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However, it is hard to find a closed analytical form of percentile life in decision variables.Problem 3 (P3):max E (x, T, W * )s.t.g i ( x ) ≤ bi ,for i = 1, K , mx∈Xormin C s (x)s.t.E ( x, T, W * ) ≥ E 0g i ( x ) ≤ bi ,for i = 1, K , mx∈XProblem 3 represents MSS optimization problems. Here, E is used as ameasure of the entire system availability to satisfy the custom demand represented by a cumulative demand curve with a known T and W*.Problem 4 (P4):8Way Kuo and Rui Wanmax z = [ f1 ( x ), f 2 ( x ),K, f S ( x )]s.t.g i ( x ) ≤ bi ,for i = 1,K, mx∈XFor multi-objective optimization, as formulated by Problem 4, a Paretooptimal set, which includes all of the best possible trade-offs betweengiven objectives, rather than a single optimal solution, is usually identified,.In all of the above formulations, the resource constraints may be linearor nonlinear or both.The literature, classified by problem formulations, is summarized inTable 2.Table 2.

Reference classification by problem formulationP1[1], [2], [3], [9], [12], [13], [15], [20], [27], [30], [35], [36], [41], [48],[49], [50], [79], [81], [83], [95], [97], [100], [101], [102], [109], [119],[120], [121], [122], [124], [127], [129], [132], [133]P2[11], [14], [39], [103], [132], [133]P3[52], [53], [54], [56], [57], [58], [59], [60], [62], [63], [64], [65], [67],[68], [72], [73], [74], [78], [79], [84], [92], [99], [105][7], [16], [28], [91], [106], [114], [115], [116], [118], [131], [132]P41.3 Brief Reviews of Advances in P1-P41.3.1 Traditional Reliability-Redundancy Allocation Problem (P1)System reliability can be improved either by incremental improvements incomponent reliability or by provision of redundancy components in parallel;both methods result in an increase in system cost.

It may be advantageousto increase the component reliability to some level and provide redundancyat that level [46], i.e. the tradeoff between these two options must beconsidered. According to the requirements of the designers, traditionalreliability-redundancy allocation problems can be formulated either tomaximize system reliability under resource constraints or to minimize thetotal cost that satisfies the demand on system reliability. These kinds ofRecent Advances in Optimal Reliability Allocation9problems have been well-developed for many different system structures,objective functions, redundancy strategies and time-to-failure distributions.Two important recent developments related to this problem are addressedbelow.1.3.1.1 Active and Cold-Standby RedundancyP1 is generally limited to active redundancy. A new optimal system configuration is obtained when active and cold-standby redundancies are bothinvolved in the design.

A cold-standby redundant component does not failbefore it is put into operation by the action of switching, whereas the failure pattern of an active redundant component does not depend on whetherthe component is idle or in operation. Cold-standby redundancy can provide higher reliability, but it is hard to implement due to the difficulties involved in failure detection and switching.In Ref [12], optimal solutions to reliability-redundancy allocation problems are determined for non-repairable systems designed with multiplek-out-of-n subsystems in series. The individual subsystems may use eitheractive or cold-standby redundancy, or they may require no redundancy.

Assuming an exponentially distributed component time-to-failure with rate λij ,the failure process of subsystem i with cold-standby redundancy can be described by a Poisson process with rate λij k i , while the subsystem reliabilitywith active redundancy is computed by standard binominal techniques.For series-parallel systems with only cold-standby redundancy, Ref [13]employs the more flexible and realistic Erlang distributed component timeto-failure.

Subsystem reliability can still be evaluated through a Poissonprocess, though ρ i (t ) must be introduced to describe the reliability of theimperfect detection/switching mechanism for each subsystem.Ref [15] directly extends this earlier work by introducing the choice ofredundancy strategies as an additional decision variable. With imperfectswitching, it illustrates that there is a maximum redundancy level wherecold-standby reliability is greater than, or equal to, active reliability, i.e.

coldstandby redundancy is preferable before this maximum level while activeredundancy is preferable after that.All three problems formulated above can be transformed by logarithmtransformation and by defining new 0-1 decision variables. This transformation linearizes the problems and allows for the use of integer programmingalgorithms.

