ARTICLE IN PRESS Reliability Engineering and System Safety 94 (2009) 830–837
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Reliability Engineering and System Safety journal homepage: www.elsevier.com/locate/ress
An efficient particle swarm approach for mixed-integer programming in reliability–redundancy optimization applications Leandro dos Santos Coelho ´, PUCPR, Imaculada Conceic- ˜ ´, Brazil Industrial and Systems Engineering Graduate Program, LAS/PPGEPS, Pontifical Catholic University of Parana ao, 1155, 80215-901 Curitiba, Parana
a r t i c l e in f o
a b s t r a c t
Article history: Received 12 November 2007 Received in revised form 29 August 2008 Accepted 1 September 2008 Available online 16 September 2008
The reliability–redundancy optimization problems can involve the selection of components with multiple choices and redundancy levels that produce maximum benefits, and are subject to the cost, weight, and volume constraints. Many classical mathematical methods have failed in handling nonconvexities and nonsmoothness in reliability–redundancy optimization problems. As an alternative to the classical optimization approaches, the meta-heuristics have been given much attention by many researchers due to their ability to find an almost global optimal solutions. One of these meta-heuristics is the particle swarm optimization (PSO). PSO is a population-based heuristic optimization technique inspired by social behavior of bird flocking and fish schooling. This paper presents an efficient PSO algorithm based on Gaussian distribution and chaotic sequence (PSO-GC) to solve the reliability– redundancy optimization problems. In this context, two examples in reliability–redundancy design problems are evaluated. Simulation results demonstrate that the proposed PSO-GC is a promising optimization technique. PSO-GC performs well for the two examples of mixed-integer programming in reliability–redundancy applications considered in this paper. The solutions obtained by the PSO-GC are better than the previously best-known solutions available in the recent literature. & 2008 Elsevier Ltd. All rights reserved.
Keywords: Reliability–redundancy optimization Particle swarm optimization Evolutionary algorithm Meta-heuristics
1. Introduction In 1952, the Advisory Group on the Reliability of Electronic Equipment defined the reliability in a broader sense: reliability indicates the probability implementing specific performance or function of products and achieving successfully the objectives within a time schedule under a certain environment [1]. A design engineer often tries to improve system reliability with a basic design, to the largest extent possible subject to constraints on any component attributes (cost, weight, and volume) of system [2]. The problem is to select the optimal combination of components and redundancy levels to meet system level constraints while maximizing system reliability. Recently, many meta-heuristics [3,4], such as evolutionary algorithms [5–13], tabu search [14,15], ant colony optimization [16–19], artificial immune system [20], fuzzy system [21], and artificial neural networks [22] have been employed in
Abbrevations: PSO, particle swarm optimization; PSO-CA, canonical particle swarm optimization; PSO-CO, particle swarm optimization with constriction factor; PSO-GC, Gaussian probability distribution and also chaotic sequences in particle swarm optimization Tel./fax: +55 41 327113 45. E-mail address:
[email protected] 0951-8320/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ress.2008.09.001
reliability–redundancy optimization problems. One of these modern meta-heuristics is the particle swarm optimization (PSO). PSO, first introduced by Kennedy and Eberhart [23,24], is a stochastic global optimization technique inspired by social behavior of bird flocking or fish schooling. It simulated the feature of bird flocking and fish schooling to configure the heuristic learning mechanism. PSO is initialized with a population of random solutions within the feasible range, called particles (individual). The learning procedure of PSO is that the solution of every individual particle is modified with the cause of its own best experience and other individuals’ best experiences. In other words, the particles fly through the search space influenced by two factors: one is the individual’s best position ever found (personal best); the other is the group’s best position (global best). In this case, each particle in PSO flies through the search space with a velocity that is dynamically adjusted according to its own and its cognitive and social behaviors. In canonical PSO, a uniform probability distribution to generate random numbers is used. However, the use of other probability distributions may improve the ability to fine-tuning or even to escape from local optima. In the meantime, it has been proposed the use of the Gaussian [25–27], Cauchy [28,29], exponential [30], Le´vy [31] probability distribution functions, and chaotic sequences [32–35] to generate random numbers to updating the velocity equation in PSO.
