Learning to face stochastic demand

(1)

Ohduqlqj wr Idfh Vwrfkdvwlf Ghpdqg Vmxu Glguln Ioèp^W Pduld Vdqgvpdun^_

Vhswhpehu 44/ 5333

Devwudfw1 Zh frqvlghu uhshdwhg lqwhudfwlrq dprqj vhyhudo surgxfhuv ri d krprjhqhrxv/ glylvleoh jrrg/ wudghg dw d frpprq pdunhw1 Ghpdqg lv xqfhuwdlq/ dqg lwv odz lv xqnqrzq1 Zh h{soruh dq dgdswlyh vfkhph ohdglqj vxfk surgxfhuv/ ryhu wlph/ wr idfh fruuhfw ghpdqg gdwd1 H{whqvlrqv lqfoxgh qrqfrrshudwlyh jdphv lq zklfk vwudwhjlf lqwhudfwlrq lv ihow yld h{dfwo|wzr uhdo sdudphwhuv1

Nh| zrugv= uhshdwhg sod|/ ohduqlqj/ Qdvk htxloleulxp/ Frxuqrw roljrsro|/ phdq0 ydoxh lwhudwhv/ glyhujhqfh/ vwrfkdvwlf dssur{lpdwlrq1

MHO fodvvlfdwlrq= F:5/ G;6/ O461

41 Lqwurgxfwlrq

Wklv sdshu frqvlghuv d {hg/ qlwh vhwU ri surgxfhuv ri d krprjhqrxv jrrg/ wudghg dw d frpprq pdunhw1 Hdfk lqglylgxdo 5 U lv txlwh frpshwhqw lq uhjxodwlqj klv rxwsxw^ fdqg dovr lq phdvxulqj klv pdujlqdo frvwE^1 Exw 0 e|dvvxpswlrq 0 kh lv olwwoh lqiruphg derxw pdunhw ghpdqg ru vwudwhjlf lqwhudfwlrq1 D pdlq lvvxh qdwxudoo|hphujhv lq wklv vhwwlqj= Krz fdq vxfk djhqwv hyhqwxdoo| frph wr lghqwli|

vdolhqw ihdwxuhv ri wkh ghpdqg fxuyhB

Lq dqdo|}lqj wklv txhvwlrq rxu lqwhuhvw frlqflghv zlwk d vxevwdqwldo olwhudwxuh rq erxqghg udwlrqdolw|dqg ohduqlqj/ gdwlqj edfn wr Vlprq +4<88,1⁴ Wkdw olwhudwxuh/ do0 uhdg|ulfk/ rhuv qhz shuvshfwlyhv rq htxloleulxp1 Lq idfw/ vxfk shuvshfwlyhv duh grp0 lqdqw lq uhfhqw errnv rq jdph wkhru|/ lqfoxglqj Zhlexoo +4<<8,/ \rxqj +4<<;,/ Ixghq0 ehuj dqg Ohylqh +4<<;,1 Wkh|duh dovr dpso|eurxjkw rxw lq pdfur0hfrqrplf vwxglhv ri krz ohduqlqj pd|frqyhujh wr udwlrqdo h{shfwdwlrqv/ vhh Hydqv dqg Krqndsrkmd +4<<:, iru d vxuyh|1

Wklv sdshu kdv d vlplodu remhfwlyh/ qdpho|wr whoo d vlpsoh vwru|derxw wkh ohduqlqj ri pdunhw ghpdqg1 Wkh ohdglqj txhvwlrq lv= krz grhv lqglylgxdo jhw wr judvs wkh dssursuldwh yhuvlrq ri klv pdujlqdo uhyhqxh -B Wkdw hqwlw|lv/ ri frxuvh/

lqglvshqvdeoh iru jrrg ghflvlrq pdnlqj 0 dqg sduw ri klv rswlpdolw|frqglwlrq

- 'E^ +4,

Qruzhjldq Vfkrro ri Hfrqrplfv dqg Exvlqhvv Dgplqlvwudwlrq> h0pdlo= vmxu1 ddpChfrq1xle1qr1 Wkdqnv duh gxh Phow}huv kû|vnrohirqg dqg Uxkujdv1

|Wklv dxwkru wkdqnv iru vxssruw iurp QIU surmhfw Txdqwhf1

4Wkh vwxg|ri ohvv wkdq udwlrqdo ehkdylru lv vxssruwhg e|hylghqfh eurxjkw iruwk e|sv|fkrorjlvwv dqg h{shulphqwdo hfrqrplvwv 0 ohw dorqh lqwurvshfwlrq 0 zklfk riwhq glvfdug wkh xqerxqghg udwlrqdolw|

k|srwkhvlv dvvxphg wr jryhuq ghflvlrq pdnhuv lq wudglwlrqdo hfrqrplf wkhru|1 Vhh Frqolvn +4<<9, iru d glvfxvvlrq dqg uhihuhqfhv1

4

(2)

Ohduqlqj wr Idfh Vwrfkdvwlf Ghpdqg 5 Suhvxpdeo|/ wkh uljkw kdqg vlgh E^ lq +4, ghulyhv gluhfwo|/ dqg pd|eh hdvlo|/

iurp d zhoo nqrzq/ frqyh{/ orzhu vhplfrqwlqxrxv/ frvw fxuyhSE1 Wkh ohiw kdqg vlgh/

-/ ri +4, uhtxluhv/ krzhyhu/ lqvljkwv wkdw duh ohvv lpphgldwh1⁵ Lqghhg/ zh vkdoo frqvlghu lqvwdqfhv zkhuh hduqv udqgrp surw

ZE1c 'c ^ G'E1c '^ SE^

dqg zruvklsv pd{lpl}dwlrq ri lwv h{shfwhg ydoxh .ZE1c 'c ^ D vdolhqw ihdwxuh khuh lv wkdw ^ pxvw eh frpplwwhg lq idfh ri ghpdqg xqfhuwdlqw|/ ehiruh wkh xqghuo|lqj udqgrp yduldeoh 1 lv xqyhlohg ru uhdol}hg1 Zh wdnh hdfk wr eh ulvn qhxwudo dqg wkh jdph wr eh uhshdwhg wlph dqg djdlq1 Wkh h{shfwdwlrq rshudwru . wkxv uh hfwv wkh pdlq frqfhuq zlwk orqj uxq dyhudjh surw1 Pd{lpl}dwlrq ri dyhudjh surw lv/ krz0 hyhu/ uhqghuhg gl!fxow vlqfh wkh pdsslqj E'c ^ :$ .ZE1c 'c ^ kdugo|lv dydlodeoh1 Dw ohdvw wzr vruwv ri kxugohv h{sodlqv lwv xqdydlodelolw|= Iluvw/ wkh dujxphqwv 0 wkh h{rjhqrxv yduldeoh 1 ru wkh hqgrjhqrxv djjuhjdwh vxsso| ' G' S

MU^ 0 frxog eh kdug wr revhuyh> vhfrqg/ wkh sulfh fxuyh E1c ' :$ E1c ' ru wkh glvwulexwlrq ri 1 pljkw uhpdlq xqlghqwlhg ru klgghq1 Vxfk revwdfohv fhuwdlqo|pdnh wkh wdvn ri gl!fxow1 Krz fdq kh rswlpl}h zkhq qr forvhg0irup/ h{solflwo|nqrzq remhfwlyh lv zlwklq uhdfkB

Wr pdnh surjuhvv lq wkhvh pdwwhuv zh srvlw wkdwZE1c ^nS

'^c ^0 ru pruh jhq0 hudoo|/ wkdw.ZE1c ^nS

'^c ^0 eh frqfdyh zlwk uhvshfw wr rzq ghflvlrq^Wkhq/

judqwhg dssursuldwh glhuhqwldelolw|ri wkh h{shfwhg pdunhw fxuyh ' :$ .E1c '/

frqglwlrq +4,/ frphv lq wkh irup

- G' Y

Y^ d.E1c '^o5YSE^ 'GE^c +5, zklfk lv erwk qhfhvvdu|dqg vx!flhqw iru rswlpdolw|1 Wkh ohiw kdqg vlgh ri +5, lv wkh sduwldo ghulydwlyh _Y^^Y ri h{shfwhg hduqlqjv.E1c '^/ dqg wkh uljkw kdqg vlghYSE^

ghqrwhv wkh jhqhudol}hg +vxe0,glhuhqwldo ri frqyh{ dqdo|vlv +Urfndihoodu/ 4<:3,1 Xq0 ghu idluo|lqqrfxrxv uhjxodulw|k|srwkhvhv rq/ zh fdq uhfdvw wkh suhfhglqj lqfoxvlrq +5, lq d pruh idploldu prog

.E1c ' n. Y

Y'E1c '^ 5YSE^c +6, zklfk doorzv xv wr uhlwhudwh wkh sureohp pruh suhflvho|= Vxssrvh hdfk sod|huv nqrzv rqo|klv rzq surgxfwlrq dqg odfnv wkh h{shulhqfh ru nqrzohgjh qhhghg wr h{hfxwh .1 Lq sduwlfxodu/ vxssrvh qr sod|hu fdq revhuyh gdwd shuwdlqlqj wr klv ulydov/ eh lw wkhlu dfwlrqv ru frvw ixqfwlrqv1 Wkhq/ krz fdq vxfk xqlqiruphg/ xqh{shulhqfhg/ ru

5Wr hvwlpdwh krz wkh sulfh uhvsrqgv wr lqfuhdvhg vxsso|/ surgxfhuv frxog ohw pdunhwlqj uhvhdufk surylgh gdwd iurp vxuyh|v ru sdqho vwxglhv/ ru iurp ehqhw0frvw vwxglhv jhduhg dw hvwlpdwlqj fxv0 wrphuv* zloolqjqhvv0wr0sd|1 Zh dvvxph khuh wkdw vxfk dqdo|vhv uhtxluh wrr pxfk hruw ru h{fhvvlyh rxwod|v1

(3)

Ohduqlqj wr Idfh Vwrfkdvwlf Ghpdqg 6 xqhgxfdwhg djhqwv hyhqwxdoo| vxffhhg lq kdylqj +6, vdwlvhg iru doo B Dqg krz fdq vxfk vdwlvidfwlrq qdoo| uhvxow iurp qrq0frruglqdwhg/ ghfhqwudol}hg dfwlrqvB

D vlpsoh revhuydwlrq eulqjv xv vrphzkdw iruzdug1 Qrwh wkdw wkh suhglfwlrq sure0 ohp/ ri hdfk dqg hyhu|up/ uhgxfhv wr wkh ohduqlqj ri htxloleulxp ydoxhv ri phuho|

wzr sdudphwhuv/ qdpho|=

wkh h{shfwhg sulfhR G'.E1c ' dqg wkh h{shfwhg vorsh R2 G'. Y

Y'E1c ' +7, Vr/ zh dvn= Fdq wkh frqfhuqhg sduwlhv 0 yld dq h{solflw surfhvv 0 qdoo|jhw wr irup d fruuhfw/ frpprq suhylvlrqR' ERc R2q Li vr/ wkhq/ lq wkh orqj uxq/ hdfk5U eulqjv iruwk d txdqwlw|^ '^ER wkdw vdwlvhv

RnR2^ 5YSE^c +8, dqg wkh dvvrfldwhg djjuhjdwh'''ER G'S

^ERfrqupvRdv uhtxluhg lq +7,1 Vxfk d glvwlqjxlvkhg sdlu Rc hperg|lqj udwlrqdo h{shfwdwlrq lq wkh phdq/ lv khuh fdoohg d vwhdg| vwdwh1 Hdfk vwhdg|vwdwh R jhqhudwhv dq dvvrfldwhg Frxuqrw0Qdvk htxloleulxp

^ER G' E^ER_MU ri wkh xqghuo|lqj roljrsro|1 Zh vkdoo uhdvrqdeo| dvvxph wkdw wkhuh h{lvwv d qrqhpsw|/ qlwh vhw ri vwhdg| vwdwhv + dqg zkhqfh ri htxloleuld,1

Rxu frqfhuq lv zkhwkhu dqg krz frqyhujhqfh wr wkdw glvfuhwh vhw pd|rewdlq lq wkh orqj uxq1 Xqghu idluo|zhdn k|srwkhvhv Vhfwlrq 5 surylghv dq d!updwlyh dqg frq0 vwuxfwlyh dqvzhu1 Zh lqyhqw dqg rhu wkhuh d udwkhu dsshdolqj/ vlpsoh vwru|derxw wkh ohduqlqj ri pdunhw ghpdqg dv fkdudfwhul}hg dw dq|vwdjh e|wkh suhydlolqj sdlu R' ERc R₂1 Pruhryhu/ zh surgxfh d wudfwdeoh dqg qryho dojrulwkp pxfk dnlq wr vr0 fdoohg fwlwlrxv sod|/ exw rqh zklfk uhtxluhv olwwoh phpru|/ qr vwdwlvwlfdo vnloov/ dqg rqo|wkh fdsdflw|wr vroyh +8, uhshdwhgo|1 Rxu dssurdfk lv wkhuhiruh glhuhqw iurp wkh dgdswlyh ohduqlqj surfhvvhv irxqg lq Eud|+4<;5,/ Wkruoxqg0Shwhuvhq +4<<3,/ dqg Ndodl dqg Ohkuhu +4<<8,1⁶ Wkh pdlq yhklfoh zloo eh vwrfkdvwlf dssur{lpdwlrq wkhru|

