A Closed-loop Supply Chain Equilibrium Model with Random Demands

Transcription

1 A Closed-loo Sly Cha Eqlb odel th Rado Deads Yoes Hadoh Deatet of Bsess Adstato Uted Aab Eates Uvesty Al A UAE Yoeshadoh@aeaae Tel: Agaha Salee The Hog og Polyteh Uvesty Deatet of Cvl ad Sttal Egeeg oloo Hog og Ceasal@olyedh Tel: POS d Aal Cofeee Reo evada USA Al 9 to ay 0

2 A Closed-loo Sly Cha Eqlb odel th Rado Deads Yoes Hadoh Agaha Salee Abstat ths ae e oose a e losed-loo sly ha eto eqlb odel osstg of a ateal sles afates etales oses ad eovey etes Ule othe stdes the extet lteate e asse that deads assoated th the dead aets ae ado We odel the otzg behavo of the dffeet deso-aes ad sho that the eqlb odtos goveg the losed-loo sly-ha eto eqlb oble a be folated as a vaatoal eqalty We dsss the exstee ad qeess of the solto to ths vaatoal eqalty ad establsh odtos de hh the oosed otatoal algoth s gaateed to ovege eyods: Closed-loo sly ha eto Eqlb Vaatoal eqaltes Rado deads todto Reet legslato has fosed atteto o the aageet of odts at the ed sefl lfe Fo exale seveal dstal otes Eoe have efoed evoet legslato hagg afates th the esosblty fo evese logsts flos Flesha et al 000 A othe exale s the eato the US of a Evoetal Poteto Agey EPA ad seveal evoetal atvst gos to ease ose aaeess of evoetal sses As a eslt oaes ae easgly beg aotable fo atos that advesely affet the evoet Davd et al 004

3 The above ases eqe afates to desg odts th eogh esable ateals eg ae glass o al ad the oses to et eylable odts to the eovey etes Ths ay lead to losed-loo sly has CLSC f the eoveed esable ateals of the ogal odts ae set to the afates the ogal sly ha Hao d ad Belles 007; Yag et al 009 ay qattatve stdes have addessed the aageet of evese flos ad losedloo sly has She et al 005 oosed a otzato-based odel to osde sed odt et ato tasfoato ate of a ateal ad oesodg sbsdes fo goveet ogazatos fo evese logsts obles of gee-sly ha aageet et al 006 develoed a xed-tege olea ogag odel fo evese logsts volvg both satal ad teoal osoldato of eted odts agey ad Toyasa 005 develoed a tegated faeo fo the odelg of evese sly ha aageet of eleto aste sg the vaatoal eqalty aoah Haod ad Belles 007 exaded the o of agey ad Toyasa 005 to eset a losed-loo sly ha eto osstg of afates ad ose aets Yag et al 009 develoed a losed-loo sly ha eto hh ldes a ateal sles afates etales oses ad eovey etes The athos sed the theoy of vaatoal eqalty to otze the obetves of the vaos deso aes ad to deve the eqlb odtos Seveal exales ee eseted to desbe the effets of aaetes et ato tasfoato ate of a ateals tasfoato ate of eylable odts o eqlb shets ad et evees ths ae e exted the o of Yag et al 009 to osde adoess at the level of dead aets Ths exteso s sgfat se ate e do ot o the

