MULTISTAGE LOT SIZING PROBLEMS VIA RANDOMIZED ROUNDING

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1 MULTISTGE LOT SIZING PROBLEMS VI RNDOMIZED ROUNDING CHUNG-PIW TEO Department f Decsn Scences, Faculty f Busness dmnstratn, Natnal Unversty f Sngapre, fbatecp@nus.edu.sg DIMITRIS BERTSIMS Slan Schl f Management and Operatns Research Center, MIT, Cambrdge, Massachusetts 039, dbertsm@mt.edu Receved January 999; revsn receved March 000; accepted June 000. We study the classcal multstage lt szng prblem that arses n dstrbutn and nventry systems. celebrated result n ths area s the 94% and 98% apprxmatn guarantee prvded by pwer-f-tw plces. In ths paper, we prpse a smple randmzed rundng algrthm t establsh these perfrmance bunds. We use ths new technque t extend several results fr the capactated lt szng prblems t the case wth submdular rderng cst. Fr the nt replenshment prblem under a fxed base perd mdel, we cnstruct a 95.8% apprxmatn algrthm t the pssbly dynamc ptmal lt szng plcy. The plces cnstructed are statnary but nt necessarly f the pwer-f-tw type. Ths shws that fr the fxed based plannng mdel, the class f statnary plces s wthn 95.8% f the ptmum, mprvng n the prevusly best nwn 94% apprxmatn guarantee.. INTRODUCTION In ths paper, we cnsder multprduct lt szng prblems that arse n dstrbutn and assembly systems. There s a set N f cmpnents. Fr each cmpnent N, there s a set called predecessrs f cmpnent fsubcmpnents cnsumed n prducng cmpnent. Wedefne the cmpnent netwr G t be a drected netwr wth nde set N and arc set. The ndes n G crrespnd t stages n the assembly prcess. In ther wrds, the netwr G crrespnds t the flw f materals n the system and cntans n crcut. The fnal prducts n ths case crrespnd t cmpnents frm the last assembly r dstrbutn stage, and hence are prduced by the sns n the cmpnent netwr G. Nte that a sn n a drected graph refers t nde wth ut-degree f zer. External demands are present nly fr the fnal prducts and are assumed t be cnstant wth rate d fr tem at stage. d 0fstage des nt crrespnd t a sn. Clearly, t satsfy the demand, rders shuld be placed fr the cmpnents dynamcally n tme. If an rder s placed fr cmpnent, anrderng cst K s ncurred. Mrever, an ncremental echeln hldng cst h s ncurred per unt tme the tem spends n nventry. The assembly rate s assumed t be nfnte. The bectve s t schedule rders fr each f the cmpnents ver an nfnte hrzn s as t mnmze lng-run average cst. Wed nt allw stcuts n the mdel. In the rest f the paper, we refer t ths prblem as P MS. Because the ptmal dynamc plcy can be very cmplcated, the research cmmunty see fr nstance Rundy 985, 986, Jacsn et al. 985., Mucstadt and Rundy 993 has fcused n the class f statnary and nested plces defned as fllws. Orders are placed perdcally n tme at equal ntervals, fr each f the cmpnents n the system.e., statnary plces. If cmpnent s used n the assembly prcess f cmpnent, then an rder s placed fr cmpnent nly when an rder s placed fr cmpnent at the same tme.e., nested plces. Nte that because the assembly rate s assumed t be nfnte, the assembly lead tme s zer, and hence, wthut lss f generalty we can assume that an rder s placed fr tem nly when the nventry level f that tem drps t zer. Ths s nwn as the zer rderng prperty f the system. Under a statnary and nested plcy, the bectve s t decde the perd T that an rder s placed. The reasn statnary and nested plces are attractve s that they are easy t mplement. Mucstadt and Rundy 993 als dscuss n detal the ratnale f usng rder ntervals T as varables. Let T dente the rderng nterval fr the tems at nde N.Inaddtn t the set f predecessrs f nde, we ntrduce the set S f successrs f nde. Furthermre, wthut lss f generalty, we may assume that t assemble a unt tem at nde, we cnsume a unt tem frm all the predecessrs n.weals assume that the hldng cst at nde s nt bgger than the hldng cst at all ts predecessrs n cmbned,.e., we have h h. Startng frm the sns f the prductn/dstrbutn system,.e., frm ndes wth S,we can defne recursvely the aggregate demand at nde by D d + S D. The average nventry and rderng cst structure under statnary and nested plces, derved Subect classfcatns: Prductn/schedulng: lt szng. Prgrammng/nteger: randmzed rundng. rea f revew: Optmzatn X/0/ $ electrnc ISSN 599 Operatns Research 00 INFORMS Vl. 49, N. 4, July ugust 00, pp

