Metodo Simplex 4.ppsx
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METODO SIMPLEX Caso en que las restricciones son del tipo > Luis Medina Aquino
PROBLEMA FORMA CANONICA Maximizar Z = 2 X1 + 4 X2 + 3 X3 Sujeto a: - X1+ X2 + 2 X3 9 6 X1 – 2 X2 + 2 X3 = 6 2 X1 + 2 X2 + X3 4 X1, X2, X3 0
PROBLEMA FORMA CANONICA Maximizar Z = 2 X1 + 4 X2 + 3 X3 Sujeto a: - X1+ X2 + 2 X3 9 (S1) 6 X1 – 2 X2 + 2 X3 = 6 (a1) 2 X1 + 2 X2 + X3 4 (-S2, a2) X1, X2, X3 0
PROBLEMA Maximizar Z = 2 X1 + 4 X2 + 3 X3 Sujeto a: - 1 X1 + 1 X2 + 2 X3 < 9 (S1) 6 X1 - 2 X2 + 2 X3 = 6 (a1) 2 X1 + 2 X2 + 1 X3 > 4 (-S2, a2) X1, X2, X3 0 FORMA ESTANDAR Maximizar Z = 2 X1 + 4 X2 + 3 X3 + 0 S1 + 0 S2 - M a1 - M a2 Sujeto a: -1 X1 + 1 X2 + 2 X3 + 1 S1 + 0 S2 + 0 a1 + 0 a2 = 9 6 X1 - 2 X2 + 2 X3 + 0 S1 + 0 S2 + 1 a1 + 0 a2 = 6 2 X1 + 2 X2 + 1 X3 + 0 S1 – 1 S2 + 0 a1 + 1 a2 = 4 X1, X2, X3, S1, S2, a1, a2 0
ITERACION 0 CB
Cj
2
4
3
0
0
-M
-M
VB
X1 -1 6 2
X2 1 -2 2
X3 2 2 1
s1
s2
a1
a2
1 0 0
0 0 -1
0 1 0
0 0 1
Zj Cj - Zj Maximizar Z = 2 X1 + 4 X2 + 3 X3 + 0 S1 + 0 S2 - M a1 - M a2 Sujeto a: -1 X1 + 1 X2 + 2 X3 + 1 S1 + 0 S2 + 0 a1 + 0 a2 = 9 6 X1 - 2 X2 + 2 X3 + 0 S1 + 0 S2 + 1 a1 + 0 a2 = 6 2 X1 + 2 X2 + 1 X3 + 0 S1 – 1 S2 + 0 a1 + 1 a2 = 4 X1, X2, X3, S1, S2, a1, a2 0
B 9 6 4 Cj-Zj < 0
ITERACION 0 R01
R02 R03
CB 0 -M -M
Cj
2
4
3
0
0
-M
-M
VB
X1 -1 6 2
X2 1 -2 2
X3 2 2 1
s1
s2
a1
a2
1 0 0
0 0 -1
0 1 0
0 0 1
s1 a1 a2
Zj Cj - Zj
B 9 6 4 Cj-Zj < 0
ITERACION 0 R01
R02 R03
CB 0 -M -M
Cj
2
4
3
0
0
-M
-M
VB
X1 -1 6 2 -8M
X2 1 -2 2 0
X3 2 2 1 -3M
s1
s2
a1
a2
1 0 0 0
0 0 -1 M
0 1 0 -M
0 0 1 -M
s1 a1 a2
Zj Cj - Zj
B 9 6 4 -10M Cj-Zj < 0
ITERACION 0 Cj R01
R02 R03
CB 0 -M -M
VB
s1 a1 a2
Zj Cj - Zj
2
4
3
0
0
-M
-M
s1 X1 X2 X3 -1 1 2 1 6 -2 2 0 2 2 1 0 -8M 0 -3M 0 2+8M 4 3+3M 0
s2
a1
a2
0 0 -1 M -M
0 1 0 -M 0
0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 0 Cj R01
R02 R03
CB 0 -M -M
VB
s1 a1 a2
Zj Cj - Zj
2
0
0
-M
-M
s1 X1 X2 X3 -1 1 2 1 6 -2 2 0 2 2 1 0 -8M 0 -3M 0 2+8M 4 3+3M 0
s2
a1
a2
0 0 -1 M -M
0 1 0 -M 0
0 0 1 -M 0
Columna pivote
4
3
B 9 6 4 -10M Cj-Zj < 0
ITERACION 0 Cj R01
R02 R03
CB 0 -M -M
VB
s1 a1 a2
Zj Cj - Zj
2
0
0
-M
-M
s1 X1 X2 X3 -1 1 2 1 6 -2 2 0 2 2 1 0 -8M 0 -3M 0 2+8M 4 3+3M 0
s2
a1
a2
0 0 -1 M -M
0 1 0 -M 0
0 0 1 -M 0
Columna pivote
4
3
B 9 6 4 -10M Cj-Zj < 0
No 6/6 = 1
4/2 = 2
ITERACION 0 Cj R01
R02 R03
CB 0 -M -M
VB
s1 a1 a2
Zj Cj - Zj
2
0
0
-M
-M
s1 X1 X2 X3 -1 1 2 1 6 -2 2 0 2 2 1 0 -8M 0 -3M 0 2+8M 4 3+3M 0
s2
a1
a2
0 0 -1 M -M
0 1 0 -M 0
0 0 1 -M 0
Columna pivote
4
3
B 9 6 4 -10M Cj-Zj < 0
No 6/6 = 1 √
4/2 = 2
ITERACION 0 Cj R01
R02 R03
CB 0 -M -M
VB
s1 a1 a2
Zj Cj - Zj
2
0
0
-M
-M
s1 X1 X2 X3 -1 1 2 1 6 -2 2 0 2 2 1 0 -8M 0 -3M 0 2+8M 4 3+3M 0
s2
a1
a2
0 0 -1 M -M
0 1 0 -M 0
0 0 1 -M 0
Columna pivote
4
3
B 9 6 4 -10M Cj-Zj < 0
Fila pivote
ITERACION 0 Cj R01
R02 R03
CB 0 -M -M
VB
s1 a1 a2
Zj Cj - Zj
2
4
3
0
0
-M
-M
s1 X1 X2 X3 -1 1 2 1 6 -2 2 0 2 2 1 0 -8M 0 -3M 0 2+8M 4 3+3M 0
s2
a1
a2
0 0 -1 M -M
0 1 0 -M 0
0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 0 R01 R02 R03
CB 0 -M -M
Cj VB s1 a1 a2
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
CB 0 2 -M
Cj VB s1 X1 a2
Zj Cj - Zj
2 X1 0 1 0
4 X2
3 X3
0 s1 1 0 0
0 s2
-M a1
-M a2 0 0 1
B
Cj-Zj < 0
ITERACION 0 R01 R02 R03
CB 0 -M -M
Cj VB s1 a1 a2
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
CB 0 2 -M
Cj VB s1 X1 a2
Zj Cj - Zj R12 = R02/6
2 X1 0 1 0
4 X2
3 X3
0 s1 1 0 0
0 s2
-M a1
-M a2 0 0 1
B
Cj-Zj < 0
ITERACION 0 R01 R02 R03
CB 0 -M -M
Cj VB s1 a1 a2
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
CB 0 2 -M
Cj VB s1 X1 a2
Zj Cj - Zj R12 = R02/6
2 X1 0 1 0
4 X2
3 X3
-1/3
1/3
0 s1 1 0 0
0 s2
-M a1
0
1/6
-M a2 0 0 1
B 1
Cj-Zj < 0
ITERACION 0 R01 R02 R03
CB 0 -M -M
Cj VB s1 a1 a2
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
CB 0 2 -M
Cj VB s1 X1 a2
2 X1 0 1 0
4 X2
3 X3
-1/3
1/3
0 s1 1 0 0
0 s2
-M a1
0
1/6
-M a2 0 0 1
Zj Cj - Zj R11= R01 + 1 R12
B 1
Cj-Zj < 0 R01
+ R12 R11
-1
1
2
1
0
0
0
9
ITERACION 0 R01 R02 R03
CB 0 -M -M
Cj VB s1 a1 a2
