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Periodic preventive maintenance policies following the expiration of warranty
Abstract
Two types of replacement policies, following the expiration of warranty, have been discussed in the literature: 1) minimal repair is applied for a fixed length of time and the unit is replaced at the end of this period, and 2) the unit is replaced at first failure following the minimal repair period. This paper considers a periodic preventive maintenance (PM) policy after the warranty period is expired. We assume that each PM, following the expiration of warranty, slows the rate of system degradation while the failure rate of the system keeps increasing monotonically. In this paper we determine the optimal number and period for the periodic PM following the expiration of warranty which minimize the expected long-run maintenance cost per unit time. Explicit solutions for the optimal periodic PM are presented for illustrative purposes.
... Planned maintenance is based on a fixed period of time for equipment maintenance activities and is done by arranging maintenance personnel and equipment in advance to save time and temporary scheduling costs. However, the formulation of the maintenance cycle does not take into account the actual state of the equipment for its subsequent operation, and often, the occurring maintenance is not enough [3][4][5]. Zhou Jian [6] proposed a failure rate model that takes into account the situation of "repair not new" and influences the formulation of preventive maintenance cycles through the influence factor, so that the maintenance plan is more in line with the failure pattern of the components and avoids the over-maintenance and under-maintenance that may be encountered in traditional preventive maintenance. He Xuehong [7] took into account the aging phenomenon of the equipment with the increase of the operation time and the amount of maintenance, so he introduced the service age regression factor and then formulated the indefinite cycle dimension strategy according to the change of the reliability of the equipment in different maintenance cycles. ...
Aiming at the problem of high failure rate and inconvenient maintenance of wind turbine gearboxes, this paper establishes a stochastic differential equation model that can be used to fit the change of gearbox oil temperature and adopts an iterative computational method and Markov-based modified optimization to fit the prediction sequence in order to realize the accurate prediction of gearbox oil temperature. The model divides the oil temperature change of the gearbox into two parts, internal aging and external random perturbation, adopts the approximation theorem to establish the internal aging model, and uses Brownian motion to simulate the external random perturbation. The model parameters were calculated by the Newton–Raphson iterative method based on the gearbox oil temperature monitoring data. Iterative calculations and Markov-based corrections were performed on the model prediction data. The gearbox oil temperature variations were simulated in MATLAB, and the fitting and testing errors were calculated before and after the iterations. By comparing the fitting and testing errors with the ordinary differential equations and the stochastic differential equations before iteration, the iterated model can better reflect the gear oil temperature trend and predict the oil temperature at a specific time. The accuracy of the iterated model in terms of fitting and prediction is important for the development of preventive maintenance.
... Oh and Bai [30] estimated reliability and lifetime distributions by gathering additional field data after a warranty had expired. Jung et al. [31] put forward a periodic maintenance strategy that considers the limits of maintenance time and changes in warranty periods. The failure distributions of warranty data have also been considered in Suzuki (1987) [32], Chou and Tang (1992) [33], Lawless et al. (1995) [34], and [35]. ...
The product warranty has become an indispensable facet of business operations. Burn-in is effective at eliminating infant mortality and improving operational reliability levels for consumers. This paper considers the influence of different failure states and different phases of product reliability and warranty policies on warranty costs from pre-delivery inspection to the end of the warranty period. We then propose a comprehensive warranty cost model (CWCM) that considers burn-in, free replacement warranty (FRW) and pro-rata warranty (PRW) as three phases for repairable products presenting two types of failure (minimal and catastrophic failure) that involve minimal repair and replacement, respectively. Warranty costs are the result of a combination of the three phases where two types of failure occur individually or simultaneously. Moreover, we developed a framework for the modeling process of warranty costs, and the effects of various parameters such as the burn-in time, warranty period and distribution function on warranty costs were analyzed. Finally, a practical case was examined by using a warranty cost model, and through an after-sales service data analysis, we obtained the failure rate distribution and optimal warranty length by minimizing the average warranty cost, which can serve as a reference for manufacturers when developing warranty policies.
... Recently, Wang and Sheu [26] and Yeh et al. [27] study the optimal production run length for a deterioration production system in which the products are sold with free minimal repair warranty. Other related studies include Berg et al. [1], Jung et al. [11], Kelle et al. [12] and their references. ...