4 deads fo a odt th etaty bt ay evetheless ossess soe foato sh as the desty fto based o hstoal data ad/o foeasted data o stdy e asse that the dead fo the odt at eah aet follos a oal dstbto hh s deedet of the dstbtos fo all othe dead aets Ths des heet vaablty the odt shets ad osts assoated th odto tasato ad eylg Wth ths exteso of ado dead e deve the otalty odtos fo all losed-loo sly ha ebes ad establsh that the goveg eqlb odtos the ado ase satsfy a ftedesoal vaatoal eqalty We ote that Dog et al 004 oosed a sly ha eqlb odel th ado deads at the level of etales Ths ae s ogazed as follos: Seto e eset the losed-loo sly ha eto odel th ado deads osstg of ltle tes of deso-aes Seto 3 deso-aes' otzg behavos ae desbed sg the theoy of vaatoal eqalty Seto 4 ovdes the eqlb atte of the losed-loo sly ha th ado deads Seto 4 also dssses the exstee ad qeess of a eqlb solto Seto 5 e otle the solto algoth ad dsss ts ovegee Fally Seto 6 oldes the ae The losed-loo sly ha eto odel ths seto e eset the losed-loo sly ha eto odel th ado deads ad old be see as a exteso of the odel of Yag et al 009 to a ado settg The odel ossts of to gos of ha ebes: foad logsts ha ebes sho at the left sde of Fge ldg a ateal sles afates etales ad dead aets; Revese logsts ha ebes llstated at the ght sde of Fge ldg dead aets eovey etes ad afates afates ad 3

5 Ra ateal sles afates afates Retales CLSC eto Reovey etes Deads aets Deads aets Foad sly ha eto Revese sly ha eto Fge : The losed-loo sly ha eto dead aets ae the odes that obe the foad sly ha eto ad the evese sly ha eto togethe to establsh the losed-loo sly ha eto The geeal losed-loo sly ha CLSC eto stte s deted Fge odel asstos ad deftos Bas asstos ae ade th the odel: The a ateal sles oly sly a ateals to the afates the foad sly ha The afates the losed-loo sly ha eto ode a hoogeeos odt the ott of oe odt aot be dstgshed fo the othes Thee s also o eeved qalty deeato of odts ade by a ad esable ateals 3 The eovey etes oly ollet the eylable odt shets fo the dead aets the ogal sly ha They oly sly esable ateals to the afates ths sly ha 4

6 4 The dead fo the odt at eah aet follos a oal dstbto hh s deedet of the dstbtos fo all othe dead aets 5 All the ha ebes oete a oooeatve fasho de the ash eqlb aoah that s eah deso ae ll detee hs otal qatty ad shet gve the otal levels of the oettos Deftos of vaables ad aaetes the losed-loo sly ha eto ae sazed belo: : be of afates the CLSC eto = { } : be of etales the CLSC eto = { } : be of dead aets the CLSC eto = { } : be of eovey etes the CLSC eto = { } : be of a ateal sles the CLSC eto = { } q : oegatve a ateal shet fo sle to afate : ooto of the a ateal shet fo sle to afate fo dead aet = 0 = = Go the ooto fo all dead aets to the veto q : oegatve esable ateal shet fo eovey ete to afate : ooto of the esable ateal shet fo eovey ete to afate fo dead aet = 0 = Go the ooto fo all dead aets to the veto 5

7 q : oegatve odt shet fo afate to etale : ooto of the odt shet fo afate to etale fo dead aet = 0 = Go the ooto fo all dead aets to the veto q : oegatve odt shet fo etale to dead aet : ooto of the odt shet fo etale to dead aet = 0 q : oegatve eylable odt shet olleted by eovey ete at dead aet : ooto of the eylable odt shet olleted by eovey ete at dead aet = 0 : fato of sable ateal that a be holly tasfoed to the e odt oe t of a ateal fo afate 0 : fato of seless dsassebled ateal oe t of a ateal fo afate These seless ateals ae set to the ladfll = : fato of sable ateal that a be holly tasfoed to the e odt oe t of esable ateal fo afate 0 6

8 : fato of seless dsassebled ateal oe t of esable ateal fo afate These seless ateals ae set to the ladfll = : fato of sable eyled odt that a be holly tasfoed to esable ateal oe t of eyled odt fo eovey ete 0 : fato of seless dsassebled ateal oe t of eyled odt fo eovey ete These seless ateals ae set to the ladfll = : sellg e e esable ateal t fo eovey ete to afate : sellg e e e odt t fo afate to etale : sellg e e e odt t fo etale to dead aet : dead e e e odt t at dead aet : by-ba e e eylable odt t fo eovey ete fo oses at dead aet : et ato of sed odts at dead aet f e : eylg fees haged by the oesodg Evoet Poteto Agey EPA fo afatg a gve ts of odts e s : t of sbsdy of evoet oteto to eovey etes offeed by the EPA : ost e t of dsosed ateals to the ladfll Rado deads ad effets o deso vaables We asse that the dead fo eah aet s ado ad follos a oal dstbto th ea = a b ad vaae hee a ad b ae ostve 7

