Warranty servicing with imperfect repair

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Int. J. Production Economics 111 (2008) 159–169

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Warranty servicing with imperfect repair

Won Young Yuna,�, D.N.P. Murthyb, N. Jackc

aDepartment of Industrial Engineering, Pusan National University, 30 Changjeon-Dong, Kumjeong-Ku, Busan 609-735, Republic of KoreabDepartment of Mechanical Engineering, The University of Queensland, St. Lucia 4072, Australia

cDundee Business School, University of Abertay Dundee, Dundee DD1 1HG, Scotland, UK

Received 24 August 2005; accepted 29 December 2006

Available online 24 February 2007

Abstract

A manufacturer of a product sold with warranty incurs additional costs from the servicing of warranty claims made by

customers. These costs depend on several factors such as the reliability of the product, the warranty terms, the maintenance

actions of the customers, and the servicing strategy used by the manufacturer. Through effective warranty servicing

strategies the manufacturer can reduce the warranty costs. In the case of repairable items, several different warranty

servicing strategies involving minimal repair and replace by new have been studied. For expensive products, servicing

strategies involving minimal and imperfect repairs should result in lower warranty costs than strategies studied in the past.

Imperfect repair results in the reliability of a repaired item being better than it was just before failure but less than that of a

new item and it is much cheaper than the cost of a new item. In this paper, we look at two new warranty servicing strategies

(strategies 1 and 2) involving minimal and imperfect repairs. In both strategies, a failed item is subjected to at most one

imperfect repair over the warranty period. In Strategy 1, the level of improvement in reliability under imperfect repair

depends on the age of the item at which imperfect repair is carried out whereas, in Strategy 2, the level of improvement

does not depend on the age. As a result, Strategy 1 involves a functional optimisation to determine the optimal

improvement in reliability when an imperfect repair is carried out. In contrast, Strategy 2 involves only a parameter

optimisation to determine the optimal reliability improvement. In a numerical example, we compare the two new servicing

strategies with two other strategies reported in the literature.

r 2007 Elsevier B.V. All rights reserved.

Keywords: Warranty servicing; Imperfect repair; Expected warranty servicing cost

1. Introduction

When a product is sold with a free replacement/repair warranty (FRW) policy the manufacturerrectifies all failures occurring during the warrantyperiod at no cost to the customer. Offering warrantyresults in additional costs to the manufacturer due

front matter r 2007 Elsevier B.V. All rights reserved

e.2006.12.058

ng author. Tel.: +8251 5102421;

7603.

ss: [email protected] (W.Y. Yun).

to servicing the warranty claims made by customers.These warranty costs depend on the productreliability, the warranty terms, the customer’smaintenance actions, and the warranty servicingstrategy used. Warranty cost models for manydifferent types of warranties can be found inBlischke and Murthy (1994, 1996). A review ofwarranty cost models from 1990 to 2000 is given inMurthy and Djamaludin (2002) and for more recentmodels (see Bai and Pham, 2006; Jain andMaheshwari, 2006; Manna et al., 2006).

.

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ARTICLE IN PRESSW.Y. Yun et al. / Int. J. Production Economics 111 (2008) 159–169160

The three ways in which the manufacturer canreduce warranty costs are as follows:

1.
Improving product reliability: This is done at thedesign and development stage through the use ofredundancy and reliability growth with test-fixcycles to eliminate some of the failure modes.Production cost is increased and the improve-ment is worthwhile only if the reduction in thewarranty costs (in an expected sense) exceeds theadditional costs incurred during design anddevelopment. For more details, see Blischkeand Murthy (1994) and Blischke and Murthy(1996), and Murthy and Blischke (2005). If theproduct has a high initial failure rate, then burnin can be used to reduce the warranty cost. SeeSheu and Chien (2005) and the references citedtherein for further details.
2.
Use of preventive maintenance: This is appro-priate if the warranty period is long and again theextra cost incurred must be less than thereduction in expected warranty costs to make itworthwhile. There are many papers dealing withthis, see for example, Kim et al. (2004) andPascual and Ortega (2006) and the referencescited therein.
3.
Warranty servicing strategy: This is applicable forrepairable products. When an item fails, themanufacturer has the choice of either repairingor replacing it with a new one. The first optioncosts less then the second but a repaired item hasa greater probability of failing during theremainder of the warranty period. It is thereforeimportant for the manufacturer to choose anappropriate servicing strategy in order to mini-mise the expected cost of servicing the warrantyper item sold. In this paper we deal with thistopic.
A brief literature review of warranty servicingstrategies is as follows. Products sold with one-dimensional warranties have received considerableattention. Biedenweg (1981) and Nguyen andMurthy (1986), and Nguyen and Murthy (1989)assumed that repaired items have independent andidentically distributed lifetimes different from thatof a new item and considered strategies wherethe warranty period is divided into distinct intervalsfor repair and replacement. Nguyen (1984) intro-duced the first servicing model with minimal repair(see Barlow and Hunter, 1960), with the warrantyperiod split into a replacement interval followed by

a repair interval. The length of the first interval isselected optimally to minimise the expected war-ranty cost.

