An Integrated Inventory Model for Three-tier Supply Chain Systems

Transcription

1 Iteatoal Joual of Opeatos eseach Iteatoal Joual of Opeatos eseach Vol. 7, No., 5 00) A Itegated Ivetoy Model fo hee-te Sply Cha Systems Mg-Jog Yao Depatmet of aspotato echology ad Maagemet, Natoal Chao ug Uvesty, 00 Uvesty oad, 3000 Hschu, awa eceved Mach 00; Accepted May 00 Abstact---I ths pape, we cosde a tegated vetoy model wth oe vedo ad multple etales. I ths thee-te sply cha system, the vedo puchases aw mateal poduces to fshed poducts, ad delves fshed poducts to the etales. We assume that the poducto at the vedo) ad the epleshmet at all the etales) of the fshed poducts shae the commo cycle tme ), but the epleshmet cycle of aw mateal fo the vedo s a tege multple of. he focus of ths study s to solve the optmal commo cycle to mmze the aveage jot total costs AJC) fo the vedo the whole sply cha system. o solve ths poblem, we deve the expesso fo the AJC, ad aalyze the theoetcal popetes of the optmal AJC cuve. We show that the optmal AJC cuve s pece-wse covex wth espect to, ad the jucto pots o the optmal AJC cuve ca be easly located by a closed-fom fomula. By utlzg ou theoetcal popetes, we popose a effcet seach algothm fo solvg ths poblem. Ou adom expemets demostate that ou seach algothm effectvely obtas the optmal soluto, ad teestgly, the vedo could ga moe cost savg whe moe etales jo the thee-te sply cha system. Keywods Sply cha, commo epleshmet epoch, seach algothm, epleshmet. INODUCION ecetly, fomato techology stogly ehaces the vetoes coodato acoss the ete sply cha. I late 980 s, EDI systems mpove vedo-etale tegato ad esult bette steamle sply cha systems. A excellet example show Seh 989) s Lev Stauss, a appael vedo, who employs LevLk a EDI system) to lk wth ts vedos to shae the fomato o vetoy ad to quck espose to the customes demad chage. Udo 993), Gottad ad Bolsa 996) ad Lambet, Stock ad Ellam 998) all emphasze that vetoy fomato shag betwee vedos ad etales leads to successful cases of vetoy maagemet. he fomato shag sply cha systems leads a ted fo the eseaches to study the coopeato betwee the vedos ad the buyes. It has bee advocatg that collaboato s a mpotat way fo ceatg w-w elatoshps amog the membes sply chas. eseaches have bee addessg lots of effots o developg effcet stateges fo the vetoes coodato acoss the ete sply cha. eseach effots have bee addessed to coodate the vetoy polces of the membes the sply cha to educe the jot vetoy costs. Oe may efe to Baejee 986), Goyal 988), Das ad Goyal 99), Baejee ad Km 995) ad Goyal ad Gta 989) fo the tegated vetoy models. Also, Baejee ad Baejee 99) ad homas ad Gff 996) povde evews o the elated eseach woks o the tegated vetoy models. he oe waehouse mult-etale OWM) lot-szg poblem s oe of the most epesetatve vetoy models that tegate pates sply cha systems. he OWM coces wth the detemato of lot szes ad schedule of etales epleshed fom the cetal waehouse. May eseaches have bee addessg the Coespodg autho s emal: myaoe@gmal.com 83-73X Copyght 00 OSW

