# a combinatorial coupon challenge

Normally I do not get that excited about grocery shopping, but this Sunday was an exception: I used OR to do my grocery shopping.

Many grocery stores have promotions that encourage shoppers to buy 10 items, such as the 10 for \$10 sales.  Oddly, there has typically been no penalty for buying fewer than ten items:  all items would be \$1 each.  The sale merely suggests to stock up on ten or more sale items.

This past year, my favorite grocery store (Kroger) has periodically offered a promotion that offers \$5 cash back for every ten items purchased (but not any ten items, just the items associated with the promotion). The items range in price from \$0.99 to \$5.99 before the discount.  The promotion essentially offers an a savings of \$0.50 per item, which is 50% off of an item costing \$0.99.  But no problem so far; I just partition my wish list into groups of ten and stock up on extra rice cakes until a reach a multiple of ten. [Check out the ad]

There is an additional twist this week: a combinatorial challenge. Most of the items in the promotion have coupons in the Sunday paper that require the purchase of two or three items.  For example, I had two coupons for \$1 off of three boxes of Kleenex.  The Kleenex are on sale for \$1.49 (not the best sale), would be effectively \$0.99 after the promotion (not too shabby), and would be \$0.66 after the coupon (a bargain!). But I can only buy them at this price if I buy three or six boxes.  Maybe this isn’t exciting to you, but my kids went through two boxes of Kleenex this week when they had the sniffles.

On Sunday morning, I cut out coupons and made a list of promotion items my family could use.  I made sure that (1) the total number of items on my list was a multiple of ten, (2) individual items were purchased in groups of two or three (according to the coupon restrictions), and (3) individual items were within the lower and upper bounds according to our needs (there are only so many cartons of orange juice we can drink).

I bought 30 items with the promotion, saving \$15 with the promotion and an additional \$12.30 with coupons. I even found some bargain-priced day old bakery bread. Needless to say, it was the most fun I have ever had at the grocery store.  Kroger has a good OR team, and I appreciate some of the savings–and challenges–they offer shoppers.

Related posts:

Have you ever had a combinatorial shopping experience?

#### 2 responses to “a combinatorial coupon challenge”

• David K Smith

Here in the UK, we don’t have the largesse of coupons that you enjoy. But there are opportunities for startegic purchasing:
(1) Many companies (especially in the run up to Christmas) have “Buy three for the price of two, cheapest item free” offers. One of the major department stores received a number of complaints that their tills treated this as: “if you buy four or five items, then the cheapest will be free, if you buy six items, then the cheapest two items will be free”. The following year, they advertised nationally that their tills had been reprogrammed so that purchases were ordered so that the free item(s) would be those ranked 3, 6, etc. So this chain of stores is OK, but not every chain has such a policy. I was recently caught by such an illogical till (two mains for the price of one) but it was socially awkward to argue at the moment of paying.
(2) Several supermarket chains run occasional promotions “Spend £50 in one visit and get a voucher for 5p reduction on fuel per litre”. (5p is curently a little less than 5% of the fuel price in the UK.) There is only one voucher per spend, so the buyer who spends £50.01 gets one, as does the buyer who spends £250.01. The vouchers last for 14 days from the date of issue. The canny shopper, spending over £100 will try to divide their purchases into £50 lots. Supermarket tills sometimes show the running total … but there may be discounts not included in this. When is the optimal time to stop?
(3) We have store loyalty cards for several chains. Currently one of these gains 2p for each whole pound spent … so if you are faced with a total of £1.99 it is (just) worth getting a couple of grapes to push the total to £2.00!

• Anonymous

According to snopes:

A UC Davis engineer professor took advantage of a Healthy Choice promotion to receive 1000 frequent flyer miles for every ten items purchased. The cost-benefit analysis revealed that pudding cups would make it a great deal. He turned 12,150 pudding cups (most of which were donated to the Salvation Army) into 1.25 million frequent-flyer miles.