We consider the problem of scheduling trains and containers (or trucks and pallets)

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1 Schedulng Trans and ontaners wth Due Dates and Dynamc Arrvals andace A. Yano Alexandra M. Newman Department of Industral Engneerng and Operatons Research, Unversty of alforna, Berkeley, alforna Dvson of Economcs and Busness, olorado School of Mnes, Golden, olorado We consder the problem of schedulng trans and contaners (or trucks and pallets) between a depot and a destnaton. Goods arrve at the depot dynamcally over tme and have dstnct due dates at the destnaton. There s a fxed-charge transportaton cost for each vehcle, and each vehcle has the same capacty. The cost of holdng goods may dffer between the depot and the destnaton. The goal s to mnmze the sum of transportaton and holdng costs. For the case n whch all goods have the same holdng costs, we consder two varatons: one n whch the holdng cost at the destnaton s less than that at the orgn, and one n whch the relatonshp s reversed. For the frst varaton, we derve propertes of the optmal soluton whch provde the bass for an O T 2 soluton procedure. For the second varaton, we ntroduce a new defnton of a regeneraton state, derve strong characterzatons of the shpment schedule wthn a regeneraton nterval, and develop an O T 4 procedure. We also analyze two mult-tem scenaros. In the frst, for each tem, the holdng cost at the orgn s less than that at the destnaton; n the second, the relatonshp s reversed for all tems. We generalze several of the structural results for the sngle-tem problem to the correspondng mult-tem case. We also show that the optmal vehcle schedule can be obtaned by solvng a related sngle-tem problem n whch the tem demands are aggregated n a partcular way. The optmal assgnment of customer orders to vehcles can then be found by solvng a lnear program. Introducton Our work s motvated by the problem of schedulng trans or trucks outbound from a sngle depot and assgnng goods to these vehcles, wth the objectve of achevng on-tme delvery at mnmum cost. We consder a fnte tme horzon durng whch customer orders dynamcally become avalable at the orgn, and have dfferent due dates at ther destnatons. We assume that the demand, dstngushed by orgn, arrval date at the orgn, destnaton, and due date at the destnaton, s known n advance, or can be forecasted accurately enough over the tme horzon for plannng purposes. We consder drect shpments between a depot and each of the varous destnatons. Assumng there s no dependence among the destnatons, we can decompose the problem by destnaton. Although our ntal motvaton was derved from tran schedulng applcatons, a smlar stuaton also arses when fnshed goods must be transported by truck from a factory to dstant markets or break-bulk warehouses. ustomer orders wth dstnct due dates and destned for a partcular market are produced at the factory and become avalable for shpment over tme. The manufacturer faces the decson of when to dspatch trucks and whch customer orders to assgn to each truck. Of course, ths problem would /01/3502/0181$ electronc ISSN Transportaton Scence 2001 INFORMS Vol. 35, No. 2, May 2001 pp

2 be smplfed f goods destned for a partcular market were manufactured close together n tme. However, there are often competng and/or overrdng consderatons n establshng the producton schedule, thus necesstatng the soluton of the transportaton schedulng problem. As s typcal n ral, truckng, and sea transport operatons, there s a (nearly) fxed charge assocated wth the drect movement of the vehcle or vessel from the orgn to the destnaton whch ncludes the cost of labor for the movement of the vehcle and any other costs that are not volume dependent, ncludng the porton of fuel and mantenance costs that do not depend on the volume of goods n the vehcle. Furthermore, the capacty of the transport vehcle s known. Because we are addressng a short-horzon problem, we assume that the fxed cost of transportaton per vehcle s constant over tme and ndependent of the number of vehcles sent. In the vast majorty of stuatons, total volumedependent costs wll dffer only slghtly, f at all, as the shppng schedule changes. For example, the ncremental cost for fuel assocated wth carryng a gven weght or volume of goods va truck from orgn to destnaton, above and beyond that requred to operate the truck(s) empty, would be roughly the same rrespectve of the allocaton of goods among trucks. It would be more effcent, of course, to shp all of the goods on as few trucks as possble, provded ther capacty s not exceeded, and ths aspect s captured by the fxed-charge transportaton cost. We assume that total volume-dependent costs are nsenstve to the detals of the shppng schedule. Smlarly, n many settngs the cost of holdng the goods at the orgn dffers lttle from the cost at the destnaton, but they may dffer wdely n other settngs. For ths reason, we also analyze stuatons n whch the holdng costs dffer among tems and between locatons. Because