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.     

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.     

Availability and maintenance modeling for systems subject to dependent hard and soft failures

This paper develops availability and maintenance models for single‐unit systems subject to dependent hard and soft failures. A hard failure stops the system immediately, whereas a soft failure only reduces the performance capacity of the system. Dependence between these 2 types of failures is reflected in the fact that each soft failure directly increases the hazard rate of the hard failure. On the basis of such interaction, we derive recursive equations for the system reliability and availability functions. To detect both types of failures, inspections are executed periodically. Furthermore, we investigate the optimal inspection policy via the minimization of the expected cost per unit time. The applicability of the developed availability and maintenance models is validated by a case study on an electrical distribution system.     

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 condition based maintenance model for a two-unit series system

This paper discusses a condition based maintenance model with exponential failures and fixed inspection intervals for a two-unit system in series. The condition of each unit, such as vibration or heat, is monitored at equidistant time intervals. The condition indicator variables for each unit are used to decide whether to repair an individual unit or to overhaul the whole system. After a maintenance action is performed the monitored condition indicator variable takes on its initial value. Each unit can fail only once within an inspection interval and when one or both units fail the system fails. The probability of failure is exponential and the failure rate is dependent on the condition. The cost to be minimized is the long-run average cost of maintenance actions and failures. We study the optimal solution to this problem obtained via dynamic programming.     

Bi-objective workforce-constrained maintenance scheduling: a case study

In this paper, a real maintenance workforce-constrained scheduling problem is formulated as a bi-objective mixed-integer programming model with the aim of simultaneously minimizing the workforce requirements and maximizing the equipment availability. The skilled workforce is provided by internal and external resources using regular time, overtime and contracting. The equipment availability is measured by the downtime required for preventive maintenance (scheduled) and failure repair (unscheduled) jobs. We also encounter imminent or potential failures whose priorities depend on the severity of the failure on the system (secondary failure). The total weighted flow time is used as a scheduling criterion to measure the equipment availability; the weight of each job directly depends on the expected downtime resulting from the associated failure. The proposed model is verified using two comprehensive numerical examples and some sensitivity analyses. We conclude by discussing the results.     

Probabilistic analysis of a system with dependent units having a single repair facility subject to preventive maintenance

The paper deals with the availability and the reliability analysis of a system with dependent units having a single repair facility subject to preventive maintenance. The system initially consists of n-identical units (connected in parallel) each with failure rate λ_n. The failure rate of a unit at any given instant of time depends upon the number of units operating at that instant. The time to repair of a failed unit and the time for maintenance of the repair- facility are arbitrarily distributed whereas the time to failure of a unit is exponentially distributed. The results obtained have been compared with those obtained when the repair facility is not subject to preventive maintenance.     

有延迟修理的两部件串联系统的预防维修策略

研究由两个部件串联组成的系统的预防维修策略, 当系统的工作时间达到T时进行预防维修, 预防维修使部件恢复到上一次故障维修后的状态. 当部件发生故障后进行故障维修, 因为各种原因可能会延迟修理. 部件在每次故障维修后的工作时间形成随机递减的几何过程, 且每次故障后的维修时间形成随机递增的几何过程. 以部件进行预防维修的间隔T和更换前的故障次数N组成的二维策略(T,N)为策略, 利用更新过程和几何过程理论求出了系统经长期运行单位时间内期望费用的表达式, 并给出了具体例子和数值分析.     

The average optimality of a repair-limit replacement policy

We consider a minimal-repair and replacement problem of a reliability system whose state at a failure is described by a pair of two attributes, i.e., the total number of its past failures and the current failure level. It is assumed that the system is bothered by more frequent and more costly failures as time passes. Our problem is to find and/or characterize a minimal-repair and replacement policy of minimizing the long-run average expected maintenance cost per unit time over the infinite time horizon. Formulating the problem as a semi-Markov decision process, we show that a repairlimit replacement policy is average optimal. That is, for each total number of past system failures, there exists a threshold, called a repair limit, such that it is optimal to repair minimally if the current failure level is lower than the repair limit, and to replace otherwise. Furthermore, the repair limit is decreasing in the total number of past system failures.     

