Extended optimal age-replacement policy with minimal repair of a system subject to shocks

An operating system is subject to shocks that arrive according to a non-homogeneous Poisson process. As shocks occur the system has two types of failure: type I failure (minor) or type II failure (catastrophic). A generalization of the age replacement policy for such a system is proposed and analyzed in this study. Under such a policy, if an operating system suffers a shock and fails at age y (⩽t), it is either replaced by a new system (type II failure) or it undergoes minimal repair (type I failure). Otherwise, the system is replaced when the first shock after t arrives, or the total operating time reaches age T (0 ⩽ t ⩽ T), whichever occurs first. The occurrence of those two possible actions occurring during the period [0, t] is based on some random mechanism which depends on the number of shocks suffered since the last replacement. The aim of this paper is to find the optimal pair (t¹, T¹) that minimizes the long-run expected cost per unit time of this policy. Various special cases are included, and a numerical example is given.     

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.     

Optimization issues in k-out-of-n systems

The k-out-of-n system is a system consisting of n independent components such that the system works if and only if at least k of these n components are successfully running. Each component of the system is subject to shocks which arrive according to a nonhomogeneous Poisson process. When a shock takes place, the component is either minimally repaired (type 1 failure) or lying idle (type 2 failure). Assume that the probability of type 1 failure or type 2 failure depends on age. First, we investigate a general age replacement policy for a k-out-of-n system that incorporates minimal repair, shortage and excess costs. Under such a policy, the system is replaced at age T or at the occurrence of the (n-k + 1)th idle component, whichever occurs first. Moreover, we consider another model; we assume that the system operates some successive projects without interruptions. The replacement could not be performed at age T. In this case, the system is replaced at the completion of the Nth project or at the occurrence of the (n-k + 1)th idle component, whichever occurs first. For each model, we develop the long term expected cost per unit time and theoretically present the corresponding optimum replacement schedule. Finally, we give a numerical example illustrating the models we proposed. The proposed models include more realistic factors and extend many existing models.     

A note on replacement policy for a system subject to non-homogeneous pure birth shocks

A system is subject to shocks that arrive according to a non-homogeneous pure birth process. As shocks occur, the system has two types of failures. Type-I failure (minor failure) is removed by a general repair, whereas type-II failure (catastrophic failure) is removed by an unplanned replacement. The occurrence of the failure type is based on some random mechanism which depends on the number of shocks occurred since the last replacement. Under an age replacement policy, a planned (or scheduled) replacement happens whenever an operating system reaches age T. The aim of this note is to derive the expected cost functions and characterize the structure of the optimal replacement policy for such a general setting. We show that many previous models are special cases of our general model. A numerical example is presented to show the application of the algorithm and several useful insights.     

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.     

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.     

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.     

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.     

A Generalized Block Replacement Policy with Minimal Repair and General Random Repair Costs for a Multi-unit System

A generalization of the block replacement policy (BRP) is proposed and analysed for a multi-unit system which has the specific multivariate distribution. Under such a policy an operating system is preventively replaced at times kT (k = 1, 2, 3,...), as in the ordinary BRP, and the replacement of the failed system at failure is not mandatory; instead, a minimal repair to the component of the system can be made. The choice of these two possible actions is based on some random mechanism which is age-dependent. The cost of the ith minimal repair of the component at age y depends on the random part C(y) and the deterministic part C_i(y). The aim of the paper is to find the optimal block interval T which minimizes the long-run expected cost per unit time of the policy.     

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.     

Analysis of a system with minimal repair and its application to replacement policy

The periodic replacement with minimal repair at failures is studied by many authors, however, there is not a clear definition for minimal repair. This paper defines a minimal repair in the term of the failure rate and devices some probability quantities and reliability properties. As an application of these results, the replacement model where a system is replaced at time T or at nth failure are considered and the optimum policies are discussed.     

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 δ-shock maintenance model for a deteriorating system

In this paper, a δ-shock maintenance model for a deteriorating system is studied. Assume that shocks arrive according to a renewal process, the interarrival time of shocks has a Weibull distribution or gamma distribution. Whenever an interarrival time of shocks is less than a threshold, the system fails. Assume further the system is deteriorating so that the successive threshold values are geometrically nondecreasing, and the consecutive repair times after failure form an increasing geometric process. A replacement policy N is adopted by which the system will be replaced by an identical new one at the time following the Nth failure. Then the long-run average cost per unit time is evaluated. Afterwards, an optimal policy N^* for minimizing the long-run average cost per unit time could be determined numerically.     

Random number of units for K-out-of-n systems

This paper takes up the reliability and preventive replacement problems for a K-out-of-n system, where K is a stochastic parameter provided. Firstly, we consider the case when K is predefined as constant numbers as is done with the traditional method, and obtain the system reliability R(t), mean time to failure (MTTF), and replacement policies, in which the number n of units for replacement and replacement time T of operation are, respectively, optimized. Secondly, we focus on the above discussions again when K cannot be predefined constantly, but it is a random variable with an estimated probability function. Furthermore, we give approximate computations in an easier way for MTTF, optimal number n* and replacement time T*, respectively.     

Optimal age-replacement time with minimal repair based on cumulative repair-cost limit for a system subject to shocks

An operating system is subject to random shocks that arrive according to a non-homogeneous Poisson process and cause the system failed. System failures experience to be divided into two categories: a type-I failure (minor), rectified by a minimal repair; or a type-II failure (catastrophic) that calls for a replacement. An age-replacement model is studied by considering both a cumulative repair-cost limit and a system’s entire repair-cost history. Under such a policy, the system is replaced at age T, or at the k-th type-I failure at which the accumulated repair cost exceeds the pre-determined limit, or at any type-II failure, whichever occurs first. The object of this article is to study analytically the minimum-cost replacement policy for showing its existence, uniqueness, and the structural properties. The proposed model provides a general framework for analyzing the maintenance policies, and presents several numerical examples for illustration purposes.     

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.     

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.