An operational policy for a single vendor multi buyer integrated inventory supply chain system considering shipping time

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1 Lousaa Sae Uversy LSU al Coos LSU Maser's Theses Grauae School 00 A operaoal polcy for a sle veor ul buyer erae veory supply cha syse coser shpp e Chraj Saha Lousaa Sae Uversy a Arculural a Mechacal Collee, chrajsaha@al.co Follow hs a aoal wors a: hps://alcoos.lsu.eu/raschool_heses Par of he Cosruco Eeer a Maaee Coos Recoee Cao Saha, Chraj, "A operaoal polcy for a sle veor ul buyer erae veory supply cha syse coser shpp e" (00). LSU Maser's Theses hps://alcoos.lsu.eu/raschool_heses/348 Ths Thess s brouh o you for free a ope access by he Grauae School a LSU al Coos. I has bee accepe for cluso LSU Maser's Theses by a auhorze rauae school eor of LSU al Coos. For ore forao, please coac rae@lsu.eu.

2 AN OPERATIONAL POLICY FOR A SINGLE VENOR MULTI BUYER INTEGRATE INVENTORY SUPPLY CHAIN SYSTEM CONSIERING SHIPPING TIME A Thess Sube o he Grauae Faculy of he Lousaa Sae Uversy a Arculure a Mechacal Collee paral fulflle of he requree for he eree of Maser of Scece Iusral Eeer The epare of Cosruco Maaee a Iusral Eeer By Chraj Saha B.S., Wes Beal Uversy of Techoloy, Slur, Ia, 006 Auus 00

3 ACKNOWLEGEMENT Frs of all, I wsh o ha he Alhy for v e he eurace o coplee y ere course. I wa o express y eepes sese of raue o r. Bhaba R. Sarer, Elo G. Yaes sushe Professor of Iusral Eeer a Charperso of y Coee, for hs valuable uace, cosa ecouraee, cooperave aue, ese paece, useful scussos ur he course of research a preparao of he auscrp. He has always bee a foua of sprao o e. Also, I a hoore o express y esee a profou sese of raue o r. Pus J. Ebelu, Coee eber a Professor of Iusral Eeer, for hs cosrucve a valuable suesos. He was always quc o exe hs help wheever I ase for, a he ever fale o offer cooperave a accooa servces. I a horouhly elhe o recoze r. Lawrece Ma Jr., Professor Eerus of Iusral Eeer epare a Avsory Coee eber, for hs valuable suesos eowe ur he course of y research wor. I a also very prou o acowlee Suaa, Raesh, Shalal a Soesh, whose cosa help a collecve effors are uoubely reflece he copleo of hs veure. Also, oh coul have bee possble whou he cessa love, affeco, sacrfce a sprao fro y pares. I feel blesse o have such supporve pares, who were a ceral force u e o hs ay. Fally, I woul also le o evoe y success o he facal asssace prove by he CMIE epare he for of asssashp ur he eure.

4 TABLE OF CONTENTS ACKNOWLEGEMENT... LIST OF TABLES... v LIST OF FIGURES... v ABSTRACT... v CHAPTER : INTROUCTION.... The Proble.... Applcaos Research Goals Research Objecves... 5 CHAPTER : LITERATURE REVIEW Sle Veor Sle Buyer Supply Cha Syse Sle Veor Mul Buyer Supply Cha Syse Shorcos of Prevous Leraure... 0 CHAPTER 3: MOEL EVELOPMENT Assupos Noaos Moel Forulao Moel I: Ie Shppe Rh Afer Prouco Averae Iveory Calculao Toal Averae Iveory Calculao Toal Cos For Moel I Orer Cos Seup Cos Iveory Carry Cos Trasporao Cos Toal Syse Cos Soluo Mehooloy Sesvy Aalyss for Moel I Effec of S o... 4 TC I 3.7. Effec of h o... 4 TC I Effec of P/ o... 4 TC I 3.8 Moel II: Ies Are Shppe I Every Pero Averae Iveory Calculao Toal Averae Iveory Calculao Toal Cos for Moel II Orer Cos Seup Cos Iveory Carry Cos Trasporao Cos... 33

5 3.9.5 Toal Syse Cos Soluo Mehooloy Sesvy Aalyss for Moel II Effec of S o TC II Effec of h o TC II Effec of P/ o TC II Moel III: Ies Shppe Uequal Baches Averae Iveory Calculao Toal Averae Iveory Calculao Toal Cos for Moel III Orer Cos Seup Cos Iveory Carry Cos Trasporao Cos Toal Syse Cos Soluo Mehooloy Sesvy Aalyss for Moel III Effec of S o TC III Effec of h o TC III Effec of P/ o TC III CHAPTER 4: RESULTS Perforace Coparso Coparso wh Alerave Approach CHAPTER 5: RESEARCH SUMMARY AN CONCLUSIONS Coclusos Research Sfcace Possble Fuure Exesos BIBLIOGRAPHY APPENIX VITA v

6 LIST OF TABLES Table : aa for sle veor 9 buyers proble... Table : Suary of resuls... Table 3: Effec of P/ ( A) rao o TC I... 6 Table 4: aa for sle veor 9 buyers proble Table 5: Suary of resuls Table 6: Effec of P/ ( A) rao o TC II Table 7: aa for sle veor 9 buyers proble Table 8: Suary of resuls Table 9: Effec of P/ ( A) rao o TC III... 5 Table 0: aa for sle veor 9 buyers proble... 5 Table : Suary of resuls... 5 Table : Perforace coparso wh alerave oels v

7 LIST OF FIGURES Fure.: Flow ara of a sle veor ul buyer erae supply cha... Fure 3. Iveory flow Moel I... 5 Fure 3.: Graphcal represeao of TC I... 3 Fure 3.3: Effec of P/ ( A) rao o TC I... 6 Fure 3.4 Iveory flow Moel II... 7 Fure 3.5: Graphcal represeao of TC II Fure 3. 6: Effec of P/ ( A) rao o TC II Fure 3.7 Iveory flow Moel III... 4 Fure 3.8: Graphcal represeao of TC III Fure 3.9: Effec of P/ ( A) rao o TC III... 5 v

8 ABSTRACT Sce s rouco, he cocep of erae veory supply cha has receve a coserable aou of aeo. The ajory of sues he las hree ecaes reveale a crease hol cos as prouc oves furher ow he cha or up he cha. A rece suy Hoque (008) cosere veor s seup cos a veory hol cos. Soe research also cosere fxe rasporao cos, whch s urealsc. Ths suy focuses o a sle-veor, ul-buyer scearo a preses hree oels. Frs, wo oels llusrae he rasferr of equally-sze baches. The, a hr oel cosers he rasferr of uequally-sze baches a lo. Ths suy relaxes he assupo ha veor s hol cos us be reaer ha or less ha all buyer s hol coss he syse. Also, hs research faclaes uequal rasporao e a cos for ffere buyers for reaer flexbly. The oal syse cos s calculae by su he aual operaoal cos for all he pares he syse. Opu values of he ecso varables are eere us a rec search eho. As presee by he hr oel, a uercal exaple eosraes ha he oal syse cos s less whe copare wh oher wo oels presee. Ths suy also preses he follow: soluo proceures o solve each oel, ay uercal exaples o suppor aheacal fs, a perforace coparsos ao hree fs. I orer o jusfy he lo-spl approach for solv he erae veory proble, alerave oels wh o lo spl are evse a ese uer he sae crcusaces. Alerave oels wh o lo spl prouce slar or beer resuls. Uer he sae crcusaces, he alerae hr oel s observe o be offer he leas oal cos for he syse. Ths suy also preses a sesvy aalyss o chec he robusess of he hree oels. The fuure exeso of hs research ay volve coser sorae capacy cosra a rao ea. v

