# Linear Programming Price Change After X Units Sold

I have a question regarding writing some formulas for LP. How would you code the price change after X number of units sold.

So lest say the base price for the first X units sold is £5 and then there is a £1 discount for units bought after the X unit.

The answer is different depending on if you are selling or buying. First, look at the case where you are selling products and want to maximize revenue.

Let

• $$w$$: the number of units sold
• $$v$$: the number units sold in excess of $$X$$
• $$r$$: the total revenue

Your constraints are then $$5 w - v = r \\ w - v \le X$$

The first constraint defines revenue as 5 times the total sold, less the number sold at a discount. The second constraint forces the discount on quantities above the threshold $$X$$. These constraints don't explicitly disallow a discount on items sold below the threshold, but If you are trying to maximize revenue there is no need. However if revenue is a cost, then you need additional constraints and a Boolean variable to indicate that you are eligible for the discount. To add this, you need the following additional variable.

• $$d$$: indicator that more than $$X$$ items have been purchased.

and the following model. $$5 w - v = r \\ v \le w - X d \\ v \le M d \\ d \in \{0, 1\}$$

Where $$M$$ is an a priori upper bound on the number of items sold at a discount.