Solving large linear optinization problens wit/ Scilab. 3. C ontents. 1 In trod ct i o n
. 5. 2 In sta 353 ing t h e codes. 6. 2.1 scilab Lipsol .
Solving Large Linear Optimization Problems with Scilab : Application to Multicommodity Problems H´ector E. Rubio Scola
To cite this version: H´ector E. Rubio Scola. Solving Large Linear Optimization Problems with Scilab : Application to Multicommodity Problems. [Research Report] RT-0227, INRIA. 1999, pp.62.
HAL Id: inria-00069944 https://hal.inria.fr/inria-00069944 Submitted on 19 May 2006
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published 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 destin´ee au d´epˆot et `a la diffusion de documents scientifiques de niveau recherche, publi´es ou non, ´emanant des ´etablissements d’enseignement et de recherche fran¸cais ou ´etrangers, des laboratoires publics ou priv´es.
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Solving large linear optimization problems with Scilab: application to multicommodity problems H´ector E. Rubio Scola
No 0227 Janvier 1999 ` THEME 4
ISSN 0249-0803
apport technique
! - #0/ " #$ % '&( %) * +#,%) . ! 1 2436587:9;@,AB7DC*347FEHG IKJMLN?OPQ R6STNUMVXW YZS\[^]_OYK[8`MYaSTN?ScbZW YaST[^] d ObHeMbHYaLN_Obfg[^N?`hV\OijOb kmla[^nHOYoOYpW^q rsW`M`F[^laYYaOtfuJM]hScv6UMO]wjx8q^q6y?Q z6W ]j{4STOl}|~8~^~Q ^q`FW 8Ob ?g
M^
6 ObJM[YaJMOST]Ffg[^lZ`F[^lpWYaST[^]S\]jYa[}YaJMORfgS\VXW O]j{4STla[8]MN?O]jY[0YaJhOfg[ d ObS\`b[^V s[^` d NW] d `0b[8V\{6O6KO d ObafglZS\FO}YaJMOS\lUFbHOla[8NRjfgS\VXW W ] d ^S\{6OW?f#[^N?`FWla ScbH[^][¡b[8N?O}laObHUMV\Y¢bK[^MY¢W ST]MO d ¡£s`M`MVTScfWYaST[^]Fb,[8lsNUMV\YZScfg[8NN[ d S\YHe`Mla[8MV\ONbKWlaO bJM[]0JMSXfuJ¤W lZOb[8V\{6O d ST]¥WUFbOlZ ¦lZS\O] d V\e§)WeDUbS\]h§oOY¢W ]MOY¨[8lYZJMO d WYpW S\]jYalZ[ d UFfgYZS\[^] W] d {jScbHUFW VTS\©WYaS\[8][ *YaJMO}laOtbUMVTYpbg ª«6¬0a®¯F°±^ 0Wla^O²bafWV\OST]MOWl.`hla[^8lpWN?N?S\]hM³´]jYaOlZS\[^l_k±[8S\]jY?N?OYZJM[ d bg oOJMlZ[^Yal¢W¨`hlaO d SXfgYa[8la fg[^lZlaOf#Ya[^l)W VT^[^lZS\YZJMNMbHS\N?`hV\OiWV\^[8laSTYaJMN6NUMV\YZScfg[8NN[ d STYeµ ¶¦·¹¸BºH»¼)¸,½±¾TºH¿ÁÀp ÃÄÅÆ±Ç.È:É4Ã0ÊMÄcËȹÅÍÌÎÐÏ#ÑHÒÁÓÎÕÔÖ#ÔÆ±Ö#×´ÎÐØ#Ì^ÖpÙ6ÔÑ¡Ç0ØgÓÖuÒÎÐØ8ÚgËÒÁÛ#ÑHÌÜÎÐÌ8Ö Ý
Unit´e de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) T´el´ephone : 01 39 63 55 11 - International : +33 1 39 63 55 11 T´el´ecopie : (33) 01 39 63 53 30 - International : +33 1 39 63 53 30
}!+, - ! -H ! ! %),}+ - ¨ ! %s '& %) ¨+ }! - +#, j ] d fg laSTY d W ]bf#OlpW`M`F[^lZYUM]O]j{4STla[8]M]MON?O]4YS\]jYaOlpW^f#YaS ZMUhYaS\VTScbZW ]jYRjfgS\VXW OY oOYpW]MOY`F[^UhlsVXWl bH[^V\UhYaS\[8] d O}`Mla[8MV\LN_Ob d [^`MYZS\N?SXbaWYaS\[8]VTS\] W S\lZObKf#laOUMi I0la[^SXb fg[ d ObbH[^]jYS\N?`hV N?O#]jY b0S\`FbH[^VMs[8` d N OY`¹sbH[^VT{6O6s]MO}W`M`MVTScfWYaS\[8]?S\VTV\UFbHYalZO V UMYaSTV\ScbZW YZS\[^] d OfgObfg[ d Otb `F[^UhlsUM]`MlZ[^MVTLN?O d O}NUhV\YaSF [^Y d ]hS`FWloOY¢W ]MOY (¯
tgp"!#'
d k lZ[^8lpW N_NW YZS\d[^] V\ST] W STlaO6¹o YaJM[ d Ob d O?k[^S\]jYpbS\]jY lZS\OUMl¢bg0WV\8[^ laSTYaJMN?O `Ml ScfgYZS\[^]h ´fg[8lalZOfgYaST[^] OoOJMlZ[^Yal¢W4FWV\^[8laSTYaJMN?O bS\N_`MV\OijO6^NUMVTYaS$ [^Y
!" #$%&('*)+% ,-.!%$ 0 1 7
A
R
6
* ! 2!3
¯0° ±^
54Á¯ 3 2!3 #
!#!4 398 5
:±« 6¯0°±«j qj\| RfgS\VXW ST`Fb[8VK}}}}}}}}}}}} qjÐq RfgS\VXW s[8` d N }}}}}}}}}}}} qj= < RfgS\VXW 0kR >@? }}}}}}}}}}} « C±¯F !« F¯ E¯ ! GH4 398 !Áh 8 « 6 !« !I4 3 «4hJ C¯ 8 B :±D @?²Uh]FfgYaST[^] }}}}}} Ph y ?mihW N?`hV\VO < }}}}}}}}}}}}} Ph yµ\| bHS\]M0kR >@?²Uh]FfgYaST[^] }}}}}} CHC !464W
4¯ 3 !cW
4Á6¯ ¯H° 4¦
pY ¬ X*¯Z ® C±¯F !« D Uj\| oUhV\YaSXfg[^N_N[ d STYHe F[ `Mla[8MV\ON b,S\][8laS\O]4YZO d ^l¢W `MJb }}}} UjÐq oUhV\YaSXfg[^N_N[ d STYHe F[ `Mla[8MV\ON b,S\]]M[8]M [^laSTO]jYaO d ^l¢W `MJb }}
ÇF\.ÌN]!^_&_&`
/
6 ;
K
y P P
T
|x | q
~ ~
|< |< | | y |~ q^x q^x q^q q< q P 7F[ q P
q^~
P
$ #.
