Inventory Models in Reverse Logistics

ovi Uiveity Bapet Falty of Bie iitatio Ivetoy Moel i evee Logiti D Dietatio by Dobo Ie Novebe 006.

otet efae 3. evee Logiti: Faewok 4. Eooi Oe atity Moel i evee Logiti 7.. evee Logiti Moel wit oeet a epai 0.. Moel wit oeet a Fiite epai ate: e Sbtittio oliy 33.3. Moel wit oeet a Fiite epai ate: e otio Sppleet oliy 46 3. Ivetoy Moel wit Wate Dipoal 55 3.. epai Moel wit ee Stokig oit 57 3.. eoveable Ite Ivetoy Syte 68 3.3. otio-eylig Moel wit Bybakig 83 4. otio laig i evee Logiti 00 5. Say a Fte eea 9 efeee 3 ppeix 9

efae Evioetal oio ateial a ivetoy aageet will be tie i ti ietatio. It i ae a evee ivee o wate ipoal logiti i te lat eae. I te Hgaia liteate tee ae o ifo efiitio fo ti ietifi aea. i eea fiel i efie i Egli peakig otie a evee logiti. foe ae of ti iea wa ivee logiti bt ti ae i e aily i Japa. e ietatio oit of tee apte. I te fit apte I efie evee logiti a it poble. I te eo apte I peet ix eteiiti evee logiti ivetoy oel. ee ivetoy oel wee otte i te lat tee eae. atal exteio of EO-type ivetoy oel wa exaie i te lat eae. e fit evee logiti ivetoy oel wa bilt i 967. e eo oel wa pblie i 979. I te eigtie o pape wa pblie o ti eea aea. e Eopea Uio a aie oe eea pojet i te ietie to ppot Eopea evioetal eglatio. e pbliatio of ew evee logiti ivetoy oel i i poge owaay. Mai poit of te eea i ow ivetoy oel wit otage. e ietatio otai all available eteiiti EOtype oel witot otage. e lat apte ivetigate otly te iflee of evee logiti o potio plaig a o ateial eqieet plaig yte. EO-type evee logiti oel a be e a a bai fo te yai lot ize evee logiti oel. e fit pbliatio appeae o ti fiel i 000. e oltio of Wage-Witi-type evee logiti oel i ot eay beae of te oplexity of yai pogaig algoit. EO-type evee logiti oel a eve a eiti to olve ki of ivetoy oel. i i a potetial appliatio of ti eea fiel. lat I aize te elt of ti ietatio. 3

. evee Logiti: Faewok.. Itotio olletio of e pot a pape bottle a battey i a kow iea i oe eooie. ee eafatig a eylig of a a eletoi appliae a ipoal of azao wate ae vey eet eea fiel. e lite ativitie ile a vey boa aea a it ee to ave iffeet aageet poble. i apte aize te evee logiti wi offe a teoetial bakgo to olve ki of bie poble. e ee i ot a ew peoeo i te patie bt a lot of pbliatio ae appeae i te iteatioal liteate i eigtie ae evee logiti. I Hgaia liteate tee ae oly a few pbliatio o ti eea fiel. e fit pbliatio i pape of ixe (995. He a alle ti fiel a ivee logiti. eléyi et al. (997 a e te expeio eylig logiti a Mike (00 a give te ae evee logiti wi i e i Egli peakig otie. ee ae oe ew pbliatio abot evee logiti i Hgaia a well. (ite a Dobo (003 Dobo (004 evee logiti ile ot oly te ateial flow fo pplie to oe bt alo te ateial flow of e pot fo oe to poe a pplie i oe to ee te be of evioet. e ai of ti apte i to peet te iteatioal (aily glo-saxo liteate o ti fiel. e evioetal eglatio beoe igoo i te Eopea Uio a i Hgay. ee ae eetly a lot of evioetal eglatio abot wate alog te life yle of a pot i te Eopea Uio. (Fo exaple abot e a. e Eopea Uio pla to olve evioetal poble i te ea fte by te elp of legal eglatio. ee ile te e of eewable evioetal eoe a eegy; avoi wate a btittio of o-eewable eoe. e Hgaia aliaet a legilate a law abot te wate aageet i 000. e ai of ti law i to potet te a ealt a evioet to ppot te atioal e of 4

eoe a to ee te evioetal loa i oe to poote taiable evelopet a eooi gowt. e law ipoe of wate a ativitie of it alig. i law oe ot to te eiio i te ai a lea azao wate. Soe of piiple ae etioe i ti law a pevetio epoibility of poe ivie epoibility polltat pay piiple bet available teiqe ot effiiey a o o. e law ipoe of epoibility of poe etaile oe a owe of wate. Step of wate aageet a ee a explaatio of iea ae ile i te law. ee ae efie olletio a tapotatio of wate ee of wate a alig. Sepaate etio peet epoibility of alig of oal a azao wate a ogaizatio of wate aageet. It i to epaize obligatio of pbliity a ifoatio. Fi t keep ti law bt appliatio of evee logiti eto a lea to ot avig i log age. Legal egitatio a ot foe etepie to poe a evioetal oio way bt eooi eaig a elt a evioetal fiely potio tte of fi. I ti apte I peet otly te evelopet of evee logiti a te I ow a oeptal faewok oieig te evelopet of ti iea i te lat eae. fte tat I look fo awe te ai qetio of evee logiti: wy-ow-wat-wo. lat I aalyze te patiipat of evee logiti exaie te ai aageet poble... bot evelopet ee wee eooi a itoi ae of evelopet of logiti a it i fo te evee logiti. etaile ave eogize te ae of takebak of pot i te Uite State at te e of eigtie a a tool of aket gowt. otol of takebak wa ot iete beae tee wa o ifo a eio eglatio of fo of et poliie of e pot. e elt of ti evelopet wa tat oe ave take bak a be of pot. e ot of ti poe ave aatially ieae at te poe a at te etaile wi a ee te pofitability a opetitivee of fi. ey ave eogize tat a effetive evee logiti yte i a ipotat itegal pat of opoate tategy of fi. e ipotae of evee logiti i ot of qetio bt te appliatio of ti oept ake oe iffilt tat ato efie evee logiti iffeetly a te oltio of 5

evee logiti oept iffe fo ea ote at fi level. Beae of ti iffilt appliability I ty to efie te iea a I eteie te potetial eea fiel of evee logiti.... Deteiatio of te oept e evee logiti wa fit efie i te eigtie. I ti tie tee wee pblie oly a few atile i te liteate o te teoetial bai of ivetigatio wa ettle. Oe of te fit pbliatio o ti fiel i te pape of Labet a Stok (98. ey ave efie evee logiti a a evee ateial flow oppoite to pply ai wi i a ba poe alog te ateial flow of fi. It ea tat til ateial flow of taitioal pply ai o i pplie-poe-woleale-etaile-oe ai evee logiti eize te et ateial flow of e pot i oe to follow ti poe bakwa fo oe to pplie. fte te egative efiitio of Labet a Stok Mpy a oit (989 ave offee a ew appoa to eteie evee logiti. ey ave efie evee logiti a a ateial flow of pot fo oe to poe i te pply ai. i efiitio i aepte by ole a Fai (99 wo pefe to apply aketig oept to evee logiti. e ipotae of tei pape i tat tey ave ae te fial oe a tey ave epaize tat te poe i evee i te pply ai. awbak of ti efiitio i tat tey ave ot eteie te ai ativitie of evee logiti wi ake oe iffilt to liit te faewok of evee logiti. I te ietie Stok (99 a give a wie efiitio wi i a bai fo wate aageet. He tee te ole of logiti wi otai eylig wate ipoal btittio of azao ateial etio of eoe a ee. i efiitio of Stok i oe aate ta tat of ealie. e oetio wit pply ai ativitie i iig i ti geeal efiitio a te evee poe i ot epaize a well. ee lat appoae ae aize by opiky et al. (993. i efiitio otai all above-etioe ativitie te evee oveet of ateial alog te pply ai oppoite to taitioal logiti. opiky et al. (993 ave itoe ifoatio flow i te 6

efiitio of evee logiti wi elp o effetive patial ftioig of evee logiti yte. ate a Ella (998 ave ollete a be of efiitio of evee logiti. I will ite oe of te efiitio. e oe geeal efiitio i: evee logiti i a ativity wi elp to otie a evioetal effetive poliy of fi wit ee of eeay ateial eafatig a wit etio of aot of eeay ateial. i effiiey toe te peoal i potio pply a optio poe. ate a Ella (998 appoa evee logiti fo poit of view of evioetal potetio. Evioetal oioe o at tee level of ativity of fi: goveetal eglatio oial pee a voltay elf etitio. ext efiitio otai bot taitioal a evee logiti. oil of Logiti Maageet efie logiti: Logiti i a efl ot-effetive plaig ealizatio a otol of aw ateial wok-i poge fial pot a oete ifoatio fo te begiig to optio i oe to pefo oe ee. oge a ibbe-lebke (999 efie evee logiti a: Logiti i a efl ot-effetive plaig ealizatio a otol of aw ateial wok-i poge fial pot a oete ifoatio fo optio to te begiig i iteet of vale egai a alig of wate. evee Logiti Exetive oil (LE a give a oe geeal efiitio of evee logiti wi aize te above efiitio: evee logiti i a oveet of ateial fo a typial fial optio i a oppoite ietio i oe to egai vale o to ipoe of wate. i evee ativity ile takbak of aage pot eewal a elageet of ivetoie tog pot takebak eafatig of pakagig ateial ee of otaie epai a eovatio of pot a alig of obolete appliae. Eopea Wokig Gop o evee Logiti (EVLOG a give a iila efiitio of evee logiti i 998. e iffeee i tat te begiig of olletio i ot oly te optio bt it a be alo potio itibtio o e. 7

e evelopet of oept of evee logiti wa peete. e oept a age aatially i te lat two eae. ill te fit appoa a oiee evee logiti a a ba ietio owaay te teoy of evee logiti otai aketig fiaial a evioetal poit of view. I ietee evee logiti a beoe a well etablie teoy. i oplex efiitio ppot te iea tat evee logiti ove all ativitie alog te pply ai..3. Fato of evee logiti: Wy? How? Wat? Wo? fte efiitio of evee logiti I exaie te fato tat ta bei ti oept. Fo qetio aie i ti otext: wy ow wat a wo. ee qetio ae awee by Bito a Dekke (00 ot opeeively..3.. Wy? i qetio otai two eea fiel. Fit wy e peo e pot bak a wy aept ote e ite? I ave etioe te ae of evee logiti i te eo etio of tee apte i.e. eooi legilative a oial oeqee. ee ae to te eeive of gop. Bito a Dekke (00 itigi iet a iiet gai iie of eooi avatage. Diet gai ae te poibility of pofit ieae tat ea a etio of e of aw ateial eeae of ot of wate ipoal a vale ae tog ee. Iiet avatage ae te gee iage of a fi wi i a fato of opetitivee fo etepie. Expeiee ave ppote tat evioetal oio ftioig of fi elt i a table oe oetio. It i a opetitive avatage of fi tat ieae i pofit ae. tit legilative eglatio i a ew aget fo patial appliatio of evee logiti poee wi eve a a eto fo evioetal potetio. e Uite State a te Eopea Uio ae leaig i evioetal legilatio wi foe te fi to keep te law. ily voltay oial epoibility of fi iet ogaizatio to potet evioet. I te patie ti voltay ativity ieae i opetitive avatage of fi. eo aea i to ivetigate te gop of ee. ey ave eie to e bak a e pot to te afate. by te eeive tee fiel ae to be aalyze: et by afate itibto a e. 8

aw ateial Sevie at fabiatio Mole aebly ot aably Ditibtio Ue 6 5 4 3 7 8 Wate 7: Iieatio 8: Lafillig ot eovey Maageet 5:aibalizatio : epai 6: eylig 3: efbiig 4: eafatig Diet ee : Diet ee/eale Fige : Itegate Spply ai Soe: iey et. al. (995 et of e pot by afate ea a e bak i te potio poe beae of aw ateial pl otage o qality of pot a by-pot. et by itibto ea a e bak of o ellable ol pot. ee pot eboy i ivetoy efetive tapot a pot a pakagig wate. et of e i gaatee evie a e-of-life pot wi ae at te e of eooi a pyial pa of life of pot. ext gop of pot i te e-of-e pot tat ave o oe vale fo tei owe bt tey a be ol fo ote oe. It i vey a to itigi tee lat two gop of pot o it i eaie to pply oe exaple. e e-of-life pot ae fo exaple wek a wi a be iatle a it pat o ole a be ee. E-of-e pot ae ete a wi a be ete afte a kow ealie. 9

.3.. How? Now I will ivetigate te ow qetio: How a be ealize a evee logiti yte? I e to awe ti qetio te pape of iey et al. (995. i poe oit of eigt tep: iet ee epai efbiig eafatig aibalizatio eylig iieatio a lafillig. Fige ow te oetio betwee te eleet of evee logiti ativitie. Diet ee: te pyial a qality popety of pot i ageable i te evee logiti poe. epai: e pot will be tafoe a afte ti tafoatio (epai te pot a be e o ol a a ew pot. epai a o at te e o i a epai op. Ue tafoatio I eta a age of pat bt ote ole o eleet of te pot ae itat. eale ee eitibtio epai efbiig eafatig eylig Iieatio Lafillig Fige : Hieaial oetio of evee logiti ativitie Soe: e Bito Dekke (00 efbiig: efbie pot ae iatle ito ole a te tey aeble e le igoo qality. ee ae epaie oly te efetive ole o te lifepa of pot ae elage. 0

eafatig: Fo eafate pot we wait a goo qality a a ew pot. eafatig i oe ta efbiig beae all ole a pat ae igooly ipete befoe e poe. Defetive ole a pat ae totally exage i te eafatig poe. aibalizatio: I ti ee poe all of te ete pot ae iatle a tee i a igoo qality ipetio. e egaie pat a ole ae ee i epai efbiig a eafatig ativitie. eylig: e pot loe te oigial ftio i ti ee eto. e objetive of eylig i to eove all able ateial. If te qality of eovee ateial i appopiate te tey a be e fo afatig of ew pot. Iieatio a lafillig: ee two ategoie belog to te wate aageet. Bot ativitie t fill igoo eqieet. Eooi avatage a be gaie fo iieatio if te iig eegy i ee. e above-etioe fiel ae aize i a pyai i fige. i pyai eate a loe oetio betwee evee logiti a evioetal potetio. e level ow wi logiti ativitie poote te potetio of evioet. Soe of te ateial a wate a pot of evee logiti a be ale wit ativitie at te botto of pyai. e objetive of a evee logiti yte i to itoe ativitie at te top of pyai. e qetio i ow if te objetive i ee o etio of eoe te te pyai i wy ot boa at te top. e ieal itatio wol be a ivee pyai bt ee i owaay ot o geeal..3.3. Wat? e ext qetio eal wit te qality of te ete pot i evee logiti. I ti ae te aotet of ete pot i exaie: wi fato aage te poibility of ee a ow will te oe e te ee pot. otet iflee te ee i two way: oogeeity a eae of te ete pot. Lifepa of pot i iflee by peiability age of eleet a

aotizatio of te pot wi ake oe iffilt te ee. typial exaple i eletoi ite wee te teial poge peee ftioig bt obolete pot. e way of e of pot iflee te ee. It epe o plae iteity a atio of e wi eteie a late eafatig. e ollete ite a be itigie wete tey oigiate fo oal o itial optio. (E.g. beae of tapotatio alig o qatity. Hee t be etioe pakagig ateial pae pat o pbli goo..3.4. Wo? e fot ipotat fiel i te ietifiatio of patiipat i te evee logiti. I ti otext I itigi te patiipat of taitioal vale ai a of evee poee a ote patiipat e.g. aity ogaizatio. ill oe of iteete peo ogaize te evee poe ote eal wit te patial ealizatio. It i vey ipotat to ooiate te oetio betwee pply ai. Oe of te ooiatio eai i a eliable ifoatio flow. Neeay ifoatio fo a efl ftioig i aize i pape of iey et al. (995. O te bai of ti atile tee ae fo gop: - Ifoatio abot pot aotet i.e. abot ateial tei obiatio qality vale aza a poibility of afatig (aalyze. - Ifoatio abot extet a etaity of evee poee: Waaty qatity a qality of ete pot i etai eeay epai ativitie ae a to pla. Off-leae a off-et otat tey a be etiate vey well i qatity a i tie bt to etiate te qality i a. Voltay by-bak it epe o te poibility of afate. e avatage of ti oltio i tat it ie iexpeive eoe fo afatig a epai. Wate ipoal ot eeae at te oe a it ake poible fo te afate to ell ew pot. - Ifoatio abot te aket of ee pot pat a ateial. It i a to fi aket o opetitive avatage ae i iffeee of qality a ot fo ew a

e pot. ee a be ae by te afate bt ote fi a ealize te ee iie a otie pply ai. - Ifoatio abot olletio of e pot a wate ipoal. e exaiatio ile ogaizatio ivolve i te poe obtale oe qatity of ete pot a ot-beefit aalyze..4. Stakeole of evee logiti atiipat of evee logiti a be appoae i aote way. teoetial bakgo i pplie i pape of ate a Ella (998 i wi tee ae iteal a exteal fato tat iflee evee logiti. I geeal tee ae fato witi ogaizatio a betwee ogaizatio wi ae exteal fato. Iteal fato belog iteete peo iie of fi tep fo potetio of evioet efl applie bie eti taa a aily toe peo wo ae epoible fo te evioet fiely opoate piloopy. lo iteal iflee ave te oe pplie opetito a goveet. ee fo eleet ae iflee alo by te ao evioet wit oial politial a eooi te tat to evee logiti iietly. e lite eto ave a iffeet effet a tey ave eveal itepetatio. og exteal fato goveetal eto a a ot eteiig iflee. It a be aepte fo evioetal potetio poit of view oieig tat evioetal poble iitiate ot of te qetio i te Eopea Uio. It t be eake tat law foe etepie till ote opetito ave to oie etepie opetitivee i te ae way. Fo ti poit of view a fi t eet te oe ee e keepig te evioetal eglatio of goveet. Witot keepig goveetal ittio a etepie a ot beoe opetitive. ee ae two view abot fi beavio. 3

