PROGRAMACIÓN LINEAL Y PROGRAMACIÓN LINEAL ENTERA

3/\3/(FRQ([FHO\/LQGR
352*5$0$&,Ï1/,1($/<352*5$0$&,Ï1/,1($/
(17(5$&21(;&(/</,1'2
$XWRUHV-DYLHU)DXOtQIIDXOLQ#XRFHGXÈQJHO$-XDQDMXDQS#XRFHGX
(648(0$'(&217(1,'26BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
3UREQRWDEOHV
&DVRVUHDOHV
3URJUDPDFLyQ
/LQHDO
3/(
(;&(/
/LQGR
([FHO
0DQHMRSUiFWLFR
6ROYHU
(;&(/YHUVXV/,1'2
,1752'8&&,Ï1BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
/D SURJUDPDFLyQ OLQHDO \ OD SURJUDPDFLyQ OLQHDO HQWHUD FXDQGR ODV YDULDEOHV GHO SUREOHPD VHDQ
Q~PHURV HQWHURV FRQVWLWX\HQ DFWXDOPHQWH HO HMH \ IXQGDPHQWR GH RWUDV PXFKDV WpFQLFDV GH
,QYHVWLJDFLyQ 2SHUDWLYD $PEDV WpFQLFDV VLUYHQ GH VRSRUWH HQ OD WRPD GH GHFLVLRQHV HQ PXFKDV
iUHDV GHO FRQRFLPLHQWR KXPDQR 3RU HOOR XQ FRQRFLPLHQWR LQPHGLDWR GH OD PDQHUD GH UHVROYHU
SURJUDPDVOLQHDOHV \ OLQHDOHVHQWHURVSURSRUFLRQDDODSHUVRQD UHVSRQVDEOHGH WRPDUGHFLVLRQHV
XQDUPDSRGHURVDSDUDVHUXVDGDHQXQDPELHQWHGHLQFHUWLGXPEUH/DUHVROXFLyQHIHFWLYDGHHVWH
WLSRGHSURJUDPDVSXHGHOOHYDUVHDFDERFRQD\XGDGHODKRMDGHFiOFXOR([FHORELHQDWUDYpVGH
SURJUDPDVHVSHFLDOL]DGRVFRPR/LQGR(OFRQRFLPLHQWRGHSUREOHPDVOLQHDOHVQRWDEOHVVLUYHSDUD
HOGLVHxRGHODUHVROXFLyQGHQXHYRVSUREOHPDVVLQSODQWHDU
/DXWLOLGDGLQIRUPiWLFD([FHOKDGHVSHUWDGRXQJUDQLQWHUpVGHVGHHOSXQWRGHYLVWDGRFHQWHSDUDOD
HQVHxDQ]D GH ODV WpFQLFDV FXDQWLWDWLYDV GH WDO IRUPD TXH VRQ QXPHURVDV ODV UHIHUHQFLDV
DFWXDOL]DGDVGHOLEURVGH(VWDGtVWLFDH,QYHVWLJDFLyQ2SHUDWLYDTXHH[SOLFDQHVWDVPDWHULDVFRQHO
XVR GH HVD KRMD GH FiOFXOR $GHPiV ([FHO VH KD FRQYHUWLGR HQ XQD KHUUDPLHQWD KDELWXDO HQ OD
*HVWLyQ GH (PSUHVDV ORJUDQGR UHVROYHU FRQ DFLHUWR QXPHURVRV SUREOHPDV GH OD HPSUHVD
FRQFHUQLHQWHV FRQ WHPDV PX\ GLYHUVRV FRPR VRQ 0DWHPiWLFDV (VWDGtVWLFD ,QYHVWLJDFLyQ
2SHUDWLYD &RQWDELOLGDG 7RPD GH GHFLVLRQHV ,QYHUVLyQ )LQDQFLDFLyQ 3URGXFFLyQ 0DUNHWLQJ
HQWUHRWURV6HKDFHSUHFLVRSXHVFRQRFHUHVWDKHUUDPLHQWDLQIRUPiWLFDLQWHJUDGRUD
3UR\HFWRH0DWK
)LQDQFLDGRSRUOD6HFUHWDUtDGH(VWDGRGH(GXFDFLyQ\8QLYHUVLGDGHV0(&'
2%-(7,926
3/\3/(FRQ([FHO\/LQGR
BBBBBBBBBBBBBBBBBBBBBBBBBBBB
•
•
•
,QWURGXFLUVH HQ HO XVR GH ([FHO SDUD OD UHVROXFLyQ GH SURJUDPDV OLQHDOHV FRQRFLHQGR VXV
YHQWDMDVHLQFRQYHQLHQWHV
$SUHQGHUDUHVROYHUSURJUDPDVOLQHDOHVXVDQGRODDSOLFDFLyQLQIRUPiWLFD0LFURVRIW([FHO
6DEHUUHVROYHUFRQODD\XGDGH/LQGRR([FHOSUREOHPDVGHSURJUDPDFLyQOLQHDOHQWHUD
&212&,0,(172635(9,26
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
(V UHFRPHQGDEOH TXH HO OHFWRU WHQJD XQRV FRQRFLPLHQWRV EiVLFRV GH 0DWHPiWLFDV D QLYHO GH
%DFKLOOHUDWRRHTXLYDOHQWH7DPELpQVRQQHFHVDULRVFRQRFLPLHQWRVEiVLFRVGHODKRMDGHFiOFXOR
([FHO $VLPLVPR HV DFRQVHMDEOH KDEHU OHtGR SUHYLDPHQWH HO PDWKEORFN ,QWURGXFFLyQ D OD
,QYHVWLJDFLyQ2SHUDWLYD
&21&(3726)81'$0(17$/(6<&$626&2162)7:$5(BBBBBBBBBBBBBB
‰
5HVROXFLyQGHSURJUDPDVOLQHDOHVFRQODKRMDGHFiOFXOR([FHO
(Q OD DFWXDOLGDG QR VyOR ORV SURJUDPDV HVSHFLDOL]DGRV FRPR HO /,1'2 SHUPLWHQ UHVROYHU ORV
SUREOHPDVGH3/7DPELpQODVKRMDVGHFiOFXORFRPR(;&(/RIUHFHQHVWDSRVLELOLGDGPHGLDQWH
ODPDFUR6ROYHU(;&(/HVFDSD]GHUHVROYHUHQFXHVWLyQGHVHJXQGRVSUREOHPDVGHKDVWD
YDULDEOHV\UHVWULFFLRQHVRIUHFLHQGRDGHPiVODVYHQWDMDVGHXQDKRMDGHFiOFXORHQFXDQWRD
SUHVHQWDFLyQ HVWXGLR GH ³HVFHQDULRV´ HWF 6L VH UHTXLHUH GH XQD FDSDFLGDG D~Q PD\RU HV
