Sequencing Optimization in k-out-of-n Cold-Standby Systems Considering Mission Cost

The k-out-of-n: G heterogeneous cold-standby system structure is a widely used fault-tolerant system design method, where the sequence in which the different system elements are initiated can greatly affect the system reliability and mission cost. This paper considers the optimal standby element sequencing problem (SESP) for such systems. Given the desired cold-standby redundancy level and fixed set of element choices, the objective of the optimal system design is to select the initiation sequence of the system elements so as to minimize the expected system mission cost while meeting a certain level of system reliability constraint. Based on a discrete approximation of time-to-failure distributions of the system elements, the system reliability and expected mission cost are evaluated simultaneously using a numerical method. A genetic algorithm is used as an optimization tool for solving the formulated SESP problem for k-out-of-n: G heterogeneous cold-standby systems. Examples are given to illustrate the considered problem and the proposed solution methodology.     

Optimization of predetermined standby mode transfers in 1-out-of-N: G systems

Warm standby redundancy is an important fault-tolerant design technique for improving the reliability of many systems used in life-critical or mission-critical applications. Traditional warm standby models aim to reduce the operational cost and failure rate of the standby elements by keeping them partially powered and partially exposed to operational stresses. However, depending on the level of readiness of a standby element, significant restoration delays and replacement costs can be incurred when the standby element is needed to replace the failed online element. To achieve a balance between the operation cost of standby elements and the replacement costs, this paper proposes a new warm standby model with scheduled (or time-based) standby mode transfer of standby elements. In particular, each standby element can be transferred from warm standby mode to hot standby mode (a mode in which the standby element is ready to take over at any time) at a fixed/predetermined time instants after the mission starts. To facilitate the optimal design and implementation of the proposed model, this paper first suggests a new algorithm for evaluating the reliability and expected mission cost of 1-out-of-N: G system with standby elements subject to the time-based standby mode transfer. The algorithm is based on a discrete approximation of time-to-failure distributions of the elements and can work with any type of distributions. Based on the suggested algorithm the problem of optimizing transfer times of standby elements to the hot standby mode and optimal sequencing of their transfer to the operation mode is formulated and solved. In this problem the expected mission cost associated with elements’ standby and operation expenses and mode transfer expenses is minimized subject to system reliability constraint. Illustrative examples are provided.     

Optimal completed work dependent loading of components in cold standby systems

This paper considers the modelling and optimization of 1-out-of-N: G cold standby (CS) systems with non-repairable components functioning at different levels of productivity or load. The productivity heterogeneity leads to difference in component failure behaviour as well as in operational and replacement costs. Thus, the choice of load or productivity of components can greatly affect the system reliability and mission cost. To make the optimal choice of component loading, we first suggest a method for analysing the reliability and expected mission cost of 1-out-of-N: G non-repairable CS systems with heterogeneous components. The optimal dynamic load distribution problem is then formulated and solved, in which the component loading is chosen depending on the amount of work completed prior to the component activation. The optimal loading is aimed at minimizing the expected mission cost, while meeting a certain system reliability constraint. Examples are given to demonstrate the proposed methodology and the improvement in the optimal design solution through introducing the component productivity’s dependence on the completed work.     

Cold-standby redundancy allocation problem with degrading components

Components in cold-standby state are usually assumed to be as good as new when they are activated. However, even in a standby environment, the components will suffer from performance degradation. This article presents a study of a redundancy allocation problem (RAP) for cold-standby systems with degrading components. The objective of the RAP is to determine an optimal design configuration of components to maximize system reliability subject to system resource constraints (e.g. cost, weight). As in most cases, it is not possible to obtain a closed-form expression for this problem, and hence, an approximated objective function is presented. A genetic algorithm with dual mutation is developed to solve such a constrained optimization problem. Finally, a numerical example is given to illustrate the proposed solution methodology.     

A PSO algorithm for constrained redundancy allocation in multi-state systems with bridge topology

Bridge topology is a commonly used structure for load balancing and control in applications such as electric power generation and transmission, transportation and computer networks, and electronic circuits. The reliability performance of engineering systems with bridge topology can be characterized by multi-state models and enhanced by allocating redundant elements. The redundancy allocation problem, which aims at finding the optimal trade-off between system performance and investment costs, is proved difficult to solve and has received much attention in the literature. This paper advances a meta-heuristic approach called particle swarm optimization and applies it to effectively solve for near-optimal solutions to the redundancy allocation problem of multi-state systems with bridge topology. Two typical redundancy allocation problem formulations, i.e., minimizing the system cost while satisfying required system availability and maximizing the system availability with a limited budget, are studied. Heterogeneous redundancy, i.e., the mixture of redundant element types in each subsystem, is allowed in the formulations. The effectiveness and efficiency of the proposed approach are validated by the case studies of a bridge-structured coal conveyor multi-state system with extra constraints. The research results have practical meaning to the design and improvement of engineering systems with bridge topology.     

