W. Kuo, R. Wan - Recent Advances in Optimal Reliability Allocation

Recent Advances in Optimal ReliabilityAllocation1Way Kuo and Rui WanDepartment of Industrial and Information Engineering, University ofTennessee, Knoxville, TN 37919, USAAcronymsGAHGASAACOTSIAGDACEANNNFTUGFMSSRB/i/jNVP/i/jLMSSWSLMCCSATNWVSACCNUMOSASMOSAgenetic algorithmhybrid genetic algorithmsimulated annealing algorithmant colony optimizationtabu searchimmune algorithmgreat deluge algorithmcellular evolutionary approachneural networknear feasible thresholduniversal generating functionmulti-state systemrecovery block architecture that can tolerate i hardwareand j software faultsN-version programming architecture that can tolerate ihardware and j software faultslinear multi-state sliding-window systemlinear multi-state consecutively connected systemacyclic transmission networkweighted voting systemacyclic consecutively connected networkUlungu multi-objective simulated annealingSuppapitnarm multi-objective simulated annealing1 C 2006 IEEE.

Reprinted, with permission, from IEEE Transactions on Systems,Man, and Cybernetics, A: Systems and Humans, 36(6), 2006.W. Kuo and R. Wan: Recent Advances in Optimal Reliability Allocation, Computational Intelligence inReliability Engineering (SCI) 39, 1–36 (2007)www.springerlink.com© Springer-Verlag Berlin Heidelberg 20072Way Kuo and Rui WanPSAPDMOSAWMOSAPareto simulated annealingPareto domination based multi-objective simulated annealingweight based multi-objective simulated annealingNotationsxjnumber of components at subsystem jrjcomponent reliability at subsystem jnmxnumber of subsystems in the systemnumber of resourcesg i (x)total amount of resource i required for xsystem reliabilitytotal system costa specified minimum RSa specified minimum C SRSCSR0C0α(x 1 ,..., x n , r1 ,...rn )g2 jsystem user’s risk level, 0<α<1system percentile life, tα , x = inf{t ≥ 0 : RS ≤ 1 − α }generalized MSS availability indexa specified minimum E Sthe set of optimal solutions of P1the set of optimal solutions of P2U-function of component ioutput performance level of component 1, i = 1,…, Ioutput performance level of component 2, j = 1,…, Jp1ilimt →∞ [Pr{g1 (t ) = g1i }]p2 jlimt →∞ [Pr{g 2 (t ) = g 2 j }]ωsω (⋅) of series componentsω sγω s for Type- γ MSSω sτωpω s in mode τω (⋅) of parallel componentsω pγω p for Type- γ MSStα , xESE0S1S2U i (z)g1iRecent Advances in Optimal Reliability Allocation3ω τpω p in mode τGτµ D (x )output performance level of the entire MSS in mode τsystem demand in mode τfunction representing the desired relation between MSSperformance level and demand in mode τmembership function of the fuzzy decisionµ f (x)membership function of ith fuzzy goalrijreliability of component j in subsystem icijcost of component j in subsystem iwijweight of component j in subsystem iτ ijpheromone trail intensity of (i, j)η ijproblem-specific heuristic of (i, j), η ij = rij /(cij + wij )transition probability of (i, j),WτFτ (Gτ ,Wτ )iPij⎧ τ ij (ηij ) 0.5⎪ ai0.5⎪Pij = ⎨ ∑τ il (ηil )⎪ l =1⎪⎩ 0TW∗PryPallj ∈ the set of availablecomponent choicesotherwiseMSS operation periodrequired MSS performance levelprobability of failure from related fault between two software versionsprobability of failure from related fault among all softwareversions due to faults in specification1.1 IntroductionReliability has become an even greater concern in recent years becausehigh-tech industrial processes with increasing levels of sophisticationcomprise most engineering systems today.

Based on enhancing componentreliability and providing redundancy while considering the trade-off between system performance and resources, optimal reliability design thataims to determine an optimal system-level configuration has long been animportant topic in reliability engineering. Since 1960, many publicationshave addressed this problem using different system structures, performance4Way Kuo and Rui Wanmeasures, optimization techniques and options for reliability improvement.Refs [45], [93] and [123] provide good literature surveys of the earlywork in system reliability optimization. Tillman, et al.

[123] were the firstto classify papers by system structure, problem type, and solution methods.Also described and analyzed in [123] are the advantages and shortcomingsof various optimization techniques. It was during the 1970s that variousheuristics were developed to solve complex system reliability problems incases where the traditional parametric optimization techniques were insufficient.

In their 2000 report, Kuo and Prasad [45] summarize the developments in optimization techniques, along with recent optimization methodssuch as meta-heuristics, up until that time. This chapter discusses the contributions made to the literature since the publication of [45]. The majorityof recent work in this area is devoted to••••••multi-state system optimization;percentile life employed as a system performance measure;multi-objective optimization;active and cold-standby redundancy;fault-tolerance mechanism;optimization techniques, especially ant colony algorithms and hybridoptimization methods.Based on their system performance, reliability systems can be classifiedas binary-state systems or multi-state systems.

