System Optimization With Component Reliability ... - IEEE Xplore

IEEE TRANSACTIONS ON RELIABILITY, VOL. 53, NO. 3, SEPTEMBER 2004

369

System Optimization With Component Reliability Estimation Uncertainty: A Multi-Criteria Approach David W. Coit, Member, IEEE, Tongdan Jin, and Naruemon Wattanapongsakorn

Abstract—Summary & Conclusions—This paper addresses system reliability optimization when component reliability estimates are treated as random variables with estimation uncertainty. System reliability optimization algorithms generally assume that component reliability values are known exactly, i.e., they are deterministic. In practice, that is rarely the case. For risk-averse system design, the estimation uncertainty, propagated from the component estimates, may result in unacceptable estimation uncertainty at the system-level. The system design problem is thus formulated with multiple objectives: (1) to maximize the system reliability estimate, and (2) to minimize its associated variance. This formulation of the reliability optimization is new, and the resulting solutions offer a unique perspective on system design. Once formulated in this manner, standard multiple objective concepts, including Pareto optimality, were used to determine solutions. Pareto optimality is an attractive alternative for this type of problem. It provides decision-makers the flexibility to choose the best-compromise solution. Pareto optimal solutions were found by solving a series of weighted objective problems with incrementally varied weights. Several sample systems are solved to demonstrate the approach presented in this paper. The first example is a hypothetical series-parallel system, and the second example is the fault tolerant distributed system architecture for a voice recognition system. The results indicate that significantly different designs are obtained when the formulation incorporates estimation uncertainty. If decision-makers are risk averse, and wish to consider estimation uncertainty, previously available methodologies are likely to be inadequate.

number of component types available to be used in subsystem number of subsystems th constrained function th constraint limit total number of constraints. reliability for th type of component in subsystem unreliability for th type of component in subsystem cost for th type of component in subsystem weight for th type of component in subsystem

,

Index Terms—Estimation uncertainty, multi-criteria optimization, redundancy allocation.

NOTATION system reliability estimate, for design vector variance of the system reliability estimate, for design reliability estimate for subsystem unreliability estimate for subsystem variance of the reliability estimate for subsystem variance of the unreliability estimate for subsystem

quantity of th type of component used in subsystem Manuscript received May 1, 2001; revised March 1, 2003. Associate Editor: R. Evans. D. W. Coit is with Rutgers University, Piscataway, NJ. T. Jin is with Teradyne, Inc, Boston, MA. N. Wattanapongsakorn is with King Mongkut’s University of Technology Thonburi, Bangkok, Thailand. Digital Object Identifier 10.1109/TR.2004.833312

variance of the unreliability estimate for th type of component in subsystem number of summation terms for RB/1/1 subsystem unreliability estimation integer coefficient for the th additive term for RB/1/1 subsystem unreliability estimation exponent in the th additive term exponent in the th additive term objective function weights Probability of failure from related fault between two software versions Probability of failure of decider (adjudication module) Probability of failure from related fault among all software versions due to faults in specification Probability of failure for software version 1 Probability of failure for software version 2 Probability of failure for hardware I. INTRODUCTION

S

YSTEM designers desire optimal design architectures which maximize a system reliability estimate, given defined performance standards, and design constraints. However, it is inadequate to simply identify the design configuration which maximizes the system reliability estimate, because a design solution with minimal reliability estimation uncertainty is also highly desirable. A system design which maximizes a system reliability estimate is not a credible solution if the constituent component reliability estimates are far from certain. This is particularly true because system designers and users often prefer a system design option with a marginally lower reliability estimate if it is known that the estimate is based

0018-9529/04$20.00 © 2004 IEEE

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TABLE I RELATED RESEARCH

on good data and sound engineering models. The problem of redundancy allocation has been studied in great detail for different formulations and optimization algorithms. For this paper, the system reliability optimization problem has been formulated with two objectives; maximization of a system reliability estimate, and minimization of the variance associated with that estimate. Estimation of system reliability is generally based on a system model, and component or lower-level (subsystem) estimates of reliability. However, the component reliability estimates are often uncertain, particular for new products with few failure data. Thus, the associated system reliability estimate is uncertain as well. The problem is further complicated because system designers and users are generally risk-averse (such as the protection system in nuclear reactors [2]). They usually prefer a system design with a high reliability estimate, and low estimation variability as well. Maximization of the system reliability, and minimization of the estimation uncertainty, become mutually important, although the latter has not been emphasized by previous research. The variance of the system reliability estimate is a good indicator to reflect the estimation uncertainty at the system-level. For redundancy allocation models, the decision variables are the component choices and the redundancy levels. Most researchers have focused on single objective optimization, while there has been some limited research which addresses multi-objective optimization. For series-parallel systems with a single objective, Chern [4] demonstrated that the redundancy allocation problem is NP-hard. Kuo et al. [14], [15] recently