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
CB 0 2 -M
Cj VB s1 X1 a2
2 X1 0 1 0
4 X2
3 X3
-1/3
1/3
0 s1 1 0 0
0 s2
-M a1
0
1/6
-M a2 0 0 1
Zj Cj - Zj R11= R01 + 1 R12
B 1
Cj-Zj < 0 R01
-1
1
2
1
0
0
0
9
+ R12
1
-1/3
1/3
0
0
1/6
0
1
R11
ITERACION 0 R01 R02 R03
CB 0 -M -M
Cj VB s1 a1 a2
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
CB 0 2 -M
Cj VB s1 X1 a2
2 X1 0 1 0
4 X2
3 X3
-1/3
1/3
0 s1 1 0 0
0 s2
-M a1
0
1/6
-M a2 0 0 1
Zj Cj - Zj R11= R01 + 1 R12
B 1
Cj-Zj < 0 R01
-1
1
2
1
0
0
0
9
+ R12
1
-1/3
1/3
0
0
1/6
0
1
R11
0
2/3
7/3
1
0
1/6
0
10
ITERACION 0 R01 R02 R03
CB 0 -M -M
Cj VB s1 a1 a2
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
CB 0 2 -M
Cj VB s1 X1 a2
2 X1 0 1 0
4 X2 2/3 -1/3
3 X3 7/3 1/3
0 s1 1 0 0
0 s2 0 0
-M a1 1/6 1/6
-M a2 0 0 1
Zj Cj - Zj R11= R01 + 1 R12
B 10 1
Cj-Zj < 0 R01
-1
1
2
1
0
0
0
9
+ R12
1
-1/3
1/3
0
0
1/6
0
1
R11
0
2/3
7/3
1
0
1/6
0
10
ITERACION 0 R01 R02 R03
CB 0 -M -M
Cj VB s1 a1 a2
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
CB 0 2 -M
Cj VB s1 X1 a2
2 X1 0 1 0
4 X2 2/3 -1/3
3 X3 7/3 1/3
0 s1 1 0 0
0 s2 0 0
-M a1 1/6 1/6
-M a2 0 0 1
Zj Cj - Zj R13= R03 - 2 R12
B 10 1
Cj-Zj < 0 R03
-2 R12 R13
2
2
1
0
-1
0
1
4
ITERACION 0 R01 R02 R03
CB 0 -M -M
Cj VB s1 a1 a2
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
CB 0 2 -M
Cj VB s1 X1 a2
2 X1 0 1 0
4 X2 2/3 -1/3
3 X3 7/3 1/3
0 s1 1 0 0
0 s2 0 0
-M a1 1/6 1/6
-M a2 0 0 1
Zj Cj - Zj R13= R03 - 2 R12
B 10 1
Cj-Zj < 0 R03
2
2
1
0
-1
0
1
4
-2 R12
-2
2/3
-2/3
0
0
-1/3
0
-2
R13
ITERACION 0 R01 R02 R03
CB 0 -M -M
Cj VB s1 a1 a2
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
CB 0 2 -M
Cj VB s1 X1 a2
2 X1 0 1 0
4 X2 2/3 -1/3
3 X3 7/3 1/3
0 s1 1 0 0
0 s2 0 0
-M a1 1/6 1/6
-M a2 0 0 1
Zj Cj - Zj R13= R03 - 2 R12
B 10 1
Cj-Zj < 0 R03
2
2
1
0
-1
0
1
4
-2 R12
-2
2/3
-2/3
0
0
-1/3
0
-2
R13
0
8/3
1/3
0
-1
-1/3
1
2
ITERACION 0 R01 R02 R03
CB 0 -M -M
Cj VB s1 a1 a2
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
CB 0 2 -M
Cj VB s1 X1 a2
2 X1 0 1 0
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
0 s1 1 0 0
0 s2 0 0 -1
-M a1 1/6 1/6 -1/3
-M a2 0 0 1
Zj Cj - Zj R13= R03 - 2 R12
B 10 1 2 Cj-Zj < 0
R03
2
2
1
0
-1
0
1
4
-2 R12
-2
2/3
-2/3
0
0
-1/3
0
-2
R13
0
8/3
1/3
0
-1
-1/3
1
2
ITERACION 0 R01 R02 R03
CB 0 -M -M
Cj VB s1 a1 a2
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
CB 0 2 -M
Cj VB s1 X1 a2
Zj Cj - Zj
2 X1 0 1 0 2
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M Cj-Zj < 0
ITERACION 0 R01 R02 R03
Cj VB s1 a1 a2
CB 0 -M -M
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
0
-M
-1/3-4M/3
0
B 10 1 2 2-2M Cj-Zj < 0
ITERACION 0 R01 R02 R03
Cj VB s1 a1 a2
CB 0 -M -M
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
Columna pivote
0 s1 1 0 0 0
0 s2 0 0 -1 M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
0
-M
-1/3-4M/3
0
B 10 1 2 2-2M Cj-Zj < 0
ITERACION 0 R01 R02 R03
Cj VB s1 a1 a2
CB 0 -M -M
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
Columna pivote
0 s1 1 0 0 0
0 s2 0 0 -1 M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
0
-M
-1/3-4M/3
0
B 10 10/(2/3)=15 1 No 2 2/(8/3) = 3/4 2-2M Cj-Zj < 0
ITERACION 0 R01 R02 R03
Cj VB s1 a1 a2
CB 0 -M -M
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
Columna pivote
0 s1 1 0 0 0
0 s2 0 0 -1 M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
0
-M
-1/3-4M/3
0
B 10 10/(2/3)=15 1 No 2 2/(8/3) = ¾ √ 2-2M Cj-Zj < 0
ITERACION 0 R01 R02 R03
Cj VB s1 a1 a2
CB 0 -M -M
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
Columna pivote
0 s1 1 0 0 0
0 s2 0 0 -1 M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
0
-M
-1/3-4M/3
0
B 10 1 2 2-2M Cj-Zj < 0
Fila pivote
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
Columna pivote
0 s1 1 0 0 0
0 s2 0 0 -1 M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
0
-M
-1/3-4M/3
0
B 10 1 2 2-2M Cj-Zj < 0
Fila pivote
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj
2 X1 0 1 0
4 X2 0 0 1
3 X3
0 s1 1 0 0
0 s2
-M a1
-M a2
B
Cj-Zj < 0