9 CHAPTER INTROUCTION Coser a sle veor supply a e o ulple buyers. The veor prouces he e baches a a a fe rae. The veor he ses he fshe es o ulple buyers. I hs process, he veor curs bach se-up a rasporao cos, a he veor a buyer boh carry he e hol cos proporoal o e. Meawhle, each buyer has hs ow eersc yearly ea. I hs scearo, he buyer has a proble wh eer he orer quay, a he veor has a proble wh eer opu prouco quay a shpp scheule, whch zes he opera cos. ur he las hree ecaes, researchers have bee search for he soluo o hese probles. Researchers have show ha by vew he veor a buyers as a syse (also ow as erae supply cha) raher ha as separae vuals, oal syse cos ca be reuce sfcaly. The bass of he erae supply cha cocep s ha each buyer has clear owlee abou her yearly ea, a buyers are reay o share hs forao wh he suppler o ejoy he beefs of coorao. Toay, rea provees elecroc forao exchae have ae hs cocep feasble. I supply cha, rasporao cos s a ajor par of operaoal cos. Trasporao e, cos, a capacy cosra play a role a ecsos. I oay s worl, shor lfe cycle a couless specales of slar proucs have ae he lobal are hhly copeve. I orer o survve are pressure, every copay has o be hhly copeve ers of prouc qualy, prce a prouc supply. The flow ara of a sle veor ul buyer supply cha syse s show Fure..

10 Buyer 3 Buyer Buyer Buyer Veor Traspor Vehcle Fure.: Flow ara of a sle veor ul buyer erae supply cha. The Proble For ecaes, he prary objecves ao research have bee eer he opal bach sze, ubers of bach szes a lo, uber of los per year, a shpp scheule of erae veory oels. The ajory of hs research s oe wh eal assupos. I boh sle a ulsae supply chas, ecso paraeers, rasporao e, cos a yearly ea vary wh e. ea of a vual buyer ca be ffere, a hol cos a shpp cos ay also vary. A vual buyer s ecooc orer quay a shpp scheule are also lely o ffer. Uwse choces of hese varables ca lea o excessve prouc coss, whch, ur, ca lea o cusoer ssasfaco a los sales. The prese research focuses o eer ecooc orer quay (EOQ) a shpp sraey of a veory syse erae wh a sle-prouc, sle-veor,

11 a ul-buyer. I a sle-veor-ul-buyer syse, a veor prouces a elvers a e o ulple buyers. epe o he ea of all buyers, prouco a shpp polces are eere. If, for he sae of splcy, we se a uversal orer quay for all buyers, he we ay fal o opze everyoe s cos. Soe buyers ay receve ore ha hey ee a parcular e spa whle ohers ay receve less ha eee. Aa, ae buyers are sprea across he coury where shpp cos a e vares sfcaly ao buyers. For exaple, buyers who are far fro he veor ee o shp earler ha hose buyers who are relavely close o veors. Bu aa ffere scheules for all buyers s ffcul o accooae, so we us f a cos-effecve polcy ha beer corols he coplexy of operao. Also, we us eere how o reulae bach sze such a way ha o oly reuces orer cos, seup cos a hol cos, bu also avos shorae.. Applcaos Auoobles are a exaple of such a e. All he showroos arou he coury receve shpe of cars fro he veor. Suppose a aufacur facly s Mcha, a soe of he showroos are Iaa, Lousaa or Alasa. Shpp cos a e s ffere for each case, a each showroo requres space for splay, whch requres hh aeace a survellace cos. I s obvous ha cos Calfora s ore ha ha Olahoa or Mssour. No ealer les o eep a excess of veory because such veory creases he hol cos. Aa, f we loo arou we oe, ulple showroos wh sae auoobles whch resuls copeo. No ealer ca affor o o hav cars ea because he cusoer has a choce o e aoher ex oor. I s pora ha he ealer es he shpe o e. 3

12 Aoher proble s every showroo has s ow ea; base o orer cos a hol cos, he showroos ecooc orer quaes ffer fro oe aoher. Aa, coser seup cos a hol cos, veor s ecooc aufacur quay a bach ca be ffere fro he EOQ of all he showroos. Now, f we arbrarly ass each showroo he sae shpp quay a scheule, he syse s lely o fal. The veor eres s o prouce as ay es a shp he o he realer as soo as possble o avo hol cos. The eres of he realer s o e he rh aou of prouc wh he rh efrae o avo hol cos a oher scellaeous expeses. Aa, rasferr es saller los resuls lower veory cos bu hher orer, seup, a rasporao cos. O he oher ha, rasferr es larer los leas o hher veory cos bu lower orer, seup, rasporao cos, a scheul erfereces ue o scarce sorae capacy for boh he veor a he buyer. Soe exaples of such usres are BMW, For, GMC, Mercees, a Toyoa, whch prouce cars, rucs, a oher oorze vehcles. The propose research wll prove he veory aaee of he supply cha syse, whch wll also sfcaly reuce he syse s opera cos a crease s profably..3 Research Goals The objecve of hs research s o suy a oel a sle-veor-ul-buyer veory syse, whch cosras he raspor capacy a aas varous rasporao es bewee ffere buyers. I realsc suaos, veory hol coss are ffere for veors a buyers; hese rasporao coss affec orer polcy, prouco polcy, a oal syse cos, a rasporao coss vary ao all buyers. Ths research preses a operaoal polcy o prouce a elver es he rh quay a he rh e whle reuc he oal syse cos. 4

13 .4 Research Objecves Ths proble aresses prouco a shpp polces of a prouc ha flows fro a veor o ulple buyers. Yearly eas for all he buyers are recore a are eersc. Veor a buyers aa a close relaoshp o reuce overall syse cos. I hs research, hree oels are presee o aress he proble uer ffere operaoal polces. The veor ses he prouc o buyers ulple baches. Baches coul be equally or uequally sze. Carry coss for he veor a each buyer ay vary epe o her eoraphc locao. Trasporao e a cos for each buyer ffers. ue o he above reasos, he aure of veory of hs supply cha syse ffers fro raoal syses. Hece, he prary objecves of hs research are: () () () (v) To suy he behavor of he veores uer hree operaoal polces. To f opal bach sze. To f opal bach uber a lo. To f bes polcy o reuce oal syse cos ao hree operaoal polces. 5

