Hybrid minimal repair and age replacement maintenance policies,with nonnegligible repair times

A system undergoes minimal repair during [0, T] with a failure replacement on first failure during [T, a], or a planned replacement if the system is still functioning at elapsed time a. Repairs and replacements are not necessarily instantaneous. An expression is obtained for the asymptotic expected cost rate, and sufficient conditions are obtained for the optimum T* > 0. Several special cases are considered. A numerical investigation for a Weibull distributed time to first failure compares this elapsed-time policy with replacement on failure only, and also a policy based on system operating time or age. It is found that in many cases the elapsed-time-based policy is only marginally worse than one based on system age, and may therefore be preferred in view of its administrative convenience. © 1994 John Wiley & Sons, Inc.     

Optimal number of minimal repairs before replacement of a system subject to shocks

A system is subject to shocks that arrive according to a nonhomogeneous Poisson process. As shocks occur a system has two types of failures. Type 1 failure (minor failure) is removed by a minimal repair, whereas type 2 failure (catastrophic failure) is removed by replacement. The probability of a type 2 failure is permitted to depend on the number of shocks since the last replacement. A system is replaced at the times of type 2 failure or at the nth type 1 failure, whichever comes first. The optimal policy is to select n* to minimize the expected cost per unit time for an infinite time span. A numerical example is given to illustrate the method. © 1996 John Wiley & Sons, Inc.     

Optimal replacement under additive damage and other failure models

A machine or production system is subject to random failure. Upon failure the system is replaced by a new one, and the process repeats. A cost is associated with each replacement, and an additional cost is incurred at each failure in service. Thus, there is an incentive for a controller to attempt to replace before failure occurs. The problem is to find an optimal control strategy that balances the cost of replacement with the cost of failure and results in a minimum total long-run average cost per unit time. We attack this problem under the cumulative damage model for system failure. In this failure model, shocks occur to the system in accordance with a Poisson process. Each shock causes a random amount of damage or wear and these damages accumulate additively. At any given shock, the system fails with a known probability that depends on the total damage accumulated to date. We assume that the cumulative damage is observable by the controller and that his decisions may be based on its current value. Supposing that the shock failure probability is an increasing function of the cumulative damage, we show that an optimal policy is to replace either upon failure or when this damage first exceeds a critical control level, and we give an equation which implicitly defines the optimal control level in terms of the cost and other system parameters. Also treated are some more general models that allow for income lost during repair time and other extensions.     

Bounding the optimal burn‐in time for a system with two types of failure

Burn‐in is a widely used method to improve the quality of products or systems after they have been produced. In this paper, we consider the problem of determining bounds to the optimal burn‐in time and optimal replacement policy maximizing the steady state availability of a repairable system. It is assumed that two types of system failures may occur: One is Type I failure (minor failure), which can be removed by a minimal repair, and the other is Type II failure (catastrophic failure), which can be removed only by a complete repair. Assuming that the underlying lifetime distribution of the system has a bathtub‐shaped failure rate function, upper and lower bounds for the optimal burn‐in time are provided. Furthermore, some other applications of optimal burn‐in are also considered. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

Replacement models under additive damage

A production system which generates income is subject to random failure. Upon failure, the system is replaced by a new identical one and the replacement cycles are repeated indefinitely. In our breakdown model, shocks occur to the system in a Poisson stream. Each shock causes a random amount of damage, and these damages accumulate additively. The failure time depends on the accumulated damage in the system. The income from the system and the cost associated with a planned replacement depend on the accumulated damage in the system. An additional cost is incurred at each failure in service. We allow a controller to replace the system at any stopping time T before failure time. We will consider the problem of specifying a replacement rule that is optimal under the following criteria: maximum total long-run average net income per unit time, and maximum total long-run expected discounted net income. Our primary goal is to introduce conditions under which an optimal policy is a control limit policy and to investigate how the optimal policy can be obtained. Examples will be presented to illustrate computational procedures.     

Two‐dimensional failure modeling with minimal repair

In this paper, we discuss two‐dimensional failure modeling for a system where degradation is due to age and usage. We extend the concept of minimal repair for the one‐dimensional case to the two‐dimensional case and characterize the failures over a two‐dimensional region under minimal repair. An application of this important result to a manufacturer's servicing costs for a two‐dimensional warranty policy is given and we compare the minimal repair strategy with the strategy of replacement of failure. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004.     

