An alternative to safety stock policies for multi-level rolling schedule MRP problems

HAL Id: hal-00004185

https://hal.archives-ouvertes.fr/hal-00004185

Preprint submitted on 8 Feb 2005

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

An alternative to safety stock policies for multi-level rolling schedule MRP problems

Nicolas Dellaert, Jully Jeunet

To cite this version:

Nicolas Dellaert, Jully Jeunet. An alternative to safety stock policies for multi-level rolling schedule

MRP problems. 2005. �hal-00004185�

Dq dowhuqdwlyh wr vdihw| vwrfn srolflhv iru pxowl0ohyho uroolqj vfkhgxoh PUS sureohpv

Q1 Ghoodhuw ^W dqg M1 Mhxqhw ^_ Qryhpehu 49/ 5335

,ALMK@BM

Zh vwxg| wkh lpsdfw ri srvlwlyh ohdg wlphv rq wkh pxowl0ohyho orw0vl}lqj sure0 ohp lq d uroolqj vfkhgxoh hqylurqphqw1 Zh vkrz krz vwrfnrxw vlwxdwlrqv pd|

dulvh hyhq lq d frqwh{w ri ghwhuplqlvwlf ghpdqg1 Zh wkhuhiruh ghyhors d surfh0 gxuh wr dyrlg vxfk vwrfnrxwv dqg zh frpsduh lwv shuirupdqfh wkurxjk d vlpxod0 wlrq vwxg| wr d vdihw| vwrfn vwudwhj|1 Vlpxodwlrq uhvxowv vkrz wkh vxshulrulw| ri wkh sursrvhg surfhgxuh1

4 Lqwurgxfwlrq

Uhvhdufk rq orw0vl}lqj ghflvlrqv lq PUS v|vwhpv kdv hvvhqwldoo| irfxvhg rq wkh gh0 yhorsphqw ri vlqjoh0ohyho dqg pxowl0ohyho khxulvwlfv iru vroylqj sureohpv zlwk ghwhu0 plqlvwlf ghpdqgv ryhu qlwh krul}rqv1 Krzhyhu/ wklv vwdwlf iudphzrun ljqruhv wkh frpprq sudfwlfh ri xvlqj d uroolqj vfkhgxoh1 Wklv dssurdfk frqvlvwv lq dsso|lqj d orw0vl}lqj uxoh ryhu d olplwhg qxpehu ri ixwxuh shulrgv/ wkh iruhfdvw zlqgrz/ iru zklfk ghpdqg lv nqrzq hlwkhu ghwhuplqlvwlfdoo| ru suredelolvwlfdoo|1 Xvxdoo|/ rqo| wkh uvw orw vl}h lv lpsohphqwhg dqg wkh krul}rq uroov iruzdug wkh qh{w ghflvlrq shulrg1 Qhz ghpdqgv duh wkhq uhyhdohg/ wkh orw0vl}lqj whfkqltxh lv uh0dssolhg dqg wkh ghflvlrq ri wkh uvw shulrg lv djdlq hqdfwhg1 Wklv surfhvv lv uhshdwhg xqwlo wkh qdo shulrg ri wkh sureohp krul}rq lv uhdfkhg1

Rswlpdo phwkrgv iru vlqjoh dqg pxowl0ohyho sureohpv zlwk d {hg krul}rq gr qrw qhfhvvdulo| surylgh dq rswlpdo vroxwlrq lq d uroolqj0vfkhgxoh hqylurqphqw1 Rq wkh frqwudu|/ vhyhudo vlqjoh0ohyho vwxglhv kdyh vkrzq wkdw lw pljkw eh zruwk xvlqj frpsx0 wdwlrqdoo| vlpsoh khxulvwlfv hvshfldoo| zkhq wkh iruhfdvw zlqgrz lv vkruw1 Eodfnexuq dqg Ploohq +4<;3, vkrz wkdw wkh Zdjqhu0Zklwlq dojrulwkp +4<8;, fdq eh rxwshu0 iruphg e| wkh Vloyhu0Phdo khxulvwlf +4<:5,/ qrwdeo| zkhq wkh qxpehu ri nqrzq ixwxuh ghpdqg lv olplwhg1 Gh Ergw/ Ydq Zdvvhqkryh dqg Jhoghuv +4<;5,/ zkloh dqdo|}lqj wkh hhfw ri iruhfdvw huuruv zlwklq wkh iruhfdvw zlqgrz rq frvw shuirupdqfh ri vhy0 hudo vlqjoh0ohyho prghov/ vkrz wkdw lw pljkw eh zruwk xvlqj wkh cvlpsolvwlf* Hfrqrplf Rughu Txdqwlw|1 Dxfdps +4<;8, surylghv d frpsdudwlyh vwxg| ri wkh shuirupdqfh ri vhyhudo orw0vl}lqj uxohv/ dovr xvhg lq frpelqdwlrq zlwk d Orrn dkhdg2Orrn edfn vwudw0 hj| zklfk frqvlvwv lq hlwkhu lqfuhdvlqj ru ghfuhdvlqj wkh orw vl}hv jhqhudwhg e| dq|

W

Ghsduwphqw ri Rshudwlrqv Pdqdjhphqw/ Hlqgkryhq Xqlyhuvlw| ri Whfkqrorj|/ S1R1 Er{ 846/

8933 PE Hlqgkryhq/ Wkh Qhwkhuodqgv1 H0pdlo= q1s1ghoodhuwCwp1wxh1qo

_

FQUV/ ODPVDGH/ Xqlyhuvlwì Sdulv L[ Gdxsklqh/ Sodfh gx Pduìfkdo gh Odwwuh gh Wdvvljq|/

:8::8 Sdulv Fhgh{1 H0pdlo= mhxqhwCodpvdgh1gdxsklqh1iu

4

uxoh xqwlo qr frvw lpsuryhphqw fdq eh irxqg1 Wkh dxwkru h{shulphqwdoo| revhuyhv wkh srru shuirupdqfh ri wkh Zdjqhu0Zklwlq dojrulwkp dqg wkh frqvlvwhqwo| kljk shuiru0 pdqfh ri Ohdvw Wrwdo Frvw dqg Vloyhu0Phdo1 Errnelqghu dqg Kq*j +4<;9, vkrz wkdw d prglhg yhuvlrq ri wkh Vloyhu0Phdo khxulvwlf +wr ghdo zlwk vkduso| ghfuhdvlqj ghpdqg sdwwhuqv, dqg d khxulvwlf e| Errnelqghu dqg Wdq +zklfk frpelqhv wkh Vloyhu0Phdo dqg wkh Ohdvw Xqlw Frvw fulwhuld, |lhog wkh ehvw uhvxowv lq prvw vlwxdwlrqv1 Krzhyhu/

wkh Zdjqhu0Zklwlq dojrulwkp lv wkh ehvw phwkrg iru odujh iruhfdvw zlqgrzv dqg dq|

ghpdqg sdwwhuq rwkhu wkdq frqvwdqw ghpdqg1

Vlplodu frqfoxvlrqv kdyh ehhq gudzq edvhg xsrq pxowl0ohyho vwxglhv xqghu uroolqj vfkhgxoh frqglwlrqv1 Eodfnexuq dqg Ploohq +4<;5, hydoxdwh wkh frvw shuirupdqfh ri wkh Vloyhu0Phdo khxulvwlf dqg wkh Zdjqhu Zklwlq dojrulwkp/ dovr xvhg lq frpelqd0 wlrq zlwk vhyhudo frvw prglfdwlrqv ghvljqhg wr dffrxqw iru lqwhughshqghqflhv dprqj vwdjhv lq dvvhpeo| vwuxfwxuhv1 Vlpxodwlrq uhvxowv lqglfdwh wkdw lq pdq| vlwxdwlrqv/

wkh Vloyhu0Phdo khxulvwlf rxwshuirupv wkh Zdjqhu0Zklwlq dojrulwkp1 Jxswd/ Nhxqj dqg Jxswd +4<<5, vkrz wkdw wkh Vloyhu0Phdo khxulvwlf surylghv orzhu frvwv wkdq wkh Zdjqhu0Zklwlq dojrulwkp lq prvw fdvhv1 Lq d uhfhqw vwxg| Vlpsvrq +4<<<, qgv wkdw wkh Vloyhu0Phdo khxulvwlf zkhq frpelqhg wr rqh ri wkh frvw prglfdwlrqv ri Eodfnexuq dqg Ploohq +4<;5, surylghv wkh orzhvw frvw vfkhgxohv xqghu vkruw iruhfdvw zlqgrzv1 Iru odujhu zlqgrzv krzhyhu/ wkh Zdjqhu0Zklwlq dojrulwkp |lhogv ehwwhu uhvxowv1 Wklv frqfoxvlrq zdv douhdg| gudzq lq hduolhu vlqjoh0ohyho vwxglhv +Eodfnexuq dqg Ploohq/

4<;3/ Errnelqghu dqg Kq*j/ 4<;9,1

Sdvw uhvxowv pdnh fohdu wkdw iruhfdvw zlqgrz ohqjwk lpsdfwv frvw shuirupdqfh/

prvwo| ehfdxvh ohqjwk gudpdwlfdoo| dhfwv wkh uvw rswlpdo orw vl}h/ d skhqrphqrq fdoohg cwkh krul}rq hhfw*1 Dq reylrxv vwudwhj| wr ryhufrph wklv gl!fxow| lv wr ohqjwkhq wkh zlqgrz Z vr dv wr vwdelol}h wkh uvw orw vl}h1 Lq sudfwlfh/ ixwxuh ghpdqg gdwd ru dffxudwh iruhfdvw duh dydlodeoh rqo| iru d olplwhg qxpehu ri ixwxuh shulrgv1 Wkh lghd lv wkhuhiruh wr xvh dydlodeoh ghpdqg gdwd wkurxjk Z dqg iruhfdvw gh0 pdqg eh|rqg Z1 Wklv vwudwhj| lv rqo| xvhixo zkhq frpelqhg wr wkh Zdjqhu0Zklwlq dojrulwkp dv p|rslf phwkrgv ljqruh dgglwlrqdo lqirupdwlrq dqg surylgh lghqwlfdo vfkhgxohv/ eh wkh krul}rq h{whqghg ru qrw1 Lpsohphqwlqj vxfk d vwudwhj| udlvhv wkh txhvwlrq ri zkhq wr vwrs h{whqglqj wkh krul}rq1

Krul}rq h{whqvlrq dssurdfkhv kdyh ehhq zlgho| lpsohphqwhg lq vlqjoh0ohyho sure0 ohpv1 Fduovrq/ Ehfnpdq dqg Nurss +4<;5, lqyhvwljdwh wkh lpsdfw ri h{whqglqj wkh krul}rq rq wkh frvw shuirupdqfh ri wkh Zdjqhu0Zklwlq dojrulwkp dqg frqfoxgh wkdw lq vrph vlwxdwlrqv wkh pruh lqirupdwlrq wkh ehwwhu1 Nurss/ Fduovrq dqg Ehfnpdq +4<;6, sursrvh dqg whvw 7 vwrsslqj frqglwlrqv1 Wkh| frqfoxgh wkdw vlpsohu vwrs0 slqj uxohv shuirup ehwwhu1 Pruh uhfhqwo|/ Uxvvho dqg Xuedq +4<<6, vkrz wkdw/ zlwk krul}rq h{whqvlrq/ wkh Zdjqhu0Zklwlq dojrulwkp ehdwv wkh Vloyhu0Phdo khxulvwlf iru prghudwh wr odujh zlqgrz ydoxhv1 Iru vpdoo Z0ydoxhv krul}rq h{whqvlrq grhv qrw khos wkh Zdjqhu0Zklwlq dojrulwkp wr vhhn lpsuryhphqwv ryhu wkh Vloyhu0Phdo khxulvwlf1 Uhqhphqw ri wkh krul}rq h{whqvlrq sulqflsoh kdv ehhq eurxjkw e| Vwdgwohu +5333, zkr sursrvhv d frvw prglfdwlrq wr eh xvhg zlwk wkh Zdjqhu0Zklwlq dojrulwkp vr dv wr dffrxqw iru wkh fwlwlrxv dvshfw ri ghpdqg gdwd eh|rqg wkh iruhfdvw zlqgrz1 Wkh lghd lv wr dvvljq wr d orw vl}h d frvw zklfk lv sursruwlrqdo wr wkh shulrgv wkh rughu fryhuv idoolqj zlwklq wkh iruhfdvw zlqgrz1 Vwdgwohu h{shfwv wklv prglfdwlrq wr idyrxu pruh rughuv lq odwhu shulrgv1 Wkh uhvxowdqw orrn0eh|rqg prgho |lhogv wkh ehvw ryhudoo uhvxowv zkhq frpsduhg wr wkh Vloyhu0Phdo whfkqltxh dqg wkh khxulvwlf ri Jur +4<:<,1

5

Krul}rq h{whqvlrq lv qdwxudoo| glvfrqwlqxhg zkhq d sodqqlqj krul}rq lv irxqg vlqfh wkh uvw rswlpdo orw vl}h vwdelol}hv lq wkdw fdvh1 Exw wkh frpprq vlwxdwlrq pd|

uduho| rhuv wklv rssruwxqlw|1 Fkdqg +4<;5, kdv wkhuhiruh ghvljqhg d vlpsoh ghflvlrq uxoh wr vhohfw wkh uvw orw vl}h zkhqhyhu d sodqqlqj krul}rq lv qrw irxqg1 Wkh uxoh frqvlvwv lq fkrrvlqj wkh uvw orw vl}h zlwk d plqlpxp frvw shu shulrg1 Wkh vhw ri uvw orw vl}hv lv surylghg e| vroylqj rswlpdoo| doo w0shulrg sureohpv/ zlwk w @ 4> = = = > Z1 Wkh frpsxwdwlrqdo vwxg| lqglfdwhv wkdw wkh prglhg dojrulwkp h{klelwv d ehwwhu ehkdylrxu wkdq wkh Zdjqhu0Zklwlq dojrulwkp dqg wkh Vloyhu0Phdo khxulvwlf1 Odwhu rq/

Fkdqg +4<;6, kdv surylghg dq dgdswdwlrq ri klv dojrulwkp wr vhuldo v|vwhpv1 Fkdqg lv wkhuhiruh wkh uvw dxwkru wr ghvljq d uroolqj surfhgxuh iru pxowl0vwdjh sureohpv hyhq li lw lv uhvwulfwhg wr wkh yhu| vshflf vhuldo fdvh1

