Production-Inventory Systems with Lost-sales and Compound Poisson Demands

Transcription

1 Podcion-Invenoy Syem wih Lo-ale and Compond Poion emand Jim (Jnmin) Shi School of Managemen, New Jeey Inie of Technology, Newak, NJ 72 J. Mack Robinon College of Bine, Geogia Sae Univeiy, Alana, GA 333 Michael N. Kaehaki ep. of Managemen Science & Infomaion Syem Rge Bine School -- Newak and New Bnwick, Newak, NJ 72 Benjamin Melamed ep. of Spply Chain Managemen & Makeing Science Rge Bine School -- Newak and New Bnwick, Picaaway, NJ 8854 YenXia epamen of Manageial Science, Geogia Sae Univeiy, Alana, Geogia 333, Abac Thi pape conide a conino-eview, ingle-podc podcion-invenoy yem wih a conan eplenihmen ae, compond Poion demand and lo-ale. Two objecive fncion ha epeen meic of opeaional co ae conideed: () he m of he expeced diconed invenoy holding co and lo-ale penalie, boh ove an infinie ime hoizon, given an iniial invenoy level; and (2) he long-n ime aveage of he ame co. The goal i o minimize hee co meic wih epec o he eplenihmen ae. I i, howeve, no poible o obain cloed fom expeion fo he afoemenioned co fncion diecly in em of poiive eplenihmen ae (PRR). To ovecome hi difficly, we conc a bijecion fom he PRR pace o he pace of poiive oo of Lndbeg fndamenal eqaion, o be efeed o a he Lndbeg Poiive Roo (LPR) pace. Thi anfomaion allow o deive cloed fom expeion fo he afoemenioned co meic wih epec o he LPR vaiable, in lie of he PRR vaiable. We hen poceed o olve he opimizaion poblem in he LPR pace, and finally ecove he opimal eplenihmen ae fom he opimal LPR vaiable via he invee bijecion. Fo he pecial cae of conan o lo-popoional penaly and exponenially diibed demand ize, we obain imple explici fomla fo he opimal eplenihmen ae. Keywod and Phae: Compond Poion aival, inego-diffeenial eqaion, Laplace anfom, Lndbeg fndamenal eqaion, lo-ale, podcion-invenoy yem, conan eplenihmen ae.

2 . Inodcion Podcion-invenoy yem wih conan podcion ae ae implemened by a vaiey of manfacing fim. Example can be fond in: () gla manfacing, whee gla fnace ofen podce a conan ae (Fedeal Regie, 29); (2) ga mill, whee aw ga i podced ilizing a conan podcion ae (Gnow e al. 27); (3) he eleconic compe indy, whee diplay ae manfaced a conan podcion ae (iplay evelopmen New, 2); and (4) he phamaceical indy, whee cell-fee poein and ohe podc ae geneally podced a conan podcion ae (Membane & Sepaaion Technology New, 997). Addiional example can be fond in he cape manfacing indy, whee he yaning and dyeing pocee opeae a conan ae ove long peiod of ime. Thee conan ae ae eleced by he manface a he podcion planning age by aking ino accon he anicipaed demand and i co ce. A he manfacing age, i podce cape oll coninoly, and pecifically, a fll capaciy fo cape dyeing. Podcion-invenoy yem wih conan podcion ae ae ypically deployed when hee ae high ep ime and high ep co, whee feqen modificaion (e.g., inepion o ae change) of he podcion line i financially o opeaionally pohibiive. Th, fo boh financial and opeaional eaon, i i ciical o eablih he pope podcion poce ealy in he planning poce. The impoance of he podcion ae i elf-eviden: an ovely high podcion ae el in high holding co de o exce invenoy, while a low podcion ae el in high penaly co de o feqen ocko and beqen lo-ale. Th, i i eaonable o expec ha hee exi an opimal podcion ae ha balance hee wo co. Fhemoe, manface ofen employ fll capaciy in podcion. Fo example, he efiney indy ha Opeable Capaciy Uilizaion Rae a 92% o even highe. Coneqenly, he podcion capaciy level coeponding o he podcion ae ha a ciical impac on he fim co ce, i invenoy policie and i evice level, a well a i managemen and aff ppo eqiemen [cf. Jacob and Chae (23)]. Thi dy hed ligh on he opimal podcion capaciy of a fim fom a long-em co minimizaion pepecive. We dy a conino-eview ingle-podc podcion-invenoy yem wih a conan podcion/eplenihmen ae and compond Poion demand, bjec o lo-ale. In he eqel, we will e he em podcion and eplenihmen inechangeably. Unaified demand may be paially flfilled fom on-hand invenoy (if any) and all exce demand (hoage) i lo; ch exce demand Accoding o a ecen pblicaion of he Independen Saiic & Analyi, U.S. Enegy Infomaion Adminiaion. - -

3 will be efeed o a he lo-ale ize. The yem inc wo ype of co: a holding co and a loale co. The holding co i inced a a fncion of he invenoy on hand, and aeed a a conan ae pe ni on-hand invenoy pe ni ime. The lo-ale co i a penaly impoed a each lo occence, and i amed o be a fncion of he lo-ale ize. The goal of hi pape i o deive he opimal eplenihmen ae ha minimize wo objecive fncion ha epeen meic of opeaional co: () he m of expeced diconed invenoy holding co and lo-ale penalie ove an infinie ime hoizon, given an iniial invenoy level; and (2) he long-n ime aveage of he ame co. The main objecive of hi pape i wofold: () o povide cloed fom expeion fo he epecive objecive fncion of he condiional expeced diconed co and of he ime-aveage co; and (2) o minimize he afoemenioned objecive fncion wih epec o he eplenihmen ae. To hi end, we fi deive an inego-diffeenial eqaion fo he condiional expeced diconed co fncion nil he fi lo-ale occence. Howeve, a cloed fom fomla fo ha co fncion i no available. To ovecome hi difficly, we obeve ha he oiginal opimizaion poblem in em of he eplenihmen ae paamee can be efomlaed and olved in a acable fom in em of anohe vaiable, and hen he eqiie opimal eplenihmen ae can be ecoveed. Moe pecifically, le he oiginal pace of all poiive eplenihmen ae be efeed o a he PRR pace, and define a elaed pace coniing of all poiive oo of he o-called Lndbeg fndamenal eqaion (ee Gebe and Shi (998) and Eq. (4.7)), o be efeed o a he Lndbeg poiive oo (LPR) pace. The wo pace, PRR and LPR, will be hown o be elaed by a bijecion (i.e., a one-one and ono mapping); ee Eq. (4.9). Indeed, he co fncion ove he PRR pace doe no have a cloed fom expeion, while he ame co fncion ove he LPR pace doe, heeby faciliaing i opimizaion. Finally, having obained he opimal olion in he LPR pace, we hall povide an algoihm o compe he eqiie opimal eplenihmen ae in he PRR pace via he invee bijecion [cf. Fige 3]. We fhe obain explici olion fo he pecial cae in which he lo-ale penaly fncion i eihe: () a conan penaly fo each lo-ale occence, o (2) a lo-popoional penaly. Finally, a nmeical dy i pefomed o illae he el and demonae addiional popeie of he yem. The mehodology employed in hi pape give ie o ineeing connecion beween invenoy managemen and qeeing and inance ik model. In paicla, hi dy i conneced o ome impoan apec of G/M/ qee in eqilibim, ch a he join diibion of he by peiod and he idle peiod [cf. Pey e al. (25), Adan e al. (25) and Pey (2)]

