The theme of this special issue of IJPE has been chosen as Multi-state system reliability and Professor Mustapha Nourelfath of Mechanical Engineering Department, Université Laval, Canada, who is on the editorial board of this journal very kindly agreed to guest-edit this special issue. Multi-state systems reliability has received importance and attention in past couple of decades as it brings in more realistic modelling of some physical systems. For example, in case of systems such as power system, computer systems, communication systems, air conditioning systems, production systems, oil transmission pipelines etc. and systems with degraded components, the usual two-state (success and failure) modelling of system does not reflect the realistic situation, where several levels of performance and several modes of failures characterized by capacity, flow, communication or data processing speed etc., ranging from perfect functioning to complete failure, could exist. It will be over simplification of the physical phenomenon to assume a two-state model of performance for such systems. Further on, some systems are also being modeled as multi-state k-out-of-n:G systems, where multiple states are interpreted as multiple levels of capacity. Reliability or availability optimization of such multi-state systems is another area in which new formulations are being suggested in the literature.

?The difficulty in analyzing such Multi-state systems (MSS) arises from the large number of states that may have to be considered for the system since each system element can have many different states (and not just two states as in case of two-state model).

Some of the approaches for multi-state system reliability assessment have been proposed based on the (1) extension of the Boolean models to the multi-valued case, (2) stochastic processes (mainly Markov and semi-Markov) approach, (3) universal generating function approach (4) fuzzy set theory etc.

The eight papers, which have been selected among the several papers that the Guest editor received in response to his call, reflect the current thinking in the area of research in the analysis of multi-state system reliability or availability. This, of course does not reflect the complete picture of the on-going research in the area but encourages the readers to think in that direction, which has been at the back of our mind and our effort. It is hoped that the readers would be benefited by this presentation which will help generate further research on the theme of the special issue.

I would like to thank the contributors of the papers included in this issue and my sincere thanks are due to the referees who helped us in improving the presentation and maintained the timeline of this special issue.

Last but not the least; I take this opportunity to thank Professor Nourelfath for his painstaking effort in bringing out this special issue.

In this special issue of IJPE, we would like to present eight impressive papers on the subject of multi-state systems reliability. In classical binary reliability analysis, a system and its constituent components and subsystems are assumed to be either in a working state or in a failed one. However, in many real-life situations, this two-state assumption may not be adequate and appropriate. The system may, in fact, have more than two levels of performance varying from perfect functioning to complete failure. For example, the presence of degradation is a common realistic situation, in which a system should be considered to be a multi-state system. Degradation can be caused by system deterioration or by variable ambient conditions. Fatigue, failures of non-essential components, and number of random shocks on the system are all examples of system degradation causes. In this case, the failure rate depends on the status of the system which can degrade gradually. The reliability analysis of such degraded systems should consider multiple operational states to take into account multiple degradation levels and a multi-state system may perform at different intermediate states between working perfectly and complete failure.

As a relatively new emerging discipline, the reliability of multi-state systems has received increasing and extensive attention during past few years. At the junction of binary reliability and performance analysis, it gives new opportunities towards many promising research directions. The papers selected for this special issue on multi-state systems cover some of the important problems that are currently being addressed by the engineering community. They demonstrate the vibrant research in the area of multi-state systems reliability analysis and optimization. The papers included in this special issue are:

Paper 1 (Defence and Attack of Multi-State Systems, by K. Hausken, Norway) develops a game-theoretical model for a two component system. Each component is protected by a defender which maximizes its reliability weighted against the defense costs, and attacked by an attacker which maximizes its unreliability weighted against the attack costs.

Paper 2 (Multi-state Reliability Modelling of a Manufacturing Cell, by M. Savsar, Kuwait) develops a stochastic model to analyze performance measures of a flexible manufacturing cell. The later is modelled as a multi-state system to take into account degraded functioning modes.??? ?

Paper 3 (A Component Importance Measure Suitable for Flow Transmission Multi-State System, by Fares Innal and Y. Dutuit, France) uses stochastic Petri net models with predicates to deal with the performance evaluation of a multi-state system and its components. A Monte Carlo simulation approach is used to show the interest of the importance factors.

Paper 4 (Models and Measures for Fuzzy Reliability and Its Relationship to Multi-State Reliability, by Z. Li and K. C. Kapur, U.S.A.) is based on the theory and methods for fuzzy sets. The authors illustrate how fuzzy modelling of the states of the components/system can capture more realistic performance of the system for its evaluation and decision making processes by the system designer as well as the customer.

