Initial inventory levels for a book publishing firm

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1 Mőhlytaulmáy Vállalatgazdaságta Itézt 93 Budapst, Fıvám tér 8. (+36 ) , Fax: Ital vtry lvls fr a b publshg frm Imr Dbs Ágs Wmmr 23. sz. Mőhlytaulmáy HU ISSN márcus Budapst Crvus Egytm Vállalatgazdaságta Itézt Fıvám tér 8. H-93 Budapst Hugary

2 Ital vtry lvls fr a b publshg frm Imr Dbs Ágs Wmmr Dpartmt f Lgstcs ad Supply Cha Maagmt Isttut f Busss Ecmcs Crvus Uvrsty f Budapst Budapst Crvus Egytm, 93 Budapst, Fıvám tér 8. Absztrat. Egy öyvadó vállalatt vzsgálu. A adó adváyat a száss értésítés lác (s- és agyrsdlm) rsztül értésít. A érdés az, hgy gy ú öyv példáyat hgya allála az értésítés lácba. Fltétlzzü, hgy a rslt smrt, Psslszlású. A észltzés öltségt szté smrt tétlzzü fl. Cél a öltség mmalzálása. Kulcsszava: Optmalzálás, Úságárusfú prbléma, Játélmélt, Készltgazdáldás Abstract. Th am f th papr s t aalyz a practcal ral wrld prblm. A publshg hus s gv. Th publshg frm has ctacts t a umbr f whlsalr / rtalr trprss ad drct ctact t custmrs t satsfy th mart dmad. Th b publshrs wr a prct dustry. Th publshr facs wth th prblm t allcat th stcs f a gv, wly publshd b t th whlsalr ad rtalr, ad t hld sm cps t satsfy th custmrs drct frm th publshr. Th dstrbut f th dmad s uw, but t ca b stmatd. Th csts csst f vtry hldg ad shrtag, bacrdr csts. Th dcs mar wats t mmz ths rlvat csts. Th prblm ca b mdld as a -warhus ad N-rtalr supply cha wth t dtcal dmad dstrbut. Th prblm structur s smlar that f a wsvdr mdl. It s assumd that th dmad dstrbut fllws a Pss dstrbut. Kywrds: Optmzat, Nwsby prblm Gam thry, Ivtry ctrl 2

3 . Itrduct Ivtry gams ar vstgatd th last yars. A umbr f publcats aalyz wsvdr-, r wsby-typ vtry gams. O f th frst paprs ths fld was publshd by Lppma ad McCardl (997). Thy hav xamd a wsby-typ vtry gam wth tw playrs ad dpdt radm dmads. Thy l fr a Nash qulbrum ad vstgat pssbl dcs ruls. Othr paprs vstgat a smlar stuat, but mr tha tw playrs. Ths paprs ar Mtrucch ad Scars (27), ad Müllr, Scars, ad Shad (22). Th last papr dals wth th cpratv slut f th wsby-typ vtry gam, ad shws that thr xsts a cr f ths gam. Athr papr xams th cct btw Nash ad Staclbrg sluts f wsvdr prblms. (Sr (27)) Sh has shw that thr xsts such a stuat, whr ths tw typ f gam sluts hav th sam. Othr applcats xam th bac-rdrs sluts (Ntss t al. (26)), ad th mult-chl prblms f vtry ctrl f wsvdr gams. (Rgrs ad Tsubata (99)) Th am f th papr s t aalyz a practcal ral wrld prblm. Th publshg frm has ctacts t a umbr f whlsalr / rtalr trprss ad drct ctact t custmrs t satsfy th mart dmad. Th b publshrs wr a prct dustry. Th publshr facs wth th prblm t allcat th stcs f a gv, wly publshd b t th whlsalr ad rtalr, ad t hld sm cps t satsfy th custmrs drct frm th publshr. Th dstrbut f th dmad s uw, but t ca b stmatd. Th csts csst f vtry hldg ad shrtag, bacrdr csts. Th dcs mar wats t mmz ths rlvat csts. Th prblm s mdld as a -warhus ad N-rtalr supply cha wth t dtcal dmad dstrbut. Th prblm structur s smlar that f a wsvdr mdl. It s assumd that th dmad dstrbut fllws a Pss dstrbut. W hav chs ths dstrbut, bcaus th dmad prcss f sllg bs ca b charactrzd wth ths radm prcss. Th papr rgazs, as fllws. I th scd sct w shw th paramtrs, dcs varabls ad th csts fucts f th mdl. Th xt sct dals wth th prprts f th mdl, ad slvs th mdl. Th furth sct suppls a umrcal xampl t charactrz th structur f a ptmal slut. Ad last w summarz th rsults f th papr. 2. Th mdl W wll us th fllwg tats. Paramtrs f th mdl: - umbr f th rtalr, - c f fxd csts f th publshg hus, - c v varabl csts f th publshg hus, - p sllg prc f th b, 3

