Multi-objective chicken swarm optimization: A novel algorithm for solving multi-objective optimization problems

Highlights

• Use an external archive to guide the chicken swarm to find the Pareto optimal solutions.

• Use an aggregate function to define the social hierarchy of chickens during the exploration of search spaces.

• Adapt the chicken’s movements to explore a multi-objective search space.

• Use the epsilon dominance and the crowding distance to preserve the diversity of chicken’s population.

Abstract

In this paper, we extend the chicken swarm optimization (CSO) to solve multi-objective optimization problems. Our extention aims to balance between diversity and convergence when searching for the optimal Pareto solutions. We use aggregation function to define the social hierarchy and simulate the behavior of chickens during the search for food in the objective search space while applying epsilon dominance and crowding distance to preserve the diversity of the solutions population. We also address the integration of the archive population that guides the chicken swarm towards the Pareto optimal solutions.

The proposed algorithm is validated on twelve test functions and compared with five well-known meta-heuristics. The results show the ability of MOCSO algorithm to provide a better spread of solutions with faster convergence.

Introduction

Many optimization problems from the industrial world (telecommunication, environment, transportation, etc.) are multidimensional in nature Coello et al., 2002, Marler and Arora, 2004a, Andersson, 2000. Indeed, the optimization problems encountered in practice are rarely single-objective and need to satisfy several contradictory criteria or objectives simultaneously. However, the complexity of multi-objective optimization problems increases according to the problem size, as well as the number of objective functions to optimize. In contrast with the single-objective optimization, in a multi-objective optimization problem, we do not search for a single optimal solution but for a set of non-dominated solutions, known as a Pareto-optimal set approximation Zitzler and Thiele, 1998, Zhou et al., 2011.

Therefore, to handle multi-objective problems there are two main approaches: a priori and a posteriori (Branke et al., 2004, Marler and Arora, 2004b). For the first approach, the problem is transformed into a single-objective by applying aggregation on the objectives; also an expert in the domain has to provide the set of weights according to the importance of each objective. This method can generate one Pareto optimal solution (Das and Dennis, 1998, Kim and de Weck, 2005, Messac and Mattson, 2002). Contrariwise, a posteriori approach generates the whole or part of the Pareto Optimal set, then the decision maker selects the most appropriate solution among the generated set (Deb, 2012, Marler and Arora, 2004b).

In the literature, a particular attention has been placed on small bi-objective instances problems using the exact methods such as the two-phase method Przybylski, Gandibleux, and Ehrgott (2008) later improved by Raith and Ehrgott, 2009, Przybylski et al., 2010, the branch and bound Przybylski et al., 2008, Vincent et al., 2013, dynamic programming Klamroth and Wiecek (2000), etc. These methods are effective for small sized problems. However, no exact method has been found effective for problems with large dimensions and more than two objectives. Unlike the exact methods, metaheuristics can approximate the Pareto set for the large-sized problem in a reasonable computational time. Basically, the existing multi-objective metaheuristics can be classified into three categories: The multi-objective evolutionary algorithms, the multi-objective swarm intelligence algorithms, and the multi-objective cooperative metaheuristics (Zouache, Moussaoui, & Ben Abdelaziz, 2018).

Multi-objective evolutionary algorithms have various instances of the algorithms, one of them is Non dominated Sorting Genetic algorithm NSGA which improve the adaptive fit of a population of candidate solutions to a Pareto front constrained by a set of objective functions. However, this algorithm has a high computational cost. For those reasons, (Deb, Pratap, Agarwal, & Meyarivan, 2002) has suggested an improved version of NSGA called NSGA-II ‘Non-dominated sorting genetic algorithm’.