Jack and Van der Duyn Schouten (2000) showedthat this strategy is sub-optimal and that theoptimal servicing strategy is in fact characterizedby three distinct intervals—[0, x), [x, y] and (y, W]where W is the warranty period. The optimalstrategy is to carry out minimal repairs in the firstand last intervals and to use either minimal repair orreplacement by new in the middle interval depend-ing on the age of the item at failure. (Refer to Jianget al., 2006 for the optimality of this strategy.) Thisstrategy is difficult to implement, so Jack andMurthy (2001) proposed a near optimal strategyinvolving the same three intervals but with only thefirst failure in the middle interval resulting in areplacement and all other failures being minimallyrepaired. (Refer to Sheu and Yu, 2005 for the casewith random minimal repair cost.)

Zuo et al. (2000) consider a replacement–repairpolicy for a deteriorating system modelled using aMarkov-chain formulation.

Servicing strategies for products sold with two-dimensional warranties have also received someattention. Iskandar and Murthy (2003) examinedtwo strategies similar to those in Nguyen andMurthy (1986), and Nguyen and Murthy (1989)but with minimal repair. Iskandar et al. (2005)discuss a servicing strategy similar to that given inJack and Murthy (2001). (Refer to Chukova andJohnston, 2006 for the case without a restrictionassumed by Iskandar et al., 2005.)

When the cost of replacement is high compared tothe cost of a minimal repair then strategies involvingreplacement are not appropriate. In this case,imperfect repair strategies (where the failure char-acteristics of the repaired item are better than thoseafter minimal repair but are not the same as a newitem) are more appropriate. The advantage of usingimperfect repair is that the degree of improvementin the reliability after repair is a decision variableunder the control of the manufacturer. The litera-ture on imperfect repair has focused mainly on itsuse in the context of preventive and correctivemaintenance of unreliable systems. See, for exam-ple, Wang (2002), Li and Shaked (2003), Doyen andGaudoin (2004), Zequeira and Berenguer (2006),Sheu et al. (2006), and Tang and Lam (2006).Chukova et al. (2004), Zequeira and Berenguer(2006) look at warranty analysis based on imperfectrepairs.

ARTICLE IN PRESSW.Y. Yun et al. / Int. J. Production Economics 111 (2008) 159–169 161

In this paper, we look at two servicing strategies(servicing strategies 1 and 2, respectively) involvingminimal and imperfect repairs. In both strategies, afailed item is subjected to at most one imperfectrepair over the warranty period. In Strategy 1, thelevel of improvement in reliability under imperfectrepair depends on the age of item at which imperfectrepair is carried out whereas, in Strategy 2, the levelof improvement does not depend on the age. As aresult, Strategy 1 involves a functional optimisationto determine the optimal improvement in reliabilitywhen an imperfect repair is carried out. In contrast,Strategy 2 involves only a parameter optimisation todetermine the optimal reliability improvement.

The outline of the paper is as follows. Modelnotation is given in Section 2 and Section 3 containsthe model formulations for Servicing Strategies 1and 2. Section 4 deals with the analysis of Strategy 1and Section 5 with that of Strategy 2. In Section 6,we consider the special case where the time to firstfailure is a Weibull distribution and present someanalytical results. In Section 7, we present anumerical example to compare the two newstrategies with other strategies studied in the earlierliterature and present some sensitivity analysis ofthe results. We conclude with a brief discussion ofsome topics for future research in Section 8.

2. Notation

The following notation is used in modelling thetwo new servicing strategies:

F ðtÞ; f tð Þ; F tð Þ; rðtÞ distribution function, densityfunction, survivor function, and hazardrate of time to first item failure.

W warranty period.[x, y] interval over which the first failure is

rectified through imperfect repair(0pxpypW ).

d(t) proportional hazard rate reduction factorunder imperfect repair at time t

(0pdðtÞp1).Dðx; yÞ :� fdðtÞ; xptpyg proportional hazard rate

reduction function over [x, y] for servicingStrategy 1.

d constant proportional hazard rate reduc-tion factor for servicing Strategy 2.

Cr average cost of each minimal repair.Ci(z, t) cost of an imperfect repair if a failure

occurs at time t and d(t) ¼ z (servicingStrategy 1).