Spply Mafatig Ditibtio Sevie Oigial e Eegy Ote e olletio ebly eafatig eitibtio ee Wate Iieatio Lafillig Fige 3: oetio of evee logiti poee Soe: ot Nagy (004 Ipotae of pply ie i epaize wit te fat tat olleage of poeet epatet pae oe e ite if te peaet goo qality of eable pot i ee. Spplie ae epoible fo olletio a eletio of e pot geeally bt te iae of expete qality of eable ite eeitate oopeatio betwee pplie a poe fi wi i a ial logiti ativity epeig o te attaiable 4

ifoatio. e qality of ete pot a a ik potetial fo te pplie o te itegatio betwee pplie a poe t be tegtee. e ole of te toe peo i a ipotat iteal fato. e owe of te fi a iflee te ftioig of evee logiti yte. ey o ot eteie te ativity of fi ietly bt tey a ie it i a log age. ei aitae i a pe-eqiite fo a efl evee poe. e ole of aageet i iila to tat of owe. Witot ay aitae of aageet a evee logiti yte a ot be ftioe effetively bt te ftioig i ae by te ile leae of te fi. ey t ave goo iploati a oiatio kill a leaig ability. ey ave te wok to peae te toe peo abot te eeity of effetive evee logiti yte. Eployee belog to te ti gop of takeole wo a elp to itoe evee logiti yte tog tei otibtio. Stilatio yte a ait te effiiey. e above-etioe exteal a iteal fato ave a yegy effet i.e. bot a ake toge tei effet togete. e oe ee t be oiee a a geeal le. lo te iteal a exteal iteet t be oiee. Witot oieatio of tee iteet a evee logiti yte a ot be ealize. Fige 3 aize te exaie oetio abot evee logiti. I epaize tat te poee t be loe. e fige peet a pape ill afatig poe. Soe of te ipotat ativitie ae eglete beae of te ipliity..5. Say ll of te peete evee logiti ativitie a ot be fo i a fi. ee ae a be of eao wy. e available teology geat vaiety of pot a eooi itatio of fi iflee te etepie eiio abot applie evee logiti yte. I o ot ivetigate wat evee logiti ea fo a peifi pot a ow a efl yte ol be itoe. ee poit eqie fte exaiatio e.g. ow a fial pot a be iatle wi eleet a ole of ti pot a be ee i te 5

afatig poe. t te ae tie it i a to follow all pat a ole fo afate to oe te tog olletio etwok to ee fabiatio. Nowaay tee i o ifoatio yte to follow te oet ateial flow alog te pply ai. I oe ae it i eay to oel te ee poe bt i geeal it i ot o. Soe of te pat a ole a ot ee a it i iffilt to fi a eooi eto wee te ee poe a bil p effetively. typial exaple i opte fo wi elatively a few pat a be eovee a te ee i eooial oly i a geat extet. ee above-etioe poble a be eliiate wit a oopeatio of iffeet itial eto a wit a ooiate eliable ifoatio flow betwee tee eto. O tatig poit wa te potetio of evioet wi i tilate by legal eglatio a by etepie epoibility. e fi ae foe to eet goveetal eglatio bt a voltay epoibility i iflee by te available fiaial oe. I a log age te ot a evee t be aalyze. Evioetal oioe i ot attative witot ay eooi gai. e ai of ti eview i to give a itotio i te teoy of evee logiti yte. i apte i a tatig poit to get aqaite wit ee poee wi aie a be of qetio. i teoetial apte give a teoetial bakgo bt te patial appliatio of evee logiti yte ee fte epiial ivetigatio. e pyial ealizatio fae wit teologial iffiltie a o te ote ie te ot t be exaie a well. efl evee logiti alog te pply ai a otibte to te etio of loa of evioet. 6

. Eooi Oe atity Moel i evee Logiti evee logiti i a exteio of logiti wi eal wit alig a ee of eable e pot witaw fo potio a optio poe. S a ee i e.g. eylig o epai of pae pat. evioetal oio ateial aageet a/o logiti a be aieve wit ee. It a a avatage fo eooi poit of view a etio of evioetal loa tog et of e ite i te afatig poe bt te exploitatio of atal eoe a be eeae wit ti ee tat ave te eoe fo extee optio fo te fte geeatio. I ti apte I peet tee evee logiti eooi oe qatity (EO. ee oel ae ot oly ow bt extee a I ow tat all of tee oel lea to te ae ateatial tte ae eta-oel aalyze i te appeix. (Dobo-ite (000 popetie of eta-oel ae peete i te appeix. e followig oel ae peete. e fit evee logiti (epai/ee/eylig oel wa fit ivetigate by Say (967 i a EO otext. e pape a exaie te ot avig of epai of ig ot ite at te U.S. Navy viatio Spply Offie i oppoite to poeet. e oitio of te bai oel i tat tee ae oly poeet a eveal epai bate. e qetio i te lot ize of poeet a epai. Moel of Naia a ivea (979 wa te eo lot izig oel. i oel a extee te elt of Say (967 wit fiite epai ate i.e. te epai poe ee tie. e epai ate i otat i tie. e poble oie wate ipoal of a ee poe. I te bai oel Naia a ivea (979 ave exaie te ae of oe epai lot ize. ee ivetigatio wee ppote by U.S. i Foe Syte oa. e lat oel i oel of o Hwag So é o (00. e ato of ti pape aalyze a oel iila to tat of Say (967. ill te fit two oel exaie a itatio wee te ew afate/poe a epaie pot a aive i a toe if te ivetoy level i eqal to zeo i ti oel te eoveable ivetoy flfil ti popety. i ivetoy tategy wa ae by Say (967 a otio ppleet poliy bt oelig of ti itatio wa ot pblie i i pape. e oel of Say 7

(967 a Naia a ivea (979 ave applie a ote ivetoy poliy ae btittio. o Hwag So a o (00 ave ot expee te bat ize bt tey ivetigate two epaate ae: be of paig bat i oe a epai bat i oe. I ow a ew folatio of te oel wi teat tee two ae i a geeal oel. o et al. (00 exaie a ote oel. I ti oel te ee apaity i ot geate ta te ea ate. I y ivetigatio I igoe ti type of oel. ee i a lti pot geealizatio of EO-type evee logiti oel pblie by Mabii itelo a Gele (998. ey ave extee te bai oel of Say (967 wit apital bget etitio. e exaie oel ave eteie te lot ize bt tey ave ot take ito aot tat be of lot i itege a te eitivity of et poe fo paaete wa ot ivetigate. fte ti bief oveview I aize te oo oitio of tee oel.. e ivetoy olig poliie ae kow i te oel. It ea tat i a ivetoy yle te ivetoy tat i give a kow i tie.. e ea fo ew a eovee pot i otat a eteiiti i tie. 3. e et ate of e ite i otat a kow i tie. It i a iila oitio to tat of lat poit. 4. e oeig ot of paig a etp ot of epai ae kow. 5. e ivetoy olig ot of eovee a ew pot a olig ot of e ite waitig fo epai ae kow. 6. ee i o otage i toe of eovee a ew pot a toe of ete ite. e fit oitio efie te ivetoy olig poliy. e vaiable of tee tategie t be eteie i a oel i.e. te lot ize fo ew a eovee pot be of bate fo ew a e pot a yle tie. e ext fo oitio ae iila to tat of taitioal oe pot EO oel i.e. ot tte a ea poe. e otage itatio i exle wit te lat oitio. oieatio of otage i ot a opliate ateatial poble bt te ai of ti apte i to give a itotio i te bai oel of EO-type evee logiti oel. 8

I peet te oel i te followig etio. 9

.. evee Logiti Moel wit oeet a epai... Itotio eteiiti EO-type ivetoy oel fo epaiable ite wa fit offee by Say (967. i oel a be ee a te fit evee logiti oel. Hi oel a exaie te U.S. Naval Spply Syte oa tok olig poble wit epaiable ite. e epaiable ite ay be appe po a faile bt te pot ae ally ete fo te e to te oveal a epai poit. e epaie ite ae et te to te eay-foie (FI ivetoy to await ea. Bae o te feaibility of epai te ite ot et bak ae ipoe of a tey ae eplae wit ew poe pot. e ete a ot epaie ite ae el i a eo tok poit i.e. te ivetoy of o-eay-fo-ie (NFI ite ae awaitig epai at te oveal a epai poit. Say a offee to ivetoy olig poliy to olve poble: te otio ppleet a btittio poliie. o ti lat poliy e a eteie te optial poeet a epai qatitie. It wa ae tat tee ae oly oe poeet qatity (bat ize a oe ta oe epai qatitie. e ai of te pape i to aalyze te itoe btittio poliy i a geeal faewok. I ti geealizatio it i allowe a oe ta oe poeet qatity. o olve te poble we e te eta-oel. (See appeix. Say a ot ivetigate te itege oltio fo te epai bat be it i exaie ow. We will ow tat te by Say offee oltio a be ipove i epeee o te eovey (et ate. e pape i ogaize a follow. e ext etio aize te paaete a ftioig of te oel. I etio 3 we ott te ivetoy olig ot ftio of te oel. e aalyzig te total aveage ot we eteie te optial poeet/epai yle. fte eliiatig te yle tie we ave attaie te oel i epeee o poeet a epai bat be wi lea to te eta-oel ivetigate by te ato a well. Setio 5 peet te bai oel of Say wit oe poeet bat. We will ow te optial itege oltio to ti oel. e followig etio olve te geealize oel wit otio bat be. 0

... aaete a ftioig of te oel e yte otai two ivetoie. e e ea a be atifie fo te FI ivetoy. e ea of te e i otat i tie. e FI ivetoy i fille p wit poe a epaie ite. Sotage i ot allowe i ti tok poit. e poeet a epai qatitie ae eqal. Fo te e te epaiable ite ae et bak to te oveal a epai poit wit a otat ate. e epaiable ite ae toe i te NFI tok poit waitig fo epai. fte epai pot ae ee a ew a tey ae et bak to te FI ivetoy. e ateial flow of te oel i epite i Fige. We efie te vaiable a paaete a follow: e eiio vaiable of te oel: - poeet qatity - be of poeet itege - epai bat ize - be of epai bate itege - poeet/epai yle tie. aaete of te oel: - ea ate it pe it tie - te eovey ate peet of te ea ate te ap ate i - - fixe poeet ot pe oe - fixe epai bat itio ot pe bat - FI olig ot pe it pe tie - NFI olig ot pe it pe tie. e followig eqalitie ow elatio betwee te i- a otflow i te tokig poit i a poeet/epai yle.

Fige. Mateial flow of te oel oeet Ivetoy of FI Ue epai Ivetoy of NFI e offee btittio poliy a te ext popety. e lea tie fo poeet a epai bate ae iegae beae i eteiiti oel it iflee a be eliiate wit a ovig away. Let ae tat a poeet/epai yle begi wit itio of a epai yle. e iitial ivetoy level i NFI tok poit i ee wit a epai bat ize. e te eaiig NFI ivetoy eeae wit a ew epai bat til it eae te zeo ivetoy level afte pply i te FI ivetoy. e tie itoy of ti poliy i ow i Fige 3. I te ext two etio we ott te ivetoy olig a aveage ivetoy ot ftio of te oel...3. e ivetoy olig ot ftio e olig ot of te oel ae allate wit te elp of te ivetoy level i tie a it i peete o Fige.

3 Lea. Let te ivetoy olig ot fo FI ite H FI a fo NFI ite H NFI. e te ot ftio ave te ext fo: FI H NFI H oof. We will pove te eo eqatio fo te NFI ite te fit eqatio a be allate i a iila way. Let ivie te aea ito - tiagle tiagle a - etagle B B... B -. See Fige 3. e legt of a epai yle i. e aea of a tiagle i. e aea of a etagle B i i eqal to i. e axi ivetoy level of NFI ite i. e aea of tiagle i [ ]. Let ow aize te aea: [ ] i H i NFI. fte oe eleetay allatio we ave te eqatio b. Exaple. Let 000 0.9 $ 750 $ 00. e fo ti ata te ivetoy olig ot ftio i 900 00 0 NFI FI H H

Fige. Ivetoy level i te FI a NFI tok poit ( 3 FI ivetoy - t NFI ivetoy t epai oeet Fige 3. e allatio of te ivetoy ot of NFI ite ( 3 B B ( t 4

5..4. Optial poeet/epai yle tie e fixe poeet a epai itio ot F e total aveage ot ae H H F NFI FI Let ow e te eqatio te balae eqatio fte btittio te eooi oe qatitie we obtai a iple ot ftio: i ftio i ovex i te yle tie te te eeay oitio of optiality ae ffiiet a well. e optial yle tie i o e iplifie ot ftio i

6 o ( ( ( ( ( E D B wee ( ( ( ( E D B Exaple. Let 000 0.9 $ 00 $ 0 $ 750 $ 00. e fo ti ata 00 93 0.9 80 0.9 35 0.9 00 0.9 33650.9 0 E D B..5. e bai oel of Say Say a ivetigate te ae wit oly oe poeet bat. e ot ftio of ti oel i ( ( ( [ ] ( ( [ ] E D B S e optial otio oltio fo ti ae i Lea. e oltio of oel of Say i a if 0 >

o te ( a S o ( ( ( ( b if ( ( ( 0 te o ( a S [ ] o ( ( ( ( ( ( oof. Let ivetigate te ftio S. i ftio i ovex i. e iial vale of te epai bat be i o ( B ( ( D(. fte btittio te optial vale of we ave te oitio a of te lea. If ti be i alle te oe te te ot ftio i ootooly ieaig fo all. i fat ppot ti oitio b. eak. e ftio F( ( ( ( i qaati a ootooly ieaig betwee zeo a oe. Vale F( 0 F( ( poitive o tee exit a eovey ate fo wi F ( 0 optial bat be i eqal to oe fo all [ 0 ] ( ]. i egative a. e te a it i geate ta oe fo all eak. e oltio fo te bat be i ot alway itege fo all ( ]. If vale o ( i itege te te poble i olve. Let ow ae tat o i ot itege. Let o ( it ( eote te axial itege ot geate ta o o a ( it ( 7

8 te iial itege ot alle ta o. e optial itege oltio a be eteie fo te followig elatio ( ( { } S S o i ag i. eoe. e optial otio te yle tie a oe qatitie of oel of Say ae ( [ ] ( ] 0 o ( [ ] ( ] 0 o a ( [ ] ( ] 0 o oof. If [ ] 0 i.e. te optial epai bat be i oe te afte btittio we ave te optial yle a oe qatitie. o eteie te ote ae we e te followig elatio ( ( o o o.

Sbtittig te optial epai bat be a yle tie i balae eqatio we get te elt of te teoe. Say i i pape a ot aalyze toe ae fo wi te optial bat be i eve oe. I ti folatio we ave ow tat te oltio pplie by Say i liite to te ae fo ( ]. e eto popoe i ti pape a te ae elt fo te eooi oe qatitie a obtaie by Say. e optial yle tie a eooi oe qatitie fo te itege bat be a be allate wit btittio a wit oe eleetay opeatio. Exaple 3. Let a i Ex.. 000 0.9 $ 00 $ 0 $ 750 $ 00. e fo ti ata te optial otio oltio a te witig poit ae 0.36 o o o o o S a 8.754 0.68 yea 6.88 30.5 $8357. 4...6. e optial be of epai a poeet bate o iiize te ot i epeee o te bat be we apply a axiliay poble (eta-oel. e poble i ( B( ( D( E( i bjet to. i poble wa exteively tie i pape [-5]. Bae o te etioe pape we exaie te otio oltio of ti oel. eoe. ee ae tee ae of optial oltio ( 3 ( i epeee o te et ate fo ti poble a te ii ot expeio 9

30 (i 0 < ( ( ( [ ] { } 3 (ii 0 ( ( ( [ ] 3 (iii > ( ( ( [ ] { } 3 It i eay to ee tat te tee egio fo te optial oltio i epeee o te et ate ae ot iteete. So we a allate te vale a ( < fo wi eite te poeet bat o te epai bat i eqal to oe bt te ote bat be i geate ta oe. Betwee tee vale bot of te bat be ae eqal to oe. Exaple 4. Let a i Ex. 3. 000 $ 00 $ 0 $ 750 $ 00. e fo ti ata 0.34 a 0.66. Let ow btitte 0.9 i te optial oltio. e te optial vale ae a allate i Ex. 3. e optial poeet a epai bat ize a te yle tie of te oel ae i epeee o te et ate: eoe 3. e oe qatitie a yle tie ae

3 (i [ 0 ( ( (. (ii [ ] [ ] ( ( [ ] ( [ ]. (iii ( ] ( (

(. e poof i eay; we t btitte te otio bat be i te oe qatitie a yle tie. Exaple 5. Let a i Ex. 3. 000 0.05 $ 00 $ 0 $ 750 $ 00. e fo ti ata te iial ot fo te bai oel: S (0.05 7589.8 a te iial ot fo te geealize oel (0.05 700.. i ea a ot avig of 3.5 peet of te total EO elate ot...7. olio I ti pape we ave efolate a olve te oel of Say. We ave ow tat fo alle eovey ate it give a bette oltio if te poeet bat be i geate ta oe a o te bai of oel of Say we a obtai a oe effetive oltio fo ige et ate. i elt a be itepete a a geealizatio of oel of Say fo te ae of oe ta oe poeet bat. 3

.. Moel wit aig a Fiite epai ate: Sbtittio oliy... Itotio e oel of Naia a ivea (979 i a atal geealizatio of oel of Say (967. i oel take ito aot tat epai poe ee tie i.e. it epe o apaity. e oel a it oltio will be peete i tee tep. Fit I ow te ftioig of te epai-poeet poe. fte tat te ot ftio will be otte a te te optial eiio vaiable ae eteie eqetially. e peete oel i a exteio of bai oel of Naia a ivea (979. e ato of ti atile ave allowe oly oe poeet bat ize. I allow i ti apte oe ta oe poeet. it will be ow te be of epai a poeet bat ize epe o te et ate.... aaete a ftioig of te oel i ivetoy yte otai two tokig poit. e ea of te e i atifie fo pply epot. Dea i otat i tie i a epai a poeet yle. Spply epot i fille p fo poeet a epai. Sotage i ot allowe i ti tokig poit o tee ae alway ew pot. oeet a epai bat ize eqal. Ue of pae pat e bak te e pot i te epai epot wit a otat et ate till tey ae waitig fo epai. I oppoite to te oel of Say (967 te apaity of te oveal epatet i fiite. It i ae tat epai ate i geate ta te ea ate. fte epai te pae pat ae et bak to te pply epot a tey ae e a ewly pae pot. e legt of epai a paig lea tie ae otat o tey o ot iflee te eiio vaiable. e ateial flow of te oel i ow i fige. e e eiio vaiable a paaete ae iila to tat of e by Say (967. i itae ake it eaie to opae tee oel. e eiio vaiable of te oel: 33