4 In mmay, he main analyical conibion of hi pape ae: () a cloed fom fomla fo he expeced diconed co fncion fo any iniial invenoy level, geneal demand ize diibion and geneal penaly fncion; (2) a chaaceizaion of he opimal conan eplenihmen ae ha minimize he expeced diconed co fncion fo geneal demand ize diibion and geneal penaly fncion; (3) cloed fom expeion fo he opimal eplenihmen ae and he aendan co fo he cae of exponenial demand ize, fo boh conan penaly and lo-popoional penaly fncion; (4) a cloed fom fomla fo he long-n ime-aveage co fncion fo geneal demand ize diibion and geneal penaly fncion; hi co fncion can alo be opimized ing he ame appoach employed fo he expeced diconed co fncion. The emainde of hi pape i oganized a follow. Secion 2 eview elaed lieae. Secion 3 fomlae he podcion-invenoy model nde dy. Secion 4 deive a cloed fom expeion fo he expeced diconed co fncion and Secion 5 ea i opimizaion. Secion 6 peen a e of nmeical die. Secion 7 examine he long-n ime-aveage co fncion and i opimizaion. Secion 8 peen idea on exenion of he model o incopoae vaiable podcion co and evice level conain. Finally, Secion 9 conclde hi pape. 2. Lieae Review Thi ecion fi eview he lieae on conino-eview podcion-invenoy yem, and hen compae he podcion-invenoy model wih elaed qeeing and inance ik model. Mo pape on conino-eview invenoy yem ame ha ode ae placed and eplenihed in bache o lo ize. One of he well-known odeing policie i he conino-eview (,S) policy; ee Scaf (96) fo a eminal wok. In cona, o dy conide a podcion-invenoy yem whee invenoy i eplenihed coninoly a a conan ae, and he goal i o find he opimal eplenihmen ae. Conan podcion o eplenihmen ae ae common in conino-eview podcion-invenoy yem. Fo example, ohi e al. (978) conide a podcion-invenoy conol model of finie capaciy ha wiche beween wo poible podcion ae baed on wo ciical ock-level. The main el of ha pape i a fomla fo he long-n ime-aveage co a a fncion of wo ciical level of he podcion ae. e Kok e al. (984) deal wih a podcion-invenoy model bjec o a evice level conain, whee exce demand i backlogged and he podcion ae can be dynamically wiched beween wo poible ae. The aho deive a efl appoximaion fo he wich-ove level. Fo he ame model, e Kok (985) conide he coeponding lo-ale cae and povide an appoximaion fo he wich-ove level. Gavih and Gave (98) conide a podcion-invenoy yem, whee he - 3 -

5 demand poce i Poion and demand ize i conan. The aho ame ha exce demand i backlogged and he podcion faciliy may be e p o h down. They ea hei yem a an M// qee and minimize he expeced co pe ni ime. Gave and Keilon (98) exend he model of Gavih and Gave (98) by conideing a compond Poion demand poce. The poblem i analyzed a a conained Makov poce, ing he compenaion mehod, and a cloed-fom expeion i deived fo he expeced yem co a a fncion of he policy paamee. Fo a imila eing, Gave (982) deive he eady-ae diibion of he invenoy level ing qeeing heoy. Moe ecenly, Pey e al. (25) dy a podcion-invenoy yem wih a fixed and conan eplenihmen ae nde an M/G (i.e., a compond Poion) demand poce and wo cleaing policie (poadic and conino) o avoid high invenoy level. The pape deive explici el fo he aociaed expeced diconed co fncion nde boh ype of cleaing policie. We noe ha while he lieae above ame he eplenihmen ae o be exogeno and fixed, o pape ea hi paamee a a deciion vaiable. The ndelying invenoy poce died in hi pape can alo be acibed a vaiey of inepeaion, dawn fom he conex of qeeing and inance ik yem. In wha follow, we povide a lieae eview on ch connecion; ineeed eade ae alo efeed o Pabh (997) fo a geneal eamen of ch model nde he heme of ochaic model. The imilaiy beween qeing and invenoy model i well ecognized in he lieae, and a nmbe of pape ea one model fom he pepecive of he ohe. Fom a qeeing vanage poin, he invenoy level can be inepeed a he aained waiing ime in a G/M/ qee, povided idle peiod ae emoved; ee Adan e al (25), Pabh (965) and efeence heein. An invenoy analyi geneally inclde an explici co ce and a olion fo opimal policie, while eeache in qeing heoy have been moe ineeed in he ndelying pobabiliic ce. Howeve, ome pape adde invenoy poblem ing qeeing heoy; wo cae in poin ae Gave (982) and Pey e al. (25). Co opimizaion ha alo been conideed in qeeing model. Sch eeach ha been dieced owad finding opimal opeaing policie fo a qeing yem bjec o a given co/ewad ce. Sch opimizaion poblem have been conideed by Bell (97), Heyman (968), Lee and Sinivaan (989) and Sobel (969). In he conex of claical inance ik model, he invenoy level can be inepeed a a pl (o capial, o ik eeve) level of an inance fim, nde a conan ae of pemim inflow and compond Poion claim aival; ee Amen (2), Gebe and Shi (998). Rik heoy in geneal, - 4 -

6 and in pobabiliy in paicla, ae adiionally conideed eenial opic in he inance lieae. Since he eminal pape by Lndbeg (932), many die have addeed hi opic; cf. Gebe and Shi (997, 998) and Rolki e al. (999). Two ypical qeion of inee in claical in heoy ae (a) he defici a in; and (b) he ime o in. To adde hoe wo qeion, Gebe and Shi (998) have inodced a compehenive penaly fncion, he o-called Gebe-Shi penaly fncion, a a fncion of pl immediaely pio o in and he defici a in; hi fncion ha been widely diced in he ecen inance lieae. Addiional exenion baed on he Gebe-Shi penaly fncion inclde baie o hehold aegie; ee Boxma e al. (2), Lin e al. (23), Lin and Pavlova (26), and efeence heein. Recenly, Boxma e al. (2) and Löpke and Pey (2) have fhe died inance ik model (ime o in, in pobabiliy, and he oal dividend) ing mehod and el fom qeeing heoy. In mo of hee die, i i noed ha he invenoy poce can be inepeed a he conen poce of a qeing o an inance ik model. In cona, he peen dy diffe fom he above in em of i objecive fncion and i condiion fo yem abiliy; in paicla, o pape ea co compaion and opimizaion while he inance lieae i pimaily ineeed in dividend and ik (e.g., ime o in and in pobabiliy), and he qeing lieae mainly foce on qaniie ch a evice level and wokload in he yem. Qeeing heoy alo p emphai on abiliy condiion: a able qee eqie he affic ineniy o be icly le han one; cf. Amen (23) and Pabh (997). Sabiliy condiion fo an inance ik model ene ha he aveage claim i le han he pemim ae (i.e., a poiive eciy-loading), ch ha he pobabiliy of limae in i le han one; cf. Eq. (2.5) in Gebe and Shi (998). In o podcion-invenoy conex, he condiion ha he aveage demand i geae han he eplenihmen ae (i.e., a negaive eciy-loading) i neceay fo he ime-aveage co opimizaion, wheea no ch eicion i eqied fo he expeced diconed co analyi. In hi pape, i i no poible o diecly olve he inego-diffeenial eqaion in Eq. (4.4). Howeve, i i poible o olve eqaion ha involve Laplace anfom [cf. Widde (959)], and hen inve he anfomed fncion o obain he eqiie fncion. We noe ha he poblem of inveing Laplace anfom i ofen difficl, o mo die foc on nmeical appoximaion, e.g., Cohen (27) and Shole e al. (24). In addiion o he conibion of analyical el lied in Secion, he main mehodology conibion of hi pape ae a follow: () we ea he oiginal poblem in em of he LPR vaiable by aking advanage of Lndbeg fndamenal eqaion and a bijecion beween poiive podcion ae (PRR vaiable) and Lndbeg poiive oo (LPR vaiable); and (2) we opimize hi co fncion - 5 -

7 in he LPR pace and hen inve he opimal LPR vaiable o obain he eqiie opimal eplenihmen ae in he PRR pace ing he invee bijecion. To he be of o knowledge, no dy in he invenoy lieae exploi ch an opimizaion echniqe. 3. Model Fomlaion We will e he following noaional convenion and eminology. Le denoe he e of eal nmbe and x max{ x, }, fo any x. Fo a andom vaiable X, i pobabiliy deniy fncion (pdf) i denoed by fx FX ( x), i cmlaive diibion fncion (cdf) by FX ( x) and i complemenay cdf by ( x). Fo wo eal fncion fx ( ) and gx ( ) on [, ), hei convolion fncion i given by f g ( ) f( x) g( x) dx. The Laplace anfom of a fncion fx ( ) i defined by L f ( z) = f( z) = e f( x) dx, z. Fo any non-negaive andom vaiable X, we hall make epeaed e of he following elaion zx z x z x FX ( z) e FX ( x) dx e dfx ( x) fx ( z), (3.) z z whee he econd eqaliy follow fom inegaion by pa. Thogho hi pape, we will acily ame he exience of a baic pobabiliy pace,,, whee i he ample pace, i a field of even, and i a pobabiliy meae on. Finally, we ame coninoly componded diconing a ae,. 3. Invenoy Poce We conide a conino-eview invenoy yem, bjec o lo ale. The demand aival eam conie a compond Poion poce wih ae and aival ime { A : i }, whee A by convenion. Th, he coeponding eqence of ine-aival ime, { Ti : i }, whee Ti Ai A i, i exponenially diibed and he eqence i idenically independenly diibed (iid). coeponding demand ize fom an iid eqence { : i } wih a common pdf f ( x ) and common mean demand, [ ], whee he demand of ize i aive a ime i i The A i. Replenihmen occ a a - 6 -

8 conan (deeminiic) ae,. Le { I( ) : } denoe he igh-conino invenoy poce, given by whee NA() i he nmbe of demand aiving ove (, ] and NA () i i i I( ) I( ) [ L( A )], (3.2) L( A ) [ I( A ) ], i,2,.. (3.3) i i i i he lo-ale ize a ime A i. Le { i : i } be he eqence of lo occence ime, given by inf { A : L( A ) }, (3.4) i j i- whee by convenion. Le { J : k } be he eqence of andom aival indexe a which a k j lo occ, namely, k A J k. Fige illae a ample pah of he invenoy poce wih lo-ale ove an infinie ime hoizon. Fige. A ample pah of he invenoy level poce, { I ( )} We noe ha he invenoy poce { I ()} of Eq. (3.2) i able nde he condiion [ ]; cf. Popoiion. in Amen (2). In cona, i i ypically amed [ ] in qeeing heoy and claical ik inance die. In paicla, qeeing yem geneally ame ha he evice ae i geae han he aival ae [cf. Adan e al. (25) and Amen (23)]; ohewie he qee lengh explode. Claical ik inance analyi ypically ame ha he pemim ae i geae han he aveage claim o ene a poiive dif; cf. Gebe and Shi (997, 998). In o model he abiliy condiion [ ] i only eqied when dying he ime-aveage co; i i no impoed fo he - 7 -

9 expeced diconed co, ince in hi cae he objecive co fncion i alway bonded de o diconing even if he invenoy poce i nable. 3.2 Co Fncion Recall ha he podcion-invenoy yem nde dy inc co in he fom of holding co and loale penalie. Specifically, a holding co i inced a ae h pe ni invenoy pe ni ime while hee i invenoy on hand, and a penaly w( x) i inced wheneve a come demand canno be flly aified fom on-hand invenoy and hee i a hoage of ize x. The penaly fncion w( x) i amed o be non-deceaing in he lo-ale ize, x, whee w( )=. Th, he oal diconed co p nil ime i given by z NA() i A C ( i ) = h e I ( z ) dz + e w ( L ( A )), (3.5) which i dependen on he iniial invenoy level I( )=. Of paicla inee i he condiional expeced diconed co fncion p nil and inclding he fi lo-ale occence, given by c ( )= [ C ( ) I( )= ]. (3.6) Fhemoe, he condiional expeced diconed co fncion ove he ineval (, ] i given by ( ) = [ C ( ) I( ) = ]. (3.7) I i eay o how ha he fncion ( ) i inceaing and nifomly bonded in, fo any given. Hence, i follow ha he condiional expeced oal diconed co fncion, ( ) = lim ( ), (3.8) i well defined. In ode o opimize ( ) wih epec o i, we nex deive he expeced diconed co fncion in Secion 4, and hen ea i opimizaion in Secion 5. All poof omied fom hee ecion ae povided in he appendice. 4. Compaion of he Expeced iconed Co Fncion To deive a cloed fom fomla fo he co fncion ( ) of Eq. (3.8), we fi eablih, in he following heoem, ha he expeced diconed co fo an abiay iniial invenoy level can be decompoed ino wo em: he diconed co p nil he fi lo-ale occence and he expeced diconed co heeafe

10 Theoem Given any iniial invenoy, ( ) and c ( ) aify he following eqaion, ( ) = c ( )+ d ( ) ( ), (4.) whee ( ) = [ ( ) = ] d e I. (4.2) Poof. Follow eadily fom he ong Makov popey of he poce { I( ) : }. In paicla, eing = in Eq. (4.), we obain c () ( ) =. (4.3) d () The following wo becion dy he componen fncion c ( ) and d ( ) of ( ). 4. The Co Fncion c ( ) In hi becion we deive an inego-diffeenial eqaion fo c ( ) in Lemma fom which we will lae obain cloed fom expeion fo c () and c ( z) in Popoiion. Lemma The fncion c ( ) defined by Eq. (3.6) i conino, diffeeniable in and aifie whee c ( ) ( ) c ( ) f c ( ) g( ), (4.4) g( ) h f ( x) w( x ) dx,. (4.5) To olve Eq. (4.4) fo c ( ), we inodce he axiliay fncion ( z), given by ( z) f( z) z, (4.6) whee by convenion, () z if f () z doe no exi. I i of inee o dy he oo of he eqaion () z, ha i, he oo of he eqaion - 9 -

11 z [ f ( z)]. (4.7) Eq. (4.7) i well known in he conex of inance model, whee i i efeed o a Lndbeg fndamenal eqaion; cf. Gebe and Shi (998). An impoan popey of he oo of ha eqaion i a follow: fo any, he eqaion () z ha wo diinc eal oo, and, whee and (ibid.). Fige 2 depic he ce of he fncion ( z ) and i wo oo. Fige 2. Illaion of he ce of he fncion ( z ) and i wo oo We noe ha eihe he negaive oo,, o he poiive one,, can be employed lae o deive he co fncion via hei one-one and ono elaionhip wih. Howeve, in he eqel, we hall employ (ahe han ) a a deciion vaiable in deiving he co fncion and opimal olion. Thee ae wo eaon fo hi pefeence. Fi, fo f ( ) o exi, he negaive oo i conained o be lage han a ceain conan (deemined by he demand diibion fncion), b ch conan i geneally difficl o idenify. In cona, he poiive oo alway gaanee he exience of f (). Second, he imeaveage co fncion, o be died in Secion 7, can be deived fom he diconed co fncion by aking he limi a he dicon ae end o zeo. In hi cae, end o zeo, while emain poiive, which can alo faciliae he dy of he ime-aveage co cae. Nex, eing z = in Eq. (4.7), i follow ha he Lndbeg poiive-oo,, aifie f( ). (4.8) Eq. (4.8) moivae he following Lemma which povide he bai fo o olion mehodology. - -

12 Lemma 2 (a) Thee i a bijecion beween and, implicily given by he eqaion F ( ). (4.9) (b) The fncion ( ), implicily defined by Eq. (4.9), i icly deceaing in and aifie: () lim ( ) and lim (2) lim ( ) and lim ( ) + ; ( ). The bijecion beween and, given by Eq. (4.9), allow o deive a cloed fom fomla fo he aendan co fncion in em of he LPR vaiable in lie of he PRR vaiable,. Fhemoe, he opimizaion of he co fncion can be pefomed wih epec o, and he coeponding opimal can be ed o ecove he opimal ( ) via he bijecion fncion given by Eq. (4.9). Fige 3 depic he idea of he olion mehodology, which we db he Bijecion Solion Mehodology. (a) Co Fncion Peenaion (b) Opimal Solion Fige 3. The Bijecion Solion Mehodology ove he LPR and PPR Space We nex eablih expeion fo c ( ) by olving Eq. (4.4). To hi end, we ake he Laplace anfom wih epec o on boh ide of Eq. (4.4), which yield z c ( z) c ( ) ( ) c ( z) f ( z) c ( z) g( z), z. (4.) Reaanging and implifying Eq. (4.), we obain - -

13 ( z) c ( z) c ( ) g( z), z, (4.) whee ( z ) i given by Eq. (4.6). The following el povide cloed fom fomla fo c () and c ( z) in em of he LPR vaiable. Popoiion Fo, c () g( ) ; (4.2) c () z g( ) g( z), z. (4.3) () z Nex, biing Eq. (4.9) ino Eq. (4.2) yield anohe expeion fo c () in em of he LPR vaiable, given by c ( ) g( ) () F ( ). (4.4) The expeion above allow o opimize c ( )() wih epec o ahe han, whee he lae i vey difficl o even impoible. The opimal can hen be ecoveed fom he opimal via he bijecion of Eq. (4.9). The minimizaion of ( ) wih epec o can be pefomed in a imila manne. We menion ha fo he limiing cae of, i can be eadily hown by Eq. (3.6) ha c () [ w ( )]. (4.5) Alenaively, he above el can be obained by aking limi on boh ide of Eq. (4.2), eling in g( ) lim g( ) g() limc ( )= lim = =, whee he econd eqaliy hold by Lemma 2, pa (b) and he hid hold by a popey of he Laplace anfom. The above eqaion can now be ewien a Eq. (4.5) by Eq. (4.5)

14 We noe ha if hee i no holding co (i.e., h ), hen c ()epeen he expeced diconed vale of he defici a in in a claical inance model. Gebe and Shi (998) have given a epeenaion analogo o Eq. (4.2) fo hi cae. If we fhe pecify w( x) o be an exponenial fncion, hen Eq. (4.3) can be inepeed in a qeeing conex a he join Laplace anfom of he by peiod and he idle peiod; cf. Pabh (997), Amen (23) and Adan e al. (25). 4.2 The Fncion d ( ) In hi becion, we deive a cloed fom fomla fo d () and povide an explici expeion fo d () z. Noe ha by Eq. (3.5) and (3.6), c ( ) can be wien a z c ( ) = h e I( z) dz + e w( L( )) I( ) =. The above eqaion implie ha d ( ), given by Eq. (4.2), i a pecial cae of c ( ) when h = and wx ( )=. The el fo d ( ) conained in he nex popoiion can be obained fom hei conepa fo c ( ). Popoiion 2 Fo, d ( ) F ( ), (4.6) d ( z) ( z) z z, z. (4.7) Noe ha he definiion of d ( ) given by Eq. (4.2) implie i coniniy in and A d ( ) [ e ] +, by vie of Eq. (4.2), whee = A when. Alenaively, hi can be veified by biing lim ( ) + (cf. Lemma 2) ino Eq. (4.6). Noe alo ha lim d ( ) in view of Eq. (4.2), ince while. Thi can be alenaively veified ing he fac ha lim (cf. Lemma 2) and Eq. (4.6). ( ) - 3 -

15 4.3 The Fncion ( ) I appea ha i i no poible o deive a cloed fom expeion fo ( ) a a fncion of. Howeve, he Bijecion Solion Mehodology allow o deive a cloed fom expeion fo ( ) = ( ) a fncion of. The main el in hi becion ae peened in Theoem 2 and ( ) Theoem 3. To keep he noaion imple, we will e and inechangeably, exploiing he bijecion beween hem. In hi fahion, ( ) and ( ) denoe he ame fncion b given in em of and, epecively. Simila noaional convenion will be adoped in he eqel fo ohe qaniie, e.g., c and c fo he ime-aveage co in Secion 7, a well a v and v fo he podcion co in Secion 8. Theoem 2 Fo a zeo iniial invenoy level, while fo an abiay iniial invenoy level, ( ) = c ( ) = g( ) ; (4.8) ( )= c ( )+ g( ) d ( ), ; (4.9) ( z)= gz ( ) g( ), z. (4.2) z z ( z) ( z) We nex obain a enewal-ype epeenaion of ( ) by inveing Eq. (4.2). Coollay Fo any iniial invenoy, ( ) aifie he eqaion, whee () i given by Eq. (4.8), G ( x ) i given by ( )= ( )+ G ( ),, (4.2) G ( x)= g( )g( x ), (4.22) and ( )i he invee Laplace anfom of ( z) a

16 In view of Coollay, ( ) can be obained by comping he convolion of ( ) and G ( x ). To deive a cloed fom expeion fo ( ), we inodce he fncion, V () z ( z )( z ). (4.23) () z We define V () and V ( ) o be he limi of V ( z ) a z end o and, epecively. Noe ha by he L Hôpial le, V () and V ( ) can be fhe implified a V ( ) ; ( ) (4.24) V ( ), ( ) (4.25) whee he deivaive () and ( ) can be obained fom Eq. (4.6). The following heoem povide an explici fomla fo ( ) and i a key el of he pape. Theoem 3 Fo any iniial invenoy level, g( ) V ( ) x ( ) + e ( ) g( x) e dx g V ( ) g( ) + e g( x) e dx + x e, (4.26) whee V () and V ( ) ae given by Eq. (4.24) and (4.25), epecively. Theoem 3 how ha he expeced diconed co ( ) depend on he iniial invenoy level,, in a complicaed way. We fhe obeve ha Eq. (4.26) edce o Eq. (4.8) when he iniial invenoy level i zeo. In he following wo becion, we inveigae wo pecial cae of he penaly fncion: conan loale penaly and lo-popoional penaly Conan Lo-Sale Penaly In hi cae we have w( x)= K, fo x, whee K i a conan. Accodingly, Eq. (4.5) become - 5 -

17 and he coeponding Laplace anfom i given by g( ) h K F ( ),, (4.27) h g( z) K 2 F ( z). (4.28) z Nex, eing z and biing F () fom Eq. (4.9) ino Eq. (4.28), we have h g() K. (4.29) 2 Now biing Eq. (4.29) ino Eq. (4.8) yield ( ) = h K. (4.3) Finally, biing Eq. (4.27) and (4.29) ino Eq. (4.26) yield V ( ) V ( ) c c ( ) = ( ) (, ) (, ), (4.3) ( ) ( ) 2 whee () i given by Eq. (4.3) and c h K (, ) = ( ) x Ke F x e dx ; h h, K F ( x) e dx. c x 2 ( ) = e e ( ) Lo-Popoional Penaly In hi cae, we have w( x)= Kx, fo x, whee K i a conan. Accodingly, Eq. (4.5) become and he coeponding Laplace anfom i given by g( ) h K x f ( x) dx,, (4.32) g() z h f() z K 2 2 z z z, (4.33) whee [ ]. Nex, eing z in Eq. (4.33) and ing f () a given by Eq. (4.8), we have h g() K. (4.34)

18 Now biing Eq. (4.34) ino Eq. (4.8) yield h ( ) = K. (4.35) Finally, biing Eq. (4.32) and (4.34) ino Eq. (4.26) yield V ( ) V ( ) ( ) = ( )+ (, )+ (, ), (4.36) ( ) p ( ) p 2 whee () i given by Eq. (4.35) and p h x h (, ) = + Ke z f ( z) e dzdx ( ) x ; p 2 h x h (, ) = + K e z f ( z) e dzdx e ( ) x. 4.4 Compaion of ( ) fo Exponenial emand-size iibion In hi becion, we deive he fncion ( ), bjec o each penaly fncion, fo he cae of exponenially diibed demand ize wih ae. Th, x f ( x ) e, x (4.37) and f ( z ) Sbiing Eq. (4.38) ino Eq. (4.6) yield z, z. (4.38) ( z )( z ) () z z, (4.39) z V () z whee Hence, he wo eal oo of he eqaion () z ae given by z V () z. (4.4) 2 4, (4.4) (4.42) 2-7 -

19 4.4. Conan Lo-Sale Penaly Recall ha in hi cae, w( x)= K, x, o Eq. (4.3) can be wien a whee ( ) = a a a e, (4.43) 2 a = h ; (4.44) a = h ; (4.45) a = K h 2 (4.46) In Eq. (4.43), he iniial invenoy level,, appea in boh a linea em and an exponenial em. Since, i follow ha when i elaively mall, he exponenial em dominae he linea em, while fo a elaively lage, he oppoie i e. A nmeical dy of ( ) wih exponenial demand diibion i peened in Secion 6. Finally, fo he pecial cae wih =, we have h ( )= a a2 = K, and a cloed fom expeion fo he opimal i povided in Table Lo-Popoional Penaly Recall ha in hi cae, w( x)= Kx, x, o Eq. (4.36) can be wien a ( ) = a a a e a e, (4.47) 4 whee a and a ae given by Eq. (4.44) and (4.45) epecively, and a 4 = K h + + ; K a 5 =. + In Eq. (4.47) he iniial invenoy level,, appea in a linea em and wo diinc exponenial em, each wih a negaive exponen. I follow ha when i elaively mall, he exponenial em dominae he linea em, while fo a elaively lage, he oppoie i e. Finally, fo he pecial cae =, we have 5-8 -

20 h K ( ) = a a4 a 5 =, + and a cloed fom expeion fo he opimal i povided in Table Opimizaion of he Replenihmen Rae In hi ecion, we opimize he expeced diconed co fncion ( ) wih epec o he eplenihmen ae,, via an opimizaion of ( ) wih epec o cal el in becion 5. fo an opimal eplenihmen ae,. We fi povide a geneal (admiing he poibiliy of mliple opimal eplenihmen ae), and hen decibe compaional implificaion in becion 5.2 fo ome eleced demand-ize diibion. 5. Opimal Replenihmen Rae Obeve ha he co fncion ( ), given by Eq. (4.26), i expeed in em of he wo oo, and. In he eqel, we hall expe ( ) in em of alone by expeing in em of. To hi end, we e z in Eq. (4.23), and dedce he elaion a follow by he fac ha () in ligh of Eq. (4.6), Sbiing Eq. (5.) ino Eq. (4.26) hen yield V ( ) /. (5.) g( ) V ( ) x ( ) = + e ( ) 2 () g( x) e dx g + V (5.2) V 2 V () / V () / V () x/ g( ) V () / + 2 e () g( x) e dx + e () + V V The bondedne of ( ) gaanee he exience of a global minimizing poin, = agmin { ( )}. Howeve, he fncion ( ) i no convex in geneal. In fac, i i challenging o pove he niqene of he global minimize, and hi ill emain an open poblem. In ligh of Theoem 3, a minimize,, can be comped in eveal way. A aighfowad, b elaively ime-conming mehod, i global each. Howeve, when ( ) i convex, he availabiliy of - 9 -

