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GHD lv dq dssurdfk irfxvhg rq iurqwlhuv lqvwhdg ri fhqwudo whqghqflhv1 Lw hydoxdwhv wkh h!flhqf| ri hdfk GPX uhodwlyh wr vlplodu GPXv1 Wkxv/ lw surylghv dq h!flhqw iurqwlhu ru hqyhorsh iru doo frqvlghuhg GPXv udwkhu wkdq wwlqj d uhjuhvvlrq sodqh wkurxjk wkh fhqwhu ri wkh gdwd1

(19)

wkdw/ wdujhw/ GPX1

Dowhuqdwlyhv iru pdnlqj hdfk lqh!flhqw GPX h!flhqw fdq eh vhhq e| surmhfwlqj wkhp rqwr wkh h!flhqw iurqwlhu1

GHD rhuv wkh iroorzlqj wkuhh srvvleoh rulhqwdwlrqv wr h!flhqf| dqdo|vlv=

+4,Lqsxw0Rulhqwhg1 Zlwk wklv rulhqwdwlrq/ zh wu| wr plqlpl}h wkh lqsxw xvdjh wr surgxfh jlyhq rxwsxw ohyhov iru hdfk GPX1

+5,Rxwsxw0Rulhqwhg1 Zlwk wklv rulhqwdwlrq/ zh wu| wr surgxfh wkh kljkhvw srvvleoh rxwsxw ohyhov iru d jlyhq lqsxw xvdjh iru hdfk GPX1

+6,Edvh0Rulhqwhg1 Zlwk wklv rulhqwdwlrq/ zh wu| wr surgxfh wkh rswlpdo xvdjh ri lqsxwv dqg wkh rswlpdo surgxfwlrq ri rxwsxwv vlpxowdqhrxvo|1 Erwk lqsxwv dqg rxwsxwv fdq eh frqwuroohg lq d Edvh0Rulhqwhg prgho1 Iljxuh 514 looxvwudwhv wkh surmhfwlrq iru hdfk rulhqwdwlrq iru wkh fdvh ri rqh lqsxw dqg rqh rxwsxw1

I: Input-Oriented Projection O: Output-Oriented Projection B: Base-Oriented Projection

O u t p u t

I n p u t

B I

O Efficient Frontier

(20)

51414 Gdwd hqyhorsphqw dqdo|vlv prghov

Wkh prvw iuhtxhqwo| xvhg GHD prghov duh wkh FFU prgho/ qdphg diwhu Fkduqhv/ Frrshu/ dqg Ukrghv ^46`/ dqg wkh EFF prgho/ qdphg diwhu Edqnhu/ Fkduqhv/ dqg Frrshu ^7`1

Lq GHD prgholqj/ zh dvvxph wkdw wkhuh duh?GPXv/ hdfk ri zklfk kdv6lqsxwv dqgo rxwsxwv ri frpprq w|shv1 Doo lqsxwv dqg rxwsxwv duh dvvxphg wr eh qrqqhj0 dwlyh/ exw dw ohdvw rqh lqsxw dqg rqh rxwsxw duh srvlwlyh1 Zh zloo xvh wkh iroorzlqj qrwdwlrq wkurxjkrxw wklv sursrvdo1

Lqglfhv= ' c2c c ?c ' c2c c 6c & ' c2c c o

Qrwdwlrq=

GPX lv wkh | GPXc

GPXJ lv wkh wdujhw GPX/

% lv wkh dprxqw ri lqsxw frqvxphg e| GPXc

' E%6f lv wkh froxpq yhfwru ri lqsxwv frqvxphg e| GPXc

J ' E%J6f lv wkh froxpq yhfwru ri lqsxwv frqvxphg e| wkh wdujhw GPX/

j ' d%o6f? lv wkh pdwul{ ri lqsxwv/

+& lv wkh dprxqw ri rxwsxw& surgxfhg e| GPXc

) ' E+&of lv wkh froxpq yhfwru ri rxwsxwv surgxfhg e| GPXc

)J ' E+&Jof lv wkh froxpq yhfwru ri rxwsxwv surgxfhg e| wkh wdujhw GPX/

v ' d+&oof? lv wkh pdwul{ ri rxwsxwv/

(21)

' E6f lv wkh froxpq yhfwru ri lqsxw zhljkwv/

& lv wkh zhljkw ri rxwsxw &c

' E&of lv wkh froxpq yhfwru ri rxwsxw zhljkwv/

J lv d vfdodu yduldeoh/ zklfk lv d gxdo yduldeoh lq wkh EFF prgho/

' Eb?fc5 -? lv wkh froxpq yhfwru ri d olqhdu frpelqdwlrq ri? GPXv/

w lv wkh remhfwlyh ydoxh +h!flhqf|,ri wkh FFU prgho/ dqg w lv wkh remhfwlyh ydoxh +h!flhqf|,ri wkh EEF prgho1

