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Luis Medina Aquino
METODO SIMPLEXCaso en que las restricciones sondel tipo >
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PROBLEMA
FORMA CANONICAMaximizar Z = 2 X1 + 4 X2 + 3 X3 Sujeto a:
- X! X" ! " X# ≤ $ % X & " X" ! " X# ' %
" X ! " X" ! X# ≥ (
X) X") X# ≥ *
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PROBLEMA
FORMA CANONICAMaximizar Z = 2 X1 + 4 X2 + 3 X3 Sujeto a:
- X! X" ! " X# ≤ $ +S, % X & " X" ! " X# ' % +a,
" X ! " X" ! X# ≥ ( +-S") a",
X) X") X# ≥ *
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PROBLEMAMaximizar Z = 2 X1 + 4 X2 + 3 X3
Sujeto a: - X ! X" ! " X# $ +S,
% X - " X" ! " X# ' % +a, " X ! " X" ! X# > ( +-S") a",
X) X") X# ≥ *FORMA ESTANDAR
Maximizar Z = 2 X1 + 4 X2 + 3 X3 + 0 S1 + 0 S2 M a1 M a2
Sujeto a:- X ! X" ! " X# ! S ! * S" ! * a ! * a" ' $
% X - " X" ! " X# ! * S ! * S" ! a ! * a" ' % " X ! " X" ! X# ! * S & S" ! * a ! a" ' ( X) X") X#) S) S") a) a" ≥ *
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ITE.ACIO/ * Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B- " * * * $% -" " * * * %" " * - * (
0 j
Cj - 0jCj-0j
*
Maximizar Z = 2 X1 + 4 X2 + 3 X3 + 0 S1 + 0 S2 M a1 M a2Sujeto a:
- X ! X" ! " X# ! S ! * S" ! * a ! * a" ' $% X - " X" ! " X# ! * S ! * S" ! a ! * a" ' % " X ! " X" ! X# ! * S & S" ! * a ! a" ' (
X) X") X#) S) S") a) a" ≥ *
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ITE.ACIO/ * Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s - " * * * $
-M a % -" " * * * %-M a" " " * - * (
0 j
Cj - 0jCj-0j
*
.*
.*"
.*#
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ITE.ACIO/ * Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s - " * * * $
-M a % -" " * * * %-M a" " " * - * (
0 j -1M * -#M * M -M -M -*M
Cj - 0jCj-0j
*
.*
.*"
.*#
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ITE.ACIO/ * Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s - " * * * $
-M a % -" " * * * %-M a" " " * - * (
0 j -1M * -#M * M -M -M -*M
Cj - 0j "!1M ( #!#M * -M * *Cj-0j
*
.*
.*"
.*#
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ITE.ACIO/ * Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s - " * * * $
-M a % -" " * * * %-M a" " " * - * (
0 j -1M * -#M * M -M -M -*M
Cj - 0j "!1M ( #!#M * -M * *Cj-0j
*
Colu2napi3ote
.*
.*"
.*#
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ITE.ACIO/ * Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s - " * * * $
-M a % -" " * * * %-M a" " " * - * (
0 j -1M * -#M * M -M -M -*M
Cj - 0j "!1M ( #!#M * -M * *Cj-0j
*
Colu2napi3ote
.*
.*"
.*#
/o
%4% '
(4" ' "
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ITE.ACIO/ * Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s - " * * * $
-M a % -" " * * * %-M a" " " * - * (
0 j -1M * -#M * M -M -M -*M
Cj - 0j "!1M ( #!#M * -M * *Cj-0j
*
Colu2napi3ote
.*
.*"
.*#
/o
%4% ' #
(4" ' "
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ITE.ACIO/ * Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s - " * * * $
-M a % -" " * * * %-M a" " " * - * (
0 j -1M * -#M * M -M -M -*M
Cj - 0j "!1M ( #!#M * -M * *Cj-0j
*
Colu2napi3ote
.*
.*"
.*#
5ilapi3ote
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ITE.ACIO/ * Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s - " * * * $
-M a % -" " * * * %-M a" " " * - * (
0 j -1M * -#M * M -M -M -*M
Cj - 0j "!1M ( #!#M * -M * *Cj-0j
*
.*
.*"
.*#
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ITE.ACIO/ * Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s - " * * * $
-M a % -" " * * * %-M a" " " * - * (
0j -1M * -#M * M -M -M -*MCj - 0j "!1M ( #!#M * -M * * Cj-0j *
.*
.*"
.*#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B
* s * *
" X * *
-M a" * * 0j
Cj - 0j Cj-0j *
ITE.ACIO/
.