Ci(d) cost of an imperfect repair (servicingStrategy 2).

J(x, y, D(x, y)) expected warranty servicing cost peritem sold (servicing Strategy 1).

J(x, y, d) expected warranty servicing cost per itemsold (servicing Strategy 2).

3. Model formulation

If the item is minimally repaired at each failurethen this type of rectification action has a negligibleimpact on its reliability. The hazard rate for itemlifetime after a minimal repair is the same as thatjust before failure. If repair times are small relativeto the mean time between failures (so that minimalrepairs can be treated as being instantaneous) thenitem failures over time follow a non-homogeneousPoisson process (NHPP) with intensity functionl(t) ¼ r(t). The intensity function is also referred toas the rate of occurrence of failures (ROCOF).

In contrast, an imperfect repair improves theitem’s operating condition and the hazard rate ofitem lifetime after a repair is smaller. This can bemodelled as follows. If the failure occurs at ti thehazard rate just before failure is rðt�i Þ and afterrepair it is given by rðtþi Þ ¼ rðt�i Þ � dðtiÞðrðt

�i Þ � rð0ÞÞ

where d(ti) can take values in the interval [0, 1]. Thereduction in the hazard rate is rðtþi Þ � rðt�i Þ ¼

dðtiÞðrðt�i Þ � rð0ÞÞ and this is a linear function of

d(ti). d(ti) is a decision variable with a higher valueindicating a greater improvement in the reliabilityafter repair. Imperfect repairs are also assumed tobe instantaneous.

The concept of imperfect repair is appropriate fora complex system containing a very large number ofcomponents. The system hazard rate can beexpressed in terms of the component hazard rates.(For example, in the case of a series configuration,the system hazard is the sum of the componenthazard rates.) The hazard rates are increasingfunctions of time (reflecting the degradation effectof age). System failure occurs due to failure of oneor more components. Under minimal repair, onlyfailed components are replaced and the effect on thesystem hazard rate is negligible. Under imperfectrepair, the failed components and also some of thenon-failed components are replaced in order toachieve the desired reduction in the system hazardrate. This implies that the cost of an imperfect repairis greater than that of a minimal repair and this costincreases as the level of hazard rate reductionincreases.


We now introduce the two new servicing strate-gies.

Servicing Strategy 1: All item failures occurring inthe interval [0, x) are minimally repaired. The firstfailure in the interval [x, y] is rectified by imperfectrepair with the proportional reduction factor in thehazard rate d(t) depending on the age t at failure. Thecost of the imperfect repair is Ci(d(t), t) All subsequentfailures in this interval are minimally repaired, as arethose that occur in the remaining interval (y, W]

This strategy is characterised by the set {x, y, D(x,y)} consisting of the two parameters x and y and thefunction Dðx; yÞ � fdðtÞ;xptpyg. These are thedecision variables that need to be selected optimallyto minimise the expected warranty servicing costJ(x, y, D(x, y)).

Comment: This strategy is similar to that pro-posed by Jack and Murthy (2001) except that wehave imperfect repair instead of replacement in theinterval [x, y]

Servicing Strategy 2: Unlike servicing Strategy 1,here the proportional reduction in the hazard ratedoes not change with time over the interval [x, y]. Itis a parameter (d) to be selected optimally. The costof an imperfect repair is then given by Ci(d). In thiscase, the strategy is characterised by the threeparameters x, y, and d and these are to be selectedoptimally to minimise the expected warranty servi-cing cost J(x, y, d).

The following two strategies have been reportedin the literature and we will compare them with thetwo new strategies in this paper through a numericalexample in Section 7.

Servicing Strategy 3: The first failure occurring in[x, y] is rectified by replacing the failed item by anew one and all other failures are minimallyrepaired. x and y are to be selected optimally tominimise the expected warranty servicing cost.

Comment: This strategy was first proposed inJack and Murthy (2001). It is important to note thatStrategy 2 with d ¼ 1 is not the same as Strategy 3.

Servicing Strategy 4: All failures under warrantyare minimally repaired.

Comment: This strategy is a special case ofStrategy 2 with d ¼ 0.

4. Analysis of servicing Strategy 1

4.1. Expected warranty servicing cost

We obtain an expression for the expected warrantyservicing cost, J(x, y, D(x, y)), using a conditional

approach by conditioning on T1 the time of the firstfailure after x. The distribution function, densityfunction, and survivor function for T1 are given by

F1ðtÞ ¼F ðtÞ � F ðxÞ

F ðxÞ; f 1ðtÞ ¼

f ðtÞ

F ðxÞand

F1ðtÞ ¼F ðtÞ

F ðxÞ; tXx,

respectively. We need to consider two cases: (i)xpT1 ¼ t1py and (ii) T1 ¼ t14y.

