how to choose between maintenance policies 1

8/13/2019 How to Choose Between Maintenance Policies 1

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ISSN 1750-9653, England, UKInternational Journal of Management Scienceand Engineering Management, 5(4): 261-267, 2010http://www.ijmsem.org/

Optimal maintenance/production policy for a

manufacturing system subjected to random failure andcalling upon several subcontractors

Sofiene Dellagi1 , Nidhale Rezg1 , Ali Gharbi2

1 Mechanical Department, University Paul Verlaine de Metz, Metz 57045, France2 Automated Production Engineering Department (GPA), Ecole de technologie superieure, Montreal (QC) H3C1K3, Canada

(Received 17 November 2009, Revised 9 March 2010, Accepted 17 May 2010)

Abstract. In this paper, we have developed an integrated maintenance policy for a manufacturing system subjected to a randomfailure and calling up on several subcontractor machines in order to satisfy a constant demand. In order to reduce the machine

failure a preventive maintenance plan is made. The machine is unable to satisfy a constant demand, thats why it called uponanother machine, comprising the so-called subcontractor machine, which produces at a certain rate the same type of product. Inorder to assure an economical objective, we have a choice between several subcontractors having some different data and to optimizethe preventive maintenance plan. Two strategies are developed and optimized in this work. The first strategy, single subcontractorstrategy, consists in choosing only one subcontractor among several ones. For this strategy we have proved analytically, that thechoice of the subcontractor machines is conditioned by the unit lost cost due an unsatisfied demand of one product; The secondstrategy, switching strategy, consists in relaying on one subcontractor machine and then switching to another. An analytical studyis developed in order to optimize firstly the preventive maintenance plan and secondly to determine the optimal switching date. Anumerical example is developed for very strategy in order to apply the analytical results.

Keywords: manufacturing system, subcontractor, decision making, economic, maintenance

1 Introduction

In todays increasingly competitive global industrialworld, the relationships between enterprises are getting im-proved towards more cooperation and collaboration. In thiscontext, many companies have recoursed to the industrialsubcontracting which became a very widespread practice toface competition (Amesse et al., 2001 [1]; Andersen, 1999[3]; Lehtinen, 1999 [14]), but also to cure the technologi-cal lack to manufacture a product with a sufficient effec-tiveness, or the incapacity to satisfy the demands withinthe deadlines (Bertrand and Sridharan, 2001 [5]). This re-course to subcontracting can also be justified by the will ofthe company to concentrate on core activities and resort toexternal sources and collaborate with external partners inorder to develop shared technological capabilities (Gomes-Casseres, 1994 [10]; Andersen and Christensen, 2000 [2]).Thus, the importance of subcontracting has increased, cre-

ating new challenges for companies (Andersen, 1999 [3];Lehtinen, 1999[14]). We find in the literature various stud-ies treating subcontracting in aeronautics (Amesse et al.,2001[1]), manufacturing (Lehtinen, 1999 [14]; Cagliano andSpina, 2002 [7]; Bertrand and Sridharan, 2001 [5]), con-struction (Lyonnet, 1988[15]), business and supply chains(Andersen, 1999[3]; Andersen and Christensen, 2000[2]).

Recently, new maintenance/production strategies by tak-ing into account the context of subcontractor are studied byDellagi et al. (2007) [9], Dellagi and Rezg (2007)[19]. A newmaintenance policy while taking into account machine sub-

contractor constraints has been developed and optimizedin Dellagi et al. (2007) [9]. A case study, which proves the

influence of the subcontractor constraints on the optimalmaintenance strategy adopted, has been presented in Del-lagi and Rezg (2007)[19]. Dealing with this frame, two casesof maintenance and production strategies, which are sub-contractor and contractor constraints, have been treatedin Dahane et al. (2008)[8]. We note that, the technical ap-proach used in these works deals with the joint optimizationof the maintenance and the production plans. We can seelately the adoption of this technical frame (Rezg et al., 2008[17]).

