Multi-state component criticality analysis for reliability improvement in multi-state systems

This paper evaluates and implements composite importance measures (CIM) for multi-state systems with multi-state components (MSMC). Importance measures are frequently used as a means to evaluate and rank the impact and criticality of individual components within a system yet they are less often used as a guide to prioritize system reliability improvements. For multi-state systems, previously developed measures sometimes are not appropriate and they do not meet all user needs. This study has two inter-related goals: first, to distinguish between two types of importance measures that can be used for evaluating the criticality of components in MSMC with respect to multi-state system reliability, and second, based on the CIM, to develop a component allocation heuristic to maximize system reliability improvements. The heuristic uses Monte-Carlo simulation together with the max-flow min-cut algorithm as a means to compute component CIM. These measures are then transformed into a cost-based composite metric that guides the allocation of redundant elements into the existing system. Experimental results for different system complexities show that these new CIM can effectively estimate the criticality of components with respect to multi-state system reliability. Similarly, these results show that the CIM-based heuristic can be used as a fast and effective technique to guide system reliability improvements.     

Monte Carlo simulation analysis of the effects of different system performance levels on the importance of multi-state components

In the present paper, we consider some frequently used importance measures, in their generalized form proposed by the authors for application to multi-state systems constituted by multi-state components. To catch the dynamics of multi-state systems, Monte Carlo simulation has been utilized. A simulation approach has been presented which allows estimating of all the importance measures of the components at a given performance level in a single simulation, provided that the components are independent. The effects of different performance demands made on the system on the importance of its multi-state components have been examined with respect to a simple multi-state series–parallel system. The results have shown that a performance level of a component may be more critical for the achievement of a system performance and less critical for another.     

Generalised importance measures for multi-state elements based on performance level restrictions

In this paper we consider some commonly used importance measures in a generalised version proposed by some of the authors for application to multi-state systems constituted by multi-state elements. Physically, these measures characterize the importance for a multi-state element of achieving a given level of performance and their definitions entail evaluating the system availability and/or performance when the functioning of the element of interest is restricted in performance.With reference to a predefined threshold of element performance, two different types of restrictions are considered. The first one limits the elements' reachable states to those corresponding to performances either larger or not larger than the threshold level. The second one allows the element to visit all its states but limits its performance to values larger or not larger than the performance threshold.An approach based on the universal generating function technique is proposed for the evaluation of the introduced importance measures. A numerical application is provided in order to highlight the informative content of the introduced measures.     

Estimation of the importance measures of multi-state elements by Monte Carlo simulation

A generalization of some frequently used importance measures has been proposed by some of the authors for application to multi-state systems constituted by multi-state elements. This paper deals with the Monte Carlo (MC) estimation of these measures, which entails evaluating the system output performance under restrictions on the performance levels of its multi-state elements. Simulation procedures are proposed according to two different performance-restriction approaches. Further, the flexibility of the MC method is exploited to account for load-sharing and operational dependencies among parallel elements. The approach is tested on a multi-state transmission system of literature.     

Importance and sensitivity analysis of multi-state systems using the universal generating function method

A method for the evaluation of element reliability importance in a multi-state system is proposed. The method is based on the universal generating function technique. It provides an effective importance analysis tool for complex series–parallel multi-state systems with a different physical nature of performance and takes into account a required performance (demand). The method is also extended for the sensitivity analysis of important multi-state system output performance measures: mean system performance and mean unsupplied demand during operating period. Numerical examples are given.     

Importance measures-based prioritization for improving the performance of multi-state systems: application to the railway industry

The railway industry is undertaking significant efforts in the application of reliability-based and risk-informed approaches for rationalizing operation costs and safety requirements. In this respect, importance measures can bring valuable information for identifying the actions to take for most effective system improvement.In this paper, the railway network is modelled within a multi-state perspective in which each rail section is treated as a component, which can stay in different discrete states representing the speed values at which the section can be travelled, depending on the tracks degradation and on the traffic conditions. The Monte Carlo method is used to simulate the complex stochastic dynamics of such multi-state system.A prioritization of the rail sections based on importance measures is then used to most effectively improve the performance of the rail network, in terms of a decrease in the overall trains delay. High-importance sections, i.e. with highest impact on the overall delay, are considered for a relaxation of their speed restrictions and the proposed changes are then verified, from the risk-informed perspective, to have negligible impact on the risk associated to the rail infrastructure.     

Reliability assessment of embedded digital system using multi-state function

This work describes a combinatorial model for estimating the reliability of the embedded digital system by means of multi-state function. This model includes a coverage model for fault-handling techniques implemented in digital systems. The fault-handling techniques make it difficult for many types of components in digital system to be treated as binary state, good or bad. The multi-state function provides a complete analysis of multi-state systems as which the digital systems can be regarded. Through adaptation of software operational profile flow to multi-state function, the HW/SW interaction is also considered for estimation of the reliability of digital system. Using this model, we evaluate the reliability of one board controller in a digital system, Interposing Logic System (ILS), which is installed in YGN nuclear power units 3 and 4. Since the proposed model is a generalized combinatorial model, the simplification of this model becomes the conventional model that treats the system as binary state. This modeling method is particularly attractive for embedded systems in which small sized application software is implemented since it will require very laborious work for this method to be applied to systems with large software.     

