Universidad Técnica de Machala

Facultad de Ciencias
Empresariales Carrera de
Comercio Exterior Taller #1
Nombre: Melany Mabel Gómez Fernández.
Curso: Cuarto Semestre – Diurna.
Fecha: 17/12/2022.
Docente: Ing. Adriano Ramírez Galeano.
Tema: Programación Lineal.
Resolver los siguientes ejercicios
-Hallar el MODELO MATEMÁTICO.
-Hallar la solución por el método gráfico, usando GEOGEBRA.
-Hallar la solución por el MÉTODO SIMPLEX.
1.- Maximizar
Z = x1 + 2x2
Sujeta a
2x1 + x2 ≤ 8 2x1 + x2 + S1 = 8
2x1 + 3x2 ≤ 12 2x1 + 3x2 + S2 = 12
x1, x2 ≥ 0
2x1 + x2 = 8 2x1 + x2 = 8
x1 = 0 x2 = 0
x2 =8 2x1 =8 x1 = 8/2 x1 = 4
A(0,8) B(4,0)
2x1 + 3x2 = 12 2x1 + 3x2 = 8
x1 = 0 x2 = 0
3x2 =12 2x1 = 12
x2 =12/3= 4 x1 = 12/2 x1 = 6
C(0,4) D(6,0)
X1 + 2X2 F(x1, x2) = x1 + 2x2
2) F(0,0) = 0 + 2(0) = 0
0 + 2(4)= 8
F(0,4) = 0 + 2(4) = 8
3)
2X1 + X2 = 8 F(3,2) = 3 + 2(2) = 7
X2 = 8 - 2X1
2X1 + 3X2 =12 2X1+X2=8 F(4,0) = 4 + 2(0) = 4
2X1+3(8-2X1) =12 2(3)+X2=8
2X1+24-6X1 =12 X2= 8-6
4X1= 24-12 X2= 2 I= 0
X1 = 12/4
X1 =3 II=8
III=7
X1 + 2X2
3+2(2)=7 IV=4
4)
4+2(0)=4

Z = x1 + 2x2
2x1 + x2 + S1 = 8
2x1 + 3x2 + S2 = 12
 Coeficiente Variable 1 2 0 0
Básico Básica X1 X2 S1 S2 Valor
0 S1 2 1 1 0 8
0 S2 2 3 0 1 12
Zj 0 0 0 0
Cj - Zj 1 2 0 0 0

Coeficiente Variable 1 2 0 0
Básico Básica X1 X2 S1 S2 Valor
0 S1 1.333 0 1 -0.333 4
2 X2 0.666 1 0 0.333 4
Zj 1.333 2 0 0.666
Cj - Zj -0.333 0 0 -0.666 8

2.- Maximizar
Z = 2x1 + x2
Sujeta a:
-x1 + x2 ≤ 4 -x1 + x2 + S1 = 4
x1 + x2 ≤ 6 x1 + x2 + S2 = 6
x1, x2 ≥ 0
-x1 + x2 = 4 -x1 + x2 = 4
x1 = 0 x2 =0
x2 = 4 -x1 = 4 x1 = -4
A(0,4) B(-4,0)
x1 + x2 = 6 x1 + x2 = 6
x1 = 0 x2 =0
x2 = 6 x1 = 6
C(0,6) D(6,0)
F(x2X
1,1x+
2)X=
2 2x1 + x2 F(0,0) = 2(0) + 0 = 0
2) = 2(0) + 4 = 4
F(0,4)
0 + 4= 4
F(1,5) = 2(1) + 5 = 7
3)
X1 + X2 = 6 F(6,0) = 2(6) + 0 = 12
X2 = 6 - X1
-X1 + X2 =4 I= 0
-X1+6-X1 =4 X1+X2=6 II=4
-X1 - X1= 4-6 1+X2=6
-2X1 = -2 X2= 6-1 III=7
X1 =-2/-2 X2= 5
X1= 1 IV=12
2X1 + X2
2(1)+5=7
4)
2(6)+0=12

Coeficiente Variable 2 1 0 0
Básico Básica X1 X2 S1 S2 Valor
0 S1 -1 1 1 0 4
0 S2 1 1 0 1 6
Zj 0 0 0 0
Cj - Zj 2 1 0 0 0
Coeficiente Variable 2 1 0 0
Básico Básica X1 X2 S1 S2 Valor
0 S1 0 2 1 1 10
2 X1 1 1 0 1 6
Zj 2 2 0 2
Cj - Zj 0 -1 0 -2 12