- poeet qatity - be of poeet itege - epai bat ize - be of epai bate itege - poeet/epai yle tie. aaete of te oel: - ea ate it pe it tie - te eovey ate peet of te ea ate te ap ate i - - epai ate pe it tie > - fixe poeet ot pe oe - fixe epai bat itio ot pe bat - olig ot i pply epot pe it pe tie - olig ot i epai epot pe it pe tie. Fige. Mateial flow of te oel oeet Spply Depot Ue epai epai Depot (- Wate Dipoal 34

e followig eqalitie ow elatio betwee te i- a otflow i te tokig poit i a poeet/epai yle. ee eqatio ake it poible to ee te be of vaiable of te oel. ( i poble otai wate ipoal bt it i ot eiio vaiable. e ateial flow a ivetoy tat ae illtate i fige a. e popoe ivetoy olig tategy of ti oel i te btittio poliy offee by Say (967. Fige peet te tategy wee a yle i beg wit oe epai bat ize a te tee lot ize ae followe by oe poeet bate. e axial ivetoy level i eqal to wi a be obtaie fo oogap of ivetoy otol. Ue ite ae epaie at a ate of it pe tie a it ae et bak to epai epot...3. e ivetoy olig ot ftio Ivetoy olig ot a be allate by te elp of fige. Lea aize ti elt. Lea. Let ivetoy olig ot ftio of pply a epai epot be a. ee two ot ftio a be witte i te followig fo:. 35

Fige. Ivetoy level i oel of Naia é ivea ( 3 Spply Depot - - t epai Depot - t epai oeet oof. We will pove te eo eqatio fo te epai epot; te fit eqatio a be allate i a iila way. Let ivie te aea ito tiagle - tiagle B tiagle D a - etagle... -. See Fige 3. e legt of a epai yle i. e aea of a tiagle i. e aea of a tiagle B i eqal to. e aea of tiagle D i (. to i (. e aea of a etagle i i eqal Let ow aize te aea: 36

37 i i. fte oe eleetay allatio we ave te eo eqatio. Fige 3. e allatio of te ivetoy ot i epai epot ( 3..4. Optial poeet/epai yle tie e fixe poeet a epai itio ot ae F e total aveage ot ae F. e oel lea to te followig oliea optiizatio poble: B B - t D

38 > > > itege. poitive 0 0 0 i ( Let ow e te balae eqatio fte btittio te eooi oe qatitie we obtai a iple ot ftio: i ftio i ovex i te yle tie te te eeay oitio of optiality ae ffiiet a well. e optial yle tie i o. e iplifie ot ftio i afte btittio o

39 ( ( ( ( ( E D B ( wee ( ( ( ( ( E D B. oble ( i iplifie a a itege optiizatio oel (. Moel ( i te etaoel of appeix...5. e bai oel of Naia a ivea Naia a ivea ave ivetigate te ae wit oly oe poeet bat. e ot ftio of ti oel i ( ( ( ( ( E D B N. e optial otio oltio fo ti ae i Lea. e otio oltio of oel of Naia a ivea i a if 0 > te o a o N

b if ( ( ( 0 o te a N o ( ( ( (. oof. Let ivetigate te ot ftio S (. i ftio i ovex i. e iial vale of te epai bat be i o B ( ( D( (. fte btittio te optial vale of we ave te oitio a of te lea. If ti be i alle te oe te te ot ftio i ootooly ieaig fo all. i fat ppot ti oitio b. eak. Ftio F( ( ( ( i a qaati ftio i a ootooly ieaig betwee zeo a oe. Vale F i poitive o tee ( 0 i egative a expeio ( F exit a ee ate fo wi F ( 0. e te optial bat be i eqal to oe fo all [ 0 ] a it i geate ta oe fo all ( ]. eak. e oltio fo te bat be i ot alway itege fo all ( ]. If vale o i itege te te poble i olve. Let ow ae tat o o i ot itege. Let eote te axial itege ot geate ta o o a te iial itege ot alle ta o. e optial itege oltio a be eteie fo te followig elatio 40

4 { } N N i ag i. e followig teoe aize te otio oltio of te bai oel ot ivetigate te itege ae. eoe. e optial otio te yle tie a oe qatitie of oel of Naia a ivea ae i epeee of ee ate ( [ ] ( ] 0 o ( [ ] ( ] 0 o a ( [ ] ( ] 0 o.

oof. If [ 0 ] i.e. te optial epai bat be i oe te afte btittio we ave te optial yle a oe qatitie. o eteie te ote ae we e te followig elatio o o ( o. Sbtittig te optial epai bat be a yle tie i balae eqatio we get te elt of te teoe. Naia a ivea i i pape a ot aalyze toe ae fo wi te optial bat be i eve oe. I ti folatio we ave ow tat te oltio pplie by Naia a ivea i liite to te ae fo ( ]. e eto popoe i ti pape a te ae elt fo te eooi oe qatitie a obtaie by Naia a ivea. e optial yle tie a eooi oe qatitie fo te itege bat be a be allate wit btittio a wit oe eleetay opeatio...6. e optial be of epai a poeet bate o iiize te ot i epeee o te bat be we apply a axiliay poble (eta-oel. e poble i ( B D E i bjet to. i poble wa exteively tie i pape (Dobo-ite (000 ite (996a ite (996b ite (997 ite-dobo (999. Bae o te etioe pape we exaie te otio oltio of ti oel. eoe. 4

43 ee ae tee ae of optial oltio a te ii ot expeio ( 3 i epeee o te et ate fo ti poble (i 0 < ( ( ( [ ] { } 3 (ii 0 ( ( ( 3 (iii > ( ( ( [ ] 3.

44 It i eay to ee tat te tee egio fo te optial oltio i epeee o te et ate ae ot iteete. So we a allate te vale a ( < fo wi eite te poeet bat o te epai bat i eqal to oe bt te ote bat be i geate ta oe. Betwee tee vale bot of te bat be ae eqal to oe...7. e itege oltio Let ow apply te elt of appeix. eoe 3. e optial itege epai a poeet bat be a yle tie ae i epeee of ee ate (i 0 < ( ( 4 (ii 0 ( ( (iii > ( ( 4.

Hee ftio eote te axial itege ot geate ta te aget. e poof of te teoe i eay wit eleetay aiplatio we get te elt. Wit tee elt I ave fiie te ivetigatio of te oel...8. olio e oel of Naia a ivea i iila to tat of oel of Say. I ti apte I ave ot peete eial exaple. Naia a ivea (979 ave etioe oe poible exteio i tei pape. S geealizatio ae e.g. otait o epot apaity. e ot of wate ipoal a et ate a eiio vaiable ae popoe by te. 45

.3. Moel wit aig a Fiite epai ate: otio Sppleet oliy.3.. Itotio e ato of ti oel ivetigate a iple oel. e poble i iila to tat of Naia a ivea (979 bt tey apply a ote ivetoy olig poliy wi i iitiate by Say (967 a alle otio ppleet poliy itea of btittio poliy. e ato o ot expe te bat ize expliitly. e oel of o et al. (00 otai a ew folatio fo otio ppleet poliy. ey exaie te ae of apaitate eafatig ate tat i ot geate ta potio a ee ate bt I o ot aalyze tat ae..3.. aaete a ftioig of te oel e ivetoy yte otai two tokig poit. Dea of e i atifie fo epot of able pot. Dea i otat i tie ig ee yle. e epot of able pot i flfille fo pae a epai. Sotage i ot allowe i ti yte o tee ae alway able pot i epot. e poeet a epai bat ize ae eqal. Ue ite et fo te optio poe wit a kow et ate. e apaity of epai epatet i fiite a it wa ae by Naia a ive (979. It i ae tat epai ate i geate ta ea ate tat i geate ta et ate. fte epai te pae pat ae et bak a tey ae e a ew pot. epleiet a epai lea tie ae iegae beae tey o ot iflee te eiio vaiable. e ateial flow of ti oel i epite o fige. Let ow efie te eiio vaiable a paaete of te oel. e eiio vaiable of te oel: - poeet bat ize - be of poeet itege - epai bat ize - be of epai bate itege 46

- poeet/epai yle tie. aaete of te oel: - ea ate it pe it tie - et ate > - p potio ate p > - o fixe poeet ot pe oe - fixe ee bat itio ot pe bat - olig ot of able pot pe it pe tie - olig ot of eable ite pe it pe tie. e ivetoy level ae peete i fige. oeet Uable ot Ue ee Ue Ite Fige. Mateial flow i oel of o Hwag So é o e followig eqatio epeet te i- a otflow of ateial i tokig poit ig paig-epai yle. I e tee tok-flow ietitie to ee te be of eiio vaiable. ( 47

e ivetoy tat i iffeet i ti oel opae to tat of Naia a ivea (979. e ato of ti oel ave ae tat te ivetoy olig of eafate a able pot i oe expeive ta tat of eable ite i.e. epleiet i eooial if te ivetoy level i zeo. I oppoite o et al. (00 ave ae tat ivetoy level of ete a eable ite t be zeo i a ee yle. e qetio aie we oel of Naia a ivea (979 a we o Hwag So a o (00 a be applie. I te followig ti qetio i awee. Fige. Ivetoy tat i oel of o Hwag So é o Uable ot p- - t eable Ite -p t ee oeet iffeee betwee two oel i tat poble of Naia a ivea (979 i oele wit oly oe poeet bat. I ave geealize ti aptio i te lat apte o te oel a be opae. o et al. (00 ave epaate te exaiatio i two pat: oeet bat be ae eqal to oe a epai bat be i oe. I o ot itigi betwee two ae; tey ae peial ae of ti geeal oel. avatage of te peete oel i tat it a be eteie we a wi patial oel i applie i 48

49 epeee of ot paaete a et ate. Of oe tee ivetigatio ae ppote by te eta-oel offee i appeix. I ott te ivetoy olig ot ftio i epeee of eiio vaiable i te ext etio. e oltio of te oel i le to a oliea pogaig poble..3.3. e ivetoy olig ot ftio allatio of ivetoy olig ot i ae by ivetoy level i fige. Ivetoy olig poliy i peeteie. e ato allate tee ot iepeet o optiality of ivetoy olig tategy. e eteiatio of ot i aize i lea. Lea. Let ivetoy olig ot ftio of able pot a eable ite be S a S. ee two ot ftio a be witte i te followig fo: p S p S oof. We will pove te eo eqatio fo te epai epot; te fit eqatio a be allate i a iila way. Let ivie te aea ito - ee yle te lat t ee yle a poeet yle. Ivetoy olig ot ae efie a itegal of a ve. Let I 0 eote te iitial ivetoy level i fige 3 wi i. Wit te elp of ti oetio te aea of - ee yle ae p. e aea of lat ee yle i. e aea of poeet yle a be witte a. See Fige 3.

50 Let ow aize te aea: p S. fte oe eleetay allatio we ave te eo eqatio. Fige 3. e allatio of te ivetoy ot of eable ite ( 3.3.4. Optial poeet/epai yle tie e fixe poeet a epai ot ae o F. e total aveage ot ae eteie i a poeet a epai yle. p p S S F o. poeet t ee - ee I 0 - eable ite t p-

5 e oel lea to te followig oliea optiizatio poble: e oel lea to te followig oliea optiizatio poble: > > > egézétékű. pozitív 0 0 0 i ( Let ow e te balae eqatio ( to iplify te poble. wo otio vaiable ae btitte i te ivetoy olig ot ftio. Fo te ake of ipliity let tee vaiable be te bat ize. Of oe te bat be a be oe bt ti oie ake oe iffilt te ivetigatio. fte btittio te eooi oe qatitie we obtai a iple ot ftio. Let te ew ot ftio eote (.. p p o I exle te yle tie eqetially fo ti ot ftio. i ftio i ovex i te yle tie te te eeay oitio of optiality ae ffiiet a well. e optial yle tie i p p o o.

5 Let ti lat expeio btitte i ot ftio (.. e te followig ot ftio (. i obtaie: p p o o ( ( ( ( ( E D B ( wee ( ( ( ( ( p p E D p p B o o o. oble ( i iplifie a a itege optiizatio oel (. Moel ( i te etaoel of appeix..3.5. e itege oltio I te lat two apte I ave otte te otio oltio fo paig a epai bat be bt ow I iega fo te. I apply te expeio of appeix iet to poe te iete oltio. eoe. e itege oltio of te oel i

53 (i 0 > p p o 4 o o o p p (ii 0 0 p p p p o o o o o (iii 0 > p p o o 4 p p o o o o. Hee ftio eote te axial itege ot geate ta te aget. e poof of te teoe i eay; wit eleetay aiplatio we get te elt. Wit tee elt I ave fiie te ivetigatio of ti oel..3.6. Say It i eay to pove tat te ot of oel of o et al. (00 ae ige ta tat of oel of Naia a ivea (979 if te it ivetoy olig ot of able pot ae ige ta tat of eable ite. I ote ae te oel of o et al. (00 ftio bette i.e.

wit lowe ot. i fat wa pove by ete (004. fte oe efolatio of oel of o et al. (00 te oel beoe poble of a Naia a ivea (979. 54

3. Ivetoy Moel wit Wate Dipoal e objetive of te oel of lat apte wa to iiize te elevat ot i oe to eteie te paig a epai lot ize a te be of tei bate. ot iiizatio i te piay aageet goal. eo qetio aie fo aageet poit of view: If te ate of pot et i a iflee vaiable te ow e pot t be eovee i a ivetoy yle. e awe will be looke fo ti qetio i tee oel of pot eovey aageet. e tee oel oit of two pat: fit te optial lot ize a be of bate ae eteie fo ee/epai a paig/afatig; eoly te optial ee ate i eteie fo kow it afatig/paig ee/epai a wate ipoal ot. So I efie two tep ivetoy optiizatio poble. e fit oel wa iitiate by ite (996a. e pot (otaie i ti ae ae afate i a op o e otaie ae epaie i ti op i oe to tapot pae pat i te to a ote op. e epty otaie ae toe at te eo op a te tey ae ollete a et bak to te fit afatig-epai op at te e of te potio peio. It i eie i te eo op ow ay otaie ae et bak to te fit op fo epai a ow ay otaie ae ipoe off a wate otie. (Wate ipoal a ea a eoay aket of otaie. e followig qetio aie i ti otext: ow peetage of otaie ae epaie a lot ize of afatig a epai if te ai of eiio ake i to iiize te elevat ot. e ext oel (ete (00 ivetigate a eafatig itatio. eafatig i followe by afatig i ti oel. kow pat of ol pot i ete fo te aket fo toe. Wate ipoal poe begi afte eig of eafatig a afte tat afatig poe tat. e qetio i ow ow ay ite t be eafate fo te ete pot a ow ay ite t be ipoe of. i oel oit of two pat: iiizatio of ivetoy ot a te eteiatio of ee ate. 55

Lat I peet a ti poble. (Dobo-ite (004 Let ae tat a fi atifie te ea fo potio a eylig. I ae tat te potio a eylig poee go o tie o potio a eylig ate i fiite. e fi pae te e pot to eylig fo aket a te fi a pae all of it afate a e pot. e qetio i te qatity of pot bogt bak fo te aket o tat te fi iiize te total elevat ot i.e. te of EO-type a o EO-type ot. e oitio of te tee oel ae oo i ivetoy olig byte. ee oitio ae fo at te e of apte. e oitio t be extee wit te aptio tat o EO-type ot ae liea. Let ow bief aize te oltio of te oel. Sie te ai of te oel i to eteie te optial ee ate te elt ae iila fo tee poble. e optial ee i o te boay i te optial oltio i.e. eite all of te ete ite t be eovee a te et ae eplae fo afatig o te ea i et fo afatig a all ete ite ae ipoe off. I pove ti popety i tee oel. ow I peet te oel. 56

3.. epai Moel wit ee Stokig oit 3... Itotio I te Eooi Oe atity oel fo oe ite te optial qatity to be oee o to be poe i eteie. It i ae tat te of fixe ot a olig ot pe tie it i iiize a oe eteiiti tatioay ea i atifie. e avatage of ti oel oit i it ipliity a well a i te poibility to expe expliitly te optial lot ize a te ii ot. Fo te ipt et-p ot pe tie it pe it olig ot tie it ea te patiable optial lot ize x * / a te ii ot * ae ie i ay textbook of potio eooi. It i a eaoable qetio to ak ow tee wiely e oltio will age if te oel i egae i te faewok of evee logiti o eafatig (Fleia et al. (997 Gpta (995 elle a Silve (989 a oe fatio β of pot avig bee poe will be ee a te ote fatio -β will be ipoe off. By of oe of te ato it wa ow i foe pape (ite (994 ite (996a b ite (997 ite a Dobo (999 tat jt eooi pee (ot iiizatio iplie a etai level of ee β (o ipoal of pot a by ti iplie a etai eologial attite o atte wi teial o teologial etitio ave to be oiee. Sop epaiig β oig Ivetoy of evieable Sop Uig β Dipoal Ivetoy of oevieable Fige. e -op yte 57

ee i a vaiety of itatio wi igt be tie a oly oe of te (ite a Dobo (999 i ie ee: I oe op ew pot ae poe a well a e pot ae epaie. e eqally goo egae ew pot a epaie pot (evieable ite ae te e jt a oet i a eo op a afte tat tey ae eite ipoe off at it ot e o toe a oevieable ite at pe tie it pe it ot p to te e of a vaiable olletio iteval [0 ] (ee Fig.. epai: etp ot pe it t k tok of epaie & afate pot wit afatig : etp ot 0 legt of te epai/potio yle tie Fige : ot ipt a ivetoy tok fo te et-p be a 3 at te fit op I te fit op te poe tat wit epaiig lot of e pot wit te ize β/ at it ot k a at et-p ot a late lot of ew pot wit te ize / ae poe at it ot b a at et-p ot. e vaiable be a ae alle et-p be. e total lot ize x of oe epai/potio yle of legt i β/ / (ee Fig.. tok of e pot wit olig ot 0 tie Fige 3. ot ipt a ivetoy tok fo te etp be a 3 at te eo op 58

e pot ae elivee aoig to te ea of te eo op a fo evey it te fatio i ipoe off a β i toe (ee Fig. 3. e paaete β ae alle wate ipoal ate a epai ate epetively. e poble kete ee ove tee level of oplexity a i ti way tee oel wi e te ftio eteie i te pevio oel: Moel I: Fo give ate a et-p be te ot-iial total lot ize x( i.e. x( ag ig( x a be fo by iple all. e te iial x ot i G( G(x(. i oel ow te ipat of te fixe eologial attite a te give et-p be o te total lot ize a o te legt of te epai/potio yle. Moel II: Fo te give ate te optial et-p be ( a ( a be eteie i.e. ( ( ag ig(. e iial ot i te G( G(((. i oel ow te ipat of te fixe eologial attite o te vale of te et-p be a to te lot ize. Moel I Give ( a G(x: Fi x( a G( x* x(*** Moel II Give a G(: Fi ( ( a G( Moel III Give G(: Fi * a G* Fige 4. ee level of te poble olvig Moel III: e optial * ag ig( a be fo. e iial ot i G* G(*. i oel ow wi eologial attite follow fo te oveall ot iiizatio. e oveall iial ot i.e. te ot egae fo all tee level i te G* i i ig( x o a peete i Fig. 4. x If te et-p be ae allowe to be o-itege te a all bae oltio olve alo te eo level poble (ite (994 ite (996a b ite (997 ite a 59