In each case, all failures over (0, x) are minimallyrepaired, so the failures occur according to anNHPP process with intensity function l(t) ¼ r(t)and the expected warranty servicing cost for thisinterval is given by Cr

R x

0 rðtÞdt.The conditional expected cost, conditional on

xpT1 ¼ t1py, is obtained as follows. The firstfailure in [x, y] occurs at t1 and is imperfectlyrepaired. All failures over the remaining interval (t1,W] are minimally repaired. As a result, the failuresover this interval occur according to an NHPP withintensity function:

lðtÞ ¼ rðtÞ � dðt1Þðrðt1Þ � rð0ÞÞ; t1otpW .

The expected cost of servicing failures over (t1, W]is given by

Cr

Z W

t1

½rðtÞ � dðt1Þðrðt1Þ � rð0ÞÞ�dt.

As a result, the conditional expected warrantycost is given by

Jðx; y;Dðx; yÞ xpt1py�� Þ ¼ Cr

Z x

0

rðtÞdt

þ Cr

Z w

t1

½rðtÞ � dðt1Þðrðt1Þ

� rð0ÞÞ� dtþ Ciðdðt1Þ; t1Þ.

ð1Þ

The conditional expected cost, conditional onT1 ¼ t14y, is obtained as follows. Note that thereis no failure in [x, y] and failures over the remaininginterval (y, W] occur according to an NHPP withintensity function

lðtÞ ¼ rðtÞ; yotpW .

As a result, the conditional expected warrantycost is given by

Jðx; y;Dðx; yÞ t14y�� Þ ¼ Cr

Z x

0

rðtÞdtþ

Z w

y

rðtÞdt

� �.

(2)

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z0 1

Ci(z,t)

ζ(t)z

a

c

b

z∼

Fig. 1. Plots of Ci(z, t) and z(t)z vs. z.

W.Y. Yun et al. / Int. J. Production Economics 111 (2008) 159–169 163

The expected warranty cost is obtained byunconditioning and is given by

Jðx; y;Dðx; yÞÞ ¼ Jðx; y; dðt1Þ t14y�� ÞF 1ðyÞ

þ

Z y

x

Jðx; y; dðt1Þ xpt1py�� Þf 1ðt1Þdt1.

ð3Þ

Define LðtÞ ¼R t

0rðuÞdu. Then (3) can be rewritten

as

Jðx; y;Dðx; yÞÞ ¼ Cðx; yÞ þ FðDðx; yÞ; x; yÞ, (4)

where

Cðx; yÞ ¼ Cr LðxÞ þ ½LðW Þ � LðyÞ�F ðyÞ

F ðxÞ

�

þ

Z y

x

LðW Þ � LðtÞ½ �f ðtÞ

F ðxÞdt

�¼ Cr LðW Þ � F1 yð Þ½ � ð5Þ

and

FðDðx; yÞ;x; yÞ ¼Z y

x

½CiðdðtÞ; tÞ � CrdðtÞfrðtÞ

� rð0ÞgðW � tÞ�f ðtÞ

F ðxÞdt. ð6Þ

4.2. Optimisation problem

The optimisation problem is given by

Minx;y;Dðx;yÞ

Jðx; y;Dðx; yÞÞ ¼ Cðx; yÞ þ FðDðx; yÞ;x; yÞ.

(7)

Note that this involves selecting optimally the twoparameters x and y (subject to the constraints0pxpypW ) and the function Dðx; yÞ �fdðtÞ;xptpyg (subject to the constraints 0pdðtÞp1).

Let x*, y* and D*(x, y) denote the optimalsolution. We obtain this using a two-stage ap-proach. In stage 1, for a fixed x and y, we obtain theoptimal D*(x, y) that minimises J(x, y, D(x, y)).Then, in stage 2, we obtain the optimal (x*, y*) byminimising J(x, y, D*(x, y)).

Stage 1: To determine D*(x, y) we need to focuson F(D(x, y)x, y) given by (6) and this can berewritten as

FðDðx; yÞ;x; yÞ ¼Z y

x

½CiðdðtÞ; tÞ � zðtÞdðtÞ�f ðtÞ

F ðxÞdt,

(8)

where

zðtÞ ¼ Cr½rðtÞ � rð0Þ�ðW � tÞ. (9)

We need to determine the optimal form for d(t) forevery point t along the time axis. This is a standardcalculus of variation problem and can also be solved asan optimal control problem (see Bryson and Ho, 1975).