9 ostats Let d ~ be the ado vaable eesetg the ealzed dead fo aet hh s deedet of the dstbtos fo all othe dead aets Gve the ootos dstbto e obta: q q q q q ad ad sg the oetes of the oal = d ~ = = = = d ~ = = = = d ~ 3 = = = d d = = ~ 4 ~ 5 ote that the esee of adoess the ootos ad ae the desos vaables of the losed-loo sly ha ebes 3 Obetves of the losed-loo sly ha eto ebes ths seto e desbe the behavo of the vaos deso-aes the losedloo sly ha 3 Ra ateal sles ad the eqlb odtos ~ Let f q deote the oeet ost of a ateal sle hh geeal = deeds o the total ateal shet Usg eqato the exeted oeet ost f s oted as follos: = 8

10 f ~ ~ = E f d = f x e dx 6 0 = = = hee = ad = = = = = We assoate th eah a ateal sle ad afate a a tasato ost ~ that ldes the ost of shg a ateals The exeted ost s q allated as: x ~ = = ~ E d x e dx 7 0 = hee = ad = = = Gve the above exeted osts eah a ateal sle sees to axze hs exeted oft: ax Z = = = = = x f 8 sbet to : = = = 0 Eqato 8 states that a a ateal sle's exeted oft s eqal to exeted sales evee s exeted osts assoated th oeet ad tasato We asse that the a ateal sles oete oooeatve fasho Also e sose that oeet ad tasato ost ftos fo eah sle ae otosly dffeetable ad ovex Hee the otalty odtos fo all sles a be exessed sltaeosly as the follog vaatoal eqalty eg agey ad Dog 00: Detee satsfyg: 9

11 Ra ateal sles Reovey etes etales Podt shet flo Ro ateal shet flo Fge : eto stte of afate 's tasatos f = 0 9 = = = 0 0 = = 3 afates ad the eqlb odtos Eah afate st ae seveal desos: a ho h of the odt to sh to etales; b ho h of a ateals to get fo sles; ho h of esable ateals to t fo eovey etes see Fge ~ afate s a odto ost fo a ateals f q ad a = ~ eafatg ost of esable ateals f q These osts deed o the eovey 0

12 level o desged to the odt Usg eqatos ad the exeted vales of these osts ae oted as follos: f f x ~ ~ = E f d = f x e dx 0 = = ~ ~ = E f d = f x e dx 0 = x hee = = = = = = = = ad = = Soe legslatve shees oose that eylg fees shold be haged fo afates ad ae sed to foste the eovey etes to eyle the sed odts As sggested by She et al 005 the aggegate eylg fee fo afate s eqal to e f q e dg a exeted eylg fee of f = We assoate th eah afate ad etale a a tasato ost ~ that ldes the ost of shg the odt Usg eqato 3 the exeted ost s allated as: x q ~ = = ~ E d x e dx 3 0 = hee = ad = = =

13 Gve the above exeted osts eah afate sees to axze hs exeted oft: ax Z = = f = f = = = f e = 4 = = = sbet to : = 5 ad the ostats = 0 = 0 ad = = = = = 0 Eqato 4 states that afate 's exeted oft s eqal to exeted sales evee s exeted osts assoated th odto ad tasato exeted ayot to a ateal sles ad eovey etes exeted eylg fees ad exeted dsosal ost Costat 5 states that the ooto of the odt shet to etale st be eqal to the s of the ootos of the odt oded/tasfoed fo a ateals ad eafated/tasfoed fo esable ateals Oe oded the seless dsassebled ateals q = of = = q old be set to the ladfll g a exeted dsosal ost = We asse that the afates oete oooeatve fasho Also e sose that odto ad tasato ost ftos fo eah afate ae otosly

