Hybrid block replacement and inspection policies for a multi-component system with heterogeneous component lives

Novel replacement policies that are hybrids of inspection maintenance and block replacement are developed for an n identical component series system in which the component parts used at successive replacements arise from a heterogeneous population. The heterogeneous nature of components implies a mixed distribution for time to failure. In these circumstances, a hybrid policy comprising two phases, an early inspection phase and a later wear-out replacement phase, may be appropriate. The policy has some similarity to burn-in maintenance. The simplest policy described is such a hybrid and comprises a block-type or periodic replacement policy with an embedded block or periodic inspection policy. We use a three state failure model, in which a component may be good, defective or failed, in order to consider inspection maintenance. Hybrid block replacement and age-based inspection, and opportunistic hybrid policies will also arise naturally in these circumstances and these are briefly investigated. For the simplest policy, an approximation is used to determine the long-run cost and the system reliability. The policies have the interesting property that the system reliability may be a maximum when the long-run cost is close to its minimum. The failure model implies that the effect of maintenance is heterogeneous. The policies themselves imply that maintenance is carried out more prudently to newer than to older systems. The maintenance of traction motor bearings on underground trains is used to illustrate the ideas in the paper.     

Condition-based maintenance optimization considering improving prediction accuracy

Condition-based maintenance (CBM) aims to reduce maintenance cost and improve equipment reliability by effectively utilizing condition monitoring and prediction information. It is observed that the prediction accuracy often improves with the increase of the age of the component. In this research, we develop a method to quantify the remaining life prediction uncertainty considering the prediction accuracy improvement, and an effective CBM optimization approach to optimize the maintenance schedule. Any type of prognostics methods can be used, including data-driven methods, model-based methods and integrated methods, as long as the prediction method can produce the predicted failure time distribution at any given inspection point. Furthermore, we develop a numerical method to accurately and efficiently evaluate the cost of the CBM policy. The proposed approach is demonstrated using vibration monitoring data collected from pump bearings in the field as well as simulated degradation data. The proposed policy is compared with two benchmark maintenance policies and is found to be more effective.     

考虑不完美检测的两阶段点检与备件订购策略联合优化

针对设备维修与备件管理相互影响与制约的问题, 在基于延迟时间理论的基础上, 提出了两阶段点检与备件订购策略联合优化。点检是不完美的, 当点检识别设备的缺陷状态时, 进行预防更新; 设备故障时, 进行故障更新。结合设备更新时备件的状态, 采用更新报酬理论建立了以第一阶段点检时间、第二阶段点检周期和备件订购时间为决策变量, 以最小化单位时间期望成本为目标的模型。最后, 通过人工蜂群算法对模型求解, 并在数值分析中将两阶段点检策略与定期点检策略进行比较, 结果表明:两阶段点检策略始终优于定期点检策略, 验证了所建模型的有效性。     

Condition-based maintenance with imperfect inspections for continuous degradation processes

A condition-based maintenance (CBM) strategy is now recognized as an efficient approach to perform maintenance at the best time before failures so as to save lifetime cycle cost. For continuous degradation processes, a significant source of variability lies in measurement errors caused by imperfect inspections, and this may lead to “false positive” or “false negative” observations, and consequently to inopportune maintenance decisions. To the best of our knowledge, researches on CBM optimization with imperfect inspections remain limited for continuous degradation processes, even though the subject is of practical interest for the implementation of a CBM policy. Imperfect inspections are indeed imperfect but still return interesting information on the system degradation level, and making them perfect can be expensive. Therefore, we analyze the economic performance of a maintenance policy with imperfect inspections, and compare it with the classical policy with perfect inspections to see which policy offers the best benefit in a given situation. Furthermore, a CBM policy with a two-stage inspection scheme is proposed to take benefit of mixing both perfect and imperfect inspections in the same maintenance policy. Through numerical experiments and a real case study, it is shown that the policy with imperfect inspections can be better than the classical one, and that the proposed policy with a two-stage inspection scheme always leads to the minimum long run maintenance cost rate.     

