Monotone optimal preventive maintenance policies for stochastically failing equipment

This paper examines various models for maintenance of a machine operating subject to stochastic deterioration. Three alternative models are presented for the deterioration process. For each model, in addition to the replacement decision, the option exists of performing preventive maintenance. The effect of this maintenance is to “slow” the deterioration process. With an appropriate reward structure imposed on the processes, the models are formulated as continuous time Markov decision processes. the optimality criterion being the maximization of expected discounted reward earned over an infinite time horizon. For each model conditions are presented under which the optimal maintenance policy exhibits the following monotonic structure. First, there exists a control limit rule for replacement. That is, there exists a number i* such that if the state of machine deterioration exceeds i* the optimal policy replaces the machine by a new machine. Secondly, prior to replacement the optimal level of preventive maintenance is a nonincreasing function of the state of machine deterioration. The conditions which guarantee this result have a cost/benefit interpretation.     

Application of Markov renewal theory and semi-Markov decision processes in maintenance modeling and optimization of multi-unit systems

In this paper, a condition-based maintenance model for a multi-unit production system is proposed and analyzed using Markov renewal theory. The units of the system are subject to gradual deterioration, and the gradual deterioration process of each unit is described by a three-state continuous time homogeneous Markov chain with two working states and a failure state. The production rate of the system is influenced by the deterioration process and the demand is constant. The states of the units are observable through regular inspections and the decision to perform maintenance depends on the number of units in each state. The objective is to obtain the steady-state characteristics and the formula for the long-run average cost for the controlled system. The optimal policy is obtained using a dynamic programming algorithm. The result is validated using a semi-Markov decision process formulation and the policy iteration algorithm. Moreover, an analytical expression is obtained for the calculation of the mean time to initiate maintenance using the first passage time theory.     

Optimal lot sizing and inspection policy for an EMQ model with imperfect inspections

An EMQ model with a production process subject to random deterioration is considered. The process can be monitored through inspections, and both the lot size and the inspection schedule are subject to control. The “in-control” periods are assumed to be generally distributed and the inspections are imperfect, i.e., the true state of the process is not necessarily revealed through an inspection. The objective is the joint determination of the lot size and the inspection schedule, minimizing the long-run expected average cost per unit time. Both discrete and continuous cases are examined. A dynamic programming formulation is considered in the case where the inspections can be performed only at discrete times, which is typical for the parts industry. In the continuous case, an optimum inspection schedule is obtained for a given production time and given number of inspections by solving a nonlinear programming problem. A two-dimensional search procedure can be used to find the optimal policy. In the exponential case, the structure of the optimal inspection policy is established using Lagrange's method, and it is shown that the optimal inspection times can be found by solving a nonlinear equation. Numerical studies indicate that the optimal policy performs much better than the optimal policy with periodic inspections considered previously in the literature. The case of perfect inspections is discussed, and an extension of the results obtained previously in the literature is presented. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 165–186, 1998     

A basically poisson queue with nonpoisson output

The output of the queueing system M/M/1 is well known to be Poisson. This has also been shown to be true for other more general models inclusive of M/M_n/1; the system in which arrivals and epochs of service completion are elements of a birth and death process with parameters Λ and nμ, respectively, when the system contains n ≥ 1 customers. We shall here show that this result is not true in M_nM/1; a system where arrival parameter is state dependent quantity Λ/n+1. Expressions will be given for the steady state joint density of two consecutive output intervals as well as the coefficient of correlation between them.     

Optimal replacement of parts having observable correlated stages of deterioration

A single component system is assumed to progress through a finite number of increasingly bad levels of deterioration. The system with level i (0 ≤ i ≤ n) starts in state 0 when new, and is definitely replaced upon reaching the worthless state n. It is assumed that the transition times are directly monitored and the admissible class of strategies allows substitution of a new component only at such transition times. The durations in various deterioration levels are dependent random variables with exponential marginal distributions and a particularly convenient joint distribution. Strategies are chosen to maximize the average rewards per unit time. For some reward functions (with the reward rate depending on the state and the duration in this state) the knowledge of previous state duration provides useful information about the rate of deterioration.     

Multi-class inventory models with demand a function of inventory level

This paper considers the problem of maintaining an inventory of an item which can deteriorate and become useless. A periodic review procedure is used and new items ordered may experience a time lag in delivery. Items are considered to deteriorate through one or two states before becoming useless. Thus the deterioration process in each period plays the role of the usual demand process and is a function of the inventory level at the beginning of each period. For the case of no time lag in delivery, one stage deterioration, and either binomial or uniform deterioration, optimal ordering policies are obtained for the n-period dynamic model with the standard cost structure. (For the shortage probability criterion see the other paper by Iglehart and Jaquette, in this issue.) These policies are of the single critical number type. For more complicated models suboptimal policies of this same type are found.     

