Soft computing approach for reliability optimization: State-of-the-art survey

In the broadest sense, reliability is a measure of performance of systems. As systems have grown more complex, the consequences of their unreliable behavior have become severe in terms of cost, effort, lives, etc., and the interest in assessing system reliability and the need for improving the reliability of products and systems have become very important. Most solution methods for reliability optimization assume that systems have redundancy components in series and/or parallel systems and alternative designs are available. Reliability optimization problems concentrate on optimal allocation of redundancy components and optimal selection of alternative designs to meet system requirement. In the past two decades, numerous reliability optimization techniques have been proposed. Generally, these techniques can be classified as linear programming, dynamic programming, integer programming, geometric programming, heuristic method, Lagrangean multiplier method and so on. A Genetic Algorithm (GA), as a soft computing approach, is a powerful tool for solving various reliability optimization problems. In this paper, we briefly survey GA-based approach for various reliability optimization problems, such as reliability optimization of redundant system, reliability optimization with alternative design, reliability optimization with time-dependent reliability, reliability optimization with interval coefficients, bicriteria reliability optimization, and reliability optimization with fuzzy goals. We also introduce the hybrid approaches for combining GA with fuzzy logic, neural network and other conventional search techniques. Finally, we have some experiments with an example of various reliability optimization problems using hybrid GA approach.     

An efficient particle swarm approach for mixed-integer programming in reliability-redundancy optimization applications

The reliability-redundancy optimization problems can involve the selection of components with multiple choices and redundancy levels that produce maximum benefits, and are subject to the cost, weight, and volume constraints. Many classical mathematical methods have failed in handling nonconvexities and nonsmoothness in reliability-redundancy optimization problems. As an alternative to the classical optimization approaches, the meta-heuristics have been given much attention by many researchers due to their ability to find an almost global optimal solutions. One of these meta-heuristics is the particle swarm optimization (PSO). PSO is a population-based heuristic optimization technique inspired by social behavior of bird flocking and fish schooling. This paper presents an efficient PSO algorithm based on Gaussian distribution and chaotic sequence (PSO-GC) to solve the reliability-redundancy optimization problems. In this context, two examples in reliability-redundancy design problems are evaluated. Simulation results demonstrate that the proposed PSO-GC is a promising optimization technique. PSO-GC performs well for the two examples of mixed-integer programming in reliability-redundancy applications considered in this paper. The solutions obtained by the PSO-GC are better than the previously best-known solutions available in the recent literature.     

Optimality Condition and Branch and Bound Algorithm for Constrained Redundancy Optimization in Series Systems

This paper considers the constrained redundancy optimization problem in series systems. This problem can be formulated as a nonlinear integer programming problem of maximizing the overall systems reliability under limited resource constraints. By exploiting special features of the problem, we derive a new necessary condition for optimal redundancy assignments. This condition leads to a new fathoming condition in the branch and bound method that may result in a significant reduction of computational efforts, as evidenced in our numerical calculation for linearly constrained redundancy optimization problems.     

Reliability Optimization by Considering Time‐Dependent Reliability for Components

Reliability optimization is an important and challenging topic both in engineering and industrial situations as its objective is to design a highly reliable system that operates more safely and efficiently under constraints. Redundancy allocation problem (RAP), as one of the most well‐known problems in reliability optimization, has been the subject of many studies over the past few decades. RAP aims to find the best structure and the optimal redundancy level for each subsystem. The main goal in RAP is to maximize the overall system reliability considering some constraints. In all the previous RAP studies, the reliability of the components is considered constant during the system's mission time. However, reliability is time‐dependent and needs to be considered and monitored during the system's lifetime. In this paper, the reliability of components is considered as a function of time, and the RAP is reformulated by introducing a new criterion called ‘mission design life’ defined as the integration of the system reliability function during the mission time. We propose an efficient algorithm for this problem and demonstrate its performance using two examples. Furthermore, we demonstrate the importance of the new approach using a benchmark problem in RAP. Copyright © 2017 John Wiley & Sons, Ltd.     

Reliability optimization using multiobjective ant colony system approaches

The multiobjective ant colony system (ACS) meta-heuristic has been developed to provide solutions for the reliability optimization problem of series-parallel systems. This type of problems involves selection of components with multiple choices and redundancy levels that produce maximum benefits, and is subject to the cost and weight constraints at the system level. These are very common and realistic problems encountered in conceptual design of many engineering systems. It is becoming increasingly important to develop efficient solutions to these problems because many mechanical and electrical systems are becoming more complex, even as development schedules get shorter and reliability requirements become very stringent. The multiobjective ACS algorithm offers distinct advantages to these problems compared with alternative optimization methods, and can be applied to a more diverse problem domain with respect to the type or size of the problems. Through the combination of probabilistic search, multiobjective formulation of local moves and the dynamic penalty method, the multiobjective ACSRAP, allows us to obtain an optimal design solution very frequently and more quickly than with some other heuristic approaches. The proposed algorithm was successfully applied to an engineering design problem of gearbox with multiple stages.     

