# Thread: Solve the linear programming problem

1. ## Solve the linear programming problem

Assume $\displaystyle x >= 0$ and $\displaystyle y >= 0$

Minimize $\displaystyle C=4x+2y$ with the constraints.

$\displaystyle { x+y>=7, 4x+3y>=4, x<=10, y<=10 }$

So, the minimum is 10, now what ! lol

2. Originally Posted by RBlax
Assume $\displaystyle x >= 0$ and $\displaystyle y >= 0$

Minimize $\displaystyle C=4x+2y$ with the constraints.

$\displaystyle { x+y>=7, 4x+3y>=4, x<=10, y<=10 }$

So, the minimum is 10, now what ! lol
1. Re-write the equations (if possible) such that the y-variable is at the LHS of the inequality:

$\displaystyle x\geq0~\wedge~y\geq0$

$\displaystyle y\geq -x+7~\wedge~y\geq-\frac43 x + \frac43~\wedge~x\leq10~\wedge~y\leq10$

2. Draw the corresponding lines which are the borders of the feasible region.

3. The line $\displaystyle y=-2x+\frac12 C$ must contain at least one point of the feasible region and the value of C must be a minimum.

4. According to the sketch you see that

$\displaystyle 7=\frac12 C ~\implies~C = 14$

5. To get an exact result calculate the coordinates of the vertices of the feasible region and check which vertex produces the minimum value of C.