W. Kuo, R. Wan - Recent Advances in Optimal Reliability Allocation, страница 4

Rarelydoes a single objective with several hard constraints adequately representa real problem for which an optimal design is required. When designing areliable system, as formulated by P4, it is always desirable to simultaneously optimize several opposing design objectives such as reliability,cost, even volume and weight. For this reason, a recently proposed multiobjective system design problem deserves a lot of attention.

The objectives of this problem are to maximize the system reliability estimates andminimize their associated variance while considering the uncertainty ofthe component reliability estimations. A Pareto optimal set, which includesall of the best possible trade-offs between the given objectives, ratherthan a single optimal solution, is usually identified for multi-objective optimization problems.When considering complex systems, the reliability optimization problem has been modeled as a fuzzy multi-objective optimization problem inRef [106], where linear membership functions are used for all of the fuzzygoals.

With the Bellman & Zadeh model, the decision is defined as the intersection of all of the fuzzy sets represented by the objectives.µ D~ ( x) = ( µ ~f ( x) ∗ L ∗ µ ~f ( x) ∗ L ∗ µ ~f ( x))1im(19)The influence of various kinds of aggregators, such as the product operator,min operator, arithmetic mean operator, fuzzy, and a convex combinationRecent Advances in Optimal Reliability Allocation17of the min and the max operator and γ -operator on the solution is also studiedprimarily to learn each advantage over the non-compensatory min operator.

It was found that in some problems illustrated in this paper, the fuzzyand the convex combination of the min and the max operator yield efficientsolutions.Refs [114, 115] solve multi-objective reliability- redundancy allocationproblems using similar linear membership functions for both objectivesand constraints. By introducing 0-1 variables and by using an add operatorto obtain the weighted sum of all the membership functions, the problem istransformed into a bi-criteria single-objective linear programming problemwith Generalized Upper Bounding (GUB) constraints.

The proposed hybrid GA makes use of the GUB structure and combines it with a heuristicapproach to improve the quality of the solutions at each generation.With a weighting technique, Ref [28] also transfers P4 into a singleobjective optimization problem and proposes a GA-based approach whoseparameters can be adjusted with the experimental plan technique.

Ref [7]develops a multi-objective GA to obtain an optimal system configurationand inspection policy by considering every target as a separate objective.Both problems have two objectives: maximization of the system reliabilityand minimization of the total cost, subject to resource constraints.P4 is considered for series-parallel systems, RB/1/1, and bridge systemsin [16] with multiple objectives to maximize the system reliability whileminimizing its associated variance when the component reliability estimates are treated as random variables.

For series-parallel systems, component reliabilities of the same type are considered to be dependent sincethey usually share the same reliability estimate from a pooled data set. Thesystem variance is straightforwardly expressed as a function in the highermoments of the component unreliability estimates [38]. For RB/1/1, thehardware components are considered identical and statistically independent, while even independently developed software versions are found tohave related faults as presented by the parameters Prv and Pall. Paretooptimal solutions are found by solving a series of weighted objective problems with incrementally varied weights.

It is worth noting that significantly different designs are obtained when the formulation incorporatesestimation uncertainty or when the component reliability estimates aretreated as statistically dependent. Similarly, [91] utilizes a multi-objectiveGA to select an optimal network design that balances the dual objectivesof high reliability and low uncertainty in its estimation. But the latterexploits Monte Carlo simulation as the objective function evaluation engine.Ref [131] presents an efficient computational methodology to obtain theoptimal system structure of a static transfer switch, a typical power electronic device.

This device can be decomposed into several components and18Way Kuo and Rui Wanits equivalent reliability block diagram is obtained by the minimal cut setmethod. Because of the existence of unit-to-unit variability, each component chosen from several off-the-shelf types is considered with failure rateuncertainty, which is modeled by a normal, or triangular, distribution.

Thesimulation of the component failure rate distributions is performed usingthe Latin Hypercube Sampling method, and a simulated annealing algorithm is finally applied to generate the Pareto optimal solutions.Ref [116] illustrates the application of the ant colony optimization algorithm to solve both continuous function and combinatorial optimizationproblems in reliability engineering.

The single or multi-objective reliabilityoptimization problem is analogized to Dorigo’s TSP problem, and a combinatorial algorithm, which includes a global search inspired by a GA coupled with a pheromone-mediated local search, is proposed. After the globalsearch, the pheromone values for the newly created solution are calculatedby a weighted average of the pheromone values of the corresponding parent solutions.

A trial solution for conducting the local search is selectedwith a probability proportional to its current pheromone trial value. A twostep strength Pareto fitness assignment procedure is combined to handlemulti-objective problems. The advantage of employing the ant colony heuristic for multi-objective problems is that it can produce the entire set ofoptimal solutions in a single run.Ref [118] tests five simulated annealing-based multi-objective algorithms – SMOSA, UMOSA, PSA, PDMOSA and WMOSA. Evaluated by10 comparisons, Measure C is introduced to gauge the coverage of two approximations for the real non-dominated set.

From the analysis, the computational cost of the WMOSA is the lowest, and it works well even whena large number of constraints are involved, while the PDMOSA consumesmore computational time and may not perform very well for problems withtoo many variables.1.4 Developments in Optimization TechniquesThis section reviews recent developments of heuristic algorithms, metaheuristic algorithms, exact methods and other optimization techniques inoptimal reliability design. Due to their robustness and feasibility, metaheuristic algorithms, especially GAs, have been widely and successfullyapplied. To improve computation efficiency or to avoid premature convergence, an important part of this work has been devoted in recent years todeveloping hybrid genetic algorithms, which usually combine a GA with heuristic algorithms, simulation annealing methods, neural network techniquesRecent Advances in Optimal Reliability Allocation19or other local search methods.

Though more computation effort is involved, exact methods are particularly advantageous for small problems,and their solutions can be used to measure the performance of the heuristicor meta-heuristic methods [45]. No obviously superior heuristic methodhas been proposed, but several of them have been well combined with exact or meta-heuristic methods to improve their computation efficiency.1.4.1 Meta-Heuristic MethodsMeta-heuristic methods inspired by natural phenomena usually include thegenetic algorithm, tabu search, simulated annealing algorithm and ant colony optimization method.

ACO has been recently introduced into optimalreliability design, and it is proving to be a very promising general methodin this field. Ref [1] provides a comparison of meta-heuristics for the optimal design of computer networks.1.4.1.1 Ant Colony Optimization MethodACO is one of the adaptive meta-heuristic optimization methods developed by M. Dorigo for traveling salesman problems in [21] and furtherimproved by him in [22-26]. It is inspired by the behavior of real life antsthat consistently establish the shortest path from their nest to food.