2 Yao: A Itegated Ivetoy Model fo hee-te Sply Cha Systems IJO Vol. 7, No., 5 00) effots to solve the optmal soluto fo the OWM. Schwaz 973) deved the ecessay codtos fo a optmal polcy ad some aalytcal bouds ude statoay-ested polcy. He also poposed a heustc that usually solves a ea-optmal soluto fo the OWM lot-szg poblem. Schwaz ad Schage 975) focused o solvg the optmal lot szes of a sgle poduct mult-echelo assembly systems ude statoay-ested polcy. Gaves ad Schwaz 977) vestgated the chaactestcs of optmal cotuous evew polces fo aboescet systems ude statoay-ested polcy. Maxwell ad Muckstadt 985) poposed a heustc fo complex mult-stage, mult-poduct systems ude statoay ested polcy. Lu ad Pose 994) solved the OWM lot-szg poblem ude so-called tege-ato polcy whch estcts each etale odes at a tege o ecpocal of a tege multple of the waehouse ode teval. Mtchell 987) exteded oudy s 985) esults to al backloggg ad toduce a class of polces, called ealy-tege-ato polces whch s dffeet fom the class of tege-ato polces by ot equg statoaty of odes placed by etales. Late, Aly ad Fedegue 990, 99), ad Hall 99) added the vehcle outg costs the OWM systems. We ote that the OWM closely elates to ths study. But t possesses some chaactestcs that make ts decso-makg sceao sgfcatly dffeet fom that ths study. Fst, the OWM does ot take the vetoy of aw mateal to accouts. Also, the waehouse does ot poduce fshed tems so that the capacty of the poducto faclty would ot be cosdeed the OWM. All of the papes evewed above ae fo the vetoy cotol poblems whee the vedo/dstbuto ad the buyes/etales play the key oles the two-echelo sply chas. Fo the ecet extesos o the two-echelo vetoy models, oe may efe to Che, Fedegue ad Zheg 00). O the othe had, some eseaches have bee devoted to vetoy models that clude aw mateal vetoy the fomulato. Most of these atcles cosde the tegato betwee the aw mateal equemet ad the poducto batch sze fo a sgle poduct. Oe may efe to Km ad Chada 987), oa, et al. 000) ad Sake ad Kha 999, 00) fo efeece. We ote that Sake ad Kha s 00) pape evews two delvey polces of aw mateal tegated poducto/vetoy system, vz., lot fo lot ad multple lot fo a lot polces. Hee, we adopt the multple lot fo a lot polcy to fomulate ou mathematcal model ths study. O the othe had, some eseaches take accout the aw mateal vetoy the ecoomc lot schedulg poblem ELSP), whch s closely elated to ou poblem ths study. Fo stace, Hwag ad Moo s 99) model cosdes the specal case wth oly two poducts, but the aw mateal s deteoatg. Gallego ad Joeja 994) fomulate the mathematcal model fo the ELSP ad cosde vaous ssues assocated wth the maagemet of aw mateals fo poducto. Sake ad Newto 00) poposes a geetc algothm to solve the ELSP wth aw mateal whch the poducto system has a lmted stoage space ad the taspotato fleet capacty s of kow capacty. Iteested eades may also efe to a excellet evew o ths categoy of poblems Goyal ad Deshmukh s 99) pape. We ote that these studes dd ot take accout the dstbuto aspect to the etales) the models. Yag ad Wee 003) popose a model that s vey smla to ous ths study. hey cosde a sply cha wth oe vedo ad multple buyes, ad the aw mateal ad the fshed poduct ae deteoatg. he key dffeece fom ou model s that they employ the cocept of JI lot-splttg fom aw mateal sply to poducto ad fom poducto to dstbuto the fomulato. ecetly, Muso ad oseblatt 00) fomulate a mathematcal model fo a thee-te sply cha system. I the study, thee s oe aw mateal sple, oe maufactue ad oly oe vedo the sply cha. he vedo makes the vetoy decso accodg to the EOQ ule. he, the maufactue detemes the tegated vetoy polcy fols ths assumpto. Iteestgly, they deve the soluto appoach by explog the optmalty stuctue of the optmal cost fucto wth espect to the odeg quatty fom the vedo. I ths pape, we exted Muso ad oseblatt s 00) study to aothe case whch thee s oe aw mateal sple, oe maufactue ad multple etales a thee-te sply cha system. Smla to Muso ad oseblatt s methodology, we deve theoetcal esults o the cuve of the optmal objectve fucto value ad popose a effectve seach algothm. We outle the est of ths pape as fols. I secto, we gve a bef toducto to the poblem fomulato ad peset the mathematcal model. Also, we deve some theoetcal aalyss o the optmalty stuctue of the mathematcal model. I secto 3, we popose ou seach algothm based o ou theoetcal esults. he, we show that ou seach algothm effectvely obtas the optmal soluto by adom expemets secto 4. Fally we addess ou cocludg emaks secto 5.. MAHEMAICAL MODEL AND HEOEICAL ANALYSIS I ths secto, we fst dscuss the sceao that the decso-make faces secto.. he, we peset the fomulato of the mathematcal model secto.. Also, we coduct full aalyss o the cuve of the optmal objectve fucto value sectos.3 to.5.

3 Yao: A Itegated Ivetoy Model fo hee-te Sply Cha Systems IJO Vol. 7, No., 5 00) 3. he decso-makg sceao I ths study, we cosde a tegated vetoy model fo a thee-te sply cha system. A sgle vedo puchases aw mateal, poduces to fshed poducts, ad delves fshed poducts to multple etales ths sply cha. We assume that the poducto at the vedo) ad the epleshmet at all the etales) of the fshed poducts shae the Commo epleshmet Epoch CE), whch s deoted by. We ote that CE has bee populaly used model fomulato fo devg coodato mechasm sply cha systems. Ad, the vedo may cosoldate seveal etales epleshmet odes ad save the ode pocessg costs by adoptg such a CE mechasm. Oe may efe to Vswaatha ad Ppla s 00) ad Msha s 003) papes fo moe dscusso o the mplemetato of CE mechasm. O the othe had, ode to save the odeg cost of + aw mateal, the vedo epleshes the aw mateal a tege multple of,.e., m whee m N. Futhemoe, we assume that aw mateal shotage s ot aled fo the vedo, ad o shotage s aled fo the etales. Also, the etales ae wllg to take the vedo s epleshmet stategy. We ote that such a decso-makg sceao apples to may of the sples etalg busess ad gocey. I these dustes, the etales gat the sellg chael to the vedo by povdg floo space o stoage acks the etales stoe ad assstg the vedo to sell the poducts to the customes. he etales pusue the poft by chagg the vedo fo the floo space ad eag the mak fom the sellg pce of the poducts. Also, the etales ofte authoze the vedos to eplesh the poducts at the wll but, usually wth a pe-specfed epleshmet quatty) such a case. Note that the CE mechasm ot oly s smply to mplemet, but also, t guaatee a feasble poducto schedule fo the vedo. heefoe, the CE mechasm could sgfcatly smplfy the poducto schedulg ad the logstcs the vedo s poducto system by usg a egula ad epettve epleshmet schedule fo each etale. Befoe pesetg ou mathematcal model, we defe the otato eeded late. Let a be the odeg cost pe aw mateals ode fo the vedo. he cayg cost pe ut of aw mateal pe ut tme s deoted as h. Deote the poducto ate of the vedo as P, whch s a kow costat. Let S be the set cost pe poducto u fo the vedo ad u be the usage ate of aw mateals fo poducg each ut of the fshed poduct. he cayg cost fo each ut of fshed poduct held pe ut tme s deoted as h f. he demad ate at etale, s deoted as D. Each ode fom etale cus fo a ode pocessg cost of a. Fo the vedo, the cayg cost fo the fshed poduct set to etale s h pe ut pe ut tme held. All of the paametes ae costats ad kow to the decso makes.. he objectve fucto I ths study, we fomulate ths tegated vetoy model fom the vedo s pot of vew. Ou focus s to get the optmal CE ad the optmal multple m to mmze the aveage jot total costs fo the vedo the whole sply cha system. he objectve fucto cludes thee categoes of cost tems: ) the vedo s aveage total costs fo the fshed poducts set to all the etales, ) the aveage total costs cued by the fshed poducts held by the vedo, ad 3) the etale s aveage puchasg ad vetoy holdg costs fo the aw mateal. hese cost tems ae deved as fols. We deote the aveage total costs fo the fshed poducts set to etale as C ). Notce that the vedo s the owe of the fshed poducts stoed at each etale. heefoe, the vedo takes the vetoy holdg costs fo the fshed poducts stoed at each etale. Sce we assume that the vedo delves the fshed poducts to etale afte a fxed cycle, C ) s gve by a C ) = + [ hd ].. ) We deote the aveage total costs cued by the fshed poducts held by the vedo as C f ). By Fgue, h f S+ D p. heefoe, oe may easly obseve that the total costs cued a epleshmet cycle s ) = C f ) s gve by S h f C f ) = + D... ) p = Next, we deve the expesso fo the vedo s aveage puchasg ad vetoy holdg costs fo the aw mateal whch s deoted as C m,). I ode to calculate the aveage vetoy holdg cost, we eed to evaluate the vetoy level wth a epleshmet cycle fo the aw mateal,.e., m. 3

4 Yao: A Itegated Ivetoy Model fo hee-te Sply Cha Systems IJO Vol. 7, No., 5 00) 4 Cost P me D = P Fgue. he vetoy level of the fshed poduct. By caefully obsevg Fgue, oe may fd that thee ae m tagles ad m ectagles the vetoy holdg aea fo the aw mateal. he total holdg costs cued a cycle of m ae gve by m m h D p u D j u D ) = = j= = Cost u D = D = P m ) u D = m me Fgue. he vetoy level of the aw mateals heefoe, oe shall have the expesso fo C m,) as fols. a uh C m, ) = + D p+ m ) D m = = By eqs. ), ) ad 4), we have the objectve fucto, deoted by Γm,), as fols... 4) Γ m, ) = C ) + C m, ) + C )... 5) f = heefoe, ou focus s to solve the poblem P) 6) as fols. a m Γ m, ) = S+ + a ℵ = m ) uh D m, m = P) ) uh D) hf D hd. p = = = whee Γm,) s the aveage jot total costs AJC) fo the vedo the whole thee-te sply cha system..3 Chaactezato of the optmalty stuctue I ode to vestgate the theoetcal popetes o the cuve of the optmal objectve fucto value, we fst solve the optmal value of m ) ad compute the optmal objectve fucto value fo each gve value of = o the -axs. hat s, m )= agm + { Γ m '), ') }.) Let Γ ) be the optmal aveage jot total costs m N fucto value wth espect to. I othe wods, ) { m ), ) + } may plot Γ ) usg a small-step as show Fgue 3. Γ Γ s a fucto of.) he, we Impotatly, Fgue 3 shows two teestg obsevatos o the Γ ) fucto: 4

5 Yao: A Itegated Ivetoy Model fo hee-te Sply Cha Systems IJO Vol. 7, No., 5 00). Γ ) s pece-wse covex wth espect to.. Let m L ad m, espectvely, ae the optmal multples of the left-sde ad ght-sde covex cuves wth egad to a jucto pot the plot of the Γ ) fucto. he, ml = m +. he fst obsevato motvates us to obta the optmal soluto fo Γ ) at ay gve value of by some close-fom calculato. he secod leads us to the dea to chage the optmal multple m at the jucto pot of two eghbog covex cuves the poposed algothm. I the folg dscusso, we wll have futhe aalyss o these two obsevatos ad wll fomally pove them as the theoetcal popetes o the Γ ) fucto. 5 Fgue 3. he cuve of the Γ ) fucto Befoe havg futhe dscussos o the fst obsevato, we peset a mpotat popety of the fucto Γ ) Lemma. Lemma Assume that a < b ad m a )= m b ). he, m )= m a ) fo all ', ). Poof:Please efe to Appedx A.. Next, we asset the optmalty stuctue of the fucto Γ ) Poposto. Poposto Γ ) s pece-wse covex wth espect to. Poof : By eq. 6), we may compute the secod devatve wth espect to ) fo Γ m, ) by Γ m, ) = a 3 m Γ m, ). heefoe, gve ay postve tege m, > 0 fo all > 0. So, we coclude that Γ m, ) s covex wth espect to. Let σ m) be the set of such that m )= m,.e., σ m) { m ) = m, > 0}... 7) By Lemma, σ m) must be a teval o the -axs. Also, dffeet values of m fom o-ovelappg tevals o the -axs. heefoe, Γ ) s pece-wse covex sce Γ m, ) s covex o ts spot set σ m)..4 Jucto pots Next, we toduce the jucto pots o the Γ ) cuve, whch s a pece-wse covex fucto wth espect to. We defe a jucto pot fo Γ ) as a patcula value of whee two cosecutve covex cuves Γ m, ) ad Γ m+, ) cocateate. hese jucto pots detemes at what value of whee oe should chage the value of m so as to obta the optmal value fo the Γ ) fucto. We fst deve a closed-fom fo the locato of the jucto pots. We defe the dffeece fucto m+, m, ) by a b 5