Lqsxw0rulhqwhg FFU prgho

Lq wkh FFU prgho/ wkh pxowlsoh0lqsxwv dqg pxowlsoh0rxwsxwv ri hdfk GPX duh djjuhjdwhg lqwr d vlqjoh yluwxdo lqsxw dqg d vlqjoh yluwxdo rxwsxw/ uhvshfwlyho|1 Wkh lqsxw0rulhqwhg FFU prgho iru wdujhw GPXJ fdq eh h{suhvvhg e| wkh iroorzlqj iudf0

wlrqdo surjudpplqj prgho=

4@ w ' +Jn2+2Jnno+oJ

%Jn2%2Jnn6%6J r| +n2+2nno+o

%n2%2nn6%6 c ' c c ? c 2c c 6 f

c 2c c o f

+514,

Ohw wW_cW _dqg W _{eh wkh rswlpdo remhfwlyh ydoxh +h!flhqf| ydoxh,/ wkh rswlpdo}

lqsxw zhljkwv dqg wkh rswlpdo rxwsxw zhljkwv/ uhvshfwlyho|1

Wkh remhfwlyh ri wklv prgho lv wr ghwhuplqh wkh lqsxw zhljkwv dqg rxwsxw zhljkwv wkdw pd{lpl}h wkh udwlr ri d yluwxdo rxwsxw wr d yluwxdo lqsxw iru GPXJ1 Wkh frq0

vwudlqwv uhvwulfw wkh udwlr ri wkh yluwxdo rxwsxwv wr wkh yluwxdo lqsxwv iru hyhu| GPX wr eh ohvv wkdq ru htxdo wr 41 Wklv lpsolhv wkdw wkh pd{lpdo h!flhqf|/ wW_c_{lv dw prvw}

(22)

Xqghu wkh dvvxpswlrq wkdw doo rxwsxwv dqg lqsxwv kdyh qrq}hur zruwk/ GPXJ

lq wkh deryh prgho zloo eh h!flhqw li wW lv htxdo wr 41 Li wW c lw lv srvvleoh wr surgxfh wkh jlyhq rxwsxw E+Jc +2Jc c +oJ xvlqj d vpdoohu yhfwru ri lqsxwv zklfk

pd| eh rewdlqhg dv d olqhdu frpelqdwlrq ri wkh lqsxw yhfwruv ri rwkhu GPXv1 Wkh h!flhqflhv ri doo GPXv duh rewdlqhg e| vroylqj wkh prgho +514,?wlphv/ rqfh iru hdfk GPX dv wkh wdujhw GPX1

Fkduqhv dqg Frrshu ^45` ghyhorshg d wudqvirupdwlrq iurp d olqhdu iudfwlrqdo sur0 judpplqj sureohp wr dq htxlydohqw olqhdu surjudpplqj sureohp1 E| xvlqj wklv wudqvirupdwlrq/ wkh iudfwlrqdo FFU prgho +514,fdq eh wudqviruphg lqwr wkh iroorz0 lqj olqhdu surjudpplqj prgho=

4@ w '+Jn2+2Jnno+oJ

r| %Jn2%2Jnn6%6J '

+n2+2nno+o %n2%2nn6%6c ' c c ?

c 2c c 6 f

c 2c c o f

+515, Wkh deryh olqhdu FFU prgho dqg lwv gxdo fdq eh zulwwhq lq wkh iroorzlqj yhfwru0 pdwul{ irup=

+FFU, 4@ A₎

J

r| A

J '

A_j_nA_v _f

f f

(23)

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v )J

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+517,

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J ' Wkhuhiruh/ wkh rswlpdo ydoxh wWri wkh GFFU prgho lv qrw juhdwhu wkdq

41 Dovr/ ehfdxvh wkh dprxqw ri lqsxwv dqg rxwsxwv lv dvvxphg wr eh qrq}hur/ wkh frqvwudlqwv)J irufhv wr eh d qrq}hur yhfwru1 Wklv dorqj zlwkwW Jjf

lpsolhv wkdw wW _:_f _Wkhuhiruh/ _f_wW _{Wkxv/ wkh GFFU prgho kdv dq rswlpdo}

vroxwlrq1 Iurp wkh vwurqj gxdolw| wkhruhp ri olqhdu surjudpplqj/ wkh FFU prgho dovr kdv dq rswlpdo vroxwlrq dqg wkh rswlpdo remhfwlyh ydoxhv ri wkh FFU dqg GFFU prghov duh htxdo1

Lqwhusuhwdwlrq ri GHD prghov

Lqwhusuhwdwlrq ri wkh FFU prgho=

Wkh wdujhw GPX EGPXJ lv ehlqj frpsduhg zlwk d olqhdu frpelqdwlrq ri rwkhu

GPXv1 Wkh remhfwlyh ri wkh FFU prgho lv wr qg d yhfwru ri zhljkwv vxfk wkdw wkh h!flhqf| ri GPXJ uhodwlyh wr rwkhu GPXv lv pd{lpl}hg/ surylghg wkdw qr rwkhu

