Mathematical Problems in Engineering

Volume 2017, Article ID 8306732, 13 pages

https://doi.org/10.1155/2017/8306732

## Application of Heuristic and Metaheuristic Algorithms in Solving Constrained Weber Problem with Feasible Region Bounded by Arcs

^{1}Faculty of Computer Science, Goce Delčev University, Goce Delčev 89, 2000 Štip, Macedonia^{2}Department of Mathematics and Informatics, Faculty of Science and Mathematics, University of Niš, Višegradska 33, 18000 Niš, Serbia^{3}Department of Systems Analysis and Operations Research, Reshetnev University, Prosp. Krasnoyarskiy Rabochiy 31, Krasnoyarsk 660037, Russia

Correspondence should be addressed to Predrag S. Stanimirović; sr.ca.in.fmp@okcep

Received 26 February 2017; Accepted 15 May 2017; Published 14 June 2017

Academic Editor: Domenico Quagliarella

Copyright © 2017 Igor Stojanović et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The continuous planar facility location problem with the connected region of feasible solutions bounded by arcs is a particular case of the constrained Weber problem. This problem is a continuous optimization problem which has a nonconvex feasible set of constraints. This paper suggests appropriate modifications of four metaheuristic algorithms which are defined with the aim of solving this type of nonconvex optimization problems. Also, a comparison of these algorithms to each other as well as to the heuristic algorithm is presented. The artificial bee colony algorithm, firefly algorithm, and their recently proposed improved versions for constrained optimization are appropriately modified and applied to the case study. The heuristic algorithm based on modified Weiszfeld procedure is also implemented for the purpose of comparison with the metaheuristic approaches. Obtained numerical results show that metaheuristic algorithms can be successfully applied to solve the instances of this problem of up to 500 constraints. Among these four algorithms, the improved version of artificial bee algorithm is the most efficient with respect to the quality of the solution, robustness, and the computational efficiency.

#### 1. Introduction

The Weber problem is one of the most studied problems in location theory [1–3]. This optimization problem searches for an optimal facility location on a plane, which satisfiesIn (1), it is assumed that , are known demand points, and are weight coefficients, and is a matrix norm, used as the distance function.

The basic Weber problem is stated with the Euclidean norm underlying the definition of the distance function. Also, many other types of distances have been used in the facility location problems [3–5]. In general, a lot of extensions and modifications of the Weber location problem are known. Detailed reviews of these problems can be found in [3, 6].

The most popular method for solving the Weber problem with Euclidean distances is given by a one-point iterative procedure which was first proposed by Weiszfeld [7]. Later, Vardi and Zhang developed a different extension of Weiszfeld’s algorithm [8], while Szegedy partially extended Weiszfeld’s algorithm to a more general problem [9]. In particular, some variants of the continuous Weber problem represent nonconvex optimization problems which are hard to be solved exactly [10]. A nonconvex optimization problem may have multiple feasible regions and multiple locally optimal points within each region [11]. Consequently, finding the global solution of a nonconvex optimization problem is very difficult.

Heuristics and metaheuristics represent the main types of stochastic methods [12]. Both types of algorithms can be used to speed up the process of finding a high-quality solution in the cases where finding an optimal solution is very hard. The distinctions between heuristic and metaheuristic methods are inappreciable [12]. Heuristics are algorithms developed to solve a specific problem without the possibility of generalization or application to other similar problems [13]. On the other hand, a metaheuristic method represents a higher-level heuristic in the sense that they guide their design. In such a way we can use any of these methods to design a specific method for computing an approximate solution for an optimization problem.

In the last several decades, there is a trend in the scientific community to solve complex optimization problems by using metaheuristic optimization algorithms. Some applications of metaheuristic algorithms include neural networks, data mining, industrial, mechanical, electrical, and software engineering, as well as certain problems from location theory [14–21]. The most interesting and most widely used metaheuristic algorithms are swarm-intelligence algorithms which are based on a collective intelligence of colonies of ants, termites, bees, flock of birds, and so forth [22]. The reason of their success lies in the fact that they use commonly shared information among multiple agents, so that self-organization, coevolution, and learning during cycles may help in creating the highest quality results. Although not all of the swarm-intelligence algorithms are successful, a few techniques have proved to be very efficient and thus have become prominent tools for solving real-world problems [23]. Some of the most efficient and the most widely studied examples are ant colony optimization (ACO) [24–26], particle swarm optimization (PSO) [15, 27–29], artificial bee colony (ABC) [19, 30–35], and recently proposed firefly algorithm (FA) [18, 36–38] and cuckoo search (CS) [17, 39–41].

