REVISTA INVESTIGACIÓN OPERACIONAL VOL., 32, NO. 2, , 2011

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1 REVISA IVESIGACIÓ OPERACIOAL VOL., 3, O., 93-6, A IEGRAED IVEORY POLICY WIH DEERIORAIO FOR A SIGLE VEDOR AD MULIPLE BUYERS I SUPPLY CHAI WHE DEMAD IS QUADRAIC ta H. Shah,Ajay S. Gor ad Chta Jhavr Dpartmt of Mathmatcs, Gujarat Uvrsty, Ahmdabad Pramukh Swam Scc ad H. D. Patl Arts Collg, Kad S. L. Isttut of Busss Admstrato, Ahmdabad-389 Gujarat, Ida ABSRAC I ths papr, a tgratd vtory polcy for dtroratg tm s dvlopd cosdrg sgl vdor ad mult buyrs supply cha systm, wh dmad of a product s crasg fucto wth tm. h modl assumd costat rat of dtrorato at ach lvl of supply cha. A umrcal xampl s show to valdat th rsults of th proposd dvlopmt. h sstvty aalyss s carrd out for crta modl paramtrs. It s obsrvd that th tgratd dcso for vtory polcy btw th playrs of supply cha lowrs th total jot cost compard wth a dpdt dcso mad by th supplr ad all buyrs. KEYWORDS: Supply cha, Sgl supplr ad mult buyrs, Itgratd vtory polcy, Dtrorato, Quadratc dmad MSC : 9B5 RESUME E st trabajo s cosdra ua polítca d vtaro tgrado d dtroro tdo cuta solo l vddor y los compradors múltpls l sstma d la cada d sumstro, cuado la dmada d u producto s crmta fucó dl tmpo. El modlo asum costat la vlocdad d dtroro cada vl d la cada d sumstro. U jmplo umérco s utlza para valdar los rsultados dl dsarrollo propusto. El aálss d ssbldad s ralza co los parámtros dl modlo dtrmado. S obsrva qu la dcsó d la polítca d vtaro tgrado tr los actors d la cada d sumstro rduc l costo total d la artculacó comparacó co ua dcsó dpdt ralzado por l provdor y los compradors.. IRODUCIO I a umbr of artcls avalabl th ltratur, svral vtory modlg problms hav b dscussd xtsvly. Most of ths rsarch artcls cosdrd th dffrt sub systm supply cha dpdtly ad drvd vtory polcy thr from vdor s or buyr s pot vws. But oft th vtory polcy drvd for o playr may ot b accptabl to th othr playrs of th supply cha. If th vtory polcy s drvd cooprato wth th all th playrs of th supply cha, th t ca b bfcal to all th playrs of th supply cha. Also, tgratd vtory systm mmzs th ovrall tgratd cost of th tr chal. Goyal(977 ad Brj(986 dvlopd a rsarch study to drv jot coomc lot sz polcy from a vdor to a buyr. h ma objctv of th study was to mmz th total rlvat costs for th vdor ad th buyr both. Goyal(988 gralzd Barj s modl by rlaxg th assumpto of th lot for lot polcy of th vdor. Goyal ad Gupta(989 ad Sarmah, Acharya ad Goyal(6 dscussd th mportac of coordato btw vdor ad buyr th supply cha maagmt. 95