+Ehqdlp/ 4<<9, frxsohg zlwk wkh Ehqgl{rq0Srlqfduì uhvxowv rq wzr0glphqvlrqdo rzv +vhh h1j1 Qhp|wvnll dqg Vwhsdqry/ 4<93,1 Wkh xvh ri vwrfkdvwlf dssur{lpdwlrq phwk0 rgv lq dqdo|}lqj frqyhujhqfh ri ohduqlqj khxulvwlfv lq hfrqrplf frqwh{wv zdv lqlwldwhg e|Pdufhw dqg Vdujhqw +4<;<,/ vhh dovr Vdujhqw +4<<6, dqg Hydqv dqg Krqndsrkmd +4<<7,1

Wkh uhvw ri wkh sdshu lv rujdql}hg dv iroorzv1 Vhfwlrq 5 suhvhqwv wkh dgdswlyh ohduq0 lqj vfkhph dqg hvwdeolvkhv frqyhujhqfh ri uhshdwhg xsgdwlqj1 Vhfwlrq 6 orrnv eulh | dw wkuhh fodvvlfdo pdunhw lqvwdqfhv= prqrsro|/ Frxuqrw roljrsro|/ dqg frpshwlwlyh ehkdylru1 Vhfwlrq 7 frqfoxghv1

51 Uhshdwhg Pdunhw Lqwhudfwlrq

D ihz zrugv rq qrwdwlrq dqg whfkqlfdolwlhv duh lq rughu1 Dq|sdlu R ' ERc R₂ pxvw ehorqj wr d qrqhpsw|/ frpsdfw/ frqyh{ vhw SUn U3 suhvfulehg d sulrul1 Wkh irup dqg qdwxuh ri wkdw vhw vkdoo qrw rffxs|xv dq|ixuwkhu1

6Rwkhu uhodwhg vwxglhv lqfoxgh Irvwhu dqg Yrkud +4<<:,/ dqg Ioèp +4<<;d,1

(4)

Ohduqlqj wr Idfh Vwrfkdvwlf Ghpdqg 7 Uhfdoo wkdw hdfk frvw ixqfwlrqS GU$ U^ in4jlv orzhu vhplfrqwlqxrxv frqyh{1 Ohw wkh lqwhuydo

_J6S G'i^ 5UGSE^ n4j

eh lwv vr0fdoohg hhfwlyh grpdlq1 Wkhq/ iru hyhu|sdudphwhu frqjxudwlrqR fc R2

fc Ht1 +8, kdv d xqltxh vroxwlrq^ '^ER5 _J6S ghshqglqj frqwlqxrxvo|rq R1 Lq wkh h{fhswlrqdo fdvh zkhqR2 ' fczh dvvxph wkdw +8, vwloo kdv d xqltxh/ frqwlqxrxvo|

ghshqghqw vroxwlrq ^ ' ^ER ' ^ERc R f Vr/ lq doo flufxpvwdqfhv/ zh zulwh 'ER G' S

MU^ER wr ghqrwh wkh djjuhjdwh vxsso|jhqhudwhg e|R Li lqglylgxdo vroyhv +8,/ dqg SE lv wzlfh glhuhqwldeoh frqyh{ rq lwv hhfwlyh grpdlq_J6S/ wkhq

Y^ER

YR f dqg Y^ER YR2 f Frqvhtxhqwo|/

Y'ER

YR f dqg Y'ER

YR2 f +9,

Wkh lqhtxdolwlhv +9, duh hdv|wr judvs= Wkh|vlpso|vd|wkdw djjuhjdwh vxsso|lq0 fuhdvhv dv wkh h{shfwhg sulfh R f ru wkh h{shfwhg vorsh R2 fc jxulqj lq +8,/

ehfrph pruh idyrudeoh1

Zh duh qrz uhdg|wr orrn dw pdunhw lqwhudfwlrq dv d g|qdplf surfhvv1 Iru rxu sxusrvhv lw ehjlqv dw wlph f dqg kdsshqv wkhuhdiwhu dw glvfuhwh wlph shulrgv | '

c2c 1 Wkh djhqwv vwduw zlwk d frpprq/ lqlwldo eholhi R^f G' ER^fc R^f₂/ ghwhuplqhg e|

dfflghqw ru klvwrulfdo idfwruv qrw glvfxvvhg khuh1 Jhqhudoo|/ wkh srlqw R^| G' ER^|c R^|₂ kdv wkh vwdwxv ri d frpprq eholhi/ ru d sxeolf suhylvlrq/ khog e|hyhu|rqh mxvw ehiruh ghfhqwudol}hg surgxfwlrq lv xqghuwdnhq dw wlph| Wkhq/ edvhg rqR^|c hdfk lqglylgxdo 5U surgxfhv d txdqwlw|^^| zklfk vroyhv

R^|nR^|₂^^| 5YSE^^|

Lq rwkhu zrugv/ hyhu|rqh lqyduldeo| surylghv d ehvw uhvsrqvh wr klv dfwxdo eholhiv1 Wkhuhdiwhu wkh pdunhw fohduv1 Wkdw lv/ djjuhjdwh ghpdqg htxdov wrwdo vxsso| '^| ' S

MU^^|cdqg doo surgxfhuv vhh wkh uhdol}hg sdlu R^|G'

E1^|c '^|c Y

Y'E1^|c '^|

c +:,

zhoo eh xqrevhuydeoh ru klgghq1, W|slfdoo|/ wkh uhdol}hg R^|ghihdwv wkh suhglfwhg R^| Wkhuhiruh/ wkh odwwhu pxvw eh frpsurplvhg zlwk wkh uvw1 Vr/ diwhu pdunhw forvxuh/

dw wkh hqg ri shulrg |/ doo lqglylgxdov xsgdwh wkhlu eholhi olnh d prylqj dyhudjh dv iroorzv=

R^|nG' Eb|R^|nb|R^| +;,

(5)

MU^ERc zlwk ^ERehlqj wkh vroxwlrq ri +8,1 Qrz ghqh d ixqfwlrq 8ERc 1 G'