penaltes for tardy delvery may be substantal, our prmary goal s to mnmze the sum of fxed-charge transportaton costs whle delverng the goods on tme. We wll, however, consder the more general problem n whch the holdng costs may dffer between the orgn and destnaton to reflect the relatve cost of storage space, as well as the extent to whch the customer s wllng to accept early shpments. In some nstances, the customer may wsh to receve the shpment as early as possble, whle n other nstances a just-n-tme shppng schedule may be preferable. The former stuaton can be modeled by mposng a hgher holdng cost at the orgn than at the destnaton, whle the latter scenaro s represented by the reverse relatonshp. We note that any opportunty costs of captal assocated wth holdng the goods n transt are usually borne by the shpper or consgnee, and not by the transporter. The transporter does, however, bear the opportunty cost of havng equpment, such as tralers and contaners, unavalable for use. We analyze the case n whch goods have homogeneous holdng costs, where the holdng cost s more expensve at the destnaton than at the orgn. The opposte holdng cost relatonshp can be addressed by the same approach usng an approprate transformaton of the problem. We also model versons of the problem wth tem-dependent holdng costs. We present results for the case n whch, for each tem, the holdng cost at the destnaton s hgher than that at the orgn. If the reverse relatonshp holds for all tems, a transformaton smlar to that mentoned above can be used to solve the problem. We do not consder the most general case n whch t s less expensve to hold some tems at the orgn than at the destnaton, and the reverse holds for the other tems. The latter stuaton may arse, for example, f the freght has dfferent temperature requrements; storage costs wll depend upon the ambent weather. Our problem s smlar to other capactated fxedcharge network flow problems but contans three mportant complcatng features: () not all goods are avalable at the begnnng of the horzon,.e., goods arrve dynamcally, () t may be necessary and/or optmal to send more than one vehcle n a gven perod,.e., to ncur multple setups, and () goods are not homogeneous and have dstnct due dates at the destnaton, and thus must be treated as dstnct customer orders. We assume, however, that the goods are homogeneous n ther use of transport capacty. Although there s a large body of lterature on lotszng problems wth a fxed-charge structure, lttle work has been done on problems n whch there s a 182 Transportaton Scence/Vol. 35, No. 2, May 2001

3 Table 1 omparson of Our Model to Related Lterature Nature of No. of Dynamc Multple Paper Demand Items Arrvals? Setups? Floran & Klen (1971) determnstc, dynamc one no no Lppman (1969) determnstc, dynamc one no yes Lppman (1971) constant one no yes Iwanec (1979) stochastc, dynamc one no yes Lee (1989) determnstc, dynamc one no yes Ben-Kheder (1990) determnstc, dynamc multple no yes Our paper determnstc, dynamc one or more yes yes capacty assocated wth each fxed charge ncurred. Floran and Klen (1971) consder a sngle-tem capactated lot-szng problem where all materals are avalable for processng (or, alternately, for shpment) at the begnnng of the horzon, and there s a fxed charge for each producton run wth fxed capacty. At most, one producton run s allowed n each perod. They show that when capacty s constant over tme, the optmal soluton has a producton level equal ether to zero or to the capacty n all perods except at most one between two consecutve regeneraton ponts. (For Floran and Klen s problem, a regeneraton pont s defned as a pont n tme wth no onhand nventory.) Ths result provdes the bass for an effcent soluton procedure based on an underlyng shortest path problem. Lppman (1969) addresses the sngle-product problem wth statc arrvals and multple setups. He derves several propertes of the optmal soluton and develops an O T 3 algorthm for the problem. We relate hs characterzaton of the optmal soluton to our structural results as our dscusson proceeds. In a later paper, Lppman (1971) treats a contnuous-tme, constant-demand verson of the problem. Iwanec (1979) studes a base stock polcy, rounded up to the next full vehcle, n the context of a dscrete-tme verson of the problem wth stochastc demand. Lee (1989) addresses the multple setup problem where all materals are avalable for shpment at the begnnng of the horzon and there s a separate setup cost per order. He presents an O T 4 procedure for the problem. All of the aforementoned artcles consder only a sngle product. The only research of whch we are aware that treats the mult-tem case (where the tems have dfferent holdng costs) n Ben-Kheder (1990). He studes the case wth dynamc demands under the assumpton that the holdng cost at the destnaton s hgher than at the orgn. He presents a soluton procedure wth an underlyng shortest path network, where the path costs are computed usng a branchand-bound algorthm. Table 1 contrasts