For each of these methods, however, no mixture of componenttypes or redundancy strategies is allowed within any of the subsystems.10Way Kuo and Rui WanIn addition, Ref [6] investigates the problem of where to allocate a sparein a k-out-of-n: F system of dependent components through minimalstandby redundancy; and Ref [110] studies the allocation of one active redundancy when it differs based on the component with which it is to beallocated. Ref [101] considers the problem of optimally allocating a fixednumber of s-identical multi-functional spares for a deterministic or stochastic mission time. In spite of some sufficiency conditions for optimality, theproposed algorithm can be easily implemented even for large systems.1.3.1.2 Fault-Tolerance MechanismFault tolerance is the ability of a system to continue performing its intended function in spite of faults.

System designs with fault-tolerancemechanisms are particularly important for some computer-based systemswith life-critical applications, since they must behave like a non-repairablesystem within each mission, and maintenance activities are performed onlywhen the system is idle [3].Ref [2] maximizes the reliability of systems subjected to imperfect faultcoverage. It generalizes that the reliability of such a system decreases withan increase in redundancy after a particular limit. The results include theeffect of common-cause failures and the maximum allowable spare limit.The models considered include parallel, parallel-series, series-parallel,k-out-of-n and k-out-of-(2k-1) systems.Similarly to imperfect fault-coverage, Ref [3] later assumes the redundancy configurations of all subsystems in a non-series-parallel system arefixed except the k-out-of-n: G subsystem being analyzed.

The analysisleads to n*, the optimal number of components maximizing the reliabilityof this subsystem, which is shown to be necessarily greater than, or equalto, the optimal number required to maximize the reliability of the entiresystem. It also proves that n* offers exactly the maximal system reliabilityif the subsystem being analyzed is in series with the rest of the system.These results can even be extended to cost minimization problems.Ref [87] considers software component testing resource allocation for asystem with single or multiple applications, each with a pre-specified reliability requirement.

Given the coverage factors, it can also include faulttolerance mechanisms in the problem formulation. The relationship betweenthe component failure rates of and the cost of decreasing this rate is modeled by various types of reliability-growth curves.For software systems, Ref [79] presents a UGF & GA based algorithmthat selects the set of versions and determines the sequence of their execution, such that the system reliability (defined as the probability of obtainingRecent Advances in Optimal Reliability Allocation11the correct output within a specified time) is maximized subject to costconstraints.

The software system is built from fault-tolerant NVP and RBcomponents.All of these optimization models mentioned above have been developedfor hardware-only or software-only systems. Ref [124] first considers several simple configurations of fault-tolerant embedded systems (hardwareand software) including NVP/0/1, NVP/1/1, and RB/1/1, where failures ofsoftware units are not necessarily statistically independent. A real-timeembedded system is used to demonstrate and validate the models solved bya simulated annealing optimization algorithm. Moreover, Ref [80] generally takes into account fault-tolerant systems with series architecture andarbitrary number of hardware and software versions without commoncause failures. An important advantage of the presented algorithm lies inits ability to evaluate both system reliability and performance indices.1.3.2 Percentile Life Optimization Problem (P2)Many diversified models and solution methods, where reliability is used asthe system performance measure, have been proposed and developed sincethe 1960s.

However, this is not an appropriate choice when mission timecannot be clearly specified or a system is intended for use as long as it functions. Average life is also not reliable, especially when the implications offailure are critical or the variance in the system life is high. Percentile life isconsidered to be a more appropriate measure, since it incorporates systemdesigner and user risk. When using percentile life as the objective function,the main difficulty is its mathematical inconvenience, because it is hard tofind a closed analytical form of percentile life in the decision variables.Ref [11] solves redundancy allocation problems for series-parallel systems where the objective is to maximize a lower percentile of the systemtime-to-failure (TTF) distribution. Component TTF has a Weibull distribution with known deterministic parameters.

The proposed algorithm uses agenetic algorithm to search the prospective solution-space and a bisectionsearch to evaluate t ' in R(t ' , x : k ) = 1 − α . It is demonstrated that the solutionthat maximizes the reliability is not particularly effective at maximizingsystem percentile life at any level, and the recommended design configurations are very different depending on the level. Later in the literature,Ref [14] addresses similar problems where Weibull shape parameters areaccurately estimated but scale parameters are random variables following auniform distribution.12Way Kuo and Rui WanRef [103] develops a lexicographic search methodology that is the firstto provide exact optimal redundancy allocations for percentile life optimization problems.

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