provide a comprehensive review on system reliability optimization. The problem has been solved using different approaches. Table I presents an overview of relevant past research. This research provided the required background and impetus to develop the new problem formulation considering the system reliability optimization problem as a multiple objective problem with a stochastic reliability estimate. The table indicates the researcher, the problem objective, and the solution methodology. II. PROBLEM FORMULATION The problem is to determine the component type, and the number of components in each subsystem, such that the system reliability estimate is maximized while the variance of the system reliability estimate is minimized. This is a new formulation of the system reliability optimization problem, and it provides for unique perspectives on reliability design. Once formulated in this manner, standard multi-objective optimality criteria and optimization approaches can be applied. It is unlikely that a single solution is optimal for both objectives, so the design problem involves a trade-off. The final solution selected depends on the decision-makers propensity for risk (tolerate high system reliability estimation uncertainty). Problem P1 is a general formulation that is common for all particular problems. The vector specifies the component choice and redundancy levels for a series-parallel system. Problem P1:

COIT et al.: SYSTEM OPTIMIZATION WITH COMPONENT RELIABILITY ESTIMATION UNCERTAINTY

subject to

for

371

where, (1)

In the Problem P1 formulation, the second objective is to . This is equivalent to minimizing the maximize reliability estimate variance. The problem was formulated with the negative variance term, because it is easier to consider a multiple objective problem where all objectives are being maximized. The problem P1 formulation includes constraints with constraint limits . The constraints correspond to physical or performance limitations on the possible design alternatives, while represents the design solution’s respective value in comparison to the constraint limit. Often the constraints limit the cost, volume, or weight of the final design; but they could also restrict the design based on fuel consumption, expected spares utilization, or many other factors. The constraints limit the possible design configurations, and thus, limit the possible reliability estimates and corresponding variance values. A. Series-Parallel Systems For series-parallel systems, the redundancy allocation problem is formulated for a system with subsystems in series. types of functionally equivalent comEach subsystem has ponents. A minimum of one component must be used for each subsystem. Due to the production cost, manufacturing practices, and associated quality assurance provisions, components may have different reliability estimates, prices, weights, and reliability, even if they provide the same function. Component reliability may suffer estimation variability, because insufficient failures have been recorded in the field or under test conditions. The objective is to maximize the system reliability esti, and to simultaneously minimize the variance, mate, . Constraints can be for weight, cost, volume, or other resource limits. They can be linear or nonlinear. For many practical redundancy allocation problems, it is not uncommon that one type of component is repeatedly used within a subsystem while they share the same reliability estimate from a pooled data set. In this case, these repeated components have dependent reliability estimates. Here, if the reliability estimate for a particular component type is high (or low), it is high (or low) everywhere copies of that component type are used. Jin & Coit [12] derived an iteration procedure to compute the variance of the system reliability estimate when arbitrarily repeated component reliability estimates are involved. The system with dependent component reliability estimates is presented as Problem P2. Problem P2:

(2) Equation (2) is a general expression to compute the variance of the system reliability estimate. The variance is expressed as a function of higher moments of the component unreliability estimates. If the underlying distribution of the component unreliability estimate is available, computation of higher moments can be straightforward. If only the mean and the variance of component unreliability estimates are available, nonparametric models (such as pseudo-binomial sampling) can be adopted to derive the higher moments. Jin & Coit [12] demonstrate how the higher-order moments and variance are estimated. Alternatively, for a series-parallel system with independent component reliability estimates, the system reliability estimate variance is given [5] by

(3) Equation (3) expresses the variance of the system reliability estimate using component unreliability and its associated variance, as proposed by Coit [5]. The objective function includes component unreliability , which is unknown. If component is used to approximate and systemunreliability estimate level constraints are for system cost and weight, then a more readily solvable problem formulation is presented as Problem P3. Problem P3:

subject to

where

subject to (4)

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Fig. 1. Recovery block (RB): RB/1/1 architecture.