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj R23= R13 x 3/8
2 X1 0 1 0
4 X2 0 0 1
3 X3
0 s1 1 0 0
0 s2
-M a1
-M a2
B
Cj-Zj < 0
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj R23= R13 x 3/8
2 X1 0 1 0
4 X2 0 0 1
3 X3
1/8
0 s1 1 0 0
0 s2
-M a1
-M a2
B
-3/8
-1/8
3/8
3/4 Cj-Zj < 0
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3
1/8
0 s1 1 0 0
0 s2
-M a1
-M a2
B
-3/8
-1/8
3/8
3/4
Zj Cj - Zj R21= R11 -2/3 R23
Cj-Zj < 0 R11
-2/3 R23 R21
0
2/3
7/3
1
0
1/6
0
10
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3
1/8
0 s1 1 0 0
0 s2
-M a1
-M a2
B
-3/8
-1/8
3/8
3/4
Zj Cj - Zj R21= R11 -2/3 R23
Cj-Zj < 0 R11
0
2/3
7/3
1
0
1/6
0
10
-2/3 R23
0
-2/3
-1/12
0
1/4
1/12
-1/4
-1/2
R21
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3
0 s1 1 0 0
1/8
0 s2
-M a1
-M a2
B
-3/8
-1/8
3/8
3/4
Zj Cj - Zj R21= R11 -2/3 R23
Cj-Zj < 0 R11
0
2/3
7/3
1
0
1/6
0
10
-2/3 R23
0
-2/3
-1/12
0
1/4
1/12
-1/4
-1/2
R21
0
0
9/4
1
1/4
1/4
-1/4
19/2
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3 9/4
0 s1 1 0 0
1/8
0 s2 1/4
-M a1 1/4
-M a2 -1/4
B 19/2
-3/8
-1/8
3/8
3/4
Zj Cj - Zj R21= R11 -2/3 R23
Cj-Zj < 0 R11
0
2/3
7/3
1
0
1/6
0
10
-2/3 R23
0
-2/3
-1/12
0
1/4
1/12
-1/4
-1/2
R21
0
0
9/4
1
1/4
1/4
-1/4
19/2
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3 9/4 1/8
0 s1 1 0 0
0 s2 1/4
-M a1 1/4
-M a2 -1/4
B 19/2
-3/8
-1/8
3/8
3/4
Zj Cj - Zj R22= R12 +1/3 R23
Cj-Zj < 0 R12
+1/3 R23 R22
1
-1/3
1/3
0
0
1/6
0
1
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3 9/4 1/8
0 s1 1 0 0
0 s2 1/4
-M a1 1/4
-M a2 -1/4
B 19/2
-3/8
-1/8
3/8
3/4
Zj Cj - Zj R22= R12 +1/3 R23
Cj-Zj < 0 R12
1
-1/3
1/3
0
0
1/6
0
1
+1/3 R23
0
1/3
1/24
0
-1/8
-1/24
1/8
1/4
R22
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3 9/4
0 s1 1 0 0
1/8
0 s2 1/4
-M a1 1/4
-M a2 -1/4
B 19/2
-3/8
-1/8
3/8
3/4
Zj Cj - Zj R22= R12 +1/3 R23
Cj-Zj < 0 R12
1
-1/3
1/3
0
0
1/6
0
1
+1/3 R23
0
1/3
1/24
0
-1/8
-1/24
1/8
1/4
R22
1
0
3/8
0
-1/8
1/8
1/8
5/4
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3 9/4 3/8 1/8
0 s1 1 0 0
0 s2 1/4 -1/8 -3/8
-M a1 1/4 1/8 -1/8
-M a2 -1/4 1/8 3/8
Zj Cj - Zj R22= R12 +1/3 R23
B 19/2 5/4 3/4 Cj-Zj < 0
R12
1
-1/3
1/3
0
0
1/6
0
1
+1/3 R23
0
1/3
1/24
0
-1/8
-1/24
1/8
1/4
R22
1
0
3/8
0
-1/8
1/8
1/8
5/4
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2
4 X2 0 0 1 4
3 X3 9/4 3/8 1/8 5/4
0 s1 1 0 0 0
0 s2 1/4 -1/8 -3/8 -7/4
-M a1 1/4 1/8 -1/8 -1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2 Cj-Zj < 0
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2 0
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2 0
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
Columna pivote
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2 0
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
Columna pivote
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
(19/2)/(1/4) No No
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2 0
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
Columna pivote
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
38 √ No No
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2 0
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
Columna pivote
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
Fila pivote
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2 0
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2 0
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
Columna pivote
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
Fila pivote
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2 0
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 3 R31 R32 R33
CB 0 2 4
Cj VB s2 X1 X2
Zj Cj - Zj
2 X1 0 1 0
4 X2 0 0 1
3 X3
0 s1
0 s2 1 0 0
-M a1
-M a2
B
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2 0
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 3 R31 R32 R33
CB 0 2 4
Cj VB s2 X1 X2
Zj Cj - Zj R31 = R21 x 4
2 X1 0 1 0
4 X2 0 0 1
3 X3
0 s1
0 s2 1 0 0
-M a1
-M a2
B
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2 0
