Imss14 Cie44 543 Fullpapers

8/19/2019 Imss14 Cie44 543 Fullpapers

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CIE44 & IMSS’14 Proceedings, 14-16 October 2014, Istanbul / Turkey

1

* Corresponding Author

Maintenance Service Contracts for A Fleet of Dump Trucks Used in Mining Industry

B. P. ISKANDAR1*, U. S. PASARIBU2, A. CAKRAVASTIA3 And H. HUSNIAH4*

1,3Department of Industrial Engineering, Bandung Institute of Technology, Ganesha 10,Bandung, 40132, [email protected]

2Department of Mathematics and Life Sience, Bandung Institute of Technology, Ganesha 10,Bandung, 40132, Indonesia

4Department of Industrial Engineering, Langlangbuana University,Karapitan 116,Bandung, 40261, [email protected]

ABSTRACT

We consider a situation where a mining company operates a number of trucks (as a fleet) for

transporting mining materials (such as coal, ores) from several mining fields to processing units.

A high availability of a fleet of trucks is critical factor for achieving a monthly production target

of the company. Performance based maintenance contracts offers an appropriate incentives to

motivate the original equipment manufacturer (OEM) or the service provider to increase the

fleet’s performance beyond the target. This leads to a win-win situation for both the owner of thefleet (high performance gained) and the service provider (incentive earned). The trucks are sold

with two dimensional warranty (e.g for maximum 3 years or 150.000 km) and PM is one package

of the warranty. After the warranty ceases, the company is fully responsible to perform

maintenance actions for the fleet. To sustain high performance of the fleet, the company usually

purchases the maintenance services from the OEM or service provider. Often, the OEM is the only

maintenance service provider for the dump trucks and offers more than one service contract

options (fully, moderate or partial maintenance coverages). In this paper, we study performance

based maintenance service contract for a group of trucks, which considers availability target and

use a non-cooperative game theoretic formulation to determine the optimal option for the owner

and the optimal number of server for the OEM. We give numerical examples to show the optimal

number of server for the OEM and the optimal strategy for the owner.

Keywords: maintenance contract, two dimensional warranty, fleet, availability, non-cooperative

game theory

mailto:[email protected]:[email protected]:[email protected]:[email protected]


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1 INTRODUCTION

Dump trucks are important equipment in an open-pit mining system. The dump trucks are used toload mining materials (such as coal, ores) at a mining site and then tranport the mining materialto an unloading site. Usually a mining company operates a number of dump trucks (as a fleet) tofullfil a daily production target. A high availability of the trucks is critical factor for achieving theproduction target.

Preventive maintenance (PM) is an effective way to keep the trucks in a high availability and thePM can be done using age based or conditioned base maintenance. When a truck fails, correctiveMaintenance (CM) action is performed, which restores the failed truck to the operationalcondition.

All dump-trucks operated in a mining industry are sold with a two-dimensional warranty. For

example, a dump truck is warranted for maximum 3 years or 150.000 km. In order to give fullassurance (the dump truck will function as promised over the warranty) to the company (or theowner), the manufacturer offers the warranty and PM in one package and this requires themanufacturer to rectify all failures under warranty as well as carry out PM.

After the warranty expires, the owner is fully responsible to carry out all maintenance (PM andCM) actions for the fleet. The PM and CM actions can be done either in house or by independentagents or the OEM. As dump-trucks and other heavy equipment used in mining sites tend to becomplex and expensive, then performing PM and CM actions in house requires expensivemaintenance facilities and skilled maintenance crews. As a result, it would not be economical todo PM and CM in house. To sustain high performance of the fleet, the company has to purchasethe maintenance services from the original equipment manufacturer (OEM) or an external agent.

Often, the OEM is the only maintenance service provider for the dump trucks and offers morethan one service contract options (fully, moderate or partial maintenance coverages), and theOEM proactively offers PM and CM to the owner just before the warranty ends.

From the owner’s perspective, maintenance programs for the fleet are aimed at not only tosustain high performance (e.g. high availability) for the fleet but also to obtain optimal businessprofitability. As a result, the owner should select the maintenance contract option that gives highavailability of the fleet with reasonable maintenance costs. Whilst the OEM needs to providemaintenance service to achieve the contracted availability. As the OEM has to performmaintenance services for a group of trucks, then maintanance capacity will affect the service rateand waiting time to get a service. This will in turn influence availability of a truck. In thiscontext, a relevant decision for the OEM is to determine maintanance capacity to attain the the

contracted dump truck availability, and the price for each contract option such that to maximisethe OEM’s profit.

Study of maintenance service contract has received a lot of attention in the literature. Severalpapers, Murthy and Ashgarizadeh [1], Ashgarizadeh and Muthy [2], and Rinsaka and Sandoh [3]studied maintenance service contract for non repairable items, and formulated decision problemsusing a Stackelberg game theory with the agent as a leader and owner as the follower. Forrepairable items, Jackson and Pascual [4], Wang [5] and Wu [11] studied maintenance servicecontract which involves preventive maintenance policies, and the optimal option is obtained tomaximize the expected profit for both the agent and the owner.

In all works on maintenance service contracts discussed above, a penalty cost is modelled based

on down time for each failure. In many cases, mining companies consider the availability of dump


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trucks is a critical performance measure for supporting their business. Hence, a maintenanceservice contract which ensures a high availability of the equipment at reasonable cost would bean attractive contract for the owner. In Iskandar et al. [7] and Iskandar et al. [8] , the authorsconsider maintenance service contracts with availability target for a dump-truck sold with a two-dimensional warranty. Pascual et al. [9] developed a model for determining jointly optimal fleetsize and maintenance capacity but this work does not deal with maintenance service contracts.

In this paper, we extend the maintenance service contracts studied in Iskandar et al. [8] to thecase of a group of trucks (fleet), and study the contracts from the manufacturer’s perspectiveand the owner’s perspective. In addition, this work can be viewed as the extention ofmaintenance service contracts studied in Murthy and Ashgarizadeh [1] for a repairable case.

This paper is organized as follows. In Sections 2 and 3 we give model formulation, and modelanalysis. We present numerical examples in Section 4, and finally conclude with topics for furtherresearch.

2 MODEL FORMULATION

A. NotationThe following notation will be used in model formulation.

W,U :Warranty time, and usage limits

ji X :Downtime caused by the i-th failure including waiting time for truck j

P G :Price of service contract for Option 2

G(x) :Distribution function of downtime, ji X K :Revenue per unit time, maintenance contract time

:Maintenance contract time

Y :Usage rateC0 :Server set up cost done by OEM

C1 :Operational server cost done by OEM

Cm :Repair cost done by OEM

Cs :Repair cost charged to the owner for each failure

C pm :Preventive maintenance cost per unit of time

C p :Penalty cost per unit of time( .) y O

:Owner’s profit( .) y O

:OEM’s profit

Cb :The annual product cost over the contract period

P 0 :PM cost done in-house over the contract period

( , ) y F t :Conditional failure distribution for a given usage rate y ( ), ( ) y yr t R t

:Hazard, and Cumulative hazard functions associated with ( , ) y F t

B. Warranty PolicyWe consider a situation where a mining company operates a number of trucks (as a fleet) fortransporting mining materials (such as coal, ores) from several mining fields to processing units.The manufacturer sales each dump truck with a two-dimensional warranty which is characterized

by a rectangle region 0, 0,W W U where W and U are the time, and the usage limits. Allfailed trucks under warranty will be fixed by the manufacturer at no cost to the buyer. The expiryof the warranty depends on the usage rate (y ) of a truck. Hence, for a given usage rate (y ), thewarranty ceases at yW W for , y U W or , yW U y for . y U W With warranty and PM in one

package, manufacturer has to perform PM and CM actions during the warranty without any charge


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to the owner. The responsibility to do PM and CM actions shifts to the owner once the warrantyends.

C. Maintenance Service ContractWe consider three maintenance service contracts and each is characterised by a time limit called

one dimensional service contract e.g. the maximum coverage for 1 year (see Fig.1). A highavailability (or a low downtime) of a fleet of trucks is critical factor for achieving a monthlyproduction target of a mining company. Performance based maintenance contracts offers anappropriate incentives to motivate the the original equipment manufacturer (OEM) to increasethe fleet’s performance beyond the target. If the downtime is below the target, then the OEMearns some incentives. Otherwise, the OEM incurs a penalty cost (i.e. if the actual downtime isabove the target). Three service contract options are considered as follows.

Option 1( 1O ): After the warranty ends, the owner performs a PM action in-house but a CM action

is outsourced to the OEM. Under this option, if the truck fails the owner calls the OEM to fix thetruck. The OEM will charge the owner a fixed cost CS for each repair (CM). No penalty cost incurs

the OEM if the downtime caused by a failure falls above the target as the OEM only performs CM.Option 2( 2O ): For a fixed price of service contract G P , the OEM agrees to perform full service

including PM and CM actions for a period of time, . The contract starts at the end ofwarranty, yW and the OEM promises that the down time for each failure is less then a target value

stated in the contract (note that downtime is repair time plus waiting time). As the maintenanceservice is full coverage (PM and CM), then a penalty cost incurs the OEM if the actual down timecaused by each failure falls above the target. But if it falls below the target, the OEM will earn anincentive. If the down time for each failure over the contract be ji X is more than the down time

target , then the OEM should pay a penalty cost. The penalty cost, CP is viewed as a

compensation received by the owner. The OEM earns some incentives if actual down time for

each failure is less then the target ji X .The incentives cost, CI is viewed as an incentive earnedby the OEM (see Mirzahosseinian and Piplani [10]).

Option 3( 3O ): For a fixed price of service contract M P , the OEM agrees to perform only PM

periodically for years but CM is done in house (when the truck fails). The repair cost for each

repair mC is considered to be less than sC of Option 1O (the repair cost charged by OEM). If not

there will be no incentive to carry out CM in house. As the owner is responsible to carry out CM – if the truck fails, then no penalty is considered for this option.

In this paper, we consider a situation where a mining company operates a number of trucks (as afleet) for transporting mining materials (such as coal, ores) from several mining fields to

processing units. It is assumed that the OEM has a limited number of maintenance servicefacilities (or servers) and hence when a truck fails, there would be a chance that the failed truckhas to wait before getting a service. To control downtime below the target, the OEM needs todetermine the number of service facilities (or servers) in order to minimise waiting time to get amaintenance service. As a result, the decision problem for the OEM is to determine (i)maintenance capacity (the number of service facilities) and (ii) the optimal price structure (i.e.repair cost for option 1, the price of a full service contract for option 2 and price of a partialservice contract for option 3) such that to maximize the expected profit.


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U

0 W W

y S

w

Usage

Age

1(a)

0 1W W 1W

y

w

S Usage

U

Age

1(b)

Figure 1.Warranty region W and service contract region S for (a) y and (b) y

D. Equipment Failures and RepairsModelling failure of a truck is essential for model formulation. One can use several approaches tomodelling failure in two dimensional warranties. Here, the one dimensional approach as inIskandar et al. [10] will be used and truck failures can be viewed as a one-dimensional pointprocess. As mentioned before that a company operates a fleet of dump trucks to support itsbusiness. Let Y be the usage rate for a given truck. Y is considered varies across the trucks but itis constant for a given truck. For Y = y , the conditional hazard function is ( ) yr t which is a non-

decreasing function of t (the age of the truck) and y . Usage rate of the truck and a land contourof a mining area where the truck is operated may strongly affect the degradation of the truck,that leads to failure.

One can use the accelerated failure time (AFT) model as in [10] to incorporate the effect of usagerate and the operating condition of the truck. Let

0 y denotes the nominal usage rate value

associated with design reliability of the truck. Using the AFT formulation, if0[ ]T T is the time to

first failure under usage rate0[ ] y y then we have 0 0 yT y y T

where is a parameter

representing the operating condition of a truck. A land contour of a mining site can be (i) a lightincline, (ii) high incline or (iii) very hilly (i)-(iii) correspond to light, moderate, heavy operatingconditions. We assign different value of for different land contour of a mining site. The small,

medium and large values of will be assigned to represent light incline, high incline and very

hilly, respectively. A larger value of gives more stress to the truck and this in turn causes alarger effect on the truck’s degradation.

We now model truck failures taking into account age, usage and operation condition. If 0 0( ; ) F x is

the distribution function for0T with scale parameter 0 then the distribution function for yT is

the same as that for 0T but with different scale parameter given by 0 0 y y y

where 1 .

Hence, we have 0 0( , ) (( ) , ) y y F t F y y t

. The hazard and the cumulative hazard functions

associated with ( , ) y F t are given by ( ) ( , ) (1 ( , )) y y yr t f t F t and0

( ) ( )t

y y R t r x dx respectively where( , ) y f t is the associated density function. Since all failures are fixed by minimal repair and repair


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times are small relative to the mean time between failures, then failures occur according to a

non-homogeneous Poisson process (NHPP) with intensity function yr t .

PM is conducted regularly for the life cycle in order to control the degradation of the truck. Weconsider that PM done by the OEM is more effective than that of in-house PM, and model theeffect of PM through the failure rate function as follows.

For in-house PM, If 1 yr t represents the failure rate function for a given usage rate y

(Option 1O ), then it is given by

1( ) 0

( )( )

y y

y

y y y

r t t W r t

r t W t W

(1)

where 1 meaning that after the warranty ceases, the failure rate increases much faster. For PM

done by OEM, the failure rate function is given by 2( ) ( ) 0 y yr t r t t W . Note that 1 meaning

that the failure rate function increases with a higher rate or the PM done in-house is less

effective than PM by OEM ( 1 ). The cumulative hazard functions for PM done in-house, and theOEM are given by 0 0

0( ) ( )

t

y y R t r x dx and 1 10( ) ( )t

y y R t r x dx , respectively.

Furthermore, we group the trucks based on the usage rate into three usage types i.e. light,medium and heavy usage. Each usage type has an unoverlapped sub-intervals given by 1[ , ]i i y y , i =

1,2, and 3, which coresponds to light, medium and heavy usage, respectively. There after, we

will represent the usage type-i by average usage rate , 1, 2, 3i y i (see Iskandar et al. [11]).

2.1 Expected value for time to wait and repair and steady state distribution for ji X

We consider a situation where a mining company operates a number of dump trucks (N) to fullfil

a daily production target. Hence, there is a chance that more than one failed trucks are waitingto get repaired. Suppose that there are k failed trucks will be repaired by the OEM with a limitednumber of service channels (or servers), S. Here, we consider that a queue system of the OEM’sservice follows a Markovian queue with a finite population (N) and finite number of servers (S).

For truck (1 ) j j N , if j is the number of failures in [0, ) , jiT is the time to failure after

( 1)i th repair (2 ) ji , jT is the time from the last repair to the end of the contract period,

and ji X (1 ) ji is time needed to make the truck back to the operational state after the i-th

failure (including waiting time and repair time), then we make the following assumptions:1.

Failed units are repaired on a first come first served basis.2. Service contract period τ is sufficiently large in relation to mean time between failures so

we can apply steady state condition for the distribution for ji X .

3.

The mean total waiting and repair times is very small in relation to the mean time to

failure or 1 ji ji E X E T where λ is failure rate. As a result total down time forgiven y for each truck is small compare with ( ) yW , hence

j j j

j1 1 1( ); ,1 j N. ji j ji ji j yi i i K T T X K T T K W

(2)

The arrival rate of failed truck and the departure (service) rate are given by

( ) for 0 ,

0 for ,k

N k k N

k N

(3)


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for 0 ,

for .k

k k S

S k S

(4)

if ji X is the total (waiting and service) time in the system with its the density function ( ) g x then,

according to Murthy and Ashgarizadeh [1], the steady state density function for ji

X is

1 1

1

0 0 0

ˆ ˆ( ) ) ) [{ } {( )( { ( ! ) }} ,( ]S N k

x k x k k j S x k

k k s

k k k

g x e P P e x e k S S S j

(5)

where ˆk P , k P and P 0 k =1, 2, ..., N – 1 are given by

0{( ) } ( )ˆ N

k k k k N k P N k P P

(6)

0

0

( ) { ! ( )! !} for 0,1,..., 1

( ) ( !){ ! ( )! !} for , 1,...,

0 for

k

k S

y

k y

N N k k P k S

S S P S N N k k P k S S N

k N

(7)

1

1 10 00 ( ) { ! ( )! !} ( ) ( !){ ! ( )! !}

S S k k y

S S S y N N k k S S S N k P N k

(8)The expected value of

ji X for given y is

1

0

( 1)[ ] 1 .

N k

y jik

P k S E X

S

(9)

Under Option 2, a penalty occurs if the down time of a truck is above the target, ji X or the total

down time caused by the i-th failure is greater than . Hence, the probability that the penalty

occurs at the i-th failure is given by P ji X . The expected penalty is given by

0, ( ) 1 ( ) y ji E Max X x g x dx G x dx

( 10)

where ( )G x is the distribution function of ji X .The expression given by (9) is depent on k P and

y , where y is estimated by the mean value of failure intensity, y (described in Section 3.2).

3.2 Expected value for number of failure times

Let y R t be the expected number of failures over [0, )t for a given y if PM is outsourced to theOEM. Following the approach used in Jackson and Pascual [4], the mean value of failure intensityis approximated as

(W, W ) ( )m y y y y R W (11)

where ( ) ( )W

m m y y

W R t r x dx

, 1, 2m referring to the expected number of failures for PM done in-

house, and outsourced to the OEM, respectively.

3 MODEL ANALYSIS

A. Owner Decision ProblemWe obtain the owner’s expected profit for three options considered, and each needs to considertwo cases –i.e. (i) y and (ii) y as the warranty ceases depends on y.

Option-1: For case y , if K is the revenue ($/hour) received by the owner as a result of

transporting mining materials from a mining site to a processing unit, bC and 0 P are the annual


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cost of the truck and PM cost over the contract period, respectively, then the expected profit forN trucks is given by

1 11 0[( ; ) (W,W ) (W] ,W ) j y si s y y b E O C N K R X C R P E C (12)

For case y , as the warranty ceases at1

W (see Fig. 1(b)) then the expected profit of the owner

is given by (12) replacing W with 1W .

Option-2:For y , the expected profit of the owner is

2 2

22

(W,W ) (W,W ) ( );

(W, W ) Incentive

[ ] y y y G

y b

ji

G

K R X R EP E O P N

R E P C

E

(13)

In (13) ( ) EP is the expected penalty viewed as a compensation received by the owner and

E [Incentive] is the expected incentive gained by the OEM. For case y , as in Option 1, the

expected profit of the owner is given by (13) replacing W with 1

W .

Option-3:For y , the expected profit of the owner is

2 23 [ ]( ;P ) (W,W ) (W,W ) ji y M y m y M b E O N K R X C P C E R (14)For y , as in options 1 and 2, the expected profit of the owner choosing 3O is given by (14)

replacing W with1

W .

B. OEM Decision Problem

As the warranty ceases depends on y , then it requires to consider two cases - i.e. (i) y and (ii)

y .

Option-1: For y , the expected profit of OEM for N trucks with S units of servers is given by

1 21 0 1; , y s s m y E O C N C C R W W C S C S (15)

where 20 1( )C S C S is the set up and operating cost for repair service servers as in [1].

For y , the expected profit of OEM is given by (15) replacing W with1

W .

Option-2: for y , when the down time of the truck is above the target, the OEM incurs a

penalty cost, and hence costs incurred by the OEM are penalty cost, repair cost, PM cost, and setup cost for maintenance facilities. Whilst the revenues of the OEM will be the price of thecontract and an incentive earned when availability above the target. Hence, the expected profitis given by

2; =N Incentive Penalty cost Repair cost PM cost [Setup cost] . y G G E O P P E E E E E

(15)

We obtain expected penalty cost, expected incentive, the expected repair cost, and expected PMcost in [W, W+ ) per truck as follows.

Expected of Penalty Cost:

Let ji X and denote the down time (including repair and waiting times) after a failure occurs and

down time target of the truck in (0,t).Then a penalty occurs the OEM if the down time of the truckis above the target, ji X . Hence, the probability that the penalty incurs is given by

P ji X with ( ) P jiG X and the expected penalty cost is given by

( ) 0, X P j ji EP t C E E Max where P C is the penalty cost and 0,X (x) . ji E Max x g dx

According to [3], we have


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2( ) (W,W ) y y p EP R C x g x dx

( 17)

Expected Incentive:

To motivate the OEM improving the truck’s performance, some incentives will be given when the

down time for each failure is below the target. Let i denote the deviation down time to thetarget. It is assumed that the incentive earned is linear function of i and given by

20

( ) (W,W ) y y I EI R C x g x dx

(18)

Expected of repair cost:

Let mC denote repair cost for each failure, then the expected repair cost is given by

2( ) , y m y EC C R W W (19)

The expected of PM cost is . pm pm EC C As a result, the total expected profit of the OEM is

2

22 0 1

0

(W, W )

( )

G pm y

I p m

P C R

E O N C S C S C x g x dx C x g x dx C

(20)

For y ,the expected profit of the OEM choosing O2 is given by (20) replacing W with W 1.

Option-3: For case y , the expected profit is given by2

3 0 1( ;P ) y M M pm E O N P C C S C S (21)

For y ,the expected profit of the OEM choosing 3O is by (21) replacing W with W 1.

Optimal Solution:We consider a situation where both the owner and the OEM want to negotiate and determinejointly the terms and condition of the service contract to achieve a win-win solution. As a result,

we can use a Nash solution of the bargaining game in Osborne and Rubinstein [12] to obtain theoptimal solution.

Lemma 1. As both parties wants to achieve a win-win solution (and hence there exists thebargaining between two parties), then the probability of negotiations break down at the end ofperiod is sufficiently small, for every preference agreements, the owner and the OEM will receivethe same expected profit, i i[ ( )] [ ( )];i 1,2,3. y y E O E O

Proof: Using the sub game perfect equilibrium approach (Osborne and Rubinstein [7]), wehave,

1 1 11 1 0max [ ( )] [ ( )] W,W W,W 2 (W,W )

s

y y ji s y bC

E O E O NR NK R E X C R P C and

then by solving the above equation we get Cs that satisfies 1 1[ ( )] [ ( )] y y E O E O . Applying the

same argument as in Option 1, this completes the proof that i i[ ( )] [ ( )],i 1,2,3. y y E O E O Based on Lemma 1 we obtain the following results.

