A Multi-Objective Gravitational Search Algorithm Based on Non-Dominated Sorting
چکیده
This paper proposes an extension of the Gravitational Search Algorithm (GSA) to multi-objective optimization problems. The new algorithm, called Non-dominated Sorting GSA (NSGSA), utilizes the non-dominated sorting concept to update the gravitational acceleration of the particles. An external archive is also used to store the Pareto optimal solutions and to provide some elitism. It also guides the search toward the non-crowding and the extreme regions of the Pareto front. A new criterion is proposed to update the external archive and two new mutation operators are also proposed to promote the diversity within the swarm. Numerical results show that NSGSA can obtain comparable and even better performances as compared to the previous multiobjective variant of GSA and some other multi-objective optimization algorithms. DOI: 10.4018/jsir.2012070103 International Journal of Swarm Intelligence Research, 3(3), 32-49, July-September 2012 33 Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. iterations and some algorithms show better performances for particular problems than the others. Hence, searching for new heuristic optimization algorithms is an open problem. Although both single and multi-agent metaheuristics have their own multi-objective variants, the multi-agent meta-heuristics such as EAs, ACO, and PSO have shown better potential to solve multi-objective optimization problems, because in a multi-objective problem, a population of solutions is going to be generated, in a single run. The multi-objective variants of the single agent meta-heuristics such as SA and TS are often working based on the definition of an aggregation function in their structure (Ulungu et al., 1999; Hansen, 1997). The use of aggregation functions may reduce the search area and provide only a subset of the Pareto front. GSA is a new multi-agent optimization algorithm, inspired from the general gravitational law (Rashedi et al., 2009). The algorithm is based on the movement of some particles under the effect of the gravitational forces, applied by the others. Hassanzadeh and Rohani (2010) have proposed the first multi-objective variant of GSA, called Multi-Objective GSA (MOGSA). MOGSA uses an external archive to store the non-dominated solutions and uses the same idea as in Simple Multi-Objective PSO (SMOPSO), proposed by Cagnina et al. (2005), to update the external archive. For this purpose, the interval of each objective function is divided into equal divisions and consequently the entire performance space is divided into hyper-rectangles. When the number of archived members exceeds the maximum archive length, one member of the most crowded hyper-rectangle is randomly chosen and removed. The main problem in proposing a multiobjective variant for GSA is updating the mass of particles based on the value of the multiple objectives. In MOGSA, the mass of all moving particles is set to one and the mass of archived particles is updated based on the distance from the nearest neighbor, within the objective space. However, no equation has been presented that relates the mass value to the distance value. This technique distributes the archived elements uniformly, similar to the niching technique. After calculating the mass of the moving particles, they move to the new positions by the forces applied from the archived members. In MOGSA, only the archived particles apply gravitational forces to the moving ones and after each movement, the well-known uniform mutation is applied to the new positions. In this paper, the authors have proposed a new multi-objective variant of GSA, called NSGSA. In single-objective GSA, proposed by Rashedi et al. (2009), the mass of each particle is taken to be proportional to its fitness. As mentioned, the main problem to propose a multiobjective GSA is to establish a relationship between the mass of each particle and its multiple objectives. In this paper, the non-dominated sorting concept, proposed in Non-dominated Sorting GA (NSGA) (Srinivas & Deb, 1994), is used to divide the particles to several layers within the performance space. In this way, the mass of each particle will depend to the rank of the layer it belongs to. NSGSA utilizes a limited length external archive to store the last found non-dominated solutions. To provide some elitism, specific members of the external archive are added to the list of moving particles and contribute in applying gravitational forces to the others. A new criterion is introduced to update the external archive, based on a new defined spread indicator. The mutation (turbulence) phenomenon is also modeled in NSGSA, not considered in the original GSA. In this regard, two new mutation operators, called sign and reordering mutations, are also proposed. The paper is organized as follows: At first, GSA is described in detail. Thereafter, the new multi-objective variant of GSA, called NSGSA, is introduced. Next, numerical results are presented. Then, the results of NSGSA are compared with those of MOGSA, SMOPSO, and NSGA-II (Deb et al., 2002) and their sensitivity to the value of tuned parameters is also investigated. Finally, the conclusions are outlined. 34 International Journal of Swarm Intelligence Research, 3(3), 32-49, July-September 2012 Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. GRAVITATIONAL SEARCH ALGORITHM GSA is a new optimization algorithm, proposed by Rashedi et al. (2009). This algorithm has been inspired from the mass interactions based on the general gravitational law. In the proposed algorithm, the search agents are a collection of masses, interact with each other based on the Newtonian laws on gravity and motion. Also, it uses some stochastic operators to increase diversity and the probability of finding the global optimum. In the following, the formulation of GSA is presented in short. The mass of each particle is calculated according to its fitness value, as follows: