Modelo de P.L en Forma de Ecuación

Modelo de P.L en Forma de Ecuación 

Método Simplex

Ejercicio 1:

Función Objetivo.-

Minimizar Z =  5X1 - 4X2 + 6X3 - 8X4  + 0S1  + 0S2  + 0S3 , Z = 0

Sujeto a:

          X1 + 2X2 + 2X3 +  4X4 +  S1 <= 40

         2X1 - X2 + X3 + 2X4 + S2  <= 8

         4X1 - 2X2 + X3 - X4 + S3 <= 10

             X1 , X2 , X3 , X4  >= 0

-Colocamos los coeficientes en la tabla:

-Encontrando la fila que va a salir:

-Cálculos nueva Fila
Punto pivote: X4
*Nueva fila pivote = Fila pivote actual / elemento pivote
0.0/2 = 0.0
2.0/2 = 1.0
-1.0/2 = -1/2 =-0.5
1.0/2 =1/2 = 0.5
2.0/2 = 1.0
0.0/2 = 0.0
1.0/2 = 1/2 = 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 ) =-5/2 = -2.5
1 - (-1 * 0.5 ) = 3/2 =1.5
-1 - (-1 * 1.0 ) = 0.0
0 - (-1 * 0.0 ) = 0.0
0 - (-1 * 0.5 ) = 1/2 = 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 : X2
*Nueva fila pivote = Fila pivote actual / elemento pivote
0.0/4 = 0.0
-3.0/4 = -3/4
4.0/4 = 1.0
0.0/4 = 0.0
0.0/4 = 0.0
1.0/4 = 1/4
-2.0/4 = -1/2
0.0/4 = 0.0
24.0/4 = 6.0

Cálculos Z
*Coeficiente: 8

1 - (8 * 0.0 ) = 1.0

-13 - (8 * - 3/4 ) = -7.0

8 - (8 * 1.0 ) = 0.0

-10 - (8 * 0.0 ) = -10.0

0 - (8 * 0.0 ) = 0.0

0 - (8 * 1/4) = -2.0

-4 - (8 * -1/2) = 0.0

0 - (8 * 0.0 ) = 0.0

-32 - (8 * 6.0 ) = -80.0

Cálculos X4
*Coeficiente: -1/2 

0 - (-1/2 * 0.0 ) = 0.0

1 - (-1/2 * -3/4 ) = 5/8

-1/2 - (-1/2 * 1.0 ) = 0.0

1/2 - (-1/2 * 0.0 ) = 1/2

1 - (-1/2 * 0.0 ) = 1.0

0 - (-1/2 * 0.25 ) = 1/8

1/2 - (-1/2 * -1/2 ) = 1/4

0 - (-1/2 * 0.0 ) = 0.0

4 - (-1/2 * 6.0 ) = 7.0

Cálculos S3
*Coeficiente -5/2
 0 - (-5/2 * 0.0 ) = 0.0

5 - (-5/2 * -3/4 ) = 25/8

-5/2 - (-5/2 * 1.0 ) = 0.0

3/2 - (-5/2 * 0.0 ) = 3/2

0 - (-5/2 * 0.0 ) = 0.0

0 - (-5/2 * 1/4) = 5/8

1/2 - (-5/2 * -1/2 ) = -3/4

1 - (-5/2 * 0.0 ) = 1.0

14 - (-5/2 * 6.0 ) = 29.0

-Cuadro Resultante:

Respuesta:
Z = -80
X2 = 6
X4 = 7
S3 = 29