A periodic maintenance model with used equipment and random minimal repair

The maintenance strategy considered in this article is in the class of block replacement policies. The working unit is replaced by new ones at instants T,2T,3T,… independently of the age and state of the unit. If a failure occurs between these instants, the random repair cost is evaluated. If it is smaller than a predetermined control limit then a minimal repair is carried out. Otherwise the unit is replaced by a used unit. The performance of this maintenance strategy is evaluated in terms of average total cost per time unit over an infinite time span. The mathematical model is defined and several analytical results are obtained. A computer program has been written, which solves the mathematical problem, and some examples are given for the cases where the underlying life distribution is gamma, Weibull or truncated normal.     

Optimal policy of continuous and discrete replacement with minimal repair at failure

This article considers combined continuous and discrete replacement with minimal repair at failure, in which a unit is replaced at time T or at number N of uses. Both optimal time T* and number N* to minimize the expected cost rate are discussed. They are found by unique solutions of equations when the hazard rates are monotonously increasing. A numerical example is given.     

Optimal burn‐in procedure for periodically inspected systems

Burn‐in is a widely used method to improve the quality of products or systems after they have been produced. In this paper, we study burn‐in procedure for a system that is maintained under periodic inspection and perfect repair policy. Assuming that the underlying lifetime distribution of a system has an initially decreasing and/or eventually increasing failure rate function, we derive upper and lower bounds for the optimal burn‐in time, which maximizes the system availability. Furthermore, adopting an age replacement policy, we derive upper and lower bounds for the optimal age parameter of the replacement policy for each fixed burn‐in time and a uniform upper bound for the optimal burn‐in time given the age replacement policy. These results can be used to reduce the numerical work for determining both optimal burn‐in time and optimal replacement policy. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Optimal replacement for fault-tolerant systems

We consider the optimal replacement problem for a fault tolerant system comprised of N components. The components are distingushable, and the state of the system is given by knowing exactly which components are operationl and which have failed. The individual component failure rates depend on the state of the entire system. We assume that the rate at which the system produces income decreases as the system deteriorates and the system replacement cost rises. Individual components cannot be replaced. We give a greedy-type algorithm that produces the replacement policy that maximizes the long-run net system income per unit time.     

Periodic replacement with minimal repair at failure and adjustment costs

We investigate periodic replacement policies with minimal repair at failure, thereby, minimizing the average expected cost per unit time over an infinite time span. The standard cost structure is modified by the introduction of a term which takes adjustment costs into account.     

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

In this article, an optimal replacement policy for a cold standby repairable system consisting of two dissimilar components with repair priority is studied. Assume that both Components 1 and 2, after repair, are not as good as new, and the main component (Component 1) has repair priority. Both the sequence of working times and that of the components'repair times are generated by geometric processes. We consider a bivariate replacement policy (T,N) in which the system is replaced when either cumulative working time of Component 1 reaches T, or the number of failures of Component 1 reaches N, whichever occurs first. The problem is to determine the optimal replacement policy (T,N)* such that the long run average loss per unit time (or simply the average loss rate) of the system is minimized. An explicit expression of this rate is derived, and then optimal policy (T,N)* can be numerically determined through a two‐dimensional‐search procedure. A numerical example is given to illustrate the model's applicability and procedure, and to illustrate some properties of the optimal solution. We also show that if replacements are made solely on the basis of the number of failures N, or solely on the basis of the cumulative working time T, the former class of policies performs better than the latter, albeit only under some mild conditions. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

Optimal maintenance policy for stochastically failing equipment: A diffusion approximation

A system receives shocks at random points of time. Each shock causes a random amount of damage which accumulates over time. The system fails when the accumulated damage exceeds a fixed threshold. Upon failure the system is replaced by a new one. The damage process is controlled by means of a maintenance policy. There are M possible maintenance actions. Given that a maintenance action m is employed, then the cumulative damage decreases at rate r^m. Replacement costs and maintenance costs are considered. The objective is to determine an optimal maintenance policy under the following optimality criteria: (1) long-run average cost; (2) total expected discounted cost over an infinite horizon. For a diffusion approximation, we show that the optimal maintenance expenditure rate is monotonically increasing in the cumulative damage level.     

