Quality Technology &
Quantitative Management
Vol. 9, No. 1, pp. 23-33, 2012
QTQM
© ICAQM 2012
Analysis of a Two-dimensional Warranty Servicing
Strategy with an Imperfect Repair Option
Rudrani Banerjee and Manish C. Bhattacharjee
Center for Applied Mathematics & Statistics, Department of Mathematical Sciences
New Jersey Institute of Technology, Newark, NJ, USA
(Received July 2010, accepted March 2011)
______________________________________________________________________
Abstract: Items sold under warranty often require post sale support in terms of repair or replacement.
Providence of such support costs the warrantor considerably and affects his profit margin. Thus reducing
warranty servicing costs is an issue of great interest in warranty studies. In this paper, we look at two
variations of a servicing strategy in two-dimensional warranty regimes involving minimal and imperfect
repairs. Our work demonstrates the modeling and analysis of costs under these servicing strategies and
compares their performance to other strategies present in the literature.
Keywords: Imperfect repairs, two-dimensional warranty, warranty servicing cost.
______________________________________________________________________
1. Introduction
W
arranty is a contract under which the warrantor/manufacturer agrees to provide some
sort of servicing to the item (in terms of repair or replacement), if it fails to meet
customer's requirement within a predetermined time period (called the warranty period).
Such servicing of failed items often cost the manufacturer considerably affecting his profit
margin. Thus different strategies are defined to reduce the total expected cost of the
manufacturer. Study related to planning and optimization of such servicing strategies is a
topic of great interest and is done by many researchers.
Blischke [3] was the first review paper on warranties and it dealt with mathematical
models for warranty cost analysis. The three-part review paper (Product Warranty
Management-I, II, III; Blischke and Murthy [4], Murthy and Blischke [13-14]) proposed a
taxonomy for new product warranties and discussed various issues. Murthy and Djamaludin
[15] review the literature over the period 1990 to 2002. Biedenweg [1] showed that the optimal
strategy is to replace with a new item at any failure occurring up to a certain time measured
from the initial purchase and then repair all other failures that occur during the remainder of
the warranty period. This technique of splitting the warranty period into distinct intervals for
replacement and repair is also used by Nguyen and Murthy [18], in which item failures
occurring in the middle interval of the warranty period are rectified using a stock of used items.
Nguyen and Murthy [19] extended Biedenweg's [1] model by adding a third interval where
failed items are either replaced or repaired and a new warranty is given at each failure. The
first warranty servicing model involving minimal repair and assuming constant repair and
replacement costs is that of Nguyen [19]. In a later paper, and with the same assumptions as
in Nguyen [19], Jack and Van der Duyn Schouten [9] investigated the structure of the
manufacturer's optimal servicing strategy over a warranty period [0, W ], using a dynamic
programming model. The optimal strategy of Jack and Van der Duyn Schouten [9] yields the
smallest expected warranty servicing cost, but the computation of the control limit policy can
24
Banerjee and Bhattacharjee
be tedious and involve considerable computational effort. The strategy also requires
continuous monitoring of the item's age by the manufacturer which is not very practical. In
the same setup Jack et al. [10] proposed a new servicing strategy that involves splitting the
warranty period [0, W ] into three distinct intervals for carrying out repairs and replacements.
In this strategy, only one replacement is allowed and there is no need to monitor the item’s
age.
On the other hand, a two-dimensional (2-D) warranty is characterized by a region in a
two-dimensional plane, with the axes representing an item's age and accumulated total usage.
