Economic production lot-sizing for an unreliable machine under imperfect age-based maintenance policy

This paper is concerned with the joint determination of both economic production quantity and preventive maintenance (PM) schedules under the realistic assumption that the production facility is subject to random failure and the maintenance is imperfect. The manufacturing system is assumed to deteriorate while in operation, with an increasing failure rate. The system undergoes PM either upon failure or after having reached a predetermined age, whichever of them occurs first. As is often the case in real manufacturing applications, maintenance activities are imperfect and unable to restore the system to its original healthy state. In this work, we propose a model that could be used to determine the optimal number of production runs and the sequence of PM schedules that minimizes the long-term average cost. Some useful properties of the cost function are developed to characterize the optimal policy. An algorithm is also proposed to find the optimal solutions to the problem at hand. Numerical results are provided to illustrate both the use of the algorithm in the study of the optimal cost function and the latter’s sensitivity to different changes in cost factors.     

The impact of inspection errors,imperfect maintenance and minimal repairs on an imperfect production system

This paper develops an integrated model of production lot-sizing, maintenance and quality for considering the possibilities of inspection errors, preventive maintenance (PM) errors and minimal repairs for an imperfect production system with increasing hazard rates. In this study, a PM activity is imperfect in that a production system cannot be recovered as good as new and might cause the production system to shift to the out-of-control state with a certain probability. Numerical analyses are used to simulate the effect of changes in various parameters on the optimal solution for which the time that the process remains in the in-control state is assumed to follow a Weibull distribution. In addition, we investigate the effects of inspection errors and PM errors on the minimum total cost of the optimal inspection interval, inspection frequency and production quantity.     

Optimal economic production quantity policy for imperfect process with imperfect repair and maintenance

This study integrates maintenance and production programs with the economic production quantity (EPQ) model for an imperfect process involving a deteriorating production system with increasing hazard rate: imperfect repair and rework upon failure (out of control state). The imperfect repair performs some restorations and restores the system to an operating state (in-control state), but leaves its failure until perfect preventive maintenance (PM) is performed. There are two types of PM, namely imperfect PM and perfect PM. The probability that perfect PM is performed depends on the number of imperfect maintenance operations performed since the last renewal cycle. Mathematical formulas are obtained for deriving the expected total cost. For the EPQ model, the optimum run time, which minimizes the total cost, is discussed. Various special cases are considered, including the maintenance learning effect. Finally, a numerical example is presented to illustrate the effects of PM, setup, breakdown and holding costs.     

A practical approach for reliability prediction of pipeline systems

Pipelines play an important role in the modern society. Failures of pipelines can have great impacts on economy, environment and community. Preventive maintenance (PM) is often conducted to improve the reliability of pipelines. Modern asset management practice requires accurate predictability of the reliability of pipelines with multiple PM actions, especially when these PM actions involve imperfect repairs. To address this issue, a split system approach (SSA) based model is developed in this paper through an industrial case study. This new model enables maintenance personnel to predict the reliability of pipelines with different PM strategies and hence effectively assists them in making optimal PM decisions.     

Optimum policy for a production system with major repair and preventive maintenance

This study applies periodic preventive maintenance (PM) to a repairable production system with major repairs conducted after a failure. This study considers failed PM due to maintenance workers incorrectly performing PM and damages occurring after PM. Therefore, three PM types are considered: imperfect PM, perfect PM and failed PM. Imperfect PM has the same failure rate as that before PM, whereas perfect PM makes restores the system perfectly. Failed PM results in system deterioration and major repairs are required. The probability that PM is perfect or failed depends on the number of imperfect maintenance operations conducted since the previous renewal cycle. Mathematical formulas for expected total production cost per unit time are generated. Optimum PM time that minimizes cost is derived. Various special cases are considered, including the maintenance learning effect. A numerical example is given.     

Model for imperfect age-based preventive maintenance with age reduction

The effect of ageing on the deterioration rate of most repairable systems cannot be ignored. Preventive maintenance (PM) is performed in the hope of restoring fully the performance of these systems. However, in most practical cases, PM activities will be only able to restore part of the performance. Bridging the gap between theory and practice in this area requires realistic modelling of the effect of PM activities on the failure characteristics of maintainable systems. Several sequential PM models have been developed for predetermined PM interval policies but much less effort has been devoted to age-based ones. The purpose of this paper is to develop an age-based model for imperfect PM. The proposed model incorporates adjustment factor in the effective age of the system. The system undergoes PM either at failure or after a predetermined time interval whichever of them occurs first. After a certain number of such PMs, the system is replaced. The problem is to determine both the optimal number of PMs and the optimal PM's schedule that minimize the total long-term expected cost rate. Model analysis relating to the existence and uniqueness of the optimal solutions is provided. Numerical examples are presented to study the sensitivity of the model to different cost function's factors and to illustrate the use of the algorithm.     