This paper studies the optimal production run length for the economic manufacturing quantity model with deteriorating production process and taking the restoration cost into account. We show that there exists a unique optimal production run length such that the expected total cost is minimized. In addition, bounds for the optimal production run length are provided to develop the solution procedure. Finally, a numerical example is given to illustrate the results and sensitivity analysis is performed to study the effects of changing parameters values on the optimal solution of the system.
... Jung et al. [81] study the optimal replacement policies during post warranty period considering the expected downtime per unit time and the expected cost rate per unit time. Jung [79] consider the optimal period for the periodic PM during the post warranty period which minimize the expected long-run maintenance cost per unit time. ...
... Different kinds of multi state systems involve minimal repair to improve system conditions 15-22 . These are warranted and guaranteed systems [23][24][25] and systems working with spare parts and maintenance tools safety stock [26][27][28] . All these models are based on total cost function such as average cost per unit time given as ...
This paper explores a reliability based heuristic mathematical model for basic and classical age replacement policy. Using new model, optimal preventive age replacement policy is determined to maximize system reliability. For special systems having exponential density function with constant hazard function, a different simple decision-making model is proposed.. A case study of two industrial machines was conducted using weibull density and exponential density function.
... Jung et al. [2] consider the maintenance strategies following the expiration of warranty. In addition, Jung and Park [3] suggest the optimal preventive maintenance policy after the warranty period is expired. ...
A model of the optimal maintenance period for repairable product after warranty expiration under renewing warranty is proposed. In the process to fulfilling warranty, the failures of product within warranty period happen randomly, the replacements of products successively failed with warranty period form a renewing process. Given failure rate of product, the period of fulfilling warranty is function of warranty period. User purchasing a product is considered an investment; the period of fulfilling warranty and the maintenance period after warranty make up the life cycle of product investment. Taking various cost factors into account, making model of the cost rate in life cycle of product investment under renewing warranty, the cost rate is function of maintenance period. Give the failure rate of product is an increasing function, it is derived that there is unique optimal maintenance period minimizing the cost rate. Finally, numerical example is given for illustration.
... Research has been extensively conducted to evaluate warranty related decision problems in consideration of periodic PM policies. For example, Park et al. [36] considered a periodic PM policy along with minimal repairs after breakdowns, and derived the optimal period and number of PM actions, Seo and Bai [32] depicted a periodic PM policy for two cases in which the time of PM can be ignored or not, and others [15], [19]- [21], [42], and [47]. ...
Warranties play an important role for marketing products in the intensively competitive market, since an attractive policy can dramatically increase market share. However, offering an unlimited warranty with intent to monopolize the market is unrealistic because of the extremely high cost for maintaining such a policy. This paper considers a decision problem under the policy of a pro-rata warranty (PRW), and a Bayesian decision model in determining the optimal warranty proportion is proposed, where a periodic preventive maintenance (PM) program is performed during the warranty term to slow down the product deterioration, and a nonhomogeneous Poisson process is employed to describe the successive breakdown times of the deteriorating product. The proposed approach provides decision-making techniques, not only for taking action in the light of all available relevant information, but also for maximizing expected profits. Finally, an application case is utilized to demonstrate the feasibility and validity of the proposed approach, and sensitivity analyses regarding the important factors that may have impacts on the warranty proportion are also discussed.
... Djamaludin et al. [184] reviews this literature and develops a framework to study warranty and maintenance. Gi et al. [60] deals with maintenance following the expiration of warranty. Yeh and Lo [61] deals with a model where preventive maintenance reduces the age of the system. ...
Warranty is an important element of marketing new products as better warranty signals higher product quality and provides greater assurance to customers. Servicing warranty involves additional costs to the manufacturer and this cost depends on product reliability and warranty terms. Product reliability is influenced by the decisions made during the design and manufacturing of the product. As such warranty is very important in the context of new products. Product warranty has received the attention of researchers from many different disciplines and the literature on warranties is vast. This paper carries out a review of the literature that has appeared in the last ten years. It highlights issues of interest to manufacturers in the context of managing new products from an overall business perspective.
