Tag Archives: modeling

Semantic Typing: When Is It Not Enough To Say That X Is Integer?

Andre Cire, John Hooker, and I recently finished a paper on an interesting, and somewhat controversial, topic that relates to high-level modeling of optimization problems. The paper is entitled “Modeling with Metaconstraints and Semantic Typing of Variables“, and its current version can be downloaded from here.

Here’s the abstract:

Recent research in the area of hybrid optimization shows that the right combination of different technologies, which exploits their complementary strengths, simplifies modeling and speeds up computation significantly. A substantial share of these computational gains comes from better communicating problem structure to solvers. Metaconstraints, which can be simple (e.g. linear) or complex (e.g. global) constraints endowed with extra behavioral parameters, allow for such richer representation of problem structure. They do, nevertheless, come with their own share of complicating issues, one of which is the identification of relationships between auxiliary variables of distinct constraint relaxations. We propose the use of additional semantic information in the declaration of decision variables as a generic solution to this issue. We present a series of examples to illustrate our ideas over a wide variety of applications.

Optimization models typically declare a variable by giving it a name and a canonical type, such as real, integer, binary, or string. However, stating that variable x is integer does not indicate whether that integer is the ID of a machine, the start time of an operation, or a production quantity. In other words, variable declarations say little about what the variable means. In the paper, we argue that giving a more specific meaning to variables through semantic typing can be beneficial for a number of reasons. For example, let’s say you need an integer variable x_j to represent the machine assigned to job j. Instead of writing something like this in your modeling language (e.g. AMPL):

var x{j in jobs} integer;

it would be beneficial to have a language that allows you to write something like this

x[j] is which machine assign(job j);

To see why, take a look at the paper ;-)

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Filed under Modeling, Research

Improving a Homework Problem from Ragsdale’s Textbook

UPDATE (1/15/2013): Cliff Ragsdale was kind enough to include the modification I describe below in the 7th edition of his book (it’s now problem 32 in Chapter 6). He even named a character after me! Thanks, Cliff!

When I teach the OR class to MBA students, I adopt Cliff Ragsdale’s textbook entitled “Spreadsheet Modeling and Decision Analysis“, which is now in its sixth edition. I like this book and I’m used to teaching with it. In addition, it has a large and diverse collection of interesting exercises/problems that I use both as homework problems and as inspiration for exam questions.

One of my favorite problems to assign as homework is problem number 30 in the Integer Linear Programming chapter (Chapter 6). (This number refers to the 6th edition of the book; in the 5th edition it’s problem number 29, and in the 4th edition it’s problem number 26.) Here’s the statement:

The emergency services coordinator of Clarke County is interested in locating the county’s two ambulances to maximize the number of residents that can be reached within four minutes in emergency situations. The county is divided into five regions, and the average times required to travel from one region to the next are summarized in the following table:

The population in regions 1, 2, 3, 4, and 5 are estimated as 45,000,  65,000,  28,000,  52,000, and 43,000, respectively. In which two regions should the ambulances be placed?

I love this problem. It exercises important concepts and unearths many misconceptions. It’s challenging, but not impossible, and it forces students to think about connecting distinct—albeit related—sets of variables; a common omission in models created by novice modelers. BUT, in its present form, in my humble opinion, it falls short of the masterpiece it can be. There are two main issues with the current version of this problem (think about it for a while and you’ll see what I mean):

  1. It’s easy for students to eyeball an optimal solution. So they come back to my office and say: “I don’t know what the point of this problem is; the answer is obviously equal to …” Many of them don’t even try to create a math model.
  2. Even if you model it incorrectly, that is, by choosing the wrong variables which will end up double-counting the number of people covered by the ambulances, the solution that you get is still equal to the correct solution. So when I take points off for the incorrect model, the students come back and say “But I got the right answer!”

After a few years of facing these issues, I decided I had had enough. So I changed the problem data to achieve the following (“evil”) goals:

  1. It’s not as easy to eyeball an optimal solution as it was before.
  2. If you write a model assuming every region has to be covered (which is not a requirement to begin with), you’ll get an infeasible model. In the original case, this doesn’t happen. I didn’t like that because this isn’t an explicit assumption and many students would add it in.
  3. If you pick the wrong set of variables and double-count the number of people covered, you’ll end up with an incorrect (sub-optimal) solution.

