Maintenance Service Contracts for Repairable Product Involving Cooperative Relationship between OEM and Agent

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1 Mainenance eice Conacs fo Repaiable Poduc Inoling Coopeaie Relaionship beween OEM and Agen H. Husniah Depamen of Indusial Engineeing, Langlangbuana Uniesi,Kaapian 6, Bandun, Indonesia Tel: (+663) -4886, B. P. Isanda and R. angsapua Depamen of Indusial Engineeing, Bandung Insiue of Technolog, anesha, Bandung, Indonesia Tel: (+663) -5855, Absac. Common, mainenance seice conacs picall hae wo paies an Oiginal Equipmen Manufacue/OEM (o an agen) and a cusome. In man siuaions whee an exenal agen has capabili o poide coecie mainenance and peenie mainenance such as Mecedes Benz, he OEM and he agen would be compeiie o coopeaie in offeing he cusome wih seice conac opions. This pape deals wih a mainenance seice conac fo a epaiable poduc (such as dump-ucs, excaaos) wih he inolemen of he coopeaie elaionship beween OEM and agen as a seice-poide (P). ame heo appoach is used o model he ineacion among OEM, agen and cusome as he owne of he poduc. Unde semi-coopeaie games, he opimal sale pice fo he P and he opimal mainenance cos o epai cos fo he agen ae obained b maximizing hei pofis. The saisfacion of he cusome is also maximized b being able o choose one of he suggesed opions fom he P and he agen, based on he is paamee. Kewods: Mainenance, aan, Thee-leel seice conac, emi-coopeaie game.

2 . INTRODUCTION As loading and anspoing mining maeial ae mao acii in mining sies, we focus ou sud o he hea equipmens inoling in hose acii such as a dump uc. As he equipmens deeioae wih age and usage, an economical wa o ca ou mainenance is o ousouce he mainenance wos o an exenal agen o o an Oiginal Equipmen Manufacue/OEM due o is complexi and is expensie cos in mainaining he equipmens. Peenie mainenance (PM) acions ae done o peen fo excessie degadaion (whehe i is an age based o condiion based mainenance), while coecie mainenance (CM) is pefomed o esoe he failed equipmen o he opeaional sae. In ode o ge he maximum pofi he owne hae o manage he equipmens mainenance cos. On he ohe hand, he agen s decision poblem o he OEM is o deemine he pice of each opion offeed ha maximises is pofi oo. Mainenance seice conacs inoling epaiable iems hae eceied aenion in he lieaue (see [], [], [3]). Fuhe moe, [4], [5], [6] and [7] has puposed an incenies o moiae he he agen o incease he equipmen s pefomance beond he age bu he hae no consideed he PM leel and numbe of PM in doing he mainenance acii duing he conac. In his pape, we assume ha he OEM and he agen coopeae and ac as an inegaed seice-poide and popose a wo dimensional seice conac whee he seice-poide poposed impefec PM in ode o educe he numbe of failue duing he conac coeage. The pape is oganised as follows. In secion we gie model fomulaion fo he seice conac sudied. ecions 3 and 4 deal wih model analsis o obain he opimal pice sucue fo he seice-poide and he opimal seice opion fo he cusome as he owne. Finall, we conclude wih opics fo fuhe eseach in ecion 5.. MODEL FORMULATION. aan Polic e conside ha each dump uc puchased is coeed b a wo-dimensional waan, and he waan also coes PM o poide moe poecion o he bue. All failues unde waan ae ecified a no cos o he bue. Hence, afe he waan ends, he esponsibili o do mainenance (CM and PM acions) shifs o he bue (o he owne). The waan coeage is chaaceized b a ecangle egion,, U whee and U ae he ime, and he usage limis. Fo a gien usage ae () of a dump uc, he waan ceases a fo U, o U, fo U (ee Fig.). The decision poblem fo he OEM is o deemine he opimal PM degee accoding o aious usage paen and he mining opeaional condiion ha minimizes he expeced waan cos.. Mainenance eice Conac e conside a wo dimensional mainenance conac whee he conac has wo limis (o paamees) epesening age and usage limis (e.g. he maximum coeage fo L (e.g. ea) o K (e.g.. m). Hence he seice conac is chaaceised b a ecangle egion (see Fig.). Fo ( U / ) he egion is gien b L U U K L U U K [,, ] and [(, ), ] fo whee / and U. U The seice conac offeed us befoe he waan ends ae consideed as follows. Hee we assume he OEM and he agen coopeae ogehe as a seice-poide (P) in offeing seice conac. Fo a fixed pice of seice conac P, he P agees o pefom boh PM and CM (full coeage) fo a peiod of ime, L o usage K, whichee occus fis. The conac sas a he end of waan,, hee he P assues a minimum down ime (epai ime and waiing ime) fo each failue as saed in he conac. As he mainenance seice is full coeage (PM and CM), hen a penal cos incus he P if he acual down ime falls aboe he age. Bu if i falls below he age, he P will ean an incenie. If he down ime fo each failue oe he conac be D() is moe han he down ime age, hen he P should pa a penal cos. The amoun of he penal cos is popoional o D (). The penal cos, CP is iewed as a penal gien b he P. The P eans some incenies C I if. The decision poblem fo he P is o deemine he opimal pice sucue (i.e. he pice of seice conac and he opimal PM degee accoding o aious usage paen and he mining opeaional condiion ha maximizes he expeced pofi..3 Failue modelling.3. Appoaches o modelling failues e use he one dimensional appoach deeloped in [9] o model poduc failues. Le Y be he usage ae fo a gien

3 uc. I is assumed ha Y aies acoss he ucs bu i is consan fo a gien uc. Fo Y =, he condiional hazad funcion fo he ime o fis failue is gien b () which is a non-deceasing funcion of (he age of he uc) and. e conside ha usage ae of he uc and a land conou of a mining aea whee he uc is opeaed ma songl affec he degadaion of he uc. To incopoae he effec of usage ae and he opeaing condiion on degadaion of he uc, we use he acceleaed failue ime (AFT) model as in [8]. Usage U K [ ( ) L ] whee [] is an inege alue..3. Modelling of PM effec Fo a gien usage ae, he effec of impefec PM acions on he inensi funcion is gien b ( ) ( ) wih i i ( ), denoes he educion of h he inensi funcion afe,, PM acion. If he PM h acion is done a, he inensi funcion is educed b, hen fo he inensi funcion is gien b ( ) ( ). i i U U K Usage U Usage U w U U U L L Age Fig. aan egion and seice conac Region fo and. If he disibuion funcion fo T is gien b F (T, α ), whee α is he scale paamee, hen he disibuion funcion fo T is he same as ha fo T bu wih a scale paamee gien b () wih whee is a paamee epesening he opeaing condiion of a uc. Hence, we hae F(, ) F (( ), ). The hazad and he cumulaie hazad funcions associaed wih F(, α ) ae gien b ( ) f (, ) ( F(, )) and R( ) (x) especiel whee f(,α ) is he associaed densi funcion. Peenie Mainenance Polic: e conside ha fo a gien Y, PM done b he OEM duing he waan peiod and and he P afe waan expied ae an impefec epai. The PM polic fo a gien, is chaaceised b single paamee [ ] duing a [ ]. The equipmen is peiodicall mainained. [ l. ]. An failue occuing beween pm is minimall epaied (ee Fig. ). Noe ( ) U U (a). PM egion fo L Age L (b). PM egion fo Fig. The wo dimensional PM egion wih. Fo simplici we assume ha fo each PM acion hen ( ) ( ) (ee Fig. 3). If an failue occuing beween pm is minimall epaied, hen he expeced oal numbe of minimal epais in ([, ), ) is gien b N ( ) d R( ). Fo hen he expeced numbe of minimal epais in [,) is defined as, N N R whee R ( ) ( ) d. And afe he waan ends, he expeced numbe of minimal epais in [, +L), wih m is gien b () Age

4 N L N(, ) R, L L, m m, R L L i i i m L whee R, L ( ) d. l (3) mainenance seices and hence he seice poide has moe bagaining powe in he conac negoiaion. As a esul, we can iew he P as he leade and he owne as he followe. Fuhemoe, we conside he owne condiion i.e is as gien b: () () e, is aese U( ), neual (4) m L L Time 3. Owne s Decision Poblem e obain he owne s expeced pofi fo wo opions and each needs o conside wo cases i.e. (i) and (ii). Fig 3. Failue ae funcion fo Y Fo case (i), Noaions:,U X i D() F(), L Y C C m C C C P ( O.) :aan ime, and usage limis :Downime caused b he i-h failue and waiing ime :Toal epai ime allowed :Toal downime in (,] :Disibuion funcion of downime :Reenue, mainenance conac legh :Usage ae :Repai cos done b OEM :Repai cos done b P :Peenie mainenance cos pe PM :Addiional cos fo leel PM ceaed pe PM :Penal cos pe uni of ime :Pofi owne ( O.) :Pofi OEM C b :The poduc cos oe he conac peiod P :PM cos done in-house oe he conac peiod F (, ) :Condiional failue disibuion fo a gien usage ae ( ), R( ) :Hazad, and Cumulaie hazad funcions associaed wih F (, ) 3. MODEL ANALYI e conside a siuaion whee he P is he onl poide of Opion O : he expeced pofi is gien b E N(, ) g( ) d C N(, ) P C s b ih he expeced uili funcion, E U ( O ) C N(, ) s e P Cb e e whee N(, ) R. Opion O : he expeced pofi pofi of he owne is gien b E Pofi) E[ upime eenue] E[ Pinal cos] E[ Incenie cos] E[ eice Conac cos] E[ Puchase cos] E N(, ) g( ) d EP( L) E Incenie P C hee EP( L) (5) (6) (7) is he expeced penal iewed as a compensaion eceied b he owne (see 3.). Fo case (i), he expeced uili funcion is ( ) ( ) Cp x C (, ) ( ) I x N e g x e g( x) P Cb E U ( O ) e e (8) b Fo case (ii),he expeced pofi of he owne is gien b (5) and (7) eplacing wih and L wih L.

5 3. eice-poide s Decision Poblem Hee, we conside on wo cases i.e. (i) and (ii) Fo. Duing case (i),, he OEM s expeced cos is gien b = PM cos CM cos E Cos E E (9) The expeced PM and epai cos condiional on Y=, is, E C R C C C Opion O : he P s expeced pofi is gien b E Cs Cm N(, ), Cs Cm C, () Opion O : duing, he P s expeced pofi is gien b = Incenie Penal E P E E E whee, E E PM E CM = Expeced of Penal Cos: () Le D () and denoe he sum of down ime afe a failue (including epai ime), and down ime age of he equipmen in (,). The expeced penal cos is gien b EP ( L ) C ( ) N, P whee, P ( ) z g( z) dz C is he penal cos and N(, ) denoes he expeced numbe of failue in ineal (, L]. Expeced Incenie Cos: The expeced of incenie eaned in (, L] is gien b EI( L) CI z dz Expeced of CM cos: Le C m is minimal epai cos hen he expeced epai is gien b EC(, L) C N(, ) m whee N(, ) is expeced numbe of failues in (, L]. Expeced PM cos ih cos of PM is gien b C C m m hen he expeced PM cos is E[ PM cos] C C L m C l m hee [ ( m ) (( m ) )] m Fo case (ii), he expeced pofi of he P is gien b (9) and () bu i needs o eplace wih and L wih L. If he owne chooses Opion O, he P.s expeced pofi becomes E 4. OPTIMAL OPTION () This secion pesens he opimal opion of he owne b maximizing he expeced pofi fo each opion, an d hen we find he opimal pice and epai cos fo he seice-poide. 4. Owne s Opimal Opion e fis obain he opimal opion fo O and hen fo O. Two cases need o be consideed i.e. (i) and (ii). Fo, Opion O is pefeed o Opion O if ( ; ) EU ( O ; P ) E U O C, and Opion pefeed if EU( O ; C ) EU ( O ; P ). The owne is indiffeen beween wo opion if ( ; ) EU ( O ; P ) E U O C, which is equialen o ( ) Cs Cp x CI ( x) P P N(, ) 4 e e g( x) e g( x) () O is Le C expess he igh-hand side of equaion (). s Then, Opion O is compaed wih Opion O. Opion O is bee han Opion O if E U( O ; C ), and if E U( O ; C ), Opion O is pefeed. B soling

6 E U( O ; C ) wih espec o C, we hae C C P ln Cb R (3) Now, we compae Opions O and O as follows. Define P as he alue ha saisfies Cp ( x ) CI ( x) P Cb N(, ) e g( x) e g( x) E U( O ; P ). e hae, (4 ) Le i i,,, ;, be defined b P C P P C C P, C ; P C, P P P, C ; P C, C C Then, he opimal opion fo he owne becomes O, if P, C * O P, C O, if P, C O, if P, C Fo, C and (5) P ae gien b (3) and (4) eplacing wih. Using (5) we ge he opimal opion fo he owne. 4. eice-poide s Opimal Opion The P s opimal opion fo P and C is obained b maximizing he P s expeced pofi based on he owne s * opimal opion, consideed i.e. (i) Fo he P s de O P C. Again, wo cases need o be and (ii). P, C, he owne opimal opion is O. Hence expeced pofi is gien b eq. (). ince, he P s expeced pofi becomes dcs maximum b a ceain poin on he cue P C. B subsiue C C o (), he expeced pofi P is gien b eq. (6). Then, he maximum expeced pofi of he P is obained * * * P C + and C C. fo s * * fo P P and C C. P, C, he owne opimal opion is O. Finall, fo The P s expeced pofi is gien b eq. (). The opimal opion fo he P is he one gies a lage posiie expeced pofi beween Opion O and Opion O. If boh ae negaie, hen he bes opion is Opion O (o do no bu an opion offeed). * Fo case (ii), he opimal epai cos, P and he opimal expeced pofi ae gien in (4) and (7) b eplacing wih and L wih L. Hee, b using acelbeg game heoeic fomulaion he opimal decision is he one ha maximizes he expeced pofi boh he seice-poide and he owne. Then, we * hae numbe of PM opimal, PM degee opimal ( ) and he opimal expeced pofi of each pa gien b he opimal leel mainenance fo he owne, and he opimal pice fo he seice-poide. 5. CONCLUION In his pape, we hae sudied mainenance conacs which aen ino accoun is aiude of he owne o he conac and deeloped models o deemine he opimal opion fo he owne and he opimal pice fo he P. In his pape, we conside onl one seice poide i.e. OEM and agen. In man cases, he OEM offes moe opions wih waan and wihou waan and he agen gies seice conac afe he expied of waan. These fuhe eseach opics ae cuenl unde inesigaion. P, C, he owne opimal opion is O. The Fo P s expeced pofi is gien b eq. () and becomes maximum if C P. ubsiue P P o (), we hae eq. (7). Then, he maximum expeced pofi of he OEM is obained

7 P C E ln C R b m R (6) Cp ( x ) CI ( x) E Cb N(, ) e g( x) e g( x) l CI ( x) g( x) Cp ( x ) g( x) CmN(, ) C Cm L m C m m (7) ACKNOLEDMENT This wo is funded b he Minis of Reseach, Technolog, and Highe Educaion of he Republic of Indonesia hough he scheme of Desenalisasi PUPT 6, No. FTI.PN REFERENCE [] E. Ashgaizadeh, and D. N. P. Muh, eice conacs, Mahemaical and Compue Modelling, ol. 3, pp.,. [] K. Rinsaa, and H. andoh, A sochasic model on an addiional waan seice conac, Compues and Mahemaics wih Applicaions, ol. 5, pp , 6. [3] C. Jacson, and R. Pascual, Opimal mainenance seice conac negoiaion wih aging equipmen, Euopean Jounal of Opeaional Reseach, ol. 89, pp , 8. [4] Isanda, B.P., Husniah, H. and Pasaibu, U.. Mainenance eice Conac fo Equipmens old wih Two Dimensional aanies. Jounal of Quaniaie and Qualiaie Managemen, 4. [5] Isanda, B.P., Pasaibu, U.. and Husniah, H. Pefomanced Based Mainenance Conacs Fo Equipmen old ih Two Dimensional aanies. Poc. of CIE43, pp.76 83, Hongong, 3. [6] Isanda, B.P., Pasaibu, U.., Caaasia, A. and Husniah, H., Pefomance-based mainenance conac fo a flee of dump ucs used in mining indus, Poc. of TIME-E, Bandung, 4. [7] Husniah, H, Pasaibu, U.., Caaasia, A. And Isanda, B.P., Pefomance-based mainenance conac fo equipmen used in mining indus, Poc. of ICMIT, ingapoe, 4. [8] Isanda, B.P., Muh, D.N.P. and Jac, N. A new epaieplace saeg fo iems sold wih a wo dimensional waan.comp.and Ope. Reseach, 3(3),669 68, 5.

STUDY OF THE STRESS-STRENGTH RELIABILITY AMONG THE PARAMETERS OF GENERALIZED INVERSE WEIBULL DISTRIBUTION

Inenaional Jounal of Science, Technology & Managemen Volume No 04, Special Issue No. 0, Mach 205 ISSN (online): 2394-537 STUDY OF THE STRESS-STRENGTH RELIABILITY AMONG THE PARAMETERS OF GENERALIZED INVERSE