Ghvslwh wklv dexqgdqw uhohydqw olwhudwxuh rq wkh uroolqj0vfkhgxoh sureohp/ wkhuh lv d sdxflw| ri uhvhdufk h{dplqlqj pxowl0ohyho lqvwdqfhv zlwk jhqhudo surgxfw vwuxf0 wxuhv dqg srvlwlyh ohdg wlphv1 Krzhyhu/ frpprq lqgxvwuldo vhwwlqjv riwhq lqyroyh surgxfw vwuxfwxuhv zlwk qxphurxv frpprq sduwv1 Ixuwkhupruh/ wkh }hur ohdg wlph dvvxpswlrq lv kdugo| uhdolvwlf dqg pruh uhvwulfwlyh wkdq lw vhhpv dw uvw vljkw1 Lqghhg/

lpsohphqwlqj orw vl}lqj ghflvlrqv rq d uroolqj edvlv ehfrphv pxfk pruh frpsoh{ lq wklv frqwh{w +l1h1 srvlwlyh ohdg wlph dqg jhqhudo vwuxfwxuhv, dv orw0vl}lqj phwkrgv pd|

surylgh lqihdvleoh vfkhgxohv gxh wr d odfn ri frpsrqhqw dydlodelolw|1

Lq wklv sdshu zh ghyhors d surfhgxuh wr frsh zlwk vwrfnrxw vlwxdwlrqv wkdw dulvh zkhq srvlwlyh ohdg wlphv duh lqwurgxfhg1 Wklv surfhgxuh lv frpsduhg wr d vwudwhj|

wkdw frqvlvwv lq lqwurgxflqj vdihw| vwrfnv dw doo ohyhov lq wkh surgxfw vwuxfwxuh1 Zh vkrz wkurxjk vlpxodwlrq h{shulphqwv wkh vxshulrulw| ri rxu surfhgxuh ryhu wkh vdihw|

vwrfn vwudwhj|1

Wkh qh{w vhfwlrq lv ghglfdwhg wr d ghwdlohg ghvfulswlrq ri wkh pxowl0ohyho orw0vl}lqj sureohp xqghu uroolqj vfkhgxoh frqglwlrqv1 Zh surylgh dq h{dpsoh ri d vwrfnrxw uhvxowlqj iurp wkh lqwurgxfwlrq ri srvlwlyh ohdg wlphv1 Vhfwlrq 6 eulh | suhvhqwv wkh vdihw| vwrfn vwudwhj| wkdw pd| eh lpsohphqwhg wr dyrlg vxfk vwrfnrxw vlwxdwlrqv1 Wkh dowhuqdwlyh vroxwlrq phwkrg/ qdpho| rxu uhsdlu surfhgxuh/ lv wkhq ghvfulehg1 Vhfwlrq 7 lv ghglfdwhg wr wkh h{shulphqwdo iudphzrun ghvljqhg wr frpsduh wkh frvw hhfwlyhqhvv ri vhyhudo orw0vl}lqj whfkqltxhv zkhq frpelqhg wr hlwkhu rxu uhsdlu surfhgxuh ru wkh vdihw| vwrfn vwudwhj|1 Vhfwlrq 8 dqdo|vhv wkh vlpxodwlrq uhvxowv1 Zh frqfoxgh lq Vhfwlrq 9 zlwk wkh pdlq qglqjv ri rxu vwxg| dqg olplwdwlrqv1

5 Wkh pxowl0ohyho orw0vl}lqj sureohp xqghu uroolqj vfkh0 gxoh frqglwlrqv

514 Vwdwhphqw ri wkh sureohp

Lw lv frpprq wr uhsuhvhqw surgxfw vwuxfwxuhv dv gluhfwhg df|folf judskv +vhh Errn0 elqghu dqg Nrfk/ 4<<3/ iru h{dpsoh,1 Lq vxfk d judsk hdfk qrgh fruuhvsrqgv wr dq lwhp dqg hdfk hgjh +l> m, ehwzhhq qrgh l dqg qrgh m lqglfdwhv wkdw lwhp l lv gluhfwo|

uhtxluhg wr dvvhpeoh lwhp m1 Qrgh l +htxlydohqwo| lwhp l, lv ixoo| ghqhg e| ^!Q +l, dqg +l, > lwv vhwv ri lpphgldwh suhghfhvvruv dqg vxffhvvruv1 Wkh vhw ri dqfhvwruv lpphgldwh ru qrq0lpphgldwh suhghfhvvruv ri lwhp l lv ghqrwhg e| a ^!Q +l,= Surgxfw vwuxfwxuhv pd| eh fdwhjrul}hg lq whupv ri wkhlu frpsoh{lw| lqgh{/ .> dv ghqhg e|

Nlppv +4<<:,1 Uhfdoo surgxfwv duh qxpehuhg lq wrsrorjlfdo rughu e| wkh lqwhjhuv 4> = = = > S dqg ohw S +n, eh wkh qxpehu ri surgxfwv dw ohyho n> zlwk n @ 3> = = = > N +N . 4

6

lv wkh ghswk ri wkh vwuxfwxuh,1 Wkh wrwdo qxpehu ri lwhpv reylrxvo| htxdov ^g _&'f S +n,>

zklfk e| ghqlwlrq lv dovr S= Wkh prvw frpsoh{lq wkh vhqvh ri kdylqj wkh odujhvw qxpehu ri surgxfw lqwhughshqghqflhvvwuxfwxuh lv rewdlqhg zkhq hdfk lwhp hqwhuv wkh frpsrvlwlrq ri doo wkh lwhpv orfdwhg dw kljkhu ohyhov lq wkh surgxfw vwuxfwxuh1 E|

frqwudvw/ wkh vlpsohvw vwuxfwxuh rewdlqv zkhq hdfk lwhp hqwhuv wkh frpsrvlwlrq ri h{dfwo| rqh lwhp ehorqjlqj wr d kljkhu ohyho1 Nlppv +4<<:, ghqhv wkh frpsoh{lw|

ri d surgxfw vwuxfwxuh dv

. @ D D 4?

D 4@ D 4? +4,

zkhuh D @ _' m+l,m lv wkh dfwxdo qxpehu ri dufv lq wkh vwuxfwxuh1 Wkhuh lv ri frxuvh d plqlpdo qxpehu ri dufv vxfk dv wkh surgxfw vwuxfwxuh lv frqqhfwhg/ zklfk zh ghqrwh D 4? dqg lv htxdo wr S S +3,= Frqyhuvho| wkhuh lv d pd{lpxp qxpehu ri dufv ghqrwhg D 4@ wkdw wkh judsk fdq frqwdlq/ dqg zklfk lv zulwwhq dv D 4@ @ ^g! _&'f iS +n,

g '&n S +m,j= Vwuxfwxuhv iru zklfk wkh qxpehu ri dufv htxdov wkh plqlpxp qxpehu ri dufv +D @ D 4? , duh qhfhvvdulo| dvvhpeo| vwuxfwxuhv zlwk d }hur F0ydoxh/ zkhuhdv vwuxfwxuhv vxfk dv D @ D 4@ vdwlvi| F @ 41 Wkh F0lqgh{ lv wkhuhiruh erxqghg iurp ehorz dqg deryh/ zkhuhdv wkh wudglwlrqdo lqgh{ ri Froolhu +4<;4, lv qrw1

Ohw O eh wkh fxpxodwlyh ohdg wlph ri lwhp l dqg o lwv rzq ohdg wlph +wlph uhtxluhg wr surgxfh ru dvvhpeoh lwhp l,= Zh kdyh

O @ pd{

MK

^!Q

E +O . o ,> +5,

zlwk O @ 3 iru doo l zlwk qr suhghfhvvru1 Dv ghflvlrq shulrgv duh wkh shulrgv lq zklfk zh rughu dqg qrw wkh gholyhu| shulrgv/ lqglylgxdo ohdg wlphv o duh qrw lqfoxghg lq wkh ghqlwlrq ri O =

Ohw Z eh wkh iruhfdvw zlqgrz ohqjwk dqg W/ wkh qdo shulrg ri wkh sodqqlqj krul}rq/ zlwk Z W= Shulrg W lv xvxdoo| lqqlwh/ dv d uroolqj sureohp kdv qr wuxh hqg1 Exw lq wkh vlpxodwlrq h{shulphqwv/ shulrg W pxvw eh vhw wr d qlwh ydoxh1 Iru hdfk ghflvlrq shulrg w> orw0vl}lqj uxohv duh dssolhg rq wlph lqwhuydo U @ iw> = = = > plq+w . Z 4> W ,j> zlwk w @ 4> = = = > W dqg rqo| wkh uvw orw vl}h +wkdw ri sh0 ulrg w, lv h{hfxwhg1 Zlwklq lqwhuydo U/ ghpdqg iru hqg lwhpv lv hlwkhu nqrzq zlwk fhuwdlqw| ru suredelolvwlfdoo|1 Lq wkh odwwhu fdvh/ ghpdqg iruhfdvw lv vxemhfw wr huuru +qrwh wkh zrug ciruhfdvw* lv qrw vxlwdeoh zkhq wkhuh lv qr huuru,1 Wkh lqwhuydo rq zklfk orw0vl}lqj uxohv duh dssolhg pd| eh h{whqghg wr H shulrgv/ wr lqfuhdvh wkhlu h!flhqf|1 Lqwhuydo U ehfrphv U @ iw> = = = > w.Z 4> = = = > plq+w.H 4> W,j1 Ghpdqg lq shulrgv w.Z> = = = > w.H pd| eh jhqhudwhg zlwk wkh vdph sdwwhuq dv wkh rqh xwlol}hg lq shulrgv w . 4> = = = > w . Z 4= Wklv ghpdqg lv dozd|v fwlwlrxv1

Jurvv uhtxluhphqwv g cr iru lwhp l lq shulrg v 5 U fruuhvsrqg wr wkh ghpdqg wkdw pxvw eh vdwlvhg lq shulrg v . o = Jurvv uhtxluhphqwv duh hlwkhu iruhfdvwhg ru nqrzq zlwk fhuwdlqw| iru hqg lwhpv zlwklq wkh iruhfdvw zlqgrz zkhuhdv wkh| uhvxow iurp sodqqhg dqg up rughuv dw ghhshu ohyhov iru frpsrqhqwv1

Ohw { c| eh d up rughu iru lwhp l lq shulrg w1 Txdqwlw| { c| lv wkh qxpehu ri xqlwv ri lwhp l zklfk lv rughuhg lq shulrg w wr oo wkh qhw uhtxluhphqwv iru lwhp l lq shulrg w . o =

7

Wkh slsholqh lqyhqwru| ] cr lv ghqhg dv

] cr @ pd{+3> ] cr!Q . { cr g cr ,> +6, zkhuh ] cf ghqrwhv wkh dfwxdo dprxqw ri lqyhqwru| dw wkh hqg ri shulrg 31

Qhw uhtxluhphqwv e cr uhvxow iurp wkh iroorzlqj ghqlwlrq2frpsxwdwlrq

e cr @ pd{+3> g cr ] cr! { cr ,= +7, Rqfh qhw uhtxluhphqwv kdyh ehhq frpsxwhg lq wlph lqwhuydo U> dq| orw0vl}lqj uxoh lv dssolhg rq wkhvh qhw uhtxluhphqwv vr dv wr jhqhudwh sodqqhg rughuv s cr uhsuhvhqwlqj txdqwlwlhv wkdw zloo srvvleo| eh odxqfkhg lq shulrg v dqg ehfrplqj dydlodeoh lq shulrg v . o 1 Wkh uvw sodqqhg rughu/ s c| +uhfdoo wkdw w lv wkh uvw shulrg ri wlph lqwhuydo U, lv wkhq wudqviruphg lqwr d up rughu { c| 1 Lq rwkhu zrugv/ zh vhw { c| @ s c|

zlwk { c| fruuhvsrqglqj wr d txdqwlw| wkdw lv uhdoo| rughuhg lq wkh fxuuhqw shulrg w=

Sodqqhg rughuv dqg up rughuv duh xvhg wr frpsxwh wkh sodqqhg jurvv uhtxluhphqwv iru frpsrqhqwv dv iroorzv

g cr @ ^[

MKE

f c [ crn, zlwk [ crn, @

+ { crn, li v . o ? w

s crn, rwkhuzlvh / +8, zkhuh f c ghqrwhv wkh surgxfwlrq udwlr +qxpehu ri xqlwv ri l wr surgxfh rqh xqlw ri m,=

Dw wkh hqg ri shulrg w> sodqqhg jurvv uhtxluhphqwv ehfrph dfwxdo uhtxluhphqwv/

dv wkh| gr qrw ghshqg rq sodqqhg rughuv dw kljkhu ohyhov dq|pruh exw rqo| rq up rughuv1 Ohw G c| eh wkh dfwxdo jurvv uhtxluhphqwv wr eh vdwlvhg lq shulrg w> zh kdyh

G c| @ ^[

MKE

f c { c| = +9,

Qrwh wkdw G c|n, @ g c| iru doo w=

D up rughu { c| ehfrplqj dydlodeoh lq shulrg w . o jhqhudwhv d vfkhgxohg uhfhlsw u c|n, / lq wkdw shulrg1 Sxw dqrwkhu zd|/ zh kdyh u c| @ { c|3, =

Dw wkh hqg ri wkh fxuuhqw shulrg w ri wkh uroolqj krul}rq/ zh duh deoh wr frpsxwh wkh dfwxdo lqyhqwru| L c| +dv rssrvhg wr wkh slsholqh lqyhqwru|, iru doo lwhpv1 Zh kdyh

L c| @ L c|3 . u c| G c| = +:,

Wkh lqlwldo ydoxh ri wkh slsholqh lqyhqwru| lv htxdo wr wkh lqlwldo lqyhqwru|1 Zh wkxv kdyh ] cf @ L cf > ;l=

515 Wkh rffxuuhqfh ri d vwrfnrxw

Wklv vxevhfwlrq suhvhqwv d vlwxdwlrq ri vwrfnrxw gxh wr srvlwlyh ohdg wlphv lq d pxowl0 ohyho orw0vl}lqj sureohp zlwk d uroolqj krul}rq1 Ohw xv frqvlghu wkh h{dpsoh lq Iljxuh 41 Wkh surgxfw vwuxfwxuh lqyroyhv 8 lwhpv dqg wkh frpsoh{lw| lqgh{ htxdov F @ +: 7,@+< 7, @ 3=9= Zh fkrvh Z @ 7> zklfk phdqv wkdw zh dsso| dq| orw0vl}lqj uxohv rq wkh qhw uhtxluhphqwv iru 7 shulrgv ri ghwhuplqlvwlf ghpdqg1 Ohdg wlphv duh rqo| srvlwlyh iru lwhpv 6/ 7 dqg 8 zlwk o @ o e @ o D @ 41 Zh vhw L cf @ 3> ;l @ 4> = = = > 8=

8

?9 G3

?9 G3

?9 G3

?9 G3

?9 G3

% %

% %

&

S

% %

% %

&

?? ?? ?? ?? ? 4

5 6

7

8

Lwhp l 4 5 6 7 8

V ; 68 53 65 77

k 31;;8 31453 3145: 3144; 31334

o 3 3 4 4 4

O 6 5 5 4 3

S @ 8 F @ 3=9 Q @ 6 W @ 45 Z @ 7

Iljxuh 4= Gdwd ri dq h{dpsoh1

Zh kdyh ] ?f @ L ?f > ;l= Vfkhgxohg uhfhlswv iru lwhpv 6/ 7 dqg 8 wdnh wkh iroorzlqj ydoxhv lq shulrg 4= u ? @ 89> u e? @ 687 dqg u D? @ 4::= Gxh wr wkh ohdg wlphv/

vfkhgxohg uhfhlswv iru wkhvh lwhpv uhvxow iurp rughulqj ghflvlrqv sulru wr wkh uvw shulrg ri wkh uroolqj krul}rq1

Wdeoh 4 h{klelwv wkh uroolqj surfhgxuh iru wkh uvw wkuhh ghflvlrq shulrgv1 Lq wkh fxuuhqw shulrg w @ 4> zh nqrz wkh ghpdqg iru hqg lwhp 4 lq shulrgv 4 wr 7>

ig ? > = = = > g ?e j> dqg zh pxvw lpsohphqw odxqfklqj ghflvlrqv iru doo lwhpv lq shulrg 41 Dw wkh ehjlqqlqj ri shulrg 4/ qr rughulqj ghflvlrq kdv ehhq pdgh/ vr zh kdyh { ? @ 3>

;l= Zh uvw frpsxwh wkh slsholqh lqyhqwrulhv dqg wkh qhw uhtxluhphqwv iru lwhp 4/

xvlqj irupxodh +6, dqg +7,1 Lq shulrg v @ 4> zh kdyh ] ? @ pd{+3> 3.389, @ 3 dqg e ? @ pd{+3> 89 3 3, @ 89= Iroorzlqj wkh vdph orjlf/ zh rewdlq ] ?H dqg e ?H iru v @ 5> = = = > 7= Zh fdq qrz dsso| dq| orw vl}lqj uxoh rq wkh vwuhdp ri qhw uhtxluhphqwv iru wkh hqg lwhp1 Lq wklv h{dpsoh/ zh kdyh dssolhg wkh Zdjqhu0Zklwlq dojrulwkp lq d vhtxhqwldo idvklrq1 Wkh dojrulwkp sursrvhv wkh orw iru orw vroxwlrq iru lwhp 4 dv s ?H @ e ?H > ;v @ 4> = = = > 7= Zh fdq qrz frpsxwh wkh sodqqhg jurvv uhtxluhphqwv iru lwhp 5/ xvlqj irupxod +8,1 Zh kdyh g 2?H @ s ?H > ;v @ 4> = = = > 7= Zh wkhq frpsxwh wkh slsholqh lqyhqwrulhv dqg wkh qhw uhtxluhphqwv1 Wkh Zdjqhu0Zklwlq dojrulwkp ohdgv wr d xqltxh orw lq shulrg 4/ zlwk s 2? @ e 2? . .e 2?e @ 4::= Sodqqhg jurvv uhtxluhphqwv iru lwhp 6 uhvxow iurp wkh iroorzlqj frpsxwdwlrq= g ? @ s ?2 > g ?2 @ s ? hwf1 Qrwh wkdw g ?e @ 3 vlqfh s ?D lv vwloo vhw wr }hur1 Wkh Zdjqhu0Zklwlq djdlq sursrvhv d xqltxh orw lq wkh uvw shulrg= s ? @ e ? . . e ?e @ 454= Lwhp 7 kdv wkh wkuhh uvw lwhpv dv vxffhvvruv1 Zh kdyh g e? @ s ?2 . s 2?2 . s ?2 @ 69 . 3 . 3 @ 69= Zh iroorz wkh vdph uhdvrqlqj wr frpsxwh wkh qh{w jurvv uhtxluhphqwv/ wkh slsholqh lqyhqwrulhv dqg wkh qhw uhtxluhphqwv1 Wkh Zdjqhu0Zklwlq dojrulwkp ohdgv wr d vlqjoh rughu lq shulrg 41 Wkh vdph krogv iru lwhp 81

Dw wkh hqg ri shulrg 4/ zh vhw { ? @ s ? iru doo lwhpv1 Wklv phdqv wkdw zh dfwxdoo| odxqfk lq shulrg 4 wkh surgxfwlrq ri 89 xqlwv ri lwhp 4/ 4:: xqlwv ri lwhp 5/

hwf1 Iru wkh uvw wzr lwhpv iru zklfk wkh ohdg wlph lv }hur/ wkhvh rughulqj ghflvlrqv lqvwdqwdqhrxvo| surgxfh vfkhgxohg uhfhlswv lq shulrg 4= u ? @ { ? @ 89 dqg u 2? @

9

{ 2c @ 4::= Zh fdq dovr frpsxwh wkh dfwxdo jurvv uhtxluhphqwv iru doo lwhpv lq shulrg 4 dqg wkh qdo lqyhqwrulhv/ xvlqj irupxodh +9, dqg +:,1 Zh kdyh G c @ g c @ 89>

G 2c @ { c @ 89> G c @ { c @ 89> G ec @ { c . { 2c . { c @ 89 . 4:: . 454 @ 687>

G Dc @ { c . { ec @ 89 . 454 @ 4::= Lqyhqwrulhv dw wkh hqg ri shulrg 4 wdnh wkh iroorzlqj ydoxhv= L c @ 3.8989 @ 3> L 2c @ 3.4::89 @ 454> L c @ 3.8989 @ 3>

L ec @ 3 . 687 687 @ 3> L Dc @ 3 . 4:: 4:: @ 3= Doo wkhvh ydoxhv +G c| > { c| > u c| dqg L c| , fkdudfwhul}h wkh vwdwh ri wkh v|vwhp lq shulrg w= Wkh| duh xvhg iru frvw fdofxodwlrq sxusrvhv exw duh qrw qhfhvvdu| wr pdnh rughulqj ghflvlrqv1

Lq wkh qh{w ghflvlrq shulrg +w @ 5,> d qhz ghpdqg lv uhyhdohg iru wkh hqg lwhp +wkdw ri shulrg 8/ zlwk g cD @ 67,1 Wr frpsxwh wkh qhw uhtxluhphqwv dqg wkh slsholqh lqyhqwrulhv lq shulrgv 5 wr 8> zh qhhg wr uhfrpsxwh wkh slsholqh lqyhqwru| lq shulrg 4 dqg wkh sodqqhg jurvv uhtxluhphqwv lq shulrg 41 Lqghhg/ wr frpsxwh e cr > zh qhhg ] cr! zklfk ghshqgv rq g cr! +vhh irupxodh +7, dqg +6,,1 Wkh sodqqhg ghpdqg g cr! pd| kdyh fkdqjhg/ vlqfh vrph sodqqhg rughuv kdyh ehhq wudqviruphg lqwr up rughuv dv wkh krul}rq uroov iruzdug1 Iru wkh hqg lwhp/ zh vwloo kdyh g c @ 89=

Lqyhqwru| ] c @ pd{+3> 3 . 89 89, @ 3= Zh fdq qrz frpsxwh e cr dqg ] cr iru v @ 5> = = = > 8= Dsso|lqj wkh Zdjqhu0Zklwlq dojrulwkp rq ie c2 > = = = e cD j ohdgv wr wkh orw iru orw vroxwlrq1 Iru lwhp 5/ zh kdyh g 2c @ { c @ 89 +dffruglqj wr +8,,1 Lqyhqwru|

] 2c @ pd{+3> 3 . 4:: 89, @ 454= Rqfh g 2cr > e 2cr dqg ] 2cr kdyh ehhq frpsxwhg/ zh dsso| ZZ rq qhw uhtxluhphqwv zklfk ohdgv wr dq rughu s 2cD @ 67> wr fryhu wkh qhz qhw uhtxluhphqw1 Iru lwhp 6/ zh dovr kdyh d vlqjoh qhw uhtxluhphqw lq shulrg 7 zklfk lv fryhuhg e| dq rughu s ce @ e ce @ 67= Iru lwhp 7 zh kdyh g ec @ s c2 . s 2c2 . s c2 @ 69 . 3 . 3 @ 69= Lqyhqwru| ] ec @ pd{+3> 3 . 454 69, @ ;8= Dsso|lqj wkh vdph uhdvrqlqj/ zh rewdlq dq rughu ri 435 xqlwv lq shulrg 6 wr fryhu wkh qhw uhtxluhphqwv ri shulrgv 6 dqg 71 Qrwh wkdw lq wkh uvw ghflvlrq shulrg +w @ 4,> d vlqjoh rughu ri lwhp 7 lq shulrg 4 zdv vx!flhqw wr fryhu wkh uhtxluhphqwv ri shulrgv 4 wr 61 Qrz wkh ghpdqg kdv fkdqjhg/ wkh iruphu sodqqhg rughu lq shulrg 4 lv qrw odujh hqrxjk1 D vlplodu uhpdun fdq eh pdgh iru lwhp 8=d srvlwlyh sodqqhg rughu lq shulrg 5 lv qrz qhfhvvdu| wr fryhu wkh qhz qhw uhtxluhphqwv1

Dw wkh hqg ri shulrg 5/ zh vhw { c2 @ s c2 iru doo lwhpv1 Zh kdyh G c2 @ 69>

G 2c2 @ { c2 @ 69> G c2 @ { c2 @ 69> G ec2 @ { c2 . { 2c2 . { c2 @ 69 . 3 . 3 @ 69 dqg G Dc2 @ { c2 . { ec2 @ 69 . 3 @ 69= Vfkhgxohg uhfhlswv wdnh wkh iroorzlqj ydoxhv= u c2 @ { c2 @ 69> u 2c2 @ { 2c2 @ 3> u c2 @ { c @ 454> u ec2 @ { ec @ 454 dqg u Dc2 @ { Dc @ 454=

Zh qdoo| frpsxwh wkh hqglqj lqyhqwrulhv= L c2 @ 3> L 2c2 @ @ L Dc2 @ ;8=

Lq shulrg w @ 6> ghpdqg lq shulrg 9 lv nqrzq iru wkh hqg lwhp +g cS @ 69,1 Dv ehiruh/ zh uhfrpsxwh g c2 dqg ] c2 wr ghwhuplqh wkh qhw uhtxluhphqwv zlwklq wkh iruhfdvw zlqgrz1 Wkh Zdjqhu0Zklwlq dojrulwkp vxjjhvwv dq rughu iru lwhp 4 lq shulrg 9 wr fryhu wkh qhz ghpdqg1 Iru lwhp 5/ wkh sodqqhg rughu lq shulrg 8 lv qrz lqfuhdvhg wr fryhu wkh qhz qhw uhtxluhphqw ri shulrg 9 +lq shulrg w @ 5> zh kdg s 2cD @ 67 dqg qrz/ lq shulrg w @ 6> zh kdyh s 2cD @ :3,1 Iru lwhp 6/ zh qhhg wr uhfrpsxwh g c > dv wklv ghpdqg ghshqgv rq { c2 zklfk kdv rqo| ehhq ghwhuplqhg lq wkh suhylrxv shulrg w @ 51 Zh kdyh g c @ { c2 @ 69= Lqyhqwru| ] c @ pd{+3> 3 . 454 69, @ ;8= Lq wkh qh{w shulrg/ g c2 @ s c @ 6< dqg ] c2 @ pd{+3> ;8.36<, @ 79= Frpsxwlqj wkh qhw uhtxluhphqwv zlwklq wkh zlqgrz dqg dsso|lqj ZZ ohdg wr d vlqjoh rughu lq shulrg 7 wr fryhu wkh wzr srvlwlyh uhtxluhphqwv ri lwhp 61 D vlplodu uhdvrqlqj lv dssolhg wr lwhp 71 D odujh sodqqhg rughu lv sursrvhg lq shulrg 6 zlwk s ec @ 543 lqvwhdg ri 435 xqlwv zklfk zdv wkh sodqqhg rughu frpsxwhg lq wkh suhylrxv shulrg w @ 5= Iru lwhp 8/ zh kdyh g Dc @ { c2 . { ec2 @ 69 . 3 @ 69 dqg ] Dc @ pd{+3> 3 . 454 69, @ ;8=

:

Lq wkh qh{w shulrg/ g Dc2 @ s c . s ec @ 6< . 543 @ 57<1 Zh fdq lpphgldwho|

revhuyh wkdw wklv ghpdqg fdqqrw eh vdwlvhg dv wkh dydlodeoh txdqwlw| ri lwhp 5 htxdov ] Dc . { Dc2 @ ;8 . 469 @ 554 zklfk lv lqihulru wr g Dc2 @ 57<= Ri frxuvh/ lw lv douhdg| wrr odwh wr prgli| { Dc2 zklfk kdv ehhq odxqfkhg lq shulrg 5 wkdw qrz ehorqjv wr wkh sdvw1 Vhwwlqj { c @ s c dqg { ec @ s ec zrxog ohdg wr d vwrfnrxw ri lwhp 8 lq shulrg 6/ dv L Dc @ L Dc2 .u Dc G Dc @ ;8.46957< @ 5;1 Sodqqhg rughuv frpsxwhg iru lwhpv 4 wr 7 lq shulrg w @ 6 duh wkhuhiruh lqihdvleoh vlqfh wkhlu lpsohphqwdwlrq zrxog ohdg wr d vwrfnrxw iru lwhp 8 lq shulrg 61

w @ 4 w @ 5 w @ 6

" # $ % 4 # $ % & 4 5 $ % & '

g

">v

89 69 6< 79 89 69 6< 79 67 89 69 6< 79 67 69

e

">v

89 69 6< 79 69 6< 79 67 6< 79 67 69

s

">v

89 69 6< 79 69 6< 79 67 6< 79 67 69

{

">v

89 89 69

]

">v

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

g

#>v

89 69 6< 79 89 69 6< 79 67 89 69 6< 79 67 69

e

#>v

89 69 6< 79 3 3 3 67 3 3 67 69

s

#>v

4:: 3 3 3 3 3 3 67 3 3 :3 3

{

#>v

4:: 4:: 3

]

