Two-dimensional replacement-repair warranty policy
M.RABBANI, A.A.ZAMANI
Department of Industrial Engineering
Faculty of Engineering, University of Tehran
Tehran, IRAN
Abstract: In this paper, a method is developed to minimize the expected warranty servicing cost per item sold.
In this method, to be decided to repair or replace a failed item. These kinds of items work in multi-state
deteriorating and they may fail in every state. Zuo, Liu and Murthy (2000), presented a paper to minimize the
expected warranty servicing cost under one-dimensional (Time) and nonrenewable warranty policy. But here,
the warranty policy has two-dimensions (Time & Usage). The decision to repair or replace a failed item
depends on two parameters, the deterioration degree of the item and the length of the residual warranty period.
The optimality of these two parameters for minimizing the expected warranty servicing cost are examined by
two methods, analytically and simulation. Then an offered algorithm is used as an approximating method for
performing the warranty policy.
Key-Words: Warranty, Fail, Repair, Replace, Maintenance
for calculation of the expected value and variance of
the manufacturer’s total warranty costs under the
combined warranty policy and weibull lifetime
distributions. Chun and Tang [8] determine the
warranty price for the FRW policy assuming
constant failure rate and constant repair cost through
out the warranty period. In these papers, they have
been considered only two situations for an item,
working state and failure state. They don’t model
multi-state equipment or items due to natural
deterioration.
Derman, Lieberman and Ross [9] study the
optimal replacement problem of a component where
there are n types of replacements available differing
only in price and the failure rates of exponential life
distributions. Zuo, Liu and Murthy [10] present a
model for a class of multi-state deteriorating with N
steps and repairable products. In all steps, the items
had exponential lifetime distribution with differing
failure rates. They present a replacement-repair
policy to minimize the expected warranty servicing
cost under FRW with fixed period of T. fig.1 shows
all the possible states of the item and the possible
transition among the states.
In this paper, we study the model of fig.1 in two
dimensions, Time and Usage. The goal of this paper
is determination of optimal replacement-repair
policy for minimizing the expected warranty
servicing cost.
1 Introduction
A warranty is a contractual obligation incurred by a
manufacturer (vendor or seller) in connection with
the sale of product. The warranty normally specifies
that the manufacturer agrees to replace or repair a
failed item with in a predetermined warranty period
[1]. The most commonly used warranty policies
include the free replacement warranty (FRW), the
pro-rata warranty, and the combined warranty [2].
The total expected cost to the manufacturer in order
to determine the warranty policy is very important
during the warranty period. The manufacturer can
determine the suitable warranty policy in order to
minimize the expected warranty cost. It depends on
characteristics of product and usage conditions.
Nguyen and Murthy [3] present models for
evaluation of expected warranty cost and the
variance of the warranty cost considering general
product lifetime distribution and time depended
repair cost. In another paper [4] for the FRW with a
fixed warranty period T, they study the warranty
servicing policy to minimize the expected warranty
servicing cost. When an item fails, the decision that
the manufacturer has to make is that, replace a failed
item by a new one or by a repaired item from the
stock of repaired items. Nguyen and Murthy [5]
examine the combined FRW with fixed and renewal
periods of T and W. Rao [6] develops an algorithm
for evaluating the cost to the manufacturer for
products with the phase-type lifetime distributions
under FRW. Bohoris and Yun [7] provide equations
1
Fig.1: State transition diagram of an item.
2.2 Assumptions
2 Description of model
(A1) (1-p1)μ1<(1-p2)μ2<...<(1-pN-1)μN-1<μN.
(A2) cm1  cm2  ...  cmN.
(A3) cr1<cr2<...<crN.
2.1 Notations
N
T
U
k
θ
cri
cmi
μi
pi
1-pi
yt
Z
u
CW
CW1
CW2
CW3
number of working states = number of
failure states,
length of time for warranty period,
amount of usage for warranty period,
a decision variable (1  k  N-1),
a decision variable (0  θ  U),
cost of repairing given that the item is fixed
in state i (i=1,2,…,N),
cost of replacing given that the item is fixed
in state i (i=1,2,…,N),
rate of usage exponential distribution in
state i (i=1,2,…,N),
probability that the item enters working
state i+1 given that it has made a transition
out of working state i (i=1,2,…,N-1),
probability that the item enters failure state i
given that it has made a transition out of
working state i (i=1,2,…,N-1),
a random variable that indicates the
customer usage during t times,
the usage along the warranty period
(Z=min{U,yt}),
the amount of residual usage to the end of
the warranty period when an item goes on
failure state (u=Z- the amount of usage item
up to it goes on failure state),
the expected warranty servicing cost per
item sold,
the expected warranty servicing cost per
item sold when yT  U,
the expected warranty servicing cost per
item sold when yT<θ,
the expected warranty servicing cost per
item sold when θ  yT<U.
(A1) states that as the item deteriorates, it becomes
more likely to make a transition to a failure state.
Assumptions (A2) and (A3) imply that the item
becomes more costly to replace and repair as the
deterioration increases [10].
2.3 Multi-state FRW policy in two
dimensions and nonrenewable warranty
period
This policy will be used for a failed item when the
usage of item is less than U and the length of
expired time from beginning of the warranty period
is less than T. Otherwise, the warranty period is
expired. In this policy, the manufacturer has to
repair or replace the failed item. The replacementrepair policy for minimization of the expected
warranty servicing cost will be defined as follow:
If the failed item is in failure state i, then it is
replaced by a new one if and only if k+1  i  N and
θ  u; otherwise, it is repaired.
However, the problem is the determination of
optimal k and θ for minimizing the CW.
3 Special case (N=2)
According to the policy is presented in the last
section, k will be equal to 1 if there are two working
states and two failure states.
2
CW  e (   / T )U CW1  (1  e (  / T ) )CW2 
3.1 yt with exponential distribution

