Applied Soft Computing 12 (2012) 3462–3471 Contents lists available at SciVerse ScienceDirect Applied Soft Computing j ourna l ho me p age: www.elsevier.com/l ocate/asoc Annealing-based particle swarm optimization to solve the redundant reliability problem with multiple component choices Nima Safaei a,∗ , Reza Tavakkoli-Moghaddam b , Corey Kiassat a a Department of Mechanical and Industrial Engineering, University of Toronto, 5 King’s College Rd., Toronto, Ontario M5S 3G8, Canada b Department of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran a r t i c l e i n f o Article history: Received 23 January 2012 Received in revised form 17 April 2012 Accepted 9 July 2012 Available online 28 July 2012 Keywords: Particle swarm optimization Annealing algorithm Series–parallel redundant reliability system Multiple component choices a b s t r a c t In this paper, the performance of a particle swarm optimization (PSO) algorithm named Annealing-based PSO (APSO) is investigated to solve the redundant reliability problem with multiple component choices (RRP-MCC). This problem aims to choose an optimal combination of components and redundancy levels for a system with a series–parallel configuration that maximizes the overall system reliability. PSO is a population-based meta-heuristic algorithm inspired by the social behavior of the biological swarms that is designed for continuous decision spaces. As a local search engine (LSE), the proposed APSO employs the Metropolis-Hastings strategy, the key idea behind the simulated annealing (SA) algorithm. In APSO, the best position among all particles in each iteration is dynamically improved using the inner loop of the SA (i.e., equilibrium loop) while the temperature is updated in the main loop of the PSO algorithm. The well-known benchmarks are used to verify the performance of the proposed APSO. Even though APSO fails to outperform the best solution obtained in the literature, the contribution of this paper is comprised of the implementation of APSO as a hybrid meta-heuristic as well as the effect of Metropolis-Hastings strategy on the performance of the classical PSO. © 2012 Elsevier B.V. All rights reserved. 1. Introduction Reliability is the mathematical probability that a component or system performs a required function for a given period of time under stated operating conditions. In most systems, it is impossible to handle component failures as quickly as the demand require- ments dictate. Hence, one of the main approaches used to enhance the system reliability utilizes redundant elements in various sub- systems. The use of this approach leads to a selection of the optimal combination of elements and redundancy levels, and the enhance- ment of the system operation, known as the redundant reliability problem or redundancy allocation problem [1,2]. However, as this approach is used, it should be kept in mind that other factors (e.g., cost, weight, volume, or time) also increase. Thus, the design of the reliability optimization problem is phrased as reliability improvement at a minimum cost. The common sense perception of reliability is the absence of failures. Therefore, reliability is some- times referred to as “quality in time dimension”, because it is determined by the failures that may or may not occur to the prod- uct during its life span [3]. Based on this approach, a series–parallel ∗ Corresponding author. Tel.: +1 416 946 3939; fax: +1 416 978 3453. E-mail address: safaei@mie.utoronto.ca (N. Safaei). redundant system consists of a number of subsystems in series, in which redundant components are used in parallel for each sub- system. If there are multiple component choices (or options) for each subsystem, the considered system is called a series–parallel redundant system with multiple component choices. In this case, it is assumed that a mix of components is allowed within a subsystem where numerous component types are available for redundancy in parallel. The redundant reliability problem with multiple compo- nent choices (RRP-MCC) aims at maximizing the system reliability by choosing the optimal combination of components and redun- dancy levels to meet the resource constraints, such as budget, weight, capacity, and/or time. 2. Literature review There are a number of studies in the literature dealing with the exact and heuristic methods to solve the redundant reliability problem with (or without) multiple component choices. Misra and Sharma [1] introduced redundant components and used a geomet- ric programming approach to solve the RRP-MCC. Chern [2] first showed that a redundancy allocation problem in series systems with linear resource constraints is an NP-hard problem. Hiller and Lieberman [4] used a dynamic programming approach to solve the RRP-MCC. Coit and Smith [5] used a genetic algorithm (GA) to solve the RRP-MCC. 1568-4946/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.asoc.2012.07.020