Reliability estimation and prediction of multi-state components and coherent systems

This paper discusses the multi-state coherent system composed of multi-state components. First, using the min cut sets or min path sets, we present our simulation algorithm, instead of the general structure function, to calculate the probability that the system is in a specified state. Second, we check the components per period, e.g. one check per year, to obtain the state sequences of all components. When the state sequences are Markovian chains, we can predict the reliability of the components in several periods, such as the probability that the components are in specified states. Also, we give two methods to compute the system reliability in a number of periods: one employs the states of the components in these periods, which can be predicted by the state transition probability matrixes of the components; the other uses the state transition probability matrix of the system obtained by the simulated states of the components.     

Structure optimization of multi-state system with time redundancy

In this article, a multi-state system with time redundancy where each system element has its own operation time is considered. In addition, the system total task must be performed during the restricted time. The reliability optimization problem is treated as finding the minimal cost system structure subject to the reliability constraint. A method for reliability optimization for systems with time redundancy is proposed. This method is based on the universal generating function technique for the reliability index computation and on genetic algorithm for the optimization. It provides a solution for the optimization problem for the complex series–parallel system and for the system with bridge topology. Two types of systems will illustrate the approach: systems with ordinary hot reserve and systems with work sharing between elements connected in parallel. Numerical examples are also given.     

Extended block diagram method for a multi-state system reliability assessment

The presented method extends the classical reliability block diagram method to a repairable multi-state system. It is very suitable for engineering applications since the procedure is well formalized and based on the natural decomposition of the entire multi-state system (the system is represented as a collection of its elements). Until now, the classical block diagram method did not provide the reliability assessment for the repairable multi-state system. The straightforward stochastic process methods are very difficult for engineering application in such cases due to the “dimension damnation”—huge number of system states. The suggested method is based on the combined random processes and the universal generating function technique and drastically reduces the number of states in the multi-state model.     

Relationships of Fussell–Vesely and Birnbaum importance to structural importance in coherent systems

The notion of node criticality was introduced by Boland, Proschan and Tong to study the problem of optimal rearrangement of components in binary coherent systems. In this paper, we use this notion to study the importance of system components. We derive various relationships between the node criticality and the component importance measures due to Birnbaum, Fussell and Vesely, respectively. Some previous results due to Boland and Proschan, and Meng have been improved.     

Birnbaum and criticality measures of component contribution to the failure of phased missions

This paper develops measures, which identify the contribution to system failure when the system operates a phased mission. The measures developed are the equivalent of Birnbaum's measure of importance and the criticality measure of importance in a conventional analysis. It is assumed that during the mission the system components cannot be repaired. In the determination of the importance measures, the contribution to phase failure is considered in two aspects: failure during the phase (in-phase importance) and failure on transition to a phase (transition importance). Component importance measures indicate the contribution to phase and overall mission unreliability.     

Approximate multi-state reliability expressions using a new machine learning technique

The machine-learning-based methodology, previously proposed by the authors for approximating binary reliability expressions, is now extended to develop a new algorithm, based on the procedure of Hamming Clustering, which is capable to deal with multi-state systems and any success criterion. The proposed technique is presented in details and verified on literature cases: experiment results show that the new algorithm yields excellent predictions.     

Redundancy analysis for repairable multi-state system by using combined stochastic processes methods and universal generating function technique

This paper discusses a type of redundancy that is typical in a multi-state system. It considers two interconnected multi-state systems where one multi-state system can satisfy its own stochastic demand and also can provide abundant resource (performance) to another system in order to improve the assisted system reliability. Traditional methods are usually not effective enough for reliability analysis for such multi-state systems because of the “dimensional curse” problem. This paper presents a new method for reliability evaluation for the repairable multi-state system considering such kind of redundancy. The proposed method is based on the combination of the universal generating function technique and random processes methods. The numerical example is presented to illustrate the proposed method.     

Algorithm for estimating reliability confidence bounds of multi-state systems

The paper suggests an effective approach for the estimation of reliability confidence bounds based on component reliability and uncertainty data for multi-state systems with binary-capacitated components. The approach presented is based on the implementation of the universal generating function technique. When compared with a pure Monte Carlo simulation approach, the universal generating function (UGF)-based approach is proven to be more effective due to a more precise reliability estimation and a considerably lower computational effort. Examples are given throughout the paper to illustrate the suggested approach.     

