-
PROGRAMACIN LINEAL MTODO DUAL (20/05/2015)
MAX
Z=5X1+6X2
s.a
X1+9X260
2X1+3X245
5X1-2X220
X230
X1,X20
1.- X1+9X2=60
2.- 2X1+3X2=45
3.- 5X1-2X2=20
X1 X2
0 -10
4 0
X1 X2
0 6,67
60 0
X1 X2
0 15
22,5 0
P(0,0)
060 Verdadero
P(0,0)
045 Verdadero
P(0,0)
020 Verdadero
-
-5X1-45X2=-300
5X1-2X2=20
-47X2=280
X2=5.96
X1+9(5.96)=60
60-53.64=6.36
X1=6.36
X2=5,96
Z=67,56
.- 6,36+9(5,96)+H160
H10
.- 2(6,36)+6(5,96)+H245
s.o.
Z=67,56
X1=6,36
X2=5,96
H1=0
H2=14
H3=0
H4=24
-
H214
.- 5(6,36)-2(5,96)+H320
H30
.- H424
PROBLEMA DUAL
Min
Z=60Y1+45Y2+20Y3+30Y4
Limitado
1.- Y1+2Y2+5Y35
2.- 9Y1+3Y2-2Y2+Y46
Y1, Y2, Y3, Y40
Y1
Y2=0
Y3
Y4=0
Y1+5Y3=5
9Y1-2Y3=6
-9y1-45y3=-45
9y1-2y3=6
+47Y3=+39
Y3= 39
47
Y1+5(39
47)=5
Y1=235195
47 =
40
47
-
Z=60 (40
47)+20(
39
47)
Z= 3180
47
Z=67,66
MTODO DE GAUSS-JORDN
EJERCICIO 1
2x1+3X2+4X3=5
2x1+X2-5X3= 6
3x1-2X2+4X3=1
1 + 0 - 19/10 = 13/10
0 + 1 + 13/5 = 4/5
0 + 0 149/10 =47/10
1 - 3/2 + 2= 5/2
0 5 13 = -4
0- 13/2 -2 = -1/2
1 + 0 + 0 = 283/149
0 + 1 + 0 = -3/149
0 + 0 + 1 = 47/149
-4 -6 -8 = -10
4 +1 -5 = 6
0 - 3/2 -39/2 =-6/5
1 +3/2 +2 = 5 /2
-3 -9/2 -6 = -15/2
3 -2 4 7
0 0 -13/5 =-611/745
0 1 13/5 = 4/5
0 - 13/2 -169/10 =26/5
1 +13/2 +2 =-1 /2
0 0 19/10 = +893/1490
1 0 -19/10 = 13/10
-
COMPROBACIN
2x1+3X2+4X3=5
2(283/149)+3(-
3/149)+4(47/149)
5 = 5
X= 283/149
Y= -3/149
Z= 47/149
EJERCICIO 2
2x+5y-2z= 10
5x-2y+3z= 7
4x+6y+8z=1
1 + 0 + 11/29 = 55/29
0 + 1 + 16/29 = 36/29
0 + 0 284/29 =-407/29
1 + 5/2 - 1= 5
0 29/2 8 = -18
0 - 4 12 = -19
1 + 0 + 0 = 693/149
0 + 1 + 0 = 32/71
0 + 0 + 1 = -407/284
-5 -25/2 5 = -25
5 -2 3 = 7
0 - 5/2 40/29 =-20/29
1 5/2 -1 = 5
-4 -10 4 = -20
4 6 8 = 1
0 0 -11/29 = 4477/8236
1 0 11/29 = 55/29
0 4 -64/29 =144/29
1 -4 12 = -19
0 0 16/29 = -1628/2059
1 0 -16/29 = 36/29
-
COMPROBACIN
2x+5y-2z= 10
2(693/284)+3(32/71)+4(-
407/284)
10 =10
X= 693/284
Y= 32/71
Z=-407/284
METODO SIMPLEX (12/06/2015)
EJERCICIO N.- 1
MAX.