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 3 R31 R32 R33
CB 0 2 4
Cj VB s2 X1 X2
Zj Cj - Zj R31 = R21 x 4
2 X1 0 1 0
4 X2 0 0 1
3 X3 9
0 s1 4
0 s2 1 0 0
-M a1 1
-M a2 -1
B 38
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0 2 0
Zj Cj - Zj
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 3 R31 R32 R33
CB 0 2 4
Cj VB s2 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3 9
0 s1 4
0 s2 1 0 0
-M a1 1
-M a2 -1
Zj Cj - Zj R32= R22 +1/8 R31
B 38
Cj-Zj < 0 R22
+1/8 R31 R32
1
0
3/8
0
-1/8
1/8
1/8
5/4
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0 2 0
Zj Cj - Zj
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 3 R31 R32 R33
CB 0 2 4
Cj VB s2 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3 9
0 s1 4
0 s2 1 0 0
-M a1 1
-M a2 -1
Zj Cj - Zj R32= R22 +1/8 R31
B 38
Cj-Zj < 0 R22
1
0
3/8
0
-1/8
1/8
1/8
5/4
+1/8 R31
0
0
9/8
1/2
1/8
1/8
-1/8
19/4
R32
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0 2 0
Zj Cj - Zj
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 3 R31 R32 R33
CB 0 2 4
Cj VB s2 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3 9
0 s1 4
0 s2 1 0 0
-M a1 1
-M a2 -1
Zj Cj - Zj R32= R22 +1/8 R31
B 38
Cj-Zj < 0 R22
1
0
3/8
0
-1/8
1/8
1/8
5/4
+1/8 R31
0
0
9/8
1/2
1/8
1/8
-1/8
19/4
R32
1
0
3/2
1/2
0
1/4
0
6
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0 2 0
Zj Cj - Zj
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 3 R31 R32 R33
CB 0 2 4
Cj VB s2 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3 9 3/2
0 s1 4 1/2
0 s2 1 0 0
-M a1 1 1/4
-M a2 -1 0
Zj Cj - Zj R32= R22 +1/8 R31
B 38 6
Cj-Zj < 0 R22
1
0
3/8
0
-1/8
1/8
1/8
5/4
+1/8 R31
0
0
9/8
1/2
1/8
1/8
-1/8
19/4
R32
1
0
3/2
1/2
0
1/4
0
6
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0 2 0
Zj Cj - Zj
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 3 R31 R32 R33
CB 0 2 4
Cj VB s2 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3 9 3/2
0 s1 4 1/2
0 s2 1 0 0
-M a1 1 1/4
-M a2 -1 0
Zj Cj - Zj R33= R23 +3/8 R31
B 38 6
Cj-Zj < 0 R23
+3/8 R31 R33
0
1
1/8
0
-3/8
-1/8
3/8
3/4
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0 2 0
Zj Cj - Zj
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 3 R31 R32 R33
CB 0 2 4
Cj VB s2 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3 9 3/2
0 s1 4 1/2
0 s2 1 0 0
-M a1 1 1/4
-M a2 -1 0
Zj Cj - Zj R33= R23 +3/8 R31
B 38 6
Cj-Zj < 0 R23
0
1
1/8
0
-3/8
-1/8
3/8
3/4
+3/8 R31
0
0
27/8
3/2
3/8
3/8
-3/8
57/4
R33
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0 2 0
Zj Cj - Zj
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 3 R31 R32 R33
CB 0 2 4
Cj VB s2 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3 9 3/2
0 s1 4 1/2
0 s2 1 0 0
-M a1 1 1/4
-M a2 -1 0
Zj Cj - Zj R33= R23 +3/8 R31
B 38 6
Cj-Zj < 0 R23
0
1
1/8
0
-3/8
-1/8
3/8
3/4
+3/8 R31
0
0
27/8
3/2
3/8
3/8
-3/8
57/4
R33
0
1
7/2
3/2
0
1/4
0
15
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0 2 0
Zj Cj - Zj
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 3 R31 R32 R33
CB 0 2 4
Cj VB s2 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3 9 3/2 7/2
0 s1 4 1/2 3/2
0 s2 1 0 0
-M a1 1 1/4 1/4
-M a2 -1 0 0
Zj Cj - Zj R33= R23 +3/8 R31
B 38 6 15 Cj-Zj < 0
R23
0
1
1/8
0
-3/8
-1/8
3/8
3/4
+3/8 R31
0
0
27/8
3/2
3/8
3/8
-3/8
57/4
R33
0
1
7/2
3/2
0
1/4
0
15
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2 0
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 3 R31 R32 R33
CB 0 2 4
Cj VB s2 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2
4 X2 0 0 1 4
3 X3 9 3/2 7/2 17
0 s1 4 1/2 3/2 7
0 s2 1 0 0 0
-M a1 1 1/4 1/4 3/2
-M a2 -1 0 0 0
B 38 6 15 72 Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2 0
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 3 R31 R32 R33
CB 0 2 4
Cj VB s2 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2 0
4 X2 0 0 1 4 0
3 X3 9 3/2 7/2 17 -14
0 s1 4 1/2 3/2 7 -7
0 s2 1 0 0 0 0
-M -M a1 a2 1 -1 1/4 0 1/4 0 3/2 0 -3/2-M -M
Ya que todos los Cj – Zj < 0 entonces la tabla es óptima. La solución es: X1 = 6, X2 = 15 y s2 = 38, y el valor de Z = 72.
