Stochastic system in discrete preventive maintenance and service contract 2012
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ABSTRACT: In recent years, there has been a dramatic increase in the out sourcing of equipment maintenance with the maintenance being carried out by an external agent rather than inhouse. Often, the agent offers more than one option and the owners of equipment (customers) are faced with the problem of selecting the optimal option. The optimal choice depends on their attitude to risk and the parameters of the different options. The agent needs to take these issues into account in the optimal selection of the parameters for the different options and this requires a game theoretic formulation. The papers deals with one such model formulation to determine the agent's optimal strategy with regards the pricing structure, the number of customers to service and the number of service channels.European Journal of Operational Research 01/1999; · 2.04 Impact Factor  SourceAvailable from: sciencedirect.com[Show abstract] [Hide abstract]
ABSTRACT: In general, a newly purchased item or system is warranted for a specific period. When the system fails during the warranty period, it is repaired free of charge. Even if the system is repairable, there exist some warranty services under which the manufacturer replaces the failed system during the warranty period. This study considers a case where a manufacturer offers an additional warranty service under which the failed system is replaced by a new one for its first failure, but minimal repairs are carried out to the system for its succeeding failures before the warranty expires. In this paper, we propose a mathematical model for setting a suitable charge of such an additional warranty service. Numerical examples assuming a personal computer are also presented.Computers & Mathematics with Applications 01/2006; 51:179188. · 2.07 Impact Factor  SourceAvailable from: Che Soong Kim
Article: Warranty and preventive maintenance
International Journal of Reliability Quality and Safety Engineering 06/2001; 8(2):89107.
Page 1
978 1 4673 1023 9/ 12/ $31. 00 ©2012 IEEE
U. S. Pasaribu1, a, H. Husniah 2 , B. P. Iskandar 3
1Department of Mathematics, Bandung Institute of Technology, Bandung, Indonesia 40132
2Graduate Program at Department of Industrial Engineering, Bandung Institute of Technology, Bandung, Indonesia 40132
3Department of Industrial Engineering, Bandung Institute of Technology, Bandung, Indonesia 40132
(audjianna@math.itb.ac.id )
Abstract— This paper deals with a stochastic system and
maintenance service contract. We consider a situation where an
agent offers more then one service contract option and a
company as the owner has to select the optimal option. This case
is typically found in a mining industry where the original
equipment manufacturer (OEM) is the only maintenance service
provider. As repair time is related to the revenue of the company,
service contract options need to consider the repair time target.
We consider the repair time target from both the owner and
OEM point of views. As an illustration, we give a numerical
example with weibull failure distribution
Keywords Maintenance, service contract, noncooperative
game theory
I.
INTRODUCTION
In mining industry, an availability of heavy equipments e.g.
dumptruck, excavator, tow truck are critical in achieving the
revenue of the company. The equipment deteriorates with
usage and age and finally fails to operate as intended. As the
result, no revenue is generated. High availability of the
equipment is needed for achieving the revenue of the company.
To get this achievement, preventive maintenance (PM) actions
are performed using age based or conditioned based
maintenance – to reduce the likelihood of failure and down
time. Beside that corrective maintenance (CM) actions are
taken after failure occurs, which restores the failed equipment
to the operational state. The maintenance program is aimed at
not only to sustain the performance (e.g. reliability) of the
equipment according to the intended function but also to obtain
optimum business profitability. In a mining industry,
availability of dump trucks is a key measure, which influences
significantly the revenue of a company.
Many companies do inhouse maintenance service with
limited skilled labors and maintenance facilities, due to
expensive investment and this affects their capability in
performing a preventive maintenance. As a result, performing a
maintenance service inhouse seems to be ineconomical
solution, and as alternative way an outsourcing maintenance
service, either preventive or corrective, could be the better
solution. The benefits of an outsourcing maintenance service
are two folds: to assure the maximum availability and to reduce
the maintenance cost.
Maintenance service contract has received attention in the
literature such as in [1], [2], [3] and [4]. They formulated
decision problems using as a Stackelberg game theory model to
obtain an optimal cost strategy with the agent as a leader and
consumer as the follower. However, they did not consider any
PM action. Further, [5], [6], [7] and [8], involved a PM policy
in the contract. Another researcher such as [9] studied the
contract that considers the periodic inspection and CM. He also
considered three contract options and the optimal strategy for
selecting the options is obtained which maximizes the expected
profit for both the agent and the owner. All those contracts
considered a penalty based on down time for each failure – i.e.
a penalty cost incurs the agent (or OEM) when the actual down
time to fix the failed equipment is greater than the target value.
In this paper, we develop [8] from the manufacturer’s
perspective and the customer’s perspective, which considers
discrete PM action and the penalty. This purposed PM is more
realistic since the company usually does PM in periodic time.
Similar with [8], the penalty is based on the repair time, if it is
lower than the target repair time the OEM incurs the penalty
cost.
The paper is organized as follows. Section II gives the
methodology which includes model formulation and model
analysis. Section III deals with the result of the solution and
numerical examples for the case where the product has a
Weibull failure distribution. The discussion presents in Section
IV. Finally, a brief conclusion and a discussion for future work
are presented in Section V.
II.
METHODOLOGY
Let L denotes as a life time product and let the OEM and
the consumer have two options.
Option
0
O : In the interval[0, )
inhouse PM to undertake a preventive maintenance. If the
product fails, OEM will charge the consumer for the cost of
repair
s
C whenever the failure is repaired by the OEM. There
is no penalty cost to the OEM if the repair time exceeds τ unit
time.
Option
1
O : For a fixed cost of service contract
OEM agrees to repair all the failures with PM and CM along
L , the consumer does an
G
P , the
Stochastic System in Discrete
Preventive Maintenance and Service Contract
Bandung, Indonesia, September 23 26, 2012
2012 IEEE Conference on Control, Systems and Industrial Informatics (ICCSII)
250
Page 2
( )
r x
L
( )
L
o
mr
( )
L
h
mr
Failure rate
the interval [0, )
reach an operational condition by τ unit time after the time it
failed, the OEM should pay a penalty cost (see Table I below).
L , without any cost. If a fail product could not
TABLE I.
OEM OPTIONS
Option in [0,L)
Option O0 Option O1
PM
In House PM Service Contract
CM Service Contract Service Contract
The consumer must choose the option
}
01
,O O . And the OEM has to determine the optimal cost
structure (service contract cost
C for option
0
O . The values of service contract cost
repair cost
m
C , will be formulated through a noncooperative
game theory using Nash equilibrium.
*
O taken from the
set{
G
P for option
1
O and repair cost
mG
P and
A. Model Formulation for Product Failures and PM Policy
We use a blackbox approach to model product failure.
And it will be modelled by its distribution function. We
consider a dump truck as a repairable product and every
failure is rectified by a minimal repair. With a minimal repair,
the failure rate after repair is the same as that before it fails. It
is assumed that the rectification time is relatively small
compared to its mean time between failures, so that it can be
neglected. As a result, the failure occurs as a Non
Homogenous Poisson Process (NHPP) with the intensity
function ( )
r x [10].
The PM policy follows [11] by assuming a PM occurs at
discrete time instants 12
,,...,
τ ττ
PM is that it results in a rejuvenation of the item so that it
effectively reduces the age of the item. The reduction in the
age depends on the maintenance effort ( )
maintenance effort is constrained so that 0
corresponds to no PM and M is the upper limit of
maintenance effort. Larger value of m corresponds to greater
maintenance effort. Let
1
−≥
, PM action. See [12] for the concept of
virtual age. If maintenance effort level m is used, the virtual
age is given by
1
,... and 0
j
τ = . The effect of
m used. The
mM
≤
.
≤
0m =
jv− denote the virtual age of the item
after ( 1) , 2
st
jj
1110
110
111
()before the PM with 0(1 )a
( )(m) after the PM for option (1 )b
( )(m) after the PM for option (1 )c
th
jjj
τ
j
τ
hhth
jjhjj
ooth
jjojj
vvjv
vvvjO
vvjO
ττ
δ
δττ
−−−
−−
−−
=+−=
==+−
=+−
⎧
⎪⎨
⎪⎩
with ( )
δ
function of m with (0)
as the PM effort is increased, the effect of aging is reduced.
With mM
=
at every PM action, then the item is restored
back to as good as new after each PM action. If
,1
jj
vj
τ=≥ .
[0,1],0( )
m
( )
m
δ
1
oh
m
δ
δ
δ∈≤<
1
≤ . ( )
δ
)
M
=
m
. This implies that
is a decreasing
= and (0
0
m = , then
We assume that the level of PM effort used is the same
throughout the life. As a result, with PMm, the item’s virtual
age at time x is given by
1110
1111
,,j=1,2,...for option (2 )
a
( )
v x
,,j=1,2,...for option (2 )
b
h
jjjj
o
jjjj
vxxO
vxxO
τττ
τττ
−−−
−−−
⎧
⎪
+ −≤<
=⎨
⎪
⎩
+ −≤<
Since failure are repaired minimally and repair times are
negligible, the rate of occurrence of failures (ROCOF) is given
by
(
=
⎨
⎪
⎩
)
()
1110
1111
,,j=1,2,...for option (3 )a
[ ( )] r v x
,,j=1,2,...for option (3 )b
h
jjjj
o
jjjj
r vxxO
r vxxO
τττ
τττ
−−−
−−−
+−≤<
+−≤<
⎧
⎪
In other words, failures over time occur according to a Non
stationary Poisson Process with intensity function is given by
the failure rate for the virtual age.
Let
L for option
ROCOF functions with PM effort m given in (4) and (5).
Their ROCOF are defined as follow:
01
,
n n denote the number of PM actions over
O and
1
O respectively. We consider two
[0, )
0
000
0
r v
11
),
1
0
0
(),
( )x 1,..., (4)
(
h
jjjj
h
m
h
nnn
r vxx
rjn
xxL
τ
τ
τ
τ
τ
−−−
+
+
−
−
≤
≤
<
≤
==
⎧⎪⎨
⎪ ⎩
111
0
r v
11
),
1
1
0
( ),
( )x1,..., (5)n
(
o
jjjj
o
m
o
nnn
r vxx
rj
xxL
τ
τ
τ
τ
τ
−−−
+
+
−
−
≤
≤
<
≤
==
⎧⎪⎨
⎪ ⎩
with
,1
j
jj
τ = Δ≥ and
01
,
n n is the largest integer less than
L Δ .
Figure 1. ROCOF for options
10,OO
251
Page 3
B. Maintenance Cost
As in [8], to get high revenue both the OEM and the
customer have to construct maintenance cost for each option.
For option
0
O , we assume that
maintenance cost per unit of time and
repair cost. We denote Cτ as the expected penalty cost of the
OEM if the repair time exceedsτ , and
repair and corrective cost in the interval [0, )
consumer does not buy a service contract, and
pricing of product.
pm
C is the average of
C
is a preventive
m
s
C is the average of
L when the
C denotes the
b
The cost of CM for option
where
m
C is the average cost of each rectification. And for
option
1
O the cost of PM is
C
function of the repair cost Y (r.v with the distribution
function ( )
G y ) and τ . Let
rectify the ith failure and for option
rectified by minimal repair and the repair time is negligible,
the expected number of failure in the interval[0, )
is given by
1
O can be expressed as
m
C ,
1 pm
n . The penalty cost is a
iY denotes the time needed to
O , since failures are
1
L ,
()
10,
NL
( )
x dx
0
L
o
m r
∫
failure in the interval [0, )
∫
()
max 0,
i
=
∑
. And for option
0
O , the expected number of
L
,
()
00,
NL
is given by
( )
x dx
0
L
h
m r
. The penalty cost in the interval [0, )
L is given
by
{}
10,N
1
L
i
CY
τ
τ−
.