21 he deivaive ( ) allow o apply he elaively fa Newon Mehod. The above dicion can be mmaized a follow. Coollay 2 Given I( )=, he opimal eplenihmen ae fo ( ) ae given by f ( ), (5.3) whee = agmin { ( )} and ( ) i given by Eq. (5.2). 5.2 Opimal Replenihmen Rae nde elayed Replenihmen Sppoe he yem opeae nde delayed eplenihmen, ha i, eplenihmen a only afe he fi lo-ale occence. Fo example, ppoe he yem ha an iniial ep peiod ding which eplenihmen i navailable (e.g., a podcion faciliy which eqie a ep ime o gea p fo podcion). Accodingly, minimizing he coeponding expeced diconed co, ˆ ( ), ove an infinie ime hoizon can be wien a ˆ ( ) = c ( )+ d ( ) ( ). (5.4) Fom Eq. (5.4), i i eadily een ha minimizing ˆ ( ) wih epec o i eqivalen o minimizing () wih epec o, ince only he econd em i a fncion of. In he following wo becion, we ea he opimizaion of () fo he pecial cae of conan lo-ale penaly and lopopoional penaly Conan Lo-Sale Penaly Recall ha in hi cae, w( x) = K, x, whee K i a conan, and () i given by Eq. (4.3). In view of Eq. (4.9), Eq. (4.3) can be ewien a h K f ( ) ( ) =. (5.5) By Eq.(5.5), he opimal i given by h agmin K f ( ). (5.6) - 2 -

22 Table exhibi he opimal, and () wih cloed-fom fomla, when available, fo eleced demand diibion; deailed deivaion ae given in Appendix B. Table. Opimal expeced diconed co bjec o conan penaly nde vaio demand diibion d d h agmin () d d e h K Ke Exp( ) h, if K h K, ohewie h K h h, ohewie a b U( a, b) h e e a agmin K e e a b ( b a ) ( b a) 2, if K h K b K K h h, if K, ohewie h K h (, ), h h K agmin K / In he able above and elewhee, he agmin opeaion coepond o a each fo he opimal, wheneve a cloed fom fomla fo i i eihe navailable o no eadily available. In paicla, fo an exponenial demand diibion, he opimal olion i available in cloed fom, and he condiion K h ene a poiive opimal eplenihmen ae; ohewie, i i opimal o have zeo eplenihmen and bea he epeaed penaly co (a degeneae cae) Lo-Popoional Penaly Recall ha in hi cae, w( x) = K x, fo x, whee K i conan, and () i given by Eq. (4.35). In view of Eq. (4.9), Eq. (4.35) can be ewien a h f ( ) ( ) = K K, (5.7) whee [ ]. Coneqenly, by Eq. (5.7), he opimal i given by ( ) agmin h f K. (5.8) - 2 -

23 Table 2 exhibi he opimal, and () wih cloed-fom fomla, when available, fo eleced demand diibion; deailed deivaion ae given in Appendix B. Table 2. Opimal expeced diconed co bjec o lo-popoional penaly nde vaio demand diibion d d h agmin K e d e d h e K () d K d Exp( ) h, if K h K, ohewie h K h, if h K, ohewie K h 2 Kh h, if K h K, ohewie U( a, b) a b a h K e e ( ba) agmin b a b e e ( b a) a b h K e e K b -a K 2 ( b a) 2 (, ), h agmin K / h K K Again, fo an exponenial demand diibion, he opimal olion i available in cloed fom, and he condiion K h ene a poiive opimal eplenihmen ae; ohewie, i i opimal o have zeo eplenihmen and bea he epeaed penaly co (a degeneae cae). 6. Nmeical Sdy Thi ecion conain wo nmeical die of podcion-invenoy yem wih eleced demand-ize diibion, bjec o conan lo-ale penaly. Boh die wee condced wih he following common paamee: =, h =, K =, and =.. Recall ha only he exponenial demand-ize diibion give ie o a cloed-fom opimal olion; in all ohe cae, opimal olion wee obained by a imple each. 6. Opimal Nmeical Solion fo Zeo Iniial Invenoie In hi dy we compe and compae he nmeical vale of () fo inceaing mean demand ize, and nde he following demand-ize diibion: conan, exponenial, nifom and Gamma. Table

24 diplay he opimal and a fncion of he mean demand, [ ] =, fo he fo afoemenioned demand-ize diibion. Table 3. Opimal () fo eleced demand-ize diibion E[ ] =/ Exp U,2 / 4,/ 4 () () () () Fom Table 3, i can be een ha he epecive and he coeponding () inceae in hi ode of diibion: exponenial, nifom, Gamma and conan. Noe ha a he aveage demand inceae, and () inceae a expeced. Fhemoe, fo each eleced demand-ize diibion, we obeve ha [ ] fo [ ] 7 (cae ), wheea [ ] fo [ ] 5 (cae 2). One poible explanaion fo hee obevaion can be deived by examining he opimal podcion aendan o a demand ae, noing ha diconing implie ha he objecive fncion i diven by he behavio of he yem in an iniial ineval (aing a ). Th, in cae, he opimal podcion ae wold be diven above he demand ae, becae ohewie, he invenoy level wold ay low, heeby incing exceive penaly co. Conveely, in cae 2, he opimal podcion ae wold be diven below he demand ae, becae ohewie, he invenoy level wold ay high, heeby incing exceive holding co. The above obevaion can be explained analyically fo he cae of exponenial demand, Exp( ), wih he aid of he explici olion given in Table. In paicla, aming ha K h hold, he

25 opimal podcion ae i given in cloed fom by K h, whence he h K diffeence [ ] i given by K h. (6.) h K Th, fo fficienly lage, i.e., fficienly mall [ ], he igh-hand ide of Eq. (6.) become poiive, implying [ ]. Conveely, fo fficienly mall b poiive lage [ ], he igh-hand ide of Eq. (6.) become negaive, implying, i.e., fficienly [ ]. Fhemoe, by Eq. (6.), he c-off poin fo [ ] i idenified by K h 2 h K 2. In hi nmeical dy wih he eleced paamee and, i how ha he c-off mean demand i [ ]= 3.7. Tha i, [ ] fo [ ] 3.7, wheea [ ] fo [ ] 3.7, which explain o obevaion. In he nex nmeical dy, we e he ame paamee a befoe, b fix [ ]= and vay he vale of he coefficien of vaiaion c v (aio of andad deviaion o mean) of he andom demand. Fo each eleced vale of c v, we choe he paamee of Unifom and Gamma diibion fo o a o keep he coeponding vale of c v he ame. Table 4 diplay eveal ch paamee vale and he coeponding, and () fo eleced cv anging beween 3 o / 4. Table 4. Opimal qaniie fo eleced demand-ize diibion wih epec o hei coefficien of vaiaion c v a b U a, b, () () / / /

26 Fom Table 4, i can be een ha he epecive and he coeponding () inceae in c v. Fo each cae, i i hown [ ]=. Noe ha alhogh he vaiaion in, and () i no ignifican compaed wih he change in c v, i eveal o wha exen he opimal ae depend on moe han he fi wo momen of he demand diibion. Fhemoe, obeve ha when he demand diibion i,, we have lage and () han hei conepa fo demand diibion U a, b. Thi phenomenon can be explained by he longe ail of he Kok (987)]., diibion [cf. e 6.2 Opimal Nmeical Solion fo Abiay Iniial Invenoy Level In hi dy we compe and compae he nmeical vale of, and ( ) fo eleced demandize diibion (conan, exponenial and nifom) wih inceaing iniial invenoy level and fo low and high aveage demand. Table 5 and Table 6 diplay, demand a fncion of he iniial invenoy level, I( ) =. and ( ) fo ample low and high Table 5. Opimal qaniie fo eleced demand-ize diibion nde a low demand wih I( ) = [ ]= / = 2 = Exp( ) U, 2/ ( ) ( ) ( )

27 Table 6. Opimal qaniie fo eleced demand-ize diibion nde a high demand wih I( ) = [ ]= / = 2 =/ Exp( ) U,2 / ( ) ( ) ( ) Table 5 and Table 6 above eveal imila behavio paen of and ( ), a fncion of I( ) =. Fo each demand-ize diibion in each able, deceae a I( ) = inceae, while he coeponding ( ) fi deceae and hen inceae in. Alo, fo any given iniial invenoy level, ( ) inceae a he aveage demand, [ ], inceae. Moeove, fo each demand-ize diibion, he opimal iniial invenoy level agmin { ( ) } inceae in he aveage demand. Fo example, = in Table 5 and [ 25, 35] in Table 6 ae cae in poin. In ohe wod, a lage demand ize i moe beneficial when he iniial invenoy level i high. Thi i iniive ince highe demand i moe likely o deplee he invenoy qickly, which edce he holding co inced de o a high iniial invenoy level. We alo obeve ha in each of hee able, deceae in he demand-ize diibion in hi ode: conan, nifom and exponenial; hi, howeve, doe no geneally hold fo ( ). 7. Time-Aveage Co and Opimizaion The long-n ime-aveage (ndiconed) co can be eaed imilaly o i diconed conepa. In hi cae, we need o ame he abiliy condiion, [ ] (o eqivalenly [ ]< );