Problem P3 is more limiting. It assumes that component reliability estimates are independent. This is often appropriate if the components in parallel (and elsewhere) are different, and the reliability is estimated independently. The significant difference between Problem P2 and P3 are demonstrated with numerical examples in Section V. The dependent nature of the reliability estimates amplifies the variance of the system reliability estimate, even if the constraints and the design conditions at the component levels are identical. B. Recovery Block Fault Tolerant Architecture for Computing Systems Recovery Block (RB) is one design approach to develop fault tolerant computing systems. RB consists of an adjudication module called an acceptance test, and at least two software components, called alternates. As the subsystem is initiated, the first or primary alternative is run, and the output is tested for acceptance. If it fails, the process rolls-back to the beginning of the process, the second alternate is executed, and its output is tested for acceptance. In general, the process continues until the output from an alternate is accepted, or all the alternates have been tested and fail. signifies a software subsystem which can tolerate hardware faults and software faults. An RB/1/1 architecture which can tolerate a single hardware fault and a single software fault is depicted in Fig. 1 for a single subsystem module, as described by Wattanapongsakorn & Levitan [24]. Each subsystem module consists of both software and hardware components. A distributed system may consist of several or many of these modules, where multiple software components/versions are running and distributed on different computer platforms. The two software versions perform the same function, but are independently developed. For the two hardware components, each is running two independent software versions (primary and secondary). The primary software version is active until it fails. If the primary software version fails, it fails in both locations. The secondary version is in cold-standby, and it is initiated once the primary version has failed. The system fails if both software versions fail, or both hardware components fail. The unreliability for a single subsystem using RB/1/1 is given as follows [24]. The software system then has several or many of these subsystems arranged in series within a distributed system (5) For the RB/1/1 problem, , , and are given in Table II. The coefficients are integer coefficients, and can be negative

or positive. The coefficients are used as multipliers for the various constituent component reliability and unreliability terms which describe the overall system configuration reliability function. definitions for RB/1/1 are given in the Nomenclature The section. Comparisons with notation from Wattanapongsakorn & Levitan [24] are also provided. Note that even though the software versions are independently developed, they are found to have related faults [24], as presented by parameters Prv (probability of failure from related fault between two software versions), and Pall (probability of failure from related fault among all software versions due to fault in specification). If Prv and Pall each has nonzero value, then the failure times of software versions 1 and 2 are not independent. This is more realistic compared with many papers that strictly require an independence assumption. or reliability Usually the exact component unreliability are not known. They are estimated from life test data or field or are used to estimate failure records. The estimates subsystem unreliability. The equation can then be re-arranged terms. by expanding the

(6) where number of summation terms after expansion, integer coefficient exponent after the expansion This expansion procedure involved multiplying all terms and grouping similar ones. For example, consider the following abbreviated form of the expansion.

For the complete expansion, . Table III lists the expansion results. (The authors’ performed the expansion using , and the component reMatlab). Given the coefficients & liability information, the variance of the subsystem unreliability, , is as follows,


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TABLE II MODEL COEFFICIENTS FOR RB/1/1 ARCHITECTURE

n

AND b

TABLE III COEFFICIENTS FOR RB/1/1 ARCHITECTURE

(7) As before, nonparametric models (such as pseudo-binomial sampling) can be adopted to derive the higher moments, based on the component reliability estimates and the corresponding variance estimates [12]. Problem P4 The problem is formulated as a multiple-objective optimization, as Problem P1, subject to a single cost constraint as follows.

subject to

are the numbers of available choices for particular hardware components and software versions, respectively. C. Bridge System Another common system configuration is a bridge system. Consider the system depicted in Fig. 2. There are five subsyscomponents which can tems. For each subsystem, there are be used individually, or in a parallel configuration. Similar to the is the number of the th type comseries-parallel example, ponent used in the th subsystem. For a bridge system, the system reliability estimates can be most readily expressed as a function of the unreliability estimates of the subsystems and the available components. System reliability can be readily determined by conditioning on the success or failure of Subsystem 3, and then, expanding and grouping terms similar to the R/B/1/1 example. The system reliability estimate and the variance estimate are given by the following equations. The coefficients are given in Table IV.

where, (8) . is equal to one if the th hardware version is used for the th subsystem. Similarly, is equal to one if the th software version is used for the th subsystem. In this example, two hardware components, which are identical but statistically indepenand dent, and two different software versions, are selected.