2 A larg umbr of otcabl studs rlatd to dtroratg vtory systm ca b foud th ltratur. Dtrorato s dfd as dcay, dcomposto, vaporato of th product ad thrby loosg % utlty of t s ffccy. Blood compots, mdcal products, fruts ad vgtabls ar som of th xampls of dtroratg commodts. Ghar ad Schradr (963 frst cosdr dtrorato of a tm th vtory systm. Aftrwards, Raafat(99, Hg t al (99, Shah ad Shah( ad Goyal ad Gr ( dscussd dtroratg vtory systm. Yag ad W(5 drvd a w-w stratgy for a tgratd systm of vdor-buyr wh uts vtory ar subjct to dtrorat at a costat rat. I th most of th abov artcls thr costat or larly trdd dmad was assumd. h vtory modls drvd wth th assumpto of larly trdd dmad by Slvr ad Mal (969, Slvr (979, Xu ad Wag (99, Chug ad g (993, 994, Bos t al. (995, Harga (995, Gr ad Chaudhur (997, L t al. ( cosdr that dmad chags uformly ovr tm, whch s ot s th markt of som products lk lctroc commodts, blood, fasho goods tc. Mhta ad Shah (3, 4 cosdr th dmad to b xpotally tm varyg whch s uralstc for wly troducd product. I ordr to hav altratv dmad pattr, quadratc dmad s cosdrd. hs typ of dmad s partally costat, partally vars larly ad partally vars xpotally wth tm. It ca b s that most of th artcl dscussd abov vtory systm s drvd cosdrg sgl vdor ad sgl buyr. But practc, o vdor-o buyr stuato ca b s vry rarly. I fact, most of th cass structur of supply cha volvs sgl vdor ad mult buyrs or mult vdor ad mult buyrs. hr ar crta ssus d to b cosdrd for sgl vdor mult buyrs systm such as jot rplacmt of tms, producto vtory polcy tc. A tgratd sgl vdor ad multpl buyrs vtory modl was studd by Woo t al ( ad Yag ad w (.hs study wll addrss th ssu of jot polcy of a sgl vdor ad multpl buyrs vtory systm wh dmad of a tm s quadratc.. ASSUMPIOS AD OAIOS. Assumptos h modl s dvlopd udr th followg assumptos:. A supply cha systm of sgl vdor ad mult buyrs s cosdrd.. A sgl tm wth costat dtrorato rat s cosdrd. Dtrorato s proportoal to o had stock vtory. Dtroratd tms ca ot b rpard or rplacd durg a cycl tm. 3. h rplshmt rat s stataous. 4. Shortags at ay stag ar ot allowd. 5. Buyr ordrs th sam ordr quatty whvr ordr s placd. 6. It s assumd that tms ca b dtrorat oly aftr thy hav b rcvd to vtory. 7. h dmad of a tm s quadratcally crasg wth th tm.. otatos h proposd modl s drvd usg th followg otatos: R (t Θ I v (t I b (t umbr of buyrs Aual dmad rat of th th buyr (uts/ut tm = a ( +b t+ c t, whr a s costat dmad, b ad c ar lar ad xpotal rat of chag of dmad wth rspct to tm. Also, a>, a > > b, c ad < b, c <, whr =,.. h lgth of vdor s cycl tm (dcso varabl umbr. of dlvry pr ordr cycl tm for th th buyr (dcso varabl h dtrorato rat Ivtory lvl for vdor s at ay tm t, t Ivtory lvl for th buyr at ay tm t, t 96

3 I mv I mb C v C b A v A b I v I b C b C v C h maxmum vtory lvl of vdor h maxmum vtory lvl of th buyr Vdor s pr ut purchas cost ($/ut Buyr s pr ut purchas cost ($/ut Vdor s ordrg cost pr ordr cycl ($/cycl Buyr s ordrg cost pr ordr cycl ($/cycl Vdor s vtory carryg charg fracto pr ut pr tm ut ($/aum Ivtory carryg charg fracto pr ut pr tm ut for buyr ($/aum otal cost of all Buyr s pr tm ut otal cost of vdor pr tm ut otal cost for vdor-buyr vtory systm wh thy tak dcso jotly 3. MAHEMAICAL MODEL I ths scto, mathmatcal modl of a tgratd vtory systm for sgl vdor ad buyrs s dvlopd. I th supply cha systm, vdor obsrvd dffrt dmads from all buyrs durg a cycl tm, whch ar satsfd from stock avalabl th vtory. h objctv of ths rsarch study s to mmz total jot cost of sgl vdor ad all buyrs. Hr, vdor s total cost pr tm ut s a compost of ordrg cost, vtory holdg cost ad dtrorato cost. Fgur shows th stock trd for vdor vtory systm. Ivtory lvl I mv m Fgur : Vdor s Ivtory systm at ay tm t. From Fgur, vdor s vtory status ca b rprstd by th followg dffrtal quato. div(t + θ I v (t = - a ( bt ct, t ( dt Durg ay cycl tm, ach buyr plac dffrt ordr quatty ad rcvd rplshmt quatty ovr a trval of tm. Hc, for dffrt buyrs supply cha ordr quatty Imb ad th umbr of dlvry ar dffrt. I a gv cycl, th vtory status for th buyr s dpctd fgur. 97