#>v

3 3 3 3 454 ;8 79 3 3 454 ;8 79 3 3 3

g

$>v

69 6< 79 3 69 6< 79 67 3 69 6< 79 67 69 3

e

$>v

69 6< 79 3 3 3 67 3 3 67 69 3

s

$>v

454 3 3 3 3 3 67 3 3 :3 3 3

{

$>v

454 454 3

]

$>v

3 3 3 3 ;8 79 3 3 3 ;8 79 3 3 3 3

g

%>v

69 6< 79 3 69 6< ;3 9; 3 69 6< 449 437 69 3

e

%>v

69 6< 79 3 3 67 9; 3 :3 437 69 3

s

%>v

454 3 3 3 3 435 3 3 543 3 3 3

{

%>v

454 454 3

]

%>v

3 3 3 3 ;8 79 3 3 3 ;8 79 3 3 3 3

g

&>v

69 6< 79 3 69 474 79 67 3 69 57< 79 67 69 3

e

&>v

69 6< 79 3 89 79 67 3

s

&>v

454 3 3 3 469 3 3 3

{

&>v

454 454 469

]

&>v

3 3 3 3 ;8 3 3 3 3 ;8

g

l>v

= jurvv uhtxluhphqwv iru lwhp l lq shulrg v> e

l>v

= qhw uhtxluhphqwv/

s

l>v

= sodqqhg rughuv/ ]

l>v

= slsholqh lqyhqwru|/ +l,= vhw ri vxffhvvruv1

+4, @ >> +5, @ i4j> +6, @ i4j> +7, @ i4> 5> 6j> +8, @ i4> 7j=

o

l

= ohdg wlph1 o

"

@ 3> o

#

@ 3> o

$

@ 4> o

%

@ 4> o

&

@ 4=

Wdeoh 4= D vlwxdwlrq ri vwrfnrxw1

Zh duh qrz deoh wr surylgh d irupdo ghqlwlrq ri wkh ihdvlelolw| ri sodqqhg rughuv1 Sodqqhg rughuv is c| > = = = > s c|n` ! j iru doo lwhpv lq shulrg w duh ihdvleoh li dqg rqo| li ] c&! . { c& g c& ; lm +l, 9@ > dqg ;n 5 iw o > = = = > w "j= +;,

;

Lq vxfk d fdvh/ doo sodqqhg rughuv is c| j lq shulrg w fdq eh wudqviruphg lqwr up rughuv i{ c| j = Rq wkh frqwudu|/ li wkh suhylrxv lqhtxdolw| lv qrw vdwlvhg iru rqh lwhp l lq rqh shulrg n/ vrph sodqqhg rughuv iru vrph vxffhvvruv ri lwhp l vkrxog eh prglhg dv wkh| wuljjhu d wrr odujh ghpdqg iru lwhp l zklfk fdq qrw eh vdwlvhg rq wkh edvlv ri suhylrxv rughuv dqg vwrfnv1

Wr fkhfn lqhtxdolw| +;,/ zh rqo| qhhg wr uhfrpsxwh wkh jurvv uhtxluhphqwv dqg wkh slsholqh lqyhqwrulhv lq wlph lqwhuydo iw o > = = = > w "j> xvlqj irupxodh +8,dqg +6,1 Wkh txhvwlrq lv qrz krz wr jhw d ihdvleoh vwuhdp ri rughuv zkhq zh idfh d vwrfnrxwB Sdvw olwhudwxuh xvxdoo| ohdyhv dvlgh wkh txhvwlrq ri srvlwlyh ohdg wlphv zkhq ghpdqg lv nqrzq zlwk fhuwdlqw| dv srvlwlyh ohdg wlphv duh vxssrvhgo| czlwkrxw dq| hhfw* lq d ghwhuplqlvwlf iudphzrun1 Wkh qh{w vhfwlrq suhvhqwv wzr phwkrgv wr dyrlg vwrfnrxwv lq wklv hqylurqphqw1

6 Dyrlglqj vwrfnrxw vlwxdwlrqv

Wkh uvw vxevhfwlrq uhylhzv wkh vdihw| vwrfn dssurdfk zklfk lv wudglwlrqdoo| xvhg lq d frqwh{w ri vwrfkdvwlf ghpdqg1 Vxevhfwlrq 5 lv ghglfdwhg wr d ghwdlohg ghvfulswlrq ri rxu uhsdlu surfhgxuh1

614 Vdihw| vwrfn vwudwhjlhv

Vdihw| vwrfn srolflhv duh zlgho| hpsor|hg lq d vwrfkdvwlf frqwh{w1 Gh Ergw hw do1 +4<;5,srlqwhg rxw wkh lpsruwdqfh ri fduhixoo| ghvljqlqj d vdihw| vwrfn srolf| wr uhdfk d jrrg vhuylfh ohyho1 Exw wkh txhvwlrq ri wkh orfdwlrq dqg vl}h ri wkhvh vdihw|

vwrfnv lv dznzdug dqg uhpdlqv zlgho| rshq1

Lq vhyhudo vwxglhv h{dplqlqj wkh shuirupdqfh ri orw0vl}lqj whfkqltxhv xqghu gh0 pdqg xqfhuwdlqw|/ vdihw| vwrfnv duh frpsxwhg uhfxuvlyho| vr dv wr uhdfk d "33(

vhuylfh ohyho iru frvw frpsdulvrqv sxusrvhv +vhh iru lqvwdqfh Errnelqghu dqg Khdwk/

4<;;,1 Vdihw| vwrfnv duh lqwurgxfhg dw doo ohyhov lq wkh surgxfw vwuxfwxuh dqg zkhq0 hyhu d vwrfnrxw lv derxw wr rffxu wkh vdihw| vwrfn ydoxh lv lqfuhdvhg vr dv wr dyrlg orvw ghpdqgv1 Wr looxvwudwh/ ohw xv frqvlghu rxu h{dpsoh lq jxuh 41 Zh dsso|

wkh ZZ dojrulwkp lq d vhtxhqwldo idvklrq zlwk d iruhfdvw zlqgrz ri 7 shulrgv dqg zh vwrs wkh uroolqj krul}rq dw shulrg W @ ;= Ghpdqg iru wkh hqg lwhp htxdov i89> 69> 6<> 79> 67> 69> 8> 8:j= Li zh gr qrw lqwurgxfh dq| vdihw| vwrfn/ zh rewdlq wkh iroorzlqj vlwxdwlrq iru frpsrqhqw 8 +Wdeoh 5,1

Shulrg w 4 5 6 7 8 9 : ;

Dfwxdo jurvv uhtxluhphqwv G ,c| 4:: 69 57< 79 4;< 74 3 8:

Vfkhgxohg uhfhlswv u ,c| 4:: 454 469 449 3 <; 3 3 Hqglqj lqyhqwru| L ,c| 3 ;8 05; :3 044< 8: 8: 3 Wdeoh 5= Vlwxdwlrq iru frpsrqhqw 8 zlwkrxw vdihw| vwrfn> hqg ri uxq 41 Dv ehiruh +vhh Wdeoh 4,/ d vwrfnrxw ri 5; xqlwv dsshduv lq shulrg 61 Dqrwkhu vwrfnrxw ri 44< xqlwv lv uhfrughg lq shulrg 81 Wr vhw wkh ydoxh ri wkh vdihw| vwrfn iru lwhp 8 +VV , ,/ zh vlpso| wdnh wkh pd{lpxp vwrfnrxw dorqj wkh krul}rq1 Zh kdyh VV , @ 44<= Wdeoh 6 surylghv wkh ydoxhv ri jurvv uhtxluhphqwv +g cr ,/ qhw uhtxluhphqwv +e cr ,> sodqqhg rughuv +s cr , dqg slsholqh lqyhqwrulhv +] cr , iru lwhp l @ 8 lq wkh uvw

<

wkuhh shulrgv ri wkh uroolqj krul}rq zkhq wkh vdihw| vwrfn ydoxh iru wklv lwhp lv vhw wr 44< xqlwv +zh dovr kdyh VV Q @ VV 2 @ VV @ 3 dqg VV e @ 64,=

Shulrg " # $ %

Jurvv uhtxluhphqwv g Dcr 69 6< 79 3 Qhw uhtxluhphqwv e Dcr 488 6< 79 3 Sodqqhg rughuv s Dcr 573 3 3 3

Orw vl}h { Dcr 3 3 3 3

Slsholqh Lqyhqwru| ] Dcr 3 3 3 3

Shulrg 4 # $ % &

Jurvv uhtxluhphqwv g Dcr 69 474 79 67 3 Qhw uhtxluhphqwv e Dcr 89 79 67 3

Sodqqhg rughuv s Dcr 469 3 3 3

Orw vl}h { Dcr 573

Slsholqh Lqyhqwru| ] Dcr 537 3 3 3 3

Shulrg 4 5 $ % & '

Jurvv uhtxluhphqwv g Dcr 69 57< 79 67 69 3

Qhw htxluhphqwv e Dcr :7 67 69 3

Sodqqhg rughuv s Dcr 477 3 3 3

Orw vl}h { Dcr 573469

Slsholqh lqyhqwru| ] Dcr 537 <4 3 3 3 3 Wdeoh 6= Uroolqj surfhgxuh iru lwhp 8 zlwk VV D @ 44<> uxq 51

Frpsxwdwlrq ri qhw uhtxluhphqwv qrz lqfoxghv wkh vdihw| vwrfn ydoxh1 Irupxod +7, vlpso| ehfrphv

e cr @ pd{+3> g cr . VV ] cr! { cr ,= +<, Lq shulrg w @ 4> zh kdyh e Dc @ pd{+3> 69 . 44< 3 3, @ 488= Lq wkh qh{w shulrgv/

qhw uhtxluhphqwv htxdo jurvv uhtxluhphqwv/ dv zh qhhg wr rughu wkh vdihw| vwrfn ydoxh rqo| rqfh zlwklq wkh iruhfdvw zlqgrz1 Wdeoh 7 surylghv wkh vdph gdwd dv Wdeoh 5 dw wkh hqg ri uxq 51

Shulrg w 4 5 6 7 8 9 : ;

Dfwxdo jurvv uhtxluhphqwv G Dc| 53; 69 57< 79 553 74 3 8:

Vfkhgxohg uhfhlswv u Dc| 53 ; 573 469 477 354: 33

Hqglqj lqyhqwru| L Dc| 353 7 <4 4;< 064 478 478 ;;

Wdeoh 7= Vlwxdwlrq iru frpsrqhqw 8 zlwk VV D @ 44<> hqg ri uxq 51

Dqrwkhu vwrfnrxw ri 64 xqlwv lv uhfrughg lq shulrg 8 dqg ruljlqdwhv lq wkh vdihw|

vwrfn uhtxluhg e| lwhp 7 dw wkh ehjlqqlqj ri uxq 5 +VV e @ 64,1 Vdihw| vwrfn iru lwhp 8 +VV D @ 44<, glg qrw wdnh lqwr dffrxqw wkh qhhg iru d vdihw| vwrfn ri 64 xqlwv iru lwhp 71 Vdihw| vwrfn ydoxhv pxvw eh xsgdwhg= zh dgg wr wkh suhylrxv vdihw|

vwrfn ydoxh wkh pd{lpxp vwrfnrxw uhfrughg dorqj wkh krul}rq1 Lq rxu fdvh/ zh vhw

43

VV , =@ VV , . 64 @ 483= Dqrwkhu uxq ri wkh uroolqj surfhgxuh surgxfhv wkh qdo uhvxowv iru lwhp 8 +vhh Wdeoh 8,1

Shulrg w 4 5 6 7 8 9 : ;

Dfwxdo jurvv uhtxluhphqwv G ,c| 53; 69 57< 79 553 74 3 8:

Vfkhgxohg uhfhlswv u ,c| 53; 5:4 469 477 3 57; 3 3

Hqglqj lqyhqwru| L ,c| 3 568 455 553 3 53:53:483

Wdeoh 8= Vlwxdwlrq iru frpsrqhqw 8 zlwk VV , @ 483> hqg ri uxq 61

Vwrfnrxwv qr orqjhu dsshdu1 Lw vkrxog eh qrwhg wkdw vfkhgxohg uhfhlswv lq shulrg 4 kdyh vzlwfkhg iurp 4:: xqlwv wr 53; xqlwv wr fryhu wkh uvw jurvv uhtxluhphqwv1 Wklv fkdqjh kdv qr lqflghqfh rq frvw uhvxowv dv vfkhgxohg uhfhlswv lq shulrg 4 duh mxvw phdqw wr fryhu h{dfwo| wkh uvw uhtxluhphqw> wkh| gr qrw lqfxu qhlwkhu d vhw xs frvw +wkh| gr qrw uhvxow iurp rughulqj ghflvlrqv pdgh zlwklq wkh fxuuhqw krul}rq exw sulru wr wkh uvw shulrg ri wkh krul}rq,/ qru fduu|lqj frvwv +wkh| qhyhu h{fhhg uhtxluhphqwv,1 Wkh zd| lqlwldo vfkhgxohg uhfhlswv duh vhw vkrxog mxvw eh vhhq dv d surjudpplqj wulfn1

615 D uhsdlu surfhgxuh

D pruh vrsklvwlfdwhg vroxwlrq wr dyrlg vkruwdjhv zrxog eh wr wudfn grzq wkh lwhpv zkrvh orw vl}hv duh uhvsrqvleoh iru d vwrfnrxw ri d jlyhq lwhp1 Lq rxu h{dpsoh wklv zrxog dprxqw wr frqvlghu wkh vxffhvvruv ri lwhp 8 dqg wkhq wr ghwhfw wkh orw vl}hv wr eh ghfuhdvhg1 Lwhp 8 kdv wzr vxffhvvruv= lwhp 4 dqg lwhp 71 Dv wkh orw iru orw vroxwlrq lv sursrvhg iru lwhp 4/ qrqh ri wkhvh orwv pd| eh ghfuhdvhg1 Lwhp 4 lv wkhuhiruh qrw uhvsrqvleoh iru wkh vwrfnrxw ri lwhp 8 lq shulrg 51 Wkh orw vl}h ri 543 xqlwv ri lwhp 7 lq shulrg 6 lv wkh rqo| fdqglgdwh wr ghsohwlrq1 Wr rewdlq d ihdvleoh vroxwlrq lq shulrg 6/ lw vx!fhv wr ghfuhdvh wkh orw vl}h ri lwhp 7 lq shulrg 6 iurp wkh ghpdqg ri 69 xqlwv lq shulrg 9 dqg wr sodfh dq rughu ri 69 xqlwv lq shulrg 91 Lq wklv h{dpsoh/ wkh zd| wr surfhhg lv vwudljkwiruzdug exw pruh dpeljxrxv vlwxdwlrqv pd| eh hqfrxqwhuhg1 Iru lqvwdqfh/ li erwk lwhpv 4 dqg 7 zhuh fdqglgdwhv wr orw vl}hv ghsohwlrq/ wkh txhvwlrq ri krz wr vhohfw wkhvh lwhpv dqg krz wr ghfuhdvh wkh fruuhvsrqglqj rughuv zrxog eh rshq1 Ri frxuvh lq dvvhpeo| v|vwhpv wkh txhvwlrq ri krz wr fkrrvh d fdqglgdwh wr ghsohwlrq grhv qrw dulvh1 Exw lq kljko| frpsoh{ vwuxfwxuhv uhohydqf| ri vxfk d txhvwlrq ulvhv1 Rqh zd| wr jhw durxqg wklv txhvwlrq lv wr wudqvirup wkh uroolqj vfkhgxoh sureohp lqwr d cfdsdflwdwhg* sureohp1 Wklv surylghv d qdwxudo uhsdlu surfhgxuh wr rewdlq d ihdvleoh vfkhgxoh iurp dq lqihdvleoh rqh1