374


Fig. 2. Bridge system example.

n

AND b

TABLE IV COEFFICIENTS FOR BRIDGE SYSTEM

(9) The problem can now be formulated as a multiple objective problem. Problem P5:

subject to

Higher moments for the component unreliability estimates are required to estimate the bridge system reliability estimate variance. Jin & Coit [12] demonstrate how the higher-order moments and variance are estimated. III. MULTIPLE-OBJECTIVE OPTIMIZATION For this problem, the optimization strategy was to determine a Pareto optimal set of nondominated solutions rather than the determination of a unique “optimal” solution. When there are multiple optimization objectives, there are well defined methods which have been adopted for different problem domains to compare and/or rank prospective solutions. Goal programming, weighted objective functions, multi-attribute utility theory, Analytic Hierarchy Procedure (AHP), and Pareto optimality are frequently used methods to select optimal solutions for multi-criteria optimization problems. Table V provides a comparison of these methods, and describes the appropriateness

for the system reliability optimization problem considered in this paper. More detailed descriptions and comparisons are provided by Giocoechea et al. [11], and Eschenauer et al. [8]. In goal programming, a designer sets goals and relative weights for each of the objective functions. An “optimal” is obtained by minimizing the deviation from the solution set goals. The optimal solution is sensitive to the choice of pre-determined goals, and the relative weight. Goal programming, weighted objective functions, and multiattributed utility theory transform a multi-objective optimization problem into a single objective problem. The interaction between the original problem and the decision-maker are made before searching for the optimal solution. These methods are effective if the decision-maker clearly & quantifiably understands objective functions, and the relative weights & utilities of each objective function. Unfortunately, in many cases, the resultant “weights” and utility functions can be very difficult to precisely quantify. If forced to select quantitative weights or utility functions, decision-makers do the best they can with approximate values; however, it is not always clear whether the final unique solution is indeed the best available. Koski [13] reported on the deficiencies of weighted objective functions. Another approach in multiple-objective programming is Pareto optimality. Pareto optimality is a well-known and established methodology for considering and comparing different prospective solutions [8], [11]. The problem is formulated to identify solutions which simultaneously maximize the system reliability estimate, and minimize the variance of the system reliability estimate. Because it is doubtful that a unique solution is available to optimize both objectives, a set of Pareto optimal solutions is identified. Dominance criteria are defined as follows. Solution dominates solutions if and and

or

Instead of identifying a single solution, a set of all nondominated solutions is the final product of the optimization process. After a set of nondominant solutions is found, the decisionmaker chooses a best-compromised solution from the set of nondominant solutions based on his or her requirements. The final decision depends on the relative importance of a high reliability estimate, and a low estimate variance. The decision depends on the system user’s needs, and their tolerance for risk. It is ultimately a subjective decision which can not always be easily accommodated mathematically. The final solution, therefore, is a Pareto optimal set. As seen in the examples, a preferred member of the Pareto optimal set can often be identified. Pareto optimality solutions become an attractive alternative for these multi-criteria optimization problems. They provide the decision-maker with all Pareto solutions, and the decision-maker must select a best-compromised solution among all nondominant solutions. IV. SOLUTION ALGORITHMS The weighted objective method with iteratively varied weights was used to identify a Pareto optimal set. When solving


375

TABLE V MULTIPLE OBJECTIVE SOLUTION METHODS

TABLE VI EXAMPLE 1 AND 2 COMPONENT PARAMETERS

a problem by weighting objectives, there are two variables in and . The algorithm assigns an the objective function: appropriate nonnegative weight , to each objective function. The problem was repeatedly solved by varying the from 1 to 0 (or from 0 to 1). The weights are value of incrementally varied to obtain different nondominant solutions of the original problem. The importance of original objective functions is controlled by the relative size of two weights. Therefore, the original multi-criteria problem is transformed into a single objective as follows,

subject to The above formation is a nonlinear integer programming problem with single objective function. Ozan [20] demonstrated that the optimal solution to a weighting problem is a nondominant solution for the multiple-objective problem as long as weights are nonnegative. Solving the single objective problem will obtain one nondominant solution. A series of nondominant and . solutions are obtained by varying the two weights, This approach is not guaranteed to produce an exhaustive list of the Pareto optimal solutions, yet it has been demonstrated to generate a meaningful sub-set of the Pareto optimal solutions. The individual problems were solved using readily available nonlinear integer programming software, or SSRP/ B&B proposed by Sung & Cho [23]. SSRP/B&B was necessary for some problems because it does not require differentiable objective functions.