B 38 6 15 72 Cj-Zj < 0
GRACIAS
Ing. Luis Medina
PROBLEMA FORMA CANONICA Maximizar Z = 2 X1 + 4 X2 + 3 X3 Sujeto a: - X1+ X2 + 2 X3 9 6 X1 – 2 X2 + 2 X3 = 6 2 X1 + 2 X2 + X3 4 X1, X2, X3 0
PROBLEMA FORMA CANONICA Maximizar Z = 2 X1 + 4 X2 + 3 X3 Sujeto a: - X1+ X2 + 2 X3 9 (S1) 6 X1 – 2 X2 + 2 X3 = 6 (a1) 2 X1 + 2 X2 + X3 4 (-S2, a2) X1, X2, X3 0
PROBLEMA Maximizar Z = 2 X1 + 4 X2 + 3 X3 Sujeto a: - 1 X1 + 1 X2 + 2 X3 < 9 (S1) 6 X1 - 2 X2 + 2 X3 = 6 (a1) 2 X1 + 2 X2 + 1 X3 > 4 (-S2, a2) X1, X2, X3 0 FORMA ESTANDAR Maximizar Z = 2 X1 + 4 X2 + 3 X3 + 0 S1 + 0 S2 - M a1 - M a2 Sujeto a: -1 X1 + 1 X2 + 2 X3 + 1 S1 + 0 S2 + 0 a1 + 0 a2 = 9 6 X1 - 2 X2 + 2 X3 + 0 S1 + 0 S2 + 1 a1 + 0 a2 = 6 2 X1 + 2 X2 + 1 X3 + 0 S1 – 1 S2 + 0 a1 + 1 a2 = 4 X1, X2, X3, S1, S2, a1, a2 0
ITERACION 0 CB
Cj
2
4
3
0
0
-M
-M
VB
X1 -1 6 2
X2 1 -2 2
X3 2 2 1
s1
s2
a1
a2
1 0 0
0 0 -1
0 1 0
0 0 1
Zj Cj - Zj Maximizar Z = 2 X1 + 4 X2 + 3 X3 + 0 S1 + 0 S2 - M a1 - M a2 Sujeto a: -1 X1 + 1 X2 + 2 X3 + 1 S1 + 0 S2 + 0 a1 + 0 a2 = 9 6 X1 - 2 X2 + 2 X3 + 0 S1 + 0 S2 + 1 a1 + 0 a2 = 6 2 X1 + 2 X2 + 1 X3 + 0 S1 – 1 S2 + 0 a1 + 1 a2 = 4 X1, X2, X3, S1, S2, a1, a2 0
B 9 6 4 Cj-Zj < 0
ITERACION 0 R01
R02 R03
CB 0 -M -M
Cj
2
4
3
0
0
-M
-M
VB
X1 -1 6 2
X2 1 -2 2
X3 2 2 1
s1
s2
a1
a2
1 0 0
0 0 -1
0 1 0
0 0 1
s1 a1 a2
Zj Cj - Zj
B 9 6 4 Cj-Zj < 0
ITERACION 0 R01
R02 R03
CB 0 -M -M
Cj
2
4
3
0
0
-M
-M
VB
X1 -1 6 2 -8M
X2 1 -2 2 0
X3 2 2 1 -3M
s1
s2
a1
a2
1 0 0 0
0 0 -1 M
0 1 0 -M
0 0 1 -M
s1 a1 a2
Zj Cj - Zj
B 9 6 4 -10M Cj-Zj < 0
ITERACION 0 Cj R01
R02 R03
CB 0 -M -M
VB
s1 a1 a2
Zj Cj - Zj
2
4
3
0
0
-M
-M
s1 X1 X2 X3 -1 1 2 1 6 -2 2 0 2 2 1 0 -8M 0 -3M 0 2+8M 4 3+3M 0
s2
a1
a2
0 0 -1 M -M
0 1 0 -M 0
0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 0 Cj R01
R02 R03
CB 0 -M -M
VB
s1 a1 a2
Zj Cj - Zj
2
0
0
-M
-M
s1 X1 X2 X3 -1 1 2 1 6 -2 2 0 2 2 1 0 -8M 0 -3M 0 2+8M 4 3+3M 0
s2
a1
a2
0 0 -1 M -M
0 1 0 -M 0
0 0 1 -M 0
Columna pivote
4
3
B 9 6 4 -10M Cj-Zj < 0
ITERACION 0 Cj R01
R02 R03
CB 0 -M -M
VB
s1 a1 a2
Zj Cj - Zj
2
0
0
-M
-M
s1 X1 X2 X3 -1 1 2 1 6 -2 2 0 2 2 1 0 -8M 0 -3M 0 2+8M 4 3+3M 0
s2
a1
a2
0 0 -1 M -M
0 1 0 -M 0
0 0 1 -M 0
Columna pivote
4
3
B 9 6 4 -10M Cj-Zj < 0
No 6/6 = 1
4/2 = 2
ITERACION 0 Cj R01
R02 R03
CB 0 -M -M
VB
s1 a1 a2
Zj Cj - Zj
2
0
0
-M
-M
s1 X1 X2 X3 -1 1 2 1 6 -2 2 0 2 2 1 0 -8M 0 -3M 0 2+8M 4 3+3M 0
s2
a1
a2
0 0 -1 M -M
0 1 0 -M 0
0 0 1 -M 0
Columna pivote
4
3
B 9 6 4 -10M Cj-Zj < 0
No 6/6 = 1 √
4/2 = 2
ITERACION 0 Cj R01
R02 R03
CB 0 -M -M
VB
s1 a1 a2
Zj Cj - Zj
2
0
0
-M
-M
s1 X1 X2 X3 -1 1 2 1 6 -2 2 0 2 2 1 0 -8M 0 -3M 0 2+8M 4 3+3M 0
s2
a1
a2
0 0 -1 M -M
0 1 0 -M 0
0 0 1 -M 0
Columna pivote
4
3
B 9 6 4 -10M Cj-Zj < 0
Fila pivote
ITERACION 0 Cj R01
R02 R03
CB 0 -M -M
VB
s1 a1 a2
Zj Cj - Zj
2
4
3
0
0
-M
-M
s1 X1 X2 X3 -1 1 2 1 6 -2 2 0 2 2 1 0 -8M 0 -3M 0 2+8M 4 3+3M 0
s2
a1
a2
0 0 -1 M -M
0 1 0 -M 0
0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 0 R01 R02 R03
CB 0 -M -M
Cj VB s1 a1 a2
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
CB 0 2 -M
Cj VB s1 X1 a2
Zj Cj - Zj
2 X1 0 1 0
4 X2
3 X3
0 s1 1 0 0
0 s2
-M a1
-M a2 0 0 1
B
Cj-Zj < 0
ITERACION 0 R01 R02 R03
CB 0 -M -M
Cj VB s1 a1 a2
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
CB 0 2 -M
Cj VB s1 X1 a2
Zj Cj - Zj R12 = R02/6
2 X1 0 1 0
4 X2
3 X3
0 s1 1 0 0
0 s2
-M a1
-M a2 0 0 1
B
Cj-Zj < 0
ITERACION 0 R01 R02 R03
CB 0 -M -M
Cj VB s1 a1 a2
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
CB 0 2 -M
Cj VB s1 X1 a2
Zj Cj - Zj R12 = R02/6
2 X1 0 1 0
4 X2
3 X3
-1/3
1/3
0 s1 1 0 0
0 s2
-M a1
0
1/6
-M a2 0 0 1
B 1
Cj-Zj < 0
ITERACION 0 R01 R02 R03
CB 0 -M -M
Cj VB s1 a1 a2
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
CB 0 2 -M
Cj VB s1 X1 a2
2 X1 0 1 0
4 X2
3 X3
-1/3
1/3
0 s1 1 0 0
0 s2
-M a1
0
1/6
-M a2 0 0 1
Zj Cj - Zj R11= R01 + 1 R12
B 1
Cj-Zj < 0 R01
+ R12 R11
-1
1
2
1
0
0
0
9
ITERACION 0 R01 R02 R03
CB 0 -M -M
Cj VB s1 a1 a2
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
CB 0 2 -M
Cj VB s1 X1 a2
2 X1 0 1 0
4 X2
3 X3
-1/3
1/3
0 s1 1 0 0
0 s2
-M a1
0
1/6
-M a2 0 0 1
Zj Cj - Zj R11= R01 + 1 R12
B 1
Cj-Zj < 0 R01
-1
1
2
1
0
0
0
9
+ R12
1
-1/3
1/3
0
0