."
.#
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ITE.ACIO/ * Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s - " * * * $
-M a % -" " * * * %-M a" " " * - * (
0j -1M * -#M * M -M -M -*MCj - 0j "!1M ( #!#M * -M * * Cj-0j *
.*
.*"
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Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B
* s * *
" X * *
-M a" * * 0j
Cj - 0j Cj-0j *
ITE.ACIO/
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.#
." ' .*"4%
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ITE.ACIO/ * Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s - " * * * $
-M a % -" " * * * %-M a" " " * - * (
0j -1M * -#M * M -M -M -*MCj - 0j "!1M ( #!#M * -M * * Cj-0j *
.*
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Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B
* s * *" X -4# 4# * * 4% *
-M a" * * 0j
Cj - 0j Cj-0j *
ITE.ACIO/
.
."
.#
." ' .*"4%
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ITE.ACIO/ * Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s - " * * * $
-M a % -" " * * * %-M a" " " * - * (
0j -1M * -#M * M -M -M -*MCj - 0j "!1M ( #!#M * -M * * Cj-0j *
.*
.*"
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Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B
* s * *" X -4# 4# * * 4% *
-M a" * * 0j
Cj - 0j Cj-0j *
ITE.ACIO/
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.
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ITE.ACIO/ * Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s - " * * * $
-M a % -" " * * * %-M a" " " * - * (
0j -1M * -#M * M -M -M -*MCj - 0j "!1M ( #!#M * -M * * Cj-0j *
.*
.*"
.*#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B
* s * *" X -4# 4# * * 4% *
-M a" * * 0j
Cj - 0j Cj-0j *
ITE.ACIO/
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.' .* ! .".* - " * * * $
! ." -4# 4# * * 4% *
.
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ITE.ACIO/ * Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s - " * * * $
-M a % -" " * * * %-M a" " " * - * (
0j -1M * -#M * M -M -M -*MCj - 0j "!1M ( #!#M * -M * * Cj-0j *
.*
.*"
.*#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B
* s * *" X -4# 4# * * 4% *
-M a" * * 0j
Cj - 0j Cj-0j *
ITE.ACIO/
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.' .* ! .".* - " * * * $
! ." -4# 4# * * 4% *
. * "4# 64# * 4% * *
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ITE.ACIO/ * Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s - " * * * $
-M a % -" " * * * %-M a" " " * - * (
0j -1M * -#M * M -M -M -*MCj - 0j "!1M ( #!#M * -M * * Cj-0j *
.*
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Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *" X -4# 4# * * 4% *
-M a" * *
0j
Cj - 0j Cj-0j *
ITE.ACIO/
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.' .* ! .".* - " * * * $
! ." -4# 4# * * 4% *
. * "4# 64# * 4% * *
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ITE.ACIO/ * Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s - " * * * $
-M a % -" " * * * %-M a" " " * - * (
0j -1M * -#M * M -M -M -*MCj - 0j "!1M ( #!#M * -M * * Cj-0j *
.*
.*"
.*#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *" X -4# 4# * * 4% *
-M a" * *
0j
Cj - 0j Cj-0j *
ITE.ACIO/
.
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.#' .*# - " .".*# " " * - * (
-" ."
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ITE.ACIO/ * Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s - " * * * $
-M a % -" " * * * %-M a" " " * - * (
0j -1M * -#M * M -M -M -*MCj - 0j "!1M ( #!#M * -M * * Cj-0j *
.*
.*"
.*#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *" X -4# 4# * * 4% *
-M a" * *
0j
Cj - 0j Cj-0j *
ITE.ACIO/
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.#' .*# - " .".*# " " * - * (
-" ." -" "4# -"4# * * -4# * -"
.#
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ITE.ACIO/ * Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s - " * * * $
-M a % -" " * * * %-M a" " " * - * (
0j -1M * -#M * M -M -M -*MCj - 0j "!1M ( #!#M * -M * * Cj-0j *
.*
.*"
.*#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *" X -4# 4# * * 4% *
-M a" * *
0j
Cj - 0j Cj-0j *
ITE.ACIO/
.
