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Our students are not fencing in farm fields, cutting wires and folding them, or designing windows, so they are often uninspired by the optimization problems we give them. They seem like something that "someone, somewhere" might use, but the examples feel distant.

What are good examples of constrained optimization problems (perhaps not simple!) that today's students might actually encounter in their lives?

If your goal is to find problems that are more easily accessible, see also the sister question What are easy examples from daily life of contrainted optimization?

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Since you are asking for examples related to calculus, the following discrete examples are only a comment: 1. Planning a trip in a city where there are mulitple ways by bus/subway/train to get from A to B. 2. Almost irrelevant, but something I encountered as a problem as a teen: Given a record tape of 2x45mins and some songs of different lengths (which aren't all required to be on the tape) - what's the best way to put songs on the tape with as few empty space at the end? –  Roland Apr 11 at 15:05
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There aren't any. There may be situations where it's possible to apply optimization to solve a problem you've encountered, but in none of these cases is it honestly worth the effort of solving the problem analytically. I optimize path lengths every day when I walk across the grass on my way to classes, but I'm not going to get out a notebook and calculate an optimal route just to save myself twelve seconds of walking every morning. Mathematics beyond basic arithmetic is simply not useful in ordinary life. But I'm not sure if that's exactly what you mean. –  Jack M Apr 11 at 15:45
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@JackM That's an extremely depressing way to think about mathematics, and I'm sorry you think that way. –  Chris Cunningham Apr 11 at 16:07
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@ChrisCunningham What's depressing is the notion that mathematics has to be relevant to practical life in order to be interesting. –  Jack M Apr 11 at 16:52
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I think you've misjudged my agenda here. Mathematics is beautiful on its own, but optimization problems could additionally be relevant. It sounds like someone once told you that the purpose of mathematics is to apply to the world, and you were very displeased. I don't think anyone here is advocating such a position -- I'm certainly not! –  Chris Cunningham Apr 11 at 19:00
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16 Answers 16

up vote 12 down vote accepted

Bad Optimization Problems

I thought that Jack M made an interesting comment about this question:

There aren't any. There may be situations where it's possible to apply optimization to solve a problem you've encountered, but in none of these cases is it honestly worth the effort of solving the problem analytically. I optimize path lengths every day when I walk across the grass on my way to classes, but I'm not going to get out a notebook and calculate an optimal route just to save myself twelve seconds of walking every morning. Mathematics beyond basic arithmetic is simply not useful in ordinary life. But I'm not sure if that's exactly what you mean. – JackM

To some extent, I agree with this comment. With few exceptions, mathematics beyond basic arithmetic is simply not useful in everyday life. Students know this, and you'll have trouble convincing them otherwise.

Because of this, I've always found "everyday"-style calculus problems a little artificial. Consider the following problem from Stewart's Calculus: Concepts and Contexts.

A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?

The proper response to this question is: who cares? Is there any reason to calculate this length precisely? Why would anyone ever use calculus to compute this? If you have an actual building and an actual ladder, you could just try it and see if the ladder fits. If you don't have a specific ladder in mind (e.g. you are buying a ladder), the thing to do would be to draw the situation on paper and then use a ruler to estimate the minimum length. Of course, it's neat that you can use calculus to solve this problem precisely, but this is more of a curiosity than a legitimate application.

Chris specifically mentions the farmer fence problem, the wire-cutting problem, and the Norman window problem as not relevant to the students' lives. I agree—none of these problems are relevant. But it's not because the students aren't farmers, or wire-cutters, or architects. Even in a class full of future farmers, the fence problem would still be bad, because farmers don't use calculus to plan their fences.

Good Optimization Problems

What calculus is useful for is science, economics, engineering, industrial operations, finance, and so forth. That is, it's useful for all the things that make our society run. Most students who take calculus at a university are planning to go into one of these fields, so calculus will be relevant in their lives—specifically in their future studies and in their professions.

