An optimal replacement policy for a multistate degenerative simple system

In this paper, a degenerative simple system (i.e. a degenerative one-component system with one repairman) with k + 1 states, including k failure states and one working state, is studied. Assume that the system after repair is not “as good as new”, and the degeneration of the system is stochastic. Under these assumptions, we consider a new replacement policy T based on the system age. Our problem is to determine an optimal replacement policy T^∗ such that the average cost rate (i.e. the long-run average cost per unit time) of the system is minimized. The explicit expression of the average cost rate is derived, the corresponding optimal replacement policy can be determined, the explicit expression of the minimum of the average cost rate can be found and under some mild conditions the existence and uniqueness of the optimal policy T^∗ can be proved, too. Further, we can show that the repair model for the multistate system in this paper forms a general monotone process repair model which includes the geometric process repair model as a special case. We can also show that the repair model in the paper is equivalent to a geometric process repair model for a two-state degenerative simple system in the sense that they have the same average cost rate and the same optimal policy. Finally, a numerical example is given to illustrate the theoretical results of this model.     

Optimal periodic replacement policy for repairable products under free-repair warranty

This paper investigates the effects of a free-repair warranty on the periodic replacement policy for a repairable product. Cost models are developed for both a warranted and a non-warranted product, and the corresponding optimal periodic replacement policies are derived such that the long-run expected cost rate is minimized. For a product with an increasing failure rate function, structural properties of these optimal policies are obtained. By comparing these optimal policies, we show that the optimal replacement period for a warranted product should be adjusted toward the end of the warranty period. Finally, examples are given to numerically illustrate the impact of a product warranty on the optimal periodic replacement policy.     

Optimal replacement strategies for repairable systems

This paper considers repair-replacement models introduced by Lam Yeh [6] and [7], and Stadje and Zuckerman [9]. Without imposing reliability theory conditions on the repair and operating distributions, the optimal replacement problem is first solved in a finite horizon setting and then extensions are given to the infinite horizon case.     

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.     

Optimal replacement policy for a deteriorating system with increasing repair times

This paper considers an optimal maintenance policy for a practical and reparable deteriorating system subject to random shocks. Modeling the repair time by a geometric process and the failure mechanism by a generalized δ-shock process, we develop an explicit expression of the long-term average cost per time unit for the system under a threshold-type replacement policy. Based on this average cost function, we propose a finite search algorithm to locate the optimal replacement policy N^∗ to minimize the average cost rate. We further prove that the optimal policy N^∗ is unique and present some numerical examples. Many practical systems fit the model developed in this paper.     

Optimal cost and policy for a Markovian replacement problem

We consider the computation of the optimal cost and policy associated with a two-dimensional Markov replacement problem with partial observations, for two special cases of observation quality. Relying on structural results available for the optimal policy associated with these two particular models, we show that, in both cases, the infinitehorizon, optimal discounted cost function is piecewise linear, and provide formulas for computing the cost and the policy. Several examples illustrate the usefulness of the results.This research was supported by the Air Force Office of Scientific Research Grant AFOSR-86-0029, by the National Science Foundation Grant ECS-86-17860, by the Advanced Technology Program of the State of Texas, and by the Air Force Office of Scientific Research (AFSC) Contract F49620-89-C-0044.     

A discussion on “A bivariate optimal replacement policy for a repairable system”

In this short paper, a bivariate optimal replacement policy for a repairable system with a geometric process maintenance model is discussed. Zhang [Zhang, Y.L., 1994. A bivariate optimal replacement policy for a repairable system. Journal of Applied Probability 31, 1123–1127] and Sheu [Sheu, S.H., 1999. Extended optimal replacement model for deteriorating systems. European Journal of Operational Research 112, 503–516] obtained different expression of the long-run average cost per unit time (i.e. average cost rate) of the system respectively. We show that both of their results are correct, therefore, Sheu’s comment (1999) on the result of Zhang (1994) is wrong.     

A sequential method for preventive maintenance and replacement of a repairable single-unit system

This paper is concerned with when to implement preventive maintenance (PM) and replacement for a repairable ‘single-unit’ system in use. Under the main assumption that a ‘single-unit’ system gradually deteriorates with time, a sequential method is proposed to determine an optimal PM and replacement strategy for the system based on minimising expected loss rate. According to this method, PM epochs are determined one after the other, and consequently we can make use of all previous information on the operation process of the system. Also the replacement epoch depends on the effective age of the system. A numerical example shows that the sequential method can be used to solve the PM and replacement problem of a ‘single-unit’ system efficiently. Some properties of the loss functions W(L? _n ,b? _n ) and W? _r (N) with respect to PM and replacement respectively are discussed in the appendix.     

Optimal policy for a general repair replacement model: average reward case

For a general repair replacement model, we study two types ofreplacement policy.Replacement policy T replaces the systemat time T since the installation or last replacement, whilereplacement policy N replaces the system at the time of Nthfailure. Let T^* and N^* be the optimal among all policies T andN respectively. Under the expected average reward criterion,then we show that the optimal policy N^* is at least as goodas the optimal policy T^*. Furthermore, for a monotone processmodel, we determine the optimal policy N^* explicitly throughtwo different approaches.     

