Holistic reliability analysis of weighted voting systems from a multi-state perspective

The computation of the reliability of weighted voting systems is an important problem in reliability theory due to its potential application in security, target identification, safety and monitoring areas. Voting systems are used in a wide variety of applications where an acceptance or rejection decision has to be made about a binary proposition presented to the system. For these systems, it is of interest to obtain the probability so that based on the vote of decision-making units, the system aggregates these votes into the right decision when presented with such a proposition. This paper presents a holistic work on weighted voting system reliability by presenting modeling, computation, estimation and optimization techniques. The modeling part takes advantage of the structure of weighted voting systems to present a model of its reliability as a multi-state system. Next, based on the multi-state view of the system, an exact computational approach based on multi-state minimal cut and path vectors is introduced. The paper then acknowledges the computational complexity of the problem and provides a Monte Carlo simulation approach that estimates system reliability accurately and in an efficient computational time. Finally, an optimization heuristic that generates quasi-optimal solutions is presented that is able to solve the problem of maximizing the reliability of a weighted voting system based on a specified number of decision-making units with known reliability characteristics.     

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.     

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.     

Reliability and risk analysis of large systems with ageing components

Applications of limit reliability functions to the reliability evaluation of large multi-state systems composed of independent components are considered. The main emphasis is on multi-state systems with ageing components because of the importance of such an approach in safety analysis, assessment and prediction, and analysing the effectiveness of operation processes of real technical systems. The results concerned with multi-state series systems are applied to the reliability evaluation and risk function determination of a homogeneous bus transportation system. Results on limit reliability functions of a homogeneous multi-state “m out of n” system are applied to durability evaluation of a steel rope. A non-homogeneous series-parallel pipeline systems composed of several lines of pipe segments is estimated as well. Moreover, the reliability evaluation of the model homogeneous parallel-series electrical energy distribution system is performed.     

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.     

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.     

New insights on multi-state component criticality and importance

In this paper, new importance measures for multi-state systems with multi-state components are introduced and evaluated. These new measures complement and enhance current work done in the area of multi-state reliability. In general, importance measures are used to evaluate and rank the criticality of component or component states with respect to system reliability. The focus of the study is to provide intuitive and clear importance measures that can be used to enhance system reliability from two perspectives: (1) how a specific component affects multi-state system reliability and (2) how a particular component state or set of states affects multi-state system reliability. The first measure unsatisfied demand index, provides insight regarding a component or component state contribution to unsatisfied demand. The second measure multi-state failure frequency index, elaborates on an approach that quantifies the contribution of a particular component or component state to system failure. Finally, the multi-state redundancy importance identifies where to allocate component redundancy as to improve system reliability. The findings of this study indicate that both perspectives can be used to complement each other and as an effective tool to assess component criticality. Examples illustrate and compare the proposed measures with previous multi-state importance measures.     

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.     

An evaluation of the multi-state node networks reliability using the traditional binary-state networks reliability algorithm

A system where the components and system itself are allowed to have a number of performance levels is called the Multi-state system (MSS). A multi-state node network (MNN) is a generalization of the MSS without satisfying the flow conservation law. Evaluating the MNN reliability arises at the design and exploitation stage of many types of technical systems. Up to now, the known existing methods can only evaluate a special MNN reliability called the multi-state node acyclic network (MNAN) in which no cyclic is allowed. However, no method exists for evaluating the general MNN reliability. The main purpose of this article is to show first that each MNN reliability can be solved using any the traditional binary-state networks (TBSN) reliability algorithm with a special code for the state probability. A simple heuristic SDP algorithm based on minimal cuts (MC) for estimating the MNN reliability is presented as an example to show how the TBSN reliability algorithm is revised to solve the MNN reliability problem. To the author's knowledge, this study is the first to discuss the relationships between MNN and TBSN and also the first to present methods to solve the exact and approximated MNN reliability. One example is illustrated to show how the exact MNN reliability is obtained using the proposed algorithm.     

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.     

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.     

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.     

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.     

A generalized multistate-based path vector approach to multistate two-terminal reliability

The two-terminal reliability problem assumes that a network and its elements are either in a working or a failed state. However, many practical networks are built of elements that may operate in more than two states i.e., elements may be degraded but still functional. Multistate two-terminal reliability at demand level d (M2TR_d) can be defined as the probability that the system capacity generated by multistate components is greater than or equal to a demand of d units. This paper presents a fully multistate-based algorithm that obtains the multistate equivalent of binary path sets, namely, Multistate Minimal Path Vectors (MMPVs), for the M2TR_d problem. The algorithm mimics natural organisms in the sense that a select number of arcs inherit information from other specific arcs contained in a special set called the “primary set.” The algorithm is tested and compared with published results in the literature. Two features of the algorithm make it relevant: (i) unlike other approaches, it does not depend on an a priori knowledge of the binary path sets to obtain the MMPVs; and (ii) the use of an information sharing approach and network reduction technique significantly reduce the number of vector analyses needed to obtain all the component levels that guarantee system success. Additionally, the complexities associated with the computation of reliability are discussed. A Monte Carlo simulation approach is used to obtain an accurate estimate of actual M2TR values based on MMPVs. Examples are used to validate the algorithm and the simulation procedure.     

