A Technical Note on the Single-Vendor Multi-Buyer Integrated Inventory Supply Chain Problem

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1 Pocd of t 4 Itatoal Cofc o Idutal E ad Opato Maat Bal Idoa Jaua A Tccal Not o t Sl-Vdo Mult-Bu Itatd Ivto Suppl Ca Pobl Md Abdul Hoqu Uvt Bu auala Bu auala Abtact Rctl two l-vdo ult-bu tatd vto odl w dvlopd b copoat o altc facto. T poducto flow wa cozd b taf t lot wt qualad/o -uqual zd batc ub-lot w t qual-zd batc w pt to b qual to t lat uqual-zd batc. A coo al total cot oluto tcqu to t odl wa dvd ad t pottal fcac w ltd wt oluto of o ucal pobl. Bcau of t flxbl tcto o t qual-zd batc t odl t fald to povd t alzd al total cot oluto to a l-vdo l-bu ucal pobl. H w xtd t odl b tct t qual-zd batc to b l ta o qual to t lat uqual-zd batc ultpld b t ato of t poducto at to t total dad at a ud aot ctl publd ac wo. T w dv a dfft plfd coo oluto tod to t xtdd odl. Tu t cotbuto of t pap t xto of t ctl dvlopd odl to o alzd o ad dvlopt of a pl coo oluto tcqu to t. T pottal fcac of t xto potd wt t oluto of o ucal pobl. T ac cotbuto of t pap t xto of two ctl dvlopd l-vdo ult-bu tatd vto odl to o alzd o ad dvlopt of a pl coo oluto tcqu to t. Kwod Sl-vdo ult-bu tatd vto qual-zd batc al total cot.. Itoducto T l-vdo ult-bu tatd vto uppl ca t dttc dad a cvd a codabl attto of t ac Vd fo dtal Hoqu 8 a b Sa ad pooo 9 Zavalla ad Zao 9 ; Sva ad Rao Gloc K ad Goal B- aa t al. 3. Ul ot Hoqu a b cozd t tatd poducto flow b taf t lot wt qualad/o -uqual zd batc. I bot ca t xt uqual-zd batc a ultpl of t pvou o b t ato of t poducto at to t total dad at. I Hoqu a t qual-zd batc a flxbl tctd to b l ta o qual to t lat uqual-zd batc ultpld b wl t latt t qual-zd batc a flxbl tctd to b qual to t lat uqual-zd batc. Bcau of t flxbl cotat o qual-zd batc Hoqu b t oluto tcqu t lad to a al total cot oluto fo a l-vdo l-bu ucal pobl oall olvd b Hll 999 ad ctl b Hoqu a dvlopd follow t cozato appoac Hll 999. Altou Hoqu b fald to ow t alzd caacttc obta t al total cot oluto fo tat pobl t a copd wt laxato of o pactcal aupto uc a fcat t up ad tapotato t ultd capact of t tapot qupt ad bu toa utctd lad t of uppl a batc ad t boudl allt batc z wc av ot b codd Hoqu a. Tfo w xtd t odl Hoqu b b ta advata of t flxbl tcto o t qualzd batc coz t poducto flow a Hoqu a but ta t altc facto a codd Hoqu b. I t cut odl t allt batc z to ac of t bu tctd to b at ta o qual to o ca of bul atal t dd allt aout of a atal ca b aud a ut a Hoqu b. T qual-zd batc to ac of t bu a pt to b at ta o qual to o but l ta o qual to t pctv lat uqual-zd batc z ultpld b. T a dfft coo pl oluto tod to t xtdd odl ptd. Tu t pl oluto tod to t tatd vdo-bu pobl wt a flxbl tcto o t qual zd batc but atf t cotat fo laxato of t tod pactcal aupto o ffctv ta t pvou o. T pottal 7