(F4 3 E 3 ^
W4Á¯ 3 j\| oUhV\YaSF[^YUM]FfgYZS\[8] }}}}}}}}}}}} jÐq kmVT[^YNUMV\YZS$F[8YmUM]FfgYZS\[^] }}}}}}}}}} j= < RµJM[} NUMVTYa$S [^YmUM]FfgYZS\[8] }}}}}}}}} j P ?mihW N?`hV\Ob }}}}}}}}}}}}} j Ph\| MsSTlaOfgYZO d 8lpW `hJFb }}}}}}}}} j PhÐq s] d S\laOtfgYaO d ^l¢W `MJb }}}}}}}}} K ¯ C±h 4¯ 3 :±¯ C° !I4IC±¯ ! !C ¯ ! G «
[ EH¯F Dj
T ! 8 ¯F 4cW
: ~j\| 0³HkKR §WV\^[8laSTYaJMN }}}}}}}}}}} ~jÐq Nk Mo W V\8[^lZS\YaJhN }}}}}}}}}}} ~jÐqj\| W^bHScfs`hlaS\N W V\ d UFWV0WV\^[8laSTYaJMN }}}}}} ~jÐqjÐq kmlaO d ScfgYZ[^lf#[^lalZOfgYZ[^lYaOtfuJM]hScv6UMO }}}}}} ~jÐqj= < oUMVTYaS\`hV\O fgO]jYalpWV\STYef#[^lalZOfgYZ[^lpb }}}}}} ~j= < 0kR >@?£sV\8[^laSTYaJMN }}}}}}}}}} ~j P M[8laYal¢W ]V\STMlpWlae?Jh[^` d N }}}}}}}}}} 1 ¯ 3 " ! 4¯ 3
A
@
W ] d d Ofg[8N?`MlZObabS\YS\YZJ S\]YaJMO bUM d STlaOfgYZ[^lZeJMOlaOe6[^UW]4YYa[S\]bYpWV\V¡ kNMo§IKJMScbSTV\VfglZOW YZOYZJMO Nk Mo bHUM d S\lZOfgYZ[^lae bYalZUFfgYZUMlaOJI´bHOOJK [8]4YZO]4Y¢bLK}FOV\[Mp rOW d W] d [8V\V\[ YaJMOr ?£Mo ?§S\]FbHYalZUFfgYaST[^]Fb# >A*@: ^:=C.$_[3 #C ` >@ W] d #$ODFG H^:=C.$_[3 #C
Ba
bHcHdegfih
KI JMOlZOWlaOY¡[KO W dMd laOtbabW^b MY¢W S\]Wfg[^`je[ ¡V\`0bH[^V\{6OS\YZJV\`MN?OiST]4YZOlW8fgOVT`0N?Oi_oW YaVXW lZ[^N YZJMO KO WY j% -*-7* k1 k1$6*1 A!1 B!6 - A*8 - 6 !l7/69G!1 MY¢W S\] Wfg[8`4e_[ mUM]fgYaST[^]RjfgSTVcW W ] d JMOV\`?lZ[^N YZJMO}KO W Y !"$# %
ÄÆÇ0ÄcË
!" #$%&('*)+% ,-.!%$
&('*)+&(,"-$. !/$0!1$2 - 3 1 #$4 -*5 276#98 - 2*/ :8 - 6 ! l7/69G!1 -9;*. l 5=9; #9B>*@
W ] d d Ofg[8N?`MlZObabS\YS\YZJ >A*@: ;!. l 5=9; #C `>@ W] d #CED(F*G ;. l 5=9; #9 S\]YZJMObHUM d S\lZOfgYZ[^laeJMOlZOe6[^UW]4YYZ[S\]FbHYpWV\VF0kR >@?jIKJMScb±STV\VhfglaOtW YaOYZJMO 0k R > ?¥bUM d STlaOfgYZ[^lZebYZlaUFfgYZUMlaOJIbOOJK [8]4YZO]jYpbLK}FOV\[M¢ rOW d W] d [8V\V\[ YaJMOr ?£Mo ?§S\]FbHYalZUFfgYaST[^]Fb $ ! , !*% ¨ ! ³´] YZJMScbslZO`F[^lZY0KOW dMd laOtbabYaJMO`MlZ[^MVTON [ bH[^VT{4ST]MVcW lZ^ObZfW VTOV\S\]hOW l`hla[^8lpWNb Oij`MlaObZbO d ST].YZJMO[^VTV\[S\]MbH[^ fW VTV\O d R6Y¢W ] d W l d M[^laN N?S\]MSTN?SÕ©O bUMMnHOf#YYa[ JMOlaO "!SXb,YaJhOUM] j]M[]0 ScbKWN W YalZS\iS\ ] $#&%'!(*),+.-±hW ] d / "# 01 ! 2 3 4DSXbYaJMO[8MnHOfgYaST{6OUM]Ff#YaS\[8]0¡W ] d YaJMOOv6UFW YZS\[8]F5b 6 478?WlaO YaJMO fg[^]bYal¢W ST]4Y¢bg¡£sV\V YaJMObHO}O]4YZS\YZS\ObmNUFbYJFW{6O}f#[^]FbHScbYZO]jY d S\N?O]FbST[^]Fb#4[ fg[8UMlpbHO6 V\S 6bOUV\W eV\VTYae[?£FOJFUMW8] b d N_Ola[^ d laO OYZfgO[8laN_V\UMS\N_]MO ]Fd bYaVTJFOWW{4]¥ST]Mla[^labgOtW WY] VXd W Ya9STYaU :d OST];YZJM.OSXfub¨Jh[^YZJMScf#OOlZOg[[^lZiOv6JhUMScS\fpYZJ O N?S\]hS\N?SÕ©O