2 600 / Te and Bertsmas n Rundy 985, 986, s as fllws: K + H e T T N wth H e h h D /. The term K /T crrespnds t the average rderng cst, and the term H et crrespnd t the average nventry cst. Jacsn et al. 985, and Rundy 985, 986 cnsder the fllwng cnvex relaxatn f the prblem P R Z R mn K + H e T T s.t. N T T f T T L fr each T L n the abve mdel refers t the fxed plannng perd, and t can be assumed t be fxed, r a varable t be ntly ptmzed. Ntce that the cnstrants T T s a relaxatn f the cndtn that plces are nested. s the bectve functn s cnvex, the relaxatn P R can be slved n plynmal tme usng nterr pnt algrthms see fr nstance, Nesterv and Nemrvs 994. Fr systems wth specal structure, the runnng tme can be mprved substantally. Fr nstance, f G s a tree, Jacsn and Rundy 99 shw that the relaxed prblem can be slved n Onlg n tme where n N. When G crrespnds t a star graph, Queyranne 987, and Lu and Psner 994 shwed that the relaxed prblem can be slved n On tme, usng a lnear tme medan fndng algrthm. Regardng apprxmatn algrthms, Rundy 985, 986, and Maxwell and Mucstadt 985, n a seres f nfluental papers, shwed hw t rund an ptmal slutn f the relaxed prblem P R t near ptmal nested plces. The plces cnstructed are called pwer-f-tw plces, where each T s f the frm p TL, where p s nteger. Let Z H be the value f the heurstc used. They btaned the fllwng bunds:. If T L s nt fxed, but subect t ptmzatn, then Z H 0 Z R lg. If T L s fxed, then Z H + / 06 Z R In bth cases the bunds are tght. These results are ften referred n the lterature as 98% /.0 and 94% /.06 effectve lt szng plces, respectvely. Rundy 985 appled the rundng technque t the ne warehuse, multretaler prblem OWMR. In ths prblem G s a star graph, wth the center nde representng awarehuse and the leaf ndes representng retalers. The retalers place ther rders wth the warehuse whch n turn rders frm an external suppler. Rundy 985 shws that the ptmal plcy need nt be nested. In fact, the ptmal nested plcy can be very bad fr ths class f prblems. See Mucstadt and Rundy 993 fr an example. Rundy 985 studed the class f nteger rat plces, where the rderng ntervals f the retalers are restrcted t be nteger multples f the rderng nterval f the warehuse, r vce versa. He prpsed smlar 94% and resp. 98% effectve algrthms t cnstruct effcent nteger-rat plces fr the fxed base perd resp. varable base mdel. He als shws that these bunds apply even wth respect t the ptmal pssbly dynamc lt szng plces because the lwer bund btaned frm the nteger-rat assumptn s a vald lwer bund fr the ptmal plces. Fndng the ptmal lt szng plces fr these prblems, thugh, s stll pen. It s nt nwn whether these prblems can be slved t ptmalty n plynmal tme r whether the prblem s NP-hard. These results have been extensvely studed and extended t ther versns f lt szng prblems: fnte assembly rates tns et al. 99, ndvdual capacty bunds f the frm l TL T u TL, but mre general cst structures Zheng 987, Fedegruen and Zheng 993, and baclg tns and Sun 995. ll these extensns use determnstc rundng t generate pwer-f-tw plces wth the same 94% and 98% bunds. Jacsn et al. 988 btaned effcen heurstcs fr the general capactated versn wth bunds f the type T C, usng the structural results fr the