The system we consider consists on a manufacturing sys-tem composed of one machine which produces a single prod-uct in order to satisfy a constant demand. This machine isunable to satisfy the demand, thats why it calls upon an-other machine, comprising the so-called subcontractor ma-

chine, which produces the same type of product. From re-liability point of view, the machine is subject to randomfailures which are reduced by making a preventive mainte-nance plan.

We have to optimize the preventive maintenance planand to choose between several subcontractor machines. Thesubcontractor machines are classified according to theiravailability rate, and their unit production cost. The avail-ability rate is defined by the rapport of the number of thedemand satisfied by the number of the total demand in aconstant period. More precisely, we assumed that, when the

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262 S. Dellagi & N. Rezg & A. Gharbi: Optimal maintenance/production policy for a manufacturing system

subcontractor machine has high availability rate, it has afeed unit production cost too.

In order to assure an economical objective, we have toselect among the offers of the several subcontractor ma-chines. The remainder of the paper is organized as follows:In Section 2 we detailed the problem and Section 3 presentsthe problem formulation. In section 4 we developed andoptimized a single subcontractor strategy. In section 5, we

presented the switching strategy applied to two subcontrac-tors. Finally we conclude in section 6.

2 Problem description

The system under consideration consists of a produc-tion enterprise PE which assures the production of a singleproduct in a JIT context. The production enterprise PEmust assure the satisfaction of constant demand d. Sincethe production enterprise PE capacity is less then the de-mand d, it became unable to satisfy the demand. Thatswhy it called upon to another production enterprise PE,called subcontractor production enterprise SPE. The sub-contractor production enterprise SPE assure at a certainrate the production of the same type of product as theproduction enterprise PE. We noted that the productionenterprise PE activity is interrupted sometimes due to themaintenance action practiced on its machines. The downperiods motivates the calling upon to of the subcontrac-tor production enterprise SPE in order to reduce the de-mand loss in these periods. The backlog is not permitted,the demand not satisfied are lost and induces a demandlost cost. The production enterprise PE must make an eco-nomical choice between several subcontractor productionenterprisesSPE i which offer their services in order to meetthe need of the production enterprise PE. Its noted thatthe subcontractor enterprises S PE differ according to theiravailability rate, and their unit production cost. We haveestimated their availability rate by i, which is defined bythe number of the demand satisfied devised by the numberof the total demand in a constant period. The industrial

problem is illustrated in Fig.1.

Machine M

Subcontractor machine M1:

SM1

Decision

demand

Subcontractor machine Mi:SMi

Subcontractor machine Mn:SMn

Set of n subcontractor

machines

Fig. 1 Industrial problem

In order to formulate analytically the industrial prob-lem, we considered that the production enterprise PE ispresented by one of its machine M. Similarly, every sub-contractor production enterprise SPEi is presented by oneof its machines noted S Mi. Since the reduced problem con-sists in a machine M which is unable to satisfy a constantdemand, thats why it called upon to another machine, com-prising the so-called subcontractor machine, which assuresat a certain rate the same type of product in order to com-plete the rest of the demand. From reliability points of view,machine M is subject to a random failure. The probability

degradation law of machine M is described by the probabil-ity density function of time to failure f(t)and for which thefailure rate (t)increases with time. Failures of machine Man be prevented by a preventive maintenance action whichis scheduled according to its history. Concerning subcon-tractor machine we have estimated its availability rate ,which is defined by the number of the demand satisfiedby the number of the total demand in a constant period.

We have several subcontractor machines which offer theirservices in order to meet the need of the machine M. Itsnoted that the subcontractor machines differ according totheir availability rate, and their unit production cost. It isobvious that, more the subcontractor machine has a highavailability rate more than the unit production cost is high.The problem is to make an economical strategy in order tochoose among the subcontractor machines. We noted byi and Cpri respectively the availability rate and the unitproduction cost of the subcontractor machine Mi. For nsubcontractor machines we can assume that: If j > ithen C prj > Cpri{i,j} {{0,1, n} {0,1, n}} with i j.

