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The continuous relaxed problem, solved by Kuhn-Tuckerconditions and a two-stage hierarchical search, is considered for obtainingan upper bound, which is used iteratively to effectively reduce searchspace. This algorithm is general for any continuous increasing lifetime distribution.Three important results are presented in [39] which describe the generalrelationships between reliability and percentile life maximizing problems.• S2 equals S1 given α t = 1 − R s (t , x ∗t ) , where x ∗t ∈ S 1 ;• S1 equals S2 given t, where X α* ∈ S 2 ;α , xα*• Let ψ (t ) be the optimal objective value of P1.

For a fixed α ,t α = inf{t ≥ 0 :Ψ (t ) ≤ 1 − α } is the optimal objective value of P2.Based on these results, a methodology for P2 is proposed to repeat solving P1 under different mission times satisfying ψ (t 0 ) ≥ 1 − α ≥ ψ (t 2 ) untilx *t =t = x *t =t and t0 − t 2 is within a specified tolerance. It is reported to becapable of settling many unsolved P2s using existing reliability-optimizationalgorithms. Without the necessity of an initial guess, this method is muchbetter than [103] at reaching the exact optimal solution in terms of execution time.021.3.3 MSS Optimization (P3)MSS is defined as a system that can unambiguously perform its task at different performance levels, depending on the state of its components whichcan be characterized by nominal performance rate, availability and cost.Based on their physical nature, multi-state systems can be classified intotwo important types: Type I MSS (e.g.

power systems) and Type II MSS(e.g. computer systems), which use capacity and operation time as theirperformance measures, respectively.In the literature, an important optimization strategy, combining UGF andGA, has been well developed and widely applied to reliability optimization problems of renewable MSS. In this strategy, there are two main tasks:According to the system structure and the system physical nature, obtainthe system UGF from the component UGFs;Find an effective decoding and encoding technique to improve the efficiency of the GA.Recent Advances in Optimal Reliability Allocation13Ref [52] first uses an UGF approach to evaluate the availability of aseries-parallel multi-state system with relatively small computationalresources.

The essential property of the U-transform enables the totalU-function for a MSS with components connected in parallel, or in series,to be obtained by simple algebraic operations involving individual component U-functions. The operator Ωω is defined by (1) - (3).J⎡ Ig ⎤Ω ω (U 1 ( z ), U 2 ( z )) = Ω ω ⎢∑ p1i z g1i , ∑ p 2 j z 2 j ⎥j =1⎦⎣ i =1IJ= ∑∑ p1i p 2 j z(1)ω ( g1i , g 2 j )i =1 j =1Ωω (U1 ( z ),L , U k ( z ), U k +1 ( z ),LU n ( z ))= Ωω (U1 ( z ), L, U k +1 ( z ), U k ( z ),LU n ( z ))Ω ω (U 1 ( z ),L,U k ( z ),U k +1 ( z ),LU n ( z ))= Ω ω (Ωω (U 1 ( z ),L,U k ( z )), Ω ω (U k +1 ( z ),LU n ( z )))(2)(3)The function ω (⋅) takes the form from (4) - (7).For Type I MSS,ω s1 ( g1 , g 2 ) = min( g1 , g 2 )(4)ω p1 ( g1 , g 2 ) = g1 + g 2(5)g1 g 2g1 + g 2(6)ω p 2 ( g1 , g 2 ) = g1 + g 2(7)For Type II MSS,ω s 2 ( g1 , g 2 ) =Later, Ref [55] combines importance and sensitivity analysis and Ref[75] extends this UGF approach to MSS with dependent elements.

Table 3summarizes the application of UGF to some typical MSS structures in optimal reliability design.With this UGF & GA strategy, Ref [84] solves the structure optimization of a multi-state system with time redundancy. TRS can be treated asa Type II MSS, where the system and its component performance are measured by the processing speed. Two kinds of systems are considered: systems14Way Kuo and Rui Wanwith hot reserves and systems with work sharing between componentsconnected in parallel.Table 3. Application of UGF approachSeries-Parallel SystemBridge SystemLMSSWSWVSATNLMCCSACCN[52], [54], [55], [59], [62], [65], [72], [73], [74], [92][53], [56], [71], [84][64], [70], [78]58], [66][63], [69][67], [68][61]Ref [57] applies the UGF & GA strategy to a multi-state system consisting of two parts:• RGS including a number of resource generating subsystems;• MPS including elements that consume a fixed amount of resources toperform their tasks.Total system performance depends on the state of each subsystem in theRGS and the maximum possible productivity of the MPS.

