A multi-objective formulation of maximal covering location problem with customers’ preferences: Exploring Pareto optimality-based solutions
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 of decision variables which optimizes the objective function vector satisfying some equality and inequality constraints (Coello, 2006, Deb, 2001, Mukhopadhyay et al., 2015). A decision vector is said to be Pareto-optimal if and only if there is no that dominates . 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.
Section snippets
Related works
MCLP was first introduced by Church and ReVelle (1974) on a network. Since the inception of MCLP, many variations of the basic MCLP have been considered based on its applications. In this section, we concisely discuss the important variants of MCLP and their solution strategies. Based on the availability and response time of ambulance, five different models for MCLP were studied for locating ambulances in Erkut, Ingolfsson, Sim, and Erdoğan (2009). The fuzzy MCLP, which considers travel time
Problem definition and mathematical formulation
The multi-objective maximal covering location problem with customers’ preferences optimizes two objective functions while satisfying several constraints. In the following subsections, at first, we describe different terminologies and decision variables used in the mathematical problem formulation of MOMCLPCP. We also mention assumptions made during the problem formulation. Subsequently, we define the two objective functions of MOMCLPCP and describe all the constraints. Note that the two
Brief description of HSA
The harmony search algorithm is a music inspired global optimization technique which was first proposed by Geem et al. (2001). HSA is based on the improvisation process of musicians (Geem et al., 2001, Lee and Geem, 2005). Like other evolutionary algorithms, HSA starts with a randomly generated pool of solution vectors. Each of these solution vectors is known as a harmony and the pool of solutions is called the harmony memory (Geem et al., 2001). HSA tries to optimize a given scalar objective
Proposed MOHSA
In this section, the proposed Pareto optimality-based multi-objective harmony search algorithm for solving MOMCLPCP is described in detail. The overall procedure of the proposed MOHSA is given in Pseudocode 1.
Experimental results and discussions
In this section, experimental results are presented and discussed in detail. Since there is no benchmark instances available for MOMCLPCP, the procedure for generating benchmark instances for MOMCLPCP is presented at first in Section 6.2. The proposed MOHSA is then applied to these instances and the results are presented. To assess the performance of MOHSA, its performance is compared with that of the well-known multi-objective evolutionary algorithm NSGA-II (Deb et al., 2002). The adaptation
Conclusion
In this article, a multi-objective variant of the maximal covering location problem with customers’ preferences (MOMCLPCP) is presented. The first objective is to maximize the sum of the demands of customers served by the open facilities and the second objective is to maximize the sum of the preferences of customers for the facilities from which they are getting services. A customized Pareto-based multi-objective harmony search algorithm (MOHSA) is proposed to solve this problem. The proposed
CRediT authorship contribution statement
Soumen Atta: Conceptualization, Methodology, Software, Writing – original draft, Writing – review & editing. Priya Ranjan Sinha Mahapatra: Supervision, Writing – review & editing. Anirban Mukhopadhyay: Supervision, Writing – review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
P. R. S. Mahapatra acknowledges the support received from MATRICS (Ref. No. MTR/2019/000792) grant of Science and Engineering Research Board (SERB), Government of India. A. Mukhopadhyay acknowledges the support received from the DST-PURSE grant of the University of Kalyani.
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