4 4 - h vtry hldg csts f th publshg hus, - h vtry hldg csts f th rtalrs, (,2,,), - D radm dmad fr rtalr, (,2,,), - paramtr f th Pss dstrbut fr rtalr, (,2,,). Th prbablty f dmad f th Pss dstrbut has th xt frm: D P ) (. Th cumulatd dstrbut s dfd, as ( ) F. W assum that th radm dmads f th rtalrs ar dpdt. Ths assumpt mas that th cumulatd prbablty f th dmad fr th publshg hus fllws a Pss dstrbut, as wll: D P!, ad th cumulatd dstrbut ( )! F. Dcs varabls: - th umbr f th bs publshd by th publshg hus, - th umbr f th rdrd bs frm th publshg hus. W ca assum that th umbr f publshd bs s t hghr tha that f bs rdrd by rtalrs frm publshg hus. It s frmulatd, as. Ths prblm ca b mdld as a -warhus ad -rtalr supply cha. Th matral flw f th systm s dpctd fgur.

5 Fgur. Matral flw f th systm. (O-warhus -rtalr.). rtalr Publshg hus 2. rtalr rtalr Lt us w mdl th cst fucts. Th mdl cssts f ( + ) csts fucts: cst fuct f th publshg hus ad cst fucts f th rtalrs. W mdl th cas frm th pt f vw f th publshg hus. Th publshg hus wats t mmz th rlvat cst, s that th hus ds t w th prcs csts f th rtalrs, but stmats thm. Frst w cstruct th cst fuct f th publshg hus. W assum that th csts fuct cssts f fxd csts, varabl csts f th publshg hus, ad th sum vtry hldg csts ad lst sals. Th frm f th cst fuct s th fllwg: C ( ) c + c K ( ). f v + Th fuct K ( ) s mdld, as K ( ) h ( ) + p ( ) +! +! Th frst part f fuct F ( ) shws th avrag vtry hldg csts. Th scd part s th avrag lst sals csts. Th cst fucts f th rtalrs ar mdld as a pur wsby prblm. W assum that th publshg hus has frmat abut th csts paramtrs f th rtalrs, ly. 5

6 stmats abut th csts. Th fxd ad varabl csts f rtalrs ar ttally uw fr th publshg hus. Th cst fucts ar C +! ( ) h ( ) + p ( )!, (,2,,). Th gal f th rtalrs s t mmz th wsby-typ vtry csts. A rtalr has frmat abut th rdrg ruls f th thr rtalrs, s ths dcss ar dpdt ach thr. W ca w summarz ur mdl, as a gam thrtc mdl. Th prblm s th fllwg: ( ) m C, (,,2,,), (). (2) I th xt sct w xam th ffrd mdl. W wll shw that ths mdl s a smpl gam ad w l fr th ptmal slut f ths play. 3. Sm prprts f th mdl ad ts slut Th prblm ()-(2) ca b trprtd, as a Nash gam wth ( + ) playr. W prv that th Nash slut f ths vtry gam s Part qulbrum, s th cmpttv slut s ths spcal cas a cpratv gam slut, as wll. Prprty. Th ptmal slut f mdl ()-(2) s a Nash qulbrum. Prf. Lt us assum that w hav calculatd th ptmal slut f th prblm, ad ptmal slut s valus { }. Lt us w frst aalyz th umbr f publshd bs, ad lt us fx th ptmal rtalr quatts { }. Th w slv th xt prblm: C ( ) c + c + K ( ) m f v,. Ths prblm ca b slvd, ad th ptmal slut s a Nash slut. If w vstgat a rtalr, th th xt prblm must b slvd: 6

7 7 ( ) ( ) ( ) m!! + + p h C,. Th slut f ths prblm s vry smpl, s fr xampl Naddr (966). Wth ths last prpst w hav prvd ths prprty. Th prprty hav shw that th qulbrum ths Nash gam ca b calculatd, as a Part ffct slut. Prprty 2. Th Nash qulbrum f mdl ()-(2) s a Part ptmal slut, as wll. Prf. Th frst prprty hlds fr th ptmal slut f prblm ()-(2). Lt us rfrmulat th prblm: ( ) ( ) m + C C,. Th ptmal slut f ths prblm s a Nash slut, ad th cumulatd cst fuct s a Part ptmum. S th ptmal slut f ths prblm s a cpratv slut f gam ()-(2). Wth ths prpst w hav prvd that th slut s a cpratv qulbrum fr ths prblm. Ths last prprty hlps us t dtrm th ptmal slut f th mdl ()-(2). Th ptmal rdrgs ca b calculatd, as th ptmal slut f ptmzat prblm (3)-(4): ( ) m C, (3). (4)

