Product warranty management — III: A review of mathematical models

European Journal of Operational Research 62 (1992) 1-34 North-Holland

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Invited Review

Product warranty m a n a g e m e n t - III: A review of m a t h e m a t i c a l m o d e l s D.N.P. Murthy

Department of Mechanical Engineering, The University of Queensland, St. Lucia, Queensland 4072, Australia W.R. Blischke

Decision Systems Department, University of Southern California, Los Angeles, CA 90089-1421, USA Received December 1991

Abstract: A variety of mathematical models have been developed to study different aspects of warranty. In this p a p e r we carry out a comprehensive review of these models by classifying them into three categories based on consumer, manufacturer and public policy decision maker perspectives. Keywords: Warranty; mathematical models; m a n a g e m e n t of warranty; cost analysis

I. Introduction In the first p a p e r of this sequence (Blischke and Murthy, 1991, hereafter called Paper 1) we (i) discussed the role and importance of product warranties to consumers, manufacturers and public policy decision makers and (ii) developed a taxonomy to group the different warranty policies that are either in use a n d / o r have been reported in the literature on warranty. There are many aspects to warranty and as a result a variety of warranty related problems have been studied, such as - (i) how do different consumers react when items fail under warranty; (ii) what is the expected cost of servicing warranty as a function of warranty duration; and (iii) how should a manufacturer choose price and warranty to maximize the expected profits, to name a few. In the second p a p e r of the sequence (Murthy and Blischke, 1991, hereafter called Paper 2), we developed a comprehensive framework to study warranty related problems using the systems approach. The systems approach is a multi-step process involving four steps. We discussed the different steps and focused our attention on the first step, system characterization. System characterization is a process of simplification and idealization and can be viewed as defining a descriptive model of the real world relevant to the study. The subsequent steps involve translating the descriptive model into a mathematical model; analysis of the mathematical model; and interpretation of the results of the analysis.

Correspondenceto. W.R. Blischke, Decision Systems Department, University of Southern California, Los Angeles, CA 90089-1421, USA 0377-2217/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

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In the context of warranty, one can define three different perspective - consumer perspective, manufacturer perspective and public policy decision maker's perspective. The system characterization to study different warranty related problems depends on the perspective. In our second p a p e r we discussed two (one simple and the other more detailed) characterizations for warranty study from consumer and manufacturer perspectives and a totally different characterization for study from a public policy perspective. In this p a p e r we focus our attention on the mathematical models that have been developed to study different warranty-related problems. A variety of mathematical models have been developed and the literature is extensive. We c a r r y out a brief but comprehensive review of this literature. The outline of the p a p e r is as follows: In Section 2 we discuss mathematical modeling of some of the key elements of the warranty process and the analysis of these models. In Sections 3 and 4 we discuss the models developed to study different aspects of warranty from the manufacturer and the consumer perspectives. In Section 5, we consider the models developed to study warranty from the public policy decision maker's perspective. We conclude with some general comments on the status of warranty analysis.

2. Modeling and analysis In this section we carry out the mathematical modeling of some of the important elements of the warranty process and discuss some aspects of the analysis of mathematical models. Modeling of other elements will be discussed in later sections. We first consider modeling for study of one-dimensional and two-dimensional warranties separately.

2.1. Modeling for study of one-dimensional warranties Modeling item failures. The system characterization for item failures under warranty is best done in a stochastic framework with time treated as a continuum. This is appropriate as item failures, and the resulting warranty claims, can be treated as random points along a time axis in the case of one-dimensional ( l - D ) warranties. One needs to differentiate the first failure from subsequent failures since the latter may depend on the type of rectification action used after the first failure. One can model the time to first failure in two different ways. The first, called 'Black-Box' modeling, models the time to first failure as a random variable with a distribution function based on the modeler's intuitive judgment or on historical data. The second, called 'Physically Based' modeling, models the physical mechanism of the item failure and from this one obtains the distribution function of the time to failure. Thus, the 'Black-Box' approach is based on a simple system characterization where an item is characterized through two states - working or failed - and the 'Physically Based' model involves a more detailed system characterization of the physics of the failure. In the Black-Box approach, the time to first failure is modeled by a random variable X (or X~) having a distribution function F(x), which is zero for x < 0 and nonnegative for x > 0. If F(x) is differentiable almost everywhere, then the density function f ( x ) is given by f ( x ) = d F ( x ) / d x . The failure rate r(x) associated with a distribution function F(x) is given by r(x) = lim [ F ( t [ x ) / t ] = f ( x ) / f f ( x ) , where F(t I x) = P ( X < t I X > x) the probability that the item will it characterizes the affect of age The failure rate r(x), density

F(x) = l-exp(-

is the conditional C D F of X, given that X > x. r(x)~x is interpreted as fail in [x, x + 6x) given that it has not failed prior to x. In other words, on item failure more explicitly. function f(x) and distribution F(x) are related as follows:

for(t) dt}

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r(x) I !

I

I x

I

x

1

2

x failure rate D

F i g u r e 1. B a t h - t u b

and

Many products exhibit a failure rate which has a 'bath-tub' shape. It is characterized by a decreasing failure rate from 0 to some point x~, a nearly constant failure rate over a range x~ to x 2 and an increasing failure rate beyond x2, as shown in Figure 1. The failures during the initial period are mainly due to defective material a n d / o r poor manufacturing processes. In the case of repairable items, such failures are called 'teething problems' and may often be fixed through some form of testing program. Failures over the middle period are due purely to chance and hence are not influenced by age. Finally, failures over the last period reflect a true aging process which results in the failure rate increasing with age. Note that in some instances x I can be zero a n d / o r equal to x 2. In the 'Physically Based' modeling approach, one models explicitly the mechanism which causes item failure. A variety of models have been developed to describe different mechanisms of failure. We indicate two such models. In the 'Shock D a m a g e ' model, the item failure occurs due to the item receiving shocks randomly over time and the magnitude of the shock itself may be a random variable. I t e m failure occurs at the first instant that a shock exceeds a critical value. In the 'Cumulative D a m a g e ' model, the item is subjected to shocks as in the previous model, but each shock does a certain amount of damage to the item and the damage is cumulative. The item fails at the first time instant the cumulative damage exceeds some critical value. Note that in both models, the time to failure is a random variable and obtaining its distribution function involves the analysis of the stochastic process characterizing the shocks. Modeling failure of multi-component items. Most items are made of more than one component. One approach to modeling multi-component item failure is to model each component failure separately either as a 'Black Box' or based on the physical mechanism causing the failure and then to relate component failures to item failure. The distribution for item failure would depend on how the components are interconnected and the effect of component failures on item failure. When component failures are statistically dependent, one cannot model each separately. In the 'Black Box' approach, one needs to model component failures by a multi-dimensional distribution function and from this obtain the time to item failure. The analysis of such formulations is, in general, fairly involved and difficult. Often when a component of a complex item or system fails, it can either induce failure of one or more components or cause damage so as to weaken them and hence accelerate their failure. These types of

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W.R. Blischke, D.N.P. Murthy / Product warrantyIII

failures are termed 'Failure Interactions' and involve interacting point processes for modeling component failures. Many complex items are designed with a modular structure with each module being a collection of components. For such items, failure of a component results in a module failing. Thus one needs to relate component failure to module failure first and then module failure to item failure in order to obtain the distribution function of item failure. The modeling of these involve more complex stochastic formulations. Modeling failure of items used intermittently. Some products are used continuously - clocks, pumps in industrial operations, air conditioning in large, closed buildings, and so forth. Many other products are used intermittently or only occasionally - e.g., a dishwasher in a home, an elevator in a building, special equipment such as an emergency generator in a hospital. H e r e usage and idle periods for the item alternate and the failure rate during usage will ordinarily be different from that when idle. In this case, in order to obtain the distribution function for the time to failure, it is necessary to model the usage pattern. In this case, the usage and idle periods are of random duration and can be modeled by a two state continuous time Markov chain (a special type of stochastic process). When the duration of each usage is very small in relation to the time interval between usages, one can model usages as points along a time continuum. For this solution, usages may be modeled as a stochastic point process. Modeling rectification actions. Whenever a repairable item fails under warranty, the manufacturer ordinarily has the option of either repairing the failed item or replacing it by a new item. For non-repairable items, the only option is to replace a failed item by a new one. We now discuss the modeling of various types of rectification actions. In the case of repairable items, a failed item can be made operational by subjecting it to repair. The behavior of the item after repair depends on the nature of repair carried out. One can define five different types of repair actions: (i) Repaired good as new: Here, after each repair, the condition of the repaired item is assumed to be as good as that of a new item. In other words, the failure distribution of repaired items is the same as that of a new item. In real life, this is seldom the case. (ii) Minimal repair: When a failed item is subjected to a minimal repair, the failure rate of the item after repair is the same as the failure rate of the item just before item failure. This type of rectification model is appropriate for repair of multicomponent items where item failure occurs due to a component failure. Suppose such a failure occurs at time X 1 from the time the item was first put into service. When the failed component is replaced by a new working one, the item becomes operational. Since all other components are of age X l, the repaired item as a whole is effectively a working item of age X 1 and hence the failure rate after repair is the same as that just before failure. (iii) Repaired items are different from new (I): Often when an item fails, not only are all the failed components replaced but also components which have deteriorated sufficiently. In other words, the item is subjected to a major overhaul which results in the failure distribution of all repaired items being Fl(x), say, which is different from the failure distribution, F(x), for new items, but is assumed to be the same after each such repair. Since repaired items are assumed to be inferior to new ones, the mean time to failure for a repaired item is taken to be smaller than that for a new item. (iv) Repaired items are different from new (II): In (ii), the failure distribution for repaired item is different from that of a new item but is independent of the number of times the item has been subjected to repair. In some instances, the failure distribution of a repaired item is a function of the number of times the item has been repaired. This can be modeled by assuming that the failure distribution of an item after j-th repair ( j > 1) is given by Fj(x) with mean ~ j . / % is assumed to be a decreasing sequence in j, implying that an item repaired j times is inferior to an item repaired (j - 1) times. (v) Imperfect repair: Minimal repair implies no change to the failure rate whereas repair action under (iii) results in a predictable failure rate associated with the distribution function Fl(x). Often, however, the failure rate of a repaired item after repair is uncertain. This is called 'imperfect repair' and can be modeled in many different ways. Figure 2 shows two different imperfect repair actions: (a) corresponds

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AS N E W

r(x) I

I ITEM FAILURE

'

X

Figure 2. Three types of repair

to the failure rate after repair being lower than that before failure and (b) corresponds to the reverse situation. The change in the failure rate is a random variable in both cases. Another form of imperfect repair is one where the item becomes operational with probability p after it is subjected to a repair action and continues to be in failed state with probability 1 - p . This implies that the item would need to be subjected to repair more than once before it becomes operational or perhaps would have to be replaced. The time duration needed to carry out a rectification action is important in the context of warranty. When warranty terms include a penalty for down time, it is in the manufacturer's interest to reduce this duration to the minimum possible. The duration is also of interest to buyers, since an item that is out of action deprives the buyer of its use and may also lead to additional costs or, in some cases, to loss of the revenue that may be generated by the item. The total time involved in rectification action consists of: (i) processing time of warranty, (ii) investigation time, (iii) r e p a i r / r e p l a c e time, (iv) testing time, and (v) time to return the item to the buyer. Processing time consists of the time needed (at the retail-outlet level) for handling the claim for rectification under warranty, the time involved in transporting the failed item to the manufacturer (or workshop), and waiting time at the workshop. Investigation time is the time needed to locate the fault and decide on the appropriate action. Repair time includes the time needed to carry out the actual repair and the waiting times that can result due to lack of spares or because of other failed items awaiting rectification actions. This time is dependent on the inventory of spares and the manning of the repair facility. For the case where the failed item is replaced by a new one, the replacement time is nearly zero as long as there is a new unit in storage. Testing time is important where the success of a rectification action demands that the item be subjected to considerable testing before it is returned to the buyer. Some of these times can be predicted precisely while others (e.g. repair time) can be highly variable depending on the type of product. The easiest approach is to aggregate all the above-mentioned times into a single time called 'service' time, )?, modeled as a random variable with a distribution function G(x) =P{)(_
=

1 - G(x) :P{g>x}.