28 ohewie he long-n ime-aveage co i infinie. In he eqel, we deive he ime-aveage co diecly fom he el fo he diconed co by aking limi a and ing he enewal ewad heoem (cf. Ro (996)). Fo, he Lndbeg fndamenal eqaion of Eq. (4.7) become f( z) z. (7.) Unde he abiliy condiion [ ], i follow ha Eq. (7.) ha wo eal oo: and. Nex, by Eq. (3.) and (7.), one ha F ( ), (7.2) which implie ha and ae conneced by a bijecion. In view of Eq. (7.2), he abiliy condiion [ ] can be wien a F( ) [ ]. Since F( z ) i monoonically deceaing in z and F ( z) [ ] a z, he abiliy condiion [ ] in he PRR pace, can be eqivalenly expeed a in he LPR pace. Unde he abiliy condiion in he LPR pace (i.e., [ ] in he PRR pace), he invenoy poce ove ime ineval of he fom i, i i a enewal poce, and he coeponding co poce can be egaded a a enewal ewad poce, wih finie expecaion of ine-enewal ime and cycle ewad. Coneqenly, by Theoem 3.6. in Ro (996), he long-n ime-aveage co i independen of he iniial invenoy level, and can be epeened by c = c (), (7.3) [ I ( )= ] whee c () i given by Eq. (3.6) wih and. Nex, we e a he deciion vaiable o deive c in cloed fom and analyze i opimal olion. Following o noaional fahion, we le c and c denoe he ime-aveage co fncion of Eq. (7.3) in em of and, epecively. To deive he ime aveage co, we e he fac ha d ( ), defined by Eq. (4.2), can be inepeed a he momen geneaing fncion of a ; cf. Ka (993). Coneqenly, by Eq. (4.6), he expeced ime o he fi hoage condiioned on he iniial invenoy level can be wien a [ I( ) = ] = f. (7.4) ( )

29 Noe alo ha Eq. (7.4) can be inepeed a he expeced vale of he ime o in in he claical inance model [cf. Gebe and Shi (998)], condiioned on a zeo iniial pl level. The following heoem povide a cloed fom expeion fo he ime-aveage co. Theoem 4 Unde he abiliy condiion [ ], he ime-aveage co i given by c = g ( ), (7.5) whee. We menion ha e Kok (987) die a coeponding podcion-invenoy yem, b wih wo wichable podcion ae, and povide an appoximaion fo he ime-aveage of invenoy holding and wiching co (cf. Eq. (2.) heein). Acally, o podcion-invenoy model can be eaed a he afoemenioned model, povided he wo podcion ae a he eqal and hee i no wiching co. In hi cae, he appoximaed caying co in e Kok (987) i exacly eqivalen o he ime-aveage holding co in Eq. (7.5). Howeve, he appoximaion popoed by e Kok (987) only accon fo he holding co b ignoe he lo-ale penaly componen. In view of Theoem 4, minimizing c wih epec o i eqivalen o minimizing c wih epec o he poiive vaiable. To hi end, we fi opimize c = g ( ) in he LPR pace o find he opimal, and hen compe he coeponding opimal in he PRR pace. The following coollay povide a geneal cal el fo he opimal eplenihmen ae,. Coollay 3 The opimal eplenihmen ae fo he ime-aveage co c nde he abiliy condiion [ ] i given by whee F ( ), (7.6) = agmin { g ( )}. (7.7)

30 8. Fhe Exenion The eeach peened in hi pape can be exended in eveal diecion. Fi, he mehodology can be exended o inclde in he objecive fncion a vaiable podcion co modeled a a nonnegaive and inceaing fncion of he eplenihmen ae. In hi cae, he expeced diconed podcion co i By Eq. (4.9), we can ewie v in Eq. (8.) a a v = a e d =. (8.) v a F ( ) =, (8.2) whee v and v denoe he ame co fncion, b of and, epecively. Finally, we can expe he oal expeced diconed co fncion a ( )+ v, whee ( ) i given by Eq. (5.2) and v by Eq. (8.2). Thi cloed fom of he objecive fncion allow one o compe he opimal diecly, fom which he opimal eplenihmen ae can be ecoveed via Eq. (5.3). Fo he cae of ime-aveage co, adding he podcion co o Eq. (7.5) yield he oal co fncion epeenaion a g( ) a F ( ) g ( ), (8.3) by vie of Eq. (7.2). The above cloed fom expeion allow one o compe he opimal diecly. The eqiie opimal eplenihmen ae can hen be obained fom Eq. (7.6). Second, we poin o ha he el of hi pape can be applied o co opimizaion (diconed o imeaveage) bjec o a given evice-level conain, e.g., a fill ae, defined a he pecenage of demand aival ha ae immediaely aified in fll fom invenoy on hand. Le he lo-ale ae be denoed by =. Then, = lim N ( )/ N ( ), whee N () and N () denoe he nmbe B of demand aival and lo-ale occence, epecively, in he ineval (, ]. The lo-ale ae,, can be alenaively epeened a [cf. Ro (996), Theoem 3.4.4] A A B

31 Sbiing [ T ]= / and Eq. (7.4) ino Eq. (8.4) yield = [ T ]. (8.4) [ ( I ) = ] = = ( ) Coneqenly, we have he following epeenaion fo he fill ae f. (8.5) = f ( ). (8.6) Fo opimizaion poblem wih objecive fncion of expeced diconed co o long-n ime-aveage co, conained by a given minimal fill ae, ', one can apply Eq. (8.6) o compe he ciical vale ' ch ha f ( ')= '. I follow ha he co opimizaion poblem (e.g., he ime aveage co died in Secion 7) wih a conained fill ae, ', can be olved by a each in he LPR pace, eiced o he ineval ', in lie of he oiginal each pace,. 9. Conclion and Fe Reeach Thi pape inveigaed a conino-eview ingle-podc podcion-invenoy yem wih a conan eplenihmen ae, compond Poion demand and lo-ale. Two objecive fncion ha epeen meic of opeaional co wee inveigaed: () he m of he expeced diconed invenoy holding co and he lo-ale penalie, ove an infinie ime hoizon, given an iniial invenoy level; and (2) he long-n ime-aveage of he ame co. A bijecion beween he PRR pace and LPR pace wa eablihed o faciliae opimizaion. Fo any iniial invenoy level, a cloed fom expeion wa deived fo he expeced diconed co, given an iniial invenoy level, in em of an LPR vaiable. The elan co fncion wa hen eadily opimized in he LPR pace, and he eqiie opimal vale of he eplenihmen ae wa ecoveed via he afoemenioned bijecion. In addiion, he ime-aveage co wa alo deived in cloed fom nde a abiliy condiion, and an opimizaion mehodology imila o he one ed fo he expeced diconed co, wa applied o opimize he eqiie ime-aveage co. Addiional wok in hi aea may inclde he following. Fi, fo he geneal model (wih geneal co fncion and geneal demand diibion), one migh admi mliple opimal eplenihmen ae, hogh i i likely ha a ingle opimal eplenihmen ae i niqe nde faily geneal condiion. The ype of condiion neceay o ene niqene i a fe eeach opic. Second, one migh inodce invenoy capaciy conain (e.g., bae ock level), ch ha eplenihmen i pended o h down when he invenoy level eache o i a capaciy. Thid, i i of inee o inveigae imila podcioninvenoy yem wih dicee eplenihmen, ha i, whee eplenihmen ode ae iggeed by - 3 -