... Pham and Zhang [17] developed a software cost model with warranty and risk costs. Jung et al. [9] considered a periodic preventive maintenance policy after the expiry of warranty period. He determined the optimal number and period for the periodic preventive maintenance following the expiry of warranty which minimizes the expected long-run maintenance cost per unit time. ...
In the present investigation, we propose a hybrid warranty model for the renewing pro-rata warranty (RPRW), in which the failure rate of units, cost of preventive maintenance and cost of replacement are assumed to be constant. The expected total discounted maintenance cost has been derived in explicit form for different life time distributions viz. exponential, Rayleigh and Weibull distributions. We determine the optimal number and optimal period for preventive maintenance after the expiry of the warranty. Numerical illustrations are provided to check the validity of the analytical results.
... The manufacturer pays the costs of rectifying failures under warranty and the buyer pays post-warranty costs. Finally, Jung et al. [47] determined the optimal number and period for periodic PMs following the expiry of a warranty by minimizing the buyer's asymptotic expected cost per unit time. Both renewing PRWs and renewing FRWs are considered. ...
Businesses use equipment to deliver their outputs (products and/or services) and individuals use consumer products to satisfy their personal needs and provide entertainment. Both types of item are getting more complex due to rapid advances in technology and to the increasing expectations of customers (businesses and individuals). These customers need to be assured that the items will perform satisfactorily over their useful lives and one way of providing this assurance is through a product warranty. Most countries have either enacted or are in the process of enacting stricter warranty legislation to protect customers’ interests. Manufacturers are required to rectify all failures that occur over the warranty period and this is referred to as warranty servicing. Warranty servicing results in additional costs to the manufacturer and it has been reported in the literature that these costs can vary between 2% and 10% of an item’s sale price depending on the product and manufacturer.
This paper develops the optimal periodic preventive maintenance policy following the expiration of non-renewing warranty. We assume that Wu and Clements-Croome's (2005) periodic PM model with random maintenance quality is utilized to maintain the system after the non-renewing warranty is expired. Given the cost structure to the user during the cycle of the product, we drive the expressions for the expected cost rate per unit time. Also, we obtain the optimal number and the optimal period by minimizing the expected cost rate per unit time. The numerical examples are presented for illustrative purpose.
Usually, reliability of products with the warranty is maintained by manufactures during the warranty period. After expiry of the warranty, however, the product is actually a secondhand product from customers’ perspective since it has operated for the warranty period. Through the review of literatures on maintenance after expiry of the warranty, there is no the literature considering imperfect preventive maintenance at a time where the warranty expires. In this paper, we integrate imperfect preventive maintenance at a time where the warranty expires with age replacement, and propose a maintenance-replacement policy after expiry of the warranty for the product with two categories of competing failure modes. The proposed maintenance-replacement policy is that an imperfect preventive maintenance is performed first at a time where the warranty expires and then an age replacement is performed at an age. The performed imperfect preventive maintenance reduces failure rate function of maintainable failure modes by a random variable. Compared with traditional age replacement policy after expiry of the warranty, although the proposed maintenance-replacement policy incurs a preventive maintenance cost at the time where the warranty expires, it can improve greatly subsequent operation time as well as decrease greatly both operation cost and failure cost. We obtain the expected cost rate from perspective of the customer, under the proposed maintenance-replacement policy and renewable warranty policies. Conditions for existence and uniqueness of optimal solutions that minimize the expected cost rate are presented. Based on a numerical example, we demonstrate that the proposed maintenance-replacement policy is superior to traditional age replacement policy by comparing respectively the optimal preventive replacement age and the expected cost rate.
Most warranty cost models based on preventive maintenance operations are assumed that products improve at each preventive maintenance (PM) operation and the failure rate is reduced to the failure rate of new products or to some specified level. To make warranty cost models more suitable to real operations, a modeling method of the PM warranty cost was proposed with the situation where each PM operation slowed the rate of product degradation. A warranty cost model was built on PM operations. On the basis of the cost model, both without and with reliability limit PM warranty policy, algorithms were presented to derive the optimal PM number and the optimal PM interval with an objective of minimizing expected total warranty cost over a finite warranty period. Finally, to demonstrate the feasibility of the presented modeling method, Weibull distribution cases were tested by numerical simulations. The simulation results indicate that the proposed modeling method is feasible and valid for resolving the optimal solution of the product warranty cost.
