RELIABILITY PAPER One-dimensional warranty cost analysis for second-hand items: an overview Stefanka Chukova School of Mathematics, Statistics and Operations Research, Victoria University of Wellington, Wellington, New Zealand, and Mahmood Shafiee Department of Industrial Engineering, Faculty of Engineering, Tarbiat Modares University, Tehran, Iran Abstract Purpose – The purpose of this paper is to review and develop stochastic models for the evaluation of the expected warranty cost, from dealer’s viewpoint, for second hand items sold under different warranty policies. Design/methodology/approach – This study makes a useful contribution to the warranty literature by developing a framework to study one dimensional warranty policies for second-hand products from dealer’s viewpoint. Also, numerical examples are provided to illustrate the application of these models. Findings – Despite the fact that warranties for second-hand products are commonly used, the accurate pricing of warranties in many situations remains an unsolved problem, for both the dealer and customer. The proposed framework allows the dealer to analyze the cost of alternative warranty policies before deciding on the policy to be offered with the sale of a second-hand item. Originality/value – The policies and models developed in this paper can be useful in managerial decisions making related to second-hand products such as automobiles, home appliances, helicopters, electronic equipment and electronics. Keywords Warranties, Cost analysis, Expected warranty cost, Free repair/replacement, Second-hand item, Warranty policy Paper type Research paper Nomenclature a ¼ past age of the second-hand item w ¼ warranty period l, b ¼ the parameters of product failure distribution F T (t) ¼ cumulative failure distribution of a new product f T (t) ¼ density function associated with F T (t) L T (t) ¼ intensity/failure rate function associated with F T (t) L I (t) ¼ intensity function for product failure due to components [ set I (included in warranty) L E (t) ¼ intensity function for product failure due to components [ set E (excluded in warranty) E [N (w,a)] ¼ the expected number of claims over the warranty E [c d (w,a)] ¼ dealer’s expected warranty cost E [c b (w,a)] ¼ buyer’s expected warranty cost c ¼ expected rectification cost The current issue and full text archive of this journal is available at www.emeraldinsight.com/0265-671X.htm Warranty cost analysis 239 Received 24 February 2011 Revised 24 February 2012 Accepted 27 February 2012 International Journal of Quality & Reliability Management Vol. 30 No. 3, 2013 pp. 239-255 q Emerald Group Publishing Limited 0265-671X DOI 10.1108/02656711311299827
Transcript
RELIABILITY PAPER
One-dimensional warranty costanalysis for second-hand items:
an overviewStefanka Chukova
School of Mathematics, Statistics and Operations Research,Victoria University of Wellington, Wellington, New Zealand, and
Mahmood ShafieeDepartment of Industrial Engineering, Faculty of Engineering,
Tarbiat Modares University, Tehran, Iran
Abstract
Purpose – The purpose of this paper is to review and develop stochastic models for the evaluation ofthe expected warranty cost, from dealer’s viewpoint, for second hand items sold under differentwarranty policies.
Design/methodology/approach – This study makes a useful contribution to the warrantyliterature by developing a framework to study one dimensional warranty policies for second-handproducts from dealer’s viewpoint. Also, numerical examples are provided to illustrate the applicationof these models.
Findings – Despite the fact that warranties for second-hand products are commonly used, theaccurate pricing of warranties in many situations remains an unsolved problem, for both the dealerand customer. The proposed framework allows the dealer to analyze the cost of alternative warrantypolicies before deciding on the policy to be offered with the sale of a second-hand item.
Originality/value – The policies and models developed in this paper can be useful in managerialdecisions making related to second-hand products such as automobiles, home appliances, helicopters,electronic equipment and electronics.
The current issue and full text archive of this journal is available at
www.emeraldinsight.com/0265-671X.htm
Warranty costanalysis
239
Received 24 February 2011Revised 24 February 2012
Accepted 27 February 2012
International Journal of Quality &Reliability Management
Vol. 30 No. 3, 2013pp. 239-255
q Emerald Group Publishing Limited0265-671X
DOI 10.1108/02656711311299827
cr(a) ¼ the replacement cost of faileditem with a used one of age a
G(c) ¼ cumulative distributionfunction of rectification cost
g(c) ¼ density function associatedwith G(c)
R(t) ¼ refund function
1. IntroductionRecently, due to the rapid technological development and fierce competition in themarketplace, product reliability and product warranty are becoming increasinglyimportant product attributes. Product warranty is a contract attached to the product atthe time of the sale. It requires manufacturers/dealers to offer a pre-specifiedcompensation to the buyers in the case of faulty product for a specified time period,providing that the product usage is according to its specification. Since the beginningof the twenty-first century, for many items the length of the warranty coverage hasbeen steadily increasing. For example, in the early 1930s the warranty period offeredby the car manufacturers was three months, becoming one year in the 1960s andcurrently it varies between three to five years (Kim et al., 2004). A longer warranty termusually attracts more customers, but at the same time it involves additional servicingcosts. For example, American manufacturers spend over $25 billion – about 2 percentof the revenue – annually on warranty services (Manna et al., 2007). According to the2002 General Motors annual report, the company had total revenues of US$186.7 billionand the future warranty cost on sold cars was estimated to be US$4.3 billion – about2.3 percent of the revenue (General Motors Corporation, 2002, annual report, p. 75).