8 Lt us w dtrm th ptmal dcss ths mdl. Th mdl ()-(2) ca b slvd wth Lagrag fuct. Th Lagrag fuct s L r L ({ } ) C ( ) +,, ({ }, ) [ C ( ) ] + [ C ( ) + ]. Th ptmal slut fr ths cas s p v + ( ) F ( ) F p + h fr th umbr f th publshd bs ad F p ( ) F ( ), (5), (,2,,), (6) p + h fr th rdrd bs. Hr th ptmal sluts ar valus, (,,2,,). I ths mdl th ptmal slut s dtrmd dpdc f th paramtr. A pssbl algrthm s shw th xt sct. 4. A umrcal xampl Th supply cha cssts f th publshg hus ad thr rtalrs, 3. I ths umrcal xampl w assum that th sllg prc s qual t 5,.. p 5. Th fxd cst f th publshg hus s,.. c f, th varabl cst s 2,.. c v 2. W assum that th vtry hldg csts ar qual fr th publshg hus ad th rtalrs, ad h h h h 2 h 3. Th dstrbut f th dmad s Pss dstrbut wth paramtrs: 5, 2 25, ad 3 3. Th prblm (3)-(4) ur cas s ( ) + 5 ( ) + ( ) ( ) + ( ) + 5 ( ) 3 ( ) + 5 ( ) 3 5! 3! 5! ! 3! 3 5! ! ! 5 + 8

9 Usg th rsults f (5)-(6), th ptmal slut s, 9, 52, 26, Th mmal cst f th publshg hus s C ( ) , C ( ) , C ( ) 2. 46, ( ) Th ttal csts f ths supply cha ar C Ccluss I ths papr w hav shw a supply cha wth wsvdr-typ vtry hldg csts. W hav vstgatd a ral publshg hus wth rtalrs. Th publshg hus wats t dtrm ptmal umbr f publshd bs ad th xpctd bs rdrg f th rtalrs. Th radm dmads fr bs f th rtalrs ar w frm frmr dmads. Th am f th publshg hus s t mmz th xpctd csts s that th rtalrs mmz th vtry hldg csts. T slv th prblm, w hav shw that ths prblm lads t a gam-thrtc mdl. Th playrs f th gam ar th publshg hus ad th rtalrs. Th structur f th prblm s vry smpl, s th Nash qulbrum f th prblm s a Part ffct slut, as wll. It mas that ths cas th Nash qulbrum s a slut f a cpratv gam. I a xt papr w ca aalyz a stuat wth t dpdt rtalrs dmad, as t s assumd by Mtrucch ad Scars (27), r Sr (27). 9

10 Rfrcs.. Lppma, S.A., McCardl, K.E. (997): Th cmpttv wsby, Oprats Rsarch 45, Mlff, P., Nhéz, K. (26): A xtdd wsvdr mdl fr custmzd mass prduct, AMO Advacd Mdllg ad Optmzat, Elctrc Itratal Jural 3. Mlff, P., Nhéz, K. (27): Evaluatg th prpr srvc lvl a cpratv supply cha vrmt, MIM 7. IFAC wrshp maufacturg mdlg, maagmt ad ctrl, Budapst, Hugary, Mtrucch, L., Scars, M. (27): Larg wsvdr gams, Gams ad Ecmc Bhavr 58, Müllr, A., Scars, M., Shad, M. (22): Th rwsvdr gam has a mpty cr, Gams ad Ecmc Bhavr 38, Naddr, E. (966): Ivtry systms, Jh Wly Ss, Ic., Nw Yr, Ld, Sydy 7. Ntss, S., Rud, N., Wag, Y. (26): Ivtry cmptt ad ctvs t bac-rdr, IIE Trasacts 38, Rgrs, D.F., Tsubata, S. (99): Nwsby-typ rsults fr mult-chl vtry prblms: Bacrdrs ptmzat wth trmdat dlays, Jural f Opratal Rsarch Scty 42, Sr, Y. (27): Cmpttv wsvdr prblms wth th sam Nash ad Staclbrg sluts, Oprats Rsarch Lttrs 35, 83-94