Multi-Item Single-Vendor-Single-Buyer Problem with Consideration of Transportation Quantity Discount

Transcription

1 ul-iem Sgle-Vedo-Sgle-Buye Poblem wh Cosdeao of Taspoao Quay Dscou Ye- WANG, Roh BHATNAGAR, Sephe C. GRAVES 3 IST Pogamme, Sgapoe-IT Allace, 5 Nayag Aveue, Sgapoe Nayag Techologcal Uvesy, Nayag Aveue, Sgapoe assachuses Isue of Techology, Cambdge, A 39 USA. Absac hs pape deals wh he poblem of shppg mulple commodes fom a sgle vedo o a sgle buye. Each commody s assumed o be cosaly cosumed a he buye, ad peodcally epleshed fom he vedo. Fuhemoe, hese epleshmes ae esced o happe a dscee me sas, e.g., a cea me of he day o a cea day of he wee. A ay such me sa, aspoao cos depeds o he shpme quay accodg o cea dscou scheme. Specfcally, we cosde wo aspoao quay dscou schemes: LTL (less-ha-ucload) cemeal dscou ad TL (ucload) dscou. Fo each case, we develop IP (mxed ege pogammg) mahemacal model whose obecve s o mae a egaed epleshme ad aspoao decso such ha he oal sysem cos s mmed. We also deve opmal soluo popees ad gve umecal sudes o vesgae he poblem. Idex Tems Vedo-Buye Poblem, Taspoao Quay Dscou, LTL Icemeal Dscou, TL Dscou I. PROBLE CONTET hs pape, we sudy he mul-em sgle-vedo- I sgle-buye poblem (a..a. sgle-waehouse-sgleeale poblem), whch aspoao quay dscou s povded by exeal cae. The poblem of ees s paally movaed by he gowh of emegg 3PL (hd-pay logscs) dusy ad applcao of VI (vedo-maaged veoy) sysems. Cosde he followg example. Supe-Vedo, a cosume goods supple, povdes oe of s cles Supe-a some does of commodes fom 5-ce dapes o 5-dolla cosmecs. Supe-a adops VI sysem, ha s, Supe-Vedo has he lbey of mag epleshme decsos fo Supe-a ahe ha fllg epleshme odes fom. A 3PL compay, Supe- Logscs, aspos he commodes. Ad accodg o he aspoao coac, ucs fom Supe-Logscs ave a 8am he mog f shpme s scheduled ha day ad o ohe aval me of a day s avalable. We cosde wo quay dscou schemes as follows. LTL cemeal dscou aspoao: Whe he shpme quay s less ha he vehcle capacy, quay dscou s appled oly o he addoal shpme quaes beyod he pedeemed beapo. I hs suao, he logscs maage of Supe-Vedo faces a basc adeoff: feque epleshmes ase veoy holdg coss, ad lowe aspoao coss, whle he coay happes wh feque epleshmes. TL dscou aspoao: Whe he shpme quay s lage ha he vehcle capacy, TL (ucload) dscou aspoao s used. I TL dscou scheme, Supe- Vedo pays LTL aspoao cos ul has pad fo he cos of a full ucload, a whch po, hee s o chage fo he emag quaes shpped ha ucload. Thus, we oba wo dffee sgle-vedo-sgle-buye poblems, whch ca be meagful ad eesg. The emade of hs pape s ogaed as follows. Nex seco peses a evew of he eleva leaue. The descpo of he poblem ude cosdeao s gve III. Cealed models fo he LTL cemeal dscou aspoao ad TL dscou aspoao ae especvely peseed IV ad V. Fally, we gve a few cocludg emas VI. II. RELEVANT LITERATURE We oe ha he leaue o VI sysem o aspoao quay dscou schemes s abuda. I he ees of bevy, ou leaue evew hs seco maly focuses o he sgle-vedo- sgle-buye poblem. The Sgle-vedo-sgle-buye poblem was fsly oduced by Goyal (976) whch suded a egaed veoy poblem of shppg a sgle commody fom a sgle supple o a sgle cusome. Goyal showed ha sysem cos savgs ca be acheved f he supple ad cusome coopeae o deeme he ecoomc o veoy polcy. oaha (984) developed a model fom a vedo s pespecve fo esablshg a opmal pce dscou schedule wh he pemse ha he vedo s ode pocessg cos s lage ha he buye s fxed ode cos. Baeee (986a) exeded ad geealed hese esuls o accou