Z= 2X1 + X2
S.A
3X1 + X2 6
X1 X2 2
X2 3
Xi 0
FORMA ESTNDAR
Z= 2X + X2 + 0H1 + 0H2 + OH3
S.A 3X1 + X2 + H1 = 6
X1 + X2 +H2 = 2
X2 +H3 = 3
Xi 0 ; Hj 0
VARIABLES
BSICAS
VARIABLES VALOR
Z X1 X2 H1 H2 H3
Z 1 -2 -1 0 0 0 0
H1 0 3 1 1 0 0 6
H2 0 1 -1 0 1 0 2
H3 0 0 1 0 0 1 3
ECUACIN
Z 2X1 X2 - 0H1 0H2 OH3 = 0
S.A 3X1 + X2 + H1 = 6
X1 + X2 + H2 = 12
X2 +H3 =3
Xi 0 ; Hj 0
-
VALOR PTIMO SOLUCIN PTIMA
Z= 5 X1= 1
X2= 3
H1= 0
H2= 4
H3= 0
Z 1 0 - 1/3 2/3 0 0 4
X1 0 1 1/3 1/3 0 0 2
H2 0 0 -4/3 - 1/3 1 0 0
H3 0 0 1 0 0 1 3
Z 1 0 0 2/3 0 1/3 5
X2 0 0 1 0 0 1 3
X1 0 0 0 1/3 0 - 1/3 1
H2 0 0 0 - 1/3 1 4/3 4
-
EJERCICIO N.-2
MAX.
Z= 400A + 300B
S.A
2A + B 60
A + 3B 40
A + B 30
A, B 0
FORMA ESTNDAR
Z=400A + 300B + OH1 + 0H2 + 0H3
S.A 2A + B + 0H1 60 A + 3B + 0H2 40
A + B + 0H3 30
A, B 0
Z A B H1 H2 A1 VALOR
Z 1 - 400 -300 0 0 0 0
H1 0 2 1 1 0 0 60
H2 0 1 3 0 1 0 40
H3 0 1 1 0 0 1 30
Z 1 0 100 0 0 400 1200
H1 0 0 -1 1 0 -2 0
H2 0 0 2 0 1 -1 10
A 0 1 1 0 0 1 30
ECUACIN
Z- 400A - 300B - OH1 - 0H2 - 0H3 = 0
S.A 2A + B + 0H1 = 60
A + 3B + 0H2 = 40
A + B + 0H3 = 30
A, B 0
-
VALOR PTIMO SOLUCIN PTIMA
Z= 12000 A = 30
B = 0
H1= 0
H2= 10
H3= 0
PROBLEMA DUAL.
Z= 60Y1+ 40Y2+ 30Y3
S.A 2Y1 + Y2 + Y3 400
Y1 + 3Y2 + Y3 300
Y1 0
(16/06/2015)
EJERCICIO N.-1
MINIMIZAR
Z = 3/2X + 2Y
S.A
2X + 2Y 8 2X + 6Y 12 X, Y 0
2Y1 + Y3 = 400
-Y1 - Y3 = -300
Y1 = 100
Y1 = 100
Y3 = 200
Z = 12000
ECUACIN
Z 3/2X 2Y - 0H1 0H2 MA1 = 0 S.A 2X + 2Y + H1 = 8
2X + 6Y + A1 H2 = 12
X, Y, H1, H2, A1 0
-
FORMA ESTNDAR
Z= 3/2X + 2Y + 0H1 + 0H2 + MA1
S.A 2X + 2Y + H1 = 8
2X + 6Y +A1-H2 = 12
X, Y, H1, H2, A1 0
Z 3/2X 2Y OH1 OH2 MA1 = 0 2XM + 6MY -MH2 + MA1 = 12M
Z 3/2X+2XM2Y+6MYOH1 MH2 = 12M (-1)
-Z + 3/2X-2XM +2Y-6MY +OH1 +MH2 = -12M
-Z +X (3/2-2M) +Y (2-6M) +OH1 +MH2 = -12M
2X + 2Y + H1 = 8
2X + 6Y + A1 H2 = 12 X, Y, H1, H2, A1 0
Z X Y H1 H2 A1 VALOR
Z -1 3/2-4M 2-6M 0 M 0 -12M
H1 0 2 2 1 0 0 8
A1 0 2 6 0 -1 1 12
Z -1 5/6 0 0 1/3 -1/3 +M -6
X 0 4/3 0 1 1/3 - 1/3 4
H1 0 1/3 1 0 -1/6 1/6 2
VALOR PTIMO SOLUCIN PTIMA
Z= - 4 Z = 4 X = 0
Y = 2
H1= 4
-
H2= 0
EJERCICIO N.-2
