paper aip 2012
TRANSCRIPT
7/30/2019 Paper AIP 2012
http://slidepdf.com/reader/full/paper-aip-2012 1/7
Renewal process in two-dimensional repairable productsUdjianna S. Pasaribu, Hennie Husniah, and Bermawi P. Iskandar Citation: AIP Conf. Proc. 1450, 37 (2012); doi: 10.1063/1.4724115 View online: http://dx.doi.org/10.1063/1.4724115 View Table of Contents: http://proceedings.aip.org/dbt/dbt.jsp?KEY=APCPCS&Volume=1450&Issue=1 Published by the American Institute of Physics. Related Articles
Estimating statistical distributions using an integral identity J. Chem. Phys. 136, 204113 (2012) Electronic coherence dynamics in trans-polyacetylene oligomers J. Chem. Phys. 136, 144501 (2012) Electronic coherence dynamics in trans-polyacetylene oligomers JCP: BioChem. Phys. 6, 04B607 (2012) Central limit theorem for fluctuations of linear eigenvalue statistics of large random graphs: Diluted regime J. Math. Phys. 53, 043501 (2012) Note: A method to isolate and detect a large number of single molecules by microdroplet fluorescencespectroscopy Rev. Sci. Instrum. 83, 036107 (2012) Additional information on AIP Conf. Proc.
Journal Homepage: http://proceedings.aip.org/ Journal Information: http://proceedings.aip.org/about/about_the_proceedings Top downloads: http://proceedings.aip.org/dbt/most_downloaded.jsp?KEY=APCPCS Information for Authors: http://proceedings.aip.org/authors/information_for_authors
Downloaded 04 Jun 2012 to 180.246.31.163. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions
7/30/2019 Paper AIP 2012
http://slidepdf.com/reader/full/paper-aip-2012 2/7
Renewal Process in Two-Dimensional Repairable Products
Udjianna S. Pasaribu∗, Hennie Husniah†,∗∗ and Bermawi P. Iskandar†
∗ Mathematics Department, Faculty of Mathematics and Natural Sciences Institut Teknologi Bandung, Bandung 40132, Indonesia.
† Industrial Engineering Department, Faculty of Industrial Technology Institut Teknologi Bandung, Bandung 40132, Indonesia.
∗∗ Industrial Engineering Department, Langlangbuana University, Bandung 40132, [email protected], [email protected], [email protected]
Abstract. One alternative to prolong a life time of a product is doing a preventive maintenance. The number of failures fromthe product can be reduced by optimal maintenance policy. This research investigates a periodic maintenance policy for aproduct sold with a two-dimensional warranty. Actually, this idea is a development of that in Murthy and Wilson (1991). Theproduct failures is modeled using a one-dimensional approach in which usage is a function of age. Furthermore, it is assumedthat the relation between these variables is linear. The slope of the line is the usage rate. The optimal periodic maintenance timecan be derived by minimizing the expected cost per unit time. We also proved that the expected cost per unit time is a concavefunction, and gave the global optimal solution. As a numerical example, we choose Weibull distribution with different levelsof usage rates. The numerical results show that the increase of usage rate will decrease the length of periodic maintenance
time.Keywords: periodic replacement policies, expected cost per unit time, global optimal solutionPACS: 02.50.Ey
1. INTRODUCTION
Most of all working systems deteriorate with age or/and
usage over time due to usage or age. This deterioration
could have a negative effect on the performance of the
systems. Preventive maintenance is an effective way to
control the deterioration. Several maintenance actions
to reduce the effect of this deterioration on the perfor-
mance of the system are inspection, repair, and/or re-
placement. Generally, actions can be classified as a cor-
rective maintenance or a preventive maintenance. The
corrective maintenance policy includes all maintenance
actions performed as a result of system failure in order to
restore the system into a specified condition. In a mean-
while, a preventive maintenance policy consists of any
maintenance activities which retain the system into its in-
tended function. In 2001, [1] estimated the cost of main-
tenance and support is 60 to 75% of the total life-cycle
cost of a system. Therefore, reducing the cost and low-
ering the risk of a catastrophic failure becomes a main
issue.
Almost all durable products are sold with warranty,
which protects the buyers with a certain level against fail-
ures during warranty period. Recently the warranty pe-
riod is getting longer. e.g automobiles is sold with 3 to 7
years and electronic appliances with 3 to 5 years. Hence,
maintenance policy for the products should consider the
existence of this warranty in order to obtain a prudent
decision. Maintenance policies following the expiry of
a warranty have been studied by [2], [3]. [4] consider
pre- and post-warranty maintenance policy. But these re-
searches deal with the case where a product is sold with
a one-dimensional warranty.
This paper deals with a periodic replacement pol-
icy (as a part of preventive maintenance policy) for a
repairable product sold with a two-dimensional non-
renewing failure replacement warranty (NFRW). For ex-
ample, in automotive industry, a dump truck is warranted
for 36 months or 30000 miles, whichever occurs first. In
NFRW, all failures under warranty are rectified by the
manufacturer at no cost to the buyers. When the prod-
uct fails under warranty, buyers only incurs the cost due
to unable to use the product while the failed product
is restored. After the warranty expires, all maintenance
costs such as the costs of each repair and replacement
are borne by the buyer.
Literatures on maintenance policies for one dimen-
sional warranty, e.g. age, can be studied in [5], [6] and
[7]. Furthermore, for two dimensional warranty, i.e. age
and usage, can be studied in [8].
This paper extends the work of [4] to the case of a two-
dimensional warranty, which considers conditions before
and after the expiry of warranty. The outline of this pa-
per is organized as follow. Section 2 discusses the model
formulation and the warranty policy. Furthermore it as-
sumes that the periodic replacement policy depends on
a given usage rate and is characterized by one parame-
ter. Section 3 and 4 deal with the analysis of the optimal
The 5th International Conference on Research and Education in Mathematics
AIP Conf. Proc. 1450, 37-42 (2012); doi: 10.1063/1.4724115© 2012 American Institute of Physics 978-0-7354-1049-7/$30.00
37
Downloaded 04 Jun 2012 to 180.246.31.163. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions
7/30/2019 Paper AIP 2012
http://slidepdf.com/reader/full/paper-aip-2012 3/7
replacement policy and numerical examples for the case
where the product has a Weibull failure distribution re-
spectively. Finally, a brief discussion for future research
is presented in Section 5.
2. MODELS FORMULATION
2.1. Modelling Failure
Two approaches can be modelled failures for products
sold with two-dimensional warranties, i.e. a one (condi-
tional) dimensional approach and a two-dimensional ap-
proach. For the first approach, product failures can be
studied in [9] or [10]. Moreover, for the second approach,
product failures is modelled as a two-dimensional ran-
dom points occurring over the maintenance region [11].
Here, we adopt the one-dimensional approach from [9].
Let U , X , and Y are the usage random variable, the
age random variable, and the usage rate random variable
respectively. Let the density function of the usage rate
Y is g( y), 0 ≤ y < ∞, e.g. in automobile Y can be the
annual distance travelled. Practically Y varies across the
customer population but it is constant for a given con-
sumer. Here, conditional on Y = y, the total usage U tot at
age X is assumed by the linear model
U tot = yX (1)
Let the conditional hazard (failure rate) function for
the time to the first failure is given by h( x| y) ≥ 0, which
is a non-decreasing function of the product age x and
product usage rate y. In fact, failures process over time
is a one-dimensional counting process. If failed products
are replaced by new ones, then this counting process is
a renewal process associated with the conditional distri-bution F ( x| y) which can be derived from h( x| y). If failed
products are repaired then the counting process is charac-
terized by a conditional intensity function λ ( x| y) which
is a non-decreasing function of x and y. Moreover, if all
repairs are minimal [12] and repair times are negligible,
then λ ( x| y) = h( x| y). [13] use a concept of the acceler-
ated failure time and proportional hazards models [see
[14] and [15]] to express the effect of usage rate on reli-
ability. Conditional on the usage rate, the time to the first
failure has distribution function
F ( x| y) = F 0
xyλ
(2)
where F 0( x) is the base failure distribution function.
Failures over time follow a non-homogeneous Poisson
process with intensity function
λ ( x| y) = yλ λ 0
xyλ −1
(3)
where λ 0 ( x) is the base intensity function.
2.2. Modelling First Failure
For a repairable product sold with a two-dimensional
warranty, one needs to model the product’s degradation
which takes into account both age and usage. [8] have in-
troduced a more appropriate model which uses the accel-
erated failure time (AFT) model, to represent the effect
of usage rate on degradation. Let y0 denotes the nominal
usage rate value associated with component reliability.
When the actual usage rate is different from this nominal
value, the component reliability can be affected and this
in turn affects the product reliability. As the usage rate
increases above the nominal value, the rate of degrada-
tion increases and this, in turn, accelerates the time to
failure. Consequently, the product reliability decreases
[increases] as the usage rate increases [decreases]. Us-
ing the AFT formulation, if X 0 denotes the time to first
failure under usage rate y0 then we have
X
X 0=
y0
y γ
(4)
Furthermore, if the distribution function for X 0 is given
by F 0( x;α 0) where α 0 is the scale parameter, then the
distribution function for X is the same as that for X 0 but
with a scale parameter given by
α ( y) =
y0
y
γ
α 0 (5)
with γ ≥ 1. Hence, we have
F ( x;α ( y)) = F 0
( y0/ y)γ x;α 0
(6)
The hazard and the cumulative hazard functions associ-
ated with F ( x,α ( y)) are given by
h ( x,α ( y)) = f ( x,α ( y))/(1−F ( x;α ( y)) (7)
and
H ( x,α ( y)) = x
0h x,α ( y)
dx (8)
respectively, where f ( x,α ( y)) is the associated density
function.
2.3. Warranty Policy
The product is sold with a two-dimensional non-
renewing failure replacement warranty (NFRW) together
with a rectangle warranty region ΩW = [0,W )× [0,U ).Here W is the warranty expiry time and U is the usage
limit. With NFRW, all failures under warranty are recti-
fied at no cost to the buyer. It is assumed that the rectifi-
cation is done through a minimal repair and the repaired
product comes with the original warranty. The warranty
38
Downloaded 04 Jun 2012 to 180.246.31.163. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions
7/30/2019 Paper AIP 2012
http://slidepdf.com/reader/full/paper-aip-2012 4/7
FIGURE 1. a). Warranty region and b).maintenance region Γ
ceases at the first instance when the age of the product
reaches W or its usage reaches U , whichever occurs first.
The two-dimensional warranty region is given by
W y =
W for y ≤U /W ,
U / y for y >U /W .(9)
3. PERIODIC REPLACEMENT POLICY
τ y FOR A PRODUCTS SOLD WITH
TWO-DIMENSIONAL WARRANTY
A periodic replacement policy for a given usage rate y
will be considered here and also the policy for various
values of usage rate. Let τ y denotes the parameter of a
periodic replacement policy for that rate. The periodic
replacement policy is defined as follows.
For a given usage rate y, the product is repaired with
minimal repair when it fails at age x, with x < τ y, and it
is replaced with a new one if its age reaches τ y. We seek
the optimal value of τ y which minimizes the long-runaverage cost to the buyer. If τ ∗ y denotes the optimal
value then, as y varies, we have a set of points (τ ∗ y , yτ ∗
y )defining a curve as indicated in Figure 2. Let Γ denotes
a closed region bounded by this curve. The replacement
policy for a various values of y is defined as follows.
For a given usage rate y, the product is repaired with
minimal repair when it fails at age x, with x < τ y , and it
is replaced with a new one if its age reaches τ y.