2 fo he suaos whee he vedo s a maufacue. He demosaed he equvalece bewee appoaches suggesed by oaha (984) ad Baeee (986b) fo lo se modfcao accompaed by a pce dscou ode o cease vedo pofs. Goyal (988) llusaed ha maufacug a bach whch s made up of a egal umbe of equal shpmes geeally poduces a lowe cos soluo. The elaed leaue up o 989 was well summaed he evew pape of Goyal ad Gupa (989). Lu (995) developed a heuscs algohm o he sgle-vedo-sgle-buye poblem wh he assumpo of a poduco bach povdg a egal umbe of equal shpmes. Ad Goyal (995) used he same umecal example fom Lu (995) o vesgae a aleave polcy volved successve shpmes wh a poduco bach ceasg by a cosa faco. Ths polcy s based o a much eale dea se ou by Goyal (977) o solve a vey smla poblem a slghly dffee seg. I Hll (997, 999), dffee ypes of polcy wee cosdeed wh he assumpo of successve shpmes o he buye wh a sgle poduco bach. Hoque ad Goyal () assumed he vedo s epleshme/poduco ae s fe. The model copoaes a capacy cosa lmg he epleshme quaes of he buye. Topal e al. (3) geealed he sgle-vedo-sgle-buye poblem o smulaeously cosde uc capacy cosas ad boud/ouboud aspoao coss. They povded boh exac soluo pocedues ad heuscs algohms. A specal sgle-vedo-sgle-buye poblem called sgle-l poblem was oduced by Speaa ad Uovch (994) o cosde shppg mulple commodes fom a og o a desao wh cosdeao of dscee shpme fequeces. They assumed he FTL (full ucload) aspoao. The model deemes he umbe of ucs o be used ad allocaes dffee commodes o ucs. The bach-ad-boud algohm of Speaa ad Uovch (996) was used fo he soluo of hs sgle-l poblem. A exeso o he sgle-l poblem was suded by Bea e al. (997) wh cosdeg oe og ad mulple desaos. They peseed dffee heuscs by solvg a sgle l poblem fo each of he gve desaos fs, ad he o mpovg he soluo hough local seach echques. Bea e al. () poposed a mpoved bach-ad-boud algohm fo he sgle-l poblem. Bea ad Speaa () gave a famewo fo he defcao of opmal couous ad dscee shppg saeges fo he sgle l poblem. I evewg he pevous wo vedo-buye poblem, we foud ha, all he papes oveloo paccal aspoao quay dscou cosdeao whe modelg he poblem. Theefoe, aalyg he mpac of aspoao quay dscou o vedo-buye poblem s oe of he ma cobuos of hs pape. III. PROBLE DESCRIPTION The followg Table dsplays he poblem oaos used hs seco, alog wh he measue us fo he cosdeed quaes. TABLE. PROBLE NOTATIONS, K: Idex ad se of commodes, ( K)., N: Idex ad se of dscee me sas, ( N). d : Demad ae of commody a he buye (quay pe me sa). v : U volume of commody (volume). h : Sysem holdg cos ae of commody ($ pe quay pe me sa)., T: Dscee value ad se of possble epleshme/shpme peods, ( T). δ : Bay coeffce abou f shpme wh peod happes a me sa. x : Faco of commody epleshed peod of, x. : Repleshme/shpme peod of commody (whe SF polcy apples). : Shpme quay ude Tme Isa Cosoldao Polcy (volume). : Shpme quay ude Fequecy Cosoldao Polcy (volume). F (): Taspoao cos fuco of LTL cemeal dscou scheme F (): Taspoao cos fuco of TL dscou scheme Cycle of oal shpme quay pae : N A. Commody Assumpos We use o deoe he dex of commodes