Applied Soft Computing 13 (2013) 2405–2411 Contents lists available at SciVerse ScienceDirect Applied Soft Computing j ourna l ho mepage: www.elsevier.com/locate/asoc Incorporating -dominance in AMOSA: Application to multiobjective 0/1 knapsack problem and clustering gene expression data Sanghamitra Bandyopadhyay a,∗ , Ujjwal Maulik b , Rudrasis Chakraborty a,1 a Machine Intelligence Unit, Indian Statistical Institute, Kolkata 700108, India b Jadavpur University, Kolkata 700032, India a r t i c l e i n f o Article history: Received 1 January 2011 Received in revised form 14 December 2011 Accepted 4 November 2012 Available online 28 December 2012 Keywords: Additive -AMOSA Multiobjective simulated annealing Multiobjective knapsack problem Multiplicative -AMOSA MOO strategy Multiobjective optimization -Dominance a b s t r a c t Recently, a new model of multiobjective simulated annealing, AMOSA, was developed which was found to provide improved performance for several multi objective optimization problems especially for prob- lems with many objectives. In this article, we aim to further improve the performance of AMOSA by incorporating the concept of -dominance which is a more generalized form of conventional dominance. This strategy is referred to as -AMOSA. The result of -AMOSA is compared with those of AMOSA, NSGA-II and -MOEA and AMOSA for several test problems with number of objectives varying from two to fifteen and the number of variables varying from one to thirty. The performance of -AMOSA is also compared with other strategies for multiobjective 0/1 knapsack problem. A real life application of -AMOSA for clus- tering genes from gene expression data is also demonstrated. The results demonstrate the effectiveness of -AMOSA. © 2012 Elsevier B.V. All rights reserved. 1. Introduction Problems with multiple objectives carry a different perspec- tive compared to one having a single objective. In the latter only one global optimum (minimum or maximum) is there whereas in case of multiobjective problems a set of non-dominated solu- tions is present forming the Pareto-Optimal set. The goal of a multiobjective optimization (MOO) strategy is to find the (near) Pareto-Optimal set. Several multiobjective evolutionary algorithms (MOEAs) have been suggested over the last decade as given in the reviews [5,7]. EAs are, in general, considered to be metaheuristic problem solvers as the top level strategies guide lower level ones to search for the feasible solutions. The first implementation of MOEA was done by David Schaffer who is credited with the invention of the first MOEA in the mid-80s. Since then a large number of publications and algorithms have been proposed. Recently a simulated annealing based MOO technique called AMOSA (archived multiobjective simulated annealing), pro- posed by Bandyopadhyay et al. [4], is found to outperform many of the existing methods, especially for many objectives. ∗ Corresponding author. E-mail addresses: sanghami@isical.ac.in (S. Bandyopadhyay), ujjwal maulik@yahoo.com (U. Maulik), rudrasischa@ieee.org (R. Chakraborty). 1 Current address: CVPR Unit, Indian Statistical Institute, Kolkata, India.. In MOO strategies, one solution is said to be non-dominated if it is not dominated by any other solutions. So the primary goal of a MOO strategy is to give non-dominated solutions as output while maintaining the following two properties. • Convergence: The algorithm should quickly converge to true Pareto-Optimal set (i.e., true solutions). • Diversity: The algorithm should maintain a spread of solutions. In all of the proposed MOEAs, the diversity or uniform spread- ing of the Pareto set is only taken under consideration but a little emphasis is given on the convergence of the Pareto set towards the true Pareto-Optimal set. In recent years several researchers have proposed alternative forms of dominance relation [13,20]. Farina and Amoto [13] introduces a new kind of fuzzy dominance namely (1 - k)-dominance to overcome some of the limitations of dominance relation while Rudolph [24,25] suggested a series of algorithms, all of which guarantee convergence but the diversity property is not ensured. Deb [1] suggested an algorithm which guarantees both the convergence and diversity properties but there is no proof for the convergence property. Another alternative form of dominance relation is proposed by Laumanns et al. [20], known as -dominance. This new form of dominance guarantees the con- vergence towards the Pareto-Optimal set while maintaining the diversity in the solutions found. It is basically an archiving strategy 1568-4946/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.asoc.2012.11.050