Warranty costs: An age-dependent failure/repair model

Warranty Costs: An Age-Dependent Failure/Repair Model

Boyan Dimitrov,1 Stefanka Chukova,2 Zohel Khalil3

1 Department of Sciences and Mathematics, Kettering University,Flint, Michigan 48504-4898

2 School of Mathematical and Computing Sciences, Victoria University of Wellington,Wellington, New Zealand

3 Department of Mathematics and Statistics, Concordia University,Montreal, Canada, H4B 1R6

Received 28 February 2002; revised 28 January 2004; accepted 9 June 2004DOI 10.1002/nav.20037

Published online 30 July 2004 in Wiley InterScience (www.interscience.wiley.com).

Abstract: An age-dependent repair model is proposed. The notion of the “calendar age” of theproduct and the degree of repair are used to define the virtual age of the product. The virtualfailure rate function and the virtual hazard function related to the lifetime of the product arediscussed. Under a nonhomogeneous Poisson process scenario the expected warranty costs forrepairable products associated with linear pro-rata, nonrenewing free replacement and renewingfree replacement warranties are evaluated. Illustration of the results is given by numerical andgraphical examples. © 2004 Wiley Periodicals, Inc. Naval Research Logistics 51: 959–976, 2004.

Keywords: warranty cost analysis; virtual age; virtual failure rate; virtual hazard function;pro-rata warranty; maintenance costs; Kijima Model I

1. INTRODUCTION

In today’s market, product warranty plays an increasingly important role. The use of warrantyis widespread and serves many purposes, including protection for manufacturers, sellers,insurance, buyers, and users. It provides indirect information about the quality of the productsand represents an important tool to influence the market. A detailed discussion of the variousaspects of warranty can be found in the handbook of Blischke and Murthy [2]. One of the mainproblems for the companies offering warranties is the prediction of the warranty costs. Forexample, the pricing of the warranty should be balanced against the expenses throughout thewarranty coverage. The evaluation of the warranty cost depends on the failure process, thedegree of repairs, and the prescribed maintenance of the item.

Correspondence to: B. Dimitrov ([email protected])

© 2004 Wiley Periodicals, Inc.

Repairable products are affected by the postfailure repairs in one of the following ways:

— A repair completely resets the failure rate of the product so that upon restart the productoperates as a new one. This is known as a complete repair, and it is equivalent to areplacement of the faulty item by a new one.

— A repair has no impact on the failure rate. The failure rate remains the same as it was priorto the failure. The repair brings the product from a down to an up state without affectingits performance. This is known as a minimal repair.

— A repair contributes to some noticeable improvement of the product. This contribution ismeasured by an age-reducing repair factor. The repair sets back the clock of the repaireditem. After the repair the performance of the item is as it was at an earlier age. This isknown as an imperfect repair.

— A repair may contribute to some noticeable degradation of the product. This contributionis measured by an age-accelerating repair factor. The repair sets forward the clock of therepaired item. After the repair the performance of the item is as it will be at some later age.This type of repair is (called by Scarsini and Shaked [17] a sloppy repair.

A repair process could be a mixture between minimal, imperfect, sloppy, and complete repair.Models of imperfect, sloppy, or mixture of repairs specify the class of age-dependent repair

models. The age-dependent models of repairable products have been discussed in reliabilityliterature (e.g., Block et al. [4]). The model we consider is Kijima Model I (Kijima [11]).Scarsini and Shaked [17] enriched this model with the concept of sloppy repair. Both papersfocus on the application of the model on a finite horizon. Stochastic inequalities for variouscharacteristics of the product are derived in terms of the intermaintenance or interrepair times.Guo and Love [9, 10] and Love and Guo [13, 14] provide an overview on statistical inferencesof the age-dependent models. For warranty purposes such models have been considered inChukova and Khalil [6, 7] and in Chukova [5]. To the best of our knowledge, models withimperfect or sloppy repairs have not been studied from warranty viewpoint.

In this paper we focus on the failure rate function of a product maintained under Kijima’sModel I scenario [11]. Following Scarsini and Shaked [17], we call it virtual failure rate. InSection 2, assuming that the age-reducing, or age-accelerating, repair factor is a constant, wefind a relationship between the virtual failure rate and the original failure rate. In Section 3 weapply the results from Section 2 to study warranty problems. In Section 4, assuming Weibulllifetime distribution, we illustrate the warranty problems studied in Section 3. We introduce theconcept of “age” of the products at the time the warranty is assigned. In addition, we refer toage-reducing or age-accelerating repair factor as an age-correcting repair factor. The length ofthe warranty period, the magnitude of the age-correcting factor and the “age” of the products atthe time the warranty is assigned are the parameters that will affect the warranty costs. In whatfollows, we reconsider the age-dependent models in Chukova and Khalil (1990), use some basicideas of Nelson [15, 16], and Block, Borges, and Savits [3], and give further development of thediscussion in Chukova [5]. In addition, our results are an useful extension of the properties ofKijima’s Model I [11] and could be useful in other reliability contexts.

2. MODEL DESCRIPTION

The most convenient description, from our point of view, of imperfect or sloppy repairs isgiven in terms of the failure rate function and corresponding hazard function of the lifetime of

960 Naval Research Logistics, Vol. 51 (2004)

the product. Then, the intensity function approach (Kotz and Shanbhag [12]) can be used tomodel the underlying random variables, and vice verse.

Let the initial lifetime X of a new product sold under warranty be a continuous randomvariable (r.v.) with probability cumulative distribution function (c.d.f.) F( x) � P(X � x) andprobability density function (p.d.f.) f( x) � (d/dx) F( x). Then its hazard function is

��x� � �ln�1 � F�x��, x � 0, (1)

and its failure rate function is

��x� �d

dx��x� �

f�x�

1 � F�x�, x � 0.

Block, Borges, and Savitz [3] and Beichelt [1] have shown that under instantaneous minimalrepairs the total number of failures within any time interval, say [u, u � v), u, v � 0, has thePoisson distribution with mean �(u � v) � �(u), u, v � 0. Let N[u,u�v) be the number offailures during the interval [u, u � v). Then the probability of no failure in [u, u � v) is

P�N�u,u�v� � 0� � e��u�v��u��, u, v � 0.

Moreover, the probability of first failure to occur after the moment u in the interval [u, u � v� v), v 0, is

P�N�u,u�v� � 0, N�u�v,u�v�v� � 1� � ��u � v�e��u�v��u�� v � o�v�. (2)

The minimal repairs have no impact on the failure rate function. They switch the product froma down to an up state. At the same time they accumulate repair cost. The amount of labor, repairtime, or money invested in the repair may have significant impact on the product improvement,which directly affects the warranty costs. We consider imperfect repairs that may affect thefuture performance of a repairable product, and are related to an age-reducing factor �. Morespecifically, following Scarsini and Shaked [17], let Xi denote the inter-repair or inter-mainte-nance times of the product. Let �i denote the lack of perfection of the ith repair. Then

T0 � 0, Ti � Ti�1 � �iXi, i � 1, 2, . . . , (3)

are the values of the virtual age of the product immediately after the ith repair. If �i � 1, thenno improvement or deterioration of the product occurs at the ith epoch of action, whereas if �i �() 1 then an improvement (deterioration) of the product occurs at that epoch. The extreme caseof �i � 0 corresponds to a complete repair. The model described in (3) is also known asKijima’s Model I [11]. In the sequel we consider this model with the assumption that �i � � �0, and refer to � as an age-correcting factor. If � � 1, we call it age-reducing factor, and if � 1 we call it age-accelerating factor. In warranty it is natural to assume that � � 1.

With no failures in [0, u), u 0 the product would have the original failure rate function�( x) for x � (0, u). Referring to Figure 1, the first age-reducing repair occurs at an instant u.After the repair, the product is improved and its performance is as it was earlier, when the ageof the product was �u. At calendar age u, which is the time of the first repair, the virtual age

961Dimitrov, Chukova, and Khalil: Warranty Costs: An Age-Dependent Failure/Repair Model

of the product is �u. From time u onwards, until the next repair, the performance of the productis modeled by modified original failure rate function �( x � (u � �u)). Assume that the nextfailure is at the calendar age u � v. The instantaneous repair improves the performance of theproduct and its virtual age is �u � �v. Physically, between the two consecutive failures, theproduct experiences age accumulation, say v, but due to the age-correcting repair, its virtual ageaccumulation is only �v. The failure rate function of the lifetime of the product maintained withage-correcting repairs is the modified original one, as shown in Figure 1.

For any particular product in a homogeneous population this function will have its jumpswhenever an age-correcting repair occurs. Therefore, future failures may reduce (or prolong) theincrements in virtual age by factor �. Hence, its virtual failure rate will be compressed (orstretched) compare to the original failure rate. For a population of products (with i.i.d. lifetimes)maintained under age-correcting repairs with identical age-correcting factors, the populationfailure rate is obtained by averaging the possible individual failure rates of all products. Thefailure rate for this population is the virtual failure rate of one randomly selected productmaintained under age-correcting repairs. We denote it by ��( x), where x is a calendar age. Itreflects the overall slow-down or acceleration of the aging process for an “average” productfrom the population. This concept is similar to Nelson [16], used to evaluate the MeanCumulative Cost Function.

Since we assume instantaneous repairs, the virtual hazard function ��( x) is a continuousnondecreasing function of the time t, and its right derivative is ��( x) � (d/dx)��( x)�x�0. Thesubscript here indicates that the value of ��( x) is considered immediately after the completionof an age-correcting repair.

In what follows we derive a relationship between the original failure rate and the virtualfailure rate of the product from a population maintained under age-correcting instantaneousrepairs of factor �.

Consider the sequence 0 � T0 � T1 � T2 � . . . � Tn � . . . of times representing thevirtual age of the product after the nth repair. To avoid technical difficulties, assume that theoriginal life time of the product, X, has a c.d.f. F( x) with F(0) � 0 and F( x) � 1 for all x �0. Denote the corresponding survival function by F� ( x) � P{X x} � 1 � F( x). Taking intoaccount the relationship between the calendar age of the item, and its virtual age after anage-correcting repair of factor �, given by (3), it follows that:

● The nth step transition probability function is

P Tn�1 � t�T1, . . . , Tn� � P�X �t

��X �

Tn

� � �F� �max�Tn/�, t/��

F� �Tn/��(4)

for n � 1, 2, . . . .

Figure 1. (a, left) Original and individual virtual failure rates under age-reducing factor � � .6. (b, right)Original and individual virtual failure rates under age-accelerating factor � � 1.2.


● The initial distribution is

P T1 � t� � P�X1 �t

�� F� � t

�� . (5)

● From (4) and (5) it is easy to check by induction that for any nonnegativemeasurable function g(t1, . . . , tn) and any n � 2 it is true that

E� g�T1, . . . , Tn�� 0

� �F� � t1

��1 �

t1

� �F� � t2

��1

· · ·�tn�2

� �F� � tn�1

� ��1

�tn�1

�

g�t1, . . . , tn� dF� tn

�� · · ·dF� t1

�� . (6)

Let {Ntv, t � 0} be the counting process corresponding to {Tn}n�0

� defined by

Ntv � �

n

I�0,t��Tn�,

where IB� is the indicator function of the set B.

THEOREM 1 (AGE TRANSFER THEOREM): {Ntv, t � 0} is a nonhomogeneous Poisson

process (NHPP) with a leading function

�v�t� � E�Ntv� � �log�1 � F� t

�� t

��, (7)

where �(t) � �log(1 � F(t)) is the leading function of the NHPP associated with the processof instantaneous minimal repairs. The proof of the theorem follows the ideas in Block, Borges,and Savits [3], and is given in the Appendix.

Equation (7) shows that the transformation between the calendar and the virtual time scalesis tv 3 tv/�; i.e., if the virtual age of a product is tv, its corresponding calendar age is tv/�. Tosee this, denote by T the r.v. representing the virtual lifetime of the product. The c.d.f. of T isFT(t) � 1 � e��v(t). Equation (7) shows that

P�T t� � FT�t� � FX� t

�� P�X t

�� .

Therefore, we have

P�T t� � P��X t�,

and this means that the virtual lifetime T, and the multiplied by � calendar lifetime X are equalin distribution, i.e.,


T �d

�X. (8)

Therefore, when the product is at calendar age x, which is the only age known, its virtual agemeasured at the calendar age scale is �x. Therefore, at calendar age x, an item maintained underage-correcting repairs of factor � operates as a new item at age �x. Thus

��x� dx � ��x� dx,

i.e., the probability to have a failure of the product from the original population within theinterval [ x, x � dx) is the same as the probability to have a failure of a product from thepopulation of products, maintained by age-reducing repairs, within the interval [�x, �x � dx).

The relations

��x� � �0

x

��u� du and ��x� � �0

x

��u� du

will lead to

��x� �1

��x�, x � 0, � � 0.

Hence, the following result holds:

THEOREM 2: The virtual failure rate ��( x) at calendar age x and the original failure rate arerelated by the equality

�*�x� � ��x�, x � 0, 0 � � 1. (9)

The virtual hazard rate ��( x) and the original hazard rate �( x) are related by

��x� �1

��x�, x � 0, � � 0. � (10)

Thus, we see that an age-reducing repair of factor � slows down the aging process of theproduct by 100(1 � �)%. The minimal repairs are age-reducing repairs with the age-reducingfactor � � 1. However, the model with replacements of the faulty product by a new one is notanalytically a particular case of the age-reducing repair model, say with � � 0. For � � 0,Theorem 1 is not valid because it reflects the failure rate immediately after an age-reducingrepair only. The values of 0 � � � 1 correspond to reliability improvement of the product.

Naturally there are certain costs associated with this improvement. Let us denote the cost ofan age-reducing repair of factor � at calendar age u of the product by Cr(u, �). A naturalassumption is that Cr(u, �) satisfies the following inequalities:

Cm�u� � Cr�u, �� Cc�u�, (11)


where Cm(u) is the cost of a minimal repair of the product at calendar age u and Cc(u) is thecost of a replacement (complete repair) of the product at age u. The expected warranty costsassociated with the product sold under warranty would depend on the warranty contract and themaintenance schedule associated with the product (Chukova [5]). Different forms of warrantycoverage with age dependent repairs will model large variety of practically feasible andimportant warranty policies. In this paper we confine out attention to some of the typicalwarranty policies.

3. WARRANTY COSTS ASSOCIATED WITH A PRODUCT DURING THEASSIGNED WARRANTY TIME

Here and onwards the time scale is the calendar age time scale. In this section we discusssome of the typical warranty policies, emphasizing the age-dependent warranty, and show howthe age of the product at the time of the sale, say t0, can affect the warranty costs. We will referto t0 as the initial age of the product. The warranty cost depends on the warranty policy and thedegree of the warranty repairs (e.g., Blischke and Murthy [2], Chap. 10). The degree of thewarranty repairs is measured by the age-correcting factor �. We assume that a sold item iscovered by warranty for a calendar time of duration T, according to certain warranty contract.The warranty coverage starts at the time the sale is completed. We consider warranty modelingfrom warranter’s point of view. The warranter covers all, or a portion of the expenses associatedwith the failures and repairs of the product within the warranty coverage.

3.1. Partial Rebate Warranty, Nonrepairable Product

First we consider a warranty cost model for a nonrepairable product with an initial age t0.Assume that the initial lifetime X of the product is known and has failure rate function �(t).During the warranty period no repairs are possible. The initial age t0 of the product at the timeof the sale, which is also the time when the warranty starts, is the parameter of interest.

If the failure is beyond the assigned warranty period, the warranter incurs no expenses. If thefailure occurs at the moment t � [t0, t0 � T), the warranter refunds the user by

C�t� � C0�t0� �T � t

T,

where C0(t0) is the purchase price of the product at age t0.

LEMMA 1: The expected warranty cost associated with an age-dependent product sold at aget0 with warranty of duration T is given by

CW�t0, T� � C0�t0� �0

T

��t0 � t�e��t0�t��t0�� T � t

Tdt, (12)

where �(t) and �(t) are the original failure rate and original hazard rate functions.

PROOF: Based on NHPP concept for the failure rate of the product and using Eq. (2) for theprobability of the first failure after t0, we find that the probability to have a failure in [t0 � t,


t0 � t � dt) is �(t0 � t)e�[�(t0�t)��(t0)] dt. Then the remaining time until the end of warrantyis max(T � t, 0), and therefore the warranter refunds the user by C0(t0)(T � t)/T. The totalexpectation rule completes (12). �

NOTE: For a new product t0 � 0. However, some used products are sold under this type ofwarranty. Thus, (12) represents the expected warranty cost under linear pro-rata warranty fornonrepairable product which performance depends on the age it has been sold. Some illustra-tions of the dependence of the expected warranty cost on the initial age t0 are given in Sec-tion 4.

3.2. Age-Reducing Repairs under Fixed Warranty

In this section we consider repairable products sold under warranty of fixed duration T. Weassume that during their usage, the products are maintained under age-reducing repairs, as inTheorem 1. We analyze a cost model for the Free Replacement Warranty (briefly FRW,Blischke and Murthy [2], Chap. 10). The warranter agrees to repair or replace, at no cost to theconsumer, any failed item up to the expiration of the warranty period of length T. This FRW isalso known as fixed free-replacement warranty or nonrenewing free replacement warranty.

LEMMA 2: The expected warranty cost CW(t0, T, �) associated with a product sold at aget0 under FRW of duration T and maintained by age reducing repairs of factor � satisfies theintegral equation

CW�t0, T, �� t0

t0�T

��u�Cr�u, �� du, (13)

where Cr(u, �) is the cost per an age-reducing repair of factor � for a product at calendar ageu and �(t) is the original failure rate function of this product.

PROOF: If there is no failure within the interval [t0, t0 � T], the warranty cost equals to 0.It equals to the cost of one repair, namely, Cr(u, �), if a failure occurs at some calendar instantu � [t0, t0 � T). A failure occurs at an instant u after t0 with probability ��(u) du ��(�u) du. Using the total expectation rule, we get (13). �

NOTE: For a new product t0 � 0. Equation (13) represents the expected warranty cost foran item with an initial age t0 covered by nonrenewing FRW of length T. In practice warrantycoverage is assigned for some future calendar time, and consumers attention is focused mainlyon the calendar age of the products. Consumers do not consider the virtual age of the product,even though it could be an important parameter of the performance of the product. From thewarranter point of view, the virtual age of the product carries an important information, and itis useful in evaluating the warranty expenses.

3.3. A Mixture of Minimal and Age-Reducing Repairs

Most technical items have some prescribed maintenance schedule such that at certain (usuallyequidistanced) moments of time preventive checkups are undertaken: for example, the recom-


mended oil change for motor vehicles after every 3000 miles; the seasonal (every 6 months)preventive checkups for house heating/cooling systems; and so on. At any of these preventivecheck ups of the product some adjustments or rectifications are made. This usually contributesto some essential age-reduction like effects. We will assume that these checkups correspond toan age-reducing repair of factor �. Meanwhile, between the scheduled checkups failures of theitem may occur. We assume that the latter are fixed by minimal repairs. Repairs and checkupsare instantaneous.

Thus, we consider the following warranty-maintenance model. The moment of purchase t0 isa free of charge maintenance checkup. At the expiration of any b units of calendar time (we callb 0 the intermaintenance time), a preventive maintenance must be performed (these are theplanned checkups). The age-dependent repair/maintenance check-up of factor � made at calen-dar age u will cost Cr(u, �). All other failures are fixed by minimal repairs (each costs Cm), andwill not change the virtual failure rate of the product. An FRW is assumed over the time [t0,t0 � T]. We call this model mixture of minimal and age reducing repairs.

LEMMA 3: The expected warranty cost associated with a product sold at age t0 under FRWof duration T and maintained under a mixture of minimal and age-reducing repairs of factor �is given by the expression

CW�t0, T, �� k�1

�T/b�

Cr�t0 � kb, �� Cm� �k�1

�T/b�

��t0 � �k � 1�b�� b� � ��t0 � kb��

� �� t0 � T

bb�� T � T

bb� � �� t0 � T

bb�� ,

where [T/b] is the smallest integer less than or equal to the number T/b.

PROOF: For a fixed value of the intermaintenance time (b 0), there will be exactly [T/b]checkups during the warranty period of duration T. The kth repair/maintenance will contributethe amount Cr(t0 � kb, �) to the expected warranty cost. This is the first sum of (14). Further,between the kth and (k � 1)st preventive checkups the virtual failure rate of the product is �(t),t � [(t0 � kb)�, (t0 � kb)� � b). Thus, the expected number of failures within thecorresponding interval is

��t0 � kb�� b� � ��t0 � kb�� t0�kb��

�t0�kb��b

��t� dt,

and each will cost Cm. This gives the second sum in (14). Similarly, the last (incomplete) periodof time, [t0 � [T/b]b, t0 � T) will contribute

�� t0 � T

bb�� T � T

bb� � �� t0 � T

bb��number of failures. �


This model is appropriate in many practical situations for calculating costs associated withservice and maintenance policies of repairable products. Information about services prior to t0

may be helpful in refining the model.

3.4. Warranty Costs under Renewing Warranty

Let us consider a cost model with renewing FRW. The warranter agrees to repair or replaceany faulty item up to time T from the time of purchase, at no cost to the consumer. At the timeof each warranty repair the item is warranted anew, i.e., the warranty period T is renewed. Thisrenewing FRW is also known as unlimited free replacement warranty (see Blischke and Murthy[2], Chap. 10). We will assume that the items have been maintained under age-reducing repairsof factor � before the time of the sale, as in Section 3.2.

LEMMA 4: The expected warranty cost CW(t0, T) associated with a product sold at age t0

under unlimited FRW of duration T and maintained under age-reducing repairs of factor �satisfies the integral equation

CW�t0, T� � �t0

t0�T

��u�e��u��t0�� C�u, �� CW�u, T�� du, (14)

with the boundary conditions CW(t0, 0) � 0. Here C(u, �) is the cost per an age-reducing repairof factor � at age u, and ��(t), and ��(t) are defined in Theorem 2.

PROOF: If there is no failure within the interval [t0, t0 � T], the warranty cost equals to 0.It equals to the cost of one repair, namely, Cr(u, �), if the first failure occurs at an instant u �[t0, t0 � T), plus the remaining expected warranty cost, CW(u, T, �), due to the renewedwarranty at age u. This first failure occurs at an instant u after t0 with probability��(u)e�[��(u)��(t0)] du. Using Theorem 2, and the total expectation rule, we obtain(14). �

NOTE: For a new item t0 � 0. Equation (14) represents the expected warranty cost for anitem with an initial age t0, covered by renewing warranty.

4. WEIBULL LIFETIME DISTRIBUTION WITH AN AGE-REDUCINGREPAIR FACTOR

As an illustration of the models in Section 3, we consider products with the Weibull life-timedistribution, i.e.,

F�t� � P T1 � t� � 1 � e��t/��

, t � 0.

Then

f�t� ��

� � t

��1

e��t/��

, t � 0,


and

��t� ��

� � t

��1

, t � 0.

The hazard rate for this lifetime is

��t� � � t

��

, t � 0.

Here � is the shape parameter and � is the scale parameter of the Weibull distribution. For � 1 the product will experience an increasing in time failure rate. For � � 1 the product hasdecreasing in time failure rate. If � � 1, the product has a constant failure rate, i.e., the life-timehas an exponential distribution.

COROLLARY 1: If a product has Weibull lifetime distribution with parameters � and � andit is maintained under age-reducing repairs of factor � 0, then its virtual failure rate and virtualhazard rate are given by the equations

��t� ��

� ��t

��1

, t � 0, 0 � � 1, (15)

and

��t� �1

� ��t

��

, t � 0, 0 � � 1. (16)

PROOF: Follows for Theorem 2. �

Figure 2a illustrates the original failure rate and the virtual failure rates for the parameters � �1.5, and � � 2, and under age-reducing factors � � .2, � � .5, and � � .85. Figure 2b illustratesthe original hazard rate and the virtual hazard rates for the parameters � � 1.5 and � � 2, andunder age-reducing factors � � .2, � � .5, and � � .85. We see that higher depth of repair

Figure 2. (a, left) Original and virtual failure rates under different age-reducing repair of factors. (b,right) Original and virtual failure rates under different age-reducing repair of factors.


(which is equivalent to smaller values of �) leads to a lower virtual failure rate and lower virtualhazard rate.

Next, assuming Weibull distribution for the lifetime of the product, we derive the warrantycosts for the models in Section 3. Assume that the purchase price C0(t0) for a product, sold atage t0, is given by the expression

C0�t0� � C0

M � t0

M. (17)

This price can be justified in the following way: There is a limiting age M after which theproduct cannot be sold, i.e., if t0 � M, no sales are possible. If the price of a new item is C0,then a “fair sale price” at age t0 should be proportional to the residual time of the sale limit.

4.1. Rebate Warranty of Nonrepairable Product

COROLLARY 2: Under the conditions of Corollary 1 and Lemma 1, the expected warrantycost is given by the expression

CW�t0, T� � C0

M � t0

M �0

T �

� � t0 � t

� ��1

e��t0�t/��t0/��T � t

Tdt. (18)

PROOF: Substitute (17), (15), and (16) in (12), and obtain (18). �

Figure 3a illustrates the dependence of CW(t0, T) on the age at the time of purchase t0 � [0,3], for the Weibull distribution with parameters � � 1.5 and � � 2 and three different valuesof the warranty period T � .5, T � .75, and T � 1. The limiting selling age of the productis M � 4. The price of a new product is assumed to be C0 � 100. The warranty cost increasesand after reaching certain threshold value of t0 decreases. This dependence is justified by theform of the sale price C0(t0) and its interaction with the increasing failure rate. There is an initialage where the warranty costs have a maximum value, and the warranter should be aware of it.For the same values of the sales/repairs related parameters, Figure 3b illustrates the dependenceof CW(t0, T) on the length of the warranty period T � [0, 3] for the same Weibull distributionand three initial ages t0 � 0, t0 � 1.2, and t0 � 2.5. We see that the warranty costs increase

Figure 3. (a, left) Dependence of CW(t0, T) on the age t0 at the time of purchase. (b, right) Dependenceof CW(t0, T) on the length T of the warranty period.


in both, the selling age t0, and the duration T of the warranty period. A comparison with thenumerical results of Patankar and Mitra (Table 11.4 in Blischke and Murthy [2]), for the pro-ratawarranty models with a Weibull distribution of same parameters, may be found useful andcurious.

4.2. Fixed Warranty

Now, let us consider repairable products. Assume that an age-reducing repair at age u by afactor � costs

Cr�u, �� C0�1 � ��M � u

M� Cm, (19)

where C0 is the purchase cost of a new item, M is the age of the sale limit, and Cm is the averagecost of a minimal repair.

COROLLARY 3: Under the conditions of Corollary 1 and the conditions of Lemma 2, theexpected warranty costs are given by the expression

CW�t0, T, �� C0�1 � ��

� �t0

t0�T M � u

M ��u

� ��1

e��u/��1/�� du

�Cm

� ��t0 � T�

� ��

� ��t0

� �� .

PROOF: Substitute (19), (15), and (16) in (13), solve an integral, and obtain (20). �

Figure 4a illustrates the dependence of CW(t0, T, �) on the age at the time of purchase t0

(t0 � [0, 3]) for the Weibull distribution with parameters � � 1.5 and � � 2. Three differentlengths of the warranty period T � .5, .75, and 1.0 are considered. The limit of the selling ageis equal to M � 4. The initial price of a new product is assumed to be C0 � 100, and a minimalrepair costs Cm � 15. The age-reducing factor is � � .95, i.e., 5% age-reduction after anage-reducing repair. We see that double the warranty period T of a new product will lead to quite

Figure 4. (a, left) Dependence of CW(t0, T, �) on the selling age t0. (b, right) Dependence of CW(t0,T, �) on the length T of the warranty period.


significant increase of the associated warranty costs. The difference between the warranty costsfor the given warranty durations is an increasing function of the initial age of the product.

Figure 4b shows the dependence of CW(t0, T, �) on the length of the warranty period T �[0, 3] for t0 � 0, 1.2, and 2.5. We observe that, even for relatively short warranty periods, theinitial age of the product has significant impact on the expected warranty costs.

Similarly, based on the expression (20), the dependence of CW(t0, T, �) on some otherparameters of the model (e.g., the selling age limit, the shape parameter �, or on the ratiobetween warranty length T and the expected lifetime E(X) � ��(1 � 1/�), where �� is theGamma function, etc.) can be studied.

4.3. The Mixture of Minimal and Age-Reducing Repairs

Let us assume that the cost of age-reducing repair at age u by a factor � is as in (19) and Mis the age of the sale limit. Then:

COROLLARY 4: Under the conditions of Corollary 1 and Lemma 3, the expected warrantycosts are given by the expression

CW�t0, T, �� C0�1 � �� k�1

�T/b� M � kb � t0

M� T

bCm

� �k�1

�T/b� � �t0 � �k � 1�b�� b

� ��

� � �t0 � kb��

� ��Cm

� � �t0 � �T/b�b�� T � �T/b�b

� ��

� � �t0 � �T/b�b��

� ��Cm.


Graphical illustration of the dependence of the expected warranty costs in (20) under mixtureof minimal and age-reducing repairs for the Weibull distribution with parameters � � 1.5 and� � 2 is shown in Figure 5. The age reduction factor is � � .95, the cost for a minimal repairis Cm � 15, and the selling age limit is M � 4. The initial price of a new product is assumed

Figure 5. (a, left) Dependence of CW(t0, T, �) on the warranty time T. (b, right) Dependence of CW(t0,T, �) on the selling age t0.


to be C0 � 100. The scheduled checkups are made in any b � .2 time units. Dependence onthe warranty period T and the initial selling age t0 are shown. Figure 5a shows how the expectedwarranty cost vary with the length of T. The stepwise-like shape is due to the additionalcomponent to the cost at the time of the checkups. Figure 5b illustrates the dependence of thecost on the initial age t0, for three different warranty durations: T � .5, .75, and 1.0. Thedecrease in the expected warranty costs is due to the impact of the selling price as a function oft0. As expected, the expected warranty costs increase as T increases.

4.4. Renewing Warranty

COROLLARY 5: Under the conditions of Corollary 1 and Lemma 4, the expected warrantycosts for a product sold under unlimited FRW are given as a solution to the integral equation

CW�t0, T, �� t0

t0�T �

� ��u

� ��1

e��1/��u/��t0/��C0�1 � ��M � u

M� Cm

� CW�u, T, �� du. (20)


Figure 6 illustrates the dependence of the expected warranty cost for a new item CW(0, T, �)on the warranty period T, under age reducing repairs of factor �. Six values of � (.05, .15, .327,.5, .75, and 1.0) were selected to illustrate the dependence of CW(0, T, �) on �. The product’slifetime is assumed to be Weibull with parameters � � 1.5 and � � 2. The age sale limit is M �4. The initial price of a new product is assumed to be C0 � 100 and the cost of a minimal repairis Cm � 15. For fixed value of �, the expected warranty cost is an increasing function of T. Onthe other hand, it depends on the value of �.

Using numerical optimization, for T � 3 (and zooming in Fig. 6), the expected warranty costsas a function of � have a maximum at �max � .327. The existence of �max can be explainedby the interruption of a renewing warranty coverage due to the improvement of the product aftereach age-reducing repair. The improvement of the product is a function of the degree of the

Figure 6. Dependence of CW(0, T, �) on the warranty duration T and on the value of age-reducing factor �.


repair �. Smaller values of � lead to shorter warranty coverage. On the other hand, the cost perage-reducing repair at an instant u, C(u, �), is a decreasing function of �.

The illustrations on Figure 7 represent the expected warranty cost related to a new product forvarious values of the age reducing factor, i.e., when the initial age is t0 � 0. We observe heresome strange behavior of the costs as function of the age reducing factor �. Equation (20) allowssimilar studies of the dependence of the expected warranty costs also as function of the initialage t0.

5. CONCLUSIONS

An age-reducing repair model is proposed, and the warranty cost for some typical warrantypolicies is analyzed. It is shown that the failure rate function and hazard function provide moreconvenient approach to age-dependent cost modeling than the approach based on the directusage of the probability distributions of the product lifetime. Numerical examples illustrate theideas of the proposed models for the Weibull distributed lifetime with an age-reducing factor.

APPENDIX

The proof of the theorem follows the ideas in Block, Borges, and Savits [3].

THEOREM 1 (Age Transfer Theorem): {Ntv, t � 0} is a nonhomogeneous Poisson process with a leading function

�v�t� � E�Ntv� � �log�1 � F� t

�� t

��,

where �(t) � �log(1 � F(t)) is the leading function of the NPP associated with the maintenance with minimal repairs.

PROOF: It suffices to show that (Cinlar [8], p. 96)

�v�T1�, �v�T2� � �v�T1�, . . . , �v�Tn�1� � �v�Tn�

is a sequence of i.i.d.’s, exponentially distributed r.v. with parameter 1.For any choice of t1 0, . . . , tn�1 0 define

g�x1, . . . , xn� � P �v�T1� � t1, �v�T2� � �v�T1� � t2, . . . , �v�Tn�1� � �v�Tn� � tn�1�T1 � x1, . . . , Tn � xn�.

Denoting the inverse function of F� (t) by G(t) and with reference to (5), we obtain

Figure 7. Dependence of CW(0, T, �) on the warranty duration T and on the value of age-reducing factor �.


P �v�T1� � t1� � P��log F��T1

� �� t1�� P�F��T1

� � e�t1�� P�T1

�� G�e�t1�� P T1 � �G�e�t1�� P�X1

�

� G�1�e�t1�� F��G�e�t1�

� �� e�t1.

In a similar way, by using equivalent sequence of transformations we obtain

g�x1, . . . , xn� � P�T1 � �G��e�t1�, T2 � �G�� e�t2F� � x1

�� , . . . , Tn�1 � �G�� e�tn�1F� � xn

�� ,

where G�(t) denotes the left-continuous inverse of F� (t/�). From (4) we get

g�x1, . . . , xn� � �0, if xi � G��e�tiF��xi�1

� ��for some i � 2, . . . , nor if x1 � G��e

�t1�,

F��max�xn

�, G��e�tn�1F��xn

��F��xn

�� e�tn�1, if xi � G��e�tiF��xi�1

� ��for all i � 2, . . . , nand x1 � G��e

�t1�.

In the last equation we have used the following sequence of equivalent representations

P �v�Ti�1� � �v�Ti� � ti�1�T1 � x1, . . . , Tn � xn� � � 0, if �v�xi�1� � �v�xi� � ti�1,1, if �v�xi�1� � �v�xi� � ti�1.

And also

P �v�Tn�1� � �v�Tn� � tn�1�T1 � x1, . . . , Tn � xn� � P��logF� �Tn�1/��

F� �xn/�� tn�1�Tn � xn�� P�F��Tn�1

� � e�tn�1F��xn

��Tn � xn�� P�Tn�1 � �G��e�tn�1F��xn

��Tn � xn��F� �max�xn/�, G��e

�tn�1F� �xn/��

F� �xn/�� e�tn�1.

For the last equality we note that

max�xn

�, G��e�tn�1F��xn

�� G��e�tn�1F��xn

��,

since it is true that

G�� e�tn�1F� � xn

�� G��F� � xn

�� xn

�.

Finally, from (6) it follows that

P �v�T1� � t1, �v�T2� � �v�T1� � t2, . . . , �v�Tn�1� � �v�Tn� � tn�1� � E�g�T1, . . . , Tn��

� e�tn�1 �0

�

�F� � x1

��1

I G��e�t1�� x1

�

�F� � x2

�� 1I G��e � t2F� �x1/�� · · · �

xn�2

�

�F� � xn�1

� �� 1

I G��e � tn�1F� �xn � 2/��

xn�1

�

I G��e � tnF� �xn � 1/�� dF� xn

� � · · ·dF� x1

� � � e � �t1 � · · · � tn � 1�.

This completes the proof of the theorem. �


ACKNOWLEDGMENTS

We are very thankful to the referees of Naval Research Logistics, whose comments contrib-uted to the improvement of the paper. We wish to thank the Department of Science andMathematics of Kettering University, the FAPESP (Sao Paulo, Brazil) for Grant 99/08263-1,and the NSERC Canada for Grant #A9095 partially funding this research, and for their moraland financial support.

REFERENCES

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Warranty costs: An age-dependent failure/repair model

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