Reliability allocation through cost minimization - IEEE Xplore

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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 components 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 optimum 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, reliability 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 constituent 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 proManuscript 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: [email protected]; [email protected]; [email protected]; [email protected]). 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 current 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 reliability 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 constraint [13], maximization of system reliability under cost constraint [7], [8], or (more generally) system reliability optimization [5], [7], [8], [14]. An overview of the methods developed during the past 3 decades for solving various reliability optimization problems has been recently published [6], [7]. Reference [7] has an up-to-date state-of-the-art for several case studies, on reliability optimization methods classified with respect to: system configuration, optimization problem, and optimization technique. The approach, in this paper, to optimize the reliability of parallel-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 evaluates the effectiveness of 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 clarified by making some relevant definitions. A. Parallel-Series System A parallel-series system is composed of blocks of subsystems or stages mounted in series. Every subsystem indexed by is made up of components mounted in parallel. All the constituent components have the same functionality but a priori not necessarily identical. The system reliability is the classical relation (1); it is proved in [1], [7], [9]:

0018-9529/03$17.00 © 2003 IEEE

(1)

ELEGBEDE et al.: RELIABILITY ALLOCATION THROUGH COST MINIMIZATION

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B. Problem Formulation Consider a parallel-series system as described in Section II-A. The problem is to determine the optimal number, , of compoof nents in subsystem and the reliability each component, to minimize the cost of overall system target . reliability Thus, for a given set of cost functions for parallel components forming a series of subsystems, one must determine the optimal parallel-series architecture and the associated set of optimal reliabilities. For that, the global-system cost must be minimized subject to reliability constraints. The decision variables are and .

and Change the variables: (4) or equivalently

with (5) Consider

C. Cost Function A cost function describes the relation between the reliability and the cost of a component. The current literature abounds with , the cost funcdiverse cost functions [1], [7], [9]. Consider tion of the components of the subsystem , where denotes reliability. The following 3 reasonable conditions are imposed on : is a positive definite function 1) is non decreasing 2) increases rapidly as gets close to 1. 3) Let us observe that in particular the third condition simply underscores the fact that the cost of a component increases very rapidly as its reliability approaches the value 1. We assume that the global cost of a system is the sum of the costs of all its constituent components and is given by relation (2):

and of (3) can be rewritten:

,

, then

(6)

and Remark: Assume throughout the paper that is a convex function, which is the case for most cost functions in the literature. , then: Let

(2) and (7)

III. 1-STAGE PROBLEM This section considers the reliability allocation problem for an arbitrary stage which constitutes a sub-problem of an -stage system. The single stage problem is: and a cost function Given a target reliability , determine an integer , corresponding to the number and the corresponding of parallel components in stage of the component such that the reliability is higher than the target reliability and the cost of stage is minimized. To solve this problem, let the number of components be known, and show that all components must have the same reliability. Based on this result, the optimal number of components and the optimal reliability of each component are determined.

Relaxing all the constraints can solve this minimization, which transforms the problem into an unconstrained one. If the optimal solution of the unconstrained problem satisfies the constraints of the constrained problem, it also gives the optimal solution for the constrained problem as well. To that end, consider the unconstrained minimization problem: (8) , the objective function of this Because of the convexity of unconstrained minimization allows a unique optimal solution of which the partial derivatives with respect to the decision variables are all zero. In particular,

A. Proof of the Necessary Condition for Optimal Allocation First consider with cost to attain , By definition:

, and let be the minimum components in parallel. Then consider ; [0, 1], ,

(3)

(9) is convex, the derivative Because creasing. Therefore (10) is equivalent to (9):

is strictly in-

(10)

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The solution of this system of equations is:

which proves that is convex with respect to . is the optimal number of comIt follows from (15) that can be obtained. ponents. A lower bound of . By definition: Let R (11a)

from which:

(11b) This means that the solution satisfies the constraints in (7), and thus yields an optimal solution to the constrained equation. This can be checked by combining (11b) with the constraints in (6). Thus:

(17) is a minimum point of for a given If , then the convexity of if with respect to ensures that verifies:

, i.e.,

(18) assuming that (18) has a solution over solution of:

; i.e.,

is the (19)

(11c) Equation (11c) means that if the components in a stage have the same reliability, then the reliability cost of the stage is optimum. This is an important result for the ensuing development.1 Thus:

Let

be the solution of ; then:

; i.e.,

(20) Furthermore, either

(12)

or By definition,

B. Optimal Number of Components in Stage I which minimizes This problem is: finding a for a given ; which comes to solving the problem:

(21) To simplify the notation, let

N

(13)

N

Then (22)

is defined only The difficulty in this problem is that . It is possible to solve it by enumeration; but this for can be very tedious if the upper limit of the cost is very large. Using the following property provides an easy solution: , there is a Property: For a given such that for any a) if , then b) if , then . defined by: Proof: Consider the extension of (14)

(23) . By definition, In summary: if the target reliability of the stage is , then , and all have the the optimal number of components is reliability: (24) IV. MULTI-STAGE PROBLEM

, for , , Obviously . To prove the property, it suffices to and for any with respect to for any prove the convexity of . Consider given

This section applies results from Sections II–III to solve the -stage problem. Now, target reliability must be assigned to each stage. Let

(15)

be the target reliability vector, then the system reliability cost and the system reliability are:

Because of the convexity of

and because

, then: (16)

1To our knowledge this has never been formally proved before. It has often only been stated as an assumption in the literature.

(25) such that the system reliability The problem is now to find . This problem can is not less than the target reliability


be transformed into a mathematical programming problem with , . linear constraints by taking: , the problem can then be equivaFor lently described as: (26)

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First, establish the convexity of for all to over From (12)

with respect , and any . (31)

From which

It can be proved that there is an optimal solution such using the fact that the functions that , for any , described in (23), are increasing functions with respect to . The previous problem can further be transformed into an unconstrained optimization problem as:

(32) Because

(27)

is convex, it follows that

And, by definition of

,

and (28) Thus Unfortunately, the objective function is not differentiable due to ” operation in the definition of [see (23)]. Otherthe “ wise, the solution might be obtained by solving a set of equations derived from partial derivatives. In light of Section III, given the target reliability of each stage, the reliability of each component can be computed. The multistage problem then consists in determining the target reliability and the number of components for each stage. and Given the decision vectors , the objective function to minimize is then:

This proves the convexity of the objective function of Problem 1. It can be solved by solving:

(33) This leads to the equivalent system:

subject to the constraints: (34) and

Because

,

, then (34) leads to:

The problem now is to determine the target reliability of each , let from which stage. For . The problem can then be written as: one derives

(29) This problem has an optimal solution with . The problem in (29) can therefore be transformed into the unconstrained minimization problem: Problem 1

(35)

is a constant. Expressed more compactly, we have:

System

(30) End of Problem 1

(36) equations with unknowns This system contains and can therefore be solved numerically. The constant, , can be shown to be positive. Starting with

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any

, the method uses the leading equations to obtain and then checks with equation . If this last equation is not satisfied, a new value for must be tried. This is repeated until a satisfactory solution is obtained for the system. , the From the first equations, for all increases with . If is too large, then . If is too small, then . Thus this system can be solved by dichotomy over . To obtain a lower bound for the optimal global cost, consider Problem 2. Problem 2 subject to

3) Initialize the loop counter . Steps 4)–7) can be computed for all 4) Compute

5) For each

.

, compute if otherwise.

