ED-problem solving by graph

Question

I got the ED/LP-problem: $$\min\ \ \ 2p_1+4p_2$$ $$Ubb\ \ \ p_1\leq 3$$ $$\ \ \ p_2\leq 4$$ $$p_1+p_2=5$$ $$p_1+p_2 \geq 0$$ I have to solve by graph. Can I solve it by this minimum-problem or do I have tor rewrite it to a max-problem? I have tried to draw a graph with the solution set, but how can I find the minimum (or maximum if I have to rewrite it to a max-problem)? See the problem here: Linear-programming, object function

Answer 1

I´ve solved the third contsraints (equality) and $z=2p_1+4p_2$ for $p_2$. Then you draw the following graphs.

$$p_2=5-p_1, \qquad p_2=\frac{z}{4}-\frac{1}{2}p_1, \qquad p_2\leq 4,$$

and the vertical line $p_1\leq 3$.

Without the equality-constraint the feasible region is the yellow one. Only the first quadrant has been regarded due $p_1,p_2\geq 0$.

With the equality constraint the feasible region reduces to the purple line.

The blue lines are $p_2=\frac{z}{4}-\frac{1}{2}p_1$ for different value of z. It starts at $z=0$ and goes up to $z=14$. The blue line touches the purple line first at $(p_1^*,p_2^*)=(3,2)$.

The graph $p_2=\frac{z}{4}-\frac{1}{2}p_1$ has been plotted for $z=0,2,4,...,14$