Dobo (999. e ftio G( a bee epaate i (ite (996a b ite (997 ite a Dobo (999 ito G( ( ( wee ( i te o alle EO-elate ot a ( ((eb(-k expee te EO-iepeet pe tie it epai afatig a ipoal ot. log a oel I /II ae egae wee i fixe te ot ( oe ot ave ipat o te ii a te ai qetio i te eteiatio of te optial oltio fo (. i eai alo te fo oel III beae of te lieaity of (. Hee oweve it will be ake wat appe if te be a t be itege. I pape ite (994 ite (996a b ite (997 ite a Dobo (999 te o-itege poble of iiizig te ot ftio G( i. e. te o-itege poble of iiizig te ot ftio ( o a bee olve by aalyzig a peial fatioal poga S( i bjet to. Siilaly te itege poble G( i [ G(] {...} ( ( i [ (] {...} a be olve if a oltio i fo fo iiizig tat axiliay ftio S( o {...}. I pape ite a Dobo (999 te itege poble ( a bee aalyze a it wa tate tat te optial oltio i boay i.e. oe of te lot be eqal oe e etai oitio oe elative eo fo te o-optiality of boay oltio a be give te ii ot ftio ae patly ovex a patly oave e vaio oitio a tee elt a be applie to te iitial oel I-III. I ti pape all popetie fo pape of ite (997 will be pove ow will be ie i geate etail a a exaple will be povie tat ow tat te optial itege oltio i ot eeaily boay. e axiliay ftio S( a bee eive i pape of ite (994 ite (996a b ite (997 ite a Dobo (999 fo te followig ot ftio 60

β ( ( ( ( ( β β wi ove te EO-elate ot fo G(. De to ootoiity oieatio te ftio S( (( / (-β /(ββ (3 a be aalyze itea of (. Fo bot ftio te elatioip ( S( ol. e paaete i te ftio (3 a be eplae by B (- β (ββ² D (ββ² E (- β (4 a t te ftio S ( B D E (5 appea. If te ii of te ftio (5 o {...} i eteie te alo te poble ( i o {...} i olve a ie G( ( (. eefoe it will be ake i te ext etio wi oltio a be fo fo te poble of iiizig te ftio (5 o te et of poitive itege. 3... ppliatio to te epai a wate ipoal oel 3... e optial oltio De to pape of ite (997 te elatio D BD E > 0 ol a te elt of te Lea 4 of appeix ae to be applie. eoe (ite (997: e otio optial oltio fo iiizig te ftio ( i (i { > } {(- β² ( ² β( β } ( β (ii ²-β(β (-β² ( ² β( β ( ( ² β( β ((( ( ( (iii ² β( β ( ( β( β 6

oig to te ae (i - (iii te egio (0 i epaate ito tee begio a it i lea tat te poble of fiig a optial itege oltio ee peial attetio oly fo (0 (. eefoe te two ae (i a (iii will be tie i etail a te elt of eoe 3 fo [3] will be extee. 054 Lea : (i 49 57 ol if 3 a 49-57 > 0 a 3 /3 i 49 054 49 03 te oppoite ae a (iii 49B 57D ol if 4 a 49 > 576 a 49 576 fo evey i te oppoite ae. oof: (i e ieqality ol if a oly if 49 57β(β o (49-57 57 054. Sie > 0 a ol te ieqality i atifie at ot by 3. (iii e ote ieqality ol if a oly if 49(-β 57(β i flfille o (49-576β 57. If 49 576 te tat ieqality ol fo evey β a. I te ote ae te ieqalitie β 57/(49-576 a 4 ol. Lea : e boaie of te two et a flfill: (i if (- > ( a ( ( (iii if > a if ( i te ote ae. oof: (i e elatio B ol if > a β²(- ( ² β( β ol. fte oe tafoatio te ieqality β [(- - (] β(- a to be aalyze a te fola e (i appea. (iii e elatio BD iplie tat (-[(-] a tat (- ( (. (6 If te tateet ol obvioly. If > te [ (] (- ( (- ( a evey atifyig ( atifie alo [ (] ( (7 eefoe te lowe bo fo obtaie fo (3 i alo a bo fo (. If < te evey atifyig ( alo atifie ( (. 6

eak: De to te Lea a boay optial itege oltio will be fo if 3 a 4 (8 epetively. I te ext eoe oitio ae folate wi ee tat te elatio (4 ol. eoe : e optial itege oltio of te EO epai a wate ipoal oel i boay (i if a (ia (- ( a 49 < 49 03 o (ib 49 > 57 a 5 49 ( o (i 49 < 49 57 a ( (- > ( o (iii if a (- > (iiia 49 576 o (iiib 49 > 576 > a 57( (49-03 o (iii 49 > 576 a 57 640 49. oof: (i Wit epet to te pevio two lea te elatio 3 ol if a oly (ia if 054 ( 054 o (ib if. e aalyi of 49 054 ( 49 054 tee ieqalitie poe te etioe elt. (iiia If 49 57 te e to Lea 7 te ieqality 4 ol obvioly. If 49 > 57 te ae (iiib a (iii ae ealt wit by aalyzig te ieqalitie 49 03 ( 49 03 4 a. 49 576 ( 49 576 eak: Let oe iple ae be oiee: : e 49 576 gaatee te boay popety of te optial itege oltio fo a wie age of ipt. : e 4 0 i.e. te fit egio i epty a fo te eo egio te boay popety i flfille. By applyig te eoe of te appeix te followig popety of te boay optial itege oltio a be fo. 63

eoe 3 (ite a Dobo (999: e boay optial oltio fo te iete EO epai a wate ipoal poble i: g B g ( i ( ( 4 (5 g g ( iii ( ( B D 4 oe etaile expeio of te optial oltio will ot be give at te oet. 3... e boay oltio a a appoxiate oltio Let ow te eoe 4 of te appeix be applie to te EO epai a wate ipoal oel. eoe 4 (ite a Dobo (999: e elative eo of a optial boay oltio b i G * * 48 wee b eote te iial vale fo boay oltio a * eote te global ii. oof: Let G b * * be etiate. Sie ( S( te elatio a be expee a G S b S* S * b S S* b ( S S* S* b S S* S * 48. Hee te elative eo of optial boay oltio i ot geate ta. %. 3...3. Mii ot fo te itege poble oig to te fit etio te ii ot g ( i{(: {...}} i g g B g ( ( g g ( ( D ( E g g ( ( 64

If te et-p be ae itege ti ftio oiie wit (. If i agig tee t be witig poit wee two eigbo et-p be ae optial. Below tee poit ae eteie a te beavio of te ot ftio betwee tee witig poit i aateize. Lea 3: If te boay popety ol tee ae two et M a N of witig be wit te popetie (i M {: B ((...} a g ( ( g ( ( g ( fo M a (iii N {: ((BD...} a g ( ( g ( ( g ( fo N. oof: It follow fo ( g ( ( g ( ieiately tat B g (( g ((. Hee te tte of M i pove. e eo ae a be ealt wit i te ae way. eak: e fiite et M {... l } a N { 3...} epaate te egio (0 a ( ito bet of ietial optial et-p be g ( a g (. Let fo itae a. e te et N {: ((-3 } {0.764 0.883 0.93 0.956 0.97 0.978...} Let fially te beavio of te ftio g ( be tie at te iffeet iteval [ i i ] a [ j j ]. Lea 4 (ite a Dobo (999: e ftio g ( i (i ovex o te iteval [ i i ] i...l (ii ovex o [ ] if 4 4-0 (iii ovex o [ j j ] if 4( ( j a (iii oave i te ote ae. I ote wo te ftio i patly pieewie ovex a 65

pieewie oave. oof: Let ( j. e aefl aalyi of g j j B( D( E j ( ow tat te ftio i ovex a oave i te appopiate itatio. eak: Lea 8 (iii ake lea tat fo all a ( te ftio g ( igt be ovex altog te o-itege otio ftio ( i oave! De to te lieaity of ( te popetie of ti ftio ol alo fo te total ot ftio G g ( g ( ( i.e. te total ot i alo patly pieewie ovex a patly pieewie oave epetively. i te otio ae if 0 < < tee igt exit oe optial (ot iial wate ipoal ate. If oweve te extee vale ae feaible te oe of te i optial. (ite (997 65 60 55 G g ( 60 58 56 54 g ( G g ( 50 0 0.5 5 0.6 0.7 0.8 0.9 Fige 5. e ii ot ftio G g ( a te oepoig et-p be g ( Exaple: Let 00 00 6 3 e 5 k b. e te optial itege oltio i obvioly boay a te pevio Lea a be e to expe te ii ot (op. Fig. 5 66

3..3. olio e EO epai a wate ipoal oel wa aalyze. e vaiable et-p be a fo potio a epai witi oe olletio tie iteval wee ppoe to be atal. Fit oitio fo oe axiliay fatioal poga wee ie to ave optial itege oltio at te boay of te feaible egio. Seoly tee oitio wee e to eteie te optial itege oltio a te ii ot fo te epai a wate ipoal oel fo a wie la of oel ipt. ily it wa ow tat te ii ot i a patly pieewie ovex patly pieewie oave ftio of te wate ipoal ate a te elative eo of optial boay oltio i ot geate ta. %. Seveal poble ae bjet to fte tie a fo itae ow to eteie te optial itege oltio if te optial oltio i ot boay ow to ile o-liea epai ot a fiite potio/epai ate i te itege oel. 67

3.. eoveable Ite Ivetoy Syte 3... Itotio atitative oel fo ivetoy yte wit pot eovey aageet povie a atal geealizatio of laial EO oel. e laial EO oel aalyze oe pot ivetoy yte. e iffilty of eovey yte i tat a be of ato ave popoe oel. i apte eal wit oe of tee popoal we ivetigate te oel of ete (999. ete i i wok a tate tat i te popoe oel tee ol be eite o oe ta oe afatig bat a o oe ta oe eafatig bat i a yle. yle i a eqee of ativitie wit a fixe be of bate. e goal of te apte i to eoie te ete` oel. Fit te expliit oel will be ie a a oltio i give fo ti oel beae te ato a eglete to eibe te expliit oel. fte olvig te poble we give a oteexaple wee te afatig a eafatig bate ae titly geate ta oe. By ti oteexaple we ow tat te ete gapial poof tat oe of te bat be eqal to oe i ot oet. I fat e pove ti popety of te bat be fo te aptio tat oly elatively pie (opie bat be of afatig a eafatig ae oiee o i ote wo fo bat be wit a geatet oo ivio geate ta oe. e pape otie te ivetigatio of te popoe oel. Now we ae tat te plaig oizo a yle tie i eiio vaiable. 3... e oel ete a ivetigate i i oel te followig ativitie: - eafatig - ipoal a - afatig. 68

Let a yle be te above-etioe ele wit fixe bat ize fo afatig a eafatig. I a plaig peio tee i oly oe yle. (i a be pove vey eaily by gopig te eafatig a afatig lot. e goal of te eiio ake i to iiize te ot fo afatig a eafatig bat be a ize a fo te ee ate. ee ae EO-oiete etp a olig ot fo eafatig a afatig liea potio a eafatig ot liea ipoal ot a olig ot fo o-evieable ite. e otatio of te oel ae te followig: Syte paaete: - et ate (0 - ate of ea. ot paaete: - etp ot fo afatig - etp ot fo eafatig - olig ot fo afate ite - olig ot fo eafate ite - olig ot fo o-evieable ite - afatig ot - eafatig ot - ot fo ipoig oe o-evieable ite. Let ae tat > i.e. te it afatig a ipoal ot ae titly geate ta te ot of eafatig. If te ateial flow of ti oel i tie te it e be ee tat te ipoe ite t be ewly afate. Fo ti eao tee two ot t be aize. If eafatig i eooial te te it eafatig ot ae lowe ta tat of afatig a ipoal. Deiio vaiable: - bat ize fo afatig 69

- bat ize fo eafatig - M be of afatig bate poitive itege - be of eafatig bate poitive itege - legt of te pot eovey yle - ee ate (0. We ae tat all paaete a te eiio vaiable ae oegative be. We will eibe te ateatial oel wit oe appliatio. Fig.. Mateial flow i te oel Mafatig M (- Fial pot Maket eafatig Ue pot (- Dipoal Fit we exaie tok-flow balae of evieable a eoveable tok. e eqatio ( ow tat te of afate a eafate pot t ove te ea i a yle. Eqatio ( i te elatio betwee te ete pot a te e of tee pot fo eafatig a ipoal. e ateial flow of te oel i ow i Fige. M ( (- ( 70

Fo te liea yte ( a ( we a wite two epaate eqatio fo te afatig a eafatig bate: Fig.. Moellig te ivetoy poliy (3 M7 Sevieable tok I (t - (- eoveable tok I (t eafatig Mafatig M ( (3 a (4 If te ee ate i eqal to zeo i.e. 0 ta te eafatig lot ize i zeo i.e. 0 i elatio (4. It ea tat all ete pat ae ipoe tee i o ee i yte a te aageet poble t ito a iple ivetoy poble. ote iteetig ae i if te et ate eqal to ee ate (. i ae ow a exaple we all ete pat ae ee a tee i o ipoal ativity. Ietity (3 a (4 will be efl to eate o ot ftio. 7

7 Now we ott te total ot ftio. We ake it i two tep. I te fit tep we ivetigate te ivetoy olig ot ftio H( M fo evieable a oevieable pat.i te eo tep we eibe te liea ot L( M of afatig eafatig a ipoal. Let ow allate te ivetoy olig ot H( M. e ivetoy olig poliy i ow i Fige. i poliy i a peeteie poliy a we look fo te optial paaete ( M of ti tategy. Let ae tat te ivetoy level ftio fo a kow tategy ae ftio I (t fo evieable tok a ftio I (t fo eoveable tok 0 t. e ivetoy olig ot ae te aea below ti ftio i.e. t t I t t I t t I M H 0 0 ( ( (. Now we e te popety of te ivetoy poliy tat te of evieable a eovaable pot i a ootoe eeaig liea a otio ftio of tie i te eafatig yle. So te ivetoy yle a be ivie ito two byle. ( te ea i atifie fo eafatig a te eoveable tok i poitive. e legt of ti iteval i eqal to. ( e ea i atifie fo te lat eafatig bat a fo afatig a te tok level of eoveable ite ae ootoe oeeaig. eafatig bat i e i a iteval legt of. e legt of ti byle i. So te ivetoy olig ot ftio a be expee wit te elp of te yle a [ ] q q q q t t I t t I t I t t I t t I t t I M H ( ( ( ( ( ( 0 0.

73 We t ow allate te five itegal. e fit itegal oit of - piee eafatig bate. e ot ae ( 0 q t t I. e eo itegal i oly a eafatig bat ( q t t I. e ti vale i te ot of ivetoy olig of afate pot wi oit of M bate ( M t t I. e optatio of te fot itegal i a little bit opliate. We ave poite ot tat te of te ivetoy level I (ti (t i a ootoe eeaig liea ftio. e taget of ti liea ftio i (-. I poit of tie ti ftio a a vale of. Wit ti aptio te vale of te itegal i [ ] q t t I t I ( ( 0. e fift lat itegal i ( q t t I. Sig p te itegal a wit eleetay allatio we ave te followig expeio fo te ivetoy olig ot: M M H. I ti expeio we ave applie fo eqatio (4 tat. e liea ot of afatig eafatig a ipoal a be allate vey eaily: L( M M (- [ ]

74 Now we a folate te aveage ot ftio a ( M ig p te total ivetoy olig ot (etp a ivetoy olig ot a te liea afatig eafatig a ipoal ot a ivie wit te legt of te yle: M M M a i way we ave otte a o-liea ixe-itege ateatial pogaig poble: > poitive itege 0 0 0 0 ( tat i M M M a ( Befoe olvig poble ( we will ow tat I te ext etio we will olve te poble fo te elevat vaiable. 3..3. Soltio of te oel I ti etio we olve poble ( i two iffeet way. e iffeee i te oe of eliiatig te otio vaiable fo te ot ftio ig eqatio (3 a (4. e fit eto offe to eliiate te itege vaiable a M i oe to expe te afatig a eafatig lot ize. Seo eto gget eliiatio te lot ize i oe to ivetigate a itege pogaig poble. Let follow ti two way.