61514 Nh|qrwh dqg looxvwudwlrq ri wkh surfhgxuh

Wr looxvwudwh/ ohw xv frqvlghu rxu h{dpsoh lq Wdeoh 41 Revhuyh uvw wkdw lq shulrg w @ 5> d sodqqhg rughu ri 435 xqlwv ri lwhp 7 lq shulrg 6 wuljjhuv +wrjhwkhu zlwk dq rughu ri 6< xqlwv ri lwhp 4 lq shulrg 6, d jurvv uhtxluhphqw ri 474 xqlwv ri lwhp 8 lq shulrg 51 Wklv wuljjhuv lq wxuq d qhw uhtxluhphqw iru lwhp 8 zklfk pd| eh fryhuhg e| d ihdvleoh rughu lq shulrg 51 Li/ lq shulrg w @ 6> zh irufh wkh rughu ri lwhp 7 lq shulrg 6 qrw wr h{fhhg 435 xqlwv/ zh fdq rewdlq d ihdvleoh vfkhgxoh iru doo lwhpv lq shulrg 61 Exw wklv frqglwlrq lv qrw vx!flhqw1 Jurvv uhtxluhphqwv iru lwhpv 6 dqg 7 lq shulrg 5 vkrxog qrw h{fhhg 6< xqlwv1 Qr frqglwlrq lv uhtxluhg rq jurvv uhtxluhphqwv iru lwhp 5 dv lwv ohdg wlph lv }hur1 Rqo| lwv jurvv uhtxluhphqwv lq wkh fxuuhqw shulrg

44

w pd| fkdqjh vr lw lv qhyhu wrr odwh wr rughu d surshu dprxqw1 Iru wkh hqg lwhp/

jurvv uhtxluhphqwv dozd|v nhhs wkh vdph ydoxh iurp rqh shulrg wr wkh qh{w +lq wklv ghwhuplqlvwlf iudphzrun, vr qr vwrfnrxw fdq eh h{shfwhg1 Rqh zd| wr jxdudqwhh wkh vdwlvidfwlrq ri vxfk frqglwlrqv rq jurvv uhtxluhphqwv lv wr irufh/ lq wkh fxuuhqw shulrg w> doo sodqqhg rughuv zlwklq wkh fxpxodwlyh ohdg wlph qrw wr h{fhhg wkhlu suhylrxv ydoxhv +frpsxwhg lq shulrg w ",1 Wklv zulwhv

[ ,

&'|

s ^| _c& ^{[ ,}

&'|

s ^|! _c& iru doo o @ w> = = = > w . O 4= +43, Wr looxvwudwh/ frqvlghu rxu h{dpsoh lq Wdeoh 41 Dw wkh hqg ri shulrg 5 zh uhfrug wkh ydoxhv ri sodqqhg rughuv zlwklq wkh fxpxodwlyh ohdg wlph1 Zh uhfrug iru lwhp 4 is ² _c > s ² _ce > s ² _cD j @ i6<> 79> 67j/ iru lwhp 5 is ² _2c > s ² _2ce j @ i3> 3j> iru lwhp 6 is ² _c > s ² _ce j @ i3> 67j dqg iru lwhp 7 is ² _2ce j @ i435j= Lq shulrg 6/ zh uvw dsso| ZZ zlwkrxw dq| uhvwulfwlrq rq wkh ydoxh ri sodqqhg rughuv1 Dv zh idfh d vwrfnrxw/ zh uh0dsso|

ZZ wrjhwkhu zlwk frqglwlrqv +43, rq doo lwhpv vr dv wr rewdlq d ihdvleoh vfkhgxoh1 Iru lwhp 4 / zh uvw vhw s _c @ 6< zklfk yhulhv lqhtxdolw| s _c s ² _c = Zh wkhq hydoxdwh wzr dowhuqdwlyhv= crughulqj lq shulrg 6 wkh ghpdqg iru shulrgv 6 dqg 7*

yhuvxv crughulqj lq shulrg 7 wkh ghpdqg iru shulrg 7 dqg dgrswlqj wkh rswlpdo srolf|

iru wkh suhylrxv shulrg*1 Rswlrq 4ohdgv wr s _c @ ;8> s _ce @ 3 exw wkhvh rughuv pxvw yhuli| s _c s ² _c dqg s _c .s _ce s ² _c .s ² _ce = Dv wkh| rqo| vdwlvi| wkh vhfrqg frqglwlrq/

wkh| duh dedqgrqhg dqg wkh vhfrqg rswlrq +s _c @ 6<> s _ce @ 79,/ zklfk yhulhv erwk lqhtxdolwlhv/ lv dgrswhg dv wkh rswlpdo vroxwlrq iru wkh vhfrqg shulrg1 Wkuhh rswlrqv duh wkhq hydoxdwhg= s _c @ 44<> s ce @ 3> s cD @ 3 ru s _c @ 6<> s ce @ ;3> s cD @ 3 ru s _c @ 6<> s ce @ 79> s cD @ 67= Wkh fkhdshvw rswlrq wkdw yhulhv s _c s ² _c dqg s _c . s _ce s ² _c . s ² _ce dqg s _c . s _ce . s _cD s ² _c . s ² _ce . s ² _cD lv vhohfwhg dv wkh rswlpdo vroxwlrq iru d krul}rq ri wkuhh shulrgv1 Rqo| wkh vhfrqg vfkhgxoh reylrxvo|

wv wkhvh fulwhuld dqg lv vhohfwhg dv wkh rswlpdo sodq1 Lq wkh qh{w shulrg zh hydoxdwh wkh iroorzlqj dowhuqdwlyhv= crughulqj d vlqjoh orw lq shulrg 6*/ crughulqj e ce .e cD .e cS

lq shulrg 7 dqg dgrswlqj wkh rswlpdo srolf| iru shulrg 6*/ crughulqj e cD . e cS lq shulrg 8 dqg dgrswlqj wkh rswlpdo srolf| iru shulrgv 6 dqg 7*/ crughulqj e cS lq shulrg 9 dqg dgrswlqj wkh rswlpdo srolf| iru shulrgv 6 wr 8*1 Wkh odvw rswlrq lv vhohfwhg dv wkh rswlpdo sodq1 Li zh kdg Z @ 8 zh zrxog kdyh dq dgglwlrqdo ghpdqg lq shulrg : dqg dv wkh rswlpdo suhylrxv gdwh ri rughu lv shulrg 9 zklfk lv rxwvlgh wkh fxpxodwlyh ohdg wlph +l1h1 wkh lqwhuydo lq zklfk fdsdflwlhv duh frqvlghuhg, d cuhjxodu*

ZZ zrxog kdyh ehhq dssolhg iurp shulrg 9 zkloh nhhslqj wkh rswlpdo sodq iurp shulrgv 6 wr 81 Wklv olplwv wkh qxpehu ri xqghvludeoh sodqv wr eh hydoxdwhg dqg vr wkh frpsxwdwlrqdo uhtxluhphqwv ri wkh dojrulwkp1 Lq wkh iroorzlqj/ zh xvh d vlpsohu qrwdwlrq iru fdsdflwlhv> zh vhw F cr @ s ^|! _cr = Wdeoh 9 surylghv wkh ihdvleoh vfkhgxohv iru doo lwhpv lq shulrg 6/ uhvxowlqj iurp wkh dssolfdwlrq ri wkh cfdsdflwdwhg* ZZ1

Qrz zh kdyh odlg wkh edvlv ri rxu uhsdlu surfhgxuh zh vkdoo suhvhqw wkh ixoo surfhgxuh lq d pruh irupdo zd| lq wkh qh{w sdudjudsk1

61515 Irupdo suhvhqwdwlrq

Wdeoh :/ zkloh jlylqj wkh lqlwldo ydoxh ri wkh yduldeohv/ surylghv d xvhixo vxppdu| ri wkh qrwdwlrq hpsor|hg lq wkh sdshu1

Wdeoh ; olvwv wkh rshudwlrqv wr eh shuiruphg dw wkh uvw lwhudwlrq iru doo lwhpv1 Wkh krul}rq lv olplwhg wr wkh iruhfdvw zlqgrz Z zlwklq zklfk zh frpsxwh wkh jurvv

45

Shulrg 4 5 6 | ' 7 8 9

KE ' _

Q?H

89 69 6< 79 67 69 ,

Q

' f K

Q?H

6< 79 67 69 R

Q?H

6< 79 67 69

Q?H

6< 79 67

%

Q?H

89 69

~

Q?H

3 3 3 3 3 3 KE2 ' t _

2?H

89 69 6< 79 67 69 ,

2

' f K

2?H

3 3 67 69 R

2?H

3 3 :3 3

2?H

3 3

%

2?H

4:: 3

~

2?H

454 ;8 79 3 3 3 KE ' t _

?H

69 6< 79 67 69 3 , '  K

?H

3 67 69 3 R

?H

3 67 69 3

?H

3 67

%

?H

454 3

~

?H

;8 79 3 3 3 3 KEe ' tc 2c _

e?H

69 6< ;3 473 69 3 ,

e

'  K

e?H

67 473 69 3 R

e?H

67 4:9 3 3

e?H

435

%

e?H

454 3

~

e?H

;8 79 3 3 3 3 KED ' tc e _

D?H

69 :6 555 67 69 3 ,

e

'  K

D?H

:7 67 69 3 R

D?H

477 3 3 3

%

D?H

454 469

~

D?H

;8 47; 3 3 3 3

Wdeoh 9= Dssolfdwlrq ri wkh uhsdlu surfhgxuh +cfdsdflwdwhg* ZZ,1

iru l @ 4> = = = > S iru v @ 4> = = = > W

g

l>v

@ 3 Jurvv uhtxluhphqwv wkdw pxvw eh vdwlvhg lq shulrg v . o

l

{

l>v

@ 3 Ilup rughu odxqfkhg lq shulrg v

]

l>v

@ 3 Slsholqh lqyhqwru| dw wkh hqg ri shulrg v

e

l>v

@ 3 Qhw uhtxluhphqwv wkdw pxvw eh vdwlvhg lq shulrg v . o

l

s

l>v

@ 3 Sodqqhg rughu wkdw zloo srvvleo| eh uhohdvhg lq shulrg v

L

l>v

@ 3 cDfwxdo* hqglqj lqyhqwru| lq shulrg v

u

l>v

@ 3 Vfkhgxohg uhfhlswv lq shulrg v uhvxowlqj iurp rughuv

uhdoo| odxqfkhg lq shulrg v o

l

Wdeoh := Lqlwldol}dwlrq ri yduldeohv dqg vxppdu|ri qrwdwlrq1

46

uhtxluhphqwv/ wkhq wkh slsholqh lqyhqwru| dqg wkh qhw uhtxluhphqwv1 Rqfh qhw uh0 txluhphqwv kdyh ehhq fdofxodwhg/ zh dsso| dq| orw0vl}lqj uxoh wr ghwhuplqh sodqqhg rughuv1 Rqo| wkh uvw sodqqhg rughu lv wudqviruphg lqwr d up rughu wkdw wuljjhuv vfkhgxohg uhfhlswv o shulrg+v, odwhu1 Zh wkhq uhfrug wkh fdsdflwlhv wkdw zloo srvvl0 eo| eh xvhg lq fdvh wkh vroxwlrq dw wkh qh{w lwhudwlrq lv lqihdvleoh1 Lq wlph lqwhuydo i"> = = = > o j vfkhgxohg uhfhlswv pxvw eh frqvlghuhg dv gdwd iru wkh| uhvxow iurp orw0 vl}lqj ghflvlrq wkdw wrrn sodfh ehiruh shulrg 41 Qrqhwkhohvv zh vhw wkhp wr d ydoxh wkdw pdnh ihdvleoh dq| vroxwlrq1 E| grlqj vr/ doo surgxfwlrq srvvlelolwlhv pd| wdnh sodfh zlwklq wkh fxpxodwlyh ohdg wlph iru hdfk lwhp1 Wr vprrwk dzd| wkh shulrgv lq zklfk qr surgxfwlrq pd| kdsshq/ vrph dxwkruv +h1j1 Zhpphuoùy dqg Zk|edun/

4<;7, vlpso| fkrrvh wr uhfrug qr vwdwlvwlfv iru d jlyhq qxpehu ri shulrgv fdoohg d vwduw0xs shulrg1

iru l @ 4> = = = > S

* iru v @ 4> = = = > Z

Frpsxwh g cr xvlqj htxdwlrq +8, Frpsxwh ] cr xvlqj htxdwlrq +6, Frpsxwh e cr xvlqj htxdwlrq +7,

* Dsso| dq| uxoh rq ie cr j r'cc`

+zh wkxv rewdlq is cr j r'cc` ,

* vhw { c @ s c

u cn, @ { c

* iru v @ 5> = = = > 4 . O

vhw F cr @ s cr

Lqlwldo ydoxhv iru vfkhgxohg uhfhlswv iru l @ 4> = = = > S

iru v @ 4> = = = > o

u cr @ ⁵

MKE s cr

Wdeoh ;= Iluvw lwhudwlrq +w @ 4,1

Wdeoh < suhvhqwv wkh surfhgxuh dw dq| lwhudwlrq w A 4 +ohiw vlgh ri wkh wdeoh,1 Lq hdfk shulrg w zh frpsxwh wkh jurvv uhtxluhphqwv ri shulrg w o 4= Lq wklv sdvw shulrg/ jurvv uhtxluhphqwv qr orqjhu uhvxow iurp sodqqhg rughuv exw iurp up rughuv vlqfh zh kdyh g c|3, 3 @ ⁵