3 Problem formulation

3.1 Notations

f : probability density function of time to failure formachine M;

F, R : failure function, reliability function ofM;p : mean time for preventive maintenance of mach-

ine M;c : mean time for corrective maintenance of mach-

ine M;W : up time period of machine M;D : down time of machine M

( i.e. D = F(m)c+R(m)p);d : demand quantity;i : S Mi availability rate, i.e., i= (demand satisfied

By S Mi/Total demand in a constant period);Cmc : corrective maintenance action cost;

Cm p : preventive maintenance action cost;Cpr : unit production cost of machineM;Cpri : unit production cost ofSMi;Cl : unit demand lost cost.

3.2 Control policy

The control production policy is defined as follows:

U(t) =

(Umax , d Umax) ifM is up and SMi is up(Umax , 0) ifM is up and SMi is down(0, Usmax) ifM is down and SMi is up

with i {0,1, , n} since the swiching state. Umax isthe maximal production rate of machine M, and Usmaxis the maximal production rate of subcontractor machineSM i(i {0, n}).

4 A single subcontractor strategy4.1 Definition

This strategy consists in making an economical choiceamong the subcontractor machines. It means that we willrelay on only one subcontractor machine. But, since thatthe subcontractor machines differ according their availabil-ity rate, and their unit production cost, we have to choosebetween.

4.2 Maintenance policy adopted for machineM

We assume for this strategy that we have a preventivemaintenance policy age type. Formally, the maintenance

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International Journal of Management Science and Engineering Management, 5(4): 261-267, 2010 263

actions practiced on the machine M are planned as follows:

To perform the preventive maintenance ofthe machine M at the age m,orTo perform a corrective maintenance of themachine M if it fails before the age m.

We assumed that the production cycle begins when the

machine M is up and the end of the production cycle cor-responds to the end of the preventive or a corrective main-tenance action practiced on the machine M.

4.3 Analytic study

In this study we begin by choosing between the two sub-contractor machines SM i and SMj and after that we willgeneralize the result for all subcontractor machines In orderto establish an analytical study for this problem, we haveconsidered two policies. The first policy, noted i, consistsin relaying only on the subcontractor SMi which has weakavailability rate i and low production cost C pri. The sec-

ond policy j(j > i ) consists at relaying only on the sub-contractor machine SMj which has high availability ratei and high production cost Cprj. We recall that j > i

and C prj > C pri. In order to decide between the two sub-contractorsS Mi and S Mj we have established the analyticexpression of the average total cost integrating maintenanceand demand loss cost of both policies i and j, noted re-spectivelyCTi andCTj. Studying the difference betweenthe two policies cost we can determine the economic strat-egy, relying on SMi and S Mj.

4.3.1 Analytic expression ofCTi

The analytic expression of average total cost integratingmaintenance action, production and demand loss of policyi is defined by:

CTi =

CprE(W)Umax+Cpri i[E(W)(dUmax)

+E(D) Usmax ] + Cl[E(W)(1 i)(d Umax)

+E(D)(diUsmax)]+CmcF(m)+CmpR(m)m

0 R(t)dt +F(m)c+R(m)p.

Component production costE(W) represents mean up time period of the machine

M. In this period the machine M produces at its maximalcadence Umax . Since that the production cost according tomachine M in this period (W) is defined by:

Cpr E(W) Umax . (1)

At this same period, when the demand arrives the subcon-tractor machine SMi is shrewd to assure the rest of thedemand quantity, i.e. (d Umax). But we recall the avail-ability rate ofSMi is i, then the quantity of the demandsatisfied by SMi in this period is: i (d Umax) E(W)

and then the production cost according to SMi in this pe-riod (W) is equal to:

Cpr i E(W) (d Umax). (2)

E(D)represents mean down time period of the machine M.In this period the subcontractor machine Mi produces at itsmaximal production rate since the control policy describedin section 3.2. Since that the production cost in this periodis defined by:

Cpr i E(D) Usmax . (3)

Summing the Eqs. (1),(2) and (3), we established the pro-duction cost during the production cycle:

Cpr E (W) Umax+ Cpri i[E (W) (d Umax)

+ E (D) Usmax ] . (4)

Component demand loss costDuring the up time period(W)the demand loss is caused

only by the subcontractor machine SMi. The demand lossin this period represents the complementary of the demandsatisfied by machine M in the period (W), i.e:

(1 i) (d Umax) E(W). (5)During the down time period(D), all demands are not sat-isfied by machine M because its under maintenance action,but the subcontractor machine Mi satisfied 0 of the totalnumber of these demands. We recall that the availabilityrate of SMi is i. If the demand is satisfied by the sub-contractor machine Mi, we have (d U

smax) demand lost;

else we have all the demand quantity d lost. Since that thequantity of the demand lost in this down period(D)is equalto:

E (D) (d Usmax) (i)+(1 i) E (D) d

=E (D) ((d Usmax) (i)+(1 i) d)

=E (D) (d iUsmax) . (6)

Using the Eq. (5) and (6) we conclude that the demand losscost is equal to:

Cl [E (W) (1 i) (d Umax)+E (D) (d iUsmax)] . (7)

Component the maintenance action costSince that we dont take into account the maintenance

action of the subcontractor machine, its proved in Jardineand Tsang (2006)[11], that the total expected maintenancecost by cycle related to machine M is equal to:

Cmc F(m) + Cmp R(m). (8)

Component the production cycle meanThe production cycle is composed by the up time period

Wand the down time period D. Analytically, in Jardine and

Tsang (2006)[11], the expected cycle length is expressed asfollowing: m0

R(t)dt +F(m)c+R(m)p. (9)

Using the Eqs. (4), (7), (8) and (9) we established the ana-lytic expression of the average total cost integrating main-tenance, production and demand loss of policy i:

CTi =

CprE(W) Umax+Cpri i[E(W)(dUmax)

+E(D) Usmax ] + Cl[E(W)(1 i)(d Umax)

+E(D)(diUsmax)]+ Cmc F(m)+CmpR(m)m

0 R(t)dt +F(m)c+R(m)p.

4.3.2 Analytic expression of policy j cost

The analytic expression of the average total cost inte-grating maintenance, production and demand loss cost ofpolicy j is similar to average total cost of policy i. Weonly replace i byj Cpri byC prj.

4.3.3 Optimal decision

In order to determine the economic strategy, relying onthe subcontractorSMior SMj, we have studied analyticallythe difference between the two policies cost expression andwe have established the following theorem.

Theorem 1. If Cl > Cl(i/j)decision then the strategy

j is more economic then the strategy i. With Cl >

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Cl(i/j)decision = (Cprj j Cpr i i+1)/(j i). Werecall thatC l represents a unit demand lost cost. Formally:If Cl > Cl(i/j)decision Ct

j Ctj 0 We selectedthe subcontractor SMj. With

Cl(i/j)decision=(Cprj j Cpri i)

(j i) .

The optimal decision is presented as Fig. 2 shows.

( )

( )( / )

j j i i

i j decision

j i

Cpr Cpr Cl

=

Choice between

SMi for i{0,1..n-1}

Cl0

SMi SMj

Fig. 2 Optimal decision

Proof. Using the expressions of the average total cost es-tablished in Section 4.3.1 we obtained:

CTj CTi =

[E(W)(d Umax) +E(D) Usmax ]

[(Cprj)j Cpri i +Cl(j i)]

m0 R(t)dt +F(m)c+R(m)p.

So its easy to see that:

[E(d Umax) +E(D) Usmax ] 0

and m0

R (t) dt +F (m)c+R (m)p > 0.

CTj CTi 0

Cprj j Cpi i

+ Clj i

< 0,

CTj CTi 0

Cprj j Cpri i

< Clj i

,

CTj CTi 0 (CprjjCpr i i )

(ji) < Cl

(remarkj i

> 0).

(CprjjCpr i i)(ji )

Cl CTi CTi 0.

4.4 Application theorem methodology

In order to apply the theorem established in Section 4.1.3we must calculate Cl(i/j)decision for

{i,j} {{0,1, n} {0,1, n}}/i < j.

Then we classify the value ofCl(i/j)decisionin order to apply

the theorem for every value ofCl (i/j)decision. Finally we can

obtain the optimal strategy of choosing one subcontractormachine between several subcontractor machines. In thenext section we will develop a numerical example which is

more detailed in the following approach.4.5 Numerical example

4.5.1 Numerical data

In this numerical example we suppose that the machineM has a degradation law characterized by a Weibull dis-tribution. The Weibull scale and shape parameters are = 100 and = 2, respectively. The times for the cor-rective and preventive maintenance actions the machineM are random variables characterized by exponential dis-tributions which means c = 30 tu and p = 20 tu, re-spectively. The following data are used for the other pa-rameters: Cpr = 150, m = 70 tu, d = 30/1 tu,Umax =

20unite/1 tu, Cmc = 2000 mu and Cmp = 500 mu, wheremuand tu denote respectively monetary unit and time unit.

We suppose that 4 subcontractor machines, SMi(i {0,1 ,2 ,3 ,4}), propose their offers to the machine M inorder to complete the rest of the demand. The maximalproduction rate of every subcontractor machine is definedbyUsmax = 20unite/1 tu. The unit production cost and theavailability rate of every subcontractor machine are defined

by:C pr0 = 100, Cpr1 = 175, Cpr2 = 190, Cpr3 = 200,0 =60%,1 = 70%,2 = 80%,3 = 90%.

4.5.2 Economical strategy

In order to apply the theorem established we beginby calculating the value of Cl(i/j)decision for ({i,j}

{{0,1, n} {0,1, n}}/i < j). Using the numerical data weobtained:

Cl(0/1)decision= 625, Cl(0/1)decision= 460,

Cl(0/3)decision= 400, Cl(1/2)decision= 295,

Cl(1/3)decision= 287, and Cl(2/3)decision= 280.

The optimal strategy is presented as Fig.3 shows.

Proof. For Cl [Cl(0/1)decision+ ]: we choose amongSM0, S M1, SM2 and SM3.

In this case (Cl [Cl(0/1)decision+]) :Cl > Cl(0/1)decision: We dont choose S M0, we choose S M1;Cl > Cl(1/2)decision: We dont choose S M1, we choose S M2;

Cl > Cl(2/3)decision: We dont choose S M2, we choose S M3.

Since that we choose SM3. We adopted the same strategyfor every interval in the decision table.

5 Switching strategy applied to two

subcontractors

5.1 Definition

The objective of this is to benefit from the advantage ofall subcontractors. It is easy to see that the subcontractorsare classified according to its availability rate and unit pro-duction cost. The proposed strategy consists in profitingsimultaneously from the less unit production cost of somesubcontractor in order to minimize the production cost andfrom the high availability rate of other subcontractor inorder to reduce the demand loss under the maintenanceperiods of the machine M. More precisely, the switchingstrategy, applied to two subcontractors, consists in rely-ing on a subcontractor machine SM1, which has less unitproduction costC pr1 and availability rate 1, and then toswitch to another subcontractor machine SM2, which hashigh unit production costC pr2and availability rate 2. The

decision of adopting firstly the subcontractor machine hav-ing less unit production cost and availability rate is justifiedby the fact that at the beginning of the production cyclethe machine M is considered as good as new. Since thatits economical to start the production cycle with relyingon subcontractor machine which has less unit productioncost and availability rate and when the failure rate of themachineM increases, we switched to another subcontractormachine having high unit production cost and availabilityrate. The optimization of this strategy consists in determin-ing firstly the optimal preventive maintenance date of themachine M and secondly to determine the optimal date ofswitching.



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Choice between

SMi for i{0,1,2,3}

Cl0 Cl(2/3)=280

SM0

Cl(1/3)=287 Cl(1/2)=295Cl(0/3)= 400 Cl(0/2)= 460 Cl(0/1)= 625

SM0 SM3 SM3 SM3SM0 SM0

Fig. 3 Optimal decision for case of four subcontractor machines

5.2 Maintenance policy adopted for machineM

For the switching strategy, the maintenance policyadopted is simply a preventive systematic maintenanceblock type. Formally:

To perform the preventive maintenance action for themachine M every K Ttime unit, K NorTo perform a corrective maintenance of the machineM at failure

with: kT represents the preventive maintenance action pe-riodicity. We still assume that the maintenance actions forthe subcontractors are not considered. The subcontractormachine unavailability, caused by maintenance action orother causes, is measured in the estimation of the availabil-ity rate .

5.3 Analytical study

The goal of the analytical study of the switching strategyis to determine the optimal preventive maintenance periodT, which minimizes the total expected cost per time unitfor preventive replacement, then the optimal switching datex minimizing the total expected cost per time unit inte-grating maintenance, production and demand loss. In thisstudy, we assume that the switching date x must be lessthan the optimal preventive date T determined. Its easyto see that it is economical to switch to the second subcon-tractor before beginning the preventive maintenance action.

The resolution of the problem is made in two steps:Step 1. Determination ofT.

Its is easy to see that T is the solution of the followingequation:

dCTm

dT |TT = 0,

with

CTm = Cmp +N(T)Cmc

T+ p.

CTm : the total expected cost per time unit for preventivereplacement,

N(T) : the expected number of failure in interval [0, T].

Determination dex

The total expected cost per time unit, integrating main-tenance, production and demand loss, according to theswitching date x is expressed as follows

CT=

Production cost+ demand loss cost+maintenance cost

the expected cycle length . (10)

The cycle adopted, in order to compute the total expectedcost per time unit, integrating maintenance, production anddemand loss, is similar to that used before in the determina-tion of the total expected cost per time unit for preventivereplacement, formally:

The expected cycle length = T+ p ,

Production cost= C pr UMmax[T N(T)c]+1

Cpr1

N(x)c USmax+(xN(x)c)

d UMmax

+2 Cpr2

(N(T

x) c) USmax+((T x)N(T x) c)

dUMmax

+pU

Smax

,

(11)

withN(x) : the expected number of failure in interval

[0, x],N(T x) : the expected number of failure in interval

[0, T x].

The demand loss cost =

Cl

(1 1)N(x) c d +(1 1) (x N(x)c)

d

UMmax

+1N(x) c (d Usmax)

(1 2) (N(T x) c d) ((1 2) ((T

x)

N(T x)) c

d UMmax

+2(d Usmax)+N(T

x) c

+2(d Usmax)

p

+(1 2)pd

(12)

From Jardine and Tsang (2006) [11], we have establishedthe total expected maintenance cost in interval [0, T]. Thetotal expected maintenance cost in interval

[0, T] =Cm p +N(T)Cmp. (13)

Proof. The production and demand loss cost in this partare similar to the cost established in Section 4.3.1. Only,in this analytical model, the computation of these costs isestablished on three different periods in the cycle: The firstperiod, x, corresponds simultaneously to the up time pe-riod of the machine M, and the period in which we hasadopted the machine MS1 as subcontractor machine. Thesecond period T x, always during up time period of themachine M, but we have switched in this period to the sec-ond subcontractor machine MS2. Finally, the third periodcorresponds to the preventive maintenance action periodwhich started at the date T. In the preventive maintenanceperiod we continue adopting machine MS2as subcontractormachine.

In order to formulate total expected cost per time unit,

integrating maintenance, production and demand loss wehave estimated every costs for every period.

The optimal switching date x is the solution of the fol-lowing equation:

dCT(x, T)

dx

x=x

= 0 withx < T. (14)

5.4 Numerical example

5.4.1 Numerical data

In this numerical example we suppose that the machineM has a degradation law characterized by a Weibull dis-tribution. The Weibull scale and shape parameters are



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= 100 and = 2, respectively. The times for the cor-rective and preventive maintenance actions the machine Mare random variables characterized by exponential distri-butions with means c = 20 tu and p = 5 tu, respectively.The following data are used for the other parameters:

Cpr= 2 , Cl = 40 mu, d= 30/1 tu,Umax = 20unite/1 tu,

Cmc = 2000 mu, and Cmp = 500 mu,

where mu and tu denote respectively monetary unit andtime unit.Concerning the two subcontractors SM1 and SM2, on

which we applied the switching strategy, we suppose thatthe maximal production rate of every subcontractor ma-chine is defined by Usmax = 20 unit/1 tu. The unit produc-tion cost and the availability rate of every subcontractormachine are defined by:C pr1 = 5.5 mu, Cpr2 = 20 mu,1 =60%,2 = 90%.

0 50 100 15010

20

30

40

50

60

70

80

90

100

T

. :

CTm

T*=42

Fig. 4 Curve ofC Tm according to T

0 50 100 150 200 250 3000

500

1000

1500

2000

2500

3000

3500

x

. :

CT

x* T*

Fig. 5 Curve ofC T according to x

5.4.2 Numerical results

Step 1.Determination ofT.Fig. 4 presents the curve of the total expected cost per

time unit for preventive replacement, CTm, according to T.Using MATLAB software we determined the optimal pre-

ventive maintenance period; T = 42tu.Step 2.Determination ofx .

Fig.5 presents the total expected cost per time unit, in-tegrating maintenance, production and demand loss, CT,according to the switching date x and with considering T

as a preventive maintenance period.Using MATLAB software we determined the optimal

switching date; x = 28,94 tu.

In order to justify the economical profit of the switchingstrategy, we will compare it with the single subcontractorstrategy. For the single strategy, considering only SM1 assubcontractor machine in the interval [0, T], is equivalentto make x = T. More precisely we dont switch to thesecond subcontractor. Naturally, considering only SM2 assubcontractor machine in the interval [0, T], is equivalentto make x = 0, we switch to the second subcontractor at

the beginning of the cycle. The numerical results of thiscompare are presented in the Tab. 1.

Table 1 Comparing between switching and single subcontractorstrategies

Single strategy Switching strategy

SM1: x = T = 42 tu SM2: x= 0 tu x= 28,94

CT 316, 14mu 340, 85mu 311, 85mu

The numerical results confirm that the switching strategyis economic comparing to the single subcontractor strategy.

5.4.3 Optimal switching strategy

This study proves that the optimal switching strategy

adopted, to make a preventive maintenance action everyT = 42tu. After the preventive maintenance actions, weconsider the subcontractor machine MS1 as subcontractormachine until x = 28, 94tu. At x = 24tu, we switch tothe second subcontractor machine MS2 until the end ofthe preventive maintenance actions.

6 Sensitivity analysis

The sensitivity study is made according to SM2 avail-ability rate 2 and the production cost Cpr2. The resultsof the sensitivity study are presented in the Tab. 2.

Table 2 Sensitivity analysis according to 2 and C pr2

Cpr 2 20 20 20 10 20 30

2 0,7 0,9 0,95 0,9 0,9 0,9x 41,02 28,94 27,02 15,27 28,94 37,19

Firstly, from the numerical results of the sensitivitystudy, we noted that, when the availability rate 2 in-creases, the value of the optimal switching date x de-creases. Its clear that, with high availability rate and thesame production cost C pr2, we have to switch early to thesecond subcontractor SM2 in order to profit of the highavailability rate of SM2 in the minimizing of the demandloss. This case is justified by the decreasing ofx value. Sec-ondly, when the unit production cost Cpr2 increases thevalue of the optimal switching date x increases. For a highvalue of unit production costC pr2and the same availabilityrate 2, we have to delay the switching date x

in order to

minimize the total production cost which became expensivesince the production cost according to SM2.

7 Conclusions

In this paper, we have reduced an industrial problemin order to make an analytical solution. The reduced prob-lem concerned an integrated maintenance production policyfor a manufacturing system subjected to a random failurecalling upon a subcontractor machine in order to satisfy aconstant demand. Points of view reliability, the manufac-turing system, which is subjected to random failure, canbe prevented by a preventive maintenance action which is



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scheduled according to its history. Several subcontractormachines can satisfy the need of manufacturing system.

Concerning subcontractor machines SMi we have esti-mated its availability rate iwhich is defined by the rapportof the demand satisfied number by the total demand num-ber in a constant period. We have to choose among severalsubcontractor machines SM i. Its noted that the subcon-tractor machines differ according to their availability rate,

and their unit production cost. It is obvious that, the morethe subcontractor machine availability rate increased, themore the unit production cost increased. Two strategiesare developed and optimized in this work. The first strat-egy, single subcontractor strategy, consists in choosing onlyone subcontractor between several subcontractors. For thisstrategy we have proved analytically, that the choice of thesubcontractor machines is conditioned by the unit lost costdue to an unsatisfied demand of one product. A numeri-cal example is made in order to prove the analytical result.The second strategy, switching strategy, consists in relyingon one subcontractor machine and then switching to an-other. For this strategy, an analytical study is developed inorder to optimize firstly the preventive maintenance planand secondly to determine the optimal switching date. A

numerical example is developed in order to apply the ana-lytical results.

References

[1] Amesse, F., Dragoste, L. and et al. (2001). Issues on part-nering: evidences from subcontracting in aeronautics. Tech-novationvol, 21(9):559569.

[2] Andersen, P. and Christensen, P. (2000). Inter-partner learn-ing in global supply chains: lessons from NOVO Nordisk.European Journal of Purchasing & Supply Management,6(2):105116.

[3] Andersen, P. (1999). Organizing international technologi-cal collaboration in subcontractor relationships: an investiga-tion of the knowledge-stickiness problem. Research Policy,28(6):625642.

[4] Beckmann, M., McGuire, C. and Winsten, C. (1956).Studies

in the Economics of Transportation. Yale University Press,New Haven, Connecticut.

[5] Bertrand, J. and Sridharan, V. (2001). A study of simplerules for subcontracting in make-to-order manufacturing. Eu-ropean Journal of Operational Research, 128(3):509531.

[6] Buzacott, J. (1967). Automatic transfer lines with bufferstocks. International Journal of Production Research, 5(3):183200.

[7] Cagliano, R. and Spina, G. (2002). A comparison of practiceperformance models between small manufacturers and sub-contractors. International Journal of Operations & Produc-tion Management, 22(12):13671388.

[8] Dahane, M., Dellagi, S. and et al. (2008). Development of

maintenance and production strategies in a subcontractingenvironment. International Journal of Production Research.Article accepted in 2008.

[9] Dellagi, S., Rezg, N., Xie, X. (2007). Preventive maintenanceof manufacturing systems under environmental constraints.International Journal of Production Research, 45(5):12331254.

[10] Gomes-Casseres, B. (1994). Group versus group: how al-liance networks compete. Harvard Business Review, 72, 6274.

[11] Jardine, A. and Tsang, A. (2006). Maintenance replacementand reliability theory and applications. Taylor & Francis.

[12] Kenne, J. and Gharbi, A. (2004). Stochastic optimal pro-duction control problem with corrective maintenance. Com-puters and Industrial Engineering, 46(4):865875.

[13] Kohl, J. (1841). Der Verkehr Und Die Ansiedelungen Der

Menschen in Ihrer Abhangigkeit Von Der Gestaltung Der Er-dorberflache. Dresden, Leipzig, Germany.

[14] Lehtinen, U. (1999). Subcontractors in a partnership envi-ronment: A study on changing manufacturing strategy. Inter-national Journal of Production Economics, 60-61:165170.

[15] Lyonnet, P. (1988). La maintenance: mathematiques etmethodes. Tec et Doc. editions TEC & DOC.

[16] Pigou, A. (1920). The economics of welfare. TransactionPublishers.

[17] Rezg, N., Dellagi, S. and Chelbi, A. (2008). Joint optimalinventory control and preventive maintenance policy. Inter-national Journal of Production Research, 46(19):53495365.

[18] Rezg, N., Xie, X. and Mati, Y. (2004). Joint optimization ofpreventive maintenance and inventory control in a productionline using simulation. International Journal of Production

Research, 42(10):20292046.[19] Dellagi, S. and Rezg, N. (2007). Production/maintenance

policies optimization in a subcontractor environment. In In-ternational Conference on Computers and Industrial Engi-neering, pages 178189, Alexandria Egypt, 20-23 October.



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