f ( yT )  e (   / T ) yT
T
P( yT  U )  e (  / T )U
P( yT   )  1  e (  / T )
P(  yT  U )  e (  / T )  e (  / T )U
(e (   / T )  e (  / T )U )CW3
(1)
(12)
For optimal policy, if the differentiating CW given
by Eq. (12) with respect to θ is calculated, the
following cases will be needed to consider.
Case 1: When
(2)
(3)
(4)
(5)
cr2  2  cr1 (1  p1 ) 1
(1  e  p11T ) ,
p1 1
we have dCW / d  0 for all  ( U ) . This
f ( yT yT   )  ( e (   / T ) yT ) /(1  e (   / T ) ) (6)
T
F ( yT   yT  U )  (1  e (   / T ) yT ) /( e (   / T )
implies that CW is a non-increasing function in θ
over interval [0,U]. Therefore the optimal θ is
U(θ*=U).
Case 2: When
F ( yT yT   )  (1  e
Then:
(   / T ) yT
) /(1  e
(   / T )
)
cm2  cr2 

(7)
 e (   / T )U )
cr2  2  cr1 (1  p1 ) 1
(1  e  p11T ) ,
p1 1
by solving dCW / d  0 we have

p11 (cm2  cr2 ) 
1
*  
ln 1 
 and
p11  cr2  2  cr1 (1  p1 ) 1 
cm2  cr2 
Then:

f ( yT   yT  U )  ( e (   / T ) yT ) /( e (   / T )
T
(   / T )U
e
)
(8)
According to [10], Eqs. (9), (10) and (11) would be
concluded:
dCW  0, if   *

d  0, if   *
CW1  [cr1 (1  p1 ) 1[  2 (U   )  ( p1 1
(1  e (  ( 2  p11 )(U  )) ) /(  2  p1 1 )  (1  e  p11 )
Now we can summarize this as follows:
(  2  p1 1e (  ( 2  p11 )(U  )) ) / p1 1 ] /(  2 
p1 1 )  cr2  2 [(  2  p1 1 )  (1  e  p11 )
If
(9)
(  2  p1 1e (  ( 2  p11 )(U  )) ) / p1 1 ] /(  2 
e (  ( 2  p11 )(U  )) ) /(  2  p1 1 )) /(  2  p1 1 )]
is equal to 
From Eq. (6):

 [( / T  e
(   / T ) yT
cr2  2  cr1 (1  p1 ) 1
(1  e  p11T ) ,
p1 1
then optimal policy is to replace the item with a new
one if it is in failure state 2 and the remaining
warranty period is longer than θ* Є (0,U), where θ*
p1 1 )  cm2 p1 1  2 (U    (1 
CW2 
cm2  cr2 
) /(1  e (   / T ) )] 