4 Ivtory lvl For vdor I mv For buyr I mb / / m Fgur : Vdor-Buyrs Ivtory systm at ay tm t From th Fgur, vtory systm for th buyr ca b xprssd th mathmatcal form usg th followg dffrtal quato dib(t dt + θ I b (t = -a ( + b t + c t, t Usg varous boudary codtos I v ( =, I b ( b = ad b = dffrtal quatos ar I v (t a ( t ( b, t (3 I b (t a ( t ( c b c ( t ( (b t (b c c c c (b (b whr =, ( c t, th solutos of th abov c t c c ( ( b t b t c t c t, t whr =, (4 Usg I v ( = I mv ad I b ( = I mb, th purchas quatts for th vdor ad th th buyr ar ( t ( t I mv = c c a ( b c (b c (b c t ( b t (5 c t 98

5 Imb a ( b c (b c c (b c whr =, (6 Durg th cycl tm [, ], th lvl for all buyrs pr tm ut s buyrs s s th buyr s vtory lvl s I b (t dt IHC b = C b I b I (t dt b h ordrg cost for th th buyr s A b. Hc, OC b = 99 I b (t dt. Hc, th avrag vtory. hrfor, th vtory holdg cost for all (7 th ordrg cost for all buyrs pr tm ut A b. (8 R( Durg cycl tm, th umbr of uts dtroratd for th th buyr s (I mb -. hrfor, dtrorato cost for all buyrs s R( C DC b = b (Imb -. (9 Hc, th buyr s total cost, C b pr tm ut s C b = IHC b + OC b + DC b ( Durg th cycl tm, th vdor s avrag vtory lvl pr tm ut s I (t dt. h vdor s vtory th jot two-chlo vtory modl s th dffrc btw th vdor-buyrs combd vtory ad all buyr s vtory. hrfor, vdor s holdg cost pr tm ut s IHC v = C v I v [ I v (t dt th vdor s ordrg cost pr tm ut s OC v = A v. I b (t dt h uts dtroratd at th vdor s vtory systm s ( I mv - dtrorato of uts s DC v = Cv ( I mv - I mb v ] ( I mb. Hc, cost du to (. (3

6 h vdor s total cost C v pr tm ut s C v = IHC v + OC v + DC v (4 h jot total cost; C s th sum of C b ad C v. C = C b + C v. (5 Hr, jot total cost C s th fucto of dscrt varabl ad cotuous varabl, whr =,.. 4. COMPUAIO PROCEDURE Hr, th objctv s to dtrm th valu of, whch mmzs th jot total cost C, whr =,,,. Sc th umbr of dlvry pr ordr cycl s a dscrt varabl, th followg stps ca b carrd out to dtrm valu of. ( For dffrt valus of, dffrtat th total cost fucto C from (5 wth rspct to dcso C varabl ad st t qual to zro.. =. For ach, dot ordr cycl by otato (, whr =,,,. ( Fd th optmal soluto of ad such that, th followg codto must satsfy: C ( -, ( - C (, ( C ( +, ( + (6 Hr, (,,,, ad (,,,, 3 5. UMERICAL EXAMPLE AD SESIIVIY AALYSIS 3 produc th optmal soluto. o llustrat proposd modl th smplst mar, th supply cha systm wth sgl vdor ad two buyrs s assumd. For umrcal aalyss, followg paramtr valus ar cosdrd propr uts: =o of buyrs = a = Aual rat of costat dmad of st buyr =8, a = Aual rat of costat dmad of d buyr =9, b = Lar rat of chag of dmad of st buyr =.5 b = Lar rat of chag of dmad of d buyr =.5 c = Expotal rat of chag of dmad of st buyr =. c = Expotal rat of chag of dmad of d buyr =. C v =, C b = 3, I v =.5, I b =.3, A v =, A b =, θ =. I tabl ad tabl, th optmal soluto s xhbtd for dpdt ad tgratd dcso. If both th buyrs follow dpdt polcy th ordrg polcy s ( =3, =3 wth total cost of $ 647. If both th buyrs agr to jo th tgratd systm wth ordrg polcy of ( =, = th th total tgratd cost s sgfcatly rducd to $ 676. h graph of th total cost for dpdt ad tgratd vtory polcy s show Fgur 3. h buyr s cost cras wh both th buyrs ad th vdor agr for makg a jot dcso. I th tgratd polcy, vdor bfts $376 ad buyr looss $5. Sc, tgratd stratgy s bfcal to vdor, buyrs do ot agr for jot dcso. o courag ad attract th buyrs to cooprat th tgratd systm, vdor should offr th buyr a prmssbl dlay paymt or som proporto of sharg of xtra bfts. hs tgratd polcy C(, - C(, rducs th tgratd total cost dfd as = by 3.59 %. C(,

