Complexity

Volume 2017 (2017), Article ID 1702506, 15 pages

https://doi.org/10.1155/2017/1702506

## Parameter Tuning for Local-Search-Based Matheuristic Methods

^{1}Pontificia Universidad Católica de Valparaíso, 2362807 Valparaíso, Chile^{2}Instituto Tecnológico Metropolitano, Calle 73 No. 76A-374, Vía al Volador, Medellín, Colombia^{3}Universidad de Playa Ancha, Casilla 34-V, Valparaíso, Chile^{4}Universidad Diego Portales, 8370109 Santiago, Chile^{5}CIMFAV-Facultad de Ingeniería, Universidad de Valparaíso, 2374631 Valparaíso, Chile

Correspondence should be addressed to Guillermo Cabrera-Guerrero; lc.vcup@arerbac.omrelliug

Received 20 June 2017; Revised 2 October 2017; Accepted 25 October 2017; Published 31 December 2017

Academic Editor: Kevin Wong

Copyright © 2017 Guillermo Cabrera-Guerrero 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

Algorithms that aim to solve optimisation problems by combining heuristics and mathematical programming have attracted researchers’ attention. These methods, also known as* matheuristics*, have been shown to perform especially well for large, complex optimisation problems that include both integer and continuous decision variables. One common strategy used by matheuristic methods to solve such optimisation problems is to divide the main optimisation problem into several subproblems. While heuristics are used to seek for promising subproblems, exact methods are used to solve them to optimality. In general, we say that both mixed integer (non)linear programming problems and combinatorial optimisation problems can be addressed using this strategy. Beside the number of parameters researchers need to adjust when using heuristic methods, additional parameters arise when using matheuristic methods. In this paper we focus on one particular parameter, which determines the size of the subproblem. We show how matheuristic performance varies as this parameter is modified. We considered a well-known NP-hard combinatorial optimisation problem, namely, the capacitated facility location problem for our experiments. Based on the obtained results, we discuss the effects of adjusting the size of subproblems that are generated when using matheuristics methods such as the one considered in this paper.

#### 1. Introduction

Solving mixed integer programming (MIP) problems as well as combinatorial optimisation problems is, in general, a very difficult task. Although efficient exact methods have been developed to solve these problems to optimality, as the problem size increases exact methods fail to solve it within an acceptable computational time. As a consequence, nonexact methods such as heuristic and metaheuristic algorithms have been developed to find good quality solutions. In addition, hybrid strategies combining different nonexact algorithms are also promising ways to tackle complex optimisation problems. Unfortunately, algorithms that do not consider exact methods cannot give us any guarantee of optimality and, thus, we do not know how good (or bad) solutions found by these methods are.

One hybrid strategy that combines nonexact methods are memetic algorithms, which are population-based metaheuristics that use an evolutionary framework integrated with local search algorithms [1]. Memetic algorithms provide lifetime learning process to refine individuals in order to improve the obtained solutions every iteration or generation; their applications have been grown significantly over the years in several NP-hard optimisation problems [2]. These algorithms are part of the paradigm of memetic computation, where the concept of meme is used to automate the knowledge transfer and reuse across problems [3]. A large number of memetic algorithms can be found in the literature. Depending on its implementation, memetic algorithms might (or might not) give guarantee of local optimality: roughly speaking, if heuristic local search algorithms are considered, then no optimality guarantee is given; if exact methods are considered as the local optimisers, then local optimality could be ensured.

To overcome the situation described above, hybrid methods that combine heuristics and exact methods to solve optimisation problems have been proposed. These methods, also known as matheuristics [4], have been shown to perform better than both heuristic and mathematical programming methods when they are applied separately. The idea of combining the power of mathematical programming with flexibility of heuristics has gained attention within researchers’ community. We can found matheuristics attempting to solve problems arising in the field of logistics [5–9], health care systems [10–13], and pure mathematics [14, 15], among others. Matheuristics have been demonstrated to be very effective in solving complex optimisation problems. Some interesting surveys on matheuristics are [16, 17]. Although there is some overlapping between memetic algorithms and matheuristic ones, in this paper we have chosen to label the studied strategy as matheuristic, as we think matheuristics definition better fits the framework we are interested in.

Because of their complexity, MIP problems as well as combinatorial optimisation problems are often tackled using matheuristic methods. One common strategy to solve this class of optimisation problems is to divide the main optimisation problem into several subproblems. While heuristics are used to seek for promising subproblems, exact methods are used to solve them to optimality. One advantage of this approach is that it does not depend on the (non)linearity of the resulting subproblem. Instead, it has been pointed out that it is desirable that the resulting subproblem would be convex [11]. Having a convex subproblem would allow us to solve it to optimality, and thus comparing solutions obtained at each subproblem becomes more senseful. This strategy has been successfully applied to problems arising in fields as diverse as logistics and radiation therapy.