."
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.#' .*# - " .".*# " " * - * (
-" ." -" "4# -"4# * * -4# * -"
.# * 14# 4# * - -4# "
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ITE.ACIO/ * Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s - " * * * $
-M a % -" " * * * %-M a" " " * - * (
0j -1M * -#M * M -M -M -*MCj - 0j "!1M ( #!#M * -M * * Cj-0j *
.*
.*"
.*#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *" X -4# 4# * * 4% *
-M a" * 14# 4# * - -4# "
0j
Cj - 0j Cj-0j *
ITE.ACIO/
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."
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.#' .*# - " .".*# " " * - * (
-" ." -" "4# -"4# * * -4# * -"
.# * 14# 4# * - -4# "
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ITE.ACIO/ * Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s - " * * * $
-M a % -" " * * * %-M a" " " * - * (
0j -1M * -#M * M -M -M -*MCj - 0j "!1M ( #!#M * -M * * Cj-0j *
.*
.*"
.*#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *" X -4# 4# * * 4% *
-M a" * 14# 4# * - -4# "
0j " -"4#-1M4# "4#-M4# * M 4#!M4# -M "-"MCj - 0j Cj-0j *
ITE.ACIO/
.
."
.#
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ITE.ACIO/ * Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s - " * * * $
-M a % -" " * * * %-M a" " " * - * (
0j -1M * -#M * M -M -M -*MCj - 0j "!1M ( #!#M * -M * * Cj-0j *
.*
.*"
.*#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *" X -4# 4# * * 4% *
-M a" * 14# 4# * - -4# "
0j " -"4#-1M4# "4#-M4# * M 4#!M4# -M "-"MCj - 0j *
+(!1M,# 64#!M4# * -M -4#-(M4# * Cj-0j *
ITE.ACIO/
.
."
.#
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ITE.ACIO/ * Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s - " * * * $
-M a % -" " * * * %-M a" " " * - * (0j -1M * -#M * M -M -M -*M
Cj - 0j "!1M ( #!#M * -M * * Cj-0j *
.*
.*"
.*#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *" X -4# 4# * * 4% *
-M a" * 14# 4# * - -4# "
0j " -"4#-1M4# "4#-M4# * M 4#!M4# -M "-"MCj - 0j *
+(!1M,# 64#!M4# * -M -4#-(M4# * Cj-0j *
ITE.ACIO/
.
."
.#
Colu2napi3ote
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ITE.ACIO/ * Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s - " * * * $
-M a % -" " * * * %-M a" " " * - * (0j -1M * -#M * M -M -M -*M
Cj - 0j "!1M ( #!#M * -M * * Cj-0j *
.*
.*"
.*#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *" X -4# 4# * * 4% *
-M a" * 14# 4# * - -4# "
0j " -"4#-1M4# "4#-M4# * M 4#!M4# -M "-"MCj - 0j *
+(!1M,# 64#!M4# * -M -4#-(M4# * Cj-0j *
ITE.ACIO/
.
."
.#
Colu2napi3ote
*4+"4#,'7/o"4+14#, ' #4(
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ITE.ACIO/ * Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s - " * * * $
-M a % -" " * * * %-M a" " " * - * (0j -1M * -#M * M -M -M -*M
Cj - 0j "!1M ( #!#M * -M * * Cj-0j *
.*
.*"
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Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *" X -4# 4# * * 4% *
-M a" * 14# 4# * - -4# "
0j " -"4#-1M4# "4#-M4# * M 4#!M4# -M "-"MCj - 0j *
+(!1M,# 64#!M4# * -M -4#-(M4# * Cj-0j *
ITE.ACIO/
.
."
.#
Colu2napi3ote
*4+"4#,'7/o"4+14#, ' 8 #
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ITE.ACIO/ * Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s - " * * * $
-M a % -" " * * * %-M a" " " * - * (0j -1M * -#M * M -M -M -*M
Cj - 0j "!1M ( #!#M * -M * * Cj-0j *
.*
.*"
.*#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *" X -4# 4# * * 4% *
-M a" * 14# 4# * - -4# "
0j " -"4#-1M4# "4#-M4# * M 4#!M4# -M "-"MCj - 0j *
+(!1M,# 64#!M4# * -M -4#-(M4# * Cj-0j *
ITE.ACIO/
.