Here's something that's closer to a real-life optimization problem:

When a critically damped RLC circuit is connected to a voltage source, the current $I$ in the circuit varies with time according to the equation $$ I \;=\; \biggl(\frac{V}{L}\biggr)te^{-Rt/(2L)} $$ where $V$ is the applied voltage, $L$ is the inductance, and $R$ is the resistance (all of which are constant).

Suppose an RLC circuit with a resistance of 30.0 volt/amp and an inductance of 0.400 volt·sec/amp is attached to a 12.0-volt voltage source. Find the maximum current that will occur in the circuit.

This is at least close to something that a physics or engineering student might actually come across in their future studies. It's real in a way that the farmer fence problem isn't, and even students who don't plan to study physics can sense that this is a legitimate application. (By the way, if you have good students, you might even ask them to come up with a formula for the maximum current, without giving them specific numbers for $V$, $L$ and $R$. This has the advantage that it can't simply be solved using a graphing calculator.)

Of course, this isn't actually a constrained optimization problem—it's just an optimization problem. I'm not actually aware of any place in science that simple constrained optimization problems arise, although there are examples from economics (maximizing utility), finance (optimal portfolios), and industrial design (e.g. shape of a can type problems). When I cover constrained optimization in calculus, I usually stick to industrial-type problems (best cans, best shipping crates/boxes, best pipeline across a river, etc.), but that's probably just because I don't know enough about economics or finance to make up problems that involve them.

Finally, I should mention that I've never found the optimization portion of Calculus I particularly compelling. It's good to introduce the idea of optimization, but setting the derivative equal to zero isn't actually a very useful optimization technique by itself. It only really works for simple formulas—for anything complicated it just replaces one essentially numerical problem (finding the maximum of a function) with another (finding roots of a function). I agree that it should be covered, but it's far from the most important application of calculus.

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Let's turn the table a bit. Suppose that it takes you 2 minutes to calculate that grass-field path that will save you 12 seconds every time you walk through the field. Then after 10 days of classes it'll pay off. You can reap the benefits for years to come. I jest but this is the kind of thing optimization problems are good for. You take the time to calculate the optimization once and you continuously benefit from it from then on. –  Aleks Vlasev Apr 30 at 4:00
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Here's the example I had which inspired me to post the question in the first place:

The game League of Legends was the most-played PC game, in number of hours played, in North America and Europe in 2012. There is a good chance that League of Legends is a part of many of your students' daily life, especially if you are teaching engineering calculus. It doesn't have any sexual or deviant themes -- it is a game about teams of "champions" fighting in an arena -- so I've found it to be reasonable to bring it up in class. The reason to go this route is that many of your students might actually be encountering this optimization problem every day. Here's the problem:

In League of Legends, a player's Effective Health when defending against physical damage is given by $E = \frac{H(100+A)}{100}$, where $H$ is health and $A$ is armor.

(1) Health costs 2.5 gold per unit, and Armor costs 18 gold per unit. You have 3600 gold, and you need to optimize the effectiveness E of your health and armor to survive as long as possible against the enemy team's attacks. How much of each should you buy?

You can actually go further if you want to make it more complicated (and more accurate to what actually happens in the game!)

(2) Ten minutes into the game, you have 1080 health and 10 armor. You have only 720 gold to spend, and again Health costs 2.5 gold per unit while Armor costs 18 gold per unit. Again the goal is to maximize the effectiveness E. Notice that you don't want to maximize the effectiveness of what you purchase -- you want to maximize the effectiveness E of your resulting health and armor. How much of each should you buy?

And the last one is much more challenging, but it is something that even professional players of League of Legends (these exist!) can get incorrect:

(3) Thirty minutes into the game, you have 2000 health and 50 armor. You have 1800 gold to spend, and again Health costs approximately 2.5 gold per unit while Armor costs approximately 18 gold per unit. Again the goal is to maximize the effectiveness E of your resulting health and armor. How much of each should you buy?


