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.     

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.     

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.     

Optimal replacement policies for multistate deteriorating systems

We consider state-age-dependent replacement policies for a multistate deteriorating system. We assume that operating cost rates and replacement costs are both functions of the underlying states. Replacement times and sojourn times in different states are all state-dependent random variables. The optimization criterion is to minimize the expected long-run cost rate. A policy-improvement algorithm to derive the optimal policy is presented. We show that under reasonable assumptions, the optimal replacement policies have monotonic properties. In particular, when the failure-rate functions are nonincreasing, or when all the replacement costs and the expected replacement times are independent of state, we show that the optimal policies are only state dependent. Examples are given to illustrate the structure of the optimal policies in the special case when the sojourntime distributions are Weibull. © 1994 John Wiley & Sons, Inc.     

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.     

Periodic replacement when minimal repair costs vary with time

A policy of periodic replacement with minimal repair at failure is considered for a complex system. Under such a policy the system is replaced at multiples of some period T while minimal repair is performed at any intervening system failures. The cost of a minimal repair to the system is assumed to be a nonde-creasing function of its age. A simple expression is derived for the expected minimal repair cost in an interval in terms of the cost function and the failure rate of the system. Necessary and sufficient conditions for the existence of an optimal replacement interval are exhibited in the case where the system life distribution is strictly increasing failure rate (IFR).     

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.     

A comparative study of three tool replacement/operation sequencing strategies in a flexible manufacturing system

This paper studies three tool replacement/operation sequencing strategies for a flexible manufacturing system over a finite time horizon: (1) failure replacement—replace the tool only upon failure, (2) optimal preventive tool replacement for a fixed sequence of operations, and (3) joint scheduling of the optimal preventive tool replacement times and the optimal sequence of operations. Stochastic dynamic decision models are used for strategies 2 and 3. The optimization criterion for strategies 2 and 3 is the minimization of the total expected cost over the finite time horizon. We will show through numerical studies that, with the same amount of information, the total expected costs can be reduced considerably by choosing an optimal strategy. Our conclusion is that in flexible manufacturing, optimal tool replacement and optimal operations sequencing are not separate issues. They should be considered jointly to minimize the expected total cost. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 479–499, 2000     

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.     

Preventive maintenance and replacement under additive damage

A system deteriorates due to shocks received at random times, each shock causing a random amount of damage which accumulates over time and may result in a system failure. Replacement of a failed system is mandatory, while an operable one may also be replaced. In addition, the shock process causing system deterioration may be controlled by continuous preventive maintenance expenditures. The joint problem of optimal maintenance and replacement is analyzed and it is shown that, under reasonable conditions, optimal maintenance rate is decreasing in the cumulative damage level and that beyond a certain critical level the system should be replaced. Meaningful bounds are established on the optimal policies and an illustrative example is provided.     

A heuristic algorithm for determining replacement policies in k-out-of-n systems

The authors study a discrete-time, infinite-horizon, dynamic programming model for the replacement of components in a binary k-out-of-n failure system. (The system fails when k or more of its n components fail.) Costs are incurred when the system fails and when failed components are replaced. The objective is to minimize the long-run expected average undiscounted cost per period. A companion article develops a branch-and-bound algorithm for computing optimal policies. Extensive computational experiments find it effective for k to be small or near n; however, difficulties are encountered when n ≥ 30 and 10 ≤ k ≤ n − 4. This article presents a simple, intuitive heuristic rule for determining a replacement policy whose memory storage and computation time requirements are O(n − k) and O(n(n − k) + k), respectively. This heuristic is based on a plausible formula for ranking components in order of their usefulness. The authors provide sufficient conditions for it to be optimal and undertake computational experiments that suggest that it handles parallel systems (k = n) effectively and, further, that its effectiveness increases as k moves away from n. In our test problems, the mean relative errors are under 5% when n ≤ 100 and under 2% when k ≤ n − 3 and n ≤ 50. © 1997 John Wiley & Sons, Inc. Naval Research Logistics 44, 273–286, 1997.     