SRVLEOH UHFXUULU D ORV FUHDGRUHV GH OD PDFUR ZZZIURQWV\VFRP SDUD REWHQHU XQD YHUVLyQ PiV
SRWHQWHGHODPLVPD
1DGDPHMRUTXHYHUXQHMHPSORSDUDLOXVWUDUFyPRIXQFLRQDHVWHSURJUDPD
(MHPSOR8QDIiEULFDGHYDVRVXWLOL]DHQHOSURFHVRGHSURGXFFLyQXQDPiTXLQDFRQKRUDV
GHGLVSRQLELOLGDG SRUVHPDQD/RV YDVRVSURGXFLGRVGXUDQWHXQD VHPDQDVHYDQDOPDFHQDQGR
KDVWD HO ILQDO GH OD PLVPD PRPHQWR HQ TXH VRQ HQYLDGRV D ODV FDVDV GH GLVWULEXFLyQ /D
HPSUHVD RIUHFH GRV WLSRV GH YDVRV SDUD ]XPR = \ YDVRV YLQR 9 /D PiTXLQD QHFHVLWD KRUDV SDUD SURGXFLU FDMDV GH = \ KRUDV SDUD SURGXFLU FDMDV GH 9 &DGD FDMD GH =
UHTXLHUHGHFFFHQWtPHWURVF~ELFRVSDUDVXDOPDFHQDPLHQWRPLHQWUDVTXHFDGDFDMDGH9
QHFHVLWD FF /RV DOPDFHQHV WLHQHQ XQD FDSDFLGDG Pi[LPD GH FF (O EHQHILFLR SRU
FDGDFDMDGH=SURGXFLGDHVGH¼\¼HOGHFDGDFDMDGH9(OGHSDUWDPHQWRGHPDUNHWLQJ
HVWLPDTXHHVSRVLEOHYHQGHUWDQWRV9FRPRVHDQSURGXFLGRVSHURVyORXQPi[LPRGHFDMDV
GH=SRUVHPDQD'HWHUPLQDUODSURGXFFLyQVHPDQDOTXHPD[LPL]DORVEHQHILFLRVGHODHPSUHVD
/ODPDQGR; ³QžFDMDVGH=SURGXFLGDV´H< ³QžFDMDVGH9SURGXFLGDV´HOSODQWHDPLHQWR
VHUi
Maximizar:
Sujeto a:
5 X
+
0.06 X
100 X
4.5 Y
+
+
0.05 Y
200 Y
X <=
X , Y
<= 60
<= 150000
800
>= 0
3UR\HFWRH0DWK
)LQDQFLDGRSRUOD6HFUHWDUtDGH(VWDGRGH(GXFDFLyQ\8QLYHUVLGDGHV0(&'
3/\3/(FRQ([FHO\/LQGR
(O VLJXLHQWH SDVR HV GLVHxDU OD KRMD GH FiOFXOR UHVXOWDQGR FRQYHQLHQWH HQ HVWH SXQWR DVLJQDU
QRPEUHV D ODV FHOGDV TXH LQWHUYHQJDQ HQ ODV UHVWULFFLRQHV PHGLDQWH OD RSFLyQ Insertar >
Nombre > DefinirGHIRUPDTXHORVUHVXOWDGRVGHOSURJUDPDVHDQPiVIiFLOHVGHLQWHUSUHWDU
$KRUDGHEHUHPRVVHOHFFLRQDUODRSFLyQ6ROYHUTXHVHHQFXHQWUDHQHOPHQ~GH+HUUDPLHQWDV
1RVDSDUHFHUiODYHQWDQDGHGLiORJRGHODPDFURHQODFXDOLQWURGXFLUHPRV)FRPRODFHOGD
D PD[LPL]DU R HO QRPEUH TXH OH KD\DPRV GDGR VLHQGR ( \ ( ODV YDULDEOHV D DMXVWDU
)LQDOPHQWHWHQGUHPRVTXHHVSHFLILFDUWRGDVODVUHVWULFFLRQHVGHOSUREOHPDLQFOXLGDVODVGHQR
QHJDWLYLGDGGHODVYDULDEOHVDPHQRVTXH\DKD\DPRVVHOHFFLRQDGRHVWDV~OWLPDVUHVWULFFLRQHV
SRUGHIHFWRHQ2SFLRQHV
<DVyORUHVWDKDFHUFOLFVREUHHOERWyQ5HVROYHU\HOSURJUDPDQRVGHYROYHUiODVROXFLyQGHHVWH
SUREOHPD KD\ TXH SURGXFLU YDVRV VHPDQDOHV GH ]XPR = \ GH YLQR 9 OR FXDO QRV
GDUiXQRVEHQHILFLRVGH¼(OSURJUDPDWDPELpQQRVSUHJXQWDUiVLTXHUHPRVDOJ~QLQIRUPH
FRPSOHPHQWDULR'HPRPHQWRQRVOLPLWDUHPRVDVROLFLWDUXQLQIRUPHVREUHODV5HVSXHVWDV
3UR\HFWRH0DWK
)LQDQFLDGRSRUOD6HFUHWDUtDGH(VWDGRGH(GXFDFLyQ\8QLYHUVLGDGHV0(&'
3/\3/(FRQ([FHO\/LQGR
9DORUySWLPRGHODIXQFLyQ
REMHWLYR
6ROXFLyQySWLPD
‰
(MHPSORVGHUHVROXFLyQGHSURJUDPDVOLQHDOHVFRQ/LQGR
(MHPSOR8QDHPSUHVDIDEULFDGRVPRGHORVGHRUGHQDGRUHVGHJUDQGHVSUHVWDFLRQHVHO$\HO
% /D HPSUHVD WLHQH FRQWUDWDGRV FLQFR WpFQLFRV FDGD XQR GH ORV FXDOHV KD GH WUDEDMDU
H[DFWDPHQWH KRUDV DO PHV HQ OD OtQHD GH HQVDPEODMH ORV GLUHFWLYRV LQVLVWHQ HQ TXH FDGD
WpFQLFRKDGHHVWDUWUDEDMDQGRGXUDQWHWRGDVODVKRUDVTXHVHOHSDJDQ(OHQVDPEODMHGHFDGD
$UHTXLHUHGHKRUDVGHWpFQLFRPLHQWUDVTXHHOGHFDGD%QHFHVLWD/DHPSUHVDTXLHUH
TXH GXUDQWHHOSUy[LPRPHVVHSURGX]FDQ QRPHQRV GH$ \ %&DGD $JHQHUDXQRV
EHQHILFLRVGH¼ \FDGD%GH¼'HWHUPLQDUHOQ~PHURGHFDGDPRGHORDSURGXFLU
GXUDQWHHOSUy[LPRPHVGHPDQHUDTXHVHPD[LPLFHQORVEHQHILFLRV
6HDQ ; ³Qž RUGHQDGRUHV GH WLSR $ D FRQVWUXLU´ H < ³Qž RUGHQDGRUHV GH WLSR % D
FRQVWUXLU´
8VDUHPRV/,1'2SDUDUHVROYHUHVWHSUREOHPDDXQTXHWDPELpQSRGUtDUHVROYHUVHFRQ(;&(/