7 abl : Optmal soluto of ad BC VC C # 3# ot that dcats th tgratd optmal soluto of ad whch mmzs C ad # dcats th buyr s optmal soluto of ad whch mmzs BC abl : Comparso of Idpdt ad Itgratd Polcy. Idpdt = 3, = 3 Itgratd =, = Buyrs Ordrg cost Holdg Cost otal cost of Buyrs Vdor Ordrg cost Holdg Cost otal cost of Vdor otal cost of Systm Chag Cost Hc, th adopto of th tgratd polcy, stad of th dpdt polcy s sgfcatly dcrasg th chal cost. h rsults obtad th umrcal aalyss dcat to prform sstvty aalyss for th modl paramtr. Its am s to dtfy th paramtrs that ar mor rlvat to th prformac of th systm. h sstvty aalyss s carrd out by crasg or dcrasg modl paramtrs by %. h valu s C(, - C(, obtad for chag th modl paramtr. Hr, = whr C(,

8 Idpdt Polcy Itgratd Polcy Fgur 3: otal cost for tgratd Vs dpdt Vdor-Buyrs Ivtory systm C(, rprsts th optmal valu of th total tgratd cost C ad ad ar th optmal valus of ad rprst optmal umbr of dlvry for both th buyrs. h rsults of sstvty aalyss for a costat rat of th dmad; a, th lar rat of chag of th dmad; b, th xpotal rat of chag of th dmad; c ut cost; C v ad C b, ordrg cost; A v ad A b, vtory holdg cost; I v ad I b ad dtrorato rat; θ, s gv abls 3 to 9. From th abls 3 to 9, t s obsrvd that th rag of s from.47 to h valu of s vry sstv to all th modl paramtrs a, b, c,c v ad C b, A v ad A b, I v ad I b ad θ,whr as (, ar ot sstv to all th paramtrs. From abl 3, th cras th fxd dmad; a chags th rag (3.69, abl 3: Sstvty aalyss for costat dmad a a a C C # (%

9 From tabl 4, t s obsrvd that cras lar rat of chag of th dmad; b, chags th valu of obsrv th rag.583 to 4.6. It s obsrvd that cras lar rat of chag of th dmad; b, crass th prctag of tgratd cost rducto ( vry sgfcatly. abl 4: Sstvty aalyss for rat of chag of lar dmad b b b C C # (% From tabl 5, t s obsrvd that chags th rag (3.389, 3.84, wh th xpotal rat of chag of d th dmad; c, s chagd. abl 5: Sstvty aalyss for rat of chag of xpotal dmad c c c C C # (% From tabl 6, t s obsrvd that wh th ut costs C v ad C b ar crasd, th valu dcrass th rag (3.3, abl 6: Sstvty aalyss for ut cost C v ad C b C v C b C C # (%

10 From tabl 7, th valu crass from 3.46 to 3.65 wth crass th ordrg costs A v ad A b. abl 7: Sstvty aalyss for ordrg cost A v ad A b A v A b C C # (% From tabl 8, wh th holdg cost s crasd, th valu bcoms smallr. h cras th holdg costs I v ad I b dcras th valu from 5.39 to.475. abl 8: Sstvty aalyss for I v ad I b I v I b C C # (% From tabl 9, t s obsrvd that chags th rag (3.57, 3.64, for chags th valu of dtrorato rat; θ. From th sstvty aalyss, w obsrv that th valu of s mor sstv to holdg costs I v ad I b ad th lar rat of chag of th dmad; b. abl 9: Sstvty aalyss for rat of dtrorato θ θ θ C C # (%