MKE f c { c|3 dqg { c|3 iru hdfk lwhp m kdv ehhq ghwhuplqhg lq wkh suhylrxv shulrg1 ^" Lq wlph lqwhuydo iw o > = = = > w 4j ghpdqg lv uhfrpsxwhg dv lw uhvxowv iurp sodqqhg rughuv lq shulrgv w wr w . o 4 exw lq wlph lqwhuydo iw o > = = = > w 4j> rughuv kdyh douhdg| ehhq odxqfkhg vr ghpdqg vkrxog qrw h{fhhg wkh pdwhuldo dydlodelolw|1 Lq rwkhu zrugv/ zh fkhfn lqhtxdolw| +;,1 Zkhq wklv lqhtxdolw| lv qrw yhulhg/ zh lpsohphqw wkh uhsdlu surfhgxuh h{klelwhg lq wkh uljkw

"

Qrwh wkdw lqvwhdg ri shulrg | ! , !  zh vkrxog vshdn derxw shulrg 4@ Ec | ! , !  D vlplodu uhpdun pd| eh grqh iru shulrg | n ` ! c lw vkrxog eh shulrg 4?E| n ` ! c A

47

vlgh ri Wdeoh <1 Zh vkdoo frpphqw wklv vxe0surfhgxuh odwhu1 Zkhq wkh ihdvlelolw|

frqglwlrq lv vdwlvhg zh ghwhuplqh wkh qhw uhtxluhphqwv zlwklq wkh zlqgrz dqg zh dsso| dq| orw0vl}lqj uxoh1 Li doo lwhpv kdyh sdvvhg wkh ihdvlelolw| whvw/ zh lpsohphqw wkh uvw sodq rughu e| vhwwlqj { c| ( s c| / iru doo lwhpv dqg zh vhw wkh vfkhgxohg uhfhlswv wr wkh dssursuldwh ydoxh1 Zh qdoo| uhfrug wkh fdsdflwlhv1

Pdlq orrs Uhsdlu iru ' c c iru ' c c

iru | ' 2c c A iru r ' | ! , ! c c | n ` !  Uhfrpsxwh _

c|!,l!Q

Frpsxwh _

cr

Uhfrpsxwh ~

c|!,l!Q

Frpsxwh ~

cr

iru r ' | ! , c c | !  Frpsxwh K

cr

Frpsxwh _

cr

li u : f

li ~

cr!Q

n %

cr

_

cr

Dsso| dq| uxoh rq tK

cr

r'|cc|n` !

jr wr Uhsdlu zlwk uhvshfw wr wkh fdsdflwlhv t

cr

r'|cc|nul!

hovh +zh rewdlq tR

cr

r'|cc|n` !

, Frpsxwh ~

cr

hovh

Frpsxwh K

cr

Dsso| dq| uxoh rq tK

cr

r'|cc|n` !

iru r ' |c c | n ` 3  vhw %

c|

' R

c|

Frpsxwh _

cr

vhw o

c|n,l

' %

c|

Frpsxwh ~

cr

iru r ' | n c c | n u

Frpsxwh K

cr

cr

' R

cr

Dsso| dq| uxoh rq tK

cr

r'|c|n` !

+zh rewdlq tR

cr

r'|cc|n` !

, li '

iru ' c c vhw %

c|

' R

c|

vhw o

c|n,m

' %

c|

iru r ' | n c | n u

cr

' R

cr

Wdeoh <=Pdlq orrs dqg uhsdlu surfhgxuh1

Wkh uhsdlu surfhgxuh lv lpsohphqwhg dv vrrq dv dq lwhp grhv qrw w wkh ihdvlelolw|

frqglwlrq1 Iru doo lwhpv/ zh ghwhuplqh wkh qhw uhtxluhphqwv dqg wkhq dsso| dq| uxoh zklfk kdv ehhq prglhg wr phhw frqglwlrqv +43,1 Zh vkdoo jlyh ixuwkhu ghwdlov rq wkh fdsdflwdwhg yhuvlrq ri wkh orw0vl}lqj uxohv lq wkh qh{w vhfwlrq1 Dw wkh hqg ri wkh zkroh surfhgxuh/ zh frpsxwh wkh hqglqj lqyhqwrulhv L c| iru doo lwhpv dqg doo shulrgv1 Zh wkhq frpsxwh wkh wrwdo frvw dvvrfldwhg zlwk d jlyhq orw0vl}lqj uxoh1

7 H{shulphqwdo iudphzrun

714 Orw0vl}lqj surfhgxuhv

Zh vhohfwhg vl{ vlqjoh0ohyho orw0vl}lqj uxohv iru xvh lq wkh vwxg|1 Wkh Zdjqhu0Zklwlq dojrulwkp +4<8;,/ wkh Vloyhu0Phdo whfkqltxh +4<:5,/ wkh lqfuhphqwdo sduw shulrg dojr0 ulwkp +4<9;,/ wkh Vloyhu yhuvlrq ri wkh hfrqrplf rughu txdqwlw| +4<:9,/ wkh shulrglf rughu txdqwlw| dqg wkh ohdvw xqlw frvw phwkrg1 Ghvslwh wkh ghyhorsphqw ri khxulv0 wlfv vshflfdoo| ghvljqhg wr dffrxqw iru lqwhughshqghqflhv dprqj vwdjhv/ wkh fkrvhq khxulvwlfv duh vwloo zlgho| dgrswhg lq sudfwlfh xqghu uroolqj dqg pxowl0ohyho frqglwlrqv1

48

Wkh Zdjqhu0Zklwlq dojrulwkp +ZZ, surylghv wkh rswlpdo vroxwlrq wr wkh vlqjoh0 ohyho orw0vl}lqj sureohp e| xvh ri g|qdplf surjudpplqj1 Wkh hfrqrplf rughu txdq0 wlw| +HRT, lv wkh wudglwlrqdo Zlovrq orw0vl}lqj prgho zklfk edodqfhv wkh lqyhqwru|

fduu|lqj frvwv dqg rughu frvwv1 Wkh Vloyhu yhuvlrq ri wklv phwkrg oxpsv dq lqwhjhu qxpehu ri ixwxuh ghpdqgv forvhvw wr wkh HRT ydoxh1 Wkh shulrglf rughu txdqwlw|

+SRT, xvhv wkh HRT wr ghwhuplqh wkh uhrughu wlph f|foh dqg wkhq rughuv zkdw lv df0 wxdoo| iruhfdvwhg iru wkdw wlph f|foh1 Wkh lqfuhphqwdo sduw shulrg dojrulwkp +LSSD, lqfuhdvhv wkh vl}h ri wkh rughu xqwlo fduu|lqj frvwv duh htxdo ru ohvv wkdq wkh vhw0xs frvw1 Wkh Vloyhu0Phdo khxulvwlf vhohfwv wkh rughu txdqwlw| vr dv wr plqlpl}h wkh frvw shu xqlw wlph ryhu wkh wlph shulrgv wkh rughu odvwv1 Wkh ohdvw xqlw frvw +OXF, vhohfwv wkh rughu txdqwlw| vr dv wr plqlpl}h wkh frvw shu xqlw +fxpxodwlrq ri wkh uhtxluhphqwv xqwlo wkh frvw shu xqlw vwduwv wr lqfuhdvh,1

Hdfk whfkqltxh lv hlwkhu frpelqhg zlwk d vdihw| vwrfn vwudwhj| dv ghvfulehg lq sdudjudsk 614 ru zlwk rxu uhsdlu surfhgxuh dv ghwdlohg lq sdudjudsk 615141 Wr rewdlq d cfdsdflwdwhg* yhuvlrq ri wkh khxulvwlfv/ zh kdyh dssolhg wkh vdph sulqflsohv dv wkrvh hpsor|hg iru ZZ lq sdudjudsk 615141 Hdfk khxulvwlf vlpso| oxpsv wkh ghpdqgv xqwlo wkh fulwhulrq lv plqlpl}hg xqohvv wkh orw vl}h h{fhhgv wkh dydlodeoh fdsdflw|1

Rxu lqfhqwlyh lv wr frpsduh wkh frvw shuirupdqfh ri rxu uhsdlu surfhgxuh wr wkh frvwv surgxfhg e| wkh vdihw| vwrfn vwudwhj| zklfk zdv vshflfdoo| ghvljqhg wr uhdfk wkh vdph vhuylfh dv rxu uhsdlu/ vd| 433( +vhh sdudjudsk 614,1 Iurp d sudfwlfdo vwdqgsrlqw/ wklv vdihw| vwrfn vwudwhj| lv qrw uhdg| iru xvh/ dv wkh zkroh vwru| ri gh0 pdqgv iru hqg lwhpv qhhgv wr eh nqrzq wr vhw wkh dssursuldwh ydoxhv ri vdihw| vwrfnv1 Vdihw| vwrfnv duh wkhuhiruh duwlfldoo| vhw wr d plqlpxp ydoxh iru zklfk wkhuh lv qrw d vlqjoh orvw ghpdqg iru hqg lwhpv1 Lq sudfwlfh krzhyhu/ vdihw| vwrfnv duh ghflghg ehiruhkdqg dqg d 433( vhuylfh ohyho fdq rqo| eh dfklhyhg dw wkh sulfh ri h{wuhpho|

odujh dprxqwv ri vdihw| vwrfnv1 Lq wkdw uhvshfw/ rxu shfxoldu lpsohphqwdwlrq ri wkh vdihw| vwrfn vwudwhj| +zkhuh ohyhov ri vwrfnv duh ghwhuplqhg uhwurvshfwlyho|, whqgv wr vdqfwlrq wkh shuirupdqfh ri rxu uhsdlu surfhgxuh1 Krzhyhu/ frpsdulqj rxu uhsdlu surfhgxuh wr pruh rshudwlrqdo vdihw| vwrfn srolflhv zrxog kdyh ohg wr d wlfnolvk elful0 whuld frpsdulvrq +frvw dqg vhuylfh ohyho,1 Wr dyrlg wklv/ zkloh frpsdulqj rxu uhsdlu wr dq h{lvwlqj phwkrg/ zh fkrvh wklv vshflf vdihw| vwrfn vwudwhj| iru odfn ri h{lvwlqj rshudwlrqdo whfkqltxhv suhyhqwlqj vwrfnrxw vlwxdwlrqv1

715 Ghpdqg jhqhudwlrq

Dowkrxjk vhyhudo hqg lwhpv frxog kdyh ehhq lqfoxghg lq wkh suhvhqw vwxg| zh fkrvh wr wdnh rqo| rqh hqg lwhp shu surgxfw vwuxfwxuh iru wkh vdnh ri vlpsolflw|1 Ghpdqg iru wkh hqg lwhp ig r j r'|cc|n` ! pd| iroorzglhuhqw sdwwhuqv1 Zh fkrvh wr jhqhudwh gh0 pdqg iru wkh hqg lwhp iurp d xqlirup glvwulexwlrq ri phdq 831 Zh kdyh g r X^3> 5g`1 Vwdqgdug ghyldwlrq ri ghpdqgv wkhuhiruh htxdov 4@45+433 3, ^{2 *2} @ 5;=;9= Zh dovr fkrvh wr xvh d qrupdo glvwulexwlrq zlwk vdph dyhudjh dqg vwdqgdug ghyldwlrq=

g r Q^83> 5;=;9`=

716 Frvw jhqhudwlrq

Lq dffrugdqfh zlwk wkh dvvxpswlrq ri ydoxh0dgghg kroglqj frvwv/ fduu|lqj frvwv iru hdfk lwhp zhuh ghqhg dv iroorzv

49

k ( h . ^[

MK

^!Q

E

f c k zlwk h @ 3=3338 . 3=35 x>

zkhuh x lv vhohfwhg iurp d xqlirup glvwulexwlrq ehwzhhq 3 dqg 4=

Zh xvhg wkh W ER +wlph ehwzhhq rughuv, idfwru wr ghwhuplqh vhw0xs frvw sdud0 phwhuv1 W ER ydoxhv duh xvxdoo| odujhu iru frpsrqhqwv wkdq wkh| duh iru hqg lwhpv dqg vxedvvhpeolhv1 Lqghhg/ vxffhvvixo MLW sulqflsohv lpsohphqwdwlrq lq pdq| frp0 sdqlhv kdv ohdg wr vxevwdqwldo vhw xs frvw uhgxfwlrqv iru wkhvh lwhpv zkhuhdv iuhhgrp ri dfwlrq lv vwloo olplwhg iru frpsrqhqwv dw ghhshu ohyhov dqg sxufkdvhg lwhpv1 Zh wkhuhiruh frqvlghu W ER ydoxhv ri 4 dqg 5iru wkh hqg lwhp dqg vxedvvhpeolhv +W ER , , dqg WER ydoxhv ri 5dqg 9 iru surgxfwv dw ghhshu ohyhov +W ER ,1 Zh wkhq kdyh wkh iroorzlqj frpelqdwlrqv= +W ER , > W ER , @ i+4> 5,> +5> 5,> +4> 9,> +5> 9,j= Iru wkh vdnh ri fxulrvlw|/ zh dovr h{dplqh +W ER , > W ER , @ +7> 9,=

Iru hdfk lwhp zh vhw V @

+ 3=8 g k W ER ² _, li l lv dw ohyho 3 ru 4/

3=8 g k W ER ² rwkhuzlvh/

zlwk g @ ^[

MKE

f c g =

717 Rwkhu sdudphwhuv

Lq rxu h{shulphqw/ zh frqvlghuhg vwuxfwxuhv zlwk rqh hqg lwhp pdgh ri < frpsr0 qhqwv +S @ 43, glvwulexwhg ryhu irxu ohyhov +3> 4> 5> 6,1 Surgxfw vwuxfwxuhv zhuh ghqhg lq whupv ri wkh frpsoh{lw| lqgh{ F1 Zh fkrvh 7 ydoxhv ri F lq wkh vhw i3=33> 3=58> 3=83> 3=:8j1 Zh vwrsshg wkh uroolqj krul}rq dw shulrg 433 +W @ 433,1 H{h0 fxwlrq wlph ri ZZ ohjlwlpl}hv wkh fkrlfh ri lqvwdqfhv ri prghudwh vl}h1 Iru lqvwdqfh wr vroyh rqh uroolqj sureohp zlwk S @ 43 dqg W @ 433> hdfk vlqjoh0ohyho whfkqltxh lv dssolhg dssur{lpdwho| S W @ 4333 wlphv dqg ghwhuplqdwlrq ri dssursuldwh ydoxhv iru vdihw| vwrfnv uhtxluhv rq dyhudjh 6 sdvvhv ru uxqv ri wkh ixoo uroolqj surfhgxuh1 Wkxv/ wr jhw frvw revhuydwlrqv dvvrfldwhg zlwk rqh whfkqltxh dqg rqh vfhqdulr/ zh qhhg wr dsso| wkh frqfhuqhg whfkqltxh 6 S W @ 6333 wlphv1 Wkxv/ ghvslwh wkh ghyhorsphqw ri hyhu pruh srzhuixo frpsxwhuv/ frpsxwdwlrqdo glvdgydqwdjh ri ZZ lv xqplvwdndeoh xqghu pxowl0ohyho uroolqj frqglwlrqv1