The objective of this paper is to review a list of important models for warranty costs, following a chronological tour and describing these studies in a general way. A classification according to their function and purpose will be also shown as well as the innovations suggested by every author, comparing those elements that form every model with the criteria considered suitable to each case. A selection of desirable features and properties for a modern and efficient warranty cost model will be possible to extract from the development of this analysis. Among these characteristics, it will be taken into account that the model, for instance, includes in itself a clear methodology for its implementation, if it makes reference to software tools of support, if the model proposes alternative process to evaluate expected costs, and many others. Scenarios as rebate programs; multiple products; items with intermittent use; modeling of warranty claims evaluating warranty expenses; inclusion of reworking cost in the warranty of failed items; differences between reworked cost before sale and warranty cost after sale; operation of a production-inventory system with products sold under warranty; assumption of repair costs depending on the product age or, for example, an age-reducing repair model…among others contexts conform the taxonomy in which it is possible to apply all these studies.
This paper considers a preventive maintenance policy, under which the repairable system undergoes preventive maintenance (PM) periodically and is minimally repaired at each failure. Most preventive maintenance models assume that each PM costs a fixed predetermined amount regardless of the effectiveness of each PM. However, it seems more reasonable to assume that the PM cost depends on the degree of effectiveness of the PM activity. In this paper, we consider a periodic preventive maintenance policy when the PM cost is an increasing function of the PM effect. The optimal number and period for the periodic PM policy with effect dependent cost that minimize the expected cost rate per unit time over an infinite time span are obtained. Explicit solutions to determine the optimal periodic PM are exemplified for the Weibull distribution case.
In this paper we present a Bayesian approach for determining an optimal maintenance policy following the expiration of warranty for a repairable system. We consider two types of warranty policies : non-renewing free replacement warranty (NFRW) and non-renewing pro-rata warranty (NPRW). The mathematical formula of the expected cost rate per unit time is obtained for NFRW and NPRW, respectively. When the failure time is Weibull distribution with uncertain parameters, a Bayesian approach is established to formally express and update the uncertain parameters for determining an optimal maintenance policy. We illustrate the use of our approach with simulated data.
This paper represents a new development on the basic classical age replacement policy, heuristically using the reliability function. First, an applied mathematical model of reliability function is obtained, and then based on this model; optimal preventive age replacement policy will be determined to maximize the reliability of the system. For the special systems which meet exponential density function with constant hazard function, a different simple decision making model will be proposed to choose optimal replacement policy based on the reliability function. A real case study will be reviewed on the maintenance data collection of two different industrial machines that one of them meets the Weibull density and other one has Exponential density function as life time pattern.
Warranty service is an important tool of market competition to producer, is also an important criterion for consumer. Along with developing of technology, information-based products with advanced technique and complex structure have been developed and produced. Improvement of technique performance and complexity not only result in increased cost of development and production, but also the cost of use and support. Using different warranty policy appropriately will reduce warranty cost, stimulate consume and advance developing of society. Correlative concepts of warranty are introduced. Origin and development and condition are analyzed, and some recommendations for problems of warranty research are proposed.
This study investigates preventive-maintenance warranty (PMW) policies for repairable products with age-dependent maintenance costs. The primary role of warranty is to offer post-sale remedy for consumers when a product fails to fulfill its intended performance during a warranty period. For repairable products, the most commonly used warranty policy is the minimal repair warranty. To reduce product failures, preventive maintenance is widely adopted in practice, especially when the product is complex and/or expensive. For most PMW policies discussed in the literature, there is a basic assumption that the cost of each maintenance action is constant. However, the cost of a maintenance action is varied as the age of repairable products increases in general. Therefore, a general cost model for repairable products with age-dependent maintenance costs is established in this study to derive the optimal PMW policy such that the expected total warranty cost is minimized.
