W. Kuo, R. Wan - Recent Advances in Optimal Reliability Allocation, страница 5

The essential trait of the ACO algorithm is the combination of a priori information about the structure of a promising solution with posteriori informationabout the structure of previously obtained good solutions [88].Ref [81] first develops an ant colony meta-heuristic optimizationmethod to solve the reliability-redundancy allocation problem for a k-outof-n: G series system. The proposed ACO approach includes four stages:1.

Construction stage: construct an initial solution by selecting componentj for subsystem i according to its specific heuristic η ij and pheromonetrail intensity τ ij , which also sets up the transition probability mass function Pij .2. Evaluation stage: evaluate the corresponding system reliability and penalized system reliability providing the specified penalized parameter.3. Improvement stage: improve the constructed solutions through localsearch4.

Updating stage: update the pheromone value online and offline giventhe corresponding penalized system reliability and the controlling parameter for pheromone persistence.20Way Kuo and Rui WanRef [95] presents an application of the ant system in a reliability optimization problem for a series system, with multi-choice constraints incorporated at each subsystem, to maximize the system reliability subject to thesystem budget. It also combines a local search algorithm and a specificimprovement algorithm that uses the remaining budget to improve thequality of a solution.The ACO algorithm has also been applied to a multi-objective reliabilityoptimization problem [116] and to the optimal design of multi-state seriesparallel power systems [92].1.4.1.2 Hybrid Genetic AlgorithmGA is a population-based directed random search technique inspired by theprinciples of evolution. Though it provides only heuristic solutions, it canbe effectively applied to almost all complex combinatorial problems, and,thus, it has been employed in a large number of references as shown inTable 1.

Ref [29] provides a state-of-the-art survey of GA-based reliabilitydesign.To improve computational efficiency, or to avoid premature convergence,numerous researchers have been inspired to seek effective combinations ofGAs with heuristic algorithms, simulation annealing methods, neural network techniques, steepest decent methods or other local search methods.The combinations are generally called hybrid genetic algorithms, and theyrepresent one of the most promising developmental directions in optimization techniques.Considering a complex system with a known system structure function,Ref [132] provides a unified modeling idea for both active and coldstandby redundancy optimization problems.

The model prohibits any mixture of component types within subsystems. Both the lifetime and the costof redundancy components are considered as random variables, so stochastic simulation is used to estimate the system performance, including themean lifetime, percentile lifetime and reliability. To speed up the solutionprocess, these simulation results become the training data for training aneural network to approximate the system performance. The trained neuralnetwork is finally embedded into a genetic algorithm to form a hybrid intelligent algorithm for solving the proposed model. Later [133] uses random fuzzy lifetimes as the basic parameters and employs a random fuzzysimulation to generate the training data.Ref [48] develops a two-phase NN-hGA in which NN is used as a roughsearch technique to devise the initial solutions for a GA.

By bounding thebroad continuous search space with the NN technique, the NN-hGA derivesRecent Advances in Optimal Reliability Allocation21the optimum robustly. However, in some cases, this algorithm may requiretoo much computational time to be practical.To improve the computation efficiency, Ref [49] presents a NN-flcGAto effectively control the balance between exploitation and explorationwhich characterizes the behavior of GAs. The essential features of the NNflcGA include:• combination with a NN technique to devise initial values for the GA• application of a fuzzy logic controller when tuning strategy GA parameters dynamically• incorporation of the revised simplex search methodLater, [50] proposes a similar hybrid GA called f-hGA for the redundancy allocation problem of a series-parallel system. It is based on• application of a fuzzy logic controller to automatically regulate the GAparameters;• incorporation of the iterative hill climbing method to perform local exploitation around the near optimum solution.Ref [33] considers the optimal task allocation strategy and hardware redundancy level for a cycle-free distributed computing system so that thesystem cost during the period of task execution is minimized.

The proposed hybrid heuristic combines the GA and the steepest decent method.Later [34] seeks similar optimal solutions to minimize system cost underconstraints on the hardware redundancy levels. Based on the GA and a local search procedure, a hybrid GA is developed and compared with thesimple GA. The simulation results show that the hybrid GA provideshigher solution quality with less computational time.1.4.1.3 Tabu SearchThough [45] describes the promise of tabu search, Ref [41] first develops aTS approach with the application of NFT [10] for reliability optimizationproblems.

This method uses a subsystem-based tabu entry and dynamiclength tabu list to reduce the sensitivity of the algorithm to selection of thetabu list length. The definition of the moves in this approach offers anadvantage in efficiency, since it does not require recalculating the entiresystem reliability, but only the reliability of the changed subsystem. Theresults of several examples demonstrate the superior performance of thisTS approach in terms of efficiency and solution superiority when compared to that of a GA.22Way Kuo and Rui Wan1.4.1.4 Other Meta-Heuristic MethodsSome other adaptive meta-heuristic optimization methods inspired by activities in nature have also been proposed and applied in optimal reliability design. Ref [9] develops an immune algorithms-based approach inspiredby the natural immune system of all animals.

It analogizes antibodies andantigens as the solutions and objection functions, respectively. Ref [109]proposes a cellular evolutionary approach combining the multimemberevolution strategy with concepts from Cellular Automata [125] for theselection step. In this approach, the parents’ selection is performed onlyin the neighborhood in contrast to the general evolutionary strategy thatsearches for parents in the whole population. And a great deluge algorithmis extended and applied to optimize the reliability of complex systems in[107]. When both accuracy and speed are considered simultaneously, it isproven to be an efficient alternative to ACO and other existing optimization techniques.1.4.2 Exact MethodsUnlike meta-heuristic algorithms, exact methods provide exact optimal solutions though much more computation complexity is involved.

The development of exact methods, such as the branch-and-bound approach andlexicographic search, has recently been concentrated on techniques to reduce the search space of discrete optimization methods.Ref [119] considers a reliability-redundancy allocation problem inwhich multiple-choice and resource constraints are incorporated. The problem is first transformed into a bi-criteria nonlinear integer programmingproblem by introducing 0-1 variables. Given a good feasible solution, thelower reliability bound of a subsystem is determined by the product of themaximal component reliabilities of all the other subsystems in the solution,while the upper bound is determined by the maximal amount of availablesources of this subsystem.

A branch-and-bound procedure, based on thisreduced solution space, is then derived to search for the global optimal solution. Later, Ref [120] even combines the lower and upper bounds of thesystem reliability, which are obtained by variable relaxation and Lagrangeanrelaxation techniques, to further reduce the search space.Also with a branch-and-bound algorithm, Ref [20] obtains the upperbound of series-parallel system reliability from its continuous relaxationproblem. The relaxed problem is efficiently solved by the greedy procedure described in [19], combining heuristic methods to make use of someslack in the constraints obtained from rounding down.

This technique assumes the objective and constraint functions are monotonically increasing.Recent Advances in Optimal Reliability Allocation23Ref [30] presents an efficient branch-and-bound approach for coherent systems based on a 1-neighborhood local maximum obtained from the steepest ascent heuristic method.

Numerical examples of a bridge system and ahierarchical series-parallel system demonstrate the advantages of this proposed algorithm in flexibility and efficiency.Apart from the branch-and-bound approach, Ref [102] presents a partialenumeration method for a wide range of complex optimization problemsbased on a lexicographic search. The proposed upper bound of system reliability is very useful in eliminating several inferior feasible or infeasiblesolutions as shown in either big or small numerical examples.