14 CHAPTER LITERATURE REVIEW Ths suy has oly cosere he eersc ea. The leraure revew orazes prevous research chroolocal orer sar wh he sples, sle-prouc, sle-veor, sle-buyer scearo, uresrce prouco rae a lo-for-lo polcy (Goyal 976). The revew he shfs o he os rece sle-veor, ul-buyers scearo, whch also volves seup a veory cos for he veor wh fe prouco rae (Hoque 008). Ao he prevous wors, ffereces are aly assupos of prouco rae a repleshe pero. Recely, oher aspecs of hs proble have also bee scusse he leraure. Oher aspecs refer o ulple buyers, rasporao capacy a cos, lea e, varable prouco cos, qualy a process falure, se up cos reuco a ore realsc ea raes.. Sle Veor Sle Buyer Supply Cha Syse Goyal (976) propose hs frs oel, aress a erae supply cha, assu a fe prouco rae, a ufor eersc ea over e. He ore lea e a resrce soc ous. I hs research, he veor prouces los a ses he ere lo o he realer. Ths process ples ha he ere lo us be prouce before shpe. Baerjee (986) ep ha lo-for-lo polcy, bu relaxe he assupo of fe prouco rae. Baerjee (986) also coe he er, JELS (jo Ecooc Lo Sz) a arue abou ecooc beefs of boh veor a realer hrouh JELS. Baerjee (986) cosere purchase ras e, seup e a elvery e accorace wh acual prouco e. Goyal (988) rouce a ore eeralze JELS oel, whch relaxe he assupo of lo-for-lo oel a propose o prouce a lo whch ca be supple eer uber of orers afer he ere lo s prouce. Goyal 6

15 (988) showe ha hs jo oal releva cos s less ha or equal o ha of he JELS oel. Goyal (995) use a ffere approach ha equally-sze shpe o coe up wh a ea of eoerc shpe sze. Ths eas ha he successve shpe sze s he prouc of he pror shpe sze relao o he rao of prouco a ea rae. Vswaaha (998) presee wo oels: oe whch he shpe szes are slar a aoher whch he shpe sze s equal o whaever veory s avalable a ha po. I hs suy Vswaaha ae Lu s (995) equal shpe sze polcy as Iecal elvery quay (IE) a Goyal s (995) a uequal shpe sze oel as elver wha s prouce (WP). Vswaaha showe ha eher polcy s beer ha he oher for all ype of probles, a he coclue ha he bes polcy epes o he proble s paraeers. Goyal a Nebebe (000) poe ou he ffcules face by he veor a realer whle apply he polcy propose by Goyal (995) a Hll (997), who poe ou he ffculy of eer bach sze for he veor, opal uber of shpes, a each shpe sze for he buyer. Goyal a Nebebe (000) propose a alerae soluo whch suess ha ao shpes, he frs shpe s saller a followe by (-) equal- sze, whch s equal o he prouc of he frs shpe sze as well as s rae of prouco over rae of ea. Goyal (000) exee he polcy propose by Hll (997). He propose ha he follow shpe szes wll be eere by frs shpe sze. The follow shpe szes ay be crease by a facor of prouco rae over ea rae ul s possble o o so. A lely rawbac of hs suy s ha he ee applcao of he oel s uclear. Hll a Oar (006) presee ervao of opal aufacur bach sze a shpe polcy whe prouc-hol cos creases as he prouc oves ow o he buyer uer o-requre 7

16 equal shpp sze. Zhou a You (007) relaxe he assupos ha veor s soc hol s always reaer ha he realer s. They showe ha her oel perfors equally well reuc averae oal cos rearless of he veor s or realer s sochol coss, whch are ever equal o each oher. I hs suy hey also allowe shoraes bu oly for buyers. They also presee a prouco veory polcy for eerora es. I hs suy hey prove ha he opal polcy for veors whose hol coss are reaer ha he realers hol cos s ha all uequally-sze shpes crease he follow shpes by rao of prouco rae o he ea rae. Pa a You (00) explore how o reuce lea-e a erae syse volv cos. They arue ha beer cusoer sasfaco levels a reuce safey soc levels ca be acheve hrouh prov lea-e; however, hese chaes occur a he expese of lea-e crash cos. Ths suy evelope a oel, whch yels lower oal cos a reuce lea e ha ha presee by Baerjee (986) a Goyal (988). Ulaely, hs research s a exeso of Goyal (988), a relaxe he fe prouco capacy assupo. Hoque a Goyal (000) cosere raspor capacy lao by prese a opal polcy for he sle-veor a sle-buyer erae supply cha, whch cosers equal a uequal shpe sze a rasporao capacy.. Sle Veor Mul Buyer Supply Cha Syse Jolear a Tharhare (990) cosere aoher area of erae supply cha,.e., sle veor a ul buyer. I ha suy hey presee a alerae soluo of he sae proble cosere by Baerjee (986) a ae as he Ivual Resposble a Raoal ecso (IRR). I IRR, hey refe JELS by brea se-up cos o veors orer process a hal cos per prouco ru seup cos. Base o he chaes, he auhors clae IRR s cossecy a free eerprse scearo a 8

17 superory over Baerjee s (986) JELS oel eal wh probles le sle-veor a o-ecal buyers. A sle-veor, ul-realer proble s also aresse by Affsco e al. (988, 99, a 993). I hese sues, researchers aresse he sle veor a ay ecal realers wh a objecve of reuc prouco seup cos a realer s orer cos. They showe ha subsaal provee ca be acheve, uer hs oel, hrouh he epee cos opzao echque; hus, a cooperave evroe, a erae veory approach s suese over epee cos opzao. Lu (995) propose hs oel coex of a sle-veor or ul-buyers scearo. Lu allowe shpes ur prouco a ore Goyal s assupo of prouc a ere lo before shpe. May oher JELS-base oels, e.. Baeerjee a K (995) a K a Ha (003), cosere equally-sze shpe polcy. Yau a Chao (004) presee a oel whch he veor prouces a supples o all he buyers; hs zes he veor s oal aual cos base o he axu cos buyers are wll o cur. They cae up wh a effce alorh o search a opal cos curve. Saja e al. (005) presee a sle-veor-ul-buyer scearo whch he veor s he sole suppler for a specfc e o all buyers. Supples are elvere equal szes, bu he shpe sze ay ffer fro oe buyer o aoher base o her ea. Supples are elvere sequece; e.., he frs buyer wll e frs supply followe by seco a hr a so o, assu prouco-cycle e a buyer s-orer cycle e are he sae. Also, he e bewee oe elvery a he ex s fxe for each buyer. Hoque (008) presee sle-veor ul-buyer syse coser a cosere veor s seup a veory hol cos. I hs research, he arue favor of rasferr of saller los over larer los whe sorae capacy s scarce for boh he veor a he buyer. Vswaaha a Ppla (00) aresse he sae proble a re o solve by us a 9