The optimal d(t) must result in ½CiðdðtÞ; tÞ �zðtÞdðtÞ� being a minimum for each t 2 x; y½ �. Asresult, d*(t) can be obtained by examining:

nðz; tÞ ¼ Ciðz; tÞ � zðtÞz (10)

for each t 2 x; y½ �. For a fixed t, Ci(z, t) is anincreasing function of z as shown in Fig. 1. z(t)z, thesecond term in n(z, t), is linear in z and so is astraight line when plotted as a function of z, asshown in Fig. 1.

We need to consider the following two cases.Case (1): The line z(t)z lies below the curve Ci(z,

t). This corresponds to (a) in Fig. 1. In this case,d*(t) ¼ 0. This is because the cost of any imperfectrepair with d(t)40 is not worth the reduction in theexpected warranty servicing cost when comparedwith only minimal repair (d(t) ¼ 0).

Case (2): The straight line z(t)z and the curve Ci(z,t) intersect. This corresponds to (b) in Fig. 1 and inthis case we have d*(t)40. Since 0pdðtÞp1 theneither d*(t) ¼ 1 (the boundary solution) or 0od �ðtÞo1 (an interior point solution). In the latter case,the optimal value is obtained from the usual first ordercondition. This yields d*(t) ¼ z* with z* given by

qCiðz; tÞ=qz ¼ zðtÞ. (11)

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t0 W

δ∗(t)

τ1 τ2

1

0

Fig. 3. Shape of the optimal proportional reduction d*(t) vs. t.

W.Y. Yun et al. / Int. J. Production Economics 111 (2008) 159–169164

Let the straight line az be a tangent to the curveCi(z, t)at z ¼ ~z. This is shown by (c) in Fig. 1. a and ~zare obtained by solving the simultaneous equationsgiven below:

Cið~z; tÞ ¼ a~z andqCiðz; tÞ

qz

��z¼~z

¼ a, (12)

where z(t) is a concave function as shown in Fig. 2with z(0) ¼ 0 and z(W) ¼ 0. Define zm ¼

max0ptpWzðtÞ We have the following proposition.

Proposition. If zmoa then d*(t) ¼ 0 for all t. If

zm4a then d*(t)40 for 0ot1otot2oW where t1and t2 are the solutions of the equation z(t) ¼ a (see

Fig. 2). For t outside the interval ½t1; t2Þ, d*(t) ¼ 0.

This implies that d*(t) has a shape as shown in Fig. 3,and note that d*(t) does not depend on x and y.

Comment: The shape of the optimal proportionalhazard rate reduction curve indicated in Fig. 3makes sense. If the failure occurs very early in thewarranty period, all the components are nearly newand hence there is no point in using imperfect repairand minimal repair is the optimal strategy. Simi-larly, when the failure occurs close to the end of thewarranty period, the risk of failure over theremainder of the warranty period is small withminimal repair and the reduction in this risk withimperfect repair is not worth the extra cost involved.However, when a failure occurs around the middleof the warranty period, the expected warranty costfor the remaining part of the warranty period is theimportant issue. Depending on the item’s hazardrate and the parameter values, this cost can bereduced by using imperfect repair (with the optimalproportional reduction) so that it is lower than thatachieved with only minimal repair.

t0 W

ζ(t)

ζm

τ1 τ2

Fig. 2. Plot of z(t) vs. t.

Stage 2: Let D � ðx; yÞ � fd � ðtÞ; 0ptpW g whichis obtained from Stage 1. x* and y*, the optimalvalues for x and y, are obtained by solving thefollowing minimisation problem;

Minx;y

Jðx; y;D � ðx; yÞÞ ¼ Cðx; yÞ þ FðD � ðx; yÞ;x; yÞ

(13)

subject to the constraint 0pxpypW . These can beobtained from the usual first-order conditions:

qJðx; y;Dðx; yÞÞqx

¼ 0 andqJðx; y;Dðx; yÞÞ

qy¼ 0

if they lie inside the interval [0, W]. It is not possibleto derive any analytical results from these condi-tions and the optimal values need to be obtainedusing a computational approach.

5. Analysis of servicing Strategy 2

5.1. Expected warranty cost

The approach is the similar to that for servicingStrategy 1 and hence we omit the details. Theexpected warranty cost is given by

Jðx; y; dÞ ¼ Cðx; yÞ þ Fðd;x; yÞ (14)

with C(x, y) given by (5) and

Fðd;x; yÞ ¼~Fðd;x; yÞ

F ðxÞ¼

F1ðx; yÞCiðdÞ � F2ðx; yÞdF ðxÞ

,

(15)

where

F1ðx; yÞ ¼

Z y

x

f ðtÞdt and F2ðx; yÞ

¼ Cr

Z y

x

½frðtÞ � rð0ÞgðW � tÞ�f ðtÞdt.


