Analysis of a Two-Echelon Inventory Model with Lost Sales and Stochastic Demand using Continuous-Time Markov chain
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1 Poceedig of te tetiol ofeece o dutil Egieeig d Opetio Mgemet li doei Juy 7 9 Alyi of Two-Ecelo vetoy Model wit Lot le d toctic Demd uig otiuou-time Mov ci m Vegi Fi d Roul Hi if Uiveity of Tecology Abtct ti ppe we tudied two-ecelo ivetoy ytem wic coit of oe upplie d oe etile. All itlltio follow be toc ivetoy mgemet policy. Te cutome ive t etile ccodig to Poio ditibutio. Te demd i tified we bot upplie d etile ve poitive o-d ivetoy. Otewie te demd i lot. Te led time of upplie d te time eeded to eplei ode of etile e umed to be expoetilly ditibuted. Te pupoe of ti tudy i to miimize te expected log-u totl cot of te ytem pe uit time. Totl cot of te ytem iclude oldig cot of upplie d etile d te cot of lot le. We modeled te poblem uig cotiuou-time Mov ci d povided ppoximtio of te tedy-tte oit pobbilitie of outtdig ode of etile i te evice of upplie d upplie o-d ivetoy. Te optiml olutio popetie of te obective fuctio well ome umeicl eult e peeted. ou umeicl eult te mximum eo of.85% w obeved. Keywod Two-ecelo ivetoy ytem e toc policy Poio demd Lot le. toductio Ti ppe focue o two-ecelo ivetoy ytem icludig oe upplie d oe etile. ot upplie d etile follow be toc ivetoy mgemet policy wit pmete d epectively. Te demd ive to etile ccodig to Poio poce wit pmete. ce of o o-d ivetoy t etile te demd i lot. Otewie etile cec te o-d ivetoy level of upplie; gi i ce of o o-d ivetoy t upplie te demd i lot. f bot etile d upplie ve poitive o-d ivetoy level fou evet occu imulteouly: (A) te demd i tified () etile follow be toc ivetoy policy etile demd oe uit of poduct to upplie () oe uit of poduct oi te queue to be ipped to te etile d (D) te upplie tt to eplei it ivetoy. Te led time of upplie d te time eeded to eplei ode of etile e expoetilly ditibuted wit pmete d epectively. Te pupoe of ti ppe i to detemie te optiml vlue of d wic miimize te expected ytem totl cot pe uit time. te followig ectio we fomulte te poblem uig cotiuou-time Mov ci d will povide ppoximtio of te tedy-tte pobbilitie. Filly we will dicu ome popetie of te obective fuctio d will peet ome umeicl exmple.. ottio Te ottio by wic te ytem w modeled i povided i tble. Tble : Model ottio ottio Defiitio Demd te Led time te Repleimet te Mximum ivetoy poitio of upplie O-d ivetoy of upplie Mximum ivetoy poitio of etile Te umbe of outtdig ode of te etile tt i te queue of ode witig to be ipped to te etile 99
2 Retile oldig cot pe uit time pe poduct upplie oldig cot pe uit time pe poduct otge cot pe poduct Retile log-u vege o-d ivetoy upplie log-u vege o-d ivetoy Log-u vege umbe of lot le. Mov Model Te tte pce of te Mov ci model i wee d e defied i tble. Let A if A i tue d A otewie. Teefoe te blce equtio fo te Mov ci model c be witte follow: () () Let ( ) deote te pobbility of cutome ( outtdig ode) i te M / M // queue d let ( ) deote te pobbility of fee eve ( ivetoy) i M / M / / queue. A i (Ro c be clculted below: 6) d t c be ow tt () (5) (6) doe ot tify () d (). Howeve we coectue tt te eltio give cceptble ppoximtio to ou ytem tedy-tte pobbilitie. Tt i: (7). Obective Fuctio Te obective fuctio deoted T i coideed te expected log-u cot of te ytem pe uit time wic i te ummtio of oldig cot of upplie d etile d te cot of lot le. Hece te obective fuctio i give by: () 99
3 99 T (8) Teoem : d e clculted follow: (9) () () Wee i Elg lo fuctio d i clculted. Poof: Uig te tedy-tte pobbilitie peeted i (7): lcultig lcultig lcultig
4 99 Rem: ote tt i clculted twice i. Hece we ubtcted fom i te clcultio of. 5. ovexity Alyi f T i covex fuctio te locl miimum of T i it globl miimum. Accodigly we peet te followig teoem: Teoem : Te obective fuctio T i covex i tem of d fo y. Poof: A i (8) te obective fuctio i defied T. f we ow tt ec compoet of T i covex i tem of d te we c pove teoem. 5. ovexity Alyi of We defie d follow: () () () Wit ome lgebic mipultio oe c ow tt: (5) (6) Lemm : Fo ll d y. Poof: wit ome implifictio we ve: (7)
