METODO SIMPLEX : (MINIMIZAR)
http://uapjosemedinaj.blogspot.com/2014/10/metodo-simplex-minimizar.html
Considere la siguiente P.L:
Minimizar Z = 5X1-4X2+6X3+8X4
Sujeto a:
X1 + 2X2 + 2X3 + 4X4 <= 40 …(1)
2X1 - X2 + X3 + 2X4 <= 8 …(2)
4X1 - 2X2 + X3 - X4 <= 10 …(3)
X1; X2; X3; X4 >= 0
Nuevos Datos:
Z = 5X1 - 4X2 + 6X3 - 8X4 + 0S1 + 0S2 + 0S3
Z - 5X1 + 4X2 - 6X3 + 8X4 - 0S1 - 0S2 - 0S3 = 0
Z - 5X1 + 4X2 - 6X3 + 8X4 - 0S1 - 0S2 - 0S3 = 0
X1 + 2X2 + 2X3 + 4X4 + S1 = 40 …(1)
2X1 - X2 + X3 + 2X4 + S2 = 8 …(2)
4X1 - 2X2 + X3 - X4 + S3 = 10 …(3)
2X1 - X2 + X3 + 2X4 + S2 = 8 …(2)
4X1 - 2X2 + X3 - X4 + S3 = 10 …(3)
X1; X2; X3; X4 >= 0
S1; S2; S3 >= 0
Llenamos nuestra tabla con los coeficientes de Z y las restricciones.
Llenamos nuestra tabla con los coeficientes de Z y las restricciones.
Ahora obtendremos la fila de pivote:
Cálculos nueva Fila
Punto pivote: 2
Nueva fila pivote = Fila pivote actual / elemento pivote
0.0/2 = 0.0
2.0/2 = 1.0
-1.0/2 = -0.5
1.0/2 = 0.5
2.0/2 = 1.0
0.0/2 = 0.0
1.0/2 = 0.5
0.0/2 = 0.0
8.0/2 = 4.0
Cálculos Z
Coeficiente: 8
1 - (8 * 0.0 ) = 1.0
-5 - (8 * 1.0 ) = -13.0
4 - (8 * -0.5 ) = 8.0
-6 - (8 * 0.5 ) = -10.0
8 - (8 * 1.0 ) = 0.0
0 - (8 * 0.0 ) = 0.0
0 - (8 * 0.5 ) = -4.0
0 - (8 * 0.0 ) = 0.0
0 - (8 * 4.0 ) = -32.0
Cálculos S1
Coeficiente: 4
0 - (4 * 0.0 ) = 0.0
1 - (4 * 1.0 ) = -3.0
2 - (4 * -0.5 ) = 4.0
2 - (4 * 0.5 ) = 0.0
4 - (4 * 1.0 ) = 0.0
1 - (4 * 0.0 ) = 1.0
0 - (4 * 0.5 ) = -2.0
0 - (4 * 0.0 ) = 0.0
40 - (4 * 4.0 ) = 24.0
Cálculos S3
Coeficiente -1
0 - (-1 * 0.0 ) = 0.0
4 - (-1 * 1.0 ) = 5.0
-2 - (-1 * -0.5 ) = -2.5
1 - (-1 * 0.5 ) = 1.5
-1 - (-1 * 1.0 ) = 0.0
0 - (-1 * 0.0 ) = 0.0
0 - (-1 * 0.5 ) = 0.5
1 - (-1 * 0.0 ) = 1.0
10 - (-1 * 4.0 ) = 14.0
Nueva tabla con los valores obtenidos:
Punto pivote: 2
Nueva fila pivote = Fila pivote actual / elemento pivote
0.0/2 = 0.0
2.0/2 = 1.0
-1.0/2 = -0.5
1.0/2 = 0.5
2.0/2 = 1.0
0.0/2 = 0.0
1.0/2 = 0.5
0.0/2 = 0.0
8.0/2 = 4.0
Cálculos Z
Coeficiente: 8
1 - (8 * 0.0 ) = 1.0