In this paper we aim to study the impact of parameter tuning on the performance of matheuristic methods as the one described above. To this end, the well-known capacitated facility location problem is used as an application of hard combinatorial optimisation problem. To the best of our knowledge, no paper has focused on parameter tuning for matheuristic methods.

This paper is organised as follows: Section 2 shows the general matheuristic framework we consider in this paper. Details on the algorithms that are used in this study are also shown in this section. In Section 3 the capacitated facility location problem is introduced and its mathematical model is described in Section 3.1. The experiments performed in this study are presented and the obtained results are discussed in Section 3.2. Finally, in Section 4 some conclusions are presented and the future work is outlined.

#### 2. Matheuristic Methods

This section is twofold. We start by describing a general matheuristic framework that is used to solve both MIP problems and combinatorial optimisation problems and how it is different from other commonly used approaches such as memetic algorithms and other evolutionary approaches. After that, we present the local-search-based algorithms we consider in this work to perform our experiments. We finish this section by introducing the parameter we will be focused on in this study.

##### 2.1. General Framework

Equations (1a) to (1f) show the general form of MIP problems. Hereafter we will refer to this problem as the MIP problem or the* main problem*.where is an objective function, is the number of inequality constraints on and , is the number of inequality constraints on , is the number of inequality constraints on , is the number of binary (≥0) decision variables , and is the number of continuous decision variables. Combinatorial optimisation problems, as the one we consider in this paper, can be easily obtained by either removing the continuous decision variable from the model or making it integer (i.e., ).

Although there exist a number of exact algorithms that can find an optimal solution for the MIP, as the size of the problem increases, exact methods fail or take too long. Because of this, heuristic methods are used to obtain good quality solutions of the problem within an acceptable time. Heuristic methods cannot guarantee optimality though.

During the last two decades, the idea of combining heuristic methods and mathematical programming has received much more attention. Exploiting the advantages of each method appears to be a senseful strategy to overcome their inherent drawbacks. Several strategies have been proposed to combine heuristics and exact methods to solve optimisation problems such as the MIP problem. For instance, Chen and Ting [18] and Lagos et al. [8] combine the well-known ant colony optimisation (ACO) algorithm and Lagrangian relaxation method to solve the single source capacitated facility location problem and a distribution network design problem, respectively. In these articles, Lagrangian multipliers are updated using ACO algorithm. Another strategy to combine heuristics and exact methods is to let heuristics seek for subproblems of MIP which, in turn, are solved to optimality by some exact method. One alternative to obtain subproblems of MIP is to add a set of additional constraints on a subset of binary decision variables. These constraints are of the form , with and being the set of index that are restricted in subproblem (see (1a) to (1f)). The portion of binary decision variables that are set to is denoted by (i.e., ), with being the number of binary variables . Then, the obtained subproblem, which we call , is

In this paper we assume that a constraint on the th resource of vector of the form , that is, , means that such a resource is not available for subproblem . We can note that, as the number of constrained binary decision variables associated with increases, that is, as ) increases, the subproblem becomes smaller. Similarly, as the number of constrained binary decision variables associated with gets smaller, that is, decreases, the subproblem gets larger, as there are more available resources. It is also easy to note that, as increases, we can obtain optimal solutions of the corresponding subproblem relatively faster. However, quality of the obtained solutions is usually impaired as the solution space is restricted by the additional constraints associated with the set . Similarly, as gets smaller, obtaining optimal solutions of the associated subproblems might take longer but the quality of the obtained solutions is, in general, greatly improved. Finally, when we have that subproblem is identical to the main problem, MIP. We assume that optimal solution of subproblem , , is also feasible for the MIP problem. Moreover, we can note that there must be a minimal set , for which an optimal solution of the associated subproblem is also an optimal solution of the main problem MIP.

In some cases, the value of might be predefined by the problem that is being solved. For instance, the problem of finding the best beam angle configuration for radiation delivery in cancer treatment (beam angle optimisation problem) usually sets the number of beams to be used in a beam angle configuration (see Cabrera-Guerrero et al. [11], Li et al. [19], and Li et al. [20]). This definition is made by the treatment planner and it does not take into account the algorithm performance but clinical aspects. Unlike this kind of problems, there are many other problems where the value of is not predefined and, then, setting it to an efficient value is important for the algorithm performance. Figure 1 shows the interaction between the heuristic and the exact method.