3.- Maximizar
Z = -x1 + 3x2
Sujeta a
x1 + x2 ≤ 6 x1 + x2 + S1 = 6
-x1 + x2 ≤ 4 -x1 + x2 + S2 = 4
x1, x2 ≥ 0

x1 + x2 = 6 x1 + x2 = 6
x1 =0 x2 = 0
x2 = 6 x1 = 6
A(0,6) B(6,0)
-x1 + x2 = 4 -x1 + x2 = 4
x1 =0 x2 = 0
x2 = 4 -x1 = 4
x1 = -4
C(0, 4) D(-4,0)
F(x2X
1,1x+
2)3X
=2-x1 + 3x2 F(0,0) = -0 + 3(0) = 0
2) = -0 + 3(4) = 12
F(0,4)
0 + 3(4)= 12
F(1,5) = -1 +3(5) = 14
3)
X1 + X2 = 6 F(6,0) = -6 + 3(0) = -6
X1 = 6 - X2
-X1 + X2 =4 I= 0
-6-X2+X2=4 X1+X2=6 II=12
2X2= 4+6 X1+5= 6
X2 = 10/2 X1= 6-5 III=14
X2 =5 X1= 1
IV=-6
-X1 + 3X2
-1+3(5)=14
4)
-6+3(0)=-6
Coeficiente Variable -1 3 0 0
Básico Básica X1 X2 S1 S2 Valor
0 S1 1 1 1 0 6
0 S2 -1 1 0 1 4
Zj 0 0 0 0
Cj - Zj -1 3 0 0 0

Coeficiente Variable -1 3 0 0
Básico Básica X1 X2 S1 S2 Valor
0 S1 2 0 1 -1 2
3 X2 -1 1 0 1 4
Zj -3 3 0 3
Cj - Zj 2 0 0 -3 12

Coeficiente Variable -1 3 0 0
Básico Básica X1 X2 S1 S2 Valor
-1 X1 1 0 0.5 -0.5 1
3 X2 0 1 0.5 0.5 5
Zj -1 3 1 2
Cj - Zj 0 0 -1 -2 14
4.- Maximizar
Z = 3x1 + 8x2
Sujeta a
x1 + 2x2 ≤ 8 x1 + 2x2 + S1 = 8
x1 + 6x2 ≤ 12 x1 + 6x2 + S2 = 12
x1, x2 ≥ 0
x1 + 2x2 = 8 x1 + 2x2 = 8
x1 = 0 x2 = 0
x2 = 8/2 x1 = 8
x2 = 4
A(0,4) B(8,0)
x1 + 6x2 = 12 x1 + 6x2 = 12
x1 = 0 x2 = 0
6x2 = 12 x1 = 12
x2 = 12/6= 2
C(0,2) D(12,0)
F(x3X
1,1x+
2)8X
=23x1 + 8x2 F(0,0) = 3(0) + 8(0) = 0
2) = 3(0) + 8(2) = 16
F(0,2)
3(0) + 8(2)= 16
F(6,1) = 3(6) +8(1) = 26
3)
X1 + 2X2 = 8 F(8,0) = 3(8) + 8(0) = 24
X1 = 8 - 2X2
X1 + 6X2 =12 I= 0
8-2X2+6X2 X1+2X2=8 II=16
=12 4X2= X1+2(1)= 8
12-8 X2= 8-2 III=26
X2 = 4/4 X2= 6
X2 =1 IV=24

3X1 + 8X2
3(6)+8(1)=26
4)
3(8)+8(0)=24
Coeficiente Variable 3 8 0 0
Básico Básica X1 X2 S1 S2 Valor
0 S1 1 2 1 0 8
0 S2 1 6 0 1 12
Zj 0 0 0 0
Cj - Zj 3 8 0 0 0

Coeficiente Variable 3 8 0 0
Básico Básica X1 X2 S1 S2 Valor
0 S1 0.666 0 1 -0.3333 4
0 S2 0.1666 1 0 0.1666 2
Zj 0 0 0 0 0
Cj - Zj 3 8 0 0 0

Coeficiente Variable 3 8 0 0
Básico Básica X1 X2 S1 S2 Valor
0 S1 0.666 0 1 -0.3333 4
8 X2 0.1666 1 0 0.1666 2
Zj 1.3328 8 0 1.3328
Cj - Zj 1.6672 0 0 -1.3328 16
 Coeficiente Variable 3 8 0 0
Básico Básica X1 X2 S1 S2 Valor
3 X1 1 0 1.5 -0.5 6
8 X2 0 1 -0.25 0.25 1
Zj 3 8 2.5 0.5
Cj - Zj 0 0 -2.5 -0.5 26

5.- Maximizar
Z = 8x1 + 2x2
Sujeta a
x1 - x2 ≤ 1 x1 - x2 + S1 = 1
x1 + 2x2 ≤ 8 x1 + 2x2 + S2 = 8
x1 + x2 ≤ 5 x1 + x2 + S3 = 5

x1, x2 ≥ 0
x1 - x2 = 1 x1 - x2 = 1
x1 = 0 x2 = 0
-x2 =1 x1 =1
X2 = -1
A(0,-1) B(1,0)
x1 + 2x2 = 8 x1 + 2x2 = 8
x1 = 0 x2 = 0
2x2 =8 x1 =8
x2 = 8/2
x2 = 4
C(0,4) D(8,0)
x1 + x2 = 5 x1 + x2 = 5
x1 = 0 x2 = 0
x2 =5 x1 =5
E(0,5) F(5,0)
 8X1 + 2X2 F(x1, x2) = 8x1 + 2x2
2) = 8(0) + 2(0) = 0
F(0,0)
8(0) + 2(4)= 8
F(0,4)
3) = 8(0) + 2(4) = 8
X1 + 2X2 = 8 F(2,3) = 8(2) + 2(3) = 22
X1 = 8 - 2X2
F(3,2) = 8(3) +2(2) = 28
8 - 2x2 + X2 =5 X1+2(3)=8
-X2 = 5-8 X1= 8-6 F(1,0) = 8(1) + 2(0) = 8
X2 = 3 X1=2
I= 0
8X1 + 2X2
8(2)+2(3)=22 II=8
III=22

