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.     

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.     

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.     

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∗.     

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.     

An age replacement policy with increasing minimal repair cost

A replacement policy for a system in which minimal repair cost increases in system age is considered. If a system fails before age T, it is minimally repaired. Otherwise, the system is replaced when if fails for the first time after age T. The mean cost rate is used as a criterion for optimization. It is shown that the optimal T minimizing the mean cost rate is finite and unique.     

Maintenance strategies following the expiration of warranty

The authors study two types of replacement policies, following the expiration of warranty, for a unit with an IFR failure-time distribution: (1) the user applies minimal repair for a fixed length of time and replaces the unit by a new one at the end of this period; and (2) the unit is replaced by the user at first failure following the minimal repair period. In addition to stationary strategies that minimize the long-run mean cost to the user, the authors also consider nonstationary strategies that arise following the expiration of a nonrenewing warranty. Following renewing warranties, they prove that the cost rate function is pseudo-convex under a fixed maintenance period policy. The same result holds under nonrenewing repair warranties, and nonrenewing replacement warranties when the optimal maintenance period of each cycle is determined as a function of the age of the item in use at the end of the warranty period     

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.     

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 geometric-process repair-model with good-as-new preventive repair

This paper studies a deteriorating simple repairable system. In order to improve the availability or economize the operating costs of the system, the preventive repair is adopted before the system fails. Assume that the preventive repair of the system is as good as new, while the failure repair of the system is not, so that the successive working times form a stochastic decreasing geometric process while the consecutive failure repair times form a stochastic increasing geometric process. Under this assumption and others, by using geometric process we consider a replacement policy N based on the failure number of the system. Our problem is to determine an optimal replacement policy N 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 analytically or numerically. And the fixed-length interval time of the preventive repair in the system is also discussed. Finally, an appropriate numerical example is given. It is seen from that both the optimal policies N** and N* are unique. However, the optimal policy N** with preventive repair is better than the optimal policy N* without preventive repair     

Optimal allocation of minimal & perfect repairs under resource constraints

The effect of a repair of a complex system can usually be approximated by the following two types: minimal repair for which the system is restored to its functioning state with minimum effort, or perfect repair for which the system is replaced or repaired to a good-as-new state. When both types of repair are possible, an important problem is to determine the repair policy; that is, the type of repair which should be carried out after a failure. In this paper, an optimal allocation problem is studied for a monotonic failure rate repairable system under some resource constraints. In the first model, the numbers of minimal & perfect repairs are fixed, and the optimal repair policy maximizing the expected system lifetime is studied. In the second model, the total amount of repair resource is fixed, and the costs of each minimal & perfect repair are assumed to be known. The optimal allocation algorithm is derived in this case. Two numerical examples are shown to illustrate the procedures.     

The branching nonhomogeneous Poisson process and its application to a replacement model

In studying and analysing the failure patterns of complex system, plausible stochastic models are needed to represent the sequence of events. A simple and frequently used model is derived by the assumption that the times-between-failures of a system are exponentially distributed and independent. Experience has shown, however, that successive times-between-failures are not exponentially distributed and not independent.These deviations are due to imperfect search of failed components. We constitute a plausible stochastic process which describes the sequence of events, and obtain the interval reliability and the expected number of failures.As an application of these results, we deal with the replacement model where a system undergoes minimal repair before time T and is replaced at time T. We discuss an optimum policy minimizing the total expected cost per unit time.     

An Optimal Decision Rule for Repair vs Replacement

This paper presents a policy for either repairing or replacing a system that has failed. The policy applies to systems whose mean residual life function is decreasing. An optimal policy is developed that minimizes the cost per unit time for repair and replacement. Results are shown graphically for a particular distribution of time to failure and are motivated in terms of an automobile replacement problem.     

Optimal replacement policy for induced draft fans

This paper derives the optimal block replacement policies for four different operating configurations of induced draft fans. Under the usual assumption of higher cost of repair or replacement on failure compared to preventive replacement, the optimal preventive replacement interval is found by minimising the total relevant cost per unit time. Specifically, this paper finds optimal preventive maintenance strategies for the following two situations.