75 3..3.. Eliiatio of lot ize a We a follow a ote way to olve te poble. Let btitte te afatig a eafatig lot ize M ( a i te ot ftio fo eqatio (3 a (4. fte btittio te te poble a te ext fo: > poitive itege 0 0 tat i ( M M M ( e ot ftio i ow ovex i te legt of te yle o te optial legt a be allate a follow: ( M M M o. fte btittio te optial legt i te ot ftio we ave te followig ot ftio I (M: M M M I ( e ftio I (M a be witte i te ext fo

76 E M D M B M M I ( ( ( ( ( (5 wee ( ( ( ( ( E D B a vaiable a M ae poitive itege a 0. e afatig a eafatig lot ae i ti ae ( ( M M M M a ( M M M. Ftio I (M i qaiovex i a M a ovex i. i popety gaatee te exitee of optial oltio a it i poove by Dobo a ite (000. Let ow itoe a axilliay ftio S(M a follow ( ( ( ( ( ( E M D M B M M S. We look fo a optial oltio of ti ftio fo eafatig a afatig bate a M. i ftio i te expeio e te qae i (5. De to ootoiity

77 oieatio te ftio S(M a be aalye fo olvig bat ize a M wee all oeffiiet ( B( ( D( a E( ae poitive. 3..3.. e otio oltio of te poble De to te elatio ( B( ( D( E( > 0 ol a te elt of te eoe of te appeix ae to be applie. eoe : e otio optial oltio fo iiizig te ftio I (M fo a M i (i ax ( M( (ii ( M( ( ( (iii

78 ( M( Fige 3. Set of i epeee o ete ate ot ftio ( a be witte i te followig fo: f f f f ( ( ( ( ( ( ( ( 0 f ( f ( 0

wee f ax a f ( (. Let ow epeet te ftio f ( a f (. i fige ow te et of poible ipoal ate fo give et ate. e ftio f ( a f ( otai te poit wee te lot be ae eqal to oe. If te et ate i alle ta te te ot ftio ( oit of ftio eibe i poit (i of teoe a te be of lot i oe fo eafatig. I ti ae te ot ftio i liea. If te et ate i betwee a te te ae (i a (ii of teoe o. if te et ate i ove te all tee ae o. oit a a be allate a oltio of eqatio f ( a f (. e ftio ( i ovex fo all et ate. Lea : ot ftio ( i ovex i a twie otioly iffeetiable. e lea a be pove vey eaily wit all. e ee ate will be eteie i te followig etio. 3..3.3. e eteiatio of optial ee ate Now te ot optial ee ate will be allate a te epeee of te optial oltio o et ate i exaie. Let ae tat (- >. 79

80 i aptio ow tat te ivetoy olig ot of ewly afate pot ae ige ta tat of e a ete a te eafate pot. et ate i a witig poit. i i te iget et ate wee te afatig a eafatig ate ae eqal to oe. e ieqality poit ot tat aafatig i oe eooial tat afatig. e poble i ow ( i 0. I olve te poble fo tee ae i epeee o et ate. (i 0 e ot ftio ( a te ext fo. It i eay to ee tat te optial ee ate i: [ ] [ ] < 0 o. It ow tat it i optial to afate witot eafatig a to ipoe of all ete it if te et ate i vey low.

8 (ii < I ti ae te ot ftio ( a two pat. f f ( ( ( ( 0. Sie ot ftio ( i a ovex ftio it i eog to exaie te ftio o poit wete te ftio i ti poit i eeai o ieaig. Let ow efolate te oitio i te follwig fo: <. It ea tat te ot ftio i ootooly eeaig o o. Fo ti ae it i optial all ete ite to eafate witaot ay wate ipoal. (iii < Now te ot ftio i f f f f ( ( ( ( ( (

Let iffeetiate ti ftio i poit a exaie te iffeetial ftio. e expeio i ( ( ( [ ( ] (. i eivate ftio i poit i eeaig. Sie ( > ( it follow tat ( < 0 o te ftio i ootooly eeaig i. e optial ee ate i o. 3..4. olio ivetoy oel wa ivetigate i ti apte. e qetio wa fo wi lot ize a ee ate ae te ot lowet. If te of it afatig a ipoal ot i ige ta eafatig ot a te ivetoy olig ot of ewly afate pot ae ige ta tat of eafate a ee ite te tee ae two ae. If te et ate i vey low te it i optial to afate a to ipoe of all etig ite. fte a give et ate it i optial all ete ite to eafate witot ipoal. I epeee of ti et ate all afatig a eafatig lot ize a be eteie. 8

3.3. potio-eylig oel wit bybakig 3.3.. Itotio laial logiti yte aage te ateial a elate ifoatioal flow fo aw ateial til te fial pot ae elivee to te toe. i ea a fowa flow. evee logiti aage bakwa poe i.e. te e a eable pat a pot et fo te toe to te poe. Evioetal oioe foe opaie to iitiate pot eovey yte. ee way atal eoe a be ave fo te fte geeatio o te fi a otibte to te taiable evelopet effot. i wok aalyze a itatio wee te ete ite ae eyle a te fi ave wit te eylig te iig of ote atal eoe. I ti pape a oel of te EO type i evelope a aalyze i wi a poe eve a tatioay pot ea oig at te ate D > 0. i ea i eve by poig o poig ew ite a well a by eylig oe pat 0 δ of te e pot oig bak to te poe at a otat et ate D 0. It i ae tat te poe i i te itatio to by bak all e pot to eyle a/o to ipoe off te. e paaete δ a ae alle agial e ate a agial bybak (et ate epetively. e eaiig pat of te o-evieable pot (-δ will be ipoe off. (-δ i alle agial ipoal ate. Fit a aalyi of te itatio i povie. e ivetoy tok fo evieable pot fo te potio a eylig poee ( a fo te o-evieable ite i eteie. O te bai of tee elt te lot ize a yle tie fo te a be fo wi iiize te pe tie it total et-p a olig ot. i elt i te expliit eteiatio of a ftio I (δ wi expee tee iial ot a ftio of te agial e a bybak ate. Seoly if liea wate ipoal potio eylig a bybak ot ae itoe te poble appea at wi δ a te total et-p olig a liea ot I (δ N (δ i iial. I ti folatio te poe ake eiio abot ow e ite by bak to eyle. 83

I ti pape we exaie a potio/eylig yte wit peeteie potioivetoy poliy a ae tat tee i o iffeee betwee ewly poe a eyle ite i.e. we apply te a-goo-a-ew piiple. e pape i ogaize a follow. e ext etio itoe te e paaete a eiio vaiable a well a te ftioig of te potio a eylig poee. I te etio 3 te ot ftio of te ivetoy olig will be otte. I te followig two etio we eteie te ot iial yle tie a bat be fo potio a eylig i epeee o te bybak a e ate. Setio 6 povie te optial bybak a e ate fo te ivetoy olig ot wile etio 7 ow te optial poliy of te total (EO a o-eo elate ot oel. I te lat etio we aize te obtaie elt a ow oe ietio of geealizatio. 3.3.. aaete a ftioig of te yte o oel te potio-eylig we e te followig paaete a eiio vaiable. e ateial flow of te oele itatio i ow i Fig. wit te itoe paaete a eiio vaiable. Lot-ize elate paaete of te oel: - D ea ate - D β potio ate (β < - D bybak ate (0 - D γ eylig ate (γ < - S etp ot of eylig - S etp ot of potio - olig ot of evieable ite - olig ot of o-evieable ite. Lot-ize iepeet ot paaete: 84

- w wate ipoal ot fo (-δd - liea potio ot fo (-δd - liea eylig ot fo δd - B bybak ot fo D. Fige. e ateial flow i te oel i a potio a eylig yle otio x (-δd x δd Sevieable tok D Maket eylig D x δd Noevieable tok (-δd Dipoal Deiio vaiable of te oel: - δ agial e ate - agial bybak ate - be of eylig lot poitive itege - tie iteval of eylig - x eylig lot ize x D - be of potio lot poitive itege - tie iteval of potio - x eylig lot ize x D - legt of potio a eylig yle. 85

Fige. Ivetoy tat i te oel ( 3 Stok of evieable ite I I D γ -D D β -D Stok of o-evieable ite I D γ D D D e ea i atifie by eylig te o-evieable pot ig tie peio a toe til te e of ti yle a well a te e pot aive at te ate D < D i o-evieable tokig poit (opae Fig.. De to te give eylig ate > D γ te poe of eylig lat fo oe γ tie it. We te eylig poe i toppe te ea a be eve by te alate tok of eyle pot. aaete of ti fige eote te legt of te eylig yle. 86

fte eylig te poe eve a ea of oe pot wi appea at a otat ate D > 0 e poe a to eteie ow of ew ite to poe at a ate D β <. Depeig o ti ifoatio e a fo ot ow log e a to toe te exe potio. e tie iteval i wi potio a ayig ew potio i aoplie i alle te potio yle a it i be eote by. e tie iteval give te legt of te potio a eylig yle. e poe of toig a ipoig off o-evieable goo a be ogaize i te followig way: te (-δ it wi ave to be ipoe ig oe iteval ae ipoe ig te tie ipoal iteval D (-δ jt we tey aive. Hee oe tok of o-evieable ite i et p ig te olletio iteval - D δ. t te e of te potio yle te ivetoy tok of o-evieable pot attai it peak I [(-(-γ] D wi i te iitial ivetoy level at te begiig of te potio a eylig yle. t te e of a eylig peio te ivetoy tok of evieable eyle pot attai it peak I (-γ D. e peak of te ivetoy tok of ewly poe ite i I (-β D. 3.3.3. Deteiatio of te ivetoy ot Let eote te ivetoy ot fo evieable ite pe it a tie it a let eote te ae ot fo o-evieable ite. If te legt of te potio a eylig yle i give te aveage ivetoy ot H H H fo te ewly poe ite eyle ite a fo te o-evieable ite oepoigly ae a ow i Lea. Let ow ae tat te et ate a te e ate δ ae poitive i.e. tee i eylig a te bybak a e ate ae ot eqal to oe i.e. tee i potio a well. Lea : e aveage ivetoy ot ae i ti oel: H H I D ( γ δ ( I D ( β ( δ ( 87

88 δ δ γ D D H (3 oof. We will pove eqality (3 te ote ae a be pove i te ae way. e ivetoy olig ot of o-evieable ite a be opte wit iviig te aea ito tiagle (- tiagle B tiagle a etagle D D D -. (See Fige 3. e aea of tiagle i D D γ γ γ γ γ. e aea of tiagle B i eqal to B D D γ γ γ. e aea of tiagle i [ ] [ ] D D D γ γ γ. Fige 3. Ivetoy tat fo te o-evieable tok e aea of etagle D i i I D B B D D

Di ( D i. e total ot ae ow B i H. Di fte oe iple allatio we get te elt of (3. Lea : e total ivetoy ot pe tie it i wit H H H H D V ( δ (4 V ( δ γ δ β δ ( δ (5 oof. Fola (4 a (5 ae obtaie if te ot a tie paaete o te left-a ie of (4 ae btitte by te expeio ( (3. Exaple : Let D 000 850 80 β /3 γ /3 / a δ /3. Fo ti ata V(//3 0.67 0.30 06.96 a H 000 06. 96 5348. ol. e ftio V ( δ expee te total ivetoy ot pe tie it a pe ea it. 3.3.4. otal ot iiizatio fo te yle tie Let te etp ot S pe potio a eylig yle a te of etp ot S a S fo te potio a te eylig epetively be give. e te etp ot pe tie it i 89

90 S S S. e aveage ivetoy ot of te oel (δ a be witte i te followig fo i ( ( δ δ V D S (6 Beae of te ovexity of te ot ftio i te potio a eylig yle tie te ot optial yle tie i ( ( δ δ V D S o (7 a te iial total etp a ivetoy ot pe tie it i ( ( ~ δ δ V S D. (8 e optial eylig a potio yle tie ae ( ( δ δ δ V D S o (9 ( ( δ δ δ V D S o. (0 e optial lot ize ae ( ( δ δ δ V DS x o ( ( ( δ δ δ V DS x o. ( Exaple : Let a i exaple D 000 850 80 β /3 γ /3 / a δ /3. It i kow fo Exaple tat V(δ 06.96 a H

9 5348. ol. Settig S 960 a S 440 te total ot pe tie it i aoig to fola (6 5348. 4360 3. e optial legt of te potio yle a eylig yle i 86 0. 3 yea o 04 ay. e iial ot pe tie it i 30445. 5348. 4360 3 ~. 3.3.5. e optial be of lot fo potio a eylig Now we will iiize te ot ftio ( ~ δ i oe to eteie te optial be of lot. fte oe allatio ti ot ftio a be witte i te followig te fo δ δ δ δ δ δ ( ~ E D B D (3 wee δ β δ γ δ δ δ δ δ δ γ δ δ β δ S S E S D S S B S o olve ti poble we a itoe a elaxe axiliay poble (eta-oel (ite (996 Dobo a ite (000: i ( E D B S. pplyig te elt of Dobo a ite (000 te optial otio oltio fo te lot be ( i

9 Lea 3: ee ae tee ae of optial otio oltio ((δ(δ a ii ot expeio I (δ fo te ftio (3: (i (δ B(δD(δ B(δ (δ(δ γ β δ δ δ δ o o S S [ ] { } γ δ β δ δ I S S D (ii (δ B(δD(δ B(δ (δ(δ δ δ o o [ ] δ δ γ δ β δ I S D S (iii (δ B(δD(δ B(δ (δ(δ δ δ β γ δ δ δ o o S S [ ] { } δ δ β γ δ δ I S S D Note tat te expeio fo te fo optial lot (bat be ae ot eeaily itege! Nevetele we all ee i te ext etio tat ti (ieiately patially ot vey efl elt will elp to pove tat te ixe tategie ae oiate by pe oe. Now we itoe te followig ftio [ ] γ β β δ S S S a

93 γ β β δ S S S S. Ftio δ ( a δ ( ae witig poit fo wi te optial be of lot ( i eqal to oe. i i ow i Fige 5. Ftio δ ( epaate te ae (i a (ii a δ ( te ae (ii a (iii. o allate te ftio we ave e te oitio i ae of eqality. It i eay to ee tat δ ( δ (. e poof i left to te eae. Let ow efie te poible et fo (δ wit te elp of ftio δ ( a δ (: { } 0 0 δ δ δ δ I { } 0 0 δ δ δ δ δ J { } 0 0 δ δ δ δ. e et I i epeete by te boe of te poible vale of (δ te ftio δ ( a te poit ( a (δ 0 wee vale i te oltio of te followig eqality fo [ ] δ γ β β S S S a γ β β δ 0 S S S. te et i epeete by te boe of te poible vale of (δ te ftio δ ( a te poit ( a (δ 0 wee vale i te oltio of te followig eqality fo

94 δ γ β β S S S S. e ivetoy ot ftio I (δ a be witte a [ ] { } [ ] [ ] { } S S D J S D S I S S D I δ δ δ β γ δ δ δ δ γ δ β δ γ δ β δ δ Exaple 3: Let a i exaple D 000 850 80 β /3 γ /3 S 960 S 440 / a δ /3. e (//3 55407.4 B(//3 675. (//3 39. D(//3 74. E(//3 6970. e optial bat be ae (//3.067 a (//3. e iial ot ae I (//3 8494.. Fige 4. e epeetatio of et I J a ( (δ 0 ( ( δ I J δ ( δ (

3.3.6. Miiizig te ivetoy olig ot fo te bybak a e ate Befoe iiizig te ivetoy olig ot I (δ we will pove a iple lea. Lea 3: Let vale a b a be poitive. e te followig eqality ol ( a b( a b. oof. Let bot ie of te ieqality aie to te eo powe. e ( a b( a b ab a afte iplifyig a b ab a ti ieqality ol fo all poitive a b a beae ( b 0 a. Let apply ti elt to te tategy wit oe-oe lot: I ( δ D S [ δ ] ( δ J ( S ( β ( δ ( γ δ (. Let ow a DS b DS ( β ( δ ( γ δ ( δ Uig te elt of lea 3 we ave te followig ieqalitie 95

[ δ ] I ( δ DS ( β ( δ DS ( γ δ ( ( δ DS ( β δ DS ( ( γ e lat ieqality ol beae we ave ee te ot wit te expeio ( δ. Wit ti eto it a be ow tat ove et I a ( δ DS ( β δ DS ( ( γ I ( δ e lat expeio i a ovex liea obiatio of te pe tategie i.e. te eylig a potio. e weigt ae te poible pot of agial e a bybak ate δ wi i o-egative a ot geate ta oe. i ot expeio i alway geate ta te alle of te ot of pe tategie: ( δ DS ( β δ DS ( ( γ i DS ( β ; DS ( ( γ { } By ti lat ieqality a poof i give fo te eoe : e optial ivetoy olig tategy i ti potio-eylig oel i a pe tategy: eite to poe to eet te ea ( o δ o 0 o to by bak a to eyle all e pot witot potio ( o δ o. Exaple 5. Let D000 β γ /3 S 960 S 440 850 a 80. e te ivetoy olig ot of eylig i 656.7 a tat of potio 3336.7. It i eooial to eyle wit bybak of all e ite. Exaple 6. Let D000 β /5 γ /3 S 360 S 440 85 a 80. e te ivetoy olig ot of potio i 6059.7 a tat of eylig 6957.0. It i oe effetive to poe a ot to eyle. 96

7. Miiizig te total lot-ize elate a lot-ize iepeet ot I ti etio we iiize te of te EO-elate a EO iepeet ot. e ot ftio i i ti ae ( δ ( δ ( δ I N wee ftio ( δ ( δ D δd ( δ D D N i te of W te liea wate ipoal eylig potio a bybak ot. B e poble to be olve a te fo ( δ i bjet to [ 0 ] [ 0] δ. I te lat etio we ave ee tat ( δ DS ( β δ DS ( ( γ I ( δ i.e. te ivetoy olig ot ae ot geate ta te ovex liea obiatio of te pe potio a eylig tategie. e o-eo elate ot a be appoxiate i te followig way N ( δ ( δ D δd (. B o get ti ieqality we ave ee te lot-ize iepeet ot wit te wate ipoal ot ( δ D a wit ot of bogt bak bt ot eyle ite ( δ D W B. Uig tee two appoxiatio we a give a lowe bo of te total ot ftio 97