My favorite part of this problem is that the game is actually more complicated than this -- in reality you need to defend against both physical and magic damage by buying both armor and magic resistance, which adds another variable. So the "punchline" of the exercise is that you can't actually play the most popular PC game properly until you understand Lagrange multipliers. I'll see you in Multivariable Calculus.


Instructors answers (not student solutions!):

(1) You do not spend equal money on A and H: $E = 3H - \frac{1}{720}H^2$ so the maximum is at $H = 1080$, plug back in for $A = 50$.

(2) One way to do this is to realize from number 1 that we know an optimal configuration is $H = 1080$ and $A = 50$, so right now we have way too much health and not enough armor. The answer to this is that we should spend all the money on 40 armor, to get exactly back to the optimized answer to #1.

(3) If H and A are the amount they plan to buy, the effectiveness is $E = \frac{(H + 2000)(100 + (50 + A))}{100}$ since they started with 2000 and 50 respectively. The critical point appears at H = -100, so the maximum actually occurs at one of the endpoints, not at the critical point at all. Again the player should spend all the money on armor.

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I'd like to point out a disadvantage of this example: PC games are, in general, played much more by the male students, so this could have the effect of exacerbating any preexisting notions that math and science classes are "boys' clubs." I'm not sure how important this is, but it is worth considering. –  Chris Cunningham Apr 11 at 19:07
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Both PC games and math have these preexisting notions and both are to a great extend true, so by disregarding one you should disregard both notions or disregard neither and accept both. Oh well, either way, a lot of examples in mathematics are about building stuff, but I haven't seen anybody complain yet that construction is a male thing to do (and the gender ratio is even far more strongly male in construction work than it is in gaming or maths) –  David Mulder Apr 11 at 22:49
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Additionally let me add that I think gaming is one of the greatest possible cases mathematics can use. Why? Because it has simpler models than (useful) physics, yet complex and relevant enough models to be useful for students. True, a part of the students will be quite alienated by the cases, but on average I think students tend to apply their mathematics skill by far the most in games (you would be surprised what crazy spreadsheets highschoolers are able to construct if it is to win some game) –  David Mulder Apr 11 at 22:51
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@ChrisCunningham - It's good to be thinking about that. In this case, I would say that it's fine, especially among engineering students. Just provide a variety of examples from different areas.. you are right on that most students will put WAY more effort into game optimization than other tasks, and see game optimization as much more relevant to their lives. –  Ian Apr 12 at 19:17
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This is from a post of mine on Math Stackexchange.

When I was in college, I owned three lamps and had a dark apartment. I kept trying to position them in different areas of the room, but it was still dark. Then I decided to model the problem with math: given three light sources of equal strength in a rectangular solid of a room, where can you place them to maximize the average value of the amount of light reaching a point in the solid?

Then I realized that the asymptotes at the light bulbs themselves would make the average diverge.

Recently, I thought of a reformulation:

Given three light sources in a rectangular solid of a room (assume height of ten feet, width of 15 and length if 25),all at height 6 ft., where should they be placed to maximize the average value (Edit: Minimum value) of the intensity of the light on the plane at height 3.5 ft.? (Since the couches are that height).

Alternatively, if someone can give a more accurate reformulation of this problem (perhaps involving reflections of the wall) and solve it, that would be much cooler.