Optimal two-layer selection policy for a flexible manufacturing system

This paper deals with the selection problem in a manufacturing system. The manufacturing system consists of a flexible manufacturing cellC ₀ which feedsM flexible manufacturing cellsC _1i ,i=1, ...,M. In addition, each cellC _1i ,i=1, ...,M, is feeding several production lines. Sufficient conditions on optimal feedback selection policies for theM+1 flexible manufacturing cells are derived. These selection policies maximize the probability of the system output reaching some demand before any of the system buffers is being overflowed. A numerical study is conducted.     

Optimal control policy for a standing order inventory system

In this paper, we consider a standing order inventory system in which an order of fixed size arrives in each period. Since demand is stochastic, such a system must allow for procurement of extra units in the case of an emergency and sell-offs of excess inventory. Assuming the average-cost criterion, Rosenshine and Obee (Operations Research 24 (1976) 1143–1155) first studied such a system and devised a 4-parameter inventory control policy that is not generally optimal. The current paper uses dynamic programming to determine the optimal control policy for a standing order system, which consists of only two operational parameters: the dispose-down-to level and order-up-to level. Either the average-cost or discounted-cost criterion can be assumed in the proposed model. Also, both the backlogged and lost-sales problems are investigated in this paper. By using a convergence theorem, we stop the dynamic programming computation and obtain the two optimal parameters.     

Optimal replacement policy for obsolete components with general failure rates

Identical components are considered, which become obsolete once new‐type ones are available, more reliable and less energy consuming. We envision different possible replacement strategies for the old‐type components by the new‐type ones: either purely preventive, where all old‐type components are replaced as soon as the new‐type ones are available; either purely corrective, where the old‐type ones are replaced by new‐type ones only at failure; or a mixture of both strategies, where the old‐type ones are first replaced at failure by new‐type ones and next simultaneously preventively replaced after a fixed number of failed old‐type components. To evaluate the respective value of each possible strategy, a cost function is considered, which represents the mean total cost on some finite time interval [0, t]. This function takes into account replacement costs, with economical dependence between simultaneous replacements, and also some energy consumption (and/or production) cost, with a constant rate per unit time. A full analytical expression is provided for the cost function induced by each possible replacement strategy. The optimal strategy is derived in long‐time run. Numerical experiments conclude the paper. Copyright © 2008 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.     

An extended replacement policy for a deteriorating system with multi-failure modes

In this paper, the maintenance problem for a deteriorating system with k + 1 failure modes, including an unrepairable failure (catastrophic failure) mode and k repairable failure (non-catastrophic failure) modes, is studied. Assume that the system after repair is not “as good as new” and its deterioration is stochastic. Under these assumptions, an extended replacement policy N is considered: the system will be replaced whenever the number of repairable failures reaches N or the unrepairable failure occurs, whichever occurs first. Our purpose is to determine an optimal extended policy N^∗ such that the average cost rate (i.e. the long-run average cost per unit time) of the system is minimized. The explicit expression of the average cost rate is derived, and the corresponding optimal extended policy N^∗ can be determined analytically or numerically. Finally, a numerical example is given to illustrate some theoretical results of the repair model proposed in this paper.     

Optimizing replacement policy for a cold-standby system with waiting repair times

This paper presents the formulas of the expected long-run cost per unit time for a cold-standby system composed of two identical components with perfect switching. When a component fails, a repairman will be called in to bring the component back to a certain working state. The time to repair is composed of two different time periods: waiting time and real repair time. The waiting time starts from the failure of a component to the start of repair, and the real repair time is the time between the start to repair and the completion of the repair. We also assume that the time to repair can either include only real repair time with a probability p, or include both waiting and real repair times with a probability 1 − p. Special cases are discussed when both working times and real repair times are assumed to be geometric processes, and the waiting time is assumed to be a renewal process. The expected long-run cost per unit time is derived and a numerical example is given to demonstrate the usefulness of the derived expression.     

Optimal spare ordering policy for preventive replacement under cost effectiveness criterion

This paper presents a spare ordering policy for preventive replacement with age-dependent minimal repair and salvage value consideration. The spare unit for replacement is available only by order and the lead-time for delivering the spare due to regular or expedited ordering follows general distributions. To analyze the ordering policy, the failure process is modelled by a non-homogeneous Poisson process. By introducing the costs due to ordering, repairs, replacements and downtime, as well as the salvage value of an un-failed system, the expected cost effectiveness in the long run are derived as a criterion of optimality. It is shown, under certain conditions, there exists a finite and unique optimum ordering time which maximizes the expected cost effectiveness. Finally, numerical examples are given for illustration.     

Optimal replacement policy based on maximum repair time for a random shock and wear model

We study a \(\delta \) shock and wear model in which the system can fail due to the frequency of the shocks caused by external conditions, or aging and accumulated wear caused by intrinsic factors. The external shocks occur according to a Bernoulli process, i.e., the inter-arrival times between two consecutive shocks follow a geometric distribution. Once the system fails, it can be repaired immediately. If the system is not repairable in a pre-specific time D, it can be replaced by a new one to avoid the unnecessary expanses on repair. On the other hand, the system can also be replaced whenever its number of repairs exceeds N. Given that infinite operating and repair times are not commonly encountered in practical situations, both of these two random variables are supposed to obey general discrete distribution with finite support. Replacing the finite support renewal distributions with appropriate phase-type (PH) distributions and using the closure property associated with PH distribution, we formulate the maximum repair time replacement policy and obtain analytically the long-run average cost rate. Meanwhile, the optimal replacement policy is also numerically determined by implementing a two-dimensional-search process.