18 ae heorec approach. They propose ha veors wll specfy he coo repleshe pero for accep he proposal, a ur, buyers wll receve a prce scou fro he veor. The prce scou wll be suffce o copesae he allevae prouc carry cos, f ay. Vswaaha a Ppla (00) erve opal repleshe pero a prce scou quay by solv Sacelber ae. Che e al. (009) scusse elvery a share rasporao cos a ulpleveors erae veory syse, a Coeaux e al. (005) scusse he prouc qualy speco polcy a erae veory syse..3 Shorcos of Prevous Leraure Ths leraure revew brefly recalls he evelope of he erae supply cha aaee syses sar fro he splsc sle-veor, sle-buyer syse a ov owar ore avace sle-veor, ulple-buyers syses. The revew pos ou ha each suy has s ow shorcos. Realscally, os of he probles are cosrae. A veor s hol cos coul be hher or lower ha oher buyers he sae syse; rasporao e o oe buyer coul be ffere fro aoher buyer, a eve rasporao cos ay ffer ao buyers. Alhouh ay researchers cosere cosras eoe above her oels oe a a e, so far, hey have ve lle aeo o bul a sle-veor-ulple-buyer erae oel, whch cosers all he above eoe cosras. Here, we prese a sle-veor-ul-buyer erae supply cha oel wh equal a uequal bach szes; hs oel cosers veor s seup cos, rasporao e, cos, capacy cosrae a uequal hol cos for buyers. Three oels are presee: he frs wo, whch coser equal bach sze a he hr oel, whch cosers uequal bach sze. 0

19 CHAPTER 3 MOEL EVELOPMENT I hs seco he forulao of he hree operaoal oels are presee o llusrae ffere prouco a shpp polces. These oels are base o soe prevously escrbe assupos a oaos a are followe by averae veory a oal syse cos ervaos. 3. Assupos The follow assupos are ae o cosruc oels: (a) ea a prouco rae are fxe a eersc. (b) Every buyer esaes her ow orer a hol cos uer varous cos facor a les be ow o he veor. (c) The cocere pares share he beefs of coorao base o a cosless way of shar. () No baclo or shoraes are allowe,.e. P. (e) Lo a bach szes are eers. (f) For boh veor a he buyer, sorae capacy s ucosrae. () All shpp vehcles are ecal, a avalably of ay uber of shpp ea s ucosrae. (h) Trasporao es are sfca a ca vary fro buyer o buyer epe o buyer s sace fro he veor. () Se-up e a cos are sfca. (j) Mu bach sze has o be reaer ha or equal o uber of buyer he syse. () For he purpose of splcy, veory carry cos ur rasporao s elece.

20 3. Noaos The follow oaos wll be requre o forulae he oel: h buyer paraeer a Orer cos ($/orer) C Cos of oe vehcle for h buyer ($/vehcle) Aual ea (u/year) aly ea of h buyer (u/ay) h Hol cos per e per year ($/u/year) q Vehcle capacy (us/vehcle) Shpp e o h buyer (ays) Veor Paraeer h Aual rae of ea (u/year) Salles bach sze Hol cos per e per year ($/u/year) P Prouco rae per year ( P >, P / ), (u/year) S Seup cos ($/seup) Varables Oral shpp sze for h buyer G G Shpp sze for h buyer whch clues rasporao e ea Salles bach sze Bach sze whch clues rasporao e ea for buyers Nuber of equal or uequal baches a lo

21 3.3 Moel Forulao I hs research, he veory oels are evelope for hree prouco a shpp polces. The frs oel as o prouce es equal sze baches a lo. Ies are shppe o he buyers as soo as he bach prouco s fshe. The seco oel also assues ha es are prouce equal-sze baches a lo a ha he veor hols he e ul buyers place a orer. The hr oel aeps o prouce es uequal-sze baches a lo, a he veor hols he e ul orer s receve. Assu he veor prouces es los, a lo cosss of baches, a baches are prouce sze of, buyers receve he e proporo o he rao of her ea o he oal ea of bach sze. We ca wre: Thus,. (3.). (3.) I hs research, we are coser rasporao e, a we acowlee ha rasporao es vary bewee buyers. Now, we wa o shp proucs o all buyers a he sae e fro he veor s e, bu sce rasporao es vary bewee buyers, soe buyers h have o receve he prouc before or afer he prevous bach s exhause. To ze hol cos, we wa o assure ha each buyer receves a ew shpe whe he prevous bach s ear epleo. To coser hs scearo, we propose o a vual rasporao e ea wh he buyer s correspo shpe. Assue rasporao e o buyer s, so he ew shpe sze for h buyer becoes where G, (3.3) s he aly ea for h buyer. The suao of oral bach sze a rasporao e ea for all he buyers leas o 3

22 G. (3.4) 3.4 Moel I: Ie Shppe Rh Afer Prouco As escrbe above, a veor shps es o buyers as soo as he bach s prouce. I he frs oel, we are assu all bach szes a lo are equal a ha here are baches a lo. Fure 3. shows he prouco a veory flow for he frs oel. I Fure 3., P,, a are he e sees, whch represe he follow: prouco e for a bach, cosupo e for h buyer a rasporao e for h buyer, respecvely. I Fure 3., rale (ABC) represes he veory for he veor. The rale (MNO) represes he eal veory for he h buyer whe a e s receve a he e of prevous veory. The area (MOPQ) wh a ashe le represes veory he buyers warehouse whle he prevous bach s be cosue. I he be of he cycle, he veor bes prouco a a fe rae, a veory bes o accuulae a a cosa rae. I Fure 3., he slope AC s he rae of bul veory ur prouco. As soo as prouco of he bach s coplee, es are shppe o each buyer, a veory of veor reuces o zero. A hs po, he veor prouces he ex bach, repea he process ul he ere lo s prouce. Alhouh es o all buyers are shppe a he sae e, hey wll receve he afer e, sce shpp e ffers ao he buyers. Ies are avalable for cosupo as soo as he buyer receves he; a, he buyer cosues he e a a cosa rae. I Fure 3., slope MN s he cosupo rae. Sce a buyer cosues es a a fxe rae, o-ha veory sars ecreas a a cosa rae. Aa, as we assue P > a / P < / 4

23 ( ) P X Y Q M BUYER BATCH SIZE P O C N NEW LOT VENOR /P A B NEW LOT TIME Fure 3.: Iveory flow Moel I o avo shorae, a buyer wll receve he ex shpe before her o-ha veory s exhause. Ths ew shpe reas he warehouse ul he prevous bach s cosue. Noce, as he cycle coues, he e ew shpe reas le he warehouse ulples, as show as MQ he Fure 3., where ( ){ ( P )} he e ew bach reas he buyer s veory. I coras o hs research, Hoque (008) o coser shpp e, a he buyer receves he e as soo as s shppe. So, he prese research, he buyer oes o have o hol veory s warehouse for aou of e Averae Iveory Calculao Ths seco erves he averae veory Moel I for he ere supply cha syse coa a sle veor a ulple buyers. Sce he ea s eersc 5

24 a prohbs baclo or shorae, he prouco rae us be reaer ha he ea rae. The oal syse veory cosss of buyer s veory, I a veor s veory, I. (a) Buyer s Averae Iveory Calculao b Each buyer receves aou of a e each shpe a aes aou of e o cosue. So, each buyer s averae veory for he frs bach s ( ). Sce he seco shpe arrves before he frs shpe s fshe, he buyer us hol he seco shpe for { P )} aoal aou of e ( ul prevous shpe s cosue. Therefore, averae veory ur he seco } shpe s [ { ( P ) ]. The hr shpe arrves before he seco shpe s fshe a acually reas he warehouse wce he e of he seco shpe. I Fure 3., we ca see QM s ouble of XY, whch are he curre es whch shpp reas le he warehouse. Ths coues ul he ere lo s supple, f here are baches a lo, he he h shpe us rea he warehouse for ) { ( P )} aou of e. Therefore, he oal averae veory for h ( buyer per cycle ca be expresse as follows: Iv cycle ( 3... ( ). (3.5) P If each bach sze s a here are baches a lo, he a coplee cycle, aou of es are prouce. So, he oal uber of cycles a year s ( ). Therefore, averae veory for h buyer per year ca be expresse as follows: Iv year ( 3... ( ) P. (3.6a) 6