According to Murthy and Blischke (2000), the two main roles of warranty are:
(1) protection – for the consumers against defective items and for the producersagainst consumers’ excessive claims; and
(2) promotion – for product differentiation by producers and by consumers.
Warranty policy is a statement on the type of compensation provided to the customers,such as a free replacement, lump-sum payment, or a pro-rata reimbursement, and onthe length of the warranty coverage. Taxonomy of different warranty policies isavailable in Blischke and Murthy (1994) and Christozov et al. (2009a). These policiescan be broadly divided into two groups: 1D and 2D warranty policies. 1D warrantypolicies are characterized by a warranty period, which is usually a time interval on theage of the item with no limitation imposed on the usage of the item. In contrast, the 2Dwarranty policies are characterized by a region on the 2D plane, where the axesrepresent the age and the usage of the item. For example, a typical warranty for anautomobile provides a free of charge service or repairs for a maximum of five years or50,000 miles, whichever occurs first.
Recently, the importance of the used/second-hand product market as a part of the totalmarket (new and used) has been steadily increasing. For instance, since 1999 in France, theused-to-new car ratio has increased from 2.4 to 2.6, whereas in the UK, the total demandfor used cars grew by 25 percent in volume and 68 percent in value from 1996 to 2006(CIRP II, 2008, pp. 3-5). In spite of the increasing importance and share of the second-handmarket, often customers of second-hand products encounter the following two problems:
(1) Uncertainty regarding the durability and performance specifications of theseproducts due to the lack of information on their past usage andmaintenance history. In order to reduce the negative impact of this issue on
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the number of sales, the dealer often offers warranty. Through this warranty,the dealer and the buyer share the risk of product malfunctioning.
(2) Despite the fact that warranties for second-hand products are commonly used,the accurate pricing of warranties in many situations remains an unsolvedproblem for both the dealer and customer. This may seem surprising sincethe fulfilment of warranty claims may cost dealers a significant amount ofmoney. Underestimating the true warranty costs can result in losses for thedealer. On the other hand, overestimating them will lead to uncompetitiveproduct prices, and as a result the amount of product sales will decrease.From the customer’s perspective, the warranty is an investment aiming toreduce the risk of post-warranty early failures and the uncertainty of relatedservicing costs.
A significant amount of academic research has been conducted in modellingwarranty policies and costs for new products. In contrast, a brief review of theliterature shows that only a few researchers have worked in the area of warranties forsecond-hand products. Table I provides a chronological literature review over the lastyears.
This study makes a useful contribution to the warranty literature by developing aframework to study 1D warranty policies for second-hand products from dealer’sviewpoint. This framework allows the dealer to analyze the cost of alternative warrantypolicies before deciding on the policy to be offered with the sale of its second-hand item.This paper reviews and develops stochastic models for estimating the expectedwarranty costs for free repair/replacement warranty (FRW), pro-rata warranty (PRW), acombination of free repair/replacement and pro-rata (FRW/PRW) and a combination offree repair/replacement and lump-sum (FRW/LSW) policies.
The rest of this paper is organized as follows. In Section 2, an overview of long-termwarranty policies is presented. The model assumptions and notations are given inSection 3. In Section 4, we discuss stochastic models for 1D warranty cost analysisand illustrate those using numerical examples. In Section 5, we present an example tocompare some of the policies discussed in Section 4. In Section 6, we conclude with abrief summary of the contributions made and offer some directions for future research.
Author/s (year) Abstract
Chattopadhyay and Murthy(2000)
Develops probabilistic models to compute the expected warranty costto the dealer and considers two policies: free-replacement warranty andPRW
Chattopadhyay and Murthy(2001)
Deals with three new cost sharing warranty policies for second-handitems (SPE, LIC and limit on individual and total cost)
Chattopadhyay and Yun(2006)
Develops models for estimation of warranty cost and best decision onprice for 2D warranty policy associated with sale of second-handproducts
Shafiee et al. (2011) Develops statistical models for estimating the dealer’s expectedwarranty cost for second-hand products sold with 2D free-repair/replacement warranty
Sarada and Mubashirunnissa(2011)
Develops two stochastic models to compute the expected warranty costfor a second-hand product using bivariate approach
Table I.A brief review of the
literature on warrantiesfor second-hand items
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241
2. Long-term warranty policiesLong-term warranties offered by manufacturers/dealers include extended warranty,warranty for second-hand products, service contract and lifetime warranty policies(Chattopadhyay and Rahman, 2008) and a diagram of their classification is shown inFigure 1.