5.2. Optimisation problem

We use a two-stage approach. In stage 1, for afixed x and y we obtain the optimal d*(x, y) thatminimises J(x, y, d). Then in stage 2 we obtain theoptimal (x*, y*) by minimising J(x, y, d*(x, y)).

Stage 1: d*(x, y) is obtained by solving thefollowing optimisation problem:

Mind x;yj

~Fðd;x; yÞ ¼ F1ðx; yÞCiðdÞ � F2ðx; yÞd. (16)

d*(x, y) can be an interior point or one of theend-points of the interval [0, 1]. If it is an inte-rior point, then it must satisfy the first-ordercondition:

q ~Fðd; x; yÞqd

¼ F1ðx; yÞdCiðdÞqd� F2ðx; yÞ ¼ 0.

Note that d*(x, y) is a function of x and y. Thiscontrasts with Section 4 where d*(t) was indepen-dent of x and y:

Stage 2: x* and y* are obtained by solving thefollowing optimisation problem:

Minx;y

Jðx; y; dÞ ¼ Cðx; yÞ þ Fðd�ðx; yÞ;x; yÞ (17)

subject to the constraint 0pxpypW . We need touse a computational approach to obtain theseoptimal values. The optimal reduction when animperfect repair is carried out is given by d*(x*, y*).

6. Special case: Weibull failure distribution

The distribution function for the time to firstfailure is a Weibull distribution with scale parameterZ40 and shape parameter b41, so

rðtÞ ¼ bt b�1ð Þ

Zb.

We assume that Ciðz; tÞ ¼ Cr þ Kzp with p41and K ¼ C0 � Cr, so the cost of an imperfect repairlies between Cr (when z ¼ 0) and C0 (when z ¼ 1).Cr is the cost of a minimal repair and C0 representsthe cost of a repair that achieves a 100% reductionin the system hazard rate.

From (9), we have

zðtÞ ¼ Crbtðb�1Þ

Zb

� �ðW � tÞ (18)

and since b41, it is easily seen that

zm ¼Crbðb� 1Þðb�1Þ

Zb

!W

b

� �b

40. (19)

From (12), we have

~z ¼Cr

Kðp� 1Þ

� �ð1=pÞ

and a ¼ KpCr

Kðp� 1Þ

� �ðp�1Þ=p

.

(20)

t1 and t2 are the solutions of the equation

Crbtðb�1Þ

Zb

� �ðW � tÞ ¼ Kp

Cr

Kðp� 1Þ

� �ðp�1Þ=p

. (21)

From (11), we have (provided zm4a)

d�ðtÞ ¼Crbtðb�1ÞðW � tÞ

KpZb

� �1=ðp�1Þ. (22)

As mentioned previously, we need to obtain x*

and y* using a computational approach.

7. Numerical example

We normalise costs so that Cr ¼ 1 and consider arange of values for C0 varying from 1.5 to 10. As C0

increases, the cost of an imperfect repair increasesfor each value of zA(0, 1]. We assume the nominalvalues W ¼ 2, Z ¼ 1, b ¼ 2.5, and p ¼ 4 for theother model parameters. For Strategy 3, we assumethat the cost of replacement is C0 (the cost of a100% reduction in the hazard rate for Strategies 1and 2). This is also necessary for a sensiblecomparison of Strategy 3 with the two newstrategies proposed in this paper.

7.1. Optimal expected warranty costs and servicing

strategies

Servicing Strategy 1: The optimal expectedwarranty servicing cost and D� � fd�ðtÞ;x�ptpy�g

for Strategy 1 were obtained using the approachdiscussed in Section 4.

Servicing Strategy 2: The optimal expectedwarranty servicing cost and fd�; x�; y�g for Strategy2 were obtained using the approach discussed inSection 5.

Servicing Strategy 3: The optimal expectedwarranty servicing cost and fx�; y�g for Strategy 3were obtained using the results given in Iskandar etal. (2005). We omit giving the details and interestedreaders should consult the cited reference.

Servicing Strategy 4: The expected warrantyservicing cost for Strategy 4 is given by

J ¼ Cr

Z W

0

rðtÞdt ¼ CrW

Z

� �b

. (23)


Table 1 shows the expected warranty servicingcosts for the four strategies together with theoptimal x* and y* for Strategies 1–3. We do notshow D* for Strategy 1 and this is easily obtainedfrom (22).