1) Customer’s decision problem
We assume that
the operational condition after the ith failure and the expected
number of failures in the interval [0, )
option
1
O is
1
N . Consumer profit upon choosing the option
O is (
,
G
O P Cτ
ω
, and the value is given by
∑
∑
iY denotes the time needed to return to
L whenever choose the
1
)
1;
()
{}
()
1
1
1
1
1
;,
max 0,
N
GibG
i
N
i
i
O P CR L
⎢
⎣
YCP
CY
τ
τ
ω
τ
=
=
⎡⎤
⎥
⎦
=−−−+
+−
(6)
Suppose that the number of failures in the interval [0, )
choosing the option
0
O is
choosing the option
0
O P C
ω
∑
L upon
0
;
N . The consumer profit upon
)
00
,
s
is given by O , (
()
0
0000
1
;,
N
sibs
i
O P C R L
⎢
⎣
YCPC N
ω
=
⎡⎤
⎥
⎦
=−−−−
(7)
2) OEM’s decision problem
OEM revenue for option
0
O is given by:
()()
00,
bsmsm
OCCCN CC
π=+−>
(8)
OEM revenue for option
1
O :
1
G
( )=
P  [Penalty cost]  [CM cost]  [PM cost]
O
π
(9)
In our model, by using Nash bargaining solution for both
options
0
O and
1
O , OEM and consumer will negotiate the
pricing of service contract
G
P and the cost of
m
C , respectively.
C. Model Analysis
We assume that OEM and consumer have the same
attitudes to risk, in order to make the solution reach equilibria.
1) Customer’s expected profit
From (6) then the expected profit of the consumer upon
choosing the
1
O option,
(
EO
ω
⎡
⎣
)
1
⎤
⎦, is given by
()()( )
() ( )
11
0
1
0,1
0,1
bG
EOR L
⎢
⎣
NLG ydy
CPC N
τ
L G ydy
τ
∫
ω
⎡
⎣
∞
∞
⎡⎤
⎥
⎦
⎧
⎨
⎩
⎫
⎬
⎭
=−−−
⎤
⎦
⎡
⎣
⎤
⎦
⎧
⎨
⎩
⎫
⎬
⎭
−+−
⎡
⎣
⎤
⎦
∫
(10)
And from (7), the expected profit of the consumer upon
choosing the option
0
O ,
(
EO
ω
⎡
⎣
)
0
,
⎤
⎦is given by
()
()( )
()
00
0
00
0,1
0,
bs
EOR L
⎢
⎣
NLG ydy
CPC NL
ω
⎡
⎣
∞
⎡⎤
⎥
⎦
⎧
⎨
⎩
⎫
⎬
⎭
⎤ =
⎦
−−−
⎡
⎣
⎤
⎦
−−−
∫
(11)
2) OEM’s expected revenue
The revenue function OEM for option
Since the failure follows the NHPP, then as in [10]
0
O is given in (8).
()
()
()
()
00,
0
0
0,
0,
!
n
NL
NLe
P NLn
n
−
⎡
⎣
⎤
⎦
==
Hence the expected revenue of the OEM whenever the
consumer chose the option
0
O is
given by
()
0
EO
π
⎡
⎣
⎤
⎦, and its value is
()()
()
000,
bsm
EOCCCNL
π
⎡
⎣
⎤ =
⎦
+−
(12)
Where
()
00,NL is given by
()
0
1
00
0
0
∑∫
1
00
1
0,
()() (13)
j
jn
n
L
h
j
h
njn
j
NL
r vxdxr vx dx
τ
ττ
ττ
+
−
=
=
=+−++−
∫
252
Page 4
From (9) then the expected revenue of the OEM,
is given by
(
[
=Penalty cost
G
PEE
−−
()
1
EO
π
⎡
⎣
⎤
⎦,
)
][][]
1
=
CM costPM cost (14)
EO
E
π
⎡
⎣
⎤
⎦
−
The expected number of failures
()
10,NL is given by
()
1
1
11
1
1
100
1
0,()() (15)
j
jn
n
∑∫
L
o
j
h
njn
j
NLr vxdx r vxdx
τ
ττ
ττ
+
−
=
=+−++−
∫
Expected cost of rectification is expressed as
(
mm
ECC NL
=
)
10,
(16)
Expected cost of penalty
()() ( )
g y
τ
−
10,ECC N
τ
Lydy
τ
τ
∫
∞
⎧
⎨
⎩
⎫
⎬
⎭
=
(17)
Using integral by part, we have
() ( )
g y dy
τ−
( )
1yG ydy
τ
∫
τ
∫
∞∞
==−
⎡
⎣
⎤
⎦
(18)
Thus (17) becomes
()( )
10,1 ECC N
τ
LG ydy
τ
τ
∫
∞
=−
⎡
⎣
⎤
⎦
(19)
Expected PM cost is
1
pmpm
ECCn
=
(20)
Then, total expected revenue of the OEM in (14) becomes
()() ( )
)
L
(
11
11
0,1
0,
G
mpm
EOPC N
τ
LG ydy
C NCn
τ
∫
π
⎡
⎣
∞
=−−−
⎤
⎦
⎡
⎣
⎤
⎦
−−
(21)
III.
RESULT
We assume that the consumer uses the product up to L , so
that there would be a negotiation between the consumer and
the manufacturer to determine the value of the service contract
cost,
G
P and the repair cost,C
obtained using the method of Nash equilibrium. In principle
this method can be used whenever there is a negotiation
between the two parties. In the presence of negotiation, for
every option, the consumer and the manufacturer receipt the
same profit if [ ]
EE
ωπ=
. Firstly we will compare the
expected profit of the consumer and manufacturer with the
respect to option
1
O and secondly we obtain
expected revenue of the OEM on option
m
. The optimal solution is
[ ]
*
G
P . Hence the
O becomes
1
()
[]
()( )
()
1
1
0
11
1
2
1
2
[ [ R L 0,{1}]]
[ 0,]
b pmm
EO
NLG ydy
CCn C NL
π
∞
=
−
−−−
⎡
⎣
⎤
⎦
++
∫
(22)
The expected profit of the OEM on option
0
O after we get
*
m
C becomes
()
()( )
()
00
0
00
(0,{1 })
0,.
sb
EOR LNLG ydy
C NL e CP
π
⎡
⎣
∞
⎧
⎨
⎩
⎫
⎬
⎭
⎤ =
⎦
−−−
⎡
⎣
⎤
⎦
−−−
∫
(23)
We consider that the failure distribution follows the Weibull
distribution with 1
α = and
β =
repair time y has a Weibull distribution with
and
0.5
β =
,
1
β <
(decreasing time of repair). And
2
. We also consider that
0.5
α =
( )m
0.04
(1),0
m
m em
η
τ =
at discrete time
δ
−
==
(years), we assume that maintenance is carried out
τ , with
0.33
Δ =
. Table II gives the pricing of
+≥
and integer. For 5L =
(years) and
j
service contract
O respectively. The sensitivity analysis of the maximum
expected revenue is carried out by varying the gradient of
failure rate latter called as degradation PM level from
0.05
η =
to
1.00
η =
. Fig. 2 shows the resulting plot of the
expected revenue [profit] for the OEM [consumer] for
O options based on the results of Table III. Solution of the
numeric is solved using MAPLE 9.5 of the Waterloo.
*
G
P and
*
m
C the repair cost for option
1
O and
0
1
O and
0
TABLE II.
SERVICE CONTRACT AND REPAIR COST
(
G
⎥
8.05 7.17
8.03 7.18
8.01 7.19
7.99 7.21
7.96 7.23
7.94 7.24
7.92 7.26
7.89 7.27
7.87 7.29
7.85 7.30
7.84 7.31
η
*
G
P
)
*
1;,
m
EO P C
ω
⎢
⎣
⎡⎤
⎦
*
m
C
()
*
m
00
;,EO P C
ω
⎢
⎣
⎡⎤
⎥
⎦
0.05
0.15
0.25
0.35
0.45
0.55
0.65
0.75
0.85
0.95
1.00
22.88
22.88
22.88
22.88
22.88
22.88
22.88
22.88
22.88
22.88
22.88
7.25
7.23
7.21
7.18
7.16
7.14
7.15
7.09
7.07
7.05
7.03
The result in Tabel II is given using parameter value as
follow.