V. ILLUSTRATIVE EXAMPLES A. Series-Parallel System Example An example series-parallel reliability optimization problem was formulated and solved to determine the Pareto optimal set. The example has three subsystems and five component choices in each subsystem. Table VI lists all component data including reliability estimate, variance of the reliability estimate, unit cost, and weight. Design constraints on the system level are a maximum cost of 29, and a maximum weight of 45. The iteratively varied weighed objective method was used to solve Problem P3, where the component reliability estimates are assumed to be statistically independent. The resultant Pareto optimal (or the nondominant) solutions are presented in Table VII. Fig. 3 depicts the relationship between the system reliability, and its associated variance for all nondominant solutions. These five nondominant solutions are depicted in Figs. 4–8. As decreases, it becomes increasingly important to minimize the variance of the reliability estimate, and less important to maximize the system reliability; and the design changes accordingly. These represent five potential solutions which can be considered by the system-designers or users. Generally, the decision-maker would start by considering the system design associated with the most reliable estimate. Then, the second highest reliability estimate is considered and compared to the first solution. The decision-makers consider whether they would be willing to tolerate a lower reliability estimate to realize a system where the estimate is known with more certainty (lower variance). For the systems depicted in Figs. 4–8, there is not a significant decrease

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TABLE VII PARETO OPTIMAL SOLUTIONS FOR INDEPENDENT COMPONENT ESTIMATES (EXAMPLE 1)

Fig. 6. Example 1 Pareto optimal solution 3 (R(t) = 0:9834, V ar (R(t)) = 0:000 353).

Fig. 3.

Pareto optimal solution for independent estimates (Example 1).

Fig. 7. Example 1 Pareto optimal solution 4 (R(t) = 0:9785, V ar (R(t)) = 0:000 319). Fig. 4.

Example 1 Pareto optimal solution 1 (R(t) = 0:9844, V ar (R(t)) =

0:000 438).



0:000 380).

in the variance, and the most reliable system estimate (Fig. 4) is likely a reasonable choice for this problem. The problem was then solved, using SSRP and B&B, with the reliability estimates of the same component type assumed to be dependent. This is a more realistic formulation. It generally leads to higher variances, and also, tends to discourage the use of the same component type in parallel. For this problem, the evaluation of the system variance estimates requires the knowledge of higher moments of component unreliability. In this example, the available information for the components is the reliability estimate, and the associated variance. Pseudo-binomial sampling is assumed for each component test, from which higher order moments are derived. The detailed procedure was given by Jin & Coit [12]. The results are presented in Table VIII. Fig. 9 depicts the relationship between the system reliability and its associated variance for the Pareto optimal set when components of the same type have dependent reliability estimates.

The six different Pareto optimal design structures are depicted in Figs. 10–15. It is interesting to note that quite different solutions are obtained. For this example, the first two Pareto optimal solutions (from Table VIII) are similar. However, the third solution ( , ) offers a 50% reduction in the estimate variance for a marginal reduction in the system reliability estimate. If the decision-maker is risk-averse, he or she would likely select the third Pareto optimal solution. Previous optimization research which ignored estimation uncertainty does not allow for this type of an assessment. The system reliability estimate variance can be significantly higher when component reliability estimates are treated as statistically dependent. For example, Tables VII and VIII have two identical nondominant solutions correand 0.9834, respectively. The sponding to variances for the independent component estimate case are and 0.000 353, respectively; while and 0.001 24, the dependent case has


377

TABLE VIII PARETO OPTIMAL SOLUTIONS FOR DEPENDENT COMPONENT ESTIMATES (EXAMPLE 2)

Fig. 13.


0:000 548).

Fig. 9. Pareto optimal solutions for dependent estimates (Example 2).

