Optimal replacement policy for a warrantied system with imperfect preventive maintenance operations

This paper determines the optimal replacement time for a system with imperfect preventive maintenance operations under the modified warranty policy. The hazard rate after preventive maintenance lies between the states as good as new and as bad as old. After minimal repair, the hazard rate remains unchanged. Modified warranty policy is a mixed type of free and pro-rata warranty policy. Numerical examples using the Weibull case are presented.     

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.     

Warranty Policies For Non-Repairable Items Under Risk Aversion

An approach is presented for analyzing full replacement and linear prorated warranty policies for items receiving renewable warranties when failure occurs during the warranty interval. The model corrects a model of Thomas and then extends the methodology by describing how sellers' risk aversity influences the policy. For constant failure intensities, linear replacement policies are more attractive to risk averse sellers than shorter term full replacement policies that result in the same average cost.     

Customer-Rush Near Warranty Expiration Limit, and Nonparametric Hazard Rate Estimation From Known Mileage Accumulation Rates

Time or mileage data obtained from warranty claims are generally more accurate for hard failures than for soft failures. For soft failures, automobile users sometimes delay reporting the warranty claim until the warranty coverage is about to expire. This results in an unusually high number of warranty claims near the end of warranty coverage. Because such a phenomenon of customer-rush near the warranty expiration limit occurs due to user behavior rather than due to the vehicle design, it creates a bias in the warranty dataset. Design improvement activities that use field reliability studies based on such data can potentially obtain a distorted picture of the reality, and lead to unwarranted, costly design changes. Research in the area of field reliability studies using warranty data provides several methods for warranty claims resulting from hard failures, and assumes reported time or mileage as actual time or mileage at failure. In this article, the phenomenon of customer-rush near the warranty expiration limit is addressed for arriving at nonparametric hazard rate estimates. The proposed methodology involves situations where estimates of mileage accumulation rates in the vehicle population are available. The claims influenced by soft failures are treated as left-censored, and are identified using information in technician comments about the repair carried out plus, if required, a more involved engineering analysis of field returned parts. Maximum likelihood estimates for the hazard function and their confidence limits are then obtained using Turnbull's iterative procedure. An application example illustrates use of the proposed methodology.     

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.     

Evaluation of average maintenance cost for imperfect-repair model

The imperfect-repair model considers units which are either perfectly-repaired or minimally-repaired, with known, fixed probabilities. Minimal repair is defined, roughly, as returning the item to the population's average state at the failure time. The average cost of maintaining the unit under this model is studied by assuming fixed amounts of cost for perfect and minimal repairs when failed. The authors' measure of such maintenance cost is an average cost per unit-time over an infinite time span. To obtain this average cost, they use the expressions for average numbers of perfect and minimal repairs under this model. They prove the few theorems necessary to derive the formulae for the average cost per unit-time. The results are applied to some examples     

Optimal age-replacement policy for nonrepairable products under renewing free-replacement warranty

This paper investigates the effects of a renewing free-replacement warranty on the age replacement policy for a nonrepairable product. For both warranted, and nonwarranted products, cost models are developed, and the corresponding optimal replacement ages are derived such that the long-run expected cost rate is minimized. Furthermore, we show that the optimal replacement age for a warranted product is closer to the end of the warranty period than for a nonwarranted product. Finally, numerical examples are given to evaluate the impact of a product warranty on the optimal replacement age.     

Optimum Warranty Policies for Nonreparable Items

An approach is presented for establishing and evaluating warranty policies for products receiving renewable warranties when failure occurs during warranty. A general rebate model is described that allows total compensation to a consumer for failures during a fixed period and prorated compensation for a remaining interval of time. Associated warranty costs are weighed against the s-expected benefit to be derived from the program. Conditions for optimum warranty intervals are provided. Closed form results are given for exponentially and uniformly distributed failure times. The more complicated case of Weibull failure times is demonstrated by example. A sensitivity analysis of the parameters is included.     

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.     

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.     

Minimizing cost-functions related to both burn-in and field-operation under a generalized model

Summary and Conclusions-Burn-in is a method used to improve the quality of products. In field operation, only those units which survived the burn-in procedure will be used. This paper considers various additive cost structures related to both burn-in procedure and field operation under a general failure model. The general failure model includes two types of failures. Type I (minor) failure is removed by a minimal repair, whereas type II failure (catastrophic failure) is removed only by a complete repair (replacement). We introduce the following cost structures: (i) the expenses incurred until the first unit surviving burn-in is obtained; (ii) the minimal repair costs incurred over the life of the unit during field use; and (iii) either the gain proportional to the mean life of the unit in field operation or the expenditure due to replacement at a catastrophic failure during field operation. We also assume that, before undergoing the burn-in procedure, the unit has a bathtub-shaped failure rate function with change points t/sub 1/ & t/sub 2/. The optimal burn-in time b/sup */ for minimizing the cost function is demonstrated to be always less than t/sub 1/. Furthermore, a large initial failure rate is shown to justify burn-in, i.e. b/sup */>0. A numerical example is presented.     

