Snapple Real Fact #804: There are 293 ways to make change for a dollar.

My first reaction was “Mmm…interesting”, but I couldn’t help wondering whether the Snapple folks did their math correctly. So after I got home and unpacked the car, I wrote a little Constraint Programming code in Comet to check this fact. It turns out that the number is indeed 293 if the following two things are allowed: (i) returning a 1-dollar coin in exchange for a dollar bill, and (ii) using half-dollar coins which, in my opinion, are rare these days. Here’s a list of the 292 ways that do not include using a 1-dollar coin which, in my opinion, isn’t really “giving change”.

If you’re wondering how many ways there are when you’re not allowed to use 1-dollar or half-dollar coins, the answer is 242. Here’s a list of all such possible ways.

Update: A friend asked me what the number would be if we considered the quarters from each of the 50 states as a different coin. In that case the number of possible ways increases to 515,184 (including the 1-dollar coin).

If you go to the gym regularly (or have been a regular gym goer at some point in your life), you might have noticed an issue that arises with some frequency: the locker you’re trying to access is very close to one in front of which someone else is standing (because their locker is right next to yours). I’ll refer to this phenomenon as interference. Interference is annoying because it creates that awkward situation in which you stand there trying to be polite and wait for the other person to finish, while at the same time getting upset because you’re wasting your precious time: “Man, I was hoping to be finished with my workout in 45 minutes. I gotta go back to the office and work on that integer programming model. Why does this guy take so long to tie his shoes?”

Interference occurs in other places, of course; hence the title of this post. When boarding planes, airlines try to be as efficient as possible, that is, they try to get everyone in their seats and ready to go in the shortest possible time. What is interference during the boarding of a plane? It’s when passengers that are standing in the aisle (e.g., because they’re still trying to put their carry-on in the overhead bin) block the passage of other passengers whose seats are further down the aisle. You might think that the obvious solution is to board everyone starting from the back of the plane towards the front, right? Well, maybe. Back-to-front boarding is intuitively good, but there are other issues at play: some passengers have priority, not everyone is there when boarding starts, etc. Another strategy that seems to work well is a hybrid of back-to-front with window-to-aisle. As you might have guessed, people have used optimization and simulation to try and come up with good boarding strategies. One of these studies was published in the journal Interfaces in 2005: “America West Airlines Develops Efficient Boarding Strategies”. This is an interesting read, and I recommend it.

Where else does interference occur? This XKCD blog post talks about the International Choice of Urinal Protocol:

…the basic premise is that the first guy picks an end urinal, and every subsequent guy chooses the urinal which puts him furthest from anyone else peeing.  At least one buffer urinal is required between any two guys or Awkwardness ensues.

Randall then proceeds to analyze this protocol and concludes that it suffers from a problem of underutilization of the available urinals, depending on how many of them there are. However, if guys are smart when picking urinals, they can achieve the optimal utilization (50%).

Now back to the locker room interference problem (which is the one that bothers me most lately). Let’s try to figure out the source of the problem and propose a solution to it. When you arrive at the University of Miami gym (known as the Wellness Center), you hand in your ID to an attendant who, in return, hands you a key that’s taken from a set of drawers that look like this (men’s lockers on the left, women’s on the right):

A key comes out of the drawer and your ID goes in. Interference is created because the attendants do not (and cannot) remember which keys they have handed out recently and what the layout of the locker room looks like. (By the way, locker numbers are not in perfect sequence in the Wellness Center; numbers jump around and you frequently see people who are lost looking for their lockers.) Ideally, what we’d like to happen is for keys to be handed out in such a way that they send the next person to a locker that is far away from the last few lockers that were given away. There are other complicating issues, of course, such as the fact that you cannot control the people who are returning from their workouts, but at least you can reduce interference among new arrivals.

We don’t need to write a mathematical model for this (or do we?). Why not pre-calculate an optimal sequence of locker hand-outs (based on the locker room layout), sort the drawers in that sequence (left to right, top to bottom), and have the attendants hand out keys in this order, cycling back to the top after they reach the last drawer? It won’t be perfect, but it sure will be better than the current system.

