An optimal repair–replacement policy for a cold standby system with use priority

In this paper, the maintenance problem for a cold standby system consisting of two dissimilar components and one repairman is studied. Assume that both component 1 and component 2 after repair follow geometric process repair and component 1 is given priority in use when both components are workable. Under these assumptions, using geometric process repair model, we consider a replacement policy N under which the system is replaced when the number of failures of component 1 reaches N. Our purpose is to determine an optimal replacement policy N1 such that the average cost rate (i.e. the long-run average cost per unit time) of the system is minimized. The explicit expression for the average cost rate of the system is derived and the corresponding optimal replacement policy N1 can be determined analytically or numerically. Finally, a numerical example is given to illustrate some theoretical results and the model applicability.     

A deteriorating cold standby repairable system with priority in use

In this paper, a cold standby repairable system consisting of two dissimilar components and one repairman is studied. In this system, it is assumed that the working time distributions and the repair time distributions of the two components are both exponential and component 1 is given priority in use. After repair, component 2 is “as good as new” while component 1 follows a geometric process repair. Under these assumptions, using the geometric process and a supplementary variable technique, some important reliability indices such as the system availability, reliability, mean time to first failure (MTTFF), rate of occurrence of failure (ROCOF) and the idle probability of the repairman are derived. A numerical example for the system reliability R(t) is given. And it is considered that a repair-replacement policy based on the working age T of component 1 under which the system is replaced when the working age of component 1 reaches T. Our problem is to determine an optimal policy T^∗ such that the long-run average cost per unit time of the system is minimized. The explicit expression for the long-run average cost per unit time of the system is evaluated, and the corresponding optimal replacement policy T^∗ can be found analytically or numerically. Another numerical example for replacement model is also given.     

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.     

An optimal replacement policy for a multistate degenerative simple system

In this paper, a degenerative simple system (i.e. a degenerative one-component system with one repairman) with k + 1 states, including k failure states and one working state, is studied. Assume that the system after repair is not “as good as new”, and the degeneration of the system is stochastic. Under these assumptions, we consider a new replacement policy T based on the system age. Our problem is to determine an optimal replacement policy T^∗ such that the average cost rate (i.e. the long-run average cost per unit time) of the system is minimized. The explicit expression of the average cost rate is derived, the corresponding optimal replacement policy can be determined, the explicit expression of the minimum of the average cost rate can be found and under some mild conditions the existence and uniqueness of the optimal policy T^∗ can be proved, too. Further, we can show that the repair model for the multistate system in this paper forms a general monotone process repair model which includes the geometric process repair model as a special case. We can also show that the repair model in the paper is equivalent to a geometric process repair model for a two-state degenerative simple system in the sense that they have the same average cost rate and the same optimal policy. Finally, a numerical example is given to illustrate the theoretical results of this model.     

Optimal repair–replacement policies for a system with two types of failures

In this paper, the optimal replacement problem is investigated for a system with two types of failures. One type of failure is repairable, which is conducted by a repairman when it occurs, and the other is unrepairable, which leads to a replacement of the system at once. The repair of the system is not “as good as new”. The consecutive operating times of the system after repair form a decreasing geometric process, while the repair times after failure are assumed to be independent and identically distributed. Replacement policy N is adopted, where N is the number of repairable failures. The system will be replaced at the Nth repairable failure or at the unrepairable failure, whichever occurs first. Two replacement models are considered, one is based on the limiting availability and the other based on the long-run average cost rate of the system. We give the explicit expressions for the limiting availability and the long-run average cost rate of the system under policy N, respectively. By maximizing the limiting availability A(N) and minimizing the long-run average cost rate C(N), we theoretically obtain the optimal replacement policies N^∗ in both cases. Finally, some numerical simulations are presented to verify the theoretical results.     

Analysis of R out of N systems with several repairmen,exponential life times and phase type repair times: An algorithmic approach

An R out of N repairable system consisting of N independent components is operating if at least R components are functioning. The system fails whenever the number of good components decreases from R to R − 1. A failed component is sent to a repair facility having several repairmen. Life times of working components are i.i.d random variables having an exponential distribution. Repair times are i.i.d random variables having a phase type distribution. Both cold and warm stand-by systems are considered. We present an algorithm deriving recursively in the number of repairmen the generator of the Markov process that governs the process. Then we derive formulas for the point availability, the limiting availability, the distribution of the down time and the up time. Numerical examples are given for various repair time distributions. The numerical examples show that the availability is not very sensitive to the repair time distribution while the mean up time and the mean down time might be very sensitive to the repair time distributions.     

