investigacion de operaciones cap4+cap5
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Modelos de Toma de Decisiones
Tarea (capitulo 4 y capitulo 5)
Jose Adrian Siles Masis
CAPITULO 4
EJERCICIO 10
(a) Resuelva el ejercicio con el simplex de dos fases.
(b) Resuelva el ejercicio con la gran M.
min z = 2 x1 + 4 x2 - 1 x4
1 x1 + 2 x2 - 1 x3 + 1 x4 = 3
x1,x2,x3+,x3+,x4 >=0
x3 sin restricciones
x3 = x3+ - x3-
a)
z - 2 x1 - 2 x2 + 1 x4 = 0
1 x1 + 2 x2 (1 x3+) + (1x3-) + 1 x4 + s5 = 2
2 x1 + 1 x2 + (2 x3+) - (2x3-) + 3 x4 + r1 = 4
1 x1 + 0 x2 (1 x3+) + (1 x3-) + 1 x4 s6 + r2 = 3
Fase I:
min w = r1 + r2
w r1 r2 = 0
BVS X1 X2 X3+ X3- X4 S5 S6 R1 R2 RHS
0 W 0 0 0 0 0 0 0 -1 -1 0
Z -2 -2 0 0 1 0 0 0 0 0
1 S5 1 2 -1 1 1 1 0 0 0 2
2 R1 2 1 2 -2 3 0 0 1 0 4
3 R2 1 0 -1 1 1 0 -1 0 1 3
Operacin elemental: f0 + f2
BVS X1 X2 X3+ X3- X4 S5 S6 R1 R2 RHS
0 W 2 1 2 -2 3 0 0 0 -1 4
Z -2 -2 0 0 1 0 0 0 0 0
1 S5 1 2 -1 1 1 1 0 0 0 2
2 R1 2 1 2 -2 3 0 0 1 0 4
3 R2 1 0 -1 1 1 0 -1 0 1 3
Operacin elemental: f0 + f3
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BVS X1 X2 X3+ X3- X4 S5 S6 R1 R2 RHS
0 W 3 1 1 -1 4 0 -1 0 0 7
Z -2 -2 0 0 1 0 0 0 0 0
1 S5 1 2 -1 1 1 1 0 0 0 2
2 R1 2 1 2 -2 3 0 0 1 0 4
3 R2 1 0 -1 1 1 0 -1 0 1 3
Entra X4 / Sale R1
BVS X1 X2 X3+ X3- X4 S5 S6 R1 R2 RHS
0 W 3 1 1 -1 4 0 -1 0 0 7
Z -2 -2 0 0 1 0 0 0 0 0
1 S5 1 2 -1 1 1 1 0 0 0 2
2 R1* 2 1 2 -2 3 0 0 1 0 4
3 R2 1 0 -1 1 1 0 -1 0 1 3
Operacin elemental: -f2+f0
Operacin elemental: f1-f2
Operacin elemental: f3-f2
Operacin elemental: f0-f2
Operacin elemental: f0-4f2
BVS X1 X2 X3* X4 S5 S6 R1 R2 RHS
0 W -1 1 5 0 0 3 4 0 -5
Z -3 -2 1 0 0 1 1 0 -3
1 S5 0 2 0 0 1 1 1 0 -1
2 X4 1 0 -1 1 0 -1 -1 0 3
3 R2 0 0 0 0 0 0 1 1 0
b)
z - 2 x1 - 2 x2 + 1 x4 - Mr1 - Mr2 = 0
1 x1 + 2 x2 - 1 x3 + 1 x4 + s5 = 2
2 x1 + 1 x2 + 2 x3 + 3 x4 + r1 = 4
1 x1 + 0 x2 1 x3 + 1 x4 s6 + r2 = 3
BVS X1 X2 X3 X4 S5 S6 R1 R2 RHS
Z -2 -2 0 1 0 0 -M -M 3M
1 S5 1 2 -1 1 1 0 0 0 2
2 R1 2 1 2 3 0 0 1 0 4
3 R2 1 0 -1 1 0 -1 0 1 3
Operacin elemental: f0+Mf3
BVS X1 X2 X3 X4 S5 S6 R1 R2 RHS
Z -2+M -2 -M M+1 0 -M -M 0 3M
1 S5 1 2 -1 1 1 0 0 0 2
2 R1 2 1 2 3 0 0 1 0 4
3 R2 1 0 -1 1 0 -1 0 1 3
Operacin elemental: Mf2+f0
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BVS X1 X2 X3 X4 S5 S6 R1 R2 RHS
Z -2+3M -2+M M 4M+1 0 -M 0 0 7M
1 S5 1 2 -1 1 1 0 0 0 2
2 R1 2 1 2 3 0 0 1 0 4
3 R2 1 0 -1 1 0 -1 0 1 3
Entra X4
Sale R1
Operacin elemental f2-2f3
BVS X1 X2 X3 X4 S5 S6 R1 R2 RHS
Z -2+3M -2+M M 4M+1 0 -M 0 0 7M
1 S5 1 2 -1 1 1 0 0 0 2
2 X4 0 1 4 1 0 2 1 -2 -2
3 R2 1 0 -1 1 0 -1 0 1 3
Operacin elemental f1-f2
BVS X1 X2 X3 X4 S5 S6 R1 R2 RHS
Z -2+3M -2+M M 4M+1 0 -M 0 0 7M
1 S5 1 1 -5 0 1 -2 -1 2 4
2 X4 0 1 4 1 0 2 1 -2 -2
3 R2 1 0 -1 1 0 -1 0 1 3
Operacin elemental f1-f2
BVS X1 X2 X3 X4 S5 S6 R1 R2* RHS
Z -2+3M -2+M M 4M+1 0 -M 0 0 7M
S5 1 1 -5 0 1 -2 -1 2 4
X4 0 1 4 1 0 2 1 -2 -2
R2* 1 -1 -5 0 0 -3 -1 3 5
Entra R2 / Sale R2
R/ Solucin Infactible
EJERCICIO 15
(a) Resuelva el ejercicio con el simplex de dos fases.
