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.     

Efficiency consideration of generalized age replacement policy

Under the generalized age replacement policy, the system is replaced either at the predetermined age or upon failure if its corresponding repair time exceeds the threshold, whichever comes first. In this paper, we investigate the optimal choice of the pre‐determined preventive replacement age for a nonwarranted system, which minimizes the expected cost rate during the life cycle of the system from the customer's perspective under certain cost structures. Furthermore, we discuss several properties of such a generalized age replacement policy in comparison with the traditional age replacement policy. An efficiency, which represents the fractional time that the system is on, is defined under the proposed generalized age replacement policy and its monotonicity properties are investigated as well. The main objective of this study is to investigate the advantageous features of the generalized age replacement policy over the traditional age replacement policy with regard to the availability of the repairable system. Assuming that the system deteriorates with age, we illustrate our proposed optimal policies numerically and observe the impact of relevant parameters on the optimal preventive replacement age.     

Quantifying the risk in age and block replacement policies

Preventive maintenance policies have been studied in the literature without considering the risk due to the cost variability. In this paper, we consider the two most popular preventive replacement policies, namely, age and block replacement policies under long-run average cost and expected unit time cost criteria. To quantify the risk in the preventive maintenance policies, we use the long-run variance of the accumulated cost over a time interval. We numerically derive the Risk-sensitive preventive replacement policies and study the impact of the Risk-sensitive optimality criterion on the managerial decisions. We also examine the performance of the expected unit time cost criterion as an alternative to the traditional long-run average cost criterion.     

Tramcar Maintenance

This article describes the work done to determine age limits for preventive replacements and overhauls in tramcar maintenance, where opportunities are provided by a failed component or an essential overhaul. A brief review of existing literature is also presented and two pair-wise suboptimal preventive replacement policies are considered. These policies are applicable to any "n" component system with subassemblies, where the cost of dismantling and assembling the various sub-units is of the same order as the cost of new parts. The results of a search to find more economical control limits are also presented.     

A multi-objective approach to simultaneous determination of spare part numbers and preventive replacement times

Different models have been proposed in the field of preventive maintenance planning for finding optimal age replacement policies. While previous studies have focused mainly on classical cost objectives, this paper presents a novel multi-objective model for preventive replacement of a part over a planning horizon. The proposed model considers different objectives and practical issues, such as corrective replacement and its consequences, residual lifetime objective, and kind of productivity index. Also, the model determines number of spare parts, required for replacement with the defected part, to be provided at the beginning of the planning horizon. The multi-objective model is applicable for machines or equipments which are repaired through replacing their defected part with new spare part.For solving the multi-objective model, regarding to ability of ε-constraint method to generate different pareto-optimal solutions, a procedure is developed based on this method. The procedure shows how the ε-constraint method can be used for finding preferred solution in situations where there is no access to decision maker. The model and solution procedure are illustrated by a numerical example.     

A multi-objective approach for solving a replacement policy problem for equipment subject to imperfect repairs

This paper proposes a multi-objective approach to model a replacement policy problem applicable to equipment with a predetermined period of use (a planning horizon), which may undergo critical and non-critical failures. Corrective replacements and imperfect repairs are taken to restore the system to operation respectively when critical and non-critical failures occur. Generalized Renewal Process (GRP) is used to model imperfect repairs. The proposed model supports decisions on preventive replacement intervals and the number of spare parts purchased at the beginning of the planning horizon. A Multi-Objective Genetic Algorithm (MOGA) coupled with discrete event simulation (DES) is proposed to provide a set of solutions (Pareto-optimum set) committed to the different objectives of a maintenance manager in the face of a replacement policy problem, that is, maintenance cost, rate of occurrence of failures, unavailability, and investment on spare parts. The proposed MOGA is validated by an application example against the results obtained via the exhaustive approach. Moreover, examples are presented to evaluate the behavior of objective functions on Pareto set (trade-off analysis) and the impact of the repair effectiveness on the decision making.     