Different heuristic methods are proposed in order to provide encouraging results for challenging continuous Weber problem with regard to solution quality and computational effort [42–46]. Also, some variants of the Weber problem have been successfully solved by different metaheuristic approaches [47–52]. In [52], the authors studied a capacitated multisource Weber problem as an extended facility location problem that involves both facility locations and service allocations simultaneously. The method proposed in [52] is based on the integration of two genetic algorithms. The problem of locating one new facility with respect to a given set of existing facilities in the plane and in the presence of convex polyhedral barriers was considered in [47]. The general strategy in [47] arises from the iterative application of a genetic algorithm for the subproblems selection. A hybrid particle swarm optimization approach was applied in solving the incapacitated continuous location-allocation problem in [48]. In [49], the authors compared performances of four metaheuristic algorithms, modified to solve the single-facility location problem with barriers. The method for solving a kind of Weber problem from [50] was developed using an evolutionary algorithm enhanced with variable neighborhood search.

The aim of this paper is to investigate the performances of some prominent swarm-intelligence metaheuristic approaches to solve the constrained Weber problem with feasible region bounded by arcs. This variant of Weber problem has a nonconvex feasible set given by the constraints that make it much harder to find the global optimum using any deterministic algorithms. Hence, metaheuristic optimization algorithms can be employed in order to provide promising results.

In this paper, four swarm-intelligence techniques are applied to solve this version of the constrained Weber problem: the artificial bee colony for constrained optimization [53], the crossover-based artificial bee colony (CB-ABC) algorithm [54], the firefly algorithm for constrained optimization [37], and the enhanced firefly algorithm (E-FA) [55]. The CB-ABC and the E-FA are two of the most recently proposed improved variants of the ABC and FA for solving constrained problems, respectively. Also, a heuristic algorithm is proposed in [44] with the aim of solving this version of the constrained Weber problem. Hence, it is also implemented for the purpose of comparison with the metaheuristic approaches. These five techniques are tested to solve randomly generated test instances of constrained Weber problem with feasible region bounded by arcs of up to 500 constraints.

The rest of the paper is organized as follows. A formulation of the constrained Weber problem with feasible region bounded by arcs and the heuristic approach developed to solve this variant of the constrained Weber problem are presented in Section 2. Section 3 presents the four metaheuristic optimization techniques used to solve this variant of the Weber problem. Description of the generated benchmark functions and comparative results of the four implemented metaheuristic techniques are given in Section 4. Concluding remarks are provided in Section 5.

#### 2. The Heuristic Method for Solving a Constrained Weber Problem

The constrained Weber problem with feasible region bounded by arcs in the continuous space was introduced in [44]. In order to complete our presentation, we briefly restate the method. It can be formulated by the goal function defined in (1) and by the feasible region which is defined on the basis of constraints of two opposite types:where is the total number of demand points and and are subsets of the set of demand point indices satisfying , , and . For the sake of simplicity, the optimization problem given by (1) with constraints (2) is denoted as the CWP problem.

Such a problem may occur if some demand points coincide with locations of some important facilities and the searched optimal location must be close to them. Other demand points may coincide with dangerous facilities and the facility must be located far from them.

The metric used in practically important location problems depends on various factors, including properties of the transportation means [44]. In the case of public transportation systems, the price usually depends on a distance. However, some minimum price is usually defined. For example, the initial fare of the taxi cab may include some distance, usually 1–5 km. Having rescaled the distances so that this distance included in the initial price is equal to 1, we can define the price function aswhere is a matrix norm.

In the case of distance function defined by (3), the problem can be decomposed into series of constrained location problems with the Euclidean metric where the area of the feasible solutions is bounded by arcs. Each of the problems has the feasible region equal to the same intersection of the discs with centers in the demand points. For more details, see [44, 56].

The Weiszfeld procedure for solving the Weber problem with a given tolerance , based on the results from [57], is presented as Algorithm 2.1 in [44].

An algorithm based on the Weiszfeld procedure for solving the CWP defined by objective (1) and constraints (2) was proposed [44]. The feasible set of our constrained optimization problems is generally nonconvex, while the objective function given by (1) is convex [58]. A solution of constrained optimization problems with convex objective functions coincides with the solution of the unconstrained problem or lies on the border of the forbidden region [59]. Thus, if is a solution of the constrained problem given by (1) with constraints (2) then it is the solution of the unconstrained problem (1) or .

Step 2.2 of Algorithm 2.1 from [44] can lead to generating a new point outside the feasible region determined by constraints (2). Let us denote this region . It is assumed that .

For an arbitrary point , let us denote the closest point in by . It can be computed using

Algorithm 1 was proposed as Algorithm 2.2 in [44], and it is based on the substitution of the point generated in Step 2.2 of Algorithm 2.1 from [44] with its closest point in the feasible region.