MINIMIZAR
Z = 4X + 5Y
S.A
2X + 2Y 10 2X + 6Y 18 X + Y = 7
X, Y 0
FORMA ESTNDAR
Z=4X + 5Y + 0H1 + 0H2 + MA1 + MA2
S.A 2X + 2Y + H1 = 10
2X + 6Y +A1-H2 = 18
X + Y + A2 = 7
X, Y, H1, H2, A1, A2 0
ECUACIN
Z 4X 5Y OH1 OH2 MA1 MA2 = 0 2XM + 6MY -MH2 + MA1 = 18M
XM + YM + MA2 = 7M
Z 4X + 3XM 5Y + 7MY OH1 MH2 = 25M (-1)
-Z + 4X - 3XM + 5Y - 7MY + OH1 + MH2= -25M
-Z +X (4 - 3M) +Y (5 - 7M) +OH1 +MH2 = -25M
2X + 2Y + H1 = 10
2X + 6Y +A1-H2 = 18
X + Y + A2 = 7
X, Y, H1, H2, A1, A2 0
Z X Y H1 H2 A1 A2 VALOR
Z -1 4-3M 5-7M 0 M 0 0 -25M
H1 0 2 2 1 0 0 0 10
-
A1 0 2 6 0 -1 1 0 18
A2 0 -1 1 0 0 0 1 7
Z -1 7/3-2/3M 0 0 5/6-1/6M -5/6+6/7M 0 -15-4M
H1 0 4/3 0 1 1/3 -1/3 0 4
Y 0 1/3 1 0 -1/6 1/6 0 3
A2 0 2/3 0 0 1/6 -1/6 1 4
Z -1 0 0 -7/4+1/2M 1/4 -1/4+M 0 -22-2M
X 0 1 0 3/4 1/4 -1/4 0 3
Y 0 0 1 -1/4 -1/4 1/4 0 2
A2 0 0 0 -1/2 0 0 1 2
VALOR PTIMO SOLUCIN PTIMA
-Z= -22-2M Z= 22 + 2M X= 3
Y= 2
H1= 0
H2= 4
A1= 0
A2= 2
NO TIENE SOLUCIN
-
MTODO ALGEBRAICO
Maximizar
F(x,y)=3x+2y
RESTRICCIONES
(1) 2x+y 10 (2) x+y 7
RESTRICCIONES DE NO NEGATIVIDAD
(3) x, y0
Despeje de x
+ = 7
= 7
= 7
=
Despeje de y
2(7 ) + = 10
14 2 + = 10
= 4
=
MTODO GRFICO
SISTEMAS ECUACIONES
(1) (2)
2x+y= 10 x+y=7
x y x y
-
COMPROBACIN
P(0,0) P(0,0)
(1) (2)
2(0)+1(0)10 1(0)+1(0)7
010 07
VERDAD VERDAD
GRFICO
0 10 0 7
5 0 7 0
-
ARCO CONVEXO
Punto X Y Z
A 0 7 14
B 3 4 17
C 5 0 15
MTODO SIMPLEX
F.O
Z= 3X+2Y
S.A
2X+Y10
X+Y7
NO NEGATIVIDAD
X,Y0
FORMA ESTANDAR
Z=3X+2Y+0H1+0H2
2X+Y+ H1 =10
X+Y - H2 =7
X,Y, H1, H2, A10
FORMA ECUACIN
Z-3X-2Y-0H1-0H2 =0
2X+Y+ H1+0=10
X + Y+0- H2=7
X,Y, H1, H2, A10
-
Z X Y H1 H2 VALOR
Z 1 -3 -2 0 0 0
H1 0 2 1 1 0 10
A1 0 4 1 0 1 7
0 -1 1
2
1
2 0 -5
0 1 1 0 1 7
0 3 3
2
3
2 0 15
1 -3 -2 0 0 0
0 0 1
2
1
2 -1 -2
0 1 1
2
1
2 0 5
0 0 1
2
1
2 1 2
Z X Y H1 H2 VALOR
Z 1 1 1
2
2
3 0 15
H1 0 1 1
2
1
2 0 5
X 0 0 1
2
1
2 1 2
Z X Y H1 H2 VALOR
Z 1 0 0 1 1 17
H1 0 1 0 1 -1 3
X 0 0 1 -1 2 4
-
0 0 1
2
3
2 0 15
METODO DUAL
MIN
f(X,Y)=10Y1+7Y2
S.A
2Y1+Y 2 3
Y1+Y 2 2
2Y1+Y2=3
- Y1-Y 2 =-3
Y1=1
Y2=1
COMPROBACIN
Z=10(1)+7(1)
Z=17
S.O
F(X,Y)=17
S.A
X=3
Y=4
H1=0
H2=0
