Optimal repair allocation in a series system expected discounted operation time criterion

Consider a series system with n repairable components maintained by a single repairman. The following assumptions are made. Component failure and repair times are independent, exponentially distributed, random variables. Component failures can occur even while the system is not functioning and it is possible to reassign the repairman among failed components instantaneously. It is shown that the policy which always assigns the repairman to the failed component with the smallest failure rate among the failed ones maximizes the expected discounted system operation time irrespective of the values of the repair rates and the discount rate     

带有多重休假及预防维修的冲击模型二维更换策略

研究了修理工带有多重休假且定期检测的累积冲击模型.为了延长系统的运行时间,在检测时考虑了预防维修.将事后维修和预防维修结合起来运用于可修系统,且假定预防维修能够"修复如新",而事后维修为"修复非新".以系统的检测周期和故障次数为二维决策变量,选取系统经长期运行单位时间内期望费用为目标函数.并通过数值分析,求出了最优策略.     

有两种故障状态的两部件串联系统的预防维修策略

本文研究的是由两个部件串联组成且有两种故障状态的系统的预防维修策略, 当系统的工作时间达到T时进行预防维修, 预防维修使部件恢复到上一次故障维修后的状态。每个部件发生故障都有两种状态, 可维修和不可维修。当部件的故障为可维修故障时, 修理工对其进行故障维修, 且每次故障维修后的工作时间形成随机递减的几何过程, 每次故障后的维修时间形成随机递增的几何过程。当部件发生N次可维修故障或一次不可维修故障时进行更换。以部件进行预防维修的间隔和更换前的可维修故障次数N组成的二维策略(T, N) 为策略, 利用更新过程和几何过程理论求出了系统经长期运行单位时间内期望费用的表达式, 并给出了具体例子和数值分析。     

A deteriorating cold standby repairable system with priority in use

In this paper, a cold standby repairable system consisting of two dissimilar components and one repairman is studied. In this system, it is assumed that the working time distributions and the repair time distributions of the two components are both exponential and component 1 is given priority in use. After repair, component 2 is “as good as new” while component 1 follows a geometric process repair. Under these assumptions, using the geometric process and a supplementary variable technique, some important reliability indices such as the system availability, reliability, mean time to first failure (MTTFF), rate of occurrence of failure (ROCOF) and the idle probability of the repairman are derived. A numerical example for the system reliability R(t) is given. And it is considered that a repair-replacement policy based on the working age T of component 1 under which the system is replaced when the working age of component 1 reaches T. Our problem is to determine an optimal policy T^∗ such that the long-run average cost per unit time of the system is minimized. The explicit expression for the long-run average cost per unit time of the system is evaluated, and the corresponding optimal replacement policy T^∗ can be found analytically or numerically. Another numerical example for replacement model is also given.     

An optimal repair–replacement policy for a cold standby system with use priority

In this paper, the maintenance problem for a cold standby system consisting of two dissimilar components and one repairman is studied. Assume that both component 1 and component 2 after repair follow geometric process repair and component 1 is given priority in use when both components are workable. Under these assumptions, using geometric process repair model, we consider a replacement policy N under which the system is replaced when the number of failures of component 1 reaches N. Our purpose is to determine an optimal replacement policy N1 such that the average cost rate (i.e. the long-run average cost per unit time) of the system is minimized. The explicit expression for the average cost rate of the system is derived and the corresponding optimal replacement policy N1 can be determined analytically or numerically. Finally, a numerical example is given to illustrate some theoretical results and the model applicability.     

Optimal preventive maintenance for equipment with two quality states and general failure time distributions

We present an economic model for the optimization of preventive maintenance in a production process with two quality states. The equipment starts its operation in the in-control state but it may shift to the out-of-control state before failure or scheduled preventive maintenance. The time of shift and the time of failure are generally distributed random variables. The two states are characterized by different failure rates and revenues. We first derive the structure of the optimal maintenance policy, which is defined by two critical values of the equipment age that determine when to perform preventive maintenance depending on the actual (observable) state of the process. We then provide properties of the optimal solution and show how to determine the optimal values of the two critical maintenance times accurately and efficiently. The proposed model and, in particular, the behavior of the optimal solution as the model parameters and the shift and failure time distributions change are illustrated through numerical examples.     

修理工可单重休假的预防维修更换策略

本文研究了一个修理工带有单重休假的单部件可修系统.为了延长系统的使用寿命,在系统故障前考虑了预防维修,且假定预防维修能够“修复如新”,而故障维修为“修复非新”时,以系统的故障次数N为更换策略.通过更新过程和几何过程理论,得出系统经长期运行单位时间内期望费用的明显表达式,并对预防维修的定长间隔时间T及更换策略N进行了讨论,最后,通过实例分析,求出最优策略N’,使得目标函数取得最优值.     

