106 IEEE TRANSACTIONS ON RELIABILITY, VOL. 52, NO. 1, MARCH 2003 Reliability Allocation Through Cost Minimization A. O. Charles Elegbede, Chengbin Chu, Member, IEEE, Kondo H. Adjallah, and Farouk Yalaoui Abstract—This paper considers the allocation of reliability and redundancy to parallel-series systems, while minimizing the cost of the system. It is proven that under usual conditions satisfied by cost functions, a necessary condition for optimal reliability allocation of parallel-series systems is that the reliability of the redundant com- ponents of a given subsystem are identical. An optimal algorithm is proposed to solve this optimization problem. This paper proves that the components in each stage of a parallel-series system must have identical reliability, under some nonrestrictive condition on the component’s reliability cost functions. This demonstration provides a firm grounding for what many authors have hitherto taken as a working hypothesis. Using this result, an algorithm, ECAY, is proposed for the design of systems with parallel-series architecture, which allows the allocation of both reliability and redundancy to each subsystem for a target reliability for minimizing the system cost. ECAY has the added advantage of allowing the optimal reliability allocation in a very short time. A benchmark is used to compare the ECAY performance to LM-based algorithms. For a given reliability target, ECAY produced the lowest reliability costs and the op- timum redundancy levels in the successive reliability allocation for all cases studied, viz, systems of 4, 5, 6, 7, 8, 9 stages or subsystems. Thus ECAY, as compared with LM-based algorithms, yields a less costly reliability allocation within a reasonable computing time on large systems, and optimizes the weight and space-obstruction in system design throughout an optimal redundancy allocation. Index Terms—Cost optimization, redundancy allocation, relia- bility allocation. NOTATION # of a series subsystem index of subsystem number of parallel components in a subsystem index of a parallel component in a subsystem index of component in subsystem reliability of component in subsystem derivative of the 1-variable function 2nd derivative of . I. INTRODUCTION R ELIABILITY allocation is an important step in system design. It allows determination of the reliability of con- stituent subsystems and components so as to obtain a targeted overall system reliability. Since the 1950s [2], several studies have been devoted to this problem and many papers have been published on this subject. But no general method has been pro- Manuscript received November 22, 2000; revised May 11, 2001 and February 19, 2002. Responsible Editor: W. Kuo. The authors are with the Laboratory of Industrial Systems Optimization; University of Technology of Troyes; 10010 Troyes Cedex, France (e-mail: Charles.Elegbede@launchers.eads.net; Chengbin.Chu@utt.fr; Kondo.Ad- jallah@utt.fr; Farouk.Yalaoui@utt.fr). Digital Object Identifier 10.1109/TR.2002.807242 posed to solve the reliability allocation problem satisfactorily. This situation is explained by the increasing complexity of cur- rent systems and the necessity to consider multiple constraints such as cost, weight, and component obstruction among others. Existing methods fall roughly into 2 categories: 1) use weighting coefficients to distribute the target value of the overall relia- bility on the components of the system [2], [9], [10]; 2) use optimization techniques to solve—redundancy allocation [3], [11], [12], minimization of system cost subject to reliability con- straint [13], maximization of system reliability under cost con- straint [7], [8], or (more generally) system reliability optimiza- tion [5], [7], [8], [14]. An overview of the methods developed during the past 3 decades for solving various reliability opti- mization problems has been recently published [6], [7]. Ref- erence [7] has an up-to-date state-of-the-art for several case studies, on reliability optimization methods classified with re- spect to: system configuration, optimization problem, and opti- mization technique. The approach, in this paper, to optimize the reliability of par- allel-series systems belongs to the exact methods in [6]. • Section II states the problem of reliability constrained cost minimization for parallel-series systems; 2 cases are considered: a system comprising 1-stage parallel components and a system comprising multi-stage components. • Section III determines the necessary optimality condition for reliability allocation and redundancy in 1-stage systems. • Section IV solves the problem of reliability-constrained cost-minimization in multi-stage parallel-series systems, and derives an appropriate algorithm, ECAY. • Section V considers a numerical implementation that evalu- ates the effectivenessof ECAY, and compares its performances with that of the LM-based algorithms. II. PROBLEM STATEMENT Before mathematically formulating the problem, the notions of parallel-series systems and associated cost functions are clar- ified by making some relevant definitions. A. Parallel-Series System A parallel-series system is composed of blocks of subsys- tems or stages mounted in series. Every subsystem indexed by is made up of components mounted in parallel. All the con- stituent components have the same functionality but a priori not necessarily identical. The system reliability is the classical rela- tion (1); it is proved in [1], [7], [9]: (1) 0018-9529/03$17.00 © 2003 IEEE