Proposition 1. For case y , there exist *S C , *

G P and * M P (which is unique and finite) such that

i i[ ( )] [ ( )];i 1,2,3 y y E O E O given by

1 1

*

1 20 1 0

( W,W ,1

2 ,

y ji m y

s

y b

K R E X C R W W C

R W W C S C S N P C

if 1 1 20 1 0W,W W,W . y ji m y b K R E X C R C S C S P C

(22)


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2 2

*

20 1

W, W 2 ( ) 2 ( ) ,1

2

y ji y y m y

G

pm b

K R E X EP EI C R W W P

C C S C S N C

if 2

W,W 0 y ji R E X .

(23)

* 2 2 20 11

W,W W,W2

M y ji pm m y b P K R E X C C S C S N C R C

if 2 2W, W W, W . y ji pm m y b K R E X C C R C

(24)

Proof: It is straight forward from Lemma 1.

Using the results from Proposition 1, we obtain the optimal expected profit of each party (theowner or the OEM) for 1O , 2O and 3O given by

* 1 1 21 0 0 1[ ( ; ) W,W W,W2

y s y ji b m y

N E O C K R E X P C C R C S C S N

(25)

* 2 2 22 0 1[ ( ; ) W,W W,W2 y G y ji pm b m y N E O P K R E X C C C R C S C S N (26)

2

*3

2 20 1

W,W;

2 W,W

y ji pm

y M

m y b

K R E X C N E O P

C R C C S C S N

(27)

For y , the optimal repair cost, *S C , *

G P and * M P and the optimal expected profit are given in

Proposition 1 and equations (25)-(27) by replacing W with 1W .

4 NUMERICAL EXAMPLE

We consider the failure distribution for Y=y given by the Weibull distribution,

( ; ) 1 exp( / ) , y y F t t and its hazard function is1( ) ( ( ) ) y yr t t where 0 0( ) y y y

.Let the parameter values be as follows. α0 =1(year),1/λ=1/300 (year), β=2.25,W =2(years),

U =2(3x104Km) (γ =U/W =1), y 0=1, η = 1.2, τ =3(years). Other parameter values are given in Table I.

TABLE I.COST PARAMETER VALUES

Parameter K Cb C pm Cm P 0 mC C p Ci

Value (103 $) 0.5 0.8 16.87 3.10-3 0.02 3τ Cpm 3Cm 3K 2 pC

*Assuming that equipment operates 2025 hours/year.

Tabel 2 shows the results (expected profit for each option, optimal option, and S*(N)) for =1.2, 1.6, and 2.0 coresponding to light incline, high incline and very hilly, respectively. For a

given usage rate (y=1.6) and land contour, S*(N) increases with N. This is to be expected sincethe set up cost for carrying out maintenance increases with S as the number of trucks increases.Note that if S > S*(N), then the OEM ‘s expected profit decreases due to increase in the set upcost and operational cost.

For the OEM’s and the owner, the maximum expected profit for Option 1, Option 2 andOption 3 decrease with the increasing in land contour, ρ . This is due to the equipment’savailability decreases as the land contour influences the degradation of the truck (which increase

the number of failures). Also shown in Table II, the optimal option for the OEM’s and the owner


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are Option 2.

TABLE II: OPTIMAL OPTION FOR 1.6 y (High usage rate )

ρ = 1.2

ρ = 1.6

ρ = 2.0

N N

N

10 30 50 70 90 10 30 50 70 90 10 30 50 70 90

*1 E O

12491

(3)*

33666

(5)

47583

(7)

51525

(9)

43221

(10)

11295

(3)

27414

(6)

30550

(9)

14571

(12)

No

Profit

9434

(4)

17314

(8)

1816

(12)

No

Profit No

Profit

*S C

13.58 14.51 15.92 17.81 17.67 8.15 9.12 10.89 13.36 - 4.99 6.10 8.05 - -

*2 E O

12907

2)*

35822

(4)

53170

(6)

62046

(8)

64528

(9)

11911

(3)

30118

(5)

39107

(8)

33961

(10)

8996

(13)

10270

(4)

22215

(7)

9661

(12)

No

Profit No

Profit

*G P

2304 1598 1403 1525 5.1010 1494 1211 1239 1172 5196 1329 1105 1168 - -

*3 E O

12411

(3)*

33287

(5)

46643

(7)

49694

(9)

32051

(10)

11133

(3)

26517

(6)

28453

(9)

10879

(12)

No

Profit

9221

(4)

16080

(8)

No

Profit No

Profit No

Profit

* M P

1327 1306 1288 1272 1038 1120 1156 1134 1120 1063 996 - - -

*O *2O *

2O *

2O *

2O *

2O *

2O *

2O *

2O *

2O *

2O *

2O *

2O *

2O None None

*S N 2 4 6 8 9 3 5 8 10 13 4 9 14 - -

*Number of optimal server in each option

5 CONCLUSION

We have studied three maintenance contracts for fleet which takes into account penalty andincentive. We model failure of the truck using the accelerated failure time (AFT) model allowingto incorporate age, usage and operating conditions into the model. The decision problems forboth the owner and OEM are formulated using a Nash game theory formulation. Numerical

examples are given to show the optimal option for the owner and the optimal number of serverfor the OEM. In this paper, we consider only one type of PM –i.e. full coverage PM. In many cases,the OEM offers more options –partial, moderate, and full coverage of service contract. Also, inthe paper, every failure is minimally repaired. One can consider a servicing strategy involvingimperfect repair. These further research topics are currently under investigation.

Acknowledgments: This work is partly funded by Hibah Desentralisasi 2014 from DGHE Indonesiaand Hibah RIK ITB 2014.

6 REFERENCES

[1] Murthy, D. N. P. and Ashgarizadeh E. 1999. Optimal decision making in a maintenance serviceoperation, European Journal of Operational Research, 31, pp. 259–273.

[2] Ashgarizadeh, E. and Murthy, D. N. P. 2000. Service contracts, Mathematical and ComputerModelling, 31, pp. 11–20.

[3] Rinsaka, K. and Sandoh, H. 2006. A stochastic model on an additional warranty servicecontract, Computers and Mathematics with Applications, 51, pp. 179–188.

[4]Jackson, C. and Pascual, R. 2008. Optimal maintenance service contract negotiation withaging equipment, European Journal of Operational Research, 189, pp. 387–398.


12/12


12

[5]Wang, 2010. A model for maintenance service contract design, negotiation and optimization,European Journal of Operational Research, 201(1), pp. 239 – 246.

[6] Wu, S. 2012. Assessing maintenance contracts when preventive maintenance is outsourced.Reliabiliy Engineering and System Safety , 98(1), 66 – 72.

[7] Iskandar, B.P., Pasaribu, U.S. and Husniah, H. 2013. Performanced Based MaintenanceContracts For Equipment Sold With Two Dimensional Warranties. Proc. of CIE43, Hongkong ,pp.176–183,.

[8] Iskandar, B.P., Husniah, H. and Pasaribu, U.S. 2014. Maintenance Service Contract forEquipments Sold with Two Dimensional Warranties. Journal of Quantitative and QualitativeManagement , (accepted).

[9] Pascual, R., Martinez, A., and Giesen, R. 2013. Joint optimization of fleet size andmaintenance in a fork-join cyclical transportation systems, Journal of Operational ResearchSociety , 64, pp.982-994.

[10]Mirzahosseinian, H., and Piplani, R. 2011. Compensation and incentive modelinginperformance-based contracts for after market service. Proc. of the 41st Int. Conf. onComputing Industrial Engineering, Singapore, pp.739–744.

[11]Iskandar, B.P., Murthy, D.N.P. and Jack, N. 2005. A new repair-replace strategy for items soldwith a two dimensional warranty.Comp.andOper. Research, 32(3),669–628.

[12]Osborne, M.J. and Rubinstein, A. 1994. A Course in game theory , Masssachusetts Institute ofTechnology.

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