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To illustrate what I said in the comments, although this is certainly an interesting problem, and easy for a student to relate to their own life, what I think would be actually harmful is telling a student that being able to solve a problem like this would be actually useful to them (although I don't think that's what the answerer is saying). Say it's interesting, but don't say with a straight face that it's useful - the student isn't that stupid, and you'll only lower their opinion of you, and cement their point of view that math is pointless. –  Jack M Apr 11 at 18:36
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@JackM Unfortunately, you are wrong. I had struggled with this problem with weeks, and after getting an answer on MathSE, I implemented their solution and significantly improved my lighting. My wife and I use it as a guide every time we rearrange furniture. –  Brian Rushton Apr 11 at 18:55
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No student is going to consider "you'll slightly (or even significantly) improve your lighting one day", or anything similar, as enough of a justification for soldiering through a class that they hate for another four years. It's better to simply try to communicate to them that the most interesting aspects in mathematics aren't in how it can solve real-world problems. Problems inspired by reality can be useful for that, but it seems counterproductive to take the attitude that the problems are interesting because they're useful. –  Jack M Apr 11 at 19:03
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@jack-m, the problems are interesting because they are useful somewhere. Some are interesting in "real life", even simple, or simplified, math can shed light on daily real problems. Those quite interest me, they seem to interest a few other people here, Chris that started this question, Brian in this reply, and so on. And I bet it interested them, as it did me, when I was a student. So simply because they do not interest you, don't extrapolate that 'no students' will enjoy them. Some will, many, or a few, I don't know, but some will for sure. –  Rolazaro Azeveires Apr 14 at 2:07
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An answer depends on what you mean with their lives. I think, it is more likely that students will face such problems in their working (not academic) life.

The Netflix Prize example

My example is the Netflix Prize, which was awarded by Netflix in the mid 2000s. At that time, Netflix was a DVD rental service and users did rate the movies they have seen. Netflix had some algorithm which gave recommendations which movies a particular user could like and awarded 1 mio $ to the team which improved the algorithm by 10%.

The rating was as follows: For all users $u\in U$ and all movies $m\in M$, there is a (possible) rating $r(u,m)\in\lbrace 0,1,2,3,4,5\rbrace$. There is a subset $K\subset U\times M$, for which the rating function is known. Netflix did a partition of $K$ into $K_{\text{known-to-everyone}}$ and $K_{\text{known-to-netflix}}$. The problem was then:

Find an approximating rating function $r_{\text{appr}}$ such that $\Vert r_{\text{appr}}(u,m)-r(u,m)\Vert_{K_{\text{known-to-netflix}}}$ subject to $r_{\text{appr}}(u,m)=r(u,m)$ on $K_{\text{known-to-everyone}}$.

That means, everyone got the known data and should make some model to make good predictions (depending on a specific given norm) on unknown data only known to Netflix (who by that data found out which algorithm was good).

The documentation of the best algorithm is published; some other teams published their approach as well and used methods from constrained optimization. I was attending a talk where someone presented his solution which uses a low rank approximations and optimization methods on manifolds.

Why could that be important to students in their future lives?

The prize was awarded and the million is gone. But I can image that such "data fitting" task are common in insurance economy where they have a lot of data and want to find models in order to derive unknown data, which will be important for the insurance company to model their prices.

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The original classic real-life example is to lay out a shop floor with multiple workstations and workers, so as to minimize travel distance of the work piece given a required sequence of operations. This can be generalized to consider a standard work load distributed amongst more than one type of workpiece, with different sequences of operations, and different times spent at each workstation.

Some links:

Any number of additional links could probably be found by Googling "Lean Kai-Zen" and "optimizing shop floor layout"

Update:
A second classic example is the optimization of paper-flow through an accounting department. Two other examples were these techniques were used to great effect in WWII are noted here (my emphasis):

While performing an analysis of the methods used by RAF Coastal Command to hunt and destroy submarines, one of the analysts asked what colour the aircraft were. As most of them were from Bomber Command they were painted black for night-time operations. At the suggestion of CC-ORS a test was run to see if that was the best colour to camouflage the aircraft for daytime operations in the grey North Atlantic skies. Tests showed that aircraft painted white were on average not spotted until they were 20% closer than those painted black. This change indicated that 30% more submarines would be attacked and sunk for the same number of sightings.[19] As a result of these findings Coastal Command changed their aircraft to using white undersurfaces.