2 fcac of t w tod of oluto to t pobl ad t ffct of t tod facto o t al total cot oluto a ow wt oluto of o ucal pobl. T outl of t ad of t pap a follow: Scto dal wt aupto ad otato ad t ptato of t odl ad t al total cot oluto tod. T follow a cto of oluto to ucal pobl ad copaatv tud of t cocd tod o t ult. T lat cto f wt dcuo ad cocluo.. Foulato of t Extdd Modl. Aupto ad Notato T aupto ad otato a t a a ud Hoqu b xcpt o addtoal otato. T otato a a follow: Fo t vdo Aual at of dad; P Aual at of poducto P > ad = P/; Ivto ca cot p t p a; S Poducto t up cot p lot; St up t ; z T allt batc pat of a lot z; T qual- zd batc fo t vdo; v Capact of t vdo toa T lot z tafd fo t vdo to t bu Total ub of qualad/o -uqual zd batc a lot; Nub of uqual-zd batc a lot ; L T lat lad t; L T allt lad t. Fo t t bu =...; Aual at of dad ; T Cot of ft a batc fo t vdo; Ivto ca cot p t p a; Cot of plac a od of a lot; t qual zd batc to t bu; t Capact of t tapot vcl; Capact of t bu toa ; t v t Ipcto load taf ad uload t ; t t Rtu t of t tapot vcl ; z T allt batc z;. Modl I ad t oluto tod Au a batc taf jut aft f t poc.. T total cot fucto T vdo taf t lot to t bu uqual zd batc of z z z z... z ad - qual zd j j batc of z = z f = ; = f = ; otw z z uc tat z z / ; / fo j j j j j j = ; = ad z z ; z / P z / z / fo j= -. Tu z z / z z z... Follow Modl I of Hoqu b but plac t patd batc z b t patd batc ad plf fd t jot total cot p a a follow: S A / ; S S S ; A T. w 7

3 .. T cotat T capact cotat o t tapot vcl a v Hoqu b z M / t [ / ] z t 3 Subttut fo z fo ad plf t lad to 4 Alo t [ / ] 5 I ca of taf t patd batc tad of z ud Hoqu b ad follow t local aut t obta z / / / 6 Subttut fo z fo ad plf t tafo to 7 Sc t z / o v. z t Slal v ca b ow. Follow Hoqu b t lad t t t up t t taf t ad t allt batc z cotat a pctvl v b l L / L / l 8 // / P / 9 z / t t ad z z / Subttut fo z fo to t lat two cotat ad plf dv ; t Max t t t Max/ t Max t t Alo z. Subttut fo z fo ad plf obta / Tfo t atatcal odl of t pobl to z t total cot fucto ubjct to t cotat ad...3. Soluto Tcqu Ca I W = t lot tafd ol wt qual zd batc I t ca = z ad c fo = z ad t jot total cot fucto tafo to S z A z Gv z t z ca b obtad a S z a z Subttut t valu of z t total cot fucto t tafo to z S A z wc covx z. T z z v b A z b Fo t valu of z z z ca b calculatd fo a. Howv t altc fabl oluto ca b foud b atf t follow qualt obtad fo cotat ad cotat alwa tu. Max /{ } z M /[ { / / }] 3 So fo t obtad z z fo b calculat t z z fo a. If t t oud dow o t oud up valu of f z < t = atf 3 t fo t z z fo b dcd t al total cot b calculat t total cot at t oud dow o t oud up valu of. If t lft ad d qualt 3 ot atfd t a b atfd b ca t oud up valu of z b at ac 7

4 73 tp. T allt of t tal valu of tat atf 3 t z t ca. If t t ad d qualt 3 ot atfd t a b atfd b dca t oud dow valu of z b at ac tp. T lat of t tal valu of tat atf 3 t z. If a tal atf 3 do ot xt t t pobl fabl. Otw calculat t valu of fo fo ow ad c t al total cot fo. Ca II W = o = t lot tafd ol wt uqual zd batc Cod t total cot fucto t z v b A S 4 So fo ac of t fabl valu of atf 8 t = ad calculat t z fabl f xt b atf t follow qualt obtad fo cotat 4 9 ad c t al total cot fo. / Max 5 T u of t al total cot obtad fo fabl valu of atf 8 alo wt t aocatd valu of ad = lad to t fal al total cot oluto t ca. Ca III W > ad z Sub-ca W I t ca v ad t z dotd b v