1/6
0
1
R11
ITERACION 0 R01 R02 R03
CB 0 -M -M
Cj VB s1 a1 a2
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
CB 0 2 -M
Cj VB s1 X1 a2
2 X1 0 1 0
4 X2
3 X3
-1/3
1/3
0 s1 1 0 0
0 s2
-M a1
0
1/6
-M a2 0 0 1
Zj Cj - Zj R11= R01 + 1 R12
B 1
Cj-Zj < 0 R01
-1
1
2
1
0
0
0
9
+ R12
1
-1/3
1/3
0
0
1/6
0
1
R11
0
2/3
7/3
1
0
1/6
0
10
ITERACION 0 R01 R02 R03
CB 0 -M -M
Cj VB s1 a1 a2
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
CB 0 2 -M
Cj VB s1 X1 a2
2 X1 0 1 0
4 X2 2/3 -1/3
3 X3 7/3 1/3
0 s1 1 0 0
0 s2 0 0
-M a1 1/6 1/6
-M a2 0 0 1
Zj Cj - Zj R11= R01 + 1 R12
B 10 1
Cj-Zj < 0 R01
-1
1
2
1
0
0
0
9
+ R12
1
-1/3
1/3
0
0
1/6
0
1
R11
0
2/3
7/3
1
0
1/6
0
10
ITERACION 0 R01 R02 R03
CB 0 -M -M
Cj VB s1 a1 a2
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
CB 0 2 -M
Cj VB s1 X1 a2
2 X1 0 1 0
4 X2 2/3 -1/3
3 X3 7/3 1/3
0 s1 1 0 0
0 s2 0 0
-M a1 1/6 1/6
-M a2 0 0 1
Zj Cj - Zj R13= R03 - 2 R12
B 10 1
Cj-Zj < 0 R03
-2 R12 R13
2
2
1
0
-1
0
1
4
ITERACION 0 R01 R02 R03
CB 0 -M -M
Cj VB s1 a1 a2
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
CB 0 2 -M
Cj VB s1 X1 a2
2 X1 0 1 0
4 X2 2/3 -1/3
3 X3 7/3 1/3
0 s1 1 0 0
0 s2 0 0
-M a1 1/6 1/6
-M a2 0 0 1
Zj Cj - Zj R13= R03 - 2 R12
B 10 1
Cj-Zj < 0 R03
2
2
1
0
-1
0
1
4
-2 R12
-2
2/3
-2/3
0
0
-1/3
0
-2
R13
ITERACION 0 R01 R02 R03
CB 0 -M -M
Cj VB s1 a1 a2
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
CB 0 2 -M
Cj VB s1 X1 a2
2 X1 0 1 0
4 X2 2/3 -1/3
3 X3 7/3 1/3
0 s1 1 0 0
0 s2 0 0
-M a1 1/6 1/6
-M a2 0 0 1
Zj Cj - Zj R13= R03 - 2 R12
B 10 1
Cj-Zj < 0 R03
2
2
1
0
-1
0
1
4
-2 R12
-2
2/3
-2/3
0
0
-1/3
0
-2
R13
0
8/3
1/3
0
-1
-1/3
1
2
ITERACION 0 R01 R02 R03
CB 0 -M -M
Cj VB s1 a1 a2
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
CB 0 2 -M
Cj VB s1 X1 a2
2 X1 0 1 0
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
0 s1 1 0 0
0 s2 0 0 -1
-M a1 1/6 1/6 -1/3
-M a2 0 0 1
Zj Cj - Zj R13= R03 - 2 R12
B 10 1 2 Cj-Zj < 0
R03
2
2
1
0
-1
0
1
4
-2 R12
-2
2/3
-2/3
0
0
-1/3
0
-2
R13
0
8/3
1/3
0
-1
-1/3
1
2
ITERACION 0 R01 R02 R03
CB 0 -M -M
Cj VB s1 a1 a2
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
CB 0 2 -M
Cj VB s1 X1 a2
Zj Cj - Zj
2 X1 0 1 0 2
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M Cj-Zj < 0
ITERACION 0 R01 R02 R03
Cj VB s1 a1 a2
CB 0 -M -M
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
0
-M
-1/3-4M/3
0
B 10 1 2 2-2M Cj-Zj < 0
ITERACION 0 R01 R02 R03
Cj VB s1 a1 a2
CB 0 -M -M
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
Columna pivote
0 s1 1 0 0 0
0 s2 0 0 -1 M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
0
-M
-1/3-4M/3
0
B 10 1 2 2-2M Cj-Zj < 0
ITERACION 0 R01 R02 R03
Cj VB s1 a1 a2
CB 0 -M -M
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
Columna pivote
0 s1 1 0 0 0
0 s2 0 0 -1 M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
0
-M
-1/3-4M/3
0
B 10 10/(2/3)=15 1 No 2 2/(8/3) = 3/4 2-2M Cj-Zj < 0
ITERACION 0 R01 R02 R03
Cj VB s1 a1 a2
CB 0 -M -M
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
Columna pivote
0 s1 1 0 0 0
0 s2 0 0 -1 M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
0
-M
-1/3-4M/3
0
B 10 10/(2/3)=15 1 No 2 2/(8/3) = ¾ √ 2-2M Cj-Zj < 0
ITERACION 0 R01 R02 R03
Cj VB s1 a1 a2
CB 0 -M -M
Zj Cj - Zj
2 X1 -1 6 2 -8M 2+8M
4 X2 1 -2 2 0 4
3 X3 2 2 1 -3M 3+3M
0 s1 1 0 0 0 0
0 s2 0 0 -1 M -M
-M a1 0 1 0 -M 0
-M a2 0 0 1 -M 0
B 9 6 4 -10M Cj-Zj < 0
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
Columna pivote
0 s1 1 0 0 0
0 s2 0 0 -1 M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
0
-M
-1/3-4M/3
0
B 10 1 2 2-2M Cj-Zj < 0
Fila pivote
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
Columna pivote
0 s1 1 0 0 0
0 s2 0 0 -1 M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
0
-M
-1/3-4M/3
0
B 10 1 2 2-2M Cj-Zj < 0
Fila pivote
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj
2 X1 0 1 0
4 X2 0 0 1
3 X3
0 s1 1 0 0
0 s2
-M a1
-M a2
B
Cj-Zj < 0
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj R23= R13 x 3/8
2 X1 0 1 0
4 X2 0 0 1
3 X3
0 s1 1 0 0
0 s2
-M a1
-M a2
B
Cj-Zj < 0
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj R23= R13 x 3/8
2 X1 0 1 0
4 X2 0 0 1
3 X3
1/8
