Optimal State-Dependent Replacement Policy for Deteriorating Systems

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.

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.

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.

Determining the optimum replacement policy for Holstein dairy herds in Iran

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*).

Age replacement policy is a commonly policy in maintenance management of spare part. It means that a spare part is always replaced at failure or fixed time after its installation, whichever occurs first. An optimal age replacement policy of spare parts concerns with finding the optimal replacement time determined by minimizing the expected cost per unit time. The age of the part was generally assumed to be a random variable in the past literatures, but in many situations, there are few or even no observed data to estimate the probability distribution of part's lifetime. In order to solve this phenomenon, a new uncertain age replacement policy has been proposed recently, in which the age of the part was assumed to be an uncertain variable. This paper discusses the optimal age replacement policies by dealing with the parts' lifetimes as different distributed uncertain variables. Several results on the optimal age replacement time are provided when the lifetimes are described by the uncertain linear, zigzag and lognormal distributions.

Condition-based maintenance using the inverse Gaussian degradation model

Optimal replacement policy for a deteriorating system with increasing repair times

Structural properties of optimal tool replacement policy in a machining center

Optimal replacement policy for the case where the damage process is a one-sided Lévy process

This paper presents a comparative study of reliability analysis of a gas production plant, before and after applying optimal age replacement policy for a specific machine. The gas production plant has six machines connected in the series configuration a sulfur (S) shaft, coal pit, reaction furnace, sulfur separation machine, adiabatic reactor, and cooling machine. Here, the optimal age replacement policy has been applied to the cooling machine as this machine is working under a wear-out period, i.e., it has completed its prescribed age (say T) and is still operational. Two different situations have been discussed to analyze the performance variation of the system (i) when the model is working with high risk because the cooling machine is operational even after completion of its prescribed age (ii) when the old cooling machine is replaced by a new one after completing the age, T. The failure of any machine can cause the model failure. Additionally, a situation of employee walkout is also considered a responsible factor for model failure. Supplementary variable technique and copula methodology have been applied to solve the Markov model.

In this paper, we consider an interval estimation of the age replacement problem, where the underlying failure time probability is given by the two-parameter Weibull distribution. The parametric bootstrap is used to obtain the probability distribution of an estimator for the optimal age replacement time minimizing the expected cost per unit time in the steady state. We focus on both cases with complete and incomplete samples of failure time data, and calculate not only the higher moments of an estimator for the optimal age replacement time but also the confidence interval.

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.

The curve of averaged prolificity was estimated in five populations from a sample of 53 commercial farms (24 Spanish, 19 Danish and 10 Cuban). The aim of the paper is to compare the curves of prolificity and analyze the impact on the optimal replacement policy of sows. The parameters of prolificity curves showed propusiedifferences among the five populations (P<0.05). As a result, an association between litter size in the first parity and maximum litter size with higher prolificity was observed. However, the slope described by the steady decline of prolificity with age did not show differences among populations. Later on, we analyzed the impact of prolificity on optimal replacement policies. For this purpose, production dynamics of five farms were simulated using identical parameters except on their prolificity. The prolificity in each simulated farm corresponded to the five estimates from previous populations. For each case, the optimal policy was calculated and corresponding herd structure, technical and economic indexes were analyzed. Indexes associated to each optimal policy revealed differences among farms (P<0.05). Hence, while the reproductive rhythm was similar, the longevity and herd structure were different. Longevity was shorter with higher prolificity and vice-versa. In fact, farms with higher prolificity concentrate more sows between 2 and 7 parities and without increasing culling rates. It was also demonstrated that net incomes in analyzed farms depended strongly on the number of piglets weaned per sow per year and that lower prolificity lead to later replacements in direct relationship with the amortization of sows

An integer linear programming model was developed to establish the optimal replacement strategy for a commercial dairy herd. Monthly revenues and costs for cows were computed based on factors such as milk production, calf and carcass values, feed expenses, replacement heifer costs, insemination charges, and veterinary expenses. The model characterized cows using five parity numbers, two decision options (retain or replace), a 5-year planning horizon, and ten cow categories. It accounted for variations in milk yield, milk prices, calf and carcass values, feed costs, and replacement expenses. The parameters and prices used were tailored to reflect the Holstein-Friesian population in a commercial dairy herd in Egypt. The optimal annual replacement rate was determined to be approximately 30%, with an average productive life of 3.33 years. The maximum annual net revenue, expressed as a yearly annuity over the 5-year planning horizon, was estimated at 1,055 LE per cow. Sensitivity analyses were conducted to assess the impact of variations in average milk yield and the prices of milk, calves, carcasses, feed, and replacement heifers on the optimal replacement decisions. Profitability declined as the replacement rate increased from 10 to 50%. Productive life, replacement rate, and annual net present value were most influenced by changes in average milk yield and the prices of milk, feed, and replacement heifers. In contrast, variations in calf and carcass prices had relatively minor effects on the optimal replacement policy.