On the application of mathematical models in maintenance

This paper may be seen as an appeal to maintenance modellers to work with maintenance engineers and managers on real problems. Such collaboration is essential if maintenance modelling is to be accepted within the engineering community. It is also particularly important in the design and building of maintenance management information systems if such systems are to be used to manage and operate maintenance policy in the new millennium. In this context, developing areas of maintenance modelling are discussed, namely: inspection maintenance; condition based maintenance; maintenance for multi-component systems; and maintenance management information systems. Some new models relating to capital replacement are also considered. Thus, we are concerned with the mathematical modelling of maintenance rather than with management processes relating to maintenance. Discussion of maintenance management information systems is included because of their importance in providing data for mathematical modelling and in implementing model-based maintenance policy.     

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.     

Optimization of stochastic maintenance policies

The present study examines various inspection policy models, also known in reliability literature as preparedness models. These types of models deal with stochastically failing systems, in which failure is detected by inspection only. The present study deals with two yet unsolved problems in the field of maintenance preparedness models. The first is the analysis of various models and various objective functions while taking into consideration a positive discount factor. The second is the analysis of those models while the maintenance costs are varying in one way or another during the period of optimization. Different modes of inspection of both pure and mixed strategies are analyzed. The objective functions are set forth and solved by both a differentiation method and a dynamic programming approach.     

Optimal maintenance scheduling for a complex manufacturing system subject to deterioration

We address the problem of determining inspection strategy and replacement policy for a deteriorating complex multi-component manufacturing system whose state is partially observable. We develop inspection and replacement scheduling models and other simple maintenance scheduling models via employing an imperfect repair model coupled with a damage process induced by operational conditions. The system state in performance of the imperfectly repaired system is modelled using a proportional intensity model incorporating a damage process and a virtual age process caused by repair. The system is monitored at periodic times and maintenance actions are carried out in response to the observed system state. Decisions to perform imperfect repair and replacement are based on the system state and crossing of a replacement threshold. The model proposed here aims at joint determination of a cost-optimal inspection and replacement policy along with an optimal level of maintenance which result in low maintenance cost and high operational performance and reliability of the system. To demonstrate the use of the model in practical applications a numerical example is provided. Solutions to optimal system parameters are obtained and the response of the model to these parameters is examined. Finally some features of the model are demonstrated. The approach presented provides a framework so that different scenario can be explored.     

Optimally maintaining a Markovian deteriorating system with limited imperfect repairs

We consider the problem of optimally maintaining a periodically inspected system that deteriorates according to a discrete-time Markov process and has a limit on the number of repairs that can be performed before it must be replaced. After each inspection, a decision maker must decide whether to repair the system, replace it with a new one, or leave it operating until the next inspection, where each repair makes the system more susceptible to future deterioration. If the system is found to be failed at an inspection, then it must be either repaired or replaced with a new one at an additional penalty cost. The objective is to minimize the total expected discounted cost due to operation, inspection, maintenance, replacement and failure. We formulate an infinite-horizon Markov decision process model and derive key structural properties of the resulting optimal cost function that are sufficient to establish the existence of an optimal threshold-type policy with respect to the system’s deterioration level and cumulative number of repairs. We also explore the sensitivity of the optimal policy to inspection, repair and replacement costs. Numerical examples are presented to illustrate the structure and the sensitivity of the optimal policy.     

Economic off-line quality control strategy with two types of inspection errors

A commonly used quality control method is to inspect products to identify their quality and to perform the related disposition of acceptance, salvage or rejection based on the findings. While the issue of finding the most economical inspection/disposition policy has been studied for a batch of units produced from an unreliable system, previous studies assumed the inspections to be perfect. In this study, we further extend the inspection/disposition model to consider two types of inspection errors in order to facilitate the adaptation of this economic inspection/disposition model to real world applications. We first describe an inspection/disposition policy for the two types of inspection errors and then obtain the related mathematical formulae. An algorithm is presented for determining the optimal inspection/disposition policy. Finally, numerical examples are given to illustrate the effect of inspection errors on the optimal inspection/disposition policy under the following three quality control policies: cost minimizing, zero-defects and perfect information policy.     