4) X1= 1+X2 IV=28

5)
8X1+2X2
8(1)+2(0)=8
 Coeficiente Variable 8 2 0 0 0
Básico Básica X1 X2 S1 S2 S3 Valor
0 S1 1 -1 1 0 0 1
0 S2 1 2 0 1 0 8
0 S3 1 1 0 0 1 5
Zj 0 0 0 0 0
Cj - Zj 8 2 0 0 0 0

Coeficiente Variable 8 2 0 0 0
Básico Básica X1 X2 S1 S2 S3 Valor
8 X1 1 -1 1 0 0 1
0 S2 0 3 -1 1 0 7
0 S3 0 2 -1 0 1 4
Zj 8 -8 8 0 0
Cj - Zj 0 10 -8 0 0 8

Coeficiente Variable 8 2 0 0 0
Básico Básica X1 X2 S1 S2 S3 Valor
8 X1 1 0 0.5 0 0.5 3
0 S2 0 0 0.5 1 -1.5 1
2 X2 0 1 -0.5 0 0.5 2
Zj 8 2 3 0 5
Cj - Zj 0 0 -3 0 -5 28

6.- Maximizar
Z = 2x1 – 6x2
Sujeta a
x1 - x2 ≤ 4 x1 - x2 + S1 = 4
-x1 + x2 ≤ 4 -x1 + x2 + S2 = 4
x1 + x2 ≤ 6 x1 + x2 + S3 = 6
x1, x2 ≥ 0

x1 - x2 = 4 x1 - x2 = 4
x1 = 0 x2 = 0
-x2 =4 x1 =4
X2 = -4
A(0,-4) B(4,0)
-x1 + x2 = 4 -x1 + x2 = 4
x1 = 0 x2 = 0
x2 = 4 -x1 = 4
x1 = -4
C(0,4) D(-4,0)
x1 + x2 = 6 x1 + x2 = 6
x1 = 0 x2 = 0
x2 =6 x1 =6
E(0,6) F(6,0)
F(x2X
1,1x-26X
) =2=0
2x1 - 6x2 F(0,0) = 2(0) - 6(0) = 0
2) = 2(0) - 6(4) = -24
F(0,4)
2(0)-6(4) = -24
F(1,5) = 2(1) - 6(5) = -28
3)
-X1 + X2 = 4 F(5,1) = 2(5) - 6(1) = 4
X2 = 4+X1
F(4,0) = 2(4) - 6(0) = 8
x1 + X2 =6
X1+(4+x1)=6 (1)+x2=6 I= 0
2x1 = 6-4 X2=6-1
X1= 2/2 X2=5 II=-24
X2 = 1
III=-28
2X1-6X2 IV=4
2(1)-6(5)= -28
V=8
4)
X1-X2=4
X1=4+X2

X1+X2=6

X1+1=6
(4+X2)+X2=6 X1=6-1
2X2=6-4 X1=5
X2= 2/2
X2= 1
2X1-6X2
2(5)-6(1)= 4

5)
Z= 2X1-6X2
2(4-6(0)= 8
Coeficiente Variable 2 -6 0 0 0
Básico Básica X1 X2 S1 S2 S3 Valor
0 S1 1 -1 1 0 0 4
0 S2 -1 1 0 1 0 4
0 S3 1 1 0 0 1 6
Zj 0 0 0 0 0
Cj - Zj 2 -6 0 0 0 0
 Coeficiente Variable 2 -6 0 0 0
Básico Básica X1 X2 S1 S2 S3 Valor
2 X1 1 -1 1 0 0 4
0 S2 0 0 1 1 0 8
0 S3 0 2 -1 0 1 2
Zj 2 -2 2 0 0
Cj - Zj 0 -4 -2 0 0 8

7.- Maximizar
Z = 2x1 – x2 + x3
Sujeta a
2x1 + x2 – x3 ≤ 4 2x1 + x2 - x3 + S1 = 8
x1 + x2 + x3 ≤ 2 x1 + x2 + x3 + S2 =
2 x1, x2, x3 ≥ 0

Coeficiente Variable 2 -1 1 0 0
Básico Básica X1 X2 X3 S1 S2 Valor
0 S1 2 1 -1 1 0 4
0 S2 1 1 1 0 1 2
Zj 0 0 0 0 0
Cj - Zj 2 -1 1 0 0 0

Coeficiente Variable 2 -1 1 0 0
Básico Básica X1 X2 X3 S1 S2 Valor
0 S1 0 -1 -3 1 -2 0
2 X1 1 1 1 0 1 2
Zj 2 2 2 0 2
Cj - Zj 0 -3 -1 0 -2 4