0 s1 1 0 0
0 s2
-M a1
-M a2
B
-3/8
-1/8
3/8
3/4 Cj-Zj < 0
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3
1/8
0 s1 1 0 0
0 s2
-M a1
-M a2
B
-3/8
-1/8
3/8
3/4
Zj Cj - Zj R21= R11 -2/3 R23
Cj-Zj < 0 R11
-2/3 R23 R21
0
2/3
7/3
1
0
1/6
0
10
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3
1/8
0 s1 1 0 0
0 s2
-M a1
-M a2
B
-3/8
-1/8
3/8
3/4
Zj Cj - Zj R21= R11 -2/3 R23
Cj-Zj < 0 R11
0
2/3
7/3
1
0
1/6
0
10
-2/3 R23
0
-2/3
-1/12
0
1/4
1/12
-1/4
-1/2
R21
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3
0 s1 1 0 0
1/8
0 s2
-M a1
-M a2
B
-3/8
-1/8
3/8
3/4
Zj Cj - Zj R21= R11 -2/3 R23
Cj-Zj < 0 R11
0
2/3
7/3
1
0
1/6
0
10
-2/3 R23
0
-2/3
-1/12
0
1/4
1/12
-1/4
-1/2
R21
0
0
9/4
1
1/4
1/4
-1/4
19/2
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3 9/4
0 s1 1 0 0
1/8
0 s2 1/4
-M a1 1/4
-M a2 -1/4
B 19/2
-3/8
-1/8
3/8
3/4
Zj Cj - Zj R21= R11 -2/3 R23
Cj-Zj < 0 R11
0
2/3
7/3
1
0
1/6
0
10
-2/3 R23
0
-2/3
-1/12
0
1/4
1/12
-1/4
-1/2
R21
0
0
9/4
1
1/4
1/4
-1/4
19/2
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3 9/4 1/8
0 s1 1 0 0
0 s2 1/4
-M a1 1/4
-M a2 -1/4
B 19/2
-3/8
-1/8
3/8
3/4
Zj Cj - Zj R22= R12 +1/3 R23
Cj-Zj < 0 R12
+1/3 R23 R22
1
-1/3
1/3
0
0
1/6
0
1
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3 9/4 1/8
0 s1 1 0 0
0 s2 1/4
-M a1 1/4
-M a2 -1/4
B 19/2
-3/8
-1/8
3/8
3/4
Zj Cj - Zj R22= R12 +1/3 R23
Cj-Zj < 0 R12
1
-1/3
1/3
0
0
1/6
0
1
+1/3 R23
0
1/3
1/24
0
-1/8
-1/24
1/8
1/4
R22
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3 9/4
0 s1 1 0 0
1/8
0 s2 1/4
-M a1 1/4
-M a2 -1/4
B 19/2
-3/8
-1/8
3/8
3/4
Zj Cj - Zj R22= R12 +1/3 R23
Cj-Zj < 0 R12
1
-1/3
1/3
0
0
1/6
0
1
+1/3 R23
0
1/3
1/24
0
-1/8
-1/24
1/8
1/4
R22
1
0
3/8
0
-1/8
1/8
1/8
5/4
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3 9/4 3/8 1/8
0 s1 1 0 0
0 s2 1/4 -1/8 -3/8
-M a1 1/4 1/8 -1/8
-M a2 -1/4 1/8 3/8
Zj Cj - Zj R22= R12 +1/3 R23
B 19/2 5/4 3/4 Cj-Zj < 0
R12
1
-1/3
1/3
0
0
1/6
0
1
+1/3 R23
0
1/3
1/24
0
-1/8
-1/24
1/8
1/4
R22
1
0
3/8
0
-1/8
1/8
1/8
5/4
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2
4 X2 0 0 1 4
3 X3 9/4 3/8 1/8 5/4
0 s1 1 0 0 0
0 s2 1/4 -1/8 -3/8 -7/4
-M a1 1/4 1/8 -1/8 -1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2 Cj-Zj < 0
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2 0
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2 0
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
Columna pivote
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2 0
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
Columna pivote
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
(19/2)/(1/4) No No
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2 0
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
Columna pivote
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
38 √ No No
ITERACION 1 R11 R12 R13
Cj VB s1 X1 a2 Zj
2 X1 0 1 0 2
Cj - Zj
0
CB 0 2 -M
4 X2 2/3 -1/3 8/3
3 X3 7/3 1/3 1/3
-2/3-8M/3 2/3-M/3 (14+8M) 3 7/3+M/3
0 s1 1 0 0 0
0 s2 0 0 -1 M
0
-M
-M a1 1/6 1/6 -1/3 1/3+M/3
-M a2 0 0 1 -M
B 10 1 2 2-2M
-1/3-4M/3
0
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2 0
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
Columna pivote
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
Fila pivote
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2 0
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2 0
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
Columna pivote
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
Fila pivote
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2 0
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 3 R31 R32 R33
CB 0 2 4
Cj VB s2 X1 X2
Zj Cj - Zj
2 X1 0 1 0
4 X2 0 0 1
3 X3
0 s1
0 s2 1 0 0
-M a1
-M a2
B
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2 0
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 3 R31 R32 R33
CB 0 2 4
Cj VB s2 X1 X2
Zj Cj - Zj R31 = R21 x 4
2 X1 0 1 0
4 X2 0 0 1
3 X3
0 s1
0 s2 1 0 0
-M a1
-M a2
B
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2 0
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 3 R31 R32 R33
CB 0 2 4
Cj VB s2 X1 X2
Zj Cj - Zj R31 = R21 x 4
2 X1 0 1 0
4 X2 0 0 1