A phase-type geometric process repair model with spare device procurement and repairman’s multiple vacations

This paper analyzes a phase-type geometric process repair model with spare device procurement lead time and repairman’s multiple vacations. The repairman may mean here the human beings who are used to repair the failed device. When the device functions smoothly, the repairman leaves the system for a vacation, the duration of which is an exponentially distributed random variable. In vacation period, the repairman can perform other secondary jobs to make some extra profits for the system. The lifetimes and the repair times of the device are governed by phase-type distributions (PH distributions), and the condition of device following repair is not “as good as new”. After a prefixed number of repairs, the device is replaced by a new and identical one. The spare device for replacement is available only by an order and the procurement lead time for delivering the spare device also follows a PH distribution. Under these assumptions, the vector-valued Markov process governing the system is constructed, and several important performance measures are studied in transient and stationary regimes. Furthermore, employing the standard results in renewal reward process, the explicit expression of the long-run average profit rate for the system is derived. Meanwhile, the optimal maintenance policy is also numerically determined.     

Two-stage generalized age maintenance of a queue-like production system

In general, the initiation of preventive maintenance should be based on the technical state as well as the operating state of a production system. Since the operating state of a production system is often subject to fluctuations in time, the planning of preventive maintenance at preset points in time (e.g. age/block replacement) cannot be optimal. Therefore, we propose a so-called two-stage maintenance policy, which - in a first stage - uses the technical state of the production system to determine a finite interval [t, t + At] during which preventive maintenance must be carried out, and - in a second stage - uses the operating state of the production system to determine the optimal starting time t̂ for preventive maintenance within that interval. A generalized age maintenance policy optimizing both t and At is formulated in the first stage. To this end, the actual starting time of preventive maintenance is modelled in terms of a uniform distribution over the maintenance interval. Moreover, the expected costs of preventive maintenance are modelled as a decreasing function of the interval size. An efficient algorithm is developed to demonstrate the optimal strategy for a queue-like production system, via numerical results that offer useful insights.     

系统最佳维修策略研究

一个复杂系统通常由多个不同部件组成，考虑到这些部件有各自不同的失效率及维修时间，本提出了一种新的维修策略模型，该模型考虑了不同部件的差异性及对系统的不同重要性，在一定可用度要求下，使系统总平均费用达到最小的最佳预防维修周期，并给出了相应的仿真算法。     

Poisson冲击下的$k/n(G)$系统的可靠性分析

本文研究了一类Poisson冲击下的$k/n(G)$系统(即$k$-out-of-$n$: $G$系统). 假定冲击的到达数形成一个参数为$\lambda$的Poisson过程, 且冲击的量服从某一分布. 当每次冲击到达时, 对系统中工作的部件独立地产生影响. 进而假定每一部件以一定的概率故障, 概率值是冲击量的函数. 且各次冲击独立地对系统造成损失, 直到工作部件数少于$k$系统故障为止. 在这些假定下, 我们获得了系统的可靠度函数和系统的平均工作时间. 进一步, 假定系统是可修的, 系统中有一个维修工, 并根据``先坏先修’’的维修规则对故障部件进行维修. 在维修时间服从指数分布的假设下, 系统状态转移服从Markov过程. 对该系统我们建立了状态转移方程, 并求得了系统可用度、稳态下的平均工作时间、平均停工时间和系统失效频率等可靠性指标. 最后, 我们还给出了一个简单例子来演示讨论的模型.     

Optimal maintenance policy in a multistate deteriorating standby system

An optimal maintenance policy for a multistate deteriorating standby system is proposed in this study. Traditionally, a system could only presume two operational states: success or failure, and the maintenance policy is to determine the optimal number of standby components, subject to factors such as maintenance capability, cost of the standby items, etc., so as to minimize the operational cost. This study considers a more general production system in which progressive deterioration is incurred during the operating time, hence resulting in degrading performance. By modeling the system as a multistate deteriorating system, an optimal maintenance policy is obtained by determining the optimal number of standby components required in the system and the optimal state in which the replacement of deteriorating components shall be made.     

A numerical method to approximate optimal production and maintenance plan in a flexible manufacturing system

The simultaneous planning of the production and the maintenance in a flexible manufacturing system is considered in this paper. The manufacturing system is composed of one machine that produces a single product. There is a preventive maintenance plan to reduce the failure rate of the machine. This paper is different from the previous researches in this area in two separate ways. First, the failure rate of the machine is supposed to be a function of its age. Second, we assume that the demand of the manufacturing product is time dependent and its rate depends on the level of advertisement on that product. The objective is to maximize the expected discounted total profit of the firm over an infinite time horizon. In the process of finding a solution to the problem, we first characterize an optimal control by introducing a set of Hamilton–Jacobi–Bellman partial differential equations. Then we realize that under practical assumptions, this set of equations can not be solved analytically. Thus to find a suboptimal control, we approximate the original stochastic optimal control model by a discrete-time deterministic optimal control problem. Then proposing a numerical method to solve the steady state Riccati equation, we approximate a suboptimal solution to the problem.     