{ D } δ { DS ( ( γ D ( } ( δ DS ( β ( δ. B e igt-a expeio i agai a ovex liea obiatio of te pe tategie o { } { ( B } { } ( δ DS ( β D δ DS γ D i DS ( β D DS ( ( γ D ( B. i elt pove te ext eoe : e optial potio-eylig tategy fo te total ot oel i eite to bybak all ol a e ite ( o δ o o to poe ew ite witot bybakig a eylig ( o δ o 0. i elt wa ow by ite [0] fo aote wate ipoal oel a by Dobo a ite [3] fo a potio/eylig oel. I te ae of liea wate ipoal potio eylig a bybak ot a fee oie of bybak a eylig ate betwee 0 a oe of te pe tategie to by bak a eyle o to poe i optial. e optial pe DS D a tategy a be fo by opaig te vale ( β ( ( γ D ( DS. B 8. olio a fte eea I ti apte we ave ivetigate a potio-eylig oel. By iiizig te ivetoy olig ot it wa ow tat oe of te pe tategie (to poe o to eyle all pot i optial. iila popoitio a be obtaie iiizig te total EO a o-eo elate ot. iila elt wa obtaie by ite (997 i a wate ipoal oel wit eafatig a by Dobo a ite (003 i a potio a eylig oel. obably tee pe tategie ae teologially ot feaible a oe e pot will ot et o eve oe a te ol oe will oe bak a oe of te will be ot 98

eylable. i ki of geealizatio of ti bai oel ol be te itotio of a ppe bo o te bybak ate wi i togly alle ta oe. I a ae a ixe tategy wol be eooial opae to te pe tategy potio. ote way to geealize ti oel i to ak fo te qality of te bogt bak pot. I te popoe oel we ave ae tat all ete ite ae evieable. Oe a be pt a qetio: Wo t otol te qality of te ete ite? If te pplie exaie te qality of te eable pot te te bybak ate i togly alle ta oe. If te e ake it te ot all ete ite ae eylable i.e. te e ate i alle ta oe. Wi oe of te otol yte ae oe ot avatageo i ti ae? 99

4. otio laig i evee Logiti 4.. Itotio Seveal aageet poble aie alog te evee ateial flow. Soe ipotat qetio ae te olletio of e pot a ateial a ogaizatio of ti poe; tapotatio toage a tokig of pot a well a itotio of pat a ole i te potio plaig poe afte ogaizatio a otol of iaebly. Oe of te ipotat eea a appliatio fiel i te itegatio of ee i te potio plaig. ee ae oly a few iteatioal pbliatio o ti fiel. Mot of te ae Gea peakig liteate. (Ieft (998 Spegle et al. (997 ateta (997 ee ae oe glo-saxo pape o ti fiel. (Fee-Wybak (000 Gie (000 I kow tee ae o Hgaia pbliatio tat ivetigate ti poble. I ti apte I o ot exaie te ogaizatio of et poee i.e. et aageet. i apte oit of followig etio. e eo etio tie to exte potio plaig wit ee. It ea tat I give oe iigt i te oetio of potio plaig a eylig plaig. I eate a oel to aalyze iaebly plaig wi a be viewe a a egative bill of ateial. e ext etio peet te itegatio of ee i te M potio plaig a otol yte. I ow a ateial eqieet plaig (M ite eo a te plaig tep till te e of ateial. e fot etio aize te ole of eylig i potio plaig. 4.. exteio of potio plaig wit ee e olletio a ee of e pot a ateial ae ew poble i te potio plaig wi eeitate a oetio betwee M a eylig plaig. Ivolveet of eylig i ateial flow ea a ew poble i ateial aageet. otio plaig a otol yte ae evelope fo taitioal potio poee wi i ot aateize by a ylial ateial flow. e ole of eylig ativitie a ieae beae of eeaig aot of aw ateial a of ie i toage pie wi 00

ave eooi a eologial ae. Stog oial pee a ieaig goveetal eglatio ake a et poble fo ee. I ti apte I efie eylig a a et of e pot fo potio poe a fo otie i oe to ee of tee e pot. Iteal eylig pot ae toe pot tat ae ot eeay i a followig potio poe i.e. by-pot o wate. Exteal eylig pot ae toe pot tat ae at te e of life a tey oigiate fo te oe poee. Bot iteal a exteal eylig pot ae ee te i te potio poe. e ai of a eylig poe i to poe ew pot fo eyle pot o to afate eable pat a ole fo fte e. e fte ot able pat a ateial a be ol i a eo a aket o ipoe otie. Mateial flow extee wit eylig poee ivolve toage of aw ateial eifiie pot e-pot a eylig pot. Uetaity of wate a ete pot i tie qality a qatity a etaity of atio of ee poe ake te eylig plaig poe etai. So te plaig beoe a oe oplex poble a tee ae a be of eiio vaiable i te eiio akig. e fit itatio i eiio abot iaebly ee a e poee. eo elevat eiio i o te fiel of afatig a paig i.e. te btittio betwee eyle a ewly poe pot a ateial a alteative poibility of ateial pply. Fo ti otext it i lea tat a itegatio of potio a eylig plaig i eeay. eylig plaig a potio plaig a well ea a tategi a tatial poit of view a te it a a opeative otet. i opeative otet a be ivie a a oigial potio plaig a otol qatity plaig tie a apaity plaig a afatig plaig. Of oe tee ativitie ae extee wit eylig ativitie. oga plaig i eylig ea a ea foeatig of type qatity a atio of eylig pot. O te bai of ti foeatig te eylig ativity a be foe a o te bai of foeate a ete pot a ative plaig a be evelope. If ti foeatig i ot take to be aot te ti poe i efie a paive eylig plaig beae te etepie eat oly o te kow aot of ete eylig pot. 0

4... oetio betwee potio a eylig plaig Itegatio i eeay betwee potio a eylig plaig beae poga a qatity pla of potio i a bai fo foeatig of poga pla of eylig pot a qatity pla of eylig iflee te aw ateial eqieet i tie a qatity. ee ae tee oept of exteio of M yte:. Itegatio of eylig a M.. Diaebly a eqieet plaig. 3. Itegate ateial ipoitio plaig. otio laig oga laig atity laig Spply oble eylig laig oga laig atity laig De Date a apaity laig otio Selig oo apaity laig De Date a apaity laig otio Selig Fige. oetio betwee potio a eylig plaig (ote-ei (99 e fit oept oe ot itegate eiio ppot yte i te M extee wit eylig i oppoite to eo a ti oept. It i a eteiiti a iet exteio of M beae it e oly te paive eylig plaig a it ile oly kow iatlig eylig a ateial pply tategie. e eee of eo a ti oept i aize i te followig etio a I i te fit oept i.e. te itegatio of M yte a eylig i a ote etio. e oetio of tee two yte i peete i fige. 0

4... Diaebly a e plaig Diaebly a e plaig ea ofte a eiio akig abot iaebly a e eae a a eteiatio of eylig pot i tatial plaig a potio plaig. Diaebly plaig ea a eiio abot te eepe of iatlig poe tep a feqey of exetio of iaebly poe. It t be eie i iaebly plaig wete te oigial pot will be eovee o ole pat a aw ateial will be egaie. It i eteie i ae of eylig of ateial a pat tat available ateial alteative iteal o exteal poibility of e ae applie. ee i alway a alteative e e taitioal eto of eylig. i eiio i eteie by teologial a politial oitio wi ae fixe by pot takebak iaebly afatig a e. e followig ata ae eeay to plaig: qality of te eable pot o tei pat iaebly ipetio afatig a toage ot a evee fo ale. Diaebly ativity State veto of iaebly Mole Mafatig oiblility of e v x y y z z VO v x y y z VO eylig pot v j x j... y i y i z i z VO v x... y y z VO Fige. Siltaeo alig of iaebly a ee ativitie (Ieft (998 Spegle et al. (997 ave eteie te exat e apaitie wit a iltaeo iaebly a e pla. e plaig poble i eibe wit a ativity aalytial oel wi i a ixe itege liea pogaig oel i ti ae. ey ave eate a iaebly gap wi otai alteative iaebly tep of a pot (v j wee 03

j... e pot a be iatle i iffeet opoet wi a be fte eopoe o ave iffeet eto of e tat a be lafillig a well. e exetio feqey of iffeet iaebly ativitie (x j wee j. tat i give by be of teatable pot eteie te be of opoet (y ji wee j wi ae able fo fte iatlig. e able qatitie eteie te be of afatig a pepaatio tep (z i wee. tat ae evee a ot i e. e objetive i to axiize te pofit tog te iaebly a e ativity vaiable x j a z i. oveview of ti plaig yte i ow i fige. 4..3. Itegate ateial ipoitio e ai poit of itegate ateial ipoitio i tat it oet te et of eable pot o opoet of tee pot wit appopiate level of afatig poe. It i a iffilt poble tat i ae by atifatio of pot eqieet of afatig a e wit ete pot wile te tie eqieet of tee poee ae iffeet. e poble of ipoitio i to o-oiate taitioal potio eylig a wate ipoal ativitie fte to iiize te expete (afatig eylig toage a tapotatio ot i a plaig peio. ee ae two poible oltio of te ipoitio poble of toage:. otio otol of eiio poe a. peioial otol of eiio poe. e etaity poble a be leae by allatio of pot eqieet a of ete eylig pot. I geeal te toage of all pot a be olve a tee i a oie betwee afatig a wate ipoal of eylig pot. Oeig etite tategie ae aateize by tee toage ipoitio:. toage etitio i taitioal potio. etitio o eylig a 3. etitio o wate ipoal i a lafillig ite. 04

If te toage of eylig pot i ot poible te etitio of eylig a wate ipoal i i keepig wit ti fat. (Ieft (998 4.3. Itegatio of ee plaig i M 4.3.. ie a gop teatable pot New pot ae afate i a potio poe ig ipt pot bt by-pot ae oigiate fo te potio poe tat i ot exle i itial potio. It ea tat goo oigiate i a afatig poe wi oe ot o i a potio pla. By-pot ae flly exle if potio of plae goo i toppe. atity of by-pot a be ee wit tep i pot plaig poe a wit appopiate tep i fiel of paig potio a qality aageet. Wate a be ategoize i two gop: bjetive a objetive wate. Sbjetive wate ae toe ateial wi ae ot fte e by te owe a tee i o ifoatio abot te eability of tee ateial. Objetive wate ae toe ateial wi ae ot eable a tey t ipoe i a lafillig ite. Wate tat ae eable ae ae by ote a ei (99 a eylig goo. ey ave gope tee goo a follow:. By-pot ae toe ateial a eegy wi o i te e-pot. Bypot a be gope i a ote way: et a wate. et ae toe ateial wi ae eable a tey a be elt of a ee poe. Wate ae ot eable o te ee poe i ot ealize i a eooial way.. Sbtaa goo oigiate by pot a by-pot i a potio poe. e ee fo of tee tee ategoie i ae a eylig. If tee ateial ae ot ietly e te tey beoe ivetoy a it lea to a ivetoy eiio poble. 3. Ue pot ae te e-of-life o e-of-e pot. falt of ti gopig i tat all by-pot ae eylig goo altog tee wate ae objetive wate a tey a ot be objet of eylig. e iea of eylig goo oe ot otai objetive wate. gopig of wate i ow i fige 3. 05

eatable pot Wate eable ateial Lafillig et Sbtaa goo Ue pot Fige 3. Gop of eable pot (Bee-oea (993 4.3.. oe of olletio a et e ee i oe befoe olletio of wate i a fi. e olletio of e pot ealize tog pyial a ifoatio oetio of oe a etiatio. 4.3... olletio e fit eleet of a et poe i olletio. olletio ea te tapotatio of e pot to a olletio plae. olletio poe i bae o te plaig ifoatio. Data olletio i a pat of olletio of e pot wi i a ifoatio poe. e olletio eqieet ae eteie i ti poe ilig ae of oe a e ate of tapotatio of ollete goo a appliae. Fte ifoatio i eeay abot type age a qality of e appliae. i ifoatio ae te bai of a veile otig plaig a iaebly a ee plaig poe i.e. tee ativitie ae te plaig bai of te olletio. ee ae tee type of olletio:. e olleto tapot te e pot to a olletio plae tat a be a iaebly fatoy o a tafe plae.. e owe of e pot tapot te ateial to te olletio plae. 3. e obiatio of te above etioe two tategie. I geeal te olletio poe i ealize by wate tapotatio opaie altog te etaile fi take bak te e pot owaay if te oe pae a ew oe. 06

olletio of ata a e pot a iffeet poble:. ee ae paallel ael fo oe to ollet ata a te pply i ot appopiate oete.. akig a oe pe telepoe o ata olletio i ot alway poible. If it i poible te te waitig tie i vey log a afte a be of tial. 3. e tie betwee egitatio a olletio a be oe ta a week beae of te attaiability of tee plae. 4. e give e ate a ot be kept. 5. e olletio ea oly te availability of pot ollete i otaie a ot te pot toe i a oe o i a ella. 6. It i e a veile wi apaity i ot flly e. 7. e ieae i olletio yte lea to a opetitio fo e pot. e goal i to axiize te tilizatio of olletio yte a of ee apaity. I ti ae te olletio ote beoe loge a ti ea geate tapotatio ot a evioetal be. 4.3... Loaig Loaig i all tapotatio a toage poe wi i loaig a loaig veile wit pot a it eege i ae of age of tapotatio failitie. Loaig i eeay i a be of ae i oe to ee te oetatio of ateial flow. e loaig i ot optioal ativity ie tee ae iet a gaal et poee. Loaig ae ealize aily wit a wi lea to ig loaig ot a to ae aage i e pot. 4.3..3. apotatio apotatio i eteie a to get e pot to a olletio plae o to a etal olletio plae. I a oe-tep et poe tapot ea tapotatio of pot to aebly fatoy a it i tapotatio of goo to ext olletio ete i a eveal-tep et poe. Ote type of veile ae e to tapot ta ollet goo i oe to ee tapotatio ot. apotatio i ot a foe ativity i te et poe ie 07

oe a etiatio ae te ae plae if te itae ae ot o log a te olletio to i eig at te etiatio. apot i flfille by tk. e aiig qetio ae te followig:. e be of atootive a oe effetive loaig poee i liite.. e e pot a be aage at loaig a loaig poee a i tapot poe. 3. Moite lea to ooio wi ee te iaebly of te pot. 4. xiliay ateial e i tapotatio ae ot toable o tee i o poibility to toe te i a plae eooial way. 4.3..4. Stoage Stoage i a plae plaig of able pot. e ai of toage i. to pevet te fltatio of paig tapotatio a potio. to balae te iffeee betwee pply a ea 3. to ee te etaity of kow pply a ea ivegee 4. to oie aotet. ee i otpt a ipt oiete toage. Otpt oiete toage oetate o te oe of e pot i.e. o owe wo will ell te e pot. e ipt oiete toage oetate o te etiatio of e pot wi i a iaebly fatoy. i fatoy poe ew ipt fo te afatig poe wit iatlig. 4.3..5. Seletio a aot Seletio of ollete e pot i te aot of pot aoig to peial iaebly o ee opeatio. e oetatio of tapotable e pot i toe i aitio to oete iatlig at ti plae i iaebly ifoatio play a ipotat ole to etiate te fatoy apaity a ale fo te iatle pat a ole. peliiay iaebly a be ae i te eletio poe o te effiiey of te tapot poe a be ieae a te iaeble qatity a be ee. ee two fat lea to a 08

bette tilizatio of apaity. ee ppleet ativitie ieae te ea fo aot fatoy poviig peial evie. ot i ot a eletio of pot afte type of aie a ole bt it i a ativity to ppot iaebly. 4.3..6. akagig akagig a a ftio to pevet to toe to tapot to ietify a to ifo ote abot te goo. i ftio eeitate te ale a e of pot. Oe of te ai of pakagig ativitie i to aieve a tate wi o ot pollte te evioet i poe of pakagig. It a be attaie by e of tapotatio failitie tat eqie le o o pakagig ateial i.e. otaie. e et poe i iflee by te ipoveet poga a evie level a qality. It i ipotat to kow te oe ee i et poee fo exaple evie qality. i eteie te e level of bilt yte a etwok i.e. te ea fo evee logiti evie wi a ee te ot of tee evie. (Walteat 00 4.3.3. Defiitio a type of eylig ee i et of oli liqi a ga tate et btaa goo a e pot i te afatig poe. ll fi ae yte tat eit goo a wate i te evioet a otpt a abob aw a ote ateial a eegy a ipt. Jake (986 itigie iteal aog fi a exteal eylig.. Iteal eylig ea tat pot itee fo eylig et to te afatig fi. ee i iet a iiet eylig. I ae of iet eylig te pot ae ete to te afatig poe tat tey poe. eylig i iiet if te et i peee by a alig befoe potio poe.. eylig aog fi i efie a ee of e pot of ote fi. 3. Exteal eylig i te ae we te pot eylig i ae by ote fi. 4. oopeative eylig a be efie if ot oly e eylig pot flow aog fi bt alo plaig a ogaizatioal ifoatio. i i a peial ae of exteal a aog fi eylig. 5. Mafatig eylig i efie a ee of eylig pot by tei eittet. 09

i i a peial ae of iteal a aog fi eylig. 6. ee of pot eate i te potio poe i alle a piay eylig i ote ae te ee i a eoay eylig. e oetio betwee piay a eoay eylig i ow i fige 4. (ateta (997 Diet Iiet iay ee Fte e Seoay eale Fte ale Fige 4. Gop of eylig (ateta (997 4.3.3.. ote gop of eylig. et o btaa goo ae ete i te ae afatig poe witot ay alig a a ipt.. et o btaa goo ae ete i a ote afatig poe witot ay alig a a ipt. 3. et a btaa goo ae ale tat i a iaebly o a tafoatio. 4. fte tafoatio te ateial ae ete i te ae afatig poe a ipt. 5. e eylig pot ae ete i a ote potio poe afte pot eovey (egai a ee. fte e tey ae toe a tapote i a lafillig ite. 6. e eylig pot ae et to a ote fi fo alig a e afte tat a ipt. 7. ee i a poibility of alig goo i a lafillig ite. 8. oit to 6 ae iteal eylig a poit 7 a 8 ae aog fi eylig. 0