3 X3 9
0 s1 4
0 s2 1 0 0
-M a1 1
-M a2 -1
B 38
Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0 2 0
Zj Cj - Zj
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 3 R31 R32 R33
CB 0 2 4
Cj VB s2 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3 9
0 s1 4
0 s2 1 0 0
-M a1 1
-M a2 -1
Zj Cj - Zj R32= R22 +1/8 R31
B 38
Cj-Zj < 0 R22
+1/8 R31 R32
1
0
3/8
0
-1/8
1/8
1/8
5/4
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0 2 0
Zj Cj - Zj
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 3 R31 R32 R33
CB 0 2 4
Cj VB s2 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3 9
0 s1 4
0 s2 1 0 0
-M a1 1
-M a2 -1
Zj Cj - Zj R32= R22 +1/8 R31
B 38
Cj-Zj < 0 R22
1
0
3/8
0
-1/8
1/8
1/8
5/4
+1/8 R31
0
0
9/8
1/2
1/8
1/8
-1/8
19/4
R32
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0 2 0
Zj Cj - Zj
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 3 R31 R32 R33
CB 0 2 4
Cj VB s2 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3 9
0 s1 4
0 s2 1 0 0
-M a1 1
-M a2 -1
Zj Cj - Zj R32= R22 +1/8 R31
B 38
Cj-Zj < 0 R22
1
0
3/8
0
-1/8
1/8
1/8
5/4
+1/8 R31
0
0
9/8
1/2
1/8
1/8
-1/8
19/4
R32
1
0
3/2
1/2
0
1/4
0
6
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0 2 0
Zj Cj - Zj
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 3 R31 R32 R33
CB 0 2 4
Cj VB s2 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3 9 3/2
0 s1 4 1/2
0 s2 1 0 0
-M a1 1 1/4
-M a2 -1 0
Zj Cj - Zj R32= R22 +1/8 R31
B 38 6
Cj-Zj < 0 R22
1
0
3/8
0
-1/8
1/8
1/8
5/4
+1/8 R31
0
0
9/8
1/2
1/8
1/8
-1/8
19/4
R32
1
0
3/2
1/2
0
1/4
0
6
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0 2 0
Zj Cj - Zj
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 3 R31 R32 R33
CB 0 2 4
Cj VB s2 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3 9 3/2
0 s1 4 1/2
0 s2 1 0 0
-M a1 1 1/4
-M a2 -1 0
Zj Cj - Zj R33= R23 +3/8 R31
B 38 6
Cj-Zj < 0 R23
+3/8 R31 R33
0
1
1/8
0
-3/8
-1/8
3/8
3/4
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0 2 0
Zj Cj - Zj
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 3 R31 R32 R33
CB 0 2 4
Cj VB s2 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3 9 3/2
0 s1 4 1/2
0 s2 1 0 0
-M a1 1 1/4
-M a2 -1 0
Zj Cj - Zj R33= R23 +3/8 R31
B 38 6
Cj-Zj < 0 R23
0
1
1/8
0
-3/8
-1/8
3/8
3/4
+3/8 R31
0
0
27/8
3/2
3/8
3/8
-3/8
57/4
R33
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0 2 0
Zj Cj - Zj
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 3 R31 R32 R33
CB 0 2 4
Cj VB s2 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3 9 3/2
0 s1 4 1/2
0 s2 1 0 0
-M a1 1 1/4
-M a2 -1 0
Zj Cj - Zj R33= R23 +3/8 R31
B 38 6
Cj-Zj < 0 R23
0
1
1/8
0
-3/8
-1/8
3/8
3/4
+3/8 R31
0
0
27/8
3/2
3/8
3/8
-3/8
57/4
R33
0
1
7/2
3/2
0
1/4
0
15
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
2 X1 0 1 0 2 0
Zj Cj - Zj
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 3 R31 R32 R33
CB 0 2 4
Cj VB s2 X1 X2
2 X1 0 1 0
4 X2 0 0 1
3 X3 9 3/2 7/2
0 s1 4 1/2 3/2
0 s2 1 0 0
-M a1 1 1/4 1/4
-M a2 -1 0 0
Zj Cj - Zj R33= R23 +3/8 R31
B 38 6 15 Cj-Zj < 0
R23
0
1
1/8
0
-3/8
-1/8
3/8
3/4
+3/8 R31
0
0
27/8
3/2
3/8
3/8
-3/8
57/4
R33
0
1
7/2
3/2
0
1/4
0
15
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2 0
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 3 R31 R32 R33
CB 0 2 4
Cj VB s2 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2
4 X2 0 0 1 4
3 X3 9 3/2 7/2 17
0 s1 4 1/2 3/2 7
0 s2 1 0 0 0
-M a1 1 1/4 1/4 3/2
-M a2 -1 0 0 0
B 38 6 15 72 Cj-Zj < 0
ITERACION 2 R21 R22 R23
CB 0 2 4
Cj VB s1 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2 0
4 X2 0 0 1 4 0
3 X3 9/4 3/8 1/8 5/4 7/4
0 s1 1 0 0 0 0
0 -M s2 a1 1/4 1/4 -1/8 1/8 -3/8 -1/8 -7/4 -1/4 7/4 -M+1/4
-M a2 -1/4 1/8 3/8 7/4
B 19/2 5/4 3/4 11/2
-M-7/4 Cj-Zj < 0
ITERACION 3 R31 R32 R33
CB 0 2 4
Cj VB s2 X1 X2
Zj Cj - Zj
2 X1 0 1 0 2 0
4 X2 0 0 1 4 0
3 X3 9 3/2 7/2 17 -14
0 s1 4 1/2 3/2 7 -7
0 s2 1 0 0 0 0
-M -M a1 a2 1 -1 1/4 0 1/4 0 3/2 0 -3/2-M -M
Ya que todos los Cj – Zj < 0 entonces la tabla es óptima. La solución es: X1 = 6, X2 = 15 y s2 = 38, y el valor de Z = 72.
B 38 6 15 72 Cj-Zj < 0
GRACIAS
Ing. Luis Medina
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