An age replacement policy with minimal repair and general random repair cost

An age replacement policy is introduced which incorporates minimal repair, replacement, and general random repair costs. If an operating unit fails at age y<T, it is either replaced by a new unit with probability p(y) at a cost c₀, or it undergoes minimal repair with probability q(y) = 1−p(y). Otherwise, a unit is replaced when it fails for the first time after age T. The cost of the i-th minimal repair of an unit 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 T which minimizes the long run expected cost per unit time of the policy. Various special cases are considered.     

Cost limit replacement policy under minimal repair

This paper presents a policy for either repairing or replacing a system that has failed. When a system requires repair, it is first inspected and the repair cost is estimated. Repair is only then undertaken if the estimated cost is less than the “repair cost limit”. However, the repair cannot return the system to “as new” condition but instead returns it to the average condition for a working system of its age. Examples include complex systems where the repair or replacement of one component does not materially affect the condition of the whole system. A Weibull distribution of time to failure and a negative exponential distribution of estimated repair cost are assumed for analytic amenability. An optimal “repair cost limit” policy is developed that minimizes the average cost per unit time for repairs and replacement. It is shown that the optimal policy is finite and unique.     

Pseudodynamic cost limit replacement model under minimal repair

This paper presents a mathematical model to evaluate pseudodynamic cost limit replacement policies for a system that follows a general time-to-failure distribution. When the failed system requires repair, it is first inspected and the repair cost is estimated. Minimal repair is only then undertaken if the estimated cost is less than the exponentially declining repair cost limit. A negative exponential distribution of estimated repair cost is assumed for analytic tractability. An example with a Weibull time-to-failure distribution is given to illustrate the computational results.     

An ordering policy with age-dependent minimal repair and age-dependent random repair costs

In this paper we consider an ordering policy for a one-unit system with age-dependent minimal repair and age-dependent random repair costs. We derive the expected cost per unit time in the steady-state as a criterion of optimality and seek the optimum policy by minimizing that cost. We show that, under certain conditions, there exists a finite and unique optimum policy. Various special cases are discussed.     

Periodic replacement with minimal repair at failure and general random repair cost for a multi-unit system

A policy of periodic replacement with minimal repair at failure is considered for the multi-unit system which have the specific multivariate distribution. Under such a policy an operating system is completely replaced whenever it reaches age T (T > 0) at a cost c₀ while minimal repair is performed at any intervening component failures. The cost of the j-th minimal repair to the component which fails at age y is g(C(y),c_j(y)), where C(y) is the age-dependent random part, c_j(y) is the deterministic part which depends on the age and the number of the minimal repair to the component, and g is an positive nondecreasing continuous function. A simple expression is derived for the expected minimal repair cost in an interval in terms of the cost function and the failure rate of the component. Necessary and sufficient conditions for the existence of an optimal replacement interval are exhibited.     

Optimal stocking for replacement with minimal repair

Joint stocking and replacement model with minimal repair at failure is considered. A recursive relationship among the optimal replacement intervals is obtained, which shows that replacement intervals are an increasing sequence due to the inventory carrying cost. Using the relationship, a procedure is given for determining how many units to purchase on each order and when to replace each unit after it has begun operating so as to minimize the total cost per unit time over an infinite time span. The problem can be simplified if equal replacement intervals are assumed, and the solution is very close to that of the unconstrained problem.     

An economic discrete replacement policy for a shock damage model with minimal repairs

A discrete replacement model for a repairable system which is subject to shocks and minimal repairs is discussed. Such shocks can be classified, depending on its effect to the system, into two types: Type I and Type II shocks. Whenever a type II shock occurs causes the system to go into failure, such a failure is called type II failure and can be corrected by a minimal repair. A type I shock does damage to the system in the sense that it increases the failure rate by a certain amount and the failure rate also increases with age due to aging process without external shocks; furthermore, the failure occurred in this condition is called type I failure. The system is replaced at the time of the first type I failure or the n-th type Il failure, whichever occurs first. Introducing costs due to replacement and mininal repairs, the long-run expected cost per unit time is derived as a criterion of optimality and the optimal number n∗ found by minimizing that cost. It is shown that, under certain conditions, there exists a finite and unique optimal number n∗.     

Periodic replacement when minimal repair costs depend on the age and the number of minimal repair for a multi-unit system

A policy of periodic replacement with minimal repair at failure is considered for the multi-unit system which have the specific multivariate distribution. Under such a policy the system is replaced at multiples of some period T while minimal repair is performed at any intervening component failures. The cost of a minimal repair to the component is assumed to be a function of its age and the number of minimal repair. A simple expression is derived for the expected minimal repair cost in an interval in terms of the cost function and the failure rate of the component. Necessary and sufficient conditions for the existence of an optimal replacement interval are exhibited.     

Multi-standby system with repair and replacement policy

This paper investigates the mathematical model of a system composed of (m + 1) non-identical units—one functioning and m standbys. Each unit of the system has four possible states—normal, partial failure, total failure and repair facility—the last one meaning that the totally failed unit is being attended to at the repair facility where it might be either repaired or eventually rejected and replaced. The normal and partial failure states are up states while the other two are down states. The system breaks down when the (m + 1)th unit after total failure is finally rejected and no standby remains to replace it. Several reliability characteristics of interest to system designers as well as operations managers have been computed. Results obtained earlier are verified as particular cases.     