11 6. COCLUSIO I ths study, a mathmatcal modl for dtroratg tms s dvlopd to study a optmal tgratd stratgy of a sgl vdor-mult buyrs vtory systm wh th dmad s quadratc. A umrcal aalyss rvals that th jot polcy lowrs th total cost of a vtory systm, v though th total cost of all th buyrs crass sgfcatly. o tc th buyr for coopratg, a promotoal ctv trms of trad crdt should b offrd by vdor to th buyrs, whch also hlps to mata log trm cotract btw all playrs of supply cha. Ackowldgmts: hs rsarch work s supportd part by UGC projct schm # 47 3/9 (WRO. REFERECES: RECEIVED MAY REVISED JULY BAERJEE, A.(986: A jot coomc lot sz modl for purchasr ad vdor. Dcso Scc, 7, 9-3. BOSE, S., GOSWAMI, A. ad CHAUDHARI K. S.(995: A EOQ modl for dtroratg tms wth lar tm- dpdt dmad rat ad shortag udr flato ad tm dscoutg. Joural of th Opratoal Rsarch Socty, 46, CHUG, K. J. ad IG, P. S.,(993: A hurstc for rplshmt of dtroratg tms wth a lar trd dmad. Joural of th Opratoal Rsarch Socty, 44, CHUG, K. J. ad IG, P. S.,(994: O rplshmt schdul for dtroratg tms wth tm proportoal dmad. Producto Plag ad Cotrol, 5, GHARE, P.M., ad SCHRADER, S.F., 963. A modl for xpotally dcayg vtory. Joural of Idustral Egrg 4, GIRI, B. C., ad CHAUDHARI K. S. (997: Hurstc modl for dtroratg tms wth shortags tm-varyg dmad ad costs. Itratoal Joural of Systms Scc, 8, GOYAL, S.K.,(977: Dtrmato of optmal producto quatty for a two-stag producto systm. Opratoal Rsarch Quartrly 8, GOYAL, S. K., 988.A hurstc for rplshmt of trdd vtors cosdrg shortags. Joural of th Opratoal Rsarch Socty, 39, GOYAL, S.K.,ad GUPA, Y.P.(989: Itgratd vtory ad modls: th buyr vdor coordato, Europa Joural of Opratoal Rsarch, 4, GOYAL, S. K., ad GIRI, B. C.(:. Rct trd modlg of dtroratg vtory. Europa Joural of th Opratoal Rsarch Socty, 34, -6. HARIGA, M. (995: A EOQ modl for dtroratg tms wth shortags ad tm-varyg dmad. Joural of th Opratoal Rsarch Socty, 46, HEG, K.J., LABBA, J., ad LI, R.L.(99: A ordr-lvl lot-sz vtory modl for dtroratg tms wth ft rplshmt rat. Computrs & Idustral Egrg,

12 3 LI, C., A, B. ad LEE, W. C.(. A EOQ modl for dtroratg tms wth tm-varyg dmad ad shortags. Itratoal Joural of Systm Sccs, 3, MEHA,. J. ad SHAH,. H.( 3: A vtory modl for dtroratg tms wth xpotally crasg dmad shortags udr flato ad tm dscoutg. Ivsgacao Opracoal, 3, MEHA,. J. ad SHAH,. H.( 4. A vtory modl for dtroratg tms wth xpotally dcrasg dmad shortags udr flato ad tm dscoutg. Idustral Egrg Joural, 33, RAAFA, F.( 99: Survy of ltratur o cotuously dtroratg vtory modls. Joural of th Oprato Rsarch Socty, 4, SARMAH, S.P., ACHARYA, D. ad GOYAL, S.K.(6: Buyr vdor coordato modls supply cha maagmt. Europa Joural of Opratoal Rsarch, 75, 5. 8 SHAH, H. ad SHAH, Y. K., (_ Ltratur survy o vtory modls for dtroratg tms. Ecoomc Aals,. 44, SILVER, E. A. ad MEAL H. C. (969: A smpl modfcato of th EOQ for th cas of a varyg dmad rat. Producto of Ivtory Maagmt,, SILVER, E. A.(979: A smpl vtory rplshmt dcso rul for a lar trd dmad. Joural of th Opratoal Rsarch socty, 3, WOO, Y.Y., HSU, S.L., ad WU, S.( :. A tgratd vtory modl for a sgl vdor ad multpl buyrs wth ordrg cost rducto. Itratoal Joural of Producto Ecoomcs, 73, 3 5. XU, H. ad WAG, H. (99: A coomc ordrg polcy modl for dtroratg tms wth tmproportoal dmad. Europa Joural of Opratoal Rsarch, 4, YAG, P. C. ad WEE, H. M. (: A sgl-vdor mult-buyrs tgratd vtory polcy. Joural of th Chs Isttut of Idustral Egrg 8,-9 4 YAG, P. C. ad WEE, H. M. (5: A w-w stratgy for a tgratd vdor-buyr dtroratg vtory systm. Mathmatcal Modllg ad Aalyss,

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