6 Yao: A Itegated Ivetoy Model fo hee-te Sply Cha Systems IJO Vol. 7, No., 5 00) 6 a m+, m, ) =Γ m+, ) Γ m, ) = + uh D mm ) ) + = We ote that m+, m, ) s the cost dffeece betwee usg m ad m+ as ts multple. Sce the fst devatve of the fucto m+, m, ) s always postve fo all > 0, m+, m, ) s a ceasg fucto wth espect to. Spose that the seach algothm poceeds fom a pe boud towad smalle values of, we evaluate m+, m, ) fom postve values, to zeo ad fally, to egatve values. Let w be the pot whee m+, m, ) eaches zeo. Assume that m s the optmal multple fo > w. hs scheme mples that oe should keep usg m utl t meets w. Fom the pot w owads, the value of Γ ) ca be mpoved by usg m+ as ts optmal multple. We ote that w s the pot whee two eghbog covex cuves Γ m+, ) ad Γ m, ) meet. Impotatly, such a jucto pot w ot oly povdes us wth the fomato o at what value of whee oe should chage the value of m so as to secue the optmal value fo the Γ ) fucto. By the atoale dscussed above, we deve a closed fom to locate the jucto pots by lettg m+, m, ) = 0 as fols. Jm ) = a [ mm + ) uh D ].... 9) = Note that Jm = ) dcates the locato of the th jucto pots. By 9), the folg equalty 0) holds Jv ) <... < Jm ) <... < J ) < J) ) whee v s a ukow) pe boud o the value of m. heoem s a mmedate esult fom 9) ad 0), ad t povdes stegthe foudato fo ou seach scheme. heoem Spose that m L ad m ae the optmal multples of the left-sde ad ght-sde covex cuves wth egad to a jucto pot w of the Γ ) fucto, the ml = m +. he folg coollay s also a by-poduct of 9) ad 0), ad t povdes a ease way to obta the optmal multple m ) fo ay gve > 0. Coollay Fo ay gve > 0, the optmal multple m ) s gve by m ) = + + 8a [ uh D ] = ) Poof: Please efe to Appedx A...5 Local optmum ecall that Γ ) s pece-wse covex wth espect to as show Poposto ). It s mpotat fo us to locate the local mma fo the Γ ) fucto sce they ae the caddates fo the optmal soluto. Let k be the local mmum caddate fo the Γ ) fucto gve m ) = k fo k N + whee a k = S+ + a k ) uh D uh D) hf D hd k + = = p ) = = = ad s deved fom solvg the equato by settg the fst devatve of Γ m, ) equal to zeo. k heefoe, oe may use the folg ule to check the exstece of a local mmum fo the Γ ) fucto: f ethe of the folg codtos holds, ) J) ad ) k [ Jk + ), Jk )), the k exsts as a local mmum fo the Γ ) fucto. 3. HE OPIMAL SEACH ALGOIHM I ths secto, we peset a seach scheme whch obtas the optmal soluto fo the poblem P) 6). Oe may efe to 6) fo the complcated expesso of Γm,). We ote that Γm,) s ot tval to solve sce t s actually a olea tege pogammg poblem. O the othe had, ecall that the Γ ) fucto s pece-wse covex wth espect to. Also, ou theoetcal esults o the jucto pots fo the Γ ) cuve ecouage us to solve the optmal soluto by seachg alog the -axs. o desg ou seach algothm, we eed to defe the seach age by settg a e ad a pe boud o the -axs, deoted by ad, espectvely. I ode to effcetly fd ad, we efe to Wldema, 6

7 Yao: A Itegated Ivetoy Model fo hee-te Sply Cha Systems IJO Vol. 7, No., 5 00) Fek ad Dekke s 997) appoach to obta the seach age fo the jot epleshmet poblem JP). We ote that they obta the seach age [, ] by solvg a elaxed veso of the JP the pape. Befoe ou devato, we fst sepaate the elaxed poblem fo Γm,) to two pats,.e., Γ m, ) = θ ) +Φ m, ) whee Φ m, ) collects all the tems cotag both the decso vaables m ad Γm,). he expessos fo θ ) ad Φ m, ) ae gve equatos 3) ad 4), espectvely. uh θ ) = D ) uh D ) + C + C = p f ) ) ) = = a Φ m, ) = + muh D,m N ) m = Smla to Wldema, Fek ad Dekke s appoach, we fst tasfom Γm,) fom a olea tege pogam to a covex pogammg poblem by elaxg the costat m N + by m, m. We deote the elaxed poblem fo Γm,) by Γ m, ). heefoe, we have Γ m, ) = θ ) +Φ m, ) ) whee Φ m, ) = { Φ m, ): m, m } ad Γ, ) m ae cotuous elaxatos fo Φ m, ) ad Γ m, ), espectvely. Let Φ ) = f { Φ m, ): m, m } ad Γ ) = f { Γ m, ): m, m }. he, by the defto of 5), we may lk the elatoshp betwee Φ ) ad Γ ) by eq. 6) as fols. Γ ) = θ ) +Φ )... 6) Iteestgly, oe may easly exploe the optmalty stuctue of Φ ) as fols. auh D,. f x = Φ ) = ) a + uh D,. f x = > whee x = a [ uh D ] =, ad x coespods to the local mmum fo the fucto Φ m =, ). By 7), oe has the secod-ode devatve fo Φ ) by d 0, f x. Φ ) =.... 8) 3 d a, f > x. heefoe, Φ ) s covex wth espect to fo >0 by 8). O the othe had, t ca be easly show that θ ) > 0, > 0,.e., θ ) s also covex wth espect to. hus, Γ ) s obvously a covex fucto. Deote the optmal soluto of Γ ) by. he folg poposto dcates the locato of Poposto Oe may locate by ', f θ x ) 0. =.... 9) ', f θ x ) < 0. whee uh h = S+ a hd + D uh D + D f ) ) ) = = p = = p =, ad uh h = S+ a + a hd + D + D Poof: Please efe to Appedx A.3. Deote f ) ) ) = = p = p =. as the optmal epleshmet cycle tme fo the fucto Γ ). I the folg poposto, we wll show that a e boud ad a pe boud o the seach age ca be obtaed by the values of whee the objectve value of Γ ) equals Γ ). Poposto 3 Let be the smallest ad be the lagest fo whch the fucto Γ ) s equal to Γ ). he. Poof: Sce the Γ ) fucto s stctly covex, we have the esult that.. Cosequetly, fo values of < the Γ ) s lage tha Γ ). Sce the Γ ) s a elaxato of Γ ), Γ ) >Γ ) fo <, s a e boud o. he poof fo that ca be doe smlaly. 7 7

