A model of imperfect preventive maintenance with dependent failure modes

Consider a system subject to two modes of failures: maintainable and non-maintainable. A failure rate function is related to each failure mode. Whenever the system fails, a minimal repair is performed. Preventive maintenances are performed at integer multiples of a fixed period. The system is replaced when a fixed number of preventive maintenances have been completed. The preventive maintenance is imperfect because it reduces the failure rate of the maintainable failures but does not affect the failure rate of the non-maintainable failures. The two failure modes are dependent in the following way: after each preventive maintenance, the failure rate of the maintainable failures depends on the total of non-maintainable failures since the installation of the system. The problem is to determine an optimal length between successive preventive maintenances and the optimal number of preventive maintenances before the system replacement that minimize the expected cost rate. Optimal preventive maintenance schedules are obtained for non-decreasing failure rates and numerical examples for power law models are given.     

Optimal warranty policies for systems with imperfect repair

We investigate a system whose basic warranty coverage is minimal repair up to a specified warranty length. An additional service is offered whereby first failure is restored up to the consumers’ chosen level of repair. The problem is studied under two system replacement strategies: periodic maintenance before and after warranty. It turns out that our model generalizes the model of Rinsaka and Sandoh [K. Rinsaka, H. Sandoh, A stochastic model with an additional warranty contract, Computers and Mathematics with Applications 51 (2006) 179–188] and the model of Yeh et al. [R.H. Yeh, M.Y. Chen, C.Y. Lin, Optimal periodic replacement policy for repairable products under free-repair warranty, European Journal of Operational Research 176 (2007) 1678–1686]. We derive the optimal maintenance period and optimal level of repair based on the structures of the cost function and failure rate function. We show that under certain assumptions, the optimal repair level for additional service is an increasing function of the replacement time. We provide numerical studies to verify some of our results.     

Determination/estimation of an optimal replacement interval under minimal repair

In this paper the problem of estimation of an optimal replacement interval for a system which is minimally repaired at failures is studied. The problem is investigated both under a parametric and a nonparametric form of the failure intensity of the system. It is assumed that observational data from n systems are available. Some asymptotic results are shown. A graphical procedure for determining/estimating an optimal replacement interval is presented. The procedure is particularly valuable for sensitivity analyses, for example with respect to the costs involved.     

A maintenance model with failures and inspection following Markovian arrival processes and two repair modes

We consider a deteriorating system submitted to external and internal failures, whose deterioration level is known by means of inspections. There are two types of repairs: minimal and perfect, depending on the deterioration level, each one following a different phase-type distribution. The failures and the inspections follow different Markovian arrival processes (MAP). Under these assumptions, the system is governed by a generalized Markov process, whose state space and generator are constructed. This general model includes the phase-type renewal process as a special case. The distribution of the number of minimal and perfect repairs between two inspections are determined. A numerical application optimizing costs is performed, and different particular cases of the model are compared.     

A general age-replacement model with minimal repair under renewing free-replacement warranty

A general age-replacement model in which incorporates minimal repair, planned and unplanned replacement, is considered in this paper for products under a renewing free-replacement warranty policy. For both warranted and non-warranted products, cost models from the user’s perspective are developed, and the corresponding optimal replacement ages are derived such that the long-run expected cost rate is minimized. The impacts of a product warranty on the optimal replacement model are investigated analytically. Furthermore, we show that the optimal replacement age for a warranted product is closer to the end of the warranty period than for a non-warranted product. Finally, numerical examples are given for illustration.     

Optimum production and inspection modeling with minimal repair and rework considerations

This paper extends an integrated model of economic production quantity (EPQ) and preventive maintenance (PM) to incorporate possibilities of minimal repair and rework. Our model determines simultaneously the optimal number of inspections, the inspection interval, the EPQ, and the PM level. Numerical examples of Weibull shock models are given to show that allowing minimal repair and rework will raise the expected profit. Our analyses demonstrate that both minimal repair and rework significantly influence the optimal policy.     

Optimal number of minimal repairs before ordering spare for preventive replacement

This paper presents a model for determining the optimal number of minimal repairs before ordering spare for preventive replacement. By introducing the costs of ordering, repair, downtime, replacement, and the salvage value of an un-failed system, the expected long-term cost rates and cost effectiveness are derived. It is shown that, under certain conditions, the optimal number of minimal repairs, which minimizes the cost rate or maximizes the cost effectiveness, is given by a unique solution of an equation. A numerical example is also given for illustration of the proposed model.     