."
.#
Colu2napi3ote
5ila
pi3ote
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ITE.ACIO/ Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *
" X -4# 4# * * 4% * -M a" * 14# 4# * - -4# "
0j " -"4#-1M4# "4#-M4# * M 4#!M4# -M "-"M
Cj - 0j *+(!1M,
# 64#!M4# * -M -4#-(M4# * Cj-0j *
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Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *" X -4# 4# * * 4% *
-M a" * 14# 4# * - -4# "
0j " -"4#-1M4# "4#-M4# * M 4#!M4# -M "-"MCj - 0j *
+(!1M,# 64#!M4# * -M -4#-(M4# * Cj-0j *
ITE.ACIO/
.
."
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Colu2napi3ote
5ila
pi3ote
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ITE.ACIO/ Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *
" X -4# 4# * * 4% * -M a" * 14# 4# * - -4# "
0j " -"4#-1M4# "4#-M4# * M 4#!M4# -M "-"M
Cj - 0j *+(!1M,
# 64#!M4# * -M -4#-(M4# * Cj-0j *
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Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B
* s * *
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( X" * *0j
Cj - 0j Cj-0j *
ITE.ACIO/ "
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ITE.ACIO/ Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *
" X -4# 4# * * 4% * -M a" * 14# 4# * - -4# "0j " -"4#-1M4# "4#-M4# * M 4#!M4# -M "-"M
Cj - 0j *+(!1M,
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Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B
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( X" * *0j
Cj - 0j Cj-0j *
ITE.ACIO/ "
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."#' .# 9 #41
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ITE.ACIO/ Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *
" X -4# 4# * * 4% * -M a" * 14# 4# * - -4# "0j " -"4#-1M4# "4#-M4# * M 4#!M4# -M "-"M
Cj - 0j *+(!1M,
# 64#!M4# * -M -4#-(M4# * Cj-0j *
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Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B
* s * *
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( X" * 41 * -#41 -41 #41 #4(0j
Cj - 0j Cj-0j *
ITE.ACIO/ "
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."#' .# 9 #41
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ITE.ACIO/ Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *
" X -4# 4# * * 4% * -M a" * 14# 4# * - -4# "0j " -"4#-1M4# "4#-M4# * M 4#!M4# -M "-"M
Cj - 0j *+(!1M,
# 64#!M4# * -M -4#-(M4# * Cj-0j *
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Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B
* s * *
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( X" * 41 * -#41 -41 #41 #4(0j
Cj - 0j Cj-0j *
ITE.ACIO/ "
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8/18/2019 Metodo Simplex 4
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ITE.ACIO/ Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *
" X -4# 4# * * 4% * -M a" * 14# 4# * - -4# "0j " -"4#-1M4# "4#-M4# * M 4#!M4# -M "-"M
Cj - 0j *+(!1M,
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Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B
* s * *
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Cj - 0j Cj-0j *
ITE.ACIO/ "
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-"4# ."# * -"4# -4" * 4( 4" -4( -4"
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8/18/2019 Metodo Simplex 4
37/63
ITE.ACIO/ Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *
" X -4# 4# * * 4% * -M a" * 14# 4# * - -4# "0j " -"4#-1M4# "4#-M4# * M 4#!M4# -M "-"M
Cj - 0j *+(!1M,
# 64#!M4# * -M -4#-(M4# * Cj-0j *
.
."
.#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B
* s * *
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Cj - 0j Cj-0j *
ITE.ACIO/ "
."
.""
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-"4# ."# * -"4# -4" * 4( 4" -4( -4"
." * * $4( 4( 4( -4( $4"
8/18/2019 Metodo Simplex 4
38/63
ITE.ACIO/ Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *
" X -4# 4# * * 4% * -M a" * 14# 4# * - -4# "0j " -"4#-1M4# "4#-M4# * M 4#!M4# -M "-"M
Cj - 0j *+(!1M,
# 64#!M4# * -M -4#-(M4# * Cj-0j *
.
."
.#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s * * $4( 4( 4( -4( $4"
" X * *( X" * 41 * -#41 -41 #41 #4(
0j
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ITE.ACIO/ "
."
.""