Optimal design of multi-state weighted k-out-of-n systems based on component design

This paper presents a study on design optimization of multi-state weighted k-out-of-n systems. The studied system reliability model is more general than the traditional k-out-of-n system model. The system and its components are capable of assuming a whole range of performance levels, varying from perfect functioning to complete failure. A utility value corresponding to each state is used to indicate the corresponding performance level. A widely studied reliability optimization problem is the “component selection problem”, which involves selection of components with known reliability and cost characteristics. Less adequately addressed has been the problem of determining system cost and utility based on the relationships between component reliability, cost and utility. This paper addresses this topic. All the optimization problems dealt with in this paper can be categorized as either minimizing the expected total system cost subject to system reliability requirements, or maximizing system reliability subject to total system cost limitation. The resulting optimization problems are too complicated to be solved by traditional optimization approaches; therefore, genetic algorithm (GA) is used to solve them. Our results show that GA is a powerful tool for solving these kinds of problems.     

Differential, criticality and Birnbaum importance measures: An application to basic event, groups and SSCs in event trees and binary decision diagrams

Recent works [Epstein S, Rauzy A. Can we trust PRA? Reliab Eng Syst Safety 2005; 88:195–205] have questioned the validity of traditional fault tree/event tree (FTET) representation of probabilistic risk assessment problems. In spite of whether the risk model is solved through FTET or binary decision diagrams (BDDs), importance measures need to be calculated to provide risk managers with information on the risk/safety significance of system structures and components (SSCs). In this work, we discuss the computation of the Fussel–Vesely (FV), criticality, Birnbaum, risk achievement worth (RAW) and differential importance measure (DIM) for individual basic events, basic event groups and components. For individual basic events, we show that these importance measures are linked by simple relations and that this enables to compute basic event DIMs both for FTET and BDD codes without additional model runs. We then investigate whether/how importance measures can be extended to basic event groups and components. Findings show that the estimation of a group Birnbaum or criticality importance is not possible. On the other hand, we show that the DIM of a group or of a component is exactly equal to the sum of the DIMs of the corresponding basic events and can therefore be found with no additional model runs. The above findings hold for both the FTET and the BDD methods.     

Common supply failures in linear multi-state sliding window systems

This paper considers a linear multi-state sliding window system (SWS) that consists of n linearly ordered multi-state elements. Each element can have different states: from complete failure to perfect functioning. A performance rate is associated with each state. The system fails if the sum of the performance rates of any r consecutive elements is lower than demand w. Different groups of elements (common supply groups (CSGs)) share some common resources. Failures in the resource supply system (common supply failures (CSF)) result in the simultaneous outage of several elements belonging to corresponding groups. Different groups of elements are affected by different CSF.This paper presents an algorithm for evaluating the reliability of SWS that is the subject of CSF. It also introduces the CSG reliability importance measure and suggests an algorithm for its estimation. Further, it formulates a problem of optimal element distribution among CSGs and presents a method for solving it.An illustrative example shows the application of the suggested algorithms.     

A joint reliability-redundancy optimization approach for multi-state series-parallel systems

In this paper, we present a practical approach for the joint reliability-redundancy optimization of multi-state series-parallel systems. In addition to determining the optimal redundancy level for each parallel subsystem, this approach also aims at finding the optimal values for the variables that affect the component state distributions in each subsystem. The key point is that technical and organizational actions can affect the state transition rates of a multi-state component, and thus affect the state distribution of the component and the availability of the system. Taking this into consideration, we present an approach for determining the optimal versions and numbers of components and the optimal set of technical and organizational actions for each subsystem of a multi-state series-parallel system, so as to minimize the system cost while satisfying the system availability constraint. The approach might be considered to be the multi-state version of the joint system reliability-redundancy optimization methods.     

A heuristic for solving the redundancy allocation problem for multi-state series-parallel systems

The redundancy allocation problem is formulated with the objective of minimizing design cost, when the system exhibits a multi-state reliability behavior, given system-level performance constraints. When the multi-state nature of the system is considered, traditional solution methodologies are no longer valid. This study considers a multi-state series-parallel system (MSPS) with capacitated binary components that can provide different multi-state system performance levels. The different demand levels, which must be supplied during the system-operating period, result in the multi-state nature of the system. The new solution methodology offers several distinct benefits compared to traditional formulations of the MSPS redundancy allocation problem. For some systems, recognizing that different component versions yield different system performance is critical so that the overall system reliability estimation and associated design models the true system reliability behavior more realistically. The MSPS design problem, solved in this study, has been previously analyzed using genetic algorithms (GAs) and the universal generating function. The specific problem being addressed is one where there are multiple component choices, but once a component selection is made, only the same component type can be used to provide redundancy. This is the first time that the MSPS design problem has been addressed without using GAs. The heuristic offers more efficient and straightforward analyses. Solutions to three different problem types are obtained illustrating the simplicity and ease of application of the heuristic without compromising the intended optimization needs.