Here’s an excerpt:

4,329 films were submitted to the 2012 Cannes Film Festival. This blog had 56,000 views in 2012. If each view were a film, this blog would power 13 Film Festivals

Click here to see the complete report.

As a matter of fact, I suggest that everyone write a post about each special date of the year (Easter, Fourth of July, Valentine’s Day, Summer Camp, etc.). Those will keep bringing you recurring visits, without any extra effort, year after year. Of course not all (in fact, most) of those visits will be here for O.R. But that’s the point! If we want to spread the word about O.R. we’ve got to take people by surprise. “I was just looking for a cute reindeer picture and this guy blew my mind by showing me that math and reindeer have something to do with each other! How cool is that!”

BTW, I just realized I do not have an Easter post. I need to take care of this…

We are very excited about our Annual conference in Phoenix this year and we hope you are too! Take a short video (under four minutes) telling or showing us why you are excited to attend the meeting and we’ll send you one of our fun INFORMS t-shirts!

I couldn’t resist; I want one of those T-shirts! So I made a short video with my views and expectations. Here it is (you can also watch it on my YouTube channel):

Jony Ive says:

Never before, have we built a product with this extraordinary level of fit and finish. We’ve developed manufacturing processes that are our most complex and ambitious.

And on Apple’s web site, they say this:

During manufacturing, each iPhone 5 aluminum housing is photographed by two high-powered 29MP cameras. A machine then examines the images and compares them against 725 unique inlays to find the most precise match for every single iPhone.

So let’s see if I understood this correctly. In a typical manufacturing operation, the multiple parts that get put together to create a product are put together without much fuss. A machine makes part A, another machine makes part B, and perhaps a robotic arm or a third machine takes any one of the many part A’s that are coming down a conveyor belt and attaches it to any one of the many part B’s that are coming down another conveyor belt. What Apple did was to improve on the “any one” choice. I don’t know if Apple pioneered this idea, I’d say probably not, but this is the first time I hear about something like this. If you’ve seen this before, let me know in the comments.

Before OR comes into play, Computer Science does its job in the form of computer vision / image processing algorithms. The photographs of the parts are analyzed and (I’m guessing) a fitness score is calculated for every possible matching pair of parts A (the housing) and B (the inlay). What happens next? How do they pick the winning match? Here are some possibilities:

1. Each part A is matched with the part B, among the 725 candidates, that produces the best matching score.
2. A 725 by 725 matrix of fitness scores is created between 725 parts of type A and 725 parts of type B, and the best 725 matches are chosen so as to maximize the overall fitness score (i.e. the sum of the fitness scores of all the chosen matches).
3. Proceed as in the previous case, but pick the 725 matches that maximize the minimum fitness score. That is, we worry about the worst case and don’t let the worst match be too bad when compared to the best match.

After these 725 pairs are put together, new sets of parts A and B come down the conveyor belt and the matching process is repeated. Possibility number 1 is the fastest (e.g. do a binary search, or build a priority queue), but not necessarily the best because every now and then a bad match will have to be made. Possibilities 2 (an assignment problem) and 3 (assignment problem with a max-min objective) are better, in my opinion, with the third one being my favorite. They are, however, more time consuming than possibility 1. Jony Ive says the choice is made “instantaneously”, which doesn’t preclude something fancier than possibility 1 from being used given the assignment problems are pretty small.

The result? In the words of Jony Ive:

The variances from product to product, we now measure in microns.

It is well-known that OR plays a very important role in manufacturing (facility layout, machine/job scheduling, etc.) but it’s not every day that people stop to think about what happens in a manufacturing plant. This highly-popular announcement being watched by so many people around the world painted a very clear picture of the kinds of problems high-tech manufacturing facilities face. I think it’s a great example of what OR can do, and how relevant it is to our companies and our lives.

The alt-text reads:

FYI: If you get curious and start trying to calculate the time adjustment function that minimizes the gap between the most-used and least-used digit (for a representative sample of common cook times) without altering any time by more than 10%, and someone asks you what you’re doing, it’s easier to just lie.