ad K o deoe he se of commodes. A he buye, commodes ae assumed o be couously cosumed a deemsc ad cosa aes. Ad o bacode s allowed. Le d deoe he demad ae fo commody. Each commody s chaaceed wh a u volume v. Ad all hese commodes ae assumed o have he same desy. Thus, we goe he shpme class caegoy ssue, whch commodes ae chaged accodg o dffee cos schemes based o he desy values. B. Dscee Peodc Repleshme Polcy We assume ha each commody s peodcally epleshed. Fuhemoe, he mmum epleshme ( hs pape, he ems shpme ad epleshme ae used echageably) eval s assumed o be a egal value, ad all possble epleshme peods mus be a mulple of hs value. Ths assumpo was usfed by Hall (985), ucsad ad Roudy(993), ad Speaa ad Uovch (994) ha he adoal EOQ-ype fomulas may poduce a mpaccal shpme polcy o mpleme such as shppg a commody evey.44 days. We use o deoe he possble dscee shpme me sa, ad o deoe he value of shpme peod. We also assume ha hee s a gve se fo : T.We assume he mmum eval bewee shpmes m s. Thus we have =,, Now, Le us cosde wo possble polces of decdg how commodes ae epleshed:

3 ulple fequecy (F) polcy: each commody ca be paally epleshed dffee fequeces. Le vaable x deoe he faco of commody epleshed peod. We have x ad x =. (3.) T Sgle fequecy (SF) polcy: each commody mus be epleshed a sgle fequecy. Theefoe, x s a bay vaable. Ude he sgle fequecy polcy, commody ca oly be assged oe shpme peod deoe by : = x. (3.) T I pcple, F polcy oupefoms SF polcy due o moe flexbly allowed. We wll show seco 4 ha hese wo polces ae equvale fo he cealed model of LTL cemeal dscou aspoao poblem. C. Taspoao Coss Fs, we have a assumpo ha all he commodes ae shpped a me sa ad o shpme saggeg s allowed. Ude hs assumpo, shpme wh peod happes a he me sas,,,3,... We use o deoe he shpme quay, ad F() o deoe he aspoao cos fuco. Le C as deoe he aveage aspoao cos. Ths cos cosdeed s defed o a fe me hoo. Howeve s obvously ha we oly eed o cosde a fe me hoo N, whch s N = lcm{: wh x > }. (3.3) I hs pape, such hoo s assumed o sa a me sa ad ed a me sa N -. Ths me sa se [,,,, N -] s deoed by N. I ems of calculao of shpme quay, we cosde wo cosoldao polces as follows: Tme sa cosoldao polcy: all he commodes shpped a he same me sa ae cosoldaed o oe shpme. We use o deoe hs oal shpme quay scheduled a me sa. Fuhemoe, we use a bay coeffce δ o deoe f he shpme wh peod happes a me sa. Thus we have = dvxδ, N (3.4) T K C = as F( ) N (3.5) N Fequecy cosoldao polcy: oly he commodes shpped he same fequecy ca be cosoldaed. We use o deoe he shpme quay wh shpme peod. = dvx, T (3.6) C T K ( ) F as = (3.7) T D. Holdg Coss We assume ha he sysem holdg cos s popooal o he oal veoy caed he sysem. Le h deoe he holdg cos ae fo commody he sysem. The we have he expesso of C C holdg holdg h dx K T = (3.8) E. Summay I hs seco, we descbe a mul-em sgle-vedosgle-buye poblem ha dffes fom he pevous wo by copoag aspoao quay dscou. Ths sgle-vedo-sgle-buye poblem models may paccal suaos such as compoe maufacue ad assembly pla, ceal waehouse ad local eale, ad so o. oeove, may cobue o he aalyss of complex supply cha ewos whe decomposo appoach s appled ad each decomposed subpoblem ca be opmed depedely as a sgle-vedo-sgle-buye poblem as cosdeed hs pape. IV. LTL INCREENTAL DISCOUNT TRANSPORTATION