2.1 Extended warrantyRecently, it is popular with the customers to purchase an additional insurance againstproduct failure in the form of an extended warranty. On average 27 percent of new carbuyers purchase an extended warranty (Lutz and Padmanabhan, 1998). Typically,these extended warranties prolong the warranty coverage of the product. Extendedwarranties are sold not only by manufacturers, but also by third parties. According tothe Consumer Electronics Association “only 20% of those who buy electronic deviceswith extended warranties ever use the service” (Maronick, 2007).
2.2 Warranty for second-hand productThe second-hand warranty contract is intended to assure the buyer that thesecond-hand item will perform its intended functions for a pre-specified period of timeor amount of usage. Nowadays warranties for second-hand items are widely used andserve many purposes including providing protection for both the dealer and the newbuyer. Also, they inform customers on the quality and reliability of the second-handitem and have a positive impact on the dealer’s reputation. The significant factors,which affect the warranty cost for a second-hand item, are:
. The age of the item: typically, the occurrence of a failure of an older item is morelikely compare to occurrence of a failure of a newer item. Second-hand items,which have past their “use-by date” time, usually require significant amount ofefforts and resources to be brought back to an acceptable working condition.
. The past usage of the item: the usage can be measured in different ways. Forexample, the usage can be measured as the number of copies produced by aphotocopier, the distance travelled by a vehicle, the number of takeoffs andlandings or the total hours flown for an aircraft and, etc.
. The warranty policy: by using an effective warranty servicing strategy thedealer can reduce the warranty costs.
Figure 1.Classification of thelong-term warranty
Long termwarrantypolicies
2.1 Extendedwarranty
2.3 Lifetimewarranty
2.4 Servicecontract
2.2 Warrantyfor second-
hand product
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. The maintenance strategy: by using an effective maintenance strategy the dealercan significantly reduce the warranty costs and prolong the lifetime of thesecond-hand item.
2.3 Lifetime warrantyLifetime warranties are defined as the manufacturer/dealer’s commitment to providefree (or cost sharing) repair/replacement of the faulty product due to its design,manufacturing defects or quality problems during the entire useful life of the product(or the buyer’s ownership period). There are various definitions of the concept ofproduct “lifetime” and these are reviewed in Chattopadhyay and Rahman (2008).
2.4 Service contractThe main difference between warranty and a service contract is that the latter isentered into voluntarily. Typically, the service contract is purchased separately and thecustomer may even have a choice on the terms of the contract. The concept of servicecontract is similar to the concept of extended warranty. On the other hand, warranty isan integral part of the sale. Murthy and Asgharizadeh (1998) model and analyze servicecontracts using a game-theoretic approach.
3. Model assumptions and notationThe proposed models are based on the following assumptions.
3.1 Assumptions. The reliability of a used item depends on its age and/or usage. The dealer has
information on the past age and/or usage of the item. This information is usuallyobtained from sources such as registration forms and log books.
. The customers’ utility is measured in terms of the length of warranty period, sothat every customer aims to maximize the length of the warranty period.
. The dealer’s disutility is measured in terms of the uncertain warranty repaircosts and aims to minimize these costs.
. Every failure results in a warranty claim and all warranty claims are valid.
. Failures are statistically independent.
. The cost of each repair is a random variable. This is because a product failure isusually due to a failure of one or more of its components and the cost of repairdepends on the type of the failed components.
. The mean time to rectify a used item is small in relation to the mean timebetween failures and it can be ignored, i.e. the repairs are instantaneous.
. Unless stated otherwise, all rectification actions have a negligible impact on thefailure rate of the item, i.e. the repairs are assumed to be minimal.
3.2 NotationThe notations given in nomenclature are used in this paper.
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4. Stochastic cost modelsNext, we focus on cost analysis of various warranty policies as shown in Figure 2. For eachof these policies, we propose a stochastic model for the failure process of a second-handproduct and derive corresponding expected warranty costs from dealer’s point of view.
We consider the following setup: first, the dealer purchases the second-hand item ofage a from an end-user. Then, he determines the expected warranty costs undervarious warranty policies and incorporates them in the sale price of the second-handitem. Lastly, the new buyer selects a second-hand item with an appropriate for him/herprice and warranty policy. So, next we review a number of stochastic models aiming toevaluate the expected warranty costs under various warranty policies.
4.1 Free repair/replacement warrantyAssume that, the warranty period w is fixed, and starts immediately after the sale tothe new user. Under a FRW with period w, the dealer agrees to rectify, free of charge tothe new buyer, any failures of the item within the warranty period (Figure 3). After theexpiration of the warranty period, all expenses are borne by the buyer.
Usually, failure-free policies fall into two categories:
(1) Non-renewing free-repair warranty (NRFRW). Under a NRFRW, the dealerrepairs any faulty item during the warranty period w (Chukova and Johnston, 2006).After the expiration of the warranty period, all expenses are borne by the buyer.For example, second-hand home appliances, such as vacuum cleaners, refrigerators,washing machines, dryers and TV sets, are covered by a non-renewing warranty.
(2) Renewing free-replacement warranty (RFRW). This policy is one of the mostcommon types of warranties applied to non-repairable items (Blischke and
Figure 2.Taxonomy for the1D-warranty forsecond-hand products
1D-Warrantyfor second-
hand product
4.1: FreeRepair-
ReplacementWarranty
4.2:Cost-Sharing
Warranty
4.3: RebateWarranty
4.2.1:Specific Parts
Exclusion
4.2.2: Limiton Individual
Cost
4.3.1:Lump-Sum
4.3.2:Pro-ratawarranty
4.4: HybridWarranty
4.4.1:FRW/PRW
4.4.2:FRW/LSW
Figure 3.FRW policy forsecond-hand item
Ownership shiftsfrom the dealer to
the new user
× × ×0 a+wa
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Murthy, 1992). Under the RFRW policy, a second-hand item, which fails withinthe warranty period, is replaced by a new/younger one, and the warranty isrenewed at no charge to the buyer. Usually, this policy is assigned toinexpensive second-hand items.
Next, we propose stochastic cost models for non-renewing as well as for RFRW.4.1.1 NRFRW. Let N(w,a) be the number of claims over the warranty period w for a
product of age a at sale. We assume that the cumulative failure distribution of the productis modeled by FT (t) with density function fT (t) ¼ dFT (t)/dt. Our assumption (8), that therepairs are minimal, means that the claims occur according to a non-homogeneousPoisson process with intensity function LT (t) ¼ fT (t)/[1 2 FT (t)]. Then, the expectednumber of claims over the warranty period is given by Chattopadhyay and Murthy (2000):
E½N ðw; aÞ� ¼
Z aþw
a
LTðtÞdt: ð1Þ
The cost of each repair (C ) is, in general, a random variable. This is because an item failureis due to the failure of one or more of its components and the cost of repair or replacementof these components vary. We denote by G(c) and g(c) the cumulative distribution functionand corresponding density of the rectification costs of the product over the warrantyperiod, i.e.:
GðcÞ ¼ PðC # cÞ: ð2Þ
Then, the expected cost of each rectification action, c̄, is given by:
�c ¼
ZcgðcÞdc: ð3Þ
Therefore, the dealer’s expected warranty cost, E[cd(w, a)], for a product of age a at sale isgiven by:
E½cdðw; aÞ� ¼ �c
Z aþw
a
LTðtÞdt: ð4Þ
Table II summarizes the form of the intensity function LT (t), the density failuredistribution fT (t), and the expected warranty costs E [cd(w,a)] for a constant, linearlyincreasing, exponentially increasing and Weibull-type intensity functions.
Example 1. Let FT (t) be given by FT (t) ¼ 1 2 exp[2 (lt)b] andf T ðtÞ ¼ ðdFðtÞ=dtÞ ¼ lbðltÞb21exp½2ðltÞb�. Therefore, LT (t) is given byLT (t) ¼ lb(lt)b21 with b . 1 and l . 0. We assume the following values for themodel parameters: l ¼ 0.443/year, b ¼ 2, as in Chattopadhyay and Murthy (2000).
LT(t) fT(t) E [cd(w,a)]
Constant l l £ e2lt clwLinearly increasing lt lt £ exp 2 lt 2
2
n o�clw w
2 þ a� �
Exponential le 2lt l exp{2lt þ e 2lt 2 1} �ce2lað1 2 e2lwÞWeibull lb £ t b21 lb £ t b21 £ e 2lt b
�cl½ða þ wÞb 2 ab�
Table II.The expected warranty
costs for constant,linearly increasing,
exponentially increasingand Weibull intensity
functions
Warranty costanalysis
245
This implies that the mean time to first failure is two years. Also assumeG(c) ¼ 1 2 exp(20.01c), therefore �c ¼ $100. Table III shows the expected warrantycosts to the dealer for a varying from one to seven years and w varying from 0.5 to twoyears.
Since we assume that L(t) is an increasing function of t, we observe that E[cd(w,a)]increases with a. The expected cost increases by a factor of 6 as a changes from one toseven years for w ¼ 0.5. This drops to 4 for w ¼ 2.0. In addition, under NRFRW policyand our assumption (8), we notice that the failure rate of the second-hand product at thetime of the expiration of the warranty coverage is LT(a þ w).
4.1.2 RFRW. Under the renewing warranty, if the used item fails before the expirationof the warranty period w, it is replaced by a new/younger one and the warranty period oflength w starts again. This process terminates as soon as the lifetime of the repaired itemexceeds w. We assume that the failure distribution of a used item of age a is given by:
F1ðtÞ ¼FTða þ tÞ2 FTðaÞ
1 2 FT ðaÞ; ð5Þ
with corresponding density function f1(w), which agrees with our assumption (8). Next,we assume that any warranty repair improves the reliability of the used item to the levelit was at the time of the sale. In other words, the warranty repairs form a renewal processwith an underlying distribution F1(t). Therefore, for RFRW, the number of warrantyrepairs within the warranty, N(w,a) is a random variable with a geometric distributionwith parameter [1 2 F1(w)]. The probability distribution of N(w,a) is:
P½N ðw; aÞ ¼ n� ¼1 2 F1ðwÞ n ¼ 0
½F1ðwÞ�n½1 2 F1ðwÞ� n $ 1
(ð6Þ
Let E [N(w,a)] be the expected number of repairs over the warranty coverage with periodw. Then:
E½N ðw; aÞ� ¼F1ðwÞ
1 2 F1ðwÞ; ð7Þ
and the expected warranty costs to the dealer, E [cd(w,a)], is:
E ½cdðw; aÞ� ¼ crðaÞE½N ðw; aÞ�: ð8Þ
where cr(a) is the replacement cost of the failed item with a used one of age a.
E [cd (w,1)] E [cd (w,2)] E [cd (w,3)] E [cd (w,4)] E [cd (w,5)] E [cd (w,6)] E [cd (w,7)]
Table III.E [cd (w,a)] for differentcombinations of w and afor NRFRW policy
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4.2 Cost-sharing warrantyUnder cost sharing warranties, the buyer and the dealer share the repair costs. In thissection, two policies are considered – specific parts exclusion (SPE) and limit onindividual cost (LIC). These are currently offered by dealers of used cars, electrical andconsumer-electronics products. Most of the used car dealers offer SPE policy and coverfailures of the engine, brakes, gearbox, suspension, steering, electrical system, ignitionsystem and cooling system, whereas they do not cover failures due to wear and tear items,as with tires. For more details on these policies, see Chattopadhyay and Murthy (2001).
4.2.1 Specific parts exclusion. The components of the product are grouped into twodisjoint sets, say I (included in the warranty) and E (excluded from the warranty).Under this policy, the dealer rectifies failed components from I at no cost to the buyerover the warranty period. The rectification of failed components from set E can becarried out either by the dealer or a third party, but the repair costs are borne bythe buyer. Let �cI denote the expected rectification cost for components in I and �cE
denote the expected rectification cost for components in set E. Then, the expectedwarranty cost to the dealer, E[cd(w, a)], is given by:
E½cdðw; aÞ� ¼ �cI
Z aþw
a
LI ðtÞdt; ð9Þ
and the expected cost to the buyer over the warranty period, E[cb(w, a)], is given by:
E½cbðw; aÞ� ¼ �cE
Z aþw
a
LE ðtÞdt; ð10Þ
where:
�cI ¼
Zi[I
cigðciÞdci and �cE ¼
Zj[E
cjgðcjÞdcj:
Example 2. Assume that LE(t) ¼ lEbE(lEt)bE 2 1 and LI(t) ¼ lIbI(lIt)bI 2 1. Let the
parameter values for failures of components from E be lE ¼ 0.441 and bE ¼ 2.31 andlet the expected cost of each rectification be �cE ¼ $30, as in case 2 in Chattopadhyayand Murthy (2001). Assume that the corresponding values for set I are lI ¼ 0.446,bI ¼ 1.8 and �cI ¼ $70. The expected warranty cost to the dealer, E[cd(w, a)], is shown inTable IV.
Table IV shows that as a changes from one to seven years, the expected warrantycost to the dealer increases by a factor of 4 for w ¼ 0.5 and by a factor of 3 for w ¼ 2.0.In addition, under SPE warranty policy and our assumption (8), we notice that thefailure rate of the components from set I (set E) at the time of the expiration of thewarranty coverage is LI(a þ w) (LE(a þ w)).
4.2.2 Limit on individual cost. Under LIC warranty policy, if the rectificationcost is below a pre-specified limit cI, it is borne fully by the dealer, but if the cost exceedscI, then the buyer is responsible for the difference between the rectification cost and cI.
E [cd (w,1)] E [cd (w,2)] E [cd (w,3)] E [cd (w,4)] E [cd (w,5)] E [cd (w,6)] E [cd (w,7)]
Let �cd denote the expected rectification cost to the dealer and �cb denote the expectedrectification cost to the buyer. Then, the expected warranty cost to the dealer,E [cd (w,a)], is given by:
E½cdðw; aÞ� ¼ �cd
Z aþw
a
LTðtÞdt; ð11Þ
and the expected cost to the buyer over the warranty period, E[cb(w,a)], is given by:
E½cbðw; aÞ� ¼ �cb
Z aþw
a
LTðtÞdt; ð12Þ
where:
�cd ¼
Z cI
0
cgðcÞdc þ cI
Z 1
cI
gðcÞdc and �cb ¼
Z 1
cI
ðc 2 cI ÞgðcÞdc:
Example 3. Let cI ¼ $125 and g(c) ¼ 0.01 exp(20.01c). Therefore, we obtain �cd ¼$71:35 �cb ¼ $28:65. The values of the remaining model parameters are as given inExample 1. Then, the expected warranty cost to the dealer, E[cd(w,a)], is shown inTable V.
Table V shows that as a changes from one to seven years, the expected warrantycost to the dealer increases by a factor of 6 for w ¼ 0.5 and by a factor of 4 for w ¼ 2.0.Under LIC warranty policy and our assumption (8), the failure rate of the second-handproduct at the time of the expiration of the warranty coverage is the same as in the caseof NRFRW, equal to LT(a þ w).
4.3 Rebate warrantyA rebate warranty (RW) is one of the most common types of warranty assigned tosecond-hand items. Under a rebate policy, if the second-hand item fails during thewarranty period [0, w ], the dealer refunds the customer some proportion of the sale price.The refund amount, R(t) may be a linear or non-linear function of the failure time t. Thetwo typical forms of RW policies are lump-sum and pro-rata (Chien and Chen, 2008).
4.3.1 Lump-sum warranty. A lump-sum RW is usually assigned for a relativelysmall period of time immediately after the sale of the product (Chukova and Johnston,2006). Under LSW policy if the item fails during the warranty period, the customer isrefunded a percentage of the replacement cost. The option “money-back guarantee”(MBG) is a common type of LSW with a full refund. The earliest known, as early asin 1777, MBG has been offered by a food manufacturer. The research on MBG andother types of refunds is relatively new. It has attracted the interest of the researchersonly for the last 30 years, and it is still mostly theoretical. Usually, dealers offer a MBG
E [cd (w,1)] E [cd (w,2)] E [cd (w,3)] E [cd (w,4)] E [cd (w,5)] E [cd (w,6)] E [cd (w,7)]
Table V.E [cd (w,a)] for differentcombinations of w and afor LIC policy
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warranty with home appliances, desktop computers, electronics excluding software,etc. For example, the department store “Target” in the USA, offers a 90-day MBG, andbuyers can return unwanted used products either by mail or within the store (www.target.com). Also in the USA, the retail giant “Wal-Mart” offers MBG with variousgrace periods from 50 days to three months depending on products (www.walmart.com), whereas “Office Depot” offers a 30-day MBG for all products in the store, butsells used computers with only 14-day MBW and no MBW for used laptops (www.officedepot.com). In contrast, second-hand products such as car tires, CDs, laptopcomputers, software and services such as massages, haircuts are usually not coveredby a MBG warranty option.
Under LSW policy, the warranty refund function can be expressed as:
RðtÞ ¼kcrðaÞ 0 # t # w
0 t . w
(; ð13Þ
where 0 , k # 1 is a refund coefficient and cr(a) is the replacement cost of the faileditem with a used one of age a. The value of k is determined by the dealer and reflectsdealer’s financial and marketing goals (Wu et al., 2007). Therefore, using equation (13),we obtain the dealer’s expected warranty costs as follows:
E½cdðw; aÞ� ¼
Z w
0
kcrðaÞf 1ðtÞdt ¼ kcrðaÞF1ðwÞ: ð14Þ
4.3.2 Pro-rata warranty. Under a PRW policy, the customer is refunded a proportion ofthe replacement that is pro-rated to the age of the item at the time of the failure. Also,the dealer may offer a rebate for the purchase of a replacement product, which dependson the age of the failed unit. Typically, the PRW is offered on relatively cheap,non-repairable second-hand items. A PRW policy encourages repeat purchases,discourages customer moral hazard, and is more suitable for items that are subject todeterioration due to aging (Polatoglu and Sahin, 1998). The refund function forwarranty under the PRW policy can be expressed by:
RðtÞ ¼crðaÞ 1 2 t
w
� �0 # t # w
0 x . w:
(ð15Þ
Therefore, using equation (15) and integrating by parts, we obtain that the dealer’sexpected warranty costs are given by:
E½cdðw; aÞ� ¼
Z w
0
crðaÞ 1 2t
w
� �f 1ðtÞdt ¼
crðaÞ
w
Z w
0
F1ðtÞdt: ð16Þ
Example 4. Let FT(t) ¼ 1 2 exp[2 (lt)b] and l ¼ 0.283/year, b ¼ 1.25. As a resultF1(t) is given by:
F1ðtÞ ¼exp½2ðltÞb�2 exp{ 2 ½lðx þ tÞ�b}
exp½2ðltÞb�;
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Table VI shows the value of the ratio E[cd(w,a)]/cr(a) for values of a varying from one toseven years and values of w varying from 0.5 to two years for LSW policy with k ¼ 0.5.Table VI shows that as a changes from one to seven years, the value of the ratioE[cd(w,a)]/cr(a) increases, and the expected warranty cost of the dealer increases by afactor of 1.5 for w ¼ 0.5 and by a factor of 1.3 for w ¼ 2.
4.4 Hybrid warrantyPure warranties, such as FRW, PRW and others, have some advantages as well assome disadvantages for the customers and dealers. Hybrid (combination) warrantiesare designed to utilize the desirable characteristics of the pure warranties anddownplay some of their drawbacks. For example, the American tire dealer “Firestone”(www.firestoneag.ca) offers a FRW/PRW policy for used tires. During the first twoyears of service, the tire is replaced without charge including normal mounting andservice charges. After this two-year free-replacement period, the replacement price ispro-rated based on years of service from the original purchase date. The US helicoptercompany “Bell” (www.bellhelicopter.com) offers a FRW/PRW policy for usedhelicopters and spare parts. It warrants its used helicopter to be free of defects inmaterial or workmanship under normal use and service within 500 hours of operationor six months after delivery, whichever occurs first. After the first 100 hours of use, thewarranty contract specifies a replacement pro-rated charges for the buyer.
4.4.1 FRW/PRW. Under the FRW/PRW policy, the dealer agrees to replace a faultyitem prior to wf after the sale, where wf , w, with a second-hand item of age a, i.e. witha used item with reliability matching the reliability of the second-hand item at the timeof the sale. Furthermore, a failure in [wf, w ] results in a pro-rated refund, see Figure 4.
The refund function for warranty under the FRW/PRW policy can be modeled by:
RðtÞ ¼
crðaÞ 0 # t , wf
kcrðaÞ 1 2 at2wf
w2wf
h iwf # t , w
0 otherwise
8>>><>>>:
; ð17Þ
where 0 # a # 1 is the proportionality coefficient related to the time of the warrantyfailure and cr(a) is the replacement cost. Using equation (17), we express the dealer’sexpected warranty costs as follows:
Table VI.E [cd (w,a)]/cr(a) fordifferent combinations ofw and a for LSW policywith k ¼ 0.5
Figure 4.FRW/PRW policy for asecond-hand item
× × ×
a+wa+wf0 a
FRW PRW
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E½cdðw; aÞ� ¼ crðaÞF1ðwf Þ
1 2 F1ðwf Þþ
Z w
wf
kcrðaÞ 1 2 at 2 wf
w 2 wf
� �f 1ðtÞdt: ð18Þ
4.4.2 FRW/LSW. Under this policy, the dealer provides a free of charge replacement ofa faulty item by a used item of age a, up to time wf after the sale, and offers a LSW overthe remaining warranty period [wf, w ]. The refund function for warranty under theFRW/LSW policy can be expressed as:
RðtÞ ¼
crðaÞ 0 # t , wf
kcrðaÞ wf # t , w
0 otherwise
:
8>><>>: ð19Þ
Using equation (19), the dealer’s expected warranty costs, are given by:
E½cdðw; aÞ� ¼ crðaÞF1ðwf Þ
1 2 F1ðwf Þþ
Z w
wf
kcrðaÞf 1ðtÞdt
¼ crðaÞF1ðwf Þ
1 2 F1ðwf Þþ kcrðaÞ½F1ðwÞ2 F1ðwf Þ�
ð20Þ
Example 5. Consider FRW/LSW policy with model parameters as given in Example 4.Table VII shows the value of the ratio E[cd(w,a)]/cr(a) for a varying from one tofour years, w varying from 0.75 to 1.5 years for FRW/LSW policy with wf ¼ 0.5 andk ¼ 0.8.
