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.     

Optimal replacement for discrete additive damage models: Algorithm

A system receives shocks at successive random points of discrete time, and each shock causes a positive integer-valued random amount of damage which accumulates on the system one after another. The system is subject to failure and it fails once the total cumulative damage level first exceeds a fixed threshold. Upon failure the system must be replaced by a new and identical one and a cost is incurred. If the system is replaced before failure, a smaller cost is incurred. In previous work, under some assumptions, we specified a replacement rule which minimizes the long-run (expected) average cost per unit time and possesses control limit property. In this paper, a general algorithm for such models is developed. This research has been jointly supported by ITDC, contract No.105-82150 and the National Natural Science Foundation of China.     

A periodical replacement model based on cumulative repair‐cost limit

This paper considers a periodical replacement model based on a cumulative repair‐cost limit, whose concept uses the information of all repair costs to decide whether the system is repaired or replaced. The failures of the system can be divided into two types. One is minor failure that is assumed to be corrected by minimal repair, while the other is serious failure where the system is damaged completely. When a minor failure occurs, the corresponding repair cost is evaluated and minimal repair is then executed if this accumulated repair cost is less than a pre‐determined limit L, otherwise, the system is replaced by a new one. The system is also replaced at scheduled time T or at serious failure. Long‐run expected cost per unit time is formulated and the optimal period T* minimizing that cost is also verified to be finite and unique under some specific conditions. Copyright © 2007 John Wiley & Sons, Ltd.     

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.     

Optimization problems of replacement first or last in reliability theory

This paper takes up age and periodic replacement last models with working cycles, where the unit is replaced before failure at a total operating time T or at a random working cycle Y, whichever occurs last, which is called replacement last. Expected cost rates are formulated, and optimal replacement policies which minimize them are discussed analytically. Comparisons between such a replacement last and the conventional replacement first are made in detail. It is determined theoretically and numerically which policy is better than the other according to the ratios of replacement costs and how the mean time of working cycles affects the comparison results. It is also shown that the unit can be operating for a longer time and avoid unnecessary replacements when replacement last is done. For further studies, expected cost rates of modified models and their applications in a standard cumulative damage model with working cycles are obtained and computed numerically. Finally, case studies on replacement last and first in maintaining electronic systems of naval ships under battle and non-battle statuses are given.     

On a scheme for predictive maintenance

An operating system contains a replaceable unit whose wear (i.e. accumulated amount of damage) can be observed over time. When the wear reaches a certain level the unit is no longer able to function satisfactorily and needs to be replaced. Although units are produced to the same nominal specification there is still some random variation among them in their wear rates. This will be expressed by incorporating a random effect, or frailty term, in the model for individual degradation. There are costs for observing the wear on a unit, for replacing a unit, and for allowing a unit to fail before being replaced. When the last cost is comparatively large replacement before failure is preferable. For some standard examples of wear processes the lifetime distributions are obtained and the cost consequences of particular maintenance schemes are investigated.     

On a Replacement Problem of a Cumulative Damage Model

Items are assumed to fail only by degradation. An appropriate stochastic model of such items is a cumulative process in which an item can fail only when the total amount of damage exceeds a prespecified failure level. This paper introduces a replacement policy in which an item is replaced at a certain level of damage before failure or at failure, whichever occurs first. The optimum replacement level of damage which will minimize the total expected cost per unit of time for an infinite time span is obtained. A numerical example is also presented. The total expected cost for a finite time span is also discussed.     

带随机启动时间与双阈值(m,N)-策略的M/G/1可修排队系统的最优控制策略

本文研究带随机启动时间与双阈值(m,N)-策略的M/G/1可修排队系统,首先讨论系统有关的排队指标,接着研究因为故障而产生的系统的下列可靠性指标,如:服务台首次失效前的寿命分布、不可用度和(0,t]时间内的平均故障次数。最后,在建立费用模型的基础上,结合实际中检测公司检测样品的这一现实情况,研究了双阈值最优控制策略(m^*,N^*),并在同一组参数下与服务台不发生故障时系统的双阈值最优控制策略进行了比较。     