A standby system with two types of repair persons

A continuously monitored one‐unit system, backed by an identical standby unit, is perfectly repaired by an in‐house repair person, if achievable within a random or deterministic patience time (DPT), or else by a visiting expert, who repairs one or all failed units before leaving. We study four models in terms of the limiting availability and limiting profit per unit time, using semi‐Markov processes, when all distributions are exponential. We show that a DPT is preferable to a random patience time, and we characterize conditions under which the expert should repair multiple failed units (rather than only one failed unit) during each visit. We also extend the method when life‐ and repair times are non‐exponential. Copyright © 2009 John Wiley & Sons, Ltd.     

New failure and minimal repair processes for repairable systems in a random environment

Most often, minimal repair is defined as a replacement of a failed item by an operable item that has the same distribution of the remaining lifetime as the failed one just prior a failure. This is the so‐called statistical minimal repair extensively explored in the literature. Another well‐known type of minimal repair takes into account the state of a system prior to a failure (the information‐based minimal repair). In this paper, we suggest the new type of minimal repair to be called conditional statistical minimal repair. Our approach goes further and deals with the corresponding minimal repair processes for systems operating in a random environment. Moreover, we also consider heterogeneous populations of items, which makes the model more realistic. Both of these aspects that affect the failure mechanism of items are studied. Environment is modeled by the nonhomogeneous Poisson shock process. Two models for the failure mechanism defined by the extreme shock model and the cumulative shock model, respectively, are considered. Some examples illustrating our findings are presented.     

Optimal control of a finite dam with a sample path constraint

In this paper, we discuss the 2-stage output procedure of a finite dam under the condition that water must be released by a fixed time. From this standpoint, the reservoir model we consider is subject to a sample path constraint and has a more general cost function than the earlier contributions. We analytically derive explicit formulas for the long-run average and the expected total discounted costs for an infinite time span and numerically calculate the optimal control policy. Finally, the optimal policy is compared with one by Zuckerman [1] and the effect of the fixed release time is discussed further.     

A geometric process maintenance model with preventive repair

In this paper, a geometric process maintenance model with preventive repair is studied. A maintenance policy (T, N) is applied by which the system will be repaired whenever it fails or its operating time reaches T whichever occurs first, and the system will be replaced by a new and identical one following the Nth failure. The long-run average cost per unit time is determined. An optimal policy (T^∗, N^∗) could be determined numerically or analytically for minimizing the average cost. A new class of lifetime distribution which takes into account the effect of preventive repair is studied that is applied to determine the optimal policy (T^∗, N^∗).     

A bivariate optimal replacement policy for a system subject to a generalized failure and repair process

Traditionally, in the studies of the optimal maintenance policies for repairable systems, the nonhomogeneous Poisson process model, which corresponds to the minimal repair process, has been intensively applied. However, in many practical situations, the repair type is not necessarily minimal. In this article, a new repair process based on a new counting process model (so‐called the generalized Polya process) is introduced. Then, the issue of the optimal replacement problem is discussed. A bivariate preventive replacement policy is developed and the properties of the optimal policy are studied. Illustrative examples are also presented. In addition, a comparison with a conventional replacement policy is performed.     

Comparison of maintenance policies with monotone failure rate distributions

We compare different maintenance policies assuming that the system lifetime has either increasing failure rate (IFR), or decreasing failure rate (DFR) distribution. We show that these assumptions yield strongly stochastic comparisons between the process' intensities. This yields weaker stochastic comparison between the processes than the stochastic comparison that hold when the system lifetime is new better than used (NBU) or new worse than used (NWU). Copyright © 2002 John Wiley & Sons, Ltd.     

A bivariate mixed policy for a simple repairable system based on preventive repair and failure repair

In this paper, a simple repairable system (i.e. a one-component repairable system with one repairman) with preventive repair and failure repair is studied. Assume that the preventive repair is adopted before the system fails, when the system reliability drops to an undetermined constant R  , the work will be interrupted and the preventive repair is executed at once. And assume that the preventive repair of the system is “as good as new” while the failure repair of the system is not, and the deterioration of the system is stochastic. Under these assumptions, by using geometric process, we present a bivariate mixed policy (R,N)$(R\text{,}N)$, respectively based on a scale of the system reliability and the failure-number of the system. Our aim is to determine an optimal mixed policy (R,N)^∗$(R\text{,}N{)}^{\ast }$ such that the long-run average cost per unit time (i.e. the average cost rate) is minimized. The explicit expression of the average cost rate is derived, and the corresponding optimal mixed policy can be determined analytically or numerically. Finally, a numerical example is given where the working time of the system yields a Weibull distribution. Some comparisons with a certain existing policy are also discussed by numerical methods.     