I hs seco, we sudy a cealed model fo he LTL cemeal dscou aspoao poblem. As dscussed I, such model s meagful he suao wh wo assumpos: () Ude cea saegc allace (e.g. VI), vedo ad buye coopeae o mme he sysem-wde cos. () The shpme quay s less ha he capacy of a vehcle. The obecve of hs cealed model s o mae opmal epleshme ad aspoao decsos such ha he oal sysem cos s mmed. A. LTL Icemeal Dscou Taspoao Cos I he LTL cemeal dscou cos scheme, quay dscou s appled oly o he addoal shpme quaes beyod he pedeemed beapo. As descbed peseed Balasha ad Gaves (989), hs cos sucue ca be modeled as a pece-wse lea ad cocave fuco F ( ) as depced Fgue 4.. F ( ) f f 4 f 3 f Fgue 4..Icemeal dscou cos fuco F()

4 Le be he dex of dffee slopes of he cos fuco, ad R be he se of ( R). Le, deoe he lowe ad uppe lms, especvely, o he h eval of R shpme quaes. Hee = ad ca be se o he possble mum shpme quay. Le f ad deoe he fxed ad vaable shpme cos assocaed wh he h eval. We ca expess he pece-wse lea cocave cos fuco as: ( ) F = f +, gve (, (4.) We assume ha me sa cosoldao polcy s used. The aveage sysem cos TC cosss of he aveage aspoao cos C as ad aveage holdg cos C holdg : TC = F ( ) + d h x (4.) N N T K B. ahemacal odel I hs seco, we develop a mxed ege pogammg model. We use a bay decso vaable y o deoe f shpme quay falls he age of (, a me sa. Ad he decso vaable s equal o shpme quay f falls he age of (,. These decso vaables ae: y =, f shpme quay (, a me sa, =, ohewse. =, f shpme quay (, a me sa, =, ohewse. A IP (mxed ege pogammg) poblem ca be fomulaed as follows. Poblem Φ ( hd x) + ( f y + ) (4.3) N K T N R s.. x =, K (4.4) T ( dvxδ ) =, N (4.5) K T R, R y, N, R (4.6) y, N, R (4.7) y N (4.8) x, T, K (4.9) y,, N, R (4.) { }, N, R (4.) The obecve fuco (4.3) peses he aveage sysem cos TC. Cosas (4.4) esue ha each commody s compleely assged shpme peods, ad cosas (4.5) se he oal shpme quay a me sa. Cosas (4.6)- (4.8) mae sue ha f cos dex s used a me sa, he he shpme quay a me sa mus fall s assocaed eval (, ]. Fally cosas (4.9) dcae ha a mos oe cos age ca be seleced a each me sa. Cosas (4.9) specfy ha mulple fequecy polcy s used. C. Opmal Soluo Popees We show wo eesg popees fo he opmal soluo of poblem Φ. Le us fs oduce he followg m oaos. Ude he mulple fequecy polcy, we use ad o deoe he mmum ad mum of he shpme peods assged o commody. m = m : x >, K (4.). ( ) ( ) = : x >, K (4.3). LEA : Le he commodes be dexed a odeceasg ode of he ao h v such ha ( h v) ( h v )... ( hk vk ). The he opmal soluo of poblem Φ, we have he followg elaoshp. m m m... K K (4.4) PROOF: See he Appedx A. Sce he poof does o eed ay pacula shpme cos sucue assumpo, Lemma s ue fo a geeal cealed model whch he aspoao cos oly depeds o he shpme quay ad he veoy cos s popooal o he oal veoy caed he sysem. LEA : I poblem Φ, mulple fequecy polcy ad sgle fequecy polcy ae equvale. The, s opmal o eplesh each commody wh a sgle peod. Fuhemoe, fo all he commodes of he same ao of hv, s opmal o eplesh hem wh he same peods. Tha s, h h =, gve =, K (4.7) v v PROOF: See he Appedx A. The poof eeds he assumpos of cocave aspoao cos sucue ad o shpme capacy esco. Lemma maes sese sce ay oe u volume of such commodes cobues he same o aspoao ad veoy coss. D. Numecal Example I hs seco, we pese a umecal example ha

