Repairable Multi-state System Research Articles

Optimal Bivariate Order-Replacement Policies for a Multi-State Repairable System with Reliability Constraint

A bivariate order-replacement policy for a multi-state repairable system with imperfect repair is put forward in this paper, where the decisions on when to order a spare part and when to place a replacement are based on the number of failures. The geometric processes are generalized to depict the characteristics that the successive working times are shorter and shorter while the consecutive repair times are longer and longer. According to the renewal reward theorem, a constrained optimization model for the order-replacement policy is formulated, in which the aim is to minimize the expected long-run cost rate (ELRCR) under the constraint that system availability is not less than a predetermined availability requirement. General representations of the ELRCR and system availability are derived and an approximation result of the optimization model is given under certain assumptions. Illustrative examples are provided to validate the applicability of the proposed order-replacement model and show the advantage of the order-replacement policy for the multi-state repairable system.

Birnbaum joint importance measures for three components of a repairable multistate systems

In this paper new measures of joint importance of two and three components for repairable multistate systems based on the classical Birnbaum measure, are proposed. By considering repairable system, first joint relevancy conditions of two and three components are given. Then probabilities of each of the relevancy are measured. The proposed method is applied on a data set. An illustrative example is given. As in the Birnbaum measure, the proposed measures are generic since they depend on the probabilistic properties of the components and the system structure. These measures are useful when consider repairable system.

Integrating Modelling of Maintenance Policies within a Stochastic Hybrid Automaton Framework of Dynamic Reliability

The dependability assessment is a crucial activity for determining the availability, safety and maintainability of a system and establishing the best mitigation measures to prevent serious flaws and process interruptions. One of the most promising methodologies for the analysis of complex systems is Dynamic Reliability (also known as DPRA) with models that define explicitly the interactions between components and variables. Among the mathematical techniques of DPRA, Stochastic Hybrid Automaton (SHA) has been used to model systems characterized by continuous and discrete variables. Recently, a DPRA-oriented SHA modelling formalism, known as Stochastic Hybrid Fault Tree Automaton (SHyFTA), has been formalized together with a software library (SHyFTOO) that simplifies the resolution of complex models. At the state of the art, SHyFTOO allows analyzing the dependability of multistate repairable systems characterized by a reactive maintenance policy. Exploiting the flexibility of SHyFTA, this paper aims to extend the tools’ functionalities to other well-known maintenance policies. To achieve this goal, the main features of the preventive, risk-based and condition-based maintenance policies will be analyzed and used to design a software model to integrate into the SHyFTOO. Finally, a case study to test and compare the results of the different maintenance policies will be illustrated.

Optimal mission abort policies for repairable multistate systems performing multi-attempt mission

Research on mission abort strategies was mostly devoted to binary systems that can be only in two states, i.e., operable or failed. However, the real-world systems can often operate in intermediate states with different levels of performance. On the other hand, if a mission has been aborted and a system has been successfully rescued, at some instances, the next attempt can be activated, thus forming the multi-attempt framework. In this paper, the possibility of multiple attempts is considered for the first time for multistate systems. After each rescue, a system is repaired to ‘as good as new’ state. The repair time depends on its state before the repair. The objective is to maximize the probability of a mission completion within the fixed time deadline for systems operating in a random environment modeled by shocks. Each shock with a given probability results in a system's transition to the states with the lower values of performance. Mission abort is activated for each attempt when the number of experienced shocks exceeds a predetermined number. This number for each attempt should be determined to maximize the mission success probability. For the considered illustrative example, the detailed sensitivity analysis is performed and the relevant discussion is provided.

A Deep Reinforcement Learning Approach to Dynamic Loading Strategy of Repairable Multistate Systems

As multistate system (MSS) reliability models can characterize the multistate deteriorating nature of engineering systems, they have received considerable attention in the past decade. The states of a multistate system/component can be distinguished by its performance capacity, which deteriorates over time and can be restored by maintenance activities. On the other hand, the deterioration of a system/component is, oftentimes, controllable by setting a loading strategy. In this article, a dynamic load optimization problem for repairable MSSs is investigated to achieve the maximum expected cumulative performance within a finite time horizon and limited maintenance resources. The degraded components in a system are dynamically maintained to recover to their better conditions, whereas the performance rate of each component can also be dynamically specified. The resulting sequential decision problem is formulated as a Markov decision process with a continuous action space and a mixed integer discrete continuous state space. The deep deterministic policy gradient algorithm, which is a specific deep reinforcement learning algorithm in the actor−critic framework, is customized to overcome the “curse of dimensionality” and mitigate the uncountable state and action spaces. The effectiveness of the proposed method is examined by two illustrative examples, and a set of comparative studies are conducted to demonstrate the advantage of the proposed dynamic loading strategy.

