Elsevier

Applied Soft Computing

Volume 38, January 2016, Pages 423-436
Applied Soft Computing

An efficient memory-based electromagnetism-like mechanism for the redundancy allocation problem

https://doi.org/10.1016/j.asoc.2015.09.043Get rights and content

Highlights

  • An efficient MBEM is proposed to solve the redundancy allocation problem.

  • MBEM employs a memory matrix in local search to save the features of good solutions.

  • Various test problems and benchmarks are used to evaluate the performance of MBEM.

  • Experimental results show that optimal solutions of all benchmark instances are obtained.

  • The computer execution times of the algorithm on all large-scale instances are reasonable.

Abstract

Meta-heuristic algorithms have been successfully applied to solve the redundancy allocation problem in recent years. Among these algorithms, the electromagnetism-like mechanism (EM) is a powerful population-based algorithm designed for continuous decision spaces. This paper presents an efficient memory-based electromagnetism-like mechanism called MBEM to solve the redundancy allocation problem. The proposed algorithm employs a memory matrix in local search to save the features of good solutions and feed it back to the algorithm. This would make the search process more efficient. To verify the good performance of MBEM, various test problems, especially the 33 well-known benchmark instances in the literature, are examined. The experimental results show that not only optimal solutions of all benchmark instances are obtained within a reasonable computer execution time, but also MBEM outperforms EM in terms of the quality of the solutions obtained, even for large-size problems.

Introduction

Optimization of system reliability is a widely surveyed subject in the field of reliability, where various types of models have been introduced so far. There are two general strategies to maximize system reliability: (1) increasing the reliability of system components and (2) finding the optimal number of redundant components used in the system. The latter that introduces the so-called redundancy allocation problem (RAP) involves a combinatorial optimization problem, in which the aim is to find optimal number of proper components [1], [2], [3]. The models proposed for RAP are applicable to design highly reliable systems that are assembled and manufactured using off-the-shelf components. Some real-world applications include transportation system design [3], telecommunication design [4], and electrical power system design [5].

Depending on system configuration, the structure of a system in the literature of RAP is categorized as series, parallel, series–parallel [6], k-out-of-n [7], and hierarchical series–parallel or complex. Among these, the series–parallel is a common structure that is used in many designs due to its wide applications and in this research; the RAP is addressed for this system structure. A series–parallel system includes a total of s independent subsystems arranged in series, in which the ith subsystem has up to nmax functionally equivalent components arranged in parallel. Each component potentially differs in reliability, weight, cost, and other features. A subsystem can work efficiently if at least one of its components is operational. Besides, the number of components in the ith subsystem, ni, can be selected from Ti available component types where multiple copies of each type can be chosen. The schematic view of a typical series–parallel system is depicted in Fig. 1.

Reliability of a series–parallel system can be improved in four ways by: (1) using components that are more reliable, (2) increasing redundant components in parallel, (3) simultaneously utilizing the previous two, and (4) enabling repeatedly the allocation of the entire system framework [8]. However, there are two types of limitations involved: (1) resources like cost, component volume, and techniques are usually confined and (2) system constraints such as weight, capacity, and other specifications are fixed and cannot be changed in short notice [9]. While it is very difficult to work out an optimal solution under multiple constraints, several optimization algorithms have been proposed in the past decades to solve the series–parallel redundancy allocation problem. Fyffe et al. [10] proposed a mathematical model for RAP without component mixing under cost and weight constraints and employed a dynamic programming approach to solve it.

Chern [11] proved that a RAP in series systems with linear resource constraints belongs to the class of Non-deterministic Polynomial-time hard (NP-hard) problems. Hence, researchers started to focus on developing meta-heuristics that can obtain near-optimum solutions of acceptable quality in reasonable computational times. Coit and Smith [12] suggested a genetic algorithm (GA) to analyze series–parallel systems and to determine the optimal design configuration when there are multiple component choices available for each of several k-out-of-n subsystems. Kulturel-Konak et al. [13] developed a Tabu search (TS) meta-heuristic to provide solutions to the system-reliability-optimization problem of redundancy allocation. Besides, an ant-colony optimization (ACO) algorithm for the RAP devised and tested on a well-known suite of problems from the literature in [14]. You and Chen [9] developed an efficient heuristic algorithm with greedy method (GM) and GA for series-parallel redundant reliability problems with separable constraints. They tested various well-known redundant allocation problems using their proposed algorithm and other competing conventional algorithms. Chena and You [15] proposed an immune algorithms-based (IA) approach for RAP with multiple component choices. They showed that the performance of IA was sensitive to value-combinations of the parameters, whose best values were case-dependent. Liang and Chen [16] presented a variable neighborhood search (VNS) algorithm to solve RAP with multiple component choices. The VNS algorithm of their research that included a newly designed neighborhood called shaking operation along with an adaptive penalty function was tested on a set of well-known benchmark problems from the literature. They showed that the shaking operation and the adaptive penalty function helped improve the performance of the algorithm.