5 99 Lemm : Fo ll d y. Poof: We pove lemm by iductio o. Let P :. Fitly we ve to ow tt P i tue: : P Wic ow te ttemet P i tue fo. ext we ve to ow tt if P old te P lo old. We wite P follow: P : (8) We dd to bot ide of P. Teefoe: Te poof of lemm i complete by owig : Te ttemet i tue ice ec of it compoet i egtive. Te bi d te iductive tep ve bee pefomed. A eult P i tue fo ll tul d y. Wit epect to lemm d lemm oe c fid tt. 5. ovexity Alyi of We defie below: (9) Te fit compoet of i lie fuctio. Tu it i covex fuctio. Te ecod compoet i Elg lo fuctio multiplied by poitive coefficiet. t i poved tt Elg lo fuctio i covex fuctio (Jge d V Doo 986). Teefoe i covex fuctio i tem of. 5. ovexity Alyi of efoe coductig te lyi we me implifictio to follow: Alo we defie te followig fuctio:
6 995 () () A i i te poduct fom of te two igle-vible fuctio mely d it i ey to ow tt te ecod fowd diffeece of equl to: () Te popetie of covex fuctio e itoduced i (Yücee ). We mut ow tt te ecod fowd diffeece of i egtive. (Jgem 97) it i ow tt Elg lo fuctio i deceig fuctio of tt i: () Wit ome imple lgeb oe c ow tt: () () d () imply tt te ecod fowd diffeece of i egtive. Hece i covex fuctio i tem of d. We defie d below: (5) (6) t i cle tt i poitive. Teefoe i covex. A ec compoet of i covex it i cocluded tt i lo covex. Filly te poof of teoem i complete te obective fuctio compoet mely d e covex fuctio of d. 6. Optiml olutio te peviou ectio we clculted. ow let ~ be defied ~. Hece:
7 996 ~ ~ (7) We defie T follow: T ~ T (8) Uig ~ ited of i te obective fuctio give dditively epble obective fuctio tt i: T T T (9) Wee: T () T () Rem: We ledy ve ow tt d T e covex fuctio. teetigly it i ey to ow tt ~ d T e lo covex fuctio. 6. Optiml vlue of d Let d be te optiml vlue of d epectively; d let T T T d T T T. A T d T e covex d e te mllet poitive itege wic tify te coditio T d T. Te fuctio T d T e clculted below: T () T () Rem: imil coditio fo bove-metioed optimlity i peeted i (Hi et l. ).
8 7. umeicl Exmple Optiml olutio d e obtied fom () d (). ome exmple peeted i tble we olved te et of blce equtio () d () fo d. Te we ued te olutio to clculte te exct vlue of te obective fuctio T. tee exmple we imed to e te pefomce of te popoed ppoximtio. ++ d Mple. ve bee utilized to obti umeicl exmple. Tble : umeicl exmple T T Eo (%) ocluio ome uto believe tt te mtemticl bcgoud of lot-le lye e moe difficult comped to toe of bcodeed (iv d Vi ). Howeve we believe tt it i moe logicl to fomulte coume puce bevio lot-le (Hu et l. 9 iv d Vi ). Exteive ttetio ve bee pid to lot-le ivetoy policy duig te pt decde. Ti ppe del wit toctic pmete be-toc ivetoy mgemet policy d lot le. Ote tudie i te e of lot-le wit viety of umptio o pmete (led time d etocig time) umbe of itlltio i ec ecelo d tei ccteitic e well ctegoized d dicued i (iv d Vi ). Te ide of focuig o ti poblem oe fom (Hi et l. ) wic coide two-ecelo ivetoy ytem imil to te oe we ivetigted i ti ppe but wit te umptio tt i ce of out of toc t etile demd e ot lot. Te uto could fid te exct tedy-tte pobbilitie of te Mov model i te poduct fom. Accodigly te exct optiml olutio fo uc ytem wee deived. ti wo we popoed ppoximtio fo te olutio to () d (). t i wot coideig tt te peeted tedy-tte pobbilitie pplie to () we. ti egd if te ou ppoximtio would be te exct olutio to blce equtio peeted i (Hi et l. ). Refeece iv M. d Vi. F. Lot-le ivetoy teoy: A eview Euope Joul of Opetiol Reec vol. 5 o. pp. -. Hi R. Hi A. d ffi M. Queueig ivetoy ytem i two-level upply ci wit oe-fo-oe odeig policy J. d. yt. Eg vol. 5 o. pp Hu W. T. Jim G. Muctdt J. A. d Rumevicietog P. Aymptotic optimlity of ode-up-to policie i lot le ivetoy ytem Mgemet ciece vol. 55 o. pp Jgem D. L. ome popetie of te Elg lo fuctio ell ytem Tec. J vol. 5 o. pp Jge A. d V Doo E. A. O te cotiued Elg lo fuctio Opetio Reec Lette vol. 5 o. pp Ro. M. toductio to pobbility model Acce Olie vi Elevie 6. Yücee Ü. Dicete covexity: covexity fo fuctio defied o dicete pce Dicete Applied Mtemtic vol. 9 o. pp
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