Gaussian quantum-behaved particle swarm optimization approaches for constrained engineering design problems Leandro dos Santos Coelho * Pontifical Catholic University of Paraná, Graduate Program in Industrial and Systems Engineering, Automation and Systems Laboratory, PUCPR/PPGEPS Imaculada Conceição, 1155, 80215-901 Curitiba, PR, Brazil article info Keywords: Particle swarm optimization Quantum computation Mechanical design Gaussian distribution Continuous optimization Engineering design Swarm intelligence abstract Particle swarm optimization (PSO) is a population-based swarm intelligence algorithm that shares many similarities with evolutionary computation techniques. However, the PSO is driven by the simulation of a social psychological metaphor motivated by collective behaviors of bird and other social organisms instead of the survival of the fittest individual. Inspired by the classical PSO method and quantum mechanics theories, this work presents novel quantum-behaved PSO (QPSO) approaches using mutation operator with Gaussian probability distribution. The application of Gaussian mutation operator instead of random sequences in QPSO is a powerful strategy to improve the QPSO performance in preventing pre- mature convergence to local optima. In this paper, new combinations of QPSO and Gaussian probability distribution are employed in well-studied continuous optimization problems of engineering design. Two case studies are described and evaluated in this work. Our results indicate that Gaussian QPSO approaches handle such problems efficiently in terms of precision and convergence and, in most cases, they outperform the results presented in the literature. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction Recently, a new class of metaheuristics, called swarm intelli- gence, was proposed (Bonabeau, Dorigo, & Theraulaz, 1999; Dorigo & Stützle, 2004; Kennedy, Eberhart, & Shi, 2001). The swarm intel- ligence is an emerging research field that presents features of self- organization and cooperation principles among group members bio-inspired on social insect societies. Swarm intelligence is in- spired by nature, based on the fact that the live animals of a group contribute with their individual experiences to the group, render- ing it stronger to face other groups. The most familiar representatives of swarm intelligence in optimization problems are: the food-searching behavior of ants (Dorigo & Di Caro, 1999), particle swarm optimization (Shi & Eberhart, 2000), bacterial colony (Sierakowski & Coelho, 2005), and spider colonies (Bourjot, Chevier, & Thomas, 2003). In this context, the development of bio-inspired methodologies based on particle swarm optimization (PSO) systems is a relevant research area with applications in areas such as control systems (Coelho, Oliveira, & Cunha, 2005), data mining (Sousa, Silva, & Neves, 2004), manufacturing (Andrés & Lozano, 2006), robotics (Coelho & Sierakowski, 2008), structural reliability (Elegbede, 2005), power systems (Chuanwen & Bompard, 2005), electromag- netics (Adly & Abd-El-Hafiz, 2004), clustering (Chen & Zhao, 2009), dynamic question generation (Cheng, Lin, & Hunag, 2009), classification (Marinakis, Marinaki, & Dounias, 2008), support vec- tor machines (Lin, Ying, Chen, & Lee, 2008), communication net- works (Huang, Chuang, & Yang, 2008), reliability-redundancy optimization (Coelho, 2009), inventory planning (Tsou, 2008), im- age segmentation (Maitra & Chatterjee, 2008), and others. The particle swarm optimization (PSO) originally developed by Kennedy and Eberhart in 1995 (Eberhart & Kennedy, 1995; Ken- nedy & Eberhart, 1995) is a population-based swarm algorithm. Similarly to genetic algorithm (Goldberg, 1989), an evolutionary algorithm approach, PSO is an optimization tool based on a popu- lation, where each member is seen as a particle, and each particle is a potential solution to the problem under analysis. Each particle in PSO has a randomized velocity associated to it, which moves through the space of the problem. However, unlike genetic algorithm, PSO does not have operators, such as crossover and mutation. PSO does not implement the survival of the fittest individuals; rather, it implements the simulation of social behavior. At the end of the 19th century, classical mechanics encountered major difficulties in describing motions of microscopic particles with extremely light masses and extremely high velocities, and the physical phenomena related to such motions. This forced scien- tists to rethink the applicability of classical mechanics and lead to fundamental changes in their traditional understanding of the nat- ure of motions of microscopic objects (Pang, 2005). The studies of Bohr, de Broglie, Schrödinger, Heisenberg and Bohn in 1920s 0957-4174/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2009.06.044 * Tel.: +55 41 3271 13 33; fax: +55 41 3271 13 45. E-mail address: leandro.coelho@pucpr.br Expert Systems with Applications 37 (2010) 1676–1683 Contents lists available at ScienceDirect Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa