Optimal Production Control of Hybrid Manufacturing/Remanufacturing Failure-Prone Systems under Diffusion-Type Demand

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1 Appled Mahemacs, 3, 4, hp://dx.do.org/.436/am Pblshed Onlne March 3 (hp:// Opmal Prodcon Conrol of Hybrd Manfacrng/Remanfacrng Falre-Prone Sysems nder ffson-type emand Samr Oare, Vladmr Polosk, Jean-Perre Kenné, Al Gharb Mechancal Engneerng eparmen, École de Technologe Spérere, Monreal, Canada Emal: samr.oare.@ens.esml.ca, vladmr.polosk@esml.ca, jean-perre.kenne@esml.ca, al.gharb@esml.ca Receved Ocober 4, ; revsed Janary 3, 3; acceped Febrary 7, 3 ABSTRACT The problem of prodcon conrol for a hybrd manfacrng/remanfacrng sysem nder ncerany s analyzed. Two sorces of ncerany are consdered: machnes are sbjec o random breakdowns and repars, and demand level s modeled as a dffson ype sochasc process. Conrary o mos of sdes where he demand level s consdered consan and fewer resls where he demand s modeled as a Posson process wh few dscree levels and exponenally dsrbed swchng me, he demand s modeled here as a dffson ype process. In parclar Wener and Ornsen- Uhlenbeck processes for cmlave demands are analyzed. We formlae he sochasc conrol problem and develop opmaly condons for n he form of Hamlon-Jacob-Bellman (HJB) paral dfferenal eaons (PEs). We demonsrae ha HJB eaons are of he second order conrary o he case of consan demand rae (correspondng o he average demand n or case), where HJB eaons are lnear PEs. We apply he Kshner-ype fne dfference scheme and he polcy mprovemen procedre o solve HJB eaons nmercally and show ha he opmal prodcon polcy s of hedgng-pon ype for boh demand models we have nrodced, smlarly o he known case of a consan demand. Obaned resls allow o compe nmercally he opmal prodcon polcy n hybrd manfacrng/ remanfacrng sysems akng no accon he demand varably, and also show ha Kshner-ype dscree scheme can be sccessflly appled for solvng nderlyng second order HJB eaons. Keywords: Sochasc Conrol; Manfacrng Sysems; Opmzaon; Falre; Random Process. Inrodcon In recen years, he reverse logscs framework allowng he nfed analyss of manfacrng plannng and he nvenory managemen has ganed a sbsanal neres among he researchers workng n he feld. In a book [] ahor descrbed he anave models o represen he acves of remanfacrng and recyclng n he conex of reverse logscs emphaszng hree sses: namely: dsrbon plannng, nvenory managemen and prodcon plannng. In he srvey [] ahors analyzed more han sxy case sdes n reverse logscs pblshed beween 984 and and dscssed nework srcres and acves relaed o he recovery of prodcs p o he end of lfe. Varos opmzaon models for spply chans wh a recovery of rerned prodcs have been proposed wh specal aenon o he prodcon conrol and nvenory managemen sng boh, deermnsc and sochasc approaches. In he majory of prevos sdes dscree me (as opposed o connos me) sengs s sed. In [3] ahors presen an effecve approach o deermne he dscree polcy of he opmal conrol for a sysem wh prodc recovery, akng no accon he ncerany n he demand of new and rerned prodcs. They model he demand and he rern as dscree ndependen random varables. In [4] a new dscree sochasc nvenory model for a hybrd sysem s proposed: new and rerned prodcs are manfacred separaely, he demands are ndependen b prodcon polces are synchronzed. Ahors of [5] develop a perodc nvenory model of on a fne plannng horzon wh consderaon of prodcon remanfacrng and dsposal acves. In [6] ahors propose a model of dscree me sochasc opmzaon for a hybrd sysem akng no accon, he prodcon, sbconracng, remanfacrng of rerned prodcs, rern marke of poor aly prodcs he prodcon lne and dsposal acves. The demand s a random varable normally dsrbed and he rern of prodcs depends on he demand. A connos me opmzaon model s consdered n [7] for he prodcon, remanfacrng and dsposal n a dynamc deermnsc sengs. Copyrgh 3 ScRes.

