Optimal Preventive Maintenance for a Fleet of Dump Trucks with Two-Dimensional Lease Contract Hennie Husniah 1, Udjianna S. Pasaribu 2 and Bermawi.P. Iskandar 3 1 Department of Industrial Engineering, Langlangbuana University, Bandung, Indonesia 2 Statistics Research Group, Bandung Institute of Technology, Bandung, Indonesia 3 Department of Industrial Engineering, Bandung Institute of Technology, Bandung, Indonesia (1
[email protected], 2
[email protected], 3
[email protected])
Abstract— This paper develops a mathematical model of optimal preventive maintenance for a two-dimensional lease contract of a fleet of dump trucks. The lease contract coverage is characterized by two parameters – age and usage. We use one dimensional approach to model the age and usage of a repairable equipment. The lessor will provide preventive maintenance and corrective (CM) actions during the contract period, and a penalty cost incurred when the time required to perform an imperfect repair (CM) exceeds a target. This strategy will reduce equipment failures and hence it decreases the penalty cost and maintenance cost during the lease contract. We find the optimal maintenance degree and time between preventive maintenance, which minimizes the expected total cost. Keywords—fleet, lease contract, imperfect repair, preventive maintenance, expected maintenance cost
I.
INTRODUCTION
In an open cut mining industry, loading and transporting mining material are the main processes in which dump trucks play a major role. The cost of these dump trucks can reach up to $ 1 billion. Mean while, the prices of ore, coal and other mining materials have fallen, and this decreases significantly the revenue of mining companies. As a result, the mining companies tend to cut back on capital expense. Leasing dump trucks to an external agent or Original Equipment Manufacturer (OEM) is an alternate way to get a function of dump trucks hauling mining materials. In most cases, the agent (or OEM) as a lessor gives a lease contract with a full coverage of the maintenance actions (Preventive Maintenance (PM) or/and Corrective maintenance (CM)). There are many literature in maintenance lease equipment have been studied. A lease contract in which PM is taken when the failure rate of the lease equipment reaches a certain threshold value is proposed by [1]. Failure rate reduction method also has been used by [2] to obtain the optimal periodical maintenance policy for lease equipment. The optimal number and degrees of PM associated with CM
introduced by [3]. Most studies in lease equipment concentrate on determining the optimal PM policy in specified contract period but none of them combine the imperfect repair and preventive maintenance during lease contract period by age and usage parameters which ever occur first. Later [4] introduced a two-dimentional lease contract with considered age and usage as contract limitation for mining equipment. Offering an equipment with a long maintenance lease contract means that the lessor may incur a greater maintenance costs for servicing the contract and this is of interest to the lessor for reducing the maintenance costs. Since in mining industry to fullfill the operational target they usually lease more than one heavy equipment. Therefore it needs to consider the number of equipment in the lease contract. In this paper, we extend the work of [4] into a two dimentional lease contract for a fleet of lease dump trucks used in a mining industry. The optimal PM (number and degree of PM) is obtained, which minimises the total cost for the lessor. This paper is composed as follows. Section 1 and 2 deal with background and model formulation for the lease contracts studied. Sections 3 and 4 give model analysis to obtain the optimal number of preventive maintenance and the lessor optimal maintenance level. In Section 5, we give numerical example to illustrate the model and finally in section 6 we conclude with topics for further research . II.
MODEL FORMULATION
A. Notation The following notation of parameters will be used in model formulation. 0, 0 0,U 0
:Lease contract coverage
y
:Preventive maintenance level
Xi
:Downtime caused by the i-th failure and waiting time :Total downtime in (0,t]
D(t)
F(t)
:Distribution function of downtime :Down time target :Usage rate :Repair cost :Preventive maintenance cost :Degree of preventive maintenance cost :Penalty cost per unit of time
S Y Cr C0 Cv
CP
: Expected lease contract cost :Hazard, and Cumulative hazard functions associated with F (t , y ) :Number of fleet
J ry (t ), Ry (t )
Z
B. A New Maintenance Lease Contract We consider that a mining company operates a number of lease dump trucks. The dump trucks is offered with a twodimensional lease contract with the lease characterised by a rectangle region 0, 0 0,U 0 where Γ0 and U0 are the time and the usage limits (e.g. the maximum coverage for 0 (e.g. 1 year) or U 0 (e.g. 50.000 km), and hence the lease contract is characterised by a rectangle region (see Fig.1). All failures under lease contract are rectified at no cost to the lessee. For a given usage rate y of the dump truck, the lease contract ceases at y 0 for y U 0 0 , or y U y for, whichever occurs first. We consider that the lease contract given by the lessor also covers PM action, and hence, during the lease period CM and PM actions are done by the lessor without any charge to the lessee (See Fig.1). As the lease contract is full coverage (PM and CM), then a penalty cost incurs the lessor if the actual down time falls above the target (S ) . If D is down time (consisting repair time and waiting time) for each failure occuring during the contract, then the lessor should pay a penalty cost when D S . The amount of the penalty cost is assumed to be proportional to D - S . The penalty cost ( CP ) is viewed as a penalty given by the lessor. The decision problem for the lessor is to determine the optimal number of PM and degree of maintenance level such that to minimize the expected cost. We use the one dimensional approach by Iskandar, et.al [5] and hence it needs to model the expected cost for a given usage rate y . Let hy (t ) 0 be the conditional hazard function for the time to first failure for a given y . It is is a nondecreasing function of the item age t and y . Furthermore, we consider the case where age, usage and operating condition where the truck is operated as major factors to influence failure. Here, the accelerated failure time (AFT) model is an appropriate model to be used as it allows to incorporate the effect of the three major factors on degradation of the truck. If the distribution function for T0 is given by F0 (T, α0), where α0 is the scale parameter, then the distribution function for Ty is the same as that for T0 but with a scale parameter given by (1) y y , y
0
0
with 1 . Hence, we have F (t , y ) F0 (( y0 y ) t , y ) . The hazard and the cumulative hazard functions associated with F(t, αy) are given by ry (t ) f (t , y ) (1 F (t , y )) and t
Ry (t ) ry (x)dx respectively where f(t,αy) is the associated 0
density function. If all failures are fixed by minimal repair and repair times are small relative to the mean time between failures, then failures over time occur according to a nonhomogeneous Poisson process (NHPP) with intensity function ry t . We will use the accelerated failure time (AFT) model to modelling failures, which allows to incorporate the effect of usage rate on degradation of the dump trucks (see [5]). Let Y be a usage rate of the truck. We consider that Y varies from customer to customer but is constant for a given customer (or a given equipment). Y is a random variable with density function g ( y ), 0 u . Conditional on Y y , the total usage u at age t is given by u yt . Within the lease coverage, a lease contract ends at y 0 for given usage rate y. Two cases need to be considered–i.e. (i) y and (ii) y . Preventive Maintenance Policy: We define periodic PM policy for a given Y y . PM policy for a given y, is characterised by single parameter y . The equipment is periodically maintained at k . y . Any failure occurring between pm is minimally repaired (See Fig. 1). Note (k 1) y T0 where k is an integer value.
y3 Usage
y2 U0
y1
0
y
k y 0
y
Age
Fig 1. The two-dimensional lease contract
Modeling of PM effect: For a given usage rate y, the effect of PM actions on the intensity function is given by r(t j ) r (t j 1 ) j with 0 j r(t j 1 )
j i 0
i , j denotes the reduction of
the intensity function after j th , j 1 , PM action. If the PM
action is done at j th , j 1 the intensity function is reduced by
j , then for t j t t j 1 the intensity function is given by rj ( t ) r ( t ) that
j i 0
for
i with 0 0 . For simplicity we assume
each
PM
j j1
action
then rj ( t ) r ( t ) j (See Fig. 2). If any failure occurring between pm is minimally repaired, then
expected
total
number
of
minimal
repairs
in
([t j 1 , t j ),1 j k y 1) is given by k y 1
ky
N rj 1 (t )dt R( 0 ) 0 jT j j 1
tj
t j 1
j 1
(2)
For t j t j 1 y then
y
Pk 1 (Z k )( )k (Z ! (Z k )!) Z 1
(Z k )( k 0
) k ( Z ! ( Z k )!)
The expected value of Y j is
k y 1
R 0 0 j y r j y r j 1 y j 1
.
(3)
r0 y t
ry t
*y
j
Z 1
0
k 0
E Y j yf ( y )dy
Pk (k 1)
(5)
The expression (5) depend on Pk and , where
is estimated
by the mean value of failure intensity, . Here, two cases are considered – case (i) y and case (ii) y . Hence, for y ,the expected total cost is given by ry t
j 1 t
( y )k (4) k! k 0 where Pk, k =1, 2, ..., Z - 1 given by the ratio Pk Pk 1 Pk 2 and Z 1
f ( y ) Pk e
Pk 2
N k y , y
0
incurs penalty cost when the down time caused by a failure exceeds the predetermined target. Suppose that there are k units failed equipment will be served by the lessor with single service channel where the first come, first served. Hence, there is queue which the model formulation is identical to a Markovian queue. The arrival rate of failed equipment is k (Z k ) for 0 k Z , where Z is number of equipment population and λ is failure rate. The service rate is µ. According to [6] and [7], the steady state density function for Y j (waiting time for truck j) is
E y = E PM cost E Repair cost E Penalty cost We obtain the expected repair and PM cost and expected penalty cost in (0, Γ0] as follows.
j 1 t
j
j 1 t Age
j 0
Fig 2. The effect of PM action on failure rate function for Y=y
As the lease contract is full coverage (PM and CM), then a penalty cost incurs the OEM if the actual down time falls above the target (S ) . If D is down time (consisting repair time and waiting time) for each failure occuring during the contract, then the OEM should pay a penalty cost when D S . The amount of the penalty cost is assumed to be proportional to D - S . The penalty cost ( CP ) is viewed as a penalty given by the OEM. The decision problem for the OEM is to determine the optimal price structure and maintenance level such that to minimize the expected cost.
Expected cost with PM and minimal repair, We obtain the expected PM and repair cost conditional on Y=y,
J k y , y C r R 0 k y C0 k y 1 Z Cr L j y Cv r j y r j 1 y j 1 where R y 0
0
0
(6)
ry (t )dt is the expected number of minimal
repairs . III.
MODEL ANALYSIS
We consider a situation where the lessor incurs repair cost for each failure and PM cost. For a larger coverage of lease contract e.g. for maximum 5 years or 250.000 km, it would require more than one PM for reducing the maintenance cost. In the proposed lease contract, the lessor expected total cost consists of PM cost, repair cost and penalty cost. The lesor
Expected penalty cost: The lessor incurs penalty cost when the down time caused by a failure exceeds the predetermined target. Let D and S denote down time (consisting repair time and waiting time) for each failure occuring during the contract and down time allowed. The expected penalty cost is given by
EP(S) CP G(S) N (k y , y ) where CP is the penalty cost and
N ( k y , y ) denotes the expected number of failure in interval (0, 0 ] .
For a given usage rate y the failure distribution is given its hazard function is ry (t ) t 1 ( y ) where αy as in [5].
E y Z J y (k y , y ) CP G(S) N (k y , y )
(5)
with CP G (S ) is defined as CP G (S ) CP
S
IV.
y S g ( y )dy.
(6)
OPTIMIZATION
In this section we will look for the optimal value of parameters k *y , *y , *y by minimizing the total cost function E y subject to constraint 0 j r(t j 1 )
j i 0
i . The
optimal values are obtained involving a two stages. In the first stage, for a fixed k, minimize E y to obtain the optimal * y ,1
Let the parameter values be as follows. α0 =1(year), β=2.25, Γ0=2(years), U=2(1x104 Km) (γ = U/W = 1), y0 = 1, Cr 100 , C0 Cv 0.5Cr , S = 80 (hours) or 4 (days) or
CP 3K and K 5.102.2025 $. The down time distribution
For case y , the expected cost of lessor is given by (6) replacing 0 with y .
j
NUMERICAL EXAMPLE
by the Weibull distribution, F (t ; y ) 1 exp(t / y ) , and
As a result, the total expected cost of the lessor is
values of
V.
j k y 1 . In the second stage, the
optimal k is obtained using the results of the first stage.
τ
Fig 3. Expected lease cost with y = 1.80 ,
is given by the exponential distribution with 1/ 1/ 300 (years). Tables 1 shows optimal number of PM and improvement level of lease contract with two usage types – medium (1.0 y 1.4) and heavy (1.4 y 1.8) The values of for three different land contours are 1.2, 2.0, and 2.2 coresponding to light incline, high incline and very hilly, respectively. For a given y and 0 (or reliability level), the optimal expected lease cost increases as the usage rate y increases. It means that under lease contract coverage, larger values of the usage rate result in shorter periods of time between PM actions *y . It indicates that the reliability of the equipment has been decreased. As a result the average cost of lease contract, E y , increases with the increasing in y, since the penalty cost increases when the number of failures increases. Figures 3 and 4 show the expected cost in lease contract region for a given usage rate y. We also observe the same behavior as the operating condition is more severe ( is bigger), since the reliability of equipment decreases as the unit deteriorates rapidly with time.
τ
Z
= 25 and k =70.
Fig 4. Expected lease cost with y = 1.8 and k = 70 to the number of dump truck and time between PM
TABLE I RESULTS FOR
Z 25
WITH
Cr 100 , S =80 HOURS AND 0 4
ρ = 1.2
ρ = 2.0
ρ = 2.2
E y
k *y
*y
*y
E y
0.32
26401.71
13
1.70
0.32
26401.71
0.89
0.47
57406.09
22
0.85
0.49
61716.88
32
0.51
0.64
1.19.105
35
0.46
0.67
1.38.105
1.03.105
45
0.32
0.86
2.34.105
51
0.28
0.91
2.89.105
1,58.105
60
0.21
1.14
4.36.105
70
0.18
1.21
5.74.105
E y
k *y
*y
*y
0.32
26401.71
13
1.70
1.04
0.41
43936.21
21
22
0.73
0.57
67428.65
1.60
28
0.50
0.68
1.80
38
0.33
0.69
k *y
*y
*y
1.00
13
1.70
1.20
18
1.40
y
MONTHS
Medium
Heavy
VI.
CONCLUSION
[5]
In this paper, we have developed a mathematical model of lease equipment foe a fleet where the lease coverage is characterized by two parameters – age and usage. Imperfect PM is performed during the lease contract. We find the optimal value of parameters k *y , *y , *y minimizing expected
[6]
cost of the lessor. For future research, it is interesting to model failures using the two dimensional approach and carry out estimation of parameters of a bivariate distribution using real data from mining industry.
[7]
ACKNOWLEDMENT This work is funded by the Ministry of Research, Technology, and Higher Education of the Republic of Indonesia through the scheme of “Desentralisasi 2015 and Hibah Bersaing 2015”. REFERENCES [1]
[2]
[3]
[4]
Yeh, R.H., and Chang, W.I., “Optimal Treshold Value of Failure-Rate for Lease Dump truckss with Preventive Maintenance Actions”, Math. and Comp. Modelling 46, 730-737, 2007. Pongpeh, J., and Murthy, D.N.P, “Optimal Periodic Preventive Maintenance Policy for Lease Equipment”, Reliability Engineering & System Safety 91, 772-777, 2006. Jaturenon, J., Murthy D.N.P and Boodiskulchok, R., “Optimal Preventive Maintenance of Lease Equipment with Corrective Minimal Repair”, European Journal of Operational Research 174, 201-215, 2006. Iskandar, B.P., Cakravastia, A. and Husniah, H., Optimal Preventive Maintenance for A Two Dimensional Lease Contract, Proc. Of CIE-45, Metz-
France, 2015. Iskandar, B.P., and Jack, N., “Warranty Servicing with Imperfect Repair Sold with a Two- dimensional Warranty”, Replacement Models with Minimal Repair, Springer, 163-174, 2011. Husniah. H, Pasaribu U.S., Cakravastia A., and Iskandar B.P., “Two dimensional maintenance contracts for a fleet of dump trucks used in mining industry”, Applied Mechanics and Material Vol 660, pp. 1026-1031, 2014. Iskandar, B.P., Pasaribu, U.S., Cakravastia, A. and Husniah, H., Performance based maintenance Contract for a fleet od dump trucks used in mining industry, Proc. Of TIME-E, Bandung, 2014.
