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.     

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.     

A general replacement of a system subject to shocks

A system is subject to shocks that arrive according to a nonhomogeneous Poisson process. The system is replaced at age T at a fixed cost c₀. If the k-th shock arrives at time S_k<T, it is either a fatal shock with probability p(S_k) or a nonfatal shock with probability 1−p(S_k). The fatal shock causes the system total breakdown, and the system is replacd at a cost c_∞. The nonfatal shock weakens the system and makes it more expensive to run. 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.     

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 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.     

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     

Replacement Policies for a Unit with Random and Wearout Failures

This paper considers three replacement models with random and wearout failures; a) the unit is replaced at failure, b) the unit undergoes minimal repair at failure, and c) the unit is replaced at failure only in a wearout failure period. Optimum replacement policies which minimize the s-expected cost rate for each model are discussed.     

Optimum Age Replacement in the Bivariate Exponential Case

An age replacement policy is considered for pairs of units which operate in parallel and which have lifetimes displaying a bivariate exponential distribution. Both units are to be replaced at the same time. The limiting expected cost per unit time is the optimization criterion. The results state that no replacements should be made until at least one of the units in the pair fails. Both units shoould then be replaced either when one fails or when both fail, depending on which procedure involves the smaller limiting expected cost per unit time.     

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.     

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.     

Cost-benefit analysis of a single server three-unit redundant system with inspection, delayed replacement and two types of repair

This paper deals with a redundant system with two types of spare units—a warm standby unit for instantaneous replacement at the time of failure of the active unit and a cold standby (stock) unit which can be replaced after a random amount of time. The type of the failure of operative or warm standby unit is detected by inspection only. The service facility plays the triple role of replacement, inspection and repair of a unit. Failure time distributions of operative and warm standby units are negative exponential whereas the distributions of replacement time, inspection time and repair times are arbitrary. The system has been studied by using regenerative points.     

Profit analysis of a cold standby system with two repair distributions

A single server two-identical unit cold standby system is analysed. Each unit has two operative modes—normal and quasi-normal. When a normal unit fails, it undergoes minor repair with probability p₁ and p₂ respectively. Upon minor repair unit works with reduced efficiency and is known as quasi-normal unit while upon major repair unit works as good as new (normal unit). When a quasi-normal unit fails, it undergoes minor or major repair with probability q₁ and q₂ respectively. Failure rates of normal and quasi-normal units are different. Failure time distributions are negative exponential whereas repair time distributions are general. Using regeneration point technique in MRP the system characteristics of interest to system designers and operations managers have been obtained.     

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     

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.     

Cost analysis of a two dissimilar-unit cold standby redundant system subject to inspection and two types of repair

This paper deals with the cost analysis of a two dissimilar-unit cold standby redundant system subject to inspection and two types of repair where each unit of the system has two modes, normal and failed. It is assumed that the failure, repair, replacement and inspection times are stochastically independent random variables each having an arbitrary distribution. The cold standby unit replaces the failed operative unit after a random amount of time. An inspection is required to decide whether it needs type I (minor repair) or type 2 (major repair). In this system the repairman is not always available with the system, but is called whenever the operative unit fails. The system is analysed by the semi-Markov process technique. Some reliability measures of interest to system designers as well as operations managers have been obtained. Pointwise availability, steady-state availability, busy period by a server and the expected cost per unit time of the system are obtained. Certain important results have been derived as particular cases.     

All opportunity-triggered replacement policy for multiple-unitsystems

A model is presented for a system which consists of n i.i.d units. Hazard rates of these units are increasing in time. A unit is replaced at failure or when the age of a unit exceeds T, whichever occurs first. When a unit is replaced, all the operating units with their age in the interval (T-w,T) are replaced. Both failure replacement and active replacement create the opportunities to replace other units preventively. This policy allows joint replacements and avoids the disadvantages resulting from replacement of new units, down time, and unrealistic assumptions for distributions of unit life. An algorithm is developed to compute the steady-state cost rate. Optimal T&W are obtained to minimize the mean total replacement cost rate. Application and analysis of results are illustrated through a numerical example     

Stochastic analysis of a two unit cold standby system with preparation time for repair

This paper deals with the cost-benefit analysis of a two unit cold standby system in which the cold standby unit replaces the failed operative unit after a random amount of time. Inspection is required to decide whether it needs type I or type II repair. Failure, repair, replacement and inspection time distributions are arbitrarily distributed. A repair man is not always available with the system, but is called for repair whenever the operative unit fails.     

A geometric-process maintenance model for a deteriorating system under a random environment

This paper studies a geometric-process maintenance-model for a deteriorating system under a random environment. Assume that the number of random shocks, up to time t, produced by the random environment forms a counting process. Whenever a random shock arrives, the system operating time is reduced. The successive reductions in the system operating time are statistically independent and identically distributed random variables. Assume that the consecutive repair times of the system after failures, form an increasing geometric process; under the condition that the system suffers no random shock, the successive operating times of the system after repairs constitute a decreasing geometric process. A replacement policy N, by which the system is replaced at the time of the failure N, is adopted. An explicit expression for the average cost rate (long-run average cost per unit time) is derived. Then, an optimal replacement policy is determined analytically. As a particular case, a compound Poisson process model is also studied.     

Extended block replacement policy of a system subject to shocks

A generalization of the block replacement (BR) policy is proposed and analyzed for a system subject to shocks. Under such a policy, an operating system is preventively replaced by new ones at times i·T (i=1,2,3,...) independently of its failure history. If the system fails in: (a) ((i-1)·T, (i-1)·T+T₀), it is either replaced by a new one or minimally repaired; or (b) ((i-1)·T+T₀, i·T), it is either minimally repaired or remains inactive until the next planned replacement. The choice of these two actions is based on some mechanism (modeled as random) which depends on the number of shocks since the latest replacement. The average cost rate is obtained using the results of renewal reward theory. The model with two variables is transformed into a model with one variable and the optimum policy is discussed. Various special cases are considered. The results extend many of the well-known results for BR policies     

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.