GPXv ru olqhdu frpelqdwlrq ri rwkhu GPXv frxog dfklhyh wkh vdph rxwsxw ohyhov zlwk vpdoohu dprxqw ri dq| lqsxw1

Lqwhusuhwdwlrq ri wkh GFFU prgho=

GPXJ lv h!flhqw li qr olqhdu frpelqdwlrq ri rwkhu GPXv fdq surgxfh wkh vdph

ru kljkhu rxwsxw ohyhov xvlqj ohvv ri doo lqsxwv1 w lqglfdwhv d srvvleoh sursruwlrqdo uhgxfwlrq lq lqsxwv + J Uhgxfwlrq lq lqsxwv J fdq eh ylhzhg dv d udgldo pryhphqw

(24)

wW _' _{lpsolhv wkdw qr olqhdu frpelqdwlrq ri rwkhu GPXv kdv} _j

J dqg

v)J1 Rwkhuzlvh/ zh fdq ixuwkhu uhgxfhwW zklohjwW J vwloo krogv1 Wkxv/wW

lv qrw dq rswlpdo vroxwlrq ehfdxvh zh fdq qgw wW _{wkdw vdwlvhv doo wkh frqvwudlqwv1}

Rq wkh rwkhu kdqg/wW lqglfdwhv wkdw wkh uhvxowlqj olqhdu frpelqdwlrq ri GPXv dfwv dv d ehqfkpdun iru GPXJ wW fdq dovr eh lqwhusuhwhg dv wkh odujhvw udwlr ri J

wrj iru zklfk rxwsxwv duh dw ohdvw htxdol}hg/ l1h1/ v)J1

Wr ghwhuplqh zklfk GPXv duh h!flhqw/ zh lqwurgxfh wkh ghqlwlrq ri Sduhwr0 Nrrspdqv h!flhqf| dv iroorzv ^44/ 48`=

Ghqlwlrq 514 +Sduhwr0Nrrspdqv H!flhqf|, D GPX lv ixoo| h!flhqw li dqg rqo| li lw lv lpsrvvleoh wr lpsuryh dq| lqsxw ru rxwsxw zlwkrxw zruvhqlqj vrph rwkhu lqsxwv ru rxwsxwv1

Iurp wkh deryh ghqlwlrq/ wkh GPXJ zlwk wW ' pd| qrw eh Sduhwr0Nrrspdqv

h!flhqw li lw lv srvvleoh wr pdnh dgglwlrqdo lpsuryhphqw +orzhu lqsxw ru kljkhu rxwsxw, zlwkrxw zruvhqlqj dq| rwkhu lqsxw ru rxwsxw1 Wkhuhiruh/ zh lqwurgxfh d yhfwru ri lqsxw h{fhvvhv Et3 _{dqg rxwsxw vkruwidoov} _Etn _{dv iroorzv=}

t3_'_w

Jjc dqgtn 'v)J

zkhuh t3 _fc _tn _f _{duh ghqhg dv vodfn yhfwruv iru dq| ihdvleoh vroxwlrq} _Ewc _ri

wkh GFFU prgho +517,1

Edvhg rq wkh vodfn yhfwruv/ d GPX lv Sduhwr0Nrrspdqv h!flhqw li lw vdwlvhv wkh iroorzlqj wzr frqglwlrqv=

+4, wW _'

(25)

Wkh uvw frqglwlrq lv uhihuuhg wr dv d zhdn h!flhqf|/ whfkqlfdo h!flhqf| ru Iduuhoo h!flhqf| diwhu P1 M1 Iduuhoo +4<8:,^5:`1

Iru wkh FFU prgho/ wkh Sduhwr0Nrrspdqv h!flhqf| lv fdoohg wkh FFU0h!flhqf|1 Zh vxppdul}h wkh FFU0h!flhqf| frqglwlrqv iru d GPX dv iroorzv1

41 Li wW _c _{wkhq wkh GPX lv FFU0lqh!flhqw1}

51 LiwW _{' c}_{dqg wkhuh lv qrq}hur vodfn/ l1h1/}_t3W _9'_{f/ ru}_tnW _9'_{f/ wkhq wkh}

GPX lv FFU0lqh!flhqw1 Iurp wkh frpsohphqwdu| vodfnqhvv frqglwlrqv ri olqhdu surjudpplqj/ wkh hohphqwv ri wkh yhfwruvW _dqgW _{fruuhvsrqglqj}

wr wkh srvlwlyh vodfnv pxvw eh }hur1 Wkxv/ wkh GPX zlwk wW _' _{lv FFU0}

lqh!flhqw li wkhuh lv qrw dw ohdvw rqh rswlpdoW _dqg W _{vxfk wkdw}W _:_f

dqgW _:_f

61 Li wW _' _{zlwk }hur vodfn/ wkhq wkh GPX lv FFU0h!flhqw1 Iurp wkh}

vwurqj wkhruhp ri frpsohphqwdulw|/ wkhuh h{lvw rswlpdo W _dqg W _vxfk

wkdw W _:_f _dqgW _:_f

Wkh lqh!flhqf| wkdw rffxuv iurp wkh vodfn yduldeohv lv fdoohg wkh pl{ lqh!flhqf| ^4;`1 Wr ghwhuplqh wkh h!flhqf| ri d GPX/ zh kdyh wr vroyh wkh iroorzlqj wzr0skdvh olqhdu surjudpplqj sureohp=

