Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 1831537, 17 pages

https://doi.org/10.1155/2017/1831537

## Reliability Optimization and Importance Analysis of Circular-Consecutive -out-of- System

^{1}Department of Industrial Engineering, School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China^{2}School of Automobile, Chang’an University, Xi’an, Shaanxi 710064, China

Correspondence should be addressed to Jiang-bin Zhao

Received 8 March 2017; Accepted 29 May 2017; Published 15 August 2017

Academic Editor: David Bigaud

Copyright © 2017 Shuai Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The circular-consecutive -out-of-: system (Cir/Con//: system) usually consists of components arranged in a circle where the system fails (works) if consecutive components fail (work). The optimization of the Cir/Con// system is a typical case in the component assignment problem. In this paper, the Birnbaum importance-based genetic algorithm (BIGA), which takes the advantages of genetic algorithm and Birnbaum importance, is introduced to deal with the reliability optimization for Cir/Con// system. The detailed process and property of BIGA are put forward at first. Then, some numerical experiments are implemented, whose results are compared with two classic Birnbaum importance-based search algorithms, to evaluate the effectiveness and efficiency of BIGA in Cir/Con// system. Finally, three typical cases of Cir/Con// systems are introduced to demonstrate the relationships among the component reliability, optimal permutation, and component importance.

#### 1. Introduction

The component assignment problem (CAP) [1] is a kind of classic problem in the optimization of system reliability. The system is composed of positions and components with different reliabilities, and then the components should be assigned into different positions to find the optimal assignment with the maximum system reliability. So, the CAP is usually a kind of combinatorial optimization and generally NP-hard problem [2]. The optimization of consecutive -out-of- system (Con// system) is a typical case of CAP, and its purpose is to ensure that the system reliability remains largest. Con// system contains linear-consecutive -out-of-: system and circular-consecutive -out-of-: system.

A linear-consecutive -out-of-: system (Lin/Con//: system) is an ordered sequence of components arranged in a line such that the system fails (works) if and only if at least consecutive components fail (work). Lin/Con// systems are used in most system designs, such as telecommunication or pipeline networks, the allocation of microwave towers and street lamps, and arrangement of spacecraft relay stations [3]. For a pipeline network, the petroleum or gas is sent to other places from the origin through the pipeline, which assumes that the pumps are arranged in the equidistant spacing. Each pump has the ability to send the gas to the following pumps. If a pump is failed, the pipeline system also can work normally unless the consecutive pumps all failed. Actually, the pipeline system is a Lin/Con//: system which is widely used in the practical engineering.

A circular-consecutive -out-of-: system (Cir/Con//: system) is an ordered sequence of components arranged in a circle such that the system fails (works) if and only if at least consecutive components fail (work). Cir/Con// systems are used in the camera system of nuclear accelerator and computer ring network [3]. For a nuclear accelerator, high-speed cameras are arranged around the accelerator to record the motion state of various particles. If and only if at least cameras work properly, the complete motion of particles can be recorded successfully. Actually, the camera system of nuclear accelerator is a Cir/Con//: system.

In recent years, many researchers have tried to optimize the Con// system by combining the heuristic algorithm with importance measure.

The concept of importance measure was first proposed by Birnbaum for binary systems in 1969 [4]. Then, it made substantial progress and was applied to broad engineering practice. Vesely [5] and Fussell [6] put forward the Fussell-Vesely importance based on the fault tree analysis. Nakashima and Yamato [7] raised the uncertainty importance for the component to discriminate the component which affected the system reliability significantly. Hong and Lie [8] proposed the joint importance to evaluate the effect on system reliability which is caused by the interaction of the components. Borgonovo and Apostolakis [9] designed the differential importance measure for the probabilistic safety assessment. Si et al. [10–12] introduced the integrated importance measure and do some research of integrated importance measure in various situations. Based on the integrated importance measure, Dui et al. [13] extended the integrated importance measure from unit time to system lifetime and to different life stages; then Dui et al. [14] also studied how the transition of component states affects system performance under the semi-Markov process. Si et al. [15] studied the component reliability importance with the changes of the optimal component sequence for Lin/Con// systems. Liu et al. [16] proposed the generalized Griffith importance measure to evaluate the accurate contribution of the components for continuous state systems.