b ; / B A S A B A 7 T fabl z atf t cotat H H f xt w H t axu of t t ad d of 9 ad H t u of t t ad d of 4 ad 7 ca b obtad ad c t aocatd total cot fo. Aa v ad t z dotd b v a follow: B A B A ; / 8 T fabl a to atf cotat ad tat M / Max Not tat otw aft ubttut fo fo lad to a cotadcto. z Tu fo ow ad t z ca b calculatd fo 8. T fabl z } /{ Max z b atf t abov qualt f xt ca b foud ad c t aocatd total cot fo. Slal v ad t z dotd b v b.5 A 9 T fabl d to atf cotat 4 7 ad tat M / Max

5 74 Not tat... fo ad c fo. So fo v ad t z tal ca b calculatd a t oud up o oud dow valu of obtad fo 9. T fabl z f xt ca b dtd b atf t qualt abov. Sub-ca W T coffct of potv. Otw o-potv coffct of pl a cotadcto. I ca of t fabl z ca b dtd follow t pocdu a Sub-ca of Ca III fo dt t a. Obv tat t total cot fucto a dca fucto v ad ad c t al at t t avalabl fabl valu t a povdd al. Alo fo v ad t total cot fucto dca f A ad c t al at t t avalabl fabl valu t a povdd al; otw t al at t lat avalabl fabl valu tat a. I ca of obv tat fo v ad t total cot fucto cocav au a al ad fo v ad t alo cocav au a al ad c pctv z ad at t lat o t fabl valu. Tu fo v ad t total cot fucto a t al valu t at = z o at t t valu of t fabl a a povdd al. Alo fo v ad t total cot fucto a t al valu t at = + o at t t valu of t fabl a a povdd al. Fo v ad f t coffct of / t total cot fucto o-potv t t cocav ad o t z xt t at t lat valu of z o at t t valu of at t fabl a a v al. Otw t total cot fucto covx ad t z v b 7 ad c t fabl z ca b dtd b follow t poc dcbd Sub-ca. Tfo tat wt ow atf 8 t = + ad } /{ Max z [atf 9 ad alwa atfd fo = z] t z ad c t fabl z ca b calculatd. U tat z ad ow ad a fabl al valu of ca b foud. Aa fo ow ad a fabl al tal valu of ca b foud. Fo ow ad t fabl z ca b calculatd t aa. Itato of t poc of calculat z ad lad to t al total cot oluto to t pobl. Mu of t al total cot obtad fo all fabl atf 8 alo wt t aocatd valu of ad lad to t fal al total cot oluto t ca..3 Modl II ad t Soluto Mtod Au a batc taf w t pvou batc f at t bu.3. T total cot Fucto Cod t batc z at t vdo ad t dtbuto to t bu a t a a Modl I t valu of z ca b obtad a. T follow Modl II of Hoqu b but plac t patd batc z b ad plf obta t total cot p a a follow: t A S

6 .3. T Cotat All t cotat xcpt t capact cotat of t bu toa a t a a Modl I. H w cod t capact cotat of t vdo toa c vto accuulatd at t vdo. I ca of taf t patd batc ft t batc tafd to t bu. T batc t t dad fo t t z z /. I t t t aufactu poduc at ot P z / z ut f qud. Aft taf ut to t bu z ut a at t vdo toa. T ut f at t bu / ut of t. At t t t aufactu ca poduc at ot P / = ut f qud. So t t accuulatd vto at t aufactu toa could b z z. If t aufactu cotu to poduc fo t t / t t xt accuulatd t vto wll b z. I t wa t t accuulatd vto at t aufactu toa fo - patd batc of ut could b z wc ut b l ta o qual to.. z / / v Subttut fo z fo ad plf t tafo to v Sc o z. t t ; t.3.3 Soluto Tcqu Ca I W = Subttut fo = z ad = z t jot total cot fucto tafo to S z A t z Fo t coffct of z potv. Fo t ca alo b ow tat t coffct of z o-atv. Gv z t z obtad a S z a z Subttut t valu of z t total