uncapactated versn. Hwever, n wrst-case bund s nwn fr ther heurstcs, and there s n guarantee f feasblty f the slutn prduced by the heurstc. Fr the case wth a lnear rderng cst, and wth a sngle capacty cnstrant, Rundy 989 btaned a 94% effcent heurstc, usng an ngenus rundng heurstc. specal case f the abve mdel s the classcal Jnt Replenshment Prblem JRP. In an nventry system f multple tems, cst savngs can be btaned when the replenshment f several tems are crdnated. Each tme an rder s placed, a mar nt rderng cst s ncurred, ndependent f the number f tems rdered. mnr rderng cst s als ncurred whenever an tem s ncluded n an rder. Ths prblem s equvalent t the multstage lt szng prblem when the cmpnent netwr G s a star graph, wth VG 0 n and EG 0 0n. The nde 0 represents the mar rderng cst. Demand fr tem ccurs at a cntnuus cnstant rate f d. Let K dente the mnr rderng cst. The nventry cst s charged at a rate f h. Let K 0 dente the mar rderng cst, and H h d /. Then a smple lwer bund t the ptmal slutn s n K P JRP Z JRP mn + H T T + K 0 T 0 st T T 0 f n T T L fr each 0 n n effcent apprxmatn algrthm s nwn fr the nt replenshment prblem by rundng ff the plces

3 btaned frm the abve relaxatn cf. Mucstadt and Rundy 993. The man nsght frm these studes s the surprsng effectveness f the class f statnary plces, whch can be shwn t be wthn 94% resp. and 98% f the ptmum fr the fxed base resp. varable base mdel. Lu and Psner 994 ntated the study f fndng mprved wrst case guarantees fr these prblems. Fr the varable fxed base perd mdel, they shw that an + -apprxmatn algrthm can be cnstructed fr the nt replenshment and ne-warehuse, multretaler system, where the runnng tme f the algrthm s a plynmal functn f /. Ther apprach, hwever, fals t prduce any mprvement fr the fxed base plannng mdel because f the dscrete nature f the prblem.e., all rderng ntervals have t be a fxed multple f a based plannng nterval. In recent years, there have been many advances n the area f apprxmatn algrthms cf. rra 998. Many ntrusly hard dscrete ptmzatn prblems can nw be apprxmated t wthn a reasnable guarantee. Many f these advances are btaned va fndng a gd fractnal relaxatn t the underlyng dscrete prblem, and devsng a sutable rundng mechansm n the ptmal fractnal slutn t btan a feasble slutn t the rgnal prblem. The analyss f the qualty f the slutn btaned s aded by an averagng argument by purpsely ntrducng a randmzatn mechansm nt the rundng step, s the qualty f the slutn btaned can be analyzed va a smple prbablstc cmputatn. Indeed, Rundy s rgnal analyss f the 98% apprxmatn algrthm fr the lt szng prblem uses a smlar averagng argument. Hwever, he dd nt elabrate n the generalty f ths apprach t the analyss f ther classes f lt szng prblems. In ths paper, we shw the generalty f ths apprach by ntrducng a new randmzed rundng analyss fr the multstage lt szng prblem. By usng an mprved frmulatn fr the JRP under the fxed base plannng perd, we als btan mprved apprxmatn results fr ths class f prblems. Our cntrbutns n ths paper are manly as fllws:. We mtvate the dea f randmzed rundng by frst prpsng new 94% and 98% randmzed rundng algrthms fr bth fxed and varable based perd mdels fr the multstage lt szng prblems. The purpse f ths dscussn s t llustrate the smplcty wth whch the apprxmatn results can be derved and t pave the way fr the mre cmplcated prf fr the mprvement t the nt replenshment prblem.. Our results generalze mmedately t several ther extensns cnsdered n the lterature. Fr resurce cnstraned prblems under submdular rderng cst functns, we prpse a rundng prcedure that btans a bund f.44 fr multple resurce cnstraned prblems, and.06 fr sngle-resurce cnstraned prblems. These extend the results f Rundy 989 fr the lnear rderng cst case. Te and Bertsmas / Fr the