our models wth others n the lterature. Note that none of the models n the lterature allows for dynamc arrvals of goods. The remander of the paper s organzed as follows. In 1, we present a dynamc programmng formulaton of the problem. In 2, we characterze the optmal soluton and present an optmal O T 4 algorthm for the case n whch all goods have the same holdng costs, and the holdng cost s larger at the destnaton than at the orgn. We also explan how the reverse holdng cost relatonshp can be handled by an approprate transformaton of the network. In 3, we extend these results to allow for tems wth dfferent holdng cost rates. Secton 4 concludes the paper wth a dscusson of the relatonshp between our results and those for related models wth statc arrvals. 1. Problem Formulaton Because of the dynamc arrvals and the nonhomogenety of the goods, we need to dstngush the goods by arrval date at the depot and due date at the destnaton. For smplcty, we assume that shpment quanttes can be treated as f they were contnuous. Practcally speakng, ths means that shpments occur n ncrements of standard pallet loads for truck travel, or standard contaner szes for ral shpments. Assumng that all shpments are n multples of the standard Transportaton Scence/Vol. 35, No. 2, May

4 load sze, and the capacty of the transport vehcle s expressed as an nteger multple of the standard load, we can treat the shpment quanttes as f they were contnuous wthout loss of generalty. We also assume that the transportaton tmes are determnstc and constant over tme, and therefore, wthout loss of generalty, can be treated as f they were nstantaneous by an approprate rendexng of the tme perods. We formulate the problem as a dynamc program. We defne the state of the system by a par of vectors. One vector represents the contaners avalable to be shpped, and contans as many entres as there are remanng due dates n the horzon. The other vector represents the nventory at the destnaton wth an entry correspondng to each remanng due date n the horzon. For ease of exposton, n the remander of the paper, we use the ral termnology of trans and contaners to represent the vehcle and the unt shpment load, respectvely. Let us defne the followng notaton: t = tme perod ndex, t = 1 T S = fxed charge per shpment = capacty of each tran (expressed as number of contaners) h o = holdng cost for one contaner for one perod at the orgn h d = holdng cost for one contaner for one perod at the destnaton D t d = quantty of contaners arrvng at the orgn n perod t that are due at the destnaton n perod d, d t D t = D t t D t t + 1 D t t + 2 D t T D t = u t D u t 0 0.e., a vector n whch the frst element s the total quantty of contaners due n perod t and the remanng T t elements are zero A t d = quantty of contaners avalable to be shpped from the orgn n perod t that are due n perod d, d t A t = A t t A t t + 1 A t t + 2 A t T I t d = nventory of contaners due n perod d held at the destnaton at the end of perod t d t I t = I t t + 1 I t t + 2 I t T x t d = number of contaners shpped n perod t that are due n perod d d t x t = x t t x t t + 1 x t t + 2 x t T decson (row) vector ft A t I t 1 = the mnmum cost for perods t T gven contaner avalablty vector A t and contaner nventory vector I t 1 f the optmal number of contaners, xt, s shpped at tme t. The dynamc programmng recurson equatons are: { f t A x t I t 1 = mn S t d +h xt o A t d x t d d t d>t [ +h d It 1 d +x t d ] where d>t ( ) } +f t+1 At +D t+1 x t I t 1 +x t D t f T +1 = 0 (1) x t t = A t t t (2) x t d A t d d t (3) x t d 0 d t (4) Equaton (1) s the boundary condton. onstrants (2) and (3) ensure that demand s satsfed on tme and that only avalable contaners are shpped, respectvely. onstrants (4) ensure nonnegatve shpment quanttes. We present a complete analyss for the case n whch the holdng cost s more expensve at the destnaton than at the orgn. Ths cost structure provdes motvaton for sendng contaners as late as possble whle accountng for economes of scale n transportaton. In 2, we derve propertes of the optmal soluton and present an O T 4 algorthm. Followng our analyss, we explan how the case wth the opposte holdng cost relatonshp can be treated by a reversal of the multcommodty network n tme and space. 2. Analyss and Algorthm for h d h o The man economc tradeoff n ths problem s between the economes of scale assocated wth sendng full trans and the addtonal nventory holdng cost ncurred f contaners are shpped early. To develop an effcent soluton procedure for ths case, we employ the concept of a regeneraton nterval, as has been used for smlar problems. However, our 184 Transportaton Scence/Vol. 35, No. 2, May 2001