7.2. Relative comparison

For all values of C0, Strategy 1 with the flexibilityto vary the level of repair is always the best strategyto use, although Strategy 2 is only slightly moreexpensive. For Strategies 1 and 2, both the locationof the imperfect repair interval and its duration donot change significantly as C0 increases. In eachcase, the interval begins near the middle of thewarranty period and ends just before the warranty iscompleted. However, increasing C0 does reduce theoptimal level of imperfect repair for Strategy 2.Strategy 3 is better than Strategy 4 for small valuesof C0 but becomes equivalent to this minimal repairstrategy as C0 increases.

7.3. Sensitivity analysis

We now vary each of the model parameters b, W,and p to investigate the effect on the optimalwarranty servicing strategies and the expected

Table 1

Optimal servicing strategies and expected warranty servicing costs (b ¼

C0 Strategy 4 Strategy 3 Strategy 2

J x* y* J(x*, y*) x* y*

1.5 5.657 0.94 1.93 3.739 0.94 1.93

2 5.657 0.95 1.84 4.230 0.94 1.91

2.5 5.657 0.96 1.74 4.712 0.94 1.91

3 5.657 0.98 1.61 5.179 0.94 1.91

5 5.657 2 2 5.657 0.94 1.91

10 5.657 2 2 5.657 0.94 1.91

Table 2

Optimal servicing strategies and expected warranty servicing costs (b ¼

C0 Strategy 4 Strategy 3 Strategy 2

J x* y* J(x*, y*) x* y*

1.5 4 0.64 1.87 2.760 0.64 1.88

2 4 0.66 1.71 3.229 0.64 1.88

2.5 4 0.71 1.5 3.668 0.64 1.88

3 4 2 2 4 0.64 1.88

5 4 2 2 4 0.64 1.88

10 4 2 2 4 0.64 1.88

warranty servicing costs. The results are shown inTables 2–6 and, in each case, Strategy 1 alwaysremains the best strategy to use.

(i) Changing the shape parameter: Tables 2 and 3show the effects of reducing and increasing b. Thesmaller (larger) value of b produces a larger(smaller) ROCOF during the first third of thewarranty period but a smaller (larger) ROCOFfrom then onwards. This accounts for the earlier(later) start and earlier (later) finish of the imperfectrepair intervals compared to those in Table 1. ForStrategy 2, the smaller (larger) values of d*

compared to those in Table 1 are caused by thesmaller (larger) ROCOF during the imperfect repairintervals.

(ii) Changing the length of the warranty period:The effects of decreasing and increasing W

are shown in Tables 4 and 5, respectively. InTable 4, the imperfect repair intervals for Strategies1 and 2 now begin approximately one quarter of theway through the warranty period and end justbefore the warranty is completed. Changing C0 doesnot have a significant effect on the location orduration of these intervals. For Strategy 2, thevalues of d* are now smaller than those in Table 1because of the smaller ROCOF during the shorterwarranty period.

2.5, p ¼ 4, W ¼ 2)

Strategy 1

d* J(x*, y*, d*) x* y* J(x*, y*, D*)

1 3.739 0.94 1.99 3.735

0.85 4.122 0.95 1.99 3.858

0.74 4.316 0.95 1.99 4.310

0.67 4.439 0.95 1.99 4.433

0.53 4.690 0.95 1.99 4.689

0.41 4.919 0.95 1.99 4.916

2, p ¼ 4, W ¼ 2)

Strategy 1

d* J(x*, y*, d*) x* y* J(x*, y*, D*)

0.96 2.757 0.64 1.99 2.747

0.77 3.013 0.64 1.99 3.005

0.67 3.138 0.64 1.99 3.131

0.61 3.217 0.64 1.99 3.211

0.48 3.378 0.64 1.99 3.374

0.37 3.526 0.64 1.99 3.522

ARTICLE IN PRESS

Table 3

Optimal servicing strategies and expected warranty servicing costs (b ¼ 3, p ¼ 4, W ¼ 2)

C0 Strategy 4 Strategy 3 Strategy 2 Strategy 1

J x* y* J(x*, y*) x* y* d* J(x*, y*, d*) x* y* J(x*, y*, D*)

1.5 8 1.16 1.96 5.106 1.16 1.96 1 5.106 1.16 2 5.106

2 8 1.16 1.91 5.605 1.16 1.92 0.95 5.590 1.16 2 5.585

2.5 8 1.16 1.85 6.102 1.16 1.92 0.83 5.894 1.16 2 5.890

3 8 1.17 1.79 6.596 1.16 1.92 0.75 6.087 1.17 2 6.083

5 8 2 2 8 1.16 1.92 0.6 6.481 1.16 1.99 6.478

10 8 2 2 8 1.16 1.92 0.46 6.841 1.16 1.99 6.839

Table 4

Optimal servicing strategies and expected warranty servicing costs (b ¼ 2.5, p ¼ 4, W ¼ 1)



