Mathematical Problems in Engineering

Volume 2016, Article ID 9109824, 10 pages

http://dx.doi.org/10.1155/2016/9109824

## Analyzing the Performance of a Hybrid Heuristic for Solving a Bilevel Location Problem under Different Approaches to Tackle the Lower Level

Facultad de Ciencias Físico-Matemáticas, Universidad Autónoma de Nuevo León, Avenida Universidad s/n, 66450 San Nicolás de los Garza, NL, Mexico

Received 19 March 2016; Accepted 18 May 2016

Academic Editor: Mónica A. López-Campos

Copyright © 2016 Sayuri Maldonado-Pinto 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 problem addressed here is a combinatorial bilevel programming problem called the uncapacitated facility location problem with customer’s preferences. A hybrid algorithm is developed for solving a battery of benchmark instances. The algorithm hybridizes an evolutionary algorithm with path relinking; the latter procedure is added into the crossover phase for exploring the trajectory between both parents. The proposed algorithm outperforms the evolutionary algorithm already existing in the literature. Results show that including a more sophisticated procedure for improving the population through the generations accelerates the convergence of the algorithm. In order to support the latter statement, a reduction of around the half of the computational time is obtained by using the hybrid algorithm. Moreover, due to the nature of bilevel problems, if feasible solutions are desired, then the lower level must be solved for each change in the upper level’s current solution. A study for illustrating the impact in the algorithm’s performance when solving the lower level through three different exact or heuristic approaches is made.

#### 1. Introduction

Location theory is an area of operational research which has attracted many researchers due to the existing necessity to abstract real day-to-day situations into mathematical models. One of the main problems is the facility location problem, which stems from Weber’s problems, which consists in determining the point that minimizes the sum of the Euclidean distances from that point to all other given points. Further explanation is detailed in [1]. In the facility location problem, there is a set of customers that are distributed in a predefined space. They desire that their demands of a particular service or product are met by one or more facilities. The problem is to determine where to locate facilities and how customers should be allocated in order to minimize location and distribution costs associated with that particular decision. This situation is known as the classic facility location problem (FLP).

The FLP has led to several extensions. For example, the simple plant location problem (SPLP) arises when the facility has an infinite capacity and can meet all the demands of any customer; the model for this problem was proposed in [2]. A taxonomy of location models including relevant issues and a classification is provided in [3]. Another variant appears when a new objective function is considered as in [4]. It includes the most popular objectives functions of location models and penalizes the distance between the customer and the facility according to the position occupied by the customer. In other words, a customer will have different penalties depending on how near or far it is regarding the facility.

Having an ordering problem within a location one increases its complexity in both the formulation and the methodology of solution. The Discrete Ordered Median Problem (DOMP) was introduced in [5], in which two formulations are proposed: as an integer linear program and as an integer nonlinear program. Then, in [6], an alternative integer linear programming formulation for the DOMP is proposed. A comparison with the existing ones is made showing that the proposed formulation is strengthened. Moreover, some properties regarding optimal solutions that allow the elimination of a subset of variables are found. Taking advantage of the properties, a branch and bound algorithm was developed and it was used for solving a set of benchmark instances.

In [7], an extension of the DOMP called the ordered capacitated facility location (OCFL) problem is proposed and it was modeled from three different points of view. In the first model, the customer’s demand can be divided; in the second model, fixed costs for locating facilities are considered; and in the third model, the shipping and locating costs are taken into account in the objective function. Also, in the latter model, they examine three approaches for incorporating shipping costs: the costs are paid (i) by the customers, (ii) by the distribution centers, or (iii) by the logistics provider. To consider the abovementioned approaches, they used the objective function proposed in [8]. The locating and shipping costs are ordered before the evaluation of the objective function is made. Two formulations of the model are proposed and some improvements for particular cases are introduced.

A different perspective regarding an ordering within the location problem can be achieved by considering the customer’s preferences towards the facilities. The first paper which considered the customer’s preferences in the simple plant location problem (SPLP) is [9], in which they assumed the location of a single facility and added restrictions to ensure that the preferences were considered. Furthermore, they presented different ways to include this set of constraints and proposed a greedy heuristic based on branch and bound for solving the problem. The problem is known as the simple plant location problem with order (SPLPO). It is shown in [10] that this problem and all its generalizations are classified as NP-Hard.