3UR\HFWRH0DWK
)LQDQFLDGRSRUOD6HFUHWDUtDGH(VWDGRGH(GXFDFLyQ\8QLYHUVLGDGHV0(&'
3/\3/(FRQ([FHO\/LQGR
MAX
1200 X + 1800 Y
ST
20X + 25Y = 800
X >= 10
Y >= 15
END
LP OPTIMUM FOUND AT STEP
2
OBJECTIVE FUNCTION VALUE
1)
55200.00
VARIABLE
VALUE
REDUCED COST
X
10.000000
0.000000
Y
24.000000
0.000000
ROW
SLACK OR SURPLUS
DUAL PRICES
2)
0.000000
72.000000
3)
0.000000
-240.000000
4)
9.000000
0.000000
NO. ITERATIONS=
2
&RPR VH DSUHFLD HQ OD YHQWDQD GH RXWSXW GHEHUtDPRV IDEULFDU RUGHQDGRUHV $ \ RUGHQDGRUHV%FRQORTXHFRQVHJXLUtDPRVXQRVEHQHILFLRVGH¼
(MHPSOR 8QD SUHVWLJLRVD HQWLGDG ILQDQFLHUD KD DQDOL]DGR \ UHFRPHQGDGR GRV SDTXHWHV GH
DFFLRQHVSHUWHQHFLHQWHVDGRVFRPSDxtDVGLIHUHQWHVDORVPLHPEURVGHXQFOXEGHLQYHUVRUHV
/RVLQYHUVRUHVHVWDEDQLQWHUHVDGRVHQIDFWRUHVWDOHVFRPRHOFUHFLPLHQWRDFRUWR\PHGLRSOD]R
GH ODV DFFLRQHV \ ODV WDVDV GH GLYLGHQGRV GH ODV PLVPDV /DV HVWLPDFLRQHV GH OD HQWLGDG VH
PXHVWUDQHQODWDEODVLJXLHQWH
3$48(7(6'($&&,21(6
)$&725(6
&RPSDxtD(OpFWULFD &RPSDxtDGH6HJXURV
3RWHQFLDO GH FUHFLPLHQWR D FRUWR SOD]R SRU
FDGDHXURLQYHUWLGR
3RWHQFLDO GH FUHFLPLHQWR D PHGLR SOD]R SRU
FDGDHXURLQYHUWLGR
7DVDSRWHQFLDOGHGLYLGHQGRV
/RVPLHPEURVGHOFOXEWLHQHQFRPRREMHWLYRVDFRQVHJXLUXQDJDQDQFLDGHDOPHQRV¼D
FRUWR SOD]R XQD JDQDQFLD GH DO PHQRV ¼ D PHGLR SOD]R \ XQRV LQJUHVRV SRU
GLYLGHQGRVGHQRPHQRVGH¼SRUDxR¢&XiOHVODFDQWLGDGPtQLPDTXHGHEHUiLQYHUWLUFDGD
PLHPEURDILQGHORJUDUVXVSUHWHQVLRQHV"
6HDQ; ³(XURVLQYHUWLGRVHQDFFLRQHVHOpFWULFDV´H < ³(XURVLQYHUWLGRV HQDFFLRQHVGH
VHJXURV´
MIN
X + Y
ST
corto) 0.36X + 0.24Y >= 720
medio) 1.67X + 1.50Y >= 5000
divid) 0.04X + 0.08Y >= 200
END
LP OPTIMUM FOUND AT STEP
1
OBJECTIVE FUNCTION VALUE
1)
3179.348
VARIABLE
VALUE
REDUCED COST
X
1358.695625
0.000000
Y
1820.652187
0.000000
ROW
SLACK OR SURPLUS
DUAL PRICES
CORTO)
206.086953
0.000000
MEDIO)
0.000000
-0.543478
DIVID)
0.000000
-2.309783
NO. ITERATIONS=
1
3UR\HFWRH0DWK
)LQDQFLDGRSRUOD6HFUHWDUtDGH(VWDGRGH(GXFDFLyQ\8QLYHUVLGDGHV0(&'
3/\3/(FRQ([FHO\/LQGR
$ SDUWLU GHO ³RXWSXW´ SRGHPRV FRQFOXLU TXH OD FDQWLGDG PtQLPD D LQYHUWLU VHUi GH ¼
UHSDUWLGRVGHODVLJXLHQWHIRUPD¼HQDFFLRQHVGHODFRPSDxtDHOpFWULFD\¼HQ
DFFLRQHVGHODFRPSDxtDGHVHJXURV
(MHPSOR 8QD HPSUHVD SURGXFWRUD GH SLHQVRV FRPSXHVWRV SDUD DQLPDOHV QHFHVLWD GHWHUPLQDU
ODV FDQWLGDGHV GH FDGD FRPSRQHQWH TXH GHEH FRPSUDU D ILQ GH FXPSOLU XQRV UHTXLVLWRV
QXWULFLRQDOHV D OD YH] TXH PLQLPL]D ORV FRVWHV WRWDOHV GH OD FRPSUD (O FRPSXHVWR SXHGH
IDEULFDUVHDSDUWLUGHWUHVWLSRVGHJUDQRVFDGDXQRGHORVFXDOHVFRQODVLJXLHQWHFRPSRVLFLyQGH
LQJUHGLHQWHVSRUNLOR
,1*5(',(17(6
$
%
&
'
7,32'(*5$12
;
<
=
(O FRVWH HQ HXURV SRU NLOR GH ORV JUDQRV ; < \ = HV UHVSHFWLYDPHQWH GH ¼ ¼ \ ¼ /D
FDQWLGDGPtQLPDUHTXHULGDSRUDQLPDO\PHVHVGHNJGHLQJUHGLHQWH$NJGHLQJUHGLHQWH%
NJGHLQJUHGLHQWH&\NJGHLQJUHGLHQWH'$GHPiVODFDQWLGDGPHQVXDOGHJUDQRGHWLSR=
TXH OD HPSUHVD SXHGH DGTXLULU GH VX SURYHHGRU HVWi OLPLWDGD D NJ 'DGR TXH HO SLHQVR
SURGXFLGRVLUYHSDUDDOLPHQWDUXQDPHGLDGHDQLPDOHVDOPHVODUHVWULFFLyQDQWHULRUVLJQLILFD
TXHQRSRGHPRVFRQWDUFRQPiVGHNJGHJUDQRGHWLSR=SRUFDGDDQLPDO\PHV3ODQWHDU\
UHVROYHUHOSUREOHPD
'HILQLPRV; ³NJDFRPSUDUGHJUDQR;SRUFDGDDQLPDO\PHV´\DQiORJDPHQWH<\=
MIN 2X + 4Y + 2.5Z
ST
ingrA)
ingrB)
ingrC)
ingrD)