E1c 'ERc Y

Y'E1c 'ER

c

zklfk uhfrugv wkh uhdol}hg sulfh dqg lwv vorsh1 Lqvhuwlqj R^| '8ER^|c 1^| lq +;,/ zh jhw d uvw0rughu/ vwrfkdvwlf/ w|slfdoo| qrq0olqhdu/ glhuhqfh htxdwlrq

R^|n' Eb_|R^|nb_|8ER^|c 1^| +<, Zh vwuhvv wkdw wkh lqlwldo eholhi 0 dqg wkh vxevhtxhqw ohduqlqj vfkhph dv zhoo 0 lv frpprq wr doo sod|huv1 Fohduo|/ wkhvh dvvxpswlrqv duh yhu|uhvwulfwlyh1 Wkh|duh mxvwlhg khuh sduwo|e|wudfwdelolw|1 Uhod{dwlrq ri hlwkhu dvvxpswlrq zrxog pdnh d 2mUm0glphqvlrqdo g|qdplfdo v|vwhp uhsodfh rxu vlpsoh rqh kdylqj sodqdu kdelwdw U²1 Kljk glphqvlrqv pdnh frqyhujhqfh dqdo|vlv pruh gl!fxow ru ohvv wudqvsduhqw1 Dgglwlrqdo mxvwlfdwlrq iru wkh frpprqdolw|ri +<, frxog eh vwdwlvwlfdo lq qdwxuh ru uhihu wr dydlodeoh h{shuwlvh/ srlqwlqj wr R^f dv wkh prvw olnho|lqlwldo jxhvv 0 dqg wr b_| wkh prvw uhdvrqdeoh dssursuldwh zhljkw dw vwdjh |

Dq|zd|/ dv pruh h{shulhqfh dffxpxodwhv/ l1h1/ zkhq pdq| revhuydwlrqv kdyh ehhq lqfrusrudwhg lqwr wkh suhydlolqj eholhi/ qhz revhuydwlrqv vkrxog eh dvvljqhg ohvv zhljkw1 Wr dffrxqw iru wklv ihdwxuh/ dqg dovr iru wkh lpsruwdqfh ri frqwlqxhg ohduqlqj/ zh srvlw wkdw

"

[

|'f

b|' n4c dqg

"

[

|'f

b²_| n4 +43,

Lq sduwlfxodu/ wkh fkrlfh b| ' _n| zrxog eh dssolfdeoh/ prvw qdwxudo/ dqg |lhog dq hpslulfdo dyhudjh R^|n ' ^R^f^{n R}_n|^{nuuun R}^| zklfk frpsolhv zlwk Ed|hvldq xsgdwlqj1 Qrwh wkdw b| $ f Qrwh dovr wkdw +<, lv dffrpsdqlhg 0 dqg gulyhq 0 e|wkh h{rjhqrxv vwrfkdvwlf surfhvv 1^fc 1c 1 Zh dvvxph wkdw wkhvh yduldeohv duh lqghshqghqw dqg lghqwlfdoo| glvwulexwhg1⁷ Zh qrz vwdwh rxu pdlq uhvxow iruwkzlwk=

Wkhruhp 4 +Frqyhujhqfh wr eholhiv wkdw duh frquphg lq wkh phdq, Vxssrvh lv frpsdfw frqyh{1 Dovr vxssrvh wkdw wkh dyhudjhg ixqfwlrq

sER G'.8ERc 1 G'

.E1c 'ERc . Y

Y'E1c 'ER

7Wklv dvvxpswlrq pd|eh uhod{hg lq whupv ri vr0fdoohg pl{lqj frqglwlrqv +Ehqdlp/ 4<<9,1 Dovr/

wr fdswxuh pxwxdo lqwhughshqghqfh ehwzhhq vhyhudo pdunhwv/ wkhvh srvvleo|ehlqj fkdudfkwhul}huhg e|>wkhuh pljkw h{lvw d Pdunryldq ihhgedfn phfkdqlvp iurpswr1 Li vr/ rqh zrxog suhvxph wkdw hdfk {hg s|lhogv dq hujrglf Pdunry fkdlq lq 1 Zh vkdoo qrw hoderudwh rq vxfk jhqhudolwlhv> vhh Ehqyhqlvwh hw do1 +4<<3,1

(6)

Ohduqlqj wr Idfh Vwrfkdvwlf Ghpdqg 9 rqo| kdv lvrodwhg {hg srlqwv lq dqg lv frqwlqxrxvo| glhuhqwldeoh wkhuh zlwk glyhu0 jhqfh

_sER G' YsER

YR nYs2ER YR2 2

Wkhq/ iru dq| lqlwldo R^f 5 c wkh vhtxhqfh iR^|j frqyhujhv doprvw vxuho| wr d {hg srlqw R'sER1

Surri1 Iru wkh vdnh ri wkh dujxphqw/ ohw f G' f dqg & G' bf n nb&3/

& ' c2c ghqh d qhz lqwulqvlf wlph vfdoh1 Revhuyh yld +43, wkdw & $ n4 dqg _&n_& 'b_&$f Dovr revhuyh wkdw +<, fdq eh uhzulwwhq lq wkh irup

R^|n 'R^|nb|

8ER^|c 1^|R^| c

ru htxlydohqwo|/ xvlqj wkh qrwdwlrq RE& 'R^&/ lw frphv dv d glhuhqfh txrwlhqw RE&nRE&

&n& '8ERE_&c 1^&RE_& 'sERE_&RE_& ne_&

zkhuh sER '.8ERc 1 lv wkh dyhudjhg ixqfwlrq/ dqg zkhuh e& G'8ERE&c 1^&sERE&

ghqrwhv dq huuru ru d ghyldwlrq iurp wkh phdq ydoxh1 Wkxv/ lq khxulvwlf whupv/

zkhq ljqrulqj wkh huurue_&csurfhvv +<, fdq eh vhhq dv d qxphulfdo lqwhjudwlrq vfkhph ã od Hxohu ri wkh ruglqdu|glhuhqwldo htxdwlrq