These improvements are obtained by adding a sixth region, changing the table of distances, and changing the population numbers as follows:

The new population numbers (in 1000’s) for regions 1 through 6 are, respectively, 21, 35, 15, 60, 20, and 37.

I am now much happier with this problem and my students are getting a lot more out of it (I think). At least I can tell you one thing: they’re spending a lot more time thinking about it and asking me intelligent questions. Isn’t that the whole purpose of homework? Maybe they hate me a bit more now, but I don’t mind practicing some tough love.

Feel free to use my modification if you wish. I’d love to see it included in the 7th edition of Cliff’s book.

Note to instructors: if you want to have the solution to the new version of the problem, including the Excel model, just drop me a line: tallys at miami dot edu.

Note to students: to preserve the usefulness of this problem, I cannot provide you with the solution, but if you become an MBA student at the University of Miami, I’ll give you some hints.


Filed under Books, Integer Programming, Modeling, Teaching

Choosing Summer Camps for Your Kids

Today I’m going to write about a decision that’s made by many American families each year: how to pick summer camps for our kids. There are several issues to take into account, such as cost, benefit, hours, and kids’ preferences. I’ll introduce an optimization model for summer camp selection through a numerical example. The example portrays a large family, but the same ideas apply if a few smaller families want to get together and solve this problem. This way they can take advantage of the discounts and take turns driving the kids around.

The Joneses have six kids: Amy, Beth, Cathy, David, Earl and Fred (yes, their first names are alphabetically sorted, matching their increasing order of age; Mr. and Mrs. Jones always knew they’d have six kids and hence named their firstborn with an ‘F’ name). This year, they’ve narrowed down their list of potential summer camps to the following ten: Math, Chess, Nature, Crafts, Cooking, Gymnastics, Soccer, Tennis, Diving, and Fishing. The Nature camp takes kids on a hike through the woods with the guidance of a biologist; they make frequent stops upon encountering specific plants and animals, during which a mini science lecture is delivered (pretty cool!). The Cooking camp involves cooking chemistry instruction, à la Alton Brown (also pretty cool).

The following table contains some data related to each camp:

The Cost column indicates the cost per child. The Discount column indicates the percentage discount that each child enrolled after the first would receive on the cost of each camp. For instance, if three children are enrolled in Math camp, the first would cost $1100, and the second and third would cost $770 each (30% less). The Hours column is self-explanatory and the last two columns indicate whether or not that particular camp develops mental and physical abilities, respectively (a value of one = yes, zero=no).

The next table shows some of the child-specific requirements:

For example, the Joneses want Fred to attend at least 3 camps that develop mental abilities, and at least 1 camp that develops physical abilities. The last two columns in the above table indicate the minimum and maximum number of camp hours for each child over the 9-week summer break.

The next thing parents need to take into account are their children’s preferences. So here they are:

The smaller the number in the above table, the more desirable a particular camp is. For example, Amy is a bit of a math nerd, and if we were to flip David’s preference scores for Math and Tennis, he could be classified as a bit of a jock. Some conflicts exist, in the sense that not all camps are compatible with each other in terms of time schedules. In this particular case, let’s assume that no child can attend both the Soccer and Tennis camps, or both the Nature and Soccer camps. Here’s how we are going to use this preference table to create a sense of fairness among the children: whenever a child that prefers camp X to camp Y goes to camp Y and doesn’t go to camp X, nobody else gets to go to camp X either. For example, if Amy goes to Nature camp and isn’t sent to either Math or Chess camp, none of her siblings are allowed to go to Math or Chess either. Conversely, if the Joneses decide to send Earl to Chess camp and Fred to Tennis camp, they must also send Earl to Tennis camp (because Earl prefers Tennis to Chess, and “Fred is going! Why can’t I go too!”). Clearly, there are other ways to use/interpret this table, such as trying to send everyone to at least one of their top N choices, but we won’t consider those alternatives here.

After taking all of the above issues and conditions into account, here’s a solution that satisfies all the requirements while resulting in the minimum cost of $22,180.00 (You guessed it…the Joneses are probably *not* among the 99%):

Amy goes to Math, Crafts, Cooking, and Tennis; Beth goes to Math, Cooking, Tennis, and Fishing; Cathy goes to Math, Crafts, Tennis, and Fishing; David goes to Math, Cooking, and Fishing; Earl goes to Math, Tennis, and Fishing; and Fred goes to Math, Crafts, Cooking, and Tennis. Mmm…interestingly, everyone goes to Math camp. I think the Joneses are on to something…

Depending on your own requirements, preferences, and costs your solution may differ, of course. But this should give you an idea of how this simple problem can easily become very complicated to solve. No need to fear, though! Operations Research is here!