Discounted warranty cost of minimally repaired series systems

Many factors should be considered in modeling DWC (discounted warranty cost) of repairable systems or products including system structure, components' failure processes, methods of discounting as well as the warranty policy itself. In this paper, we present DWC models for repairable series systems. In particular, a free repair warranty policy and a pro-rata warranty policy are studied. The impact of repair actions on components' failure times is assumed to be minimal, hence NHPPs are used to describe the failure processes. Two types of discounting methods are considered in this paper: a continuous discount function and a discrete discount function. Expressions for both the expected value and variance of DWC are derived. The applications of our findings can be seen in warranty design, warranty reserve determination and risk analysis. Our approach incorporates the information of system structure, the value of time and the impact of repair actions, which are of great importance to warranty cost prediction and evaluation, but have not been sufficiently studied in the literature of warranty analysis.     

Optimal Age-Replacement Policy Under an Imperfect Renewing Free-Replacement Warranty

This paper investigates the effects of an imperfect renewing free-replacement warranty (RFRW) on the classical age-replacement policy for a product with an increasing failure rate. Under the imperfect RFRW, whenever a product fails during the warranty period, it is replaced by a repaired one from an infinite stock of refurbished items, at no cost to the purchaser, with a new full warranty. We assume that the failure potential of the repaired product is inferior to that of a new product; that is, the repaired product is less reliable than a new one. Long-run expected cost rates for the age-replacement policy are developed for two cases: when the preventive replacement age occurs before, and when it occurs after the warranty expires. The optimal replacement ages that minimize the cost rates are determined, and the impact of an imperfect RFRW on the optimal replacement age is illustrated with a numerical example. We review the literature dealing with warranty and maintenance against this framework, and conclude with some discussions on topics for research in the future. The imperfect RFRW proposed in this paper is practical, and novel; and this study represents a useful extension of Yeh that promises to be of interest to reliability engineers, managers, and theoreticians.     

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.     

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.     

General Distributional Properties of Discounted Warranty Costs With Risk Adjustment Under Minimal Repair

We study the distributional properties (mean, variance, characteristic function) of the discounted warranty cost (DWC) for general warranty programs including free replacement (FRW), pro rata (PRW), and FRW/PRW in the context of minimal repair. Because only failure times & types are needed in the derivation of these properties, the reliability of the systems is modeled according to a general competing risk model. Under these assumptions, our results extend those obtained in 2004 by Bai & Pham. By obtaining the characteristic function of the DWC, we can consider some risk management issues, an area that has not yet been extensively studied in this context. More precisely, we show how risk adjustment principles considered in the economics and actuarial science literature can be applied to the determination of a warranty reserve. Because some of these risk adjustment calculations require the probability mass or density function of the DWC, we show how to numerically invert the characteristic function of the DWC to obtain risk adjusted warranty costs with MATLAB in an appendix.      

Optimal operation of a markovian queueing system with a removable and non-reliable server

This paper deals with the optimal operation of a single removable and non-reliable server in a Markovian queueing system under steady-state conditions. The server can be turned on at arrival epochs or off at departure epochs. We assume that the server may break down only if working and requires repair at a repair facility. Interarrival and service time distributions of the customers are assumed to be exponentially distributed. Breakdown and repair time distributions of the server are assumed to be exponentially distributed. The following cost structure is incurred to the system: a holding cost for each customer in the system per unit time, costs per unit time for keeping the server on or off, a breakdown cost per unit time when a server fails, and fixed costs for turning the server on or off. The total expected cost function per unit time is developed to obtain the optimal operating policy at minimum cost.     

Optimal repair-cost limit for a consumer following expiry of a warranty

Previously, repair-cost limits have been determined only for the producer; or the consumer has been assumed always to replace the product on failure outside the warranty period without the possibility of a minimal repair. This paper discusses the optimal repair-cost limit to the warranty literature. The objective is to find an algorithm to obtain the optimal repair-cost limit for the consumer which minimizes his long-run expected 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.      

Optimal policies with decreasing probability of imperfect maintenance

This study applies periodic preventive maintenance to three repair models: major repaired, minimal repaired, or fixed until perfect preventive maintenance upon failure. Two types of preventive maintenance are performed, namely imperfect preventive maintenance, and perfect preventive maintenance. The probability that preventive maintenance is perfect depends on the number of imperfect maintenance operations performed since the last renewal cycle. Mathematical formulas for the expected cost per unit time are obtained. For each model, the optimum preventive maintenance time T/sup */, which would minimize the cost rate, is discussed. Various special cases are considered. A numerical example is presented.