5 defnton of a regeneraton state dffers sgnfcantly from the tradtonal one. The tradtonal regeneraton state n a lot-szng settng s defned as a state wth no on-hand nventory. In the context of our problem, ths would correspond to havng nothng remanng to be shpped. Ths state would rarely occur n our problem due to the economc ncentve to shp as late as possble. Usng the tradtonal defnton would often lead to regeneraton ponts only at the begnnng and end of the horzon, and the soluton procedure would degenerate to the enumeratve dynamc programmng procedure descrbed earler. Defnton 1. We say that a regeneraton occurs at the begnnng of perod t (equvalently, at the end of perod t 1) f no contaners due n perods t, t + 1 T have been shpped n perods 1 2, t 1,.e., not contaners due n perod t or later have been shpped before perod t. Such a state clearly defnes a pont n tme such that the problem can be separated nto two dstnct problems: () shppng contaners due n perods 1 2 t 1, and () shppng contaners due n perods t, t + 1 T. Thus, ths defnton provdes the same type of decomposton as n earler models. Our defnton of a regeneraton nterval affords the advantage of permttng multple regeneratons durng the horzon, whch, n turn, reduces the computatonal effort requred. Note that n addton to any regeneraton ponts that may occur between perods 2 and T 1 regeneratons occur at the begnnng and end of the horzon. Hence, the optmal soluton for the entre horzon can be obtaned by fndng the optmal soluton for each potental regeneraton nterval, and then solvng a shortest path problem over the entre horzon to determne the optmal set of regeneraton ntervals haracterstcs of the Optmal Soluton Usng our defnton of a regeneraton nterval, we derve several propertes of the optmal soluton. We note that f h d = h o the optmal soluton generally s not unque, so alternate schedules consdered n our proofs may not be strctly domnant but may nstead represent alternate optmal solutons. Proposton 1. There exsts an optmal soluton n whch the number of contaners sent ahead of schedule n any ndvdual tme perod s strctly less than. In other words, tems are shpped early only for the purpose of fllng up a tran ether completely or partally. Proof. Omtted. A proof appears n Yano and Newman (1998). Note that Proposton 1 does not necessarly mply that the cumulatve number of contaners shpped early must be less than. It may be optmal for the cumulatve number of contaners shpped early to be greater than, especally when transportaton costs domnate holdng costs, makng t desrable to send trans full or nearly full. Proposton 2. There exsts an optmal schedule n whch, whenever contaners are shpped early, they are shpped n ncreasng order of ther due dates among the contaners avalable to be shpped. Proof. We brefly sketch the proof here. The proof reles on two observatons wth respect to feasble solutons: () the cost n the current perod depends only on the total number of contaners shpped (assumng that all contaners due n the current perod are ncluded among the shpped contaners), and () the cost to go for the remanng perods s the same or smaller for an earlest due date (EDD) contaner shppng schedule than for any non-edd shppng schedule. Part () s self-evdent. Part () can be proved by showng that an arbtrary change toward an (EDD) shppng schedule n the current perod reduces the quanttes n the vector of cumulatve shpments requred to ensure on-tme delvery for the remanng perods, whch, n turn, reduces the cost to go. See Yano and Newman (1998) for a complete proof. Proposton 3. There exsts an optmal schedule n whch, f any contaners due n perod t are shpped n perod t <t, trans sent n perods t + 1 t + 2 t are full. Proof. Suppose that we have an optmal soluton n whch we shp contaners that are due n perod t <t and that all trans (f any) sent n perods t +1, t + 2 t are not full. Then, because these contaners are not due untl perod t, a feasble shpment Transportaton Scence/Vol. 35, No. 2, May

6 plan exsts n whch some or all of these early contaners are sent n some perod(s) n t + 1 t + 2 t. Furthermore, because h o h d, a lower- or equalcost shpment plan exsts n whch some or all of these contaners are sent n these subsequent perods. Hence, the schedule cannot be optmal, whch contradcts our orgnal hypothess. Theorem 1. The optmal schedule wthn a regeneraton nterval has full trans n every perod, except possbly the frst. Proof. Let s be the frst perod n the regeneraton nterval and be the last perod n the regeneraton nterval. Let t be a perod such that s<t.e., an arbtrary perod ether n the mddle of, or at the end of, the regeneraton nterval. Because the system does not regenerate at the begnnng of t by assumpton, we must have: I t 1 d > 0 d t by our defnton of a regeneraton nterval. Suppose that f any trans are sent n perod t, they are not full. Then t would be possble to delay some or all of the earler-sent contaners wth due date t or later, and thereby reduce the holdng costs wthout ncreasng the transportaton cost. Thus, f any trans are sent n perod t, they must be sent full. Because t s an arbtrary perod between s + 1 and, the result follows for all perods except the frst. Theorem 1 provdes a stronger characterzaton of shpment quanttes durng a regeneraton nterval than that gven by Lppman (1969), and t does so for the more general