8 Yao: A Itegated Ivetoy Model fo hee-te Sply Cha Systems IJO Vol. 7, No., 5 00) It s show how the bouds ad ae secued Fgue 4. We ote that the bouds ad ca be obtaed by ay le seach algothm e.g., bsecto seach, o quadatc ft seach, see Bazaa, et al., 993). Now, we ae eady to eucate the seach algothm. he seach algothm. Secue by Poposto. he, obta the bouds ad by some le seach algothm, e.g., bsecto seach.. Use 9) to get all the jucto pots [, ]. Stat the seach fom the lagest jucto pot, say Jk), by settg = Jk ), m = k ad Γ =.e., the optmal objectve value at = Jk )). he, move to the ext jucto pot, ad go to Step At each jucto pot Jk), do the folg tems: a) If all the jucto pots [, ] ae examed, go to Step 4; othewse, go to Step 3b). b) Compute Γ k, Jk )) ad check: f Γ k, Jk )) <Γ, the Γ =Γ k, Jk )), = J k) ad m = k. c) Check the exstece of a local mmum by Coollay. If the local mmum k exsts, the compute Γ k, k ) ad check: f Γ k, k ) <Γ, the Γ =Γ k, k ), = k ad m = k. 4. epot the optmal soluto: the optmal epleshmet cycle, the optmal multple m, ad the optmal objectve value Γ. Stop NUMEICAL EXPEIMENS Fgue 4. he seach age of [, ] the seach algothm I ths secto, we peset a umecal example to demostate the mplemetato of the poposed seach algothm. Also, we show that ou seach algothm effectvely obtas the optmal soluto by adom expemets. 4. A Demostatve Example Fst, we use a smple example wth oe vedo ad thee etales to demostate the seach pocess of the poposed algothm. he data set fo ths example s show able as fols. We summaze the seach pocess as fols.. By Poposto, we secue by = he, obta the bouds ad by some le seach algothm, e.g., bsecto seach. We have = 3.59 ad = 5.77 fo ths example. 8

9 Yao: A Itegated Ivetoy Model fo hee-te Sply Cha Systems IJO Vol. 7, No., 5 00). We locate the lagest jucto pot J)=4.330, ad compute the optmal objectve value at J) by Γ m =, J)) = $, Set = J), m = ad Γ =Γ, J)). Sce [ J), ), exsts as a local mmum. We evaluate Γ, ) =$,4.7 whch s less tha Γ. heefoe, we eplace the optmal soluto wth that at by settg =, ad Γ = Γ, ). 3. he, we move to the ext jucto pot J) =.5. Sce J) <, t s the last jucto pot that we eed to exame. Now, we locate,.e., the local mmum fo m=, by = Sce locates betwee ts eghbog jucto pots,.e., [ J ), J)), t exsts as a local mmum. We evaluate Γ, ) =$,4.5 whch s less tha Γ. heefoe, we eplace the optmal soluto wth that at by settg =, m =, ad Γ = Γ, ). 4. Sce J) s the last jucto pot that we eed to exame, we stop the seach algothm. We shall epot the optmal soluto by =, m =, ad Γ = Γ, ) =$,4.5. able. he data set of the demostatve example. etales Paametes 3 Odeg cost:a Holdg cost:h Demad ate aual):d Vedo aw mateal odeg cost:a 750 aw mateal holdg cost:h 0.0 Fshed tems holdg cost:h f 0.07 Set cost:s 300 Poducto ate:p 700 usage ate of aw mateals:u adom expemets I ths secto, we pefom some expemets usg adomly geeated examples to aalyze the chaactestcs of the optmal solutos fo dffeet poblem settgs. Fo example, we would lke to obseve how dffeet umbe of etales ad dffeet utlzato ates of the vedo s poducto system may affect the u tme ad soluto qualty of the poposed algothm. Fst, we dscuss how to adomly geeate the staces ou umecal expemets. able pesets the mea ad age,.e., the two ecessay paametes of ay ufom dstbuto, fo all of the paametes of the expemetal poblems. he, all the paametes fo a expemetal poblem ae ufomly geeated fom [ mea age, mea + age ]. Sce the poducto s a value-added pocess, we assume that the vetoy holdg cost ate fo the aw mateal shall be o lage tha that fo the fshed poduct,.e., h h. We dvde ou expemets to 6 settgs by combg dffeet umbe of etales ad dffeet levels of. Hee, we desgate fou levels of P = D at.,.,.3 ad.4 that coespod to utlzato ates of the vedo s poducto system at 0.9, 0.83, 0.77, ad 0.7, espectvely). We ote that P = D must be geate tha to guaatee a feasble poducto schedule fo the vedo. Also, the umbe of etales the sply cha systems s set to be 0, 5, 0 ad 5, espectvely. Fo each combato of P = D ad the umbe of etales, we adomly geeate 5 staces. Afte solvg each stace, we collect the u tme ad the eo estmate of the poposed algothm. We defe the f 9

