# Category Archives: Exam Fun

## Big Bang Theory Party Planning

Penny is organizing a party at her apartment, but she is on a tight budget. Having a working knowledge of all of the important things in the universe, Sheldon knows everything about linear programming and offered to help her. He postulates that it’s ideal to have two kinds of mixed nuts: a plain party mix, and a luxury mix (for those with a distinct taste like himself). Based on the expected number of guests, Howard quickly calculates that they’ll need a total of at least 10 pounds of snacks, but no more than 6 pounds of each kind of mix. On his white board, Sheldon has already come up with the following table:

Raj wants to dip the hazelnuts into liquor, but that’s not in the budget, so he gives up. Leonard reminds everyone that, because of their allergies, it’s important to keep the average allergenicity level per pound in both mixes to no more than 3. Write an optimization model to help Penny prepare the two kinds of snacks at minimum cost. But be careful: Sheldon will check it later for correctness!

This post is part of my series “Having Fun with Exam Questions”. Previous questions dealt with Farmville, vampires, and (potentially) Valentine’s day.

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Filed under Big Bang Theory, Exam Fun, Linear Programming, Modeling, Teaching

## Transporting Flowers with Love

Mr. Lovr, a lonely gentleman, does not want to spend Valentine’s day alone in 2011. As one of his New Year’s resolutions, he intends to send roses to nine of his lady friends. Being an avid procrastinator, however, he waits until the last minute and finds out that only eight flower shops in his city still have roses available. For lack of better names, let’s call those flower shops F1, F2, F3, …, F8. The number of bouquets of roses available at each shop are, respectively, 4, 4, 3, 2, 2, 2, 2, and 1. Based on how well he knows each of his potential valentines, whom we’re going to call V1, V2, V3, …, V9, he calculates how many bouquets he needs to send to each of them to increase his chances of going on at least one date. The numbers are, respectively, 3, 2, 2, 2, 2, 2, 2, 2, and 3. At this point, a light bulb goes off in Mr. Lovr’s head, and he remembers from his Operations Research class that this is a transportation problem. But there’s something missing…Ah! The costs! He calls each of the eight flower shops and asks how much it would cost to ship one bouquet of roses to the addresses of each of his nine lady friends. He then compiles the following table of costs (in dollars):

He also remembers that because the total supply is equal to the total demand, he can write all of the constraints in this problem as equalities. Essentially, he has to say that, for each flower shop, the number of bouquets that it ships has to be equal to the number of bouquets that it has. Similarly, he needs one constraint for each valentine saying that the number of bouquets that they receive has to be equal to the number that they want (according to his estimates above). To avoid suspicion, he also decides that it’s better for each flower shop to send no more than one bouquet to the same person. So far, so good, but he needs a specific shipment plan because he’s running out of time.

He opens up an Excel spreadsheet and creates the following layout of cells (he chose pink to make it more romantic):

Somewhere else in his spreadsheet he also typed the table of costs shown above. To his surprise, he even remembered to use the SUMPRODUCT function to calculate the total cost expression. He clicks “Solve” and finds out that the cheapest way to send all 20 bouquets will cost him \$38. Not bad…but wait a second…something amazing happened! He cannot believe his eyes! The optimal solution exhibits a very curious pattern! Could it be a Valentine’s Day miracle? Could it be the power of love? If you want to see for yourself, download Mr. Lovr’s spreadsheet, open Excel Solver, and solve the model (no setup necessary, just click the “Solve” button). (NOTE: this spreadsheet has been tested with Excel 2007 under Windows XP, and with Excel 2008 under Mac OS X Snow Leopard. I’m not sure the “trick” will work in earlier Excel versions. For a free download of Excel Solver for Mac OS X, go here.)

Thanks to my love for giving me the idea for this blog post.

## Twilight Network Flow

Part of a professor’s job is to come up with new exam questions each year. That may be a time-consuming (and sometimes tedious) task. You want the question to be just right: not too easy, not too difficult, and capable of testing whether or not the students understood a given concept. This year, I figured I might as well have some fun while doing it. Inspired by Laura McLay’s insightful (and very popular) post on vampire populations, I decided to create a Twilight-themed network flow question:

Alice is in charge of planning Edward and Bella’s wedding and she ordered 2000 roses to be delivered to three locations as shown in the network below. The Cullen’s house (node 2) needs 1000 roses, Charlie’s house (node 4) needs 800 roses, and Billy’s house (node 7) is supposed to get 200 roses (just to tease Jacob). The numbers next to the arcs of the network represent shipping costs per rose (in cents); they’re proportional to the distance between each node. The roses are coming from two local growers in Forks (nodes 1 and 3). Each of them can supply 1000 roses. Arrows with two heads indicate that shipments can be made in both directions.

Write down the supply and/or demand values next to each node and write a linear programming model to determine the shipment plan that minimizes the total cost of delivering all the roses (include all the necessary constraints). (Note: Alice already knows whether you’re going to get the right answer.)

The second half of the fun is to see if any students react to it. In fact, I got a couple of interesting written comments: “How dare you incorporate Twilight into Management Science?”, and a Harry Potter enthusiast wrote “Team Harry!”, while at the same time substituting the names of Hermione, Harry, Ginny and Malfoy for Alice, Edward, Bella, and Jacob.