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." * * $4( 4( 4( -4( $4"
8/18/2019 Metodo Simplex 4
39/63
ITE.ACIO/ Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *
" X -4# 4# * * 4% * -M a" * 14# 4# * - -4# "0j " -"4#-1M4# "4#-M4# * M 4#!M4# -M "-"M
Cj - 0j *+(!1M,
# 64#!M4# * -M -4#-(M4# * Cj-0j *
.
."
.#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s * * $4( 4( 4( -4( $4"
" X * *( X" * 41 * -#41 -41 #41 #4(
0j
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ITE.ACIO/ "
."
.""
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.""
8/18/2019 Metodo Simplex 4
40/63
ITE.ACIO/ Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *
" X -4# 4# * * 4% * -M a" * 14# 4# * - -4# "0j " -"4#-1M4# "4#-M4# * M 4#!M4# -M "-"M
Cj - 0j *+(!1M,
# 64#!M4# * -M -4#-(M4# * Cj-0j *
.
."
.#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s * * $4( 4( 4( -4( $4"
" X * *( X" * 41 * -#41 -41 #41 #4(
0j
Cj - 0j Cj-0j *
ITE.ACIO/ "
."
.""
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!4# ."# * 4# 4"( * -41 -4"( 41 4(
.""
8/18/2019 Metodo Simplex 4
41/63
ITE.ACIO/ Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *
" X -4# 4# * * 4% * -M a" * 14# 4# * - -4# "0j " -"4#-1M4# "4#-M4# * M 4#!M4# -M "-"M
Cj - 0j *+(!1M,
# 64#!M4# * -M -4#-(M4# * Cj-0j *
.
."
.#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s * * $4( 4( 4( -4( $4"
" X * *( X" * 41 * -#41 -41 #41 #4(
0j
Cj - 0j Cj-0j *
ITE.ACIO/ "
."
.""
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!4# ."# * 4# 4"( * -41 -4"( 41 4(
."" * #41 * -41 41 41 74(
8/18/2019 Metodo Simplex 4
42/63
ITE.ACIO/ Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *
" X -4# 4# * * 4% * -M a" * 14# 4# * - -4# "0j " -"4#-1M4# "4#-M4# * M 4#!M4# -M "-"M
Cj - 0j *+(!1M,
# 64#!M4# * -M -4#-(M4# * Cj-0j *
.
."
.#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s * * $4( 4( 4( -4( $4"" X * #41 * -41 41 41 74(( X" * 41 * -#41 -41 #41 #4(
0jCj - 0j Cj-0j *
ITE.ACIO/ "
."
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!4# ."# * 4# 4"( * -41 -4"( 41 4(
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ITE.ACIO/
8/18/2019 Metodo Simplex 4
43/63
ITE.ACIO/ Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *
" X -4# 4# * * 4% * -M a" * 14# 4# * - -4# "0j " -"4#-1M4# "4#-M4# * M 4#!M4# -M "-"M
Cj - 0j *+(!1M,
# 64#!M4# * -M -4#-(M4# * Cj-0j *
.
.
"
.#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s * * $4( 4( 4( -4( $4"" X * #41 * -41 41 41 74(( X" * 41 * -#41 -41 #41 #4(
0j " ( 74( * -64( -4( 64( 4"Cj - 0j Cj-0j *
ITE.ACIO/ "
."
.""
."#
ITE.ACIO/
8/18/2019 Metodo Simplex 4
44/63
ITE.ACIO/ Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *
" X -4# 4# * * 4% * -M a" * 14# 4# * - -4# "0j " -"4#-1M4# "4#-M4# * M 4#!M4# -M "-"M
Cj - 0j *+(!1M,
# 64#!M4# * -M -4#-(M4# * Cj-0j *
.
.
"
.#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s * * $4( 4( 4( -4( $4"" X * #41 * -41 41 41 74(( X" * 41 * -#41 -41 #41 #4(
0j " ( 74( * -64( -4( 64( 4"Cj - 0j * * 64( * 64( -M!4( -M-64( Cj-0j *
ITE.ACIO/ "
."
.""
."#
ITE.ACIO/
8/18/2019 Metodo Simplex 4
45/63
ITE.ACIO/ Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *
" X -4# 4# * * 4% * -M a" * 14# 4# * - -4# "0j " -"4#-1M4# "4#-M4# * M 4#!M4# -M "-"M
Cj - 0j *+(!1M,
# 64#!M4# * -M -4#-(M4# * Cj-0j *
.