Practical solutions for multi-objective optimization: An application to system reliability design problems

For multiple-objective optimization problems, a common solution methodology is to determine a Pareto optimal set. Unfortunately, these sets are often large and can become difficult to comprehend and consider. Two methods are presented as practical approaches to reduce the size of the Pareto optimal set for multiple-objective system reliability design problems. The first method is a pseudo-ranking scheme that helps the decision maker select solutions that reflect his/her objective function priorities. In the second approach, we used data mining clustering techniques to group the data by using the k-means algorithm to find clusters of similar solutions. This provides the decision maker with just k general solutions to choose from. With this second method, from the clustered Pareto optimal set, we attempted to find solutions which are likely to be more relevant to the decision maker. These are solutions where a small improvement in one objective would lead to a large deterioration in at least one other objective. To demonstrate how these methods work, the well-known redundancy allocation problem was solved as a multiple objective problem by using the NSGA genetic algorithm to initially find the Pareto optimal solutions, and then, the two proposed methods are applied to prune the Pareto set.     

Efficiently Solving the Redundancy Allocation Problem Using Tabu Search

A tabu search meta-heuristic has been developed and successfully demonstrated to provide solutions to the system reliability optimization problem of redundancy allocation. Tabu search is particularly well-suited to this problem and it offers distinct advantages compared to alternative optimization methods. While there are many forms of the problem, the redundancy allocation problem generally involves the selection of components and redundancy levels to maximize system reliability given various system-level constraints. This is a common and extensively studied problem involving system design, reliability engineering and operations research. It is becoming increasingly important to develop efficient solutions to this reliability optimization problem because many telecommunications (and other) systems are becoming more complex, yet with short development schedules and very stringent reliability requirements. Tabu search can be applied to a more diverse problem domain compared to mathematical programming methods, yet offers the potential of greater efficiency compared to population-based search methodologies, such as genetic algorithms. The tabu search is demonstrated on numerous variations of three different problems and compared to integer programming and genetic algorithm solutions. The results demonstrate the benefits of tabu search for solving this type of problem.     

Reliability optimization of series-parallel systems with a choice of redundancy strategies using a genetic algorithm

This paper proposes a genetic algorithm (GA) for a redundancy allocation problem for the series-parallel system when the redundancy strategy can be chosen for individual subsystems. Majority of the solution methods for the general redundancy allocation problems assume that the redundancy strategy for each subsystem is predetermined and fixed. In general, active redundancy has received more attention in the past. However, in practice both active and cold-standby redundancies may be used within a particular system design and the choice of the redundancy strategy becomes an additional decision variable. Thus, the problem is to select the best redundancy strategy, component, and redundancy level for each subsystem in order to maximize the system reliability under system-level constraints. This belongs to the NP-hard class of problems. Due to its complexity, it is so difficult to optimally solve such a problem by using traditional optimization tools. It is demonstrated in this paper that GA is an efficient method for solving this type of problems. Finally, computational results for a typical scenario are presented and the robustness of the proposed algorithm is discussed.     

Recent advances in system reliability optimization driven by importance measures

System reliability optimization problems have been widely discussed to maximize system reliability with resource constraints. Birnbaum importance is a well-known method for evaluating the effect of component reliability on system reliability. Many importance measures (IMs) are extended for binary, multistate, and continuous systems from different aspects based on the Birnbaum importance. Recently, these IMs have been applied in allocating limited resources to the component to maximize system performance. Therefore, the significance of Birnbaum importance is illustrated from the perspective of probability principle and gradient geometrical sense. Furthermore, the equations of various extended IMs are provided subsequently. The rules for simple optimization problems are summarized to enhance system reliability by using ranking or heuristic methods based on IMs. The importance-based optimization algorithms for complex or large-scale systems are generalized to obtain remarkable solutions by using IM-based local search or simplification methods. Furthermore, a general framework driven by IM is developed to solve optimization problems. Finally, some challenges in system reliability optimization that need to be solved in the future are presented.     