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Analogous to the concept of failure rate function, we can define a 'service' rate function p(x) given by

p(x) =g(x)/G(x). p ( x ) f x is interpreted as the probability that the service activity will be completed in [x, x + 6x), given that it has not been completed in [0, x). In general, r(x) would be a decreasing function of x, indicating that the probability of the service being completed in a short time increases with the duration that the service has been going on. In other words, p(x) has a 'decreasing service rate' a concept similar to decreasing failure rate. If the variability in the service time is small in relation to the mean time for service, then one can approximate the service time as being deterministic. If this mean value is very small in comparison to time between failures, then we can view service time as being nearly zero. This point of view allows a much simpler characterization of failures over time, as will be demonstrated in the next section. Modeling subsequent failures. As mentioned earlier, the modeling of subsequent failures depend on the nature of the rectification action. In the case of nonrepairable items, if the failure is detected immediately and the failed item is replaced by a new one, then the first and the subsequent failures can be modeled by a renewal process with time between renewals distributed according to F(x). In the case of repairable items, if the failed item is always subjected to minimal repair immediately after failure and the time to repair is negligible, then the first and subsequent failures can be modeled by a non-stationary Poisson process with intensity function given by the failure rate r(x). Modeling repair cost. When an item is returned under warranty, the manufacturer incurs a variety of costs. These are as follows: (i) administration cost, (ii) transportation cost, (iii) r e p a i r / r e p l a c e m e n t cost - comprising material cost and labor cost, (iv) transportation cost to return the repaired item, (v) handling costs of retailer, and (vi) spare parts inventory costs. One can aggregate all of these costs into a single cost termed 'service' cost for each warranty claim. Since some of the costs are uncertain (e.g. repair cost being dependent on the type of repair), the service cost is a random variable and needs to be modeled by a suitable distribution function. If the variability in the service cost is small, it can be treated approximately as a deterministic quantity. Modeling item sales. For the manufacturer, item sales are important in the context of warranty study for purposes of planning repair facilities and for deciding on warranty reserve levels. Item sales can be divided into the following three categories: (i) first purchases, (ii) replacement purchases under warranty, and (iii) repeat purchases outside warranty. Repeat purchases are important in the context of evaluating total cost over the product life cycle. Modeling of first and repeat purchases of a product has received a good deal of attention and the literature on the subject is extensive. In this section, we discuss a fairly simple model formulation which has formed the basis for many extensions. For certain types of products (e.g., defense products) a sale can be treated as taking place over a very short time interval. In these cases, the transaction can be approximated as a point sale in which a lot of items is sold to a buyer. The more interesting, and more general, case, however, is that in which sales occur over time, with a changing sales rate. One approach to this problem is to model aggregate sales. The advantage of such a n approach is that a static formulation can be used to model aggregate quantities. When modeled explicitly, sales over time can be modeled in two ways. The first approach involves a discrete time formulation in which sales aggregated over small time intervals (e.g. monthly, quarterly or yearly) are the variable being modeled. In the second approach, sales over time are modeled as continuous functions over time.

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We define a repeat purchase as any purchase, other than the first purchase, where the buyer pays full price. As such, replacements bought at reduced price and under warranty are not considered to be repeat purchases. The time intervals between the first purchase and the first repeat purchase, and between successive repeat purchase, are random variables. The distribution functions for these random variables are dependent on many factors: product reliability, type of warranty policy and consumers' reactions to the product and to the warranty services offered. In the case of non-repairable products, a potential repeat purchase situation occurs at the first time instant the item fails outside the warranty period. Whether a purchase eventuates or not depends on factors such as advertising, competition, the general economy, buyer satisfaction with regard to the product and its performance, and the warranty service provided by the manufacturer. This uncertainty can be modeled by a binary-valued random variable assuming the value 1 (implying a repeat purchase) with probability p and 0 (implying no repeat purchase sale) with probability (1 - p ) . In the case of a repairable product, when an item fails outside warranty, the buyer has the option of either repairing it or replacing it with a new item. In this case, the probability of a repeat purchase depends, in addition, on the buyers economic evaluation of the choice between repair and replacement. In addition, for both repairable and non-repairable products, a buyer may sometimes decide to replace a working item by a new one. In this case, we have an additional feature - namely, the market for second-hand items. A typical example is the market for cars. When a change of ownership occurs before the warranty has expired, the original warranty terms are, in many cases, invalidated. Modeling for cost analysis. For both manufacturer and consumer, the following costs are of importance: (i) warranty cost per unit sale, (ii) cost of operating a unit over its lifetime (LCC-I), and (iii) cost of operation over the product life cycle (LCC-II). The last two costs are often called Life Cycle Costs (LCC). As such, we shall denote them as LCC-I and LCC-II respectively. We discuss some issues related to obtaining these cost measures. Since the number of claims over the warranty period is also a random variable, the total warranty cost (i.e., the cost of servicing all warranty claims for an item over the warranty period) is a random sum of these individual costs. The Life Cycle Cost LCC-I is the cost of operating a single item over its life and consists of the following cost elements: (i) acquisition (or purchase) cost (CA), (ii) maintenance and repair cost of operating the item beyond the warranty period (CM), (iii) operating cost (energy, labor, etc.) (Co), (iv) incidental ownership costs (C~), and, if necessary, (v) disposal cost (CD). C A is a fixed cost. C M is a function of the time period beyond warranty for which the item is in use. This time period usually depends on the warranty period and will be denoted by g(W). The remaining costs are functions of W+ g(W), the life of the unit. The Life Cycle Cost LCC-II depends on the life cycle of the product, that is, the time interval over which buyers buy the product. After this time period, sales of the product cease, often because of the introduction of a new and better replacement. Let L denote the product life cycle. We assume that the buyer continues repeat purchases over this period. The number of repeat purchases is a random variable. The total life cycle cost is given by the product of this random variable and the life cycle cost per item, {LCC-I}. As a result, the total cost over the product life cycle is also random. Other cost elements, such as marketing, product development, design, manufacturing, consumer learning, servicing, discounting, etc., will be discussed in later sections.

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2.2. Modeling for study of 2-D warranties Modeling item failures. In this case, item failures can be treated as random points on a two-dimensional l~lane with one axis representing time as a continuum and the other representing usage as a continuum. Two different approaches have been used to model item failures. In the first approach, the item failure is modeled by an intensity function which is a function of age and usage of the item. The usage in turn is modeled as a function of age so that the intensity function is effectively a function of a single variable - age of the item. As a result, the modeling of item failures over the 2-D plane is done effectively in terms of a one-dimensional point process formulation. In the second approach, item failures are modeled by a 2-D point process formulation. Thus, if the items are nonrepairable and replaced immediately with replacement time negligible, then the second approach allows one to model item failures by a two-dimensional renewal process similar to the one-dimensional case. Note that the above two approaches assume a 'Black-Box' approach to modeling. The authors are not aware of any 'Physically Based' modeling for study of 2-D warranties. Modeling of other elements of the warranty process is similar to the one-dimensional case. 2.3. Analysis of mathematical models The final form of the mathematical model to study a specific warranty related problem depends on the modeling of the different elements of the warranty process relevant to the problem. This will be discussed in the next three sections. Once the modeling is completed, the next steps in the procedure are analysis of the model and proper interpretation of the analysis to obtain a solution to the problem under consideration. There are essentially two approaches to analysis, analytical and computational. In the analytical approach the solution is obtained as a closed-form analytical expression. This is possible only for a very limited class of mathematical models. For the majority of mathematical models, the computational approach is necessary. In this context one needs to differentiate numerical methods from simulation methods. Numerical methods are used in conjunction with the analytical approach when it is impossible to obtain the final result in closed form. Here, the solutions are obtained as numerical values for specific model parameters (e.g., solving the renewal integral equation associated with the Weibull distribution function using cubic splines). The simulation method involves simulating the time histories of changes in variables on a computer. The solution to the problem is obtained in terms of the inferences drawn based on the statistics obtained from a large number of repetitions of such time histories.

3. Review of c o n s u m e r and manufacturer cost m o d e l s

In the last section we discussed briefly the cost analysis associated with warranties. The expected cost per unit sale and the expected cost of operating a unit over its life time [LCC-I] are of interest to both the consumer and the manufacturer. From a manufacturer's perspective, such cost analysis allows for evaluation of the warranty costs for different policies and the effect of policy terms on the costs; from a consumer's perspective, they help in determining the total cost of ownership and in deciding whether to buy a product with or without warranty, when such an option exists. A variety of models based on the simple system characterization of Section 4 of Paper 2 have been developed for such cost analyses. This simple system characterization is shown in Figure 3. In this section we carry out a review of such models. The majority of the models to be reviewed assume the following: (i) Item failure is modeled by a suitable failure distribution, i.e., a 'Black-box' approach is used for modeling item failure.

W.R. Blischke, D.N.P. Murthy / Product warranty III

I MANUFACTURER ]

9

I CONSUMER}

( P"OOU0OH''OT"'ST'OS 1 WARRANTY]

1

PRODUCT 1 • r/PERFORMANCE / l NO }-'~~ I WARRANTY I SATISFACTORY COST

b WARRANTY COSTSI Figure 3. Simplified system characterization of the warranty process.

(ii) Failed items are replaced by new ones if the product is nonrepairable. In the case of a repairable product, failed items can be either repaired or replaced by new or used ones. Alternate forms of repairs have been modeled. (iii) The time to repair a n d / o r replace is assumed to be sufficiently small in relation to the mean time between failures so that it is approximated as being zero. (iv) All claims under warranty are exercised and are valid. (v) The cost of servicing each warranty claim is modeled by a single variable which aggregates all the relevant costs. We first consider models for cost analysis of 1-D warranties and then proceed to the 2-D case. In both cases, costs depend on the precise warranty structure, i.e., on the compensation to the buyer in case of failure, and on the life distributions of the item sold under warranty of and its replacements. 3.1. One-dimensional warranties Many one-dimensional warranties were described in Paper 1. The most common are the free-replacement warranty (FRW) and the pro rata warranty (PRW). The basic terms of these are as follows: Under the FRW, replacements are supplied to the buyer free of charge on failure of an item within the warranty period of length W. Under the PRW, the items are supplied at pro-rated cost. There are many variations of these policies and, all told, they comprise the vast majority of both consumer and commercial warranties in common use. Most other warranties are basically extensions of these ideas. The most common is a warranty that is often called a combination warranty and includes features of both the F R W and PRW. The usual combination warranty begins with a free-replacement term, followed by a period in which replacements will be supplied at pro-rata cost. Other combinations, with different terms in more than two periods (and decreasing compensation through time), are also used. An extension that has been proposed for use in commercial and governmental transactions is the cumulative warranty. Under warranties of this type, batches of items are warrantied for a total service time, with compensation being given the buyer on that basis and not on the basis of any individual item. If the total service time does not exceed an agreed-upon amount at the end of the warranty' period, replacements are provided either free or at pro-rated cost until the total service requirement is achieved.

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W.R. Blischke, D.N.P. Murthy / Product warranty III

Finally, a fifth type of 1-D warranty, which generalizes some of the above ideas in a number of ways, is the Reliability Improvement Warranty (RIW). This, too, is used only in commercial and governmental applications. In these transactions, the buyer has considerably more power (often more than the seller), and is therefore able to negotiate the warranty terms. Under RIW, the basic idea is to reward the seller for improvements in product quality, as measured, for example, by the mean time to failure (MTTF). In the remainder of this section, we look at a few of the basic models for each of these five types of warranties. Details of the models depend on a number of other factors relating to the warranty structure and context. One important consideration with regard to structure is whether or not the warranty is renewing. A renewing warranty is one under which the warranty terms begin anew (in essence, the clock is set to zero) on failure of an item within the warranty period. A non-renewing warranty is one under which a replacement item is covered only for the time remaining in the warranty period of the item it replaces. Most often, the PRW is renewing, while the F R W is not. By context is meant whether or not items are repairable, whether or not repaired items or other replacement items have the same life distributions as the original new items, and so forth. We will restrict attention primarily to contexts in which the modeling is relatively straightforward (though not necessarily simple). Extensions of these results may be found in the references cited below and in Blischke and Murthy (1993). 3.1.1. Cost models for the free replacement warranty policy The FRW is widely used for consumer durables. Many examples are given in Paper 1, in which the non-renewing FRW is Policy 1 and the renewing version is Policy 5. Since the non-renewing FRW, also called the 'standard FRW', is the most important in applications, most attention will be given to that version. Some cost models for the rebate and renewing forms of the F R W will be included as well. Standard F R W Seller's expected cost. We look first at the cost to the manufacturer of selling a non-repairable item under non-renewing FRW. Suppose that c S is the average cost to the seller of supplying an item, including manufacturing, advertising, distribution, and all other costs of doing business. Then the total expected cost to the seller of selling an item under warranty of length W is c~ times the expected number of items that must be supplied. If we denote this total cost Cs(W), we have (Blischke and Scheuer, 1975) E [ C s ( W ) ] =cs[1 + M ( W ) ] ,

(1)

where M(-) is the ordinary renewal function associated with the distribution function F ( . ) and E[-] denotes expected value. Equation (1) expresses the total expected cost simply in terms of the cost of the original item supplied plus the expected number of replacements. A problem is that renewal functions are difficult to evaluate except for a few simple distributions. However, fairly extensive tables are available. (See Baxter et al. (1981, 1982) for tables of renewal functions and Blischke and Scheuer (1981) for discussion of their use in cost analysis of the FRW.) If items are repairable, at a constant cost of c r, and repair is 'good-as-new', (1) is easily modified to obtain the supplier's total expected cost as E [ C s ( W ) ] = Cs + C r M ( W ) .

(2)

Models for other types of repair are given by Nguyen and Murthy (1984a). Buyer's expected cost. In terms of the cost of acquisition alone (ignoring costs associated with comparison shopping, taking delivery and other incidental costs), the buyer's cost is simply the purchase price of the item, which we denote c b. Under FRW, no replacement costs are incurred. The total cost of ownership, however, includes many other cost elements, as discussed previously in Section 2.

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Life-Cycle Cost (LCC-II). Here we assume that after initial purchase, replacement items are obtained (instantaneously replaced) throughout a life cycle of length L from initial purchase. Models for buyer's and seller's costs over a life cycle L are more complex, generally involving renewal functions associated with the random variable Y = W + y(W), where y(W), the 'excess random variable', is the remaining life of the item in service at the end of the warranty period, that is, on expiration of the warranty coverage. Realizations of the random variable Y correspond to purchase instances for the buyer, and hence to income events for the seller. LCC-II for the buyer is expressed directly in terms of the renewal function of Y, M r ( . ) , as LCCb(L, W) = %[1 + M r ( L ) ] .

(3)

This expresses the expected life-cycle cost simply as a function of the expected number of warranty cycles in the period [0, L). The difficulty is in evaluation of M r ( L ) . An integral-equation expression can be obtained for this renewal function (see Nguyen and Murthy, 1988a), but closed-form solutions exist for only a few distributions. In other cases, numerical solution, approximations, or simulation are required. For the seller, LCC-II is obtained as the product of the expected number of warranty cycles times the expected cost per cycle. The result is

LcC~(L, W) =Cs[1 + M ( W ) ] [ 1 + M r ( L ) ] .

(4)

Note that the seller's life-cycle expected profit can be obtained as the difference between the right-hand sides of (3) and (4). We note that discounting to present value can be incorporated into the cost models in a relatively straightforward manner. This may be of particular importance in the case of life-cycle costs and is also discussed in several of the references cited.

Rebate FRW Under the rebate FRW, a full refund is given on failure of an item under warranty rather than a free replacement. Whether or not to purchase a replacement is then the buyer's option. (Under the standard FRW, the buyer's choice is a free replacement or nothing at all.) This warranty can be considerably more costly to the seller, because the buyer's compensation on failure of an item under warranty is based on the selling price rather than the cost of supplying an item. The expected cost to the seller per item sold under rebate F R W is

E[Cs(W)] = c~ + CbF(W ) .

(5)

Although the buyer's initial cost is Cb, his net expected cost is

E[Cb(W)] = %[1 - / ( W ) ] .

(6)

Because of the simple structure of this warranty, other factors, such as repairability, discounting, and life-cycle costs, are not important considerations.

Renewing FRW Under the renewing FRW, free replacements are provided until an item having a life of at least W is found. The cost to the buyer remains c b. The seller's cost, however, can be substantial, unless the item is quite reliable relative to the warranty period W. We consider only the seller's cost for non-repairable items and a single purchase. The number of replacements required to satisfy the warranty conditions is a random variable having a geometric distribution with parameter F(W). From this it is easily seen that the expected cost to the seller is

E[C~(W)] = Cs[1 + F ( W ) / f f ( W ) ] where F ( W ) = 1 - F ( W ) . This is easily extended to repairable items, under any repair regime.

(7)

12

W.R. Blischke, D.N.P. Murthy / Product warrantyIll

3.1.2. Cost models for the pro-rata warranty policy The pro-rata warranty is also commonly used for consumer durables, especially for non-repairable items. The rationale is that the consumer has received some use of the item and should be willing to pay for the service received. This is a compromise between a full refund and no warranty coverage at all. Under PRW, replacements are provided at pro-rated cost, with pro-ration some function of service time, which may be actual calendar time (commonly used, for example, for automobile batteries) or usage (e.g., miles driven, for tires). The pro-ration function is most often straight linear or proportional linear pro-ration, but, in principle, it may be any nondecreasing function of service time. In applications, the PRW is usually renewing, but rebate forms are used as well. (We know of no instances in which the PRW is non-renewing in the sense that replacement items are warrantied only for the time remaining in the warranty period of the item being replaced, as is often the case under FRW.) Policies 2, 2a-c, and 6 of Paper 1 and the examples given are illustrative of the many forms of the PRW that may be used. Below, cost models will be given for the rebate and renewing forms of the PRW. In these models, we use the same notation as above, i.e., c~, c~, C S, etc., with the understanding that here these are interpreted as costs under PRW.

Rebate PRW In general, the PRW may be characterized by a rebate function, q(t), which expresses either the amount of rebate as a function of service time t, or, in the case of a renewing PRW, the amount of discount on the purchase of a replacement. The two most common rebate functions, for straight linear and proportional linear pro-ration, respectively, are

ql(t)={(1--t/W)Cb,

O<_t
0,

otherwise,

(8)

and q2(t)

=l"(1-t/W)cb, o<_t
otherwise,

(9)

where 0 < a < 1. Nonlinear rebate functions may be used as well; for example,

q3(t) =

(1--t/W)Zcb, 0,

O
(10)

represents quadratic pro-ration. Seller's expected cost. We consider only non-repairable items, since repairability is irrelevant as far as the seller is concerned when compensation to the buyer for item failure is in the form of a rebate. The actual cost to the manufacturer of a single item sold under PRW is Cs(W) = c~ + q(XL), so his expected cost per item, say

(11)

E[Cs(W; q)], is

E[Cs(W; q)] = c~ + fffq(t) d F ( t ) .

(12)

For the rebate functions of (8)-(10), expected costs are given by E[Cs(W;

ql)] =c~ + c , [ F ( W ) - I ~ w / W ]

(13)

dF(t)

(14)

where /Zw=

f0Wt

W.R. Blischke, D.N.P. Murthy / Product warranty III

is the partial expectation of

13

X;

E[Cs(W; q2)] = cs +acb[F(W) -tzw/W];

(15)

E[Cs(W; q3)] =Cs+Cb F(W) - -21Zw ~- + -~ ]

(16)

and

where/z(~) is the partial expectation of X 2, given by tZ~2w ) = foWt2 d F ( t ) .

(17)

Note that, since income is c b at each sale, expected profit can be calculated as the difference between c u and the above expected costs. Buyer's expected cost. Buyer's expected per-unit cost under PRW for the three rebate functions given above are easily obtained by the same methods. The results for straight linear pro-ration are as follows: Buyer's actual cost of a single unit purchased under PRW is given by Cb(W)

= (Cb,

X1 >- W,

c b - (1

--XI//W)Cb,

0

<_X1 < W,

(18)

so buyer's expected unit cost is E[Cb(W)]

~-

cb['-W

-[- 1 - F ( W )

].

(19)

Similar expressions for the proportional and quadratic PRW forms are easily obtained. If meaningful, models for repairable items can be derived by the same methods.

Renewing PRW For the seller, expected costs can be obtained simply as a function of the expected number of renewals as before. Models for buyer's costs (and hence seller's profit) for the renewing PRW, however, are considerably more complex. (See Mamer (1982, 1987) and Nguyen and Murthy (1984b).) The per-item costs are the same as for the rebate PRW given above. Because of the nature of the warranty, it is more meaningful to look at expected cost over a fixed period of time or long-run average cost. Thus, we consider LCC-II for the renewing PRW. To analyze LCC-II, it is convenient to look at the replacement cost to the buyer, say Rb(L, W), over a life cycle L, with L > W. This may be written as Cbx/W+Rb(L-x

Rb(L'W)=Iob+Ru(L-x'W)

W)

if0_L.

(20)

Analysis of this expression, based on conditioning on the first failure time, leads to a renewal-type equation, the solution of which yields LCCb(L, W) =Cb{1

+ F( L ) - [ F ( W ) - txw/W ] [1 + M( L

W)]

+ foL-WF(L-x)aM(x)+ f?_wfot-X(u/W)dE(u)dM(x)).

(21)

(This result is given incorrectly in several articles. Derivation of the correct result and a discussion can be

W.R. Blischke, D.N.P. Murthy / Product warranty III

14

found in Nguyen and Murthy (1984b).) This provides a formal solution to the problem. In practice, the integral expressions in (21) cannot be evaluated analytically except for a few simple life distributions. Seller's total life-cycle expected profit may be calculated as the difference between this expression and the total expected cost to the seller over the life cycle, given by Cs[1 + M(L)]. Discounted costs and profit can be derived in a similar fashion.

3.1.3. Combined F R W and P R W policies The most common combination warranty is free-replacement up to time, say W~, from purchase, followed by linear pro-rata coverage from W~ until some later time, W2. We again consider the rebate and renewing forms. In the first paper of the series, these were Policies 3 and 7, respectively. Several other versions of combined policies and a number of examples are also given. Whether the terms renew or a rebate is given, this warranty can also be analyzed by means of a rebate function. For the combination in question, the rebate function is c b,

0 < t < W~,

q(t)=]Cb(W 2-t)/(W I 0,

z-W1) ,

Wl _ < t < W 2 , otherwise.

(22)

Note, incidentally, that this combination reduces to the FRW if W 1 = W2 = W, and to the linear PRW if W 1 = 0. Rebate combination warranty. The expected cost to the seller is given by oo

E[Cs(W,, W2) ] =Cs + f0 q ( t ) d F ( t ) = c s + Cb[Wzr (W2) - WIF ( W l ) - IXw2 + IXw,] / ( W z - W,).

(23)

Buyer's expected cost is easily calculated from this as c b minus the expected rebate. Other combination warranties can be analyzed similarly. Renewing combination warranty. The introduction of the concept of renewing leads to many more possible combination warranties. Attention will be restricted to the combination just considered, for which two reasonable approaches to renewing are (Nguyen and Murthy, 1984b): (i) Renew only in the PRW portion (i.e., in the F R W period of the warranty, replacement items assume only the remaining time in the warranty period); and (ii) Always renew. Nguyen and Murthy (1984b) analyze both policies from buyer and seller points of view. Long-run and average costs and LCC-II are modeled. Additional renewal-type equations are encountered in the analysis and the renewal function M r ( . ) again plays an important role in the solutions. Because of the intractability of many of the expressions obtained, the authors present a number of approximation for use in the cost models.

3.1.4. Cumulative warranty policies Under a cumulative warranty, items are sold in lots, with the warranty applicable to the lot as a whole, and not to individual items. For example, a lot of N items may be warrantied for a total time of W ' = NW, with either FRW of PRW (or some other) warranty terms. The advantage to the seller is that early failures of a few items can be offset by some long-lived items. Since warranty costs are thereby reduced, items can be sold at a lower price, which would be the appeal to the buyer. A distinct disadvantage is the bookkeeping that would be required in implementing such a policy. Cumulative warranties were proposed in the mid-70's for use in the U.S. in military acquisition. Whether or not they were ever implemented is not known. There are numerous ways in which a cumulative policy may be structured. Policies 14 through 18 of Paper 1 illustrate the many possibilities. The only analyses of cumulative policies are in Guin (1984) and Blischke and Scheuer (1984). Here we look briefly the basic cumulative versions of F R W and PRW.

W.R. Blischke,D.N.P.Murthy / ProductwarrantyIII

15

Cumulative free replacement warranties. Suppose that a lot of N units is sold at cost Nc b and warrantied for a total service time of W ' = NW, and that the cost to the seller is c S per unit. Let X1,..., X u be the service times of the respective items and S u = E~= 1Xi denote the total service time. If S u < W', new items are supplied free-of-charge until the total service time exceeds W'. We consider seller's cost for the rebate and non-renewing versions of the warranty. For the rebate warranty, we assume that a full refund of Nc b if given if the warranty condition is not met. It is easy to see that the expected cost to the seller is then E[Cs(W, U ) ] = Uc~ + UcbF(U)( u w )

(24)

where F ( N ) ( • ) is the distribution function of S N (the N-fold convolution of F ( - ) w i t h itself). For the non-renewing version, the expected cost to the seller may be expressed in terms of the 'delayed renewal function', say MN('), namely, E [ C s ( W , U ) ] = cs[1 + MN(NW)]

(25)

where

mN(t ) = F(N)(t) + foM(t - x ) d F ( x ) .

(26)

Other policies have been analyzed by Guin (1984). Some analytical results are obtained. The policies have been found to be conceptually and mathematically difficult, but quite amenable to numerical investigation, and a number of simulation results have been obtained as well. All of these policies have the property that their expected cost to the seller is less than that of the standard FRW. The same is true, of course, of the PRW, to which we turn next. Cumulative pro rata warranties. Cumulative PRW's are described in Policies 16 - 18 of Paper 1. The setup is the same as above, except that the terms are linear PRW rather than FRW and we look at the rebate and renewing versions. In either case, one can again express the costs in terms of a rebate function. Here (8) becomes

q(t) = I cb(1 - t / U W ) , 0,

O
(27)

otherwise.

For the rebate version of the warranty, the analysis proceeds as before, witl~ F(N)(.) replacing F ( ' ) . Equation (13) becomes

E[Cs( W, U ) ] = c s + cb[ F(N)( N W ) -- I~uw/UW ]

(28)

where tZNW is the partial expectation with respect to F(N)('). The life-cycle cost analysis is the same as that of (20) and (21), except that F ( - ) is replaced by F (N)("), W by NW, and M ( . ) by the renewal function associated with F ( N ) ( " ) (which is different from Mu(.) defined above).

3.1.5. Reliability improvement warranties The RIW or warranties with similar features have been used in certain commercial and government transactions since about 1970. Applications include aircraft sales to airlines and the military as well as many other complex military equipment, such as radar units, avionics, and so forth. The intent of the RIW is to motivate the seller to provide high quality equipment as well as to provide an opportunity to the seller to increase system reliability through engineering design changes after one or more units have actually been put into service. To this end, contracts often include field service provided by the supplier and incentive fees for demonstrated reliability improvements. Policies 19 and 20 of Paper 1 are relatively simple illustrations of the RIW concept. Most are much more complicated, with many features covering service requirements, spares availability, repair constraints, and so forth.

16

W.R. Blischke, D.N.P. Murthy / Product warranty III

The types of items covered by R I W are ordinarily repairable and usually not replaced repeatedly over long periods of time. Thus cost models generally involve LCC-I considerations and the analysis concentrates on identifying and estimating the various cost elements at a detailed level. Also of concern are means of assessing and demonstrating reliability improvements. As an illustration of the types of models used in analyzing RIW, we use the profit model for R I W with M T B F guarantee derived by Gates et al. (1977). U n d e r this model, total profit ~- on a lot of N items is given by = C w - Cf - N U W S r / t x . - I(/Za)

-

Cs(~g//.t~

a -

1)N~ - D t ,

(29)

where C w = fixed price paid by buyer for a warranty of length W, Cf = seller's fixed warranty cost, U = usage rate of item (operating t i m e / c a l e n d a r time), Sr = seller's expected repair cost per unit, ~a = achieved MTBF, I ( ' ) = cost of improvement actions to achieve a given MTBF, C s = cost per consignment spare, /zg = guaranteed MTBF, N c = number of consignment spares, and D t = damages paid to buyer for failure to meet a guaranteed turn-around time. Because of the complexity of the models and the many elements that may be included, it is useful to develop a computerized system for tracking warranty and other cost factors. A computerized cost model of this type is given by Rotz (1986). For further examples and discussion, see Blischke (1990) and the references cited therein. 3.2. Two dimensional warranties

As mentioned in Paper 1, two-dimensional warranties are natural extensions of one-dimensional warranties. A two-dimensional warranty is characterized by a region in two dimensions with one axis representing age and the other usage. For example, in the case of automobiles usage corresponds to miles driven and in the case of airplanes it corresponds to number of hours flown. One can define a family of warranty policies based on the shape of the warranty region in the two-dimensional plane. The only two-dimensional warranty policy that is currently offered by manufacturers is Policy 8 of Paper 1. This policy is characterized by a rectangular warranty region [0, W ) × [0, U). U n d e r this policy, all failures occurring within a time period W subsequent to the sale are covered as long as the usage at failure is less than U. In other words, the warranty ceases either when the sold item reaches an age W or earlier if the usage exceeds U. Policies 9 - 11 of Paper 1 are three other two-dimensional warranty policies with warranty regions different from the simple rectangle. As in the case of one-dimensional warranties, two-dimensional warranties can be subdivided into many types. In this section, we will confine our attention to simple two-dimensional policies comprising the F R W and the P R W policies as only these have been studied so far. As mentioned earlier, one can model item failures using two different approaches. We first outline and compare the two approaches before proceeding to a review of cost models for the F R W and the P R W policies. 3.2.1. Item failures Approach 1: one-dimensional approach

Let t = 0 correspond to the time instant of a sale. Let Xc(t) and Yc(t) denote the age and usage of the item currently in use at time t. Let Y ( t ) denote the total usage that a buyer has had from the current plus earlier items over the interval [0, t). If no item failure occurs in [0, t), then X ~ ( t ) = t and Yc(t) = Y ( t ) . This is also true for the case where all failed items are repaired minimally and the repair time is assumed to be zero. In contrast, if the item is not repairable and there have been one or more failures in [0, t), then X c ( t ) < t and Y~(t) < Y ( t ) . In the one dimensional approach, one models Yc(t) as a function of Xc(t). This relationship characterizes item usage as a function of the age of the item. We assume that the relationship is linear with a nonnegative coefficient R, that is, Y c ( t ) = RX~(t).

(30)

W.R.Blischke,D.N.P.Murthy / ProductwarrantyII1

17

R represents the average usage per unit time, or usage rate, and may vary from user to user. As a result, R is modeled as a non-negative random variable with a distribution G(r). Conditional on R = r, let A(tlr)6t denote the probability that the current working unit at time t will fail in the small interval [t, t + 6t) - i.e., failures occur according to a Poisson process with intensity function M t l r ) , t > 0. A(t I r) can be modeled by a relationship of the form A(t I r ) =

qJ(Sc(t ), Y~(t)),

(31)

where the function qJ(x, y) is an increasing function of both x and y. This implies that the probability of item failure increases with its age and usage. Using (30) and (31) results in item failures, conditional on the usage rate R = r, occurring according to a one dimensional point process. The duration for which the item is covered under warranty, ~'r, depends on r. Murthy and Wilson (1991) examine the case A(t I r ) = 00 +

Olr+ OzSc(t ) + 03Yc(t),

(32)

with the parameters Oi>O for 0 < i < 3 . For repairable items with minimal repair, X c ( t ) = t and Yc(t) = Rt. As a result, (32) becomes ,t(t I r) = 00 +

01r + (02 + 03r)t.

(33)

The model formulation of Moskowitz and Chun (1988) is a special case of this. For the case of non-repairable items, failed items need to be replaced by new ones. The time between failures depends on the usage rate. The failure distribution F(t I r), conditional on R = r, is given by

F(tlr)= l-exp(-

fotA(t'lr)} dt',

(34)

with A(t I r) given by (33).

Approach 2: two-dimensional approach In the second approach, the item failures are characterized by a two-dimensional distribution function. Let (T~, X 1) denote the time to first failure and the item usage at first failure. Similarly, let (T/, Xi), i > 2, denote the time interval between i-th and (i - 1)st failure and the item usage between the two failures. (T,, X~) is modeled through a bivariate distribution function F,(t, x), i.e.,

Fi( t, x ) : e { T i < t , Xi
(35)

The form of Fi(t, x) depends on the nature of the rectification actions. For non-repairable items, (T~, X ) , i >__1, is a sequence of independent and identically distributed random variables with a common two-dimensional joint distribution function F(t, x). For the repairable case, the characterization of (T/, X,) depends on whether a failed item is repaired or not and the type of repair. The form of the distribution function F(t, x) (or the density function f(t, x)) appropriate for modeling would require a preliminary analysis of the failure data. However, it must have the property that E [ X I T = t] is an increasing function of t. In other words, the greater the time between failures, the greater the usage. Murthy, Iskandar and Wilson (1990) suggest three distribution functions which have this property. They are (i) Beta Stacy distribution, (ii) Multivariate Pareto distribution and (iii) Multivariate Pareto distribution of the Second Kind (Type 2). For (i) and (ii) E [ X I T = t] is a linear function of t and for (iii) it is a nonlinear function of t.

RelatiL~e comparison Approach 1 requires the specification of the distribution function G(r) for R (the variability in the usage rate across the buyer population) and the function ~b(., • ) (the conditional failure rate). Once this is done, the modeling and the analysis essentially involve results from the theory of one-dimensional

W.R.Blischke,D.N.P.Murthy/ ProductwarrantyIII

18

point processes. However, from an application point of view, this approach poses some problems which we discuss briefly. In real life, the item failure data available to the manufacturer is in the form of pairs of numbers (tij, xij), corresponding to the realized values of (T#, Xij), where the first subscript refers to consumer i (1 < i < NC), the number of consumers who have bought the product, and the second subscript refers to the j-th failed item (1 < j < J ) returned by consumer i to the manufacturer under warranty. The manufacturer seldom receives any information about the age and usage of the item at failure for failures outside the warranty period. Based on this data, the manufacturer can only obtain a crude estimate for the usage rate R for each consumer. Any parameter in the distribution of R then needs to be estimated using this information. In general, the estimates will not be reliable. Similarly, the estimation of the parameters of q~(', • ) poses problems. In contrast, this data is highly suited to estimating model parameters for Approach 2 for the case where the item is non-repairable. Differentiation between consumers is not as critical and, based on the failure data for each item in the form of (ti, x;), one can estimate the parameters of the distribution function F(t, x) once the form of the function is specified. We now discuss some of the different forms that are suited for modeling item failures.

3.2.2. Cost models for the free replacement warranty We first consider models based on Approach 1 and then discuss those based on Approach 2.

Approach 1 For Policy 8 of Paper 1, conditional on the usage rate r, the warranty ceases at ~'r" rr is equal to W if results.

r < ~ = U/W and to U/r when r >_?. From Murthy and Wilson (1991) we have the following Repairable items: minimal repair. The expected seller's cost per unit sold is given by E[Cs(W,

U)] = C s +

[fofoW~(tlr)dt

dG(r)+

ref=ffrA(tlr)~0

dt dG(r)]Cr,

(36)

where c S is the cost of each item to the seller and c r is the expected cost of each repair. Non-repairable items. The seller's expected cost per unit sold is given by

E[Cs(W, where

U)]= [1+

M(t Jr) is the

foM(Wlr) dG(r)

+

~M(~'r[ r)dG(r)]c~,

renewal function associated with

M(tlr) =F(tlr) + ~ M ( t - t ' l r )

F(t

(37)

I r) and given by

dF(t'lr).