A Warwick in the revised RAF Coastal Command green/dark grey/white finish Other work by the CC-ORS indicated that on average if the trigger depth of aerial-delivered depth charges (DCs) were changed from 100 feet to 25 feet, the kill ratios would go up. The reason was that if a U-boat saw an aircraft only shortly before it arrived over the target then at 100 feet the charges would do no damage (because the U-boat wouldn't have had time to descend as far as 100 feet), and if it saw the aircraft a long way from the target it had time to alter course under water so the chances of it being within the 20-foot kill zone of the charges was small. It was more efficient to attack those submarines close to the surface when the targets' locations were better known than to attempt their destruction at greater depths when their positions could only be guessed. Before the change of settings from 100 feet to 25 feet, 1% of submerged U-boats were sunk and 14% damaged. After the change, 7% were sunk and 11% damaged. (If submarines were caught on the surface, even if attacked shortly after submerging, the numbers rose to 11% sunk and 15% damaged). Blackett observed "there can be few cases where such a great operational gain had been obtained by such a small and simple change of tactics".[20]

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Edit (4/29/14): A popular media piece in the New York Times on rent division gives a great example of how Sperner's Lemma can be used for fair division problems (as related to cake cutting, the history of which can be found in a pasted excerpt at the end of my response here). The AMM article drawn upon is:

Su, F. E. (1999). Rental Harmony: Sperner's Lemma in Fair Division. American mathematical monthly, 106(10), 930-942. Link (no paywall).


A March 2014 PLOS paper describes a mathematical model using optimal control theory and a system of ODEs to build an app that combats jet-lag. A popular media piece can be found here, with the headline: A new mathematical model can cut jet-lagged time in half. Even if the details elude students, this is certainly a problem that today's students might run into; moreover, they can now take part in testing the app to determine if it works.

Excerpt from the latter link:

[Biological mathematician] Forger admits that the schedules his model spits out may sometimes seem counterintuitive, but intuition, he says, is not the point.

"We're trying to move the science beyond your grandmother's advice of 'wake up late' or 'avoid carbohydrates' to something that can be rigorously tested," he said. "All I know is these schedules are optimal according to the mathematics."

Forger spent 10 years building his model based on data collected from sleep studies done at Harvard and the University of Michigan. He had no idea what type of schedules the model would come up with when he first started, so he was pleasantly surprised that according to the math, the best way to beat jet lag is to adjust the time of your dawn and dusk each day.

See more in the LA Times write-up or in the methods section (p. 10) of the actual paper.


In case the cake-division discussion below does not suffice, there is the question of how one physically carries out the division for a different food: matzah.

See the short Wired article here, and perhaps the question of how to break matzah optimally (for two people: in half) can provide a real-life optimization problem.


I would probably start by looking to literature on mathematical modeling. An example source is the COMAP book "For All Practical Purposes" which includes sections on optimal entry (in elections), optimal production policies, and optimal schedules. There are plenty of other problems that don't have the word "optimal" in their name (e.g., bin packing). The book provides a wealth of examples; another nice one is fair division, which can be applied to scenarios ranging from company mergers to inheritance to organ transplant policies. Some of these are probably a bit removed from students' daily lives, but others are run into frequently. For instance, you could read over the literature on cake-division.

For three cake eaters, there is the Steinhaus Proportional Procedure; for four or more, there is the Banach-Knaster proportional procedure. The book contains a sixty-year history of cake cutting. I have pasted an image of this below (written up as p. 421 in the eighth edition); the word "California" is, for some reason, omitted between the two columns:

enter image description here

I have included only one actual excerpt here, but I recommend this book - and other materials on mathematical modeling, e.g., COMAP's Mathematical Modeling Handbook - as sources for optimization problems encountered (either literally, such as when you want to divide up a cake, or mathematically, when you see another problem to which you can apply these models/methods).