.
"
.#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s * * $4( 4( 4( -4( $4"" X * #41 * -41 41 41 74(( X" * 41 * -#41 -41 #41 #4(
0j " ( 74( * -64( -4( 64( 4"Cj - 0j * * 64( * 64( -M!4( -M-64( Cj-0j *
ITE.ACIO/ "
."
.""
."#
Colu2napi3ote
ITE.ACIO/
8/18/2019 Metodo Simplex 4
46/63
ITE.ACIO/ Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *
" X -4# 4# * * 4% * -M a" * 14# 4# * - -4# "0j " -"4#-1M4# "4#-M4# * M 4#!M4# -M "-"M
Cj - 0j *+(!1M,
# 64#!M4# * -M -4#-(M4# * Cj-0j *
.
.
"
.#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s * * $4( 4( 4( -4( $4"" X * #41 * -41 41 41 74(( X" * 41 * -#41 -41 #41 #4(
0j " ( 74( * -64( -4( 64( 4"Cj - 0j * * 64( * 64( -M!4( -M-64( Cj-0j *
ITE.ACIO/ "
."
.""
."#
Colu2napi3ote
+$4",4+4(,/o/o
ITE.ACIO/
8/18/2019 Metodo Simplex 4
47/63
ITE.ACIO/ Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *
" X -4# 4# * * 4% * -M a" * 14# 4# * - -4# "0j " -"4#-1M4# "4#-M4# * M 4#!M4# -M "-"M
Cj - 0j *+(!1M,
# 64#!M4# * -M -4#-(M4# * Cj-0j *
.
.
"
.#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s * * $4( 4( 4( -4( $4"" X * #41 * -41 41 41 74(( X" * 41 * -#41 -41 #41 #4(
0j " ( 74( * -64( -4( 64( 4"Cj - 0j * * 64( * 64( -M!4( -M-64( Cj-0j *
ITE.ACIO/ "
."
.""
."#
Colu2napi3ote
#1 /o/o
ITE.ACIO/
8/18/2019 Metodo Simplex 4
48/63
ITE.ACIO/ Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * "4# 64# * 4% * *
" X -4# 4# * * 4% * -M a" * 14# 4# * - -4# "0j " -"4#-1M4# "4#-M4# * M 4#!M4# -M "-"M
Cj - 0j *+(!1M,
# 64#!M4# * -M -4#-(M4# * Cj-0j *
.
.
"
.#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s * * $4( 4( 4( -4( $4"" X * #41 * -41 41 41 74(( X" * 41 * -#41 -41 #41 #4(
0j " ( 74( * -64( -4( 64( 4"Cj - 0j * * 64( * 64( -M!4( -M-64( Cj-0j *
ITE.ACIO/ "
."
.""
."#
Colu2napi3ote
5ilapi3ote
ITE.ACIO/ "
8/18/2019 Metodo Simplex 4
49/63
ITE.ACIO/ " Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * * $4( 4( 4( -4( $4"
" X * #41 * -41 41 41 74(( X" * 41 * -#41 -41 #41 #4(0j " ( 74( * -64( -4( 64( 4"
Cj - 0j * * 64( * 64( -M!4( -M-64( Cj-0j *
."
."
"
."#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s * * $4( 4( 4( -4( $4"" X * #41 * -41 41 41 74(( X" * 41 * -#41 -41 #41 #4(
0j " ( 74( * -64( -4( 64( 4"Cj - 0j * * 64( * 64( -M!4( -M-64( Cj-0j *
ITE.ACIO/ "
."
.""
."#
Colu2napi3ote
5ilapi3ote
ITE.ACIO/ "
8/18/2019 Metodo Simplex 4
50/63
ITE.ACIO/ " Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * * $4( 4( 4( -4( $4"
" X * #41 * -41 41 41 74(( X" * 41 * -#41 -41 #41 #4(0j " ( 74( * -64( -4( 64( 4"
Cj - 0j * * 64( * 64( -M!4( -M-64( Cj-0j *
."
."
