Solving the Set Covering Problem Using the Binary Cat Swarm Optimization Metaheuristic Broderick Crawford, Ricardo Soto, Natalia Berrios, Eduardo Olgu´ ın Abstract—In this paper, we present a binary cat swarm optimization for solving the Set covering problem. The set covering problem is a well-known NP-hard problem with many practical applications, including those involving scheduling, production planning and location problems. Binary cat swarm optimization is a recent swarm metaheuristic technique based on the behavior of discrete cats. Domestic cats show the ability to hunt and are curious about moving objects. The cats have two modes of behavior: seeking mode and tracing mode. We illustrate this approach with 65 instances of the problem from the OR-Library. Moreover, we solve this problem with 40 new binarization techniques and we select the technical with the best results obtained. Finally, we make a comparison between results obtained in previous studies and the new binarization technique, that is, with roulette wheel as transfer function and V3 as discretization technique. Keywords—Binary cat swarm optimization, set covering problem, metaheuristic, binarization methods. I. I NTRODUCTION T HE Set Covering Problem (SCP) [15], [14], [26] is a classic problem that consists in finding a set of solutions which allow to cover a set of needs at the lowest cost possible. There are many applications of these kind of problems, the main ones are: location of services, files selection in a data bank, simplification of boolean expressions, balancing production lines, among others. In the field of optimization, many algorithms have been developed to solve the SCP. Examples of these optimization algorithms include: Genetic Algorithm (GA) [25], [1], Ant Colony Optimization (ACO) [3], [30], Particle Swarm Optimization (PSO) [14], [16], Firefly Algorithm [17], [18], Shuffled Frog Leaping [19], and Cultural Algorithms [15] have been also successfully applied to solve the SCP. Our proposal of algorithm uses cat behavior to solve optimization problems, it is called Binary Cat Swarm Optimization (BCSO) [32]. BCSO refers to a serie of heuristic optimization methods and algorithms based on cat behavior in nature. Cats behave in two ways: seeking mode and tracing mode. BCSO is based in CSO [29] algorithm, proposed by Chu and Tsai in 2006 [12]. The difference is that in BCSO the vector position consists of ones and zeros, instead the real numbers of CSO. Broderick Crawford is with the Pontificia Universidad Cat´ olica de Valpara´ ıso, Universidad San Sebasti´ an, and Universidad Central de Chile, Chile (e-mail: broderick.crawford@ucv.cl). Ricardo Soto is with the Pontificia Universidad Cat´ olica de Valpara´ ıso, Universidad Aut´ onoma de Chile, Chile, and Universidad Cientifica del Sur, Lima, Per´ u (e-mail: ricardo.soto@ucv.cl). Natalia Berrios is with the Pontificia Universidad Cat´ olica de Valpara´ ıso, Chile (e-mail: natalia.berriosp.p@mail.pucv.cl). Eduardo Olgu´ ın is with the Universidad San Sebasti´ an, Chile. This paper is an improvement of previous work [13], this seeks to get better results for each instace of OR-Library. We use a new method of setting parameters, which we choose different parameters for each instances set. Moreover, we tested 40 new techniques binarization [20], then we analyze the results to choose the method with which the best results. To use the binarization technique, we change the technique usually proposed for tracing mode, and we discover if this could help to improve results. This paper is structured as follows: In Section II, a brief description of what SCP is given. Section III gives: what BCSO is, the explanation and algorithm of behaviors. In Section IV, an explanation of how was BCSO used for solving the SCP is presented. Section V gives an analysis and results table. Finally, conclusions are given in Section VI. II. SET COVERING PROBLEM The SCP [9], [6], [28] can be formally defined as follows. Let A =(a ij ) be an m-row, n-column, zero-one matrix. We say that a column j can cover a row if a ij =1. Each column j is associated with a nonnegative real cost c j . Let I ={1,...,m} and J ={1,...,n} be the row set and column set, respectively. The SCP calls for a minimum cost subset S ⊆ J , such that each row i ∈ I is covered by at least one column j ∈ S.A mathematical model for the SCP is v(SCP)= min  j∈J c j x j (1) subject to  j∈J a ij x j ≥ 1, ∀ i ∈ I, (2) x j ∈{0, 1}, ∀ j ∈ J (3) The objective is to minimize the sum of the costs of the selected columns, where x j = 1 if column j is in the solution, 0 otherwise. The constraints ensure that each row i is covered by at least one column. The SCP has been applied to many real world problems such as crew scheduling [2], location of emergency facilities [35], production planning in industry [34], vehicle routing [4], ship scheduling [22], network attack or defense [7], assembly line balancing [23], traffic assignment in satellite communication systems [31], simplifying boolean expressions [8], the calculation of bounds in integer programs [10], information retrieval [21], political districting [24], stock cutting, crew scheduling problems in airlines [27] and other important real life situations. Because it has wide applicability, we deposit our interest in solving the SCP. World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:10, No:3, 2016 113 International Scholarly and Scientific Research & Innovation 10(3) 2016 scholar.waset.org/1307-6892/10004050 International Science Index, Mathematical and Computational Sciences Vol:10, No:3, 2016 waset.org/Publication/10004050