Parameter
b
C
R
m
C
s
C
pm
CCτ
0P
Value
(103 $)
5 4 0.1 1 0.05 0.5 2
253
Page 5
Figure 2. Expected OEM revenue for various degradation PM levels based
on Table II
IV. DISCUSSION
Table II presents the decreasing value of expected
revenue [profit] of OEM [consumer] as the degradation PM
level η increases (worsen). In Fig. 2, we show that option
is the optimal option for the consumer. It also shows in Fig. 2,
that option
1
O gives a higher expected value for the OEM
(consumer) compared to option
the degradation preventive maintenance level the greater the
discrepancy between the expected revenues of the different
option.
1
O
0
O . It is clear that the larger
V.
CONCLUSION
In this paper, we have studied a maintenance service
contract for a warranted product with discrite PM. Some
insights are discussed and one can extend to service contract
for product sold with twodimensional warranty. This topic is
currently under investigation.
ACKNOWLEDGMENT
Part of this work is funded by the DGHE, Ministry of
National Education, the Republic of Indonesia through Hibah
Desentralisasi 2012 .
References
[1] D. N. P. Murthy, and E. Ashgarizadeh, “Optimal decision making in a
maintenance service operation,” European Journal of Operational
Research, vol. 62, pp. 1–34, 1999.
[2] E. Ashgarizadeh, and D. N. P. Murthy, “Service contracts,”
Mathematical and Computer Modelling, vol. 31, pp. 11–20, 2000.
[3] K. Rinsaka, and H. Sandoh, “A stochastic model on an additional
warranty service contract,” Computers and Mathematics with
Applications, vol. 51, pp. 179–188, 2006.
[4] D. N. P. Murthy, and E. Ashgarizadeh, “A stochastic model for service
contract,” International Journal of Reliability, Quality and Safety
Engineering, vol. 5, pp. 2945, 1998.
[5] C. Jackson, and R. Pascual, “Optimal maintenance service contract
negotiation with aging equipment,” European Journal of Operational
Research, vol. 189, pp. 387–398, 2008.
[6] I. Djamaludin, D. N. P. Murthy, and C.S. Kim, “Warranty and
preventive maintenance,” International Journal of Reliability, Quality
and Safety Engineering, vol. 8, no. 2, pp. 89–107, 2001.
[7] D. N. P. Murthy, and V. Yeung, “Modelling and analysis of maintenance
service contracts,” Mathematical and Computer Modelling, vol. 22, pp.
219–225, 1995.
[8] H. Husniah, U. S. Pasaribu, A. H. Halim, and B. P. Iskandar, “A
Maintenance service contract for a maintenance product,” IEEE IEEM.
Singapore, vol. 11, pp. 1577–1581, December 2011.
[9] Wang, “A model for maintenance service contract design,negotiation
and optimization,” European Journal of Operational Research,
vol.201(1), pp. 239 – 246, 2010.
[10] R. E. Barlow, and L. Hunter, “Optimum preventive maintenance
policies,” Operational Research, vol. 8, pp. 90100, 1960.
[11] C.S. Kim, I. Djamaludin and D. N. P. Murthy, “Warranty and discrite
preventive maintenance,” Reliability Engineering & System Safety, vol.
84, no. 3, pp. 301– 309, 2004.
[12] M. Kijima, H. Morimura and Y. Suzuki “Periodical replacement
problem without assuming minimal repair,” EJOR, vol. 37, no.2,pp.
194–203, 1988.
η η
Ε[π(Ε[π(Ο Οι ι)] )]
1
O
0
O
254
Supplementary to (1)

Stochastic System in Discrete Preventive Maintenance and Service Contract