Onishi et al. [17] developed an exact solution method, based on the improved surrogate constraint (ISC) [18], and used it to solve benchmark problems from the literature. They showed the ISC-based method to be effective in terms of computational speed, as all computational times were less than one second. Nahas et al. [19] hybridized an ACO with a degraded ceiling local search technique to develop efficient solutions to the RAP, called ACO/DC. Tavakkoli-Moghaddam et al. [20] utilized a GA to solve RAP of a series–parallel system when either active or cold standby redundancy could be selected for individual subsystems. Beji et al. [21] developed a hybrid particle swarm optimization (PSO) algorithm to design an optimal reliability series–parallel system where the number of redundancy component and component reliability in each subsystem were to be decided simultaneously. Yeh and Hsieh [22] and Hsieh and Yeh [23] showed that the solution of RAP in reliability of series–parallel systems found by artificial bee colony (ABC) was better than some other meta-heuristics. Wu et al. [24] developed an improved particle swarm optimization algorithm for reliability problems. An annealing-based PSO, called APSO was proposed by Safaei et al. [25] to solve RAP with multiple component choices. They employed the metropolis-hastings, the main strategy behind the simulated annealing (SA), as a local search engine to improve the best position among all particles. Sadjadi and Soltani [26] developed a honey bee mating optimization (HMBO) algorithm for RAP with several constraints and different alternatives for components. Soltani et al. [27] adapted Sadjadi and Soltani [26] heuristic in order to enable it to enhance the reliability of the serial–parallel systems with component mixing. Chambari et al. [28] proposed a simulated annealing algorithm (SA) for the redundancy allocation problem with a choice of redundancy strategies, consisting of active and cold-standby components.

It is usually difficult to judge which meta-heuristic performs the best among all for a given design problem. These algorithms perform variously in different design problems or even in a given problem with different parameters [29]. However, on the one hand, there are opportunities for improved effectiveness and efficiency of reported GA, ACO, TS, and IA for example through combining these algorithms with other meta-heuristic algorithms. On the other hand, some meta-heuristic algorithms such as ABC, HMBO, VNS, and VND may offer excellent solutions for reliability optimization problems. Recently, Soltani [30] offered a comprehensive survey and classification of reliability optimization models and methods.

The electromagnetism-like mechanism (EM) algorithm is known as a new population-based meta-heuristic to tackle complex optimization problems [31], [32]. It has been used in different cases such as capacitated vehicle routing problems [33], nonlinearly constrained global optimization [34], scheduled trucks in cross-docking systems [35], PID controller optimization [36], periodic job-shop scheduling problem [37], and layout design of reconfigurable manufacturing system [38]. EM imitates the attraction–repulsion of the electromagnetic theory that is based on Coulomb's law for obtaining the optimal solution. One of the advantages of this method is its small number of adjusting parameters. EM algorithm shows significant performance compared with other meta-heuristics in some NP-hard problems. However, it is still in its infancy and intensive studies are needed to improve its performance.

To enhance the quality of solution in EM, several approaches have been proposed in literature. Xing et al. [39] proposed a revised electromagnetism-like mechanism for layout design of reconfigurable manufacturing system. They adopted the VNS strategy in the local search procedure of EM. Ali and Golalikhani [34] presented an electromagnetism-like method for a nonlinear constrained global optimization problem, in which a charge calculation of a point based on both the function value and the total constraint violations was adopted. Therefore, the calculation of the total force vector is different from the original EM method. In addition, Yurtkuran and Emel [33] developed a new hybrid electromagnetism-like algorithm for a capacitated vehicle routing problem, in which the iterated swap procedure (ISP) was applied as a new fast technique instead of the local search of ordinary EM.

Although EM and its variations have been utilized to solve various optimization problems so far, to the best of the authors’ knowledge it has not been utilized to solve RAP of series–parallel systems yet. The algorithms available in the literature include some local search-based meta-heuristics [16] and one way of improving the performance of meta-heuristics as a hybrid approach using a local search procedure as well [19], [21], [25]. In addition to a local search that may improve the performance of EM, it will be modified in this research to suit RAP of series–parallel systems (instead of its original intends to solve continuous optimization problems). More specifically, this paper intends to develop a learning method for EM and proposes a memory-based electromagnetism-like mechanism (MBEM) for RAP. To do this, a memory matrix is used in the local search procedure to separate positive variations from negative variations, in a hope to generate better results. Using the available benchmarks in the literature, we will show that using a memory matrix in the local search procedure will significantly improve both the efficiency and the robustness of EM.

The remainder of the paper is arranged as follows: Section 2 describes the problem in details. Section 3 presents the proposed memory-based electromagnetism-like mechanism (MBEM). The computational and comparison results are presented in Section 4 and finally, the conclusion comes in Section 5.

Section snippets

Problem formulation

In this section, the problem along with its assumptions is first described. Then, nomenclature used to model the problem is stated. Finally, the mathematical formulation of the problem is given.

The memory-based electromagnetism-like mechanism (MBEM)

EM is a population based meta-heuristic introduced by Birbil and Fang [31]. EM imitates attraction and repulsion of the electromagnetic theory to obtain optimal solution. The four procedures used in this algorithm are: (1) initialization, (2) neighborhood search to exploit local minima, (3) total force calculation exerted on each particle, and (4) movement along the direction of the force. The pseudo-code of EM is shown as follows:

Procedures
1. Initialize
2: While (the stop criterion is not met)

Experimental results

Computational experiments are conducted in this section to evaluate the performance of the proposed MBEM algorithm to solve the RAP at hand. The algorithm has been coded in MATLAB software and the runs are performed for some test problems described in Section 4.1 on a personal computer with Intel (R)-64 Core (TM) i7 CPU, [email protected] GHz having 4 GB of memory.

The main parameters in MBEM and EM algorithms are: the number of particles (popsize), the maximum number of iterations (MAXITER), maximum

Conclusion and future research directions

In this paper, an efficient memory-based electromagnetism-like mechanism called MBEM was presented for RAP. The proposed MBEM employs a memory matrix to guide EM in each iteration in order to improve the solutions obtained. This memory matrix separates positive variations from negatives variations and generates better results in the local search. The first set of experiments that conducted to compare the performance of the proposed algorithm to the ones of three other methods available in the

Acknowledgments

The authors are thankful for constructive comments and suggestions of the anonymous reviewers. Taking care of the comments improved the presentation of the manuscript.

References (42)

Cited by (0)

View full text