10 Yao: A Itegated Ivetoy Model fo hee-te Sply Cha Systems IJO Vol. 7, No., 5 00) eo estmate by Γ Γ) Γ whee Γ, expessed 3), s a e boud o the optmal objectve fucto value of the poblem P) 6). Oe may efe to Appedx A.4 fo the devato of Γ.) o exame the effectveess of the poposed seach algothm, we fst evew the u tme data able 3. he aveage u tme of the 400 staces s aoud oly secods. Also, though the aveage u tme does cease wth the umbe of etales, ts gowth ate s ot sgfcat. heefoe, ou u tme data able 3 vefy that ou seach algothm effectvely obtas the optmal soluto fo the staces ou umecal expemets. 0 able. he ages fo the paametes used the adom expemets Mea age Odeg cost of the etales [0,600] Holdg cost of the etales [0,0.] Demad ate of the etales [500,500] aw mateals odeg cost of the vedo [0,00] aw mateals holdg cost of the vedo [0,0.07] Fshed tems holdg cost of the vedo h = max{ h } f Set cost of the vedo [0,000] Poducto ate of the vedo Deped o the sum of etales demad Usage ate aw mateals [0.6,.] of the vedo Next, we evew the soluto qualty of the poposed algothm by examg the data of the eo estmates. he aveage value of the eo estmates fo the 400 staces able 3 s aoud 3.565% that could be easoable fo most of the decso makes the eal wold. Accodg to ou expemets, the aveage eo estmate does ot sgfcatly vay wth the utlzato ate of the vedo s poducto system. But, teestgly, the aveage eo estmate deceases as the umbe of the etales ceases. It mples that the vedo could ga moe cost savg whe moe etales jo the thee-te sply cha system the sceao of ths study. able 3. he summay of the adom expemets 0 etales 5 etales 0 etales 5 etales P D = Eo u tme sec.) Eo u tme sec.) Eo u tme sec.) Eo u tme sec.). 5.00% % % % % % % % % % % % % % % % 0.7 Aveage 5.% % % %

11 Yao: A Itegated Ivetoy Model fo hee-te Sply Cha Systems IJO Vol. 7, No., 5 00) 5. CONCLDING EMAKS I ths pape, we cosde a tegated vetoy model fo a thee-te sply cha system wth oe vedo ad multple etales. I ths sply cha, the vedo puchases aw mateal, poduces to fshed poducts, ad delves fshed poducts to the etales. We assume that the poducto at the vedo) ad the epleshmet at all the etales) of the fshed poducts shae the commo cycle tme ), but the epleshmet cycle of aw mateal fo the vedo s a tege multple of,.e., m, m N +. he focus of ths study s to secue the optmal commo cycle ad the optmal multple m to mmze the aveage jot total costs fo the vedo the whole sply cha system. o appoach ths poblem, we deve the expesso fo the aveage jot total costs, ad aalyze the theoetcal popetes of the optmalty objectve fucto value wth espect to,.e., Γ ). We show that the Γ ) fucto s pece-wse covex wth espect to, ad the jucto pots o the Γ ) cuve ca be easly located by a closed-fom fomula. By utlzg ou theoetcal popetes, we popose a effcet seach algothm to solve ths poblem. Ou adom expemets demostate that ou seach algothm effectvely obtas the optmal soluto, ad teestgly, the vedo could ga moe cost savg whe moe etales jo the thee-te sply cha system. EFEENCE. Aly, S. ad Fedegue, A. 990). O Waehouse Multple etale Systems wth Vehcle outg Costs, Maagemet Scece, 36, Aly, S. ad Fedegue, A. 99). ejode to Commets O Oe-Waehouse Multple etale Systems wth Vehcle outg Costs, Maagemet Scece, 37, Baejee, A. 986). A jot ecoomc-lot-sze model fo puchase ad vedo, Decso sceces, 7, Baejee, A. ad Baejee, S. 99). Coodated, odeless vetoy epleshmet fo a sgle vedo ad multple buyes though electoc data techage, Iteatoal Joual of echology Maagemet,, Baejee, A. ad Km, S. 99). A tegated JI vetoy model, Iteatoal Joual of Opeatos & Poducto Maagemet, 7, Bazaa, M., Sheal, H. D. ad Shetty, C.M. 993). Nolea Pogammg: heoy ad Algothms, d ed. Joh Wley& Sos Ic. 7. Che, F. Fedegue, A. ad Zheg, Y. S. 00). Coodato mechasms fo a dstbuto system wth oe vedo ad multple vedos, Maagemet Scece, 47, Das, C. ad Goyal, S. K. 99). Ecoomc odeg polcy fo detemstc two-echelo dstbuto systems, Egeeg Costs ad Poducto Ecoomcs,, Gallego, G. ad Joeja, D. 994). Ecoomc lot schedulg poblem wth aw mateal cosdeatos, Opeatos eseach, 4, Gottad, G. ad Bolsa, E. 996). A ctcal pespectve o fomato techology maagemet: the case of electoc data techage, Iteatoal Joual of echology Maagemet,, Goyal, S. K. 988). A jot ecoomc-lot-sze model fo puchase ad vedo:a commet, Decso sceces, 9, Goyal, S. K. ad Deshmukh, S. G. 99). Itegated pocuemet - poducto systems: a evew, Euopea Joual of Opeatoal eseach, 6, Goyal, S. K. ad Gta, Y. P. 989). Itegated vetoy ad models: the buye-vedo coodato, Euopea Joual of Opeatoal eseach, 4, Gaves, S. C. ad Schwaz, L. B. 977). Sgle Cycle Cotuous evew Polces fo Aboes cet Poducto/Ivetoy Systems, Maagemet Scece, 3, Hall,.W. 99). Commets o Oe-Waehouse Multple etale Systems wth Vehcle outg Costs, Maagemet Scece, 37, Hwag, H. ad Moo, D.-H. 99). Poducto vetoy model fo poducg two-poducts at a sgle faclty wth deteoatg aw mateals, Computes & Idustal Egeeg, 0, Km, S.-H. ad Chada, J. 987). Itegated vetoy model fo a sgle poduct ad ts aw mateals, Iteatoal Joual of Poducto eseach, 5, Lambet, D. M., Stock, J. M., ad Ellam, L. M. 998). Fudametals of Logstcs Maagemet, McGaw-Hll, NY. 9. Lu, L. ad Pose, M. E. 994). Appoxmato Pocedues fo the Oe-Waehouse Mult-etale System, Maagemet Scece, 40, Maxwell, W. L. ad Muckstadt, J. A. 985). Establshg Cosstet ad ealstc eode Itevals

