So i have this LP problem:
And i need to find the optimal solutions of x1 and x2.
On the answer sheet it states that x1=a and x2=0.
Now i was thinking of doing it this way:
x1+ax2= a
then
x2=1-(x1/a)
The only way i could solve this problem as to get the same results from the answer sheet would be by logic. Then if x1=a
a/a would be equal to 1 and
x2=1-(x1/a)
would become x2=1-1=0.
which indeed satisfies the answer sheet.
However if x2=0 then it is not the same as x2=1 on the second constraint.
Then i was thinking of doing it another way :
from the second constraint i know that x2=1, then i substitute it into the first constraint to get that x1=0.
Then the second constraint would be satisfied it x2=1 and not 0.
Is this method right? WHat a i doing wrong? Is there any other method ? Thanks!
In some way you´re right. But I´ve taken into account that the constraints are inequalities rather than equations.
From the second constraint we know that $x_2-1\leq 0$. Next I´ve transformed the first constraint: $x_1+ax_2\leq a\Rightarrow x_1+a(x_2-1)\leq0$. We know that $x_1\geq 0$. Thus $a(x_2-1)$ must be $\leq 0$. Since $(x_2-1)\leq 0$ the factor $a$ must be $\geq 0$.
To maximize $x_1$ we look at the first constraint $x_1+ax_2\leq a$. We maximize $x_1$ if we minimize $ax_2$. For that purpose we set $x_2=0$.