A minimal repair replacement model with two types of failure and a safety constraint

Consider a system subject to two types of failures. If the failure is of type 1, the system is minimally repaired, and a cost C₁ is incurred. If the failure is of type 2, the system is minimally repaired with probability p and replaced with probability 1−p  . The associated costs are _C2,_m$C_{2\text{,}m}$ and _C2,_r$C_{2\text{,}r}$, respectively. Failures of type 2 are safety critical and to control the risk, management has specified a requirement that the probability of at least one such failure occurring in the interval [0, A] should not exceed a fixed probability limit ω. The problem is to determine an optimal planned replacement time T, minimizing the expected discounted costs under the safety constraint. A cost C_r is incurred whenever a planned replacement is performed. Conditions are established for when the safety constraint affects the optimal replacement time and causes increased costs.     

Optimal production control of a dynamic two-product manufacturing system with setup costs and setup times

This paper deals with the optimal control of a one-machine two-product manufacturing system with setup changes, operating in a continuous time dynamic environment. The system is deterministic. When production is switched from one product to the other, a known constant setup time and a setup cost are incurred. Each product has specified constant processing time and constant demand rate, as well as an infinite supply of raw material. The problem is formulated as a feedback control problem. The objective is to minimize the total backlog, inventory and setup costs incurred over a finite horizon. The optimal solution provides the optimal production rate and setup switching epochs as a function of the state of the system (backlog and inventory levels). For the steady state, the optimal cyclic schedule is determined. To solve the transient case, the system's state space is partitioned into mutually exclusive regions such that with each region, the optimal control policy is determined analytically.     

Optimal replacement model with age-dependent failure type based on a cumulative repair-cost limit policy

This paper presents a replacement model with age-dependent failure type based on a cumulative repair-cost limit policy, whose concept uses the information of all repair costs to decide whether the system is repaired or replaced. As failures occur, the system experiences one of the two types of failures: a type-I failure (minor), rectified by a minimal repair; or a type-II failure (catastrophic) that calls for a replacement. A critical type-I failure means a minor failure at which the accumulated repair cost exceeds the pre-determined limit for the first time. The system is replaced at the nth type-I failure, or at a critical type-I failure, or at first type-II failure, whichever occurs first. The optimal number of minimal repairs before replacement which minimizes the mean cost rate is derived and studied in terms of its existence and uniqueness. Several classical models in maintenance literature are special cases of our model.     

The Availability and Hazard of a System Under a Cumulative Damage Process With Replacements

Zacks (Failure distribution associated with general renewal damage processes. In: Nikulin M, Commenges D, Haber C (eds) Probability statistics and modelling in public health. Springer, Berlin, pp 465–475, 2006) studied the reliability function, the hazard function and the distribution of the failure time when a system is subject to a cumulative, compound renewal damage process. The failure occurs when the damage process crosses a threshold β. In the present paper these results are generalized to the model where the system is replaced after failures. Two cases are considered: instant replacement and random positive replacement time. The distribution of the age of the current renewal cycle, as well as its excess life, and the availability function are studied. We derive also the distribution of total time in (0, t) at which the system has been operational.     

An imperfect maintenance model with block replacements

We consider a model in which when a device fails it is either repaired to its condition prior to failure or replaced. Moreover, the device is replaced at times kT, k = 1, 2, … The decision to repair or replace the device at failure depends on the age of the device at failure. We find the optimal block time, T₀, that minimizes the long-run average cost of maintenance per unit time. Our results are shown to extend many of the well known results for block replacement policies.     

An ideal way of obtaining an optimal inspection permutation for a system with components connected in series

The problem of an inspection permutation or inspection strategy (first discussed in a research paper in 1989 and reviewed in another research paper in 1991) is revisited. The problem deals with an N‐component system whose times to failure are independent but not identically distributed random variables. Each of the failure times follows an exponential distribution. The components in the system are connected in series such that the failure of at least one component entails the failure of the system. Upon system failure, the components are inspected one after another in a hierarchical way (called an inspection permutation) until the component causing the system to fail is identified. The inspection of each component is a process that takes a non‐negligible amount of time and is performed at a cost. Once the faulty component is identified, it is repaired at a cost, and the repair process takes some time. After the repair, the system is good as new and is put back in operation. The inspection permutation that results in the maximum long run average net income per unit of time (for the undiscounted case) or maximum total discounted net income per unit of time (for the discounted case) is called the optimal inspection permutation/strategy. A way of determining an optimal inspection permutation in an easier fashion, taking advantage of the improvements in computer software, is proffered. Mathematica is used to showcase how the method works with the aid of a numerical example. Copyright © 2016 John Wiley & Sons, Ltd.     