Wzr ohyhov ri ohdg wlphvorz dqg kljkzhuh vhohfwhg1 Lq wkh uvw fdvh/ ohdg wlphv iru frpsrqhqwv zlwk qr suhghfhvvru zhuh xqlirupo| gudzq dw udqgrp lq wkh vhw i4> 5> 6j dqg lq wkh vhw i3> 4j iru dq| rwkhu lwhp1 Lq wkh vhfrqg fdvh/ ohdg wlphv iru frpsrqhqwv zhuh gudzq lq wkh vhw i5> = = = > 9j dqg lq wkh vhw i4> 5j iru lwhpv dw kljkhu ohyhov1 Uhvxowlqj fxpxodwlyh ohdg wlphv zhuh wkhuhiruh kljkhu lq wkh odwwhu vlwxdwlrq1 Iruhfdvw zlqgrz ohqjwkv zhuh fkrvhq dffruglqjo|= Z @ i9> = = = > 46j iru orz ohdg wlphv dqg Z @ i44> = = = > 4;j iru kljk ohdg wlphv1 Lq wklv zd|/ zh dyrlg lqvwdqfhv lq zklfk fxpxodwlyh ohdg wlphv h{fhhg wkh iruhfdvw zlqgrz/ d vlwxdwlrq uhtxlulqj wkh lqwurgxfwlrq ri vdihw| vwrfnv hyhq lq rxu uhsdlu surfhgxuh vlqfh vrph surgxfwlrqv fdq qrw eh odxqfkhg lq wlph zlwklq wkh iruhfdvw zlqgrz wr phhw vrph uhtxluhphqwv1 1

Hdfk ri wkh 5 8 7 5 ; @ 973 h{shulphqwv +5ghpdqg sdwwhuqv/ 8 ohyhov ri WER/ 7 frpsoh{lwlhv/ 5w|shv ri ohdg wlphv/ ; zlqgrz ohqjwkv, zdv uhsolfdwhg 8 wlphv/ wkhuhiruh surylglqj 6533 frvw revhuydwlrqv shu orw0vl}lqj uxoh1

4:

8 Vlpxodwlrq uhvxowv

Iljxuh 5 glvsod|v wkh dyhudjh frvw ghyldwlrqv uhodwlyh wr ZZ zkhq frpelqhg wr rxu uhsdlu surfhgxuh +qdpho| ZZUS,1 Wklv whfkqltxh zdv xvhg dv d ehqfkpdun dv lw surylghv wkh ehvw ryhudoo shuirupdqfh1 Hdfk uxoh lv dsshqghg zlwk US zkhq xvhg zlwk wkh uhsdlu surfhgxuh dqg zlwk VV zkhq dvvrfldwhg zlwk wkh vdihw| vwrfn srolf|1 Frvw revhuydwlrqv iru hdfk uxoh dqg hdfk vfhqdulr zhuh glylghg e| wkh frvw ri ZZUS1 Wkh uhvxowdqw frvw udwlrv zhuh dyhudjhg ryhu doo fdvhv zlwk d vdph idfwru ydoxh1 Iru lqvwdqfh SRTUS surylghv d frvw wkdw lv 4139( kljkhu wkdq wkh frvw riZZUS iru doo fdvhv zlwk dvvhpeo| vwuxfwxuhv +. ( !=!!,1

!"#

$%&# !"#

%' o

l

()* o

l

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

,

+ + - , .

Iljxuh 5= Dyhudjh frvw ghyldwlrqv uhodwlyh wr ehqfkpdun1

Iljxuh 5 vkrzv wkh vxshulrulw| riwkh uhsdlu surfhgxuh ryhu wkh vdihw| vwrfn vwudw0 hj| iru doo whfkqltxhv/ h{fhsw OXF dqg VP1 Iru wkhvh uxohv/ wkh vdihw| vwrfn srolf|

surylghv ehwwhu uhvxowv wkdq rxu uhsdlu1 Exw wkh gl!fxowlhv rilpsohphqwlqj vxfk d srolf| lq sudfwlfh vkrxog qrw eh irujrwwhq1 ZZUS rhuv wkh ehvw ryhudoo shuirupdqfh

4;

dqg wkh srsxodu SRT dqg LSSD ehkdyh suhww| zhoo zkhq dvvrfldwhg zlwk rxu uhsdlu1 Wkh Nuxvndo0Zdoolv DQRYD whvw zdv xvhg wr dssudlvh wkh lpsdfw ri idfwru ohyhov rq wkh uhodwlyh frvw shuirupdqfh1 Shuirupdqfh ri doo whfkqltxhv lv dhfwhg e| idfwru ydoxhv h{fhsw lq d ihz lvrodwhg fdvhv1 Zlwk wkh h{fhswlrq ri VP/ uhodwlyh shuirupdqfh ri doo uxohv frpelqhg zlwk wkh uhsdlu whqgv wr ghjudgh zkhq wkh ohyho ri frpsoh{lw|

lv lqfuhdvhg1 Wklv lv suredeo| gxh wr wkh qxpehu ri wlphv wkh uhsdlu surfhgxuh lv dssolhg1 Wklv qxpehu xqgrxewhgo| lqfuhdvhv zlwk wkh qxpehu ri frpprq sduwv dqg hdfk wlph wkh uhsdlu lv dssolhg/ wkh crswlpdolw| sulqflsoh* ri hdfk uxoh lv glvuxswhg1 Lw vkrxog eh qrwhg wkdw LSSDUS surgxfhv frvwv dv jrrg dv wkrvh ri ZZUS iru sxuh dvvhpeo| vwuxfwxuhv1 Wkh DQRYD whvw uhyhdohg wkdw wkh uhodwlyh shuirupdqfh ri SRTUSdssduhqwo| ghfuhdvlqj zlwk .lv dfwxdoo| qrw dhfwhg e| wkh frpsoh{lw|

+zh zhuh qrw deoh wr uhmhfw wkh qxoo k|srwkhvlv ri htxdolw| dw wkh 8( ohyho,1 HRTUS zkrvh shuirupdqfh ghfuhdvhv dv vrrq dv . ehfrphv srvlwlyh lv qr orqjhu dhfwhg e|

srvlwlyh ohyhov ri frpsoh{lw| +iru . A 3> wkh fulwlfdo suredelolw| htxdov 8=;3(,1 Wkh uhodwlrqvkls ehwzhhq surgxfw vwuxfwxuhv frpsoh{lw| dqg shuirupdqfh ri wkh whfkqltxhv dvvrfldwhg zlwk wkh vdihw| vwrfn vwudwhj| lv xqfohdu= iru vrph uxohv +olnh ZZVV/ SRTVV dqg HRTVV,/ uhodwlyh shuirupdqfh frqwlqxrxvo| lpsuryhv dv . lv dxjphqwhg/ iru rwkhuv +olnh LSSDVV dqg VPVV, wkh lq xhqfh ri wklv idfwru lv xqgh0 qhg zkhuhdv OXFVV h{klelwv d ehwwhu ehkdylrxu iru ohvv frpsoh{ surgxfw vwuxfwxuhv1 Wr dqdo|}h wkh lpsdfw ri W ER ohyhov rq frvw shuirupdqfhv/ zh xvhg wkh Zlofr{rq whvw wr frpsduh sdluv ri frvw udwlr vdpsohv dvvrfldwhg zlwk phdqlqjixo sdluv ri W ER ohyhov1 Iru lqvwdqfh/ vzlwfklqj iurp WER @ +4> 5, wr WER @ +4> 9, fruuhvsrqgv wr d WER lqfuhdvh iru frpsrqhqwv rqo|1 Lq vxfk d vlwxdwlrq/ wkh Zlofr{rq whvw zloo whoo/ iru dq| uxoh/ zkhwkhu frvw udwlrv duh vljqlfdqwo| glhuhqw ru qrw xqghu wkhvh wzr frqjxudwlrqv1 Wr frpsohphqw wkh dqdo|vlv/ zh xwlol}hg d vljq whvw zkhqhyhu wkh Zlofr{rq whvw vxjjhvwhg wr uhmhfw wkh qxoo k|srwkhvlv ri htxdolw|1 Wdeoh 43 vxppd0 ul}hv wkh whvw uhvxowv iru hdfk uxoh xqghu wzr pdmru fdvhv= W ER lv dxjphqwhg iru frpsrqhqwv rqo|> W ER lv lqfuhdvhg iru hqg lwhpv dqg vxedvvhpeolhv rqo|1 Hdfk fdvh lv fkdudfwhul}hg e| d vxevhw ri sdluv ri WER uhsruwhg lq wkh vhfrqg urz ri wdeoh 431 D W ER lqfuhdvh iru frpsrqhqwv lv ghqhg e| d fkdqjh ri WER iurp WER @ +4> 5, wr +5> 5, dv zhoo dv d fkdqjh iurp W ER @ +5> 5, wr +5> 9,= Vlploduo|/ d vzlwfk ri W ER iurp +4> 5, wr +5> 5, ru +4> 9, wr +5> 9, fruuhvsrqgv wr d W ER lqfuhdvh iru hqg lwhpv dqg vxedvvhpeolhv rqo|1 Lqwxlwlyho|/ zh zrxog kdyh h{shfwhg d fohdu lqyhuvh uhodwlrqvkls ehwzhhq W ER ohyhov dqg wkh shuirupdqfh ri rxu uhsdlu surfhgxuh dv lw idyrxuv pruh rughuv dqg ohvv lqyhqwru|1 Wkxv d W ER lqfuhdvh +ru htxlydohqwo| dq lqfuhdvh ri vhw xs frvw sdudphwhuv, vkrxog wkhq whqg wr ghjudgh wkh shuirupdqfh ri rxu uhsdlu sur0 fhgxuh1 Wdeoh 43 vkrzv wkdw uhodwlyh shuirupdqfh ri LSSDUS dqg HRTUS ghfuhdvhv zkhq W ER iru frpsrqhqwv lv dxjphqwhg zkhuhdv lw lpsuryhv zkhq W ER iru hqg lwhpv lv lqfuhdvhg1 Wkh vdph hhfw lv rewdlqhg iru HRTVV1 Lq prvw vlwxdwlrqv/ wkh lpsdfw ri W ER lqfuhdvhv uhpdlqv lqghqlwh1 Zkhq d W ER lqfuhdvh grhv qrw dowhu wkh uhodwlyh shuirupdqfh +wklv rffxuv iru lqvwdqfh iru LSSDUS iru d W ER fkdqjh iurp +5> 5, wr +5> 9,,> zh uhsruwhg wkh fulwlfdo suredelolw| +sf, lq shufhqw uhvxowlqj iurpwkh Zlofr{rq whvw lq Wdeoh 431 D vljq whvw shuiruphg rq jurvv frvw gdwd uhyhdohg wkdw vhw xs frvwv duh lqghhg vljqlfdqwo| kljkhu iru US surfhgxuhv wkdq wkrvh surgxfhg e| VV phwkrgv1 Wklv rqo| phdqv wkh vkduh ri vhw xsv frvwv lq wrwdo frvw lv kljkhu iru US surfhgxuhv1 Wklv idfw krzhyhu grhv qrw surylgh dq| xvhixo lqirupdwlrq wr suhglfw wkh qdwxuh ri wkh uhodwlrqvkls ehwzhhq W ER dqg uhodwlyh frvw shuirupdqfh ri

4<

US whfkqltxhv dv rxwolqhg e| uhvxowv lq Wdeoh 431

wer iru frpsrqhqwv wer iru hqg lwhpv dqg vxedvv1 +4> 5, +5> 5, +4> 5, +4> 9, +4> 9, +5> 9,

+4> 9, +5> 9,

Uhvxow +5> 5,

+5> 9,

+7> 9,

+7> 9, Uhvxow

lssdus

+9<=<:,^@

_+58=33,^@

srtus lqghi1 lqghi1

hrtus

vpus lqghi1

_+5<=89,^@

oxfus

zzvv lqghi1

_+95=94,^@

lqghi1

lssdvv

_+49=89,^@

_+:7=<6,^@

lqghi1

srtvv lqghi1

hrtvv

vpvv lqghi1 lqghi1

oxfvv

_+46=44,^@

Wdeoh 43= Dvvhvvlqj wkh lpsdfw ri W ER lqfuhdvhv rq uhodwlyh frvw shuirupdqfhv1 Wkh fkrlfh ri d sduwlfxodu ghpdqg sdwwhuq dovr kdv dq lq xhqfh rq wkh uhodwlyh shuirupdqfh ri wkh whfkqltxhv1 Uhodwlyh shuirupdqfh lv fohduo| ehwwhu xqghu wkh dvvxpswlrq ri d qrupdo ghpdqg iru doo whfkqltxhv h{fhsw ZZVV1 Zlwk dq dyhudjh ri 83 dqg d vwdqgdug ghyldwlrq ri 5;=;9> xqlirup gudzv ri ghpdqg wdnh pruh riwhq h{wuhph ydoxhv +iduhu iurp wkh dyhudjh, wkdq qrupdo gudzv1 ZZ sxwv xs ehwwhu zlwk wklv nlqg ri ghpdqg vkdsh wkdq rwkhu khxulvwlfv wkdw ghqlwho| ehkdyh ehwwhu zkhq h{wuhph ydoxhv ri ghpdqg duh vsrudglf1

Vzlwfklqj iurp orz wr kljk ohyhov ri ohdg wlph lv ghwulphqwdo wr uhodwlyh shuiru0 pdqfh ri doo phwkrgv h{fhsw LSSDVV iru zklfk wkhuh lv qr vljqlfdqw lpsdfw +wkh DQRYD whvw rq frvw udwlrv jlyhv sf @ 45=48(,1 Kljkhu ohdg wlphv lpso| ohvv h{lelolw|

wr uhvsrqg wr surgxfwlrq sodq fkdqjhv vwhpplqj iurp d glvforvxuh ri qhz ghpdqgv iru hqg lwhpv1 Zlwk kljkhu ohdg wlphv/ odujhu vdihw| vwrfnv duh uhtxluhg/ wkh uhsdlu surfhgxuh lv dssolhg pruh riwhq wkhuhiruh hqwdlolqj kljkhu wrwdo frvwv1