JRP under the fxed base perd mdel, we prpse an mprved 95.8% apprxmatn algrthm. The best nwn bund prr t ths wr s the 94% guarantee usng the pwer-f-tw plces. Mre mprtantly, the bund s vald wth respect t the pssbly dynamc ptmal lt szng plces. The mprvement s btaned usng a new and mprved relaxatn f the prblem. Because ur man bectve s t demnstrate the smplcty f the randmzed rundng methds, we fcus n the analyss f the qualty f the relaxatns and d nt dscuss the detals f slvng these relaxatns effcently n practce. In the next sectn, we establsh the well nwn bund f 94% and 98% usng the new randmzed rundng dea. In 3, we study the capactated lt szng prblem wth submdular rderng cst. In 4, we descrbe the mprved apprxmatn algrthm fr JRP under the fxed rder perd mdel. The technque can be used n a varety f ther lt szng prblems ne warehuse multretaler system fr example t btan mprved apprxmatn guarantee under the fxed base plannng perd mdel when restrcted t statnary rderng plcy. We refer the readers t Te and Bertsmas 996 and Te 996 fr detals.. RNDOMIZED ROUNDING ND LOT SIZING PROBLEMS In ths sectn, we ntrduce the ey randmzed rundng deas n the cntext f nested plces. T see hw ur rundng algrthm departs frm the classcal ne, we frst revew the basc dea behnd the classcal apprach. Cnsder the sngle-tem nventry lt szng prblem. The graph G n ths case s smply a sngletn. It s well nwn that the ptmal slutn t ths prblem satsfes the fllwng prperty: K T HT where T s the ptmal slutn. Ths frmula s the well-nwn ecnmc rder quantty EOQ slutn t the prblem. The ptmal slutn T s surprsng elastc because f T T, then the devatn frm ptmal cst s ust + /. Fr r/, the bund s.06. Ths prperty s essentally the bass fr the 94% and 98% guarantee f the pwer-f-tw plcy cnstructed by Rundy 985, 986 Jacsn et al These papers cnstruct a pwer-f-tw plcy that runds each T t T wth T p TL fr sme p, and T L s ether fxed r ptmally selected. By a sutable chce f p,we can ensure that T 0 les n the nterval [ T ] T Furthermre, by studyng the structure f the ptmal slutn, the prevus papers essentally establshed that fr the mre cmplcated mdel P R, the ptmal slutn satsfes an EOQ type prperty wth the same elastc structure n the ptmal slutn. Snce T / T T,

4 60 / Te and Bertsmas the 94% bund hlds mmedately. By ptmzng the chce f T L, the bund can be mprved t 98%. The abve rundng technque wrs, hwever, nly when the plcy T used n the rundng prcedure s the ptmal slutn t the cnvex prgrammng relaxatn, as the argument depends crtcally n the EOQ structure n Equatn. The 94% bund des nt apply f T, the nput t the rundng prcess, s smply a feasble slutn t P R,even thugh T mght be clse t ptmal... Fxed Base Perd Mdel: 94% pprxmatn lgrthm Cnsder the relaxatn P R Z R mn K + H e N T T st T T f T T L fr each s the plces cnstructed frm the relaxatn may nt satsfy the nested prperty, the average nventry cst structure mght be cmplcated. s n Rundy 985, 986, we rund the slutn t ne that satsfes the pwer-ftw prperty. Rundy shwed that the ptmal slutn t P R has smlar structure as n the smple EOQ mdel cf. Equatn, and hence the.06 bund hlds als fr the prblem P R.Weperfrm a randmzed rundng analyss and shw that the dependence n the structure f the ptmal slutn can be remved fr the.06 bund t hld. Thus ths analyss allws us t analyze mre cmplcated lt szng mdels, wthut havng t characterze the structure f the ptmal slutns. Cnsder the fllwng randmzed rundng algrthm: lgrthm.. Let T L be fxed. Let T T T n be a feasble slutn t relaxatn P R.. Wrte T p z T L, where z < 3. Generate a pnt Y n the nterval, wth prbablty dstrbutn PY y Fy y + y 4. Fr all, fz <Y, then T p TL ; therwse T p+ T L. Nte that the abve rundng scheme always generates a nested slutn T T T n the slutn btaned s f the pwer-f-tw type. Therem. Let