Periodic replacement when minimal repair costs depend on the age and the number of minimal repairs for a multi-unit system

A policy of periodic replacement with minimal repair at failure is considered for a multi-unit system which has a specific multivariate distribution. Under such a policy the system is replaced at multiples of some period T while minimal repair is performed for any intervening component failure. The cost of a minimal repair to the component is assumed to be a function of its age and the number of minimal repairs. A simple expression is derived for the expected minimal repair cost in an interval in terms of the cost function and the failure rate of the component. The necessary and sufficient conditions for the existence of an optimal replacement interval are found.     

Improvement, deterioration, and optimal replacement underage-replacement with minimal repair

Improvement and deterioration for a repairable system are studied, in particular in terms of the effect of ageing on the distribution of the time to first failure under a nonhomogeneous Poisson process. For a repairable system undergoing minimal repair, the optimal replacement time under the age replacement policy is discussed     

A multi-standby multi-failure mode system with repair and replacement policy

This paper investigates the mathematical model of a system composed of (m + 1) non identical units—one functioning and m standby. Each unit of the system has three possible states—normal, degraded and failed. We consider two types of repair facilities—overhaul and minor repair. The system breaks down when the (m + 1)th unit after total failure is finally rejected and no standby remains to replace it. Several reliability characteristics of interest to system designers as well as operations managers have been computed. Results obtained earlier are verified as particular cases.     

Optimal periodic preventive repair and replacement policy assuming geometric process repair

In this paper, a simple deteriorating system with repair is studied. When failure occurs, the system is replaced at high cost. To extend the operating life, the system can be repaired preventively. However, preventive repair does not return the system to a "good as new" condition. Rather, the successive operating times of the system after preventive repair form a stochastically decreasing geometric process, while the consecutive preventive repair times of the system form a stochastically increasing geometric process. We consider a bivariate preventive repair policy to solve the efficiency for a deteriorating & valuable system. Thus, the objective of this paper is to determine an optimal bivariate replacement policy such that the average cost rate (i.e., the long-run average cost per unit time) is minimized. The explicit expression of the average cost rate is derived, and the corresponding optimal replacement policy can be determined numerically. An example is given where the operating time of the system is given by a Weibull distribution.     

Optimum policy of one-unit system with two types of maintenance and minimal repair

This paper deals with a one-unit system with minimal repair. Two policies (new Policy IV and Policy IV′) are considered. Under these policies, the Laplace transform of the point-wise availability and the stationary availability of the system are obtained using not the renewal theory but the supplementary variable method. And under new Policy IV, the optimum policy in the sense of the availability is discussed.     

Cost-benefit analysis of a one-server two-unit cold standby system with repair and age replacement

This paper deals with the cost-benefit analysis of a one-server two-identical-unit cold standby system with repair and preventive maintenance (PM). The PM is of the age replacement type, where, if a unit has been in operation for a certain period of time, which may be a random variable, and if the other unit is in standby, the operating unit is taken off for PM. The expected net revenue in the interval [0,t) is obtained using two different approaches. The first approach is more general and allows nonlinearities in the revenue and costs. It is assumed that the revenue obtained by operating a unit for an uninterrupted interval of time is some function of the length of that interval. Similarly, the cost of a repair or PM action is function of the length of the repair or PM time, respectively, for that action. The second approach assumes that the revenue, repair cost and PM cost vary linearly with time. The pointwise availability is derived. The busy period of the server is divided into time spent in performing repair and time spent on PM. The expected net revenue in [0,t) is obtained. Both techniques make use of regeneration points. It is finally shown that the results of the first approach under assumptions of linear revenue and cost functions reduce to those of the second approach.     

On computing the repair cost of a maintained reliability system with known repair cost function and probability of repair

In this paper, a simplified analytic cost model for maintained reliability system under opportunistic repair scheme is discussed. Life cycle cost curves under various operating life cycle times and linear repair cost function is derived.     

A note on minimal repair

The cumulative function for the number of failures for a unit which is subjected to minimal repair after each failure is a nonstationary Poisson process. In this note, a simple method for obtaining this result is presented. A conditional probability approach is used for the derivation     

An optimal replacement policy for a three-state repairable system with a monotone process model

In this paper, a deteriorating simple repairable system with three states, including two failure states and one working state, is studied. Assume that the system after repair cannot be "as good as new", and the deterioration of the system is stochastic. Under these assumptions, we use a replacement policy N based on the failure number of the system. Then our aim is to determine an optimal replacement policy N/sup */ such that the average cost rate (i.e., the long-run average cost per unit time) is minimized. An explicit expression of the average cost rate is derived. Then, an optimal replacement policy is determined analytically or numerically. Furthermore, we can find that a repair model for the three-state repairable system in this paper forms a general monotone process model. Finally, we put forward a numerical example, and carry through some discussions and sensitivity analysis of the model in this paper.