12 Yao: A Itegated Ivetoy Model fo hee-te Sply Cha Systems IJO Vol. 7, No., 5 00) Poducto- Dstbuto Systems, Opeatos eseach, 33, Msha, A. K. 003). Selectve dscout fo sple-buye coodato usg commo epleshmet epochs, Euopea Joual of Opeatoal eseach, 53, Mtchell, J. S. B. 987). 98%-Effectve Lot-Szg fo Oe-Waehouse, Mult-etale Ivetoy Systems wth Backloggg, Opeatos eseach, 35, Muso, C. L. ad oseblatt, M. J. 00). Coodatg a thee-level sply cha wth quatty dscout, IIE asacto, 33, oa, J., Gog, L. ad ag, K. 000). Jot detemato of pocess mea, poducto u sze ad mateal ode quatty fo a cotae-fllg pocess, Iteatoal Joual of Poducto Ecoomcs, 63, oudy,. O. 985). 98%-Effectve Itege-ato Lot-Szg fo Oe-Waehouse Mult-etale Systems, Maagemet Scece, 3, Sake,. A. ad Kha, L.. 999). Optmal batch sze fo a poducto system opeatg ude peodc delvey polcy, Computes ad Idustal Egeeg, 37, Sake,. A. ad Kha, L.. 00). A optmal batch sze ude a peodc delvey polcy, Iteatoal Joual of Systems Scece, 3, Sake,. ad Newto, C. 00). A geetc algothm fo solvg ecoomc lot sze schedulg poblem, Computes ad Idustal Egeeg, 4, Schwaz, L. B. 973). A Smple Cotuous evew Detemstc Oe-Waehouse N-etale Ivetoy Poblem, Maagemet Scece, 9, Schwaz, L. B. ad Schage, L. 975). Optmal ad System Myopc Polces fo Mult-Echelo Poducto /Ivetoy Assembly Systems, Maagemet Scece,, Seh, B. 989). Commucatos ovatos: Lev Stauss stegthes custome tes wth electoc data techage, Compute Wold, Jauay, homas, D. J. ad Gff, Y. P. 996). Coodated sply cha maagemet, Euopea Joual of Opeatoal eseach, 94, Udo, G. J. 993). he mpact of telecommucatos o vetoy maagemet, Poducto ad Ivetoy Maagemet Joual, 34, Vswaatha, S. ad Ppla,. 00). Coodatg sply cha vetoes though commo epleshmet epochs, Euopea Joual of Opeatoal eseach, 9, Wldema. E., Fek, J. B. G. ad Dekke,. 997). A effcet optmal soluto method fo the jot epleshmet poblem, Euopea Joual of Opeatoal eseach, 99, Yag, P.C. ad Wee, H.M. 003). A tegated mult-lot-sze poducto vetoy model fo deteoatg tem, Computes ad Opeatos eseach, 30, APPENDIX A. he poof fo Lemma Poof: Let m < m whee m ad m ae postve teges. We defe a fucto m, m, ) by m m ) a m m ) m m m m uh D. he, by,, ) =Γ, ) Γ, ) = mm = m m) a m, m, ) 0 3 mm = > fo all >0), we asset that m, m, ) s a covex fucto wth espect to. heefoe, Case ) Fgue 5 eve exsts sce t cotadcts ou asseto that m, m, ) s a covex fucto. Also, oe may locate the tesecto pot of Γ m, ) ad Γ m, ) at m m = a mmuh D > 0 by settg m, m, ) = 0 as show Case ) of Fgue 5., ) t = Next, we wll pove Lemma by cotadcto. Assume that thee exsts a value of ', ) such that ') ). So, m m b ') ), ad m m < <. hat s, we would have the folg m m a t '), a)) ' esult, Γ m '), b) <Γ m b), b) =Γ m a), b), whch cotadcts ou assumpto that the optmal multple at b s m b )= m a ). b a b