Maintenance of a device with age-dependent exponential failures

We consider a device that deteriorates over time according to a Markov process so that the failure rate at each state is constant. The reliability of the device is characterized by a Markov renewal equation, and an IFRA (increasing failure rate on average) property of the lifetime is obtained. The optimal replacement and repair problems are analyzed under various cost structures. Furthermore, intuitive and counterintuitive characterizations of the optimal policies and results on some interesting special problems are presented. © 1992 John Wiley & Sons, Inc.     

A unifying treatment of replacement policies with minimal repair

This article presents the mathematical background for analyzing maintenance policies with minimal repair. Standard maintenance policies are summarized. A general failure model is introduced which allows a unifying treatment of more recent maintenance policies with minimal repair. In particular, repair cost limit maintenance policies are considered. © 1993 John Wiley & Sons, Inc.     

A modified block replacement policy

A well known preventive replacement policy is the block replacement policy (BRP). In such a policy the item undergoes a planned replacement at a sequence of equally spaced time points independent of failure history. The main advantage of a BRP is its simplicity, because under this policy it is unnecessary to keep detailed records about times of failures or ages of items. The main drawback of a BRP is that at planned replacement times we may be replacing practically new items. In this paper we study a modified BRP which is free of this drawback. We calculate the expected cost of following a modified BRP for lifetime distributions possessing a special structure and illustrate it for the case of an Erlang distribution. A numerical comparison is made between a modified BRP and a standard BRP for the special case of a two stage Erlang distribution.     

Bayesian enhanced decision making for deteriorating repairable systems with preventive maintenance

Since a system and its components usually deteriorate with age, preventive maintenance (PM) is often performed to restore or keep the function of a system in a good state. Furthermore, PM is capable of improving the health condition of the system and thus prolongs its effective age. There has been a vast amount of research to find optimal PM policies for deteriorating repairable systems. However, such decisions involve numerous uncertainties and the analyses are typically difficult to perform because of the scarcity of data. It is therefore important to make use of all information in an efficient way. In this article, a Bayesian decision model is developed to determine the optimal number of PM actions for systems which are maintained according to a periodic PM policy. A non‐homogeneous Poisson process with a power law failure intensity is used to describe the deteriorating behavior of the repairable system. It is assumed that the status of the system after a PM is somewhere between as good as new for a perfect repair and as good as old for a minimal repair, and for failures between two preventive maintenances, the system undergoes minimal repairs. Finally, a numerical example is given and the results of the proposed approach are discussed after performing sensitivity analysis. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

A cumulative damage shock model with imperfect preventive maintenance

In this article we consider a cumulative damage shock model under a periodic preventive maintenance (PM) policy. The PM is imperfect in the sense that each PM reduces the damage level by 100(1 – b)%, 0 < b < 1. A system suffers damage due to shocks and fails when the damage level exceeds some threshold. We derive a sufficient condition for the time to failure to have an IFR distribution. We also discuss the associated problem of finding the number of PM's that minimizes the expected cost rate.     

基于二维量度的复杂设备预防性维修决策优化

针对使用与维修具有两个测量维度的复杂设备,开展了其预防性维修决策的优化研究。基于二维量度的故障模式,给出了二维故障率的定量描述方法;分析了其预防性更换过程的基本过程,探讨了二维量度下更换周期对维修效果的影响,并从经济性角度建立了二维工龄更换费用模型;最后,采用算例的形式,对某设备维修决策同时考虑日历使用时间和行驶里程的情况,进行了二维更换间隔期的优化求解,从而验证了所建立方法与模型的实用性。     

Optimal replacement rules based on different information levels

We consider the following replacement model in reliability theory. A technical system with random lifetime is replaced upon failure. Preventive replacements can be carried out before failure. The time for such a replacement depends on the observation of a random state parameter and is therefore in general a random time. Different costs for preventive and failure replacements are introduced which may depend on the age of the working system. The optimization criterion followed here to find an optimal replacement time is to minimize the total expected discounted costs. The optimal replacement policy depends on the observation of the state of the system. Results of the theory of stochastic processes are used to obtain the optimal strategy for different information levels. Several examples based on a two-component parallel system with possibly dependent component lifetimes show how the optimal replacement policy depends on the different information levels and on the degree of dependence of the components. © 1992 John Wiley & Sons, Inc.