Inference on a future reliability parameter with the weibull process model

An inferential procedure is presented which provides confidence intervals for a future reliability parameter when reliability growth testing is only partially completed. Hypothesis tests based on this method are uniformly most powerful unbiased. These results are applicable if (1) the system failure rate can be modeled as the intensity function of a Weibull process and (2) efforts to improve reliability are assumed to continue at a steady rate throughout the intervening period of testing. The usefulness of this methodology is illustrated by evaluating the risk of not reaching some future reliability milestone. If such risk is unacceptably high, program management may have time to identify problem areas and take corrective action before testing has ended. As a consequence, a more reliable system may be developed without incurring overruns in the scheduling or cost of the development program.     

Nonperiodic preventive maintenance for repairable systems

As a complex system with multiple components usually deteriorates with age, preventive maintenance (PM) is often performed to keep the system functioning in a good state to prolong its effective age. In this study, a nonhomogeneous Poisson process with a power law failure intensity is used to describe the deterioration of a repairable system, and the optimal nonperiodic PM schedule can be determined to minimize the expected total cost per unit time. However, since the determination of such optimal PM policies may involve numerous uncertainties, which typically make the analyses difficult to perform because of the scarcity of data, a Bayesian decision model, which utilizes all available information effectively, is also proposed for determining the optimal PM strategies. A numerical example with a real failure data set is used to illustrate the effectiveness of the proposed approach. The results show that the optimal schedules derived by Bayesian approach are relatively more conservative than that for non‐Bayesian approach because of the uncertainty of the intensity function, and if the intensity function are updated using the collected data set, which indicates more severe deterioration than the prior belief, replacing the entire system instead of frequent PM activities before serious deterioration is suggested. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

On a new approach to the analysis of stationary inventory problems

The intent of this paper is to demonstrate that the theory of stationary point processes is a useful tool for the analysis of stationary inventory systems. In conventional inventory theory, the equilibrium distributions for a specified inventory policy are obtained, whenever possible, by recursive or limiting procedures, or both. A different and more direct approach, based on stationary point processes, is proposed here. The time instants at which stock delivery is effected are viewed as points of the stationary point process, which possesses uniform statistical properties on the entire real axis; hence the equilibrium statistics of the inventory process can be calculated directly. In order to best illustrate this approach, various examples are given, including some that constitute new results.     

On‐line parameter estimation for a partially observable system subject to random failure

In this paper, we study the on‐line parameter estimation problem for a partially observable system subject to deterioration and random failure. The state of the system evolves according to a continuous time homogeneous Markov process with a finite state space. The system state is not observable, except for the failure state. The information related to the system state is available at discrete times through inspections. A recursive maximum likelihood (RML) algorithm is proposed for the on‐line parameter estimation of the model. The RML algorithm proposed in the paper is considerably faster and easier to apply than other RML algorithms in the literature, because it does not require projection into the constraint domain and calculation of the gradient on the surface of the constraint manifolds. The algorithm is illustrated by an example using real vibration data. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006     

舰艇隐身对反舰导弹末制导 雷达目标截获能力的影响

以法国海军的拉菲特级隐身舰为典型对象,就水面舰艇目前的隐身水平进行了评估,分析和计算了典型反舰导弹末制导雷达对隐身舰艇的检测概率和理想状态下对隐身舰艇的截获概率。指出在反舰导弹的攻击过程中,如果能够保证末制导雷达对目标的多次捕捉,舰艇隐身的效果将被抵销;隐身舰艇利用电子干扰产生假目标将可以抑制末制导雷达对真实目标的捕捉。因此,反舰导弹及其末制导雷达要对付舰艇隐身,其关键不是采用新体制雷达来提高检测能力,而在于提高现体制末制导雷达的目标识别能力。     

Optimal replacement under additive damage in randomly varying environments

A system is subject to a sequence of randomly occurring shocks. Each shock causes a random amount of damage which accumulates additively. Any of the shocks might cause the system to fail. The shock process is in some sense related to an environmental process in order to describe randomly varying external factors of an economical and/or technical nature as well as internal factors of a statistical nature. A discrete time formulation of the problem is given. Sufficient conditions are found for optimality of a generalized control-limit rule with respect to the total cost criterion: Whenever the accumulated damage s is not less than a specified critical number t(i), depending on the environmental state i, replace the system by a new one; otherwise do not replace it. Moreover, bounds are given for these critical numbers.     

A Markovian model for process setup and improvement

We examine the setup and improvement policies for a production process with multiple performance states. Assume that the production process deteriorates randomly over time, following a Markovian process with known transition probabilities. In order to reduce the production cost incurred because of process deterioration, the process is inspected at the end of each period. Then one of three actions may be taken: do nothing, perform routine process setup, or perform routine setup and process improvement. The routine setup operation returns the process to its best performance state, whereas the process improvement action may reduce future production and setup costs and improve the process-state transition probabilities. A discounted Markovian model is formulated to find the strategy that minimizes the total cost of operating the production process. © 1997 John Wiley & Sons, Inc. Naval Research Logistics 44: 383–400, 1997     