Skdvh 4= Vroyh wkh GFFU prgho +517,1 wW _{lv htxdo wr wkh rswlpdo remhfwlyh ydoxh}

EWA₎

Jri wkh FFU prgho +516,1

Skdvh 5= Xvh wW _{iurp skdvh 4 wr vroyh wkh iroorzlqj OS zlwk} _E_c_t3_c_tn _dv

(26)

4@ iA_t3_n_iA_tn

r| t3 _'_wW

Jj

tn _'_v₎ J

f

t3 _fc _tn_fc

+518,

zkhuh i ' Ec cA_/ _iA_t3 _' S6 'r

3

dqgiAtn ' o

S

&'r 3

&c r3 lv wkh lqsxw h{fhvv ri wkh

| _{lqsxw/ dqg} _r3

& lv wkh rxwsxw vkruwidoo ri wkh &| rxwsxw1

Dq rswlpdo vroxwlrq EW_c_t3W_c_tnW _{ri Skdvh 5 lv fdoohg wkh pd{0vodfn vroxwlrq1 Li}

wkh pd{0vodfn vroxwlrq vdwlvhv t3W _' _f _dqg _tnW _' _fc _{wkhq lw lv fdoohg }hur vodfn1}

Skdvh 5 qgv dq rswlpdo vroxwlrq wkdw pd{lpl}hv wkh vxp ri lqsxw h{fhvvhv dqg rxwsxw vkruwidoov rewdlqdeoh zlwk wW iurp Skdvh 41 Li d GPX kdv wW ' c t3W _' _f

dqgtnW _'_fc _{lw lv FFU0h!flhqw1}

Iru dq lqh!flhqw GPXJ/ d uhihuhqfh vhw/ .Jc lv ghqhg edvhg rq wkh pd{0vodfn

vroxwlrq dv iroorzv=

.J 'imbW :fc ' c c ?j

Wkh olqhdu frpelqdwlrq ri wkh uhihuhqfh vhw lv wkh surmhfwhg srlqw rq wkh h!flhqw iurqwlhu ri wkh lqh!flhqw GPXJ1 Wkh uhodwlrqvkls ehwzhhq wkh rswlpdo vroxwlrq ri

GPXJ dqg lwv uhihuhqfh vhw fdq eh jlyhq dv=

wW J ' S M.J

W nt3W

)J ' S M.J

)W tnW

Iurp wklv uhodwlrqvkls/ wkh h!flhqf| ri wkh GPXJzlwkE Jc)Jfdq eh lpsuryhg e|

(27)

lqsxw h{fhvvhv e| t3W_{Iurp wkh rxwsxw ylhzsrlqw/ wkh h!flhqf| fdq eh lpsuryhg e|}

lqfuhdvlqj wkh rxwsxwv )J e| wkh rxwsxw vkruwidoov/ tnW

OhwE R

Jc)RJeh wkh surmhfwhg srlqw ri wkh lqh!flhqw GPXJ rq wkh h!flhqw iurqwlhu1

Wkh surmhfwhg srlqw lv rewdlqhg iurp=

R

J 'wW Jt3W dqg)RJ ')JntnW

Wkh srlqw E R

Jc)RJ surylghv d uhihuhqfh iru vshfli|lqj krz wr lpsuryh wkh shu0

irupdqfh ri wkh GPXJ Ehfdxvh lw surmhfwv E Jc)J rqwr wkh h!flhqw iurqwlhu/ wkh

lpsuryhphqw phwkrg lv fdoohg wkh FFU surmhfwlrq1

Wkh FFU prgho +516,lv ghyhorshg rq wkh dvvxpswlrq ri frqvwdqw uhwxuqv wr vfdoh ri GPXv1 Iru wkh orqj0uxq dqdo|vlv/ wkh vfdoh ri up*v rshudwlrqv vkrxog eh frqvlghuhg1 Wkh dprxqw ri lqfuhdvhg rxwsxwv dvvrfldwhg zlwk lqfuhdvhg lqsxwv lv ixqgdphqwdo wr wkh orqj0uxq qdwxuh ri wkh up*v surgxfwlrq surfhvv1 Iurp wkh hfrqrplf wkhru|/ wkhuh duh wkuhh w|shv ri uhwxuqv wr vfdoh=

+4,Frqvwdqw uhwxuqv wr vfdoh +FUV,= dq lqfuhdvh lq wkh dprxqw ri lq0 sxwv frqvxphg ohdgv wr d sursruwlrqdo lqfuhdvh lq wkh dprxqw ri rxwsxwv surgxfhg1