Stationary availability of a semi‐Markov system with random maintenance

We consider a reparable system with a finite state space, evolving in time according to a semi‐Markov process. The system is stopped for it to be preventively maintained at random times for a random duration. Our aim is to find the preventive maintenance policy that optimizes the stationary availability, whenever it exists. The computation of the stationary availability is based on the fact that the above maintained system evolves according to a semi‐regenerative process. As for the optimization, we observe on numerical examples that it is possible to limit the study to the maintenance actions that begin at deterministic times. We demonstrate this result in a particular case and we study the deterministic maintenance policies in that case. In particular, we show that, if the initial system has an increasing failure rate, the maintenance actions improve the stationary availability if and only if they are not too long on the average, compared to the repairs ( a bound for the mean duration of the maintenance actions is provided). On the contrary, if the initial system has a decreasing failure rate, the maintenance policy lowers the stationary availability. A few other cases are studied. Copyright © 2000 John Wiley & Sons, Ltd.     

Optimal maintenance strategy under renewable warranty with repair time threshold

In this paper, we consider a periodic preventive maintenance model, from the manufacturer's perspective, which can be implemented to reduce the maintenance cost of a repairable product during a given warranty period. The product is assumed to deteriorate with age and the warranty policy we adopt in this paper takes into account the two factors of failure time and repair time of the product when the product failure occurs. Under the proposed two-factor warranty, a repair time threshold is pre-determined and if the repair takes more time than that of the threshold, the failed product is replaced with a renewed warranty policy. Otherwise, the product is only minimally repaired to return to the operating state. During such a renewable warranty period, preventive maintenance is conducted to reduce the rate of degradation periodically while the product is in operation. By assuming certain cost structures, we formulate the expected warranty cost during the warranty period from the manufacturer's perspective when a periodic preventive maintenance strategy is adapted. Although more frequent preventive maintenance increases the warranty cost, the chance of product failures would be reduced. The main aim of this paper is to accomplish the optimal trade-off between the warranty cost and the preventive maintenance period by determining the optimal preventive maintenance period that minimizes the total expected warranty cost during the warranty period. Assuming the power law process for the product failures, we illustrate our proposed maintenance model numerically and study the impact of relevant parameters on the optimal preventive maintenance policy.     

Optimum preventive maintenance policies maximizing the mean time to the first system failure for a two-unit standby redundant system

A two-unit standby redundant system with repair and preventive maintenance is considered under the following assumptions: (I) the inspection of an operative unit is made only if the other unit is in standby; and (II) an operative unit, which forfeited inspection due to assumption (I), undergoes inspection just upon repair completion of the failed unit (or inspection completion). We derive the Laplace-Stieltjes transform of the cumulative distribution function of the time to the first system failure and the mean time to the first system failure. Further, we obtain the necessary and sufficient conditions for an optimum preventive maintenance policy to exist with respect to the mean time to the first system failure. More importantly, under certain conditions, we find the analytical form of an optimum inspection time maximizing the mean time to the first system failure. A numerical example is presented.The work reported in this article was supported by the National Institutes of Health under Grant No. GM-16197-05. The authors would like to express their appreciation to Professor D. L. Jaquette and Professor R. Vasudevan, University of Southern California, for their advice and encouragement.     

有延迟修理的两部件串联系统的预防维修策略

研究由两个部件串联组成的系统的预防维修策略, 当系统的工作时间达到T时进行预防维修, 预防维修使部件恢复到上一次故障维修后的状态. 当部件发生故障后进行故障维修, 因为各种原因可能会延迟修理. 部件在每次故障维修后的工作时间形成随机递减的几何过程, 且每次故障后的维修时间形成随机递增的几何过程. 以部件进行预防维修的间隔T和更换前的故障次数N组成的二维策略(T,N)为策略, 利用更新过程和几何过程理论求出了系统经长期运行单位时间内期望费用的表达式, 并给出了具体例子和数值分析.     

A novel replacement policy for a linear deteriorating system using stochastic process with dependent components

In this study, a mechanical system with linear deterioration and preventive maintenance is considered. The state of the system over time is represented by a semicontinuous stochastic process with dependent components. The system cycles through on and off periods during its lifetime. The state of the system deteriorates linearly as a function of the usage time during on periods. When the system is offline, preventive maintenance is conducted, which improves the system state by a random amount. The system's on and off times and random improvement amounts are assumed to have general distributions. For such a system, our objective is to determine the expected value and variance for the number of preventive maintenance activities needed during the system lifetime and to propose a novel replacement policy for the system based on delay‐time modeling. Finally, the effectiveness of the obtained asymptotic results and the proposed replacement policy are tested through simulation.     

A deteriorating system with its repairman having multiple vacations

This paper considers a repairable system with a repairman, who can take multiple vacations. If the system fails and the repairman is on vacation, it will wait for repair until the repairman is available. Assume that the system cannot be repaired “as good as new” after failures. Under these assumptions, using the geometric process and the supplementary variable technique, some important reliability indexes are derived, such as the system reliability, availability, rate of occurrence of failures, etc. According to the renewal reward theorem, the explicit expression of the expected profit per unit time is obtained. Finally, a numerical example is given to illustrate that there exists an optimal replacement policy N∗, which maximizes the value of the expected profit rate after a long time run.