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On a similar note, it may be a good idea to provide students with an open-ended mathematical modeling problem which seeks the maximal or minimal value of some quantity. Students will ultimately be forced to use calculus to determine the maximum or minimum of this function. –  Draksis Apr 12 at 3:49
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This answer is turning into an incredible resource. Thanks so much for this. –  Chris Cunningham Apr 29 at 14:34
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@ChrisCunningham Surprising to see fair-division problems repeatedly cropping up: cake-division, then matzah-division, now rent-division. (The Baader-Meinhof Phenomenon in action...) If I start to accrue other (non-division) examples, then I may have to post another response in order to keep this one from becoming unwieldy! –  Benjamin Dickman Apr 29 at 14:43
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As I mentioned in my other answer, it highly depends what is meant by their lives. Here is a every day problem which I, at least, often encounter:

If you do the dishes and finish, there is a lot of foam in your sink. Arrange the position and strength of the water-tap in order to let the foam disappear as fast as possible.

The problem sounds easy, but is indeed an optimal control problem with partial differential equations (Navier-Stokes) and boundary control, which is not very easy to solve :)

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Ha! This is amazing. Maybe I should remove the calculus tag from my question to encourage more such answers. –  Chris Cunningham Apr 11 at 19:03
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Not at all relevant to the question, but there is a non-mathematical answer to your foam problem. Dissolve one or two Gas-X tablets in the dish water. The simethicone in the Gas-X is an anti-foaming agent. It will prevent formation of foam, and it will make any existing foam disappear quickly. –  Tristan Apr 11 at 19:51
    
Or sprinkle some salt on top. –  JPBurke Apr 12 at 20:56
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Here are a couple you see every day:

  • Light appears to follow a path of least time. Determine the angle subtended at the students eye by the apparent position and actual position of an object.

  • Stable positions are at minima of potential energy.

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A lot of examples exist in industry.

  • Manufacturing plants use optimization to figure out how to best run their machinery, buy raw materials, ship finished goods, etc.
  • Airlines and other passenger transportation services use optimization to determine their schedules.
  • The cargo transportation industry (trucking, trains, etc.) uses optimization to determine how to transport goods as quickly and inexpensively as possible.

I had an entire series of classes in my undergraduate program centered around an example of optimization in transportation. (I was part of a program that combined computer science and business, so the administrators had a lot of leeway designing classes that went together.) We had to write a program for a fictitious train company that determined how to transport orders in the cheapest way possible using the simplex algorithm. We had to write our own implementation of the simplex solver in a programming class and learned about the algorithm in detail in a math-heavy business course.

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Here's a problem that I found interesting: I listen to audiobooks. When you import them from CD, you get lots of tiny chapters (often 2 or 3 minutes) which is not very practical. Also, audiobooks can be very long - some are 30 hours - which is also not very practical. I prefer audiobooks to be split into individual volumes of not more than 8 hours, and chapters around 10 minutes. Assume that you can combine consecutive chapters into one chapter (chapters are indivisible), and that you can combine consecutive chapters into volumes.

Problem 1: Define an optimisation problem. Find a function that, if minimised, produces audiobooks in the way that I (or you) prefer them. I'd want volumes as long as possible, but never more than 8 hours (unless a chapter is longer than 8 hours and it becomes unavoidable), and volumes of approximately equal length. I'd want chapter length around ten minutes, and preferably equal lengths. Defining a function to minimise is not trivial.

Problem 2: Find an optimal solution according to the function to be minimised. Decide how important execution speed is (hint: Actually creating an audiobook according to the solution takes a long time). Obviously the way you combine chapters affects the possible lengths of volumes.

Problem 3: Does the optimal solution actually match my/your preferences? If not, is it because finding a good solution was hard, or because the optimisation problem wasn't chosen well? If needed, go back to problem 1.

