A multi-objective formulation of maximal covering location problem with customers’ preferences: Exploring Pareto optimality-based solutions

Highlights

• A new multi-objective maximal covering location problem is proposed.

• Customers’ preferences for different potential facility locations are considered.

• A Pareto optimality-based multi-objective harmony search algorithm is proposed.

• NSGA-II and CPLEX optimizer are used for performance comparisons.

Abstract

The maximal covering location problem (MCLP) is a well-known combinatorial optimization problem with several applications in emergency and military services as well as in public services. Traditionally, MCLP is a single objective problem where the objective is to maximize the sum of the demands of customers which are served by a fixed number of open facilities. In this article, a multi-objective MCLP is proposed where each customer has a preference for each facility. The multi-objective MCLP with customers’ preferences (MOMCLPCP) deals with the opening of a fixed number of facilities from a given set of potential facility locations and then customers are assigned to these opened facilities such that both (i) the sum of the demands of customers and (ii) the sum of the preferences of the customers covered by these opened facilities are maximized. A Pareto-based multi-objective harmony search algorithm (MOHSA), which utilizes a harmony refinement strategy for faster convergence, is proposed to solve MOMCLPCP. The proposed MOHSA is terminated based on the stabilization of the density of non-dominated solutions. For experimental purposes, 82 new test instances of MOMCLPCP are generated from the existing single objective MCLP benchmark data sets. The performance of the proposed MOHSA is compared with the well-known non-dominated sorting genetic algorithm II (NSGA-II), and it has been observed that the proposed MOHSA always outperforms NSGA-II in terms of computation time. Moreover, statistical tests show that the objective values obtained from both algorithms are comparable.

Introduction

The covering location problem is one of the well-studied facility location problems (Drezner and Hamacher, 2001, Farahani and Hekmatfar, 2009, ReVelle and Eiselt, 2005), where the locations of a set of customers and a set of potential facilities are given. The objective is to select a subset of facility locations from the given set of potential facility locations such that all the customers are covered by these open facilities (Drezner and Hamacher, 2001, Farahani and Hekmatfar, 2009). Sometimes, it is not possible to cover all the customers with a limited number of open facilities. This happens due to mainly scarcity of resources and budgetary constraints. This leads the researchers to consider the maximal covering location problem (MCLP). The objective of MCLP is to find a fixed number of facilities from the given set of potential facility locations (sites) such that the sum of the demands of the customers which are covered by these open facilities is maximized (Adenso-Diaz and Rodriguez, 1997, Atta et al., 2018, Church and ReVelle, 1974, Galvão et al., 2000, Galvão and ReVelle, 1996, Lorena and Pereira, 2002, Resende, 1998). Note that a customer is said to be covered by an open facility if it is within the service distance (or coverage area) of the facility (Atta et al., 2018, Lorena and Pereira, 2002). In the traditional MCLP, it may happen that a customer is within the service distance of more than one open facilities, and then the customer is assigned, in general, to its nearest open facility (Atta et al., 2018). In real-life, a customer may have different preferences for different potential facility locations. A customer always wants to get service from a facility for which it has the maximum preference. In this article, a new multi-objective variant for MCLP is proposed, where customers’ preferences for different facilities are considered. We call this variant as the multi-objective maximal covering location problem with customers’ preferences (MOMCLPCP). In MOMCLPCP, a set of customers and a set of potential facility locations are given. Each customer has certain demand and each facility has its service distance beyond which it cannot serve any customer. MOMCLPCP involves finding a subset of fixed number of facilities from the given potential facility locations such that (i) the sum of the demands of customers and (ii) the sum of the preferences of the customers covered by the open facilities are maximized.

Several objective functions, often conflicting with each other, are optimized simultaneously using multi-objective optimization (MOO) techniques to solve a real-world problem (Coello, 2006, Deb, 2001, Mukhopadhyay et al., 2015). The goal of MOO is to find a solution vector ${\bar{x}}^{\ast }={[x_1,x_2,\dots ,x_n]}^T$ of $n$ decision variables which optimizes the objective function vector $\bar{f}\left(\bar{x}\right)={[f_1\left(\bar{x}\right),f_2\left(\bar{x}\right),\dots ,f_m\left(\bar{x}\right)]}^T$ satisfying some equality and inequality constraints (Coello, 2006, Deb, 2001, Mukhopadhyay et al., 2015). A decision vector ${\bar{x}}^{\ast }$ is said to be Pareto-optimal if and only if there is no $\bar{x}$ that dominates ${\bar{x}}^{\ast }$. In general, Pareto-optimum consists of a set of non-dominated solutions (Coello, 2006, Deb, 2001, Mukhopadhyay et al., 2015). A survey on the multi-objective evolutionary algorithms can be found in Zhou et al. (2011).

In this article, a Pareto-based multi-objective harmony search algorithm (MOHSA) is proposed to solve MOMCLPCP. The harmony search algorithm (HSA) is a music-inspired metaheuristic algorithm developed by Geem, Kim, and Loganathan (2001). HSA, as well as MOHSA, have been successfully applied to a wide range of engineering disciplines for solving many different real-world optimization problems such that the maximal covering species problem (Geem & Williams, 2007), the flow shop problem (Wang, Pan, & Tasgetiren, 2010), the power flow problem (Sivasubramani & Swarup, 2011), the university time tabling problem (Al-Betar & Khader, 2012), the ecological diversity planning (Geem, 2015a), the tool indexing problem (Atta, Mahapatra, & Mukhopadhyay, 2019b), the multi-objective time-cost trade-off problem (Geem, 2010a), the multi-objective optimization of water distribution networks (Geem, 2015b), to name a few. A comprehensive detail of the applications and development of the harmony search algorithm can be found in Geem, 2010b, Ingram and Zhang, 2009, Manjarres et al., 2013 and Moh’d Alia and Mandava (2011). The proposed MOHSA incorporates a harmony refinement technique to improve the harmonies (i.e., the candidate solutions), and a stability measure (Roudenko & Schoenauer, 2004) based on the density of the non-dominated solutions is used as the terminating criterion.

Since MOMCLPCP is considered for the first time, there exists no benchmark data set for it. Hence, we have generated MOMCLPCP instances from the benchmark instances available for the traditional single objective MCLP instances. We have made these instances public, and these instances are available at the supplementary web-page (https://soumenatta.github.io/momclpcp/) of this article. The proposed MOHSA is applied to these MOMCLPCP instances, and we have compared the performance of the proposed MOHSA with the popular NSGA-II algorithm. Statistical tests show that the performance of the proposed MOHSA and NSGA-II is comparable. Moreover, the computation time of MOHSA is found to be far better than NSGA-II. Note that we have also incorporated the same refinement technique and used the same measure of the terminating criterion for NSGA-II to make both the algorithms comparable. We have also compared the obtained sum of coverage (i.e., the first objective value) with the same available for the traditional single objective MCLP in the literature. Moreover, IBM’s CPLEX optimizer is used to get a bound on the sum of preferences of customers (i.e., the second objective value), and we use this bound for the performance analysis of the proposed algorithm MOHSA.

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 et al., 2011 and Máximo, Nascimento, and Carvalho (2017). In Section 4, a brief description of the harmony search algorithm (HSA) is given. In Section 5, the proposed Pareto-based MOHSA is discussed in detail. The results obtained using the proposed MOHSA and its comparison with NSGA-II are presented in Section 6. Finally, Section 7 concludes the article. Moreover, all the generated benchmark instances of MOMCLPCP along with their solutions, including the assignment of customers to the open facilities, and visualization of each solution are available at the supplementary web-page (https://soumenatta.github.io/momclpcp/) of this article.