Optimal replacement policy for a deteriorating system with increasing repair times

In this paper, a cold standby repairable system consisting of two dissimilar components and one repairman is studied. When failures occur, the repair of both component 1 and component 2 are not ‘as good as new’. The consecutive operating times of component 1 after repair constitute a decreasing geometric process, while the repair times of component 1 are independent and identically distributed. For component 2, its failure is rectified by minimal repair, and the repair time is negligible. Component 1 has priority in use when both components are good. The replacement policy N is based on the failure number of component 1. Under policy N, we derive the explicit expression of the long-run average cost rate C(N) as well as the average number of repairs of component 2 before the system replaced. The optimal replacement policy N*, which minimises the long-run average cost rate C(N), is obtained theoretically. If the failure rate r(t) of component 2 is increasing, the existence and uniqueness of the optimal policy N* is also proved. Finally, a numerical example is given to validate the developed theoretical model. Some sensitivity analyses are provided to show the influence of some parameters, such as the costs for replacement and repair, and the parameters of the lifetime and repair time distributions of both components, to the optimal replacement policy N* and corresponding average cost rate C(N*).

ABSTRACTThis paper proposes an extended geometric process repair model (EGPRM) to generalise the geometric process repair model (GPRM), and studies a repair–replacement problem for a cold standby repairable system consisting of two dissimilar components and one repairman. Assume that each component after repair in the system is not always successive degenerative, and the failure component in the system is not always immediately repaired. Under these assumptions, using the extended geometric process, we consider a replacement policy N based on the failure number of Component 2 under which the system is replaced when the failure number of Component 2 reaches N. Our problem is to choose an optimal replacement policy N* such that the average cost rate (i.e. the long-run average cost per unit time) of the system is minimised. The explicit expression of the average cost rate of the system is derived, and the corresponding optimal replacement policy N* can be determined by minimising the average cost rate of the system. Finally, a numerical example is given to illustrate some theoretical results and the model applicability.

A geometrical process repair model for a repairable system with delayed repair

In this paper, a cold standby repairable system consisting of two identical components and one repairman is studied. Assume that each component after repair is not “as good as new”, by using a geometric process, we consider two kinds of repair replacement policy, one based on the working age T of component 1 under which the system is replaced when the working age of component 1 reaches T, and the other based on the failure number N of component 1 under which the system is replaced when the failure number of component 1 reaches N. Our problem is to choose optimal replacement policies T* and N* respectively such that the long-run average cost per unit time of the system is minimized. And we can prove under some mild conditions that the optimal policy N* is better than the optimal policy T*. Finally, a numerical example for policy N is given.

Condition-based maintenance using the inverse Gaussian degradation model

This paper investigates the effects of unsuccessful repairs on the optimal periodic replacement policy for a repairable product with multiple attempts of minimal repair under the free-repair warranty. The motivation for this study is based on the novel notion of ‘multi-attempt minimal repair’, first introduced by Cha and Finkelstein (2024a, 2024b), in which they highlighted various practical reasons and examples to illustrate that minimal repair attempts may sometimes fail and need to be repeated until success is achieved. Their repair model and useful properties served as the motivation for this study. For both the warranted and non-warranted products, cost models are developed from the user’s perspective, and the corresponding optimal replacement policies are derived to minimize the long-run expected cost rates. By comparing these optimal policies, we show that the optimal planned time for preventive replacement should be adjusted toward the end of the warranty period, and the effects of unsuccessful minimal repair produce different degrees of impact, respectively, on the warranted and non-warranted product. Finally, detailed numerical examples and computation results illustrate our findings, and some insightful observations about practical and managerial implications are concluded.

In this paper, a two-unit system with failure interactions is studied. The system is subject to two types of shocks (I and II) and the probabilities of these two shock types are age-dependent. A type I shock, rectified by a minimal repair, causes a minor failure of unit A and type II shock causes a complete system failure that calls for a replacement. Each unit A minor failure also results in an amount of damage to unit B. The damages to unit B caused by type I shocks can be accumulated to trigger a preventive replacement or a corrective replacement action. Besides, unit B with cumulative damage of level z may become minor failed with probability \(\pi (z)\) at each unit A minor failure and rectified by a minimal repair. We consider a more general replacement policy. Under this policy, the system is preventively replaced at the Nth type I shock, or at the time when the total damage to unit B exceeds a pre-specified level Z (but less than a failure level K where \(K>Z\)) or is replaced correctively at first type II shock or when the total damage to unit B exceeding a failure level K, whichever occurs first. To minimize the expected cost per unit time, the optimal replacement policy \((N^{*}\), \(Z^{*})\) is derived analytically and determined numerically. We also show that several previous maintenance models in the literature are special cases of our model.

Optimal replacement policy with replacement last under cumulative damage models

Structural properties of optimal tool replacement policy in a machining center

It is well-known in the standard age replacement policy that a finite preventive replacement time does not exist when the failure time is exponential and the optimal preventive replacement time is nonrandom. It is shown that when the failure time is exponential, a finite time exists by introducing the shortage and excess costs. In addition, the random age replacement is proposed and similar discussions are made. Furthermore, the periodic and random inspection policies are taken up, and their optimal policies are shown to correspond theoretically to those of the age replacement ones. It is shown finally that when the random inspection cost is the half of the periodic one, two expected costs are almost the same.