1.5 1 1 1 1 0.27 0.96 0.57 0.909 0.26 0.99 0.907

2 1 1 1 1 0.27 0.96 0.45 0.928 0.26 0.99 0.927

2.5 1 1 1 1 0.26 0.96 0.39 0.937 0.26 0.99 0.936

3 1 1 1 1 0.27 0.96 0.36 0.943 0.26 0.99 0.942

5 1 1 1 1 0.26 0.96 0.28 0.955 0.26 0.99 0.954

10 1 1 1 1 0.27 0.96 0.22 0.965 0.26 0.99 0.965

Table 5




1.5 15.588 1.65 2.94 8.947 1.65 2.94 1 8.947 1.65 2.93 8.941

2 15.588 1.64 2.89 9.441 1.64 2.89 1 9.441 1.65 2.91 9.441

2.5 15.588 1.64 2.85 9.941 1.64 2.85 1 9.941 1.65 2.96 9.941

3 15.588 1.64 2.81 10.441 1.64 2.79 0.96 10.423 1.65 2.78 10.423

5 15.588 1.64 2.59 12.441 1.64 2.82 0.76 11.491 1.65 2.8 11.490

10 15.588 3 3 15.589 1.64 2.76 0.58 12.462 1.64 2.9 12.460

Table 6




1.5 5.657 0.94 1.93 3.739 0.94 1.93 1 3.739 0.94 1.99 3.738

2 5.657 0.95 1.84 4.229 0.95 1.84 1 4.229 0.95 1.99 4.224

2.5 5.657 0.96 1.74 4.712 0.95 1.79 0.82 4.670 0.96 1.98 4.658

3 5.657 0.98 1.61 5.179 0.95 1.79 0.62 4.916 0.96 1.99 4.907

5 5.657 2 2 5.657 0.95 1.79 0.31 5.287 0.96 1.98 5.282

10 5.657 2 2 5.657 0.95 1.79 0.14 5.492 0.96 1.95 5.490

W.Y. Yun et al. / Int. J. Production Economics 111 (2008) 159–169 167


Because of the short warranty period, no replace-ment is ever carried out under Strategy 3 and so it isequivalent to the minimal repair Strategy 4 for allvalues of C0.

In Table 5, the imperfect repair intervals nowbegin during the second half of the warranty period.Increasing C0 does not change where these intervalsbegin but it does produce a general reduction intheir duration. For Strategy 2, the values of d* arelarger than they were in Table 1 due to the largerROCOF in the longer warranty period.

(iii) Changing the rate of increase in imperfect

repair costs: Increasing the value of p from 2 to 4produces a slower rate of increase in imperfectrepair costs as the proportional hazard rate reduc-tion varies from 0 to 1. The effect of this change isshown in Table 6. Comparing Tables 1 and 6 we cansee that, for fixed C0, there is no significant changein the starting time of the imperfect repair intervalfor Strategies 1 and 2. For Strategy 1, the finishingtime of the interval is also approximately unchangedbut this time is significantly smaller in the case ofStrategy 2.

8. Conclusions and topics for future research

In this paper, we introduced two new warrantyservicing strategies (Strategies 1 and 2) involvingminimal and imperfect repair. In both strategies afailed item is subjected to at most one imperfectrepair over the warranty period. In Strategy 1 thelevel of improvement in reliability under imperfectrepair depends on the age of item at which imperfectrepair is carried out whereas in Strategy 2 the levelof improvement does not depend on the age. Twoother servicing strategies (Strategies 3 and 4) studiedby earlier researchers were also considered forcomparison purposes.

We found that the expected warranty servicingcosts for Strategy 1 were always at least as small asthose for Strategy 2, but the difference (for theparameter values considered) was often negligible.This suggests that Strategy 2 with a constant level ofimperfect repair is almost as good as Strategy 1where a variable level is used. Both of thesestrategies yield lower expected warranty servicingcosts compared to Strategies 3 and 4. The differencein costs is significant when the cost of replacement(perfect repair) is high.

The type of imperfect repair that we discuss inthis paper needs to be generalised to two-dimen-sional failure modelling. Servicing strategies with

imperfect repair for products sold with two-dimen-sional warranties can then be studied.

Another promising future research topic in war-ranty servicing is to consider preventive maintenanceduring warranty period. In most of the existingwarranty servicing studies, the manufacturer onlyconsiders repair or replacement when the item failsduring the warranty period and the customer makes awarranty claim. It is also possible to reduce the totalwarranty servicing cost by maintaining warranteditems preventively and proactively if the effect ofpreventive maintenance is better than minimal repair.

Acknowledgement

The authors thank the reviewers for their con-structive comments on an earlier version of this paper.

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Warranty servicing with imperfect repair