Another common algorithm is the Strength Pareto Evolutionary Algorithm (SPEA-II), proposed by Zitzler, Laumanns, and Thiele (2001). SPEA-II algorithm represents a modified version of the SPEA algorithm. In contrast with NSGA-II, SPEA-II is based on the use of an external population called archive. This external population contains a limited number of non-dominated solutions generated by the algorithm during the optimization phase. At any iteration, the new non-dominated solutions of the population are compared to members of the archive using the dominance criterion. To ensure a diversity of the obtained solutions with a lower computational cost, Deb et al. proposed $∊$-MOEA algorithm (Deb, Mohan, & Mishra, 2005) which uses a stationary evolution scheme and is based on the $∊$-dominance relation. To provide a better compromise to the ‘exploration vs exploitation’ criteria, Indicator-Based Evolutionary Algorithm (IBEA) has been developed by (Zitzler & Künzli, 2004). In the last decade, several extensions and improvements have been proposed in the area of multi-objective evolutionary algorithms including the multi-objective evolutionary algorithm based on decomposition (MOEA/D) (Zhang & Li, 2007), the multi-objective evolutionary algorithm based on decision variable analyses (Ma et al., 2016) and the multi-objective Evolutionary Algorithm Based on the Generational Distance Indicator (Rodríguez Villalobos & Coello Coello, 2012).

For multi-objective metaheuristics based on the swarm intelligence, we can cite the two most popular methods: Multi-objective particle swarm optimization (Lin, Li, et al., 2015, Reyes-Sierra and Coello, 2006), and multi-objective ant colony optimization (García-Martínez et al., 2007, Rada-Vilela et al., 2013). Over the last few years, several swarm intelligence methods have been invented and extended to solve multi-objective optimization problems (MOP): multi-objective firefly algorithm (Yang, 2013), multi-objective cat swarm optimization (Bilgaiyan, Sagnika, & Das, 2015), multi-objective harmony search algorithm (Salcedo-Sanz et al., 2013), multi-objective teaching learning-based optimization algorithm (Lin, Yu, et al., 2015), multi-objective artificial bee colony algorithm (Akbari, Hedayatzadeh, Ziarati, & Hassanizadeh, 2012), multi-objective bat algorithm (Xinshe, 2011), multi-objective artificial bee colony algorithm (Xiang, Zhou, & Liu, 2015) and multi-objective discrete cooperative swarm intelligence algorithm (Zouache et al., 2018).

Several hybrid metaheuristics have been also proposed to take advantage of more than one metaheuristics at the same time. Jaszkiewicz (2002) proposed the multi-objective genetic local search (MOGLS) which combines a genetic algorithm with tabu search operator. Abdelaziz, Krichen, and Chaouachi (1999) proposed one of the first hybrid heuristics based on the combination of a tabu search algorithm and a genetic algorithm dedicated to the multi-objective knapsack problem. Numerous evolutionary algorithms that integrate a hyper-volume indicator calculation have also been introduced (Auger et al., 2012, Jiang et al., 2015). Recently, a new swarm intelligence algorithm, called chicken swarm optimization (CSO), has been proposed by Meng, Liu, Gao, and Zhang (2014). The first studies have proven that the CSO algorithm provides better results compared to well-known evolutionary algorithms for solving global optimization problems. In this work, we propose a Multi-Objective Chicken Swarm Optimization (MOCSO) to optimize multi-objective problems with the following five majors contributions:

• An archive is integrated into CSO for saving and retrieving the Pareto optimal solutions.

• The use of the aggregate function to define the social hierarchy and simulate the behavior of chickens during the search for food in the multi-objective search spaces.

• The adaptation of the different movements of the chickens to explore a multi-criteria search space

• The use of epsilon distance and the crowding distance to manage efficiently the archive population that contains the best non-dominated solutions during the exploration of the space search.

• The repair diversity operator is introduced in CSO to avoid the premature convergence in case of a large number of local optima.

The rest of the paper is organized as follows. A preliminary view of the used concepts is provided in Section 2. It includes a brief overview of the multi-objective optimization problem (Section 2.1), the main definitions of the Pareto dominance relation (Section 2.2) and a short view of the chicken swarm optimization (Section 2.3). Section 3 is devoted to the description of the MOCSO algorithm: the global description of MOCSO algorithm, the update of the hierarchical order between the chickens, the movement of the chickens, the update of the archive, the improvement of the diversity of solutions population and the pseudo-code of our algorithm. The experimental results are presented and discussed in Section 4. Finally, Section 5 concludes our work and suggests some directions for future research.