This paper deals with the optimal replacement policies following the expiration of warranty: renewing warranty and non-renewing warranty. If the system fails during its warranty period, it is replaced with a new one and if the system fails after the warranty period is expired, then it is minimally repaired at each failure. The criterion used to determine the optimality of the replacement period is the overall value function, which is established based on the expected downtime and the expected cost rate combined. Firstly, we develop the expected downtime per unit time and the expected cost rate per unit time for our replacement model when the cost and downtime structures of maintaining the system are given. The overall value function suggested by Jiang and Ji [Age replacement policy: a multi-attribute value model. Reliab Eng Syst Saf 2002;76:311–8] is then utilized to determine the optimal maintenance period based on the expected downtime and the expected cost rate. Numerical examples are presented for illustrative purpose.
This paper studies the optimal periodic PM policies of a second-hand item following the expiration of warranty. The criterion used to determine the optimality of the PM is the expected maintenance cost rate per unit time from the customer's perspective. The periodic PM of the item starts right after the warranty is expired and all maintenance costs are paid by the customer. Given the cost structures of maintaining the item, we determine the optimal number of PM's before replacing the item by a new one and the optimal length of period for the periodic PM following the expiration of warranty.
This paper develops the optimal periodic preventive maintenance policies following the expiration of warranty. We consider two types of warranty policies to discuss such optimum maintenance policies: renewing warranty and non-renewing warranty. From the user's perspective, the product is maintained free of charge or with prorated cost on failure during the warranty period. However, the users will have to repair or replace the failed product at their own expenses during the post-warranty period. Given the cost structure to the user during the cycle of the product, we derive the expressions for the expected maintenance costs for the periodic preventive maintenance following the expiration of warranty when applying two types of warranty policies and obtain the optimal number and the optimal period for such post-warranty maintenance policies by minimizing the expected long-run maintenance cost per unit time. Explicit solutions for the optimal periodic preventive maintenance are presented for illustrative purposes.
This article adopts a Bayesian approach to derive an optimal maintenance policy following the expiration of non renewing warranty. If the system fails during its warranty period, it is replaced with a new one. If the system failure occurs after the warranty period is expired, then it is minimally repaired at each failure. As the criteria to determine the optimal replacement period, we use the expected cost and the expected downtime during the life cycle of the system. Under the replacement model considered, we first derive the formulas to compute the expected downtime per unit time and the expected cost rate per unit time in general. When the failure times are assumed to follow a Weibull distribution with unknown parameters, we propose an optimal maintenance policy based on the Bayesian approach, under which such unknown parameters are updated using the observed data. The overall value function suggested by Jiang and Ji (20023.
Jiang , R. , Ji , P. ( 2002 ). Age replacement policy: a multi-attribute value model . Reliab. Eng. Sys. Safety 76 : 311 – 318 . [CrossRef], [Web of Science ®]View all references) is utilized to combine the expected downtime and the expected cost rate and to determine the optimal maintenance period. Numerical examples are presented for illustrative purpose.
In dealing with the effects of product deterioration in the context of reliability analysis, it may not be satisfactory to consider only the effects of time or age because usage is often another essential factor that accounts for deterioration. A two-dimensional warranty with consideration of both time and usage for deteriorating products would be more advantageous for manufacturers. In this paper, a two-dimensional warranty model in which the customer is expected to perform appropriate preventive maintenance is analyzed and the warranty policy that maximizes the manufacturers' profits is determined. The proposed approach provides manufacturers with guidelines on how to offer customers two-dimensional warranty programs with proper time and usage limits. A numerical example shows the effectiveness of the proposed approach. Sensitivity analyses are conducted to investigate the robustness of the derived optimal warranty policy.
A number of optimal maintenance policies have been proposed and studied based on several types of warranty policies. As the criteria for optimality, the expected cost rate per unit time during the life cycle of the system is quite often used by many authors. However, the expected cost rate may depend on the length of life cycle and so the definition of life cycle plays a significant role in optimizing the maintenance policy. This paper considers a system maintenance policy during the post-warranty period under the renewing warranty policy and the life cycle is defined from the user's perspective. The life cycle starts with the installment of a new system and ends when the system is replaced by a new one at the expense of the user. In many renewing warranty models, the life cycle is defined as the lifelength of the new system installed initially, which is quite different from our definition. The expected cost rate per unit time is evaluated based on the life cycle newly defined and is compared with the existing results.