Food for Thought: Here’s an interesting question that helps illustrate how high-quality solutions can be counterintuitive: by looking at the preferences table, we see that everyone prefers Soccer to Tennis. In addition, Soccer camp is less expensive than Tennis camp. So how come we send almost everyone to Tennis camp? Isn’t that strange? Let me know what you think in the comments below! That’s one of the advantages of using an analytical approach to decision making: it helps us find solutions we wouldn’t even consider otherwise because they don’t seem to make sense (at least not at first).

If you’re curious about how I managed to find the optimal solution, read on!

Details of the Analysis:

To find the minimum-cost solution, we can create a mathematical representation of the problem, a.k.a. a model, and then solve this model with the help of a computer. Let’s see how.

The first obvious decision to make is who goes where. So let the binary variable x_{ij} equal 1 when child i goes to camp j, and equal to 0 otherwise. We’ll also need another binary variable y_j that is equal to 1 when at least one child goes to camp j and equal to 0 when none of the children go to camp j. We are now ready to write our objective function and constraints. I’ll refer to the problem data using the column headings of the tables above. The subscript i will always refer to a child, and the subscript j will always refer to a camp.

To minimize the total cost, we write the following objective function:

\displaystyle \min \sum_i \sum_j (1-\mathrm{Discount}_j)\mathrm{Cost}_j x_{ij} + \sum_j \mathrm{Discount}_j \mathrm{Cost}_j y_j

Note how we are using the y_j variable to handle the discount for sending more than one child to camp j: we charge every child the discounted price in the double summation and add the discount back in only once if y_j=1.

Now we have to deal with the four requirements: minimum and maximum hours, mental activity, and physical activity. For every child i, we have to write the following four constraints:

\displaystyle \sum_j \mathrm{Hours}_j x_{ij} \geq \mathrm{MinTimeReq}_i

\displaystyle \sum_j \mathrm{Hours}_j x_{ij} \leq \mathrm{MaxTimeReq}_i

\displaystyle \sum_j \mathrm{IsMental}_j x_{ij} \geq \mathrm{MentalReq}_i

\displaystyle \sum_j \mathrm{IsPhysical}_j x_{ij} \geq \mathrm{PhysicalReq}_i

Next, we enforce the preference rules. Let’s recall the example involving Earl and Fred: if Earl goes to Chess camp and someone else (it doesn’t matter who) goes to Tennis camp, then Earl has to go to Tennis camp as well. Here’s what this constraint would look like:

x_{\mathrm{Earl},\mathrm{Chess}} + y_{\mathrm{Tennis}} - x_{\mathrm{Earl},\mathrm{Tennis}} \leq 1

Of course, we have to repeat this constraint for every child i and every pair of camps j_1 and j_2 such that child i prefers j_1 to j_2 in the following way:

x_{ij_2} + y_{j_1} - x_{ij_1} \leq 1

The camp compatibility constraints say that no child i can attend both Soccer and Tennis, or both Nature and Soccer, therefore:

x_{i,\mathrm{Soccer}} + x_{i,\mathrm{Tennis}} \leq 1

x_{i,\mathrm{Nature}} + x_{i,\mathrm{Soccer}} \leq 1

Finally, we need to relate the x_{ij} and y_j variables by stating that unless y_j=1 , no x_{ij} can be equal to 1 . So we write the following constraint for all values of i and j :

x_{ij} \leq y_j

And that’s the end of our model. Here’s a representation of this mathematical model in AMPL in case you want to play with it yourself. This is the model I used to obtain the numerical results reported above. Enjoy!


Filed under Applications, INFORMS Monthly Blog Challenge, Integer Programming, Mathematical Programming, Modeling, Motivation, Promoting OR, Summer camp

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

Find Out What Happens to Mr. Lovr

I’ve had a number of people tell me that they like my Valentine’s Day post, but many of them did not solve the Excel Spreadsheet. That’s the most important part of the post! You have to see what happens to Mr. Lovr! There’s no set-up necessary; just open Excel Solver (it’s an add-in you have to enable in Windows and a separate program in the Mac), and click on the “Solve” button. You’ll be glad you did.

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Filed under INFORMS Monthly Blog Challenge, Love, Valentine's Day

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.


