This problem is driving me mad, it's supposed to be really simple but I hate word problems such as this. I have the answer and can formulate other problems fine but for the life of me can't figure this out. I don't understand where the numbers in the solution are coming from. Can someone help me out with this?
It's supposed to be this:
I just don't understand where the constraints are coming from. The numbers make literally no sense. They can only produce 8000 and 7000 of each individually if the other is 0, yet the constraint is 56000? What?
The problem has an implicit assumption that production and packing of chocolates both take some fixed amount of effort based on what kind of chocolate it is, and that the two sections have a fixed amount of effort they can spend on that task.
So on that basis, if we say that the total production effort is $P$ and the effort it takes to produce almond and hazelnut chocolates are $p_1$ and $p_2$ respectively, then:
$$\begin{eqnarray}P & = & \frac{x_1}{p_1} + \frac{x_2}{p_2} & \mbox{total effort spent on production} \\ & = & \frac{8000}{p_1} + \frac{0}{p_2} & \mbox{producing only almond} \\ \implies p_1 & = & \frac{8000}{P} \\ P & = & \frac{0}{p_1} + \frac{7000}{p_2} & \mbox{producing only hazelnut} \\ \implies p_2 & = & \frac{7000}{P} \\ P & = & \frac{x_1}{8000/P} + \frac{x_2}{7000/P} \\ & = & P \left(\frac{x_1}{8000} + \frac{x_2}{7000} \right) \\ 1 & = & \frac{x_1}{8000} + \frac{x_2}{7000} \\ \therefore 7 x_1 + 8 x_2 & = & 56000 & \mbox{multiplying by 56000} \end{eqnarray}$$
That gives us equality, but this is assuming that all possible effort is expended, so it's an upper limit, hence the $\leq$ sign in the constraint. And you get a similar result for the packing effort.