(b) Resuelva el ejercicio con la gran M.
max z = 5 x1 - 2 x2 + x3
1 x1 + 4 x2 + 1 x3 = 2
x1,x3 >= 0
x2 sin restricciones
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a)
z -5 x1 + 2 x2 - 1 x3 = 0
1 x1 + 4 x2 + 1 x3 + s4 = 6
2 x1 + 1x2 + 3 x3 s5 + r1 = 2
Fase I:
min w = r1 = 2
w r1 = 0
BVS X1 X2 X3 S4 S5 R1 RHS
0 W 0 0 0 0 0 -1 0
Z -5 2 -1 0 0 0 0
1 S4 1 4 1 1 0 0 6
2 R1 2 1 3 0 -1 1 2
Operacin Elemental: f0+f2
BVS X1 X2 X3 S4 S5 R1 RHS
0 W 2 1 3 0 -1 0 2
Z -5 2 -1 0 0 0 0
1 S4 1 4 1 1 0 0 6
2 R1 2 1 3 0 -1 1 2
Entra X3 / Sale R1
Operacin elemental: f2-2f1
Operacin elemental: f1-f2
Operacin elemental: f0+f2
Operacin elemental: f0-3f2
BVS X1 X2* X3 S4 S5 R1 RHS
0 W 2 22 0 6 2 -3 32
Z -5 -5 0 -2 -1 1 -10
1 S4 1 11 0 3 1 -1 16
2 X3* 0 -7 1 -2 -1 1 -10
Entra X2 / Sale X3
Operacin elemental: (-1/7)f2
Operacin elemental: f1-11f2
Operacin elemental: f0+5f2
Operacin elemental: f0-22f1
BVS X1 X2 X3* S4 S5 R1 RHS
0 W 2 0 22/7 -2/7 -8/7 1/7 0.57
Z -5 0 -5/7 -4/7 -2/7 2/7 -20/7
1 S4* 1 0 11/7 -1/7 -4/7 4/7 0.29
2 X2 0 1 -1/7 2/7 1/7 -1/7 10/7
Entra X3 / Sale S4
Solucin Infactible (ya haba entrado X3)
b)
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z -5 x1 + 2 x2 - 1 x3 + Mr1 = 0
1 x1 + 4 x2 + 1 x3 + s4 = 6
2 x1 + 1x2 + 3 x3 s5 + r1 = 2
BVS X1 X2 X3 S4 S5 R1 RHS
Z -5 2 -1 0 0 M 0
1 S4 1 4 1 1 0 0 6
2 R1 2 1 3 0 -1 1 2
Operacin elemental: Mf2-f0
BVS X1 X2 X3 S4 S5 R1 RHS
Z 2M-5 M-2 3M+1 0 -M 0 2M
1 S4 1 4 1 1 0 0 6
2 R1 2 1 3 0 -1 1 2
Entra S5 / Sale (NO HAY)
Solucin Infactible.
EJERCICIO 16
(a) Resuelva el ejercicio con el simplex de dos fases.