Reliability Evaluation for Multi-State Repairable Systems with Hybridization the Markov Stochastic Process and the Universal Generating Function

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A repairable multi-state system with a general α-series process and an order-replacement policy

This article considers an one-unit multi-state system with a general α-series process and an order-replacement policy. It is supposed that the working times of the system after repair become shorter and shorter, while the repair times of the system become longer and longer. Further, an ordering policy N − 1 and a replacement policy N is adopted. After the Nth repair, the system is replaced by an identical new one, and the unit for replacement is immediately ordered at the end of the th repair. Some important properties of the system are derived by using the probability theory and the stochastic process theory. In addition, the explicit expression for the long-run expected cost function is developed to search the optimal order-replacement policy.

Modeling of Repairable Multi-State Systems with time-varying failure and repair rates using Equivalent Systems Dynamics Models

This paper treats with the reliability assessment of a Repairable Multi-State System (RMSS) by means of a Nonhomogeneous Continuous-Time Markov Chain (NH-CTMC). A RMSS run on different operating conditions that may be considered acceptable or unacceptable according to a defined demand level. In these cases, the commonly used technique is Homogeneous Continuous-Time Markov Chain (H-CTMC), since its solution is mathematically tractable. However, the H-CMTC involve that the time between state transitions is exponentially distributed, and the failure and repair rates are constants. It's certainly not true if the system components age with the operation or if the repair activities depend on the instant of time when the failure occurred. In these cases, the failure and repair rates are time-varying and the NH-CTMC is needed to be considered. Nevertheless, for these models the analytical solution may not exist and the use of others techniques is required. This paper proposes the use of an Equivalent Systems Dynamics Model (ESDM) to model a NH-CTMC. A ESDM represent the Markov Model (MM) by means of the language and the tools of the Systems Dynamics (SD), and the results are obtained by simulation. As an example, an RMSS with three components, failure rates associated with the Weibull distribution and repair rates associated with the Log-logistic distribution is developed. This example serves to identify the advantages and disadvantages of a ESDM to make model a RMSS and evaluate some reliability measures.

BIRNBAUM CRITICALITY AND IMPORTANCE MEASURES FOR MULTISTATE SYSTEMS WITH REPAIRABLE COMPONENTS

We suggest four new measures of importance for repairable multistate systems based on the classical Birnbaum measure. Periodic component life cycles and general semi-Markov processes are considered. Similar to the Birnbaum measure, the proposed measures are generic in the sense that they only depend on the probabilistic properties of the components and the system structure. The multistate system model encodes physical properties of the components and the system directly into the structure function. As a result, calculating importance is easy, especially in the asymptotic case. Moreover, the proposed measures are composite measures, combining importance for all component states into a unified quantity. This simplifies ranking of the components with respect to importance. The proposed measures can be characterized with respect to two features: forward-looking versus backward-looking and with respect to how criticality is measured. Forward-looking importance measures focus on the next component states, while backward-looking importance measures focus on the previous component states. Two approaches to measuring criticality are considered: probability of criticality versus expected impact. Examples show that the different importance measures may result in unequal rankings.

Improved Availability Index for Repairable Fuzzy Multi-State Systems

Fuzzy multi-state systems (FMSSs) comprise FMS components (FMSCs) aims at analyzing non-probabilistic uncertain circumstances defined by fuzzy state-transition rate and performance numbers. The triangular fuzzy number that features intuitive and algebraic operations is widely used for quantifying fuzziness. This study aims to overcome the drawbacks of the existing FMSS availability index that either does not conform to a normal convex set or is inconsistent with the crisp-circumstance multi-state system (MSS) availability beyond summation over state probabilities that do no equal unity. The proposed method first evaluates the instantaneous state probability (ISP) of FMSCs by establishing the FMSCs Markov model. Subsequently, the FMSSs ISP is evaluated by integrating the FMSCs ISP using the fuzzy universal generating function based on the system-structure function. A “constrained nonlinear parameter programming model” has been developed to evaluate the improved FMSSs availability. The corresponding availability index features the normal convex set whilst fulfilling the MSS theory, which requires state probabilities to equal unity during availability evaluation. The effectiveness of the proposed approach has been verified using an illustrated community-based smart-grid system comprising three FMSCs—solar- and wind-energy systems as well as diesel-generator equipment. Moreover, the results of an independent sensitive analysis provide further insights into the improved FMSSs availability with regard to the time and circumstance fuzziness.

