Reliability optimization of series-parallel systems with a choice of redundancy strategies using a genetic algorithm
Introduction
The primary goal of reliability engineering is to improve the reliability system. In the initial design activity, the redundancy allocation is a direct way of enhancing system reliability. The redundancy allocation problem involves the simultaneous selection of components and a system-level design configuration, which can collectively meet all design constraints in order to optimize some objective functions such as system cost and/or reliability [1]. This reliability design problem has generally been formulated by considering active redundancy. To deal with these problems, a large number of models and solution methods have been proposed such as dynamic programming, Lagrangean multiplier [2], heuristic approach [3], integer programming [4], and genetic algorithms (GAs) [5]. Comprehensive overviews of these methods have been addressed by Kuo et al. [6] and Gen et al. [7]. Furthermore, the redundancy allocation problem is considered for various system structures such as series, parallel, network, parallel-series [8], k-out-of-n [9] and the like. The series-parallel system, as depicted in Fig. 1, is a common system structure that is used in most system designs. Thus, the series-parallel redundancy allocation problem is considered in this paper.
The above-mentioned solution methodologies for series-parallel systems are not applicable when the system design involves active and cold-standby redundancies. Coit [9] presented a new problem formulation and solution method to determine the optimal system design configuration when a system design includes multiple subsystems that are designed with either active or cold-standby redundancy. This solution method assumes that the redundancy strategy (active or cold-standby) for each subsystem is predetermined. However, the choice of these two redundancy strategies for each subsystem is much more realistic and it provides a better tool for the designers. Coit [10] also presented an optimal solution to redundancy allocation problems when there are some subsystems using active redundancy, cold-standby redundancy, or selecting the best redundancy strategy. This becomes an additional decision variable in redundancy allocation problems. Coit [10] solved this problem by first formulating it and then applying a zero-one integer programming method.
Chern [11] showed that even a simple redundancy allocation problem in series systems with linear constraints is NP-hard. This has prompted recent researchers to develop metaheuristic methods to achieve approximate solutions of acceptable quality in a reasonable computational time. Metaheuristic methods can be used to solve complex discrete optimization problems. These methods provide more flexibility and require fewer assumptions on the objective function and the associated constraints. A GA is one of the metaheuristic methods trying to imitate the biological phenomenon of evolutionary production through the parent–children relationship [12]. Recently, GAs have been designed to solve a variety of reliability optimization problems. Coit et al. [1] successfully applied a GA for redundancy allocation problems.
The structure of this paper is organized as follows. 2 Redundancy strategies, 3 Problem formulation present a review on the redundancy strategies and the problem formulation, respectively. In Section 4, a GA is proposed for solving the redundancy allocation problem when either active or cold-standby redundancy can be selected for individual subsystems. Section 5 considers a numerical example to demonstrate the efficiency of the proposed methodology through computational results. Finally, Section 6 presents conclusion.
Section snippets
Redundancy strategies
There are two types of redundancy strategies, namely, active and standby. If all the redundant components operate simultaneously from time zero, even though the system needs only one at any given time, such an arrangement is called active redundancy. There are three variants of the standby redundancy, namely, cold, warm, and hot. In the cold standby redundancy, the component does not fail before it operates. In the warm standby redundancy, the component is more prone to failure before operation
Problem formulation
The mathematical model of the series-parallel redundant reliability system with s subsystems and two separable linear constraints is considered and presented as the following integer nonlinear programming problem [10]. In this model, the components within the same subsystem are of the same type.
Genetic algorithms
A GA is a probabilistic search method for solving optimization problems. Holland [14] first made pioneering contributions to the development of GAs that can be effectively adopted for complex combinatorial problems. For a detailed description of applications of GAs to reliability optimization problems, one may refer to Gen et al. [12]. The proposed GA developed for this general problem is described in the following subsections.
A numerical example
To evaluate the performance of the proposed GA, a typical example taken from Coit et al. [10] is first solved. In this example, the series-parallel system is connected by 14 parallel subsystems and each system has three or four components of choice. Component cost, weight, and Gamma distribution parameters (λij, kij) are given in Table 1. The objective is to maximize system reliability at t=100 h, given the constraints for the system cost (C=130 max) and the system weight (W=170 max). For each
Conclusion
In this paper, a GA has been proposed for solving the redundancy allocation problem when either active or cold-standby redundancy can be selected for individual subsystems. These reliability design problems are usually formulated as a nonlinear integer programming model under a number of constraints. In general, these problems are not easy to solve in real cases, and especially for large systems. This is the motivation of using genetic algorithms (GAs). The proposed GA is more flexible in the
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