+5,Lqfuhdvlqj uhwxuqv wr vfdoh +LUV,= dq lqfuhdvh lq wkh dprxqw ri lqsxwv frqvxphg ohdgv wr d odujhu wkdq sursruwlrqdo lqfuhdvh lq wkh dprxqw ri rxwsxwv surgxfhg1

(28)

Lqfuhdvlqj uhwxuqv wr vfdoh dqg ghfuhdvlqj uhwxuqv wr vfdoh duh uhihuuhg wr dv ydul0 deoh uhwxuqv wr vfdoh ehfdxvh wkh rxwsxwv surgxfhg lqfuhdvh pruh ru ohvv wkdq sur0 sruwlrqdoo| wr wkh lqfuhdvh lq wkh lqsxwv ^5/ 95`1

LUV rffxuv zkhq wkh odujh vl}h ri rshudwlrq doorzv pdqdjhuv dqg zrunhuv wr vsh0 fldol}h dqg pdnh xvh ri pruh vrsklvwlfdwhg/ odujh0vfdoh idfwrulhv dqg htxlsphqw1 Wklv kdsshqv lq/ iru h{dpsoh/ dq dxwrpreloh dvvhpeo| olqhv1 LUV lv dq lpsruwdqw lvvxh iurp d sxeolf srolf| shuvshfwlyh1 Li wkhuh duh lqfuhdvlqj uhwxuqv/ wkhq lw lv hfrqrpl0 fdoo| dgydqwdjhrxv wr kdyh rqh odujh up surgxflqj/ dw d uhodwlyho| orz frvw/ udwkhu wkdq wr kdyh pdq| vpdoo upv surgxflqj dw d uhodwlyho| kljk frvw1

Lq frqwudvw/ GUV dulvhv zkhq gl!fxowlhv ri pdqdjhphqw dvvrfldwhg zlwk wkh frp0 soh{lw| ri rujdql}lqj dqg uxqqlqj d odujh0vfdoh rshudwlrq ohdg wr wkh ghfuhdvhg surgxf0 wlylw| ri erwk oderu dqg fdslwdo1 GUV lv olnho| wr eh dvvrfldwhg zlwk wkh sureohpv ri frruglqdwlqj wdvnv dqg pdlqwdlqlqj d xvhixo olqh ri frppxqlfdwlrq ehwzhhq pdqdjhuv dqg zrunhuv1

FUV lv d vshfldo fdvh ri yduldeoh uhwxuqv wr vfdoh1 Iru FUV/ wkh vl}h ri wkh up*v rshudwlrq grhv qrw dhfw wkh surgxfwlylw| ri lwv idfwruv1 Wkh dyhudjh dqg pdujlqdo surgxfwlylw| ri wkh up*v lqsxwv uhpdlq frqvwdqw qr pdwwhu zkhwkhu wkh up lv vpdoo ru odujh1 Wklv rffxuv lq d sodqw xvlqj d sduwlfxodu surgxfwlrq surfhvv/ zklfk fdq hdvlo| eh uhsolfdwhg/ vr wkdw wzr sodqwv surgxfh wzlfh wkh rxwsxwv ri rqh1

Pdqxidfwxulqj lqgxvwulhv duh pruh olnho| wkdq duh vhuylfh lqgxvwulhv wr h{klelw lqfuhdvlqj uhwxuqv wr vfdoh ehfdxvh pdqxidfwxulqj uhtxluhv d vljqlfdqw lqyhvwphqw lq fdslwdo htxlsphqw ehiruh upv fdq rshudwh prvw h!flhqwo|1 Iru pruh ghwdlov rq uhwxuqv wr vfdoh/ zh uhihu wkh uhdghu wr Slqg|fn ^83` dqg Pdqvhog ^76`1

(29)

zklfk lv wkh EFF prgho e| Edqnhu/ Fkduqhv/ dqg Frrshu ^7`1 Wkh surgxfwlrq iurqwlhu +hqyhorsh,iru wkh EFF prgho lv vsdqqhg e| wkh frqyh{ kxoo ri doo GPXv1 Wkh iurqwlhu lv slhfhzlvh olqhdu dqg frqfdyh/ zklfk ohdgv wr yduldeoh uhwxuqv wr vfdoh1 Iljxuh 515 dqg Iljxuh 516 vkrz wkh h!flhqw iurqwlhuv ri wkh FFU dqg wkh EFF prghov/ uhvshfwlyho|/ zkhq wkhuh lv rqo| rqh lqsxw dqg rqh rxwsxw1

Output

Input Production Frontier

Production Possibility Set

Iljxuh 515= Surgxfwlrq Iurqwlhu +Hqyhorsh,ri wkh FFU prgho

Output

Input Production Possibility Set Production Frontier

IRS

DRS

CRS

(30)

Iurp Iljxuh 516/ LFU rffxuv rq wkh uvw olqh vhjphqw ri wkh iurqwlhu iroorzhg e| GFU rq wkh vhfrqg olqh vhjphqw1 Ilqdoo|/ FUV rffxuv dw wkh srlqw ri wudqvlwlrq iurp LFU wr GFU1

Lqsxw0rulhqwhg EFF prgho

Wkh lqsxw0rulhqwhg EFF prgho phdvxuhv wkh h!flhqf| ri GPXJ e| vroylqj wkh

iroorzlqj sdlu ri sulpdo dqg gxdo olqhdu surjudpplqj prghov=

+EFF, 4@ A₎

JJ

r| A

J '

A_j_nA_v_iA

J f

f f

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xvlqj % _dqg +_{/ erwk IFFU dqg GIFFU prghov zloo eh uhgxfhg wr wkhlu fulvs}

frxqwhusduwv +FFU dqg GFFU prghov,1 Wkhuhiruh/ wkh h!flhqf| ydoxhv zloo eh fulvs qxpehuv1 Wkh fulvs frxqwhusduwv ri wkh IFFU dqg GIFFU prghov duh

4@ A_U+ J

A_U% J '

A%_nA+ _f

f f

(57)

4? w r| wU%

J %f

+_U+ J

f

+617,

Wkh ghix}}lfdwlrq dssurdfk zloo eh looxvwudwhg e| xvlqj dq h{dpsoh wdnhq iurp Jxr dqg Wdqdnd ^5<`1 Wdeoh 614 surylghv wkh gdwd iru wkh h{dpsoh1 Lw frqvlvwv ri wzr ix}}| lqsxwv dqg wzr ix}}| rxwsxwv/ doo ri zklfk kdyh v|pphwulfdo wuldqjxodu phpehuvkls ixqfwlrqv1 Wkh phpehuvkls ixqfwlrqv duh ghqrwhg e| ESc _ zkhuh S lv wkh fhqwhu dqg _ lv wkh vsuhdg ri wkh phpehuvkls ixqfwlrqv1

Wdeoh 614= GPXv zlwk wzr ix}}| lqsxwv dqg wzr ix}}| rxwsxwv1

GPXE 4 5 6 7 8

lqsxw 4 +713/318,+51</313,+71</318,+714/31:,+918/319, lqsxw 5 +514/315,+418/314,+519/317,+516/314,+714/318, rxwsxw 4 +519/315,+515/313,+615/318,+51</317,+814/31:, rxwsxw 5 +714/316,+618/315,+814/31;,+81:/315,+:17/31<,

Iurp Wdeoh 614/ wkh ghix}}lfdwlrq ri wkh ix}}| lqsxwv dqg rxwsxwv surgxfhv wkh fulvs lqsxwv dqg rxwsxwv dv vhhq lq Wdeoh 6151

Wdeoh 615= Fulvs ydoxhv iurp ghix}}lfdwlrq ri ix}}| lqsxwv dqg rxwsxwv ri doo GPXv1

GPX E 4 5 6 7 8

lqsxw 4 713 51< 71< 714 918

lqsxw 5 514 418 519 516 714

rxwsxw 4 519 515 615 51< 814

rxwsxw 5 714 618 814 81: :17

(58)

iurp Wdeoh 616 vkrz wkdw wkh h!flhqw GPXv +l1h1/ wW _' _{zlwk }hur vodfn,duh GPXv}

5/ 7 dqg 8/ zkloh wkh lqh!flhqw GPXv duh GPXv 4 dqg 61 Wkh uhihuhqfh vhw ri GPX 4 lv frpsrvhg ri GPXv 5 dqg 8/ zkloh wkh uhihuhqfh vhw ri GPX 6 lv frpsrvhg ri GPXv 5/ 7 dqg 81

Wdeoh 616= Rswlpdo vroxwlrqv iru doo GPXv xvlqj fulvs lqsxwv dqg fulvs rxwsxwv1

GPXE wW b b2 b be bD

4 31;88 3 413<; 3 3 31369

5 4W ₃ ₄ ₃ ₃ ₃

6 31;94 3 4156< 3 31384 31397

7 4W ₃ ₃ ₃ ₄ ₃

8 4W ₃ ₃ ₃ ₃ ₄

- Vodfn yduldeohv/ Er3W_{c r}nW_c _{ri GPXv 5/ 7 dqg 8 duh }hur1}

Frpsduhg zlwk wkh ix}}| udqnlqj phwkrg ri Jxr dqg Wdqdnd ^5<`/ wkh dgydqwdjhv ri wkh ghix}}lfdwlrq dssurdfk duh vlpsolflw| lq lpsohphqwdwlrq dqg lqwhusuhwdwlrq ri wkh uhvxowv1 Krzhyhu/ wkh xqfhuwdlqw| lq lqsxwv dqg rxwsxwv +l1h1/ srvvleoh udqjh ri ix}}| lqsxwv dqg rxwsxwv dw glhuhqw k ohyhov,lv hhfwlyho| ljqruhg1