14 dffeetable ad ovex Hee the otalty odtos fo all afates a be exessed sltaeosly as the follog vaatoal eqalty: Detee satsfyg: = = f = f = e f 0 6 = = 7 = 0 8 = = = 0 9 = = 0 0 = 33 Retales ad the eqlb odtos Eah etale s faed th a hadlg ost hh ay lde the dslay ad stoage ost assoated th the odt We deote ths ost by ~ q hh s a fto of ho h of the odt etale has obtaed fo the vaos afates Usg eqato 3 the exeted hadlg ost s oted as follos: = = = = = x ~ = = ~ E d x e dx 0 3

15 hee = ad = = = = = Gve the above exeted ost the otzato oble of a etale s gve by: ax Z = sbet to : = = = = = 3 = ad the ostats = 0 ad = 0 = Obetve fto states that the dffeee betee the exeted evee ad the exeted hadlg ost ls the exeted ayot to the afates shold be axzed Costat 3 exesses that fo eah dead aet the ooto of the odt shet fo etale st be eqal to the s of ootos of the odt shed fo all afates to etale Asse that the etales oete oooeatve ae Also e sose that hadlg ost ftos fo eah etale ae otosly dffeetable ad ovex Theefoe the otalty odtos fo all etales a be exessed sltaeosly as the follog vaatoal eqalty: Detee satsfyg: = = 0 4 = 5 = = 0 6 = = 0 7 4

16 34 Coses at dead aets ad the eqlb odtos The oses tae to aot ag the deso ag: a ho h of the odt to hase fo the etales; b ho h they ae llg to ay fo the odt; ho h of the sed odt they ae llg to et to the eovey etes the foad sly ha let ~ deote the t tasato ost assoated th q obtag the odt by oses at dead aet fo etale that deeds o the odt shet betee a Usg eqato 4 the exeted tasato ost a be exessed as: = E ~ d hee ad = = x ~ = x e dx 8 0 Follog agey ad Dog 00 ad Yag et al 009 the eqlb odtos fo oses at dead aet ae dtated by the follog eqatos: fo all etales : = > 0 f 9 f = 0 ad fo eah dead aet : = f > 0 = f = 0 30 Codto 9 states that oses at dead aet ll hase the odt fo etale f the e haged by the etale ls the exeted tasato ost does ot exeed the e that oses ae llg to ay fo the odt Codto 30 o the othe had 5

17 states that f the e oses ae llg to ay at dead aet s ostve the the ooto of the odt osed at dead aet s esely eqal to the evese sly ha ose aveso at dead aet s odeled by a ootoe easg fto g~ q that deeds o the total aot of sed odt eted to all eovey etes The oe aot of odt to be olleted the CLSC the oe a eovey ete has to offe as a by-ba e easg the by-ba e des oe oses a dead aet to et at of the sed odt Usg eqato 5 the exeted vale of the ose aveso fto g s oted as: x ~ = = ~ g E g d g x e dx 3 0 hee = ad = As stated by Haod ad Belles 007 fo a gve by-ba e the aveso odel segets oses at dead aet to to gos: oses that ll be esaded to et at of the sed odt ad those ho ll ot Ths segetato a be exessed as follos: = f > 0 g 3 = f = 0 sbet to = 33 Codto 3 states that oses at dead aet ll hoose to et a at of the eylable odt shets oesodg to the vale of the by-ba e Costat 33 6

18 states that the ooto of the eylable odt shet oses at dead aet dede to et to all eovey etes st be eqal to the ooto of the odt shet fo the vaos etales ltly by the et ato Cobg oses behavos both the foad ad the evese sly has the eqlb odtos of the oses a be folated as the follog vaatoal eqalty: Detee satsfyg: g 0 34 o eqvaletly by ltlyg by : g Reovey etes ad the eqlb odtos Reovey etes ae assed to by-ba the aot of sed odt fo oses at vaos dead aets Let ~ deote the tasato ost assoated th eah q eovey ete ad dead aet a Usg eqato 5 the exeted tasato ost s exessed as follos: = E ~ d x ~ = x e dx

