A new multi-objective particle swarm optimization method for solving reliability redundancy allocation problems Kaveh Khalili-Damghani a,n , Amir-Reza Abtahi 1,b , Madjid Tavana 2,c a Department of Industrial Engineering, South-Tehran Branch, Islamic Azad University, Tehran, Iran b Department of Knowledge Engineering and Decision Sciences, Faculty of Economic Institutions Management, University of Economic Sciences, Tehran, Iran c Business Systems and Analytics, Lindback Distinguished Chair of Information Systems and Decision Sciences, La Salle University, Philadelphia, PA 19141, USA article info Article history: Received 28 October 2011 Received in revised form 1 October 2012 Accepted 26 October 2012 Available online 5 November 2012 Keywords: Multi-objective redundancy allocation problem Meta-heuristics Dynamic self-adaptive multi-objective particle swarm optimization e-constraint method NSGA-II abstract In this paper, a new dynamic self-adaptive multi-objective particle swarm optimization (DSAMOPSO) method is proposed to solve binary-state multi-objective reliability redundancy allocation problems (MORAPs). A combination of penalty function and modification strategies is used to handle the constraints in the MORAPs. A dynamic self-adaptive penalty function strategy is utilized to handle the constraints. A heuristic cost-benefit ratio is also supplied to modify the structure of violated swarms. An adaptive survey is conducted using several test problems to illustrate the performance of the proposed DSAMOPSO method. An efficient version of the epsilon-constraint (AUGMECON) method, a modified non-dominated sorting genetic algorithm (NSGA-II) method, and a customized time-variant multi- objective particle swarm optimization (cTV-MOPSO) method are used to generate non-dominated solutions for the test problems. Several properties of the DSAMOPSO method, such as fast-ranking, evolutionary-based operators, elitism, crowding distance, dynamic parameter tuning, and tournament global best selection, improved the best known solutions of the benchmark cases of the MORAP. Moreover, different accuracy and diversity metrics illustrated the relative preference of the DSAMOPSO method over the competing approaches in the literature. & 2012 Elsevier Ltd. All rights reserved. 1. Introduction The utilization of redundancy is one of the most important attributes in meeting high-level reliability. The problem is to select the feasible design configuration that optimizes the mea- surement functions such as reliability, cost, weights, and risk [10]. This is called the reliability redundancy allocation problem (RAP) which was first introduced by Misra and Ljubojevic [17]. A series- parallel system is characterized through a predefined number of sub-systems which are connected serially. Multiple component choices and redundancy levels are available to connect in parallel for each sub-system [10]. A given component may have a binary- state or a multi-state in the RAPs [13]. In binary-state RAP, the problem of a proper structure can be handled by increasing the reliability of components or supplying parallel redundant compo- nents at some stages [10]. In some other cases, called multi-state systems, the states of a given component may follow more than two different levels, ranging from perfectly working to completely failed [1]. The RAP is assumed to be a NP-hard (non-deterministic polynomial-time hard) problem [3]. The application and the development of the meta-heuristic procedures are assumed to be useful to properly solve NP-hard problem. Different heuristic and meta-heuristic methods such as Evolutionary Computation methods, variable neighborhood search, ant colony optimization, and particle swarm optimization (PSO) were proposed in this area ([4,7,10,14,15,19,21,25]). Gen and Yun [7] surveyed the Genetic Algorithm-based (GA-based) approaches for various reliability optimization problems. Konak et al. [11] presented an overview and tutorial describing GA developed specifically for problems with multiple objectives. Li et al., 2009 [15] proposed a two-stage approach for solving multi-objective system reliability optimiza- tion problems. In the first stage, a Multi-Objective Evolutionary Algorithm (MOEA) generated non-dominated solutions. Then, a Self-Organizing Map (SOM) was used to cluster similar solutions. Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/ress Reliability Engineering and System Safety 0951-8320/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ress.2012.10.009 Abbreviations: ANOVA, analysis of variance; AUGMECON, efficient epsilon-constraint method; CI, confidence intervals; cTV-MOPSO, customized time-variant multi-objective particle swarm optimization; DM, decision maker; DiM, diversification metric; DSAMOPSO, dynamic self-adaptive multi-objective particle swarm optimization; ER, error ratio; GA, Genetic Algorithms; GD, generational distance; MODM, multi-objective decision making; MOEA, Multi-Objective Evolutionary Algorithm; MORAP, multi-objective reliability redundancy allocation problem; NNS, number of non-dominated solutions; NSGA-II, non-dominated sorting genetic algorithm; PSO, particle swarm optimization; RAP, redundancy allocation problem; RS, reference set; SM, spacing metric; SOM, self-organizing map; Std. Dev., standard deviation n Corresponding author. Tel.: þ98 912 3980373; fax: þ98 21 77868749. E-mail addresses: kaveh.khalili@gmail.com (K. Khalili-Damghani), amir_abtahi@yahoo.com (A.-R. Abtahi), tavana@lasalle.edu (M. Tavana). 1 Tel.: þ98 912 1887920; fax: þ98 21 44453640. 2 Tel.: þ215 951 1129; fax: þ267 295 2854. Reliability Engineering and System Safety 111 (2013) 58–75