1. (i)|Both the time to failure and time to carry out minimal repair or replacement are exponentially distributed.

2. (ii)|The time to failure follows the Weibull distribution and there is no possibility of on-line repair or replacement.

For both situations closed form expressions are derived whose solutions give optimum preventive maintenance intervals.     

A failure-repair model with minimal and major maintenance

A Markov model for a continuously operating service device whose condition deteriorates with time in service is proposed. The model incorporates deterioration and Poisson failures, minimal repair, periodic minimal maintenance, and major maintenance after a given number of minimal maintenances. An exact recursive algorithm computes the steady-state probabilities of the device. A cost function is defined using different cost rates for the different types of outages. Based on minimal unavailability or minimal costs, optimal solutions of the model are derived. Major maintenance is seldom beneficial if optimal maintenance intervals are used. If a maintenance policy is based on nonoptical intervals between maintenances, periodic major maintenance can reduce costs     

Joint optimization of block-replacement and periodic-review spare-provisioning policy

This paper considers the problem of joint optimization of "preventive maintenance" and "spare-provisioning policy" for system components subject to wear-out failures. A stochastic mathematical model is developed to determine the jointly optimal "block replacement" and "periodic review spare-provisioning policy." The objective function of the model represents the s-expected total cost of system maintenance per unit time, while the preventive replacement interval and the maximal inventory level are chosen as the decision variables. The objective function of the model is in an analytic form with parameters easily obtainable from field data. The model has been tested using field data on electric locomotives in Slovenian Railways. The calculated optimal values of the model decision variables are realistic. "Sensitivity analysis of the model" shows that the model is relatively insensitive to moderate changes of the parameter values. The results of testing and of sensitivity analysis of the model prove that a trade-off exists between the replacement related cost and the inventory related cost. The jointly optimal preventive replacement interval defined by this model differs appreciably from the corresponding interval determined by the conventional model where only replacement related costs are considered. Also, the results of the sensitivity analysis show that even minor modification of the value of each model decision variable (without the appropriate adjustment of the value of the other decision variable) can lead to important increase of the s-expected total cost of system maintenance. This indicates that separate optimization of preventive maintenance policy and spare-provisioning policy does not ensure minimal total cost of system maintenance. This model can be readily applied to optimize maintenance procedures for a variety of industrial systems, and to upgrade maintenance policy in situations where block replacement preventive maintenance is already in use.     

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.     

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.     

A maintenance policy for repairable systems based on opportunisticfailure-rate tolerance

An opportunistic hazard rate replacement policy for a repairable system with several types of units is presented. A unit is repaired at failure when the hazard rate falls in (0, L-u). A unit is replaced at failure when the hazard rate falls in (L-u , L). An operating unit is replaced when its hazard rate reaches L. When a unit is replaced because its hazard rate reaches L, all operating units with their hazard rates falling in (L-u, L) are replaced. The long-run mean cost rate as a function of L and u is derived. Optimal L and u are obtained to minimize the total maintenance cost rate. Application and analysis of results are demonstrated through a numerical example. The maintenance model is designed for a system with multitype units. Each type has its own increasing hazard rate. Units are repaired or replaced depending on their hazard rate at a failure or active replacement of another unit. The repair interval, replacement limit, and replacement tolerance are determined to yield the optimal total maintenance cost rate     

An Extended Periodic Imperfect Preventive Maintenance Model With Age-Dependent Failure Type

This paper presents a generalized periodic imperfect preventive maintenance (PM) model for a system with age-dependent failure type. The imperfect PM model proposed in this study incorporates improvement factors vis-À-vis the hazard-rate function, and effective age. As failures occur, the system experiences one of the two types of failure: type-I failure (minor), and type-II failure (catastrophic). Type-I failures are rectified with minimal repair. In a PM period, the system is preventively maintained following the occurrence of a type-II failure, or at age $T$ , whichever takes place first. At the $N$th PM, the system is replaced. An approach that generalizes the existing studies on the periodic PM policy is proposed. Taking age-dependent failure type into consideration, the objective consists of determining the optimal PM & replacement schedule that minimize the expected cost per unit of time, over an infinite horizon.