? &@H B A*bedeDC
0³Hk)R HIS\]MOWl`Mla[8^l¢W N?N?ST]M³´]jYaOlZS\[8la k±[^ST]4YR {6Olpb MKSXbW¨`FW8f 8W^OYZJFW YUFbHOb oW YZVcW bb`W lpbHO NW YZlaS\i d W Y¢W bHYalaUfgYaUMlZO W ] d o ?FE I M[^lZYalpW]M oW YaVXW S\]jYaOlHW^fgOb V\Otb MFUMYZS\V\STYaSTObMYa[W^fpJMSTO{6O±F[8YaJ}`Mla[8^l¢W N bSTN?`MV\SXfgS\YHe¡W ] d fg[8N?`MUMY¢W YaST[^]FWVOHGf#S\O]Ff#e6г´Y JFW^b}FOO]§lZS\YaYZO]S\]YaJhOoW YZVcW VcW]M^UW ^O_S\YaJ M[^lZYalpW]M oW YaVXW S\]jYaOlW^f#O VTOb 4Je IL KFJFW]M Ðq^x IKJMSXbK{6OlpbHS\[8].JW^b:OO]lZS\YaYZO] I´W d W `MYZO d M¡S\]VcW]M^UW ^O?RfgS\VXW \| ´MYZJFW YUFbHOb RjfgSTVcW bbH`FW l¢bO NWYalaSTi d W Y¢WbYZlaUFfgYZUMlaO W ] d :2 U /*6Uh]FfgYaST[^H ] I b`FWlpbHOfpJM[8V\Ob 4e W^fgYZ[^lZS\©WYaS\[8
? &@H
N deb
aBa
>
? &@H
bHcHdegfih0C
$ #.
C
kNMo I´sST^JMOl l d OlkmlZS\NWV\ "MsUFW V oOYaJM[ d M}Scb}W`FW^f 8W 8O[^l b[^VT{4ST]MVXW lZ^O bafWV\OVTS\]MOWlW ] d fg[8]4{6Oi'v6UFW d lpWYaScf`MlZ[^^l¢W N?N_S\]M`MlZ[^MVTONb# IKJMO fg[ d OSXb?W ] S\N?`hV\ON?O#]jYpW YZS\[8][ }YZJMOS\]4OW8bSTMV\O?`hlaS\N W V\ d UFWVS\]jYaOlZS\[^l`F[^S\]jYN?OYaJh[ d §³´YUFbHOb NUhV\YaST`MV\O¡fgO]4YZlpW VTS\YHefg[^lZlaOf#Ya[^l¢b 6YaJhOS\lm]jUMNFOlmScbfpJM[6bHO]W `h`Mla[8`MlaSXW YaOV\e[^l¡W}^S\{6O] `Mla[8MV\ONST][8l d Ol Ya[?lZO d UFfgO}YZJMO[{6Ol¢W V\VbH[^VTUMYaST[^]YaSTN?O6 kNMo W UMYZ[^NWYaScfW V\VTe fuJh[6[6bOtb,YaJhOsN?[µbY¡W `h`Mla[8`MlaSXW YaOW^f#Ya[^lZS\©WYaST[^]?N?OYaJM[ d [8lKW}^ST{6O]`Mla[8MV\ON IÁOSTYaJMOl ]M[^lZNW V¡Ov6UFW YZS\[^]b[^lW]W UM8N?O]jYaO d bHeMbHYaO] N M¢IKJhOf#[ d Ofg[8N?`FW lZObsW{6[^UMl¢W MVTe d kD`W^f ^W^Ob# S\YZJf#[^N?N?OlpfgSXW VF0k¥W] Nk Mo )W^bslZS\YZYaO] 4e z6W^f#O s[8] d ©ST[M}³´Ypb k¥OijYaO]bS\[8])W^b d O{6OV\[8`:O d Ya[8^OYZJMOlS\YaJ £s]h]FW£sV\YZNW ]¹ Nk Mo I´sST^JMOl l d OlkmlaSTNW VT MUFW V¹oOYaJM[ d ScbYaJMO}VTS\Ml¢W lae_[ YZJMO [8laYZlpW ] la[8UMYaST]MObK[^lb[8V\{jS\]MVXW lZ^O bafWV\O}V\ST]MOWl`Mla[8^lpWNbKS\YZJS\]jYaOlZS\[^l¡`F[^S\]jYN?OYZJM[ d b# IKJMOf#[ d OS\N?