61515

k

,ee,

edvhg dssurdfk

(59)

α µ

input 0

X~

[ L

] U

Iljxuh 616= Lqwhuydo ydoxh iru d ix}}| lqsxw dw k ohyho1

Phwkrg 4= Ehvw0Ehvw

Zlwk wklv phwkrg/ hyhu| GPX lv frqvlghuhg lq dq rswlplvwlf zd|1 Iurp wkh lqsxw dqg rxwsxw lqwhuydov dw hdfk k ohyho/ wkh vpdoohvw lqsxwv dqg wkh odujhvw rxwsxwv duh xvhg iru hyhu| GPX1 Wklv uhsuhvhqwv wkh ehvw srvvleoh frpelqdwlrqv ri lqsxwv dqg rxwsxwv iru hyhu| GPX1 E| xvlqj wklv phwkrg/ wkh IFFU prgho dw dq| jlyhqk ohyho ehfrphv wkh iroorzlqj olqhdu surjudpplqj prgho1

4@ w 'A_E₎_h JLk

A_E_h

Juk '

A_E_h

JuknAE)hJLk f iru GPXJ

A_E_h

uk nAEh)Lk fc ' c c ? dqg9'J

f fc

+618,

zkhuhEhu

k lv wkh froxpq yhfwru ri wkh plqlpxp ydoxhv ri wkh fruuhvsrqglqj ix}}| vhwv

dfklhyhg dw wkh jlyhqkohyho/ dqg vlploduo|/EhL

(60)

wkh GIFFU prgho iru dq| k ohyho fdq eh rewdlqhg lq d vlplodu zd|/ dqg wkhuhiruh lv qrw vkrzq khuh1

Phwkrg 5= Zruvw0Zruvw

Xvlqj wklv phwkrg/ hyhu| GPX lv frqvlghuhg lq d shvvlplvwlf zd|1 Iurp wkh lqsxw dqg rxwsxw lqwhuydov dw hdfk k ohyho/ wkh odujhvw lqsxwv dqg wkh vpdoohvw rxwsxwv duh xvhg iru hyhu| GPX1 Wklv uhsuhvhqwv wkh zruvw srvvleoh frpelqdwlrqv ri lqsxwv dqg rxwsxwv iru hyhu| GPX1 E| xvlqj wklv phwkrg/ wkh IFFU prgho dw d jlyhq k ohyho ehfrphv wkh iroorzlqj olqhdu surjudpplqj prgho1

4@ w'A_E_h₎ Juk

A_E_h

JLk '

A_E_h

JLk nAE)hJuk f iru GPXJ

A_E_h

Lk nAE)huk fc ' c c ? dqg9'J

f f

Phwkrg 6= Ehvw0Zruvw

Xqghu wklv phwkrg/ GPXJ lv frqvlghuhg lq dq rswlplvwlf zd| exw doo rwkhu GPXv

duh frqvlghuhg lq d shvvlplvwlf zd|1 Iurp wkh lqsxw dqg rxwsxw lqwhuydov dw hdfk k ohyho/ wkh vpdoohvw lqsxwv dqg wkh odujhvw rxwsxwv duh xvhg iru GPXJczkloh wkh odujhvw

lqsxwv dqg vpdoohvw rxwsxwv duh xvhg iru doo rwkhu GPXv1 Wkxv/ zlwk wkhvh lqsxwv dqg rxwsxwv/ GPXJ zloo kdyh wkh odujhvw srvvleoh h!flhqf| ydoxh zkhq frpsduhg

(61)

4@ w 'A_E₎_h JLk

A_E_h

Juk '

A_E_h

JuknAE)hJLk f iru GPXJ

A_E_h

Lk nAE)huk fc ' c c ? dqg9'J

f f

+619,

Phwkrg 7= Zruvw0Ehvw

Lq frqwudvw wr Phwkrg 6/ wkh qdo phwkrg lv edvhg rq d shvvlplvwlf ylhzsrlqw iru GPXJ1 Iurp wkh lqsxw dqg rxwsxw lqwhuydov dw hdfkk ohyho/ wkh odujhvw lqsxwv dqg

vpdoohvw rxwsxwv duh xvhg iru GPXJc zkloh wkh vpdoohvw lqsxwv dqg odujhvw rxwsxwv

duh xvhg iru doo rwkhu GPXv1 Wkxv/ GPXJ zloo kdyh wkh vpdoohvw srvvleoh h!flhqf|

ydoxh dprqj wkh irxu phwkrgv1 E| xvlqj wklv phwkrg/ wkh IFFU prgho dw d jlyhqk ohyho ehfrphv wkh iroorzlqj olqhdu surjudpplqj prgho1

4@ w'A_E_h₎ Juk

A_E_h

JLk '