Optimal maintenance strategy under renewable warranty with repair time threshold

In this paper, we consider a periodic preventive maintenance model, from the manufacturer's perspective, which can be implemented to reduce the maintenance cost of a repairable product during a given warranty period. The product is assumed to deteriorate with age and the warranty policy we adopt in this paper takes into account the two factors of failure time and repair time of the product when the product failure occurs. Under the proposed two-factor warranty, a repair time threshold is pre-determined and if the repair takes more time than that of the threshold, the failed product is replaced with a renewed warranty policy. Otherwise, the product is only minimally repaired to return to the operating state. During such a renewable warranty period, preventive maintenance is conducted to reduce the rate of degradation periodically while the product is in operation. By assuming certain cost structures, we formulate the expected warranty cost during the warranty period from the manufacturer's perspective when a periodic preventive maintenance strategy is adapted. Although more frequent preventive maintenance increases the warranty cost, the chance of product failures would be reduced. The main aim of this paper is to accomplish the optimal trade-off between the warranty cost and the preventive maintenance period by determining the optimal preventive maintenance period that minimizes the total expected warranty cost during the warranty period. Assuming the power law process for the product failures, we illustrate our proposed maintenance model numerically and study the impact of relevant parameters on the optimal preventive maintenance policy.     

Planning performance based contracts considering reliability and uncertain system usage

This paper investigates a novel quantitative approach for planning and contracting performance-based logistics in the presence of uncertain system usage. Our efforts focus on an integrated service delivery environment where the manufacturer develops capital-intensive systems and also provides after-sales support. We propose an analytical model to characterize system operational availability by comprehending five performance drivers: inherent failure rate, usage rate, spare parts inventory, repair time, and the fleet size. This analytical insight into the system performance allows the service supplier to minimize the total cost across system design, production, maintenance, and repair. Two contracting schemes are investigated under cost minimization and profit maximization schemes. For the first time in literature, reliability design and service parts logistics are seamlessly integrated into one decision support model for improving operational availability while lowering the lifecycle cost. Numerical examples are provided to demonstrate the applicability and the effectiveness of the proposed decision support tool.     

The combined manpower planning and preventive maintenance strategies by aggregate planning for the service division of vending machines

This paper presents a mathematical model of aggregate planning for the service division of vending machines, which determines its workforce size and the preventive maintenance level that may affect the failure rate. The unique aspect of this problem is that it exhibits two types of failures requiring different repair and preventive maintenance. The practice is to divide the service division into two groups, within which there are hierarchies according to these types of failures, and in each group, there are full-time workers with annual contracts and temporary workers hired only during the peak season. A case study based on actual operational data demonstrates how effective the introduced model and the policy are in reducing the total cost and improving the quality of service, and thus enhancing the overall system performance of the service division. We perform sensitivity analysis on key parameters, which helps us change maintenance strategies under diverse operating situations.     

Cost of ownership model for spare engines purchase for the Korean navy acquisition program

Major weapon system acquisition programmes often require high initial purchase cost which can be a burden for the procurement of a highly reliable system. In order to avoid the tendency of acquiring a less expensive weapon system with lower performance, a cost of ownership (COO) model can be applied to assess the lifetime cost of the weapon system. In many existing cost estimation models for weapon systems, the failure rate of the system is assumed to be constant and the functional relationship between the initial purchase cost and maintenance cost is not well defined. In this paper, we propose a revised COO model where random effects models are employed to accommodate the variations of the system failure frequency and repair time. It is expected that our model can contribute to the cost-effective procurement of spare engines for the Korean Navy acquisition programme within the limited national defence budget.     