19 hee ad = = Coesodg to the eylg fees eovey ete old obta a aggegate sbsdy eqal to e s q = fo the EPA fo fosteg ts eylable atvtes see She et al 005 Befoe tasatg th the afates the ogal sly ha fo sellg the esable ateals eah eovey ete st lea set ad dsasseble the aot of sed odt g a eylg ost ~ q that s a fto of q The exeted vale of eylg ost a be oted as: = = x ~ = = ~ E d x e dx 37 0 = hee = ad = = = Eah eovey ete otats th afates the ogal sly ha to sell the esable ateals eg ae glass o al Let deote ~ the tasato q ost assoated th eovey ete tasatg th afate The exeted tasato ost s allated as: = E ~ ~ d = x e dx 38 0 = hee = ad = = = Gve the above exeted osts eah eovey ete shes to axze hs exeted oft: x = 8

20 ax = Z = = = = s e = = 39 = = sbet to : = 40 = ad the ostats = 0 ad = 0 = Obetve 39 exesses that the dffeee betee the exeted sales evee ls the exeted sbsdes ad the exeted tasato eylg ad dsosal osts ls the exeted ayot to the oses shold be axzed Costat 40 states that fo eah dead aet the s of the ootos of the esable ateals to be sold to all afates st be eqal to the ooto of the eylable odt shet olleted fo oses ltly by the tasfoato ate of eyled odt Oe dsassebled the exeted dsosal ost of eovey ete fo sedg the sable ateals to the ladfll s desbed as = Asse that the eovey etes oete oooeatve ae Also e sose that tasato ost ad eylg ost ftos fo eah eovey ete ae otosly dffeetable ad ovex Theefoe the otalty odtos fo all eovey etes a be exessed sltaeosly as the follog vaatoal eqalty: Detee satsfyg: 9

21 = s e 0 4 = = = 4 = 0 43 = = Eqlb odtos of the CLSC 4 Eqlb odtos eqlb e st have that the s of the otalty odtos fo all a ateal sles as exessed by eqalty 9 the otalty odtos fo all afates as exessed by eqalty 6 the otalty odtos fo all etales as exessed by eqalty 4 the otalty odtos fo all dead aets as exessed by eqalty 35 ad the otalty odtos fo all eovey etes as exessed by eqalty 4 st be satsfed We state ths exltly the follog defto: Defto Closed-loo sly ha eto eqlb th ado deads The eqlb state of the losed-loo sly ha th ado deads s oe hee the foad ad evese flos betee tes of the eto ode ad the shet ootos ad es satsfy the s of the otalty odtos ad 4 The sato of eqaltes ad 4 afte algeba slfato yelds the follog eslt: 0

22 Theoe The eqlb odtos goveg the losed-loo sly ha odel th ado deads ae eqvalet to the solto of the vaatoal eqalty oble gve by: fo eah dead aet detee satsfyg = = = = f f = = = f = = = e f = = = = a e s 0 = 45 hee s the feasble set defed as: } { = satsf Fo easy efeee the sbseqet setos vaatoal eqalty 45 a be ette stadad vaatoal eqalty fo as follos: detee X sh that

23 F X X X 0 X 46 hee = X s a veto th X as ts ooets ad F = s the veto th F x F F F F F F as ts ooets th the sef ooets of F x beg gve by the esetve ftoal tes eedg the ltlato sgs 45 4 Qaltatve oetes Fo agey 999 a vaatoal eqalty adts at least oe solto f the fto F that etes the vaatoal eqalty 46 s otos ad feasble ego s oat Theoe Exstee: vaatoal eqalty 46 adts at least oe solto de the follog odtos: f f f ae otosly dffeetable fo all ; g ae otos fo all ; thee exsts a ostve ostat sh that < fo all Uqeess s also a otat oety to vestgate Fo agey 999 f the veto fto that etes the vaatoal eqalty 46 s sttly ootoe the ts solto s qe Theoe 3 Uqeess: Ude the follog odtos of stt oototy thee st exst a qe eqlb solto of vaatoal eqalty 46: Oe of the fales of ovex ftos f f f fo all s a faly of sttly ovex ftos; g ae sttly ootoe fo all