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End of Problem 2. With the same argument as in Problem 1, Problem 2 can be transformed to:

(38) Recall that (39)

If the Stop criterion is verified, then stop; otherwise go to step 6). by substituting for 6) Solve System . Let be the solution. ; go to step 4). 7) Increase Stop criterion The convergence of the algorithm can be proved using for . By definition for all . Thus

Hence (40) is convex with respect to bewhich shows that . Consequently, the objective function cause is convex. The optimal solution of Problem 2 is found by solving the system of equations:

(41) leading to the equivalent system: (42) Put more compactly, there is a system of unknowns:

equations with

and

Thus . End Stop criterion. ECAY can be either used as an optimization or an approximation algorithm. For optimization, the stopping criterion sets the from to be within a prescribed error relative deviation tolerance. For approximation, the stopping criterion is chosen and the lower bound so that once the relative gap between is less than a prescribed value, the associated solution is satisfactory. V. NUMERICAL IMPLEMENTATION

System

(43)

This system can be solved by using the same dichotomy method over as for System ( ). be the solution obtained for System Let ( ). An optimal algorithm for solving the reliability allocation problem is: Optimal Algorithm: ECAY , compute by solving: 1) For . for . Let by solving 2) Compute System ( ).

Consider the reliability optimization problem for a system with series subsystems: , and • Find the optimal value of , • Calculate:

subject to: reliability of the each component in subsystem containing components. The cost of each component is defined by:


TABLE I COST-FUNCTION PARAMETERS

TABLE II COST COMPARISON FOR R

= 0:994 05

From (4), it is obvious that the associated is convex is 0.994 05. Table I with respect to . The target value of presents the parameters used for this cost function in the benchmarks described in the remainder of this section. For comparison, the ECAY algorithm was used, and the LM-based algorithm [8] was used, to solve this problem for a set of 6 systems comprising 4 to 9 series-subsystems. In the LM-based algorithm, the number of components in subsystem is given, for a given , by

Table II compares the effectiveness of the two algorithms. Contrary to the ECAY algorithm, the LM-based algorithm does (0.994 05). Indeed, the not attain the optimal reliability , which in LM-based algorithm provides a system reliability . every case exceeds the target reliability obOn the other hand, whenever the system reliability tained by the LM-based algorithm is taken as the target reliability, then the ECAY algorithm produces the minimum cost, which in every case is lower than the cost obtained with the is LM-based algorithm. The same holds when the same considered for the 2 algorithms. In the latter case, the relative cost difference of the two algorithms is very important and becomes more pronounced as the number of stages in the subsystems increases as shown in Table II. ACKNOWLEDGMENT The authors would like to thank the referees for their suggestions, which have allowed for appreciable improvement of the paper. REFERENCES [1] K. K. Aggarwal, Reliability Engineering: Z. Kellers, 1993, edition A. [2] S. Arold and Balanba, “Allocation of system reliability,”, Tech. Rep., ASD-TDR-62-20, 1962. [3] C. S. Chern and R. H. Jan, “Reliability optimization problems with multiple constraints,” IEEE Trans. Rel., vol. R-35, no. 4, pp. 431–436, 1986. [4] W. Kuo, H. H. Lin, Z. Xu, and W. Zhang, “Reliability optimization with the Lagrange-multiplier and branch-and-bound technique,” IEEE Trans. Rel., vol. R-36, no. 5, pp. 624–630, 1987. [5] W. Kuo, Z. Xu, and H. H. Lin, “Optimization limits improving system reliability,” IEEE Trans. Rel., vol. R-39, no. 1, pp. 51–60, 1990. [6] W. Kuo and V. R. Prasad, “An annotated overview of systems reliability optimization,” IEEE Trans. Rel., vol. R-49, no. 2, pp. 176–187, 2000.