Heteogeeity of et ateial a low oetatio of et ateial ae poble i ee poe wi ea te eae of e ooio a level of azaoe. Bot fato ake oe iffilt te olletio toage tapotatio aot a alig of ateial. (ote H. ei M. 99 4.3.3.. Gop of eylig afte poee. otio wate eylig i a iteal eylig tat toe te et a btaa pot of afatig poe.. eylig ig pot e i a e of pot i oe to ake te pot patly eable. 3. Ue ateial eylig i a iffeet fo eylig ig pot e. e iffeee i tat te pot a ot be a ew eyle. ee pot ae iatle a eovee a a aw o ote ateial ae ete i te potio poe. 4.3.4. Objetive oitio tool a etitio of eylig 4.3.4.. Objetive e objetive of eylig ae te etio of aw ateial a eegy eqieet loa of evioet a avig of toage apaity tog eeae a liqiatio of wate a et ateial. e ai of a pivate etepie i to ee te eqieet of aw ateial a to legte te lifetie of eylig goo i a log i oe to low ow te eatio ate of tee goo a to eeae te etaity. Miiizatio of expeive apital ivetet ae te eeae of lifetie i evelopet of potio poe. i iiizatio ile te eylig logiti afatig plaig a taatio ot. 4.3.4.. oitio e fi t oie a be of exogeo fat i ee poe wi etit te ativity of te etepie fo exaple:

. te fll eovey of aw ateial fo eylig goo i ofte ot poible. eylig goo ae ot ee wit a itee feqey 3. ot all eylig pot ae eable eooially 4. evioet ijig pot oigiate i a eylig poe 5. ee of etai goo i eglate by a law o te e of tee pot i ot a eiio poble. eylig a be ee a a tepoay elieve of e o oe of piay aw ateial i oe to legte te lifetie of a pot. 4.3.4.3. ool o ivetigate te tool e i eylig tee i a iffeee betwee appliatio a alig of eylig pot:. Soe of te eylig goo ae eable witot ay eafatig by te elp of aot tapotatio a toage.. eletio o tafoatio poee t be ae i a alig poe wi a be a biologial-teial o eial-teial poe i oe to ake te eylig pot eable. eylig eiio oel a be applie a a eiio a plaig oel if elevat ifoatio abot te eylig goo ae available e.g. pie qatity a qatity of eylig pot. Oe of te fato ifleig te eiio oplexity i te faetal goal wi a be a ltiieioal goal ftio. I ae of ltiieioal goal tee i a oflit aog te goal a ot oly oflit betwee eologial a eooi objetive. e iffeet gop of fato eteie te ext level of oplexity.. otio poe: a te lowe a ppe bo ae give fo a able qatity of eylig goo b te fittig eqieet of potio a eylig poe ae kow

te et of eylig pot i te ae a ote afatig poee i kow.. eylig pot: a oly et btaa goo o e pot o bot of te ae exaie b te begiig of eylig pot a be otio o iotio i tie te toability of goo afte eteogeeity i.e. leae fo olo o eateitae et. epaability of pat a ole ateial btittio. 3. ee poe: a ept of ee poe i.e. level of iatlig a poeig b by-pot of a ee poe i.e. able a able aage a loe i ee loe i qality. e ai of a potio plaig a otol yte i to eteie te potio a paig qatitie i tie e oieatio of apaity etitio a alig tep fo eviatio of plaig ealizatio a otol i oe to at te faetal goal. 4.3.4.4. etitio. eial etitio: eylig pot ae ot able witot ay feqey beae te qality of goo eteioate wit te feqey of ee. Fte eylig goo ae ot flly eable beae tei eopoability i etite a iatlig i teially ot poible.. Eooi etitio: ot ae by eylig a exee it elt a te avig i piay ateial. 3. Eologial etitio: eylig eeitate eegy to tapot eylig goo a ofte piay ateial to efbi tei qality. eylig i ot efl fo eologial poit of view ie eylig a ae evioetal be wi exee te evioetal tility. 4. yologial etitio: pot afate fo e pot ee to ave a lowe qality ta ew oe o te aket eat abtaie wit tee pot. (ateta (998 3

4.3.5. M yte Faetal objetive of a M yte i to iiize te ifleeable ot i.e. potio tapotatio toage a apaity ot. e tie a qatity ai of tee yte ae te followig: a iial togpt tie b geat aay low ivetoy level axial apaity tilizatio. Fige 5 ow te eogeo a ifleeable eleet of a M yte. Goal Objet apaity Oe ie Utilizatio of apaity ogpt tie atity Wok foe a aet apotability ot ot of apaity ot of otage a toage Fige 5. Objetive of M (ote-ei (99 I oe to itegate ee poee i a M yte it i eeay to ollet te elevat ifoatio to plaig abot eylig pot a eylig poee. etepie evioetal ifoatio yte t be bilt to attai ti ifoatio i a appopiate fo. Law a ittio t be followe wit attetio i ti ifoatio yte; tep to ee eiio t be itoe wi otai evioetal tatiti a ifoatio abot wate ipoal paig eto qality a ateial a eegy balae oieig iffeet ipt a otpt. Evioetal polltio a qatity of ti polltio ae exaie i a potio poe ie te polltio tax i pai o ti bai. 4

Hoizotal exteio of a M yte wit a ee poe ea tee poible geealizatio: eylig poga plaig eylig apaity plaig 3 eylig togpt plaig. I. Beat a ept of a eylig poga i peifie. e followig exteio poibilitie ae i a M yte: apotatio a toage apaitie t be oiee a pioity le ave to eteie i oe to e te etitely toable pot fit. eviio of tapotatio i eeay to oie te qatity a e ate of tapotable eylig pot. Ieiate ilio of ot o oly etitely toable eylig pot i te potio poe i vey ipotat. It i eeay fte to oie te afatig tep qatity of eylig pot a e ate i a afatig poe. Eiio liit ave to be pai attetio i ae of alig of ot eable by-pot. atitative a qalitative iteia t be kept i a fte afatig. II. atity plaig Exteio a tafoatio of a M yte i eeay wit qatity plaig. Net eqieet of pat a aw ateial ae ee by et a wate a tee ee ipt goo ae e a ipt i te potio poe bt ie of eylig goo ea a geat etaity. Go eqieet ae eteie fo ateial eqieet plaig. ee i a exteio eqieet i ata alig a poeig. It otai olletio toage efeig a poeig of ata oete wit: a eliability (potio tie qatity a qality b aie (waitig tie wok foe (abee a peee tie a ateial (otage a availability of ateial at potio plae. 5

Of oe tee ateial t be available ot oly i te potio bt alo i te ee poe. It i eeay a pla to gop of eylig goo afte qatity a type. III. eylig togpt plaig e plaig level epe o oe-ie ea ote i a M yte tey follow ea ote oetively. eylig poe a a ila aate i.e. tei poee ae iepeet fo ea ote. Diffeet tep of potio plaig ae bilt p liealy o iepeee o plaig level i ot oiee. efoability of ea tep epe o peeet i.e. ea eiio level ae oitio fo te followig eiio level. otio poga a apaity ae oete wit ea ote. If ativitie ae ealize wit iepeet togpt tie fo apaity te ti lea to a ioitey i plaig ie qatitie ae fixe i qatity pla a e ate a apaity pla ae ot el fo a give e ate beae e ate give i otat o ot oepo wit eeay e ate. Sie et a wate ateial et i ot otat bt yteatial qatitie eylig ake oe oplex te etiatio of e ate i a taitioal M yte. Lieaity of plaig a yliality of plaig objet ake oe iffilt te tielie of plaig. ee pot ae e fo et eqieet. Fatoy a oee pot pae pat a afety tok ae e to ove go eqieet e eylig goo. e eae of etalize eiio t be etioe i plaig. If et wate btaa goo a e pot appea i a eylig poe te te eylig poe i a ltiple poe a te M yte beoe oe etalize. ee i a poitive oelatio betwee oplexity a etalizee. If a eylig poe i oe etai te a extee M yte i le etalize. (ote-ei (99 4.3.6. M ite eo extee wit eylig e fit pat of a M ite eo extee wit eylig i iila to tat of taitioal M ite eo altog tee i a eyle pot tok lie wi ea tat taitioal ivetoy lie i extee wit a alteative ivetoy. ee tok otai te pae pat 6

a ateial eovee fo ete ite a tey beoe ivetoy. e tee i o iffeee betwee ee a ewly afate ateial. Let ae tat te legt of plaig oizo i 6 week afety tok i 5 it a lea tie i week. e ata of ti exaple otai fige 6. e ateial flow i ow i fige 7. e tok lie i eqal to te of poe eyle a iitial tok level ee wit go eqieet. e afety tok i 5 it wi i oiee at te tok. e expete level of ete ite i 4 it. eylig tok level i give. eylig eqieet ae eqal to 4 it wi ae te be of ete ite. eylig oe oe fo eylig eqieet wit a leg of week lea tie. Halig eqieet i te tok level of eylig poe ee wit eylig oe. lae oe eeipt i te et eqieet ee wit eylig eqieet. lae oe eleae i plae oe eeipt wit a leg of lea tie tat i week i o exaple. 0 3 4 5 6 Go eqieet 0 0 0 0 0 0 oe tok 8 4 eyle tok 5 4 Stok 9 0 5 5 5 5 Net eqieet 3 0 5 0 0 0 lae et 4 4 4 4 4 4 eylig tok 7 4 4 4 4 4 4 eylig eqieet - - 5 4 4 4 eylig oe 5 4 4 4 - - Halig eqieet 0 0 0 - - lae oe eeipt - - 0 6 6 6 lae oe eleae 0 6 6 6 - - Fige 6. M ite eo extee wit eylig (Ieft-Jee (998 7

otio eylig tok eylig E-pot tok Dea Fige 7. Mateial flow i M ite eo 4.4. Say Evioetal potetio i beoe oe ipotat i etepie paxi owaay. ill evioetal potetio i ot a geat bie fo fi i.e. evee logiti i ot a fato of opetitive avatage; it a ot be teate at a tategi level. It beoe opetitive if oiety akowlege evioetal oio ativity ppote wit evioetal ait a te ebe of oiety pae evioet fiely pot. Soe of eto ae peete i ti apte wi elp fi ee te e of piay aw ateial a eegy a well evioetal polltio. Fi a ooe te bet appopiate eto fo ti eklit. View of oiety t age i iteet of evioet. e goveet play a ipotat ole to iflee te opoate tategy of fi. e avig of atal eoe a legte of e of atal eoe ppot to aieve a taiable evelopet pat i iteet of fte geeatio. 8

5. Say a Fte eea evee logiti a it appliatio i potio plaig ae ivetigate i ti wok. evee logiti a opletely bilt i ateial eqieet plaig (M yte. e exteio i tat laial M ite eo otai te file of ete a eable pot i aitio to taitioal ifoatio. Lat ow of M ite eo ow te eqieet of pevio potio pae a/o paig. e ivetoy poble o at ti level wete te eiio ake t ite potio a/o paig lot o ot. It i e eiti i taitioal M yte to eteie te lot ize a Goff-algoit o Silve-Meal eiti a o o. ee eiti apply te optiality oitio of EO oel. i popety i tat te oeig a/o etp ot ae eqal to te ivetoy olig ot i ot ii. e qetio i ow wete EO-type evee logiti ivetoy oel ae applie to lot izig i extee M yte. o awe te lat qetio I ave peete ix evee logiti EO-type ivetoy oel. e oel ave oo oitio tat ae te exlio of otage. e ot tte i iila to tat of laial EO oel i.e. tee ae yle fix paig/potio ot ivetoy olig ot fo ew pot a fix a olig ot fo eable ite. I ave ivetigate te exaie oel e tee oitio I ave ow tat te oel lea to a eta-oel peete i appeix. I a iplify te ot ftio afte it ottio wit two eqalitie. Eite te be of lot o te lot ize a be btitte i te ot ftio. If te be of bate i btitte i te ot ftio te te vaiable ae te lot ize a te poble tat te be of lot i itege a ot be tie i te oel fte. i i te eao wy it i oe iply to btitte te lot ize i ot ftio. Wit te elp of ti eto I ave ow a exaple wee paig a ee lot be ae titly geate ta oe. I ave exaie toe ae we EO-type a o EO-type liea paig/afatig ee a wate ipoal ot ae ile i te ot ftio. Fo ti type of oel I ave ow tat wate ipoal i eglete i te optial oltio i.e. all ete a eable ite ae eooial to ee. e eeay a ffiiet 9

oitio fo eability of all e ite i tat oe of te pe tategie i.e. oly paig/afatig a oplete ee i ot iial. e peete ivetoy oel a be te bai fo toe eiti wi ae appliable i extee M yte. I kow tee i o pbliatio o ti fiel. e geealizatio of Wage-Witi yai lot ize oel wit ee wa etake by ite-sobtzki (000 ite-webe (00 a ite-gob (005. I peet te geealize oel of ite-sobtzki (000 wi i te exteio of oel of Say (967 fo te ae of tie vayig ea a et. I ti oel tee i o wate ipoal ativity. e paaete a vaiable ae iffeet tat of pblie atile a tey ae te ae a tey wee itoe by Say (967. e tok-flow ietity elatio ae te followig: I i t t I i t t t t t t Dt (t I t 0 t 0 it 0 (t 0. t wee I 0 i 0 0. e fit eqality i te balae oetio i tt peio fo ew pot i.e. te ivetoy level at te e of peio t i eqal to te of iitial ivetoy level paig a epai ee wit ea. e eo eqatio ile te ete ite a te ite take i ee poe. e lat ieqalitie ae te oegativity oitio of vaiable. e ot ftio i ( igt I t igt it i t. 0

e ot ftio ile te oeig etp a ivetoy olig ot. Ftio ig i zeo if it aget i opoitive a oe i ote ae. Let ow aize te paaete a vaiable of te oel. aaete of te oel: - D t ea fo ew pot i peio t oegative - t ete e ite i peio t oegative - I 0 iitial ivetoy level of ew pot - i 0 iitial ivetoy level of ete ite - fixe poeet ot pe oe - fixe epai bat itio ot pe bat - olig ot of ew pot - olig ot of e ite - legt of plaig oizo. Vaiable of te oel: - I t ivetoy level of ew pot i peio t oegative - i t ivetoy level of e ite i peio t oegative - poeet qatity oegative - epai bat ize oegative. ite a Sobtzki (000 ave pove oe popetie of te oel. Lea (ite-sobtzki (000: It ol i optial oltio: i t t 0 (t ii I t ( t t 0 (t.

I o ot pove tee popetie it a be fo i above etioe pape. oitio (i ay tat bot epai a paig a ot o i a peio i optial oltio. e eo eqatio expee tat paig o epai o oly toe peio we iitial ivetoy level i zeo. If te iitial ivetoy level i zeo te paig o epai t o. i popety i iila tat of Wage-Witi yai lot ize oel i.e. poeet/potio o i peio wit zeo iitial ivetoy level. we ee te ivetoy olig poliy offee by Say (967 applie tee two popetie. i yai oel a be olve wit eto of yai pogaig bt te oltio i vey tie oig fo elatively all poble o ottio of eiti to eate boptial oltio i eeay. ee fo fte eea i obvio bt te qetio i wete EO-type evee logiti oel ae appopiate to ott a boptial oltio of extee Wage-Witi oel i.e. to eate wit it atifatoy lot ize fo epai a paig. ow a qetio i ow to ott a algoit. e ext gop of qetio i; if tee ae effetive eiti te ow tey ftio. Fo wi ot a yte paaete offe te algoit a atifatoy oltio? o tet te effetivee of algoit it i eeay to ake ilatio. Witot ay eial aalyi we a ot awe ti qetio. ee offee eiti a be e i potio plaig a M yte.

efeee. ow.j. Etove.. (96: ai-oave pogaig Eooetia Vol. 9 No. 4 779-800. ow.j. ali S. (958: otio ove ie wit Ieaig Magial ot I:.J. ow S. ali H.Saf (E.: Stie i te Mateatial eoy of Ivetoy a otio Stafo Uiv. e Stafo 6-69 3. Bee J. oea M. (993: Logitik IM Spige-Velag Beli et al. 4. ate..- Ella L. M. (998: evee logiti: eview of te liteate a faewok fo fte ivetigatio Joal of Bie Logiti. No.. 85-0. 5. ote H. ei M. (99: eylig i S-Sytee Die Betiebwitaft 65-67 6. eléyi J.- Mag B.- Báyaié ót Á.- Báyai. (997: eylig logiztika it a logiztikai ktatáok iaika fejlőő egyik új iáya Logiztika. z. 8-3. 7. e Bito M.. Dekke. (004: faewok fo evee logiti I: Dekke.- Fleia M. Ieft. va Waeove L. (004 E.: evee Logiti: atitative Moel fo loe-loop Spply ai Spige- Beli et al. 3-7 8. Dobo I. (999: otio-ivetoy Stategie fo a Liea evee Logiti Syte Diio ape 43 Falty of Eooi a Bie iitatio Uiveity Bielefel. 9. Dobo I. (00: Optial ivetoy tategie fo EO-type evee logiti yte Wokig ape N. Depatet of Bie Eooi Bapet Uiveity of Eooi a bli iitatio 0. Dobo I. (00: e geealizatio of SHDY oel: a oel wit epai Wokig ape N. 7 Depatet of Bie Eooi Bapet Uiveity of Eooi a bli iitatio. Dobo I. (00: e optiality of ite oel of epai a wate ipoal Wokig ape N. 0 Depatet of Bie Eooi Bapet Uiveity of Eooi a bli iitatio. Dobo I. (003a: opaio of ipoal tategie i liea evee logiti oel Wokig ape N. 4 Depatet of Bie Eooi Bapet 3

Uiveity of Eooi a bli iitatio 3. Dobo I. (003b: Optial potio-ivetoy tategie fo a HMMS-type evee logiti yte Iteatioal Joal of otio Eooi 8-8 35-360 4. Dobo I. (004: ézletoellek a vizta logiztikába I: zakó E. Dobo I. őegyi. (Szek.: Vállalatai veeyképeég logiztika kézletek: aláyok iká ttila tizteletée BÁE Vállalatgazaágta tazék (004 Bapet 90-303 5. Dobo I. ite.-. (000: otio-ivetoy otol i a evee Logiti Syte oeeig of te t It. Wokig Se. o o. Eo. Igl/Ibk 000 tia e-it Vol.. 67-86 6. Dobo I. ite. (999a: opaio of Deteiiti Oe-ot evee Logiti Moel i: Hill. Sit D. (E.: Ivetoy Moellig: Seletio of eea ape eete at te Fot ISI Se Sool (999 Exete 999 69-78 7. Dobo I. ite. (999b: eafatig Moel eoiee: eial Note Diio pape N. 8 (999 Witaftwieaftlie Fakltät Eopa-Uiveität Viaia Fakft (Oe 8. Dobo I. ite. (999: e be of bat ize i a eafatig oel Diio pape 3 Viaia Eopea Uiveity of Fakft (Oe Falty of Eooi a Bie iitatio. 9. Dobo I. ite. (000: e itege EO epai a wate ipoal oel fte aalyi. etal Eopea Joal of Opeatio eea 8 73-94 0. Dobo I. ite. (003: potio/eylig oel wit tatioay ea a et ate etal Eopea Joal of Opeatio eea 35-46. Dobo I. ite. (004: extee potio/eylig oel wit tatioay ea a et ate Iteatioal Joal of otio Eooi 90 3-33. Dobo I. ite. (006: potio/eylig oel wit qality oieatio It. J. of otio Eooi to appea 3. Feitige G. Hatl.F. (986: Optiale otolle ökooie ozee: wege e Maxipizip i e Witaftwieafte e Gyte Beli 4