Filed under Exam Fun, Linear Programming, Modeling, Network Flows, Teaching

## Facebook’s Farmville: What’s the Fastest Way to Get Rich?

One can’t help but notice that a lot of people have been playing Farmville on Facebook lately. Some of my relatives, friends, students and co-workers have taken the plunge. I don’t have time to play Farmville, but I thought it would be interesting to study it. It’s a simple and fun game, easy to explain, and apparently addictive.

You begin the game with a 12 by 12 piece of land. That gives you 144 one by one squares in which to plant crops. To simplify the explanation, I’m going to (1) restrict my attention to the four crops initially available (strawberries, eggplant, wheat, and soybeans); (2) assume crops will be harvested as soon as they are ripe (that is, ignore wilting); (3) disregard the random bonuses and gifts that show up every now and then. Blogger Mark Newheiser has a nice post about Farmville in which he includes a useful table. The cost column includes the cost of plowing the land (15 coins) plus the cost of purchasing the seeds. Because the harvest times of all four crops are multiples of four hours, we can think in terms of 4-hour periods.

Question: What’s the largest amount of cash one can obtain in 6 days? (need to rest on the 7th day).

You initially have 200 coins, one ripe and one half-grown square of strawberries, one ripe and one half-grown square of eggplant. By harvesting the two ripe squares, your fortune grows to 323 coins. I’ll also go ahead and clean (delete) the two half-grown squares so that we can begin with a clean slate. It turns out that the answer to the above question is to plant strawberries only, and as much as you can. Here’s the planting schedule:

 Period Strawberry Squares 0 12 1 17 2 24 3 34 4 47 5 66 6 92 7 129

At period 8 (after 32 hours), you’ll have enough money to fill all your 144 squares with strawberries and you just continue to plow, plant and harvest them all every 4 hours. After 6 days (36 four-hour periods), you’ll have 44,913 coins (adjusted to take into account that 4 squares were already plowed at time zero). This solution is not very exciting and, moreover, easy to predict. Taking a second look at the cost table, we can see that the profit per hour of strawberries, eggplant, wheat, and soybeans are 2.5, 1, 0.903, and 1.375, respectively. This fact, combined with the fact that strawberries are the fastest to grow, yields the above result. As one progresses through the game, other crops become available and strawberries’ dominance goes away. However, we can show that things can get very complicated by changing the numbers a little bit. If we increase the cost of strawberries to 31 (including plowing), their profit per hour changes to 1 (worse than soybeans). Here’s the new optimal planting schedule for 6 days, which yields a profit of 15,725 coins:

 Period Strawberry Squares Soybean Squares 0 2 8 1 3 6 1 15 7 7 1 8 6 2 9 7 12 5 26 13 8 14 13 15 29 16 25 8 17 2 27 18 57 22 16 23 75 24 69 29 16 59 30 85 35 59

Note that the best course of action now is to mix strawberries and soybeans and, at least to me, the proper way of doing it isn’t so obvious (hence the value of using Operations Research). If you’re now wondering whether there would be a situation in which the best idea is to use three different crops, the answer is yes! If we decrease the cost of eggplant to 19 (including plowing), for instance, the optimal planting schedule results in a profit of 18,157 coins and looks like this:

 Period Strawberry Squares Soybean Squares Eggplant Squares 0 17 12 1 37 18 13 1 14 1 1 15 1 16 1 17 2 18 123 24 18 26 1 27 1 28 1 29 3 30 123 35 3

In conclusion, I believe that Farmville is a rich decision-making environment that can be used to illustrate some of the analytical techniques from the field of Operations Research. This simple example shows how complicated things can become, even with only four crops!

One possibility of extending this analysis would be to consider that one isn’t willing to log in every 4 hours or so in order to harvest crops, so here’s another question:

Question: Given a schedule of off-line hours (i.e. not available for harvesting), how much money can you make in 6 days?

Finally, I haven’t actually answered the question that appears in the title of this blog post: what’s the fastest way to obtain X coins? I’ll leave these last two questions as an exercise.

Technical Details:

This section of the post is dedicated to those who want to know more about how this analysis was conducted. I used an integer programming model written in AMPL (I think dynamic programming would work too). My variables are: $P_{it}$ = number of squares planted with crop $i$ in period $t$, $H_{it}$ = number of squares of crop $i$ harvested in period $t$, $A_t$ = area in use in period $t$, $M_t$ = money available in period $t$. Periods go from $0$ to $T$ (in our example, $T=36$). So we are interested in maximizing $M_T$.

The constraints update the values of $A_t$ and $M_t$ based on what you do in period $t$. Here they are (I’m omitting the details of boundary conditions):

$\displaystyle A_t = A_{t-1} + \sum_{i} P_{it} - \sum_{i} H_{it} \enspace \forall \; t$

$\displaystyle M_t = M_{t-1} - \sum_{i} c_i P_{it} + \sum_{i} r_i H_{it} \enspace \forall \; t$

where $c_i$ and $r_i$ are, respectively, the cost and revenue of crop $i$. Because we don’t consider wilting, $H_{it}=P_{i(t-g_i)}$, where $g_i$ is the time it takes crop $i$ to grow and ripen. Therefore, the $H_{it}$ variables can be eliminated. Finally, we require $M_t \geq 0$ and $0 \leq A_t \leq 144$. Here’s the mixed-integer programming (MIP) model in AMPL in case anyone wants to play with it. When it is optimal to use more than one crop, the MIP can get pretty hard to solve, considering its size.