5 llusaes he poblem Φ. The ma pupose s o show how ou model acually wos. We also cosde a smple shpme saegy whch all he commodes mus be shpped wh he same fequecy. We efe hs saegy as Ufed-T polcy. We vesgae he suaos whch ou model oupefoms he smple Ufed-T model. I ou example, commodes ae shpped. The vehcle wll ave a he vedo evey oday mog, ha s, he basc dscee shpme peod s oe wee. The possble shpme peod se T s [,, 3, 4, 6, 8, (wees)]. Cosequely, we oly eed o cosde a plag hoo TABLE NUERICAL EAPLE OF INCREENTAL DISCOUNT CASE of 4 wees. We assume demad ae d s 3 (quay pe wee) ad u volume v s (volume pe u commody) fo evey commody. Fo he LTL cemeal dscou cos sucue cosdeed. Thee ae fou cos ages whch ae [5, 5,, ]. The fxed cos of each shpme s dollas ad u shpme cos aes a fou cos ages ae [, 8, 7, 6 (dollas)]. The holdg cos aes (h ) of commodes ae fom.5 dolla pe u pe wee o 5 dollas pe u pe wee. (see alble ). ` holdg cos ae h, epleshme peod T ad sysem cos UfedT Savg # Cos Cos h % T h % T 3 h % T 4 h % T 3 5 h % T 6 h % T 4.5. Summay I hs seco, we suded he cealed model whose obecve s o mme he sum of LTL cemeal dscou aspoao coss ad holdg coss. Ths model accous fo he specfces of a paccal suao whee vedo-buye saegc allace exss ad shpme quay s less ha he vehcle capacy. We fs oduced he cos sucue of he LTL cemeal dscou aspoao, ad developed a IP fomulao fo he poblem cosdeed. The we showed wo eesg opmal soluo popees of he poblem. Fally, we gave a umecal example o show how ou model acually wos. V. TRUCKLOAD DISCOUNT TRANSPORTATION I las seco, we dscuss he sgle-vedo-sgle buye poblem wh cosdeao of LTL cemeal dscou aspoao mode, whch s wdely used whe shpme quay s smalle compaed wh he capacy of a aspoao vehcle (e.g. uc, coae ec). I hs seco, we assume mulple vehcles ae eeded o delve lage shpme quay. Ths suao was suded Speaa ad Uovch (994) whee a so-called sgle-l model was oduced o cosde FTL (full-ucload) aspoao mode. They showed ha sysem cos savgs ca be obaed hough mag epleshme ad aspoao decsos smulaeously. Howeve, he assumpo of exclusve FTL aspoao opo he sgle-l model may o be paccal some suaos because sedg a almos empy uc coss he same as a full oe FTL aspoao mode. I hs seco, we vesgae he TL (TucLoad) dscou aspoao mode whch ca model wo paccal suaos as follows. () Wh TL dscou cos sucue, cae ca gve ceves o shppes o pacce less-ha-ucload shpme so ha veoy cos s deceasg. Cosequely, he cae ca also ga pofs by chagg hghe aspoao cos. () Boh FTL cae ad LTL cae ae avalable he mae. The shppe uses FTL cae o shp he quaes of ucloads. Whle fo he delvey of he lefove quaes, he shppe chooses FTL cae o LTL cae based o he cos chaged. I he umecal case aalyss, ou model esuls moe sysem cos savgs ha he sgle-l model does.

6 A. TucLoad Dscou Taspoao Cos We ae ow oducg he TL dscou aspoao cos sucue. We use P o deoe he ucload capacy volume. Ad a pedeemed umbe P ' ( < P' < P ) dvdes he ucload P o wo segmes: (, P '] ad ( P', P ]. Taspoao cos chaged fo ay vehcle depeds o shpme quay caed o he vehcle: (). Whe (, P'] : f he shpme quay falls o he fs segme (, P '], aspoao cos cosss of a fxed ad a vaable compoe. The fxed cos, deoed by c, s cued depede of he shpme quay as log as s o eo. The vaable cos, deoed by, s cued o a