`hV\ONO]jYpbW`MlZS\NWV\ d UFWV6V\[^µW lZS\YaJhN?ScfmFWlalaSTOl IS\]jYaOlZS\[^l`F[^ST]4Y M0N?O YaJM[ d S\YZJ'NUMV\YZS\`MVTOfg[8lalaOtfgYa[8lpb[f#O]4YZlpWV\S\YHe6mS O6¥YZJMOW VT^[^lZS\YZJMN `Mla[8`:[µbO d je z6W^fgO [^] d ©S\[ =P £ dhd S\YZS\[^]W V\VTe6mYaJhOfg[ d O [ Ol¢b?bO{6OlpW VUbOgUMV[8`MYaST[^]Fb#O6 h Wb[8`MJMSXbYaSXfW YZO d `MlaOtb[^VT{6O}la[^UhYaS\]hO6FW W8bY b`FWlpbHS\YHe4 Oij`MV\[8S\YZS\]M JM[8V\Ob 4e d Ofg[8N?`F[6bSTYaST[^]0jOYpf^TMYaJFWY W lZOW VTV0fWlaOgUhV\V\e d [jfgUhN?O]jYaO d ST]YaJMO}laO#OlaO]fgObg 0 k R > ? SXbW ST]MOW l_kmla[8^lpWN?N?ST]M`FW^f 8W 8O[ YaJMO oS\ijO d ³´]4YZO^OlS\]MOWl kmla[^8lpWN b[^VT{6Ol ³´Ys)W^bslZS\YaYZO]S\] YaJMO £ Rj³ VXW ]M8UFW 8Oje oScfpJMOVKOl6OVXW^W l [ YZJMO?mS\] d Jh[ {6O] s]MST{6OlpbHS\YHe[¡IÍOtfuJM]h[^V\[8^eo[µbY[mYZJMOfuJFW]M^OtbK:OY¡OO]|^ U W ] d q4 x¨¡OlaO}f#[^]jYalaSTMUMYZO d 4e_z^Ola[6O] MsSTlhbg z^O lZOe W]4YZ[^lJFW8b,laSTYaYZO]?W]0ko ?&E ST]4YZOlW8fgOK[^l¡YaJMO oW YZVcW R6ehbYZON 0ko ?FE ScbW V\[ ÁVTO{6OVST]4YZOlW8fgOYa[ YaJhO VT`0b[^VT{6OYa[6[8V jS\Y ³´YNWe :O fWV\V\O d d S\lZOfgYZV\e6[^lNWe§FOUFbHO d YZ[DhUMS\V d JMST^JMOlV\O{6OVKUM]FfgYZS\[8]Fb¨[^l b[^VT{4ST]M{^WlaST[^UFb 4ST] d bm[ VTS\]MOtW l¡W ] d N?STi4O d ÁST]4YZO^OlV\ST]MOWlm`Mla[8^l¢W Nb#³BYUFbHObW ]?ST]4YZO^OlmJFW ] d V\OYZ[ `F[^S\]jYYa[WV\S\]hOW l¡`Mla[8^lpWN £s]S\]jYaOlHW^fgORjfgSTVcW W8b N W d O Ya[§ST]4YZOla`MlZOYYaJMO N?Oi V\Ob[ YaJMOoWYaVXW R6ehbYaONmIKJMSXbW VTV\[b0k R > ?'Ya[:O}UFbHO dd S\lZOfgYZV\ela[^NRjfgS\VXW 0
ÄÆÇ0ÄcË
!" #$%&('*)+% ,-.!%$
" m +!Í £sV\VYaJhOb[^VT{6OlpbfW]¥:OUbO d¥d STlaOf#YaV\e 4efWV\V\ST]MYaJMO 6 $/6RjfgSTVcW 'Uh]FfgYaST[^]0 U / X Rjf#S\VcW?UM]FfgYZS\[^]¹: 6 $/69G"1RjfgSTVcW Uh]FfgYaST[^]0*IKJMO d S OlZO]FfgO ScbKJM[ YaJMO VTO ScbKlZOW d W ] d JM[ YZJMO d W Y¢WWlaOUbO d S\]YZJMORjfgS\VXW ?O]4{jS\lZ[^]MN?O]4Y OYUFb]M[ d Obaf#laS\FO?ST]§N?[8laO d OYpWS\V¡YaJMO?N W S\]UM]Ff#YaS\[8]Fbg IKJMObHOUM]FfgYZS\[8]Fb JFW{6OW V\V0FOO].STN?`MVTONO#]jYaO d W] d JFW{6O}W U 16 N W ]jUFW V
?
@ ! @!
uf JFWlpW8fgYaOlZScbHYaScfb VTS\`FbH[^V VcW ]h^UFW^O RfgS\VXW S\]jYaOlHW^fgO Yej`FO fg[^]FbHYal¢W S\]jYpb . Yej`FO b[^VTUMYaST[^] lZOW V laOW d N?`FbH ¦[8laNWY TV O W ] d b[^VT{6O h] [ B
JM[^` d N [8laYZlpW ] [8laYZlpW ]+\^RjfgSTVcW
V\`0b[8V\{6O £ Rj³
oOi V\O oW YaVXW
laOWV
.
laOWV0[ S\]jYaO8Ol e6Ob
e6Ob
NO @ 7@L? !
