I am having difficulty understanding how you represent a specific point and write it as a convex combination of the linear programming bfs.
According to an example found in operations research textbook, it gives an example where you represent a point $G=$ $(20,10)$ is within the feasible region of four extreme points $(0,40)$ , $(0,0)$ , $(30,0)$ , and $(20,20)$.
Given the contraints:
$max$ $z$ = $4$ $x_3$ + $3$$x_2$
$st.$ $x_1$ + $x_2$ $ \leq 40$
$2x_1$ + $x_2$ $ \leq 60$
$x_1$ , $x_2$ $ \geq 0$
From this I can see that the feasible region is bounded, and that there is some optimal solution. I also know that point $G$ is a combination of $F$ and $H$, and a combination of $E$ and $C$. Therefore point $G$ is convex combination of $FEC$. However, i don't know how they got this result where:
point $G$ can be written as $1/6F + 5/6H $ , where $H= (24, 12)$ (Comes from intersection between line segments $FH$ and $EC$). They say that $H$ may be written as $.6E + .4C $. By putting these two relationships together, they write point $G$ as $1/6F + 5/6 (.6E +.4C) = 1/6F + 1/2E + 1/3C $
Based on the functions they've written I can't see where are these fractions coming from?
Best Answer
We have $F=(0,0)$ and $H$ is the intersection of the line between $FG$ and $EC$. We can solve for the intersection and find out that $H=(24,12).$
and we can see that $$G=(20,10)=\frac56(24,12)=\frac16(0,0)+\frac56(24,12)=\frac16F+\frac56H$$
Now, we just have to figure out where is $H$ located along the line $EC$.
Here $E=(20,20)$ and $C=(30,0).$
From the $y$ coordinate, $12=\frac35 \cdot 20$
We have $$H=\frac35E+\frac25$$