Methodology for the optimal component selection of electronic devices under reliability and cost constraints

The objective of this paper is to present an efficient computational methodology for the reliability optimization of electronic devices under cost constraints. The system modeling for calculating the reliability indices of the electronic devices is based on Bayesian networks using the fault tree approach, in order to overcome the limitations of the series–parallel topology of the reliability block diagrams. Furthermore, the Bayesian network modeling for the reliability analysis provides greater flexibility for representing multiple failure modes and dependent failure events, and simplifies fault diagnosis and reliability allocation. The optimal selection of components is obtained using the simulated annealing algorithm, which has proved to be highly efficient in complex optimization problems where gradient‐based methods can not be applied. The reliability modeling and optimization methodology was implemented into a computer program in Matlab using a Bayesian network toolbox. The methodology was applied for the optimal selection of components for an electrical switch of power installations under reliability and cost constraints. The full enumeration of the solution space was calculated in order to demonstrate the efficiency of the proposed optimization algorithm. The results obtained are excellent since a near optimum solution was found in a small fraction of the time needed for the complete enumeration (3%). All the optimum solutions found during consecutive runs of the optimization algorithm lay in the top 0.3% of the solutions that satisfy the reliability and cost constraints. Copyright © 2007 John Wiley & Sons, Ltd.     

A hybrid ‘dynamic programming/depth-first search’ algorithm, with an application to redundancy allocation

A hybrid 'dynamic programming/depth-first search' algorithm has been developed to solve non-linear integer programming problems arising in the reliability optimization of redundancy allocation. Initially, the technique solves the knapsack relaxation of the original mathematical programming problem using dynamic programming. Then, all solutions in some range of the relaxation problem are obtained via an enumerative depth-first search technique. The solutions are ranked and the optimal solution is given by the best one that satisfies the remaining constraints of the given problem. Computational complexity of the algorithm is also discussed. The salient features of our hybrid algorithm are its simplicity and ease of programming. Our algorithm also has an advantage over the traditional Lagrangian and surrogate dual approaches. It does not have to deal with the issue of 'duality gap' as in classical dual approaches, which is responsible for the failure to identify optimal solutions to the primal integer optimization problems. Of most importance, it guarantees to succeed in identifying an optimal solution.     

A bidirectional evolutionary structural optimization algorithm for mass minimization with multiple structural constraints

A bidirectional evolutionary structural optimization algorithm is presented, which employs integer linear programming to compute optimal solutions to topology optimization problems with the objective of mass minimization. The objective and constraint functions are linearized using Taylor's first-order approximation, thereby allowing the method to handle all types of constraints without using Lagrange multipliers or sensitivity thresholds. A relaxation of the constraint targets is performed such that only small changes in topology are allowed during a single update, thus ensuring the existence of feasible solutions. A variety of problems are solved, demonstrating the ability of the method to easily handle a number of structural constraints, including compliance, stress, buckling, frequency, and displacement. This is followed by an example with multiple structural constraints and, finally, the method is demonstrated on a wing-box, showing that topology optimization for mass minimization of real-world structures can be considered using the proposed methodology.     

An entropy-based three-stage approach for multi-objective system reliability optimization considering uncertainty

This article presents a three-stage approach for solving multi-objective system reliability optimization problems considering uncertainty. The reliability of each component is considered in the formulation as a component reliability estimate in the form of an interval value and discrete values. Component reliability may vary owing to variations in the usage scenarios. Uncertainty is described by defining a set of usage scenarios. To address this problem, an entropy-based approach to the redundancy allocation problem is proposed in this study to identify the deterministic reliability of each component. In the second stage, a multi-objective evolutionary algorithm (MOEA) is applied to produce a Pareto-optimal solution set. A hybrid algorithm based on k-means and silhouettes is performed to select representative solutions in the third stage. Finally, a numerical example is presented to illustrate the performance of the proposed approach.     

System Reliability Optimization Using Gray Wolf Optimizer Algorithm

For the past two decades, nature‐inspired optimization algorithms have gained enormous popularity among the researchers. On the other hand, complex system reliability optimization problems, which are nonlinear programming problems in nature, are proved to be non‐deterministic polynomial‐time hard (NP‐hard) from a computational point of view. In this work, few complex reliability optimization problems are solved by using a very recent nature‐inspired metaheuristic called gray wolf optimizer (GWO) algorithm. GWO mimics the chasing, hunting, and the hierarchal behavior of gray wolves. The results obtained by GWO are compared with those of some recent and popular metaheuristic such as the cuckoo search algorithm, particle swarm optimization, ant colony optimization, and simulated annealing. This comparative study shows that the results obtained by GWO are either superior or competitive to the results that have been obtained by these well‐known metaheuristic mentioned earlier. Copyright © 2016 John Wiley & Sons, Ltd.     

Probabilistic optimal design of laminates using improved particle swarm optimization

A new approach to the particle swarm optimization (PSO) is proposed for the solution of non-linear optimization problems with constraints, and is applied to the reliability-based optimum design of laminated composites. Special mutation-interference operators are introduced to increase swarm variety and improve the convergence performance of the algorithm. The reliability-based optimum design of laminated composites is modelled and solved using the improved PSO. The maximization of structural reliability and the minimization of total weight of laminates are analysed. The stacking sequence optimization is implemented in the improved PSO by using a special coding technique. Examples show that the improved PSO has high convergence and good stability and is efficient in dealing with the probabilistic optimal design of composite structures.     

