# Retail Discount Design – Threshold Discount I

Setup

• Qualifying products. We will aggregate (discounted) prices of all qualifying products and compare the total (discounted) prices against threshold tier(s).
• Applying products. If qualified, we will apply the discount to the applying products. In many cases, they are the same as qualifying products, but they can be different.
• Qualifying threshold and discount method. In general, we use amount to define the threshold, and discount method can be either amount off or percentage off. It is also possible to have multiple threshold tiers.

One of key difference between threshold discounts and non-threshold discounts is that you can apply threshold discount only once per checkout cart. For example, say you see a threshold discount of \$10 off if you buy \$100+. If you have \$200+ in the cart that can be split into two groups, each of \$100+, we still give you \$10 total. In other words, we do not split the products to apply them more than once.

How to apply the threshold discount when it is the only discount for the checkout cart?

It is straightforward to figure out the discount and allocation, following the threshold discount definition.

Threshold discount competes with non-thresholds.

Threshold discount is not fit for knapsack problem in that it would drag down the performance of any knapsack problem because it can generate tons of basic discount applications. Take a checkout cart of 10 different products, each of quantity 1, and ignore the threshold qualification for a second, how many basic discount applications can you have?

Secondly, the result may not be intuitive.  For example, two exclusive discounts: a simple discount of \$1 off for one keyboard, and a threshold discount of \$10 off if you spend more than \$100. If you buy 10 keyboards with \$20 each. To get the best deal out of the knapsack problem, we would split 20 keyboards into two groups: one group of quantity 15 for simple discount and another group of quantity 5 for threshold discounts.

The recommendation is to leave threshold discounts out of non-threshold discount knapsack problem, and evaluate them after we have evaluated non-threshold discounts.

Threshold discount competes with other threshold discounts.

Strictly speaking, the competition between threshold discounts is a knapsack problem, but it is different from the one between non-threshold discounts.

1. Competition is rare, and is often between two threshold discounts.
2. When there is a competition, functionally more often than not, they cover same set of products, so it is really picking one over the other.
3. On the functional side, it is not about splitting products to accommodate all threshold discounts to maximize the deal, unlike the case with non-threshold discounts.

I do not think the traditional knapsack algorithm is a good fit. Instead, functionally, it make more sense to rank them in some way, and apply in that order. Technically, it is superior too.

Compounding

Let me just say it is big relief for threshold discount compounding, comparing to non-threshold compounding.

Rounding

Rounding is not easy. Forgive me for just leaving it like that for now.