limitz)
3X
2X
X
6X
+ 2Y + 4Z >= 4
+ 3Y + Z >= 5
+
2Z >= 1
+ 8Y + 4Z >= 8
Z <= 5
END
LP OPTIMUM FOUND AT STEP
3
OBJECTIVE FUNCTION VALUE
1)
5.000000
VARIABLE
VALUE
REDUCED COST
X
2.500000
0.000000
Y
0.000000
1.000000
Z
0.000000
1.500000
ROW
SLACK OR SURPLUS
DUAL PRICES
INGRA)
3.500000
0.000000
INGRB)
0.000000
-1.000000
INGRC)
1.500000
0.000000
INGRD)
7.000000
0.000000
LIMITZ)
5.000000
0.000000
NO. ITERATIONS=
3
$GTXLULUHPRVVyORNJGHWLSR;SRUFDGDDQLPDO\PHVFRQORTXHHOFRVWHSRUDQLPDO\PHV
VHUiGH¼
3UR\HFWRH0DWK
)LQDQFLDGRSRUOD6HFUHWDUtDGH(VWDGRGH(GXFDFLyQ\8QLYHUVLGDGHV0(&'
3/\3/(FRQ([FHO\/LQGR
(MHPSOR 0ROOHW (OHFWURQLFV 6$ IDEULFD FXDWUR OtQHDV GH SURGXFWRV GH DOWD WHFQRORJtD ORV
FXDOHV VRQ XWLOL]DGRV HQ OD LQGXVWULD DHURVSDFLDO &DGD SURGXFWR GHEH SDVDU SRU GLIHUHQWHV
GHSDUWDPHQWRVGXUDQWHVX HODERUDFLyQ (QODVWDEODV VLJXLHQWHV VHGDLQIRUPDFLyQVREUHD HO
WLHPSR HQ KRUDV TXH XQD XQLGDG GH FDGD FODVH KD GH SHUPDQHFHU HQ FDGD XQR GH ORV
GHSDUWDPHQWRV \ ORV EHQHILFLRV TXH GLFKD XQLGDG UHSRUWD \ E OD FDSDFLGDG SURGXFWLYD
GLVSRQLEOH SRU GHSDUWDPHQWR \ PHV DVt FRPR ODV FDQWLGDGHV PtQLPDV D SURGXFLU 8WLOL]DQGR
GLFKDLQIRUPDFLyQGHWHUPLQDORVQLYHOHVGHSURGXFFLyQPHQVXDO
(MHPSOR/DHPSUHVD%LFLFOHWDV&DVWDOOD6$RIUHFHDVXVFOLHQWHVXQRGHORVSURGXFWRVPiVGH
PRGD HQ OR UHIHUHQWH D MXJXHWHV SDUD QLxRV ODV QXHYDV ELFLFOHWDV HUJRQyPLFDV GH FXDGUR GH
DOXPLQLR \ GLVHxR IXWXULVWD HQ YHUVLRQHV SDUD FKLFR \ SDUD FKLFD /D FRPSDxtD VDEH TXH SRGUi
YHQGHUWRGDVODVELFLFOHWDVTXHVHDFDSD]GHIDEULFDUD¼ODVGHFKLFR\D¼ODVGHFKLFD
/RVFRQWDEOHVGHODHPSUHVDKDQFDOFXODGRTXHORVFRVWHVGHPDQRGHREUDVXSRQHQHOGHO
SUHFLRGHYHQWDHQODVGHFKLFR\HOGHOSUHFLRGHYHQWDHQODVGHFKLFD/RVGHPiVFRVWHV
GHSURGXFFLyQH[FOX\HQGRODSLQWXUD \ HO HPSDTXHWDGRDVFLHQGHQD¼ SRU FDGDELFLFOHWDGH
FKLFR\¼SRUFDGDELFLFOHWDGHFKLFD)LQDOPHQWHORVFRVWHVGHSLQWXUD\HPSDTXHWDGRVRQGH
¼SRUELFLFOHWDVHDGHFKLFRRGHFKLFD
/D FDSDFLGDG SURGXFWLYD GH OD SODQWD HV GH ELFLFOHWDV SRU GtD &DGD ELFLFOHWD GH FKLFR
QHFHVLWD GH KRUDV GH PDQR GH REUD SRU KRUDV FDGD XQD GH FKLFD (Q OD DFWXDOLGDG
%LFLFOHWDV &DVWDOOD WLHQH XQD SODQWLOOD GH WUDEDMDGRUHV FDGD XQR GH HOORV FRQ XQD MRUQDGD
ODERUDO GH KRUDV GLDULDV /D HPSUHVD QR WLHQH LQWHQFLyQ GH YDULDU VX SODQWLOOD GDGR TXH VLJXH
XQDSROtWLFDEDVDGDHQODHVWDELOLGDGGHODPLVPD
'HWHUPLQDU OD SURGXFFLyQ ySWLPD TXH PD[LPLFH EHQHILFLRV GH FDGD XQR GH ORV PRGHORV GH
ELFLFOHWDV
6HDQ; ³QžELFLFOHWDVGHFKLFRDSURGXFLU´H
< ³QžELFLFOHWDVGHFKLFDDSURGXFLU´
3UR\HFWRH0DWK
)LQDQFLDGRSRUOD6HFUHWDUtDGH(VWDGRGH(GXFDFLyQ\8QLYHUVLGDGHV0(&'
3/\3/(FRQ([FHO\/LQGR
,QJUHVRV ;<
&RVWHV ;<;<;< ;<
%HQHILFLRV ,QJUHVRV±&RVWHV ;<
MAX
57X + 55Y
ST
X +
Y <= 390
2.5X + 2.4Y <= 960
END
LP OPTIMUM FOUND AT STEP
2
OBJECTIVE FUNCTION VALUE
1)
VARIABLE
X
Y
ROW
2)
3)
21930.00
VALUE
240.000000
150.000000
REDUCED COST
0.000000
0.000000
SLACK OR SURPLUS
0.000000
0.000000
DUAL PRICES
7.000000
20.000000
NO. ITERATIONS=
2
$VtSXHVORySWLPRVHUiSURGXFLUELFLFOHWDVGHFKLFR\GHFKLFDFRQORTXHVHORJUDUiQ
XQRVEHQHILFLRVGH¼
(MHPSOR )RUMDGRV 6$ KD VXVFULWR XQ FRQWUDWR SDUD VXPLQLVWUDU FKDVLV GH DFHUR SDUD
DXWRPyYLOHV TXH VHUiQ IDEULFDGRV HQ XQD QXHYD SODQWD GH SURGXFFLyQ FHUFDQD /D SROtWLFD GH