ARTICLE IN PRESS L.S. Coelho / Reliability Engineering and System Safety 94 (2009) 830–837
831
tmax ud
Nomenclature a constant of He´non map a constant of He´non map the cognitive learning rate the social learning rate the objective function for the overall system reliability g the set of constraint functions gbest the global best particle gi the ith constraint function k the iteration number in He´non map l the vector of resource limitation m the number of subsystems in the system n ¼ (n1,n2,n3, y, nm) the vector of the redundancy allocation for the system ni the number of components in the ith subsystem pbest the personal best particle pi ¼ [pi1,pi2, y, pin]T the best previous position of the ith particle r ¼ (r1,r2,r3, y, rm) the vector of the component reliabilities for the system ri the reliability t the iterations (generations)
a b c1 c2 f( )
This paper employs the Gaussian probability distribution and also chaotic sequences in PSO (PSO-GC) design to solve the reliability–redundancy optimization problems. In this context, two examples in reliability–redundancy design are evaluated. The results of proposed PSO-GC algorithm, canonical PSO (PSO-CA), and PSO with constriction factor (PSO-CO) are compared. The novel PSO-GC algorithm outperforms and provide solutions when compared with PSO-CA, PSO-CO, and also other techniques presented in literature for the two reliability–redundancy optimization examples [36–38]. The remaining content of this paper is organized as follows. In Section 2, the reliability–redundancy optimization problem is introduced, while the concepts of PSO approaches are explained in Section 3. Section 4 presents the simulation results for two reliability–redundancy optimization problems. Finally, Section 5 contains the concluding remarks and further research.
2. Description of reliability–redundancy optimization problem
the maximum number of allowable iterations a uniformly-distributed random number within the range [0,1] vi the volume of each component in subsystem i vmax the maximum velocity that each particle can make at each iteration xi ¼ [xi1,xi2, y, xin]T the position of the ith particle of population wi the weight of each component in subsystem i y1 the output of He´non map y2 a signal of state in He´non map C the upper limit on the cost of the system F the feasible region Rs the system reliability S the search space V the upper limit on the sum of the subsystems’ products of volume and weight Ud a uniformly-distributed random number within the range [0,1] W the upper limit on the weight of the system li ¼ [li1,li2, y, lin]T the velocity of the ith particle j a parameter of design o the inertia weight w the constriction coefficient
r ¼ (r1,r2,r3, y, rm) is the vector of the component reliabilities for the system, n ¼ (n1,n2,n3, y, nm) is the vector of the redundancy allocation for the system; ri and ni are the reliability and the number of components in the ith subsystem, respectively; f( ) is the objective function for the overall system reliability; and l is the vector of resource limitation; m is the number of subsystems in the system. The goal is to determine the number of component and the components’ reliability in each system so as to maximize the overall system reliability. The problem belongs to the category of constrained nonlinear mixed-integer optimization problems. 2.1. Example 1: complex (bridge) system The first example problem used to demonstrate the efficiency of PSO approaches were proposed in [38,40,41]. Fig. 1 represents the complex (bridge) system analyzed in this paper. The complex (bridge) system optimization problem can be stated as follows [38]: maximize
The goal of reliability engineering is to improve the reliability system. The reliability–redundancy optimizations are useful for system designs that are largely assembled and manufactured using off-the-shelf components, and also, have high reliability requirements [39]. A reliability–redundancy optimization problem can be formulated with system reliability as the objective function or in the constraint set. In this work, the reliability–redundancy allocation problem of maximizing the system reliability subject to multiple nonlinear constraints can be stated as a nonlinearly mixed-integer programming model in general form as follows: Maximize
Rs ¼ f ðr; nÞ,
(1)
subject to
gðr; nÞpl
(2)
0pr i p1;
r i 2