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     

Fluctuations of the optimal advertising and inventory policy: A qualitative analysis

This paper discusses the properties of the inventory and advertising policy minimizing the expected discounted cost over a finite horizon in a dynamic nonstationary inventory model with random demand which is influenced by the level of promotion or goodwill. Attention is focused on the relation between the fluctuations over time of the optimal policies and the variations over time of the factors involved, i.e., demand distributions and various costs. The optimal policies are proved to be monotone in the various factors. Also, three types of fluctuations over time of the optimal policies are discussed according to which factor varies over time. For example, if over a finite interval, the random demand increases (stochastically) from one period to the next, reaches a maximum and then decreases, then the optimal inventory level will do the same. Also the period of maximum of demand never precedes that of maximum inventory. The optimal advertising behaves in the opposite way and its minimum will occur at the same time as the maximum of the inventory. The model has a linear inventory ordering cost and instantaneous delivery of stocks; many results, however, are extended to models with a convex ordering cost or a delivery time lag.     

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.     

Age replacement policies in two time scales

We develop and estimate optimal age replacement policies for devices whose age is measured in two time scales. For example, the age of a jet engine can be measured in the number of flight hours and the number of landings. Under a single‐scale age replacement policy, a device is replaced at age τ or upon failure, whichever occurs first. We show that a natural generalization to two scales is to replace nonfailed devices when their usage path crosses the boundary of a two‐dimensional region M, where M is a lower set with respect to the matrix partial order. For lifetimes measured in two scales, we consider devices that age along linear usage paths. We generalize the single‐scale long‐run average cost, estimate optimal two‐scale policies, and give an example. We note that these policies are strongly consistent estimators of the true optimal policies under mild conditions, and study small‐sample behavior using simulation. © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 592–613, 2003.     

A note on the costly surveillance of a stochastic system

We consider the costly surveillance of a stochastic system with a finite state space and a finite number of actions in each state. There is a positive cost of observing the system and the system earns at a rate depending on the state of the system and the action taken. A policy for controlling such a system specifies the action to be taken and the time to the next observation, both possibly random and depending on the past history of the system. A form of the long range average income is the criterion for comparing different policies. If R ^Δ denotes the class of policies for which the times between successive observations of the system are random variables with cumulative distribution functions on [0, Δ], Δ < ∞, we show that there exists a nonrandomized stationary policy that is optimal in R ^Δ. Furthermore, for sufficiently large Δ, this optimal policy is independent of Δ.     

Monotone optimal preventive maintenance policies for stochastically failing equipment

This paper examines various models for maintenance of a machine operating subject to stochastic deterioration. Three alternative models are presented for the deterioration process. For each model, in addition to the replacement decision, the option exists of performing preventive maintenance. The effect of this maintenance is to “slow” the deterioration process. With an appropriate reward structure imposed on the processes, the models are formulated as continuous time Markov decision processes. the optimality criterion being the maximization of expected discounted reward earned over an infinite time horizon. For each model conditions are presented under which the optimal maintenance policy exhibits the following monotonic structure. First, there exists a control limit rule for replacement. That is, there exists a number i* such that if the state of machine deterioration exceeds i* the optimal policy replaces the machine by a new machine. Secondly, prior to replacement the optimal level of preventive maintenance is a nonincreasing function of the state of machine deterioration. The conditions which guarantee this result have a cost/benefit interpretation.     

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     

Application of Markov renewal theory and semi-Markov decision processes in maintenance modeling and optimization of multi-unit systems

In this paper, a condition-based maintenance model for a multi-unit production system is proposed and analyzed using Markov renewal theory. The units of the system are subject to gradual deterioration, and the gradual deterioration process of each unit is described by a three-state continuous time homogeneous Markov chain with two working states and a failure state. The production rate of the system is influenced by the deterioration process and the demand is constant. The states of the units are observable through regular inspections and the decision to perform maintenance depends on the number of units in each state. The objective is to obtain the steady-state characteristics and the formula for the long-run average cost for the controlled system. The optimal policy is obtained using a dynamic programming algorithm. The result is validated using a semi-Markov decision process formulation and the policy iteration algorithm. Moreover, an analytical expression is obtained for the calculation of the mean time to initiate maintenance using the first passage time theory.     

Optimal replacement policies with two or more loaded sliding standbys

This article defines optimal replacement policies for identical components performing different functions in a given system, when more than one spare part is available. The problem is first formulated for two components and any number of spare parts and the optimal replacement time y(x) at time x is found to have a certain form. Sufficient conditions are then provided for y(x) to be a constant y* for x > y*, and y(x) = x for x > y* (single-critical-number policy). Under the assumption that the optimal policies are of the single-critical-number type, the results are extended to the n-component case, and a theorem is provided that reduces the required number of critical numbers. Finally, the theory is applied to the case of the exponential and uniform failure laws, in which single-critical-number policies are optimal, and to another failure law in which they are not.