A_E_h

JLk nAE)hJuk f iru GPXJ

A_E_h

uk nAEh)Lk fc ' c c ? dqg9'J

f f

+61:,

Wr looxvwudwh wkh k,ee, edvhg dssurdfk/ wkh vdph h{dpsoh dv jlyhq lq Wdeoh 614 lv xvhg1 Wkh uhvxowv iurp wkh diruhphqwlrqhg irxu phwkrgv iru yh glhuhqw k ohyhov +3/ 3158/ 318/ 31:8/ 4,duh surylghg lq Wdeohv 617/ 618/ 619/ dqg 61:/ uhvshfwlyho|1

(62)

Wdeoh 617= Uhvxowv ri Ehvw0Ehvw phwkrg ri k,ee, edvhg dssurdfk1

k wW

GPX4 GPX5 GPX6 GPX7 GPX8 3133 31;<< 31<;8 41333 41333 41333 3158 31<4< 41333 41333 41333 41333 3183 31<3< 41333 31<8; 41333 41333 31:8 31;;: 41333 31<3: 41333 41333 4133 31;88 41333 31;94 41333 41333

Wdeoh 618= Uhvxowv ri Zruvw0Zruvw phwkrg ri k,ee, edvhg dssurdfk1

k wW

GPX4 GPX5 GPX6 GPX7 GPX8 3133 31:;; 41333 319<< 41333 31;4: 3158 31;33 41333 31:67 41333 31;9; 3183 31;46 41333 31::5 41333 31<54 31:8 31;59 41333 31;47 41333 31<:9 4133 31;88 41333 31;94 41333 41333

Dw hdfkk ohyho/ wkh lqwhuydo h!flhqf| fdq eh frqvwuxfwhg iurp wkhwW _{ydoxhv iurp}

Phwkrg 6 dqg Phwkrg 71 Wkhq/ wkhvh lqwhuydov iru glhuhqw k ohyhov fdq eh xvhg wr exlog wkh fruuhvsrqglqj ix}}| h!flhqf| phpehuvkls ixqfwlrqv iru hdfk GPX1 Wkh frqvwuxfwlrq ri wkh phpehuvkls ixqfwlrq xvlqj wkh wW _{ydoxhv iurp Phwkrg 6 dqg}

Phwkrg 7 lv vkrzq lq Iljxuh 6171 Wkh phpehuvkls ixqfwlrqv lq Iljxuh 617 vkrz wkh odujhvw srvvleoh vsuhdgv iru doo GPXv ehfdxvh zh frpelqh wkh prvw shvvlplvwlf dqg rswlplvwlf ylhzsrlqwv1

(63)

Wdeoh 619= Uhvxowv ri Ehvw0Zruvw phwkrg ri k,ee, edvhg dssurdfk1

k wW

GPX4 GPX5 GPX6 GPX7 GPX8 3133 41333 41333 41333 41333 41333 3158 41333 41333 41333 41333 41333 3183 31<96 41333 41333 41333 41333 31:8 31<37 41333 31<65 41333 41333 4133 31;88 41333 31;94 41333 41333

Wdeoh 61:= Uhvxowv ri Zruvw0Ehvw phwkrg ri k,ee, edvhg dssurdfk1

k wW

GPX4 GPX5 GPX6 GPX7 GPX8 3133 31957 31;69 318:4 31;88 3196; 3158 31:35 31<3; 31975 31<76 31:68 3183 31:8; 31<<3 31:49 41333 31;78 31:8 31;3: 41333 31:<4 41333 31<9< 4133 31;88 41333 31;94 41333 41333

E| xvlqj wkh k,ee, edvhg dssurdfk/ irxu dowhuqdwlyhv +phwkrgv,iru d ghflvlrq pdnhu wr hydoxdwh wkh h!flhqf| ri GPXv duh=

Li wkh ghflvlrq pdnhu lv rswlplvwlf derxw hdfk GPX/ wkhq Phwkrg 4 fdq eh xvhg wr hydoxdwh wkh h!flhqf|/

Li wkh ghflvlrq pdnhu lv rswlplvwlf derxw wkh wdujhw GPX +GPXJ, / wkhq

Phwkrg 6 fdq eh xvhg wr hydoxdwh wkh h!flhqf|/ +l1h1/ wkh prvw rswlplvwlf phwkrg,

Li wkh ghflvlrq pdnhu lv shvvlplvwlf derxw hdfk GPX/ wkhq Phwkrg 5 fdq eh xvhg wr hydoxdwh wkh h!flhqf|/ dqg

Li wkh ghflvlrq pdnhu lv shvvlplvwlf derxw wkh wdujhw GPX +GPXJ, / wkhq

(64)

ylhz,1

Wkhk,ee, edvhg dssurdfk surylghv ghflvlrq pdnhuv zlwk dowhuqdwlyhv wr hydo0 xdwh wkh h!flhqf| ri GPXv1 Lw fdq surylgh ix}}| h!flhqf| exw uhtxluhv wkh udqnlqj ri ix}}| h!flhqf| vhwv1

(65)