-5 - (8 * 1.0 ) = -13.0
4 - (8 * -0.5 ) = 8.0
-6 - (8 * 0.5 ) = -10.0
8 - (8 * 1.0 ) = 0.0
0 - (8 * 0.0 ) = 0.0
0 - (8 * 0.5 ) = -4.0
0 - (8 * 0.0 ) = 0.0
0 - (8 * 4.0 ) = -32.0
Cálculos S1
Coeficiente: 4
0 - (4 * 0.0 ) = 0.0
1 - (4 * 1.0 ) = -3.0
2 - (4 * -0.5 ) = 4.0
2 - (4 * 0.5 ) = 0.0
4 - (4 * 1.0 ) = 0.0
1 - (4 * 0.0 ) = 1.0
0 - (4 * 0.5 ) = -2.0
0 - (4 * 0.0 ) = 0.0
40 - (4 * 4.0 ) = 24.0
Cálculos S3
Coeficiente -1
0 - (-1 * 0.0 ) = 0.0
4 - (-1 * 1.0 ) = 5.0
-2 - (-1 * -0.5 ) = -2.5
1 - (-1 * 0.5 ) = 1.5
-1 - (-1 * 1.0 ) = 0.0
0 - (-1 * 0.0 ) = 0.0
0 - (-1 * 0.5 ) = 0.5
1 - (-1 * 0.0 ) = 1.0
10 - (-1 * 4.0 ) = 14.0
Nueva tabla con los valores obtenidos:
Encontrando la fila que va a salir:
Cálculos nueva Fila
Punto pivote: 4
Nueva fila pivote = Fila pivote actual / elemento pivote
0.0/4 = 0.0
-3.0/4 = -0.75
4.0/4 = 1.0
0.0/4 = 0.0
0.0/4 = 0.0
1.0/4 = 0.25
-2.0/4 = -0.5
0.0/4 = 0.0
24.0/4 = 6.0
Punto pivote: 4
Nueva fila pivote = Fila pivote actual / elemento pivote
0.0/4 = 0.0
-3.0/4 = -0.75
4.0/4 = 1.0
0.0/4 = 0.0
0.0/4 = 0.0
1.0/4 = 0.25
-2.0/4 = -0.5
0.0/4 = 0.0
24.0/4 = 6.0
Cálculos Z
Coeficiente: 8
Coeficiente: 8
1 - (8 * 0.0) = 1.0
-13 - (8 * -0.75) = -7.0
8 - (8 * 1.0) = 0.0
-10 - (8 * 0.0) = -10.0
0 - (8 * 0.0) = 0.0
0 - (8 * 0.25) = -2.0
-4 - (8 * -0.5) = 0.0
0 - (8 * 0.0) = 0.0
-32 - (8 * 6.0) = -80.0
Cálculos S4
Coeficiente: -0.5
Coeficiente: -0.5
0 - (-0.5 * 0.0) = 0.0
1 - (-0.5 * -0.75) = 0.625
-0.5 - (-0.5 * 1.0) = 0.0
0.5 - (-0.5 * 0.0) = 0.5
1 - (-0.5 * 0.0) = 1.0
0 - (-0.5 * 0.25) = 0.125
0.5 - (-0.5 * -0.5) = 0.25
0 - (-0.5 * 0.0) = 0.0
4 - (-0.5 * 6.0) = 7.0
Cálculos S3
Coeficiente -2.5
Coeficiente -2.5
0 - (-2.5 * 0.0) = 0.0
5 - (-2.5 * -0.75) = 3.125
-2.5 - (-2.5 * 1.0) = 0.0
1.5 - (-2.5 * 0.0) = 1.5
0 - (-2.5 * 0.0) = 0.0
0 - (-2.5 * 0.25) = 0.625
0.5 - (-2.5 * -0.5) = -0.75
1 - (-2.5 * 0.0) = 1.0
14 - (-2.5 * 6.0) = 29.0
Respuesta:
Z = -80
X2 = 6
X4 = 7
S3 = 29
Z = -80
X2 = 6
X4 = 7
S3 = 29
