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. 
 A: 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.