2 S. OUARET ET AL. 55 Manfacrng sysems sbjec o random breakdowns and repars were sysemacally analyzed n [8-] n connos me sng sochasc opmzaon echne. Recenly n [] hs mehodology has been exended o address he global performances of he manfacrng sysem wh he spply chan n closed loop. A sochasc dynamc sysem conssed of wo machnes dedcaed respecvely o manfacrng and remanfacrng; he random phenomena are breakdowns and repars of he machnes, he demand of new prodcs was consdered deermnsc and known, he rerned prodc was a poron of hs demand. The consan demand s a prevalng assmpon n he large body of he research devoed o sochasc connos me opmzaon of prodcon managemen n falre-prone sysems. Some papers develop opmaly condons and se hem for searchng nmercal solons []; ohers presen analycal solons as he recen arcle [3]. In mch fewer sdes where he random demand s analyzed s mos ofen modeled as a Posson process. Ths approach allows o keep he sal framework of random dscree evens changng he sae of he sysem for boh machne breakdowns and demand jmps [4]. Posson-ype demand s sed more sysemacally n nvenory opmzaon problems [5]. A combned model: Posson process copled wh he dffson process has been recenly proposed n [6] for modelng he demand n nvenory problem. In fac dffson-ype processes were sed for modelng he demand n he classcal paper [8] were opmaly condons have been obaned, however was he only sorce of random behavor snce he machne breakdowns were no consdered. The sysem consdered n hs paper conans reverse logscs loop wh manfacrng and remanfacrng branches revsng he model proposed n [4]. We se connos me sochasc conrol approach and adop he dffson-ype componen no he demand model mergng hs sorce of random behavor wh random machne breakdowns descrbed by Posson process as n [9,]. As a drec conseence of an adoped demand model he opmaly condons lead o he Hamlon- Jacob-Bellman (HJB) eaon of he second order. Second order HJB s ofen me n opon prce modelng, b for sochasc conrol n manfacrng sysems he HJB s sally of he frs order [9,,4]. Analyzng he second order HJB we se he Kshner fne dfference approxmaons and he polcy mprovemen algorhm [7]. The paper s srcred as follows. In Secon we descrbe he model of he hybrd sysem conssng of machnes. The frs machne ses prmary prodc, and he second rerned prodc; boh are sbjec o breakdowns and repars consng he frs sorce of ncerany. We descrbe n deals or demand model sng dffson ype random processes consrced as an op of shapng fler exced by he whe nose. We sdy versons of sch model smple Brownan moon and frs order Markovan process. Laer verson seems more realscally f he real world saons. In Secon 3 we derve opmaly condons n he form of Hamlon-Jacob-Bellman (HJB) eaons whch are second order paral dfferenal eaons (PEs) for he chosen demand model. In Secon 4 we descrbe he nmercal mehod based on fne dfference approxmaons and polcy mprovemen approach followng he mehodology proposed n [7] and also n []. In Secon 5 we apply he developed mehodology o he manfacrng sysem descrbed n Secon, compe he opmal prodcon polcy and show ha s of classcal hedgng pon ype. In conclson we dscss he proposed mehodology and obaned resls, and olne he possble drecons for fre works.. Model of a Hybrd Manfacrng Sysem Sable for Sochasc Conrol We consder a hybrd manfacrng/remanfacrng sysem conssng of wo parallel machnes denoed M and M respecvely, prodcng he same ype of prodc. Sochasc phenomena are demand level and machne breakdowns/repars. We ake no accon he acvy of prodcon n forward drecon and he acvy of relzaon of rerned prodcs n reverse logscs. The demand ms be sasfed by nvenory for servceable ems. Ths nvenory wll be bl by he prodcs manfacred or resed. The rerned prodcs wll be n he second nvenory namely recovery, hey can be remanfacred, or be hold on sock for fre remanfacrng. In or problem, we assme ha he maxmal prodcon raes for each machne are known and he machne M s prodcng a average spply for rern rae, whch s also s maxmal rae. Ths saon s llsraed n Fgre. Sae of he machne M wh, s modeled as a Markov process n connos me wh dscree sae, wh B, ( B machne s operaonal, B machne s o of order). We may he defne j B BB,,,,,,,,,3,4 Sae ranson dagram s shown n Fgre. Hybrd sysem s n prodcon whle n modes, and 3. Transon probables from sae o sae for machne M P. o f (), B. o f Copyrgh 3 ScRes.