T T T n be any feasble slutn t Prblem P R wth cst GT N K H e T + T. lgrthm returns a pwer-f-tw plcy T T T T n wth expected cst at mst.06 GT. Prf. It s easy t see that ET p T L Pz <Y+ p+ T L Pz Y p T L Fz + p+ T L Fz T + Fz /z.e., ET T 3z + / z + T 06 T The bund fllws because the maxmum value f the functn 3z s at mst 3 /4. z Smlarly, + E/T Fz / p TL + Fz / p+ T L Fz /z /T,.e., E 3z + / T 0 T z + T The therem fllws as EGT 06GT. Nte that the dstrbutn functn Fy s chsen s that + Fy/y y Fy/ 3y/y +. The maxmum s attaned at the pnt y wth a value f 3 /4 06. Furthermre, usng the ptmal slutn t P R as nput t the rundng prcess, we btan EGT 06Z R, whch s a /06 94% apprxmatn algrthm t the rgnal lt szng prblem. Derandmzatn. The abve randmzed algrthm can be made determnstc: Wthut lss f generalty, assume that the z s are n nn-decreasng rder,.e., z z z n.frall y n z z +, the randmzed algrthm returns the same slutn. Hence, there are at mst n + dstnct slutns btaned frm the randmzed rundng prcedure. These dstnct slutns can be btaned n a determnstc manner, nce the z s are srted. Thus, the best slutn can be btaned n tme Onlg n... Varable Base Perd Mdel: The 98% pprxmatn lgrthm The same nsenstvty result can als be mprved t a 98% guarantee, f ne allws the base perd T L t vary,.e. wth T L as a varable n P R.Infact, Rundy s 98% algrthm 985, 986 already has ths feature. We recast Rundy s algrthm nt the fllwng randmzed rundng algrthm: lgrthm B.. Let T T T n T L be a feasble slutn t P R, wth T L > 0.. Let T p TL z, where z < 3. Generate a pnt Y n the nterval, wth prbablty dstrbutn Fy lg y lg 4. Fr all, fy>z, then T p+ Y. Let T L Y TL. p Y ; therwse T Nte that T les n the nterval T T. Ths prperty s useful when we are dealng wth capactated systems see 3. Furthermre, t s clear that T T T n T L s nested. Nte that df y/dy /y lg.

5 Therem. Let T T n T L be any feasble slutn t P R wth cst GT. lgrthm B returns a pwer-ftw plcy T T T n T L wth expected cst at mst GT lg 0 GT. Prf. Wthut lss f generalty, we may assume T L Then, ET z p + y y lg dy + p y z y lg dy p z dy + dy lg p z + z T lg lg Smlarly, E/T z z p /y dy + z p /y dy lg p / z / + z T lg T lg and the therem fllws as / lg 0 Derandmzatn. Suppse z z z n.fry n z z +, suppse the algrthm returns a plcy wth cst /y + By,then fr all ther y n the same nterval, the algrthm returns a plcy wth cst /y + By.Bychsng a y n the nterval that mnmzes ths term, and dng the same fr each nterval parttned by the z s, we btan an Onlg n determnstc algrthm, whch s exactly Rundy s rundng prcedure. The argument used abve can easly be adapted t analyze mre cmplcated bectve cst functns. Fr nstance, we have the fllwng: Therem 3. Under lgrthm B, T E /T 3 E 4 lg 08 lg E T T lg ET T E T 06 T T T T T T T 3 4 lg 3 4 lg Therem 3 wll be used later t analyze a capactated versn f the lt szng mdel..3. Submdular Orderng Csts In the abve mdels, the rderng cst s determned by the set f tems rdered n a lnear fashn. Fr mst realstc assembly envrnments, a fxed charge K 0 s ncurred whenever an rder s placed, regardless f the number f Te and Bertsmas / 603 tems rdered at the tme. Ths gves rse t an rderng cst KS K 0 + S K fr a set S f tems that are rdered at the same nstance. Mre generally, when K s submdular and nndecreasng, Federgruen et al. 99 and Zheng 987 have shwn that the crrespndng lt szng prblem has average cst bunded belw by G T max /T + H e T n where T T T n s feasble lt szng plcy, and the vectr n ranges ver the plymatrd { KS } KN 0 S N Nte that G T s a cnvex functn n T. Furthermre, f T s a pwer-f-tw plcy, then the average lt szng cst s exactly G T. lwer bund t the lt szng prblem, under a submdular rderng cst functn, can be btaned by slvng P SUB Z SUB mn max + H e T T T N T T f T T L fr each Fr each fxed