The consideration of customer’s preferences may be seen as an optimization problem within the constraints of the location problem. These kind of problems can be modeled with bilevel programming. The first bilevel model for the SPLPO was proposed in [11]. The bilevel model is reformulated as a single-level problem by introducing pseudo-Boolean functions. Then, valid bounds are obtained through a relaxation of the lower level problem. Additional assumptions were made during that research: the optimal solution of follower is unique for any arbitrary solution of leader and all values of the customer’s preferences are different.

Most articles that addressed the bilevel model of the SPLPO reformulate the bilevel model into a single-level one and solve the reformulation. For example, for the aforementioned purposes in [12], valid inequalities are used and in [13] a combinatorial formulation is considered. The main motivation for avoiding solving the bilevel version of the problem relies in the difficulty of solving an optimization problem for each configuration of facilities located.

Therefore, it is convenient to emphasize that it is extremely important to obtain the optimal solution of the lower level problem in order to obtain a bilevel feasible solution. For the uncapacitated facility location bilevel problem (UFLBP), the lower level consists of an allocation problem, which can be solved in an efficient manner by an optimizer. Moreover, due to the nature of the allocation problem, alternative exact methods can be easily adapted for solving it. For instance, in [14, 15], an exact method based on the ordered matrix of preferences is considered for solving the bilevel version of UFLBP. However, the construction of efficient exact methods cannot be always obtained for solving the lower level problem; it clearly depends on its structure.

For clarifying the latter idea, some papers in which the lower level is solved in a heuristic manner are described. For example, in [16], a new formulation for the ring star problem as a bilevel model is proposed, in which they considered the existence of a leader and two independent followers. One of the followers must solve a travelling salesman problem and the use of a greedy algorithm with 2-opt and 3-opt local searches is applied. Another example is found in [17] where a bilevel urban transportation network design model is studied. In order to avoid finding the optimal solution for the follower, a local approach was designed for getting the follower’s response. Then, in [18], a topological design of Local Area Networks bilevel problem is considered. The lower level must construct a capacitated spanning tree and it is solved by a greedy constructive algorithm similar to Kruskal’s algorithm. Also, in [19], a competitive facility location problem is considered; a branch and bound method is applied for solving a nonlinear programming relaxation of the lower level problem. Finally, in [20], an ant colony optimization algorithm for solving a bilevel production-distribution problem was implemented. They illustrated the behavior of the algorithm when the lower level is solved in an exact manner or by a heuristic procedure. In the case when the heuristic procedure was applied to the lower level, it corresponds to a differential evolution algorithm.

As it is mentioned above, in order to obtain bilevel feasible solutions, the lower level problem must be optimally solved by an optimizer or by an exact method. But this is not always possible and the problem needs to be solved somehow. Hence, a heuristic procedure that balances efficiency and computational effort would be desired. Commonly, the utilization of heuristic procedures for solving the bilevel problem results very costly in terms of computational time due to the number of times that the lower level problem is solved. Then, the exploration of methodologies in which the search is guided without the resolution of the lower level turns out to be a matter of interest.

In this paper, we proposed a hybrid algorithm for solving the bilevel version of the uncapacitated facility location problem with preferences of the customers. Also, a discussion is presented about the effects that result from solving the allocation problem with an optimizer, by an alternative exact procedure or by a methodology that avoids solving it at each step. The paper is organized as follows: in Section 3, we present the classical mathematical bilevel formulation of the problem. Section 3 describes the proposed algorithm that hybridizes an evolutionary algorithm with path relinking. The computational experimentation and its corresponding interpretation of the results are shown in Section 4. Finally, Section 5 states the final remarks that arose from the findings derived from the previous section and some ideas for further research.

#### 2. Description of the Problem

In this section, the problem and its mathematical formulation are described. The bilevel model considered in this paper is proposed in [12], where the leader aims to minimize the total cost, that is, the locating and distributing costs. On the other hand, the follower aims to minimize the customer’s preferences. The sets, parameters, decision variables involved in the mathematical formulation, and the assumptions considered during the research are presented next.

Let and be the indexes of the facilities and customers, respectively. Let represent the costs of supplying all the customer’s demand for the facility . Let denote the cost of locating facility . Finally, represents the preference that customer places for being served by facility .

The decision variables of the problem are two and are of binary nature, whereThe following two assumptions are considered in the problem:(1)Customers delivered a list of ordered preferences which reflected their desire of being served by each facility; the first position in the list, that is, the 1st, indicates the most preferred facility and therefore th represents the least preferred facility.(2)The facilities have no capacity; therefore, a facility can supply multiple customers but a customer must be served by a single facility.