SHyFTOO, an object-oriented Monte Carlo simulation library for the modeling of Stochastic Hybrid Fault Tree Automaton

Dependability assessment is a crucial activity to ensure the correct operation of complex systems. The output of dependability assessment activities include the quantification of reliability, availability, maintenance and safety related metrics. These metrics can assist in the identification of the system weak points or in the conception of mitigation strategies to increase the system dependability level. The development of advanced computer-aided methodologies to support dependability assessment activities is essential to automate and reduce the efforts implied by this process and similarly, the development of accurate dependability assessment methods is very important to increase the quality of the results. In this context, it is possible to identify different contributions that improve the dependability assessment through general-purpose modeling methodologies. However, existing solutions are ad-hoc applications specified with low-level stochastic formalisms and this complicates their adoption in the industry. Accordingly, this paper presents Stochastic Hybrid Fault Tree Automaton (SHyFTA) based simulation algorithm that allows the accurate dependability analysis of repairable multi-state systems. SHyFTA integrates the stochastic and deterministic operation of the system under study as well as their interactions. The algorithm is formalized through an object-oriented software architecture, which is developed as a software library for the modeling and simulation of repairable SHyFTA models. Following the proposed architecture, a Matlab® implementation of this library, SHyFTOO, has been developed and validated with a thorough test campaign. In order to provide a guideline to the end-users and show the potential of the SHyFTOO library, the case study of a feed-water pumping system is implemented in detail and it is used to evaluate different preventive maintenance policies. The SHyFTOO library can open the way to further investigations that address the interactions between the failure behavior and the functional operation of a system and their combined effect on system dependability.

A computational method for finding the availability of opportunistically maintained multi-state systems with non-exponential distributions

Availability is one of the most important performance measures of a repairable system. Among various mathematical methods, the method of supplementary variables is an effective way of modeling the steady-state availability of systems governed by non-exponential distributions. However, when all the underlying probability distributions are non-exponential (e.g., Weibull), the corresponding state equations are difficult to solve. To overcome this challenge, a new method is proposed in this article to determine the steady-state availability of a multi-state repairable system, where all the state sojourn times, as well as the maintenance times, are generally distributed. As an indispensable step, the well-posedness and stability of the system’s state equations are illustrated and proved using C0 operator semigroup theory. Afterwards, based on the generalized Integral Mean Value Theorem, the expression for system steady-state availability is derived as a function of state probabilities. Then, the original problem is transformed into a system of linear equations that can be easily solved. A simulation study and an instance studied in the literature are used to demonstrate the applications of the proposed method in practice. These numerical examples illustrate that the proposed method provides a new computational tool for effectively evaluating the availability of a repairable system without relying on simulation.

Reliability and maintenance strategy for systems with aging‐dependent components

AbstractAn innovative approach is presented for the reliability analysis of aging multistate systems that considers the subsystems and their components' dependency. A reliability function is determined for an aging series system with the component dependency following the local load‐sharing rule, and a reliability function is determined for an aging “m out of n” system with the component dependency following the equal load‐sharing rule. Linking the results of those load‐sharing models, a mixed‐dependency model for multistate “m out of l”‐series systems is constructed by assuming the dependence between subsystems connected in series under the local load‐sharing rule and the dependence between their components under the equal load‐sharing rule. As a special case, the reliability of this system, modeled using piecewise exponential reliability functions, is considered, and the results are applied to characterize shipyard rope elevator reliability. Finally, the maintenance of this elevator as a repairable multistate system is analyzed with the time of renovation ignored.

Reliability analysis of semi-Markov systems with restriction on transition times

In this paper, a competing risk model with a restriction on transition times for semi-Markov multi-state repairable systems is proposed. The states of semi-Markov systems can be divided into three categories: a normal working subset, a defective working subset and a breakdown subset which contains an absorbing state where the system cannot escape once entering it. If the number of transitions between the normal and defective working subsets exceeds a given value, the system will be abandoned due to the high maintenance costs. The theory of aggregated stochastic process is employed to obtain the closed-form formulas of the probabilities of competing risks, distributions of survival times and point-wise availabilities. An integer nonlinear programming is presented which enables the system to satisfy the prescribed performance requirements and control its economic cost. A corresponding algorithm for solving the integer nonlinear programming is provided to find the optimal number of transition times. Two numerical examples including Markov process and semi-Markov process are conducted to illustrate the obtained results and proposed methods finally.