Maintenance of continuously monitored degrading systems

This paper considers a condition-based maintenance model for continuously degrading systems under continuous monitoring. After maintenance, the states of the system are randomly distributed with residual damage. We investigate a realistic maintenance policy, referred to as condition-based availability limit policy, which achieves the maximum availability level of such a system. The optimum maintenance threshold is determined using a search algorithm. A numerical example for a degrading system modeled by a Gamma process is presented to demonstrate the use of this policy in practical applications.     

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.     

An extended replacement policy for a deteriorating system with multi-failure modes

In this paper, the maintenance problem for a deteriorating system with k + 1 failure modes, including an unrepairable failure (catastrophic failure) mode and k repairable failure (non-catastrophic failure) modes, is studied. Assume that the system after repair is not “as good as new” and its deterioration is stochastic. Under these assumptions, an extended replacement policy N is considered: the system will be replaced whenever the number of repairable failures reaches N or the unrepairable failure occurs, whichever occurs first. Our purpose is to determine an optimal extended policy N^∗ such that the average cost rate (i.e. the long-run average cost per unit time) of the system is minimized. The explicit expression of the average cost rate is derived, and the corresponding optimal extended policy N^∗ can be determined analytically or numerically. Finally, a numerical example is given to illustrate some theoretical results of the repair model proposed in this paper.     

Optimal replacement policies with minimal repair and age-dependent costs

In this paper, we study a modified minimal repair/replacement problem that is formulated as a Markov decision process. The operating cost is assumed to be a nondecreasing function of the system's age. The specific maintenance actions for a manufacturing system to be considered are whether to have replacement, minimal repair or keep it operating. It is shown that a control limit policy, or in particular a (t, T) policy, is optimal over the space of all possible policies under the discounted cost criterion. A computational algorithm for the optimal (t, T) policy is suggested based on the total expected discounted cost.     

Reliability-based measures for a retrial system with mixed standby components

In this paper, we consider a retrial and repairable multi-component system with mixed warm and cold standby components. It is assumed that the failure times of primary (operating) and warm standby components follow exponential distributions. When a component fails, it is sent to a service station with a single server (repairman) and no waiting space. The failed component is repaired if the server is idle and it has to enter an orbit if the server is busy. The failed component in the orbit will try to get the repair service again after an exponentially distributed random time period. The repair time also has an exponential distribution. The mean time-to-failure, MTTF, and the steady-state availability, A_T(∞), are derived in this retrial and repairable system. Using a numerical example, we compare the systems with and without retrials in terms of the cost/benefit ratios. Sensitivity analysis for the mean time-to-failure and the steady-state availability are investigated as well.     

Optimal replacement policy for a deteriorating system with increasing repair times

This paper considers an optimal maintenance policy for a practical and reparable deteriorating system subject to random shocks. Modeling the repair time by a geometric process and the failure mechanism by a generalized δ-shock process, we develop an explicit expression of the long-term average cost per time unit for the system under a threshold-type replacement policy. Based on this average cost function, we propose a finite search algorithm to locate the optimal replacement policy N^∗ to minimize the average cost rate. We further prove that the optimal policy N^∗ is unique and present some numerical examples. Many practical systems fit the model developed in this paper.     

A geometric process maintenance model with preventive repair

In this paper, a geometric process maintenance model with preventive repair is studied. A maintenance policy (T, N) is applied by which the system will be repaired whenever it fails or its operating time reaches T whichever occurs first, and the system will be replaced by a new and identical one following the Nth failure. The long-run average cost per unit time is determined. An optimal policy (T^∗, N^∗) could be determined numerically or analytically for minimizing the average cost. A new class of lifetime distribution which takes into account the effect of preventive repair is studied that is applied to determine the optimal policy (T^∗, N^∗).     

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.     

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.     