FDOLGDGWRWDOTXHVHHVWiLPSODQWDQGRHQGLFKDSODQWDH[LJHTXHODFRPSRVLFLyQGHFDGDFKDVLV
VLJDODVVLJXLHQWHVHVSHFLILFDFLRQHV
0$7(5,$/
0Ë1,02
0È;,02
0DQJDQHVR
6LOLFLR
&DUEyQ
)RUMDGRV6$PH]FODRFKRPDWHULDOHVSDUDSURGXFLUXQDWRQHODGDGHDFHURGHVWLQDGRDFKDVLV
0DWHULDO
GLVSRQLEOH
$
$
$
,
,
&
&
&
0DQJDQHVR
6LOLFLR &DUEyQ .LORVGLVSRQLEOHV &RVWHSRUNLOR
6LQOtPLWH
6LQOtPLWH
6LQOtPLWH
6LQOtPLWH
3UR\HFWRH0DWK
)LQDQFLDGRSRUOD6HFUHWDUtDGH(VWDGRGH(GXFDFLyQ\8QLYHUVLGDGHV0(&'
¼
¼
¼
¼
¼
¼
¼
¼
3/\3/(FRQ([FHO\/LQGR
'HWHUPLQDUTXpFDQWLGDGGHFDGDPDWHULDOGHEHUtDXVDUVHSDUDSURGXFLUXQDWRQHODGDGHDFHUR
GHIRUPDTXHVHFXPSODQORVUHTXLVLWRVGHFDOLGDG\VLPXOWiQHDPHQWHVHPLQLPLFHQORVFRVWHV
MIN
ST
1.2A1 + 1.3A2 + 1.5A3 + 0.9I1 + 0.7I2 + 1.0C1 + 1.2C2 + 0.9C3
A1 + A2 + A3 + I1
0.70A1 + 0.55A2 +
0.70A1 + 0.55A2 +
0.15A1 + 0.30A2 +
0.15A1 + 0.30A2 +
0.03A1 + 0.01A2 +
0.03A1 + 0.01A2 +
A2
C1
C2
C3
<=
<=
<=
<=
+ I2 +
0.12A3
0.12A3
0.26A3
0.26A3
0.03I1
0.03I1
C1 + C2 + C3 = 1000
+0.01I1 + 0.05I2 >=
+0.01I1 + 0.05I2 <=
+0.10I1 + 0.025I2 +
+0.10I1 + 0.025I2 +
+ 0.18C1 + 0.20C2 +
+ 0.18C1 + 0.20C2 +
21
45
0.24C1
0.24C1
0.25C3
0.25C3
+ 0.25C2 + 0.23C3 >= 43
+ 0.25C2 + 0.23C3 <= 46
>= 25.5
<= 53.5
300
50
200
200
END
LP OPTIMUM FOUND AT STEP
6
OBJECTIVE FUNCTION VALUE
1)
720.4000
VARIABLE
VALUE
REDUCED COST
A1
0.000000
0.476000
A2
0.000000
0.592000
A3
0.000000
0.800000
I1
0.000000
0.176000
I2
898.000000
0.000000
C1
0.000000
0.156000
C2
0.000000
0.340000
C3
102.000000
0.000000
ROW
SLACK OR SURPLUS
DUAL PRICES
2)
0.000000
-0.700000
3)
23.900000
0.000000
4)
0.100000
0.000000
5)
2.910000
0.000000
6)
0.090000
0.000000
7)
0.000000
-0.800000
8)
28.000000
0.000000
9)
300.000000
0.000000
10)
50.000000
0.000000
11)
200.000000
0.000000
12)
98.000000
0.000000
NO. ITERATIONS=
6
‰
8VRGH/LQGRYHUVXVXVRGH([FHO
(Q OD UHVROXFLyQ GH ORV SURJUDPDV DQWHULRUHV VH KD KHFKR XVR SULQFLSDOPHQWH GHO SURJUDPD
/,1'2 (Q HVWH VHQWLGR HO HPSOHR GH XQ SURJUDPD X RWUR SDUD OD UHVROXFLyQ GH XQ SURJUDPD
OLQHDO GHSHQGH GH OR TXH HO XVXDULR TXLHUD REWHQHU FRQ ORV UHVXOWDGRV GHO SUREOHPD 6L
FRPSDUDPRV ODV YHQWDMDV H LQFRQYHQLHQWHV GH DPERV SURJUDPDV SRGHPRV HVWDEOHFHU OR
VLJXLHQWH
• 9HQWDMDVGH/,1'2IUHQWHD(;&(/
/,1'2SUHVHQWDGRVYHQWDMDVIXQGDPHQWDOHVIUHQWHD(;&(/
(O SODQWHDPLHQWR GHO SURJUDPD OLQHDO HQ /,1'2 HQ FXDQWR D VX LQWURGXFFLyQ HQ HO
SDTXHWHLQIRUPiWLFRHVH[DFWDPHQWHLJXDODFRPRVHHVFULEHRULJLQDOPHQWHHQXQDKRMD
GHSDSHO
3UR\HFWRH0DWK
)LQDQFLDGRSRUOD6HFUHWDUtDGH(VWDGRGH(GXFDFLyQ\8QLYHUVLGDGHV0(&'
3/\3/(FRQ([FHO\/LQGR
(O XVXDULR SRVHH XQ FRQWURO PX\ VHQFLOOR GH ODV YDULDQWHV GHO SURJUDPD OLQHDO TXH VH
TXLHUDQKDFHUDQiOLVLVGHVHQVLELOLGDGGXDOLGDGUHVWULFFLRQHVGHLQWHJULGDG
(VWRVIXHURQORVJUDQGHVPRWLYRVGHVXSRSXODULGDGDILQDOHVGHORVRFKHQWD\SULQFLSLRVGH
ORV QRYHQWD $O ILQ \ DO FDER pVRV HUDQ ORV REMHWLYRV TXH EXVFDED VX FRQVWUXFWRU /LQXV
6FKUDJH HQ FXDQGR VH GLVHxD OD SULPHUD YHUVLyQ GH /,1'2 ,QLFLDOPHQWH HVWDED
SUHVHQWDGDHQIRUPDWR'26SDUD3&3RVWHULRUPHQWH\DHQORVDxRVQRYHQWDVHSXEOLFDVX
YHUVLyQEDMR:LQGRZVWDO\FRPRVHODFRQRFHDKRUD
•
9HQWDMDVGH(;&(/IUHQWHD/,1'2
(OHPSOHRGH(;&(/SUHVHQWDYHQWDMDVIUHQWHD/,1'2TXHVHH[SOLFLWDUiQDFRQWLQXDFLyQ
D
E
(;&(/ HV XQ SURJUDPD FRQWHQLGR HQ HO SDTXHWH LQIRUPiWLFR 0LFURVRIW 2IILFH GH