基于产品质量控制的设备维修优化模型

根据产品质量和生产该产品的设备退化状态之间的相关性,设计了周期性设备检测与产品质量控制相结合的设备维修策略。该策略是在对设备进行周期性检测的基础上,利用控制图进行产品质量异常波动的检测,结合对设备退化状态的检测选择设备应采取的维修活动。根据这一设备维修策略,利用更新过程理论和统计过程控制方法,构建了基于产品质量控制的设备维修优化模型,并用遗传算法对其进行求解。通过实例仿真验证了该模型的可行性与有效性。     

A sequential condition‐based repair/replacement policy with non‐periodic inspections for a system subject to continuous wear

This paper studies a condition‐based maintenance policy for a repairable system subject to a continuous‐state gradual deterioration monitored by sequential non‐periodic inspections. The system can be maintained using different maintenance operations (partial repair, as good as new replacement) with different effects (on the system state), costs and durations. A parametric decision framework (multi‐threshold policy) is proposed to choose sequentially the best maintenance actions and to schedule the future inspections, using the on‐line monitoring information on the system deterioration level gained from the current inspection. Taking advantage of the semi‐regenerative (or Markov renewal) properties of the maintained system state, we construct a stochastic model of the time behaviour of the maintained system at steady state. This stochastic model allows to evaluate several performance criteria for the maintenance policy such as the long‐run system availability and the long‐run expected maintenance cost. Numerical experiments illustrate the behaviour of the proposed condition‐based maintenance policy. Copyright © 2003 John Wiley & Sons, Ltd.     

Maintenance scheduling of a protection system subject to imperfect inspection and replacement

An inspection and replacement policy for a protection system is described in which the inspection process is subject to error, and false positives (false alarms) and false negatives are possible. We develop two models: one in which a false positive implies renewal of the protection system; the other not. These models are motivated by inspection of a protection system on the production line of a beverage manufacturer. False negatives reduce the efficiency of inspection. Another notion of imperfect maintenance is also modelled: that of poor installation of a component at replacement. These different aspects of maintenance quality interact: false alarms can, in a worst case scenario, lead to the systematic and unnecessary replacement of good components by poor components, thus reducing the availability of the system. The models also allow situations in which maintenance quality differs between alternative maintainers to be investigated.     

The availability model and parameters estimation method for the delay time model with imperfect maintenance at inspection

The delay time model (DTM) is widely used to model the two-stage failure process and is helpful for developing cost-effective inspection/maintenance plans. Imperfect maintenance is common in practice, but seldom considered in DTM. An improved DTM with imperfect maintenance at inspection has been developed based on the assumption of imperfect inspection maintenance and perfect failure maintenance. The model of the long-run availability for the improved DTM is established. Parameters estimation method and the test for goodness of fit method are given. Numerical simulations are performed to study the influence of imperfect maintenance on the long-run availability and to validate the credibility of the parameters estimation method. The results show that imperfect maintenance will decrease the long-run availability. The existence of the optimal inspection interval regarding the maximum long-run availability is tightly related to the improvement factor, which denotes the maintenance effect. The parameters estimation method proves credible. The maximum likelihood estimations of the reliability parameters can be easily achieved by the Genetic Algorithms (GAs) searching tool.     

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.     

Age-dependent production planning and maintenance strategies in unreliable manufacturing systems with lost sale

In this paper, we formulate an analytical model for the joint determination of an optimal age-dependent buffer inventory and preventive maintenance policy in a production environment that is subject to random machine breakdowns. Traditional preventive maintenance policies, such as age and periodic replacements, are usually studied based on simplified and non-realistic assumptions, as well as on the expected costs criterion. Finished goods inventories and the age-dependent likelihood of machine breakdowns are usually not considered. As a result, these policies could significantly extend beyond the anticipated financial incomes of the system, and lead to crises. In order to solve this problem, a more realistic analysis model is proposed in this paper to consider the effects of both preventive maintenance policies and machine age on optimal safety stock levels. Hence, a unified framework is developed, allowing production and preventive maintenance to be jointly considered. We use an age-dependent optimization model based on the minimization of an overall cost function, including inventory holdings, lost sales, preventive and corrective maintenance costs. We provide optimality conditions for the manufacturing systems considered, and use numerical methods to obtain an optimal preventive maintenance policy and the relevant age-dependent threshold level production policy. In this work, this policy is called the multiple threshold levels hedging point policy. We include numerical examples and sensitivity analyses to illustrate the importance and the effectiveness of the proposed methodology. Compared with other available optimal production and maintenance policies, the numerical solution obtained shows that the proposed age-dependent optimal production and maintenance policies significantly reduce the overall cost incurred.     