Extended great deluge algorithm for the imperfect preventive maintenance optimization of multi-state systems

This paper deals with preventive maintenance optimization problem for multi-state systems (MSS). This problem was initially addressed and solved by Levitin and Lisnianski [Optimization of imperfect preventive maintenance for multi-state systems. Reliab Eng Syst Saf 2000;67:193–203]. It consists on finding an optimal sequence of maintenance actions which minimizes maintenance cost while providing the desired system reliability level. This paper proposes an approach which improves the results obtained by genetic algorithm (GENITOR) in Levitin and Lisnianski [Optimization of imperfect preventive maintenance for multi-state systems. Reliab Eng Syst Saf 2000;67:193–203]. The considered MSS have a range of performance levels and their reliability is defined to be the ability to meet a given demand. This reliability is evaluated by using the universal generating function technique. An optimization method based on the extended great deluge algorithm is proposed. This method has the advantage over other methods to be simple and requires less effort for its implementation. The developed algorithm is compared to than in Levitin and Lisnianski [Optimization of imperfect preventive maintenance for multi-state systems. Reliab Eng Syst Saf 2000;67:193–203] by using a reference example and two newly generated examples. This comparison shows that the extended great deluge gives the best solutions (i.e. those with minimal costs) for 8 instances among 10.     

Homogeneous redundancy optimization in multi-state series-parallel systems: A heuristic approach

This paper presents a heuristic for a series-parallel system, exhibiting a multi-state behavior, minimizing the cost in order to provide a desired level of reliability. System reliability is defined as the ability to satisfy consumer demands which is presented as a piecewise cumulative load curve. The components are binary and chosen from a list of products available on the market, and are characterized in terms of their feeding capacity, reliability and cost. The solution approach makes use of a homogeneous collection of components to provide redundancy in a subsystem. The algorithm is applied to power systems found in the literature for various levels of reliability requirement. The heuristic offers a straightforward analysis and is efficient both in terms of solution quality and computational time, as compared to existing genetic algorithms and heuristics. Thus, the developed heuristic is attractive, and it can be easily and efficiently applied to numerous real-life systems.     

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.     

Block diagram method for analyzing multi-state systems with uncovered failures

The paper suggests a modification of the generalized reliability block diagram (RBD) method for evaluating reliability and performance indices of multi-state systems with uncovered failures. Such systems (or their subsystems) can fail to perform their task in the case of undetected failure of any one of their elements. Examples of this effect can be found in computing systems, electrical power distribution networks, pipe lines carrying dangerous materials etc. The suggested method based on a universal generating function technique allows performance distribution of complex multi-state series-parallel system with uncovered failures to be obtained using a straightforward recursive procedure. Illustrative examples are presented.     

Multi-state systems with multi-fault coverage

The paper introduces a new model of fault level coverage for multi-state systems in which the effectiveness of recovery mechanisms depends on the coexistence of multiple faults in related elements. Examples of this effect can be found in computing systems, electrical power distribution networks, pipelines carrying dangerous materials, etc. For evaluating reliability and performance indices of multi-state systems with imperfect multi-fault coverage, a modification of the generalized reliability block diagram (RBD) method is suggested. This method, based on a universal generating function technique, allows performance distribution of complex multi-state series–parallel system with multi-fault coverage to be obtained using a straightforward recursive procedure. Illustrative examples are presented.     

Reliability analysis of universal joint of a compliant platform

The paper describes a methodology for computation of fatigue reliability of universal joint in an articulated offshore tower. Failure criteria were formulated using the conventional Palmgren‐Miner rule (S‐N curve approach) and the fracture mechanics (F‐M) principle. The dynamic analysis of double hinged articulated tower under wind and waves is carried out in time domain. The response histories of hinge shear stresses are employed for the reliability analysis. Advanced first‐order reliability method and Monte Carlo simulation method were used to estimate the reliability. Various parametric studies were carried out, which yield important information for the reliability based design. The S‐N curve approach yields a significantly conservative estimate of probability of failure when compared to the F‐M approach.