_

_RE 'sERR +44,

Zh qh{w glvfxvv wkh dv|pswrwlf ehkdylru ri +44,1 Qrwh uvw wkdw dq| vroxwlrq wud0 mhfwru|ri +44,/ vwduwlqj lq S/ zloo uhpdlq lqvlgh wkdw vhw iruhyhu1 Wkh vlpsoh uhdvrq lv wkdw vlqfh sER5S/ wkh yhorflw|sERR srlqwv lqzdugv dw hyhu|erxqgdu|srlqw R5S1 Lq sduwlfxodu/ vlqfhSlv frpsdfw/ hyhu|ruelw pxvw eh erxqghg1 Wkh Ehqgl{rq0 Srlqfduì wkhru|ri erxqghg sodqdu rzv vd|v wkdw dq|vroxwlrq wudmhfwru|ri +44, pxvw frqyhujh hlwkhu wr d vwdwlrqdu|srlqw ru wr d shulrglf vroxwlrq +ru d wr olplw f|foh,/

fi1 Qhp|wvnll dqg Vwhsdqry +4<93, dqg Ndwrn dqg Kdvvhoeodww +4<<8,1 Zh vhh wkdw _dsERRo '_sER2 fc

dqg khqfh/ e|Ehqgl{rq*v fulwhulrq/ +44, kdv qr shulrglf vroxwlrq ru olplw f|foh lq wkh erxqghg grpdlq S Wkh xsvkrw lv wkdw dq|vroxwlrq RE ri +44, frqyhujhv wr d vwhdg|vwdwh dv $ n4/ wkdw lv/ wkh v|vwhp lv joredoo|dv|pswrwlfdoo|vwdeoh zlwk vwhdg|vwdwhv dv wkh rqo|srvvleoh olplwv1

Nqrzlqj qrz wkh jrrg olplw surshuwlhv ri +44,/ wkh uhvw iroorzv iurp vwrfkdvwlf dssur{lpdwlrq wkhru|/ Ehqdlp +4<<9, Wkhruhp 415 dqg Fruroodu| 61:1 Wkdw wkhru|

(7)

Ohduqlqj wr Idfh Vwrfkdvwlf Ghpdqg : vd|v wkdw/ xqghu rxu frqglwlrqv/ +<, dqg +44, kdyh wkh vdph dv|pswrwlf olplw vhwv1 Fruroodu| 41 +Frqyhujhqfh wr Frxuqrw0Qdvk roljrsrolvwlf pdunhw htxloleulxp, Vxs0 srvh _Y'^Y .E'c 1f/ dqg _Y'^Y²2.E'c 1fcdqg wkdw hyhu| frvw ixqfwlrq S lv wzlfh glhuhqwldeoh frqyh{ rq _J6S Li hyhu| lqglylgxdo xsgdwhv klv suhylvlrq

R'

R derxw wkh h{shfwhg sulfh .E'c 1c R₂ derxw wkh h{shfwhg vorsh _Y'^Y .E'c 1

dffruglqj wr +<,/ wkhq lw iroorzv wkdw hdfk surgxfhu*v surgxfwlrq frqyhujhv doprvw vxuho| wr d orqj0uxq Frxuqrw0Qdvk htxloleulxp ohyho iru wkh jdph lq zklfk kdv remhfwlyh .E1c ^nS

'^^ SE^1 Surri1 Vlpso|qrwh yld +9, wkdw

_sER ' Y

Y'.E'c 1Y' YR n Y²

Y'².E'c 1Y' YR₂ f dqg lqyrnh Wkhruhp 4

Vlplodu dujxphqwv duh irxqg lq Ndqlryvnl dqg \rxqj +4<<8, dqg Ioèp +4<<;e,1 Vhh dovr Nhhqdq dqg Udghu +4<;8, dqg Frufkrq dqg Pdv0Frohoo +4<<9,1

6 1 H{dpsohv

Zh vkdoo frqvlghu wkuhh pdunhw lqvwdqfhv/ udqnhg lq whupv ri wkh qxpehu mUm ri roljrsro|phpehuv1

6141 Prqrsro|1 Dq h{wuhph vhwwlqj kdvUdv d vlqjohwrq/ l1h1/mUm' cdqg^ '' Wkhq/ prvw olnho|/ wkh surgxfhu douhdg| nqrzv/ ru txlfno| ghwhfwv/ wkdw kh lv wkh rqo|

vxssolhu1 Vxfk nqrzohgjh lv qrw qhfhvvdu|wkrxjk1 Lqghhg/ kh pd|zhoo eholhyh/ iru txlwh d zkloh ru hyhq shuvlvwhqwo|/ wkdw ulydo djhqwv dovr vxsso| wkh pdunhw/ exw kh uhiudlqv iurp h{sorulqj wkdw pdwwhu1 Lw vx!fhv lqvwhdg iru klp wr revhuyh +:, dw wkh hqg ri hdfk wlph shulrg |/ dqg wr nhhs rq uhvsrqglqj dv ghvfulehg deryh1 Lq sduwlfxodu/ kh qhhg qrw dvfhuwdlq ru uhfrug wkh xqghuo|lqj surfhvvE1^|c '^|

6151 Frxuqrw Roljrsro|1 Qrz/ ohwmUm 2 Rxu vfhqdulr wkhq wv uhshdwhg sod|

ri d Frxuqrw roljrsro|/ jhqhudol}hg khuh wr frpsulvh ghpdqg xqfhuwdlqw|1 Lqvwhdg ri orrnlqj dw lwhudwhg ehvw uhvsrqvhv wr revhuyhg vwudwhj|surohv/ zh irfxv rq uhfxuvlyh lghqwlfdwlrq ri phuho| wzr hqgrjhqrxv sdudphwhuv/ qdpho|wkh dyhudjh sulfh dqg lwv vorsh lq htxloleulxp1 Zkhq mUm : 2 wklv lv d xvhixo vlpsolfdwlrq1 Lq idfw/ zh dyrlg wkh glyhujhqfh wkdw pd|sodjxh fwlwlrxv sod|ri qrq}hur0vxp jdphv1 Ehvlghv/

(8)

frqglwlrq

E' nE'^ 5 Y

Y^SE^c

lq h{dfw qxphulfdo irup1 Lqvwhdg/ qrw nqrzlqj Ec dqg srvvleo|qrw revhuylqj djjuhjdwh vxsso|/ doo surgxfhuv hqwhuwdlq d eholhiR' ERc R₂derxwEE'c E'dqg vroyh wkhlu vxuurjdwh rswlpdolw|frqglwlrqv +8, iru doo Dowhuqdwlyho|/ exw vrphzkdw ohvv dsshdolqj/ rqh pljkw vd|wkdw hdfk surgxfhu eholhyhv kh rewdlqv surw

ZE^c R ' ERnR2

2^^SE^

Wkhq klv ehvw uhvsrqvh vdwlvhv +8,1

Wr vroyh +8, lv riwhq hdv|1 Lw dprxqwv wkrxjk/ zkhq hdfk S lv vprrwk rq_J6S ' dfcn4c wr vroyh wkh v|vwhp

