2012 IEEE Conference on Control, Systems and Industrial Informatics (ICCSII) Bandung, Indonesia, September 23-26, 2012
Stochastic System in Discrete Preventive Maintenance and Service Contract U. S. Pasaribu1, a, H. Husniah 2 , B. P. Iskandar 3 1
2
Department of Mathematics, Bandung Institute of Technology, Bandung, Indonesia 40132 Graduate Program at Department of Industrial Engineering, Bandung Institute of Technology, Bandung, Indonesia 40132 3 Department of Industrial Engineering, Bandung Institute of Technology, Bandung, Indonesia 40132 (
[email protected] ) 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.
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, non-cooperative game theory
I.
INTRODUCTION
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.
In mining industry, an availability of heavy equipments e.g. dump-truck, 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.
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.
Let L denotes as a life time product and let the OEM and the consumer have two options. Option O0 : In the interval [0, L) , the consumer does an in-house PM to undertake a preventive maintenance. If the product fails, OEM will charge the consumer for the cost of repair Cs 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 O1 : For a fixed cost of service contract PG , the OEM agrees to repair all the failures with PM and CM along
Many companies do in-house 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 in-house seems to be in-economical solution, and as alternative way an out-sourcing maintenance service, either preventive or corrective, could be the better solution. The benefits of an out-sourcing maintenance service are two folds: to assure the maximum availability and to reduce the maintenance cost.
978-1-4673-1023-9/12/$31.00 ©2012 IEEE
METHODOLOGY
250
We assume that the level of PM effort used is the same throughout the life. As a result, with PM m , the item’s virtual age at time x is given by
the interval [0, L) , without any cost. If a fail product could not reach an operational condition by τ unit time after the time it failed, the OEM should pay a penalty cost (see Table I below). TABLE I.
⎧v hj−1 + x − τ j −1 , τ j −1 ≤ x < τ j ,j=1,2,...for option O0 ⎪ v( x ) = ⎨ ⎪v o + x − τ , τ ≤ x < τ ,j=1,2,...for option O 1 j −1 j −1 j ⎩ j −1
OEM OPTIONS Option in [0,L)
Option O0
Option O1
PM
In House PM
Service Contract
CM
Service Contract
Service Contract
⎧r ( v ⎪ r[v ( x )] = ⎨ ⎪r ( v ⎩
structure (service contract cost PG for option O1 and repair cost Cm for option O0 . The values of service contract cost PG and repair cost Cm , will be formulated through a non-cooperative game theory using Nash equilibrium.
h
(1a )
after the j
PM for option O0
(1b )
after the j
th
PM for option O1
(1c )
j −1
τ j −1 ≤ x < τ j ,j=1,2,...for option O0
(3a )
)
τ j −1 ≤ x < τ j ,j=1,2,...for option O1
(3b)
+ x − τ j −1 ,
⎧⎪ r0 (v hj−1 + x − τ j −1 ), τ j −1 ≤ x < τ j j = 1,..., n0 h ⎪⎩ r0 (vn + x − τ n ), τ n ≤ x ≤ L
(4)
⎧⎪ r0 (v oj−1 + x − τ j −1 ), τ j −1 ≤ x < τ j j = 1,..., n1 o τn ≤ x ≤ L ⎪⎩ r0 (vn + x − τ n ),
(5)
0
o
0
0
rm ( x) = ⎨
1
1
1
with τ j = j Δ, j ≥ 1 and n0 , n1 is the largest integer less than L Δ.
r ( x)
rmh ( L )
Failure rate
after ( j − 1) st , j ≥ 2 , PM action. See [12] for the concept of virtual age. If maintenance effort level m is used, the virtual age is given by PM with v0 = 0
o
)
+ x − τ j −1 ,
rm ( x ) = ⎨
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 (m) used. The maintenance effort is constrained so that 0 ≤ m ≤ M . m = 0 corresponds to no PM and M is the upper limit of maintenance effort. Larger value of m corresponds to greater maintenance effort. Let v j −1 denote the virtual age of the item
th
j −1
Let n0 , n1 denote the number of PM actions over [0, L) for option O0 and O1 respectively. We consider two ROCOF functions with PM effort m given in (4) and (5). Their ROCOF are defined as follow:
The PM policy follows [11] by assuming a PM occurs at discrete time instants τ1 ,τ 2 ,...,τ j ,... and τ1 = 0 . The effect of
th
h
In other words, failures over time occur according to a Nonstationary Poisson Process with intensity function is given by the failure rate for the virtual age.
A. Model Formulation for Product Failures and PM Policy We use a black-box 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 NonHomogenous Poisson Process (NHPP) with the intensity function r ( x ) [10].
before the j
(2b)
Since failure are repaired minimally and repair times are negligible, the rate of occurrence of failures (ROCOF) is given by
The consumer must choose the option O* taken from the set {O0 , O1} . And the OEM has to determine the optimal cost
⎧ v j −1 = v j −1 + (τ j − τ j −1 ) ⎪ h h v = ⎨v j = v j −1 + δ h ( m )(τ j − τ j −1 ) ⎪ v o = v o + δ ( m )(τ − τ ) j −1 o j j −1 ⎩ j
(2a)
rmo ( L )
L
with δ (m) ∈ [0,1], 0 ≤ δ o (m) < δ h (m) ≤ 1 . δ (m) is a decreasing function of m with δ (0) = 1 and δ ( M ) = 0 . This implies that as the PM effort is increased, the effect of aging is reduced. With m = M at every PM action, then the item is restored back to as good as new after each PM action. If m = 0 , then vj =τ j, j ≥1 .
Figure 1. ROCOF for options O 0 , O1
251
π ( O 0 ) = Cb + ( Cs − Cm ) N 0 , Cs > Cm
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 O0 , we assume that C pm is a preventive
OEM revenue for option O1 :
π (O1 )=
maintenance cost per unit of time and Cm is the average of repair cost. We denote Cτ as the expected penalty cost of the OEM if the repair time exceeds τ , and Cs is the average of repair and corrective cost in the interval [0, L) when the consumer does not buy a service contract, and Cb denotes the pricing of product.
PG - [Penalty cost] - [CM cost] - [PM cost]
pricing of service contract PG and the cost of Cm , respectively. C. Model Analysis We assume that OEM and consumer have the same attitudes to risk, in order to make the solution reach equilibria.
option O1 the cost of PM is C pm n1 . The penalty cost is a
∫
L
0
rmo
( x ) dx .
1) Customer’s expected profit From (6) then the expected profit of the consumer upon choosing the O1 option, E ⎡⎣ω ( O1 ) ⎤⎦ , is given by
⎡ ⎧ E ⎡⎣ω ( O1 ) ⎤⎦ = R ⎢ L − N1 ( 0, L ) ⎨ ⎩ ⎣
∫
0
And for option O0 , the expected number of
by Cτ
∑
i =1
∑ + Cτ ( ∑
i =1
max {0, Yi − τ }
)
n
⎡ N 0 ( 0, L ) ⎤⎦ e P ( N0 ( 0, L ) = n ) = ⎣ n!