DPSOLD GLIXVLyQ FRPHUFLDO HQ XQLYHUVLGDGHV \ HPSUHVDV 3RU HVWD UD]yQ QR HV
QHFHVDULR KDFHU XQD LQYHUVLyQ HVSHFtILFD SDUD OD UHVROXFLyQ GH HVWH SURJUDPD
+DELWXDOPHQWH VH GLVSRQH GH pO SRUTXH VX XVR HV VREUDGDPHQWH FRQRFLGR (Q HVWH
VHQWLGR/,1'2KDGHDGTXLULUVHH[FOXVLYDPHQWHSDUDRSWLPL]DU SURJUDPDVOLQHDOHV
$GHPiV PXFKRV XVXDULRV FRQRFHQ SHUIHFWDPHQWH (;&(/ \ QR HV QHFHVDULR XQ
HQWUHQDPLHQWRHVSHFtILFRSDUDVXHPSOHRHQODRSWLPL]DFLyQGHSURJUDPDVOLQHDOHV
$XQTXHODLQWURGXFFLyQGHORVGDWRVGHXQSURJUDPDOLQHDOHQ([FHOHVLQLFLDOPHQWH
PiVFRVWRVDTXHHQ/,1'2ODUDSLGH]GHORVFiOFXORV\ODIDFLOLGDGGHFRPSUHQVLyQ
GHODYHQWDQDGHLQIRUPHVILQDOHVSXHGHQFRPSHQVDUHVDGLILFXOWDGLQLFLDO
F
(;&(/SHUPLWHODFRQVWUXFFLyQGHFRPSOHPHQWRVRDGGLQVTXHSRWHQFLDQODIXHU]D
UHVROXWRULD GHO SURJUDPD (VWRV FRPSOHPHQWRV SHUPLWHQ GLVHxDU RSHUDFLRQHV GH
FiOFXOR TXH RULJLQDULDPHQWH QR HVWDEDQ SHQVDGDV SDUD (;&(/ 'H HVWH PRGR HO
6ROYHU GH (;&(/ HV XQ FRPSOHPHQWR TXH SHUPLWH UHVROYHU SURJUDPDV OLQHDOHV
QRUPDOPHQWH6ROYHUQRDSDUHFHFRQXQDLQVWDODFLyQWtSLFDGHO (;&(/VLQRTXHHV
QHFHVDULRXQDLQVWDODFLyQSHUVRQDOL]DGD
‰
3URJUDPDFLyQ/LQHDO(QWHUD3/(FRQ/LQGR\([FHO
(QPXFKDVVLWXDFLRQHVGHODYLGDUHDOGHELGRDODLQGLYLVLELOLGDGGHODPD\RUtDGHORVSURGXFWRV
QRVHUiVXILFLHQWHFRQREWHQHUFRPRVROXFLyQD XQSUREOHPDGH3/YDORUHVGHFLPDOHV$VtSRU
HMHPSORQRWLHQHPXFKRVHQWLGRTXHODVROXFLyQDQXHVWURSUREOHPDGHPD[LPL]DUEHQHILFLRVVHD
IDEULFDUOiPSDUDVGHWLSR,\OiPSDUDVGHWLSR,,QLTXHODIRUPDGHPLQLPL]DUORVFRVWHV
GH WUDQVSRUWH VHD KDFLHQGR YLDMHV FRQ HO FDPLyQ $ \ FRQ HO % (Q WDOHV VLWXDFLRQHV
GHEHUHPRVLQFOXLUHQHOSODQWHDPLHQWRODUHVWULFFLyQDGLFLRQDOGHTXHWRGDVODVYDULDEOHVKDQGH
VHU YDORUHV HQWHURV SRU OR TXH HVWDUHPRV DQWH XQ SUREOHPD GH 3URJUDPDFLyQ /LQHDO (QWHUD
3/(
8QD SULPHUD DSUR[LPDFLyQ D OD VROXFLyQ GH XQ SUREOHPD 3/( SRGUtD REWHQHUVH UHVROYLHQGR HO
SUREOHPD 3/ DVRFLDGR HV GHFLU ROYLGiQGRVH GH OD UHVWULFFLyQ VREUH OD QR GLYLVLELOLGDG GH ODV
YDULDEOHV 'H KHFKR VL OD VROXFLyQ GHO 3/ UHVXOWD VHU HQWHUD HQWRQFHV pVWD VHUi WDPELpQ OD
VROXFLyQGHO3/(6LDOJXQDGHODVYDULDEOHVGHODVROXFLyQQRHVHQWHUDSRGUtDPRVSHQVDUHQ
UHGRQGHDU HO YDORU REWHQLGR SRU HO HQWHUR PiV SUy[LPR \ TXH HVWp HQ OD UHJLyQ IDFWLEOH (VWH
SURFHGLPLHQWRSXHGHVHUUHODWLYDPHQWHEXHQRFXDQGRORVYDORUHVUHGRQGHDGRVVRQPX\JUDQGHV
SHURUHVXOWDPX\SHOLJURVRVLHVWRVYDORUHVVRQSHTXHxRVHQWDOFDVRHVPX\SUREDEOHTXHOD
VROXFLyQGHO3/(VHDPX\GLIHUHQWHDODTXHVHREWLHQHUHGRQGHDQGRORVYDORUHVGHO3/
/RV SURFHGLPLHQWRV WHyULFRV TXH VH HPSOHDQ HQ OD UHVROXFLyQ GH HVWRV SUREOHPDV GH 3/( VRQ
EiVLFDPHQWH GRV HO PpWRGR %UDQFK%RXQG \ HO GH 3ODQRV GH &RUWH GH *RPRU\ 'HVGH HO
3UR\HFWRH0DWK
)LQDQFLDGRSRUOD6HFUHWDUtDGH(VWDGRGH(GXFDFLyQ\8QLYHUVLGDGHV0(&'
3/\3/(FRQ([FHO\/LQGR
SXQWR GH YLVWD FRPSXWDFLRQDO WDQWR HO /,1'2 FRPR OD KRMD GH FiOFXOR ([FHO IDFLOLWDQ
VREUHPDQHUDODUHVROXFLyQGHXQ3/(WDQVyORHVQHFHVDULRLQGLFDUOHDOSURJUDPDTXpYDULDEOHV
KDQGHWRPDUYDORUHVHQWHURV9HDPRVXQHMHPSORFRQ/,1'2
11 X + 10 Y
2X + Y < 12
X – 3Y > 1
X
Y
MAX
ST
END
GIN
GIN
LP OPTIMUM FOUND AT STEP
2
OBJECTIVE VALUE =
72.4285736
NEW INTEGER SOLUTION OF
66.0000000
AT BRANCH 0
PIVOT 7
BOUND ON OPTIMUM: 66.00000
ENUMERATION COMPLETE. BRANCHES=
0 PIVOTS=
7
LAST INTEGER SOLUTION IS THE BEST FOUND
RE-INSTALLING BEST SOLUTION...