T, t s well nwn frm the wr f Edmnds 970 that the slutn K that maxmzes G T can be btaned by a greedy algrthm. The greedy prcedure srts the ndces n decreasng rder f and selects T the values fr greedly n that rder. Hence f T s anther lt szng plcy that preserves the rder f T,.e., T T T n and T T T n fr sme rderng f the ndces, and f maxmzes G T, then K s als an ptmal slutn t G T. Let K T be an ptmal slutn t P SUB. Let T be the slutn btaned by rundng T usng lgrthm r B. Snce the randmzed rundng algrthm preserves the rder f the rgnal slutn, t fllws that EG T E /T + H et. Furthermre, T s a pwer-f-tw plcy. In ths way, lgrthms and B can be used t rund the fractnal ptmal slutn n P SUB t 94% and 98% ptmal pwer-f-tw slutns. Therem 4. Let T T n T L be any feasble slutn t P SUB wth cst G T, lgrthm returns a pwerf-tw plcy wth fxed base T L wth an expected cst f nt mre than.06 G T. Smlarly, lgrthm B returns a pwer-f-tw plcy T T T n T L wth expected cst at mst.0 G T. 3. RESOURCE CONSTRINED LOT SIZING PROBLEMS Fr the lt szng mdel P MS wth resurce cnstrants f the type a /T m

6 604 / Te and Bertsmas added, Rundy 989 shws that there s a pwer-f-tw plcy varable base wth cst at mst.44 tmes the ptmal slutn. We generalze ths result t the lt szng prblems wth submdular nt rderng cst functn. Cnsder the fllwng prcedure. lgrthm C.. Let K T be an ptmal slutn t P SUB wth the resurce cnstrants added.. Use T T fr all n lgrthm B t btan a pwer-f-tw plcy T. Nte that n lgrthm B, T T, and hence T T. Hence, T satsfes the resurce cnstrants as T satsfes. ls, because scalng by des nt affect the rderng f T, the result fllws drectly frm ET T T 44T lg lg and E T lgt T The slutn K s als a maxmum slutn t G T. Hence, we have Therem 5. Therem 5. Let T be an ptmal slutn t the resurce cnstraned versn f P SUB. lgrthm C return a pwerf-tw plcy wth expected cst at mst.44 tmes f the ptmal cst. 3.. Sngle-Resurce Cnstrant In the rest f ths sectn, we shw that fr the sngleresurce cnstraned prblem, the bund can be mprved t.06. We prve that bund fr submdular rderng cst case, generalzng a result btaned frst by Rundy 989 fr the resurce cnstraned versn f P MS. Ths prblem s nterestng, snce t cnsttutes a vald relaxatn fr the well-nwn ecnmc lt schedulng prblem. See Dbsn 987 and Rundy 989 fr a revew f ths mdel and ts cnnectn wth the ecnmc lt schedulng prblem. Cnsder the fllwng algrthm. lgrthm D.. Let T be the slutn btaned frm sngle resurce cnstrant f the type a /T added t P SUB.. Use lgrthm B wth T t btan T. Let max a /TJ, where a /T. 3. Use T as the lt szng slutn. It can be shwn that the abve algrthm reduces t Rundy s 989 algrthm, thugh the ntal plcy T befre the scalng s dfferent. Furthermre, Rundy 989 cntans a mre careful chce f t mprve the plcy, but tdes nt seem t be useful n mprvng the wrstcase bund. Therem 6. lgrthm D returns a pwer-f-tw plcy T whch s wthn.06 frm the ptmal cst. Prf. We nte that scalng by des nt alter the rder T. Hence, T has the same rder as T and T. Let K n be ptmal slutn t G T. EG T E /T + E H T If, then the bund fllws frm Therem. Cnsder the case /T.Wedente by E the cndtnal expectatn E a /T. Let X T /T wth prbablty a /T fr all. Wehave E X T a T T By Jensen s nequalty, E /X /E X, and therefre S, E G T E E +E E By Therem 3, E G T a /T T T +E T T H T a /T H T a T T T by Jensen s nequalty a /T a /T + T T T T 06 /T + a H +E + T a H a /T H T H T a /T T T T T 4. JOINT REPLENISHMENT PROBLEMS In ths sectn, we fcus n an mprved apprxmatn algrthm fr the nt replenshment prblem under the fxed base perd mdel. Recall that each tme an rder s placed, a nt rderng cst K 0 s ncurred, ndependent f the number f tems rdered. n addtnal rderng cst K s als ncurred whenever an tem s ncluded n an rder. The nventry cst s charged at a rate f h and H h d /.