pe-u-volume bass. We also efe o c as he seup cos ad as he popooal shpme cos. (). Whe ( P', P] : f he shpme quay falls o he secod segme ( P', P ], aspoao cos s a cosa value c depede of he shpme quay. We also efe o c as full-ucload cos. Fuhemoe, he TL dscou aspoao fuco F ( ) s couous ove he age of (, P ], ha s o say, we have he equao elaoshp of c + P' = c. The geeal TL dscou aspoao cos fuco F ( ) ca be descbed as follows: () f ( η ) P< ( η ) P+ P' ( ) F( ) = ( η ) c+ c + η P () f ( η ) P+ P' < ηp F ( ) = ηc (5.) Whee η deoes he umbe of ucs used o cay shpme quay. The cos fuco F ( ) ca be modeled as a pece-wse lea cos sucue as below. F () c c P' P P'+P P Fgue 5.. Tucload dscou aspoao cos B. ahemacal odel I hs seco, we develop a mxed ege pogammg model. We assume fequecy cosoldao polcy s used. We use bay decso vaable y o deoe f he lefove quay of falls he segme of (, P ']. The decso vaable s equal o he lefove quay of f (( η ),( η ) ' P P+ P, ad he decso vaable s equal o he lefove quay of f ( ( ηt ) P+ P', ηp. These decso vaables ae: y =, f ( ( η ) P, ( η ) P+ P'. =, f ( ( ηt ) P+ P', ηp. = ( η ) P, f ( ( η ) P, ( η ) P+ P' =, ohewse = ( η ) P, f ( ( ηt ) P+ P', ηp =, ohewse Accodg o he decso vaables gve as above, he TL dscou aspoao fuco F ( ) ca be fomulaed as: ( η ) f y c+ c + = F ( ) = η =,,3... (5.) η f y c = We also have he elaoshp c = c + P'. Thus (5.) ca be efomulaed as: ηc ( P' ) f y = F ( ) = η =,,3... (5.3) η f y c = Fuhemoe, we ca use he bay vaable y o deve a geeal expesso: F ( ) ( ' ) = ηc P y (5.4) A mxed ege pogammg poblem ca be fomulaed as follows. Poblem Φ ( hd x) + ηc ( P' y ) (5.5) K T T s.. x =, K (5.6) T vdx = ( η ) P + +, T (5.7) K ', P y T (5.8) P'( y ) P( y ), T (5.9) x, T, K (5.) y {, }, T (5.), T (5.) η ege T (5.3) The obecve fuco (5.5) expesses he mmao of aveage sysem cos: he fs em peses he aveage sysem holdg cos, ad he secod em peses he aveage sysem aspoao cos. Cosas (5.6) esue ha, fo each commody, he whole quay s shpped ad assged o dffee shpme peods. The quay elaoshps bewee vaables x, η, ad

7 ae defed cosas (5.7). Cosas (5.8) ad (5.9) specfy ha f shpme wh peod s full-ucload chaged, he bay vaable y T mus be ; ohewse y T mus be. C. Numecal Example I hs seco, we cosde a poblem of shppg 5 commodes. The u volume v fo each commody s (volume pe u commody) ad he demad ae d s he same fo all commodes. The holdg cos aes h of each commody s [,.,.5,.8, ]. The vehcle capacy P s (volume u) ad he aspoao cos of full ucload c s 3 (dolla). The seup cos c s 4 ad vaable cos s 4. The possble epleshme peods ae [,, 3, 4, 5, 6, 7, 8] d = TL Dscou odel = = =3 =4 =5 =6... = =... =3... =4... =5... η... Toal Cos 56.8 d = Sgle-L odel = = =3 =4 =5 =6... = =... =3... =4... =5... η... Toal Cos d = 7 TL Dscou odel = = =3 =4 =5 =6... =... =... =3... =4... =5... η... Toal Cos d = 7 Sgle-L odel = = =3 =4 =5 =6... = =... =3... =4... =5... η... Toal Cos d = 3 TL Dscou odel = = =3 =4 =5 =6... = =... =3... =4... =5... η... Toal Cos d = 3 Sgle-L odel = = =3 =4 =5... =8 = =... =3... =4... =5... η... Toal Cos D. Summay I hs seco, we cosdeed he TL dscou aspoao mode. Ths model ca be seemed as a exeso of he sgle-l model oduced by Speaa ad Uovch (994). I he umecal example, we vesgaed boh TL dscou