p11 (cm2  cr2 ) 
ln 1 
,
p11  cr2  2  cr1 (1  p1 ) 1 
1
and to repair the item otherwise.
yT  0
[cr2  2 yT  [cr1 (1  p1 )  cr2  2 / 1 ]
(1  e
 p11 yT
(10)
3.2 yt with uniform distribution
1
f ( yT ) 
aT  yT  bT
bT  aT
bT  U
P( yT  U ) 
bT  aT
  aT
P ( yT   ) 
bT  aT
U  aT
P(  yT  U ) 
bT  aT
y  aT
F ( yT yT   )  T
  aT
) / p1 ]  dyT
From Eq. (8):
U
CW3 
 [(  / T  e
(   / T ) yT
) /( e (  / T )  e (  / T )U )].
yT 
[cr1 (1  p1 ) 1 [  2 ( yT   )  ( p1 1 (1 
e
(  (  2  p11 )( yT  ))
) /(  2  p1 1 )  (1  e
 p11
)(  2 
p1 1e (  (  2  p11 )( yT  )) ) / p1 1 ] /(  2  p1 1 ) 
(11)
cr2  2 [(  2  p1 1 )  (1  e  p11 )(  2 
p1 1e
(  (  2  p11 )( yT  ))
) / p1 1 ] /(  2  p1 1 ) 
(13)
(14)
(15)
(16)
(17)
Then:
cm2 p1 1  2 ( yT    (1  e (  (  2  p11 )( yT  )) )
f ( yT yT   ) 
/(  2  p1 1 )) /(  2  p1 1 )]  dyT
From Eqs. (2), (3), (4), (9), (10) and (11), CW is
calculated in the following form:
3
1
  aT
(18)
F ( yT   yT  U ) 
yT  aT
U 
(19)
1
U 
(20)
From Eq. (18):
EyT yT  U  
Then:
f ( yT   yT  U ) 
 
E  *  * 
After calculation of CW, θ could have been
obtained just like as section 3.1.
With considering of 3.1 and 3.2, the manufacturer
will replace the failed item when k=1 and θ*  u.
Otherwise, it will be repaired. Since yT is not
determinable for a failed item, so the usage along
the warranty period (Z) is unknown and u is not
calculable.
If u΄ define as the residual warranty period from
U (u΄=U- The amount of usage item up to it goes on
failure state), then always it will calculable. u΄ can
be written in the following form:
yT  U
yT  U
yT  U
4 General case (N > 2)
In general case that N  3, could be assigned more
than one value to k. In this situation, there is no
analytical solution for determination of optimal k
and β. By using computerize simulation method
(fig.2), CW can be calculated with all combinations
of k and β. Then considering to minimum value of
CW, optimal value of k and β will be determined.
One of advantage of using the simulation method
is ability to termination of some warranty model
assumptions. For example, every kind of
distribution function could be used for the lifetime
or the customer usage during t times.
(21)
(22)
3.3.1 yT with exponential distribution like Eq. (1).
E    e (   / T )U  (  U  E yT yT  U )
(1  e (   / T )U )
(23)
4.1 Distribution function of yT is known
From Eq. (6):
(U  T  Te ( / T )U )
EyT yT  U  
(  e (  / T )U )
Example 1: If μi is rate of exponential distribution
function and yt~Normal(2t,t), N=4, μ1=0.5, μ2=2,
μ3=3, μ4=3.5, p1=0.9, p2=p3=0.6, cr1=40, cr2=50,
cr3=300, cr4=400, cm1=300, cm2=500, cm3=600,
cm4=800, U=3 and
T=2, then the result of
simulation is shown in table 1.
The model is simulated for three times by k=1, 2
and 3. The minimum value of CW is 397 with k*=2
and β*=0.34.
(24)
From Eqs. (23) and (24):
 
E  *   *  U  T (1  e (  / T )U ) / 
(25)
3.3.2 yT with uniform distribution like Eq. (13).
bT  U
)  (  U 
bT  aT
U  aT
E yT yT  U )(
)
bT  aT
(28)
When an item fails and its warranty period has not
been expired yet, if k=1 and u΄>E[β*], then the
failed item will be replaced with new one.
Otherwise, it will be repaired.
From Eq. (21), β is defined in Eq. (22).


 
  U  yT
(U  aT ) 2
2(bT  aT )
According to Eqs. (25) and (28), the replacementrepair policy will be performed in the following
form:
3.3 Comment
yT  U
(27)
From Eqs. (26) and (27):
*
u