Fig. 14. Example 2 Pareto optimal solution 1 (R(t) = 0:9844, V ar (R(t)) = 0:001 34).

Fig. 10.


Fig. 12.


0:000 569).

respectively. The later is approximately 3 to 3.5 times larger than the former. B. Recovery Block Architecture for Distributed Systems The problem is formulated where there are four subsystems (Audio A/D, Filter, Decode Speech, and Text) that collectively form a speech recognition software system, as depicted in


0:000 509).


0:000 487).

Fig. 16. Each subsystem has three alternative hardware options and four alternative software options. To implement RB/1/1 for each module, two hardware and two software versions must be selected for each subsystem. One hardware selection is made, and then, two of that type are used. Alternatively, two different software versions are selected. The hardware and software alternatives are presented in Table IX. , , and Other design conditions are . Their corresponding variances are , , and . The design optimization problem is solved for problems with a maximum cost of 1600, and 1952. The problems were solved using SSRP/B&B for iteratively and to determine a Pareto optimal set. varied values of The Pareto optimal solution for the first problem is given in Table X, and Fig. 17. As before, consideration of estimation uncertainty yields significantly different solutions. Examination of the solutions indicates that the third Pareto optimal solution seems to provide the best trade-off between reliability

378

Fig. 16.


Processor-type subsystems. TABLE IX RB/1/1 INPUT PARAMETERS

TABLE X PARETO OPTIMAL SOLUTIONS FOR RB/1/1EXAMPLE 1

Fig. 18. Fig. 17.

Pareto optimal solutions.

Pareto optimal solutions for RB/1/1 Example one.

and variance. If this had been formulated as a single objective problem, it is unlikely that this solution would have been identified. The Pareto optimal solution for the next problem is given in Table XI, and Fig. 18. For this problem, the third Pareto optimal solution also seems to provide the best compromise between reliability and variance for risk-averse decision-makers. It is noted that in the figures, there is often a pronounced “knee.” This is the point where very small decreases in variance are associated with relatively large decreases in the reliability

estimate. Solutions below the “knee” can generally be discarded prior to the consideration and selection of a final recommended design. VI. DISCUSSION The paper pertains to stochastic optimization of system reliability, where risk is considered in decision making. Table XII provides an overview or “roadmap” of different solution methodologies for different problem domains, considering the decision-makers propensity for risk, and the availability of defensible quantitative weights to compare solutions with different reliability estimates and variance values.


379

TABLE XI PARETO OPTIMAL SOUTION FOR RB/1/1 EXAMPLE 2

TABLE XII RELIABILITY OPTIMIZATION ANALYSIS SUMMARY

This new formulation provides the reliability analyst a more realistic tool to select a recommended system design which more faithfully reflects the needs of the user community. The method was demonstrated on sample problems, and as expected, very different designs were obtained. The decision-maker selects the best compromised solution from a series of nondominant solution sets based on his or her experiences and ingenuity, or assisted by decision-making analytical tools, e.g., utility theory, AHP.

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[5] D. W. Coit, “System reliability confidence intervals for complex systems with estimated component reliability,” IEEE Trans. Rel., vol. 46, pp. 487–493, 1997. [6] D. W. Coit and A. Smith, “Reliability optimization for series-parallel systems using a genetic algorithm,” IEEE Trans. Rel., vol. 45, pp. 895–904, Sept. 1996. [7] A. Dhingra, “Optimal apportionment of reliability & redundancy in series systems under multiple objectives,” IEEE Trans. Rel., vol. 41, no. 4, pp. 576–582, 1992. [8] H. Eschenauer, J. Koski, and A. Osycza, Eds., Multicriteria Design Optimization. Berlin, New York: Springer-Verlag, 1990. [9] D. E. Fyffe, W. W. Hines, and N. K. Lee, “System reliability allocation problem and a computational algorithm,” IEEE Trans. Rel., vol. 17, pp. 64–69, June 1968. [10] P. M. Ghare and R. E. Taylor, “Optimal redundancy for reliability in series system,” Oper. Res., vol. 17, pp. 838–847, Sept. 1969. [11] A. Goicoechea, D. R. Hansen, and L. Duckstein, Multiobjective Decision Analysis With Engineering and Business Applications. New York: John Wiley, 1982. [12] T. Jin and D. W. Coit, “Variance of system reliability estimates with arbitrarily repeated components,” IEEE Trans. Rel., vol. 50, September 2001. [13] J. Koski, “Defectiveness of weighting method in multicriterion optimization of structures,” Communications in Applied Numerical Methods, vol. 1, pp. 333–337, 1985. [14] W. Kuo, V. Prasad, F. Tillman, and C.-L. Hwang, Optimal Reliability Design: Fundamentals and Applications, UK: Cambridge University Press, 2000.