7 Fr any t nteger, we let L t K t + H t If <t<+ fr sme nteger, let L t t L + + tl + We cnsder the fllwng relaxatn fr the JRP: P JR Z JR mn L T + K 0 T 0 s.t. T T 0 T 0 0 Nte that the functn Lt s pecewse lnear and cnvex. Let Z dente the ptmal slutn t the JRP under the fxed base perd mdel, ver all pssblty dynamc plces. Prpstn. In the ptmum replenshment plcy, all rderng ntervals are bunded by sme cnstant M fr sme large enugh M that depends nly n the rderng and hldng cst rates f the tems. Prf. It s clear that the ptmum replenshment plcy wll place an rder fr an tem f and nly f the nventry level f the tem drps t zer. Ths fllws frm the assumptn that replenshement s nstantaneus and the lead tme s zer. Thus, we nly need t specfy the rderng nterval as ths wll autmatcally determne the rder quantty. Suppse that fr tem, there s an rderng nterval f fr sme nteger nte that T L. In the ptmum plcy, f we replace the sngle rder by smaller rders, each wth enugh demand t last thrugh a sngle tme unt, the new plcy wll ncrease the rderng cst by at mst K 0 + K whereas the hldng cst f tem decreases by h d h d H By the ptmalty f the rgnal plcy, we must have H <K 0 + K.e., < + K 0 + K H Ths prves the prpstn, wth M max + K 0+K. H Prpstn. The value Z JR f Prblem P JR s a lwer bund n the ptmal slutn value Z f the nt replenshment prblem under general dynamc plces,.e., Z Z JR Te and Bertsmas / 605 Prf. Cnsder an ptmal plcy ver an nterval 0. Durng ths nterval, numerus rders wll be placed fr each tem. By a slght abuse f ntatn, we wll say that an rder s f rder nterval f the tme perd between the rder and the next rder s. Let m be the number f rders placed fr tem wth rder ntervals m s thus the ttal number f rders placed durng 0. Let I t be the nventry level f tem at tme t. The ttal number f dstnct rders placed s at least max m. The ttal nt replenshment cst ver 0 s thus at least K 0 max m + K m + h I tdt 0 Let E dente the nventry level f tem at tme.by Prpstn, E /d M. Nte that h I tdt h m d 0 E d Let, dente the errr term E. Then, d K m + h I tdt 0 K m m +H K m m m +H m where 0 h d M Nte that 0as. Rewrtng the expressn, we have K m + h I tdt 0 K m m + H m m m L m m m m 3 By the cnvexty f L t, tfllws that m m L m L m 4 Let and T m m m m T 0 mn m max m

8 606 / Te and Bertsmas Nte that m T m m m as m m m m <m + m <m m m m It thus fllws that T T 0 fr all >. Nw, K L m + m m L 0 h I tdt m frm 3 m m m frm 4 m L T and max K m mn m 0 K 0 T 0 S, we have K 0 max m + K m h I tdt + mn m 0 K 0 + L T T + 0 mn m Z JR + Let, then 0. On the ther hand, because all rder ntervals are bunded by M, M m s mn m Hence Z lm K 0 max m + K + h I tdt Z JR 0 m Therefre, Z JR s a vald lwer bund fr the ptmal replenshment slutn. Prblem P JR can be vewed as a cnvex nteger prgrammng prblem wth separable bectve functn, and a ttally unmdular cnstrant matrx. It can be slved effcently usng the algrthm prpsed by Hchbaum and Shantumar 990. In fact, fr the mprved apprxmatn algrthm, we d nt need t utlze the lwer bund n ts full generalty. We need nly t ensure that the rderng nterval T, whenever T 3T L,snthe set T L T L 3T L. Let T be an ptmal slutn t Prblem P JR. The pecewse lnear cst structure f L t ensures that the crdnates T are ntegers fr all assumng T L. Hence the plces n T satsfy the fxed base rderng perd cndtn. If the plces T s are nested,.e., there exsts T such that T dvdes T fr all ther, then ths relaxatn s exact. We descrbe next hw t rund the plces btaned n T nt a nested plcy. The man dea behnd the rundng heurstc s as fllws. Cnsder the class f pwer-f-tw plces called Class plces. We bserved that the 94% wrst-case bund