A preventive maintenance policy with sequential checking procedure for a Markov deteriorating system

We consider a repairable system subject to a continuous-time Markovian deterioration while running, that leads to failure. The deterioration degree is measured with a finite discrete scale; repairs follow general distributions; failures are instantaneously detected. This system is submitted to a preventive maintenance policy, with a sequential checking procedure: the up-states are divided into two parts, the “good” up-states and the “degraded” up-states. Instantaneous (and perfect) inspections are then performed on the running system: when it is found in a degraded up-state, it is stopped to be maintained (for a random duration that depends on the degradation degree of the system); when it is found in a good up-state, it is left as it is. The next inspection epoch is then chosen randomly and depends on the degradation degree of the system by time of inspection. We compute the long-run availability of the maintained system and give sufficient conditions for the preventive maintenance policy to improve the long-run availability. We study the optimization of the long-run availability with respect to the distributions of the inter-inspection intervals: we show that under specific assumptions (often checked), optimal distributions are non-random. Numerical examples are studied.     

Optimal scheduling of non-perfect inspections

^** Corresponding author. Email: romulo.zequeira{at}utt.fr^*** Email: christophe.berenguer{at}utt.fr In this paper, we study the determination of optimal inspectionpolicies when three types of inspections are available: partial,perfect and imperfect. Perfect inspections diagnose withouterror the system state. The system can fail because of threecompeting failure types: I, II and III. Partial inspectionsdetect without error type I failures. Failures of type II canbe detected by imperfect inspections which have non-zero probabilityof false positives. Partial and imperfect inspections are madeat the same time. Type III failures are detectable only by perfectinspections. If the system is found failed in an inspection,a repair is made which renders the system in a good-as-new condition.The system is preventively maintained following an age-basedpolicy. Preventive maintenance actions return the system toa good-as-new condition. We consider cost contributions of inspections,repairs, preventive maintenance and periods of unavailability.The model presented permits to determine the optimal (constant)inter-inspection period for partial, imperfect and perfect inspectionsand the optimal times of preventive maintenance actions.     

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。     

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.     

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．     

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.     

Economic optimization of off-line inspection with inspection errors

In this paper, we find the optimal inspection policy for the production process of a finite batch of items with inspection errors. We study the effects of inspection errors on the optimal solution and determine which unit should be inspected to minimize the expected total costs. We also find the expected number of inspections for a given batch size. Comparisons of total costs are investigated numerically among the cost minimizing policy, the policy of perfect information, and the policy of zero-defects.     

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。     

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.     

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.     

A two-phase inspection policy with imperfect testing

This paper presents an inspection policy to detect failures of a single component system that remain hidden otherwise. Inspection reveals whether the unit is in good or failed state. The possibility of non perfect testing is assumed, thus, successive inspections may fail detecting a failure or result in a false alarm. The occurrence of false alarms is reported in optical fire detectors and inspection of printing circuit boards which are on the basis of electronic systems. A two-phase inspection schedule takes into account the changes in component’s aging. The system may undergo different inspection frequencies to detect both early failures or those due to the natural deterioration in the system as time goes by. The examples reveal the advantages of a two-phase inspection when comparing with the unique interval inspection.     

Prototype modelling of irregular condition monitoring of production plant

The previously reported modelling of inspection-maintenanceproblems utilizing the delay-time concept is here extended toembrace condition-monitoring checks. The main requirements arethe relaxation of regular periodic inspections, and the requirementthat the initial point of a defect be measured as the time fromthe as-new condition and not, as previously, as the time sincea previous inspection. An expression for the probability ofa defect arising as a breakdown and as an inspection repairis constructed for a general inspection policy, under the assumptionof both perfect and imperfect repair. The revised role of theseprobability expressions in modelling the process of maintenancefor cost-effective production is discussed, various modellingalternatives are outlined, and a numerical example is presented.Parallels are drawn between condition monitoring in an industrialsetting and in the delivery of health care.     