Different shapes of the warranty region have been proposed (see Blischke and Murthy [4-5]
and Singpurwalla and Wilson [20-21]), the most common and easily implemented being a
rectangular one. The expected warranty costs for a variety of policies can be found in
Moskowitz and Chun ([12] and [6], Chapter 13), Singpurwalla and Wilson [21], Blischke and
Murthy ([5], Chapter 8), Murthy et al. [16] and Chun and Tang [8]. Jack et al. [11] considered
the three distinct intervals strategy of [10] and extended the servicing strategy from 1-D to the
2-D case.
In this article, a new 2-D warranty servicing strategy (Section 3.1) is proposed in the
spirit of Jack et al. [11]. We demonstrate the modeling, analysis and a numerical illustration
(Section 3.2-3.4) followed by conclusions and future works (Section 4).
2. Usage Rate Based Servicing Strategies
In 2-D warranty, since the item degrades due to both age and usage rate, failures need to
be modeled accordingly. Here we assume the usage rate Y is constant for a given customer.
Thus conditional on Y  y, the total usage u of an item at age x is u  yx , 0  u  .
This assumption is valid mainly for expensive items (e.g., commercial air-craft, which
usually have a fixed schedule and thus a known usage rate) whose replacement is very costly
and the warrantor has to keep track of the usage intensity of every sold item. It is possible
since the number of consumers of such items are limited compared to other goods and could
be monitored through some institution (e.g., airline company). Also unlike other consumer
durable goods, the usage rates on these items have a significant effect on length of the
warranty period and the corresponding servicing strategy and thus cannot be ignored.
2.1. Modeling Failures
An appropriate distribution of failure time conditional on y is considered to model
item failures, with corresponding conditional hazard rate denoted by h ( x ; y ).
2.1.1. Modeling First Failure
We use an ‘Accelerated Failure Time (AFT) model’ ([7, 17]) to describe the impact of a
given usage rate y on the item's time to failure. If y0 ( y, respectively) represent the nominal
(typical, respectively) usage rate with corresponding time to failure T0 ( T y , respectively);
then the standard AFT model postulates,
Ty
T0
(
y0 
),
y
where   1 is the acceleration parameter. If F (;  0 ) with a scale parameter  0 denote the
baseline cdf of T0 , then the accelerated failure time T y has cdf F (;  ( y )) with scale
parameter given by
Analysis of a Two-dimensional Warranty Servicing Strategy
 ( y)  (
25
y0 
) 0 ,
y
and conditional hazard rate h(;  ( y )). Note  ( y0 )   0 .
2.1.2. Modeling Subsequent Failures
For a repairable product, if the subsequent failures are rectified via minimal repairs, then
the number of failures over time occur according to a non-homogeneous Poisson process with
intensity function same as the hazard function h( x ;  ( y )) for time to first failure [2].
We further assume (i) h( x ;  ( y )) is a co-ordinatewise non-decreasing function of age x
and usage rate y, (ii) no preventive maintenance is carried out on the item during the
warranty period, (iii) all item failures are detected immediately and result in immediate
claims by the consumer, (iv) all claims are valid and must be rectified by the manufacturer
immediately, (v) repair and replacement times are small relative to the mean time between
item failures and therefore can be ignored.
2.1.3. Warranty Policy and Coverage
Consider a repairable item sold with a 2-D non-renewing free replacement warranty of
period W and maximum usage limit U . Then the 2-D warranty region is the rectangle
[0, W )  [0, U ). Fixed y, the usage sensitive warranty expires when the item currently in use
reaches an age W y  min(W , U / y ).
3. Servicing Strategies for 2-D Warranties
The 2-D strategy of Jack et al. [11] is described via three disjoint intervals
[0, K y ), [ K y , L y ) and [ L y , W y ) with 0  K y  L y  W y , along the age (time) scale where
failures in [0, K y ) undergo only minimal repairs; the first failure (if any) in [ K y , L y ),