A number-dependent replacement policy for a system with continuous preventive maintenance and random lead times

This paper considers a number-dependent replacement policy for a system with two failure types that is replaced at the nth type I (minor) failure or the first type II (catastrophic) failure, whichever occurs first. Repair or replacement times are instantaneous but spare/replacement unit delivery lead times are random. Type I failures are repaired at zero cost since preventive maintenance is performed continuously. Type II failures, however, require costly system replacement. A model is developed for the average cost per unit time based on the stochastic behavior of the system and replacement, storage, and downtime costs. The cost-minimizing policy is derived and discussed. We show that the optimal number of type I failures triggering replacement is unique under certain conditions. A numerical example is presented and a sensitivity analysis is performed.     

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.     

Maintenance optimization for second‐hand products following periodic imperfect preventive maintenance warranty period

The maintenance policy for a product's life cycle differs for second‐hand and new products. Although several maintenance policies for second‐hand products exist in the literature, they are rarely investigated with reference to periodic inspection and preventive maintenance action during the warranty period. In this research, we study an optimal post‐warranty maintenance policy for a second‐hand product, which was purchased at age x with a fixed‐length warranty period. During the warranty period, the product is periodically inspected and maintained preventively at a prorated cost borne by the user, while any product failure is only minimally repaired by the dealer. After the warranty expires, the product is self‐maintained by the user for a fixed‐length maintenance period and the costs incurred during this time are fully borne by the user. At the end of the maintenance period, the product is replaced with a product of the user's choice. This study is focused on the determination of an optimal length for the maintenance period after the warranty expiration. As a criterion for the optimality, we adopt the long‐run mean cost during the second‐hand product's life cycle from the user's perspective. Finally, our results are analyzed numerically for sensitive analysis of several relevant factors, assuming that the failure distribution follows a Weibull distribution.     

Mean time to failure and availability of semi-Markov missions with maximal repair

We analyze mean time to failure and availability of semi-Markov missions that consist of phases with random sequence and durations. It is assumed that the system is a complex one with nonidentical components whose failure properties depend on the mission process. The stochastic structure of the mission is described by a Markov renewal process. We characterize mean time to failure and system availability under the maximal repair policy where the whole system is replaced by a brand new after successfully completing a phase before the next phase starts. Special cases involving Markovian missions are also considered to obtain explicit formulas.     

General Distribution Function for Periods of Downtime of a Single System Exceeding a Limiting Downtime

A reparable two-state system whose components upon failure are replaced is considered. The time to failure and the time to repair of the components are a pair of renewal processes. The distribution of the random variable D(τ, t), which is defined as the random sum of those repair times of the system in the interval of time (0, t) that are greater than or equal to a constant time τ, is derived.     

Stability and queueing time analysis of a reader-writer queue with alternating exhaustive priorities

This paper considers a reader-writer queue with alternating exhaustive priorities. The system can process an unlimited number of readers simultaneously. However, writers have to be processed one at a time. Both readers and writers arrive according to Poisson processes. Writer and reader service times are general iid random variables. There is infinite waiting room for both. The alternating exhaustive priority policy operates as follows. Assume the system is initially idle. The first arriving customer initiates service for the class (readers or writers) to which it belongs. Once processing begins for a given class of customers, this class is served exhaustively, i.e. until no members of that class are left in the system. At this point, if customers of the other class are in the queue, priority switches to this class, and it is served exhaustively. This system is analyzed to produce a stability condition and Laplace-Stieltjes transforms (LSTs) for the steady state queueing times of readers and writers. An example is also given.