integer programming with indicators

Question

I have the following question, and I need to write it as an integer programming problem:
A manager of a company wants to give presents to his 1000 employers. He can buy the presents from two different suppliers:

Every present costs 110$. For any present above 500, the manager would have to pay 5$ extra for insurance (for example, for 502 presents, there will be 10$ extra for insurance).
Every present costs 120$. For any present above 300, there is a discount of 30% (each present above 300 costs 84$).

I need to find how many presents he should buy from each supplier, in order to spend the minimal amount of money.
I defined:
$x_1, x_2$ - The number of presents to buy from each supplier.
$z_1$ - Indicator which represents whether or not $x_1 ge501$
$z_2$ - Indicator which represents whether or not $x_2 ge301$
I know how to represents the indicators using linear functions, but I don't know how to find a linear objective function.
My best suggestion is:
$$ 110x_1+5z_1(x_1-500)+120x_2-36z_2(x_2-300) $$
However, this is not a linear function, because I multiply the decision variables.
How can I turn it into a linear function?

RobPratt · Answer

Introduce nonnegative integer decision variables $y_1$ and $y_2$ to represent "extra" presents.  The problem is to minimize
$$110x_1+5y_1+120x_2-36y_2$$
subject to
begin{align}
x_1 &le 500 + y_1 tag1 \
y_2 &le (1000-300) z_2 tag2 \
x_2 &ge 300 z_2 tag3
end{align}
Constraint $(1)$ and the objective enforce $y_1 = max{x_1-500, 0}$.  You don't need $z_1$.
Constraint $(2)$ enforces $y_2 > 0 implies z_2 = 1$.
Constraint $(3)$ enforces $z_2 = 1 implies x_2 ge 300$.

integer programming with indicators

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