Á³ }YaJMO`MlZ[^MVTON Scb d O ]MO d S\] YaJMORjfgSTVcW O]j{jS\la[8]MN?O]jY d S\lZOfgYZV\e4e fW V\VTS\]MYZJMO 6! [/*6M U / !X0!1 !#$!#
E 3 ^
W4Á¯ 3
IKJMOlaO}WlaOYZJMlaOORjfgS\VXW Uh]FfgYaST[^]Fb#4JMSXfuJ b[^VT{6OsYZJMOVTS\]MOWlK`MlZ[^8lpW N_N?O6^JMOlaO
6 +17F ScbW R fgS\VXW `MlaSTN?S\YZS\{6O}S\]jYaOlHW^fgOUM]Ff#YaS\[8]0 6 [/*67G"1JFW^bW bSTN?S\VXW l}bHe4]jYpWi U / X W ] d 6"l[/*67G"1}W] d 6 "l +#7Y"1CW lZOW d W`MYaO d oW ZY VcW ?UM]fgYaST[^]Fb# F /780 6 $/69G:1 , 82 V # #7A /$2#*6$1 9/ 80 FX$A:#*6 !" 6"l[/*67G"1 k## 8 1
G:698
698"/7A*!X G*A8
A*8:/9AX F+
#9>"1 BFB"#
DD 9:# :1 /980 F
#
D #
#
#
#
#
?«jM°
EB¯: 4
3 ° ¯ ! G«?® c4
W: : ¯C°
DD F /980 U / * X& 1F:# " 6$1 & /980 F #
D #
#
#
#
#
R 6 A
?«jM° !«
4 398
E 3 ^
W4Á¯ 3
EH¯F Dj
6 :l U #9X 6*1 E6 + 17F&!1$#$X!l 6 "l U #7X!61
& & 19F:# " 6*1
&
$
KI JMOlaO}Scb]M[?`F[6bZbSTMS\VTS\YHe?Ya[_laObHYa[^lZOWV\V0YaJhO{8W lZScW hV\ObKla[8NYZJMO 19F"# :6*1 V\OS\]jYa[YaJMORjfgSTVcW d O]j{4STla[^]hN?O]jY6[^]MVTeYaJMONW YZlaS\i £ fW]FOlaOYZlaSTO{6O d STYaJYZJMO [^VTV\[S\]Mf#[^N?NW] b
ÇF\.ÌN]!^_&_&`
|P
$ #.
2*/6*l7G"1*2" 6 +19F'& >"1 l$2/*6A & 6 "l U #7X!61 2/*6 !/ V l7G"1*2" 6 +19F'& >"1 l9 / V & 6 :l U #7X 6*1 !/ V G"#69A:1$E6 +71 F'& +# l$1*6 &k6"l U #!X!6$1#/ V 2*/$6
³´YKSXbK`F[6babHS\MVTOYZ[`MlaST]4YYZJMO d W Y¢W lZ[^N YZJMOfg[^]FbH[^VTOSTYaJ
6 B"# :1 !# + 1 $4 U / !X& 1F"# "$6 14 & "!
ÄÆÇ0ÄcË
!" #$%&('*)+% ,-.!%$
DD "A 66 , #9
# #
#
$
#
#
$ #
$
#
#
$
D
DD 98& #9 DD *2#& #9
#
$ #
$
#
#
DD #7G"#C& #9 #
#
DD C 9# >"1& #9
#
DD *678"/7A*X & #9 $ D # DD 9A8"/7A*X & #9 DD C V # & #9
$
$$$
4
#
$ #
4
#
$
#
DD 9:# :1 /780 F
#
#
#
$ #
#
#
IKJMObZW N?OlZObUhV\YScbK[8MYpWS\]MO d je?YZJMO[^V\VT[S\]M F /780!6 7 /69G!1 &19F!# !6*1$4#& ! IKJMSXb,UM]FfgYZS\[8].UbObKYZJMO 1*#$X sUM]fgYaST[^]Ya[laOW d YaJMO VTO6
DD 6 :l U #7X61$" 6 = ]jYK {6 O(fg YZ [^lsMp [8lfg[^N?N_[ d S\YH)e eIÁS O6 8 9 (
#*67>'
ÄÆÇ0ÄcË
!" #$%&('*)+% ,-.!%$
b[^UhlpfgOb bS\] Mb fg{ W V\ f `MJMS
/
{6Of#Ya[^lMbH[^UMl¢fgObK]Md [ d Otb ]jUMN:Olpb {6Of#Ya[^lMbHS\] hbK]M[ Ob]jUMNFOl¢b d {68lpOW f#`hYa[^Jl4V\SX{8bW Y VTUMObK[fg[8]FbYZlpWS\]MO F [b