(38)

Life-Cycle Costs (LCC-II). We assume that the first time an item fails outside the warranty region the buyer purchases a replacement item at full price. Conditional on R = r, the time between repeat purchases Z r is given by Z r = zr + ~, where ~ is the time to first failure after z r. In the case of non-repairable items, ~ is the excess age or residual life of the item in use at % for a renewal process associated with the distribution F(tl r). The distribution Fz(t I r) of Z r is given by Fz(tlr)=(F~(t-~'r)

for / >'r~, otherwise,

(39)

with

F ~ ( t l r ) = F ( T ~ + t i r ) - for[1-F('r~+t-u]r)] dM(ulr),

(40)

W.R. Blischke, D.N.P. Murthy / Product warranty III

19

where M(t I r) is the renewal function associated with F(t I r). Let K(LI r) denote the number of repeat purchases over the product life cycle L, conditioned on R = r. The expected value of this is given by Mz(z r I r) where Mz(tlr) is the renewal function associated with Fz(tlr). As a result, the expected value of K(L), the number of repeat purchases over the product life cycle, is given by

E[K(L)] =

f;Mz(LIr)

da(r).

(41)

The total life cycle cost to the seller is given by LCCs(L; W, U) = {ElK(L)] + 1}E[Cs(W, U)],

(42)

with E[Cs(W, U)] given by (34). The total life cycle cost for the buyer is given by LCCb(L; W, U) = {E[K(L)] + 1}Cb,

(43)

where c b is the sale price per unit. Similar results for Policies 9 - 11 can be found in Murthy, Wilson and Iskandar (1991).

Approach 2 We shall confine our attention to the case where the items are nonrepairable, in which case all failed items are replaced by new ones. Seller's cost. The seller's expected costs are derived (Murthy, Iskandar and Wilson, 1991) using various counting processes related to the joint distribution function F(t, x) and the two marginal distribution functions FrO)and Fx(x ). For Policy 8 of Paper 1, the seller's expected cost is given by

E[Cs(W, U)] = [1 +M(W, U)]Cs,

(44)

where M(t, x) is the two dimensional renewal function associated with F(t, x) and given by

M(t, x ) = F ( t , x) + fotfoxM ( t - t ' , x - x ' ~, fj t, t ' , x') dt' dx'.

(45)

For Policy 9 of Paper 1, it is given by

E[Cs(W, U)] = [1 +Mr(W ) + Mx(U ) - M ( W , U)]cs,

(46)

where Mr(t) and Mx(x) are the one-dimensional renewal functions associated with the two marginal distribution functions FrO) and Fx(x). For Policy 10 of Paper 1, it is given by [Cs(W 1, W2, U1, U2) ] = [1 + M ( W 2, /_]1) + M ( W l, U2) -M(W~, U~)]c s.

(47)

For Policy 11 of Paper 1, it is given by

E[Cs(W, U)] = [1 +Mv(U)lcs,

(48)

where My(t) is the renewal function associated with the distribution Fv(t) , given by

ev(t )

=

f~f(t-")W/udV(

u, v ) .

(49)

JO " 0

Life-Cycle Costs (LCC-II). Whenever an item fails outside the warranty region, we assume that the buyer purchases a replacement item at full price. Let K(L) denote the number of repeat purchases over

W.R. Blischke, D.N.P. Murthy / Product warranty III

20

the product life cycle. Let Z denote the time between purchases. The distribution function for Z,

H(t; W, U), depends on the policy. For Policy 8 of Paper 1, H(t; W, U) = 1 - H(t; W, U) is given by H ( t ; W, U) =

f0TdF(x, y) + f02-H ( t - x ;

W - x , U - y ) dF(x, y)

(50)

for 0_
H(t; W, U) =

,Td F(x, y) + f0fo

( t - x ; W - x , U - y ) dF(x, y)

(51)

for W_< t < oo. The expected number of repeat purchases over the life cycle is given by E[K(L)] = Mz(L), where Mz(t) is the renewal function associated with H(t; W, U). The expected life cycle cost to the seller is given by LCCs(L; W, U) = [1 +Mz(L)]E[Cs(W, U)]

(52)

with E[Cs(W, U)] given by (44). The expected life cycle cost to the buyer is given by LCCb(L ; W, U) = [1 +Mz(L)]c b

(53)

where c b is the selling price per unit. Expressions for H(t; W, U) for Policies 9 - 11 of Paper 1 can be found in Murthy, Iskandar and Wilson (1991).

3.2.3. Cost models for the pro-rata warranty Let the warranty region be denoted by 12 and let R(x, y) denote the refund conditional on X~ = x and Y1 = Y- R(x, y) is non-increasing in x and y for (x, y) ~ 12 and 0 for (x, y) ~ g2. Different forms of g2 and R(x, y) define different policies. We consider Policy 12 of Paper 1 where the warranty region 12 is given by 12 = [0, W) × [0, U) and the rebate function by

R(x, y ) =

(a-x/W)(l-y/U)c 0

b

if(x,y)~12, if(x, y)~12,

(54)

where c b is the buyer's purchase price per unit. As in the case of one-dimensional warranties, the policy can be either renewing or non-renewing. A unified approach. Iskandar, Wilson and Murthy (1991a, 1991b) suggest a unified approach for treating both the renewing and non-renewing policies together using a binary valued random variable U/ to denote the buyer's decision at the failure of the i-th item, namely US =

1,

(Xi, Yi) e 12 and the buyer buys a replacement,

0,

(X/, Y/) ~ 12 and the buyer does not buy a replacement or ( X i, Y,) ~ 12.

(55)

The probability that Us = 1 conditional on X i = x, Y~= y is given by P{U/= II g i = x , Y/=y} =p(x, y)

(56)

and given {X i = x, Y, = y}, US is independent of previous failure history. In other words, the buyer's decision to replace items which fail under warranty by a new one is uncertain. The two limiting cases are as follows: (1) p(x, y ) = 1 for (x, y ) ~ 12: This implies that whenever an item fails under warranty, the buyer always uses the refund for purchasing a replacement item. This case corresponds to a renewing rebate policy.

W..R. Blischke, D.N.P. Murthy / Product warranty III

21

(2) p(x, y) = 0 for (x, y) e O: This implies that the buyer always collects the refund and never buys a replacement item. Hence, this corresponds to a nonrenewing rebate policy. With p(x, y) = p (a constant), the expected warranty cost to the seller per item sold is given by u

w

F(W' U) - fo fo {xU + Wy-xy} dF(x, y) E[Cs(W, U)] =Cs+Cb

W U { 1 - p F ( W , U)}

(57)

From this we have the following results: Seller's expected cost for the nonrenewing PRW (p = 0):

e[Cs(W,

WU F(W, U ) - f v fW{xu + Wy-xy} dF(x, y) u)] =Cs+Cb wv

(58)

Seller's expected cost for the renewing PRW (p = 1):

WU F(W, U) - f v fW{xu + Wy - xy} d F ( x , y) E[Cs(W, U)] =Cs+C b

WU{I'°'°-F(W, U)}

(59)

4. Detailed models of the warranty process

The simple system characterization of the previous section ignored many other elements of the warranty process. A more detailed system characterization of the warranty process was done in Section 5 of Paper 2 and is shown schematically in Figure 4. It indicates the different elements of the warranty process and the interactions between them. This characterization allows one to build models to solve a variety of warranty related problems of interest to consumer and to manufacturer. There is no single model which incorporates all the factors of the detailed system characterization. Even if such a model were to be built, it would be of limited use due to its complexity. Instead, a variety of models have been built to study specific issues or topics of the warranty process and incorporating one or more of the factors discussed.

4.1. Warrantyand purchases decision On the consumer side, as indicated in Paper 2, product purchase decisions involve four stages and warranty is an important issue in the last three stages - viz., search for information; evaluation of alternatives; and, finally, the product choice. Although the consumer literature contains many mathematical models for each of the above three stages (e.g. Kotler, 1980), the authors are aware of only a few models which incorporate warranty explicitly. The general 'search for information models' (e.g., Gastwirth, 1976) are applicable for the warranty case when warranty is viewed as a signal providing some information about product reliability. When consumers have a choice of buying a product with or without warranty, utility theory (Varian, 1990) provides a basis for modeling product evaluation and the purchase decision. This approach forms the starting point for most of the models dealing with public policy issues and will be discussed in the next section. Suppose the price per unit is (c b + 3) with warranty and c b without warranty. ~ is the premium charged by the manufacturer for providing the warranty. Let U(x) denote the utility function, with x measured in monetary terms. Let I denote the initial wealth of the consumer. If the consumer does not buy the product, his utility is given by U(I).

22

W.R. Blischke, D.N.P. Murthy / Product warranty III

I DESIGN/MANUFACTURE<

1

L

~"

I PRODUCTCHARACTERIffI'ICS I

I

II

CONSUMER~ LEARNING

T

lcoM,. r,o.,sI_ ACTIONS I

1'

rP,oo,.,CTOOI .

PRICE& ADVERTISING

I''OT"'LESl

~, [ PRODUCTPERFORMANCEl---r~

YES l M~C'' I ~

II EXPECTATION CONSUMERI

NO

I OTHERACTIONSI ' ~ - ~

DISSATISFACTION]

I LONGTERMGOALSI

+

~-~--tRESOLUTION ~ CONSUMER SATISFACTION/ DISSATISFACTION

MANUFACTURER'S ACTONS l,~ - -

'~ DISPUTE < I SERVICING

I=I WARRANTYCOST

'~ JWARRA~L'C¥ " ~,

I I

,L ¥

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DESIGNAND MANUFACTURINGCOSTS

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Figure 4. Detailedsystemcharacterizationfor warrantystudy Let Z,, be the monetary return (or benefit) to the consumer if he buys a single unit and the item does not fail in the warranty period. Let Zf denote the monetary return should be item fail within the warranty period. Note that Zf can be negative. Let 0 denote the perceived probability (by the consumer) that an item will not fail in the warranty period. If the consumer buys a single unit without warranty, then his expected utility is given by

E[U INo warranty] = 0 U( I - c b + Zw) + ( 1 - O)U( I - c b + Ze).

(60)

Consider the case where the product is sold with lump sum rebate policy which requires the manufacturer to refund an amount Z r should an item fail within the warranty period. If the consumer buys a single unit with warranty, then his expected utility is given by E [ U [Warranty] = 0 U( I - c b -- 8 + Z w) + ( 1 - 0) U( I - c b - 6 + I f "~-I r ).

(61)

W.R. Blischke, D.N.P. Murthy / Product warranty III

23

A rational consumer would choose to buy the product with warranty if

E[U tWarranty] > E[U [No warranty]

(62)

and without warranty if E [ U IWarranty] < E [ U [No warranty].

(63)

If the two expected utilities are the same, then the consumer is indifferent between the two. Note also that the critical variable is 0, the perceived probability that an item will not fail in the warranty period. Let 7r denote the true probability that an item will not fail in the warranty period. When the consumer is not fully informed, then in general 0 would not equal 7r. If the consumer over (under) estimates the risk, then0< (>)Tr.

4.2. Warranty as a marketing tool The two most important marketing variables for the manufacturer are (i) the selling price i b and (ii) the warranty period W (in the case of one-dimensional warranties). The warranty period is a critical factor; a longer period not only assures greater protection to the consumer, it also signals a better quality product. The total sales Q is a function of these two variables. Glickman and Berger (1976) model Q as follows:

Q(c b, w ) = klCba( k2 + W) b,

(64)

where k~ > 0, k 2 > 0, a > 1 and 0 < b < 1. k 1 is an amplitude factor and k 2 is a constant to allow for nonzero demand with no warranty. Note that Q(c b, W) increases as c b decreases a n d / o r W increases, implying that a lower price a n d / o r a longer warranty results in greater sales, a and b are the price and warranty elasticities. The seller's expected profit per unit sale, ~-(c b, W), is given by -w(c b, W) = c b - e m - E [ f s ( W ) ] ,

(65)

where c m is the manufacturing cost per unit and E[Cs(W)] is the expected warranty cost per unit and depends on the type of warranty. As a result, the seller's total expected profit, II(%, W), is given by

Fl(c b, W) = kl{c b - c m - E[Cs(W)] }cffa( W+ k2) b.