Let τ ∗ y denotes the optimal value of τ y which mini-
mizes the expected cost time per unit to the buyer. The
problem is to find a pair of points (τ ∗ y , yτ ∗
y ) defining a
curve as indicated in Figure 2. Let Γ denotes a closed re-gion bounded by this curve. The replacement policy for
a various values of y can be seen as follows.
The product is repaired with minimal repair when
failure occurs in region Γ and it is replaced with a new
one when its age at τ y . Furthermore, the expected cost
per unit time, denotes as J (τ y), is obtained as follows.
J (τ y) =E [Cost per cycle]
E [Cycle length](10)
From above equation, the expected cycle length for
this replacement policy is τ y. Since all failures, either
occur within (0,W y) or (W y,τ y), are rectified with min-
imal repairs, then they will follow a non-homogeneous
Poisson process in the interval (0,τ y) with the inten-
sity function λ y( x) = h( x;α ( y)). For a product with two-
dimensional NFRW, the consecutive times between re-
placement form a failure process with a renewal cycle.
Product replacement may be carried out either after or
prior the warranty ceases, i.e. when τ y ≥W y or τ y <W y.
Next the replacement time both for τ y ≥W y and τ y <W ywill be modelled.
Let C r , C m (with C m > C r ) and C d the cost of each
repair, the cost of a replacement and the cost incurred
by a buyer for any minimal repair done respectively.
When τ y ≥W y, the expected cost per unit time for a two-
dimensional warranted product is
C d
W y
0λ y( x)dx +C r + (C d +C m)
τ yW y
λ y( x)dx (11)
And hence the expected cost per cycle is given by
J (τ y) =C d R(W y) +C r + (C d +C m)[ R (τ y)− R (W y)]
τ y(12)
with R(W y) = W y
0 λ y( x)dx.
On the other hand, when τ y < W y the expected cost
per unit time for a two-dimensional warranted product is
given by
C d
τ y
0λ y( x)dx +C r (13)
with the expected cost per unit time
J (τ y) =C d
τ y0 λ y( x)dx +C r
τ y(14)
39
Downloaded 04 Jun 2012 to 180.246.31.163. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions
7/30/2019 Paper AIP 2012
http://slidepdf.com/reader/full/paper-aip-2012 5/7
In the case τ y = W y, the expected cost per unit time
in equations (12) is equivalent to that in (14). In other
words, the optimal solution of τ y time can be derived for
each case.
3.1. Local Optimal SolutionHere, for a warranted product, optimal strategies τ ∗ y
are found by separating two cases: after the expire of the
warranty τ y ≥ W y and prior the expiry of the warranty
τ y <W y.
The investigation for the local optimal solution will be
done by differentiating the cost function J i(τ y), i = 1,2
where condition 1 and 2 represent τ y ≥W y and τ y < W yand set the resulting derivative to zero. Following the
proof in [4]; Lemma 1, Theorems 1, and 2 are derived
for each condition.
Lemma 1. Let λ y(τ y) is an Increasing Failure Rate
(IFR), then L(τ y) is a non-negative increasing function
of τ y with limτ y→0 L(τ y) = 0 and limτ y→∞ L(τ y) > 0.
Proof. Note that is λ y(τ y) IFR, then λ y(τ y) > 0. This
means that
dL(τ y)
d τ y= λ (τ y) +τ yλ
y(τ y)−λ y(τ y) = τ yλ
y(τ y) > 0
Then we have
limτ y→0
L(τ y) = limτ y→0
L(0) = 0
and
limτ y→∞
L(τ y) = limτ y→∞
L(∞) > 0
Theorem 1. Let λ y(τ y) is an IFR, is an IFR, then for
a product sold with two-dimensional NFRW policy, the
optimal replacement strategy after the expiry of the war-
ranty, τ y ∈ [W y,∞) is given by:
1. τ ∗ y = ∞ , whenever L(∞) ≤C r −C m R(W y
(C m+C d )
2. a finite and unique τ ∗ y > W y satisfying
L(τ ∗ y ) =C r −C m R(W y
(C m+C d )with the expected cost per
unit is given by J (τ ∗ y ) =C r +C m[ R(τ y)− R(W y)]+C d R(τ y)
τ y
whenever L(W y) <C r −C m R(W y
(C m+C d )< L(∞).
Proof. If L(∞) ≤C r −C m R(W y
(C m+C d )then
dJ (τ y)d τ y
< 0. This means
that J (τ y) is a decreasing function with the respect to
τ y in the interval [W y,∞), and consequently τ ∗ y = ∞. On
the other hand, if L(W y) <C r −C m R(W y
(C m+C d )< L(∞), then from
Lemma 1. L(τ y) is a non-negative increasing function
with the respect to τ y. Consequently, there exist a finite
and unique τ ∗ y , satisfyingdJ (τ y)
d τ y= 0, such that L(τ ∗ y ) =
C r −C m R(W y)(C m+C d )
. The resulting expected cost per unit time is
given by J (τ ∗ y ) =C r +C m[ R(τ y)− R(W y)]+C d R(τ y)
τ y.
Theorem 2. Let λ y(τ y) is an IFR, then for a product
sold with two-dimensional NFRW warranty policy, the
optimal replacement strategy prior the expiry of the war-
ranty, τ y ∈ [0,W y) is given by:
1. τ ∗ y = W y , whenever L(W y) ≤ C r C d
2. a finite and unique τ ∗ y < W y satisfying L(τ ∗ y ) = C r C d
with the expected cost per unit is given by J (τ ∗ y ) =C d R(τ ∗ y )+C r
τ ∗ ywhenever L(W y) > C r
C d .
Proof. Since L(τ y) is a non-negative increasing function
in τ y thendJ (τ y)
d τ y≤ 0 for τ y ∈ [0,W y) whenever L(W y) ≤
C r C d
. Hence, J (τ y) is a decreasing function in τ y within the
interval [0,W y), which means that τ ∗ y = W y with J (τ ∗ y ) =
J (W y) =C d R(W y+C r
W y. On the other hand, if L(W y) > C r
C d ,
then there exist a finite and unique τ ∗ y < W y, satisfyingdJ (τ y)
d τ y= 0, such that L(τ ∗ y ) = C r
C d . The associated expected
cost per unit time is given by J (τ ∗ y ) =C d R(τ ∗ y +C r
τ ∗ y.
3.2. Global Optimal Solutions
In practice, some experts obey the separation of re-
placement time in τ y ≥ W and τ y > W , and the appro-priate strategy is a global optimal solution. If τ ∗G y
is the
global optimal solution, with the expected cost per unit
time J (τ ∗G y), then it can be obtained by combining the
local optimal solution τ ∗ y in each case. The following
Corollary is a direct consequences of Theorems 1 and
2 related to τ ∗G yand J (τ ∗G y
).
Corollary 1. Assume that a product sold with two-
dimensional NFRW warranty policy and the hazard rate
for the product, h y( x) , which follows an IFR for x∈ (0,τ y]with C r , C d > 0 , then the global optimal replacement so-
lution is given by:
1. τ ∗
G y
= τ ∗
y
> W y with the expected cost per unit time
is given by (12) whenever L(W y) <C r −C m R(W y)
(C m+C d ).
2. τ ∗G y= τ ∗ y ≤W y with the expected cost per unit time is
given by (14) whenever C r −C m R(W y)
(C m+C d )≤ L(W y) ≤ C r
C d .
40
Downloaded 04 Jun 2012 to 180.246.31.163. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions
7/30/2019 Paper AIP 2012
http://slidepdf.com/reader/full/paper-aip-2012 6/7
TABLE 1. The optimal replacement solution, τ ∗G ywith C r = 5, C d = C m = 1, α 0 = 1, y0 = 2 for various usage rates y.