Paper 5 (Analytical Uncertainty Propagation for Availability Assessment of Stochastic Multi-State Systems, by M. El Falou and E. Chatelet, France) generalizes uncertainty propagation methods to multi-state systems. The proposed method is based on an analytical calculation of the probability of the system states. It is able to deduce analytically the availability of the system with the associated uncertainty.

Paper 6 (Optimal Assignment of a Two Types of Components in a Multi-state Stochastic-flow Network, Y.-K. Lin and C.-T. Yeh, Taiwan) models a system as a network with edges and nodes, in which each component (edge/node) is multi-state due to complete failure, partial failure, maintenance, etc. To maximize the system reliability according to the optimal two-type component assignment under a budget constraint, a genetic algorithm is developed. The system reliability is evaluated in terms of minimal paths and recursive sum of disjoint products (RSDP). ?

Paper 7 (Planning Inspections in the Case of Damage Tolerance Approach to Service of Fatigued Aircraft Structures, by N. Nechval et al., Latvia) uses a damage tolerance approach to determine appropriate decreasing intervals between inspections of fatigue-sensitive aircraft structures modelled as multi-state systems. The proposed stochastic model evaluates the crack probability, and the probability of random time to reach a specified crack size.???

Paper 8 (Sustainable Reuse Approaches based on Residual Lifetime : A Case Study in Assistive Technology, S. H. Mohammadian et al., Canada) deals with the selection of qualified-for-reuse components for returned systems. Physical properties and age are used to assess the multi-state degraded components’ residual lifetimes. The proposed approach has been applied to the end of life of electric-powered wheelchair batteries.

In concluding this editorial, I would like to thank all contributing authors for their outstanding papers and my sincere thanks are due to referees whose timely help in reviewing these papers was always available. I am also grateful to the Editor-in-Chief, Professor Krishna B. Misra, for his support and help in realizing this project.?????

Mustapha Nourelfath is a full professor of Industrial Engineering at Université Laval (Canada), in the Department of Mechanical Engineering at the Faculty of Science and Engineering. Before his current position, he was Professor at UQAT (Université du Québec en Abitibi-Témiscamingue, Canada). After graduating from ENSET-Mohammedia (Morocco), Professor Nourelfath obtained a DEA and a Ph.D. in industrial automation and industrial engineering from INSA (National Institute of Applied Science) of Lyon (France), in 1994 and 1997, respectively. Dr. Nourelfath is member of the organizing and scientific committees of different international conferences, and regularly acts as a referee for many scientific journals. He is a member of CIRRELT (Interuniversity Research Centre on Enterprise Networks, Logistics and Transportation). His specific topics of interest are operations research and artificial intelligence applications in reliability, logistics, manufacturing, and supply chain management problems. He serves editorial boards of International Journal of Performability Engineering and Reliability Engineering and System Safety.

A two component system can be fully operational, in two states of intermediate degradation, or fail. Each component is protected by a defender which maximizes its reliability weighted against the defense costs, and attacked by an attacker which maximizes its unreliability weighted against the attack costs. The system reduces to the series system when the values of the two intermediate states equal zero, and reduces to the parallel system when the values of the two intermediate states equal the value of the fully operational state. The attacker prefers the series system, the defender prefers the parallel system. The defender benefits from increasing the values of the two intermediate states. The article illustrates how the agents’ efforts and utilities depend on the values of the intermediate states, assuming these are equal, that one value is zero and the other varies, or that one value equals the value of the fully operational state while the other value varies.
Received on September 17, 2007, revised on July 1, 2010
References: 29

A Manufacturing cell consists of a machine served by a loading and unloading robot and a pallet handling system, which moves a batch of parts into and out of the system. In this study, a stochastic model is developed to analyze performance measures of a cell, which is allowed to operate under multi-states including degraded mode. The model is used to determine state probabilities of the system, which are used to determine reliability and productivity of the cell, as well as the utilization of its components, under various operational conditions, including equipment failures and fault-tolerant states. The model and the results can be useful for design engineers and operational managers to analyze performance of a system at the design or operational stage.
Received on November 29, 2009, revised July 08,2010
References: 21