RnR₂^ 'SE^>c

^c > f dqg ^> ' f

iru doo 5Uc

> ehlqj wkh Odjudqjh pxowlsolhu dvvrfldwhg zlwk wkh frqvwudlqw^ fWr dyrlg wkhvh pxowlsolhuv/ dqg wr dffrpprgdwh qrqvprrwk frvw ixqfwlrqv/ lqvwhdg ri dgguhvvlqj +8,/

zh xvh wkh iroorzlqj dowhuqdwlyh surfhgxuh=

Vroyh wkh vwdjh jdph dv iroorzv= Ohw| #|n dqgb# _|Jlyhq wkh dfwxdo frpprq eholhi Rc vroyh wkh dx{loldu|rswlpl}dwlrq sureohp

pd{lpl}h RS

MU^ n^R₂²S

MU^²S

MUSE^

vxemhfw wr ^ firu doo Ohw 'G'S

MU^ER eh wkh hqvxlqj wrwdo vxsso|1 Xsgdwh eholhiv e|vhwwlqj

R # EbRnbE'c R2 # EbR2nbE' Frqwlqxh wr vroyh wkh vwdjh jdph xqwlo frqyhujhqfh1

D qxphulfdo looxvwudwlrq= Wr vlpsoli|vxssrvh wkhuh lv qr xqfhuwdlqw|1 Ohw U frpsulvh upv ' c cDc kdylqj frvw ixqfwlrqv SE^ ' k^n f^² zlwk k ' 2c k2 ' ec k ' Dc ke ' .c dqg kD ' Hc uhvshfwlyho| Wkhuh lv jlyhq dq xqnqrzq ghpdqg fxuyh E' ' fff2' dqg d frpprq lqlwldo eholhi R^f ' EDfc Wkh jdph vwduwv dw wlph | ' f dqg eulqjv wkhq rxw

E'^f ' S2SSSH dqgE'^f '2

Wkh xsgdwhg eholhi ehfrphvR ' ES2SSHHc2czklfk lq lwv wxuq |lhogv wkh uhdol}dwlrq E' ' f dqgE' '2c

(9)

Ohduqlqj wr Idfh Vwrfkdvwlf Ghpdqg <

dqg wkh qhz eholhiR² ' Eec21 Wkh ghyhorsphqw lq vxsso| 'dqg lqglylgxdo rxwsxw ^ lv jlyhq lq Wdeoh 4 ehorz1

Lq wklv h{dpsoh/ wkh fruuhfw vorsh lv txlfno|ohduqhg zkhuhdv lghqwlfdwlrq ri wkh htxloleulxp sulfh uhtxluhv pruh wlph1 Dq|zd|/ dv wkh wdeoh vkrzv/ frqyhujhqfh frphv idluo|txlfno|/ dqg zh fkrrvh wr vwrs diwhulwhudwlrqv1 Wkh uhdol}hg sulfh dqg vorsh duh wkhq

E' ' HeDbf dqgE' '2c zklfk irup d vxevhtxhqw eholhi R^e' EHecDbfc2

Wdeoh 4

Ghyhorsphqw lq 'dqg ^

| q ' ^ ^₂ ^ ^_e ^_D

3 4;91999 731333 6;1666 6:1833 681;66 681333 4 4745175: 5;61<73 5;61364 5;518:9 5;4199: 5;41546 5 :331637 4741848 4731939 4731485 46<1576 46;1:;;

6 7951<5< <71373 <61464 <519:: <41:9; <41646 7 6;9169; :;1:5; ::1;4< ::1698 :91788 :91334 8 7431365 ;61794 ;51885 ;513<: ;414;; ;31:67 9 73:1;;5 ;61364 ;51455 ;4199: ;31:8; ;31637 : 73:1:75 ;61336 ;513<7 ;4196< ;31:63 ;315:9

; 73:1:49 ;51<<; ;513;< ;41967 ;31:58 ;315:3

< 73:1:3< ;51<<9 ;513;: ;41966 ;31:57 ;3159<

43 73:1:3: ;51<<9 ;513;: ;41965 ;31:56 ;3159<

44 73:1:39 ;51<<9 ;513;: ;41965 ;31:56 ;3159;

45 73:1:39 ;51<<9 ;513;: ;41965 ;31:56 ;3159;

46 73:1:38 ;51<<9 ;513;9 ;41965 ;31:56 ;3159;

Wr fkhfn wkh deryh uhvxow/ zh fdq xvh wkh phwkrg lq Pxusk|hw do1 +4<;5,= Li '^W 'S

MU^^W ' ef..fD lv vxfk wkdw vroylqj wkh sureohp pd{lpl}h E'^WS

MU^n₂E'^WS

MU^²S

MUSE^

vxemhfw wr S

MU^ ''^W dqg^ firu doo

uhvxowv lq wkh pxowlsolhu >E'^Wc dvvrfldwhg zlwk wkh frqvwudlqw S

MU^ ' '^Wc ehlqj }hur/ wkhq E^^W_MU pxvw eh d Frxuqrw htxloleulxp1 Lq rxu h{dpsoh >Eef..fD ' ffff2c zklfk zh ghhp vx!flhqwo|forvh wr }hur1

6161 Sulfh0wdnlqj ehkdylru1 Vxssrvh doo 5 U ehkdyh dv wkrxjk qrqh ri wkhp dhfw wkh sulfh1 Dffruglqjo|/ doo upv uhpdlq frqylqfhg wkdw R2 f lq dq|flufxp0 vwdqfh1 Li vr/ +8, uhgxfhv wr

R 5YSE^c +45,

(10)

Ohduqlqj wr Idfh Vwrfkdvwlf Ghpdqg 43 zlwk vroxwlrq ^ '^ER Ohduqlqj/ wkhuhiruh/ surfhhgv zlwk uhvshfw wr wkh h{shfwhg sulfh R '.E1c 'c dqg wkh wzr0glphqvlrqdo surfhvv +<, vlpsolhv wr

R^|n ' Eb_|R^|nb_|8ER^|c 1^| +46, zkhuh

8ERc 1 G'E1c[

MU

^ER Frqyhujhqfh qrz rewdlqv xqghu uhpdundeo|plog frqglwlrqv=

Wkhruhp 51 +Frqyhujhqfh wr frpshwlwlyh htxloleulxp lq rqh pdunhw, Vxssrvh wkh dyhudjhg ixqfwlrq

sER G'.8ERc 1

lv Olsvfklw} frqwlqxrxv dqg kdv rqo| lvrodwhg htxloleuld1 Wkhq +46, frqyhujhv doprvw vxuho| iru dq| lqlwldo R^f wr d {hg srlqw R ' sER dw zklfk doo sulfh0wdnlqj upv kdyh pd{lpl}hg wkhlu h{shfwhg surw1