rectified by a replacement and all subsequent failures therein, and in [ L y , W y ) are repaired
minimally. For a given y, the optimal K y and Ly minimize the expected warranty
servicing cost. As y varies, the set of points ( K y , Ly ) defines a closed region defined as the
‘Gamma’ region (see Figure 1 [11]).
In the 1-D repair/replacement warranty, Yun et al. [22] have investigated the impact of
allowing ‘imperfect repair’ as a mode of rectifying the first failure in the middle interval
[ K y , L y ) to restore the unit to a working condition. The degree of ‘imperfect repair’ is
described by a parameter   [0, 1] with   0 (1, respectively) being equivalent to minimal
repair (replacement, respectively). This strategy assumes that it is possible to restore a failed
item with any chosen degree ( ) of repair.
3.1. A New Servicing Strategy for 2-D Warranties
We propose and investigate a new strategy that extend Jack et al.’s [11] strategy by
allowing an ‘imperfect repairs’ to the first failure in the middle interval [ K y , L y ). Our
warranty servicing strategy is formally defined as follows.
Partition the warranty period [0, W y ] into three intervals [0, K y ), [ K y , L y ) and
[ L y , W y ], with 0  K y  L y  W y . The first failure, if any, in the middle interval is rectified
with an ‘imperfect repair’. All other failures are rectified by minimal repairs.
The justification for using a replacement in Jack et al. [11] rather than the less costly
minimal repair for the first failure in the middle interval when the failed unit is neither too old
nor too young rests on the possible trade-off between higher cost of replacement vs. improved
26
Banerjee and Bhattacharjee
degradation profile (as a consequence of replacement by a new unit) leading to possibly less
failures and hence an overall reduction in the total expected servicing costs.
Our motivation in allowing an ‘imperfect repair’ in the spirit of Yun et al. [22] is the same,
and additionally provides not just one but a spectrum of ‘degree of repair’ options with
corresponding cost of rectification monotonically increasing as the degree of repair is closer
to replacement. The corresponding minimal cost obtained by optimizing the choice of
partitioning interval end points, also achieves a further reduction compared to Jack et al. [11]
as shown in Table 1(b) (Section 2.4.3).
Table 1(a). Optimal servicing strategies and expected servicing costs.
Strategy 1
Strategy 2
y
Wy
K y
Ly
J ( K y , Ly ,  y ( x ))
K y
Ly
 y
J ( K y , Ly ,  y ( x ))
0.1000
0.3000
0.5000
0.7000
0.9000
1.0000
1.2000
1.4000
1.6000
1.8000
2.0000
2.5000
3.0000
3.5000
4.0000
4.5000
5.0000
2.0000
2.0000
2.0000
2.0000
2.0000
2.0000
1.6600
1.4200
1.2500
1.1100
1.0000
0.8000
0.6700
0.5700
0.5000
0.4400
0.4000
1.9004
1.9000
1.8000
1.4301
0.8048
0.6360
0.5900
0.5329
0.5029
0.4730
0.4470
0.3777
0.3199
0.3045
0.2717
0.2437
0.2068
1.9000
1.9000
1.8000
1.9860
1.9770
1.768
1.4502
1.1929
0.9951
0.9080
0.7719
0.5542
0.4130
0.3420
0.3206
0.2650
0.2467
0.0004
0.0324
0.2500
0.9602
2.1880
3.1950
4.0250
5.1022
6.1899
7.6135
8.7756
13.2400
18.7374
21.6691
33.2484
35.8719
60.1243
1.2500
0.2990
0.2080
0.2380
0.5400
0.6470
0.6057
0.5690
0.4790
0.4790
0.4430
0.3690
0.3150
0.2750
0.2430
0.2190
0.1980
1.3100
1.8900
1.9200
1.9200
1.8800
1.8470
1.5300
1.3900
1.0120
1.0100
0.8740
0.6490
0.4910
0.4190
0.3420
0.2609
0.2423
0.0360
0.1440
0.2840
0.4520
0.6560
0.7670
0.8762
0.9780
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.0004
0.0324
0.2460
0.8870
2.1361
3.0130
4.0210
5.1010
6.1290
7.3985
8.5730
11.4020
13.8050
15.4690
16.0890
15.3930
13.0792
Table 1(b). Comparison of costs with respect to Jack et al. (2009).
y
0.9000
1.0000
1.2000
1.4000
1.6000
1.8000
2.0000
2.5000
3.0000
3.5000
4.0000
4.5000
5.0000
Strategy 1
2.1880(11.11)
3.1950(1.12)
4.0250(1.78)
5.1022(0.02)
6.1899(1.42)
7.6135(0.03)
8.7756(3.77)
13.2400(4.05)
18.7374(5.32)
21.6691(21.69)
33.2484(11.21)
35.8719(25.67)
60.1243(02.51)
Strategy 2
2.1361(13.22)
3.0130(6.75)
4.0210(1.88)
5.1010(0.04)
6.1290(2.39)
7.3985(2.85)
8.5730(5.99)
11.4020(17.37)
13.8050(30.24)