The maximumpossible productivity of the MPS is determined by an integer linear programming problem related to the states of the RGS.Ref [59] develops an UGF & GA strategy for multi-state series-parallelsystems with two failure modes: open mode and closed mode. Two optimal designs are found to maximize either the system availability or theproximity of expected system performance to the desired levels for bothmodes. The function ω (⋅) and the conditions of system success for both twomodes are shown as follows.For Type I MSS,ω sO ( g 1 , g 2 ) = ω sC ( g 1 , g 2 ) = min(g 1 , g 2 )(8)ω Op ( g1 , g 2 ) = ω Cp ( g1 , g 2 ) = g1 + g 2(9)FC (GC , WC ) = GC − WC ≥ 0(10)FO (G O , WO ) = WO − G O ≥ 0(11)For Type II MSS,ω sO ( g 1 , g 2 ) = min( g 1 , g 2 )(12)Recent Advances in Optimal Reliability Allocation15ω Op ( g1 , g 2 ) = max( g1 , g 2 )(13)ω sC ( g1 , g 2 ) = max( g1 , g 2 )(14)ω Cp ( g1 , g 2 ) = min( g1 , g 2 )(15)FC (GC , WC ) = WC − GC ≥ 0(16)FO (GO , WO ) = WO − GO ≥ 0(17)Thus, the system availability can be denoted byAs (t ) = 1 − Pr{FC (GC (t ), WC ) < 0} − Pr{FO (GO (t ), WO ) < 0}(18)Later [65, 71] introduces a probability parameter of 0.5 for both modesand [71] even extends this technique to evaluate the availability of systemswith bridge structures.To describe the ability of a multi-state system to tolerate both internalfailures and external attacks, survivability, instead of reliability, is proposed in Refs [56, 60, 67, 72-74].

Ref [56] considers the problem of howto separate the elements of the same functionality between two parallelbridge components in order to achieve the maximal level of system survivability, while an UGF &GA strategy in [60] is used to solve the more general survivability optimization problem of how to separate the elements ofa series-parallel system under the constraint of a separation cost. Ref [72]considers the problem of finding structure of series-parallel multi-state system (including choice of system elements, their separation and protection)in order to achieve a desired level of system survivability by minimal cost.To improve system’s survivability, Ref [73, 74] further applies a multilevel protection to its subsystems and the choice of structure of multi-levelprotection and choice of protection methods are also included.

Other thanseries-parallel system, Ref [67] provides the optimal allocation of multistate LCCS with vulnerable nodes. It should be noted that the solution thatprovides the maximal system survivability for a given demand does notnecessarily provide the greatest system expected performance rate and thatthe optimal solutions may be different when the system operates under different vulnerabilities.In addition to a GA, Ref [92] presents an ant colony method that combines with a UGF technique to find an optimal series-parallel power structure configuration.16Way Kuo and Rui WanBesides this primary UGF approach, a few other methods have beenproposed for MSS reliability optimization problems.

Ref [105] develops aheuristic algorithm RAMC for a Type I multi-state series-parallel system.The availability of each subsystem is determined by a binomial technique,and, thus, the system availability can be obtained in a straightforwardmanner from the product of all subsystem availabilities without usingUGF. Nevertheless, this algorithm can only adapt to relatively simple formulations, including those with only two-state component behavior and nomixing of functionally equivalent components within a particular subsystem.A novel continuous-state system model, which may represent realitymore accurately than a discrete-state system model, is presented in Ref[86]. Given the system utility function and the component state probabilitydensity functions, a neural network approach is developed to approximatethe objective reliability function of this continuous MSS optimal designproblem.1.3.4 Multi-Objective Optimization (P4)In the previous discussion, all problems were single-objective.

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