cot fucto ad plf t z z v b A z b T al total cot oluto to t pobl ca b obtad b follow t oluto pocdu of Ca I of Modl I but u t total cot fucto dvlopd foula a b tad of a ad b pctvl ad t cotat tad of 7. Ca II W = o = I t ca t total cot fucto ca b tafod to t o a obtad ca II of Modl I ad c follow t a oluto pocdu t but u cotat tad of cotat 7. Ca III W > ad z Sub-ca W T coffct of ca b ow potv. T dot t coffct of / ad b A ad B pctvl t al ca b v a A / B. T fabl ca b foud b atf t cotat H H w H t axu of t t ad d of 9 ad H t u of t t ad d of 4 ad. A Sub-ca of Ca III of Modl I t al fo ow ad v b A / B A ; B T fabl ca b foud b atf t follow cotat obtad fo cotat ad 75 v

7 v / Max M Gv ad t al v b A.5 3 T d to atf t cotat blow foud fo 4 ad. v Max M Eac of t poc of dt t fabl z ad v t a a ub-ca of Ca III of Modl I but qu t u of t foula fo ad t fabl a dvd t ubcto. Sub-ca W Gv atf 8 t poc of dt t fabl z ad a pctvl t a a ub-ca of Ca III of Modl I but u t objctv fucto ad t fabl a dvlopd. Tu fo a v t of valu of t paat of a pobl t al total cot oluto b cod ac of t ca of t Modl I ad Modl II ca b foud. T lat of t aocatd wt t valu of t vaabl t fal oluto. 3. Nucal Illutato Altou Hoqu b obtad t a al total cot oluto fo t of t fou l-vdo lbu ucal pobl oall foud b Hll 999 ad Hll ad Oa 6 t ld to a al total cot oluto fo t a o oall olvd b Hll 999. T ucal pobl a olvd b t tod dvlopd t pap ad ac of t oluto foud to b t a a foud oall. T ucal pobl of uppl a poduct b a vdo to fv bu oall olvd b Hoqu a b olvd u t oluto tod dvlopd t pap. T cut oluto tod povd t a lat al total cot oluto to t ucal pobl a obtad b Hoqu a. Aot l-vdo 5-bu ucal pobl oall olvd b Hoqu 8 ad t b Hoqu b alo olvd b appl t cut tod ad foud to povd t a oluto a obtad b t. T tod dvlopd t pap alo appld to olv t l vdo two-bu ucal pobl oall olvd b Zavalla ad Zao 9. T data fo t pobl a a v Tabl. Tabl : ata fo t l-vdo -bu pobl of Zavalla ad Zao 9 Fo vdo Bu T t t S P ad o lt o ac of L L t ad v. Copaatv al total cot oluto a v t Tabl. Tabl : Copaatv ult fo a l-vdo -bu pobl of Zavalla ad Zao 9 Mtod Modl & Lot Sz Batc Sz fo t vdo T. Cot Z & Z * 9 Jot Optu = = x.5+ 3x Hoqu a Modl II x Hoqu b Modl I x T Pap Modl I x * Z & Z dot Zavalla ad Zao 76

8 Cod t altc tuato t allt batc z to t bu a pt at ta o qual to o ad Hoqu b ad c t lad to t a al total cot oluto. But uc a cotat ad ot b pod Hoqu a ad c t ld to t lat al total cot oluto b dtbut ac batc of ut lad fo t vdo to t bu a follow. x5/5 = ut to bu ad x/5.666 ut to bu. Rcall tat Modl I of Hoqu b Modl II of Hoqu a ad Modl I of t pap a dvlopd b accuulat t vto at t bu wl Modl II of Hoqu b Modl I of Hoqu a ad Modl II of t pap a dvlopd b accuulat t vto at t vdo. Tu t cut oluto tod foud to lad to t lat al total cot oluto to all t ucal pobl tudd. T pvou l-vdo two-bu ucal pobl alo wt o addtoal paat valu a v Tabl 3 oall olvd b Hoqu b aa olvd b appl t tod dvlopd t pap. Copaatv oluto a v t Tabl 4: Tabl 3: Addtoal data fo t o vdo two-bu ucal pobl t t t = 3. = = 3.94 = 3.5 T L 5.6 a.6 5 = 5.6 ou =.9 da L. = 73 da v 8 Tabl 4: Gal oluto of t