Coordinated stocking,clearance sales,and return policies for a supply chain

The results of a study of inter-organizational coordination effect in inventory control, return, and clearance sales policies for a distribution channel consisting of a supplier, a retailer, and a Discount Sales Outlet (DSO) are reported here. We first study the retailer–DSO coordination issues. The products are initially sold in a retail outlet. After the selling season, the leftovers are moved to a DSO for a permanent clearance sale. When the retailer and the DSO coordinate, they share information on the demand forecast and jointly decide the stocking, markdown sales, and return policies to maximize mutual profit. In the absence of coordination, the demand information is not shared by the two parties, and the decisions are decentralized to optimize the individual party’s objective function. For the supplier–retailer–DSO cooperation issue, two models are considered: namely, the Joint Optimal Model (JOM) and the Individual Optimal Model (IOM), respectively. In the IOM, the stocking, markdown sales, and return policies are individually designed by the retailer–DSO, while the supplier independently sets the terms of the return policy. We compare this approach with the JOM, in which the supplier–retailer–DSO jointly designs a mutually beneficial plan so as to maximize the supply chain joint profit. Optimal coordination policies are analyzed, and the factors that make coordination an effective approach are studied.     

修理工单重定期休假可修系统更换模型研究

针对修理工带有单重休假的单部件可修系统,提出了一种新的维修更换模型.假定系统是可修的,逐次故障后的维修时间构成随机递增的几何过程,系统工作时间构成随机递增的几何过程,在修理工休假时间为定长的情况下,分别选取系统的总工作时间T和故障维修次数N为更换策略,以长期运行单位时间内的期望效益为目标函数,通过更新过程和几何过程理论建立数学模型,导出了目标函数的解析表达式,通过最大化目标函数来获取系统最优的更换策略T*和N*.并在一定条件下给出了策略N比策略T优的充分条件.最后,通过数值例子验证了方法的有效性.     

Maintenance optimal control three-machine replacement model under technological breakthrough expectations

This paper addresses a finite-horizon profit maximization three-machine replacement problem. More precisely, a model is formulated allowing for preventive maintenance to slow down machine quality and profit reduction caused by obsolescence, to determine the timing of replacing an existing machine by another available machine with improved technology. This decision is considered under uncertainty regarding the introduction time of a machine with a not-yetachieved technology. Given an exponential probability distribution function of the introduction time, the optimality of a bang-bang nonincreasing preventive maintenance control is shown.Moreover, subproblems maximizing the expected discounted profit are analyzed. Closed-form solutions are provided to compare machines of different technologies and to derive an analytical sensitivity analysis concerned with many issues related to the problem. The results are not necessarily intuitive and simple. For example, different relationships between the planning horizon and the preventive maintenance switching time are presented for the three-machine problem versus the single-machine problem.The focus of this paper is on the formulation and the analytical analysis of the problem rather than on its computational aspects.     

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.     

Multi-parameter maintenance optimisation via the marginal cost approach

In this paper we show how the marginal-cost approach can be used to optimise multi-parameter replacement rules. We will illustrate this for an opportunity-based age replacement rule that consists of two parameters. The first parameter is a control limit t, which indicates from what age on a unit is replaced preventively at the first arising opportunity. The second parameter is a planned replacement age T, which indicates at what age the unit is replaced if it has not been replaced yet. The unit can fail and is immediately replaced upon failure. It can be shown that this replacement rule belongs to a class of policies for which the long-run average-cost function is unimodal. The marginal cost approach is based on the following assertion: any point, in which the marginal cost(s) of deferring maintenance equals the average-cost, is an average-cost minimum. Assuming unimodality the minimisation problem can be solved as a root-finding problem, for which there are numerous efficient routines. It appears that the marginal cost approach is very practical for the optimisation of the considered replacement rule, especially because a quick assessment can be made of the optimal parameter values. The marginal cost approach can be used for many other multi-parameter problems, insofar as they can be modelled as a regenerative process.     

Resource-constrained management of heterogeneous assets with stochastic deterioration

We consider a collection of heterogeneous assets which exhibit independent stochastic behavior, but are economically interdependent due to resource constraints on management decisions. We represent the collection of assets as a network of nonhomogeneous Markov decision processes linked by side constraints. To facilitate a procedure for obtaining nonrandomized decision policies, we express the resource limitations as integrated chance constraints and relax them into the objective function within a Lagrangian penalty term. We utilize subgradient optimization to establish upper bounds and a greedy randomized Lagrangian repair heuristic to obtain feasible solutions. We empirically validate the tightness of these a posteriori upper and lower bounds with computational experiments on a pair of applications. For pavement maintenance, we examine the effect that the reward structure of each constituent MDP has on the maintenance policy in the presence of budget constraints. For equipment replacement, we consider constraints on two resource types.     