Also, it shows that as a changes from one to four years, the expected warrantycost to the dealer increases by a factor of 1.35 for w ¼ 0.5 and by a factor of 1.26 forw ¼ 2.0.
5. Numerical resultsNext, we present an example to illustrate and compare some of the policies discussed inSection 4. Consider a dealer who sells non-repairable second-hand products of agea ¼ 1 and of age a ¼ 2. Assume that the product lifetime is five years, and the productis subject to random failures with the time to failure following the Weibull distributionwith parameters b ¼ 2 and l ¼ 0.354. This implies that the mean time to first failure is2.5 years. The model parameters under study include: refund coefficient (k),proportionality coefficient related to the time of the failure (a) and warranty durations(wf, w) where wf is 0.25 and w varies from 0.5 to 3. We examine and compare thefollowing five warranty policies:
. Policy IV. Combination FRW/PRW with wf ¼ 0.25, k ¼ 0.5, a ¼ 0.5.
. Policy V. Combination FRW/LSW with wf ¼ 0.25, k ¼ 0.8.
In Table VIII we present our findings from the analysis of the above Policies I-V.From Table VIII, as expected, it is easy to see, that a longer warranty period results
in a higher expected warranty cost. Figure 5 compares the E[cd(w,a)]/cr(a) underPolicies I-V for a second-hand item with past age of one year. Figure 5 shows that thehighest expected warranty cost is related to Policy I – RFRW, whereas Policy II –PRW leads to the lowest expected warranty cost.
Of course, this expected warranty cost will be included one way or another in the saleprice of the second-hand item. Therefore, it is expected that the sale price for an item soldwith RFRW will be higher than the sale price of the same item sold with PRW. Finally,based on our analysis, we can provide the buyer with a list of options – (sale price,warranty policy) – to choose from when making a decision regarding the purchase of
Figure 5.E[cd(w,a)]/cr(a) for variouswarranty policie
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the second-hand product. Also, while making his/her decision regarding the “best”(sale price, warranty policy) option, the buyer should take into account the level of theproduct failure rate at the time of the expiration of the warranty contract, which willaffect the post-warranty performance of the product and related servicing costs.
In all of our examples from the previous section and in this section there was a needto identify the values of the model parameters and we assumed that these are known.Unfortunately this is not the case in reality when we deal with real life warrantyproblems. Estimating the reliability and warranty parameters from real warranty datais usually a challenging problem. Most of the well known statistical techniques andapproaches need a significant modification before they can be used. One of the mainreasons for this difficulty is due to the fact that the warranty data entry are not evenclose in resembling results of a controlled experiment, so the assumption of havingindependent identically distributed observations is rarely valid. Moreover, in mostwarranty databases the data are doubly censored (or truncated) – first, due to thelimits of the window of observations (usually the window of observation is thewarranty period or, in multidimensional case – the warranty region) and second, someof the warranty measures might be not observable (e.g. the odometer readings in 2Dautomotive warranty). In addition, due to the fact that the warranty data are highlycommercially sensitive, most of the manufacturers are reluctant to share theirwarranty data with researchers outside of their company and, if they do, the data areappropriately masked, which introduces additional difficulties in the data analysis.Some initial ideas on how to handle real warranty data for estimating the meancumulative functions can be found in Chukova and Robinson (2005) andChristozov et al. (2009b).
6. Conclusion and extensionsThe use of warranty for second-hand items is a relatively new strategy dealers use toshare the risk of malfunctioning with their customers. Providing warranty leads toadditional servicing costs that the dealers of second-hand items need to estimate andappropriately include in the sale price structure. This paper provides an overview ofstochastic models that are useful for modelling the product failure process and forevaluating the expected warranty costs for second-hand items sold under 1D warrantypolicies. We present several numerical examples to illustrate the application of thesemodels. The policies and models developed in this paper can be useful in managerialdecisions making related to second-hand products such as automobiles, homeappliances, helicopters, electronic equipment and electronics.
There is a broad scope of topics for future research in the area of second-handproduct warranty, such as:
. Cost analysis for MBG option assuming that the number of failures in thewarranty period exceeds some pre-specified value (this policy is one of the mostcommon types of MBG in second-hand markets).
. Constructing and analyzing stochastic models that incorporate the past usageand maintenance history of a second-hand item. This could be done similarly toJack et al. (2009).
. Modelling the failures and warranty costs for 3D warranties, which are offered,for example, for used acquisitions (used fleet of tanks or jet flight engines).
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. Developing models for, so-called, flexible warranty. Under this type of warranty,different 2D warranties, based on different 2D regions, are offered to a new buyerof a second-hand product to choose from, so that the buyer can select the bestwarranty for his/her needs.
The authors are currently working on some of above topics.
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Corresponding authorStefanka Chukova can be contacted at: [email protected]
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