model ad sgle-l model. We foud ha TL dscou model wll lead o moe sysem cos savgs ad moe feque epleshmes. VI. CONCLUSION Ths pape vesgaes he sgle-vedo-sgle- buye poblem wh copoao of aspoao quay dscou. We sudy cosde wo dscou schemes: LTL cemeal dscou ad TL dscou. IP models ae developed ad umecal examples ae caed ou. The fuue wo ca be developg cealed models fo he poblems cosdeed hs pape. APPENDICES A.. Poof of Lemma To pove Lemma, s suffce o pove he followg saeme: fo ay wo commodes ad, f we have he elaoshp of ( h v) ( h v), he commody should o be epleshed moe feque ha commody, ha s m If ( h v) ( h v), he, K (A.) Ths ca be poved by coadco. Suppose hee exss a opmal soluo whee we have m <. m Le δ = m ( vdx, vdx ) feasble soluo ' decal o. We cosuc a ew excep fo he

8 shpme peods of commody ad. Cosde δ ( vd ) of commody wh shpme peod m δ vd of commody wh shpme peod he ogal soluo. We chage he shpme peods o each ' ohe s o ge he ew soluo. The he quaes shpped a m ad ad ( ) ema he same, so does he aveage sysem aspoao cos. Theefoe, he cos dffeece bewee soluos ad ' s oly abuable o he aveage holdg cos fo δ of commody ad δ of commody : ' TC( ) TC( ) δ δ m = d h+ d h vd vd δ m δ d h+ d h vd vd δ m δ m = ( ) h + ( ) h v v h h m = δ ( ) (A.) v v The fs em A. peses he aveage holdg cos vd δ vd of commody δ of commody ad ( ) fo ( ) ', ad he secod em peses he he ew soluo couepa he ogal soluo due o he assumpos of ad ( h v) ( h v). Thus he ogal soluo o oupefom he ew soluo poved. Cosequely, Lemma s ue.. The equaly s m < does '. Theefoe, A. s A.. Poof of Lemma To pove Lemma, s suffce o pove he followg saeme: ude sgle fequecy polcy, fo ay wo commodes ad wh he elaoshp of ( h v) = ( h v), s opmal o le hem shpped he same fequecy, ha s If ( h v) = ( h v), he =, K (A.3) Ths ca be poved by coadco as follows. Suppose hee s a uque opmal soluo, whch. Le us defe a oao N o deoe he se of me sas a whch commody s shpped. Thus commody ad have he dffee shpme me sa ses: N ad N. Suppose ha, a me sa, he oal shpme quay (, ] ad le f = f, =. Thus he coespodg shpme cos a me sa ca be descbed as F ( ) = f +. The aveage sysem cos assocaed wh soluo s: TC( ) = hd + F ( ) K N N (A.4) = hd + [ f + ] N K N Le s cosuc wo ew feasble soluos decal o commody ad : ' ' ' : =, ad = (A.5) '' '' '' : =, ad = ' ad '' excep fo he shpme peod of (A.6) Noe ha, we eed he assumpo of o aspoao capacy esco o mae sue ha hese wo ew soluos ae feasble. The aveage sysem cos assocaed ' wh he ew soluo s: ' TC( ) ' ' = hd + ( ) F K N N = hd + hd + { F ( ) K\{ } N N\( NUN) + F( + v d ) + F( v d ) N\ N N\ N + F( + v d ( ))} NI N hd + hd + { F ( ) K\{ } N N\( NUN) + [ F ( ) + v d ] + [ F( ) + ( v d )] N\ N N\ N + [ F( ) + v d ( )]} NI N hd ( ) + hd + { F ( ) K N N + vd vd} N N (A.7) The above equaly holds because of he cocavy ad mooocy of he cemeal dscou shpme cos fuco F ( ) : F( + δ ) F( ) + δ. The he ' '' dffeece sysem cos bewee soluo ad s: ' ' = TC( ) TC( ) hd ( ) + vd vd N N N h dv ( ) + ) (A.8) v N N N