Simple probability models for project contracting

There are two aspects to the process of price setting using sealed bids. The project owner is interested in deploying the contract mechanism that will secure reliable service at the cheapest cost. On the other hand, each contractor bidding for the project is interested in winning the contract but at a price that assures him a reasonable profit margin. We use a parsimonious stochastic model to compare and contrast some commonly used contracts from the point of view of the project owner. We show that if the bidders are risk neutral, a Fixed Price contract results in the smallest expected procurement cost for the project owner. We introduce and analyze Menu contracts and show that the expected price of a Menu contract lies in between the prices of the Fixed Price and Cost Plus contracts for the same project. We analyze how risk aversion and collusion, which we model using concepts of stochastic dependence, impacts the average winning bid price.     

修理设备可更换且有修理延迟的N部件串联系统分析

假定部件的寿命服从指数分布,修理延迟时间和修理时间均服从任意分布,并且修理设备的寿命服从指数分布,其更换时间服从任意分布的情况下,利用马尔可夫更新过程理论和拉普拉斯变换工具,研究了修理有延迟且修理设备可更换的n部件串联可修系统,求得了系统的可用度和(0,t]时间内的平均故障次数.进一步,在定义修理设备“广义忙期”下,利用全概率分解,提出了一种新的分析技术,讨论了修理设备的可靠性指标,得到修理设备的一些重要可靠性结果.     

系统最佳维修策略研究

一个复杂系统通常由多个不同部件组成，考虑到这些部件有各自不同的失效率及维修时间，本提出了一种新的维修策略模型，该模型考虑了不同部件的差异性及对系统的不同重要性，在一定可用度要求下，使系统总平均费用达到最小的最佳预防维修周期，并给出了相应的仿真算法。     

Availability optimization of systems subject to competing risk

This paper considers a competing risk (degradation and sudden failure) maintenance situation. A maintenance model and a repair cost model are presented. The degradation state of the units is continuously monitored. When either the degradation level reaches a predetermined threshold or a sudden failure occurs before the unit reaches the degradation threshold level, the unit is immediately repaired (renewed) and restored to operation. The subsequent repair times increase with the number of renewals. This process is repeated until a predetermined time is reached for preventive maintenance to be performed. The optimal maintenance schedule that maximizes the unit availability subject to repair cost constraint is determined in terms of the degradation threshold level and the time to perform preventive maintenance.     

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.     

Optimizing maintenance service contracts under imperfect maintenance and a finite time horizon

When a company decides to outsource a service, the most important reasons for doing so usually are to focus on core business, to be able to access high‐quality services at lower costs, or to benefit from risk sharing. However, service contracts typically follow a structure whereby both owner and contractor attempt to maximize expected profits in a noncoordinated way. Previous research has considered supply chain coordination by means of contracts but is based on unrealistic assumptions such as perfect maintenance and infinite time‐span contracts. In this work, these limitations are overcome by defining the supply chain through a preventive maintenance strategy that maximizes the total expected profit for both parties in a finite time‐span contract. This paper presents a model to establish such conditions when maintenance is imperfect, and the contract duration is fixed through a number of preventive maintenance actions along a significant part of the asset life cycle under consideration. This formulation leads to a win–win coordination under a set of restrictions that can be evaluated a priori. The proposed contract conditions motivate stakeholders to continually improve their maintenance services to reach channel coordination in which both parties obtain higher rewards. Copyright © 2012 John Wiley & Sons, Ltd.     

A replacement model for an additive damage model with restoration

A system existing in a random environment receives shocks at random points of time. Each shock causes a random amount of damage which accumulates over time. A breakdown can occur only upon the occurrence of a shock according to a known failure probability function. Upon failure the system is replaced by a new identical one with a given cost. When the system is replaced before failure, a smaller cost is incurred. Thus, there is an incentive to attempt to replace the system before failure. The damage process is controlled by means of a maintenance policy which causes the accumulated damage to decrease at a known restoration rate. We introduce sufficient conditions under which an optimal replacement policy which minimizes the total expected discounted cost is a control limit policy. The relationship between the undiscounted case and the discounted case is examined. Finally, an example is given illustrating computational procedures.