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[15] W. Kuo and V. Prasad, “An annotated overview of system-reliability optimization,” IEEE Trans. Rel., vol. 49, pp. 487–493, June 2000. [16] D. Li, “Interactive parametric dynamic programming and its application of large system reliability,” J. Math. Analy. Appl., vol. 191, pp. 589–607, 1995. [17] K. Misra and U. Sharma, “An effective approach for multiple criteria redundancy optimization problems,” Microelectron. Reliab., vol. 31, no. 2/3, pp. 303–321, 1991. , “Multi-criteria optimization for combined reliability and redun[18] dancy allocation in systems employing mixed redundancies,” Microelectron. Reliab., vol. 31, no. 2/3, pp. 323–335, 1991. [19] Y. Nakagawa and S. Miyazaki, “Surrogate constraints algorithm for reliability optimization problems with two constraints,” IEEE Trans. Rel., vol. 30, pp. 175–180, 1981. [20] T. M. Ozen, Applied Mathematical Programming for Production and Engineering Management: Prentice-Hall, 1986. [21] L. A. Painton and J. E. Campbell, “Genetic algorithms in optimization of system reliability,” IEEE Trans. Rel., vol. 44, pp. 172–178, 1995. [22] R. Rubinstein, G. Levitin, A. Lisniaski, and H. Ben-Haim, “Redundancy optimization of static series-parallel reliability models under uncertainty,” IEEE Trans. Rel., vol. 46, pp. 503–511, 1997. [23] C. S. Sung and Y. K. Cho, “Branch-and-bound optimization for a series system with multiple-choice constraints,” IEEE Trans. Rel., vol. 48, no. 2, pp. 108–117, June 1999. [24] N. Wattanapongsakorn and S. P. Levitan, “Reliability optimization models of fault-tolerant distributed systems,” in Reliability and Maintainability Symp. (RAMS), Philadelphia, PA, Jan. 22–25, 2001, pp. 193–199.


David W. Coit is an Associate Professor in the Industrial and Systems Engineering Department at Rutgers University. He received a B.S. degree in Mechanical Engineering from Cornell University, an M.B.A. from Rensselaer Polytechnic Institute and M.S. and Ph.D. in Industrial Engineering from the University of Pittsburgh. He also has over ten years of experience working for IIT Research Institute (IITRI), Rome NY where he was a Reliability Analyst and Project Manager, and in his final position, the Manager of Engineering at IITRI’s Assurance Technology Center. In 1999, he was awarded a CAREER grant from NSF to study reliability optimization. His current research involves reliability prediction and optimization, and multi-criteria optimization considering uncertainty. He is a member of the IEEE and IEEE Reliability Society.

Tongdan Jin obtained his Ph.D. from the Industrial and Systems Engineering Department at Rutgers University in 2001. He received his M.S. (1994) in Electromechanical Engineering from Beijing Institute of Technology, specializing in Computer-Aided Design. He received a B.S. (1991) in Electrical Engineering from Northwest Institute of Light Industry, PR China. He is currently working as a Reliability Engineer at Digital Process Lab at Teradyne to improve field semiconductor test equipment reliability via corrective actions and accelerated life testing. His research interests are system and component reliability analysis and prediction, reliability optimization.

Naruemon Wattanapongsakorn is an Assistant Professor in Computer Engineering at the King Mongkut University of Technology Thonburi (KMUTT), Thailand. She received the B.S. degree in Computer Engineering (1994) and the M.S. degree in Electrical Engineering (1995), both from The George Washington University, and Ph.D. degree in Electrical Engineering (2000) from the University of Pittsburgh. Her research interests include distributed system dependability analysis, optimization algorithms, embedded system modeling, software fault-tolerance, and statistical analysis of system reliability. She is a member of the IEEE and IEEE Reliability Society.