This paper deals with the performance evaluation of both a production system through the measurement of its average production availability over a given period and its constitutive parts through their respective contributions to the overall production loss. To perform such measures, a model of the system behaviour must be built and exploited. In this paper this behaviour (nominal, degraded, failed) is modelled by means of Petri nets (PN), by taking into account the upstream and downstream effects induced by the failure of a given component on the production level of other components, and then, on the production level of the whole system, i.e., on its performance level. The magnitude of these effects is closely linked to the so-called importance of the components which have created them. The notions of production availability and component importance are presented and illustrated in this paper on the basis of a simple but not trivial example of an oil production system.
Received on November 08, 2009, revised on December 22, 2010
References: 24

This paper proposes a new perspective and methodology to model the behavior of the components as well as the system using the theory and methods for fuzzy sets. We propose to use the indicator or performance or substitute variable which is very well understood by the customer to fuzzify the states of the component or the system. Both component and system’s fuzzy reliability definitions and models are constructed under fuzzy binary states and multi-state assumptions. Numerical examples are given to compute fuzzy reliability measures which also illustrate their advantages over the classical reliability measures.
Received on November 08, 2009, revised on May 23, 2010
References: 30

The proposed method generalizes uncertainty propagation methods for complex stochastic multi-states system modeling. Until now, analytical methods for uncertainty propagation are only used in simple system modeling as fault tree model. These methods are usually based on moment calculation or variance evaluation. In complex system cases or when complex mathematic factors affect the uncertainties, asymptotic calculation or Monte Carlo simulation are often used to estimate the resulting availability. The proposed method is based on an analytical calculation of the probability of the system states to propagate analytically the uncertainty in the model. It allows a better assessment of the availability of stochastic multi-state systems. The analytical methods used are based on calculus of conjoint probability density function, Universal Generating Function model and associated operators. The method consists in an analytical propagation of uncertainty in the model and an output uncertainty assessment. Particularly, it is able to deduce analytically the output availability of the system with the associated uncertainty. For complex UGF modeling, the analytical calculation to the sub-systems and Monte Carlo simulation on the resulting UGF system model are applied. Consequently, the computing time for the simulation could be drastically reduced.
Received on October 18, 2009, revised April 21, 2010
References: 15

A real-life system is usually modeled to be a network with edges and nodes for the performance evaluation. This paper devotes to finding out the optimal two-type component assignment with maximal system reliability, in which the components are separated to be two sets: one assigned to edges and the other set assigned to nodes. Each component should be multistate due to complete failure, partial failure, maintenance, etc. Such a system according to a two-type component assignment is a stochastic-flow network. Furthermore, each component has a transmission cost in practice. Therefore, the system reliability is the probability that d units of demand are transmitted through the network successfully subject to a budget. A genetic algorithm based algorithm is developed to solve the proposed problem in which the system reliability according to a two-type component assignment is evaluated in terms of minimal paths and Recursive Sum of Disjoint Products. The experimental results show that the proposed algorithm can be executed in reasonable time.
Received on March 17, 2010 and revised on December 18, 2010
References: 31

Fatigue is one of the most important problems of aircraft arising from their nature as multiple-component structures, subjected to random dynamic loads and representing Multi-State Systems. For guaranteeing safety, the structural life ceiling limits of the fleet aircraft are defined from three distinct approaches: Safe-Life, Fail-Safe, and Damage Tolerance approaches. The common objectives to define fleet aircraft lives by the three approaches are to ensure safety while at the same time reducing total ownership costs. In this paper, the Damage Tolerance approach is considered and the focus is on the inspection scheme of the fatigued Multi-State System (MSS) with decreasing intervals between inspections. The paper proposes an analysis methodology to determine appropriate decreasing intervals between inspections of fatigue-sensitive aircraft structures (as alternative to constant intervals between inspections often used in practice), so that risk of catastrophic accident during flight is minimized. The suggested approach is unique and novel in that it allows one to utilize judiciously the results of earlier inspections of fatigued aircraft structures for the purpose of determining the time of the next inspection and estimating the values of several parameters involved in the problem that can be treated as uncertain. An illustrative example is given.
Received on March 17, 2010 and revised on December 18, 2010
References: 10

This paper presents two approaches for selecting qualified-for-reuse components for returned systems. Physical properties and age are used to assess the components’ residual lifetimes. Only components in the acceptable state are qualified for reuse. For a large number of components to be sorted only the ones whose ages are below a certain threshold are evaluated. This age threshold is estimated so that the proportion of components in the unacceptable states does not exceed the standard significance level. The proposed approach has been applied to the end of life of electric-powered wheelchair batteries.
Received on November 8, 2009, revised November 11, 2010
References: 15