case n whch dynamc arrvals are permtted. It also generalzes the result of Floran and Klen (1971) to the case of multple setups. Observaton 1. From our defnton of a regeneraton nterval, the frst perod of the regeneraton nterval satsfes: I s 1 d = 0 d s.e., there are no contaners due n perod s or later that have been shpped pror to perod s. Thus, we may have a less-than-full tran. The followng corollary generalzes a result of Lppman (1969) to the case of dynamc arrvals. orollary 1. There exsts an optmal soluton n whch for all t, [ ] d t I t 1 d [ d t ] x t d mod = 0 Proof. The result follows from Theorem 1 and Observaton Test for Feasblty of a Perod as the End of a Regeneraton Interval The results n the prevous subsecton can be used to determne whether a perod s elgble to be the last perod n a regeneraton nterval, and could substantally reduce computaton tmes n practcal applcatons. We frst descrbe the logc underlyng the procedure, and then descrbe the procedure. Recall that n the last perod of any regeneraton nterval,, the trans must be full. Because t s optmal to delay contaners as much as possble, the optmal shpment quantty n perod s: x = u D u The balance of perod s demand s gven as follows: D u u u D u = u D u mod Ths balance must be shpped early, that s, n perod 1 or earler. If ths quantty s not avalable at the begnnng of perod 1 because all or nearly all of the demand for perod arrves n, then we cannot have all trans full n perod and have some of ts demand shpped early. If 1 u=1 D u < D u D u u=1 u=1 cannot be the last perod n any regeneraton nterval, because we cannot send all trans full n perod. In addton to the nfeasbltes noted above, some regeneraton ntervals may not be feasble because contaner arrvals may not allow full-tran shpments 186 Transportaton Scence/Vol. 35, No. 2, May 2001

7 accordng to the pattern descrbed n Theorem 1. Although we can test for ths n advance, t also can be done as effcently wthn the context of the procedure for fndng the optmal schedule wthn a regeneraton nterval. If no feasble schedule exsts, we assgn an nfnte cost to the correspondng arc n the shortest path network Algorthm for Determnng the Optmal Schedule: h d h o For any potentally feasble regeneraton nterval, the optmal schedule can be constructed as follows: Step 1. Let [s ] denote a regeneraton nterval, where s s the frst perod n the nterval, and s the last perod n the nterval. ompute the quantty (f any), L, to be sent n the less-than-full tran n the frst perod of the regeneraton nterval: L = D u d mod d=s u d Step 2. In perod s, send u s D u s trans f u s D u s mod L Otherwse, send u s D u s + 1 tran(s). Fll all but one tran; the last tran wll be flled wth L< contaners. ontaners should be assgned to the trans n ncreasng order of due date. If there are nsuffcent contaners avalable, termnate the algorthm. The regeneraton nterval s nfeasble. Otherwse, go to Step 3. Step 3. For t = s + 1to (a) Update A t. (b) Send A t t trans, fllng them wth contaners n ncreasng order of due date. If there are nsuffcent contaners avalable to shp the trans full, termnate the algorthm. The regeneraton nterval s nfeasble. Otherwse, contnue Step 3 (ncrementng t). Ths procedure has a complexty of O T 2 and must be performed for each possble regeneraton nterval, of whch there are O T 2. Hence, the computaton of arc costs for the shortest path problem s O T 4. The shortest path problem tself s O T 2. Thus, the overall procedure has complexty O T 4. Note that n computng the cost of each arc n the shortest path network, one needs to nclude the orgn holdng cost for contaners that are due durng the regeneraton nterval but arrve at the orgn before the begnnng of the regeneraton nterval. If few contaners arrve early, many regeneraton ntervals wll be nfeasble. On the other hand, for the statc case n whch all contaners are avalable to be shpped at the begnnng of the horzon, all regeneraton ntervals are feasble because the avalablty of goods does not lmt the shpments n Steps 2 and 3(b) above Modfcaton for h d <h o We can solve the case n whch holdng costs are hgher at the orgn than at the destnaton by reversng the network n tme and space. That s, we treat the problem as f goods arrve at the destnaton at ther respectve due dates and are due at ther orgns on ther respectve arrval dates. Trans and contaners flow from destnaton to orgn and backward n tme. osts on all arcs reman the same. The soluton procedure descrbed n the prevous secton can be appled to ths problem. When solvng the problem forward rather than backward, the regeneraton state s defned as a state n whch there are no further contaners avalable to be shpped. All trans are full, except possbly n the last perod of the regeneraton nterval. 3. Multple Items wth Dfferent Holdng osts A great deal of lterature treats mult-tem lot-szng problems wth ether fxed-charge jont replenshment costs or a sngle capacty constrant. However, lttle research has been done on mult-tem problems wth multple setup costs. The only work of whch we are aware s that of Ben-Kheder (1990), who consders the case of h d for all tems, gven statc arrvals. He employs the tradtonal defnton of a regeneraton state and characterzes propertes of the shpment schedule wthn a regeneraton nterval, such as the tmng of full and partally full trans wthn the regeneraton nterval. From ths, he develops an O T 3 soluton procedure. We generalze our results n 2 to the case of multple tems, each wth dfferent holdng costs. The dynamc programmng formulaton s essentally the same as that gven earler, except that tems are also dstngushed by holdng cost. Thus, the state and Transportaton Scence/Vol. 35, No. 2, May