24 The last oety s Lshtz otty hh s eqed to gaatee ovegee of the oosed algoth Theoe 4 Lshtz otty: The fto that etes the vaatoal eqalty 46 s Lshtz otos that s ' '' ' '' F X F X L X X ' '' X X hee L > 0 47 de the follog odtos: f f f ae addtve ad have boded seod-ode devatves fo all ; have boded seod-ode devatves fo all ; g have boded fst-ode devatves fo all 5 Solto Algoth The algoth that ll be sed to ote the solto of vaatoal eqalty 46 s the odfed oeto ethod th ostat ste legth oosed by oelevh 977 hh a be aled to solve ay vaatoal eqalty oble stadad fo 5 odfed oeto ethod Ste 0 talzato Set 0 X Let = ad let be a sala sh that 0 < / L hee L s the Lshtz otty ostat Ste Cotato Cote X by solvg the the vaatoal eqalty sboble: X T F X X X X 0 X 48 3

25 Ste Adatato Cote X by solvg the the vaatoal eqalty sboble: X Ste 3 Che fo ovegee T F X X X X 0 X 49 f ax X fo all th > 0 a esefed toleae the sto; X othese set := ad go to Ste 5 Cotato of the eqlb es Follog agey ad Dog 00 e o dsss ho to eove the es ad 4 ad fo the soltos of vaatoal eqalty 46 o Tae the es Se the obetve fto 8 s otosly dffeetable oave ad the feasble set s ovex the ash-h-te otalty odtos hh ae both eessay ad sffet fo tae the fo: f f = 0 = = 0 deed f thee s a shet ooto > 0 the = f = 50 Usg the sae aget the othe es ad ae obtaed as follos: 4

26 = = a s e 5 = = 5 = 53 = a 54 6 Colso ths ae e have develoed a e losed-loo sly ha eto odel th a eqlb otext the ase of ado deads assoated th the oses at dead aets atla e assed a losed-loo syste osstg of a ateal sles afates etales ad eovey etes eah of ho sees to axze exeted ofts The dead fo the odt at eah aet follos a oal dstbto hh ded adoess the odt shets Usg shet ootos as deso vaables e deved the goveg eqlb odtos ad the shoed they satsfy a vaatoal eqalty oble Exstee ad qeess of the eqlb solto ee dsssed ad a solto algoth based o the odfed oeto ethod as oosed to solve the odel Ths o establshes the fodato fo losed-loo sly ha eto obles the ase of ado deads th a eqlb faeo Fte eseah ay lde the odelg of oe systes ad the tegato of losed systes ad oe systes to the gee sly ha eto 5

27 Refeees Davd B Stehe L oe BH 004 Logsts Pete Hall Dog Zhag D agey A 004 A sly ha eto eqlb odel th ado dead Eoea oal of Oeatoal Reseah Flesha e HR Dee R Flae SDP 000 A haatezato of logsts etos fo odt eovey Oega Haod D Belles P 007 Closed-loo sly ha eto eqlb de legslato Eoea oal of Oeatoal Reseah oelevh G 977 The extagadet ethod fo fdg saddle ots ad othe obles ateo H o CS o H 006 The satal ad teoal osoldato of eted odts a losed-loo sly ha eto Cotes ad dstal Egeeg agey A 999 eto eoos: a vaatoal eqalty aoah seod ad evsed ed le Aade Pblshes Dodeht The ethelads agey A Toyasa F 005 Revese sly ha aageet ad eleto aste eylg: a ltteed eto eqlb faeo fo E-ylg Tasotato Reseah E 4-8 agey A Dog Zhag D 00 A sly ha eto eqlb odel Tasotato Reseah E She B Cho YH H CC 005 A tegated logsts oeatoal odel fo geesly ha aageet Tasotato Reseah E Yag GF Wag ZP L XQ 009 The otzato of the losed-loo sly ha eto Tasotato Reseah E