(6)
⎣⎢
∑
Y ⎤ − Cb − P0 − Cs N 0 ⎥ i =1 i ⎦
− N 0 ( 0, L )
given by E ⎡⎣π ( O0 ) ⎤⎦ = Cb + ( Cs − Cm ) N 0 ( 0, L )
(12)
Where N 0 ( 0, L ) is given by
choosing the option O0 , ω ( O0 ; P0 , Cs ) is given by N0
(11)
Hence the expected revenue of the OEM whenever the consumer chose the option O0 is E ⎡⎣π ( O0 ) ⎤⎦ , and its value is
Suppose that the number of failures in the interval [0, L) upon choosing the option O0 is N 0 . The consumer profit upon
ω ( O0 ; P0 , Cs ) = R ⎡ L −
(10)
OEM’s expected revenue The revenue function OEM for option O0 is given in (8). Since the failure follows the NHPP, then as in [10]
N1
N1
∫τ
⎫ ⎡⎣1 − G ( y ) ⎤⎦ dy ⎬ ⎭
2)
O1 is ω ( O1 ; PG , Cτ ) , and the value is given by ⎣⎢
∞
∫
1) Customer’s decision problem We assume that Yi denotes the time needed to return to the operational condition after the i-th failure and the expected number of failures in the interval [0, L) whenever choose the option O1 is N1 . Consumer profit upon choosing the option
Y ⎤ − Cb − PG + ⎥ i =1 i ⎦
0
⎫⎤ ⎡⎣1 − G ( y ) ⎤⎦ dy ⎬ ⎥ − ⎭⎦
⎡ ⎧ ∞ ⎫⎤ E ⎡⎣ω ( O0 ) ⎤⎦ = R ⎢ L − N 0 ( 0, L ) ⎨ ⎡⎣1 − G ( y ) ⎤⎦ dy ⎬⎥ − 0 ⎩ ⎭⎦ ⎣ − Cb − P0 − Cs N 0 ( 0, L )
max {0, Yi − τ } .
ω ( O1 ; PG , Cτ ) = R ⎡ L −
∞
And from (7), the expected profit of the consumer upon choosing the option O0 , E ⎡⎣ω ( O0 ) ⎤⎦ , is given by
rmh ( x ) dx . The penalty cost in the interval [0, L) is given N1 ( 0, L )
∫
⎧ -Cb − PG + Cτ N1 ( 0, L ) ⎨ ⎩
failure in the interval [0, L) , N 0 ( 0, L ) is given by L
(9)
In our model, by using Nash bargaining solution for both options O0 and O1 , OEM and consumer will negotiate the
The cost of CM for option O1 can be expressed as Cm , where Cm is the average cost of each rectification. And for function of the repair cost Y (r.v with the distribution function G ( y ) ) and τ . Let Yi denotes the time needed to rectify the i-th failure and for option O1 , since failures are rectified by minimal repair and the repair time is negligible, the expected number of failure in the interval [0, L) , N1 ( 0, L ) is given by
(8)
N 0 ( 0, L ) = n0 −1
=
(7)
∑∫ j =1
2) OEM’s decision problem OEM revenue for option O0 is given by:
252
τ j +1
τj
h
r0 (v j + x − τ j )dx +
∫
L
τ n0
h
r0 (vn0 + x − τ n0 ) dx
(13)
From (9) then the expected revenue of the OEM, E ⎡⎣π ( O1 ) ⎤⎦ , is given by
E [π ( O1 ) ] =
1 2
∞
∫
[ R[ L − N1 ( 0, L ) {
0
⎡⎣1 − G ( y ) ⎤⎦ dy}]] −
1
(22)
− [Cb + C pm n1 + Cm N1 ( 0, L )] 2
E ⎡⎣π ( O1 ) ⎤⎦ =
=PG − E [ Penalty cost ] − E [ CM cost ] − E [ PM cost ]
(14) The expected profit of the OEM on option O0 after we get Cm* becomes
The expected number of failures N1 ( 0, L ) is given by N1 ( 0, L ) =
n1 −1
∑∫ j =1
τ j +1
τj
o
r0 (v j + x − τ j ) dx +
∫
L
h
τ n1
r0 (vn1 + x − τ n1 ) dx
(15)
⎧ ∞ ⎫ E ⎡⎣π ( O0 ) ⎤⎦ = R( L − N 0 ( 0, L ) {⎨ ⎡⎣1 − G ( y ) ⎤⎦ dy ⎬}) − ⎩ 0 ⎭ − Cs N 0 ( 0, L ) e − Cb − P0 .
(16)
We consider that the failure distribution follows the Weibull distribution with α = 1 and β = 2 . We also consider that repair time y has a Weibull distribution with α = 0.5 and β = 0.5 , β < 1 (decreasing time of repair). And
∫
Expected cost of rectification is expressed as ECm = Cm N1 ( 0, L ) Expected cost of penalty
⎧ ECτ = Cτ N1 ( 0, L ) ⎨ ⎩
∞
∫τ
( y − τ ) g ( y )⎫⎬ dy ⎭
η = δ (m) = (1 + m)e − m , m ≥ 0 and integer. For L = 5 (years) and
(17)
τ = 0.04 (years), we assume that maintenance is carried out at discrete time τ j , with Δ = 0.33 . Table II gives the pricing of
Using integral by part, we have ∞
∫τ
∞
( y − τ ) g ( y )dy == ∫τ
⎡⎣1 − G ( y ) ⎤⎦ dy
service contract PG* and C m* the repair cost for option O1 and O0 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 η = 0.05 to η = 1.00 . Fig. 2 shows the resulting plot of the expected revenue [profit] for the OEM [consumer] for O1 and O0 options based on the results of Table III. Solution of the numeric is solved using MAPLE 9.5 of the Waterloo.
(18)
Thus (17) becomes ECτ = Cτ N1 ( 0, L )
∞
∫τ
⎣⎡1 − G ( y ) ⎦⎤ dy
(19)
Expected PM cost is EC pm = C pm n1
(23)
(20) TABLE II.
Then, total expected revenue of the OEM in (14) becomes E ⎡⎣π ( O1 ) ⎤⎦ = PG − Cτ N1 ( 0, L )
∞
∫τ
⎡⎣1 − G ( y ) ⎤⎦ dy −
− Cm N1 ( 0, L ) − C pm n1
III.
η
PG*
SERVICE CONTRACT AND REPAIR COST
(
)
E ⎡⎢ω O1; PG* , Cm ⎤⎥ ⎣ ⎦
0.05 22.88 8.05 0.15 22.88 8.03 0.25 22.88 8.01 0.35 22.88 7.99 0.45 22.88 7.96 0.55 22.88 7.94 0.65 22.88 7.92 0.75 22.88 7.89 0.85 22.88 7.87 0.95 22.88 7.85 1.00 22.88 7.84 The result in Tabel II is given follow.
(21)
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, PG and the repair cost, Cm . The optimal solution is 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 E [ω ] = E [π ] . Firstly we will compare the expected profit of the consumer and manufacturer with the respect to option O1 and secondly we obtain PG* . Hence the expected revenue of the OEM on option O1 becomes
253
C m*
(
)
* ⎤ E ⎡⎢ω O0 ; P0 , Cm ⎣ ⎦⎥
7.17 7.25 7.18 7.23 7.19 7.21 7.21 7.18 7.23 7.16 7.24 7.14 7.26 7.15 7.27 7.09 7.29 7.07 7.30 7.05 7.31 7.03 using parameter value as
Parameter
Cb
R
Cm
Cs
C pm
Cτ
P0
Value (103 $)
5
4
0.1
1
0.05
0.5
2
[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. 90-100, 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.
Ε[π(Οι)]
O1
O0
η
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 O1 is the optimal option for the consumer. It also shows in Fig. 2, that option O1 gives a higher expected value for the OEM (consumer) compared to option O0 . It is clear that the larger the degradation preventive maintenance level the greater the discrepancy between the expected revenues of the different option. 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 two-dimensional 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]
[2] [3]
[4]
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. E. Ashgarizadeh, and D. N. P. Murthy, “Service contracts,” Mathematical and Computer Modelling, vol. 31, pp. 11–20, 2000. 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. D. N. P. Murthy, and E. Ashgarizadeh, “A stochastic model for service contract,” International Journal of Reliability, Quality and Safety Engineering, vol. 5, pp. 29-45, 1998.
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