Alternatives and challenges in optimizing industrial safety using genetic algorithms

Safety (S) improvement of industrial installations leans on the optimal allocation of designs that use more reliable equipment and testing and maintenance activities to assure a high level of reliability, availability and maintainability (RAM) for their safety-related systems. However, this also requires assigning a certain amount of resources (C) that are usually limited. Therefore, the decision-maker in this context faces in general a multiple-objective optimization problem (MOP) based on RAMS+C criteria where the parameters of design, testing and maintenance act as decision variables. Solutions to the MOP can be obtained by solving the problem directly, or by transforming it into several single-objective problems. A general framework for such MOP based on RAMS+C criteria is proposed in this paper. Then, problem formulation and fundamentals of two major groups of resolution alternatives are presented. Next, both alternatives are implemented in this paper using genetic algorithms (GAs), named single-objective GA and multi-objective GA, respectively, which are then used in the case of application to solve the problem of testing and maintenance optimization based on unavailability and cost criteria. The results show the capabilities and limitations of both approaches. Based on them, future challenges are identified in this field and guidelines provided for further research.     

Optimization of series–parallel multi-state systems under maintenance policies

In the redundancy optimization problem, the design goal is achieved by discrete choices made from components available in the market. In this paper, the problem is to find, under reliability constraints, the minimal cost configuration of a multi-state series–parallel system, which is subject to a specified maintenance policy. The number of maintenance teams is less than the number of repairable components, and a maintenance policy specifies the priorities between the system components. To take into account the dependencies resulting from the sharing of maintenance teams, the universal generating function approach is coupled with a Markov model. The resulting optimization approach has the advantage of being mainly analytical.     

Dynamic Programming for QFD Optimization

Quality function deployment (QFD) is a useful method in product design and development and its aim is to improve the quality and to better meet customers' needs. Due to cost and other resource constraints, trade‐offs are always needed. Many optimization methods have been introduced into the QFD process to maximize customer satisfaction under certain constraints. However, current optimization methods sometimes cannot give practical optimal results and the data needed are hard or costly to get. To overcome these problems, this paper proposes a dynamic programming approach for the optimization problem. We first use an extended House of Quality to gather more information. Next, limited resources are allocated to the technical attributes using dynamic programming. The value of each technical attribute can be determined according to the resources allocated to them. Compared with other optimization methods, the dynamic programming method requires less information and the optimal results are more relevant. Copyright © 2005 John Wiley & Sons, Ltd.     

MONSS: A multi-objective nonlinear simplex search approach

This article presents a novel methodology for dealing with continuous box-constrained multi-objective optimization problems (MOPs). The proposed algorithm adopts a nonlinear simplex search scheme in order to obtain multiple elements of the Pareto optimal set. The search is directed by a well-distributed set of weight vectors, each of which defines a scalarization problem that is solved by deforming a simplex according to the movements described by Nelder and Mead's method. Considering an MOP with n decision variables, the simplex is constructed using n+1 solutions which minimize different scalarization problems defined by n+1 neighbor weight vectors. All solutions found in the search are used to update a set of solutions considered to be the minima for each separate problem. In this way, the proposed algorithm collectively obtains multiple trade-offs among the different conflicting objectives, while maintaining a proper representation of the Pareto optimal front. In this article, it is shown that a well-designed strategy using just mathematical programming techniques can be competitive with respect to the state-of-the-art multi-objective evolutionary algorithms against which it was compared.     

Maximization of System Reliability with a Choice of Redundancy Strategies

Optimal solutions to the redundancy allocation problem are determined when either active or cold-standby redundancy can be selectively chosen for individual subsystems. This problem involves the selection of components and redundancy levels to maximize system reliability. Previously, solutions to the problem could only be found if analysts were restricted to a predetermined redundancy strategy for the complete system. Generally, it had been assumed that active redundancy was to be used. However, in practice both active and cold-standby redundancy may be used within a particular system design and the choice of redundancy strategy becomes an additional decision variable. Available optimization algorithms are inadequate for these design problems and better alternatives are required. The methodology presented here is specifically developed to accommodate the case where there is a choice of redundancy strategy. The problem is formulated with imperfect sensing and switching of cold-standby redundant components and k -Erlang distributed time-to-failure. Optimal solutions to the problem are found by an equivalent problem formulation and integer programming. The methodology is demonstrated on a well-known test problem with interesting results. The optimal system design is distinctly different from the corresponding design obtained with only active redundancy. The availability of this tool can result in more reliable and cost-effective engineering designs.