The conception rate in dairy cows is dependent on a number of cow factors such as days in milk and insemination number. Unfortunately, some of these factors were not accounted for in optimal insemination and replacement decision models. By using wrong estimates of the conception rate, the calculated optimal insemination and replacement policy might differ from the real optimal insemination and replacement policy. The objective of this study was to evaluate different sets of conception rates with an increasing level of accuracy to determine the insemination policy. An existing dynamic program for optimal insemination and replacement was used to compare three different scenarios in the estimation of conception rates, based on the reproductive performance of Dutch dairy cattle: (i) constant conception rate throughout lactation, (ii) conception rate dependent on parity and months in milk, and (iii) conception rate dependent on parity, months in milk and insemination number. The time step of the model was 30.4 days (1 month). The discounted future cash flow of culling a cow at each time step (replace a heifer immediately) was compared with keeping that cow under optimal future decisions. The difference between immediate culling and optimal decisions is defined as the retention pay-off. The insemination value was calculated as the difference between the future cash flow between immediate insemination of a cow and waiting one time-step. The results show that the difference in the insemination values and the optimal time to stop insemination depend on parity, lactation stage and the relative milk yield. In older cows with equal milk yields and at the same months in milk, the insemination value was lower than in younger cows. Within a parity, the insemination value was higher for cows with a higher milk yield. On individual cow level, using more accurate conception rate as input in the optimal insemination and replacement model might reduce miscalculation of the economic consequences for at least of €20-€38 per cow per year. Basing insemination decisions on less accurate input of the probabilities of conception, however, did not have an economic consequence at the herd level. In conclusion, using the appropriate conception rate as input in the optimal insemination and replacement model would increase the precise decision for the optimal time to stop insemination and hence improve the reproductive management efficacy.

This paper deals with continuous-time Markov decision processes in Polish spaces, under the discounted and average cost criteria. All underlying Markov processes are determined by given transition rates which are allowed to be unbounded, and the costs are assumed to be bounded below. By introducing an occupation measure of a randomized Markov policy and analyzing properties of occupation measures, we first show that the family of all randomized stationary policies is ‘sufficient’ within the class of all randomized Markov policies. Then, under the semicontinuity and compactness conditions, we prove the existence of a discounted cost optimal stationary policy by providing a value iteration technique. Moreover, by developing a new average cost, minimum nonnegative solution method, we prove the existence of an average cost optimal stationary policy under some reasonably mild conditions. Finally, we use some examples to illustrate applications of our results. Except that the costs are assumed to be bounded below, the conditions for the existence of discounted cost (or average cost) optimal policies are much weaker than those in the previous literature, and the minimum nonnegative solution approach is new.

Burn-in is a widely used engineering method which is adopted to eliminate defective items before they are shipped to customers or put into field operation. In order to shorten the burn-in process, burn-in is most often accomplished in an accelerated environment. However, there have been few probabilistic or stochastic models for the burn-in procedures in accelerated environment. In this article, under a new stochastic model for accelerated burn-in procedure, the problems of determining both optimal accelerated burn-in time and optimal replacement policy are considered. Components are burned-in under an accelerated environment, then those surviving the burn-in procedure are put into field operation and they are maintained under a replacement policy. The properties of the optimal accelerated burn-in time and optimal replacement policy are obtained and a numerical example which illustrates the usage of obtained results will be presented.

In this paper, a combined shock maintenance model is proposed for the deteriorating system, and the imperfect maintenance decision based on the replacement policy N is studied. This approach assumes that the system has two kinds of failure conditions, and the combined shock model is established by combining with the δ-shock model and accumulative shock model. The optimal replacement policy N is determined by using this combined model, and the average cost in life cycle is minimized. Finally, a numerical case study is provided to validate the feasibility of the proposed method.

Caching web pages at proxies and in web servers' memories can greatly enhance performance. Proxy caching is known to reduce network and both proxy and server caching can significantly decrease latency. Web caching problems have different properties than traditional operating systems caching, and replacement can benefit by recognizing and exploiting these differences. We address two aspects of the predictability of traffic patterns: the overall experienced by large proxy and web servers, and the distinct access patterns of individual pages. We formalize the notion of cache load under various replacement policies, includingLRU andLFU, and demonstrate that the trace of a large proxy server exhibits regular load. Predictable allows for improved design, analysis, and experimental evaluation of replacement policies. We provide a simple and (near) optimal replacement policy when each page request has an associated distribution function on the next request time of the page. Without the predictable assumption, no such online policy is possible and it is known that even obtaining an offline optimum is hard. For experiments, predictable enables comparing and evaluating replacement policies using partial traces, containing requests made to only a subset of the pages. Our results are based on considering a simpler caching model which we call the interval caching model .W e relate traditional and interval caching policies under predictable load, and derive (near)-optimal replacement policies from their optimal interval caching counterparts.