3 55 S. OUARET ET AL. Fgre. Sysem srcre. (, ) (, ) (, ) (, ) Fgre. Sae ranson dagram. o Wh,, lm. Sae ranson (4 4) marx Q s herefore gven by Q Sae eaons can be wren n he smplfed form x d x R () (3) R raes are Snce he demand d and rern consdered as sochasc processes he more rgoros Io form of Eaons (3) wll be sed laer. Namely le d() be a saonary Gassan process wh he consan mean and varance d ~ N,, where ~ N,. Below we frher specfy n one of wo ways: eher an ncremen of a sandard Brownan moon, or an ncremen of he frs order Markov process defned laer sng he shapng fler. For he rern (remanfacrng) rae, an assmpon s made ha s proporonal o he csomer demand rae R rd R R wh r s a percenage of rern. Sochasc sae dfferenal Eaons (3) can be rewren n Io form sng noaon z x x R z R (4) Eaons (4) wll be also sed n he followng generc form: x x f x,,, w, x f x,,, g z f x,,, g For he Case A he np z o Eaons (4) s specfed as a sandard Brownan moon ncremen z W. For he Case B he np z o (4) s specfed as an ncremen of he shapng fler op (Ornsen-Uhlenbeck process) (5) z az bw where a, b (6) Process z s a frs order Markovan, s correlaon fncon s k b aexp a. Addonal nsgh o he proposed demand model can be gven by consderng he cmlave demand: V. where s a consan demand ramp, V. s a randomly varyng poron of he demand. For he case A. V W (Wener process), for he case B. V z (Ornsen-Uhlenbeck process). Also case A can be obaned from B seng a, b. Followng consrans have o be added o (5)-(6) x Le he cos rae fncon o be defned as follows: G x, x, cx cx cx c p c c wh B, r Here x, max, x,, x max x, c, c : nvenory holdng and backlog coss for manfacred prodc (per me n); c : nvenory holdng cos for remanfacred prod- (7) (8) Copyrgh 3 ScRes.

4 S. OUARET ET AL. 553 c (per me n); c, p c r : prodcon coss for manfacrng and remanfacrng processes (per n); c : manenance cos for nonoperaonal sae of he machnes: where r 4, c c Ind c Ind 3 r cr cr Ind Ind P. f P. s re oherwse The objecve s o deermne he prodcon raes. and. n order o mnmze he expeced dsconed cos ( s he dscon rae): J, x, x, E e G x, x, d,,.,. x x x x (9) The doman of admssble conrols s defned as.,. R. UmaxInd,. UmaxInd.,. () efned hybrd sysem s sad o be meeng feasbly condon f 4 U U U U U max max max max 3 max E d () where e U max are lmng probables and maxmal prodcons raes. We recall ha he vecor of lmng probables s defned as an egenvecor of he ranson marx Q(.) 4 e. Q. () 3. Opmaly Condons for Sochasc Conrol Problem Le s defne he vale fncon s defned as a mnmm (nfmm) of expresson (9) over all possble conrol nps:.,., x, x nf J, x, x, B (3) Le s brefly recall he gdelnes for obanng opmaly condons. Inrodcng me-dependan α-dependan cos fncon and vale fncon we have: s,, J (, ) E e G x s, x s, s d s x x x x (4) Accordng o Bellman opmaly prncple for cos fncon a we can wre, x, x, s e,,, x, mn E e G x s, x s, s ds mn E G x, x, e (, x, x x (5) Usng Taylor expanson for he erm e and he vale fncon, x, x, over las 3 argmens, and keepng lnear erms over and p o second order erms over x we ge:,,, e, x, x, x x x x x x x x x x x x xx x x xx x x o (6) Second order erms over x are kep for frher analyss becase of dffson-ype processes affecng sysem dynamcs. One more echncal sep consss of compng he vale fncon, x, x, sng Markov chan-ype machne dynamcs () defned hrogh ranson probables, x, x,, x, x,, x, x, Mergng Eaons (6) and (7) we ge e, x, x, x xx x x o x, x, x,, x, x, xx x x xx x x x x x x (7) (8) Copyrgh 3 ScRes.

5 554 S. OUARET ET AL. Averagng over random realzaons of he demand drven by he Brownan np w, sng Eaons (5) and applyng Io s lemma we ge: E g g w g Ew g Ew x x x x E x x E x x,,, E w E x g E x g E xx g g Now neglecng all erms of order hgher han over, akng lm, and consderng he saonary re- gme ge HJB eaons n he followng form:, x, x,, x, x, x, x.,. mn G x, x, x x, we fnally, x, x f f (9) g g g g, x x xx, B 4. Nmercal Mehod Polcy Improvemen A nmercal approach proposed by Kshner n [7] and sccessflly sed n he seres of works [9,] consss of nrodcng he grd n he sae space x, x for approxmang he vale fncon, x, x approxmang he frs dervaves by p wnd fne dfferences, hen se polcy mprovemen dscree analog of a graden descen n polcy (conrol) space. Use of p-wnd dervaves resls n condonal compaons b grealy mprove convergence of he nmercal. 4.. Compaons of Frs ervave To descrbe condonal compaons of he dervaves le s nrodce he followng noaon: K Ind, K Ind f P. s re wh Ind P. oherwse The frs dervaves of he vale fncon wh respec o x and x are:, x, x x, x, x, x, xk, x, x, x, xk, x, x x, x, x, x, x () () I worh emphaszng ha here s no condonal compaon for x snce R all he me de o assmpon descrbed n Secon. 4.. Compaons of Second ervaves For xx, x, x and for xx, x, x, x, x xx, x, x we have, x, x, x, x xx, x, x, x, x, x, x, x, x () (3) Boh expressons above do no need condonal compaons, conrary o he cross dervave xx, x, x whch mgh need p o for dfferen schemes. Snce he rern nvenory s always posve we wll se js wo schemes (man nvenory can be posve or negave). If and R we have xx, x, x, x, x (4), x, x, x, x, x, x If R, x, x, x, x and we have xx, x, x, x, x, x, x (5) As a resl we oban for he case of Brownan moon he followng dscree HJB eaons: Copyrgh 3 ScRes.