Optimal threshold values of age and two-phase maintenance policy for leased equipments using age reduction method

This paper investigates the optimal threshold values of age to perform preventive maintenance (PM) actions for leased equipment within the lease period. In this paper, we use age reduction method to describe the degree of PM and construct the maintenance cost function. For repairable leased equipment, two maintenance models are proposed: (i) maintenance policy of single-phase and (ii) maintenance policy of two-phase. During the lease period, PM actions are carried out when the age of equipment reaches a certain threshold value. Any failure of the leased equipment is rectified by a minimal repair within the lease period. Under these maintenance schemes, the mathematical models of the expected total cost for maintenance policies of single-phase and two-phase are established, and the optimal maintenance policies are derived such that the expected total cost is minimized. Finally, the features of the optimal maintenance policy are illustrated through numerical examples.     

Imperfect maintenance

The maintenance of a deteriorating system is often imperfect: the system after maintenance will not as good as new, but younger. Imperfect maintenance study indicates a significant breakthrough in reliability and maintenance theory. Research activities in maintenance engineering have been conducted over the past 30 years and more than 40 mathematical imperfect maintenance models have been proposed for estimating the reliability measures and determining the optimum maintenance policies. Various treatment methods and optimal policies on the imperfect maintenance are discussed and summarized in this paper. Several important results for imperfect maintenance are also presented.     

Optimal production time and number of maintenance actions for an imperfect production system under equal-interval maintenance policy

This paper deals with the optimal production/maintenance (PM) policy for a deteriorating production system which may shift from the in-control state to the out-of-control state while producing items. The process is assumed to have a general shift distribution. Under the commonly used maintenance policy, equal-interval maintenance, the joint optimizations of the PM policy are derived such that the expected total cost per unit time is minimized. Different conditions for optimality, lower and upper bounds and uniqueness properties on the optimal PM policy are provided. The implications of another commonly used policy, to perform a maintenance action only at the end of the production run, are also discussed. Structural properties for the optimal policy are established so that an efficient solution procedure is obtained. In the exponential case, some extensions of the results obtained previously in the literature are presented. A numerical example is provided to illustrate the solution procedure for the optimal production and maintenance policy.     

Optimizing Preventive Maintenance Models

We deal with the problem of scheduling preventive maintenance (PM) for a system so that, over its operating life, we minimize a performance function which reflects repair and replacement costs as well as the costs of the PM itself. It is assumed that a hazard rate model is known which predicts the frequency of system failure as a function of age. It is also assumed that each PM produces a step reduction in the effective age of the system. We consider some variations and extensions of a PM scheduling approach proposed by Lin et al. [6]. In particular we consider numerical algorithms which may be more appropriate for hazard rate models which are less simple than those used in [6] and we introduce some constraints into the problem in order to avoid the possibility of spurious solutions. We also discuss the use of automatic differentiation (AD) as a convenient tool for computing the gradients and Hessians that are needed by numerical optimization methods. The main contribution of the paper is a new problem formulation which allows the optimal number of occurrences of PM to be determined along with their optimal timings. This formulation involves the global minimization of a non-smooth performance function. In our numerical tests this is done via the algorithm DIRECT proposed by Jones et al. [19]. We show results for a number of examples, involving different hazard rate models, to give an indication of how PM schedules can vary in response to changes in relative costs of maintenance, repair and replacement. Part of this work was carried out while the first author was a Visiting Professor in the Department of Mechanical Engineering at the University of Alberta in December 2003.     

基于累积损伤过程的不完全预防维护决策

基于累积损伤过程研究旧系统的不完全预防维护策略,冲击服从非时齐Poisson过程,并产生随机的损伤量,损伤量是加法累加的.系统在累积损伤量达到k或系统运行年龄达到T时进行计划内预防维护.在两次计划内预防维护之间,当累积损伤量达到预定水平K (k K)时,对系统进行计划外维护,其费用高于计划内预防维护,利用再生过程理论得到单位时间维护成本,讨论在时齐Poisson过程下的时间预防维护策略与水平预防维护策略,同时给出算例.     

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.     

多部件系统故障预防工作的组合优化

研究了实践中常见的四种预防工作组合类型,即定时维修、功能检测、使用检查及检查与定时维修的组合策略,以单位时间维修费用最小为目标建立了多部件故障预防工作的组合优化模型.在此基础上,提出了两个故障预防工作的组合优化算法,并给出了计算示例以说明模型和算法的性能.     

Integrated production and quality model under various preventive maintenance policies

In this paper, we investigate the effect of various preventive maintenance policies on the joint optimisation of the economic production quantity (EPQ) and the economic design of control chart. This has been done for a deteriorating process where the in-control period follows a general probability distribution with increasing hazard rate. In the proposed model, preventive maintenance (PM) activities reduce the shift rate of the system to the out-of-control state proportional to the PM level. For each policy, the model determines the EPQ, the optimal design of the control chart and the optimal preventive maintenance level. The effects of the three PM policies on EPQ and quality costs are illustrated using an example of a Weibull shock model with an increasing hazard rate.     