9 Now we cosde he ohe ew soluo ''. By a smla devao as A.7, we ca show he dffeece aveage sysem cos bewee soluos ad s: '' '' = TC( ) TC( ) hd ( ) + vd vd N N N h dv ' dv ( ) + = (A.9) v N N N d v The las equaly A.9 holds because of he assumpo ( h v) = ( h v). Thus, we have ' '' ' dv ' m(, ) = m(, ), whch meas ha he dv og soluo does o oupefom boh of he ew ' '' soluos ad. Ad hs coadcs he al assumpo ha s he uque opmal soluo. Theefoe, saeme A.3 s ue. To ea commody ad as he dffee amou of he same commody, we ca pove Lemma. REFERENCES Balasha, A., S. Gaves (989). A compose algohm fo a cocave-cos ewo flow poblem. Newos 9, 75-. Baeee, A. (986a). O a quay dscou pcg model o cease vedo pof. aageme Scece 3, Baeee, A. (986b). A o ecoomc lo-se model fo puchase ad vedo. Decso Sceces 7, 9 3. Bea, L.,. G. Speaa (999). mao of veoy ad aspoao coss: a suvey. Lecue Noes Ecoomcs ad ahemacal Sysems 48, Bea, L.,. G. Speaa, W. Uovch (). Exac ad heusc soluos fo a shpme poblem wh gve fequeces. aageme Scece 46, Bea, L.,. G. Speaa (). Couous ad dscee shppg saeges fo he sgle l poblem. Taspoao Scece Blumefeld, D. E., L. D. Bus, C. F. Dagao,. C. Fc, R. W. Hall ( 987). Reducg logscs coss a Geeal oos. Iefaces 7, Blumefeld, D. E., L. D. Bus, J. D. Dl, C. F. Dagao (985). Aalyg adeoffs bewee aspoao, veoy ad poduco coss o fegh ewos. Taspoao Reseach 9B, Bus, L. D., R. W. Hall, D. E. Blumefeld, C. F. Dagao (984). Dsbuo saeges ha mme aspoao ad veoy cos. Opeaos Reseach 33, Ceaya, S., C.-Y. Lee (). Soc epleshme ad shpme schedulg fo vedo maaged veoy sysems. aageme Scece 46, 7-3. Goyal, S. K. (987). Comme o geealed quay dscou pcg model o cease supple s pofs. aageme Scece 33, Goyal, S. K. (988). A o ecoomc-lo-se model fo puchase ad vedo: a comme. Decso Sceces 9, Goyal, S. K. (995). A oe-vedo mul-buye egaed veoy model: a comme. Euopea Joual of Opeaoal Reseach 8, 9. Goyal, S. K., F. Nebebe (). Deemao of ecoomc poduco-shpme polcy fo a sgle-vedo-sgle-buye sysem. Euopea Joual of Opeaoal Reseach, ' '' Hall, R.. (99). Roue seleco o fgh ewos wh wegh ad volume cosas. Taspoao Reseach 5B, Hll, R.. (997). The sgle-vedo sgle-buye egaed poduco-veoy model wh a geealed polcy. Euopea Joual of Opeaoal Reseach 97, Hll, R.. (999). The opmal poduco ad shpme polcy fo he sgle-vedo sgle-buye egaed poduco veoy poblem. Ieaoal Joual of Poduco Reseach 97, Hoque,. A., S. K. Goyal (). A opmal polcy fo sgle-vedo sgle-buye egaed poduco-veoy sysem wh capacy cosa of aspo equpme. Ieaoal Joual of Poduco Ecoomcs 65, Joglea, P. N. (988). Commes o a quay dscou pcg model o cease vedo pofs. aageme Scece 34, Joglea, P. N., S. Thahae (99). The dvdually esposble ad aoal decso appoach o ecoomc lo ses fo oe vedo ad may puchases. Decso Sceces, Lee, H. L.,. J. Rosebla (986). A geealed quay dscou pcg model o cease supple s pofs. aageme Scece 3, Lu, L. (995). A oe-vedo mul-buye egaed veoy model. Euopea Joual of Opeaoal Reseach 8, oaha, J. P. (984). A quay dscou pcg model o cease vedo pofs. aageme Scece 3, ucsad, J., R. O. Roudy (993). Aalyss of mulsage poduco sysems. S. C. Gaves, A. H. G. Rooy Ka, P. H. Zp, eds., Hadboo Opeaos Reseach ad aageme Scece, Vol. 4: logscs of Poduco ad Iveoy. Noh Hollad, Amsedam Pa, J. C., J. Yag (). A sudy of a egaed veoy wh coollable lead me. Ieaoal Joual of Poduco Reseach 4, Speaa,. G., W. Uovch (994). mg aspoao ad veoy coss fo seveal poducs o a sgle l. Opeaos Reseach 4, Topal, A., S. Ceaya, C. Y. Lee (3). The buye-vedo coodao poblem: modelg boud ad ouboud cago capacy ad coss. IIE Tasacos 35, 987-.