OBJECTIVE FUNCTION VALUE
1)
66.00000
VARIABLE
X
Y
ROW
2)
3)
VALUE
6.000000
0.000000
SLACK OR SURPLUS
0.000000
5.000000
NO. ITERATIONS=
7
BRANCHES=
0 DETERM.=
1.000E
REDUCED COST
-11.000000
-10.000000
DUAL PRICES
0.000000
0.000000
0
$TXt OH KHPRV LQGLFDGR DO SURJUDPD TXH DPEDV YDULDEOHV HUDQ HQWHUDV PHGLDQWH HO FRPDQGR
*,1\/,1'2QRVKDGHYXHOWRHQHORXWSXWODVROXFLyQHQWHUD; < FRQODTXHHOYDORUGH
ODIXQFLyQREMHWLYRHVGH3RGHPRVYHUODGLIHUHQFLDHQWUHHVWHSUREOHPD3/(\VXDVRFLDGR
3/HQHOVLJXLHQWHRXWSXW
MAX 11 X + 10 Y
ST 2X + Y < 12
X – 3Y > 1
END LP OPTIMUM FOUND AT STEP
2
OBJECTIVE FUNCTION VALUE
1)
72.42857
VARIABLE
X
Y
ROW
2)
3)
VALUE
5.285714
1.428571
SLACK OR SURPLUS
0.000000
0.000000
NO. ITERATIONS=
REDUCED COST
0.000000
0.000000
DUAL PRICES
6.142857
-1.285714
2
2EVHUYDPRVTXHODVROXFLyQGHO3/HV; H< VROXFLyQTXHQRWHQGUiVHQWLGRVL
;H<UHSUHVHQWDQREMHWRVLQGLYLVLEOHVFRQODFXDOVHREWLHQHXQYDORUSDUDODIXQFLyQREMHWLYRGH
1RWDUILQDOPHQWHTXHVLUHGRQGHiVHPRVHVWRVYDORUHVWRPDUtDPRVFRPRVROXFLyQ; H< FRQORTXHQXHVWUDIXQFLyQREMHWLYRYDOGUtDLHOD³VROXFLyQGHUHGRQGHR´QRVGDXQ
YDORUSHRUTXHHOORJUDGRXVDQGR3/(HVWRHQHOVXSXHVWRGHTXHHVWDVROXFLyQVHDIDFWLEOH(Q
HVWHHMHPSORSXHGHSDUHFHUTXHODGLIHUHQFLDHVSRFDSHUREDVWDFRQLPDJLQDUHOFDVRH[WUHPR
GHTXHFDGDXQLGDGVXSXVLHVHPLOOyQGH¼SDUDGDUVHFXHQWDGHODVJUDQGHVSpUGLGDVTXHHVWH
³UHGRQGHR´FDXVDUtDDQXHVWUDHPSUHVD
3UR\HFWRH0DWK
)LQDQFLDGRSRUOD6HFUHWDUtDGH(VWDGRGH(GXFDFLyQ\8QLYHUVLGDGHV0(&'
3/\3/(FRQ([FHO\/LQGR
(QRFDVLRQHVSXHGHRFXUULUTXHODYDULDEOHQRVyORWHQJDTXHVHUHQWHUDVLQRTXHDGHPiVGHED
VHUELQDULDLH~QLFDPHQWHSXHGDWRPDUORVYDORUHVy(VWDUHPRVSXHVDQWHXQSUREOHPD
GH3/(%LQDULD
(VWHWLSRGHYDULDEOHVHVWtSLFRGHODVVLWXDFLRQHV³WRGRRQDGD´FRPRSRUHMHPSORHOWHQHUTXH
GHFLGLUVLFRQVWUXLURQRXQDQXHYDIDFWRUtDRFRPSUDURQRXQORWHJUDQGHGHDOJ~QUHFXUVRSDUD
REWHQHUGHVFXHQWRVGHEHUHPRVWHQHUHQFXHQWDHOFRVWHGHPDQWHQHUHOUHFXUVRHQVWRFN
/,1'2 RIUHFH OD SRVLELOLGDG GH LQGLFDU TXH XQD YDULDEOH HV ELQDULD PHGLDQWH HO FRPDQGR ,17
9HDPRVXQHMHPSOR
MAX -100X
+ 20A + 12B
ST
A - 10X
< 0
B < 11
A +
B < 7
END INT X
LP OPTIMUM FOUND AT STEP
1
OBJECTIVE VALUE =
124.000000
SET
X TO >=
1 AT
1, BND= 112.0
TWIN=
84.00
NEW INTEGER SOLUTION OF 112.000000
AT BRANCH
BOUND ON OPTIMUM: 112.0000
DELETE
X AT LEVEL
1
ENUMERATION COMPLETE. BRANCHES=
1 PIVOTS=
7
1 PIVOT
7
7
LAST INTEGER SOLUTION IS THE BEST FOUND
RE-INSTALLING BEST SOLUTION...