Optimal policies for a delay time model with postponed replacement

We develop a delay time model (DTM) to determine the optimal maintenance policy under a novel assumption: postponed replacement. Delay time is defined as the time lapse from the occurrence of a defect up until failure. Inspections can be performed to monitor the system state at non-negligible cost. Most works in the literature assume that instantaneous replacement is enforced as soon as a defect is detected at an inspection. In contrast, we relax this assumption and allow replacement to be postponed for an additional time period. The key motivation is to achieve better utilization of the system’s useful life, and reduce replacement costs by providing a sufficient time window to prepare maintenance resources. We model the preventive replacement cost as a non-increasing function of the postponement interval. We then derive the optimal policy under the modified assumption for a system with exponentially distributed defect arrival time, both for a deterministic delay time and for a more general random delay time. For the settings with a deterministic delay time, we also establish an upper bound on the cost savings that can be attained. A numerical case study is presented to benchmark the benefits of our modified assumption against conventional instantaneous replacement discussed in the literature.     

考虑非周期检测的视情维修与备件订购联合策略优化

针对单部件系统/关键部件提出视情维修与备件订购联合策略,其中系统退化服从两阶段延迟时间过程且采用非周期检测策略,退化初期以检测间隔T₁检查系统状态,而在第一次识别缺陷状态时,缩短检测周期为T₂、订购备件且进行不完美维修;若系统在随后的退化中被识别处于缺陷状态,执行不完美维修直至超过阈值次数N_max并采取预防性更换,但若在检测周期内发生故障则进行更换。根据系统状态和备件状态分析各种可能更新事件及相应的联合决策,利用更新报酬理论构建最小化单位时间内期望成本的目标函数,优化T₁,T₂, N_max。与对比模型策略相比,算例结果表明所提出的联合策略能有效降低单位时间内的期望成本。     

考虑不完全检测的冲击模型最优维修策略

针对制造系统中设备检测不完全的情形,研究基于不完全检测的冲击模型的周期检测、维修联合策略．通过定期检测获知系统的劣化状态以进行必要的预防性维修．在假设系统是退化的且有k个不同故障状态的条件下,以最小化系统运行成本为目标,以检测周期T、系统更换前故障次数Ⅳ为联合决策变量,利用更新过程理论建立了系统平均费用率C（T,N）的数学模型,并且给出最优联合策略的数值算法．最后借助数值例子演示了该模型,分析了检测水平对系统运行成本的影响．     

Inspection scheduling for imperfect production processes under free repair warranty contract

The paper considers scheduling of inspections for imperfect production processes where the process shift time from an ‘in-control’ state to an ‘out-of-control’ state is assumed to follow an arbitrary probability distribution with an increasing failure (hazard) rate and the products are sold with a free repair warranty (FRW) contract. During each production run, the process is monitored through inspections to assess its state. If at any inspection the process is found in ‘out-of-control’ state, then restoration is performed. The model is formulated under two different inspection policies: (i) no action is taken during a production run unless the system is discovered in an ‘out-of-control’ state by inspection and (ii) preventive repair action is undertaken once the ‘in-control’ state of the process is detected by inspection. The expected sum of pre-sale and post-sale costs per unit item is taken as a criterion of optimality. We propose a computational algorithm to determine the optimal inspection policy numerically, as it is quite hard to derive analytically. To ease the computational difficulties, we further employ an approximate method which determines a suboptimal inspection policy. A comparison between the optimal and suboptimal inspection policies is made and the impact of FRW on the optimal inspection policy is investigated in a numerical example.