15.4690(44.10)
16.0890(57.03)
15.3930(68.10)
13.0792(78.79)
Jack et al. (2009)
2.4614
3.2312
4.0980
5.1032
6.2790
7.6157
9.1197
13.7987
19.7906
27.6721
37.4444
48.2577
61.6739
Analysis of a Two-dimensional Warranty Servicing Strategy
27
It may be noted that if K y is set to zero in Jack et al. [11], then the first failure after sale of
the item is rectified with a replacement. This will reduce the three interval strategy to two
intervals and is relevant for items which fail sooner due to some stress (like extreme
operating/atmospheric condition). Our strategy confirms the same i.e.,
y    K y  0   y ( x )  1.
This implies that for product functioning under heavy usage (extreme operating
conditions), it is nearly optimal to rectify the first failure after sale by a replacement, since a
minimal/imperfect repair in this scenario will not result in the minimal cost.
3.2. Model Formulation
Notationally we follow Yun et al. [22] and Jack et al. [11], although ours can be followed
independently. In the spirit of the latter, let K y and Ly denote the optimal values of K y
and L y respectively which minimizes the overall expected cost of our proposed servicing
strategy.
Let C r (C 0 ) denote the cost of minimal (perfect) repair and C i ( y ( x ), x ) (  C r ) denote
the cost of imperfect repair conditioned on y. Here, the chosen degree of  y ( x )  [0, 1] also
represents the conditional proportional reduction factor in the hazard rate after failure at age
x imperfectly repaired. We consider two different strategies: (i)  y ( x ) depends on both age
( x ) and usage rate ( y ) and (ii)  y ( x )(   y ) depends on usage rate ( y ) only.
Unlike minimal repairs, an imperfect repair improves the item’s operating condition and
the hazard rate of item lifetime after a repair is smaller. Thus given y, if the failure occurs at
age x the conditional hazard rate just before failure is h( x ;  ( y )) and after repair it is
given by h( x ;  ( y ))  h( x ;  ( y ))   y ( x )(h( x ;  ( y ))  h (0;  ( y ))), where  y ( x )  [0, 1].
Thus  y ( x ) is a decision variable with higher value indicating greater improvement in the
reliability after repair.
3.3. Analysis of Servicing Strategy
Conditional on y, we proceed to derive the total expected servicing cost, by examining
the costs of rectification incurred over the constituent intervals and the imperfect repair
option defining our strategy.
3.3.1. Conditional Expected Warranty Cost
All failures over [0, K y ) are minimally repaired, so the expected cost is
Ky
C r  h( x ;  ( y ))dx .
0
Let T1 denote the time of the first failure under usage rate y after age K y . i.e.,
T1 ~ F1 (t ;  ( y ))  [ F (t ;  ( y ))  F ( K y ;  ( y ))]/ F ( K y ;  ( y )). For failures occurring after age
K y , we need to consider two cases:
(1) K y  T1  L y and
(2) T1  L y .
Case 1 ([T1  L y )) : Since there is no failure in [ K y , L y ), and failures in [ L y , W y ] occur
according to a NHPP, the corresponding contribution to expected costs is
Wy
C r  h ( x ;  ( y ))dx .
Ly
28
Banerjee and Bhattacharjee
Case 2 ([ K y  T1  L y )) : If the first failure in [ K y , L y ) is observed at T1  t ; then there are
no failures in [ K y , t ), the failure at t is ‘imperfectly repaired’ choosing ‘degree of
repair’   y (t ). All subsequent failures in [t , W y ] are minimally repaired. Accordingly
conditional on T1  t and usage rate y  0; failures over [t , W y ] occur as a NHPP with
intensity function
y ( x )  h( x ;  ( y ))   y (t )(h (t ;  ( y ))  h(0;  ( y ))), t  x  W y ,
where the second term captures the constant reduction in the item's hazard rate as a
consequence of imperfect repair at t . The corresponding contribution to expected servicing
cost is
Wy
C r  [h ( x ; ( y ))   y (t )(h(t ; ( y ))  h(0; ( y )))]dx .
t
Let K y J  J ( K y , L y ,  y ( K y , L y )) be the total cost of our policy where  y ( K y , L y )
 { y ( x ) : K y  x  L y } is the set of ‘degree of repairs’ chosen and hence
Ly
EJ  E ( J |T1  t  L y ) F1 ( L y ;  ( y ))   E ( J | K y  T1  t  L y ) f 1 (t ;  ( y ))dt ,
Ky
(1)
  ( K y , L y )   ( ( K y , L y ), K y , L y ),
where