y τ ∗G y J (τ ∗G y
) y τ ∗G y J (τ ∗G y
) y τ ∗G y J (τ ∗G y
)
0.60 9.518014110 1.027945524 1.5 3.442651863 2.904737510 8.0 0.2795084972 35.777087640.65 8.415852999 1.155601815 2.0 2.236067977 4.472135955 8.5 0.2552123033 39.183063950.70 7.503934828 1.286924823 2.5 1.600000000 6.250000000 9.0 0.2342427896 42.690748420.75 6.738501867 1.421402737 3.0 1.217161239 8.215838362 9.5 0.2159954425 46.29727314
0.80 6.087897831 1.558501845 3.5 0.965890576 10.35313962 10.0 0.2000000000 50.000000000.85 5.528712317 1.697660226 4.0 0.790569415 12.64911064 10.5 0.1858857282 53.796491520.90 5.043296764 1.838281670 4.5 0.662538660 15.09345885 11.0 0.1733568344 57.684486650.95 4.618118188 1.979729541 5.0 0.565685424 17.67766953 11.5 0.1621747493 61.661880431.00 4.242640687 2.121320344∗ 5.5 0.490327172 20.394545841.05 3.908548576 2.262316773 6.0 0.430331482 23.237900081.10 3.609196035 2.401919961 6.5 0.381645337 26.202337681.15 3.339210183 2.539260644 7.0 0.341493888 29.283100931.20 3.094200070 2.673388861 7.0 0.341493888 29.283100931.25 2.870540019 2.803261737 7.5 0.307920143 32.47595264
∗ y ≤ 1
Remark 1. If the cost of down time, C d , is relatively
smaller than the replacement cost C r , then the corollary
reveals that optimal replacement period would be longer than the warranty period. In contrast, if that cost is
relatively greater than the cost of replacement, then it
is optimal to replace the product before the warranty
ceases. Logically, this is acceptable to avoid unnecessary
additional cost of down time. In this case, if L(W y) > C r C d
then an early replacement before the warranty expires
also optimal.
Remark 2. Beside L y( x) , the quantity h y( x) also influ-
ences the optimal replacement period. This is due to the
monotonic property of h y( x) , which in turn cause the
monotonic property of L y( x).
Remark 3. The model discussed in [4] is nested in the
current model, that is when y = y0. In this case, if x ≥W then (12) reduces to equation (4) in [4] , meanwhile if
x <W, then (14) collapses to equation (6) in [4].
4. NUMERICAL EXAMPLE
A numerical example is presented to illustrate the prop-
erties of the optimal replacement discussed in the pre-
vious section. Assume that the time to the first fail-
ure under usage rate, X 0, follows a Weibull distribution
with the conditional distribution function F 0( x;α 0) =1 − exp(− x/α 0)β . The conditional failure distribution
function for a given usage rate y is given by F ( x;α ( y)) =
1−exp(− x/α ( y))β with α ( y) as in (5). The hazard func-tion associated with F ( x;α ( y)) is given by F ( x;α ( y)) =
β
y y0
γβ xβ −1
α β 0
.
Moreover, since all repairs are minimal, then λ y( x) =h( x;α ( y)) with cumulative hazard function R(W y) =
W β
y / [α 0( y0/ y)γ ]β . The parameter values chosen are: (1)
warranty policy: W = 2 (year) and U = 2×104 km,
so U /W = 1; (2) reliability design: (×104 km per year),α 0 = 1 (year) and β = 2; (3) AFT parameter: γ = 1.5; (4)
costs: C m = 1, C r = 5, and C d = 1. Table 1. shows the op-
timal solution τ ∗G yfor warranty area ΩW = [0,2)× [0,2).
Figure 2 and 3 show the visual solutions of the optimal
replacement period τ ∗G yfor various usage rates.
Tables 1 shows that
1. Numerical examples conform with the properties
predicted by Theorem 1 and 2 in Section 3.1 and the
corollary 1 in Section 3.2, e.g. the increase of usage
rate shortening the length of periodic replacement
interval τ ∗G y. This is plausible since a product with a
high usage rate would be replaced more often than
that with a lower usage rate.2. The increase of down time cost C d widening the
length of periodic replacement interval τ ∗G y.
FIGURE 2. Maintenance region Γ for y ≤ 1.