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[7] W. Kuo, V. R. Prasad, F. A. Tillman, and C. L. Hwang, Optimal Reliability Design: Cambridge University Press, 2001. [8] K. B. Misra and M. Lbjubojevic, “Optimal reliability design of systems: A new look,” IEEE Trans. Rel., vol. R-22, no. 5, pp. 255–258, 1973. [9] K. B. Misra, Reliability Analysis and Predictions: Elsevier, 1992. [10] Y. Nakagawa and K. Nakashima, “A heuristic method for determining reliability allocation,” IEEE Trans. Rel., vol. R-26, no. 3, pp. 31–38, 1977. [11] V. R. Prasad and M. Raghavachari, “Optimal allocation of interchangeable components in a series-parallel system,” IEEE Trans. Rel., vol. R-47, no. 3, pp. 255–260, 1998. [12] V. R. Prasad, W. Kuo, and O. K. M. Kim, “Optimal allocation of s-identical multi-functional spares in a series system,” IEEE Trans. Rel., vol. R-48, no. 2, pp. 118–126, 1999. [13] G. Sasaki, T. Okada, and O. S. Sihingai, “A new technique to optimize system reliability,” IEEE Trans. Rel., vol. R-32, no. 2, pp. 175–182, 1983. [14] C. S. Sung and Y. K. Cho, “Reliability optimization of a series system with multiple-choice and budget constraints,” Eur. J. Oper. Res., vol. 127, pp. 159–171, 2000. [15] F. A. Tillman, C. L. Hwang, and W. Kuo, Optimization of System Reliability: Marcel Dekker, 1985.

A. O. Charles Elegbede received the Ph.D. in 2000 from Compiegne University of Technology in System Dependability. His Ph.D. topic concerns reliability allocation on complex systems. He graduated in Civil Engineering in 1996 from the C/U/S/T of Clermont-Ferrand II University (France). The same year he received a D.E.A. in Material Science, Mechanical and Civil Engineering Structure Reliability. He was a member of the Laboratory of Industrial Systems Optimization (LOSI) of Troyes University of Technology. He rejoined European Aeronautical Defense and Space Company in 2000 in Launch Vehicles Division (EADS LV), in the Department of Analysis and Justification in the reliability office. Chengbin Chu received the B.Sc. in 1985 from Hefei University of Technology (China) in Electrical Engineering, and the Ph.D. in 1990 from Metz University (France) in Computer Science. He worked at INRIA (National Research Institute in Computer Science and Automation), France, for 9 years as a Ph.D. student, Expert Engineer, and Research Officer. He is a Professor at Troyes University of Technology, France, and is also Director of the Industrial Systems Optimization Laboratory (LOSI). He is interested in research areas related to operations research and modeling, analysis and optimization of discrete event systems. He has authored or coauthored one book and more than 30 papers in journals such as Operations Research, IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, and SIAM Journal of Computing. For his research and application activities, he received the first Robert Faure Award in 1996. He also received the 1998 Best Transactions Award from IEEE Robotics and Automation Society. He is a Member of the IEEE and French Association of Operation Research and Decision Aid (ROADEF). Kondo H. Adjallah obtained the M.Sc. in 1989 in Metrology, Automatics, and Electrical Engineering at University de Nancy 1 (France), followed by the Ph.D. in 1993 in Automatics from the National Institute of Polytechnic of Lorraine (France). His Ph.D. research focused on diagnosis of dynamic systems and was conducted at the Automation Research Center in Nancy. From 1993 to 1994, he served as a Research Assistant at the Nancy University 1. He is an Associate Professor at Troyes University of Technology where, in 1994, he initiated the project on RAMS requirements allocation in system design (which involves the work in this paper) in the Laboratory of Systems Modeling and Dependability (LM2S). He is now with the Laboratory of Industrial Systems Optimization (LOSI) for research interests including production systems maintenance and reliability optimization, and maintenance resources allocation. Farouk Yalaoui is an Assistant Professor at Troyes University of Technology (France). He received the Ph.D. in 2000 in Production Management from Troyes University of Technology (UTT). His research topic focused on the problems of parallel machine scheduling under various criteria and on optimization problems in general. He obtained the D.E.A. in Industrial Systems Engineering at National Institute of Polytechnic of Lorraine (France) after having obtained the M.S. degree in Industrial Engineering from the Polytechnic School of Algeria. He is pursuing his research activities in the Laboratory of Industrial Systems Optimization (LOSI) of UTT.