Dynamics AX Overlapped Discounts | Optimal Combinations

 Introduction

This documents describes how Microsoft Dynamics AX for Retail determine which combination of discounts to apply to transactions when more than one discount applies to one or more items in the transaction and the total quantity of the items with multiple possible discounts increases beyond a few (2-5) items. The exact number of items depends on how many discounts can be applied to each item and how complex the discounts are.

Overlapping discounts is the term we’ll use when two or more discounts can be applied to the same set of items in a sales transaction. Determining the combination of discounts to use that would results in the lowest transaction total is a combinatorial optimization problem. This is especially true when a discount requires 2 or more items to be purchased together and the effective discount amount varies with the price of the two items, as is the case in the common retail discount Buy 1 Get 1 X% off. To determine the best combination of discounts to apply quickly and continue to provide the rich discount functionality available in Dynamics AX for Retail a new methodology has been introduced for determining the best combination of discounts by evaluating the discounts relative to each other as applied to all the items in a transaction with overlapping discounts.

Discount Example

Let’s look at an example to understand the problem of how to choose the best combination of discounts.

We’ll buy 2 items and there are two possible discounts which cannot be combined 

• Discount 1: Buy 1 item get a 2^nd item of equal or lesser value at 50% off.
• Discount 2: Buy 2 items and get 20% off both.

For any 2 items the better of these two discounts depends on the price of the two items with respect to each other. When the price of both items is equal or close to equal then Discount 1 is better.  When the price of one item is significantly less than the other item then Discount 2 is better. The rule for evaluating these two specific discounts against each other works out to be: If     Then Discount 1 is the better value otherwise Discount 2 is the better value. Note, when the price of item 1 is equal to 2/3 the price of item 2 then the two discounts are equal.

In this example the effective discount percent for Discount 1 varies from a few percent when the price of the two items is far apart to a maximum of 25% when the two items have the same price. The effective discount percent for Discount 2 is fixed. It is always 20%. Because the effective discount for Discount 1 is ranges above and below Discount 2 the best discount can only be chosen based on the price of the items to be discounted.

If we buy 4 items and both discounts could be applied to all 4 items then only two applications of Discount 1 or Discount 2 could be applied to the transaction because each discount requires two items to qualify and the discounts cannot be combined on any single item.  These are ‘Exclusive’ discounts.

Let’s look at three sets of items with different price ranges to illustrate the possible combinations. 

1. All items have the same Rs. 15 price. The best discount combination is two applications of Discount 1. You will have 2 full priced items and two 50% off items. The transaction total will be Rs. 45 which is net 25% off the total of Rs. 60. (Rs. 15, Rs. 15, Rs. 7.5, Rs. 7.5). Discount 2 is only 20% off.
2. Now let’s say two items are Rs. 20 each and two items are Rs. 10 each. Now the best discount combination is two applications of Discount 2. All 4 items would have 20% off each. The transaction total will be Rs. 48 which is net 20% off the total of Rs. 60. (Rs. 16, Rs. 16, Rs. 8, Rs. 8). Discount 1 would be only 16.6% off. (Rs. 10 / Rs. 60).
3. Finally let’s say two items are Rs. 20 each one item is Rs. 15 and one item is Rs. 5. Now the best discount combination is one application of Discount 2 and one application of Discount 1. Since the effective discount for Discount 2 is always 20% if we choose two items that eliminates the least effective combinations for Discount 1 we can maximize the effective discount of Discount 1. The least effective application of Discount 1 is a Rs. 20 item and a Rs. 5 item. So we use these two items for Discount 2. That leaves a Rs. 20 item and a Rs. 15 item. Discount 1 give a slightly better value so we’ll use it. The transaction total is Rs. 47.50 which is 20.8% off the total of Rs. 60. (Rs. 16, Rs. 4, Rs. 20, Rs. 7.5).

To make it easier to visual the 3^rd example two tables below showing the effective discount percentage for all possible combinations of items for both Discount 1 and Discount 2. I’ve marked the lowest effective discount in red.

Discount 1 (50%)

	20	20	15	5
20	25.0%	25.0%	21.4%	10.0%
20	25.0%	25.0%	21.4%	10.0%
15	21.4%	21.4%	25.0%	12.5%
5	10.0%	10.0%	12.5%	25.0%

Discount 2 (20%)

	20	20	15	5
20	20%	20%	20%	20%
20	20%	20%	20%	20%
15	20%	20%	20%	20%
5	20%	20%	20%	20%

Now an example that falls in between. When prices vary and two or more discount competes the only way to guarantee the best combination of discounts is to evaluate the discounts and compare them.

 

Combination Examples

Continuing the example from the previous section let’s add more items and an additional discount that also requires 2 items to qualify. You’ll see the total number of possible combinations that need to be evaluated very quickly grows beyond what can possibly be calculated quickly.

# of items in the transaction	One 2-item discount can be applied	Two 2-item discounts can be applied	Three 2-item discounts can be applied
	# of possible pair combinations	# of possible pair combinations	# of possible pair combinations
4	4	8	18
5	15	60	135
6	15	90	315
7	90	540	1890

 

When additional multi-item discounts are applied to a transaction or larger quantities are added to the transaction then the total number of possible discount combinations grows quickly into the millions and the time required to calculate the best possible combination of discounts will quickly become a noticeable amount of time. Some optimizations can be done to reduce the total number of combinations that need to be evaluated, however if the number overlapping discounts or the quantities in the transaction are not limited you will always end up with a very large number of combinations to evaluate whenever you have overlapping multi-item discounts.

Marginal Value

As mentioned at the beginning, to solve this problem we have added an optimization that calculates the value per shared item of each discount on the set of items that can have two or more discounts applied to them.  We refer to this as the Marginal value of the discount for the shared items. The marginal value is the average per item increase in the discount amount when the shared items are included in each discount. Marginal value is calculated by taking the total discount amount (D_Total) and subtracting the discount amount without the shared items (D_{Minus Shared}) and dividing that difference by the number of shared items (Items_Shared).

((D_Total)  -  (D_{Minus Shared})) / (Items_Shared) = Marginal value_D

 

After we calculate the marginal value of each discount on a shared set of items we then apply the discounts to the shared items in order from highest to lowest marginal value