8 decson vectors n the sngle-tem case become a state matrx and decson matrx, respectvely, n the multtem case. We frst treat the case n whch h d for all tems, and then explan how to adapt our results to the case n whch h d <h o for all tems. In most practcal settngs, the goods beng shpped are owned by ether the shpper or consgnee, so the value of the goods does not play an mportant role n the determnaton of holdng costs borne by the transporter. However, the holdng costs ncurred by the transporter may be locaton related. For example, the cost of electrcty may be more expensve at one locaton than another, leadng to hgher costs for holdng refrgerated contaners. For such realstc stuatons, we show that wthn a regeneraton nterval, the followng characterstcs developed for the sngletem case do not change when multple tems are consdered: () the structure of the tran schedule wth respect to full versus partal loads; and () the optmal soluton regardng how many trans to send n each perod. We also show that gven () and (), the problem of allocatng contaners to trans can be solved usng lnear programmng. In the nterest of brevty, we state some results wthout detaled proofs; n all of these nstances, the logc follows n a straghtforward manner from that of the sngle-tem case. The case n whch h d for some tems and h d <h o for other tems proves to be very dffcult because the constructon of an optmal tran schedule s much more complex. We elaborate on these mplcatons n the concludng secton Multple Items wth h d Although we do not use these results drectly, we note that Propostons 1 through 3 and Theorem 1 all generalze to the case of multple tems. If the holdng cost at the destnaton s greater than that at the orgn for each tem, the proofs can be constructed n a smlar way. We now extend the defnton of a regeneraton state: Defnton 2. We say that a regeneraton occurs at the begnnng of perod t (equvalently, at the end of perod t 1) f no contaners of any type due n perod t t + 1 T have been shpped n perods 1 2 t 1.e., no contaners of any type due n perod t or later have been shpped before perod t. We frst construct a soluton consstng of a tran schedule derved from an adaptaton of the sngletem soluton procedure combned wth an optmal allocaton of contaners for ths tran schedule obtaned by solvng a lnear program. We frst show how to construct ths soluton, then demonstrate that t s optmal for the mult-tem problem. onsder a varant of the mult-tem problem n whch, for each (arrval date, due date) par, we aggregate demands across holdng costs. For ths aggregate tem, choose an arbtrary postve holdng cost at the destnaton and assume, wthout loss of generalty, that the nventory holdng cost at the orgn s zero. We now use the algorthm descrbed n 2.3 to fnd the optmal soluton for ths revsed problem wthn the regeneraton nterval. From the soluton to the revsed problem, we can construct a feasble schedule for the orgnal problem wth the same tran schedule and the same aggregate shpment quanttes. We show how to construct an optmal detaled, multtem schedule from ths aggregate schedule. onsder an assgnment of (avalable-to-shp) contaners n whch contaners are assgned to trans startng n the frst perod of the regeneraton nterval n ncreasng order of ther due dates. Such a schedule would be comparable to the sngle-tem schedule n that dfferences n holdng costs among the tems are gnored. learly, we can mprove the soluton by mantanng the same tran schedule and modfyng the shpment of contaners to mnmze nventory holdng costs subject to satsfyng on-tme delvery. Ths can be done by solvng a lnear program for each regeneraton nterval. Defne: s = frst perod n the regeneraton nterval = last perod n the regeneraton nterval z t = number of trans scheduled n perod t L = D u d mod N = = d=s u d u s D u s f D u s mod L u s u s D u s otherwse 188 Transportaton Scence/Vol. 35, No. 2, May 2001