25 By subsu equao (3.6a) a upo splfcao, yels P Iv year Therefore, he oal averae veory for all buyers per year s. (3.6) Iv b ( ) P. (3.7) (b) Veor s Averae Iveory Calculao The veor prouces es baches a a fe rae a hols he ul prouco of he frs bach s fshe. If each bach sze s a requres P pero o prouce, he averae veory ur he frs cycle s ( P ). Sce here are baches a cycle, he averae veory per cycle ca be expresse as Iv cycle v. (3.8a) P Sce here are uber of cycles per year, averae veory per year ca be expresse as follows: Iv yearv. (3.8) P P 3.4. Toal Averae Iveory Calculao I a erae supply cha syse, he oal averae veory s calculae by su averae yearly veores of he veor, equao (3.8), a all buyers, equao (3.7). Therefore, he yearly averae veory of he syse ca be wre as follows: Iv yearsyse P ( ) P. (3.9) 7

26 3.5 Toal Cos For Moel I I he curre a prevous secos, veores for he veor a all buyers are calculae uer he assupos of frs oel. Geerally, he oal cos of he syse cosss of ajor coss such as (a) orer cos, (b) seup cos, (c) veory hol cos, a () rasporao cos. The oal syse cos, coss of orer cos, seup cos, veory hol cos, a rasporao cos, ca be calculae Orer Cos Each e a buyer places a orer o he veor curs a cos, whch ay coss of paper wor, elephoe coversaos, ec., he each buyer presuably places a orer before he cycle sars. Hece, each buyer wll place uber of orers a year. Therefore, ( ) buyers a year wll place uber of orers. The cos of plac oe orer for h buyer s a. Therefore, he oal orer cos, A, for all buyers ca be expresse as A a, (3.0) where s he ea rae (us/year), s he uber of baches a cycle a s bach sze Seup Cos I each ew prouco cycle ha a veor sars for a ew lo, a seup cos, such as cha ye, pu raw aerals ec s requre. If he aufacur process requres seup for every ew lo, he oal uber of seups requre s cos ( S ) ca be calculae as,. Hece, he oal seup S S, (3.) 8

27 where s he ea rae (us/year), s he uber of baches a cycle, s he bach sze, a S he seup cos per lo Iveory Carry Cos Whle he bach s be prouce, veory buls up. Thus, he veor curs he veory carry cos ul prouco s fshe a he es are shppe. Slarly, each buyer receves es a hols he ul all he es are cosue. Therefore, each buyer also curs e carry coss. The oal syse veory carry cos ca be calculae. (a) Iveory Carry Cos for Buyers Fro equao (3.7) we ow he averae yearly veory for all buyers. If h s he carry cos for h buyer, veory hol cos for all buyers per year ca be calculae as Iv b h ( ) h h P, (3.) where s yearly ea of h buyer, s bach sze, s ea rae (us/year), s uber of baches a lo; P s prouco capacy (us/year) a s rasporao for h buyer. (b) Iveory Carry Cos for Veor Fro equao (3.8) we ow averae yearly veory for he veor. If h s he carry cos for veor, veory hol cos ( h ) for he veor per year ca be calculae as h h. (3.3) P 9

28 (c) Toal Syse Carry Cos Calculao Toal syse s carry cos cosss of he buyer s carry cos per year a veor s carry cos per year. Equaos (3.) a (3.3) represe all buyers carry coss a he veor s carry cos, respecvely. Hece, he oal syse s carry cos ( h sys ) ca be expresse as h sys h h h P h. (3.4) P Trasporao Cos Every e he veor ses a shpe o a buyer, he buyer curs a rasporao cos. Realscally, he capacy of a rasporao vehcle s le. Aa, eve f a coveyace s parally flle, he buyer has o pay he prce of a full loa. Aoher coserao s rasporao cos for oe shpe, whch ay ffer ao buyers epe o her saces fro he veor. If q s he carry capacy of a coveyace a s he shpe sze, he recev buyer has o pay for ( q ) uber of loas. If he coveyace cos s C for h buyer a here are uber of baches a cycle, he rasporao cos per cycle ( T cycle ), pa by h buyer, ca be calculae as T C. (3.5) cycle q h Sce here are ( ) uber of cycles a year, rasporao cos o buyer per year ca be calculae as T year C C. (3.6) q q 0

29 By equa ( ), where s bach sze (es/bach) a s he yearly ea (es/year). Sce here are buyers he syse, oal rasporao cos ( buyers a year ca be expresse as T b ) for all T b C q. (3.7) Toal Syse Cos The oal syse cos, TC I cosss of orer cos, seup cos, veory carry cos, a rasporao cos. Hece, he oal syse cos for Moel I ca be calculae by a equaos (3.0), (3.), (3.4), a (3.7) as TC I S h P a C q h h P h.(3.8) 3.6 Soluo Mehooloy A hs po, oe us uersa he aure of he oal syse cos ( TC I ) fuco for opzao purposes. If he oal cos fuco s fou o be covex, he TC I 0 leas o opaly. By loo a q C equao (3.8), s clear ha TC I s o a covex fuco ve cosa. A opal bach quay * ca be calculae by a rec search eho wh a bouary of (,0), a (, ). Slarly, a opal uber of baches * ca also be calculae by a rec search eho wh a bouary of (,0), a (, ) us equao (3.8).

30 Exaple 3. Ie shppe equally-sze baches upo prouco copleo Assue, P 900 us/year, 50 us/year, S 90, h. 3, q 30. aa s ve Table. For eer, he u oal cos s presee Table. Necessary u coverso s perfore pror o soluo. For eale resuls, see Appex. Fure 3., a plo us MaLab 009b; he u oal cos s are o he fure. I Fure 3., a sue cu occurs he plo because a ha po he uber of vehcle requre chaes for soe buyers. Buyer Table : aa for sle veor 9 buyers proble (ays) a SUM 4 50 h C Table : Suary of resuls N TC I

31 TC TC Fure 3.: Graphcal represeao of TC I 3.7 Sesvy Aalyss for Moel I The oal cos s a soluo of he oel where oel paraeers (veor hol cos, seup cos, a prouco o oal yearly ea rao) are presuably fxe. I s useful o carry ou sesvy aalyss of he oel. For hs purpose, effec of chaes syse paraeers us be verfe o chec f he curre soluo. Reas uchae.. Becoes sub-opal, ec. 3