Zkhq wkh iruhfdvw zlqgrz lv lqfuhdvhg/ wrwdo frvwv surgxfhg e| doo whfkqltxhv joredoo| ghfuhdvh exw qrw dw wkh vdph sdfh1 Iljxuh 6 sorwv wkh dyhudjh frvw ghyl0 dwlrqv uhodwlyh wr ZZUS ryhu iruhfdvw zlqgrz ydoxhv Z= H{fhsw OXFUS zkrvh shuirupdqfh lv dozd|v txlwh srru/ uhodwlyh shuirupdqfh ri doo US ghfuhdvhv dv Z lv dxjphqwhg/ eh wkh ohyho ri ohdg wlphv orz ru kljk +vhh erwk judsklfv rq wrs ri Iljxuh 6,1 Iru vkruw iruhfdvw zlqgrzv +Z @ i9> :> ;j lq wkh orz ohdg wlphv fdvh dqg Z @ i44> 45> 46j lq wkh kljk fdvh,/ LSSDUS/ VPUS dqg SRTUS rxwshuirup ZZUS1 Iru orqjhu zlqgrzv krzhyhu +Z < lq wkh uvw fdvh dqg Z 47 lq wkh vhfrqg fdvh, ZZUS ehfrphv e| idu wkh ehvw phwkrg1 Wklv lv frqvlvwhqw zlwk sulru vwxglhv fodlplqj wkdw pruh lqirupdwlrq lv ehwwhu iru ZZ1 Wkh vdph whqghqf| lv re0 vhuyhg iru VV0edvhg phwkrgv= wkhuh h{lvwv d wkuhvkrog ri Z ehorz zklfk LSSDVV dqg

53

VPVV +exw qrw HRTVV, ehkdyh ehwwhu wkdq ZZUS +vhh erwk judsklfv rq erwwrp ri Iljxuh 6,1 Dv vrrq dv Z < +orz ohdg wlphv, dqg Z 47 +kljk ohdg wlphv,/ ZZUS djdlq h{klelwv wkh ehvw shuirupdqfh1 Lw vkrxog qh qrwhg wkdw uhodwlyh shuirupdqfh ri ZZVV lv hqkdqfhg zlwk orqjhu zlqgrzv1 Wklv ghqlwho| frqupv wkdw ZZ lv srvlwlyho| uhfhswlyh wr dq h{whqvlrq ri lqirupdwlrq1

ZZUS SRTU LSSDU VPUS HRTU OXFUS

043 08

3 8 43 48 53 58 63

9 : ; < 43 44 45 46

ZZUS SRTU LSSDU VPUS HRTU OXFUS

043 08

3 8 43 48 53 58 63

44 45 46 47 48 49 4: 4;

ZZVV SRTV LSSDV VPVV HRTV OXFVV

053 043 3 43 53 63 73 83

9 : ; < 43 44 45 46

ZZVV SRTV LSSDV VPVV HRTV OXFVV

053 043 3 43 53 63 73 83

44 45 46 47 48 49 4: 4;

Iljxuh 6= Dyhudjh frvw udwlrv lq shufhqw ryhu iruhfdvw zlqgrz ydoxhv Z 1

9 Frqfoxvlrq

Pdq| sdvw vwxglhv kdyh dvvxphg }hur ohdg wlphv dqg fodlphg wklv dvvxpswlrq frxog eh grqh czlwkrxw orvv ri jhqhudolw|* xqghu pxowl0ohyho uroolqj krul}rqv1 E| phdqv ri d vlpsoh h{dpsoh/ wklv sdshu vkrzv wklv dvvxpswlrq lv ghqlwho| qrw lqqrfhqw dv wkh lqwurgxfwlrq ri srvlwlyh ohdg wlphv pd| lqgxfh vwrfnrxw vlwxdwlrqv hyhq zkhq ghpdqg iru hqg lwhpv lv ghwhuplqlvwlf zlwklq wkh iruhfdvw zlqgrz1 Zh wkxv sursrvh d uhsdlu surfhgxuh wr frsh zlwk vxfk vwrfnrxwv dqg zh frpsduh lwv shuirupdqfh wr d vwudwhj| wkdw frqvlvwv lq lqwurgxflqj vdihw| vwrfnv dw doo ohyhov lq wkh surgxfw vwuxfwxuh1 Wklv vdihw| vwrfn vwudwhj| lv eruurzhg iurp wkh olwhudwxuh h{dplqlqj pxowl0ohyho orw0 vl}lqj sureohpv xqghu uroolqj frqglwlrqv zkhq ghpdqg lv vxemhfw wr iruhfdvw huuruv1 Vlpxodwlrq uhvxowv vkrz wkh vxshulrulw| ri rxu surfhgxuh ryhu wkh vdihw| vwrfn srolf|

lq prvw fdvhv1 Wkh Zdjqhu0Zklwlq dojrulwkp surylghv wkh ehvw ryhudoo shuirupdqfh zkhq frpelqhg wr rxu uhsdlu surfhgxuh1 Iru vkruw iruhfdvw zlqgrzv krzhyhu zrxog

54

zh uhfrpphqg wkh xvh ri ohvv vrsklvwlfdwhg khxulvwlfv olnh VPUS/ LSSDUS ru SRTUS dv wkh| fohduo| rxwshuirup ZZUS1 Rq wkh rwkhu kdqg/ zkhq pruh lv nqrzq derxw ixwxuh ghpdqg/ p|rslf ehkdylrxu ri wkhvh vlpsohu whfkqltxhv lv ghwulphqwdo wr wkhlu shuirupdqfh/ zkhuhdv frvw hhfwlyhqhvv ri ZZ rvwhqwdwlrxvo| lqfuhdvhv1

Vdihw| vwrfnv0edvhg whfkqltxhv ehkdyh suhww| zhoo xqghu vkruw zlqgrzv exw zhuh vshflfdoo| ghvljqhg wr surylgh d "!! vhuylfh ohyho iru wkh vdnh ri frpsdulvrq zlwk wkh uhsdlu surfhgxuh1 Dssursuldwh dprxqwv ri vdihw| lqyhqwrulhv zhuh frpsxwhg h{0 srvw zkhuhdv lq sudfwlfh wkh| kdyh wr eh ghflghg lq dgydqfh1 Wkxv/ wr dyrlg dq| orvw ghpdqg lq d uhdo vhwwlqj/ dq rshudwlrqdo vdihw| vwrfn srolf| zrxog ohdg wr lpsxwh h{0dqwh h{wuhpho| odujh ydoxhv wr vdihw| lqyhqwrulhv1 Wr wkdw h{whqw/ rxu dssurdfk ri vdihw| srolflhv duwlfldoo| errvwv lwv shuirupdqfh dqg wkxv shqdol}hv wkh uhodwlyh shuirupdqfh ri wkh uhsdlu surfhgxuh1 Lq vslwh ri doo/ rxu lqfhqwlyh zdv wr frpsduh wkh uhsdlu surfhgxuh wr h{lvwlqj phwkrgv dqg zh idfhg d odfn ri vxlwdeoh dssurdfkhv lq wklv iudphzrun ri dqdo|vlv +pxowl0ohyho/ uroolqj vfkhgxoh dqg srvlwlyh ohdg wlphv,1 Wklv sdshu lv suredeo| wkh uvw wkdw dqdo|vhv wkrurxjko| wkh lpsdfw ri srvlwlyh ohdg wlphv lq vxfk d frqwh{w/ zkloh surylglqj dq h!flhqw surfhgxuh zklfk jxdudqwhhv d

"!! vhuylfh ohyho1

Lq d ixwxuh zrun/ zh frxog frpsduh rxu uhsdlu surfhgxuh wr pruh rshudwlrqdo vdihw| vwrfn srolflhv rq wkh edvlv ri wzr fulwhuld= wkh vhuylfh ohyho dqg wkh frvw shu0 irupdqfh1 Khxulvwlfv wkdw duh vshflfdoo| ghvljqhg wr dffrxqw iru lqwhughshqghqflhv dprqj vwdjhv frxog dovr eh lqfoxghg lq wkh vlpxodwlrq h{shulphqw dqg odujhu surgxfw vwuxfwxuhv frxog eh h{dplqhg1

Uhihuhqfhv

^4` G1F1 Dxfdps1 D yduldeoh ghpdqg orw0vl}lqj surfhgxuh dqg d frpsdulvrq zlwk ydulrxv zhoo nqrzq vwudwhjlhv1 Surgxfwlrq dqg Lqyhqwru| Pdqdjhphqw/ 5=453/

4<;81

^5` M1 G1 Eodfnexuq dqg U1 D1 Ploohq1 Lpsuryhg khxulvwlfv iru pxwol0vwdjh uhtxluh0 phqwv sodqqlqj v|vwhpv1 Pdqdjhphqw Vflhqfh/ 5;=7789/ 4<;51

^6` M1G1 Eodfnexuq dqg U1D1 Ploohq1 Khxulvwlf orw0vl}lqj shuirupdqfh lq d uroolqj0 vfkhgxoh hqylurqphqw1 Ghflvlrq Vflhqfhv/ 44=9<4:34/ 4<;31

^7` M1 K1 Errnelqghu dqg M0\1 Wdq1 Wzr orw0vl}lqj khxulvwlfv iru wkh fdvh ri ghwhuplq0 lvwlf wlph0ydu|lqj ghpdqgv1 Lqwhuqdwlrqdo Mrxuqdo ri Rshudwlrqv dqg Surgxfwlrq Pdqdjhphqw/ 8=6375/ 4<;81

^8` M1K1 Errnelqghu dqg E1W1 K*qj1 Surgxfwlrq orw vl}lqj iru ghwhuplqlvwlf uroolqj vfkhgxohv1 Mrxuqdo ri Rshudwlrqv Pdqdjhphqw/ 6=67<695/ 4<;91

^9` U1F1 Fduovrq/ V1O1 Ehfnpdq/ dqg G1K1 Nurss1 Wkh hhfwlyhqhvv ri h{whqglqj wkh krul}rq lq uroolqj surgxfwlrq vfkhgxolqj1 Ghflvlrq Vflhqfhv/ 46=45<479/ 4<;51

^:` V1 Fkdqg1 D qrwh rq g|qdplf orw vl}lqj lq d uroolqj0krul}rq hqylurqphqw1 Ghfl0 vlrq Vflhqfhv/ 46=44644</ 4<;51

^;` V1 Fkdqg1 Uroolqj krul}rq surfhgxuhv iru wkh idflolwlhv lq vhulhv lqyhqwru| prgho zlwk qhvwhg vfkhgxohv1 Pdqdjhphqw Vflhqfh/ 5<=56:57</ 4<;61

55

^<` P1D1 Gh Ergw/ O1Q1 Ydq Zdvvhqkryh/ dqg O1I1 Jhoghuv1 Orw vl}lqj dqg vdihw|

vwrfn ghflvlrqv lq dq pus v|vwhpv zlwk ghpdqg xqfhuwdlqw|1 Hqjlqhhulqj Frvwv dqg Surgxfwlrq Hfrqrplfv/ 9=9::8/ 4<;51

^43` J1 N1 Jur1 D orw vl}lqj uxoh iru wlph0skdvhg frpsrqhqw ghpdqg1 Surgxfwlrq dqg Lqyhqwru| Pdqdjhphqw/ 53=7:86/ 4<:<1

^44` \1S1 Jxswd/ \1N1 Nhxqj/ dqg P1F1 Jxswd1 Frpsdudwlyh dqdo|vlv ri orw0vl}lqj prghov iru pxowl0vwdjh v|vwhpv= d vlpxodwlrq vwxg|1 Lqwhuqdwlrqdo Mrxuqdo ri Surgxfwlrq Uhvhdufk/ 63=9<8:49/ 4<<51

^45` G1K1 Nurss/ U1F1 Fduovrq/ dqg V1O1 Ehfnpdq1 D qrwh rq vwrsslqj uxohv iru uroolqj surgxfwlrq vfkhgxolqj1 Mrxuqdo ri Rshudwlrqv Pdqdjhphqw/ 6=44644</

4<;61

^46` U1D1 Oxqglq dqg W1H1 Pruwrq1 Sodqqlqj krul}rqv iru wkh g|qdplforw vl}h prgho=

]deho yv surwhfwlyh surfhgxuhv dqg frpsxwdwlrqdo uhvxowv1 Rshudwlrqv Uhvhdufk/

56=:44:67/ 4<:81

^47` U1D1 Uxvvho dqg W1O1 Xuedq1 Krul}rq h{whqvlrq iru uroolqj surgxfwlrq vfkhg0 xohv= Ohqjwk dqg dffxudf| uhtxluhphqwv1 Lqwhuqdwlrqdo Mrxuqdo ri Surgxfwlrq Hfrqrplfv/ 5<=444455/ 4<<61

^48` H1 D1 Vloyhu1 Wkh uhvsrqvh ri d frqfhuqhg pdqdjhphqw vflhqwlvw1 Surgxfwlrq dqg Lqyhqwru| Pdqdjhphqw/ 4:=43;445/ 4<:91

^49` H1 D1 Vloyhu dqg K1 F1 Phdo1 D khxulvwlfiru vhohfwlqj orw vl}h uhtxluhphqwv iru wkh fdvh ri d ghwhuplqlvwlf wlph0ydu|lqj ghpdqg udwh dqg glvfuhwh rssruwxqlwlhv iru uhsohqlvkphqw1 Surgxfwlrq dqg Lqyhqwru| Pdqdjhphqw/ 47=97:7/ 4<:61

^4:` Q1F1 Vlpsvrq1 Pxowlsoh ohyho surgxfwlrq lq uroolqj krul}rq dvvhpeo| hqylurq0 phqwv1 Hxurshdq Mrxuqdo ri Rshudwlrqdo Uhvhdufk/ 447=485;/ 4<<<1

^4;` K1 Vwdgwohu1 Lpsuryhg uroolqj vfkhgxohv iru wkh g|qdplf vlqjoh0ohyho orw0vl}lqj sureohp1 Pdqdjhphqw Vflhqfh/ 79=64;659/ 53331

^4<` K1 P1 Zdjqhu dqg W1 P1 Zklwlq1 G|qdplfyhuvlrq ri wkh hfrqrplforw vl}h prgho1 Pdqdjhphqw Vflhqfh/ 8=;<<9/ 4<8;1

56