Surri1 Wkh glhuhqwldo htxdwlrq

R 'sERR +47,

lv dv|pswrwlfdoo| vwdeoh1 Lqghhg/ frqvlghu wkh ixqfwlrq uE G'

] _R_E

REf dsERRo_R

Lw lv O|dsxqry ehfdxvh _{_}^_uE ' dsERRo² fzlwk vwulfw lqhtxdolw|xqohvvR

lv dq htxloleulxp1 Vwrfkdvwlf dssur{lpdwlrq wkhru|+Ehqdlp/ 4<<9, qrz vkrzv wkdw +46, dqg +47, kdyh wkh vdph dv|pswrwlf olplwv1

71 H{whqvlrqv

Zh frqfoxgh e|devwudfwlqj iurp wkh pdunhw frqwh{w1 Wr vlpsoli|zh vxssuhvv xq0 fhuwdlqw|khuh1 Wklv pd|khos wr vhh rwkhu dssolfdwlrqv ri wkh dssurdfk xvhg deryh1

Vxssrvh sod|hu5U rewdlqv sd|rZERc %0 ru uhfrugv pdujlqdo sd|r6ERc % 0 wkdw ghshqgv rq d wzr0glphqvlrqdo sdudphwhu R 5 S dqg rzq fkrlfh % 5 [1 Wkh yhfwru R lv hqgrjhqrxv/ ehlqj wkh uhvxow ri vwudwhjlf lqwhudfwlrq1 Vshflfdoo|/

wkhuh h{lvwv dq djjuhjdwlrq pdsslqjwkdw dvvrfldwhv wr hdfk vwudwhj|suroh%G' E%MUc lq wkh dpelhqw vsdfh [ G' MU[c d wzr0glphqvlrqdo sdudphwhu R ' E%/

ehorqjlqj wr vrph vshflhg vxevhw Sri wkh Hxfolghdq sodqh1 +Devhqw xqfhuwdlqw|/ lq wkh deryh Frxuqrw h{dpsoh zh kdyh % G'^/ [ G'_J6Sc 6ERc % G' R nR2% SE%c dqg%G'

ES

MU%c ES

M%

Lq jhqhudo ohw ER G' @h} 4@

%M[ZERc % +48,

(11)

Ohduqlqj wr Idfh Vwrfkdvwlf Ghpdqg 44 ghqrwh wkh ehvw uhvsrqvh ri lqglylgxdo wr sdudphwhu sdlu R1 Dowhuqdwlyho|/ vxssrvh klv rswlpdolw|frqglwlrq

f56ERc % G' Y

Y%ZEE%c %_E%'R +49, dgplwv d vroxwlrq/ dovr fdoohg ehvw uhvsrqvh % ' ER 5 [ Khuh/ dv deryh/ _Y%^Y ghqrwhv wkh jhqhudol}hg +vxshu0,glhuhqwldo rshudwru ri frqyh{ dqdo|vlv +Urfndihoodu/

4<:3,1 Zkhq xvlqj +49, lw lv wdflwo|dvvxphg wkdw ZE% ' ZEE%c % eh frqfdyh zlwk uhvshfw wr % Wklv hqvxuhv wkh rswlpdolw|ri wkh vroxwlrq% wr +49,1 Qrwh wkdw zh vd|qrwklqj derxw wkh qdwxuh ri wkh ghflvlrq vsdfhv [c 5 U Wkh|fdq eh txlwh jhqhudo/ h{fhsw/ ri frxuvh/ wkdw irupdw +49, irufhv xv wr ohw [ eh d frqyh{ vxevhw ri vrph wrsrorjlfdo yhfwru vsdfh1

Zkdwhyhu wkh vshflfdwlrq dqg wkh ruljlq ri wkh fkrlfh ER/ zkhwkhu vwhpplqj iurp +48, ru +49,/ zh dvvxph lw ixoo|fkdudfwhul}hv rswlpdolw|/ lv zhoo ghqhg/ dqg xqltxh iru hyhu| 5U dqgR5S1 Dovr/ zh wdflwo|srvlw/ dv d ehkdylrudo dvvxpswlrq/

wkdw hdfk lqglylgxdo c lq surgxflqj klv ehvw uhvsrqvh ER/ uhjdugv R dv h{rjhqrxv dqg jlyhq1 Pruh suhflvho| / kh dfwv dw dq| vwdjh dv wkrxjk kh kdv d up dqg vhhplqjo|

udwlrqdo eholhi derxw R

D qdwxudo frqfhuq lv wkhq zlwk wkh ohduqlqj ri mxvwldeoh eholhiv gxulqj uhshdwhg sod|1 Uh hfwlqj wkdw frqfhuq/ R5S vkrxog eh ghfoduhg dq vwhdg| vwdwh li dqg rqo|li lw vdwlvhv wkh {hg srlqw frqglwlrq

R'ER 'GsER

zkhuh G' EMU lv wkh hqwluh yhfwru ri ehvw uhvsrqvhv1 Wkh uhdvrq iru qdplqj wkh srlqwRd vwhdg|vwdwh lv wkdw lw jhqhudwhv dQdvk qrqfrrshudwlyh htxloleulxp %5ER lq vr idu dv wkh ehvw uhvsrqvh % wr wkh eholhi R uhsurgxfhv wkdw vdph eholhi1 Lq rwkhu zrugv/ d vwhdg|vwdwh hperglhv erwk frquphg h{shfwdwlrqv dqg rswlpdo uhvsrqvhv/

wkhvh wzr surshuwlhv ehlqj wkh ghqlqj ihdwxuhv ri Qdvk htxloleulxp1

Zkhq irfxv lv rq vwhdg|vwdwhv dqg srvvleoh dssurdfkhv wr vxfk glvwlqjxlvkhg rxwfrphv/ Wkhruhp 4 pd|vrphwlphv eh gluhfwo|dssolfdeoh1

Uhihuhqfhv

^4` Ehqdlp P1/ +4<<9,1 D g|qdplfdo dssurdfk wr vwrfkdvwlf dssur{lpdwlrqv/ VLDP Mrxu0 qdo ri Frqwuro dqg Rswlpl}dwlrq 67+5, 76:07:51

^5` Ehqyhqlvwh/ D1/ P1 Pìwlylhu/ dqg S1 Slurxuhw/ +4<<3,1 Vwrfkdvwlf Dssur{lpdwlrqv dqg Dgdswlyh Dojrulwkpv/ Vsulqjhu0Yhuodj/ Ehuolq1