A model of imperfect preventive maintenance with dependent failure modes

Consider a system subject to two modes of failures: maintainable and non-maintainable. A failure rate function is related to each failure mode. Whenever the system fails, a minimal repair is performed. Preventive maintenances are performed at integer multiples of a fixed period. The system is replaced when a fixed number of preventive maintenances have been completed. The preventive maintenance is imperfect because it reduces the failure rate of the maintainable failures but does not affect the failure rate of the non-maintainable failures. The two failure modes are dependent in the following way: after each preventive maintenance, the failure rate of the maintainable failures depends on the total of non-maintainable failures since the installation of the system. The problem is to determine an optimal length between successive preventive maintenances and the optimal number of preventive maintenances before the system replacement that minimize the expected cost rate. Optimal preventive maintenance schedules are obtained for non-decreasing failure rates and numerical examples for power law models are given.     

The development of a hybrid intelligent maintenance optimisation system (HIMOS)

This paper reports on the development of a hybrid intelligent maintenance optimisation system (HIMOS) for decision support. It is a follow-up to an earlier paper published in the Journal of the Operational Research Society in 1995. Both papers refer to systems where there are very many components which may break down independently. When a component breaks down, corrective action (CO) is required. The problem is to determine the optimal maintenance policy, essentially the frequency of preventive maintenance (PM) which minimises the sum of down time due to PM and CO.HIMOS, like its predecessor IMOS, uses an ‘intelligent’ decision support system to carry out an automated analysis of the maintenance history data. Maintenance data are presented to the system and the most suitable mathematical model from a model-base is identified utilising a hybrid knowledge/case based system (KBS/CBR). Thus initially a rule base is applied to select a model, as in the case of IMOS. If no model is matched, the system reverts to its historical case-base to match the current case with a similar case that has been previously modelled. This double reasoning adds to the system's true learning capabilities (intelligence) and increases the rate of success of model selection. A prototype system is written in Visual Basic® for an IBM compatible PC. The study results include optimal PM intervals for a sample of industrial data sets. The results of the validation exercise of HIMOS against expert advice has shown that the system functions satisfactorily.     

An overview of some classical models and discussion of the signature-based models of preventive maintenance

In reliability engineering literature, a large number of research papers on optimal preventive maintenance (PM) of technical systems (networks) have appeared based on preliminary many different approaches. According to the existing literature on PM strategies, the authors have considered two scenarios for the component failures of the system. The first scenario assumes that the components of the system fail due to aging, while the second scenario assumes the system fails according to the fatal shocks arriving at the system from external or internal sources. This article reviews different approaches on the optimal strategies proposed in the literature on the optimal maintenance of multi-component coherent systems. The emphasis of the article is on PM models given in the literature whose optimization criteria (cost function and stationary availability) are developed by using the signature-based (survival signature-based) reliability of the system lifetime. The notions of signature and survival signature, defined for systems consisting of one type or multiple types of components, respectively, are powerful tools assessing the reliability and stochastic properties of coherent systems. After giving an overview of the research works on age-based PM models of one-unit systems and $k$-out-of-$n$ systems, we provide a more detailed review of recent results on the signature-based and survival signature-based PM models of complex systems. In order to illustrate the theoretical results on different proposed PM models, we examine two real examples of coherent systems both numerically and graphically.     

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.     

Determining the optimal probing lot size for the wafer probe operation in semiconductor manufacturing

In this study, we reformulated the problem of wafer probe operation in semiconductor manufacturing to consider a probe machine (PM) which has a discrete Weibull shift distribution with a nondecreasing failure rate. To maintain the imperfect PM during the probing of a lot of wafers, a minimal repair policy is introduced with type II inspection error. To increase the productivity of the PM, this paper aims to find an optimal probing lot size that minimizes the expected average processing time per wafer. Conditions and uniqueness for the optimal lot size are explored. The special case of a geometric shift distribution is studied to find a tighter upper bound on the optimal lot size than in previous study. Numerical examples are performed to evaluate the impacts of minimal repair on the optimal lot size. In addition, the adequacy of using a geometric shift distribution is examined when the actual shift distribution has an increasing failure rate.     

Availability and maintenance of series systems subject to imperfect repair and correlated failure and repair

The series system is one of the most important and common systems in reliability theory and applications. This paper investigates availability, maintenance cost, and optimal maintenance policies of the series system with n constituting components under the general assumption that each component is subject to correlated failure and repair, imperfect repair, shut-off rule, and arbitrary distributions of times to failure and repair. Imperfect repair is modeled through the basic idea of the quasi renewal processes introduced by H. Wang, H. Pham, A quasi renewal process and its applications in imperfect maintenance, International Journal of Systems Science 27(10) (1996) 1055–1062; 28(12) (1997) 1329. System availability, mean time between system failures, mean time between system repairs, asymptotic fractional down time of the system, etc., are derived, and a numerical example is presented to compare with the existing models by R.E. Barlow, F. Proschan, Satistical Theory of Reliability of Life Testing, Holt, Renehart & Winston, NY, 1975. Then two classes of maintenance cost models are proposed and system maintenance cost rates are modeled. Finally, properties of system availability and maintenance cost rates are studied. Optimization models to optimize system availability and/or system maintenance costs are developed, and optimum system maintenance policies are discussed through a numerical example.