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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.

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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.

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