s acheved nly f there s sme tem wth T fr sme ntegral.fr these ntervals, rundng ff t plces f the Type r 3 p called Class plces can be mre effcent. Hwever, fr the case T, the Class plces can be neffectve as we are rundng T tr3.by prperly tradng ff the tw classes f plces and carefully handlng the case wth T, we can acheve a better guarantee. Let p 07q 03, and let p + 3q/ a p + q/3 b p + 3q/ p/ + q/3 and Fpz p qz 3 q p p + 8 z F 9 p z q pz 4 p q 3 q + 8 z Nte that <a<b<4, and 3 <b<a<6. Plcy. Let T p z, where z s n the nterval, and p nteger. Let Y be a randm number generated n the nterval a b wth dstrbutn functn PY y Fpy. Let { T p f z <Y p+ f z Y Nte that f z <a/rz >b/, then z s always runded t r, regardless f the value f Y.Sfz falls nt ths range, the rundng s determnstc. Plcy. Fr all tems wth T 3, let T 3 p z,sn the nterval [,. Let Y be a randm number generated n

9 the nterval b a wth dstrbutn functn PY y F p y. Let { T 3 p f 3z <Y 3 p+ f 3z Y Nte that f z <b/3rz > a/3, then z s always runded t r, regardless f the value f Y.Sfz falls nt ths range, the rundng s determnstc. Fr all tems wth T, we rund all f them n the same manner t T 3 wth prbablty 9 and t T 4 wth prbablty 5. 4 Fnally, f T T. Nte that n ths way, fr T, ET 8 7 E T ls, the functns Fpz and F p z are selected s we have Fpa 0 F p b, and Fpb F p a. Furthermre, F and F are nndecreasng and are vald dstrbutn functns. lgrthm E.. Select Plcy wth prbablty p, and Plcy wth prbablty q.. Let T dente the plcy selected. Therem 7. lgrthm E returns a plcy wth expected cst at mst.043 Z JR 043z. Remar. In essence, the abve therem says that ne f the tw plces cnstructed abve wll attan a bund f at mst.043. The frst plcy has the classcal pwer-ftw structure and s therefre easly mplemented n practce. The secnd plcy has rder ntervals f the type 3 6 and can als be easly mplemented n practce. Remar. In Plcy, we lse a bt n rundng rder ntervals f type T tr3.the bund can be tghtened slghtly f the ptmal slutn t the relaxatn des nt have rder ntervals f the type T L. Ths dffculty als precludes the pssblty f extendng the technque t nclude rder ntervals f type 5T L and abve. Prf. If T, then ET E T p q 048 Thus we need nly t cnsder the case when T s greater than 3. Suppse T les n a p a p b r b p b p + a. InCase a, Plcy always runds T t 3 p, whereas n Case b, Plcy always runds T t p +. Case a. T les n p a p b,.e., T p w p+ z, where w a b and z. Then ET qet + pet q3 p + p p+ Pz <Y+ p+ Pz Y 3q T + p + Fw w w and E T qe T T + pe q w 3 + p Te and Bertsmas / 607 T Fw We have chsen Fp such that q 3 w + p + Fw w q w 3 + p w Fw w Wth ths chce f F, and p 07q 03, we can ptmze the bund ver the range f w a b t btan ET T T E T 043 Case b. T les n p b p + a,.e., T p w 3 p z, where w b a and z. Then E py + qt and E p T p p + + q 3 p P3z <Y + 3 p+ P3z Y T p 4w + q 3 + F w + q T We have chsen F such that p 4 + q 3 + F w w T w w p w T 4 + q F w p w 4 + q Wth ths chce f F,agan we have E pt + qt T p + q max w b a 043 Hence the result fllws. T T 3 w F w w 3 p 4w + q 3 + F w 5. CONCLUDING REMRKS In ths paper, we prpsed a new randmzed rundng apprach t several multstage nventry/dstrbutn lt szng prblems. The apprach smplfes and extends the prfs t several well-nwn results n ths area, especally n the case f cnstraned lt szng prblems. Mre mprtantly, thrugh the use f a strnger frmulatn we btan an mprved apprxmatn bund fr the nt replenshment prblem under the fxed base plannng perd mdel. Our result shws that the class f statnary plces s wthn 95.8% f the pssbly dynamc ptmal plces under ths mdel. w

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