OBJECTIVE FUNCTION VALUE
1)
112.0000
VARIABLE
X
A
B
ROW
2)
3)
4)
VALUE
1.000000
10.000000
1.000000
SLACK OR SURPLUS
0.000000
0.000000
6.000000
NO. ITERATIONS=
BRANCHES=
9
1 DETERM.=
REDUCED COST
20.000000
0.000000
0.000000
DUAL PRICES
8.000000
12.000000
0.000000
1.000E
0
'H QR KDEHU H[LJLGR TXH OD YDULDEOH ; IXHVH ELQDULD KXELpVHPRV REWHQLGR FRPR VROXFLyQ ORV
YDORUHV; $ \% FRQXQDIXQFLyQREMHWLYRGH6LDKRUDUHGRQGHiVHPRVHO
YDORU REWHQLGR SDUD ; HQ HVWD VROXFLyQ WRPDUtDPRV; FRQ OR TXH QXHVWUD IXQFLyQ REMHWLYR
VHUtDGH3RUVXSXHVWRODVROXFLyQTXHKHPRVREWHQLGRXVDQGR,17HVPHMRUSXHVQRVGDXQ
YDORUSDUDODIXQFLyQREMHWLYRGH
(OODGRQHJDWLYRGHHVWHFRPDQGRHVTXHVLVHXWLOL]DFRQPXFKDVYDULDEOHVHQXQSUREOHPDPX\
H[WHQVRDXPHQWDUiHOWLHPSRGHFRPSXWDFLyQQHFHVDULRSDUDREWHQHUODVROXFLyQGHOSURJUDPD
5HVROYHUXQSUREOHPDFRQYDULDEOHVHQWHUDVRELQDULDVXVDQGR([FHOHVWDQVHQFLOORFRPRDxDGLU
ODV FRUUHVSRQGLHQWHV UHVWULFFLRQHV HQ OD PDFUR GH 6ROYHU &RQYLHQH WHQHU FXLGDGR FRQ OD
QRWDFLyQ GH DPERV SURJUDPDV SXHV SDUD ODV YDULDEOHV HQWHUDV /,1'2 XVD HO FRPDQGR *,1 \
6ROYHUODH[SUHVLyQLQWPLHQWUDVTXHSDUDODVYDULDEOHVELQDULDV/,1'2XVDHOFRPDQGR,17\
6ROYHU ODV OHWUDV ELQ $ FRQWLQXDFLyQ VH PXHVWUDQ ODV YHQWDQDV FRUUHVSRQGLHQWHV DO HMHPSOR
ELQDULRDQWHULRU
3UR\HFWRH0DWK
)LQDQFLDGRSRUOD6HFUHWDUtDGH(VWDGRGH(GXFDFLyQ\8QLYHUVLGDGHV0(&'
3/\3/(FRQ([FHO\/LQGR
3UR\HFWRH0DWK
)LQDQFLDGRSRUOD6HFUHWDUtDGH(VWDGRGH(GXFDFLyQ\8QLYHUVLGDGHV0(&'
%,%/,2*5$)Ë$
3/\3/(FRQ([FHO\/LQGR
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
>@ $QGHUVRQ '5 6ZHHQH\ ' - \ :LOOLDPV 7$ &RQWHPSRUDU\ 0DQDJHPHQW 6FLHQFH
ZLWK6SUHDGVKHHWV,QWHUQDWLRQDO7KRPVRQ3XEOLVKLQJ&RPSDQ\
>@ &DPP-\(YDQV-50DQDJHPHQW6FLHQFHDQG'HFLVLRQ7HFKQRORJ\6RXWK:HVWHUQ
&ROOHJH3XEOLVKLQJ
>@ (SSHQ *' *RXOG )- 6FKPLGW &3 0RRUH-+ :HDWKHUIRUG /5 ,QWURGXFWRU\
0DQDJHPHQW6FLHQFH'HFLVLRQ0RGHOLQJZLWK6SUHDGVKHHWV3UHQWLFH+DOO
>@ +LOOLHU )6 +LOOLHU 06 \ /LHEHUPDQQ *- ,QWURGXFWLRQ WR 0DQDJHPHQW 6FLHQFH $
0RGHOLQJDQG&DVH6WXGLHV$SSURDFKZLWK6SUHDGVKHHWV,UZLQ0F*UDZ+LOO
>@ /DZUHQFH$/\3DVWHUQDFN%$$SSOLHG0DQDJHPHQW6FLHQFH$&RPSXWHU,QWHJUDWHG
$SSURDFKIRU'HFLVLRQ0DNLQJ(G:LOH\
>@:LQVWRQ:\$OEULJKW6&3UDFWLFDO0DQDJHPHQW6FLHQFH6SUHDGVKHHW0RGHOLQJDQG
$SSOLFDWLRQV'X[EXU\3UHVV
(1/$&(6 BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
‰
KWWSZZZOLQGRFRP
3iJLQDZHEGHOVRIWZDUH/,1'2
KWWSZZZPDWKQLXHGXaUXVLQNQRZQPDWKLQGH[;;KWPO
:HEFRQUHFXUVRVVREUHSURJUDPDFLyQOLQHDO
‰ KWWSZZZSHUVRQDOSVXHGXIDFXOW\WPWPFWPFOLQNVKWPO
:HEFRQUHFXUVRVVREUHSURJUDPDFLyQOLQHDO
‰
‰
KWWSZZZRSVPDQDJHPHQWFRP
:HEGH2360$1$*(0(17&20UHFXUVRVVREUHGLUHFFLyQGHRSHUDFLRQHV
‰
KWWSZZZUSLHGXaPLWFKMVLWHVBRUKWPO
(QODFHVDZHEVVREUHLQYHVWLJDFLyQRSHUDWLYD
‰
KWWSOLRQKUWSXEFRP2506KWPO
2506-RXUQDO
‰
KWWSZZZSLWWHGXaMUFODVVRURULQWURGRF
$UWtFXORLQWURGXFWRULRDOD,QYHVWLJDFLyQ2SHUDWLYD\VXVDSOLFDFLRQHV
KWWSZZZNHPDHSR]QDQSO%RRNV([FHO6ROYHU77KWP
7XWRULDOVREUHRSWLPL]DFLyQFRQ([FHO6ROYHU
‰ KWWSZZZIDTVRUJIDTVOLQHDUSURJUDPPLQJIDT
:HEGHGLFDGDDSUHJXQWDVPiVFRPXQHVDFHUFDGH3URJUDPDFLyQ/LQHDO
‰
‰
KWWSFDUERQFXGHQYHUHGXaKJUHHQEHFRXUVHZDUH/3VKRUWLQWURKWPO
6HWUDWDGHXQFXUVREUHYHGH3URJUDPDFLyQ/LQHDO
3UR\HFWRH0DWK
)LQDQFLDGRSRUOD6HFUHWDUtDGH(VWDGRGH(GXFDFLyQ\8QLYHUVLGDGHV0(&'