A bivariate optimal replacement policy for a cold standby repairable system with preventive repair

In this paper, the repair-replacement problem for a deteriorating cold standby repairable system is investigated. The system consists of two dissimilar components, in which component 1 is the main component with use priority and component 2 is a supplementary component. In order to extend the working time and economize the running cost of the system, preventive repair for component 1 is performed every time interval T, and the preventive repair is “as good as new”. As a supplementary component, component 2 is only used at the time that component 1 is under preventive repair or failure repair. Assumed that the failure repair of component 1 follows geometric process repair while the repair of component 2 is “as good as new”. A bivariate repair-replacement policy (T, N) is adopted for the system, where T is the interval length between preventive repairs, and N is the number of failures of component 1. The aim is to determine an optimal bivariate policy (T, N)^∗ such that the average cost rate of the system is minimized. The explicit expression of the average cost rate is derived and the corresponding optimal bivariate policy can be determined analytically or numerically. Finally, a Gamma distributed example is given to illustrate the theoretical results for the proposed model.     

Replacement times and costs in a degrading system with several types of failure: The case of phase-type holding times

A reliability system submitted to external and internal failures, that can be repairable or non-repairable, with degradation levels, and with sojourn times phase-type distributed, is considered. Repair is not as good as new, and the repair of internal failure follows policy N, that is, after N completed repairs the system is replaced by a new one to the following failure, repairable or not. For this system, a Markov model is constructed, and the stationary probability vector is calculated. It is shown that the distribution of the time between two consecutive replacements follows a phase-type distribution, whose representation is determined. The costs of these periods are calculated. An optimization problem involving the costs, the availability, and the number of internal repairs is illustrated by a numerical example.     

Hybrid block replacement and inspection policies for a multi-component system with heterogeneous component lives

Novel replacement policies that are hybrids of inspection maintenance and block replacement are developed for an n identical component series system in which the component parts used at successive replacements arise from a heterogeneous population. The heterogeneous nature of components implies a mixed distribution for time to failure. In these circumstances, a hybrid policy comprising two phases, an early inspection phase and a later wear-out replacement phase, may be appropriate. The policy has some similarity to burn-in maintenance. The simplest policy described is such a hybrid and comprises a block-type or periodic replacement policy with an embedded block or periodic inspection policy. We use a three state failure model, in which a component may be good, defective or failed, in order to consider inspection maintenance. Hybrid block replacement and age-based inspection, and opportunistic hybrid policies will also arise naturally in these circumstances and these are briefly investigated. For the simplest policy, an approximation is used to determine the long-run cost and the system reliability. The policies have the interesting property that the system reliability may be a maximum when the long-run cost is close to its minimum. The failure model implies that the effect of maintenance is heterogeneous. The policies themselves imply that maintenance is carried out more prudently to newer than to older systems. The maintenance of traction motor bearings on underground trains is used to illustrate the ideas in the paper.     

Modeling past-dependent partial repairs for condition-based maintenance of continuously deteriorating systems

We are interested in the stochastic modeling of a condition-based maintained system subject to continuous deterioration and maintenance actions such as inspection, partial repair and replacement. The partial repair is assumed dependent on the past in the sense that it cannot bring the system back into a deterioration state better than the one reached at the last repair. Such a past-dependency can affect (i) the selection of a type of maintenance actions, (ii) the maintenance duration, (iii) the deterioration level after a maintenance, and (iv) the restarting system deterioration behavior. In this paper, all these effects are jointly considered in an unifying condition-based maintenance model on the basis of restarting deterioration states randomly sampled from a probability distribution truncated by the deterioration levels just before a current repair and just after the last repair/replacement. Using results from the semi-regenerative theory, the long-run maintenance cost rate is analytically derived. Numerous sensitivity studies illustrate the impacts of past-dependent partial repairs on the economic performance of the considered condition-based maintained system.     

A Closed Queueing Maintenance Network with Two Repair Centres

In this paper we introduce a model to determine the maintenance float needed to maximize the availability of an operating system with N number of circulating units. An implicit enumeration algorithm is used as a solution technique to the closed queueing maintenance network with two types of repairs: minor and major repairs. It is shown that when there is no differentiation of repair type, this special case is obtained as a by-product of the two-repair-centre model. This paper assumes exponential failure times and exponential repair times with load-independent servers. The approach followed in this paper provides an approximate and simple way to solve the maintenance-float problem of this complex closed-network system.