Reliability allocation problem in a series–parallel system

doi:10.1016/j.ress.2004.10.007

Reliability Engineering & System Safety

Volume 90, Issue 1, October 2005, Pages 55-61

https://doi.org/10.1016/j.ress.2004.10.007 Get rights and content

Abstract

In order to improve system reliability, designers may introduce in a system different technologies in parallel. When each technology is composed of components in series, the configuration belongs to the series–parallel systems. This type of system has not been studied as much as the parallel–series architecture. There exist no methods dedicated to the reliability allocation in series–parallel systems with different technologies. We propose in this paper theoretical and practical results for the allocation problem in a series–parallel system. Two resolution approaches are developed. Firstly, a one stage problem is studied and the results are exploited for the multi-stages problem. A theoretical condition for obtaining the optimal allocation is developed. Since this condition is too restrictive, we secondly propose an alternative approach based on an approximated function and the results of the one-stage study. This second approach is applied to numerical examples.

Introduction

Reliability engineering attracts many researchers due to reliability's critical importance in a variety of systems. A considerable amount of work has been done for design problems with reliability requirements, and a variety of techniques have been used. As explained by Tzafestas in 1980 [1], one of the undeniable steps in the design of such multi-component systems is the problem of using the available resources in the most effective way so as to maximize the overall system reliability, or so as to minimize the consumption of resources while achieving specific reliability goals. The diversity of system structures, resource constraints, and options for reliability improvement has led to the construction and analysis of several optimization models [2]. Surveys on reliability optimization have regularly appeared (e.g. [1], [3], [4], [5]). One of the most recent publications on this subject is the book of Kuo et al. [6].

According to Kuo et al. [6], several investigations have been done for the optimization of parallel–series systems, but few are devoted to the series–parallel systems. Most of these works deal with redundancy allocation as the method developed by Jensen 1968 [7] for series–parallel–series networks using dynamic programming. He assumed that the studied systems are composed of blocks in series and each block is composed of identical technologies in parallel. Recently, Marquez and Coit [8] have proposed an heuristic for the redundancy allocation in multi-state series–parallel system. The new solution methodology that they proposed offers several distinct benefits compared to traditional formulations of this problem. Levitin et al. [9], [10], Zuo [11], Hoang Pham [12] and Venugopal [13] suggest solutions for more general redundancy optimization problem.

In this paper, we are interested in the reliability allocation problem in which the reliability of the components have to be determine in order to minimize the consumption of a resource under a reliability constraint, in a series–parallel systems. This problem arises when different technologies which have the same function are used in parallel.

Two resolution approaches for this problem are proposed, in the case of convex cost functions. First, we present a theoretical study, which is composed of two steps: the optimal resolution study of a one-stage problem and the one of the multi-stages problem. We obtain a convexity condition on the cost functions for guaranteeing the optimality of the global solution. Unfortunately, this theoretical condition is a too restrictive for being applied. Then, alternatively, an approximation resolution method is proposed. This approach provides an approximated reliability allocation to each technology. Then, the reliability allocation for each of them is realized with the results on the one-stage theoretical study. This paper is organized as follows. In Section 2, we present the problem description, with the assumptions and mathematical formulation. Section 3 is dedicated to the theoretical study of a one-stage problem. The optimality condition for the multi-stages problem is developed in Section 4. We show that this theoretical condition is too restrictive. Section 5 is dedicated to the approximation resolution method. Section 7 describes the numerical experiments for investigating the performances of this second approach.

Section snippets

Problem description

A series–parallel system is composed of k subsystems in parallel, where each subsystem i (i=1,…,k) is composed of n_i components in series (Fig. 1). The reliability of the system is $R_{s} = 1 - \prod_{i - 1}^{k} (1 - \prod_{j = 1}^{n_{i}} r_{i j})$ with r_ij, the reliability of the jth component of subsystem i (i=1,…,k and j=1,…,n_i). We assume that c_ij is the cost associated to r_ij with c_ij=f_ij(r_ij). Then, the total cost of the system is: $C_{s} = \sum_{i = 1}^{k} C_{i} = \sum_{i = 1}^{k} \sum_{j = 1}^{n_{i}} f_{i j} (r_{i j})$

We assume that the decision variables r_ij may be any real value between 0

Theoretical study of the one-stage problem

In this section, we assume that we know the reliability target R_i for the technology i (i=1,…,k). We develop the optimal resolution of the one stage problem corresponding to the technology. Then, this subproblem i is to find the reliability allocation of the n_i components in a series structure. The model is: $(M 2) \min \sum_{j = 1}^{n_{i}} f_{i j} (r_{i j})$

$\prod_{j = 1}^{n_{i}} r_{i j} = R_{i}$

$0 < r_{i j} < 1 \forall j = 1, \dots, n_{i}$

We introduce a substitution of variable: y_ij=ln(r_ij) which is equivalent to r_ij=exp(y_ij). We define the function h_ij(y_ij)=f_ij(r_ij). Then, we

Theoretical study of the multi-stage problem

We propose in this section a theoretical study of the multi-stage problem. We obtain a convexity condition that must stratify the cost function. We show that this condition is too restrictive.

The approximation method

Since we are unable to exploit the convexity condition of Section 4, we propose in this section an alternative approximation method in order to obtain reliability allocation for each technology.

We define Z_i=ln(R_i), for i=1,…,k. Then, we have the cost function of subsystem i, in function of the reliability level Z_i required which is $h_{i} (Z_{i}) = min \sum_{j = 1}^{n_{i}} h_{i j} (y_{i j} (Z_{i}))$ so that $y_{i j} (Z_{i}) = {h^{'}}_{i j}^{- 1} (A (Z_{i}))$ and $\sum_{j = 1}^{n_{i}} y_{i j} (Z_{i}) = Z_{i}$ . The global problem is: $(M 5) min \sum_{i = 1}^{k} h_{i} (Z_{i})$

$1 - \prod_{i = 1}^{k} (1 - exp (Z_{i})) = R_{min}$

$Z_{i} \in] - \infty; 0 [\forall i = 1, \dots, s$

Numerical experiment

We propose in this section a detailed example for the approximation method in the Truelove case and several numerical experiments.

Conclusion

Reliability allocation in series–parallel systems is one of the less studied problems in reliability optimization. We have proposed in this paper a study of this reliability allocation problem. First, we have developed the optimal resolution of a one stage-problem, assuming the cost functions satisfy a convexity condition. Then, we have established a convexity condition on the component cost functions in order to obtain the global optimal solution. We have considered two of the most used cost

Acknowledgements

We are grateful to the Associate Editor and the referee for their constructive comments and suggestions which led to an improvement of the paper.

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View full text

Reliability allocation problem in a series–parallel system

Abstract

Introduction

Section snippets

Problem description

Theoretical study of the one-stage problem

Theoretical study of the multi-stage problem

The approximation method

Numerical experiment

Conclusion

Acknowledgements

Optimization of system reliability: a survey of problems and techniques

Int J Syst Sci

An annotated overview of system reliability optimization

IEEE Trans Reliab

Optimization techniques for systems reliability with redundancy

IEEE Trans Reliab

Optimization of systems reliability

On optimal reliability design: a review

Syst Sci

Optimal reliability design: fundamentals and applications

Optimization of series–parallel–series networks

Oper Res