Analyzing the effects of binarization techniques when solving the set covering problem through swarm optimization

Highlights

• We study the impact of binarization methods when solving the set covering problem.

• We apply a metaheuristic inspired by the behavior of cats for solving the problem.

• We consider forty binarization methods and a freely available dataset.

• We conclude that it is crucial to select an adequate binarization method.

Abstract

The Set Covering Problem (SCP) is one of the classical Karp’s 21 NP-complete problems. Although this is a traditional optimization problem, we find many papers assuming metaheuristics for solving the SCP in the current literature. However, while the SCP is a discrete problem, most metaheuristics are defined for solving continuous optimization problems, specially Swarm Intelligence Algorithms (SIAs). Hence, such algorithms should be adapted for working on the discrete scope, but most authors did not perform any study to select a concrete binarization approach. This situation might lead to the conclusion that selecting a concrete binarization technique does not influence the behavior of the algorithm, but rather the general approach of the metaheuristic. This circumstance led us to write this paper focusing on the inherent difficulty in binarization of metaheuristics designed for continuous optimization, when solving a discrete optimization problem, concretely the SCP. To this end, we consider a recent SIA inspired by the behavior of cats and adapted to the discrete scope, which is called Binary Cat Swarm Optimization (BCSO). We replace the binarization technique assumed in the original BCSO by forty different approaches from the current literature. The results obtained while solving a standard SCP benchmark are analyzed through a widely accepted statistical method, concluding that it is crucial to select an adequate binarization approach to ensure that the solving algorithm reaches its full potential. Thus, the task of adapting a metaheuristic to the discrete scope is not a simple matter and should be carefully studied. To this end and as a result of this study, we give some recommendations to perform this task.

Introduction

The Set Covering Problem (SCP) is one of the classical 21 problems shown to be NP-complete by Karp (1972) and whose optimization version is NP-hard (Garey & Johnson, 1979). Although the SCP is a traditional optimization problem, it is widely considered in the current literature for designing expert systems, which emulate the decision-making ability of human experts in a given field (Reggia, Nau, & Wang, 1983). For example, we find works considering the SCP for facility location (Farahani, Asgari, Heidari, Hosseininia, & Goh, 2012), ship scheduling (de Mare, Spliet, & Huisman, 2014), production planning (Adulyasak, Cordeau, & Jans, 2015), crew scheduling (Chen, Shen, 2013, Juette, Thonemann, 2012 Chen, Shen, 2013, Juette, Thonemann, 2012), vehicle routing (Bai, Xue, Chen, Roberts, 2015, Cacchiani, Hemmelmayr, Tricoire, 2014, Vidal, Crainic, Gendreau, Prins, 2013), musical composition (Simeone, Nouno, Mezzadri, & Lari, 2014), information retrieval (Zhang, Wei, & Chen, 2014), and territory design (Elizondo-Amaya, Rios-Mercado, & Diaz, 2014), among many others.

Chvatal (1979) defined the SCP as follows. Given a set M of m objects and a collection S of n sets of these objects, each set with a non-negative cost associated. The goal is to find a minimum cost family of subsets C ⊆ S, such that each element i ∈ M belongs to at least one subset of the family C.

Some authors solved the SCP by applying exact techniques, such as branch-and-bound and branch-and-cut algorithms. However, such methods are not recommended for solving this type of complex problems, because computational times rise exponentially with the problem dimension.

Instead, approximate techniques should be considered, such as metaheuristics. This type of techniques is successfully considered in the literature for solving NP-hard problems from different fields, including the SCP (Dasgupta & Michalewicz, 2013). However, while the SCP is defined as a discrete optimization problem, many metaheuristics are designed for solving continuous optimization problems, specially Swarm Intelligence Algorithms (SIAs). Thus, such metaheuristics should be adapted for working on the discrete search space.

There are many SIAs adapted for solving general discrete optimization problems, such as Binary Gravitational Search Algorithm (BGSA) (Rashedi, Nezamabadi-Pour, & Saryazdi, 2010), Binary Magnetic Optimization Algorithm (BMOA) (Mirjalili & Hashim, 2012), and Binary Cat Swarm Optimization (BCSO) (Sharafi, Khanesar, & Teshnehlab, 2013). Usually, the algorithms are adapted by following the two-step binarization method introduced by Kennedy and Eberhart (1997) in their approach of Binary Particle Swarm Optimization (BPSO) for transforming real numbers into binary ones. In this case, the authors explained how to get a new binary solution according to the particle velocity, which is a real number. The method followed by the authors is as follows. Firstly, we map the real value to a number in the interval [0, 1] through a transfer function. Secondly, we transform the number in the interval [0, 1] into a binary value through a discretization function. In this line, there are eight major transfer functions and five major discretization functions in the current literature, denoted as $S_1,S_2,\dots ,S_4,V_1,V_2,\dots ,V_4$ and $D_1,D_2,\dots ,D_8,$ respectively (Crawford, Soto, Peña, Riquelme-Leiva, Torres-Rojas, Johnson, et al., 2015c, Mirjalili, Lewis, 2013).

Most authors did not do any study to select a concrete binarization approach when adapting a metaheuristic. This situation might lead to the conclusion that selecting a concrete binarization technique does not influence the behavior of the algorithm, but rather the general approach of the metaheuristic. To the best of our knowledge, this is the first work performing this study in the literature. We do the following three main tasks throughout this study:

• We select a recent SIA from the current literature, which was initially designed for continuous optimization and later adapted to the discrete scope. Specifically, the BCSO algorithm inspired by the behavior of cats, whose original continuous approach was proposed by Chu, Tsai, and Pan (2006).

• The authors of BCSO considered a transfer and discretization function, without performing any formal study to this task. We change the original formulation of BCSO by combining the eight transfer functions and the five discretization functions introduced before, i.e., we get forty BCSO approaches.

• We apply the forty BCSO approaches for solving two freely available SCP sets. We study the results obtained through an accepted statistical methodology to analyze if selecting a binarization technique influences the behavior of the metaheuristic.

The rest of this paper is structured as follows. We list the acronyms considered in Table 1. In Section 2, we discuss the related work, including the major motivations for performing this work. In Section 3, we give a formal SCP definition, including a problem example. In Section 4, we explain the BCSO metaheuristic. In Section 5, we describe the transfer and discretization functions in this study. In Section 6, we discuss the experimental method followed and the results obtained. In Section 7, we give some implementation details. Finally, conclusions are left for Section 8.