W fuJW lpW8fgYaOlbYZlaST]M NWYalZS\i[ YaJMO}{8W VTUMObK[ F[bK[^]YZJMOW l¢fb 3 A:6 6/ fg[^N_`MUMYaOtbYaJhO.N_S\]MSTNUMNfg[6bHY F[ S\]§YZJMO]MOYHK[8lMS\YZJDYZJMO {8W V\UhO[ 0YZJMO F[§la[8N b[^UhlpfgOb]M[ d Ob $/7A 2*1KYa[}bHS\] hb±]M[ d Ob Y fg[^]bYal¢W ST]MO d Ya[FOOtvUFWV0Ya[_fg{ 3 A:6 6/ laOYaUMlZ]Fb YZJMOYa[8YpWVmfg[µbY[¡YaJMO F[bs[^]YZJMOW l¢fbsf8YaJMONW YZlaSTi?[ YaJMO F[b [8]YZJMOW lpfbK`MJMS´ U Y 5 (W N_[^UM]jY[ fg[^N_N?[ d S\YHe f#S\lpf#UMVcWYaS\]h[^] O d 8OS IKJMO d S OlZO]4YWV\^[8laSTYaJMN+ b &6 $/6 &} & U / !X & I d OgW UhV\LY M [ &6 [/*67G"1 &sfW ] FOfpJM[6bHO]4e?YZJMO}[^`MYZS\[8]FW V0{8W lZScWMV\O #*67>
{8W VTUMO}[ fg[6bHY
B b @ c S @ 0 @ ) 6 @! :6*/
l !A"6 6 / `MV\[8Y[ F[bS\]?YZJMO}^l¢W `MJ
:6*/ l !A"6 6 / U B>
KI JMOUM]fgYaST[^]:6*/ l !A"6 6/ `MV\[8Ypb^l¢W `MJST]WRjfgSTVcW.8lpW`MJMScfW VS\] d [ S\YZJ d S OlZO]jYfg[8V\[^UhlpbgIKJMOlaO_Scb}Wfg[^VT[^UMl W ] d W]4UMNFOl[^lOW^fpJ F[} ?W^fpJ b[8UMlpfgO ScbbJM[]S\YZJSTYpb29G`:[µbSTYaS\{6O{8W VTUMO66OW^fpJ_bS\]ScbKbHJM[]?S\YZJ?S\Y¢bm]MO6WYaST{6O {8W V\UhOW] d YaJMO baW YZUMlpWYaO d fW`FW^f#S\YHe O d ^ObW lZOWVcbH[bJM[]0IKJMSXb 4ST] d [ RjfgSTVcW S\] d [ N W 6Ob,STYK`F[6bZbS\hV\O}Ya[`MlZS\]jYW ] d fglZOW YZO d S OlaO]jYKYHej`:OtbK[ k±[6bHYpfglZS\`MY VTObg Ba ? c S '@ @) 6 @! U / V l !A"6 6/ d ScbH`MVcWehbW^lpW`MJW] d bHJM[ YaJhO [ b
ÇF\.ÌN]!^_&_&`
!!# U W ] d YaJhO]KOfW ]`MlZS\]jY ^UhlaOq4 2 U !!A"6 *6 / 7 /7A2*1 $(*Y k27GB> 2 $
DD "#$ $ 1 U #9 44 4' :#C [ 1 +# "(F
ÄÆÇ0ÄcË
!" #$%&('*)+% ,-.!%$
//
'4 4 4 '4 4 '4 4 4$ '4 4 44 4 :6*/ l9>!!# U >1 k!1 !
fgSTlpfgUMVXW YZS\]M[^]O d ^O}S´ U Y (WN?[^UM]jYK[¡fg[8N?N?[ d STYe
³´] ^UMlZOq4KO_fW ]²bOOS\]MV\UhO60S\YaJ YaJMO]4UhNFOl|8±YZJMO [ FO8S\]Fbla[8N ]M[ d O |W ] d ]MSXbJMOtbW Y]M[ d O ~j§IKJMOlZO?Scb}YaJMO8laOO] F[},STYaJ§YZJMO]4UMNFOlq JMScfpJFO^ST]FbsWY]M[ d O* # U Ya[bHOO YaJMO}lZObUMVTY sOlaO6W^bST]§YaJhO.VXW^bHYfW8bO6m¡OfW ]bHOO?YaJhO?YaJMlZOO F[bST] d S OlZO]jYf#[^V\[8UMlpb# MV\UhO Ia| Mp8laOO] I´$q MpFV\ST^JjYsMVTUMJO I?KO d [ >'&X:9 1*2 1CX#& +
]FfgO
W 6WS\]KO}UFbHO6 A"6 6*/ Ya[ ] d YaJMOb[8V\UMYZS\[8]W] d : 6*/ l9>!!# U Ya[bHOO YaJMO}lZObUMVTY
C Á! « K
O`Mla[jfgOO d ST]YZJMObZW N?O)Wte W^b±ST]YZJMOKVXW^bHY±OihW N_`MV\O6#MUMY]M[§KOUFbHOKOihW N_`MV\O U¨YZ[[8MYpWS\] W ] UM] d STlaOf#YaO d 8lpW`MJ§
ÄÆÇ0ÄcË
!" #$%&('*)+% ,-.!%$
/?
1(3), 1, 1(-3),
2,Sat 1,
1, Sat
1, 2,Sat
1, Sat
1, Sat 2,Sat
2(-4),
2,Sat
2(4),
±S\8UMlaOUQ?mihW N?`MVTOUM] d S\laOtfgYaO d ^lpW`MJbH[^VTUMYaST[^]
ÇF\.ÌN]!^_&_&`
P6x
$ #.
