Use of chaotic sequences in a biologically inspired algorithm for engineering design optimization Leandro dos Santos Coelho a, * , Viviana Cocco Mariani b a Pontifical Catholic University of Parana ´ , Industrial and Systems Engineering Graduate Program, PUCPR/CCET/PPGEPS, Imaculada Conceic ¸a ˜o, 1155, Curitiba 80215-901, PR, Brazil b Pontifical Catholic University of Parana ´ , Graduate Program in Mechanical Engineering, PUCPR/CCET/PPGEM Imaculada Conceic ¸a ˜o, 1155, Curitiba 80215-901, PR, Brazil Abstract Recent computational developments in ant colony systems have proved fruitful for transforming discrete domains of application into continuous ones. In this paper, new combinations of an ant colony inspired algorithm (ACA) and chaotic sequences (ACH) 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 ACA and ACH handle such problems efficiently in terms of precision and convergence and, in most cases, they outperform the results presented in the literature. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Metaheuristics; Ant colony; Chaos theory; Mechanical design; Chaotic sequences; Continuous optimization 1. Introduction Recently, a new class of heuristic techniques, called swarm intelligence, was proposed (Bonabeau, Dorigo, & Theraulaz, 1999; Kennedy, Eberhart, & Shi, 2001; Dorigo & Stu ¨ tzle, 2004). In this context, researchers in the field of ‘‘artificial life’’ have lately been turning to insects for ideas that can be applied in heuristics. Many aspects of the col- lective activities of social insects, such as ant foraging, bird flocking and fish schooling, are self-organizing, meaning that complex group behavior emerges from the interactions of individuals that exhibit simple behaviors by themselves (Bonabeau et al., 1999; Kennedy et al., 2001). Swarm intelligence is an emerging research area that emphasizes cooperative behavior among group members. Swarm intelligence methodologies are used to solve optimi- zation problems and also cooperative tasks among intelli- gent agents, mainly in artificial neural network training (Van den Bergh & Engelbrecht, 2001), multi-objective opti- mization problems (Hu & Eberhart, 2002), economic dis- patch of electrical energy (Victoire & Jeyakumar, 2004), cooperative and/or decentralized control (Baras, Tan, & Hovareshti, 2003), electromagnetic optimization (Baum- gartner, Magele, & Renhart, 2004), and others. Swarm intelligence is inspired by nature, based on the fact that, among the living animals of a group, each indi- vidual contributes with its own experience to the group, making it stronger in relation to other groups. The most familiar representatives of swarm intelligence in optimiza- tion problems are: the food-searching behavior of ants (Dorigo & Di Caro, 1999), particle swarm optimization (Shi & Eberhart, 2000), bacterial foraging (Sierakowski & Coelho, 2005), spider colonies (Bourjot, Chevier, & Thomas, 2003) and artificial immune systems (Castro & Timmis, 2002). The main ant algorithms presented in the literature are: the ant system, elitist ant system, ant-Q, ant colony system, Max–Min ant system, rank-based ant system, ANTS, and hyper-cube ant system (Dorigo & Stu ¨tzle, 2004). 0957-4174/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2007.02.002 * Corresponding author. Tel.: +55 41 3271 13 33; fax: +55 41 3271 13 45. E-mail addresses: leandro.coelho@pucpr.br (L.d.S. Coelho), viviana. mariani@pucpr.br (V.C. Mariani). www.elsevier.com/locate/eswa Available online at www.sciencedirect.com Expert Systems with Applications 34 (2008) 1905–1913 Expert Systems with Applications