海洋拖曳系统运动方程的变步长有限差分数值仿真

有限差分算法是常用的海洋拖曳系统运动仿真计算方法,传统的有限差分算法在对拖缆进行离散时,采用固定空间步长,为了提高数值计算的精度,通常需要减小空间步长,这既增加了计算时间,又浪费了计算机内存,甚至造成无法仿真一些工况。针对这一问题,提出了变步长有限差分数值算法,给出了步长变化的设定原则与算法,在建立海洋拖曳系统运动模型的基础上,系统分析了这一方法的求解思想与过程,并分别采用传统固定步长有限差分算法(大空间步长和小空间步长)和变步长有限差分算法,对模型进行了数值仿真,结果表明:变步长有限差分算法不仅保证了仿真计算精度,而且降低了计算机内存需求量,减少了计算时间。此外,变步长有限差分算法具有较高的灵活性,可根据实际情况,综合考虑计算时间、内存需求量和计算结果精度,从而选择合适的空间步长变化规律。     

Optimal inspection under semi-markovian deterioration: Basic results

This article develops a model for determining the optimal inspection schedule for a system which deteriorates according to a semi-Markov process that progresses through three states: good, defective, and bad. A binary test is used, and false positives may occur. A true positive results in an action that reduces the likelihood of entering the bad state, but at most one such corrective action can occur during the lifetime of the system. Costs are associated with each inspection, each false positive, the corrective action, and the entrance into the bad state. Dynamic programming is used to compute the minimum expected cost, which is a function of the age of the system. The optimal inspection schedule is readily derived from this value function. Computational examples are provided. This model is appropriate for medical screening or for a mission where there is only one spare part.     

Transient analysis of Markov models of fault‐tolerant systems with deferred repair using split regenerative randomization

The (standard) randomization method is an attractive alternative for the transient analysis of continuous time Markov models. The main advantages of the method are numerical stability, well‐controlled computation error, and ability to specify the computation error in advance. However, the fact that the method can be computationally very expensive limits its applicability. In this paper, we develop a new method called split regenerative randomization, which, having the same good properties as standard randomization, can be significantly more efficient. The method covers reliability‐like models with a particular but quite general structure and requires the selection of a subset of states and a regenerative state satisfying some conditions. For a class of continuous time Markov models, model class C₂, including typical failure/repair reliability‐like models with exponential failure and repair time distributions and deferred repair, natural selections are available for both the subset of states and the regenerative state and, for those natural selections, theoretical results are available assessing the efficiency of the method in terms of “visible” model characteristics. Those results can be used to anticipate when the method can be expected to be competitive. We illustrate the application of the method using a large class C₂ model and show that for models in that class the method can indeed be significantly more efficient than previously available randomization‐based methods. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006     

多阶段任务系统任务持续能力数学评价模型研究

分析了多阶段任务系统(PMS)及任务可靠度、可信度和任务效能等任务持续能力评价参数。结合实际装备系统大部分属于可用马尔可夫过程进行描述的可修复系统的特点,为简化模型复杂程度提出了一些合理的假设条件。针对两状态PMS,通过分析其状态转移关系,从阶段任务成功概率和阶段任务间转换概率的概念出发,分析给出其计算方法。结合对可信度和任务效能等参数的分析结果,建立其计算模型,从而建立了多阶段任务系统任务持续能力数学评价模型。最后结合常见的“靶场打靶”任务,通过对比仿真结果进行了模型的实例验证。     

Determining optimum reject allowances for deteriorating production systems

Since most manufacturing processes inevitably produce some defective items, it is common practice to produce a quantity larger than the actual order size. This excess is called a reject allowance. In this paper we consider production processes which undergo stochastic deterioration and demonstrate that under appropriate conditions a smallest optimum reject allowance exists and can be easily computed. We also investigate the optimality of periodic process inspections and show that in some cases one can do better by following an (r, R) inspection-processing policy.     

Optimal control of a production‐inventory system with both backorders and lost sales

We consider the optimal control of a production inventory‐system with a single product and two customer classes where items are produced one unit at a time. Upon arrival, customer orders can be fulfilled from existing inventory, if there is any, backordered, or rejected. The two classes are differentiated by their backorder and lost sales costs. At each decision epoch, we must determine whether or not to produce an item and if so, whether to use this item to increase inventory or to reduce backlog. At each decision epoch, we must also determine whether or not to satisfy demand from a particular class (should one arise), backorder it, or reject it. In doing so, we must balance inventory holding costs against the costs of backordering and lost sales. We formulate the problem as a Markov decision process and use it to characterize the structure of the optimal policy. We show that the optimal policy can be described by three state‐dependent thresholds: a production base‐stock level and two order‐admission levels, one for each class. The production base‐stock level determines when production takes place and how to allocate items that are produced. This base‐stock level also determines when orders from the class with the lower shortage costs (Class 2) are backordered and not fulfilled from inventory. The order‐admission levels determine when orders should be rejected. We show that the threshold levels are monotonic (either nonincreasing or nondecreasing) in the backorder level of Class 2. We also characterize analytically the sensitivity of these thresholds to the various cost parameters. Using numerical results, we compare the performance of the optimal policy against several heuristics and show that those that do not allow for the possibility of both backordering and rejecting orders can perform poorly.© 2010 Wiley Periodicals, Inc. Naval Research Logistics 2010