A binary-state system andits components may exist in only two possible states – either working orfailed. Binary system reliability models have played very important rolesin practical reliability engineering. To satisfactorily describe the performance of a complex system, however, we may need more than two levels ofsatisfaction – for example, excellent, average, and poor [46]. For this reason, multi-state system reliability models were proposed in the 1970s, although a large portion of the work devoted to MSS optimal design hasemerged since 1998. The primary task of multi-state system optimizationis to define the relationship between component states and system states.Measures of system performance are basically of four kinds: reliability,availability, mean time-to-failure and percentile life. Reliability has beenwidely used and thoroughly studied as the primary performance measurefor non-maintained systems.

For a maintained system, however, availability, which describes the percentage of time the system really functions,should be considered instead of reliability. Availability is most commonlyemployed as the performance measure for renewable MSS. Meanwhile,Recent Advances in Optimal Reliability Allocation5percentile life is preferred to reliability and mean time-to-failure when thesystem mission time is indeterminate, as in most practical cases.Some important design principles for improving system performance aresummarized in [46]. This chapter primarily reviews articles that addresseither the provision of redundant components in parallel or the combinationof structural redundancy with the enhancement of component reliability.These are called redundancy allocation problems and reliability-redundancyallocation problems, respectively. Redundancy allocation problems are welldocumented in [45] and [46], which employ a special case of reliabilityredundancy allocation problems without exploring the alternatives of component combined improvement.

Recently, much of the effort in optimalreliability design has been placed on general reliability-redundancy allocation problems, rather than redundancy allocation problems.In practice, two redundancy schemes are available: active and coldstandby. Cold-standby redundancy provides higher reliability, but it is hardto implement because of the difficulty of failure detection. Reliability design problems have generally been formulated considering active redundancy; however, an actual optimal design may include active redundancyor cold-standby redundancy or both.However, any effort for improvement usually requires resources.

Quiteoften it is hard for a single objective to adequately describe a real problemfor which an optimal design is required. For this reason, multi-objectivesystem design problem always deserves a lot of attention.Optimal reliability design problems are known to be NP-hard [8]. Finding efficient optimization algorithms is always a hot spot in this field.Classification of the literature by reliability optimization techniques issummarized in Table 1.

Meta-heuristic methods, especially GAs, havebeen widely and successfully applied in optimal system design because oftheir robustness and efficiency, even though they are time consuming, especially for large problems. To improve computation efficiency, hybridoptimization algorithms have been increasingly used to achieve an effective combination of GAs with heuristic algorithms, simulation annealingmethods, neural network techniques and other local search methods.This chapter describes the state-of-art of optimal reliability design. Emphasizing the foci mentioned above, we classify the existing literaturebased on problem formulations and optimization techniques.

The remainder of the chapter is organized as follows: Section 2 includes four mainproblem formulations in optimal reliability allocation; Section 3 describesadvances related to those four types of optimization problems; Section 4 summarizes developments in optimization techniques; and Section 5 provides6Way Kuo and Rui Wanconclusions and a discussion of future challenges related to reliabilityoptimization problems.Table 1. Reference classification by reliability optimization methodsMeta-heuristic AlgorithmACO[82], [92], [95], [116][1], [7], [11], [14], [28], [35], [52], [53], [54], [56], [57], [58], [59],GA[60], [62], [63], [64], [65], [67], [68], [72], [73], [74], [78], [79],[84], [91], [128]HGA[33], [34], [48], [49], [50], [114], [115], [121], [122], [132], [133]TS[41]SA[1], [118], [124], [131]IA[9]GDA[107]CEA[109]Exact Method[20], [27], [30], [83], [102], [103], [119], [120]Max-Min Approach[51], [104]Heuristc Method[105], [129]Dynamic Programming[97], [127]1.2 Problem FormulationsAmong the diversified problems in optimal reliability design, the following four basic formulations are widely covered.Problem 1 (P1):max R S = f (x)s.t.g i (x) ≤ bi , for i = 1, K , mx∈XorRecent Advances in Optimal Reliability Allocation7min C S = f (x)s.t.Rs ≥ R0g i ( x ) ≤ bi ,for i = 1, K , mx∈XProblem 1 formulates the traditional reliability- redundancy allocationproblem with either reliability or cost as the objective function.

Its solutionincludes two parts: the component choices and their corresponding optimalredundancy levels.Problem 2 (P2):max t α , xs.t.g i (t α , x ; x) ≤ bi ,for i = 1,..., mx∈XProblem 2 uses percentile life as the system performance measure instead of reliability. Percentile life is preferred especially when the systemmission time is indeterminate.