The mathematical model for the UFLBP is as follows:The problem in the upper level is defined by (2)–(4), where (2) represents the leader’s objective function who seeks to minimize both location and distribution costs, (3) establish the binary constraints for each variable , and finally (4) is the constraint that indicates that the variables are controlled by the follower; these variables are implicitly determined by the optimal solution of the lower level. This problem is defined by (4)–(7); the follower’s objective function is defined in (4) where she/he tries to minimize the ordered preferences of the customers, (5) ensures that each customer is supplied by a single facility, (6) indicates that the customer’s allocation can be made only to the located facilities, and, finally, (7) indicates the binary constraints for the decision variables .

The existence and uniqueness of the lower level’s solution are guaranteed due to the way the preferences are given; that is, they are ordered and are different from each other. In other words, it is not allowed to assign the same preference value to more than one facility. The proof which validates the latter is shown in [12]. By assuming this, the bilevel problem is well defined.

#### 3. A Hybrid Evolutionary Algorithm with Path Relinking

In this section, the hybrid algorithm and its components are described. Hybrid algorithms combine advantages of two or more heuristics to solve a problem, where the best part of each heuristic is taken in order to obtain either better solutions or a quick convergence. Some applications of hybrid algorithms are shown in [21]. They identify connections and contrasts between heuristics (genetic algorithms and Tabu search) that offer an almost untapped area for empirical research. In [22], a genetic algorithm and particle swarm optimization algorithms are hybridized showing good performance in some applications.

Considering the discussed improvements for hybrid algorithms, in this paper, hybridization between an evolutionary algorithm and path relinking is proposed. The main idea for including the path relinking scheme is basically in order to substitute the common random crossover. On the other hand, the motivations for developing an evolutionary algorithm (EA) are as follows: first, a population based algorithm gives more information about the interaction between leader and follower due to the large number of solutions being explored; second, it is well known that EAs can handle* not-easy-to-solve* problems in an efficient manner; and, finally, the EAs perform random movements allowing a wider exploration of the solution space while the quality of solutions is intended to be improved.

The EA consists in three phases: in the first one, the construction of the initial population containing feasible solutions is made; the second phase concerns the genetics operators, crossover and mutation; and the last one is the selection mechanism of survival solutions, which depends on two features, quality and diversity. Commonly, the criteria selected in the genetic operators are chosen in a random fashion. The latter might affect the algorithm’s performance due to the lack of exploration within the current neighborhood between both solutions. Hence, we decided to implement a combinatorial method called path relinking (PR) for reaching the local optimum in the current neighborhood. Path relinking begins with a pair of good quality solutions, parts from the first one, and changes its components once a time to convert this initial solution into the second one. Furthermore, PR explores all the trajectory between leader’s solutions. If the second solution is identical to the first one, then the method stops and returns the best solution found in the trajectory. It is important to highlight the notion that since the aim of the proposed algorithm is to solve a bilevel problem, when computing the corresponding objective function value, the follower’s problem is solved for each movement. This issue clearly affects the algorithm’s efficiency and it is discussed in the next section.

The addition of PR in other algorithm’s frameworks has shown a good performance and has attracted the attention of researchers. For example, in [23], there is a description of the elements and methods that conform Scatter Search (SS), including the most recent elements incorporated in successful applications in both global and combinatorial optimization. They also described the hybridization of PR and genetic algorithms (GA) and displayed the notion that the use of PR almost always improves the performance of the heuristic. Also, in [24], a project scheduling problem is considered, where hybridization of PR and a GA is developed to solve it. The performance of the hybrid algorithm is examined on an available test set, and it is shown that PR is more efficient than GA in this specific problem. Moreover, the combination of PR with SS is analyzed in [25]; they solved the capacitated -median problem (CMP). In the CMP, the intention is to make a partition over a set of costumers, each of them has known demands, in exactly facilities. They conducted a series of experiments on different sets, and the results were satisfactory in terms of the quality of the solutions found. Furthermore, they declared that the combination of PR with SS gave the best results. Recently, in [26, 27], path relinking is also hybridized with other heuristics showing good results. Based on these results, we were motivated for hybridizing this procedure within the evolutionary framework.

Next, the principal components and the phases involved in the hybrid algorithm are detailed and depicted in Pseudocode 1.