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.     

Un modéle d'lnspections optimales

We consider the following model: we inspect the motion of a Markov process with which an “evolution cost” is associated. We inspect the process at times T ₁…, T _n ,…. If when we inspect, its value is in a given set A, it continues its evolution, otherwise we kill it. At each inspection we associate an "inspection cost" and a "killing cost". The problem consists of finding a sequence of optimal inspections. After the modelization we construct the value function by an iterative procedure as in impulse control theory, by using the theory of analytic functions and theorems of section. Thanks to the criteria of optimality we get a sequence of optimal inspections under very general hypotheses.     

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.     

Optimal overhaul intervals with imperfect inspection and repair

Author for correspondence.Email:m.j.newby{at}city.ac.uk This paper is motivated by the idea of a maintenance-free operatingperiod whose objectives are to improve mission reliability andcarry out as much maintenance as possible as a second-line activity.The system may be in one of three states (good, faulty, andfailed), and expressions are developed for the average costper unit time until failure. The system is periodically inspected,the inspection being imperfect in the sense that it can resultin both false-positive and false-negative results. Simple faultscan be fixed, but a repair is imperfect, in that there is anon-zero probability of a fault remaining after a repair. Aftera fixed number of inspections, the system is overhauled. Ifthe system fails during operation, it is replaced at increasedcost. The sojourn time in each state has non-constant failurerate, and discretization and supplementary variables are usedto give a Markovian structure which allows easy computationof the average costs. Minimizing the average cost gives theoptimal number of inspections before overhauling the system.     

An Optimal Complete Inspection Plan for Critical Multicharacteristic Components

In this paper, a new inspection plan for critical multicharacteristic components is presented. A mathematical model that depicts and represents the plan has been developed. An algorithm is proposed to determine the optimal number of repeat inspections and sequence characteristics for inspection that minimizes the expected total cost. The expected total cost consists of the cost of false acceptance (cost of Type II error), cost of false rejection (cost of Type I error), and the cost of inspection. Empirical comparisons with the Raouf et al. model on randomly generated problems have been conducted. The results have shown that the proposed plan performs better in terms of the expected total cost on 82% of the generated problems for the assumed specific parameters. The reduction in the expected total cost ranges from 0.1-10%.     

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.     

A Markov decision algorithm for optimal inspections and revisions in a maintenance system with partial information

This paper deals with a gradually deteriorating equipment whose actual degree of deterioration can be revealed by inspections only. An inspection can be succeeded by a revision depending on the system's degree of deterioration. In the absence of inspections and revisions, the working condition of the system evolves according to a Markov chain whose changes of state are not observable with the possible exception of a breakdown. Examples of this model include production machines subject to stochastic breakdowns, and maintenance of communication systems. The cost structure of the model consists of inspection, revision and operating costs. It is intuitively reasonable that in many applications a simple control-limit rule will be optimal. Such a rule prescribes a revision only when inspection reveals that the degree of deterioration has exceeded some critical level. A special-purpose Markov decision algorithm operating on the class of control-limit rules is developed for the computation of an average cost optimal schedule of inspections and revisions.     

A condition-based maintenance model

Condition-based maintenance (CBM) is very appealing becauseit enables one to make maintenance decisions based on the currentinformation about the system. Various monitoring techniqueshave been developed to obtain this information, but there arestill very few mathematical models capable of utilizing it foreffective maintenance decision making. In this paper, we propose a CBM model for situations where onlypartial information is available through monitoring a signalprocess that does not necessarily exhibit monotone behaviour.The evolution of the signal process is determined by randomfactors and minor maintenance actions between inspections. Theobjective is to find the replacement policy that maximizes thetotal expected profit during the lifetime of the system. Wewill show that, under weak monotonicity assumptions, the optimalpolicy is of a control-limit type, and we will develop an algorithmfor finding the limit for an -optimal policy.