9 Note that N represents the number of full tranloads and L s the fractonal tranload sent n the frst perod of the regeneraton nterval. The remanng notaton parallels that of the sngle-tem case, wth denotng the tem (or contaner) type. Omttng the sunk nventory holdng costs from the objectve functon, the problem for fxed z t, t = s, can be formulated as: mn h d h o x t d t=s d>t subject to: x s d = N + L (5) d s t x u d = N + L + u=s d u t u=s+1 z u t s + 1 (6) x u d D u d d s t < d (7) u t u t x u d = D u d d s (8) u d u d x t d 0 t d (9) The objectve s to mnmze total nventory holdng costs. The objectve functon represents the holdng cost ncurred by the contaners shpped early, summed across all tems and perods. onstrants (5) and (6) ensure that no more contaners are shpped than the capacty of the scheduled trans wll allow. onstrants (7) ensure that for each due date, cumulatve shpments n each perod do not exceed cumulatve arrvals. onstrants (8) ensure that all contaners are shpped on tme. onstrants (9) ensure nonnegatvty of shpment quanttes. The lnear program smply reorganzes the contaners wthn the confnes of a fxed tran schedule to mnmze the combned holdng costs ncurred at the orgn and at the destnaton. We note that the problem has the structure of a mnmum cost network-flow problem, and thus, there exsts an optmal ntegral soluton. The total quantty shpped n each perod s the same as n the related sngle-tem problem, but the mx of contaners now dffers. For convenence, let us refer to the optmum soluton from the lnear program, along wth the assocated tran schedule, as the reference schedule. In the reference schedule all trans are full n each regeneraton nterval, except possbly one tran n the frst perod. Thus, wthn each regeneraton nterval, the transportaton cost cannot be reduced. Because the tran schedule for the related sngle-tem problem s feasble for the mult-tem problem, t s also the mnmum transportaton cost schedule for the multtem problem. We show that no other tran schedule, along wth ts optmal allocaton of contaners to trans (from the lnear program), wth the same or greater number of trans n the regeneraton nterval, yelds lower overall (transportaton and nventory) costs. In dong so, we also show that the optmal tran schedule wthn a regeneraton nterval has the same propertes for the mult-tem case as for the sngletem case. Theorem 2. For each regeneraton nterval, the reference schedule produces an optmal soluton to the multtem problem n whch h d for all. Proof. The proof contans three parts. We show that (a) t s not feasble to reduce the cumulatve number of trans n any perod n the reference schedule whle mantanng the same total number of trans n the regeneraton nterval; (b) ncreasng the cumulatve number of trans n any perod whle mantanng the same total number of trans n the regeneraton nterval ncreases costs; and (c) ncreasng the total number of trans n the regeneraton nterval, whch requres ncreasng the cumulatve number of trans n at least one perod, ncreases costs. Together, these results demonstrate that the number and tmng of trans n the reference schedule s optmal for the mult-tem problem. Part (a). Let us consder reducng the cumulatve number of trans n some perod, retanng the same total number of trans n the regeneraton nterval. In the reference schedule, the contaners are sent as late as possble whle mantanng the full-tran property n all but the frst perod of each regeneraton nterval, and whle satsfyng on-tme delvery. Thus, t s not possble to reduce the cumulatve number of trans n any tme perod wthout causng some contaners to be tardy. Transportaton Scence/Vol. 35, No. 2, May

10 Part (b). We now consder the case n whch the cumulatve number of trans sent on or before some arbtrary perod ncreases but the total number of trans durng the regeneraton nterval remans the same. Suppose that n some perod t there are suffcent contaners avalable to ncrease the cumulatve number of tran movements, e.g., to shp one tran one perod earler. Then, the revsed lnear program wll have the same structure as that above, except that the rght-hand sde of (5) or (6) for the gven perod t s ncreased accordngly. The holdng costs are hgher at the destnaton than at the orgn; thus all coeffcents on the x t (d) terms n the objectve functon are postve. Hence, we can replace the equaltes n onstrants (5) and (6) wth greater than or equal to relatons. Increasng the rght-hand sde of any of these revsed constrants (.e., ncreasng the cumulatve number of trans) would ncrease the objectve functon value. Thus, t s not possble to modfy the tran schedule to reduce nventory costs by ncreasng the cumulatve number of trans n one perod whle retanng the same total number of trans. Part (c). The proof for the case n whch the cumulatve number of trans shpped on or before some arbtrary perod ncreases whle the total number of trans n the regeneraton nterval ncreases parallels that n Part (b). The procedure descrbed above produces an optmal soluton for a gven regeneraton nterval. The test for determnng the feasblty of a perod as the end of a regeneraton nterval for the sngle-tem case (descrbed n 2.2) also can be appled to the multtem case usng the aggregate tem data. After the contaner shpment schedule s determned usng the lnear program for each potental regeneraton nterval, a shortest path problem must be solved to fnd the best set of regeneraton ntervals. The lnear program allows the