Stochastic Comparison of Age-Dependent Block Replacement Policies

The age-dependent block replacement policy is a modified block replacement policy with an age limit for preventive replacements. Under this policy, any failed component is repaired, but only the components whose ages exceed a fixed age limit are replaced preventively at the scheduled maintenance times. Using the compensator method, we compare stochastically the failure counting processes of the age-dependent block replacement policies with different parameters, and show that the age-dependent block replacement policy, although cost effective, leads to more failures than the age and block replacement policies. AMS 2000 Subject Classification 60K10     

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.     

Repair Replacement Modelling over Finite Time Horizons

In this paper an integral equation approach is given for evaluating the expected cost of repair replacement policies over finite time horizons. An asymptotic estimate of this expected cost is also obtained. The policy involving imperfect repair on failure with replacement after N failures is taken as an illustrative example and optimal policies N* are found for both infinite and finite time horizons of use.     

A Markov model for single-leg air cargo revenue management under a bid-price policy

In this paper, we consider the capacity allocation problem in single-leg air cargo revenue management. We assume that each cargo booking request is endowed with a random weight, volume and profit rate and propose a Markovian model for the booking request/acceptance/rejection process. The decision on whether to accept the booking request or to reserve the capacity for future bookings follows a bid-price control policy. In particular, a cargo will be accepted only when the revenue from accepting it exceeds the opportunity cost, which is calculated based on bid prices. Optimal solutions are derived by maximizing a reward function of a Markov chain. Numerical comparisons between the proposed approach and two existing static single-leg air cargo capacity allocation policies are presented.     

基于两类失效的产品保修策略与出保维修优化

基于消费者对保修的需求,设计消费者偏爱的保修策略是制造商巩固市场地位、提升市场竞争力的一种战略决策。与此同时,以较低费用制定较长出保服役时间的出保维修策略也是消费者一直追求的目标。本文以两类失效产品为研究对象,首先从制造商角度将消费者偏爱的更新免费更换保修策略与产品定价相融合,开展了产品保修策略设计。其次,从消费者角度将预防维修与经典周期更换策略相融合,提出了维修—周期更换策略,且将其作为出保维修策略并对相应的性能进行了说明。通过数值实验发现,利润模型可对保修开展设计;与总费用模型作为目标函数相比,费用率模型作为目标函数可降低寿命周期费用;与经典周期更换策略相比,提出的维修—周期更换策略能使出保服役时间更长、费用率更低。     

Marginal cost analysis of single-item maintenance policies with several decision variables

The marginal cost approach for the analysis of repair/replacementmodels was introduced by Berg in 1980 and has since been appliedto many maintenance policies of various complexity. All modelshitherto analysed in the literature by the marginal cost approachhave one single decision variable only, this being, typically,the age of the current item at the time of ordering or replacement.This paper is concerned with the extension of the marginal costtechnique to maintenance policies with several decision variables.After addressing the general framework appropriate for the multi-parametercase, we exemplify the workings of the technique by analysinga two-variable maintenance model involving replacement and minimalrepair. We demonstrate that the marginal cost approach is anattractive and intuitively appealing technique also for modelswith several decision variables. Just as in the single-parametersituation, the approach is amenable to economic interpretation,a welcome feature for users of maintenance models with a primeinterest in its economic (rather than its mathematical) aspects.As an added bonus of the marginal cost approach, in our example,some otherwise necessary tools from the theory of stochasticprocesses are dispensable.     

Optimizing System Availability Under Minimal Repair with Non-Negligible Repair and Replacement Times

Availability measures are given for a repairable system under minimal repair with constant repair times. A new policy and an existing replacement policy for this type of system are discussed. Each involves replacement at the first failure after time T, with T representing total operating time in the existing model and total elapsed time (i.e. operating time + repair time) in the new model. Optimal values of T are found for both policies over a wide range of parameter values. These results indicate that the new and administratively easier policy produces only marginally smaller optimal availability values than the existing policy.     

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.