"."#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B
* s" * *
" X * *
( X" * *0j
Cj - 0j Cj-0j *
ITE.ACIO/ #
.#
.#"
.##
ITE.ACIO/ "
8/18/2019 Metodo Simplex 4
51/63
ITE.ACIO/ " Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * * $4( 4( 4( -4( $4"
" X * #41 * -41 41 41 74(( X" * 41 * -#41 -41 #41 #4(0j " ( 74( * -64( -4( 64( 4"
Cj - 0j * * 64( * 64( -M!4( -M-64( Cj-0j *
."
."
"."#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B
* s" * *
" X * *
( X" * *0j
Cj - 0j Cj-0j *
ITE.ACIO/ #
.#
.#"
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.# ' ." 9 (
ITE.ACIO/ "
8/18/2019 Metodo Simplex 4
52/63
ITE.ACIO/ " Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * * $4( 4( 4( -4( $4"
" X * #41 * -41 41 41 74(( X" * 41 * -#41 -41 #41 #4(0j " ( 74( * -64( -4( 64( 4"
Cj - 0j * * 64( * 64( -M!4( -M-64( Cj-0j *
."
."
"."#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s" * * $ ( - #1
" X * *
( X" * *
0j
Cj - 0j Cj-0j *
ITE.ACIO/ #
.#
.#"
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.# ' ." 9 (
ITE.ACIO/ "
8/18/2019 Metodo Simplex 4
53/63
ITE.ACIO/ " Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * * $4( 4( 4( -4( $4"
" X * #41 * -41 41 41 74(( X" * 41 * -#41 -41 #41 #4(0j " ( 74( * -64( -4( 64( 4"
Cj - 0j * * 64( * 64( -M!4( -M-64( Cj-0j *
."
."
"."#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s" * * $ ( - #1
" X * *
( X" * *
0j
Cj - 0j Cj-0j *
ITE.ACIO/ #
.#
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.#"' ."" !41 .#."" * #41 * -41 41 41 74(
!41 .#
.#
"
ITE.ACIO/ "
8/18/2019 Metodo Simplex 4
54/63
ITE.ACIO/ " Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * * $4( 4( 4( -4( $4"
" X * #41 * -41 41 41 74(( X" * 41 * -#41 -41 #41 #4(0j " ( 74( * -64( -4( 64( 4"
Cj - 0j * * 64( * 64( -M!4( -M-64( Cj-0j *
."
."
"."#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s" * * $ ( - #1
" X * *
( X" * *
0j
Cj - 0j Cj-0j *
ITE.ACIO/ #
.#
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.#"' ."" !41 .#."" * #41 * -41 41 41 74(
!41 .# * * $41 4" 41 41 -41 $4(
.#"
ITE.ACIO/ "
8/18/2019 Metodo Simplex 4
55/63
ITE.ACIO/ " Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * * $4( 4( 4( -4( $4"
" X * #41 * -41 41 41 74(( X" * 41 * -#41 -41 #41 #4(0j " ( 74( * -64( -4( 64( 4"
Cj - 0j * * 64( * 64( -M!4( -M-64( Cj-0j *
."
."
"."#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s" * * $ ( - #1
" X * *
( X" * *
0j
Cj - 0j Cj-0j *
ITE.ACIO/ #
.#
.#"
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.#"' ."" !41 .#."" * #41 * -41 41 41 74(
!41 .# * * $41 4" 41 41 -41 $4(
.#" * #4" 4" * 4( * %
ITE.ACIO/ "
8/18/2019 Metodo Simplex 4
56/63
ITE.ACIO/ " Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * * $4( 4( 4( -4( $4"
" X * #41 * -41 41 41 74(( X" * 41 * -#41 -41 #41 #4(0j " ( 74( * -64( -4( 64( 4"
Cj - 0j * * 64( * 64( -M!4( -M-64( Cj-0j *
."
."
"."#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s" * * $ ( - #1" X * #4" 4" * 4( * %
( X" * *
0j
Cj - 0j Cj-0j *
ITE.ACIO/ #
.#
.#"
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!41 .# * * $41 4" 41 41 -41 $4(
.#" * #4" 4" * 4( * %
ITE.ACIO/ "
8/18/2019 Metodo Simplex 4
57/63
ITE.ACIO/ " Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * * $4( 4( 4( -4( $4"
" X * #41 * -41 41 41 74(( X" * 41 * -#41 -41 #41 #4(0j " ( 74( * -64( -4( 64( 4"
Cj - 0j * * 64( * 64( -M!4( -M-64( Cj-0j *
."