(66)

Glickman and Berger (1976) discuss the optimal selling price and warranty to maximize the total expected seller's profit. Ritchken and Tapiero (1986) develop a model for warranty price schedules under buyer and seller risk aversion for non-repairable items.

4.3. Cost accounting of warranty Balachandran et al. (1981) develop a continuous time Markov chain model for warranty costing. Grimlund (1985) formulated a model based on a 'revised beta-normal' (RBN) approximation to calculate warranty liabilities. They highlight the accounting aspect. However, both analyses require very restrictive assumptions. The cost models of Section 3 provide better estimates for accounting purposes then these two models.

4.4. Warranty servicing Warranty reserves. For items sold with nonrenewable pro-rata warranty, the manufacturer refunds a fraction of the sale price should the item fail within the warranty period. For one-dimensional warranties, the amount refunded depends on the age of the item at failure relative to the warranty period (W) and the rebate function (q(t)). To meet these payouts, the manufacturer must set aside a fraction y ( W ) of the sale price of each item to create warranty reserves.

W.R.Blischke,D.N.P.Murthy/ ProductwarrantyIII

24

In the case of a single lot sale of size N, if the reserves equal the expected payouts, then y ( W ) must equal {E[Cs(W, q)]--Cm}/C b, where E[Cs(W, q)] is the sellers expected cost (given by (13)-(16), depending on the rebate function q(t)). Since the payouts occur over a period W subsequent to the sale whilst the reserves are created at the instant of sale, the reserves can be invested (e.g., in short term money market or as interest bearing deposits) to generate income and thus reduce the fraction y ( W ) needed for warranty reserves. Let S(t) denote the value at time t of an investment of value S o at time t = 0. Two models for S(t) are as follows: (i)

S( t ) = SOexp(4,t),

(67)

and (ii)

S(t)

= S0(1 + ~b)'

(68)

where 4~ ( > 0) is the rate of return on investment. Let ~/(W; ~b), denote the fraction to be set aside (and invested), such that the reserves plus the returns equal the expected total payout over the warranty period. This quantity is given by

y(W;

d~)= f 0 W q ( x ) e - e ' X f ( x )

dx

(69)

for the first model and by

y(W;

d~) = f0Wq(x)(1 + 4~)-Xf(x)

dx

(70)

for the second model. Note that 7(W; 4)) decreases as ¢b increases for both the models. This follows since a larger value of ~b implies a greater return on investment and hence the need for a smaller reserve to begin with. Thomas (1989) compares the effect of ¢b for a range of failure distributions. See also Menke (1969) and Amato and Anderson (1976). In the case of continuous sales, one needs to model the sales over time. Let s(t), 0 < t < L, denote the sales rate (i.e., sales per unit time) over the product life cycle L. This includes first and repeat purchases for the total consuming population. Since the manufacturer has to provide a refund for items which fail before they reach age W and since the last sale occurs at or before time L, the manufacturer has to service warranty claims over the interval [0, L + W]. The refund rate (i.e., the amount refunded per unit time) is a random variable since item failures and the resulting claims occur randomly. Let v(t) denote the expected value of the refund rate at time t. Note that an item sold at time t - r will fail in the interval [t, t + & ) with probability f(r)6t. As a result,

u(t) = {f:s(.r)f(t-r)q('c) d~)

(71)

where 0 = max(0, t - W). The fraction of the sale price, 7(W), to be set aside to form the warranty reserves so that the warranty reserves over the product life cycle equal the total expected payout, is given by

= ( f: +wu(t) dt} / (cb f~s( t ) a t } .

(72)

If the warranty reserves are invested to generate income, then the fraction to be set aside is given by

W.R. Blischke, D.N,P. Murthy / Product warranty III

25

or

(74) depending on the model for return on investment. For an alternative approach to modeling of warranty reserves, see Tapiero and Posner (1988). Demand for spares. When a non-repairable product is sold with a nonrenewing free replacement policy, then the manufacturer has to replace all item failures occurring within warranty period W. In the case of single lot sale of size N, for each item the failures over the warranty period occur according to a renewal process since failed items are replaced by new ones. Let p(t) denote the expected demand rate for spares at time t, 0 < t < W. This is given by p(t) = m(t), where m(t) is the renewal density function associated with the failure distribution F(t). With continuous sales given by s(t), 0 < t < L (the product life cycle), the expected demand rate of spares at time t, p(t), is given by

p ( t ) = f t s ( ' c ) m ( t - ~') dr,

(75)

where m(t) is the renewal density function and ~0= max(0, t - W). Demand for repairs. For repairable items sold with free replacement policy, the manufacturer can either repair the failed item or replace it by a new one. We confine our attention to the case where the manufacturer repairs all failed items returned under warranty. The demand for repairs depends on the type of repair carried out. For items repaired minimally, the item failures occur according to a non-stationary Poisson process with intensity function h(t) given by h ( t ) = r(t), where r(t) is the failure rate of the item. Let Pr(t) denote the expected return rate at time t, i.e., the expected number of items returned over the interval [t, t + fit) for repair. For a single lot sale of size N, this is given by pr(t) =Nr(t), 0 < t < W. In the case of continuous sales s(t) over the product life cycle L, Pr(t) is given by

pr(t)

= fts(7")r(t-~')

d7",

O
< L + W.

(76)

For the case where repaired items are all identically distributed with a distribution function G(t) different from F(t), we have the expected return rate for a single lot sale given by pr(t) = Nmd(t), where md(t) is the renewal density function for the modified renewal process, given by md(t ) = f ( t ) + f o t m g ( t - - x ) f ( x )

dx,

(77)

with mg(t) the renewal density function associated with G(t). For the continuous sale case, we have, for t ~ [ 0 , L + W], p~(t) =

f's(~)md(t -

~) d.r,

(78)

with ~b = max(0, t - W) and md(t) given by (77). Repair vs replace. When a repairable item is returned to the manufacturer for repair under free replacement warranty, the manufacturer has the option of either repairing it or replacing it by a new one. The optimal strategy is one which minimizes the expected cost of servicing the warranty over the warranty period. Nguyen and Murthy (1986) consider the following two simple strategies: Strategy 1: An item is replaced by a new one if it fails in (0, W - a] and subjected to minimal repair if it fails in ( W - a, W]. The parameter a (0 < a _< W) is selected to minimize the expected cost of servicing the warranty.

W.R. Blischke, D.N.P.Murthy / ProductwarrantyIII

26

Strategy 2: An item is subjected to minimal repair if it fails in (0, a] and is replaced by a new one if it fails in (a, W]. The parameter a (0 < a < W) is selected to minimize the expected cost of servicing the warranty. Strategy 1 is more appropriate where the initial failure rate is high due to a small fraction of the items being of an inferior quality (not conforming to design specification and having a relatively high failure rate). As a result, replacing items which fail early by new ones can be viewed as an effective way of weeding out inferior quality items. Strategy 2 is more appropriate when items have a decreasing failure rate in the early stages of their life and failed items can be repaired relatively cheaply. As the item ages, the failure rate increases and hence it is more sensible to replace failed items by new ones when they are old. Under Strategy 1, the expected cost of servicing the warranty per item sale, to(a; W), is given by to(a; W ) = Cm[1 + M ( W - a ) ]

+ c r In P~(ol),

(79)

where c m and c r are cost of each new replacement and repair respectively,

P~( x) = 1 - F~( x) = 1 - F ( W - a + x) - foW-'~F(W - a + x - y )

dM(y),

(80)

for t > 0, and M(t) is the renewal function associated with F(t). Under Strategy 2, the expected cost of servicing the warranty is given by to(a; W ) = C m [ 1 + M d ( W - a ) ]

+ c r In F ( a ) ,

(81)

where Md(t) is the renewal function associated with a modified renewal process with the distribution for first failure given by [F(t + a ) - F(a)]/F(a) and subsequent failures occurring according to F(t). a*, the optimal a, is obtained by minimizing E[Cs(W; a)] subject to the constraint that 0 < a _< W. Other models for choice between repair and replacement can be found in Nguyen and Murthy (1989) and Biedenweg (1981). Cost repair limit strategy. In the case of repairable items, the cost to repair a failed unit, Cr, is a random variable characterized by a distribution function which we denote H(z). Similar to the failure rate characterization, one can define a repair cost rate given by {h(z)/H(z)}, where h(z) is the derivative of H(z). Depending on the form of H(z), the repair cost rate can increase, decrease or remain constant with z. A decreasing repair cost rate is usually an appropriate characterization of the repair cost distribution (see, e.g., Mahon and Bailey, 1975). As a result, when repairable items are sold under FRW, the manufacturer must choose a repair strategy based on an estimate of the repair cost. Murthy and Nguyen (1988) deal with a repair limit strategy where the failed item is minimally repaired and returned to the owner if the estimate is less than a specified limit O. If not, the failed item is junked and the customer supplied with a new one at no cost. The optimal limit O* is the one which minimizes the expected seller's cost, given by

E[Cs(W; O)] where Mg(t; O) is the renewal function associated with G(t; O) and is given by

G(t, O) = 1 - [F(t)] "(°'.

(83)

Numerical methods are required to obtain O*.

4.5. Warranty and product design Warranty and design. In the engineering of a product, the first step is product design. We first consider the case where no product development is required. The manufacturer can design the product so that its

W.R. Blischke, D.N.P. Murthy / Product warranty II1

27

reliability can assume any value within a specified interval. The manufacturing cost depends on the reliability chosen. For example, higher item reliability might require components with tighter functional specifications or more reliable components - in either case, the resulting cost is higher. Later we consider the case where the desired product reliability is higher than the actual reliability of the product. In this case, it is necessary to improve the reliability of the product through a research and development program. Optimal reliability choice. Let F(x; O) be the product failure distribution. Suppose that the manufacture can choose the value for the parameter 0 (within limits 0-_< 0 < 0 ÷) at the design stage, with smaller values of 0 corresponding to more reliable products. Let Cm(0) denote the manufacturing cost per unit, with dc m ( 0 ) / d 0 < 0 implying greater reliability and hence greater manufacturing cost. The seller's expected cost E[Cs(W; 0)] depends on the type and duration W of warranty offered and the design parameter 0. 0", the optimal 0, is the 0 which minimizes E[Cs(W; 0)] subject to the constraint 0 - < 0 < 0 ÷. This is a standard nonlinear programming problem and can be solved by the usual methods. Murthy (1990) deals with such a model where, in addition to the design parameter, the price and the warranty terms are also decision variables to be selected optimally. The problem of reliability allocation in the context of warranties is discussed in Nguyen and Murthy (1988b). Optimal product development. One way of reducing the expected warranty cost is to improve product reliability. This involves research and development effort in which the product is subjected to a sequence of t e s t - f i x - t e s t - f i x cycles. During this process, the product is tested until a failure mode appears. Design a n d / o r engineering modifications are then made as attempts to eliminate the failure mode and the product is tested again. As this continues, the product reliability improves The development program costs money and the process of improvement is uncertain. To determine optimal development programs in this context, one needs to build models for product improvement which take into account the underlying uncertainty. By use of such models, one can decide on optimal development plans which minimize the total expected cost of development plus the servicing cost for warranty. Murthy and Nguyen (1987) propose three different models (Model I - III) for product development in the context of product warranty. If ~- is the duration of the development program and N(~-) is the number of design modifications made, then the cost of the development program is given by Cd~"+ CaN(r), where C d is the cost per unit time of running the development program and C a is the expected cost of each design modification to fix a failure mode. In Model I, ~- is deterministic and N(~-) is a random variable; in Model II it is the reverse. In both Model I and II, the failure rate and the end of development is a deterministic function of the program development period. In contrast, in Model III the failure rate at the end of the development period is a random variable. In these models, the failures (and hence modifications to eliminate them through design) are assumed to occur according to a nonhomogeneous Poisson process with intensity u(t). u(t) is called the modification rate. u(0) is the failure rate before the start of the development program and the aim of the program is to reduce this quantity, u(t) is a decreasing function of t. Various forms for u(t) have been proposed. As an example, Crowe (1974) proposed the model u ( t ) = A/3t ~t3-1),

(84)

where A > 0 and 0
28

W.R. Blischke, D.N.P. Murthy / Product warranty III

in use and the system functions as long as one of them is working. Thus, if the individual failure times of the two are XI and X 2 respectively, then the failure time, as a pair, is given by X12 = max{ X 1, X2}. In contrast, in the case of passive redundancy, at the start only one component is in use. When it fails, the duplicate (or spare) is automatically switched on. As a result, the failure time, as a pair, is given by X12 = X 1 -}- X2.