32 3.7. Effec of S o TC I The veor seup cos S plays a ajor role eer oal cos of he syse, a also affecs he opal bach sze a uber of baches a lo. The effec of S o oal syse cos, TC I s expresse aheacally equao (3.9), TC I S, (3.9) whch s a cosa er. The rae a reco of chae of TC I wh respec o S epe o he values of paraeers use he uercal exaple. If we pu values of,, a fro Table 0 a Table equao (3.9), he we ca observe ha for u chae S, TC I wll crease by $ Effec of h o TC I The veor s hol cos h s a pora paraeer evelop he oel. The effec of h o oal syse cos TC I s expresse aheacally equao (3.0), TC I, (3.0) h P whch s also a cosa er. The rae a reco of chae of TC I wh respec o h epe o he values of paraeers use he uercal exaple. If we pu values of,, a fro Table 0 a Table (presee laer), he we ca wre, ha for every u crease (ecrease) h, TC I wll crease (ecrease) by $ Effec of P/ o TC I The rao bewee prouco rae a oal yearly ea P/ s o oly a pora paraeer o eere he oal cos; bu also, cojuao wh he veor hol cos h, he rao eeres whch oel o choose for a parcular scearo ao he hree. The effec of P/ o oal syse cos, equao (3.), TC I s expresse aheacally 4

33 TC h I A A A h. (3.) The effec of P/ over TC s represee by Fure 3.3 us equaos (3.) I a P [ 0,5]. A suy s perfore wh respec o P/ where he paraeer values are he sae excep for he values of P/, whch chaes fro o 5, a he resuls are presee Table 3. Fure 3.3 llusraes ha P/ rao affecs TC I chaes raply up o., a beyo hs po, TC I becoes less sesve o a chae P/. We ca also wre ha TC I creases as he rao creases. Table 3 suarzes he chae he chae P/ rao. 3.8 Moel II: Ies Are Shppe I Every Pero TC I for As escrbe above, a veor hols es a shps he every pero. I he seco oel we are assu all bach szes a lo are equal a here are baches a lo. Fure 3.4 shows he prouco a veory flow for he seco oel. I Fure 3., G P, G, a, are he e sees, whch represe he prouco e for a bach clu rasporao e ea, cosupo e for h buyer a h rasporao e for buyer, respecvely. I Fure 3.4, rale (MNO) represes he eal veory for he veor. The rale (ABC) represes he eal veory for he h buyer whe he e s receve a he e of prevous veory. The area (XYZ) represes he veory bul up he veor s warehouse ue o cosa prouco of es. 5

34 Table 3: Effec of P/ ( A) rao o A TC I A TC I TC I A A Fure 3.3: Effec of P/ ( A) rao o TC I 6

35 P A BATCH SIZE O G Y P X Z Q T R S Q B C BUYER VENOR M G/P N ( G P ) NEW LOT TIME Fure 3.4: Iveory flow Moel II I he be of he cycle, he veor sars prouco a a fe rae a veory buls up a a cosa rae. I Fure 3.4, he slope MO s he rae of accuula veory ur prouco. As soo as prouco of he frs bach s coplee, es are shppe o each buyer, a veory of veor s eplee. A hs po, he veor bes prouc he ex bach. Sce we assue P > o avo shorae, veory bes bul up for he veor beyo G as G / P < G /, as show Fure 3.4. Whe a buyer s veory rops o rasporao e ea, ew es are shppe, a he veor s veory s reuce by G. Ths process s repeae ul he ere lo s prouce. Ies are shppe o all buyers a he sae e, bu buyers wll receve es afer e sce shpp e for each buyer vares. Ies are realy avalable o cosue as soo as he buyer receves he, a he buyer cosues he es a a cosa rae. I Fure 3.4, slope AC s he cosupo rae. Sce buyer cosues es a 7

36 cosa rae, o-ha veory bes ecreas cosaly. Whe he ew es arrve, veory rops o zero. Hece, hey buyer oes o have o hol ew es for exra e, ule he frs oel. Noce ha he cycle propels he e so ha ew es he veor s warehouse ulply. Sce he prouco s couous ul he ere lo s prouce, he veory level also rses wh e. Fure 3.4 proves hs oo wh he area of rale (QRS), whch s ouble ha of rale (XYZ) Averae Iveory Calculao Ths seco erves he averae veory Moel II for he ere supply cha syse coa a sle veor a ulple buyers. Aa, he ea s eersc a prohbs baclo a shorae; he prouco rae s presuably reaer ha he ea rae o avo shorae. The oal syse veory cosss of buyer s veory, a veor s veory, I. (a) Buyer s Averae Iveory Calculao Each buyer receves G aou of e each shpe a aes G aou of e o cosue. So each buyer s averae veory for he frs bach s ( G ). If h here are baches a lo, averae veory,, for buyer per cycle ca be expresse as, I b G. (3.) I b We eere ha G, a upo subsu Equao (3.) a splfy, we oba 8

37 9 b I. (3.3) Sce here are uber of cycles per year, he he averae veory, per year ca be expresse as I year G I year. (3.4) We eere, so subsu ( G ) Equao (3.4), we e, G G G I year ) ( ) (. (3.5) Upo splfcao hs resuls, G G G G I year, (3.6) Therefore, averae veory, for all ( buyers) buyers per year ca be expresse as I b b G G I. (3.7) (b) Veor s Averae Iveory Calculao The veor prouces es baches a a fe rae a hols he ul he prouco of he frs bach s fshe, a he, he shps every / pero. Reeber, o save o orer cos, buyers oly place orer he be of he lo a every bach s auoacally shppe afer a / pero. If each bach sze s G a requres G/P peros o prouce, he averae veory ur he frs bach s P G. Sce he seco bach s shppe afer was prouce, he veor hols hs bach for ) ( P G pero of e. Therefore, he averae veory ur he seco bach s [ ) ( P G G P G ]. The hr shpe also reas for a whle before s shppe a acually reas relavely

38 loer (wce) ha he seco bach. I Fure 3.4, we ca see ha QS s ouble YZ, whch represes he es ha prevous baches are rea he warehouse. Ths coues o occur ul he ere lo s exhause. If here are baches a lo, he h bach wll have o rea warehouse for ( )( G P) aou of e. Therefore, he oal averae veory for he veor, per cycle, ca be expresse as G G v G{ 3... ( ) }. (3.8) P P I cycle If each bach sze s G a here are baches a lo, he a coplee cycle, G aou of es are prouce. So, he oal uber of cycles a year s. Therefore, averae veory for he veor per year ca be expresse as I v G G G{ 3... ( ) }, (3.9) ' P P Q where he lo sze Q ' G. We efe G, by subsuo expresso ers of G, so G G G I v G{ 3... ( ) }, (3.30) G P P hs, afer splfcao, yels I v ( ) G ( ) ( ) G P G. (3.3) P 3.8. Toal Averae Iveory Calculao I a erae supply cha syse he oal averae veory s calculae by a averae yearly veores for all buyers us equao (3.7) a veor equao (3.3). Therefore, he yearly averae veory of he syse ca be wre as follows: 30