1(5),2(-4),3(-6), 2,3, 2, Sat 2,
2,3,
2, Sat
2,3,
1, 2,
3(6),
2,
1(-5),
2,
2(4),
±S\8UMlaOQ?mihW N?`MVTOyUM] d S\laOtfgYaO d ^lpW`MJbH[^VTUMYaST[^]
ÄÆÇ0ÄcË
!" #$%&('*)+% ,-.!%$
K! $ - !Í !Í [^N?`hUMYpWYaS\[8]FW VlZObUMVTYpb [8lb[^N_O`Mla[8MV\ON bs[ )YaJMO ?I0³ YZObY bOYIKJhScbYaObHY bOY fW].FO}[^hYpW ST]MO d [^]YaJMO}¡O W Y YZ` \9\gYa`¹Ðb©YpW jS´ J4+U \`MUM+\[^`hVcW +\0kI ?KRjIsR?I \ ?I0³ \ JjYaYZ` \9\} N fbg W ]MV ^["{ \[8Yp(f \ sUhS d 7O \tIÍOtbYpkmlZ[^MVTONb\t0kmYaObHYL\ST] d Oi JjYaNV
0
0$ & +& )*+% $ )%+' $ ! ) "%$!'?#+0" $ ) " '?#+0""# $%# && ('*)+-,*.+/ 01) 23 54 + 64 870 92 :2+ $;< '*=>-,*7=/ >> 23 821 298 )1 92 :24 $??@= 'A)4 +-,B7@2=/ 0)4 ) )3 82= 4.9 8= 4 92 :7> $??@7 'A)4 +-,B7@2=/ 0)74 )3 8=) 4.9 5 4 ) 92 :7> C $ED7/ )=4 23 54 . 46 0. 92 I44 C= 64 I4 + DQ@;-,*7=/ >> 7)3 8=) >@>9 82. 42468 7= % ?P F= 'T00)-,B)@70/ 000. 0 871 4=9 07 70L 01 % ?P F7 '54 )20H, 4 >4 >P/ =1=@72 .4J 8.= 4 0E+9 01 46 :> 1
==@1 'A==0H,B=@>=/ =+1+ 4J 8== +9 54 . 46 :> 1
&$@$D;-,B1#4 0/ =+4 0 4J 0+ 4.9 8)1 =9 :2 ) &!4% '*=)-, 4 2=1P/ 4 00E72 73 8>7 479 54 > >9 I4 &!4 'A1=>-, 4 1++P/ 4 2>@.0 4 729 81@= =.@=9 8>> 7=9 8)1 ?$FE? '54 74 2-, 4 1>4/ +2=4 13 8)+ 7@79 82. 4 0L 8>4 ?;P% FDU'A14 +-, 4 2.=P/ 701+ 4J 54 = 4=9 8+. =9 ?; 2/ +)= 23 54 > 46 8)4 29 :27 D42) '4 2)-,K427/ =>2 23 54 ) 46 8=. 29 :2+ D=@2) 'A=2)-,B=@27/ ))4 23 8=. =9 822 29 :>4 D)@2$ '*)2-,X0>/ 4 72 23 82> 29 8> 29 :2= D)@2C '*)2-,X0>/ 44 > 23 82+ 29 8+= 29 :2= D$E?;=) 'T0+=-,B)@22/ =2=. 23 8>. )9 87. 79 :1 D$E?;+ '4 72-,K402/ ))7 23 8== 46 81> 29 I4 ) D@Y4 'A774!,Z0E)+/ =14 = 4J 8== +9 8.) 46 :1 ÇF\.ÌN]!^_&_&`
P6q
D@Y= 'A114!,B.#4 0/ D>2P/ D&@$7 '54 0>4!,R=0>2P/ C@$ 'A)4 1-, 4 2=>P/ $@; 4 C '44 >-,B=@=)/ $@; =C '*.+-,*+./ @ " 'A)7+-, 4 ++)P/ ?$!'20 'T027-,R=44 >P/ 20 'T027-, 4 0)>P/ 4 = '544 )=-,R)0=+P/ 4 = '544 )=-,R=+17P/ @;;$ '54 ==>-,R=271P/ &$@F%U'T01>-, 4 2+)P/ &-,R=274/ V<< %V '54 2..-,R>02)P/
)==. 4 +2> =2)= 4 2+@70 0>+0 44 >= +72 0.22 >0)2 )>4 2 =4 )@.+ 4 2.P04 .=)= 727> 0+0 .0.= 700E+>
3= 8+) 23 8)7 0 8)= )3 8== 23 8). 23 8)) 23 877 =3 0. =3 82) 4J 87) )3 81= =3 04 0 8>1 4J 8>4 23 8= >3 8=> =0 8>1
4 .9 8.= 0L 54 = =@>9 82= 0E29 8=0 44.9 8+) 79 04 46 81. 4@46 8+2 .9 8=1 19 8)7 )@.9 8>4 =@=9 54 ) 0E+9 871 40L 81) =9 8=4 )@)9 54 2 ==@=9 0.
$ #.
+9 :)1 29 :+ >9 :1 4 0L 8). 46 N0+ 29 :.1 29 I4 = 79 :7 79 N0= =9 :2+ =)9 87) 4 29 827 )9 :2= =9 :2+ 29 :20 7)9 54 7 079 81.
´³ ]W VTVMYaJhOsYZObY¢bg^¡O ] d `Ml¢W^fgYZScfWV\VTeYZJMObaWN?O[^MnHOYaS\{6O¡{8W V\UhOW ] d {6OlaebSTN?S\VXW l {8W V\UhOb [F Ib[^VTUMYaST[^] {6OfgYa[8lLM¢KIKJMO d S OlZO]FfgO KOfW ][8FbOla{6O}ScbKYZJMOfWVcfgUhVcW YZS\[8] YaSTN?O6 ³´] W dMd S\YaST[^] Ya[YaJMO`MlaObHO]jYaO d YaOtbYFKO}NW d OfW VXfgUMVXW YaST[^]FbKSTYaJo³k0³ MST]FW lZe?{8W laSXW MVTO i x[^l| d ±STOV qHI´fg[8V\UMN?]bJU a|q[MbH`FOfgS Ob?YaJMOF[^Uh] d b.lZ[ VcWFOV´ o[8laOYaJFW] [8]MO F[^UM] d bla[ NWe FOO]jYaOlZO d mMUMYYZJMO d W Y¢W NUFbHYFO^la[8UM`FO d YZ[^8OYaJMOlfg[^]jYaST ^UM[8UFbVTe?