13 Yao: A Itegated Ivetoy Model fo hee-te Sply Cha Systems IJO Vol. 7, No., 5 00) 3 Γ m, ) Γ m, ) Γ m, ) Γ m, ) a t b Fgue 5. he two possble cases the m, m, ) fucto A. he poof fo Coollay Poof: By the defto of jucto pot Jm), m ) = m whe Jm ) < Jm ). By the closed fom fo the jucto pot, we have a mm + ) uh D.0) = ad < a mm ) h D.. ) = Oe may smplfy 0) by m + m a uh D 0. Sce m must be a postve tege, we have = m + + 8a uh D ) = Smlaly, oe may smplfy ) ad get 3) as fols. m < + + 8a uh D.....3) = By combg ) ad 3), we have + + 8a uh D m < + + 8a uh D.... 4) = = Sce the dffeece betwee the two equaltes 4) s equal to, t mples that a tege exsts betwee Jm) ad Jm-) o both sdes of 4) ae both teges. I ethe case, takg the pe-ete of the expesso the ght-had sde satsfes ). A.3 he poof fo Poposto ' Poof: I Poposto, we use the value of x ) θ to dchotomze to two possble cases as fols. ecall that Γ ) = θ ) +Φ ) whee θ ) ad Φ ) ae expessed Secto 3. ' Case ): x ) 0 d θ. By the defto of Φ ), Φ ) > 0 fo ' θ ' ) ad 6), t s obvous that Γ ) 0 fo must exst fo Γ ) whe x. ecall that asset that the mmum value of Γ ) must exst at d > x whe ) 0 = uh h = S+ a hd + D uh D + D > x. Also, by the covexty of θ ' x. heefoe, the local mmum Φ ) = auh D whe f ) ) ) = = p = = p = ' Case : x ) 0 Case ) Case ). d θ <. By the defto of Φ ), Φ ) = 0 fo d x. Hece, we may x. By the covexty of θ ) ' ' ad 6), Γ ) < 0 fo x whe θ x ) < 0. heefoe, the local mmum must exst fo Γ ) a whe > x. ecall that g ) = + uh D = whe > x. Hece, we may asset that the mmum value of Γ ) must exst at 3

14 Yao: A Itegated Ivetoy Model fo hee-te Sply Cha Systems IJO Vol. 7, No., 5 00) 4 uh h = S+ a + a hd + D + D f ) ) ) = = p = p =. A.4 he devato of the e boud o the objectve fucto value I ode to deve the e boud o the objectve fucto value, we sepaate t to thee pats ad deve a e boud fo each pat coespodgly.. A e boud o the aveage total costs fo the buyes: We may expess the aveage total costs fo the buyes by C ) [ ] = a + hd =. Obvously, the EOQ fomula.e., = a hd) ) povdes a easy e boud o the total costs fo each buye. heefoe, we have a e boud o the aveage total costs fo the buyes by C ) = ahd = = ). A e boud o the aveage total costs fo the vedo: By ), we have the aveage total costs fo the vedo expessed by C ) S h p) D = +. We may f f = solve ts optmal value of by takg ts fst devatve wth espect to, ad settg t equal to zeo by = ps h D. heefoe, we have a e boud o the aveage total costs fo the vedo by f f = C ) = psh D. 6) f f f = 3. A e boud o the aveage total costs fo the aw mateal: ecall that we expess the aveage total costs fo the aw mateal as fols. a uh C m, ) = + D + m ) uh D m p =.... 7) = I ode to utlze ou aalyss, we collect the tems C m, ) wth the vaable m the fucto a Φ m, ) = + muh D, as we dd 4). Also, we defe a ew fucto ψ ) fo those tems m = C m, ) wthout the vaable m as fols. uh = = uh ψ ) = D D p 8) So, we have C m, ) =Φ m, ) + ψ ). o mmze the objectve fucto value of Φ m, ), we would have the optmal soluto by x = a uh D. = Let us defe a ew fucto ) ω by ) { m, ): m } ω = Φ. Obvously, the objectve fucto value of Φ m, ) s bouded fom be by ω ) sce ω ) s a cotuous elaxato of Φ m, ). Futhemoe, we have the exact expesso fo the fucto ω ) as fols. auh D, f < x. = ) Φ, ), f x. heefoe, we have a e boud o the aveage total costs fo the aw mateal by C ) = ω ) + ψ ) ) By the expesso of ω ), we ote that thee ae two possble cases to obta the mmze fo ) ω = C m, ), amely, ethe Case : < x : < x o x. Now, we have futhe aalyss fo these two cases as fols. By 5), we kow that ' ω ) = 0 Case. Also, we have must hold that ψ ' ) = D p uh D p < 0 = = sce t D < p fo the feasblty of the poducto capacty utlzato. heefoe, we coclude that = C ' ) = ω' ) + ψ ' ) <0. Impotatly, t mples that thee exst o optmal soluto Case. 4

15 Yao: A Itegated Ivetoy Model fo hee-te Sply Cha Systems IJO Vol. 7, No., 5 00) 5 Case : x : uh uh a uh C D D D p I Case, we have ). We may easly solve the ) = + + = = = optmal soluto by pluggg the mmze ) = a p uh = D the fucto C ). Summazg ou dscusso above, we have a e boud fo the aveage total costs the whole sply cha system, deoted as Γ, by Γ = = C ) + C ) + C )... 3) f f 5