In general, a warranty is an obligation attached to products that require the manufacturer to provide compensation for customer
(buyer) according to the warranty terms when the warranted products fail to perform their intended functions. A warranty is
important to the manufacturer as well as the customer of any commercial product since it provides protection to both parties.
As for the customer, a warranty provides a resource for dealing with items that fail due to the uncertainty of the product’s
performance and unreliable products. For the manufacturer, it provides protection since the warranty terms explicitly limit
the responsibility of a manufacturer in terms of both time and type of product failure. Because of the role of the warranty,
manufacturers have developed various types of warranty policies to grab the interest of the customers. However, manufacturers
cannot extend the warranty period without limit and maximize warranty benefits because of the cost related to it. Many researchers
have studied in the last several decades on various warranty modeling and policies along with its maintenance policies. This
chapter focuses on the developments of warranty modeling with various maintenance policies as well as the methodologies with
various aspects that can be used to derive the mathematical warranty modeling. The concepts of warranty and review of the
overall information about the warranty policy such as warranty’s role, concept and different types will be discussed. The
basic mathematical maintenance concepts including counting processes such as renewal process, quasi-renewal process, non-homogeneous
Poisson process, compound and marked Poisson process and bivariate exponential distribution also will be provided.
In this paper, a general periodic preventive maintenance (PM) policy for a repairable revenue-generating system is developed and studied. We define ‘ageing losses’ as the difference in revenues generated by an ideal system (no ageing) and a real system that ages over the same period of consideration. It is assumed that preventive maintenance slows the system deterioration process and therefore reduces ageing losses. The proposed model is general in the sense that (1) both the warranty contracts and system ageing losses are incorporated in the maintenance cost modeling and (2) the implementation of PM actions does not have to be strictly periodic. A cost model is developed for the buyer under two decision variables—the calendar time of the first PM and the degree of each PM. Numerical examples are then presented to show the effectiveness of the proposed model. Sensitivity analyses are further conducted to investigate the impact of model parameters on optimal solutions.
A device is repaired at failure. With probability p, it is returned to the 'good-as-new' state (perfect repair); with probability 1 minus p, it is returned to the functioning state, but it is only as good as a device of age equal to its age at failure (imperfect repair). Repair takes negligible time. The distribution F//p of the interval between successive good-as-new states is obtained in terms of the underlying life distribution F. It is shown that if F is in any of the life distribution classes IFR, DFR, IFRA, DFRA, NBU, NWU, DMRL, or IMRL, then F//p is in the same class. Finally, a number of monotonicity properties are obtained for various parameters and random variables of the stochastic process. The results are of interest in the context of stochastic processes in general, as well as being useful in the particular imperfect repair model studied.
A policy of periodic replacement with minimal repair at failure is considered for a complex system. Under such a policy the system is replaced at multiples of some period T while minimal repair is performed at any intervening system failures. The cost of a minimal repair to the system is assumed to be a nonde-creasing function of its age. 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 system. Necessary and sufficient conditions for the existence of an optimal replacement interval are exhibited in the case where the system life distribution is strictly increasing failure rate (IFR).
Most preventive maintenance(PM) models assume that the hazard function of a system after each PM occurrence is restored to like new or to some specified level. Thus, there is no provision for system degradation with time. Operation of many systems causes stress which results in system degradation and hence an increase in the level of the hazard function with time. PM is assumed to relieve stress temporarily and hence slow the rate of system degradation. However, this type of activity does not reverse degradation; so the hazard function is monotone. A hazard function is developed in this paper consistent with this concept of PM effect. The special case for which PM reduces the operational stress to that of a new system is considered in greater detail (eg, a new system begins operation with the full benefit of a PM activity; each subsequent PM completely restores the benefits). It is shown for this case that the hazard function under PM is approximately a 2-parameter Weibull with shape parameter 2 for systems with strictly increasing hazard without PM. Cost optimization of the PM intervention interval is obtained by determining the average cost-rate of system operation. When the hazard function without PM is unknown, optimization may be achieved through an iterative process. This avoids the necessity of estimating system failure characteristics without PM; such testing can be destructive or use expensive equipment.
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