[^lOW^fpJ?F[^UM] d bsla[} ±STOV d < Ifg[^VTUMN?]Fb?|!U q^q[M¨b`FOfgS ObYaJMO fg[8V\UMN?] VXW FOVfg[8lalaOtb`F[^] d S\]M§YZ[ YZJMO {8W laSXW MVTO ±STOV d P Ifg[^VTUMN?]Fbq U ST] YZJMO W F[{6O[8`MYaSTNW VTS\YHefg[8] d STYaS\[8]FbgMJhS\V\O 6OO`MS\]MYaJMO}STYaOl¢W YaOtb ( `:[µbSTYaS\{6OJIYaJjUFb YaJMOST]4YZOlaST[^lZ Á`F[^ST]4YN?OYZJM[ d M¢IKJMO W V\8[^lZS\YaJhNfW]:ObZfuJhONW YZScfWV\VTe d Obaf#laS\FO d W^b [^VTV\[bg OYUFbfg[^]bS d OlnHUbY[^]MOSTYaOl¢W YaST[^]0¡bUh`M`MlaOtbabST]M§YZJMOSTYaOl¢W YaST[^]¤fg[^UM]jY ²[8l bSTN?`MV\SXfgS\YHe8WYSTYaOl¢W YaO 4 ' ( ( ´JMOlZ) O ' ( @ j? O l¢bYf#[^N?`MUhYaOYaJhO?bH[^ d d d fW VTV\O `MlZO SXfgYaST[^] STlaOf#YaS\[8] > (
JMSXfuJSXbs]M[8YaJMST]MMUhYYZJMO sOYZ[^] b d S\lZOfgYaST[^] YaJMO]YaJhO?bH[^ fW VTV\O d fg[8lalZOfgYa[8l d S\lZOfgYZS\[^] > JMOlaO §ScbfWV\V\O d YaJhOfgO]jYaOlZS\]M`FW l¢W N?OYaOlJMScfpJ'JW^b?Ya[§FOfpJM[6bHO] fW lZOgUMVTV\e6FW ] d @ } SXbsW©OlZ[^ [^]MO{6OfgYa[8lsSTYaJ[^]hObfg[8lalaOtb`F[^] d S\]M?YZ[YZJMO]h[^]MVTS\]MOWl Ov6UFW YZS\[^] b S\! ] a O fg[^NMS\]hOYaJhO YHK[ d STlaOf#YaS\[8]Fb?S\YZJ WbYaO`M ÁVTO]M8YaJ `FW l¢W N?OYaO#l "DW] d Uh` d W Ya$O ¨YZ[[^MY¢W ST]YaJhO]hO STYaOl¢W YZS\[^]
%
& "
&
ÄÆÇ0ÄcË
!" #$%&('*)+% ,-.!%$
JMOlaOYZJMObYZO`M V\O]h^YaJ¥`W lpWN?OYZOl " SXb?fpJM[6bHO] b[§YZJFW Y ' ( ( ab WYaScb Ob) ' ( @ 4 IKJhOWV\8[^laSTYaJMN YaJMO] lZO`FOW Y¢b¨YZJMObO bYaO`FbWYOW^fpJ ]MO S\YZOlpWYaS\[8]UM]4YZS\V¹fg[^]j{6Ola8O]FfgOµ IKJMOW:[{6O W V\8[^lZS\YaJhN?ScfbZfuJhON?OSXbW¨{8W lZScW]4Y[ YZJMO`hlaO d SXfgYa[8la fg[^lZlaOf#Ya[^lWV\8[^ laSTYaJMN `Mla[8`F[6bO d je?oOJhla[^YZlpWj[^]FfgO`MYaUFWV\VTe64STYKScb¡WlaOVXW YZS\{6OVTebSTN?`MVTOW V\8[^lZS\YaJhN YaJM[8UM^JW] W^fgYZUFW V¡S\N?`hV\ONOg]jYpWYaST[^]fW ]§FONUFfpJN?[8laOfg[8N?`MV\SXfW YZO d FOb`FOf#ScW VTV\e [^l}VcWla8Ob`FWlpbO`MlZ[^MVTONbg}oOJMla[8YalpW bW`M`MlZ[6W^fpJUFbObYaJMlZOO d S\lZOfgYZS\[^]bsWYOW^fpJ S\YZOlpWYaS\[8] sYZJMO`MlZO d Scf#Ya[^lYaJhOfg[^lZlaOfgYZ[^l W ] d YaJhOfgO]jYaOlZS\]M d S\lZOfgYZS\[^]bg}IKJMO lpbHY YHK[ d STlaOf#YaS\[8]FbKOlZON?[8YaS\{8WYaO d jefg[8]FbS d OlaST]M_W?fgOlZYpW ST]YZlpW nHOf#Ya[^lZeV\S\] 4ST]MYZJMO fgUMlZlaO]jYSTYaOl¢W YZS\[^]¥YZ[ W§bH[^V\UhYaS\[8]0 IKJMO`MlaO d ScfgYZ[^lScbYZJMO l¢bY d OlaS\{8WYaS\{6O[YZJMO Yal¢W nHOfgYZ[^lae W ] d YaJMOfg[8lalaOtfgYa[8lYaJMObOtfg[^] d M[^lN?[8laO d OY¢W S\VXb¨[8]¥oOJhla[^YZlpW b`MlaO d ScfgYZ[^la fg[8lalaOtfgYa[8l?WV\^[8laSTYaJMNS\Y¢b¨STN?`MVTO N?O]jYpWYaS\[8]FbKW] d `hla[^`FOlZYaSTObgFbHOO \| x ´ \|^| ¹W ] d Ðq8x ´
P ! '@ ? S
N deb
B
KI JMOWV\8[^laSTYaJMN STN?`MV\ONO]jYaO d S\]W{6Ol¢bS\[8] q4\|q[ , k M o ScbW]hO {8W lZScW ]jY [ ¡W`MlZS\NWV\ d UFW VFN?OYaJM[ d `MlZ[^`F[6bHO d ST] Õy T 7 I1
M4 ±C W4# !¦¢° !m ! 8 ¯FW4¦
W:
=P
³´]§YaJMSXbbHOfgYaST[^]0KO?[8UMYaVTS\]MOYaJhOW VT^[^lZS\YZJMN UbO d ST] Jh[^` d N M[8lbSTN?`MVTScfgSTY´e6FKO fg[^]bS d OlYaJMO[8V\V\[ST]MbY¢W ] d Wl d c[^lZN [mVTS\]MOWl`Mla[8^l¢W N
)(
) '- /
(*. (
JMOlZOB #