Behavioral study and availability optimization of a multi-state repairable system with hot redundancy

PurposeThe purpose of this paper is to identify the criticality of various sub-systems through the behavioral study of a multi-state repairable system with hot redundancy. The availability of the system is optimized to evaluate the optimum combinations of failure and repair rate parameters for various sub-systems.Design/methodology/approachThe behavioral study of the system is conducted through the stochastic model under probabilistic approach, i.e., Markov process. The first-order differential equations associated with the stochastic model are derived with the use of mnemonic rule assuming that the failure and repair rate parameters of all the sub-systems are constant and exponentially distributed. These differential equations are further solved recursively using the normalizing condition to obtain the long-run availability of the system. A particle swarm optimization (PSO) algorithm for evaluating the optimum availability of the system and supporting computational results are presented.FindingsThe maintenance priorities for various sub-systems can easily be set up, as it is clearly identified in the behavioral analysis that the sub-system (A) is the most critical component which highly influences the system availability as compared to other sub-systems. The PSO technique modifies input failure and repair rate parameters for each sub-system and evaluates the optimum availability of the system.Originality/valueA bottom case manufacturing system is under the evaluation, which is the main component of front shock absorber in two-wheelers. The input failure and repair rate parameters were parameterized from the information provided by the plant personnel. The finding of the paper provides the various availability measures and shows the grate congruence with the system behavior.

Asymptotic behavior of repairable device systems under warranty and beyond warranty periods

In this paper, we present a new mathematical model generalizing the model of a single unit repairable device system, with preventive maintenance under warranty period, and the model of a repairable multi-state device system. To study the time dependance solution of the resulting system, we present in this paper a new analytic approach. So, by using the $$C_0$$ -semigroup theory of linear operators and some spectral properties, we proof that the dynamic solution of the system converges to steady state solution when the time go to infinity.

Reliability performance for dynamic multi-state repairable systems with K regimes

ABSTRACTIt is well known that many factors can influence the rate at which a machine degrades. In this article, we study a multi-state repairable system subject to continuous degradation and dynamically evolving regimes. The degradation rate of the system depends not only on the system states but also on the regimes driven by the varying external environments. Movements between the system states are governed by continuous-time Markov processes but with different transition rate matrices due to different regimes; meanwhile, the evolution of regime is also governed by a Markov process. Such a system can be developed by the Markov regime-switching model. To derive the system performance such as the first passage time distribution, a Markov renewal process is introduced by giving its semi-Markov kernel. We also consider the system in the context of periodic inspections and maintenance and give the limiting average availability. Finally, some numerical examples are given to demonstrate and validate the proposed framework.

Modeling and analysis for multi-state systems with discrete-time Markov regime-switching

The main focus of this paper is on the development of reliability measures for a repairable multi-state system which operates under dynamic regimes under the discrete-time hypothesis. The switching process of regimes is governed by a Markov chain, and the functioning process of the system follows another Markov chain with different transition probability matrices under different regimes. In terms of two chains as above, a new Markov chain is constructed to depict the evolution process of the dynamic system. For the regime consideration, some novel reliability indices are essential and firstly introduced in this paper. By means of hierarchical partitions for the new state space, Ion-Channel modeling theory and discrete-time Markov chain, the traditional and novel reliability and availability functions for the system under random regimes are easily obtained with the closed form solutions, such as two types of system reliabilities, two types of system point availabilities, two types of system multiple-point availabilities and the associated system multi-interval availabilities and so on. In addition, some probability distributions of sojourn times we are interested in are discussed and computed here. Finally, a numerical example is given to illustrate the results obtained in the paper.

Reliability analysis of multi-state repairable system with dynamic transition probabilities

Inadequate maintenance decisions lead to incremental overall costs. In order to minimize costs in maintenance of the multi-state repairable system, we model a preventive maintenance (PM) scheme of the multistate repairable system using non-Markov process. The periodically decreasing reliability model of the non-Markov dynamic system with dynamic transition probabilities is established to satisfy the probability change. The diesel engine system is taken as an example to illustrate the model. The reliability of the diesel engine is analyzed and its PM scheme is worked out. RENO software is used to simulate the diesel engine system. The maintenance cost of components and the optimal PM interval data of the system are obtained by using the minimal average cost as the objective function. The adaptability of PM is judged, and the optimal PM scheme is presented.

Availability equivalence analysis of a repairable multi-state series-parallel system

The availability equivalence of different designs for a repairable multi-state series-parallel system (RMSPS) is discussed in this paper. The system components are assumed to be independent, and their failure and repair rates to be constant. The system availability is defined as the ability of the system to satisfy consumer demand. Factor improvement method and standby redundancy method are used to improve the system design. To evaluate availability of the both original and improved systems, a fast technique, based on universal generating function, is adopted. The availability equivalence factor is introduced to compare different system designs. Two types of availability equivalence factors of the system are derived. A numerical example is provided to illustrate how to utilize the obtained results.