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+1 For adding Problem 3, that's a very important part of "real world" optimization that often gets left out of exercises. –  Ian Apr 12 at 19:13
    
How critical is it that the chapters stay in the correct order? I wouldn't want an audiobook that jumps from chapter 3 to chapter 10. –  Nate Kerkhofs Apr 14 at 12:08
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Here's an optimisation problem that really should have solutions readily available, but hasn't: Consider a SatNav that tells you the best way to get from A to B. Now assume the distance is quite long, and you won't be able to get from A to B without going to a petrol station. So what I want: Find the shortest way from A to B visiting any petrol station within X miles of my current location. They all are capable of showing you nearby petrol stations, but that's not what I want if I'm 200 miles away from B and have fuel for 150 miles in the tank.

By the way: Finding the shortest path from A to B is actually an interesting problem in itself. SatNavs (at least the cheap ones that you will tend to use) are not capable of actually finding what I would consider good solutions, which take into account time, fuel consumption, wear and tear on the car, driver stress. (For example a short journey with many turns where I have to give way to other drivers means I stop and accelerate a lot and have more stress).

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Even more interesting, find me a petrol station between A and B taking into account the cost of petrol at each station, how much petrol/time is used to get to the station, how much petrol I have at present and how a full tank reduces MPG. –  Ian Apr 19 at 13:11
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Finance has lots of good examples. Here is one version of the portfolio optimization problem.

Suppose two assets have mean excess returns $\mu, \mu'$, standard deviations $\sigma, \sigma'$ and a correlation $\rho$. You have cash to distribute among these two assets in proportions $a, a'$ where $a+a'=1$. What values of $a,a'$ maximize the Sharpe ratio of your investment: $$\frac{a\mu + a'\mu'}{\sqrt{a^2 \sigma^2 + a'^2\sigma'^2 + 2\rho\, aa'\sigma\sigma'}}?$$

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Creating a timetable, all students should understand the issues. However mathematical solutions don’t work as well as computer sci solutions.

Choosing a set of options, so as to get a degree while having the lease dead time, the dead time being when you are waiting for the next activity to start, but don’t have long enough to do someone useful.

In the US when degrees can take a undefined number of years, deciding how much part time work to do, so as to minimise the debt level at age 30. More part time work, increase the time to get the degree, but decreases the debt earn per year. Completing the degree sooner, allows the repayment of the debts to start sooner.

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Can't believe no one's mentioned the coke-can problem!

You are the designer of drinks cans for Coke, how can you make them the most money?

Pretty quickly it comes out that you want a cylinder of volume 330ml with minimal surface area (use the least metal possible to enclose your beverage).

It's a classic problem and one kids can very easily relate to on all number of levels, especially if you hype it a little bit.

The interesting punchline is that Coke cans don't look at all like your design (which is predictably as close to a cube as a cylinder can get), for reasons that transpire to be aesthetic. But paint cans totally do though... This is because the metal must be significantly thicker in this case.

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Profit maximation!

It's easy to understand, not so hard to calculate and even testable. Let them sell lemonade or whatever at the next school festival and see, if they can find the optimal price for the lemonade.

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This isn't really a math problem, as it involves a lot of knowledge about customer habits and preferences. –  Jack M Apr 11 at 18:38
    
@JackM That's no reason not to apply a mathematical model. And there are easy enough models usable for constrained optimization problems. –  Toscho Apr 11 at 19:09
    
Sure. I'd be interested in a longer answer if you have any specific ideas. –  Jack M Apr 11 at 19:12
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The most simple, usable model is: $\text{Numbers sold}=f(\text{price})$ with a linear function $f$ with negative linear coefficient. The result will be quadratic function $\text{Profit}=g(\text{price})$ with negative main coefficient. –  Toscho Apr 11 at 19:15
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A real-life optimization problem: It takes about 20 clicks and 2 minutes to shoot a wolf. 200 dead wolves will get you enough experience points to get to the next level. It takes about 50 clicks and 10 minutes to steal gold from a palace. 50 burglaries will get you enough experience points to get to the next level. However, you would get carpal tunnel syndrome after 3000 clicks which would force you to stop playing for a day. What is the shortest possible time you can get to the next level?

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