6 S. OUARET ET AL. 555 In mode : ; we oban: Mode : ; and Mode 3: 3 ; and, x, x R mn G x, x,, x, x v 3, x, x.,. Q 3, x, xk, x, xk R v, x, x, x, x, x, x, x, x, x, x R, x, x, x, x, x, x K, x, x, x, x, x, xk, x, x mn G x, x,, x, x v 4, x x.,. Q R 4 v, x, x, x, x, x, x R, x, xk, x, xk, x, x, x, x R, x, x, x, x, x, x K, x, x, x, x, x, xk 3, x, x R mn G x, x,, x, x v 4, x x.,. 3Q , x, xk3 3, x, xk 3 R 3 v 3, x, x 3, x, x 3, x, x R 3, x, x 3, x, x 3, x, x 3, x, x 3, x, x K 3 3,, 3,, 3,, x x x x x x K 3 Mode 4: 4 ; and, and he vale fncon (6) (7) (8) Copyrgh 3 ScRes.

7 556 S. OUARET ET AL. v 4, x, x mn G x, x,, x, x v 3, x, x ,. 4Q4 4, x, x 4, x, x 4, x, x 4, x, x R 4, x, x 4, x, x R 4, x, x 4, x, x 4, x, x (9) where: Q R 3 R R K K R 4 R R K K Q 3 R R R K3 K3 Q Q R R R (3) (3) (3) (33) In he second case (fler demand) we have smlar HJB eaons n for modes, b wh slghly dfferen parameers n he frs dervave of he vale fncon namely: f a and f ra. 5. Opmal Prodcon Polcy for Hybrd Sysem-Smlaon Resls The frs case we have analyzed corresponds o he hybrd sysem wh manfacrng coss se relavely hgh n order o enforce prodcon n remanfacrng loop. The (cmlave) demand s modeled as a Brownan process. The resls are shown n Fgres 3-6. Fgre 3 ll- sraes he shape of he vale fncon, x, x dependng on he sock levels of manfacred x and remanfacred x prodcs n mode. Vale fncons n oher modes have smlar shapes and are no shown. Fgres 4 and 5 llsrae he opmal polcy for he machne (manfacrng) n mode (boh machnes n operaon) and mode (remanfacrng machne n falre) respecvely. The opmal polces for he machne are of hedgng-pon ype, namely: maxmal prodcon f he sock level x s below he hreshold, zero prodcon above he hreshold and prodcon on demand a he hreshold-level. Comparng Fgres 3 and 4 one can observe ha he hreshold level n mode when machne s n falre s hgher han n mode. Fgre 6 llsraes he opmal polcy for he machne Fgre 3. Vale fncon n mode. Fgre 4. Prodcon polcy: Machne, mode. Copyrgh 3 ScRes.

8 S. OUARET ET AL. 557 Fgre 5. Prodcon polcy: Machne, mode. Fgre 7. Brownan demand and rern raes. Fgre 6. Prodcon polcy: Machne, mode. (remanfacrng) n modes when boh machnes are n operaon (opmal polcy of he machne n mode 3 s dencal). Machne s ms prodce a average spply (proporonal o demand) rae whch s also s maxmal rae as explaned n he secon. Fgres 7 and 8 show he realzaons of Brownan and Markov-ype (flered) demand raes respecvely (he varance s se o he same vale). Comparng wo graphs one can see ha n Brownan case (Fgre 7) he varaon rae s mch faser han n Markov case (Fgre 8). Fgres 9 and llsrae he opmal polcy for he machne (manfacrng) n mode (boh machnes n operaon) and mode (remanfacrng machne n falre) respecvely. The resls are o be compared wh hose shown n Fgres 4 and 5. One can observe ha he hreshold vales for he case of slower varyng Markov demand are lower as compared o he case of Brownan demand. Parameers sed for smlaons are smmarzed n Table. In he second case of Markov-ype (flered) demand he parameers of he fler may be sed o f he model Fgre 8. Flered demand and rern raes. Fgre 9. Prodcon polcy: Machne, mode. o he characerscs observed n he real lfe applcaons. A classcal assmpon of he consan demand n hs conex means ha he varably of he demand s gnored and only s average rae s aken no accon. A Copyrgh 3 ScRes.

