A friend of mine is working on this problem for a course. I will post the problem in full and tell you what my/our thoughts are on it.
The problem then asks for a model that can be used to develop a daily production schedule for each plant as well as the optimal solution.
Here is the rest of the problem in the book. I hope it's legible.
I am familiar with problems in which there are two variables which I can then graph and produce a feasibility region. I can then test the vertices of the region against the optimization equation.
$C_1$: the total produced of Component 1. $C_2$: the total produced of Component 2.
$$C_1 = C_2$$
$B_1$: the total of Component 1 produced in Buffalo.
$B_2$: the total of Component 2 produced in Buffalo.
$D_1$: the total of Component 1 produced in Dayton.
$D_2$: the total of Component 2 produced in Dayton.
$$C_1=B_1+D_1=B_2+D_2=C_2$$
$$B_1\le 2,000$$ $$B_2\le 1,000$$ $$D_1\le 600$$ $$D_2\le 1,400$$ $$C_1\le 2,600$$ $$C_2\le 2,400$$
In the example given, $60\%$ of Buffalo's production time is used on component one. I could express this as $R_{B1} = 0.60$, signifying the amount of resources Buffalo puts into Component 1. Then:
$$R_{B1}+R_{B2}=R_{D1}+{R_D2}=1$$
Furthermore:
$$2,000R_{B1}+1,000R_{B2}=B_1+B_2\le 3,000$$ $$600R_{D1}+1,400R_{D2}=D_1+D_2\le 2,000$$
And
$$2,000R_{B1}+600R_{D1}=1,000R_{B2}+1,400R_{D2}$$ $$2,000R_{B1}+600R_{D1}=1,000(1-R_{B1})+1,400(1-R_{D1})$$
It's also noted that we only need to maximize one plant.
EDIT: I said that, but it wasn't a part of the problem. I'm doubting now that it's so.
If this is the optimization equation...
$$C_1=2,000R_{B1}+600R_{D1}=C_2=1,000R_{B2}+1,400R_{D2}$$
... can I rewrite everything in terms of $R_{B1}$ and $R_{D1}$, and graph those as variables? How can I go about this?
EDIT:
Another way of looking at it is this. Any thoughts?
I'm thinking that the factories won't operate at 100% efficiency. If they did, then $C_1$ would not equal $C_2$ and that would be inefficient. (I'm reminded of The Goal by Elyahu Goldratt.)
So, do we need to test the vertices of that small triangle? How?
EDIT (again): I'm inclined to think that we find $C_1$ and $C_2$ by taking the intersection between the $C_1 = C_2$ line and the Dayton line. Then we could feed that results back in to the equations
$$C_1=2,000R_{B1}+600R_{D1}=C_2=1,000R_{B2}+1,400R_{D2}$$
to solve for $R_{B1}$ and its kin. Am I on the right track?