o-vdo two-bu ucal pobl Modl Lot Sz Batc z fo t vdo Total cot Hoqu b Modl I x Hoqu b Modl II x T pap Modl I T pap Modl II % ducto..86 T Modl II t pap a ducd t al total cot fo Hoqu b b.86% a aoabl ducto. T ultd bcau of flxbl tcto o t qual-zd batc t cut tod. T dffc of t t ad t lowt lad t affct t al total cot oluto. T dotatd b ca t t lad t of t pvou pobl to.3 a ad t olv t pobl aa. T oluto a v Tabl 5. Tabl 5: Soluto of t o-vdo two-bu ucal pobl wt L =.3 Modl Lot Sz Batc z fo t vdo Total cot T pap Modl I T pap Modl II Not tat ac of t odl lad to t low al total cot copao wt t copod cot Tabl 4 bcau of t cad t lad t. Tapot capact of a tapot vcl alo a a ffct o t al total cot oluto. To dotat t ffct t valu of t = 4 Tabl 6 cad to t = b p all ot paat valu t a ad t t olvd. T obtad oluto a v Tabl 6. Tabl 6: Soluto of t o-vdo two-bu ucal pobl wt t = Modl Lot Sz Batc z fo t vdo Total cot T pap Modl I x T pap Modl II T al total cot oluto Tabl 6 a to b dfft fo t copod oluto Tabl 4 dotat t ffct of t low tapot capact. Bu toa capact alo affct t al total cot oluto. To dotat t t l-vdo two-bu ucal pobl olvd b plac t valu of t ad Tabl 6 b ad 5 pctvl but p all ot paat valu t a. T oluto a v Tabl 7. 77

9 Tabl 7: Soluto of t o-vdo two-bu ucal pobl wt = Modl Lot Sz Batc z fo t vdo Total cot T pap Modl I T pap Modl II It ca b otd fo Tabl 6 ad Tabl 7 tat Modl I lad to t low al total cot oluto bcau of o tcto o t cod bu toa capact. 4. cuo ad Cocluo H w av xtdd two l-vdo ult-bu odl dvlopd b Hoqu b b copoat t oto of coz t poducto flow wt a flxbl tcto o t qual zd batc a Hoqu a. I t latt o two l-vdo ult-bu odl w dvlopd b taf t lot wt qualad/o -uqual zd batc but t qual-zd batc w pt l ta o qual to t lat uqual-zd batc ultpld b. I t odl ad ot pod a ot cotat. But Hoqu b ptd two l-vdo ult-bu odl b taf t lot wt qualad/o -uqual zd batc wc t qual-zd batc w pt to b qual to t lat uqual-zd batc. T allt batc to t bu w alo pt at ta o qual to o. Bd cotat o t capact of tapot vcl ad bu toa t allt ad t t lad t t up ad tapotato t w pod. Altou t odl Hoqu a a o alzd pct of d t batc to coz t poducto flow t av ot ta to accout t altc cotat. Wl t odl Hoqu b a o altc pct of cod altc cotat but t lac of coz t poducto flow popl. Bcau of t lac of cozato t fald to lad to t alzd al total cot oluto to a lvdo l-bu ucal pobl a obtad b Hll 999. So t odl Hoqu b av b xtdd b taf t lot wt batc a Hoqu a but ta t altc cotat t. I addto t capact cotat o t vdo toa pod. T a dfft coo pl oluto tod to t odl dvlopd. Follow t oluto tod ucal pobl a olvd ad copaatv tud a cad out. T pt tod foud to povd t a oluto to ac of fou l-vdo l-bu ucal pobl oall olvd b Hll 999 ad Hll ad Oa 6. Alo two lvdo fv-bu ucal pobl oall olvd b Hoqu 8 ad t b Hoqu a a olvd b appl t tod dvlopd. I bot t ca t cut tod lad to t lat al total cot oluto. T cut tod alo foud to v t lat al total cot oluto to t l-vdo twobu pobl of Zavalla ad Zao 9. T ucal pobl of Zavalla ad Zao 9 aa olvd b po t lt o t capact of t tapot vcl t vdo ad t bu toa t t lad t ad t allt lt o t allt lad t alo wt potv t up t ad tapotato