Optimal elements separation in non-repairable phased-mission systems

Enhancement of the survivability of a system can be achieved by separating its elements. Since different elements can play different roles in fulfilling the system mission, the way in which they are separated strongly affects the achieved level of the mission survivability. Thus, the optimal separation problem arises. This paper formulates a new problem of optimal element separation in non-repairable phased-mission systems (PMSs) where the mission consists of multiple, consecutive and non-overlapping phases of operation. An accurate survivability analysis of a PMS must consider the statistical dependence of element states across phases as well as dynamics in system structure function and element failure behaviour. We suggest a method for finding the separation of the system elements that provides a maximal possible level of the PMS survivability while satisfying a certain separation cost constraint. A backward recursion algorithm is applied for determining the survivability of a vulnerable PMS. A genetic algorithm is used as an optimization tool in solving the newly formulated separation problem for PMSs.     

k-out-of-n: G system with repair: the D-policy

In this paper we consider a k-out-of-n: G system with repair under D-policy. According to this policy whenever the workload exceeds a threshold D a server is called for repair and starts repair one at a time. He is sent back as soon as all the failed units are repaired. The repaired units are assumed to be as good as new. The repair time and failure time distributions are assumed to be exponential. We obtain the system state distribution, system reliability, expected length of time the server is continuously available, expected number of times the system is down in a cycle and several other measures of performance. We compute the optimal D value which maximizes a suitably defined cost function.Scope and purposeThis paper considers a repair policy, called D-policy, of a k-out-of-n: G system. In a k-out-of-n: G system, the system functions as long as there are atleast k operational units. The server activation cost is high once it becomes idle due to all failed units repaired. Hence it is activated when the accumulated amount of work reaches D. This paper examines the optimal D-value by bringing in costs such as the cost of system being down, the server activation cost. Activating the server the moment the first failure takes place may involve very heavy fixed cost per cycle (a cycle is the duration from a point at which the server becomes idle to the next epoch at which it becomes idle after being activated). The other extreme of server activation only after n−k+1 units fail leads to the system being down for a long duration in each cycle. Hence the need for the optimal D-policy. A brief account of k-out-of-n: G system can be had in Ross (Ross, SM. Introduction to probability models, 6th ed., New York: Academic Press, 1997). The results obtained here find direct applications in reliability engineering, Production systems, Satellite communication, etc.     

串并联系统中支持实时替换的混合冗余策略优化

在需要长时间可靠运行的软件系统中,由于持续运行时间和任务响应速度的要求增加,工作组件在被探测到失效后将被冗余组件实时替换.但现有可靠性优化研究通常假设冷备份冗余在所有积极冗余组件失效后才使用.针对支持实时替换的混合冗余策略,对其冗余度优化分配进行研究.该策略不仅能够保障系统可靠性,而且能够保障系统性能,故选用实时可用性和任务完成效率两类约束条件,建立冗余配置代价最小化模型.基于马尔可夫链理论对可靠性及性能两类系统指标进行定量分析;采用数值计算方法对非线性的状态分析模型进行计算;改进二元组编码遗传算法对上述优化问题进行求解.采用实例对串并联系统中实时可用性及任务完成效率的分析进行了说明,并对优化冗余分配模型进行了验证.实验结果表明,在相同冗余度下,支持实时替换的混合冗余策略在任务完成效率方面优于传统的混合冗余策略.所以,在相同约束条件下不同混合冗余策略需要采用不同的冗余优化配置方案.     

Reliability evaluation and optimal design in heterogeneous multi-state series-parallel systems

This paper addresses the heterogeneous redundancy allocation problem in multi-state series-parallel reliability structures with the objective to minimize the total cost of system design satisfying the given reliability constraint and the consumer load demand. The demand distribution is presented as a piecewise cumulative load curve and each subsystem is allowed to consist of parallel redundant components of not more than three types. The system uses binary capacitated components chosen from a list of available products to provide redundancy so as to increase system performance and reliability. The components are characterized by their feeding capacity, reliability and cost. A system that consists of elements with different reliability and productivity parameters has the capacity strongly dependent upon the selection of constituent components. A binomial probability based method to compute exact system reliability index is suggested. To analyze the problem and suggest an optimal/near-optimal system structure, an ant colony optimization algorithm has been presented. The solution approach consists of a series of simple steps as used in early ant colony optimization algorithms dealing with other optimization problems and offers straightforward analysis. Four multi-state system design problems have been solved for illustration. Two problems are taken from the literature and solved to compare the algorithm with the other existing methods. The other two problems are based upon randomly generated data. The results show that the method can be appealing to many researchers with regard to the time efficiency and yet without compromising over the solution quality.     