A note on the exponentiality of total hazards before failure

It is well known that a univariate counting process with a given intensity function becomes Poisson, with unit parameter, if the original time parameter is replaced by the integrated intensity. P. A. Meyer (in Martingales (H. Dinges, Ed.), pp. 32–37. Lecture Notes in Mathematics, Vol. 190, Springer-Verlag, Berlin) showed that a similar result holds for multivariate counting processes which have continuous compensators. Even more is true in the multivariate case: If each coordinate process is transformed individually according to a convenient time change, the resulting Poisson processes become independent. Our aim is to show that the continuity assumption of the compensators can be relaxed and, when the jumps of the compensator become small, we obtain the independent Poisson processes as a limit. An application for testing goodness-of-fit in survival analysis is given.     

A bivariate optimal replacement policy for a cold standby repairable system with preventive repair

In this paper, the repair-replacement problem for a deteriorating cold standby repairable system is investigated. The system consists of two dissimilar components, in which component 1 is the main component with use priority and component 2 is a supplementary component. In order to extend the working time and economize the running cost of the system, preventive repair for component 1 is performed every time interval T, and the preventive repair is “as good as new”. As a supplementary component, component 2 is only used at the time that component 1 is under preventive repair or failure repair. Assumed that the failure repair of component 1 follows geometric process repair while the repair of component 2 is “as good as new”. A bivariate repair-replacement policy (T, N) is adopted for the system, where T is the interval length between preventive repairs, and N is the number of failures of component 1. The aim is to determine an optimal bivariate policy (T, N)^∗ such that the average cost rate of the system is minimized. The explicit expression of the average cost rate is derived and the corresponding optimal bivariate policy can be determined analytically or numerically. Finally, a Gamma distributed example is given to illustrate the theoretical results for the proposed model.     

An LDQBD process under degradation,inspection, and two types of repair

A warm standby n-system with operational and repair times following phase-type distributions is considered. The online unit goes through degradating levels, determined by inspections. Two types of repairs are performed, preventive and corrective, depending on the degradation level. The standby units undergo corrective repair. This systems is governed by a level-dependent-quasi-birth-and-death proces (LDQBD process), whose generator is constructed. The availability, rate of occurrence of failures, and other quantities of interest are calculated. A numerical example including an optimization problem and illustrating the calculations is presented. This system extend other previously studied in the literature.     

A sequential condition‐based repair/replacement policy with non‐periodic inspections for a system subject to continuous wear

This paper studies a condition‐based maintenance policy for a repairable system subject to a continuous‐state gradual deterioration monitored by sequential non‐periodic inspections. The system can be maintained using different maintenance operations (partial repair, as good as new replacement) with different effects (on the system state), costs and durations. A parametric decision framework (multi‐threshold policy) is proposed to choose sequentially the best maintenance actions and to schedule the future inspections, using the on‐line monitoring information on the system deterioration level gained from the current inspection. Taking advantage of the semi‐regenerative (or Markov renewal) properties of the maintained system state, we construct a stochastic model of the time behaviour of the maintained system at steady state. This stochastic model allows to evaluate several performance criteria for the maintenance policy such as the long‐run system availability and the long‐run expected maintenance cost. Numerical experiments illustrate the behaviour of the proposed condition‐based maintenance policy. Copyright © 2003 John Wiley & Sons, Ltd.     

On some applications of a discrete-time model of failure and repair

Research on a discrete-time model of failure and repair studied by Rocha-Martinez and Shaked (1995) is continued in this paper. Among various related results, we prove that if for one point x∈]0,1[ the probability generating function of a non-negative integer valued random variable S satisfies Φ_S(x)⩽x^m for some integer m⩾0, then E(S)⩾m. We use these results to show that for any M (the ‘input’ lifetime of a unit in the model) the R_m's (the allowed number of repairs on the unit at time m, m⩾0) can be chosen such that M_u (the ‘output’ lifetime of the unit through the model) is in hazard rate ordering (therefore in stochastic ordering) arbitrarily large and such that E(R_m) is a minimum in some sense. As a first application, we see how a low-quality item (car, computer, washing machine, etc.) might fulfil strict durability regulations under an appropriate imperfect repair strategy (and be able to compete against the existing leading brand in the market) in such a way that the mean number of repairs be a minimum in some sense. As a second application we show how it can be easily proven that if M is of class: NBU, NWE, DMRL, IMRL, NBUE or NWUE, then M_u is not necessarily of the same class. © 1998 John Wiley & Sons, Ltd.     

Economic production quantity model with repair failure and limited capacity

We develop an economic production quantity (EPQ) model with random defective items and failure in repair. The existence of only one machine results with limited production capacity and shortages. The aim of this research is to derive the optimal cycle length, the optimal production quantity and the optimal back ordered quantity for each product so as to minimize the total expected cost (holding, shortage, production, setup, defective items and repair costs). The convexity of the model is derived and the objective function is proved convex. Two numerical examples illustrate the practical usage of the proposed method.