The design of passive redundancy is more complex as it requires a switching mechanism. If the switch is itself imperfect, then the failure time, as a pair, is given by Xj X12 =

X 1 +X

with probability q, 2

with probability (1 - q ) ,

where q is the probability that the switch is in a failed state when needed. Introducing redundancy increases the manufacturing cost but results in reduced warranty servicing cost. Optimal redundancy minimizes the sellers expected cost, which is the sum of the two costs.

4.6. Warranty and manufacturing Warranty and quafity control. Manufacturing deals with processes and input material needed to translate a conceptual design into a physical end-product for sale. Because of inherent variations in input material, and in the process itself, not all end-product items conform to the design specification. Items which do not conform to the required design specification are called defective items and are items which do not perform satisfactorily. Defective items have higher failure rates than non-defective items and have a significant impact on the warranty cost. By proper design of the process and effective control, the fraction of defective items produced can be reduced. However, it is not possible to completely eliminate the production of defective items. One way of reducing defective items reaching the market is to inspect and test items at the final stage. These actions reduce expected warranty cost but increase the manufacturing cost. Djamaludin, Murthy and Wilson (1991a) deal with the case where items are produced in batches. The quality control scheme consists of life testing, for a period T, of a small sample of n items from each batch. If no item in the sample fails, the batch is released for sale. If one or more items in the sample fail, the complete batch is subjected to life testing for period T and only times which survive the test are released for sale. The details of this scheme for different types of warranties can be found in Murthy, Djamaludin and Wilson [1991]. Currently under investigation are quality control schemes when items are produced in lots with product quality dependent on lot sizing. For more on this, see Djamaludin, Murthy and Wilson (1991b). Pre-sale testing. For products with bathtub failure rate, the probability of an item failing at a young age is high due to 'infant mortality'. Once the item survives this initial period, the failure rate becomes small. Warranty costs for such items can be high. To reduce this cost, the manufacturer can use 'burn-in'. This involves putting the item into use on a test bed for a certain period of time before it is sold. Should the item fail during burn-in, it is repaired in the case of repairable items and junked in the case of non-repairable items. Thus, items sold are ones which have either survived the burn-in or have early failures fixed before sale. As a result, the items have progressed beyond the infant mortality period to the period of lower failure rate. Thus, the expected number of returns under warranty and hence the expected warranty costs are reduced. The burn-in is worthwhile only if the reduction in costs due to failures is greater than the cost of the burn-in program. If an item is subjected to a burn-in for a period r, the failure distribution of the item after burn-in, F~(t), is given by F~( t) = [F(~- + t) - F(-c)]/ff('r).

(85)

W.R. Blischke, D.N.P. Murthy / Product warrantyIII

29

As a result, the sellers expected cost E[Cs(W; r)] is a function of r. Let ~O(r) be the expected cost per item when the burn-in is of duration r. This depends on whether the product is repairable or not. In an analysis of repairable products, Nguyen and Murthy (1984) assume that all failures during burn-in are fixed minimally. As a result,

~(7") =Cm q-Cl q-C2Tq-Crf0Tr(t) dt,

(86)

where c m is the manufacturing cost per item and c r is the expected cost of each repair; c l is the fixed setup cost of burn-in per unit and c 2 is the cost per unit time of burn-in per unit. Note that qJ(r) is an increasing function of r. For a non-repairable product, the model for burn-in cost per unit is given by: (i) c I + c2t, 0 < t < r, if the items fail during burn-in, and (ii) cl + Czr, if the item survives the burn-in. Thus, the expected burn-in cost per unit is given by c~ + c2f~ff(t) dt. Since the probability of a unit surviving the burn-in is F ( r ) , and since costs must be recovered from items sold, we have the expected cost per unit given by

tlt( r) = [Cm + Cl + c2 f(]ff( t ) dt ] / f f ( r).

(87)

Note that ~0(r) is an increasing function of r. Nguyen and Murthy (1982) consider both F R W and PRW policies and derive results for optimal burn-in to minimize the total expected seller's cost, comprising warranty servicing cost and the burn-in cost. For a comprehensive treatment of the modeling of burn-in, see Leemis and Beneke (1990).

4. 7. Dispute resolution In some instances, a manufacturer might refuse to admit a claim under warranty. In this case, we have a warranty dispute and the resolution of the dispute is of interest to both the consumer and the manufacturer. Although the literature on dispute resolution is vast, the treatment of the subject from a mathematical modeling viewpoint is very limited. Palfrey and Romer (1983) describe models for alternate dispute resolutions in the context of warranty disputes. An important variable in dispute resolution is 'negligence' and the outcome depends on the negligence rule. For a mathematical treatment of different negligence rules, see Brown (1973). Also, in this context, the literature on the mathematical treatment of the economics of law is highly relevant and offers scope for study of dispute resolution using mathematical models.

5. Review of models: Public policy perspective The system characterization for study of topics from the public policy perspective was done in Section 6 of Paper 2 and the main elements are shown schematically in Figure 5. Here, one is interested in the aggregate behavior of consumers rather than in an individual consumer. In a monopolistic situation, we have a single manufacturer and, in the case of a more competitive situation, there are many manufacturers. A variety of mathematical models have been developed to study the effect of warranties on market behavior and the resulting social welfare implications. With symmetric information, risk neutral and homogeneous consumers, and specified product quality, warranties do not matter. In this case, from a social welfare point of view, it does not matter whether manufacturers or consumers carry the cost of warranty, price will adjust accordingly. The effects of warranties on the market become important when

30

W.R. Bl~chk¢ D.N.P. Murthy / Product warran~ III MARKET

CONSUMERS

MANUFACTURERS

m,

~m

mLEGISLATIVE PROCESS

,ram JUDICIAL PROCESS

Figure 5. Main factors for study of warranty at the aggregate level

information is asymmetric or consumers are risk averse. Product quality and consumers being heterogeneous complicates the picture considerably. One can classify the models dealing with public policy issues into four categories based on the role assigned to warranties. These are: Warranties as Insurance, Warranties as Signals, Warranties as Incentives, and Warranties as Marketing Devices. In nearly all the models, warranty is viewed as a lump sum payment to the buyer should the product fail within the warranty period. In terms of the taxonomy of our earlier paper, this corresponds to a rebate policy with lump sum payment. Item failures are characterized by the probability of an item failing during the warranty period. Some models deal with monopolistic situations while others deal with non-monopolistic situations. The informational aspects (the information available to consumers and manufacturers) vary from complete information to very little information. Most models start with a statement of consumer behavior and manufacturer behavior in the form of axioms. Using fairly standard techniques of analysis (see Varian, 1990), the market equilibria are first analyzed (in terms of model variables such as price, warranty, quality, etc.) and then the social welfare aspects and policy recommendations are examined. Rather than attempting to discuss all of the different models, we confine ourselves to a small illustrative sample, categorized into the four groups indicated earlier. Warranties as Insurance

The underlying assumption for this viewpoint is that consumers are more risk averse than are manufacturers. Warranty is viewed as an insurance against product failure or a product not performing as expected.

W.R. Blischke, D.N.P. Murthy / Product warranty III

31

Heal (1977) deals with a model formulation where both manufacturers and consumers have the same information. Consumers are assumed to be risk averse and manufacturers are assumed to be risk neutral. The model examines the risk sharing aspects. Epple and Raviv (1978) present a model to study the effect of two different liability rules - (i) consumer liability (no warranty) and (ii) manufacturer liability (full warranty) - on product quality under two different market structures - (i) perfect competition and (ii) monopoly. An interesting topic in the context of warranties as insurance is the 'Adverse Selection' problem (Emmons, 1989). The problem arises when there are high-risk and low-risk consumers, and, for a given product quality, the probability of product failure is higher for the high-risk consumer. Viewed as a risk, manufacturers need to charge high-risk consumers a higher premium for warranty. When manufacturers cannot discriminate high-risk users from low-risk consumers, two situations arise. The first is a pooling situation in which manufacturers choose their premium based on an average breakdown probability obtained by pooling the two types of consumers. In this case, the warranty terms are unfair to the low-risk consumers and high-risk consumers will try to purchase this warranty so as not to reveal their identity. The second is a separating solution, in which the manufacturer offers two different warranties, with high risk-consumers getting full coverage at a higher premium and low risk-consumers getting a coverage less than full at a lower premium. The lower-level coverage is set up so that high-risk consumers have no incentive to opt for this and purchase the warranty designed for them. A similar problem arises when product failure depends on the intensity of usage, with consumers being either low-intensity or high-intensity users and manufacturers unable to distinguish between the two. Emmons (1988a) explains limited warranty duration as the outcome of low-intensity users preference in both pooling and separating situations. Warranties as Signals

When consumers cannot discern quality, then low quality products can drive out high quality products from the market, leading to a market breakdown due to adverse selection (Akerlof, 1970). In this context, warranties serve as signals to indicate quality and hence facilitate consumers ability to differentiate product quality based on the warranty offered. The basis for this inference is that manufacturers of high quality products can offer warranties which cannot be matched by manufacturers of low quality products. Spence (1977) deals with both risk neutral and risk averse consumers and studies the impact of warranty as a signal to communicate information. Schwartz and Wilde (1982) deal with a model where consumers are differentiated as shoppers and non-shoppers depending upon the effort made to obtain information about product price and warranty. Non-shoppers are those who buy the product from the first manufacturer they contact whereas shoppers contact at least two manufacturers before they purchase the product. The product market consists of two markets - one for products with warranty and the other for those without. Each market can be non-existent, competitive or noncompetitive. Warranties as Incentives

The incentive viewpoint is related to the signaling viewpoint but product quality is determined by each manufacturer (hence treated as an endogenous rather than as an exogenous variable) and warranties serve as incentives for manufacturers to provide consumers with high quality products. Without warranties, there would be no incentive for manufacturers to produce high quality products. However, when the probability of item failure is dependent on consumer maintenance efforts, we have a consumer moral hazard problem unless manufacturers can monitor each consumer. The study of warranty incorporating moral hazard consideration has received considerable attention. Cooper and Ross (1985) deal with a model formulation where the probability of the product not failing is a function with both product quality and consumer maintenance effort. They analyze the problem as a two-stage game. In the first (cooperative) stage, manufacturers and consumers sign a

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binding agreement with respect to price and degree of warranty protection which compensates the consumer should the product fail. The second (noncooperative) stage takes price and warranty terms as given and manufacturers choose product quality to maximize expected profits while consumers choose a maintenance effort to maximize expected utility. Emmons (1988b) deals with a model in which the double moral hazard problem is analyzed as a two-stage game. In the first stage, manufacturers offer quality levels and price-warranty combinations. In the second stage, consumers observe price-warranty combinations but not quality level. Consumers choose their maintenance effort and the product offering to maximize their utility. Warranties as Marketing Devices

When consumers are heterogeneous, different price-warranty schedules can be used as a device by a monopolist to segment consumers and extract consumer surplus. (See Mathews and Moore, 1987; Kubo, 1986.) The price, warranty and quality combinations that the monopolist chooses for maximizing profit have the following properties: All consumers except the one with the highest willingness to pay get quality and warranty terms which are less than the socially efficient levels (levels which maximize social welfare). The monopolist provides less than the efficient warranty and quality level to consumers with a low valuation to discourage high demand buyers from switching to low-margin products. When consumers are risk averse with identical tastes but different incomes, the optimal strategy for the monopolist is to offer products without warranty at low price and with warranty at high price. Items purchased with warranty are replaced free of charge should they fail under warranty. (Emmons, 1989.) Courville and Hausman (1979) deal with a model which shows that a monopolist will choose quality level and warranty terms that are the same as under a competitive market but would charge a higher price. This implies that monopolists will use warranties to credibly bond themselves to high quality production rather than reduce quality or warranty. Grossman (1981) deals with a model formulation where the monopolist has an incentive to reveal his quality even when it is low and warranties serve as devices to communicate product quality.