39 G I ( ) G G. P G G P (3.3) 3.9 Toal Cos for Moel II I he curre a prevous secos, veores for he veor a all of he buyers are calculae uer assupos of he seco oel. Usually, he oal cos of he syse cosss of he ajor coss, such as (a) orer cos, (b) seup cos, (c) veory hol cos, a () rasporao cos. The oal syse cos coss of hese four elees ca be calculae Orer Cos Each e a buyer places a orer wh he veor, he curs a cos, whch ay be relae o paperwor, elephoe calls, ec. We assue each buyer places a orer before he cycle sars. Hece, each buyer wll place G uber of orers a year. Therefore, buyers a year wll place G uber of orers. The cos of plac oe orer for he h buyer s. Therefore, oal orer cos A for all buyers ca be expresse as a A G a, (3.33) where s he ea rae (us/year), s he uber of baches a lo a G s bach sze Seup Cos Each e he veor sars a ew lo, prouco requres a seup such as cha e, se raw aerals, ec. If he aufacur process requres seup for every ew lo, he he oal uber of seup requre s G. Thus, he oal seup cos ( ) ca be S calculae as 3

40 S S, (3.34) G where s he ea rae (us/year), s he uber of baches a cycle, G s bach sze, a S s seup cos per lo Iveory Carry Cos Whle he bach s be prouce, veory buls up. Thus, he veor curs he veory carry cos ul prouco s coplee a es are shppe. Slarly, each buyer receves es a hols he ul all of he es are cosue. Therefore, each buyer also curs he e carry cos, a he oal syse veory carry cos ca be calculae. (a) Iveory Carry Cos for Buyers Fro equao (3.7) we ow he averae yearly veory for all buyers. If h s he carry cos for h buyer, veory hol cos for all buyers per year ca be calculae as H b G h h h G, (3.35) where s he yearly ea of h buyer, G s bach sze, s ea rae (us/year), s uber of baches a lo, a P s prouco capacy (us/year). (b) Iveory Carry Cos for Veor Fro equao (3.3) we ow he averae yearly veory for he veor. If h s he carry cos for veor, veory hol cos ( calculae as h ) for he veor per year ca be I v Gh ( ) Gh ( ) h P ( ) Gh. (3.36) P 3

41 (c) Toal Syse Carry Cos Calculao Toal syse s carry cos cosss of he buyer s carry cos per year a veor s carry cos. Equaos (3.35) a (3.36) represe all buyers carry cos a veor s carry cos, respecvely. Hece, he oal syse s carry cos H ca be expresse as Trasporao Cos G H h h G Gh ( ) h G G. P P h (3.37) Every e he veor ses a shpe o a buyer, he buyer curs rasporao cos. I realy, he capacy of a rasporao vehcle s le. Aa, eve f a vehcle s parally flle, he buyer has o pay he ere prce of full loa. Aoher facor s rasporao cos for oe shpe ca be ffere ao buyers epe o her saces fro he veor. If q s he carry capacy of a vehcle a s he shpe sze, he recev buyer has o pay for C G q uber of vehcles. If he cos/vehcle s for h buyer, a here are uber of baches a cycle, he, rasporao cos per cycle ( T cycle ) o he h buyer ca be calculae as, T cycle G q C. (3.38) Sce here are h uber of cycles a year, rasporao cos o buyer per year ca be calculae as, T G G year C C. (3.39) G q G q 33

42 Equa G G, where G s bach sze (es/bach) a s yearly ea (es/year). Sce here are buyers he syse, oal rasporao cos T for all buyers a year ca be expresse as T G G q C. (3.40) Toal Syse Cos The oal syse cos TC II, cosss of orer cos, seup cos, veory carry cos, a rasporao cos. Hece, he oal syse cos for Moel I ca be calculae by a equaos (3.33), (3.34), (3.37), a (3.40) as TC II G G a S G G h G q C G h h Gh ( ) h G P G. P (3.4) 3.0 Soluo Mehooloy A hs po, s ecessary o uersa he aure of he oal syse cos TC II fuco for opzao purposes. If he oal fuco TC II s covex, he, * TC II 0, whch allows for opaly a opal bach quay evaluao. By loo a G G q C equao (3.4), s clear ha TC II s o a covex fuco ve reas cosa. A opal bach quay * ca be calculae by a rec search eho wh a bouary of (,0) a (, ). Slarly, a opal uber of baches * ca also be calculae by a rec search eho wh a bouary of (,0) a (, ) us equao (3.4). 34

43 Exaple 3. Orer shppe afer every / pero Assue, P 300 us/year, 300 us/year, S $50, h., a q 40. aa s ve Table 4. For eer, he u oal cos s $377. (for, a 506), presee Table 4. Necessary u coverso s perfore pror o soluo. For eale resul see Appex. I Fure 3.5, a sue cu occurs he plo because a ha po he uber of vehcle requre chaes for soe buyers. Buyer Table 4: aa for sle veor 9 buyers proble (ays) a h C 3. Sesvy Aalyss for Moel II Table 5: Suary of resuls TC II Sesvy aalyss has bee perfore hs seco o es he robusess of he propose oel, whch s subjec o chae ve paraeers, seup cos, veor hol cos, a prouco o ea rao. 35

44 3.. Effec of S o TCII The veor seup cos plays a ajor role eer oal cos of he syse, a also affecs he opal bach sze. So, s pora o uersa he effec of chae S o TC. The effec of S o oal syse cos TC s expresse equao (3.4), II TC II S G II, (3.4) whch s a posve cosa er. I ca be wre ha TC II creases (ecreases) wh a crease (ecrease) seup cos S. If we plee values of, a fro Table 0 a Table (show laer) for a u chae S, he wll crease by $.. TC II TC TC Fure 3.5: Graphcal represeao of TC II 36

45 3.. Effec of h o TCII The veor s hol cos s a pora paraeer eer he oal cos. Also, o a exe, he veor s hol cos overs whch oel shoul be chose ao he hree presee. I s pora o uersa how veor hol cos h affecs TC II. The effec of h o TC II oal syse cos s expresse equao (3.43), TC h II G ( ) G P G, (3.43) P whch s also a posve, cosa er. I ca be wre ha TC II creases (ecreases) wh a crease (ecrease) hol cos h. By ser values of,, P, a G fro Table 0 a Table (show laer), here wll be a u chae h a wll crease by $ Effec of P/ o TCII The rao bewee prouco rae a oal yearly ea o oly eeres he oal cos, bu also, eeres, cojuao wh veor hol cos, whch oel o choose for a parcular scearo ao he hree presee. Hece, s crucal o uersa he effec of P/ o oal syse cos TC. The effec of P/ o oal syse cos TC, s expresse equao (3.44). II TC II II TCII A Gh A ( )h G A. (3.44) The effec of P/ over TC II s represee by Fure 3.6 us equaos (3.44) a P [,5]. A suy s perfore wh respec o P/ where he paraeer values are he sae excep for he values of P/, whch shfs fro o 5; he resuls are presee Table 6. Fure 3.6 reveals ha P/ rao affecs TC II a chaes raply up o.0. 37

