Problem: A Company can use 3 different procedures to produce a product, for the production of every product are necessary 3 machines as below:
The numbers relate the hours necessary. every machine is avaible for 50 hours. The profict of the product depends of the procedure that has been used:
With:
Proc 1=7
Proc 2=9
Proc 3=5
this is how I will proceed:
MAXIMIZE profit
$Xi$= product with procedure i=1,2,3;
max $7x_1+9x_2+5x_3$ subject to:
2$ x_1+ x_2+3 x_3 ≤50$
4 $x_1+2 x_2+3 x_3 ≤50$
3 $x_1+4 x_2+2 x_3 ≤50$
$x_1,x_2,x_3≥0$
MINIMIZE hours of usage of machine 2 with the obligation than profit must be at least 100
$X_i,_j$ where i=machine and j=proc
min $X_2,_1+X_2,_2+X_2,_3$
subject to
$2X_1,_1+X_1,_2+3X_1,_3≤50$
$4X_2,_1+2X_2,_2+3X_3,_3≤50$
$3X_3,_1+4X_3,_2+2X_3,_3≤50$
$7(2X_1,_1+4X_2,_1+3X_3,_1)+9(2X_1,_2+2X_2,_2+4X_3,_2)+ 5(3X_1,_3+4X_2,_3+2X_3,_3)≥100$
Is my doing correct for the resolution of the problem?