41
Downloaded 04 Jun 2012 to 180.246.31.163. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions
7/30/2019 Paper AIP 2012
http://slidepdf.com/reader/full/paper-aip-2012 7/7
FIGURE 3. Maintenance region Γ for y > 1.
3. The cost of down time, C d , is relatively smaller than
the replacement cost C r , then the corollary reveals
that optimal replacement period would be longer
than the warranty period (Remark 1). However, if
the cost of down time is relatively greater than the
cost of replacement, then it is optimal to replace theproduct before the warranty ceases.
5. CONCLUSION
In this paper, a periodic replacement policy for a prod-
uct with a two-dimensional warranty has been studied
particularly for product with Non Renewing Warranty
using one dimensional approach. One can study other
two-dimensional replacement policies such an age re-
placement policy or an opportunity-based age replace-
ment [16] incorporates with a two-dimensional approach
where product failures is modelled by two-dimensional
random points occurring over the maintenance region[11]. These topics are currently under investigation.
REFERENCES
1. S. C. Yang, and J. A. Nachlas, “Bivariate Reliability andMaintenance Planning Models”, IEEE Transactions on Reliability, 2001, Vol. 50, No.1, pp. 26–35.
2. G. M. Jung, and D. H. Park, “Optimal maintenancepolicies during the post-warranty period”, Reliability
Engineering and System Safety, 1999, 82, pp. 123–185.3. K. M. Jung, S. S. Han, and D. H. Park, “Optimization
of cost and downtime for replacement model followingthe expiration of warranty”, Reliability Engineering and
System Safety, 2008, 93, pp. 995–1003.4. R. H. Yeh, M. Y. Chen, and C. Y. Lin, “Optimal periodicreplacement policy for repairable products under free-repair warranty”, European Journal of Operational Research, 2007, 176, pp. 1678–1686.
5. J. S. Dagpunar, “Hybrid minimal repair and agereplacement maintenance policies with non-negligible
repair time”, Naval Research Logistics, 1994, 41, pp.1029–1037.
6. T. Nakagawa, “Modified Periodic Replacement withminimal Repair at failure”, IEEE Transactions on Reliability, 1981, 30, pp. 165–168.
7. J. E. Goulionis, and V. K. Benos, “A maintenancepolicy under markovian deterioration”, Advances and Applications in Statis tics, 2007, 7(3), pp. 357–388.
8. J. Nat, B. P. Iskandar, and D. N. P. Murthy, “A repair-replace strategy based on usage rate for items sold with atwo-dimensional warranty”, Reliability Engineering and System Safety, 2008.
9. D. N. P. Murthy and R. J. Wilson, “Modelling two-dimensional warranties,” in Proceedings of the Fifth International Symposium on Applied Stochastic Modelsand Data Analysis, Granada, Spain, 1991, pp. 481–492.
10. B. P. Iskandar, R. J. Wilson, and D. N. P. Murthy,“A new repair-replace strategy for items sold with atwo-dimensional warranty”, Computers & Operations Research, 2005, 32, pp 669–682.
11. D. N. P. Murthy, and W. R. Blischke, Warranty Cost Analysis, Marcel Dekker Inc., New York,1993.
12. R. E. Barlow, and L. Hunter, “Optimal preventivemaintenance policies”, Operations Research, 1960, 8, pp.
90–100.13. J. Lawless, J. Hu, and J. Cao, “Methods for estimation of
failure distributions and rates from automobile warrantydata”, Lifetime Data Analysis, 1995, 1, pp. 227–240.
14. J. Lawless, Statistical Models and Methods for Lifetime Data, John Wiley, New York, 1982.
15. W. R. Blischke, and D. N. P. Murthy, Reliability:
Modeling,Prediction, and Optimisation, John Wiley, NewYork, 2000.
16. B. P. Iskandar, and H. Sandoh, “An opportunity-based agereplacement policy considering warranty”, International Journal of Reliability, Quali ty and Safety Engineering,1999, 1(3), pp. 229–236.
42