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