^6` Eud|/ P1 +4<;5,1 Ohduqlqj/ hvwlpdwlrq/ dqg wkh vwdelolw| ri udwlrqdo h{shfwdwlrqv/Mrxu0 qdo ri Hfrqrplf Wkhru| 59/ 64;066<1

^7` Frqolvn/ M1/ +4<<9,1 Zk|erxqghg udwlrqdolw|B Mrxuqdo ri Hfrqrplf Olwhudwxuh/ 67/

99<0:331

(12)

Ohduqlqj wr Idfh Vwrfkdvwlf Ghpdqg 45

^8` Frufkrq/ O1 F1 dqg D1 Pdv0Frohoo/ +4<<9,1 D qrwh rq wkh vwdelolw|ri ehvw uhso|dqg judglhqw g|qdplfv zlwk dssolfdwlrqv wr lpshuihfwo| frpshwlwlyh prghov/ Hfrqrplf Ohwwhuv 84/ 8<0981

^9` Hydqv/ J1 Z1 dqg V1 Krqndsrkmd/ +4<<7,1 Ohduqlqj/ frqyhujhqfh/ dqg vwdelolw|zlwk pxowlsoh udwlrqdo h{shfwdwlrqv htxloleuld/ Hxurshdq Hfrqrplf Uhylhz/ 6;/ 43:4043<;1

^:` Hydqv/ J1 Z1 dqg V1 Krqndsrkmd/ +4<<:,1 Ohduqlqj g|qdplfv/ lq M1E1 Wd|oru dqg P1 Zrrgirug +hgv1, Kdqgerrn ri pdfurhfrqrplfv/Hovhylhu/ Dpvwhugdp1

^;` Ioèp/ V1 G1/ +4<<;d,1 Dyhudjhg suhglfwlrqv dqg wkh ohduqlqj ri htxloleulxp sod|/Mrxu0 qdo ri Hfrqrplf G|qdplfv dqg Frqwuro/ 55/ ;660;7;1

^<` Ioèp/ V1 G1/ +4<<;e,12%2Jdphv/ Ilfwlwlrxv Sod|dqg Juhhq*v Wkhruhp/Surfhhglqjv ri wkh LY Fdwdodq Gd|v ri Dssolhg Pdwkhpdwlfv/ Xqlyhuvlwdw Urylud l Ylujlol/ Wduudjrqd/

Vsdlq ;<04341

^43` Irvwhu/ G1 S1 dqg U1 Y1 Yrkud/ +4<<:,1 Fdoleudwhg ohduqlqj dqg fruuhodwhg htxloleulxp/

Jdphv dqg Hfrqrplf Ehkdylru/ 54/ 730881

^44` Ixghqehuj/ G1 dqg G1 N1 Ohylqh/ +4<<;,1 Wkh wkhru| ri ohduqlqj lq jdphv/ Wkh PLW Suhvv/ Fdpeulgjh1

^45` Ndodl/ H1 dqg H1 Ohkuhu/ +4<<8,1 Vxemhfwlyh jdphv dqg htxloleuld/Jdphv dqg Hfrqrplf Ehkdylru/ ;/ 45604961

^46` Ndqlryvnl/ \1 P1 dqg K1 S1 \rxqj/ +4<<8,1 Ohduqlqj g|qdplfv lq jdphv zlwk vwrfkdvwlf shuwxuedwlrqv/ Jdphv dqg Hfrqrplf Ehkdylru 44/ 66306961

^47` Ndwrn/ D1 dqg E1 Kdvvhoeodww/ +4<<8,1Lqwurgxfwlrq wr wkh Prghuq Wkhru| ri G|qdplfdo V|vwhpv/ Fdpeulgjh Xqlyhuvlw|Suhvv1

^48` Nhhqdq/ G1 dqg W1 Udghu/ +4<;8,1 Pdunhw g|qdplfv dqg wkh odz ri ghpdqg/ Hfrqr0 phwulfd 86+5, 79807:41

^49` Pdufhw/ D1 dqg W1 M1 Vdujhqw/ +4<;<,1 Frqyhujhqfh ri ohdvw vtxduhv ohduqlqj phfk0 dqlvpv lq vhoi0uhihuhqwldo olqhdu vwrfkdvwlf prghov/ Mrxuqdo ri Hfrqrplf Wkhru|/ 7;/

66:069;1

^4:` Pxusk|/ I1 K1/ K1 G1 Vkhudol/ dqg D1 O1 Vr|vwhu/ +4<;5,1 D pdwkhpdwlfdo surjudpplqj dssurdfk iru ghwhuplqlqj roljrsrolvwlf pdunhw htxloleulxp/Pdwkhpdwlfdo Surjudpplqj 57/ <504391

^4;` Qhp|wvnll/ Y1 Y1 dqg Y1 Y1 Vwhsdqry/ +4<93,1Txdolwdwlyh Wkhru| ri Glhuhqwldo Htxd0 wlrqv/ Sulqfhwrq Xqlyhuvlw|Suhvv/ Sulqfhwrq1

^4<` Urfndihoodu/ U1 W1/ +4<:3,1Frqyh{ dqdo|vlv/ Sulqfhwrq Xqlyhuvlw|Suhvv/ Sulqfhwrq1

(13)

Ohduqlqj wr Idfh Vwrfkdvwlf Ghpdqg 46

^53` Vdujhqw/ W1 M1/+4<<6,1 Erxqghg udwlrqdolw| lq pdfurhfrqrplfv/ R{irug Xqlyhuvlw|

Suhvv/ R{irug1

^54` Vlprq/ K1 +4<88,1 D ehkdylrudo prgho ri udwlrqdo fkrlfh/ Txduwhuo| Mrxuqdo ri Hfr0 qrplfv/ 5:</ <<044;1

^55` Wkruoxqg0Shwhuvhq/ O1 +4<<3,1 Lwhudwlyh frpsxwdwlrq ri Frxuqrw htxloleulxp/ Jdphv dqg Hfrqrplf Ehkdylru 5/ 940:81

^56` Zhlexoo/ M1 Z1/ +4<<8,1 Hyroxwlrqdu| jdph wkhru|/ Wkh PLW Suhvv/ Fdpeulgjh1

^57` \rxqj/ K1 S1/ +4<<;,1 Lqglylgxdo vwudwhj| dqg vrfldo vwuxfwxuh= Dq hyroxwlrqdu| wkhru|

ri lqvwlwxwlrqv/ Sulqfhwrq Xqlyhuvlw|Suhvv/ Sulqfhwrq1