For the optimal assignment, Kontoleon [17] gave an iterative algorithm to calculate the optimal permutation. This algorithm could assign the component with the lowest reliability into all the positions in the system and order the component based on the Birnbaum importance and finally place the component into the system through iterative allocation. Zuo and Kuo [18] proposed two similar heuristic algorithms, called ZKA and ZKB, which could be applied into the Lin/Con// systems. Then, Zhu et al. [19] put forward two new algorithms, referred to as ZKC and ZKD, to improve the ZK algorithms. Lin and Kuo [20] established a greedy algorithm (which is called LKA) based on the Birnbaum importance by assigning components one by one into the system. On the basis of LKA, Yao et al. [21] built the LK type algorithms, which included LKA, LKB, LKC, and LKD, and designed a Birnbaum importance-based two-stage approach (BITA) according to the numerical experimentation of ZK and LK algorithms. BI is one of the most widely investigated importance measures and has been applied to the CAP especially in [17–21]; therefore, this article focuses on the BI applied to Cir/Con// systems for the CAP.

Evolutionary algorithm is a kind of advanced heuristic algorithm, which combines the random algorithm with local search. Problem-independent technique is used to solve various complex problems, such as genetic algorithm [22], simulated annealing algorithm [23], particle swarm optimization [24], tabu search algorithm [25], and neural net algorithm [26]. The research of combining metaheuristics with importance measure is also developed to solve CAP. Cai et al. [27] proposed an improved genetic algorithm based on heuristic method (BGA) to deal with the CAP, and the research illustrates that BGA is more effective for the systems with arbitrary reliable components. Yao et al. [28] constructed a Birnbaum importance-based genetic local search algorithm (BIGLS), which is a comprehensive genetic algorithm to reduce the solution space of the optimal solution based on the local search. When the components of CAP are less, local search could improve the accuracy and convergence speed of the algorithm, but it will take longer time. Cai et al. [29] proposed a Birnbaum importance-based genetic algorithm (BIGA) to analyze the performance of the algorithm in the Lin/Con// systems, which is stable and feasible to solve the general CAP with stronger robustness.

The rest of this paper is organized as follows. In Section 2, BIGA is introduced to optimize the reliability of Cir/Con// systems which can break through the limitation of local optimal solution. By comparing with BITA and BIGLS, the numerical examples are implemented to discuss the optimization results of BIGA in the small and large systems. In Section 3, three typical cases with different , , and in Cir/Con// system are implemented, and the relationships among the Birnbaum importance and the optimal assignment are discussed. Finally, conclusions of the research are summarized in Section 4.

#### 2. Reliability Optimization Method for Cir/Con// System

##### 2.1. BIGA for Cir/Con// System

In order to optimize the reliability of Cir/Con// systems efficiently, the BIGA [29] is introduced in this section. The detailed process of BIGA is as follows, and the flowchart is shown as in Figure 1.(1)For the optimization problem, the objective function is system reliability of Cir/Con// system, and the solution is the permutation of the components with maximum reliability. The system reliability can be calculated based on the literature [30].(2)Choose the real-number encoding method and determine the encoding space for individuals.(3)Generate an initial parent population which contains individuals. Perform Birnbaum importance-based local search on all the individuals, and update the initial parent population .(4)For the population , calculate the fitness of each individual.(5)When the generation meets the termination condition 1, which is the limit of the generation scale, the algorithm will be terminated and the optimal solution will be output. If not, go to Step (6).(6)Selection is performed on the current population , and the best chromosome will be selected and saved.(7)Measure the fitness scaling of each individual.(8)For the current population, when the termination condition 2 is satisfied, the process will be terminated and the optimal solution will be output. The termination condition refers to the convergence degree of populations; that is, , where , , and represent the minimum, average, and maximum fitness of chromosomes, respectively. is conversion factor, and is a very small positive number. If not, go to Step (9).(9)For the current population , select new individuals according to the fitness of each individual.(10)Perform the crossover on the previous selected individuals to generate offspring population which contains individuals.(11)Perform mutation on the previous selected offspring population to generate the new offspring population .(12)Measure the fitness of every individual in the offspring population.(13)Perform the BITA on the offspring chromosomes. If the fitness of individual is larger than that of the optimal chromosome, the individual will not perform the BITA.(14)Perform the elitist strategy, replace the lowest fitness chromosome of offspring population by the optimal chromosome, and then update the offspring population .(15)Replace the initial population by the new offspring population , and turn back to Step (5).