Numbers represent items in the grocery basket.

Ask pupils to devise a formula for the time it will take a queue to be processed, including suitable variables and constants.

Which queue should you join?

It seems to me there are two major questions, or at least two major categories of question.

The first is about reality: How long will each line take? Which one will go faster?

The second is about perception: If you walked up right now, which line would you choose? Or, which line *seems* like it would go faster? Why is it not necessarily the one that actually will go faster? It’s hard to tell exactly how many items someone has, plus not all items take an equal time to scan. Are there visual cues that indicate potential slowdowns, for example, a distracted mom whose kids are bothering her, making it hard for her to find her credit card quickly.

I guess the first question might be more mathematical in some sense, but both are worthy classroom topics, I would say.

Maybe I need some coffee, but I had a little trouble interpreting what the things in the photo were (red thing, black and white thing). The grocery baskets from top view looked like cough drops or something. Once I read the comments, I got it though.

Nice concept, there’s a formula I would use to estimate how long each line would take, and this might motivate the students to derive it themselves. Of course, there’s some extra variability (are there baggers at each counter, is each customer paying with the new tap-n-go credit cards or are they counting out change) but the basic idea is nice, since this is a common decision people make everyday.

P.S. I got the photo, but it took a moment. The red things are hand baskets, I think? Maybe cart-shaped would work better. Maybe a view from an angle, instead of directly overhead.

I really like where Dan is going with this.

I know this isn’t necessarily the point of WCYDWT, but there are tremendous opportunities for extension here. There is a whole field of mathematics devoted to waiting line models. The calculations are deceptively simple, but there is a tie-in to probability with arrival and service times involved. You can start to look at when it makes more sense to open additional lines from a management perspective vs. when it makes more sense to add baggers to existing lines etc.

I learned all of this stuff in a graduate math course in applied math, but have had incredible success with showing it to AP students post-exam. They eat up the mathematics where the application to real-life is blatantly evident.

The book I have is “An Introduction to Management Science: Quantitative Approaches to Decision Making”, although I’m sure 99% of the info is available online.

Considering I worked as a cashier for a grocery store for 3 years and am a math teacher now, I could probably write a book on this topic, but I won’t. I’d estimate time per item scan to be about 4 seconds (Most items would be less, but items with warped bar codes, produce and bulky items bring it up). Average time per customer would probably be around 40 seconds. Bagging opens up a whole new can of worms though. It’s usually trivial for small orders (everything in one bag), but increases quickly with larger orders (What do I want to keep together or separate, how heavy should I make each bag, uh oh I’m running out of room in the cart…). Assuming a good bagger and the person/cashier help at appropriate times, however, it’s probably irrelevant for the order sizes above.

As for payment, I’d say it would go card/cash/check, but I’d actually say the difference is probably pretty small unless the person is able to pay with their card while the cashier is still checking.

Oh geez, I really missed the boat. I thought it was about prime numbers.

I was baffled by the image. Those carts don’t look like anything like ones I’ve ever seen. :-) Also, I forgot to notice the key word: groceries.

Great problem! If anybody does this problem with kids I’d love to read about/see it.

Here’s an interesting extension.

http://blog.cleveland.com/business/2008/12/which_retail_system_is_better.html

-Ihor

That’s OK. What was misleading for me (after I knew what it was) was that I’ve never seen more than one person holding a basket on the same line. I guess I just don’t get out enough.

I was a cashier for Longs Drugs and Barnes and Noble during high school and college. We didn’t have any form of rating system, like what is described above for Target.

I’d say either 1 or 2 seconds to scan an item. With a lot of stuff it would be less than a second. Books were especially quick since the ISBN was in the same place for any book.

There is a lot of variability in the payment method time. Since we’re only going to care about the time that is additional to “scanning time”, I’ll talk about that: the time spent after a cashier presses [Total].

Check: customer writes amount (or entire check, which seemed to be the case a lot) and takes out ID for cashier to look at. 10-40 seconds.
Check must be read and endorsed by computer/printer: +15 seconds.

Cash: customer hands bills modestly over the amount: 20 seconds for cashier to make change.
OR customer tries to count exact change: 30-45 seconds
OR customer needs $80+ worth of change (rare) 1-5 minutes for a manager to arrive with key.

Credit: sometimes this can overlap almost entirely with the scanning. If the customer is ready: 5-10 seconds to accept charge and sign.
OR if the customer hands card to cashier, 10-20 seconds.

Other line-hazards:
1) pricechecks (dealbreaker)
2) people returning items (dealbreaker)

Bagging:
Not a big deal at either Longs or Barnes and Noble. At grocery stores, sometimes there are people who are there specifically to bag. If that’s the case, they are pretty much done before the transaction is finished.

As for a different visual: what about a 3/4 shot of the queues at eyelevel (or perhaps 8 ft.) Something that would simulate how they might arrive at the lines in a real store.

@Aaron: I had a physics professor who would always go off on tangents from the material we were studying, usually relating to classes that several students would later take (for example, “Here, we’re assuming a perfectly rigid body. Of course, there’s no such thing and so to really describe this accurately, you need to deal with stress and strain. If you’re a mechanical engineer, you’ll learn about that next semester in [whatever course]” and so suddenly you’re not just learning “first semester mechanics” — you’re getting the first steps toward something that’s ahead of you, and your current class begins to become part of a bigger picture.

On the other hand, teachers going off on tangents annoys some students. I think there’s definitely a right way and a wrong way to do it. But anyway, even if it’s beyond the scope of the particular class, it can’t hurt to mention it, and guide students in the right direction if they’re interested to learn more.

As a cashier at a grocery store I can’t resist posting SOMETHING here. Here it is:

Being the first customer in an empty checkout lane does not ensure the fastest possible checkout time. It all depends on the customer. The reason is that the cashier’s scanning speed is limited by the number of items on the belt. If the customer unloads the cart one item at a time and the cashier begins scanning immediately, then the customer’s unloading speed equals the maximum cashier scanning speed. The fastest customers, as far as ringing up is concerned, are the ones who have already placed their groceries on the belt.