32 demand aival ha dop he invenoy level below ome pecibed bae ock level. Finally, egading he dicee-ime veion of hee poblem, we noe ha he inego-diffeenial eqaion obained in Lemma i no longe valid, ince i deivaion i baed on ime coniniy. Theefoe, a diffeen appoach which ilize Makov chain and/o enewal heoy migh be employed o ea he coeponding dicee-ime model. Acknowledgmen We ae indebed o he Aea Edio, he Aociae Edio and hee anonymo efeee fo many concive commen and efl ggeion. Refeence [] Adan, I., O. Boxma, and. Pey (25) The G/M/ qee eviied, Mahemaical Mehod of Opeaion Reeach, 62(3), [2] Amen, S. (2) Rin Pobabiliie, Advanced Seie on Saiical Science & Applied Pobabiliy, Wold Scienific. [3] Amen, S. (23), Applied Pobabiliy and Qee, Spinge, 2nd Ediion. [4] Bell, C.E. (97) Chaaceizaion and compaion of opimal policie fo opeaing an M/G/ qeing yem wih emovable eve, Opeaion Reeach, 9(), [5] Boxma, O.J., A. Löpke and. Pey (2) Thehold aegie fo ik pocee and hei elaion o qeeing heoy, Jonal of Applied Pobabiliy, 48A (Special volme), [6] Chchill, R. V. (97) Opeaional Mahemaic, McGaw-Hill Book Company. [7] Cohen A. M. (27) Nmeical Mehod fo Laplace Tanfom Inveion, Spinge. [8] e Kok, A.G., (985) Appoximaion fo a lo-ale podcion/invenoy conol model wih evice level conain, Managemen Science, 3, [9] e Kok, A.G. (987), Appoximaion fo opeaing chaaceiic in a podcion-invenoy model wih vaiable podcion ae, Eopean Jonal of Opeaional Reeach, Vol. 29 (3), [] e Kok, A.G., H.C. Tijm, and F.A. Van e yn Schoen (984) Appoximaion fo he ingle podc podcion-invenoy poblem wih compond Poion demand and evice level conain, Advance in Applied Pobabiliy. 6, [] ohi, B.T., F.A. Van de yn Schoen, and A.J.J. Talman (978) A podcion invenoy conol model wih a mixe of back-ode and lo-ale, Managemen Science, 24, [2] Gavih, B. and S.C. Gave, (98) A One-podc podcion/invenoy poblem nde conino eview policy, Opeaion Reeach, 28(5), [3] Gebe H.U. and S.W. Shi (997) The join diibion of he ime of in, he pl immediaely befoe in and he defici a Rin, Inance: Mahemaic and Economic. 2, [4] Gebe, H.U. and E.S.W. Shi (998) On he ime vale of in, Noh Ameican Acaial Jonal, 2, [5] Gave, S.C. and J. Keilon (98) The compenaion mehod applied o a one-podc podcion invenoy model, Mahemaic of Opeaion Reeach, 6,

33 [6] Gave, S. C. (982) The applicaion of qeeing heoy o conino peihable invenoy yem, Managemen Science, 28, [7] Gnow, M., H. O. Günhe, and R. Weinne (27) Spply opimizaion fo he podcion of aw ga, Inenaional Jonal of Podcion Economic,, [8] Heyman,. P. (968) Opimal opeaing policie fo M/G/ qeing yem, Opeaion Reeach, 6(2), [9] Jacob, F. R. and R. B. Chae (23). Opeaion and pply managemen. New Yok, NY: McGaw-Hill, 4 h Ediion. [2] Lee, H and M. M. Sinivaan (989) Conol policie fo he M x /G/ qeeing yem, Managemen Science, 35(6), [2] Lin, X.S. and K.P. Pavlova (26) The compond Poion ik model wih a hehold dividend aegy, Inance: Mahemaic and Economic, 38(), [22] Lin, X.S., G.E. Willmo and S. ekic (23) The claical ik model wih a conan dividend baie: analyi of he Gebe-Shi diconed penaly fncion, Inance: Mahemaic and Economic. 33, [23] Löpke, A. and. Pey (2) The idle peiod of he finie G/M/ qee wih an inepeaion in ik heoy, Qeeing Syem, 64, [24] Lndbeg, F. (932), Some pplemenay eeache on collecive Rik heoy, Skandinavik Akaieidkif 5: [25] Mandaoy Repoing of Geenhoe Gae - Pa 3 of 9, Fedeal Regie, Apil 29, 74(68). [26] Pey,. 2. The G/M/ Qee. Wiley Encyclopedia of Opeaion Reeach and Managemen Science. [27] Pey,., W. Sadje and S. Zack (25) Spoadic and conino cleaing policie fo a podcion/invenoy yem nde an M/G demand poce, Mahemaic of Opeaion Reeach, 3(2), [28] PixTech Hi Podcion Mileone, iplay evelopmen New, Nov., 2. [29] Pabh, N.U. (965) Qee and Invenoie: A dy of hei baic ochaic pocee, John Wiley & Son, Inc. [3] Pabh, N.U. (997) Sochaic Soage Pocee: Qee, Inance Rik, am and aa Commnicaion. 2nd Ed, Spinge. [3] Ro, S.M. (996) Sochaic Pocee, Wiley Seie in Pobabiliy and Mahemaical Saiic, 2nd Ediion. [32] Saff, E. B. and A.. Snide (993) Fndamenal of Complex Analyi fo Mahemaic, Science, and Engineeing, Penice Hall. [33] Scaf, H. (96) The opimaliy of (, S) policie fo he dynamic invenoy poblem, Poceeding of he Sanfod Sympoim on Mahemaical Mehod in he Social Science, Sanfod Univeiy Pe, Sanfod, CA. [34] Shole, J. F., P.H. Bill, M. J. Fiche,. Go and. M.B. Mai (24) An algoihm o compe he waiing ime diibion fo he M/G/ qee, INFORMS Jonal on Comping, 6, 2. [35] Sobel, M. J. (969) Opimal aveage-co policy fo a qee wih a-p and h-down co, Opeaion Reeach, 7(), [36] Widde,. V. (959) The Laplace anfom, Pinceon Univeiy Pe. [37] Bioech/Biomedical: Cell-Fee Poein Podcion. Membane & Sepaaion Technology New. Apil

34 Appendix A A. Poof of Lemma Fo any given iniial invenoy level, conide a ime ineval (, ],. We have he following wo dijoin even and he coeponding condiional expeced diconed co fncion on hoe even: () On he even { A }, he coeponding condiional expeced diconed co i [ C ( ) I( ) = ] { A } = z e h ( z) e dz c ( ) e d (A.) z e h ( z) e dz c ( ) e. Hee, he fi em in he inegal above i he diconed holding co ove (, ], he econd i he diconed eidal co ove ( ], and we e he elaion { A } { }. (2) On he even{ A }, he coeponding condiional expeced diconed co can be expeed a C ( ) I( ) = e M(, ) d, (A.2) { A } whee M(, ) = C ( ) A, I( ) = i given by z M(, ) h ( z) e dz e f ( x) c ( x) dx + e f ( x) w( x ( ) ) dx (A.3) Nex, adding Eq. (A.) and (A.2) yield c ( ) h e ( z) e dz h e ( z) e dz d z ( ) ( ) ( ) c ( ) e e f ( x) c ( x) dx d + e f ( x) w( x ( ) dx d z (A.4)

35 In he following, we hall pove he coniniy and diffeeniabiliy of c ( ) fo. To pove coniniy, le in Eq. (A.4), and obeve ha he igh hand ide vanih. c ( ) lim c ( + ) ince all he inegal on We poceed o pove diffeeniabiliy by definiion. By Eq. (A.4), one ha c ( ) c ( ) z z h e ( z) e dz h e ( z) e dz d ( ) ( ) c ( ) e e f( x) c ( x) dx d ( ) e f ( x) w( x ( ) ) dx d The eqaion above how ha c ( ) i igh diffeeniable by aking he limi (eqivalenly, a ) and he igh deivaive i h c ( ) c ( ) f( x) c ( x) dx d f ( x) w( x ) dx d (A.5) Fo /, eplacing wih in Eq. (A.4) yield c ( ) h e ( z) e dz h e ( z) e dz d z ( ) ( ) c ( ) e e f ( x) c ( x) dx d ( ) + e f ( x) w( x ( ) )dx d z Fom he eqaion above, one fhe ha

36 c ( ) c ( ) h z h z e ( z) e dz e ( z) e dz d ( ) ( ) c ( ) e e f( x) c ( x) dx d e ( ) f( x) w( x ( ) ) dx d. In a imila vein, he eqaion above implie ha c ( ) i lef diffeeniable by aking he limi (eqivalenly, a ), which yield he ame expeion a ha of Eq. (A.5). Th, c ( ) i diffeeniable fo (i deivaive a = i given by he igh deivaive). Finally, we conclde ha Eq. (A.5) hold fo boh he lef and igh deivaive fo, and he emainde of he poof of Eq. (4.4) e Eq. (4.5) and imple eaangemen of em. A.2 Poof of Lemma 2 To pove pa (a), noe ha by Eq. (4.8), f (). (A.6) Eq. (4.9) now follow fom he above eqaion wih he aid of Eq. (3.). Fhemoe, Eq. (4.9) define implicily a mapping fom he PRR pace o he LPR pace. Thi mapping i one-one, ince by Eq. (4.9), ( ) = ( 2) implie 2; i i ono, ince evey invee map o ome, again by Eq. (4.9). Thi mapping i, heefoe, a bijecion. To pove pa (b), we diffeeniae Eq. (4.9) wih epec o, yielding x ( ) ( ) x e F ( ) 2 x dx. ( ) The eqaion above implie ha ( ) ince each em in he backe on he igh-hand ide above i icly poiive fo all, which in n implie he eqiie el

