The optimization objective is to maximize the total covered reward (Maximo, Nascimento, & De Carvalho, 2017). Numerous studies have been conducted on the MCLP and its variants, such as those by Current and Storbeck (1988), Pirkul and Schilling (1990) and Haghani (1996), who investigate a capacitated MCLP; Kerkkamp and Aardal (2016), who research a weighted MCLP; Revelle and Hogan (1989), who examine a probabilistic MCLP; Berman and Krass (2002) and Berman, Krass, and Drezner (2003), who study a partial MCLP; and Murray and Tong (2007), who research a continuous MCLP.
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Intelligent-guided adaptive search for the maximum covering location problem

Highlights

Abstract

Introduction

Section snippets

Related works

The MCLP formulation

Proposed solution method

Computational experiments

Final remarks

Acknowledgments

Pap Reg Sci Assoc

Comput Ind Eng

Oper Res Lett

Neural Netw

Oper Res Lett

Comput Oper Res

Comput Ind Eng

Omega

Eur J Oper Res

Eur J Oper Res

Comput Oper Res

Comput Oper Res

Comput Ind Eng

A decomposition heuristic for the maximal covering location problem

Adv Oper Res

Iterated greedy algorithms for the maximal covering location problem

Computers and intractability: a guide to the theory of np-completeness

Recent applications of the maximal covering location planning (MCLP) model

J Oper Res Soc

A review of covering problems in facility location

Locat Sci

The hierarchical service location problem

Manag Sci