Filed under Exam Fun, INFORMS Monthly Blog Challenge, Love, Valentine's Day

Political Districting and OR

When I think about how Operations Research can contribute to the world of politics, one of the first things that come to mind is the redistricting process which, of course, has been receiving a lot of attention after the 2010 census. Here’s an excerpt from the census.gov web site:

Another major use for decennial census data is for geographically defining state legislative districts, a “redistricting” process that begins in 2011. The census data allow state officials to realign congressional and state legislative districts in their states, taking into account population shifts since the last census and assuring equal representation for their constituents in compliance with the “one-person, one-vote” principle of the 1965 Voting Rights Act.

One of my favorite articles on the topic is “An Optimzation-Based Heuristic for Political Districting” by Anuj Mehrotra, Ellis Johnson and George Nemhauser, which appeared in Management Science in 1998 (henceforth referred to as the MJN paper). For a districting plan to be acceptable, it has to be fair/unbiased (to avoid gerrymandering), where fairness can be defined in terms of criteria that districts have to satisfy, such as population equality, contiguity, and compactness. Contiguity means that the district cannot have holes in it. Compactness is a bit more difficult to define, but it essentially means that districts should look more like a circle or square than like a snake or a tree branch. For example, would you say that this state senate district in Florida is compact?

Another important issue is that the methodology used to create the districts be transparent. This is a strong point in favor of using mathematical techniques such as those of Operations Research. The algorithm and software code used to create the districts can be made publicly available for anyone to inspect. Specific questions about the methodology can be answered precisely and unambiguously. There’s no beating around the bush. I applaud efforts such as the Public Mapping Project, which attempts to make the districting process more transparent and increase public participation. Nevertheless, given how difficult the problem is (in both practical and mathematical terms), I believe that it would also be important to consider plans created with the help of OR. As no mathematical model is perfect, those plans will probably need to be fine-tuned by a human being using tools such as this district builder. Of crucial importance, however, is the need for clear rules to guide such modifications. When is it OK to move part of a district’s border? by how much? etc. Given that current OR software tools are capable of producing multiple high-quality solutions at the end of a run, perhaps an even better alternative is to present the decision makers with a myriad of OR-generated and man-made plans, and let them pick the most adequate one. This practice is common in sports scheduling, where the leagues typically ask schedulers to provide them with many alternative solutions to choose from.

At this point, I’d like to switch gears and explore the districting problem in more detail from an OR point of view (using some ideas from the MJN paper). Districts are made up of smaller geographical units, for example counties (or pieces of counties). For illustration purposes, if a state like Ohio, which has 88 counties, is creating 18 congressional districts and those districts are to be made up of counties only, the potential number of distinct districts is

{88 \choose 18} = 2,\!418,\!561,\!960,\!739,\!869,\!780

That number is greater than the number of miles I’d cover if I could travel at the speed of light for 400,000 years! (Without a bathroom break!) Obviously, many of those districts are not legitimate because they violate contiguity and/or compactness constraints. However, even if those invalid districts are eliminated, the remaining number of possible districts is still very large. Ideally, however, we’d like to use district building blocks that are much smaller than counties, such as zip codes, but that is still very challenging (and would increase the above number even more!). In the MJN paper, the authors create an initial plan using counties as building blocks and then fine tune the districts’ borders to equalize populations. Here are the “before” and “after” districting plans for South Carolina:

Pretty good improvement, huh? Their basic methodology is to associate one decision with each potential district indicating whether or not it is part of the final plan (a binary variable). The optimization model then selects a subset of the districts that covers each county exactly once (a set partitioning problem). Because the number of variables is huge (as we’ve seen above), they resort to a technique called branch-and-price. In short, branch-and-price solves an optimization problem by considering only a subset of its variables at a time and by adding in promising variables little-by-little as the search progresses. It’s possible to define mathematically what a missing “good” variable is, and to construct one (i.e. a new district) on-the-fly. During this process, many set partitioning problems have to be solved. Because these intermediate problems are not always solved to optimality, the final solution obtained in MJN, albeit very good, isn’t technically an optimal solution, and that’s why they use the word “heuristic” in the title of their paper.

Acknowledgment: I’d like to thank Brady Hunsaker for providing a nice collection of links in the comments section of this blog post by Michael Trick.

Update: Here is another post I came across this week that also talks about using OR for political districting.


Filed under Applications, INFORMS Monthly Blog Challenge, Integer Programming, Modeling, Motivation, Politics, Promoting OR