Constrained (k,d)-out-of-n systems

Two new coherent system reliability models, which generalise k-out-of-n:F and consecutive k-out-of-n:G systems, are proposed and explicit formulae for the reliabilities of these systems are derived when the components are independent and identical and when they are Markov dependent. Our method of deriving reliability functions is based on the use of classical combinatorial arguments. These extensions consider an additional constraint on the number of working components between successive failures. More explicitly, in addition to the working conditions of k-out-of-n:F and consecutive k-out-of-n:G systems there must be at least d consecutive working components between any of two successive failures. This type of consideration might be useful in some situations including the analysis of constrained binary sequences arising in communications systems, and particular infrared detecting systems.     

Standby redundancy optimization problems with fuzzy lifetimes

Three types of system performances—the expected system lifetime, α-system lifetime, and system reliability—characterized in the context of credibility are investigated in this paper. Some fuzzy simulations are designed to estimate these system performances. In order to formulate general standby redundancy optimization problems with fuzzy lifetimes, a spectrum of standby redundancy fuzzy programming models are proposed. Fuzzy simulation, neural network, and genetic algorithm are also integrated to produce a hybrid intelligent algorithm for solving those models. Finally, some numerical experiments on multi-stage system and network system are provided.     

Warm standby in hierarchically structured process-control programs

We classify standby redundancy design space in process-control programs into the following three categories: cold standby, warm standby, and hot standby. Design parameters of warm standby are identified and the reliability of a system using warm standby is evaluated and compared with that of hot standby. Our analysis indicates that the warm standby scheme is particularly suitable for long-lived unmaintainable systems, especially those operating in harsh environments where burst hardware failures are possible. The feasibility of warm standby is demonstrated with a simulated chemical batch reactor system     

Sequential testing policies for complex systems under precedence constraints

We study the problem of sequentially testing the components of a multi-component system to learn the state of the system, when the tests are subject to precedence constraints and with the objective of minimizing the expected cost of the inspections. Our focus is on k-out-of-n systems, which function if at least k of the n components are functional. A solution is a testing policy, which is a set of decision rules that describe in which order to perform the tests. We distinguish two different classes of policies and describe exact algorithms (one branch-and-bound algorithm and one dynamic program) to find an optimal member of each class. We report on extensive computational experiments with the algorithms for representative datasets.     

Cost-oriented task allocation and hardware redundancy policies in heterogeneous distributed computing systems considering software reliability

Task allocation policy and hardware redundancy policy for distributed computing system (DCS) are of great importance as they affect many system characteristics such as system cost, system reliability and performance. In recent years, abundant research has been carried out on the optimal task allocation and/or hardware redundancy problem, most of which took a reliability-oriented approach, i.e., the optimization criterion was system reliability maximization. Nevertheless, besides system reliability, other system characteristics such as system cost may be of great concern to management. In this paper, we take a cost-oriented approach to the optimal task allocation and hardware redundancy problem for DCS, which addresses both system cost and system reliability issues. A system cost model which could reflect the impact of system unreliability on system cost is developed, and by minimizing the total system cost, a satisfactory level of system reliability could be reached simultaneously. In the reliability modeling and analysis of DCS, we take both hardware reliability and software reliability into account. Two numerical examples are given to illustrate the formulation and solution procedures, in which genetic algorithm is used. Results show that based on the developed system cost model, appropriate decision-makings on task allocation and hardware redundancy policies for DCS could be made, and the result obtained seems to be a fairly good trade-off between system cost and system reliability.     

基于遗传算法的信息系统可靠性优化设计

设备冗余是信息系统进行可靠性优化设计的常用策略之一,其主要问题在于冗余设备的选择和配置,以达到满足一定可靠性要求下实现成本最小化的目的。这是一类结构复杂的规划问题,很难采用传统的数值算法进行求解,遗传算法提供了有效的解决方法。首先运用信息系统Petri网模型的层次结构分析结果,给出区分结点重要度的系统可靠性度量公式。在此基础上提出优化模型,给出遗传算法求解优化问题的步骤,并通过实例证明了方法的有效性及实用性。     

复杂多态系统的冗余备份元件优化

在复杂的多态系统中,系统可靠性非常重要,最常见的是冷热备份模式来实现系统的可靠性.本文中我们提出了混合冗余备份模式,计算复杂系统的可靠性和任务成本,解决复杂系统中的备份元件优化分布和初始化问题.本文主要是通过离散数学的概率分布计算复杂系统中元件的可靠性和任务成本,利用量子遗传算法来解决冗余备份元件的优化分布问题.最后同时通过仿真实验来计算出系统的可靠性和预期的任务成本,以及冗余备分元件的优化分布,得出了复杂系统可靠性与成本的平衡关系.     