It seems Randall is trying to find (or has already found) a closed-form function to accomplish this task. I don’t endorse number discrimination either; unlike my wife who insists on adjusting restaurant tips so that the cents portion of the total amount is either zero or 50, but I digress… I’m not sure exactly how to find these adjusted cook times with a closed-form function, but I can certainly do it with OR, more specifically with an integer program. So here we go. For simplicity, I’ll restrict myself to cook times under 10 minutes.

Let’s begin with an example. As I think about the microwave usage in my house, I end up with the following cook times and how often each one is used:

So let’s first calculate the usage frequency of each digit from zero to nine. The above table can be interpreted in the following way. For every 100 times I use the microwave, 20 of those times I type a 3 followed by a zero (to input a 30-second cook time), 30 of those times I type a 1 followed by two zeroes, etc. Therefore, during these 100 uses of the microwave, I type a total of 280 digits. Out of those 280 digits, 160 are zeroes, 40 are 1′s, 30 are 2′s, and so on. Hence, the usage frequency of zero—the most-used digit—is . (Usage frequencies for the remaining digits can be calculated in a similar way.) Digits 5 through 9 are apparently never used in my house, so the current difference between the most-used and least-used digit in my house is (57.1-0)%.

If, as Randall suggests, I’m allowed to adjust cook times by no more than 10% (up or down) and I want to minimize the difference in usage between the most-used and least-used digit, here’s one possible adjustment table:

Now let’s compare the usage frequency of each digit before and after the adjustment:

After the adjustment, the most frequently used digits are 0, 1, 2, 3, 5, 8, and 9 (12% of the time), whereas the least frequently used digits are 6 and 7 (4% of the time). The difference now is 12-4=8%, which is significantly less than 57.1%. In my household, there’s absolutely no way to do better than that. Guaranteed! (Note: this doesn’t mean there aren’t other adjustment tables that achieve the same 8% difference. In fact, there are many other ways to achieve the 8% difference.)

If you’re curious about how I computed the time-adjustment table, read on. I’ll explain the optimization model that was run behind the scenes and I’ll even provide you with an Excel spreadsheet that allows you to compute your own adjusted cook times. Let the fun begin!

Let be the set of typical cook times for the household in question. In my case, . For each , let be the set of cook times that fall within 10%—or any other range you want—of . For example, . In addition, for each , let be the usage frequency of cook time . In my example, , , and so on.

For each and , create a binary variable that is equal to 1 if cook time is to be adjusted to cook time , and equal to zero otherwise. There are such variables. In my example, 119 of them. Because each original cook time has to be adjusted to (or mapped to) a unique (likely different) cook time, the first constraints of our optimization model are

To be able to calculate the difference between the most-used and least-used digit (in order to minimize it), we need to know how many times each digit is used. Let this quantity be represented by variable , for all . In my house, before the adjustment, and . We now need to relate variables and .

For each  and , let equal the number of times digit appears in cook time . For example, , , and . We are now ready to write the following constraint

Once the adjusted cook times are chosen by setting the appropriate variables to 1, the above constraint will count the total number of times each digit is used, storing that value in .

Our goal is to minimize the maximum difference, in absolute value, between all distinct pairs of variables. Because the absolute value function is not linear, and we want to preserve linearity in our optimization model (why?), I’m going to use a standard modeling trick. Let be a new variable representing the maximum difference between any distinct pair of variables. The objective function is simple: . To create a connection between and , we include the following constraints in the model

With 10 digits, we end up with 90 such constraints, for a grand total of 105 constraints, plus 130 variables. This model is small enough to be solved with the student version of Excel Solver. I would, however, recommend using OpenSolver, if you can.

Here’s my Excel sheet that implements the model described above. It shouldn’t be too hard to understand, but feel free to ask me questions about it in the comments below. Variable cells are painted gray. Variables are in column E, variables are in the range K3:T3, and variable is in cell K96. The values and the formulas relating with are calculated with the help of SUMIF functions in the range K3:T3. The differences between all pairs of variables are in column U. All Solver parameters are already typed in. The final values assigned to variables (range K3:T3) represent absolute counts. To obtain the percentages I list above, divide those values by the sum of all ‘s. (The same applies to the value of in cell K96.)

Feel free to modify this spreadsheet with your own favorite cook times and help end number discrimination in the microwaving world! Enjoy!