INVESTIGACION DE OPERACIONES I 1.− Considere el problema de programación lineal:

INVESTIGACION DE OPERACIONES I
1.− Considere el problema de programación lineal:
Máx. P =
Sujeta a:
3X +
2X +
3X +
X+
X"0
2Y
2Y
2Y
½Y
Y"0
"8
" 12
"3
1
MAX P: 3X+2Y
SUJETA A: 2X+2Y " 8
3X+2Y " 12
X+1/2Y" 3
X"0
Y"0
2X+2Y = 8
3X+2Y = 12
X+1/2Y = 3
X=0
Y= 0
2X+2Y = 8
2(0)+ 2Y = 8
2Y = 8
Y = 8/2 = 4
2X+2Y = 8
2X+2(0) = 8
2X = 8
X = 8/2 PUNTOS(4,4)
1
3X+2Y = 12
3(0)+2Y = 12
2Y = 12
Y= 12/2 = 6
3X+2Y = 12
3X+2(0) = 12
3X = 12
X = 12/3 = 4 PUNTOS(X4,Y6)
X+1/2Y = 3
0+1/2Y = 3
Y = 3 / 1/2 = 6
X+1/2(0) = 3
X = 3 PUNTOS (3,6)
2X+2Y = 8
−2 (X+1/2Y = 7)
2X−2Y = 6
−2X−Y = 6
Y=2
2X+2(2) = 8
2X = 8 − 4
X = 4/2
X=2
SUSTITUCIÓN DE P
3X +
2Y
3(2) + 2(2)
6+4 = 10
2
6
(4,4)
(4,6)
(3,6)
Punto de
intersección
(2,2)
5
4
3
2
1
1
2
3
4
5
2.− Considere el problema de programación lineal:
Min. P =
Sujeta a:
2X +
4X +
2X +
X+
X
Y
5Y
Y
Y
3Y
"0
"0
" 40
" 30
" 30
MIN P: 2X+5Y
SUJETA A: 4X+Y " 40
2X+Y " 30
X+3Y" 30
X"0
Y"0
4X+Y = 40
2X+Y = 30
X + 3Y = 30
4X+Y = 40
4(0)+Y = 40
Y = 40
3
4X+Y = 40
4X+(0) = 40
4X = 40
X = 40/4 = 10 PUNTOS (10,40)
2X+Y = 30
2(0)+Y = 30
Y = 30
2X+Y = 30
2X+(0) = 30
2X = 30
X = 30/2 = 15 PUNTOS (15,30)
X+3Y = 30
(0)+3Y = 30
3Y = 30
Y = 30
Y = 30/3 = 10
X+3(0) = 30
X = 30 PUNTOS (30,10)
4X+Y = 40
−4 ( X+3Y = 30)
4X+Y = 40
− 4X−12Y = −120
0−11Y = 80
Y = −80/−11 = 7.2
4X+(7.2) = 40
4X = 40 − 7.2
4
X = 32.8 / 4 = 8.2
SUSTITUCIÓN DE P
Min. P =
2X +
5Y
2(8.2)+5(7.2)
52.4
40
30
25
20
15
10
10
15
20
25
30
35
40
3 Considere el problema de programación lineal:
Min. P =
Sujeta a:
10X +
X+
3X +
−2X +
X
Y
15Y
Y
Y
3Y
"0
"0
" 10
" 12
"3
MIN P: 10X + 15Y
SUJETA A: X+Y " 10
3X+Y " 12
−2X + 3Y " 3
X"0
Y=0
X+Y = 10
3X +Y = 12
−2X+3Y = 3
5
X=0
Y=0
X+Y=10
O+Y = 10
Y = 10
X+Y = 10
X+0 = 10
X = 10 PUNTOS (10,10)
3X+Y = 12
3(0)+Y = 12
Y = 12
3X+Y = 12
3X+(0) = 12
3X = 12
X = 12/3 = 4 PUNTOS (4,12)
−2X+3Y = 3
−2(0)+3Y = 3
Y = 3/3 = 1
−2X+3Y = 3
−2X + 3(0) = 3
−2X = 3
X = 3/−2 = −1.5 PUNTOS (−1.5, 1)
(−3)X+Y = 10
−2X+3Y = 3
−3X−3Y = −30
−2X+3Y = 3
6
−5 = −27
X = −27/−5 = 5.4
5.4+Y = 10
5.4 = 10
Y =10−5.4
4.6
Min. P =
10X +
15Y
SUSTITUCIÓN DE P
10(5.4)+15(4.6)
123
11
(10,
10)
7