46 Beyo hs po, TC II becoes less sesve o chae P/. We ca also wre ha TCII creases as he rao ecreases. Table 6 suarzes he chae TCII for he chae P/ rao. 3. Moel III: Ies Shppe Uequal Baches As escrbe above, he veor hols es ul he orer s place a he shps es o buyers. I he hr oel, we assue ha bach szes a lo are uequal, ha hey crease by a facor (where P ) a ha here are baches a lo. Fure 3.7 shows he prouco a veory for he hr oel. A lo s prouce wh bach szes of,,,,. I Fure 3.7, ( ) P ( ),, a, are he e sees, whch represe he bach prouco e, clu rasporao e ea, cosupo e for h buyer a rasporao e for h buyer, respecvely. I Fure 3.7, rales MNO, OPQ a QRS represe veores for he veor for subseque baches. Trales ABC, CE a EFH represe veores for he h buyer whe he e s receve eaely a he e of prevous veory. I he be of he cycle, he veor bes prouco a a fe rae a veory buls up a a cosa rae. I Fure 3.7 he slope MN s he rae of veory bul-up ur prouco. As soo as prouco of he frs bach s coplee, es are shppe o each buyer, a veor veory reuces o zero. A hs po, he veor bes prouc he ex bach of sze. Reeber, we assue P > o avo shorae. Ule he frs wo oels, he hr oel s bach sze creases wh a ulplcao facor of. Hece, o avo shorae, 38

47 A Fure 3.6: Effec of P/ ( A) rao o TC II Table 6: Effec of P/ ( A) rao o A TC II A TC II / P /. Whe he buyer s veory falls o he rasporao e ea, ew es are shppe, a he veor s veory s reuce o zero. Ths s repeae ul he ere lo s prouce. Ies are shppe o all buyers a he sae e, bu buyers wll receve es afer e sce shpp es vary ao 39

48 buyers. Ies are realy avalable o cosue as soo as he buyer receves he, a buyer cosues he es a a cosa rae. I Fure 3.7, slope BC s he cosupo rae. Sce he buyer cosues es a a cosa rae, o-ha veory bes o ecrease cosaly. Whe he ew es arrve, veory rops o zero. Hece, he buyer s o requre o hol ew es for exra e, ule he frs oel. Ule he oher wo oels, hs oel, eher he veor or he buyers ee o hol a ew bach s warehouse whle he prevous bach s be cosue. 3.. Averae Iveory Calculao Ths seco erves he averae veory Moel III for he ere supply cha syse coa a sle veor a ulple buyers. Slar o he oher wo oels, he ea s eersc a o baclo or shorae s allowe, he prouco rae has o be reaer ha he ea rae. The oal syse veory cosss of buyer s veory I b a veor s veory. (a) Buyer s Averae Iveory Calculao I Each buyer receves aou of es for he frs shpe, followe by,,, es. Each buyer aes ( ) e o cosue he bach fully so each buyer s averae veory for he frs bach s ( ). For he seco bach, he averae veory s ( ) ). Slarly, for he h bach, he averae veory s ( ). If here are h baches a lo, he averae veory ( ) for buyer per cycle ca be expresse as I b I b [( ) ( ) ( )... ( ) ].. (3.45) 40

49 4 Fure 3.7: Iveory flow Moel III Upo splfcao hs leas o b I. (3.46) We ow, a subsu equao (3.46), we e, BUYER A B C E F H M N O P Q R S TIME BATCH SIZE ) ( ) ( ) ( ) ( ) ( ) ( P ) ( P ) ( P ) ( VENOR NEW LOT NEW LOT

50 4 b I. (3.47) Assue he su of all baches a lo equals, so he lo sze s ' Q G Q '. (3.48) Therefore, here wll be uber of los a year. Hece, he averae veory for buyer per year ca be expresse as ' / Q year I h ' Q I year. (3.49) If here are buyers he syse, he averae veory for all buyers per year ca be expresse as b Q I '. (3.50) (b) Veor s Averae Iveory Calculao The veor prouces bach es a a fe rae wh a bach of szes,,,...,. The veor hols es ul prouco s fshe a shps as soo as he bach prouco s coplee. Sce bach szes are uequal, we ca calculae he wor--process (WIP) veory for he veor. Hece, he WIP veory per cycle ca be expresse as: P P P P WIP cycle... ) (. (3.5)

51 43 If he lo sze s a here are uber of cycles a year, he averae WIP veory (WIP ) per year ca be expresse as ' Q year ' / Q year P Q WIP ', (3.5) Where P/, s yearly ea (e/year), s he u bach sze, s uber of baches a lo, s lo sze (e/lo) a P s prouco rae (e/year). ' Q 3.. Toal Averae Iveory Calculao I a erae supply cha syse, he oal averae veory s calculae by a averae yearly veores of all buyers a veors us equaos (3.50) a (3.5). Therefore, yearly averae veory I of he syse ca be wre as P Q I '. (3.53) 3.3 Toal Cos for Moel III I he curre a prevous secos, veores for he veor a all buyers are calculae as he seco oel. Usually, he oal cos of he syse cosss of ajor coss: (a) orer cos, (b) seup cos, (c) veory hol cos, a () rasporao cos. The oal syse cos coss of hese coss ca be calculae Orer Cos Each e a buyer places a orer o he veor, he buyer curs a cos, whch ay coss of paperwor, elephoe calls, ec. Assue each buyer places a orer before he cycle sars. Hece, each buyer wll place uber of orers a year. Therefore, buyers a year wll place uber of orers. The cos of plac oe orer for h ' / Q ' / Q

52 buyer s. Therefore, oal orer cos A for all buyers ca be expresse as a A a, (3.54) ' Q where s he ea rae (us/year), s he uber of baches a cycle, a ' Q s lo sze (es/lo) Seup Cos Each e he veor bes prouco of a ew lo, he curs a seup cos for cha e, se raw aerals, ec. If he aufacur process requres seup for every ew lo, he oal uber of seup requres s ca be calculae as, ' / Q. Hece, he oal seup cos S S S, (3.55) ' Q where s he ea rae (us/year), s he uber of baches a cycle, ' Q s lo sze (es/lo) a S seup cos per lo ($/lo) Iveory Carry Cos Whle he bach s be prouce, veory buls up. Thus, he veor curs a veory carry cos ul prouco s coplee, a he he shps he es. Slarly, each buyer receves es, hol he ul all es are cosue. Therefore, each buyer also curs a carry cos. The oal syse veory carry cos ca be calculae. (a) Iveory Carry Cos for Buyers Fro equao (3.50), we ow he averae yearly veory for all buyers. If h s he carry cos for h buyer, he veory hol cos ca be calculae as h b for all buyers per year 44

53 45 b h h h Q h ', (3.56) where s he yearly ea of h buyer, s bach sze, s ea rae (us/year), s uber of baches a lo a P s prouco capacy (us/year). (b) Iveory Carry Cos for Veor Fro equao (3.5), we ow he averae yearly veory for he veor. If s he carry cos for he veor, he veory hol cos for he veor per year ca be calculae as follows: h h v v P h Q h '. (3.57) (c) Toal Syse Carry Cos Calculao Toal syse s carry cos cosss of he buyer s carry cos per year a he veor s carry cos. Equaos (3.56) a (3.57) represe he carry coss for all of he buyers as well as for he veor, respecvely. Hece, he oal syse s carry cos H ca be expresse as h P h h P h h P h Q H '. (3.58) Trasporao Cos Every e he veor ses a shpe o a buyer, he buyer curs rasporao cos. I realy, capacy of a rasporao vehcle s le. Aa, eve f a vehcle s parally flle, he buyer has o pay accor o he full-loa prce. Aoher h o coser s he rasporao cos for oe shpe because ay vary ao ffere buyers epe o her saces fro he veor. If s carry capacy of a vehcle q