Consider this example problem:
Suppose the cost for setting up a factory to generate a pencil is 1000 and to generate a pen is 2000. The profit for each pencil is 10 and the profit for each pen is 12. For each month, the factory can generate 100 items (pen or pencil). How can I include the initial cost in order to maximize the profit of the factory?
I can think of:
$$Maximize 10x_1 + 12x_2 - (1000b_1 + 2000b_2)$$ $$x_1+x_2 \leq 100$$
Where $b_1$ and $b_2$ are binary variables indicating whether to generate a pen or pencil. But I don't know how to relate them to $x_1$ and $x_2$, so that, for example, if $x_1 \ge 0$ then $b_1 =1$, otherwise $b_1=0$