Optimal two-threshold replacement policy in a cumulative damage 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 bivariate replacement policy (n, T) for a cumulative shock damage process is presented that included the concept of cumulative repair cost limit. The arrival shocks can be divided into two kinds of shocks. Each type-I shock causes a random amount of damage and these damages are additive. When the total damage exceeds a failure level, the system goes into serious failure. Type-II shock causes the system into minor failure and such a failure can be corrected by minimal repair. When a minor failure occurs, the repair cost will be evaluated and minimal repair is executed if the accumulated repair cost is less than a predetermined limit L. The system is replaced at scheduled time T, at n-th minor failure, or at serious failure. The long-term expected cost per unit time is derived using the expected costs as the optimality criterion. The minimum-cost policy is derived, and existence and uniqueness of the optimal \( n^{*} \) and \( T^{*} \) are proved. This bivariate optimal replacement policy (n, T) is showed to be better than the optimal \( T^{*} \) and the optimal \( n^{*} \) policy.

Optimal replacement policy with replacement last under cumulative damage models

A (τ, T) replacement policy is commonly used to prevent system failure. In previous research, it was assumed that the maintenance time can be ignored since it is negligibly short compared with operation time. However, neglecting maintenance time when optimizing maintenance policies for systems for which the maintenance time is not relatively short can lead to sub-optimal or even incorrect solutions to the problem, resulting in higher maintenance costs. An optimal (τ, T) replacement policy is presented for series systems in which the units deteriorate following different stochastic processes independently. The failure of each unit in the time interval [τ, T) occurs in accordance with a nonhomogeneous Poisson process, the failure is immediately removed by a minimal repair and the system continues operating. The failure of any unit in time interval [τ, T) causes the entire system to fail, which is immediately detected, and corrective maintenance is performed on the whole system. If a failure does not occur during time interval [τ, T), the system receives preventive replacement at T A (τ, T) replacement policy was formulated that considers both preventive and corrective maintenance times and that determines an optimal preventive replacement interval that minimizes the expected long-run cost per unit time. A numerical example illustrates that implementing the proposed (τ, T) replacement policy can effectively reduce maintenance costs.

In reality, most operating systems are repaired or replaced when they have failed. For such systems, it would be reliable to maintain them in a control-limit consideration of repairs or operations. This article addresses a system scheduling policy for preventive replacement with discrete control limits of minimal repair and random work. An operating system which works at random times for multiple jobs (N tandem jobs or N parallel jobs) is considered. The system subjects to two failure mechanisms and maintained imperfectly: type-I (minor) failure rectified by a minimal repair and type-II (catastrophic) failure removed by a corrective replacement. To control the deterioration process, the preventive replacement is planned to undergo before catastrophic failure at number n of minimal repairs or the completion of N multiple jobs. The objective is to determine the optimal discrete scheduling replacement parameters (n* or N*) that minimizes the mean cost rate function in a finite time horizon. Model analysis relating to the existence and uniqueness of optimal solutions is provided. A Numerical example is presented for illustrative purposes.

WITHDRAWN: A trivariate optimal replacement policy for a deteriorating system based on cumulative damage and inspections

Optimal replacement policy under a general failure and repair model: Minimal versus worse than old repair

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*).

Structural properties of optimal tool replacement policy in a machining center

This paper proposes a bivariate (T, L2) replacement policy with double repair-cost limits for a system subject to increasing failure rate model. External shocks are divided into two types: type-I shock and type-II shock. Each type-II shock makes the system suffer a minor failure, while each type-I shock increases the failure rate of the system by a certain amount and it will induce the system into critical failure. When a minor failure occurs, the repair cost is evaluated and minimal repair is executed if the repair cost is less than a single repair-cost limit L1 and the accumulated repair cost is less than a cumulative repair-cost limit L2, otherwise, the system is replaced by a new one. In addition, the system is replaced at scheduled time T or at critical failure. By formulating the optimisation problem, the optimal T* and L2* which minimise the long-run expected cost per unit time are found, respectively. We develop the corresponding computational algorithm to obtain the optimal replacement policy and present a numerical example to illustrate the effectiveness of the proposed model.

A general lost-sales Economic Manufacturing Quality (EMQ) model with machine breakdowns, minimal repairs and planned replacements is considered. When the machine fails, minimal repair is performed in negligible time and production is resumed immediately. It is assumed that resumption cost is lower than the set-up cost incurred at the beginning of each production cycle. Other cost components include inventory holding cost, operating cost, lost-sales cost and the preventive replacement (overhaul) cost. Both time to failure and the time to perform an overhaul are assumed to be generally distributed random variables. The objective is to find the optimal lot-sizing/ preventive replacement policy minimising the long-run expected average cost per unit time. The problem is formulated in the framework of semi-Markov decision processes. It is shown that the optimal lot size is a function of the operating age of the machine and the optimal preventive replacement is an age replacement. An EMQ model with lost sales is studied under the assumption of a constant production lot size. It is shown that a good approximation to the optimal policy can be found by solving two nonlinear equations.

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.

Condition-based maintenance using the inverse Gaussian degradation model

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.

Optimal replacement policy for a deteriorating system with increasing repair times