6. Conclusions In this three part sequence, we have proposed a taxonomy for classifying the different policies that have been studied in the past in Paper 1; developed a unified framework to study different aspects of warranty from three different perspectives in Paper 2; and, in this paper, carried out a brief review of the mathematical models that have been developed and studied in the past. In this section we comment on topics for further study and research in the future. As can be seen from Paper 3, the bulk of the models developed to date deal with cost analysis of warranties which can be called one-dimensional and one-variate warranties. (See Wilson and Murthy, 1991.). The cost analysis of two-dimensional and one-variate warranties has received some attention in the past and is a topic of active research. In contrast, multi-dimensional and multi-variate warranty policies have received very little attention and as such offer scope for new research. From the manufacturer's perspective, there is need for further research into linking design and manufacturing with warranty. This would need better models for product development and quality variations in manufacturing. From the consumer's perspective, there is need for modeling of the buying decision which incorporates search for warranty, evaluation of warranty and the exercising of warranty claims. Models for dispute resolution are very limited and highly simplified. There is need for more refined models which more adequately model the process. From the public policy perspective, most of the models developed are very stylized and model the dynamic nature of the process in a very simplified manner. Also, no two models are built on the same set

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of assumptions. This makes any comparative evaluation difficult. As a result, there is a strong need to build more complex models which yield the models studied so far as special cases. These are just a few of the topics for further study and research. Many other are apparent in the detailed discussions in these papers and in the many references cited.

References Akerlof, G. (1970), "The market for 'Lemons': Quality uncertainty and the market mechanism", Journal of Econometrics 84, 488-5O0. Amato, H.N., and Anderson, E.E. (1976), "Determination of warranty reserves: An extension", Management Science 22, 1391-1394. Balachandran, K.R., Maschmeyer, R.A., and Livingstone, J.L. (1981), "Product warranty period: A Markovian approach to estimation and analysis of repair and replacement costs", Accounting Review LVI, 115-124. Baxter, L.A., Scheuer, E.M., Blischke, W.R., and McConologue, D.J. (1981), "Renewal tables: tables of functions arising in renewal theory", Technical Report, Decision Systems Dept., Univ. of Southern California, Los Angeles, CA. Baxter, L.A., Scheuer, E.M., Blischke, W.R., and McConologue, D.J. (1982), "One the Tabulation of the Renewal Function", Technometrics 24, 151-156. Biedenweg, F.M. (1981), "Warranty analysis: consumer value vs manufacturers cost", Unpublished Ph.D. Thesis, Stanford University, USA. Blischke, W.R. (1990), "Mathematical models for analysis of warranty policies", Mathematical and Computer Modelling 13 (7), 1-16.

Blischke, W.R., and Murthy, D.N.P. (1992), "Product warranty management - I: A taxonomy for warranty policies", European Journal of Operational Research 62, 127-148. Blischke, W.R., and Murthy, D.N.P. (1993), Mathematical Models for Warranty Cost Analysis, Marcel Dekker, New York, to appear. Blischke, W.R., and Scheuer, E.M. (1975), "Calculating the cost of warranty policies as a function of estimated life distributions", Naval Research Logistics Quarterly 22, 681-696. Blischke, W.R., and Scheuer, E.M. (1981), "Applications of renewal theory in analysis of the free-replacement warranty", Naval Research Logistics Quarterly 28, 193-205. Blischke, W.R., and Scheuer, E.M. (1984), "On cumulative warranties", TIMS XXVI, Copenhagen. Brown, J.P. (1973), "Toward an economic theory of liability", Journal of Legal Studies 2, 323-349. Cooper, R., and Ross, T.W. (1985), "Product warranties and double moral hazard", Rand Journal of Economics 16, 103-113. Courville, L., and Hausman, W.H. (1979), "Warranty scope and reliability under imperfect information and alternate market structures", Journal of Business 52, 361-378. Crowe, L.H. (1974), "Reliability analysis for complex repairable systems", in: F. Proschan and R.J. Serfling (eds.), Reliability and Biometry, SLAM, Philadelphia, 397-410. Djamaludin, I., Murthy, D.N.P., and Wilson, R.J. (1991a), "Product Warranty and Quality Control", Presented at the ORSA/TIMS Meeting, Los Angeles, CA. Djamaludin, I., Murthy, D.N.P., and Wilson, R.J. (1991b), "Quality control through lot sizing for products sold under warranty", under preparation. Duane, J.T. (1964), "Learning curve approach to reliability monitoring", IEEE Transactions on Aerospace 2, 563-566. Emmons, W. (1988a), "On the limitation of warranty duration", Journal of Indian Economics 37, 287-301. Emmons, W. (1988b), "Warranties, moral hazard, and the lemons problem", Journal of Economic Theory 46, 16-33. Emmons, W. (1989), "The theory of warranty contracts", Journal of Economics Surveys 3, 43-57. Epple, D., and Raviv, A. (1978), "Product safety: Liability rules, market structure, and imperfect information", American Economic Review 68, 80-90. Gastwirth, J. (1976), "Models of consumer search for information", The Quarterly Journal of Economics 38, 38-50. Gates, R.K., Bortz, J.E., and Bicknell, R.S. (1977), "Quantitative models used in the RIW decision process", Proceedings Annual Reliability and Maintainability Symposium, 16, 229-238. Glickman, T.S., and Berger, P.D. (1976), "Optimal price and protection period for a product under warranty", Management Science 22, 1381-1390. Grimlund, R.A. (1985), "A proposal for implementing the FASB's "Reasonably Possible" disclosure for product liabilities", Journal of Accountancy Research 23, 575-594. Grossman, S. (1981), "The informational role of warranties and private disclosure about product quality", Journal of Law and Economics 24, 461-483. Guin, L. (1984), "Cumulative warranties: Conceptualization and analysis", doctoral dissertation, Univ. of Southern California. Heal, G., (1977), "Guarantees and risk sharing", Review of Economic Studies 44, 549-560. Iskandar, B.P., Murthy, D.N.P. and Wilson, R.J. (1991a), "Two dimensional rebate warranty policies", Proceedings of the llth ASOR Conference, Gold Coast, Australia, 249-265.

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lskandar, B.P., Murthy, D.N.P., and Wilson, R.J. (1991b), "Two dimensional pro-rata policies", under preparation. Kotler, P. (1980), Marketing Decision Making. A Model Building Approach, Holt, Rinehart and Winston, New York. Kubo, Y. (1986), "Quality uncertainty and guarantee - A case of strategic segmentation by a monopolist", European Economic Review 30, 1063-1079. Leemis, L.M., and Beneke, M. (1990), "Burn-in models and methods: A review", liE Transactions 22, 172-180. Mahon, B.H., and Bailey, R.J.M. (1975), "A proposed improvement replacement policy for army vehicles", Operations Research Quarterly 26, 477-494. Mamer, J.M. (1982), "Cost analysis of pro-rata and free replacement warranties", Naval Research Logistics Quarterly 29, 345-356. Mamer, J.M. (1977), "Discounted and per unit costs of product warranty", Management Science 33, 916-930. Mathews, S., and Moore, J. (1987), "Monopoly provision of quality and warranties: An exploration in the theory of multidimensional screening", Econometrica 55, 441-467. Menke, W.W. (1969), "Determination of warranty reserves", Management Science 15,542-549. Moskowitz, H., and Chun, Y.H. (1988), "A Bayesian approach to the two-attribute warranty policy", Paper No. 950, Krannert Graduate School of Management, Purdue University, West Lafayette, IN. Murthy, D.N.P. (1990), "Optimal reliability choice in product design", Engineering Optimization 15, 281-294. Murthy, D.N.P., and Blischke, W.R. (1991), "Product warranty management - II: An integrated framework for study", European Journal of Operational Research 62, 261-281. Murthy, D.N.P., Djamaludin, I., and Wilson, R.J. (1991), "Quality control for warranties items", under preparation. Murthy, D.N.P., lskandar, B.P., and Wilson, R.J. (1990), "Two-dimensional warranty policies: A mathematical study", Proceedings of the lOth ASOR Conference, Perth, Australia, 89-104. Murthy, D.N.P., Iskandar, B.P., and Wilson, R.J. (1991), "Two-dimensional failure free warranties: Two-dimensional point process models", submitted for publication in Operations Research. Murthy, D.N.P., and Nguyen, D.G. (1987), "Optimal development testing policies for products sold with warranty", Reliability Engineering 19, 113-13. Murthy, D.N.P., and Nguyen, D.G. (1988), "An optimal repair cost limit policy for servicing warranty", Mathematical and Computer Modelling 11,595-599. Murthy, D.N.P., and Wilson, R.J. (1991), "Modelling two dimensional warranties", Proceedings of the 5th International Conference on Applied Stochastical Models and Data Analysis, Granada, Spain, 481-492. Murthy, D.N.P., Wilson, R.J., and Iskandar, B.P. (1991), "Two-dimensional failure free warranties: One-dimensional point process models", under preparation. Nguyen, D.G., and Murthy, D.N.P. (1982), "Optimal burn-in time to minimize cost for products sold under warranty", liE Transactions 14, 167-174. Nguyen, D.G., and Murthy, D.N.P. (1984a), "A general model for estimating warranty costs for repairable products", liE Transactions 16, 379-386. Nguyen, D.G., and Murthy, D.N.P. (1984b), "Cost analysis of warranty policies", Naval Research Logistics Quarterly 31, 525-541. Nguyen, D.G., and Murthy, D.N.P. (1986), "An optimal policy for servicing warranty", Journal of the Operations Research Society 37, 1081-1088. Nguyen, D.G., and Murthy, D.N.P. (1988a), "Failure free warranty policies for non-repairable products: A review and some extensions", Operations Research 22, 205-220. Nguyen, D.G., and Murthy, D.N.P. (1988b), "Optimal reliability allocation for products sold under warranty", Engineering Optimization 13, 35-45. Nguyen, D.G., and Murthy, D.N.P. (1989), "Optimal replace-repair strategy for servicing items sold under warranty", European Journal of Operational Research 39, 206-212. Palfrey, T., and Romer, T. (1983), "Warranties, performances, and the resolution of buyer-seller disputes", Bell Journal of Economics 14, 97-117. Ritchken, P.H., and Tapiero, C.S. (1986), "Warranty design under buyer and seller risk aversion", Naval Research Logistics Quarterly 33, 657-671. Rotz, A.O. (1986), "Warranty pricing with a life cycle cost model", Proceedings Annual Reliability and Maintainability Symposium 25, 454-456. Schwartz, A., and Wilde, L.L. (1982), "Competitive equilibria in markets for heterogeneous goods under imperfect information: A theoretical analysis with policy implications", Bell Journal of Economics 13, 181-193. Spence, M. (1977), "Consumer misperceptions, product failure and product liability", Review of Economic Studies 44, 561-572. Tapiero, C.S., and Posner, M.J. (1988), "Warranty reserving", Naval Research Logistics Quarterly 35, 473-479. Thomas, M.U. (1989), "A prediction model for manufacturer warranty reserves", Management Science 35, 1515-1519. Varian, H.R. (1984), Microeconomic Analysis, Norton, New York. Wilson, R.J., and Murthy, D.N.P. (1991), "Multi-dimensional and multi-variate warranty policies", presented at the ORSA/TIMS Joint National Meeting, Los Angeles, CA.