cot ad t. T tod dvlopd t pap foud to lad to a total cot ducto of.86% copao wt t lat al total cot obtad b Hoqu b. T ultd bcau of t flxbl tcto o t qual-zd batc a Hoqu a tad of p t qual zd batc jut qual to t lat uqual zd batc a Hoqu b. I addto t pottal ffct of t cotat o t lad t ad o t capact of t tapot vcl ad bu toa a dotatd wt t oluto of ucal pobl. Tu t cut tod a t ablt to povd t alzd lat al total cot oluto to t lvdo ult-bu pobl wt a flxbl wa of tct t qual-zd batc but cod lt o t lad t ad capact of t tapot vcl ad t vdo ad t bu toa alo wt t up ad tapotato t ad tapotato cot. I t ac dad of t bu ad lad t of dlv batc to t a aud to b dttc cotat. Howv pactc t dad a va bcau of t fluc of vaou facto. Lad t of dlv batc to t bu alo a va du to vaato t t of pcto load tapotato ad uload. T vaato t dad ad t lad t a ult xta vto o ota of t cocd poduct to t bu. Hc fut ac o t topc t b cad out b cod vaabl 78

10 dad ad lad t of batc ad alo clud t ult xta vto ad ota cot. It a b potd b ac tat t tatd vto t ca lad to a ducd jot total cot. I t tuato o pat a ca xcd t ot pat lo ad c qu a aoabl wa of dtbuto of t poft. Gall dtbuto of t poft ao t cocd pat bad o otato utd b ac. Notl ac ca focu o fd a aoabl wa of dtbuto of t ad poft fo t tatd t to t cocd pat. Rfc B-aa M. Ha E. Haa M. ad Al uaa M. 3 Cot ad vdo aad vto l-vdo ultpl bu uppl ca Itatoal Joual of Poducto Rac Gloc C.H. T jot cooc lot z odl: A vw Itatoal Joual of Poducto Ecooc Hll R. M. 999 T optal poducto ad pt polc fo t l-vdo l-bu tatd poducto-vto pobl Itatoal Joual of Poducto Rac Hll R. M. ad Oa M. 6 Aot loo at t l-vdo l-bu tatd poducto-vto pobl Itatoal Joual of Poducto Rac Hoqu M. A. 8 Scozato t l-aufactu ult-bu tatd vto uppl ca Euopa Joual of Opatoal Rac Hoqu M. A. a Galzd l-vdo ult-bu tatd vto uppl ca odl wt a btt cozato Itatoal Joual of Poducto Ecooc Hoqu M. A. b A optal oluto tcqu to t l-vdo ult-bu tatd vto uppl ca b copoat o altc facto Euopa Joual of Opatoal Rac K T. ad Goal S.K. Ecoocal poducto ad tapt polc fo coodat ultpl poducto t Itatoal Joual of St Scc Sa B. R ad pooo A. 9 Optal poducto pla ad pt cdul a uppl-ca t wt ultpl uppl ad ultpl bu Euopa Joual of Opatoal Rac Sva C. Rao ad C.S.P. Optzato of uppl ca fo l-vdo ultbu cot toc polc wt tc alot Itatoal Joual of Advacd Maufactu Tcolo Zavalla L. ad Zao S. 9 A o-vdo ult-bu tatd poducto-vto odl: T Cot Stoc ca Itatoal Joual of Poducto Ecooc Zavalla L. ad Zao S. Eatu to 'A o-vdo ult-bu tatd poducto-vto odl: t ''Cot Stoc'' ca' Itatoal Joual of Poducto Ecooc 5-3. Boap Moaad Abdul Hoqu cutl a Aocat Pofo t patt of Matatc at Uvt Bu auala. H obtad B. Sc. Ho. ad M. Sc. Matatc fo aa Uvt Balad. T wa awadd t d of M. Sc. ad P.. Maat Scc fo Lacat Uvt Elad. H wa a Pofo t patt of Matatc at Jaaaa Uvt Sava aa Balad. H alo wod a a cotact taff at Uvt Sa Malaa. H ajo ac aa Poducto ad Ivto Cotol. H ac pap av appad Euopa Joual of Opatoal Rac Joual of t Opatoal Rac Soct Itatoal Joual of Poducto Ecooc Itatoal Joual of Poducto Rac Coput ad Opato Rac Itatoal Joual of St Scc tc. H a dtoal boad b of val Itatoal joual. 79