4. Fee G. Wybak D.. (000: Mateial laig fo a eafatig Faility otio a Opeatio Maageet Vol. 0-4 5. Fleia M. Bloeof-waa J.M. Dekke. va e Laa E. va Ne J..E.E. va e Waeove L.N. (997: atitative oel fo evee logiti: a eview. Eopea Joal of Opeatioal eea 03-7 6. Gpta M..(995 Evioetal aageet a it ipat o te opeatio ftio It. Joal of Opeatio & Deiio Maageet 5(8: 34-5 7. Gí. Dobo I. (006: eelétevezé a vizta logiztikába Vezetétoáy egjeleé alatt 8. Gie V.D.. (000: otio plaig a otol fo eafatig: ity patie a eea ee Joal of Opeatio Maageet 8 467-483 9. Hill.M. (996: Optiizig a potio yte wit fixe elivey ele Joal of te Ope. eea Soiety 47 954-960 30. Holt.. Moigliai F. Mt J.F. Sio H.. (960: laig otio Ivetoie a Wok Foe etie-hall Eglewoo liff N.J. 3. Ieft. (998: Nee fgabe Lögaatze e oktioplag bei okteylig epit N. 6 Fakltät fü Witaftwieafte Ottovo-Geike Uiveität Magebg 3. Ieft. Jee. (998: alyi of M poliie wit eovey optio 0 t It. Wokig Se. o otio Eooi Ibk/Igl tia e- it Vol.. 65-300 33. Jake B. (986: Betieblie eylig: oktiowitaftlie oblee betiebwitaftlie oeqeze Gable-Velag Wiebae 34. elle. Silve E.. (989: aig poliy of ew otaie oieig te ao et of pevioly ie otaie IIE aatio (4: 349-354 35. ite.-. Dobo I. (000: Optial otio-ivetoy Stategie fo a evee Logiti Syte I: Doke E. J. Hatl. F. Lptaik M. Soge G. (E.: Optiizatio Dyai a Eooi alyi: Eay i Hoo of Gtav Feitige yia-velag Heielbeg New Yok 46-58 36. lebe. Mie S. ieülle G. (00: otio tie ivetoy oel fo a pot eovey yte wit ltiple optio Iteatioal Joal of otio Eooi 79-4 37. o S.-G. Hwag H. So.-I. o.-s. (00: optial oeig a 5

eovey poliy fo eable ite opte & Itial Egieeig 43 59-73 38. ot Z. Nagy. (004: logiztika köyezetvéeli voatkozáa: vizta logiztika Özagba az elélet é a gyakolat? D - olgozat BÁE Vállalatgazaágta azék Bapet 39. ot Z. Nagy. Dobo I. (005: Vizta logiztika: Egy fogali keet Vezetétoáy 36 47-54 40. opiky. J. Beg M. J. Legg L. Daappa V. Maggioi. (993: ee a eylig: evee logiti oppotitie oil of Logiti Maageet Oak Book IL 4. wiee Dei W.-Se we: oel fo evee logiti ety by tipaty povie Oega 30 00 35-333 4. Labet D.M. Stok J.. (98: Stategi yial Ditibtio Maageet Iwi Hoewoo IL 43. Mabii M.. itelo L.M. Gele L.F. (998: EO type folatio fo otollig epaiable ivetoie. Iteatioal Joal of otio Eooi 54 73-9 44. Mike G. (00: logiztika köyezetvéeli kééei é a evee Logiti 9. z. űelytaláy BÁE Vállalatgazaágta azék Bapet 45. Mie S. lebe. (00: Optial otol of otio a eafatig i a Siple eovey Moel wit Liea ot Ftio O Spekt 3 3-4 46. Mpy.. oit.. (989: Maagig of logiti etooveet: epiial aalyi of liteate ggetio apotatio eea Fo Vol. 9 No. 77-84 47. Naia N. ivea H.: (979: eteiiti oel fo epaiable ite ivetoy yte wit a fiite epai ate. Iteatioal Joal of otio eea 7(3 5-48. ole. L.- Fai M. (99: evee logiti i plati eylig Iteatioal Joal of yial Ditibtio a Logiti Maageet Vol. No. 7. pp. 35-47. 49. ateta. (997: Fakozept fü ei itegiete oktio- eyligplag- Steegyte (S Walte e Gyte Beli 50. ite. (994: EO epai a wate ipoal oel oeeig of te Eigt Iteatioal Wokig Seia o otio Eooi Vol. 3 83-9 Igl/Ibk 6

5. ite. (996a: e EO epai a wate ipoal oel wit vaiable et-p be Eopea Joal of Opeatioal eea 96 33-34 5. ite. (996b: e extee EO epai a wate ipoal oel Iteatioal Joal of otio Eooi 45 443-447 53. ite. (996: Moellieg vo kobiiete Wieeveweg- Etoggpozee i: H. Wilea (E.: oktio- Zliefeetzweke W afe-et-velag Müe 79-9 54. ite. (997: e a ixe tategie fo te EO epai a wate ipoal poble O Spekt 9 3-9 55. ite. Dobo I. (996: Solvig te itege EO epai a wate ipoal poble I: Flappe S. D. e o. J. (E.: oeeig: Fit Iteatioal Wokig Seia o ee (996 Eiove 47-55 56. ite. Dobo I. (999: alyi of te EO epai a wate ipoal oel wit itege etp be Iteatioal Joal of otio Eooi 59 463-467 57. ite. Dobo I. (003a: z újaazoítá atáa a gazaági oozatagyága Sziga XXXIV. (003 45-63 58. ite. Dobo I. (003b: evee Logiti Moel wit Itege Setp Nbe I: Leopol-Wilbge U. el F. Wäe G. (E.: Opeatio eea oeeig 00: Selete ape of te Iteatioal ofeee o Opeatio eea (SO 00 Spige-Velag Beli Heielbeg New Yok 95-0 59. ite. Dobo I. (004: otio-ivetoy otol i a EO-type evee logiti yte I: Dykoff H. Lake. eee J. (E.: Spply ai Maageet a evee Logiti Spige Velag Beli et al. 39-60 60. ite. Gob B. (005: eilaf-logitik it Logöeetiktioe Zeitift fü Betiebwitaft-Speial Ie 4/005 57-78 6. ite. Sobtzki M. (000: eafatig plaig by evee Wage/Witi oel Eopea Joal of Opeatioal eea 304-35 6. ite. Webe J. (00: e evee Wage/Witi oell wit vaiable afatig a eafatig ot It. J. of otio Eooi 7 447-456 63. ixe. (995: z ivez logiztika é a logiztika it köfolyaat özlekeétoáyi Szele XLV. 66-75 7

64. oge D. S. -ibbe-lebke. S. (999: Goig Bakwa: evee Logiti e a atie evee Logiti Exetive oil ittbg 65. Say D.. (967: eteiiti ivetoy oel fo epaiable ite Naval eea Logiti ately 4 39-398 66. Seieta. Syaete. (987: Optial otol eoy wit Eooi ppliatio Not-Holla tea 67. Spegle. üet H. ek. etz O. (997: Evioetal Itegate otio a eylig Maageet Eopea Joal of Opeatioal ea 97 308-36 68. Stok J. (998: Developet a Ipleetatio of evee Logiti oga oil of Logiti Maageet Oak Book IL 69. Stok J.. (99: evee Logiti oil of Logiti Maageet Oak Book IL 70. akayaa. (985: Mateatial Eooi abige Uiveity e abige et al. 7. ete. va e Laa E. (00: O te o-optiality of te aveage ot appoa fo ivetoy oel wit eafatig Iteatioal Joal of otio Eooi 79 67-73 7. ete.h. (00: Eooi Oeig atitie fo eoveable Ite Ivetoy Syte Naval eea Logiti Vol. 48 484-495 73. ete. (004: Lot-izig fo ivetoy yte wit pot eovey opte & Itial Egieeig 46 43-44 74. iey M. - Saloo M. J. - va Ne J. - va Waeove L. (995: Stategi Ie i ot eovey Maageet alifoia Maageet eview Vol. 37. No.. pp. 4-35. 75. opo G.L. Seti S.. (980: pike Hoizo fo otio laig Maageet Siee 6 9-4 76. Vööaty Gy. Dobo I. (003: aig Stategie i evee Logiti Syte I: iká. (E.: oeeig of te t Iteatioal ISE ofeee Bapet (003 08-088 77. Walteat.. M. (00: ltpoktükfüg al logitie Dietleitg Dietatio eie Uiveität Beli 8

ppeix. e eta-oel e eta-oel i a fatioal poga of iiizig te followig ftio fo abitay eal ipt S( i ( G {(: {...}} i. e. te poble of fiig a optial ( i ie below. e poble tie by Dobo a ite (999 will be otly alle itege poble. e elaxe fatioal poga S( i ( {(: ( } a bee tie by ite (994 996a 996b 996 997 a ite a Dobo (999. It will be alle te otio poble. Fit oe popetie fo by ite (994 996a 996b 996 997 will be peete ee... e exitee of optial oltio fo te otio a te itege poble Bot te poble ave optial oltio at te ae tie. Lea (ite (997: e ftio S( i boe o a o G if a oly if 0 D 0 0 BD 0. ( Let below te elatio ( be flfille: Lea (ite (997: ovie te elatio (6 ol te itege a te elaxe 9

poble ave a optial oltio if a oly if { 0 B 0} { > 0 BD > 0}. (.. e tte of te optial oltio fo te otio poble Let ae tat paaete a B ae poitive. Lea 3 (ite (997: ee ae two ve M( B a N( B D of loal iia i o fo a epetively wit te vale S(M( ( B D E a S(N( ( B D E fo te ftio S(. e ftio S( i ootooly ieaig alog tee two ve. e level et of a ftio i efie a lev F S {( > 0: S( F} fo a abitay F. e ftio S( i alle qai-ovex if te level et lev F S ae ovex fo all feaible F. eqivalet efiitio of qai-ovexity fo a ftio f i ( { } f ( x x ax f ( x f ( x ( 0 xx X (ee ow a Etove (96 akayaa (985. ftio f i titly qai-ovex if ( { } f ( x x < ax f ( x f ( x ( 0 xx X 30

Below oe exaple of te level et fo a qai-ovex ftio i give. eoe : If te elatio > 0 B > 0 D 0 ol te ftio S( i titly qai-ovex. oof: o pove te teoe let ek te oitio of titly qai-ovexity. It follow fo te efiitio of qai-ovexity tat te ftio F( f ( x ( x x a o axi betwee 0 a. Let ivetigate o poble i te followig fo: G ( B ( D ( E wee (>(00 i a abitay poit a ( a feaible ietio. We t ow pove tat ftio G( a o axi fo evey ( a (. (i < 0 > 0 I ti ae te ftio i ovex beae te eato i poitive a te ftio i alo ovex beae vale i egative. e ote ftio ae liea a ovex a te ftio G( i ovex a a o axi fo evey >0. e ae >0 <0 a be ale iilaly. 3

(ii > 0 > 0 - > 0 Let ow ivetigate te eivative of ftio G(. We will ow tat te ftio a a ii if ay. G B ( D ( ( e ae i oiee we te ftio G( i ootooly o-eeaig. e ( ( ( D ( B( e left-a ie ftio i ootooly ieaig a te qaati ftio i ootooly eeaig beae of te o-egativity of paaete a D. (It i eay to ek wit eivatio. ee exit oe a oly oe 0 atifyig te eqality if ( ( D B(. It ea tat o te iteval > 0 te ftio i ootooly ieaig a ootooly eeaig i ote ae. If vale 0 oe ot exit te te ftio G( i ootooly ieaig fo evey oegative. we ave pove of te teoe. I ae of qai-ovex pogaig te followig teoe povie a eeay a ffiiet oitio of optiality. vaiable i alle elevat if it a take o a poitive vale witot eeaily violatig te otait. eoe. (ow a Etove (96 akayaa (985: Let f(x be a iffeetiable qai-ovex ftio of te -ieioal veto x a let g(x be a -ieioal iffeetiable qai-ovex veto ftio bot efie fo x 0. Let x 0 a 0 atify te -ke-lagage oitio a let oe of te followig oitio be atifie: 3

33 (a f xi0 0 > fo at leat oe vaiable x i0 ; (b f xi 0 < fo oe elevat vaiable x i ; ( f x 0 a f(x i twie iffeetiable i te eigbooo of x 0 ; ( f(x i ovex. e x 0 iiize f(x bjet to te otait g(x 0 x 0. oof i oitte; ee ow a Etove (96. Let ow ek oitio ( of eoe to o poble. Lea 4: Let poit ( 0 0. e (a 0 0 0 0 0 0 0 0 0 0 D B B S S ; (b 3 0 0 0 0 0 0 3 0 0 0 0 B B B S S S S. e poof of te lea i eay a i left to te eae. it i ow o otio axiliay poble i a qai-ovex a te ftio S( atifie te oitio ( of eoe. i oitio gaatee te optial oltio. exaple of te ftio S( i ow i Fig..

N( 0.5 M( Fig..: e ve of loal iia a te level et of te ftio S( wit 5 B 0 0 D 5 E 0 a F 48.73 Soe of te popetie of te ftio S( povie by ite (997 ae ollete i ab.. ae B D BD popetie of S( a > 0 > 0 0 ovex i a i titly qai-ovex i ( b 0 > 0 > 0 > 0 ieaig i ovex i > 0 0 > 0 > 0 ieaig i ovex i 0 0 0 0 ieaig i a i ab.. opetie of te ftio S( e expliit oltio of te otio poble i give by eoe 3 (ite (996a If te elatio (6 - (7 ol tee ae tee ae of optial oltio ( a ii ot expeio S fo te ftio (5 o : (i B (** B S* B ( D E 34

(ii -D B (** ( S* BDE (iii BD (** S* ( B D E. B D.3. e optial oltio fo te itege poble.3.. e ae a i ab. Lea 5: Let i eoe to te oitio (i aitioally 49 57 o to (iii aitioally 49B 57D be flfille. e te optial itege oltio i o te lie o o te lie epetively. oof: Let te ae (iii be oiee. Let ae tat S( S( a te optial otio oltio i (*. Let *δ. It a be ow by eleetay opeatio tat S B D E B D δ ( ( ( * * ( δ wee 0 < δ < 0.5. Let te followig poble be ivetigate: δ p. * 5. 0< δ < 05. *( * δ e ftio i ootooly ieaig i δ a ootooly eeaig i *. e δ ( δ * * a 35

5 S( ( B D E. (0 y ote itege oltio wit attai o alle vale ta S( wee M( a S( ( B D E aoig to Lea 3. e ieqality B D 5 S ( ( B D E ( B D E S( ol if a oly if 5 B ( D B ( D i flfille. e iiate oitio of te lea ee ti ieqality. If S( S( te *-δ a te ae etiatio will be fo. eak: Let fo itae 0.5 B 0.04 D0.000 E5. e te optial otio oltio i give by (* (;4.5 wit S(* 4.04 wile S(4 4.03 > S(5 4.09 > S(9 4.08 i. e. te optial itege oltio i ot o te lie. It i alo lea tat te ae b fo ab. te ae effet igt o (ee Fig... S(408 S(404 0 0 3 4 5 6 7 8 9 Fig. : e level et of te ftio S( fo otio a itege oltio wit 0.5 B 0.04 D 0.000 E 5 36

.3.. e ae b i ab. Lea 6: If te otio optial oltio i ot itege te te optial itege oltio i (i B ( g wee g i oe of eaet itege of B (iii BD ( g wee g i oe of eaet itege of. B D oof: (i e 0 < B. Sie te ftio S( i at te ae tie ovex i a oave a ieaig i oe of te etioe oltio i optial..3.3. e ae i ab. i ae o oly fo -D B i eoe a te otio optial oltio i atoatially itege. optial itege oltio o te lie o o te lie fo i te pevio Lea will be alle boay a below te qetio if boay optial itege oltio a be fo fo a igifiat pat of te epai a wate ipoal poble will be ie. eoe 4 (ite a Dobo (999 Let te oitio of te Lea 4 o 5 be flfille. e te followig boay optial oltio fo te iete poble a be fo: g B g ( i B 4 g g ( ii D B g g ( iii B D B D 4 ( oof: (iii It i lea fo te lea tat oe of te two itege oltio ( ( wit < * < i optial. e S( S( ol if a oly if B D ( 37

o if 0 o if B D B D 4 B D 4. i igt a ie i ot le ta * a ot geate ta *. If oe ot flfill tat ieqality te oe a it i optial. e fit ae a be tie iilaily. eoe 5: Let S G eote te iial vale fo te itege poble. e fo te boay oltio ( te elative eo S G S g g ( S S G G 4 ol. oof: Let te ae (iii be oiee a let te oltio ( ot be optial. a g *. e S( g - S G S( g - S(* a g g S( S S( ( B D E G S B ( D E G. e ig ieqality (0: S G B ( D B ( D E 4 B ( D. B ( D E b If g * te te ae etiatio will be fo. e ae (i a be teate iilaly. If te boay popety i ot gaatee te followig Lea ol. Lea 7: e optial oltio of te itege poble flfill (i g g * o g * a (iii g g * o g *. oof: (i If g < * a F g S( g g te ( lev S bt ( lev S a ay ote oltio i ot optial beae of qai-ovexity of ftio S. e ae (iii a be ie iilaly. F g g F g 38

eak: a oeqee fo tat Lea it a be ote tat te optial itege oltio follow te age of te optial otio oltio. If * (* ieae te appopiate lowe bo of te opoet of te optial itege oltio will ot eeae! 39