A preference ordered classification for a multi-objective max–min redundancy allocation problem

In this study, we consider a bi-objective redundancy allocation problem on a series–parallel system with component level redundancy strategy. Our aim is to maximize the minimum subsystem reliability, while minimizing the overall system cost. The Pareto solutions of this problem are found by the augmented ε-constraint approach for small and moderate sized instances. After finding the Pareto solutions, we apply a well known sorting procedure, UTADIS, to categorize the solutions into preference ordered classes, such as A, B, and C. In this procedure, consecutive classes are separated by thresholds determined according to the utility function constructed from reference sets of classes. In redundancy allocation problems, reference sets may contain a small number of solutions (even a single solution). We propose the τ-neighborhood approach to increase the number of references. We perform experiments on some reliability optimization test problems and general test problems.     

考虑多维修台异步多重休假的温贮备冗余系统可靠性模型

考虑多维修台保障多个系统时维修力量存在调度与分配的情况,引入多维修台异步多重休假策略;以温贮备冗余系统为研究对象,针对以往研究利用指数分布等典型分布导致模型约束条件过于严格的问题,采用连续phase-type(PH)分布描述系统中工作部件寿命、温贮备部件寿命以及维修台休假时间和维修时间,建立通用性更好的系统可靠性解析模型,给出系统可靠度、系统稳态可用度等冗余系统可靠性指标和稳态忙期维修台数量等维修台稳态指标;利用算例验证模型适用性,演示了维修台数量、系统温贮备部件数量变化以及修理工休假速率、维修速率变化对系统各可靠性指标和维修台稳态指标的影响.算例计算结果表明,所提出的可靠性模型能够有效复现多维修台调度对冗余系统可靠性的影响,从而为维修台数量的合理安排及系统部件数量的优化配置提供理论基础和实践参考.     

Solving reliability redundancy allocation problems using an artificial bee colony algorithm

This paper proposed a penalty guided artificial bee colony algorithm (ABC) to solve the reliability redundancy allocation problem (RAP). The redundancy allocation problem involves setting reliability objectives for components or subsystems in order to meet the resource consumption constraint, e.g. the total cost. RAP has been an active area of research for the past four decades. The difficulty that one is confronted with the RAP is the maintenance of feasibility with respect to three nonlinear constraints, namely, cost, weight and volume related constraints. In this paper nonlinearly mixed-integer reliability design problems are investigated where both the number of redundancy components and the corresponding component reliability in each subsystem are to be decided simultaneously so as to maximize the reliability of the system. The reliability design problems have been studied in the literature for decades, usually using mathematical programming or heuristic optimization approaches. To the best of our knowledge the ABC algorithm can search over promising feasible and infeasible regions to find the feasible optimal/near-optimal solution effectively and efficiently; numerical examples indicate that the proposed approach performs well with the reliability redundant allocation design problems considered in this paper and computational results compare favorably with previously-developed algorithms in the literature.     

Multilevel redundancy allocation using two dimensional arrays encoding and hybrid genetic algorithm

With the popularity of multilevel design in large scale systems, reliability redundancy allocation on multilevel systems is becoming attractive to researchers. Multilevel redundancy allocation problem (MLRAP) is not only NP-hard, but also qualifies as hierarchy optimization problem. Exact method could not tackle MLRAP very well, so heuristic and meta-heuristic methods are often used to solve it. To improve the effectiveness of current algorithms on MLRAP, this paper proposes a hybrid genetic algorithm (HGA) based on the two dimensional redundancy encoding mechanism. Instead of hierarchical genotype representation, a two dimensional array is used to represent the solutions to MLRAP. Each row of the array contains the redundancy information of a certain unit in the system and each element in one row stands for the redundancy value of one element of that unit. The number of rows of this array is fixed and equals to the number of distinct units in the system. Each row of the array is an unfixed-length vector whose length depends on the redundancy of all elements of its parent unit. On top of this two dimensional arrays, a local search operator employing simulated annealing strategy is used to generate new population for the next generation instead of the traditional genetic operators. Experimental results have shown that our two dimensional arrays based HGA outperforms the state-of-the-art approaches using two kinds of multilevel system structure.