(b) Resuelva el ejercicio con la gran M.
max z = 4 x1 + 5 x2 + 7 x3 - x4
x1 + x2 + 2 x3 + -1 x4 >= 1
2 x1 + 6 x2 + 3 x3 + x4 = 3
x1 + 4 x2 + 3 x3 + 2 x4 = -5 // -1 x1 - 4 x2 - 3 x3 - 2 x4 = 5
x1,x2,x4 >= 0
x3 sin restriccion
a)
z 4 x1 5 x2 -7 x3 + 1 x4 = 0
1 x1 + 1 x2 + 2 x3 + -1 x4 s5 + r1 = 1
-2 x1 - 6 x2 3 x3 -1 x4 s6 + r2 = 3
-1 x1 - 4 x2 - 3 x3 - 2 x4 + r3 = 5
min w = r1 + r2 + r3
w r1- r2 r3 = 0
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BVS X1 X2 X3 X4 S5 S6 R1 R2 R3 RHS
0 W 0 0 0 0 0 0 -1 -1 -1 0
Z -4 -5 -7 1 0 0 0 0 0 0
1 R1 1 1 2 -1 -1 0 1 0 0 1
2 R2 -2 -6 -3 -1 0 -1 0 1 0 3
3 R3 -1 -4 -3 -2 0 0 0 0 1 5
Operacin elemental: f0+f1
Operacin elemental: f0+f2
Operacin elemental: f0+f3
BVS X1 X2 X3 X4 S5 S6 R1 R2 R3 RHS
0 W -2 -9 -4 -4 -1 -1 0 0 0 9
Z -4 -5 -7 1 0 0 0 0 0 0
1 R1 1 1 2 -1 -1 0 1 0 0 1
2 R2 -2 -6 -3 -1 0 -1 0 1 0 3
3 R3 -1 -4 -3 -2 0 0 0 0 1 5
Entra (no hay)
b)
BVS X1 X2 X3 X4 S5 S6 R1 R2 R3 RHS
Z -4 -5 -7 1 0 0 M M M 0
1 R1 1 1 2 -1 -1 0 1 0 0 1
2 R2 -2 -6 -3 -1 0 -1 0 1 0 3
3 R3 -1 -4 -3 -2 0 0 0 0 1 5
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CAPITULO 5
EJERCICIO 4: Encuentre el dual
min z = 1 x1 - 1 x2
2 x1 - 1 x2 >= 2
-1 x1 - 1 x2 >= 1
x1,x2 >= 0
PRIMAL MATRICIAL
min z = 1 x1 - 1 x2 Sujeto: 2 x1 - 1 x2 >= 2 -1 x1 - 1 x2 >= 1 x1,x2 >= 0
X1 X2
Y1 2 -1 >=2
Y2 -1 -1 >=1
= 0
PRIMAL MATRICIAL
min z = 8 x1 - 2 x2 - 4 x3 1 x1 - 4 x2 - 2 x3 >= 2 1 x1 + 1 x2 - 3 x3 >= -1 x1,x2,x3 >= 0
X1 X2 X3
Y1 1 -4 -2 >=2
Y2 1 1 -3 >=1
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EJERCICIO 11: Encuentre el dual
min z = 1 x3 + 1 x4 + 1 x5
1 x1 - 1 x3 + 1 x4 - 1 x5 = -2
1 x2 - 1 x3 - 1 x4 + 1 x5 = 1
x1,x2,x3,x4,x5 >= 0
PRIMAL MATRICIAL
min z = 1 x3 + 1 x4 + 1 x5 Sujeto a: 1 x1 - 1 x3 + 1 x4 - 1 x5 = -2 1 x2 - 1 x3 - 1 x4 + 1 x5 = 1 x1,x2,x3,x4,x5 >= 0 [0]max z = 1 x3 + 1 x4 + 1 x5 (1a) 1 x1- 1 x3 + 1 x4 - 1 x5 = 0 -1 y1 + 1 y2 + -1 y3 + 1 y4 >= 1 1 y1 + - 1 y2 + -1 y3 + 1 y4 >= 1 -1 y1 + 1 y2 + 1 y3 + -1 y4 >= 1 y1, y2, y3, y4 >= 0
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EJERCICIO 13: Encuentre el dual
max z = 6 x1 + 4 x2 + 6 x3 + 1 x4
4 x1 + 4 x2 + 4 x3 + 8 x4 = 21
3 x1 + 17 x2 + 80 x3 + 2 x4 = 0
x3,x4 sin restricciones
PRIMAL MATRICIAL
max z = 6 x1 + 4 x2 + 6 x3 + 1 x4 Sujeto a: 4 x1 + 4 x2 + 4 x3 + 8 x4 = 21 3 x1 + 17 x2 + 80 x3 + 2 x4 = 0 x3,x4 sin restricciones [0]max z = 6 x1 + 4 x2 + 6 x3 + 1 x4 (1a) 4 x1 + 4 x2 + 4 x3 + 8 x4 = 6 4 y1 + -4 y2 + 17 y3 >= 4 4 y1 + -4 y3 + 80 y3 >= 6 8 y1 + -8 y2 + 2 y2 >= 1 y1, y2, y3 >= 0
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