tradeoff between urgency of the contaners and ther holdng costs to be made optmally n assgnng contaners to trans. It may seem ntutve to follow a nave approach wth respect to a contaner shpment order,.e., shp contaners n such a way that the holdng costs are mnmzed. However, because of the competng factor of due-date requrements, such an approach may result n more tran shpments. The fxed costs assocated wth these shpments may more than offset any nventory holdng costs saved. In general, nether urgency nor costs can be consdered alone Modfcaton for h d <h o The mult-tem problem wth h d <h o for all can be solved by () aggregatng the demands across tems for each (arrval date, due date) combnaton, () reversng the network n tme and space to fnd the optmal tran schedule for the aggregated sngletem problem, then () usng a lnear program analogous to the one descrbed for the case of h d for all to assgn contaners to trans. 4. onclusons and Dscusson We now relate the results n ths paper to the specal case n whch arrvals are statc,.e., all contaners are avalable at the begnnng of the horzon. 1. Sngle-tem case wth h d h o : The specal case of statc arrvals reduces to the problem consdered by Lppman (1969), who provdes an O T 3 algorthm for the cost structure that we use here (n addton to more complex algorthms for more general cost structures). We extend certan propertes of the optmal soluton to the case of dynamc arrvals, present tests that elmnate some perods as the last perod n a regeneraton nterval, and devse an optmal algorthm wth O T 4 complexty. The statc-arrval verson of our problem s also a specal case of Lee (1989), who ncludes an order setup cost n addton to the multple setup cost for transportaton. Hs algorthm has O T 4 complexty. 2. Sngle-tem case wth h d <h o : In the case of statc arrvals, the problem s trval; everythng s sent n the frst perod. When arrvals are dynamc, the problem s more complcated and can be solved wth a varant of the algorthm presented for the case n whch h d h o. 3. Multple tems wth h d : The specal case of statc arrvals reduces to a problem dscussed n Ben-Kheder (1990). He shows that wthn a regeneraton nterval, the structure of the optmal polcy has full trans n all perods except possbly the frst. He presents an O T 3 soluton procedure. We extend ths structural result to the case of dynamc arrvals, and 190 Transportaton Scence/Vol. 35, No. 2, May 2001

11 show that the optmal tran schedule can be obtaned by solvng a related sngle-tem problem n place of the more complex mult-tem problem. Due to the dynamc arrval pattern, however, the allocaton of contaners to the trans requres the soluton of a lnear program. 4. Multple tems wth h d <h o : We are not aware of any pror work on ths problem. Most work on lot szng s motvated by manufacturng applcatons where nventory holdng costs typcally ncrease as the product becomes more complete. However, locaton-dependent holdng costs arse frequently n transportaton applcatons. In the statc case, the problem s trval as all contaners are shpped n the frst perod. For the case of dynamc arrvals, the problem can be solved wth a varant of the algorthm presented for the case n whch h d for all. The ablty to fnd an optmal tran schedule usng an aggregate tem can provde consderable reductons n computaton tme when there are many types of tems and/or many trans to be scheduled. We are not aware of any pror observatons of ths phenomenon. Of course, the result apples only when h d <h o,orh d >h o for all. When the relatonshps are mxed, both the tran schedule and the contaner shpment schedule may be qute complex. Some tems should be shpped as soon as possble whle others should be shpped just n tme, makng t dffcult to characterze when full and partal tranloads should be sent. Ths remans a topc for future research. Acknowledgments Ths research was supported by Natonal Scence Foundaton Grant GER/HRD to the Unversty of alforna, Berkeley. The authors gratefully acknowledge the helpful comments of the referees and assocate edtor on earler versons of the paper. References Ben-Kheder, N Economc lot szng for mult-suppler, multtem procurement systems. Unpublshed Ph.D. dssertaton, Department of Industral and Operatons Engneerng, Unversty of Mchgan, Ann Arbor, MI. Floran, M., M. Klen Determnstc producton plannng wth concave costs and capacty constrants. Management Sc. 18(1) Iwanec, K An nventory model wth full load orderng. Management Sc. 25(4) Lee,. Y A soluton to the multple set-up problem wth dynamc demand. IIE Trans. 21(3) Lppman, S. A Optmal nventory polcy wth multple setup costs. Management Sc. 16(1) Economc order quanttes and multple setups. Management Sc. 18(1) Yano,. A., A. M. Newman Schedulng trans and contaners wth due dates and dynamc arrvals. Workng paper, Department of Industral Engneerng and Operatons Research, Unversty of alforna, Berkeley, A. (Receved: September 1998; revsons receved: September 1999, Aprl 2000; accepted: June 2000). Transportaton Scence/Vol. 35, No. 2, May