F ( L y ;  ( y ))
 ( K y , L y )  C r  H ( K y ;  ( y ))  [ H (W y ;  ( y ))  H ( L y ;  ( y ))]

F ( K y ;  ( y ))

Ly
  [ H (W y ;  ( y ))  H (t ;  ( y ))]
Ky

f (t ;  ( y ))
dt  ,
F ( K y ;  ( y )) 
Ly
 (  ( K y , L y ), K y , L y )   [C i ( y (t ), t )  C i  y (t ){h(t ;  ( y ))  h(0;  ( y ))}(W y  t )]
Ky
f (t ;  ( y ))

dt .
F ( K y ;  ( y ))
(2)
3.3.2. Optimization Problem
Strategy 1: The optimization problem
min
K y , Ly ,  y ( K y , Ly )
EJ 
min
{ ( K y , L y )   (  ( K y , L y ), K y , L y )}.
K y , Ly ,  y ( K y , Ly )
involves optimally selecting the parameters K y , L y and the set of functions
 y ( K y , L y )  { y ( x ) : K y  x  L y }, subject to the constraints 0  K y  L y  W y and
0   y ( x )  1. Let K y and Ly denote the optimal values. We obtain this using a two-stage
approach. In stage 1, for a fixed K y and L y , we obtain the optimal y ( K y , L y ) that
minimizes J ( K y , L y ,  y ( K y , L y )). Then, in stage 2, we obtain an optimal ( K y , Ly ) by
minimizing J ( K y , L y , y ( K y , L y )).
Analysis of a Two-dimensional Warranty Servicing Strategy
Stage 1: To determine
y ( K y ,
29
L y ) we need to focus on  (  y ( K y , L y ), K y , L y ) given by (2)
and this can be rewritten as
Ly
 (  y ( K y , L y ), K y , L y )   [C i ( y ( x ), x )   y ( x ) y ( x )]
Ky
f ( x ;  ( y ))
dx
F ( K y ;  ( y ))
(3)
where  y ( x )  C r {h( x ;  ( y ))  h (0;  ( y ))}(W y  x ), K y  x  L y .
Assume the baseline survival distribution F0 of the product's lifetime is such that  y ( x )
is concave in the item's age x . This postulate is satisfied by many parametric lifetime models
that are increasingly degrading with age. In particular, the following is a sufficient condition
for such concavity.
Proposition 1: If h( x ;  0 ) is increasing and concave (i.e., baseline survival time T0 with df
F (;  0 ) is IFR with concave hazard rate), implies g ( x )  {h( x ;  0 )  h(0;  0 )}(W  x ) is
concave in {0  x  W }.
Assuming h( x ;  0 ) is twice differentiable, it can be seen that g ( x ) is concave in
{0  x  W }. If h( x ;  0 ) does not exist, then the proof follows from the general definition of
concavity. Note: Since F ( x ;  ( y ))  F (( y / y0 ) x ;  0 )  h ( x ;  ( y ))  ( y / y0 ) h (( y / y0 ) x
;  0 ) and  y ( x )  C r ( y / y0 ) {h(( y / y0 ) x ;  0 )  h (0;  0 )}(W y  x ) is also concave in
x  [0, W y ].
We need to determine the optimal form for  y ( x ) for every point x along the time axis.
The optimal  y ( x ) must result in [Ci ( y ( x ), x )   y ( x ) y ( x )] being a minimum for each
x  [ K y , L y ]. As result,  y ( x ) can be obtained by examining:
v ( z y , x )  [Ci ( z y , x )   y ( x ) z y ],
for each x  [ K y , L y ]. For a fixed x , C i ( z y , x ) is an increasing function of z y .  y ( x ) z y ,
the second term in v ( z y , x ), is linear in z y and so is a straight line when plotted as a function
of z y . We need to consider the following two cases.