."
"."#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s" * * $ ( - #1" X * #4" 4" * 4( * %
( X" * *
0j
Cj - 0j Cj-0j *
ITE.ACIO/ #
.#
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!#41 .#
.#
#
ITE.ACIO/ "
8/18/2019 Metodo Simplex 4
58/63
ITE.ACIO/ " Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * * $4( 4( 4( -4( $4"
" X * #41 * -41 41 41 74(( X" * 41 * -#41 -41 #41 #4(0j " ( 74( * -64( -4( 64( 4"
Cj - 0j * * 64( * 64( -M!4( -M-64( Cj-0j *
."
."
"."#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s" * * $ ( - #1" X * #4" 4" * 4( * %
( X" * *
0j
Cj - 0j Cj-0j *
ITE.ACIO/ #
.#
.#"
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!#41 .# * * "641 #4" #41 #41 -#41 764(
.##
ITE.ACIO/ "
8/18/2019 Metodo Simplex 4
59/63
ITE.ACIO/ " Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * * $4( 4( 4( -4( $4"
" X * #41 * -41 41 41 74(( X" * 41 * -#41 -41 #41 #4(0j " ( 74( * -64( -4( 64( 4"
Cj - 0j * * 64( * 64( -M!4( -M-64( Cj-0j *
."
."
"."#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s" * * $ ( - #1" X * #4" 4" * 4( * %
( X" * *
0j
Cj - 0j Cj-0j *
ITE.ACIO/ #
.#
.#"
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!#41 .# * * "641 #4" #41 #41 -#41 764(
.## * 64" #4" * 4( * 7
ITE.ACIO/ "
8/18/2019 Metodo Simplex 4
60/63
ITE.ACIO/ " Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * * $4( 4( 4( -4( $4"
" X * #41 * -41 41 41 74(( X" * 41 * -#41 -41 #41 #4(0j " ( 74( * -64( -4( 64( 4"
Cj - 0j * * 64( * 64( -M!4( -M-64( Cj-0j *
."
."
"."#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s" * * $ ( - #1" X * #4" 4" * 4( * %( X" * 64" #4" * 4( * 7
0jCj - 0j Cj-0j *
ITE.ACIO/ #
.#
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!#41 .# * * "641 #4" #41 #41 -#41 764(
.## * 64" #4" * 4( * 7
ITE.ACIO/ "
8/18/2019 Metodo Simplex 4
61/63
ITE.ACIO/ " Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * * $4( 4( 4( -4( $4"
" X * #41 * -41 41 41 74(( X" * 41 * -#41 -41 #41 #4(0j " ( 74( * -64( -4( 64( 4"
Cj - 0j * * 64( * 64( -M!4( -M-64( Cj-0j *
."
."
"."#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s" * * $ ( - #1" X * #4" 4" * 4( * %( X" * 64" #4" * 4( * 7
0j " ( 6 6 * #4" * 6"Cj - 0j Cj-0j *
ITE.ACIO/ #
.#
.#"
.##
ITE.ACIO/ "
8/18/2019 Metodo Simplex 4
62/63
ITE.ACIO/ " Cj " ( # * * -M -M
CB !B X1 X2 X3 "1 "2 a1 a2 B* s * * $4( 4( 4( -4( $4"
" X * #41 * -41 41 41 74(( X" * 41 * -#41 -41 #41 #4(0j " ( 74( * -64( -4( 64( 4"
Cj - 0j * * 64( * 64( -M!4( -M-64( Cj-0j *
."
."
"."#
Cj " ( # * * -M -MCB !B X1 X2 X3 "1 "2 a1 a2 B* s" * * $ ( - #1" X * #4" 4" * 4( * %( X" * 64" #4" * 4( * 7
0j " ( 6 6 * #4" * 6"Cj - 0j * * -( -6 * -#4"-M -M Cj-0j *
ITE.ACIO/ #
.#
.#"
.##
;a que todos los Cj & 0j * entonces la ta
8/18/2019 Metodo Simplex 4
63/63
@.ACIAS
Ing.Luis
Medina
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