37 Nex, we pove lim Then, he following poiive limi exi, ( ) in pa () of (b) by conadicion. Sppoe L lim + F( ( )) + F( )>. ( ) lim ( ) exi. Howeve, aking he limi on boh ide of Eq. (4.9) a yield L, which conadic he above, heeby eablihing he eqiie el. To pove lim ( ) + in pa () of (b), noe ha Eq. (4.8) can be ewien a Since lim ( ) (j poven befoe) implie by aking he limi a on boh ide of Eq. (A.7). ( ) + f ( ( )). (A.7) lim f ( ( )), he eqiie el now follow Nex, we pove lim in pa (2) of (b) by conadicion. Sppoe lim ( ) exi. Then, he following finie limi exi, L lim + F( ( )) + F( )< ( ). Howeve, ending on boh ide of Eq. (4.9) yield L, which conadic he above, heeby eablihing he eqiie el. Finally, o pove lim ( ) in pa (2) of (b) we again e Eq. (A.7). Since lim ( ) (j poven befoe) implie lim f ( ( )), he eqiie el now follow by aking he limi a on boh ide of Eq. (A.7)

38 A.3 Poof of Popoiion Fo, Eq. (4.2) follow by eing z in Eq. (4.) and noing ha i fi em now vanihe by vie of he eqaion (). Eq. (4.3) follow immediaely by biing Eq. (4.2) ino Eq. (4.) and dividing he elan eqaion by () z. A.4 Poof of Popoiion 2 To pove Eq. (4.6), conide he pecial cae whee h and wx ( ) = fo x. In hi cae, Eq. (3.6) implie and Eq. (4.5) become while in view of Eq. (A.8) and (A.9), Eq. (4.2) become By vie of Eq. (3.) and (4.6), we fhe have c ( ) = d ( ), (A.8) g( ) f ( x) dx F ( ), (A.9) d ( ) F ( ). (A.) F( z) z ( z). (A.) z Coneqenly, eing z above, noing ha (), and biing he elan F () ino Eq. (A.) yield Eq. (4.6). To pove Eq. (4.7), we ake he Laplace anfom of Eq. (A.9) and bie i ino Eq.(4.3) o obain d ( z) F( ) F( z) ( z) Finally, Eq. (4.7) follow by biing Eq. (A.) ino Eq. (A.2)., z. (A.2) A.5 Poof of Theoem 2 Seing = in Eq. (4.) and eaanging yield

39 ( ) c () ( ) ( )=. (A.3) d () Eq. (4.8) now follow by biing Eq. (4.2) and (4.6) ino Eq. (A.3). Finally, Eq. (4.9) follow eadily by biing Eq. (4.8) ino Eq. (4.), while Eq. (4.2) obain by aking Laplace anfom of boh ide of Eq. (4.9) and biing c () z fom Eq. (4.3) and d () z fom Eq. (4.7). ( ) A.6 Poof of Coollay Eq. (4.2) can be ewien a g( ) g( ) ( z) = ( ) ( ) g z. z z z Eq. (4.2) now follow by inveing he eqaion above, noing ha g( ) = ( ) by Eq. (4.8) and g( ) g( z) = G ( z) by Eq. (4.22). z A.7 Poof of Theoem 3 To pove Theoem 3, we fi how ha he invee Laplace anfom of ( z) i given by V ( ) V ( ) ( ) L ( ) e e, (A.4) () z whee he vale of V () and V ( ) ae he limi of V () z a and, given by Eq. (4.24) and Eq. (4.25), epecively. Thi invee Laplace anfom i obained by a andad applicaion of he Reide Theoem [Chchill (97)] and cono inegaion, and by aking advanage of he fac ha Lndbeg fndamenal eqaion () z ha wo diinc oo, and. Accodingly, by Eq. (4.23), V () z, (A.5) ( z) ( z )( z ) whee and ae inglaiie of. Taking he invee Laplace anfom of Eq. (A.5) yield ( z )

40 ir zx V () z L ( x) = lim e dz ( z) R 2 i ir ( z )( z ) fo any eal. To ee ha, define fo R a cone-clock cono pah CR HR L R (ee Fige 4), whee HR= ( x, iy) : ( x ) y R, R x L ( x,iy) : x, R y R. R Fige 4. Cono inegal fo he invee Laplace anfom Hence, he cono inegal can be wien a z x z x ir z x e dz e dz e dz, (A.6) 2 i CR ( z) 2 i HR ( z) 2 i ir ( z) and by he Reide Theoem and Eq. (A.5), he lef-hand ide of Eq. (A.6) become V ( ) V ( ) e dz e e 2 i CR () z z x Eq. (A.4) now follow by biing Eq. (A.7) ino Eq. (A.6) and ending R. (A.7), ince he fi em on he igh-hand ide of Eq. (A.6) vanihe in view of he fac ha fo any x, and we have [cf. Saff and Snide (993)], lim R HR e zx R dz lim R () z ( Re ) Rx

41 Nex, in view of Eq. (A.4), he convolion em in Eq. (4.2) become V ( ) ( ) ( ) V x ( ) x G ( ) = e G ( x) e dx e G ( x) e dx, (A.8) whee G ( x ) i given by Eq. (4.22). Now, biing Eq. (4.8) and (A.8) ino Eq. (4.2) yield g( ) V ( ) ( ) ( ) V x ( ) x ( ) = + e G ( x) e dx e ( ) G x e dx, (A.9) Finally, Eq. (4.26) follow by biing Eq. (4.22) ino Eq. (A.9) and implifying. A.8 Poof of Theoem 4. Fo, Eq. (4.2) become c () g ( ). Eq. (7.5) eadily follow by biing c () g ( ) and [ I ( )= ]= (cf. Eq. (7.4) ) ino Eq. (7.3) and implifying

42 Appendix B B. Poof of Table Fomla Conan emand Size. Conide he fi diibion ow of Table, whee d i a conan, o ha f ( z) exp{ zd}. (B.) The coeponding follow by biing Eq. (B.) ino Eq. (5.6); he coeponding follow by biing hi and Eq. (B.) ino Eq. (5.3); and he coeponding () follow by biing hee and ino Eq. (5.5). Exponenially-iibed emand Size. Conide he econd diibion ow of Table, whee Exp( ). Sbiing Eq. (4.38) ino Eq. (5.6) yield h K agmin. (B.2) Finally, he coeponding i obained fom Eq. (B.2) by aking he fi deivaive wih epec o ; he coeponding follow by biing hi ino Eq. (5.3); and he coeponding () follow by biing hi ino Eq.(5.5). Unifomly-iibed emand Size. Conide he hid diibion ow of Table, whee U( a, b), o ha f () z e a z e bz ( b a) z. (B.3) The coeponding follow by biing Eq. (B.3) ino Eq. (5.6); he coeponding follow by biing hi and Eq. (B.3) ino Eq. (5.3); and he coeponding () follow by biing hee and ino Eq. (5.5). Gamma-iibed emand Size. (, ), o ha Conide he foh diibion ow of Table, whee - 4 -

43 ( ) z f z. (B.4) The coeponding follow by biing Eq. (B.4) ino Eq. (5.6); he coeponding follow by biing hi and Eq. (B.4) ino Eq. (5.3); and he coeponding () follow by biing hee and ino Eq. (5.5). B.2 Poof fo Table 2 Fomla Conan emand Size. Conide he fi diibion ow of Table 2, whee = d. The coeponding biing hi follow by biing Eq. (B.) ino Eq. (5.8); he coeponding follow by and Eq. (B.) ino Eq. (5.3); and he coeponding () follow by biing hee and ino Eq. (4.35). Exponenially-iibed emand Size. Conide he econd diibion ow of Table 2, whee Exp( ). Sbiing Eq. (4.38) ino Eq. (5.8) yield h K agmin. (B.5) The coeponding i obained fom Eq. (B.5) by aking he fi deivaive wih epec o ; he coeponding follow by biing hi ino Eq. (5.3); and he coeponding () follow by biing hi ino Eq. (4.35). Unifomly-iibed emand Size. Conide he hid diibion ow of Table 2, whee U( a, b). The coeponding follow by biing Eq. (B.3) ino Eq. (5.8); he coeponding follow by biing hi and Eq. (B.3) ino Eq. (5.3); and he coeponding () follow by biing hee and ino Eq.(4.35). Gamma-iibed emand Size. Conide he foh diibion ow of Table 2, whee (, ). The coeponding follow by biing Eq. (B.4) ino Eq. (5.8); he coeponding follow by biing hi and Eq. (B.4) ino Eq. (5.3); and he coeponding () follow by biing hee and ino Eq. (4.35)