Case 1: If the line  y ( x ) z y lies below the curve C i ( z y , x ),  y ( x )  0. This is because the cost
of any imperfect repair with  y ( x )  0 is not worth the reduction in the expected warranty
servicing cost when compared with only minimal repair  y ( x )  0.
Case 2: If the straight line  y ( x ) z y and the curve Ci ( z y , x ) intersect, we have  y ( x )  0.
Since we know 0   y ( x )  1 therefore either  y ( x )  1 (the boundary solution) or
0   y ( x )  1 (an interior point solution). In the latter case, the optimal value is obtained from
the first order condition. This yields  y ( x )  z y for a given y with z y given by
 Ci ( z y , x )
  y ( x ).
 zy
Let the straight line  z y be a tangent to the curve Ci ( z y , x ) at z y  z.  and z are
obtained by solving the simultaneous equations given below:
C i ( z, x )  kz and
 C i ( z y, x )
z y  z  k.
zy
where  y ( x ) is a concave function with  y (0)  0 and  y (W y )  0. Define  y ( max )
 max 0 x W y  ( x ).
30
Banerjee and Bhattacharjee
Proposition 2: If  y (max)   then  y ( x )  0 for all x . If  y (max)   then  y ( x )  0 for
0   1 y  x   2 y  W y where  1y and  2 y are the solutions of the equation  y ( x )   . For
x outside the interval [ 1 y ,  2 y ),  y ( x )  0.
Stage 2: Let y ( K y , L y )  { y ( x ) : 0  x  W y } which is obtained from Stage 1. K y and Ly ,
the optimal values for K y and L y , are obtained by solving the following minimization
problem
min J ( K y , L y , y ( K y , L y ))  min{ ( K y , L y )   ( y ( K y , L y ), K y , L y )}.
K y , Ly
K y , Ly
subject to the constraint 0  K y  L y  W y . We have used a grid-search approach to find the
optimal values of K y and Ly .
Strategy 2: The optimization problem is same as strategy 1, except here for each y, the
optimal  y will be a function of K y and L y .
3.3.3. Special Case: Weibull Failure Distribution
Let the time to first failure under the nominal usage rate y0 denoted by T0 follow a
Weibull distribution with scale parameter  0  0 and shape parameter   1, i.e.,
F ( x ;  0 )  1  F ( x ;  0 )  exp(  x /  0 )  . We use the corresponding AFT model survival
function F ( x ;  ( y )) as specified in (Section 2.1.1 to derive the special forms of equations (1)
and (2), and compute the optimal values K y , Ly with the corresponding minimal cost
J ( K y , Ly , p ).
Strategy 1: The cost of imperfect repair for usage rate y is given by the expression
C i ( z y )  C r  (C 0  C r ) z yp , ( p  1), where z y  [0, 1] is the proportional reduction factor,
with a greater value indicating a greater improvement in the reliability of the item after
repair. Then maximizing the concave function

 1

 y  x
(W y  x ),


 
 y 0   0 
 y ( x )  C r{h( x ;  ( y ))  h (0;  ( y ))}(W y  x )  C r  
We get,
  1
 C r ( y / y0 )    1 
 y  max    y  
W
y


  
 0


  

 1
W y (  0).
Computations following Section 3.3.2. in this case yields
1

p


Cr
Cr
z  
 and k  (C 0  C  )p  (  )( p  1) 


(
)(
1)
p
 C0 Cr

 C0 Cr

p 1
p
,
and  1 y ,  2 y (for each y ) as solutions to the equation:

 1

 y  x


Cr

C r      (W y  x )  (C 0  C r ) p 

 (C 0  C r )( p  1) 
 y 0   0 
The optimum  y ( x ) for strategy 1 is
 p 1 
p 

 1
 C r 
 

 y   x

 y ( x )  
     (W y  x ) 
 (C 0  C r ) p   y 0    0 

p 1
p
 0.
Analysis of a Two-dimensional Warranty Servicing Strategy
31
(provided  y (max)   ) which is a concave function, indicating that if failure occurs too early
(or, too late) in the warranty period when the item is very new (or, very old), an imperfect
repair is not worth the cost and minimal repair is the optimal strategy. Note  y ( x ) does not
depend on the values of K y and L y .
Strategy 2: Here the cost function reduces to Ci ( y )  C r  (C 0  C r ) yp . The first order
condition

 ( , K y , L y )  0

gives
1
2 ( K y, L y )

 p 1
 
 ,
 (C 0  C r ) p1 ( K y, L y ) 

y
L
where 1 ( K y, L y )  K yy f ( x ,  ( y ))dx and
2 ( K y, L y )  C r [{h ( x ;  ( y ))  h(0;  ( y ))}(W y  x )] f ( x ;  ( y ))dt .
Unlike strategy 1, here the optimum reduction proportion  y depends on K y and
L y . As mentioned previously, the values K y and Ly are obtained using a computational
method.
3.4. Numerical Example
The following example is considered from Jack et al. [11], where C r  1, C 0  2, W  2
(years), U  2(10000 miles),  0  1,   2, y0  1,   2 and p  4. The optimal values of
parameters and the corresponding minimal cost is shown in Table 1(a). In Table 1(b), the
figures in brackets are percentage cost savings relative to the strategy of Jack et al. [11].
Qualitative interpretation of results: (i) Strategy 2 is more cost-effective compared to
Strategy 1, since maintaining a setup that can execute any degree of repair  y ( x ),
K y  x  L y is more expensive compared to the fixed  y case. (ii) But Strategy 1 is more
consumer friendly in the sense that the degree of repair being dependant on age has a greater
appeal to the customer and signals higher reliability of the item after repair. (iii) Finally, for
Strategy 2, it can be seen that when y is large enough ( 1.6), the optimal  y is 1
(equivalent to replacement), since any repair of degree less than 1 will not result in the
minimization of the total warranty cost.
4. Conclusion and Future Work
Our proposed servicing strategy extends the work of Jack et al. [11] by introducing at
most one imperfect repair in the middle interval. Since a replacement is costlier than a repair;
the manufacturer/warranty provider has a natural incentive to do repairs rather than a
replacement. However practising an imperfect repair in the middle interval with reduce the
expected warranty cost without completely trading-off the reliability of the item. A possible
future work is to study the reliability of the item at the end of the warranty period. Secondly,
for life-distributions other than IFR Weibull, it might be interesting to study the cost behavior
under the three interval setup.
Acknowledgements
The authors thank the reviewers for the constructive comments on the earlier version of
the manuscript.
32
Banerjee and Bhattacharjee
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Analysis of a Two-dimensional Warranty Servicing Strategy
33
Authors’ Biographies:
Rudrani Banerjee received the M.Sc. degree in Statistics from Visva-Bharati University, India.
As a doctoral student in Statistics; she is currently finishing her Ph.D. degree, in
Mathematical Sciences, jointly offered by the New Jersey Institute of Technology and Rutgers
University - Newark. Her current research interests include warranty modeling, reliability
and Bayesian analysis. A related paper appeared in Advanced Reliability Modeling IV, Edited
by S. Chukova, J. Haywood and T. Dohi, McGraw Hill, Taiwan. She is now a Senior
Biometrician with NOVARTIS, in Hyderabad, India.
Manish C. Bhattacharjee received his Ph.D. in Statistics from University of California at
Berkeley, USA. He is currently a Professor in the Department of Mathematical Sciences,
New Jersey Institute of Technology, USA. His current research interests include applied
probability, stochastic modeling, reliability, repairable systems, inference of aging models.
His research has been published in various international journals including the Journal of
Applied Probability, Journal of Statistical Planning and Inference, Naval Research Logistics,
and has also appeared in various research monographs including the IMS (Institute of
Mathematical Statistics) Collections.