9 558 S. OUARET ET AL. Fgre. Prodcon polcy: Machne, mode. Table. Parameers of he nmercal example. c c c c c r max 3, 4 c p,, 34 3, 4, r a b second order erms n HJB eaons reflecng demand varably are n ha case negleced and he opmal polcy s fond sng he frs order approxmaon of HJB eaon. Opmal polcy for he man (manfacrng) machne s of hedgng pon n boh sded (Brownan and Markov) cases as s for he consan demand. Accordng o Fgres 4, 5, 9 and one can see ha more he demand vares rapdly n me, more he hedgng sock level ncreases n order o respond o he demand varably. In addon, he average oal cos also ncreases from 939 (Markov) o 36.5 (Brownan) as he demand varably ncreases. 6. Conclson and Fre Work We have shown ha he problem of sochasc conrol correspondng o opmzaon of prodcon plannng n falre prone hybrd manfacrng/remanfacrng sysems wh random demand can be sccessflly analyzed for dffson-ype demands. We nvesgae hs problem n connos me whch seems o be he mos naral seng. We develop opmaly condons n he form of HJB eaons and show ha de o he Brownan componen n he demand he HJB eaons are he second order PEs, conrary o he case of a consan demand max where hey are of he frs order. We se fne dfference approxmaons for HJB eaons redcng a connos me opmzaon problem o he dscree me, dscree sae, nfne horzon dynamc programmng problem, and se polcy mprovemen echne [7] for solvng. Vale fncons of he sochasc opmzaon problems are sally non-smooh and correspondng HJB eaons have o be addressed sng generalzed approaches sch as vscosy solons [8]. Theorecal sdes of he convergence of dscree approxmaons o an exac (vscosy-ype) solon of HJB eaons when he sze of he grd ends o zero s addressed n [9,]. Sch heorecal analyss s o of he scope of hs paper where we propose a nmercal approach argeng he new model for he nceran demand ha allows addressng more narally he growng nmber of ndsral applcaons. Consderng possble exensons of he sdy presened n hs paper we con o explore a compond demand model of Posson and dffson-ype process hs allowng boh he jmps and connos random varaon of he demand. REFERENCES [] M. Fleschmann, Qanave Models for Reverse Logscs, Sprnger Verlag, New York,. do:.7/ [] M. P. e Bro, R. ekker and S.. P. Flapper, Reverse Logscs: A Revew of Case Sdes, ERIM Repor Seres Reference No. ERS-3--LIS, 3. [3] G. P. Kesmüller and C. W. Scherer, Compaonal Isses n a Sochasc Fne Horzon One Prodc Recovery Invenory Model, Eropean Jornal of Operaonal Research, Vol. 46, No. 3, 3, pp do:.6/s377-7()49-7 [4] K. Inderfrh, Opmal Polces n Hybrd Manfacrng/Remanfacrng Sysem wh Prodc Sbson, Inernaonal Jornal of Prodcon Economcs, Vol. 9, No. 3, 4, pp do:.6/s95-573( )47-X [5] M. E. Nkoofal and S. M. M. Hssen, An Invenory Model wh ependen Rerns and sposal Cos, Inernaonal Jornal of Indsral Engneerng Compaons, Vol., No.,, pp [6] S. Oscar and F. Slva, Sbopmal Prodcon Plannng Polces for Closed-Loop Sysem wh Unceran Levels of emand and Rern, The 8h IFAC World Congress, Mlano, 8 Ags- Sepember. [7] I. obos, Opmal Prodcon-Invenory Sraeges for HMMS-Type Reverse Logscs Sysem, Inernaonal Jornal of Prodcon Economcs, Vol. 8-8, 3, pp do:.6/s95-573()77-3 [8] W. H. Flemng, H. M. Soner and S. P. Seh, A Sochasc Prodcon Plannng Problem wh Random emand, SI Jornal on Conrol and Opmzaon, Vol. 5, No. 6, 987, pp do:.37/358 Copyrgh 3 ScRes.

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