u'  
u  U  yT
U 2  a 2T 2
2(U  aT )
E     (
(26)
4
Fig.2: Visual SLAM simulation model for calculation of CW.
Table 1: Results for simulation of example 1.
k
Min ( CW )
β
1
411
0.53
2
397
0.34
3
433
0.16
expired yet, is suggested to perform the following
algorithm for a failed item.
4.2 Distribution function of yT is unknown
In order to minimizing the expected warranty
servicing cost when the warranty period is not
5
Fig.3: Visual SLAM simulation model for determination of the algorithm precision that is presented in section 4.2.
Step 5: Calculate v  u '(U  V )(1  V / U ) . If
Algorithm:
Step 1: Solve the problem in one dimension with U
as a period of warranty. Then get k* and β* [10].
Step 2: Repair the failed item if i  k*+1. Otherwise,
go to next step.
Step 3: Calculate V  T
v   * , then repair the failed item. Otherwise,
repair it.
Example 2: Consider the example 1 where the
distribution of yT is not known. The failure item has
these conditions: (u΄=2, t=1, i=3)
1. Step 1: k *  2,  *  0.39
2. Step 2: 3  2  1
3. Step 3: V  2  V  U
U  u'
. If U  V , go to
T t
step 4. Otherwise, go to step 5.
Step 4: If u '   * , then replace the failed item.
Otherwise, repair it.
*
4. Step 5: v  1.6  v   Replace the failed
item.
6
Example 3: Consider the example 2. The failure
item has these conditions: (u΄=2, t=1.6, i=3)
1. Step 1: k *  2,  *  0.39
2. Step 2: 3  2  1
3. Step 3: V  5  V  U
(6) Presenting more efficient algorithms
performing the warranty policy.
for
References:
[1] Murthy D.N.P, Blischke W.R., Warranty
management I: A taxonomy for warranty
policies, European Journal of Operational
Research, Vol.62, 1992, pp. 127-148.
[2] Murthy D.N.P, Blischke W.R., Warranty
management III: A review of mathematical
models, European Journal of Operational
Research, Vol.62, 1992, pp. 1-34.
[3] Neguyen D.G, Murthy D.N.P., A general model
for estimating warranty costs for repairable item,
IIE Transactions, Vol.16, 1984, pp. 379-386.
[4] Neguyen D.G, Murthy D.N.P., An optimal
policy for servicing warranty, Journal of
Operational Research Society, Vol.11, 1986, pp.
1081-1088.
[5] Neguyen D.G, Murthy D.N.P., Optimal replacerepair strategy for servicing items sold under
warranty, European Journal of Operational
Research, Vol.39, 1989, pp. 206-212.
[6] Rao B.M., Algorithms for the free replacement
warranty with phase-type lifetime distributions,
IIE Transactions, Vol.27, 1995, pp. 348-357.
[7] Bohoris G.A, Yun W., Warranty costs for
repairable products under hybrid warranty,
IMA Journal of Mathematics Applied in
Business and Industry, Vol.6, 1995, pp. 13-24.
[8] Chun Y.H, Tang K., Determining the optimal
warranty price based on the producer’s and
customer’s risk preferences, European Journal of
Operational Research, Vol.85, 1995, pp. 97-110.
[9] Derman C, Lieberman G.J, Ross S.M., A
renewal decision problem, Management
Science, Vol.24, 1978, pp. 554-561.
[10] Zuo M.J, Liu B, Murthy D.N.P., Replacementrepair policy for multi-state deteriorating
products under warranty, European Journal of
Operational Research, Vol.123, 2000, pp. 519530.
[11] Ross S.M., Introduction to probability and
statistics for engineers and scientists, New
York: John Wiley, 1987.
[12] Ross S.M., Stochastic Processes, New York:
John Wiley, 1983.
[13] Banks J, Carson J.S, Nelson B., Discrete-Event
system simulation, New Jersey: Prentice-Hall,
1996.
[14] Pritsker A.A.B, O’Reilly J.J, La Val D.K.,
Simulation with Visual SLAM and AWESIM,
New York: John Wiley, 1997.
4. Step 4: v  2  v   * Replace the failed
item.
The precision measure of this algorithm can be
calculated by simulation method for a known
distribution function of yT. If yT has a uniform
distribution function, the expected warranty cost of
the algorithm performing will be calculated by
simulation model that is shown in fig.3.
The value of expected warranty cost is gotten
from this method can be compared with the
expected warranty cost when the distribution
function of yT is known (section 4.1).
By a little change on the simulation model, could
be measured the algorithm efficiency for other
distribution function of yT.
5 Conclusion
In this paper has been presented a warranty policy
for those items that have N working states and N
failure states. In this policy has been presented a
method for decision to repair or replace the failed
item, in order to minimizing the expected warranty
servicing cost. This method is developing of Zuo,
Liu and Murthy (2000) [10] method with two
dimensions. In this method, if N=2, decision
variables (k*,β*) have been calculated analytically.
Then the problem has been solved in general case
(N  3) by using computerize simulation method. At
last, it has been presented the practical algorithm for
performing the warranty policy.
The model reported in this paper can be extended
in several ways. Some of the many issues that need
further study are listed bellow.
(1) Determination of β* for other distribution
function of yt, when N=2.
(2) Changing the deterioration of an item from
discretely to continuously and developing the
analytical and simulation methods.
(3) Changing the simulation model when an item
goes from working state j to working state i for
i=j+1,j+2,…,N.
(4) Using the other warranty policies.
(5) Releasing the independent and constant repair
cost assumption so that repair cost is a function
of the failed state.
7